Handbook of Applied Hydrology [2 ed.] 9780071835091, 0071835091

Table of contents :
00_Singh_FM_pi-lvi
01_Singh_ch01_p1.1-1.10
02_Singh_ch02_p2.1-2.8
03_Singh_ch03_p3.1-3.12
04_Singh_ch04_p4.1-4.8
05_Singh_ch05_p5.1-5.8
06_Singh_ch06_p6.1-6.14
07_Singh_ch07_p7.1-7.10
08_Singh_ch08_p8.1-8.6
09_Singh_ch09_p9.1-9.8
10_Singh_ch10_p10.1-10.6
11_Singh_ch11_p11.1-11.6
12_Singh_ch12_p12.1-12.6
13_Singh_ch13_p13.1-13.4
14_Singh_ch14_p14.1-14.8
15_Singh_ch15_p15.1-15.8
16_Singh_ch16_p16.1-16.8
17_Singh_ch17_p17.1-17.10
18_Singh_ch18_p18.1-18.12
19_Singh_ch19_p19.1-19.6
20_Singh_ch20_p20.1-20.10
21_Singh_ch21_p21.1-21.12
22_Singh_ch22_p22.1-22.20
23_Singh_ch23_p23.1-23.10
24_Singh_ch24_p24.1-24.8
25_Singh_ch25_p25.1-25.8
26_Singh_ch26_p26.1-26.16
27_Singh_ch27_p27.1-27.10
28_Singh_ch28_p28.1-28.6
29_Singh_ch29_p29.1-29.12
30_Singh_ch30_p30.1-30.10
31_Singh_ch31_p31.1-31.8
32_Singh_ch32_p32.1-32.8
33_Singh_ch33_p33.1-33.12
34_Singh_ch34_p34.1-34.6
35_Singh_ch35_p35.1-35.10
36_Singh_ch36_p36.1-36.10
37_Singh_ch37_p37.1-37.10
38_Singh_ch38_p38.1-38.8
39_Singh_ch39_p39.1-39.10
40_Singh_ch40_p40.1-40.6
41_Singh_ch41_p41.1-41.14
42_Singh_ch42_p42.1-42.18
43_Singh_ch43_p43.1-43.4
44_Singh_ch44_p44.1-44.12
45_Singh_ch45_p45.1-45.10
46_Singh_ch46_p46.1-46.14
47_Singh_ch47_p47.1-47.4
48_Singh_ch48_p48.1-48.8
49_Singh_ch49_p49.1-49.12
50_Singh_ch50_p50.1-50.10
51_Singh_ch51_p51.1-51.6
52_Singh_ch52_p52.1-52.16
53_Singh_ch53_p53.1-53.8
54_Singh_ch54_p54.1-54.14
55_Singh_ch55_p55.1-55.10
56_Singh_ch56_p56.1-56.12
57_Singh_ch57_p57.1-57.10
58_Singh_ch58_p58.1-58.10
59_Singh_ch59_p59.1-59.8
60_Singh_ch60_p60.1-60.12
61_Singh_ch61_p61.1-61.12
62_Singh_ch62_p62.1-62.8
63_Singh_ch63_p63.1-63.10
64_Singh_ch64_p64.1-64.6
65_Singh_ch65_p65.1-65.12
66_Singh_ch66_p66.1-66.10
67_Singh_ch67_p67.1-67.10
68_Singh_ch68_p68.1-68.8
69_Singh_ch69_p69.1-69.10
70_Singh_ch70_p70.1-70.4
71_Singh_ch71_p71.1-71.8
72_Singh_ch72_p72.1-72.10
73_Singh_ch73_p73.1-73.4
74_Singh_ch74_p74.1-74.16
75_Singh_ch75_p75.1-75.8
76_Singh_ch76_p76.1-76.8
77_Singh_ch77_p77.1-77.8
78_Singh_ch78_p78.1-78.10
79_Singh_ch79_p79.1-79.12
80_Singh_ch80_p80.1-80.10
81_Singh_ch81_p81.1-81.12
82_Singh_ch82_p82.1-82.6
83_Singh_ch83_p83.1-83.4
84_Singh_ch84_p84.1-84.6
85_Singh_ch85_p85.1-85.8
86_Singh_ch86_p86.1-86.8
87_Singh_ch87_p87.1-87.10
88_Singh_ch88_p88.1-88.8
89_Singh_ch89_p89.1-89.14
90_Singh_ch90_p90.1-90.12
91_Singh_ch91_p91.1-91.6
92_Singh_ch92_p92.1-92.8
93_Singh_ch93_p93.1-93.8
94_Singh_ch94_p94.1-94.4
95_Singh_ch95_p95.1-95.6
96_Singh_ch96_p96.1-96.10
97_Singh_ch97_p97.1-97.6
98_Singh_ch98_p98.1-98.8
99_Singh_ch99_p99.1-99.4
100_Singh_ch100_p100.1-100.6
101_Singh_ch101_p101.1-101.6
102_Singh_ch102_p102.1-102.10
103_Singh_ch103_p103.1-103.4
104_Singh_ch104_p104.1-104.4
105_Singh_ch105_p105.1-105.6
106_Singh_ch106_p106.1-106.4
107_Singh_ch107_p107.1-107.6
108_Singh_ch108_p108.1-108.12
109_Singh_ch109_p109.1-109.6
110_Singh_ch110_p110.1-110.4
111_Singh_ch111_p111.1-111.6
112_Singh_ch112_p112.1-112.6
113_Singh_ch113_p113.1-113.12
114_Singh_ch114_p114.1-114.6
115_Singh_ch115_p115.1-115.6
116_Singh_ch116_p116.1-116.4
117_Singh_ch117_p117.1-117.6
118_Singh_ch118_p118.1-118.10
119_Singh_ch119_p119.1-119.6
120_Singh_ch120_p120.1-120.10
121_Singh_ch121_p121.1-121.6
122_Singh_ch122_p122.1-122.10
123_Singh_ch123_p123.1-123.6
124_Singh_ch124_p124.1-124.8
125_Singh_ch125_p125.1-125.14
126_Singh_ch126_p126.1-126.18
127_Singh_ch127_p127.1-127.8
128_Singh_ch128_p128.1-128.6
129_Singh_ch129_p129.1-129.6
130_Singh_ch130_p130.1-130.8
131_Singh_ch131_p131.1-131.4
132_Singh_ch132_p132.1-132.6
133_Singh_ch133_p133.1-133.10
134_Singh_ch134_p134.1-134.8
135_Singh_ch135_p135.1-135.8
136_Singh_ch136_p136.1-136.10
137_Singh_ch137_p137.1-137.8
138_Singh_ch138_p138.1-138.10
139_Singh_ch139_p139.1-139.6
140_Singh_ch140_p140.1-140.8
141_Singh_ch141_p141.1-141.6
142_Singh_ch142_p142.1-142.8
143_Singh_ch143_p143.1-143.10
144_Singh_ch144_p144.1-144.10
145_Singh_ch145_p145.1-145.10
146_Singh_ch146_p146.1-146.8
147_Singh_ch147_p147.1-147.8
148_Singh_ch148_p148.1-148.12
149_Singh_ch149_p149.1-149.10
150_Singh_ch150_p150.1-150.8
151_Singh_ch151_p151.1-151.14
152_Singh_ch152_p152.1-152.6
153_Singh_ch153_p153.1-153.4
154_Singh_ch154_p154.1-154.6
155_Singh_ch155_p155.1-155.12
156_Singh_ch156_p156.1-156.8
157_Singh_INDEX_pI.1-I.22

Citation preview

Handbook of Applied Hydrology

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Editor-in-Chief Vijay P. Singh, Ph.D., D.Sc., D. Eng. (Hon.), Ph.D. (Hon.), D. Sc. (Hon.), P.E., P.H., Hon. D. WRE, Academician (GFA), is a Distinguished Professor and Caroline & William N. Lehrer Distinguished Chair in Water Engineering in the Department of Biological and Agricultural Engineering and Zachry Department of Civil Engineering at Texas A&M University. He holds a B.Tech degree from U.P. University of Agriculture and Technology, a Master’s Degree from the University of Guelph, a Ph.D. from Colorado State University, and a D.Sc. from the University of the Witwatersrand. One of today’s leading experts in the field of hydrology, Dr. Singh specializes in surface water hydrology, groundwater hydrology, hydraulics, irrigation engineering, environmental quality, and water resources. He has published 25 books and has edited over 58 books, and has published hundreds of journal articles. He has been the Editorin-Chief of the Journal of Hydrologic Engineering, ASCE; is currently serving as Editor-inChief of Open Agriculture, and Journal of Agricultural research, and Journal of Groundwater Research and is on the editorial boards of numerous journals. He is also serving as Editor-inChief of Water Science and Technology Book Series as well as World Water Resources Book series. He has received more than 75 national and international awards.

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Handbook of Applied Hydrology EDITED BY VIJAY P. SINGH

Second Edition to replace the classic 1963 edition edited by Ven Te Chow

New York  Chicago  San Francisco  Athens  London   Madrid  Mexico City  Milan  New Delhi   Singapore  Sydney  Toronto

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Cataloging-in-Publication Data is on file with the Library of Congress. McGraw-Hill Education books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please visit the Contact Us page at www.mhprofessional.com. Handbook of Applied Hydrology, Second Edition Copyright ©2017 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 6 7 8 9 DOW 21 20 19 18 17 16 ISBN 978-0-07-183509-1 MHID 0-07-183509-1 The pages within this book were printed on acid-free paper. Sponsoring Editor Lauren Poplawski

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Information contained in this work has been obtained by McGraw-Hill Education from sources believed to be reliable. However, neither McGraw-Hill Education nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill Education nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill Education and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

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Dedicated to Hydrologists and Water Scientists

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Contents in Brief Part 1               Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 Part 2               Data Collection and Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 Part 3               Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1 Part 4               Hydrologic Processes and Modeling . . . . . . . . . . . . . . . . . . . . . 37-1 Part 5            Sediment and Pollutant Transport . . . . . . . . . . . . . . . . . . . . . . . 63-1 Part 6              Hydrometeorologic and Hydrologic Extremes . . . . . . . . . . . . 72-1 Part 7               Systems Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-1 Part 8              Hydrology of Large Rivers and Lake Basins . . . . . . . . . . . . . . . 93-1 Part 9         Applications and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-1 Part 10  Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151-1

vii

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For online supplements and color versions of images, please go to www.mhprofessional.com/ handbookofappliedhydrology

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Contents Preface   xxxv Foreword   xxxix Acknowledgments   xli Contributors   xliii International Advisory Board   liii Practitioner Advisory Board   lv

Part 1.  Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 Chapter 1. The Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 1.1  Characteristics of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 1.2  Definition of Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 1.3  Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 1.4  Components of the Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 1.5  Schematic Representation of the Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 1.6  Scales in Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6 1.7 Impact of Climate Change on the Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 1-6 1.8  Influence of Human Activities and Land Use Changes on Hydrologic Cycle . . . 1-6 1.9  Relation Between Hydrologic Cycle and Carbon and Nitrogen Cycles . . . . . . . . 1-7 1.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9

Chapter 2. Watersheds, River Basins, and Land Use . . . . . . . . . . . . . . . . 2-1 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.2  Components of Watersheds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.3  Delineation of a Watershed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 2.4  Watershed Hydrological Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 2.5  Characteristics of a Watershed That Impact on Hydrological Processes . . . . . . 2-4 2.6  River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 2.7  River Basin Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 2.8  Major River Basins in the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7 2.9  Land Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7 2.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8

Chapter 3.  Water Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.2  Hydrologic Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.3  Water on the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.4  Water Balance Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 3.5  Natural and Anthropogenic Effects on the Water Balance . . . . . . . . . . . . . . . . . . . 3-7 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

PART 2.  Data Collection and Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 Chapter 4.  Hydrometeors and Quantitative Precipitation Estimation

4-3

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 4.2  Types of Hydrometeorological Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 4.3  Remote Sensing of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 4.4  Hydrometeorological Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 4.5  Hydrometeorological Data Quality Assurance and Control . . . . . . . . . . . . . . . . . . 4-5 4.6  Quantitative Precipitation Estimate, Data Use, Archiving, and Accessibility . . 4-6 ix

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x    Contents

Chapter 5.  Streamflow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 5.1 Streamflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 5.2  Types of Streamflow Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 5.3  Streamgage Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 5.4  Quality Assurance of Streamgage Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 5.5  Derived Streamflow Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7

Chapter 6.  Streamflow Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 6.2  Rating Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 6.3  Simple Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4 6.4  Complex Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6 6.5  Slope Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8 6.6  Rate of Change of Stage Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8 6.7  Dynamic-Flow Model Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10 6.8  Index-Velocity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10 6.9  Shifting-Control Method for Dealing with Rating Complexities . . . . . . . . . . . . 6-11 6.10  Uncertainty in Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

Chapter 7.  Hydrologic Information Systems . . . . . . . . . . . . . . . . . . . . . . 7-1 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1 7.2  Hydrologic Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1 7.3  Service-Oriented Architectures for Integrating Distributed Hydrologic Data and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 7.4 The CUAHSI Hydrologic Information System as an Example HIS . . . . . . . . . . . 7-3 7.5  HydroShare as a Next-Generation HIS Software . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8

Chapter 8. Remote Sensing Techniques and Data Assimilation for Hydrologic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 8.2  Remote Sensing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 8.3  Remote Sensing in Hydrological Sciences: A Historical Perspective . . . . . . . . 8-1 8.4  Remote Sensing: Methods and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2 8.5  Data Assimilation: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

Chapter 9.  Geographic Information Systems . . . . . . . . . . . . . . . . . . . . . . 9-1 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1 9.2  Basic Principles of GIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1 9.3  Data Sources and Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 9.4  Representation of Model Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5 9.5  Model/GIS Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7 9.6  Current Status and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8

Chapter 10.  Design of Hydrologic Networks . . . . . . . . . . . . . . . . . . . . . . 10-1 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1 10.2  Hydrologic Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1 10.3  Necessity of Hydrologic Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1 10.4  Impact of Hydrologic Network Density on Streamflow Estimates . . . . . . . . . . 10-2 10.5  Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 10.6  Design of Hydrologic Networks: Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 10.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

Part 3.  Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1 Chapter 11.  Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 11.2  Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 11.3  Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4 11.4  ANN Training and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4

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11.5  Drawbacks of ANN Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5 11.6  Shortcomings in ANN Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5 11.7  Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

Chapter 12.  Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1 12.1  Fuzzy Logic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1 12.2  Function of Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 12.3  Fuzzy Rule-Based Modeling (Fuzzy Inference) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5

Chapter 13.  Evolutionary Computing: Genetic Algorithms . . . . . . . . . 13-1

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-1 13.2  Evolutionary Computing in Hydrology: An Overview . . . . . . . . . . . . . . . . . . . . . . 13-1 13.3  Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-2 13.4  GA Applications in Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3 13.5  Conclusion and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3

Chapter 14.  Relevance Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-1 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-1 14.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-1 14.3  Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-2 14.4  Application of RVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-3 14.5  Examples from Hydrology and Future Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-4 14.6  Miscellaneous Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-6

Chapter 15.  Harmonic Analysis and Wavelets . . . . . . . . . . . . . . . . . . . . . 15-1 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1 15.2  The Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1 15.3  Discrete Time Wavelet Transform and Multiresolution Analysis . . . . . . . . . . . . 15-3 15.4  Signal Energy Repartition in the Wavelet Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 15-3 15.5  Wavelet Analysis of the Time-Scale Relationship Between Two Signals . . . . . 15-4 15.6  Wavelet Cross Spectrum and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-4 15.7  Applications of Wavelet Transforms in Hydrology and Earth Sciences . . . . . . 15-5 15.8 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-5

Chapter 16. Outlier Analysis and Infilling of Missing Records in Hydrologic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-1

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-1 16.2 Concepts and Methods for Outlier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-2 16.3  Concepts and Methods for Handling Missing Records . . . . . . . . . . . . . . . . . . . . . 16-3 16.4 Discussion and Concluding Remarks on Methods for Outliers and Infilling of Missing Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   16-5 16.5  Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-6

Chapter 17.  Linear and Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . 17-1 17.1  Linear and Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-1 17.2  Measures for Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-4 17.3  Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-5 17.4  Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-9

Chapter 18.  Time Series Analysis and Models . . . . . . . . . . . . . . . . . . . . . 18-1 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-1 18.2  Properties of Hydrological Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-1 18.3  Time-Series Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-2 18.4  Modeling of Continuous Time Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-2 18.5  Univariate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-2 18.6  Univariate Periodic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-4 18.7  Multivariate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-6 18.8  Disaggregation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-7 18.9  Nonparametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-7

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18.10  Stochastic Simulation, Forecasting, and Uncertainty . . . . . . . . . . . . . . . . . . . . . . 18-8 18.11  Conceptual Stochastic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-8 18.12  Final Remarks and Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-9

Chapter 19. Statistical Detection of Nonstationarity: Issues and Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-1 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-1 19.2  Exploratory Methods for Detection of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-2 19.3  Statistical Exploration of Nonstationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-2 19.4  Effect of Nonconstant Error Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-3 19.5  Effect of a Priori Filtering of Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-4 19.6  Distribution of a Breakpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-4 19.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-5

Chapter 20.  Spatial Analysis and Geostatistical Methods . . . . . . . . . . 20-1 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1 20.2  Data Types and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1 20.3  Spatial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1 20.4  Heterogeneous Field Estimation and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 20-3 20.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-7

Chapter 21.  Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1 21.2  Discrete Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1 21.3  Classification of Continuous Frequency Distributions . . . . . . . . . . . . . . . . . . . . . 21-2 21.4  Continuous Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-2 21.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10

Chapter 22. Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation . . . . . . . . 22-1

22.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1 22.2  Parameter Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1 22.3  Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-3 22.4  Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-5 22.5  Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-10 22.6  Validation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-15

Chapter 23.  Bayesian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-1 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-1 23.2  The Bayesian Inference Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-1 23.3  Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-4 23.4  Diagnostics to Scrutinize Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-6 23.5  Applications in Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-7 23.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-9

Chapter 24. Optimization Approaches for Integrated Water Resources Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-1 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-1 24.2 Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-2 24.3  Challenges and Research Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-4 24.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-5 24.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-5 24.6  Appendix: Literature Trend Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-5

Chapter 25.  Nonparametric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-1 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-1 25.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-1 25.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-2 25.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-3 25.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-5

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Chapter 26. Predictive Uncertainty Assessment and Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-1 26.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-1 26.2  Forecasting in Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-1 26.3  Motivations for Converting Deterministic to Stochastic Prediction . . . . . . . . . 26-1 26.4  Predictive Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-2 26.5  Techniques Aimed at Assessing Predictive Uncertainty . . . . . . . . . . . . . . . . . . . . 26-4 26.6  Verification of the Estimated Predictive Density . . . . . . . . . . . . . . . . . . . . . . . . . . 26-9 26.7  Major Reasons Undermining the Operational Use of Predictive Uncertainty 26-9 26.8  Examples of Proper Use of Predictive Uncertainty to Improve Decisions . . . . 26-13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-15

Chapter 27.  Risk-Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-1 27.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-1 27.2  Measures of Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-1 27.3  Performance Function and Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-2 27.4  Direct Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-2 27.5  First-Order Second-Moment Reliability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 27-3 27.6  Time-Dependent (Dynamic) Reliability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-5 27.7  Time-to-Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-5 27.8  Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-9

Chapter 28.  Scaling and Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-1 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-1 28.2  Scale-Invariant Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-1 28.3  Some Properties of H-SSSI Processes and MF Cascades . . . . . . . . . . . . . . . . . . . . 28-2 28.4  Inference of Scaling for Stationary Multifractal Measures . . . . . . . . . . . . . . . . . 28-4 28.5  Processes with Limited Scale Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-4 28.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-5

Chapter 29.  Nonlinear Dynamics and Chaos . . . . . . . . . . . . . . . . . . . . . . 29-1 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-1 29.2  Chaos Theory: A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-2 29.3  Chaos Concepts and Identification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-2 29.4  Issues in Chaos Identification and Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-5 29.5  Hydrologic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-7 29.6  Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-8

Chapter 30.  Copula Modeling in Hydrologic Frequency Analysis . . . 30-1 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-1 30.2  Description of Copula Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-1 30.3 Overview of Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-2 30.4 Multivariate Quantile and Return Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-5 30.5  An Illustration: The Fraser River at Hope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-5 30.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-7 30.7  Resources and Further Specific References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-9

Chapter 31.  Entropy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-1 31.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-1 31.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-1 31.3  Forms of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-2 31.4  Directional Information Transfer Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-3 31.5  Entropy under Transformation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-3 31.6  Informational Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-3 31.7  Total Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-3 31.8  Theory of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-4 31.9  Methodology for Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-4 31.10  Hydrologic Modeling Using Entropy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-7 31.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-8

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Chapter 32.  Entropy Production Extremum Principles . . . . . . . . . . . . . 32-1 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-1 32.2  Background and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-1 32.3  Maximum Entropy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-4 32.4  Review of Applications in Hydrology and Hydraulics . . . . . . . . . . . . . . . . . . . . . . 32-5 32.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-5

Chapter 33.  Data-Based Mechanistic Modeling . . . . . . . . . . . . . . . . . . . 33-1 33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-1 33.2  The Main Stages of DBM Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-2 33.3  Linear DBM Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-3 33.4  Time Variable and State-Dependent Parameter Models . . . . . . . . . . . . . . . . . . . 33-6 33.5  Hypothetico-Inductive DBM Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-7 33.6  DBM Emulation Modeling of High-Order Simulation Models . . . . . . . . . . . . . . . 33-8 33.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-11

Chapter 34.  Decomposition Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-1 34.1  Introduction: Adomian’s Decompositions Method . . . . . . . . . . . . . . . . . . . . . . . . 34-1 34.2  Regional Flow in an Unconfined Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-2 34.3  Propagation of Nonlinear Kinematic Flood Waves in Rivers . . . . . . . . . . . . . . . . 34-2 34.4  Nonlinear Infiltration in Unsaturated Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-4 34.5  Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-5

Chapter 35.  Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-1 35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-1 35.2  Network Theory: Concept and History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-2 35.3  Network Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-3 35.4  Network Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-4 35.5  Applications in Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-6 35.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-8

Chapter 36.  Hydroeconomic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-1 36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-1 36.2  Estimating the Economic Value of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-1 36.3  Water Demand Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-2 36.4  Considerations in the Design of Hydroeconomic Analysis Studies . . . . . . . . . . 36-3 36.5  Applications and Implementation of Hydroeconomic Analysis for Management and Decision Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-5 36.6  Discussion of Challenges, Limitations, and Future Directions . . . . . . . . . . . . . . 36-6 36.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36-7

Part 4.  Hydrologic Processes and Modeling . . . . . . . . . . . . . . . . . . . . . . . 37-1 Chapter 37.  Weather and Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-3 37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-3 37.2  Hydrologic Engineering and Intersection with Weather and Climate . . . . . . . 37-3 37.3 Weather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-5 37.4 Observing Weather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-7 37.5 Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-7 37.6  Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-10

Chapter 38.  Hydroclimatology: Global Warming and Climate Change 38-1 38.1  Introduction: The Ambiguity of Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38-1 38.2  Natural and Human Influences on Present-Day Climate . . . . . . . . . . . . . . . . . . . 38-1 38.3  Impacts of Climate Change on the Hydrological Cycle in the Twentieth and Twenty-First Centuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38-2 38.4  Global Climate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38-3 38.5  Working with Climate Model Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38-4 38.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38-6

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Chapter 39. Spatial and Temporal Estimation and Analysis of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39-1

39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-1 39.2  Estimates of Mean Areal Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-1 39.3  Missing Precipitation Data Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 39-4 39.4  Limitations of Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-5 39.5  New Methods for Missing Data Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-5 39.6  Summary of Issues for Missing Precipitation Data Estimation . . . . . . . . . . . . . . 39-6 39.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-8

Chapter 40.  Snow Distribution and Snowpack Characteristics . . . . . 40-1 40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-1 40.2  Processes Controlling Snow Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-1 40.3  Spatial Patterns of Snow at Various Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-2 40.4  Snowpack Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-2 40.5  Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-3 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-4

Chapter 41.  Time-Space Modeling of Precipitation . . . . . . . . . . . . . . . . 41-1 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41-1 41.2  Stochastic Modeling of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41-1 41.3  Deterministic Numerical Modeling of Time-Space Precipitation . . . . . . . . . . . . 41-3 41.4  Remote Sensing for the Modeling of Time-Space Precipitation . . . . . . . . . . . . 41-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41-10

Chapter 42.  Evapotranspiration and Evaporative Demand . . . . . . . . . 42-1 42.1  Introduction and History of Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42-1 42.2  Relevant Concepts and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42-3 42.3 Outstanding Problems and Directions for Future Work . . . . . . . . . . . . . . . . . . . . 42-10 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42-14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42-14

Chapter 43. Rainfall Interception, Detention, and Depression Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-1 43.1  Canopy Interception Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-1 43.2  Forest Floor Interception Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-3 43.3  Detention and Depression Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-3 43.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-3

Chapter 44.  Watershed Geomorphological Characteristics . . . . . . . . . 44-1 44.1  Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44-1 44.2  Watersheds and Drainage Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44-2 44.3 Outstanding Problems and Directions for Future Work . . . . . . . . . . . . . . . . . . . . 44-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44-11

Chapter 45.  Infiltration Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-1 45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-1 45.2  Basic Equations for Vertical Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-1 45.3  Classical Models for Point Infiltration into Vertically Homogeneous Soils . . . 45-2 45.4  Modeling of Point Infiltration into Vertically Nonuniform Soils . . . . . . . . . . . . 45-4 45.5  Models for Rainfall Infiltration over Heterogeneous Areas . . . . . . . . . . . . . . . . . 45-5 45.6  Soil Conservation Service Runoff Curve Number Model . . . . . . . . . . . . . . . . . . . 45-6 45.7 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-7

Chapter 46.  Soil Moisture and Vadose Zone Modeling . . . . . . . . . . . . . 46-1 46.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46-1 46.2  Continuum-Scale Models for Partially Saturated Flow in the Vadose Zone . . 46-2 46.3  Numerical Vadose Zone and Land Surface Models . . . . . . . . . . . . . . . . . . . . . . . . 46-3 46.4  Soil Moisture across Spatial-Temporal Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46-5 46.5  Inverse Modeling—Soil Hydraulic Properties at the Model Grid Scale . . . . . . 46-8 46.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46-11

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Chapter 47.  Hydrogeologic Characterization . . . . . . . . . . . . . . . . . . . . . 47-1 47.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47-1 47.2  Borehole Samples and Groundwater Monitoring Wells . . . . . . . . . . . . . . . . . . . 47-1 47.3  Investigation of Borehole Drilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47-1 47.4  Cone Penetrometry, Permeametry, and Electrical Conductivity Logging . . . . 47-2 47.5  Electrical Resistivity Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47-2 47.6  Hydraulic Methods for in Situ Conductivity Measurement . . . . . . . . . . . . . . . . . 47-2 47.7  Characterization of the Hydraulic Gradient and Flow Rates . . . . . . . . . . . . . . . . 47-3 47.8  Recharge Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47-3 47.9  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47-3

Chapter 48.  Groundwater Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-1 48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-1 48.2  Groundwater Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-1 48.3  Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-2 48.4  Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-3 48.5  Case Study: Groundwater Modeling in Baton Rouge, Southeastern Louisiana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-3 48.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-7

Chapter 49. Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . 49-1 49.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-1 49.2    Dominant Runoff Generation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-3 49.3  Infiltration Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-6 49.4  Factors Affecting Runoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-6 49.5  Poorly Understood Factors Affecting Runoff Generation . . . . . . . . . . . . . . . . . . 49-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-11

Chapter 50.  Snowmelt Runoff Generation and Modeling . . . . . . . . . . 50-1 50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-1 50.2  Snow Accumulation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-1 50.3  Energy Budget of Snow Pack and Snowmelt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-2 50.4  Simulation of Snow Accumulation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-3 50.5  Heat and Water Transfer in Melting Snow Cover . . . . . . . . . . . . . . . . . . . . . . . . . . 50-3 50.6  Spatial Variability of Snow Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-4 50.7  Constructing General Model of Snowmelt Runoff Generation . . . . . . . . . . . . . . 50-4 50.8  Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50-9

Chapter 51.  Glacial Melting and Runoff Modeling . . . . . . . . . . . . . . . . . 51-1 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51-1 51.2  Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51-1 51.3  Application of Remote Sensing in Glacier Quantification . . . . . . . . . . . . . . . . . . 51-1 51.4  Glaciated Versus Nonglaciated Watersheds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51-3 51.5  Application in Streamflow Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51-3 51.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51-4

Chapter 52.  Reservoir and Channel Routing . . . . . . . . . . . . . . . . . . . . . . 52-1 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-1 52.2  Reservoir Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-1 52.3  River Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-2 52.4  The Classical Muskingum Flood Routing Method . . . . . . . . . . . . . . . . . . . . . . . . . 52-3 52.5  Nash Cascade Model for River Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-4 52.6 Other Linear Storage Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-4 52.7  Linear Diffusion Analogy Routing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-5 52.8  Nonlinear Routing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-5 52.9  Flow Routing Using Hydraulic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-6 52.10  Basis for the Development of Simplified Momentum Equations . . . . . . . . . . . 52-7 52.11  Simplified Hydraulic Flood Routing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-7 52.12  Kalinin–Milyukov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-9 52.13  Variable Parameter Muskingum Stage Routing Method . . . . . . . . . . . . . . . . . . . 52-12

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52.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-14 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52-15

Chapter 53.  Waterlogging and Salinzation . . . . . . . . . . . . . . . . . . . . . . . . 53-1 53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-1 53.2  Salinity Features and Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-1 53.3    Irrigation Induced Rises of the Watertable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-2 53.4  Irrigation Induced Land Salinization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-3 53.5  Land Salinization Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-4 53.6  Salt Balance and Leaching Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-4 53.7  Monitoring and Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-5 53.8  Remedial Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-5 53.9  New Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-6 53.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53-7

Chapter 54. Surface Water–Groundwater Interactions: Integrated Modeling of a Coupled System . . . . . . . . . . . . 54-1 54.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-1 54.2  Surface Water Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-2 54.3  Subsurface Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-4 54.4  Soil Plant Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-5 54.5  Coupling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-6 54.6  Scale Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-7 54.7  Data Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-9 54.8  Integrated Models and Watershed Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 54-10 54.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-12

Chapter 55. Seawater Intrusion in Coastal Aquifers: Concepts, Mitigation, and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 55-1 55.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55-1 55.2  Assumptions and Approaches for Modeling Seawater Intrusion . . . . . . . . . . . 55-1 55.3  Mitigation of Seawater Intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55-3 55.4  Case Study: The Nile Delta Aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55-3 55.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55-9

Chapter 56. Regional Land Subsidence Caused by the Compaction of Susceptible Aquifer Systems Accompanying Groundwater Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-1 56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-1 56.2  Detection and Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-2 56.3  Analysis and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-9

Chapter 57.  Hydraulic Fracturing and Hydrologic Impacts . . . . . . . . . 57-1 57.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57-1 57.2  Hydraulic Fracturing Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57-2 57.3  Risk of Groundwater Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57-4 57.4  Potential for Induced Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57-6 57.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57-8

Chapter 58.  Catchment Classification and Regionalization . . . . . . . . . 58-1 58.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-1 58.2  Catchment Classification: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-2 58.3  Classification Based on Chaos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-2 58.4  Classification Based on Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-4 58.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58-7

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Chapter 59.  Rainfall-Runoff Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-1 59.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-1 59.2  A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-1 59.3  Computation of Runoff Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-1 59.4  Determination of Peak Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-3 59.5  Runoff Hydrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-3 59.6  Computation of Runoff Hydrograph by Hydraulic Approaches . . . . . . . . . . . . . 59-6 59.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59-8

Chapter 60.  Continuous Watershed Modeling . . . . . . . . . . . . . . . . . . . . . 60-1 60.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-1 60.2  Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-2 60.3  Concepts and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-3 60.4 Outstanding Problems/Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-7 60.5  Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-10

Chapter 61.  Calibration and Evaluation of Watershed Models . . . . . . 61-1 61.1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-1 61.2  Calibration and Evaluation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-2 61.3  Elements of a Calibration/Evaluation Strategy  . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-2 61.4  Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-2 61.5  Model Calibration and Evaluation Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-3 61.6  Strategies for Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-3 61.7  Desirable Properties of a Successful Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 61-4 61.8  Preparation Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-4 61.9  Calibration Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-8 61.10  Evaluation Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-8 61.11  Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-9

Chapter 62. Feasibility, Engineering, and Operations Models: Using the Decision Environment to Inform the Model Design 62-1 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-1 62.2  The Decision Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-1 62.3  USACE Decision Environment Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-2 62.4  Precision Dimensions of Decision Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-2 62.5  Process Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-3 62.6  Information Content Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-5 62.7  Case Study: Picayune Strand-Restoration Project . . . . . . . . . . . . . . . . . . . . . . . . . 62-5 62.8  Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-6 62.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-7

Part 5.  Sediment and Pollutant Transport . . . . . . . . . . . . . . . . . . . . . . . . . 63-1 Chapter 63.  Water Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-3 63.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-3 63.2  Water Pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-3 63.3  Water Quality Standards and Water Use Designations . . . . . . . . . . . . . . . . . . . . . 63-3 63.4  Restoration of Water Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-5 63.5  Sensor-Based Water Quality Monitoring Technologies . . . . . . . . . . . . . . . . . . . . 63-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-9

Chapter 64.  Soil Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-1 64.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-1 64.2  Erosion by Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-1 64.3  Erosion by Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-1 64.4  Gravity-Induced Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-2 64.5  Tillage Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-3 64.6  Snowmelt Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-3 64.7  Irrigation-Induced Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-3 64.8  Erosion by Wind-Driven Rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-3 64.9  Erosion Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-4 64.10  Erosion Assessment—Field and Laboratory Measurements . . . . . . . . . . . . . . . 64-4

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64.11  Erosion Assessment—Equations and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-5 64.12  Erosion Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-5 64.13  Erosion Control and Soil Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-5 64.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-6

Chapter 65.  Channel Erosion and Sediment Transport . . . . . . . . . . . . . 65-1 65.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-1 65.2  Sediment Production and Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-1 65.3  Partitioning of Sediment Loads in Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-2 65.4  Bank Profile and Erosional Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-3 65.5  Modes of Sediment Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-5 65.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65-10

Chapter 66.  Sedimentation of Floodplains, Lakes, and Reservoirs . . . . . . 66-1 66.1 Floodplain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-1 66.2 Lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-1 66.3  Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-2 66.4  The Sedimentation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-5 66.5  Prediction of Sediments in Floodplains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-7 66.6  Reservoir Trap Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-7 66.7  Estimation of Sediment in Lakes and Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . 66-7 66.8  Protective Measure Against Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-9 66.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-9 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66-10

Chapter 67.  Pollutant Transport in Surface Water . . . . . . . . . . . . . . . . . 67-1 67.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-1 67.2  Fundamental Processes Controlling Pollutant Transport . . . . . . . . . . . . . . . . . . 67-1 67.3  Pollutant Transport in Rivers and Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-3 67.4  Pollutant Transport in Lakes and Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-4 67.5  Pollutant Transport in Coastal Waters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-9

Chapter 68.  Pollutant Transport in Vadose Zone . . . . . . . . . . . . . . . . . . 68-1 68.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68-1 68.2  Water Potential in the Unsaturated Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68-1 68.3  Governing Equation of Flow in Vadose Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68-1 68.4  Deterministic Approach to Solute Transport in the Vadose Zone . . . . . . . . . . . 68-3 68.5  Codes for Numerical Solution of Vadose Zone Flow and Transport . . . . . . . . . 68-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68-6

Chapter 69.  Pollutant Transport in Groundwater . . . . . . . . . . . . . . . . . . 69-1 69.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-1 69.2  Learn from the Field Work—A Case of Saltwater Intrusion Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-3 69.3  Reactive Solute Transport Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-3 69.4  Dispersion Processes in Groundwater (Sato and Lwasa, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-5 69.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-9

Chapter 70.  Salinization and Salinity Management in Watersheds . . . . 70-1

70.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-1 70.2 Salinization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-1 70.3  Salinity-Related Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-2 70.4  Salinity Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-2 70.5  Salinity Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70-4

Chapter 71.  Transport of Biochemicals and Microorganisms . . . . . . . 71-1 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-1 71.2  Biochemicals and Microorganisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-1

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71.3  Mathematical Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-2 71.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-6

Part 6.  Hydrometeorologic and Hydrologic Extremes . . . . . . . . . . . . . . 72-1 Chapter 72.  Atmospheric Rivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-3 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-3 72.2  Large-Scale Circulation and Moisture Sources and Pathways . . . . . . . . . . . . . . 72-4 72.3  Precipitation and Flooding Associated with ARs . . . . . . . . . . . . . . . . . . . . . . . . . . 72-4 72.4  Modeling of ARs and Associated Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-5 72.5  Projection of Future Changes in ARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-6 72.6  Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-8

Chapter 73. Hydrometeorological Extremes (Hurricanes and Typhoons) . . . . . . . . . . . . . . . . . . . . . . . . . . 73-1 73.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-1 73.2  Climatology of Heavy Rainfall and Flooding from Tropical Cyclones . . . . . . . . 73-1 73.3  Remote Rainfall Associated with Tropical Cyclones: Predecessor Rain Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-2 73.4  Projected Increases in Rainfall Associated with Tropical Cyclones . . . . . . . . . . 73-2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-3

Chapter 74.  Extreme Rainfall: Global Perspective . . . . . . . . . . . . . . . . . 74-1 74.1  Introduction: The Importance of Studying Extreme Rainfall and Related Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-1 74.2  A Global Survey of Record Rainfall Depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-3 74.3  Approaches in Estimating Extreme Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-7 74.4  The Concept of Probable Maximum Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . 74-10 74.5  Probabilistic Approach to Extreme Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-12 74.6 Ombrian (Intensity-Duration-Frequency) Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 74-13 74.7  Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-15 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-16

Chapter 75.  Floods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-1 75.1  Introduction to Floods and Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-1 75.2  Flood Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-1 75.3  Flood Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-2 75.4  Estimation of Flood Magnitudes and Design Floods . . . . . . . . . . . . . . . . . . . . . . . 75-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-5

Chapter 76.  Flood Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-1 76.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-1 76.2  Describing the Chance of Flood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-1 76.3  Looking at the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-3 76.4  Fitting Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-4 76.5  Index Flood Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-5 76.6  Method of Moments, Bulletin 17B, and Bulletin 17C, with the LP3 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-6 76.7  Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-6 76.8  Estimation at Ungaged Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-7 76.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-7

Chapter 77.  Regional Flood Frequency Modeling . . . . . . . . . . . . . . . . . 77-1 77.1  Introduction: The Regional Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-1 77.2  Regional Estimation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-1 77.3  Delineation of Homogeneous Regions and Homogeneity Testing . . . . . . . . . . 77-1 77.4  Regional Transfer Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-3 77.5 One-Step Regional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-4 77.6  Nonlinear Models in Regional Flood Frequency Modeling . . . . . . . . . . . . . . . . . 77-4 77.7  Multivariate Regional Flood Frequency Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 77-5

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77.8  Nonstationary Regional Flood Frequency Approaches . . . . . . . . . . . . . . . . . . . . 77-5 77.9  Regional Flood Frequency Analysis Based on Seasonality Measures . . . . . . . 77-6 77.10  Combination of Local and Regional Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 77-6 77.11  Daily Streamflow Estimation at Ungauged Sites . . . . . . . . . . . . . . . . . . . . . . . . . . 77-6 77.12  Discussion and New Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-7

Chapter 78. Risk, Reliability, and Return Periods and Hydrologic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78-1

78.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78-1 78.2  Probabilistic- and Risk-Based Approaches to Hydrologic Design . . . . . . . . . . 78-2 78.3  Multivariate Probabilistic- and Risk-Based Approaches to Hydrologic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78-9

Chapter 79.  Drought Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-1 79.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-1 79.2  Drought, Aridity, and Water Scarcity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-2 79.3  Drought Occurrences in the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-3 79.4  Drought Properties Based on Statistical Techniques . . . . . . . . . . . . . . . . . . . . . . 79-3 79.5  Numerical Characterization of Drought Properties . . . . . . . . . . . . . . . . . . . . . . . 79-4 79.6  Indices for Drought Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-6 79.7 Outstanding Problems and Direction for Future Work . . . . . . . . . . . . . . . . . . . . . 79-8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79-10

Chapter 80.  Low Flow and Drought Analysis . . . . . . . . . . . . . . . . . . . . . . 80-1 80.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-1 80.2  Need for Low Flow Hydrology Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-1 80.3  Factors Affecting Low Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-2 80.4  Low Flow Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-2 80.5  Methods of Low Flow Estimation in Ungaged Catchments . . . . . . . . . . . . . . . . . 80-4 80.6  Drought Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-4 80.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80-9

Part 7.  Systems Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-1 Chapter 81.  Isotope Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-3 81.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-3 81.2 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-3 81.3  Groundwater Dating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-5 81.4  Sampling Methods and Isotope Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-7 81.5  Isotope Applications in Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81-12

Chapter 82.  Lake Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-1 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-1 82.2 Origin of Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-1 82.3  Water Balance of Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-1 82.4  Thermal Regime of Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-2 82.5  Ice Growth on Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-4 82.6  Circulation Processes in Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82-5

Chapter 83.  Urban Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83-1 83.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83-1 83.2  The Effects of Urbanization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83-2 83.3 Other Aspects of Urban Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83-3 83.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83-4

Chapter 84.  Agricultural Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84-1 84.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84-1 84.2  Water Movement in the Root Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84-1 84.3  Evaporation and Transpiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84-4

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Chapter 85.  Forest Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-1 85.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-1 85.2  Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-1 85.3  Principles of Forest Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-2 85.4  Research Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-3 85.5  Key Findings in Forest-Stream Water Quantity and Quality Relationships . . . 85-4 85.6  Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-5

Chapter 86.  Coastal Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86-1 86.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86-1 86.2 Overview of Coastal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86-1 86.3  Movement of Water and Sediment in Coastal Areas . . . . . . . . . . . . . . . . . . . . . . . 86-2 86.4  Mathematical Models of Water and Sediment Movement . . . . . . . . . . . . . . . . . . 86-6 86.5  Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86-8

Chapter 87.  Wetland Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-1 87.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-1 87.2  Importance of Hydrology on Wetland Functioning . . . . . . . . . . . . . . . . . . . . . . . . 87-1 87.3 Hydroperiod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-2 87.4  Wetland Hydrologic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-2 87.5  Wetland Water Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-3 87.6  Wetland Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-5 87.7  Modeling Groundwater-Surface Water Interactions . . . . . . . . . . . . . . . . . . . . . . . 87-6 87.8  Wetland Hydrology at the Watershed Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-7 87.9  Anthropogenic and Climate Change Impacts on Wetlands . . . . . . . . . . . . . . . . . 87-7 87.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-8

Chapter 88.  Arid Zone Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-1 88.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-1 88.2 Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-1 88.3 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-1 88.4 Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-1 88.5 Runoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-2 88.6  Transmission Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-3 88.7  Change in Storage (ΔS)-Groundwater Recharge . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-3 88.8  Evapotranspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-4 88.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88-6

Chapter 89.  Karst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89-1 89.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89-1 89.2  Investigation and Characterization of Karst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89-4 89.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89-10 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89-11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89-11

Chapter 90.  Cryospheric Hydrology: Mountainous Environment . . . 90-1 90.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-1 90.2  Alpine Runoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-1 90.3  Monsoon Dominated Asian Mountain Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-1 90.4  Glacier Runoff as a Resource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-1 90.5  Glacier Melting Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-4 90.6  Glacier Melt Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-5 90.7  Drainage and Storage Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-5 90.8 Diurnal and Seasonal Variations in Glacier Meltwater . . . . . . . . . . . . . . . . . . . . . 90-6 90.9  Cryospheric Hydrology and Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-7 90.10  Glacier Lake Outburst Floods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-9 90.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-10

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Chapter 91.  Hydrology of Transportation Systems . . . . . . . . . . . . . . . . 91-1 91.1  Pathways in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91-1 91.2  Scales of Movement and Accumulation Processes . . . . . . . . . . . . . . . . . . . . . . . . . 91-2 91.3  Soil and Groundwater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91-3 91.4 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91-4 91.5 Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91-6

Chapter 92.  Large-Scale and Global Hydrology . . . . . . . . . . . . . . . . . . . 92-1 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92-1 92.2  The Distribution of Water on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92-1 92.3  The Global Water Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92-1 92.4  Numerical Modeling and Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92-2 92.5  Global Water Cycle Variability, Predictability, and Change . . . . . . . . . . . . . . . . . 92-4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92-6

Part 8.   Hydrology of Large River and Lake Basins . . . . . . . . . . . . . . . . . 93-1 Chapter 93.  Amazon River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-3 93.1  93.2  93.3  93.4  93.5 

Main Geographical Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-3 Amazon Hydrological Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-3 Environment, Economics and Potentialities of the Basin . . . . . . . . . . . . . . . . . . . 93-5 Impact of Anthropic Activities in the Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-6 Amazon Basin in the Context of Its Water Footprint and Environmental Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-6

Chapter 94.  Paraná (Rio de la Plata) River Basin . . . . . . . . . . . . . . . . . . . 94-1 94.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94-1 94.2  Geographical Features and Main Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94-1 94.3 Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94-1 94.4  Hydrological Features and Water Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94-2 94.5  Variability and Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94-2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94-4

Chapter 95. Orinoco River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-1 95.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-1 95.2  Regional Geological and Topographical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 95-1 95.3  Hydroclimatic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-1 95.4  The Main Stem and Its Major Tributaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-2 95.5  Floodplains and Seasonal Sediment Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-2 95.6 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-3 95.7  Regional Vegetation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-4 95.8  Human Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-4 95.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95-4

Chapter 96.  Nile River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96-1 96.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96-1 96.2 Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96-3 96.3  Hydrology of the Nile Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96-9

Chapter 97.  Congo River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-1 97.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-1 97.2  The State of Hydrological Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-1 97.3  Climate Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-2 97.4  Catchments Characteristics and Hydrological Similarities . . . . . . . . . . . . . . . . . 97-2 97.5  Hydrogeochemical Processes and Sediment Transport . . . . . . . . . . . . . . . . . . . . 97-4 97.6  Hydrological Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-4 97.7  Climate and Land Use Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-5 97.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-5 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97-5

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Chapter 98.  Zambezi River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-1 98.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-1 98.2  Physical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-1 98.3  Main Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-1 98.4 Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-3 98.5  Runoff Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-3 98.6  Past Hydrological Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-4 98.7  Hydrological Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-5 98.8  Current Concerns and Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98-6

Chapter 99.  Euphrates and Tigris River Basin . . . . . . . . . . . . . . . . . . . . . 99-1 99.1  General Characteristics of the Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-1 99.2  Hydrology and Water Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-1 99.3  Water Resources Developments in the Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-3 99.4  Environmental Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-3 99.5  Climate Change Impacts on Basin Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-3 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-4

Chapter 100.  Yangtze River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-1 100.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-1 100.2  Climate and Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-2 100.3  Station Network and Water Conservancy Projects . . . . . . . . . . . . . . . . . . . . . . . 100-4 100.4  Significant Water Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-5 100.5  Research on the Yangtze River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-5 100.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100-6

Chapter 101.  Yellow River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-1 101.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-1 101.2  Climate and Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-3 101.3  Station Network and Water Conservancy Projects . . . . . . . . . . . . . . . . . . . . . . . 101-4 101.4  Significant Water Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-5 101.5  Research on the Yellow River basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-5 101.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-6

Chapter 102.  Mekong River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102-1 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102-1 102.2  Upper Mekong River (Langcang Jiang) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102-3 102.3  Lower Mekong River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102-5 102.4  Floods and Flood Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 6 102.5  Mekong Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 8 102.6 Biodiversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 8 102.7  Agriculture and Aquaculture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 8 102.8  Mekong River Commission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 9 102.9  Environmental Threats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 9 102.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102- 9

Chapter 103.  Yenisei River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103-1 103.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103-1 103.2  Central and Lower Sections of the Yenisei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103-1 103.3  Upper Section of the Yenisei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103-1 103.4  Fluvial System of the Angara River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103-1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103-4

Chapter 104.  Lena River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104-1 104.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104-1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104-4

Chapter 105.  Brahmaputra River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . 105-1 105.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105-1 105.2 Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105-1

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105.3  Drainage Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105-2 105.4 Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105-4 105.5  Problems Faced, Projects Undertaken, and Future Scope . . . . . . . . . . . . . . . . 105-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105-6

Chapter 106.  Ganga River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-1 106.1  Ganga River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-1 106.2  Ganga River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-1 106.3  Climate and Hydrology of the Ganga Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-2 106.4  Floods and Droughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-2 106.5  Groundwater Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-2 106.6 Hydropower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-2 106.7 Sediments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-3 106.8  Water Quality Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-3 106.9  Major Water Resources Development Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-3 106.10  Social and Environmental Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-3 106.11  Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106-4

Chapter 107.  Narmada Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-1 107.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-1 107.2 Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-2 107.3  Major Tributaries and Sub-Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-2 107.4  Climate in Narmada Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-3 107.5  Soils and Land Use in Narmada Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-4 107.6  Water Resources of Narmada Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-4 107.7  Major Water Resources Projects in Narmada Basin . . . . . . . . . . . . . . . . . . . . . . . 107-5 107.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107-6

Chapter 108.  Indus River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-1 108.1  Introduction to Indus River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-1 108.2  Physiography of Indus River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-1 108.3  River Network and Principal Hydrologic Units of Indus River Basin . . . . . . . 108-1 108.4  Climate of Indus River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-2 108.5  Hydrological Characteristics of Indus River Basin . . . . . . . . . . . . . . . . . . . . . . . . 108-3 108.6  Indus Water Treaty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-5 108.7  Major Water Resources Development Projects in Indus River Basin . . . . . . . 108-5 108.8  Groundwater Resources of Indus River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-9 108.9  Climate Change in Indus River Basin and Its Hydrologic Consequences . . . 108-9 108.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-10

Chapter 109.  The Mississippi River Basin . . . . . . . . . . . . . . . . . . . . . . . . . 109-1 109.1  Mississippi River Basin Physiography and Hydrology . . . . . . . . . . . . . . . . . . . . 109-1 109.2  Mississippi River Basin Climatology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109-1 109.3  Anthropogenic Changes in the River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109-2 109.4  Future Critical Challenges in the Mississippi River Basin . . . . . . . . . . . . . . . . . . 109-4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109-5

Chapter 110.  Colorado River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-1 110.1  Introduction and Basin Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-1 110.2  History of Water Resources Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-1 110.3  Reservoirs and Other Water Management Facilities . . . . . . . . . . . . . . . . . . . . . . 110-2 110.4  Development and Use of the Colorado River Simulation System . . . . . . . . . . 110-3 110.5  Hydrologic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-3 110.6  Generating Projected Future Flow Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-4 110.7  Future Priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-4

Chapter 111.  Columbia River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-1 111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-1 111.2  Basin History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-1 111.3  River Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-2

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111.4  Current and Future Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-4 111.5  Future Opportunities and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-5

Chapter 112.  St. Lawrence River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112-1 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112-1 112.2  Characteristics of the St. Lawrence River and Its Basin . . . . . . . . . . . . . . . . . . . 112-1 112.3  Hydrological Characteristics of the River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112-4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112-5

Chapter 113.  River Rhine Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-1 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-1 113.2 Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-1 113.3  Water Balance in the Rhine Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-4 113.4  Long-Term Variability of Hydrometeorological Variables in the Rhine Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-4 113.5  Discharge Characteristics in Longitudinal Profile . . . . . . . . . . . . . . . . . . . . . . . . 113-5 113.6  The Runoff Regime of the Rhine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-6 113.7  Changes in the Runoff Characteristics of the Rhine Since the Beginning of the Twentieth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-9 113.8  Changes in the Runoff Regime of the Upper Rhine . . . . . . . . . . . . . . . . . . . . . . . 113-10 113.9  Changes in the Runoff Regime of the Middle and Lower Rhine . . . . . . . . . . . 113-10 113.10  Development in Extreme Runoff Situations: Flood . . . . . . . . . . . . . . . . . . . . . . . 113-10 113.11  Development in Extreme Runoff Situations: Low Water . . . . . . . . . . . . . . . . . . 113-10 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-10

Chapter 114.  Danube River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114-1 114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114-1 114.2  History of the River System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114-1 114.3  Climate, Drainage Characteristics, and Hydrology . . . . . . . . . . . . . . . . . . . . . . . 114-2 114.4  Problem Faced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114-3 114.5  Scope of Future Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114-5

Chapter 115. Ob River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-1 115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-1 115.2 Ob River Basin and Discharge Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-1 115.3  Streamflow Characteristics and Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-1 115.4  Water Temperature and Thermal Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-5

Chapter 116.  Po River Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116-1 116.1  116.2  116.3  116.4 

River Basin Morphology and Geology, Geometry of the River Network . . . . 116-1 Climate and Meteorology. Genesis of Extreme Events and Droughts . . . . . . 116-1 Monitoring Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116-3 Hydrological Balance in the Po River Basin: Exploitation of Water Resources and Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116-3 116.5  The River Regime: Variability, Seasonality, Long-Term Patterns . . . . . . . . . . . 116-3 116.6  History of Po River Floods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116-3 116.7  Flood Hazard Mitigation Along the Course of the Po River . . . . . . . . . . . . . . . 116-4 116.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116-4

Chapter 117.  River Thames Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-1 117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-1 117.2  The Thames Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-1 117.3  Hydrometric Measurement in the Thames Basin . . . . . . . . . . . . . . . . . . . . . . . . . 117-1 117.4  Droughts and Alleviation Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117- 2 117.5  Floods and Flood Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117- 3 117.6  Research and Forecasting Initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-4 117.7  Trends in Runoff Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-4 117.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-6

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-6 Useful Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117-6

Chapter 118. Managing Water in an Arid Land: The Murray Darling Basin, Australia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-1 118.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-1 118.2  The Murray Darling Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-2 118.3  The Changing Hydrology of the Murray Darling Basin . . . . . . . . . . . . . . . . . . . . 118-3 118.4  Potential for Ecosystem Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-5 118.5  The Policy Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-5 118.6  What Future Do We Want for the Basin Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . 118-6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-7 Useful Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-9

Chapter 119.  The Great Lakes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-1 119.1  Introduction to the Great Lakes Hydrological System . . . . . . . . . . . . . . . . . . . . 119-1 119.2  The Great Lakes Water Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-2 119.3  Great Lakes Water levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-5

Chapter 120.  The East African Great Lakes . . . . . . . . . . . . . . . . . . . . . . . . 120.1 120.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.2  Lake Victoria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.3  Lake Tanganyika . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.4  Lake Malawi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.5  Lake Albert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.6  Lake Turkana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.7  Lake Kivu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120.8  Lake Edward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120. 1 120. 1 120. 3 120. 4 120. 5 120. 6 120. 7 120. 8 120. 9

Chapter 121.  Aral Sea Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121-1 121.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121-1 121.2  Subsurface Flux Changes and Interactions with the Shrinking Sea . . . . . . . . 121-1 121.3  Surface Flux Changes and Interactions with Climate Change . . . . . . . . . . . . . 121-2 121.4 Opportunities and Challenges for Water Quantity and Quality Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121-3 121.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121-4

Chapter 122.  Baltic Sea Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-1 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-1 122.2  Physiography and Hydroclimatology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-1 122.3  Hydrology and Water Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-4 122.4  Water Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-6 122.5  Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-8 122.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122-9

Chapter 123.  Black Sea Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123-1 123.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123-1 123.2  Geographical Location and Basic Morphometric Characteristics . . . . . . . . . . 123-1 123.3  Hydrological Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123-1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123-5

Chapter 124.  The Caspian Sea Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-1 124.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-1 124.2  The Caspian Sea Watershed Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-1 124.3  The Volga River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-1 124.4  Physicogeographical Conditions of the Caspian Sea . . . . . . . . . . . . . . . . . . . . . 124-3 124.5  Hydrometeorology and Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-3 124.6  Physical Oceanography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-3 124.7  Sea Level Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-4

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124.8  Marine Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-4 124.9  Marine Biology and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-5 124.10  Ecological Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-5 124.11 Oil Pollution of the Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-6 124.12  Seismic Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-7 124.13 Desertification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-7 124.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124-7

PART 9.  Applications and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-1 Chapter 125.  Design Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-3 125.1  Purpose of Design Rainfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-3 125.2  Constructing Databases of Rainfall to Derive Design Rainfalls . . . . . . . . . . . . 125-4 125.3  AMS Versus PDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-5 125.4  Appropriate Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-5 125.5 Regionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-6 125.6  Deriving Sub-Daily and Sub-Hourly IDF Relationships . . . . . . . . . . . . . . . . . . . 125-7 125.7  Scaling Relationships in Design Rainfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-7 125.8  Developing Design Rainfall Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-8 125.9  Uncertainty in Design Rainfall Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-8 125.10  Design Temporal Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-8 125.11  Design Spatial Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-9 125.12  Implications of Temperature Linked Non-Stationarity on the Design Rain . 125-10 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-11

Chapter 126.  Probable Maximum Precipitation . . . . . . . . . . . . . . . . . . . 126-1 126.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-1 126.2  Deterministic Method of Estimation of PMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-1 126.3  Probabilistic Method of Estimation of PMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-10 126.4  Generalized Versus Basin-Specific PMP Estimates . . . . . . . . . . . . . . . . . . . . . . . . 126-13 126.5  All-Season Versus Seasonal PMP Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-13 126.6 Orographic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-13 126.7  Spatial Variation of PMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-13 126.8  Temporal Distribution of PMP—Development of PMS . . . . . . . . . . . . . . . . . . . . 126-14 126.9  Seasonal Variation of PMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-16 126.10  Cautionary Notes on the Procedures for Estimation of New Set of PMP . . . . 126-16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126-17

Chapter 127.  Runoff Prediction in Ungauged Basins . . . . . . . . . . . . . . 127-1 127.1  The Prediction in Ungauged Basins Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-1 127.2  Best Practice Recommendations for Predicting Runoff in Ungauged Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-1 127.3  Prediction of Floods in Ungauged Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-2 127.4  Prediction of Low Flows in Ungauged Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-3 127.5  Prediction of Runoff Hydrographs in Ungauged Basins . . . . . . . . . . . . . . . . . 127-4 127.6  Where to Go from Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127-7

Chapter 128. Stochastic Streamflow Simulation and Forecasting . . . 128.1 128.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.2  Stochastic Simulation of Streamflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.3  Nonparametric Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.4  Multisite Streamflow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.5  Nonstationary Streamflow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.6  Streamflow Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.7  Stochastic Weather Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.8  Software and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 129. Flood Forecasting and Flash Flood Forecasting— Special Considerations in Hydrologic Modeling for the Expressed Purpose of Flood and Flash Flood Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-1 129.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-1 129.2  Real-Time Data Requirements and Forecasts and Data Quality Control . . . . 129-1 129.3  Computational Efficiency and Latency Requirements . . . . . . . . . . . . . . . . . . . . . 129-3 129.4  Data Assimilation and/or Adjusting Model Inputs, States, and Outcomes . 129-3 129.5  Use of Future Weather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-3 129.6  Requirements for Regulation Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-3 129.7  Flood-Control and Water-Supply Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.8  Reliability and Stability Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.9  Understanding and Conveying Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.10  Lead Time Considerations and Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.11  Temporal (Time Step) Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.12  Dissemination and Coordination with Customers, Partners, and the General Public . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.13  Applications Related to Flash Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.14  Special Temporal (Time Step) Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-4 129.15  Automated Data-Analysis Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-5 129.16  Automated Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-5 129.17  Ungaged Watershed Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-5 129.18  Identification of Highly Vulnerable/at Risk Locations . . . . . . . . . . . . . . . . . . . . . 129-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-5

Chapter 130.  Reservoir Operation Design . . . . . . . . . . . . . . . . . . . . . . . . 130-1 130.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-1 130.2  Reservoir Planning and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-1 130.3  Reservoir Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-3 130.4  Future Trends in Reservoir Operation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-6 130.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-7

Chapter 131.  Floodplain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . 131-1 131.1  Responses to Flood Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131-1 131.2  Evolution of Floodplain Management Practices . . . . . . . . . . . . . . . . . . . . . . . . . . 131-2 131.3  Concepts and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131-2 131.4  Certification in Floodplain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131-3 131.5  Summary of Issues and Needed Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131-3



Chapter 132.  Storm Water Management, Best Management Practices, and Low-Impact Development . . . . . . . . . . . . 132-1 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-1 132.2  The Need for Stormwater Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-2 132.3  Specific BMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-3 132.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-4 132.5 Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-4

Chapter 133.  Flood Proofing and Infrastructure Development . . . . . 133-1 133.1  Introduction to Flood Proofing and Infrastructure Development . . . . . . . . . 133-1 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133-9

Chapter 134.  Environmental Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-1 134.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-1 134.2  Evolution of Environmental Flow Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-2 134.3  Trade-offs in Development and Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-2 134.4  Estimation of Environmental Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-2 134.5  Methodologies for Assessment of Environmental Flow Requirement . . . . . 134-3 134.6  Implementation of EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-6 134.7  Environmental Flows in IWRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-7 134.8  Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134-7

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Chapter 135.  Drainage and Culvert Design . . . . . . . . . . . . . . . . . . . . . . . 135-1 135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-1 135.2  Fundamentals of Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-1 135.3  Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-4 135.4  Potential Impact of Climate Change on Culvert Design and Operation . . . . 135-6 135.5  Sustainable Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-6

Chapter 136.  Wetland and River Restoration . . . . . . . . . . . . . . . . . . . . . . 136-1 136.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-1 136.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-1 136.3  The Restoration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-2 136.4  Approaches to Wetland and River Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-3 136.5  Advancing the Science and Practice of Stream and Wetland Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-6 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136-7

Chapter 137.  Institutional Framework for Water Management . . . . . 137-1 137.1  Sustainable Integrated Water Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-1 137.2  Water Management in Texas: An Illustrative Case Study . . . . . . . . . . . . . . . . . . 137-1 137.3  Water Management Communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-1 137.4  Federal Agency Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-3 137.5  State Water Resources Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-4 137.6  Water Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-4 137.7  Environmental Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-5 137.8  Flood Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-6 137.9  Institutional Aspects of Computer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-6 137.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137-7

Chapter 138. Peak Water, Virtual Water, and Water Footprints: New Definitions and Tools for Water Research and Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-1 138.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-1 138.2  Peak Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-2 138.3  Comparison of Peak Production in Oil and Water . . . . . . . . . . . . . . . . . . . . . . . . 138-3 138.4  Water Transfers and the Concept of Virtual Water . . . . . . . . . . . . . . . . . . . . . . . 138-4 138.5  Water Footprints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-6 138.6  Soft Water Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-7 138.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138-8

Chapter 139.  Transboundary Water Management . . . . . . . . . . . . . . . . . 139-1 139.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-1 139.2  Water Conflict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-1 139.3  Sources of Water Conflict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-2 139.4  International Water Conflicts Versus National Water Conflicts . . . . . . . . . . . . 139-2 139.5  Resolving Water Conflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-2 139.6  The Importance of Institutional Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-3 139.7 Hydro-Hegemony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-3 139.8  International Water Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-3 139.9  Third Party Involvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-3 139.10  Future Directions and Ways to Address New Problems . . . . . . . . . . . . . . . . . . . 139-4 139.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-4 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139-4

Chapter 140.  Integrated River Basin Management . . . . . . . . . . . . . . . . 140-1 140.1  Integrated River Basin Management: A Framework and Process . . . . . . . . . . 140-1 140.2  Elements of IRBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-1 140.3  Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-2 140.4  Institutional Arrangements for IRBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-2 140.5  Technical Concepts and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-4 140.6  Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-5 140.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140-7

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Chapter 141.  Conflict Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-1 141.1  Conflict Exists Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-1 141.2  Game Theoretic Models for Conflict Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-1 141.3  Graph Model for Conflict Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-1 141.4  Fair Water Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-2 141.5  Compliance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-4 141.6  Agent-Based Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-5 141.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141-5

Chapter 142.  Long-distance Water Transfers . . . . . . . . . . . . . . . . . . . . . . 142-1 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142-1 142.2  Transfers Among Basins, Regions and Countries—Achievements and Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142-1 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142-6

Chapter 143.  The Indian River-Linking Program . . . . . . . . . . . . . . . . . . . 143-1 143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143-1 143.2  India’s Water Resources and the River-Linking Plan . . . . . . . . . . . . . . . . . . . . . . 143-1 143.3  Prognosis and Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143-5 143.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143-8 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143-8 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143-8

Chapter 144.  Irrigation Scheduling and Management . . . . . . . . . . . . . 144-1 144.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-1 144.2  Soil-Plant-Atmosphere Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-2 144.3 On-Farm Irrigation Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-2 144.4  Canal Irrigation Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-7 144.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-8 144.6  Research Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144-8

Chapter 145.  Rainwater Harvesting and Groundwater Recharge . . . 145-1 145.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145-1 145.2  Ancient Methods of Water Harvesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145-1 145.3  Watershed Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145-2 145.4  Rainwater-Harvesting Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145-3 145.5  Assessment of Rainwater Harvesting Using Remote Sensing and GIS . . . . . 145-7 145.6  Rainwater Harvesting Studies a Global Perspective . . . . . . . . . . . . . . . . . . . . . 145-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145-9

Chapter 146. Reuse-Reclaimed Water in Managed Aquifer Recharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-1 146.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-1 146.2  Treatment Mechanisms in Natural Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-1 146.3  Managed Aquifer Recharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-2 146.4  Water Quality Considerations for Managed Aquifer Recharge . . . . . . . . . . . . 146-3 146.5  Surface Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-3 146.6  Injection Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-4 146.7  Recovery of Reclaimed Water through Aquifer Storage and Recovery . . . . . 146-5 146.8  Subsurface Geochemical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-6 146.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146-6

Chapter 147.  River Bank Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-1 147.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-1 147.2  River Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-1 147.3  Potential of River Bank Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-2 147.4 Simulating RBF Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-3 147.5 Optimizing Distance of Well from the River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-6 147.6  River Bank Filtration Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-6 147.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147-7

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Chapter 148. Assessment of Climate Change Impacts on Water Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-1 148.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-1 148.2  Climate Change Observations and Model-Based Projections . . . . . . . . . . . . . 148-1 148.3 Observations and Projections of Climate Change Impact on Water Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-2 148.4  Aim of Modeling of Climate Change Impact on Freshwater Resources . . . . 148-3 148.5  Methodology of Modeling Climate Change Impacts on Water Resources . . 148-3 148.6  Hydrological Models for Climate Change Impact Assessment . . . . . . . . . . . . 148-4 148.7  Model Selection, Calibration, and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-6 148.8  Examples of Applications of Hydrological Models for Climate Change Impact Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-7 148.9  Propagation of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-7 148.10  Gaps and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148-10

Chapter 149.  Human Impacts on Hydrology . . . . . . . . . . . . . . . . . . . . . . 149-1 149.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149-1 149.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149-2 149.3  Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149-4 149.4    Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149-7 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149-8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149-8

Chapter 150. Climate Change and Its Impacts on Hydrologic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-1 150.1  150.2  150.3  150.4  150.5 

Climate Change: What Does It Mean? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-1 Causes of Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-1 Measure of Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-3 Impacts of Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-3 Impacts of Climate Change on The Hydrologic Cycle   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-3 150.6 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-7

Part 10.  Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151-1 Chapter 151.  Human-Hydrology Systems Modeling . . . . . . . . . . . . . . . 151-3 151.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151-3 151.2  Evolution of Human-Hydrology Systems Modeling . . . . . . . . . . . . . . . . . . . . . . 151-3 151.3  Methods for Modeling Human-Hydrology Systems . . . . . . . . . . . . . . . . . . . . . . 151-4 151.4  Applications of Human-Hydrology Systems Modeling . . . . . . . . . . . . . . . . . . . 151-6 151.5 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151-9 151.6  Future Directions and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151-11

Chapter 152. Variability of Hydrological Processes and Systems in a Changing Environment . . . . . . . . . . . . . . . . 152-1 152.1  Climate Change and the Water Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152-1 152.2  Human Activities and the Water Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152-1 152.3  Intensification of the Hydrological Cycle in a Changing Environment . . . . . 152-2 152.4  Sustainability, Hydrologic Risk, and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 152-3 152.5  Tracking Hydrologic Change: Trend and Predictability . . . . . . . . . . . . . . . . . . . 152-3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152-4

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Chapter 153.  Extraterrestrial Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-1 153.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-1 153.2  The Origin of the Earth’s Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-1 153.3  Water in Our Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-1 153.4  Water Beyond Our Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-3 153.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153-4

Chapter 154.  Water Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-1 154.1  Availability of Freshwater Around the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-1 154.2  Current State of Water Affairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-1 154.3  Water Security—the Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-3 154.4  Global Availability of Groundwater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-3 154.5  Water Security as Part of Economic Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-4 154.6  Managing Water in a Changing World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-4 154.7  Meeting the Water Gap: Unlocking the Potential of Green Water . . . . . . . . . . 154-4 154.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154-5

Chapter 155.  Social Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-1 155.1  155.2  155.3  155.4  155.5 

Water and Human Beings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-1 What is Social Hydrology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-3 Great Hydraulic Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-4 Nature’s Talk Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-4 Evolution of Social Hydrology as a New Discipline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-6 155.6  Challenges ahead in Water Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-7 155.7  Recent Trends and Developments in Social Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-8 155.8  Climate Change and Future Issues in Social Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-8 155.9  Future Direction in Social Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-10

Chapter 156.  Grand Challenges Facing the Hydrologic Sciences . . . . 156-1 156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-1 156.2  Quality-Differentiated Water Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-1 156.3  Conversion Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-2 156.4  Scoping the Greater Hydrologic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-2 156.5  Scoping Water Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-3 156.6  Emerging Infrastructure Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-3 156.7  Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-4 156.8 Prioritization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-4 156.9  Selected Priority Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-5 156.10  From Research to Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-6 156.11  Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156-6

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1

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Preface

The original Handbook of Applied Hydrology, edited by V.T. Chow, was published in 1964. The Handbook has had a major influence on hydrology. Since then it has been widely used as a reference book and has inspired numerous hydrologists by its depth and breadth of coverage. During the intervening period of over 50 years hydrology has witnessed phenomenal development in both depth and breadth, with the result that it has become a full-fledged geophysical science while still serving as basis for engineering applications. The development has occurred along four main lines. First, the 1960s witnessed the birth of a digital revolution and the computing prowess has since expanded exponentially. Now-a-days portable laptops are manifold more powerful than a large computer that then filled an entire room. The enormous computing capability permitted the birth of digital or numerical hydrology. Beginning with the development of the Stanford Watershed Model in 1966, a large number of watershed models, simulating virtually the entire hydrologic cycle, were developed. One can easily count in the world over one hundred watershed hydrology models in use these days. The development of watershed models is still continuing, encompassing simulation of other aspects that are associated with hydrology, such as atmospheric or hydrometeorological processes, hydraulics, climate change, ecosystems, geo-biochemistry, and human interactions. Another area has been the solution of hydrodynamic equations, be those of surface water, subsurface water, groundwater, water quality, and so on. Because of computers, it is now possible to simulate any component of the hydrologic cycle in great detail and do inverse modeling as well. Second, a tremendous growth in the development of new methods of solution, both deterministic and stochastic, has occurred. On the deterministic side, new transform techniques, optimization techniques, artificial intelligence methods, data mining, and numerical schemes were developed. On the stochastic side, developments have been even greater. Examples are copula theory, entropy theory, chaos theory, network theory, fractals, scaling theory, Monte Carlo simulation, and so on. Third, techniques for novel data collection, processing, storage, archiving, retrieval, and sharing have reached unprecedented heights. Remote sensing methods, such as radar and satellite technologies are now providing spatial data that were not even imagined prior to 1964. Information technology is providing means for dissemination of information in seconds, permitting hydrology to get closer to people as for example is happening in applications at farming scalwe. Another example is social hydrology, an emerging area that is gaining wide recognition these days. Fourth, new theories and new concepts have been developed, facilitated greatly by the aforementioned three areas. We have witnessed the evolution of new concepts in geomorphology and, more importantly, application of these concepts in watershed hydrology, hydrogeological characterization, hydrometeorology, cryosphere, glaciology, geo-bio-chemical transport, land use change, climate change, ecosystems science, medical hydrology, and the list goes on. In some areas, developments have been so large that together they have given rise to new branches of hydrology that were not even heard of prior to 1964. It is, therefore, now appropriate to pause and take a stock of where we were, where we have been and where the multiple disciplines that make hydrology should be going. This philosophy served as the motivation for preparing the layout for the second version of the Handbook of Applied Hydrology. The word “Applied” is important here. The emphasis in this version has been on the application at the expense of deep science and mathematics. With these considerations in mind, the subject matter of the handbook is divided into ten parts each containing a xxxv

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number of chapters. The First Part is on fundamentals and contains three chapters that deal with the hydrologic cycle, watersheds, river basins, land use, and water balance. This part attempts to set the context for the handbook. For the science and engineering of hydrology to progress, data and their collection and accessibility are the foundation. Therefore, the Second Part of this handbook deals with data collection and processing. It comprises seven chapters that discuss hydrometeorological data, streamflow data, rating curves, hydrologic information systems, remote sensing and geographical information systems. This part is concluded with a discussion on design of hydrologic networks. Hydrologic analyses and syntheses entails tools from mathematics, statistics, physics, biology, chemistry, information science, and social-economic science. Like hydrology, these areas have witnessed enormous developments. Fortunately, hydrologists have been quite savvy and open minded to borrow techniques from other disciplines and apply them in hydrology. Thus, Part Three spans sixteen chapters and presents these methods. These chapters deal with artificial neural networks; fuzzy logic; evolutionary computing; relevance vector machines; harmonic analysis and wavelets; outlier analysis; infilling of missing records; regression analysis; time series analysis; change detection and nonstationarity; spatial analysis and geostatistical methods; frequency distributions; model calibration and validation; sensitivity analysis; Bayesian methods; optimization methods; nonparametric methods; uncertainty analysis and decision making; risk-reliability assessment; scaling and fractals; nonlinear dynamics and chaos; copula modeling; entropy theory; entropy production extremum principles; data-based mechanistic modeling; decomposition methods; network theory; and hydroeconomic analysis. Experimental and field data and a variety of methods to analyze them have helped uncover hydrologic mysteries and better understand and model hydrologic processes. Therefore, Part Four containing twenty six chapters is focused on hydrologic processes and modeling. Beginning with a discussion of weather and climate, these chapters go on to discuss hydroclimatology; spatial and temporal analysis of precipitation; snowpack characteristics; precipitation modeling; evapotranspiration; interception; detention and depression storage; geomorphological characteristics; infiltration; soil moisture and the vadose zone; hydrogeologic characterization; groundwater modeling; watershed runoff; streamflow generation; snowmelt runoff generation; glacial melting; reservoir and channel routing; water logging and salinization; surface water-groundwater interaction; saltwater intrusion; land subsidence; hydraulic fracturing; catchment classification; rainfall-runoff modeling; continuous watershed modeling; and calibration and evaluation of watershed models. The concluding chapter deals with feasibility, engineering, and process models. With growing awareness of the environment, water, air and soil quality started to receive increasing attention in the 1970s and onwards. To that end, sediment and pollutant transport are dealt with in Part Five that comprises nine chapters. Beginning with a discussion of water quality, the chapters go on to discuss soil erosion; channel erosion and sediment transport; sedimentation of floodplains, lakes and reservoirs; pollutant transport in surface water, vadose zone and ground water; and salinization and salinity management in watersheds. The concluding chapter discusses transport of biochemicals and microorganisms. The availability of data and appropriate statistical methods facilitated analysis of extremes. Part Six, therefore, includes nine chapters that discuss hydrometeorological and hydrologic extremes. The subject matter includes atmospheric rivers; hurricanes and typhoons; extreme rainfall; floods; flood frequency analysis; regional hydrological modeling and regional frequency analysis; risk, reliability, return periods and hydrologic design; drought characteristics; and analysis of low flow and drought. In hydrology, many schools of thought are applied. One school emphasizes systems approach where intrinsic system details are not considered in detail. This school of thought was pervasive in the 1950s through the 1970s but the subject is still of great importance. Therefore, systems hydrology constitutes the theme of Part Seven which contains twelve chapters. These chapters discuss isotope hydrology; lake hydrology; urban hydrology; agricultural hydrology; forest hydrology; coastal hydrology; wetland hydrology; arid zone hydrology; karst hydrology; cryospheric hydrology; hydrology for transportation systems; and large-scale and global hydrology.

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Part Eight deals with the hydrology of large river basins and basins of lakes and seas spanning thirty-two chapters. The large river basins included are those of the Amazon River; Parana (Rio de la Plata) River; Orinoco River; Nile River; Congo River; Zambezi River; Euphrates and Tigris Rivers; Yangtze River; Yellow River; Mekong River; Yenisei River; Lena River; Brahmaputra River; Ganga River; Narmada River; Indus River; Mississippi River; Colorado River; Columbia River; St. Lawrence River; Rhine River; Danube River; Ob River; Po River; Thames River; and Murray Darling River. The basins of the Great Lakes System and the African Great Lakes are described in two chapters. The last four chapters on the basins of the Aral Sea; the Baltic Sea; the Black Sea; and the Caspian Sea conclude this part. Because the Handbook emphasizes application, Part Nine comprising twenty four chapters is devoted to applications and design. Beginning with a discussion of design rainfall, the chapters go on to discuss probable maximum precipitation; prediction in ungaged basins; stochastic streamflow simulation and forecasting; flood forecasting and flash and real-time flood forecasting; reservoir design, regulation and operation; floodplain management; stormwater management and low impact development; flood proofing and infrastructure development; environmental flows; drainage and culvert design; wetland and river restoration; institutional framework for water management; virtual water, water footprint, and peak water; transboundary water management; integrated river basin management; conflict resolution; long distance water transfer; the Indian river linking program; irrigation scheduling and management; rainwater harvesting and groundwater recharge; reuse-reclaimed water in managed aquifer recharge; and river bank filtration. This part is concluded with a discussion on the assessment of climate change impacts and water resources. Where the multiple disciplines of hydrology progress to is partly conjectured in the concluding Part Ten that deals with the future in seven chapters. These chapters discuss human impacts on hydrology, climate change and its impact on the hydrologic cycle, human-hydrology systems modeling, variability of hydrological processes and systems in a changing environment, extraterrestrial water, and water security. This part is concluded with a discussion of grand challenges. It is hoped that the Handbook will become useful to college faculty, graduate students, and researchers as well as practitioners in hydrology, water science and engineering, water resources management, urban development, hydrometeorology, geosciences, environmental and ecological sciences, and agricultural and forest sciences. Vijay P. Singh Texas A&M University College Station, Texas

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Foreword

Looking back when Ven Te Chow published his Handbook of Applied Hydrology in 1964, it was without any doubt a landmark event for the water sector. Using his personal status as one of the world’s leading water experts, he successfully managed to convince leading world authorities on different aspects of water to contribute chapters in their areas of expertise. After the book was published, it was universally acknowledged to be the most definitive and authoritative text available on water resources management for well over a decade. This mammoth book at 1,467 pages summarized and elaborated all the latest thinking and developments in different areas of water. It was a tribute to Chow’s encyclopaedic knowledge and versatility that he wrote several chapters on very different topics himself. Later on, he confided to me that a few of the chapters did not meet his high standards and thus he had to substantially rewrite them. Looked from any direction, the depth and quality of the Handbook was a major contribution which ensured that almost a generation of professionals used the text whenever they needed reliable information on any aspect of water. The Handbook was directly responsible for forging a lifelong friendship between Chow and myself. When the book was published, I was starting my career at the University of Strathclyde, Glasgow, as a lecturer. One of the British water journals requested me to write a review of all the water texts available and suggest 3 to 4 that I considered the best and most useful. In the mid-Sixties, it was possible to read all the texts on water that were available since the number was rather limited. After I published my assessment in late 1964, I received a letter from a somewhat annoyed Chow since this list did not include his latest tome Handbook of Applied Hydrology. I explained to him that I did not omit the book deliberately. I fact, I knew the book was published and available in the United States. Thus, I had written to McGraw-Hill asking if I could get a copy of the book for my assessment purposes. McGraw-Hill responded by saying that the book was indeed available in the United States but it would not be available in the United Kingdom for another six months. Chow, being the gentleman he was, immediately apologized for his letter. Shortly thereafter, I moved to Canada. In the sixties, when I was a Director in the Government of Canada, Chow invited me to come to the University of Illinois and give a talk during a meeting of the hydrology professors in Urbana. In early 1970, when Chow was considering the possibility of creating an interdisciplinary and multisectoral professional water association, the International Water Resources Association, he contacted me and requested me if I could be a founding member and help him to establish a Canadian chapter of the new Association. We remained very good friends until his untimely death. There have been exponential scientific and technological developments in all water-related areas, especially during the post-1980 period. Thus, advances in knowledge had made the Handbook obsolete by the early 1980s. It is indeed a tribute to Chow’s genius that the book had remained useful for such a long period.

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xl    Foreword

I am delighted that Prof. Vijay P. Singh, Caroline and William N. Lehrer Distinguished Chair in Water Engineering, Texas A&M University, College Station, Texas, is completely rewriting and updating Chow’s Handbook. Prof. Singh is a prominent international authority in hydrologic sciences, and, in my view, he is easily within the top 0.1% of all global waterrelated researchers. Only a person of Prof. Singh’s stature can undertake such a Herculean task. Following Chow’s footsteps mean these are big shoes to fill. From the manuscript, it is obvious to me that Prof. Singh has not only filled the big shoes completely but he has achieved almost the impossible with panache and style. Even with knowledge advancing at a significantly higher rate at present, compared to when Chow first published the book in 1964, I have no doubt that the new and updated version of this Handbook will remain the authoritative text for at least a decade. The new text is a fitting memory and tribute to Chow’s enduring legacy, and also a fitting confirmation of Prof. Singh’s global stature. Singapore October 18, 2015 Asit K. Biswas Distinguished Visiting Professor Lee Kuan Yew School of Public Policy National University of Singapore

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Acknowledgments This Handbook is a result of the collective effort of the authors who have contributed the chapters. It has been my privilege and honor to have known most of the authors for a long time. They have long been at the forefront of hydrologic research and teaching or practice and their rich experience and knowledge are hopefully reflected in the chapters they have written. Their contribution symbolizes their love of labor and their desire to contribute to advancing hydrology, for their time and effort are ad honorem. I am much grateful to the authors for accepting my invitation to write the chapters and their cooperation during the preparation of the Handbook. Without them, this Handbook would not have come to fruition, and it clearly belongs to them. I would like to express my deep gratitude to the members of two advisory boards: academic and practitioners. When the Handbook was being conceptualized, I often turned to them for their advice on the chapter layout and identifying potential chapter authors. They were more than willing to help and responded to my request promptly. The chapters were peer reviewed and finding three or more reviewers for each of 156 chapters was a challenge. They are too numerous to mention here, but I am grateful for the reviewers’ constructive reviews. I would like to acknowledge my family (wife Anita, son Vinay, daughter Arti, and grandsons Ronin, Kayden and Davin) for their support and affection while I worked on this handbook project. Without them, my task of completing the handbook would have been much more difficult. Finally, I want to express my gratitude to the McGraw-Hill staff, particularly Lauren Poplawski, who were always willing to help and provide advice and suggestions.

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Contributors Benjamin Abban Department of Civil and Environmental Engineering University of Tennessee Knoxville, Tennessee

M. Babbar-Sebens School of Civil and Construction Engineering Oregon State University Corvallis, Oregon

M. C. Acreman Centre for Ecology and Hydrology Crowmarsh Gifford Wallingford, United Kingdom

Daniel W. Baker Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado

Lauren Adams Center for Watershed Sciences University of California, Davis Davis, California Pradeep Aggarwal Isotope Hydrology Section Division of Physical and Chemical Sciences Department of Nuclear Sciences and Applications International Atomic Energy Agency (IAEA) Vienna, Austria Mohammad Z. Al-Hamdan Universities Space Research Association NASA/Marshall Space Flight Center National Space Science and Technology Center Huntsville, Alabama Doug Alsdorf Byrd Polar and Climate Research Center The Ohio State University Columbus, Ohio Devendra M. Amatya Center for Forested Wetlands Research Southern Research Station, USDA Forest Service Cordesville, South Carolina Daniel P. Ames Department of Civil and Environmental Engineering Brigham Young University Provo, Utah Michael Anderson State Climatologist, California Department of Water Resources Sacramento, California Mazdak Arabi Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado Luis J. Araguas-Araguas Isotope Hydrology Section Division of Physical and Chemical Sciences Department of Nuclear Sciences and Applications International Atomic Energy Agency (IAEA) Vienna, Austria

Emanuele Baratti Department DICAM University of Bologna Bologna, Italy Donald E. Barbe Department of Civil and Environmental Engineering University of New Orleans New Orleans, Louisiana Henry Barousse Louisiana Department of Transportation and Development, Retired Robert Bastian U.S. Environmental Protection Agency Washington, D.C. Peter Bauer-Gottwein Department of Environmental Engineering Technical University of Denmark Kongens Lyngby, Denmark Hiroko Kato Beaudoing Earth System Science Interdisciplinary Center University of Maryland College Park, Maryland Beijing Normal University Research and Development Centre Nippon Koei Co. Ltd. (Consulting Engineers) Tsukuba, Japan Kati Bell Water Reuse Global Practice Leader, MWH Global Nashville, Tennessee Joerg Uwe Belz Department of Hydrometry and Hydrological Survey Federal Institute of Hydrology Koblenz, Germany Lars Bengtsson Department of Water Resources Engineering Lund Institute of Technology Lund University Lund, Sweden Ronny Berndtsson Department of Water Resources Engineering and Center for Middle Eastern Studies Lund University Lund, Sweden

Mustafa M. Aral School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta, Georgia

Daniela Biondi Institute of Hydraulic Construction University of Bologna Bologna, Italy

Bhavna Arora Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California

Brian P. Bledsoe College of Engineering University of Georgia Athens, Georgia

xliii

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xliv    Contributors Günter Blöschl Centre for Water Resource Systems Institut für Wasserbau und Ingenieurhydrologie Technische Universität Wien Wien, Austria Istvan Bogardi Professor, Department of Meteorology Eotvos University Budapest, Hungary Department of Civil Engineering University of Nebraska Lincoln Lincoln, Nebraska John D. Bolten Hydrological Sciences Branch NASA Goddard Space Flight Center André Bouchard Program Manager, National Hydrological Service Meteorological Service of Canada Environment and Climate Change Canada Québec, Canada Armando Brath Department of Civil, Chemical, Environmental and Materials Engineering - DICAM ALMA MATER STUDIORUM - Università di Bologna Bologna, Italy Juan Martín Bravo Instituto de Pesquisas Hidráulicas Universidade Federal do Rio Grande do Sul IPH-UFRGS Porto Alegre, Brazil Dmitry A. Burakov Head of the Department of Environmental Engineering, Krasnoyarsk State Agrarian University Krasnoyarsk, Russia Donald H. Burn Department of Civil and Environmental Engineering University of Waterloo Waterloo, Canada Aaron R. Byrd Research Civil Engineer, Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi Ximing Cai Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Urbana, Illinois Darryl Carlyle-Moses Department of Geography and Environmental Studies Thompson Rivers University Kamloops, Canada Attilio Castellarin Department of Civil, Chemical, Environmental and Materials Engineering - DICAM ALMA MATER STUDIORUM - Università di Bologna Bologna, Italy Serena Ceola Department of Civil, Chemical, Environmental and Materials Engineering - DICAM ALMA MATER STUDIORUM - Università di Bologna Bologna, Italy Cem P. Cetinkaya Dokuz Eylül University Water Resources Management Research & Application Center (SUMER) Izmir, Turkey

Yang Cheng Department of Civil and Environmental Engineering Syracuse University Syracuse, New York Nannan Cheng State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau Institute of Soil and Water Conservation Chinese Academy of Sciences and Ministry of Water Resources Northwest Agriculture and Forest University Yangling, China Ekaterina V. Chuprina Water Problems Institute of the Russian Academy of Sciences Moscow, Russia Theodore G. Cleveland Associate Professor, Department of Civil, Environmental, and Construction Engineering Texas Tech University Lubbock, Texas Walter Collischonn Institute of Hydraulic Research Universidade Federal do Rio Grande do Sul IPH-UFRGS Porto Alegre, Brazil Zhentao Cong Department of Hydraulic Engineering Tsinghua University Beijing, China James Connaughton Richmond, California Corrado Corradini Department of Civil and Environmental Engineering University of Perugia Perugia, Italy Alva Couch Department of Computer Science Tufts University Medford, Massachusetts Norman H. Crawford President, Hydrocomp, Inc. Menlo Park, California James F. Cruise Department of Civil and Environmental Engineering University of Alabama in Huntsville Huntsville, Alabama Allegra da Silva Water Reuse Practice Leader - Rocky Mountain Region, MWH Global Denver, Colorado Baptiste Dafflon Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California Kumer Pial Das Department of Mathematics Lamar University Beaumont, Texas Claus Davidsen Department of Environmental Engineering Technical University of Denmark Kongens Lyngby, Denmark

Maria A. Charina Water Problems Institute of the Russian Academy of Sciences Moscow, Russia

Rodrigo Cauduro Dias de Paiva Institute of Hydraulic Research Federal University of Rio Grande do Sul Porto Alegre, Brazil

Fateh Chebana Institut national de la recherche scientifique Centre Eau-Terre-Environnement Québec, Canada

Zhi-Qiang Deng Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, Los Angeles

Ji Chen Department of Civil Engineering The University of Hong Kong Hong Kong, China

Georgia Destouni Department of Physical Geography Stockholm University Stockholm, Sweden

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Contributors    xlv  Michael DeWeese National Weather Service North Central River Forecast Center Chanhassen, Minnesota Daniel H. Doctor Eastern Geology and Paleoclimate Science Center U.S. Geological Survey Reston, Virginia Alessio Domeneghetti Department of Civil, Chemical, Environmental and Materials Engineering - DICAM ALMA MATER STUDIORUM - Università di Bologna Bologna, Italy A. S. Donigian, Jr. Principal Consultant, AQUA TERRA Consultants Mountain View, California Charles W. Downer Research Civil Engineer, Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi André Dozier Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado Jim Dumont Water Infrastructure Specialist Salt Spring Island, Canada Dipankar Dwivedi Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California Walter L. Ellenburg Department of Civil and Environmental Engineering University of Alabama in Huntsville Huntsville, Alabama A. Ercan Hydrologic Research Laboratory Department of Civil and Environmental Engineering University of California Davis, California Alvar Escriva-Bou Research Fellow, Water Policy Center Public Policy Institute of California San Francisco, California James S. Famiglietti NASA Jet Propulsion Laboratory California Institute of Technology Pasadena, California Koren Fang School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia Liping Fang Department of Mechanical and Industrial Engineering Faculty of Engineering and Architectural Science Ryerson University Toronto, Canada Bonifacio Fernandez Departamento de Ingeniería Hidráulica y Ambiental Pontificia Universidad Católica de Chile Santiago, Chile

Donald K. Frevert Hydraulic Engineer and Team Leader, Retired Technical Services Center Bureau of Reclamation Lakewood, Colorado Devin L. Galloway Water Mission Area U.S. Geological Survey Indianapolis, Indiana Timothy S. Gambrell Executive Director, Mississippi River Commission U.S. Army Corps of Engineers Mississippi Valley Division Vicksburg, Mississippi Christian Genest Department of Mathematics and Statistics McGill University Montréal, Canada Timothy R. Ginn Washington Sate University Pullman, Washington Jorge Gironás Departamento de Ingeniería Hidráulica y Ambiental Pontificia Universidad Católica de Chile Santiago, Chile Peter H. Gleick Pacific Institute Oakland, California Narendra Kumar Gontia Principal and Dean, College of Agricultural Engineering and Technology, Junagadh Junagadh Agricultural University, Junagadh Gujarat, India Jonathan L. Goodall Department of Civil and Environmental Engineering University of Virginia Charlottesville, Virginia David C. Goodrich USDA-ARS Southwest Watershed Research Center Tucson, Arizona R. S. Govindaraju Delon and Elizabeth Hampton Hall of Civil Engineering Purdue University West Lafayette, Indiana Robert E. Griffin Atmospheric Science Department Earth System Science Program University of Alabama in Huntsville Huntsville, Alabama Neil S. Grigg Professor, Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado Andrew D. Gronewold NOAA (Great Lakes Environmental Research Laboratory) University of Michigan (Civil and Environmental Engineering) Ann Arbor, Michigan Georges Gulemvuga Commission Internationale du bassin Congo-Oubangui-Sangha Kinshasa-Gombe, Democratic Republic of the Congo

Stefan Finsterle Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California

Orhan Gunduz Dokuz Eylul University Department of Environmental Engineering Izmir, Turkey

Faith A. Fitzpatrick Research Hydrologist, U.S. Geological Survey​​Wisconsin​ Water Science Center

Hoshin V. Gupta Department of Hydrology and Water Resources University of Arizona Tucson, Arizona

Darrell Fontane Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado

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Semyon M. Guziy Engineer, Krasnoyarsk Hydroproject Institute Krasnoyarsk, Russia

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xlvi    Contributors Alan F. Hamlet Department of Civil and Environmental Engineering and Earth Sciences University of Notre Dame Notre Dame, Indiana Cameron Handyside Earth System Science Center University of Alabama in Huntsville Huntsville, Alabama

Justin Huntington Desert Research Institute Division of Hydrologic Sciences Western Regional Climate Center Reno, Nevada John C. Imhoff Senior Scientist, AQUA TERRA Consultants Ouray, Colorado

Mohamed M. Hantush Research Hydrologist, National Risk Management Research Laboratory U.S. Environmental Protection Agency Cincinnati, Ohio

K. Ishida Hydrologic Research Laboratory Department of Civil and Environmental Engineering University of California Davis, California

Umesh Haritashya Department of Geology University of Dayton Dayton, Ohio

Sharad K. Jain Water Resources Systems Division National Institute of Hydrology Roorkee, India

Bekki Ward Harjo Senior Hydrologist, National Weather Service Arkansas-Red Basin River Forecast Center Tulsa, Oklahoma

L. Douglas James Fairfax, Virginia

Nilgun B. Harmancioglu Dokuz Eylul University Water Resources Management Research & Application Center (SUMER) Izmir, Turkey Julien J. Harou School of Mechanical, Aerospace and Civil Engineering The University of Manchester Manchester, United Kingdom Robert Hartman Hydrologist in Charge, California-Nevada River Forecast Center National Weather Service Sacramento, California Hongming He Institute of Soil and Water Conservation Chinese Academy of Sciences and Ministry of Water Resources Yangling, China Janet S. Herman Professor, Department of Environmental Sciences University of Virginia Charlottesville, Virginia Kith W. Hipel Department of Systems Design Engineering University of Waterloo Waterloo, Canada Yoshinari Hiroshiro Department of Urban & Environmental Engineering Graduate School of Engineering Kyushu University Fukuoka, Japan Michael T. Hobbins Research Hydrologist, Earth System Research Laboratory & Cooperative Institute for Research in Environmental Sciences NOAA-Earth System Research Laboratory Boulder, Colorado Robert R. Holmes, Jr. Hydrologist and National Flood Hazard Coordinator, Office of Surface Water U.S. Geological Survey Rolla, Missouri Jeffery S. Horsburgh Department of Civil and Environmental Engineering Utah State University Logan, Utah Chi Hua Huang National Soil Erosion Research Laboratory West Lafayette, Indiana

A. W. Jayawardena Department of Civil Engineering The University of Hong Kong Hong Kong, China Research and Development Center Nippon Koei Co. Ltd. (Engineering Consultants) Tsukuba, Japan Shanhu Jiang Lecturer, College of Hydrology and Water Resources Hohai University Nanjing, China Kenji Jinno Department of Urban & Environmental Engineering Graduate School of Engineering Kyushu University Fukuoka, Japan Fiona Johnson School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia V. Jothiprakash Department of Civil Engineering Indian Institute of Technology Bombay Mumbai, India Tijana Jovanovic Department of Civil and Environmental Engineering The Pennsylvania State University State College, Pennsylvania Latif Kalin Professor, School of Forestry and Wildlife Sciences Auburn University Auburn, Alabama Jagath Kaluarachchi Professor of Civil and Environmental Engineering College of Engineering Utah State University Logan, Utah Raghupathy Karthikeyan Department of Biological and Agricultural Engineering Texas A&M University College Station, Texas Laila Kasuri Lead Analyst and Senior Lecturer, Centre for Water Informatics and Technology Lahore University of Management Sciences Lahore, Pakistan Dmitri Kavetski School of Engineering University of Newcastle Callaghan, Australia

Denis Arthur Hughes Institute for Water Research Rhodes University Grahamstown, South Africa

M. Levent Kavvas Hydrologic Research Laboratory Department of Civil and Environmental Engineering University of California Davis, California

Rui Hui Center for Watershed Sciences University of California, Davis Davis, California

Akira Kawamura Department of Civil and Environmental Engineering Tokyo Metropolitan University Tokyo, Japan

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Contributors    xlvii  Soksamnang Keo State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau Institute of Soil and Water Conservation Chinese Academy of Sciences and Ministry of Water Resources Northwest Agriculture and Forest University Yangling, China Greg Kerr Office of Water Programs University of Wyoming Laramie, Wyoming Martin Keulertz Agricultural and Biological Engineering Purdue University West Lafayetter, Indiana Jongho Keum Department of Civil Engineering McMaster University Hamilton, Canada C. Prakash Khedun Department of Biological and Agricultural Engineering Texas A&M University College Station, Texas Max Kigobe Department of Civil Engineering College of Engineering, Design, Art and Technology (CEDAT) Makerere University Kampala, Uganda Thomas R. Kjeldsen Department of Architecture and Civil Engineering University of Bath Bath, United Kingdom Barbara A. Kleiss Director, Mississippi River Science & Technology Program U.S. Army Corps of Engineers Mississippi Valley Division Vicksburg, Mississippi Randal Koster Global Modeling and Assimilation Office NASA Goddard Space Flight Center Greenbelt, Maryland Andrey Kostianoy Chief Scientist, Ocean Experimental Physics Laboratory P.P. Shirshov Institute of Oceanology Moscow, Russia Demetris Koutsoyiannis Department of Water Resources and Environmental Engineering School of Civil Engineering National Technical University of Athens Zographou, Greece Peter Krahe Contact person of subject group River Basin Modelling Department Water Balance, Forecasting and Predictions Federal Institute of Hydrology Rhineland-Palatinate, Germany Valentina Krysanova Potsdam Institute for Climate Impact Research Potsdam, Germany Lev Kuchment Laboratory of Water Cycle Water Problems Institute Russian Academy of Sciences Moscow, Russia George Kuczera School of Engineering University of Newcastle Callaghan, Australia Bhishm Kumar Ex Professional Staff, Isotope Hydrology Section International Atomic Energy Agency (IAEA) Vienna, Austria Ex Head, Hydrological Investigations Division National Institute of Hydrology Roorkee, India

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Zbigniew W. Kundzewicz Institute of Agricultural and Forest Environment Polish Academy of Sciences Poznan, Poland John Labadie Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado David Labat Géosciences Environnement Toulouse Toulouse, France Venkataraman (Venkat) Lakshmi Department of Earth and Ocean Sciences University of South Carolina Columbia, South Carolina Upmanu Lall Department of Earth and Environmental Engineering and Department of Civil Engineering and Engineering Mechanics Columbia Water Center International Research Institute for Climate and Society Columbia University New York, New York Alain Laraque GET - UMR CNRS / IRD / UPS - UMR 5563 du CNRS, UMR234 de l'IRD Toulouse, France Magnus Larson Department of Water Resources Engineering Lund Institute of Technology Lund University Lund, Sweden Se-Yeun Lee Climate Impacts Group, College of the Environment University of Washington Seattle, Washington Stanley A. Leake Water Mission Area U.S. Geological Survey Tucson, Arizona Chiara Lepore Ocean and Climate Physics Lamont-Doherty Earth Observatory of Columbia University Palisades, New York L. Ruby Leung Atmospheric Sciences and Global Change Division Pacific Northwest National Laboratory Richland, Washington Benjamin Lord Raleigh, North Carolina Yajie Lu State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau Institute of Soil and Water Conservation Chinese Academy of Sciences and Ministry of Water Resources Northwest Agriculture and Forest University Yangling, China Jay R. Lund Center for Watershed Sciences Department of Civil and Environmental Engineering University of California Davis, California Gil Mahe IRD, Laboratoire Hydro Sciences Montpellier, France D. R. Mailapalli Agricultural and Food Engineering Department Indian Institute of Technology Kharagpur Kharagpur, India Terry Marsh Centre for Ecology & Hydrology Wallingford, United Kingdom Philip Marsh Water Science Wilfrid Laurier University Waterloo, Canada José Pedro Matos Laboratoire de constructions hydrauliques (LCH) Ecolepolytechniquefédérale de Lausanne (EPFL) Lausanne, Switzerland

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xlviii    Contributors Larry W. Mays Arizona State University Tempe, Arizona Richard H. McCuen The Ben Dyer Professor of Civil & Environmental Engineering Civil and Environmental Engineering Department University of Maryland College Park, Maryland Steven G. McNulty Eastern Forest Environmental Threat Assessment Center Southern Research Station, USDA Forest Service Raleigh, North Carolina Alfonso Mejia Department of Civil and Environmental Engineering The Pennsylvania State University State College, Pennsylvania Tarek Merabtene Department of Civil and Environmental Engineering College of Engineering University of Sharjah Sharjah, United Arab Emirates Venkatesh Merwade Lyles School of Civil Engineering Purdue University West Lafayette, Indiana Guziy Semyon Mikhailovich Institute “Krasnoyarskgidroproekt” Krasnoyarsk State Agricultural University Krasnoyarsk branch of JSC “SibENTC” Krasnoyarsk, Russia Pavol Miklanek Institute of Hydrology Slovak Academy of Sciences Bratislava, Slovakia

Dauren Mussabek Department of Water Resources Engineering Lund University Lund, Sweden Kei Nakagawa Nagasaki University Graduate School of Fisheries Science and Environmental Studies Nagasaki University Nagasaki, Japan Mohamed K. Nassar Department of Civil and Environmental Engineering University of California Davis Davis, California Environmental Studies and Research Institute University of Sadat City Sadat, Egypt Grey S. Nearing NASA/GSFC Hydrological Sciences Branch Greenbelt, Maryland WaiWah Ng Department of Civil Engineering Lakehead University Thunder Bay, Canada John Nielsen-Gammon Department of Atmospheric Sciences Texas A&M University College Station, Texas Jun Niu College of Water Resources and Civil Engineering China Agricultural University Beijing, China Robert K. Niven School of Engineering and Information Technology The University of New South Wales Canberra, Australia

Srikanta Mishra Institute Fellow & Chief Scientist, Energy & Environment Battelle Memorial Institute Columbus, Ohio

P. E. O’Connell School of Civil Engineering and Geosciences Newcastle University Newcastle upon Tyne, United Kingdom

Ashok K. Mishra Glenn Department of Civil Engineering Clemson University Clemson, South Carolina

Greg O’Donnell School of Civil Engineering and Geosciences Newcastle University Newcastle upon Tyne, United Kingdom

Natalia N. Mitina Water Problems Institute of the Russian Academy of Sciences Moscow, Russia

Robert Occhipinti United States Army Corps of Engineers, Retired

Iliana E. Mladenova Hydrological Sciences Branch NASA Goddard Space Flight Center Greenbelt, Maryland Binayak P. Mohanty Biological and Agricultural Engineering Texas A&M University College Station, Texas Rabi H. Mohtar Biological and Agricultural Engineering and Zachry Department of Civil Engineering Texas A&M University College Station, Texas Kazuro Momii Professor, Faculty of Agriculture Kagoshima University Korimoto, Japan Alberto Montanari Department of Civil, Chemical, Environmental and Materials Engineering - DICAM ALMA MATER STUDIORUM - Università di Bologna Bologna, Italy R. Morbidelli Department of Civil and Environmental Engineering Perugia University Perugia, Italy Biswajit Mukhopadhyay National Practice Leader, Water Resources North American Infrastructure Jacobs Engineering Group, Inc. Dallas, Texas

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Fred L. Ogden Department of Civil and Architectural Engineering and Haub School of Environment and Natural Resources University of Wyoming Laramie, Wyoming N. Ohara Department of Civil and Architectural Engineering University of Wyoming Laramie, Wyoming C. S. P. Ojha Department of Civil Engineering Indian Institute of Technology Roorkee Roorkee, India Taha B. M. J. Ouarda iWATER Center, Masdar Institute of Science and Technology Abu Dhabi, United Arab Emirates Hydro-Climate Modeling Lab INRS-ETE Quebec, Canada Hisashi Ozawa Graduate School of Integrated Arts and Sciences Hiroshima University Higashi-Hiroshima, Japan Umed S. Panu Department of Civil Engineering Lakehead University Thunder Bay, Canada Simon Michael Papalexiou Department of Water Resources and Environmental Engineering School of Civil Engineering National Technical University of Athens Zographou, Greece

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Contributors    xlix  Thanos Papanicolaou Department of Civil and Environmental Engineering University of Tennessee Knoxville, Tennessee

Manuel Pulido-Velazquez Research Institute of Water and Environmental Engineering (IIAMA) Universitat Politècnica de València Valencia, Spain

Sandra Pavlovic University Corporation for Atmospheric Research Boulder, Colorado

Narendra Singh Raghuwanshi Agricultural and Food Engineering Department Indian Institute of Technology Kharagpur Kharagpur, India

Pavla Pekarova Institute of Hydrology Slovak Academy of Sciences Bratislava, Slovakia Angela Pelle Department of Civil and Environmental Engineering University of Alabama Tuscaloosa, Alabama Mauri Pelto Department of Environmental Science Nichols College Dudley, Massachusetts Silvio J. Pereira-Cardenal COWI A/S Lyngby, Denmark Magnus Persson Department of Water Resources Engineering Lund Institute of Technology Lund University Lund, Sweden Muthiah Perumal Professor, Department of Hydrology Indian Institute of Technology Roorkee Roorkee, India Jacob Petersen-Perlman Oregon State University Corvallis, Oregon Christa D. Peters-Lidard Hydrological Sciences Laboratory NASA Goddard Space Flight Center Greenbelt, Maryland Hai V. Pham Postdoctoral Fellow, Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, Louisiana Mikołaj Piniewski Warsaw University of Life Sciences Warsaw, Poland Paulo Rógenes Monteiro Pontes Instituto de Pesquisas Hidráulicas Universidade Federal do Rio Grande do Sul IPH-UFRGS Porto Alegre, Brazil Erik C. Porse UCLA Institute of the Environment and Sustainability Los Angeles, California Nawa R. Pradhan Research Civil Engineer, Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi James Prairie Hydrologic Engineer, Upper Colorado Region Bureau of Reclamation Boulder, Colorado K. S. H. Prasad Department of Civil Engineering Indian Institute of Technology Roorkee Roorkee, India

Ataur Rahman Water and Environmental Engineering School of Computing, Engineering and Mathematics Western Sydney University Penrith, Australia Balaji Rajagopalan Department of Civil, Environmental and Architectural Engineering University of Colorado Boulder, Colorado Harji D. Rank Professor, Department of Soil and Water Engineering, College of Agricultural Engineering and Technology, Junagadh Junagadh Agricultural University, Junagadh Gujarat, India Liliang Ren Professor, State Key Laboratory of Hydrology, Water Resources and Hydraulic Engineering Hohai University Nanjing, China Ben Renard School of Engineering University of Newcastle Callaghan, Australia Pedro J. Resptrepo National Weather Service North Central River Forecast Center Chanhassen, Minnesota Mehdi Rezaeianzadeh School of Forestry and Wildlife Sciences Auburn University Auburn, Alabama Niels Riegels DHI Hørsholm, Denmark Matthew Rodell Laboratory Chief, Hydrological Sciences Laboratory NASA Goddard Space Flight Center Greenbelt, Maryland Renata del Giudice Rodriguez CAPES Foundation Ministry of Education of Brazil, Brasilia, Distrito Federal, Brazil A. Charles Rowney Manager, ACR, llc Longwood, Florida Albert. I. Rugumayo Faculty of Engineering Ndejje University Kampala, Uganda College of Engineering, Design Art and Technology Makerere University Kampala, Uganda Udisha Saklani Former Research Assistant at the Institute of Water Policy, Lee Kuan Yew School of Public Policy, National University of Singapore, Singapore Jose D. Salas Professor Emeritus, Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado

Roland K. Price Emeritus Professor, UNESCO-IHE, Institute for Water Education Delft, The Netherlands

Samuel Sandoval-Solis University of California Davis Department of LAWR Davis, California

Fernando Falco Pruski Agricultural Engineering Department Federal University of Viçosa Viçosa, Brazil

Sankar Sarkar Physics and Applied Mathematics Unit Indian Statistical Institute Kolkata, India

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l    Contributors Arup K. Sarma Professor, Department of Civil Engineering Indian Institute of Technology Guwahati Guwahati, India Shinji Sato Department of Civil Engineering University of Tokyo Tokyo, Japan John Schaake Annapolis, Maryland William A. Scharffenberg CEIWR-HEC Hydrologic Engineering Center Davis, California Anton Schleiss Laboratoire de constructions hydrauliques (LCH) Ecolepolytechniquefédérale de Lausanne (EPFL) LCH - ENAC - EPFL Lausanne, Switzerland

Jery R. Stedinger Dwight C. Baum Professor of Engineering School of Civil and Environmental Engineering Cornell University Ithaca Ithaca, New York R. Subbaiah Department of Soil and Water Engineering College of Agricultural Engineering and Technology Junagadh Agricultural University Junagadh, India Caroline A. Sullivan School of Environment, Science and Engineering Lismore Campus Marine Ecology Research Centre Southern Cross University New South Wales, Australia Ge Sun Research Hydrologist, Eastern Forest Environmental Threat Assessment Center Southern Research Station, USDA Forest Service Raleigh, North Carolina

Sergio E. Serrano Department of Civil and Environmental Engineering Temple University Philadelphia, Pennsylvania

Liqun Sun Department of Civil Engineering The University of Hong Kong Hong Kong, China

Yury V. Shan’ko Scientific Researcher, Institute of Computational Modeling SB RAS Krasnoyarsk, Russia

Óli G. B. Sveinsson Landsvirkjun (The National Power Company of Iceland) Reykjavík, Iceland

Ashish Sharma School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia

David G. Tarboton Department of Civil and Environmental Engineering Utah State University Logan, Utah

Jack Sharp University of Texas Austin, Texas

Gokmen Tayfur Department of Civil Engineering Izmir Institute of Technology Izmir, Turkey

Mohsen Sherif Civil and Environmental Engineering Department College of Engineering UAE University United Arab Emirates Haiyun Shi State Key Laboratory of Hydroscience and Engineering Tsinghua University Beijing, China Joel S. Sholtes Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado Lucas Siegfried Ahtna Engineering Services, LLC Marina, California Vijay P. Singh Department of Biological and Agricultural Engineering & Zachry Department of Civil Engineering Texas A&M University College Station, Texas Bellie Sivakumar School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia Department of Land, Air and Water Resources University of California Davis, California Brian E. Skahill Research Civil Engineer, Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi Lambert K. Smedema Independent Irrigation/Drainage Consultant Arnhem, The Netherlands Soroosh Sorooshian Center for Hydrometeorology and Remote Sensing Department of Civil and Environmental Engineering University of California Irvine Irvine, California

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Charles J. Taylor Water Resources Section Kentucky Geological Survey University of Kentucky Lexington, Kentucky Ramesh S. V. Teegavarapu Associate Professor, Department of Civil, Environmental and Geomatics Engineering Florida Atlantic University Boca Raton, Florida A. K. Thakur Darbhanga College of Engineering Darbhanga, India Reed Thayer Center for Watershed Sciences University of California Davis, California Mark Thyer School of Engineering University of Newcastle Callaghan, Australia Ezio Todini Retired Professor, Dipartimento di Scienze Biologiche, Geologiche e Ambientali University of Bologna Bologna, Italy Glenn Tootle Associate Professor, The University of Alabama Department of Civil, Construction and Environmental Engineering (CCEE) Tuscaloosa, Alabama Cecilia Tortajada Institute of Water Policy Lee Kuan Yew School of Public Policy National University of Singapore Singapore Shivam Tripathi Department of Civil Engineering Indian Institute of Technology Kanpur Kanpur, India

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Contributors    li  Frank T.-C. Tsai Professor, Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, Los Angeles

Ralph A. Wurbs Zachry Department of Civil Engineering Texas A&M University College Station, Texas

Raphael M. Tshimanga Faculty of Agronomic Sciences, Department of Natural Resources Management & CB-HYDRONET University of Kinshasa Kinshasa, Democratic Republic of the Congo

Yi Xiao Department of Systems Design Engineering University of Waterloo Waterloo, Canada

Yeou-Koung Tung Chair Professor, Disaster Prevention & Water Environment Research Center National Chiao Tung University Hsinchu, Taiwan Kamshat Tussupova Department of Water Resources Engineering Lund University Lund, Sweden Julie A. Vano Research Applications Laboratory National Center for Atmospheric Research Boulder, Colorado Jennifer C. Veilleux Oregon State University Corvallis, Oregon Daniele Veneziano Department of Civil and Environmental Engineering Massachusetts Institute of Technology Cambridge, Massachusetts R. Vignesh Department of Civil Engineering Indian Institute of Technology Bombay Mumbai, India Gabriele Villarini IIHR-Hydroscience and Engineering The University of Iowa Civil and Environmental Engineering Iowa City, Iowa Richard M. Vogel Department of Civil and Environmental Engineering Tufts University Medford, Massachusetts Haruko M. Wainwright Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California Glenn Warner Department of Natural Resources and the Environment The University of Connecticut Storrs, Connecticut Wallace A. Wilson Advisor, ASFPM Foundation Association of State Floodplain Managers, Inc. Madison, Wisconsin Fitsum M. Woldemeskel School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia

Dawen Yang Department of Hydraulic Engineering Tsinghua University Beijing Beijing, China Daqing Yang National Hydrology Research Center Environment Canada Saskatoon, Canada Xiaoli Yang Associate Professor, College of Hydrology and Water Resources Hohai University Nanjing, China Sooyeon Yi University of California Davis, California Peter C. Young Professor Emeritus, Systems and Control Group, Lancaster Environment Centre & Integrated Catchment Assessment and Management Centre Fenner School of Environment & Society Lancaster University, UK & Australian National University, Canberra, ACT Australia S. Yu Witte Moscow University Moscow, Russia Fei Yuan Associate Professor, State Key Laboratory of Hydrology Water Resources and Hydraulic Engineering Hohai University Nanjing, China Edith Zagona Research Professor, Department of Civil, Environmental and Architectural Engineering University of Colorado Boulder, Colorado Kaveh Zamani Department of Mechanical and Aerospace Engineering University of California San Diego, California Ilya Zaslavsky Spatial Information Systems Laboratory San Diego Supercomputer Center University of California, San Diego La Jolla, California Qiang Zhang Department of Water Resources and Environment Sun Yat-sen University Guangzhou, China

Aaron T. Wolf Oregon State University Corvallis, Oregon

Lan Zhang Assistant Professor, Department of Civil Engineering University of Akron Akron, Ohio

Ming-ko Woo School of Geography and Earth Sciences McMaster University Hamilton, Canada

Igor S. Zonn Director, Engineering Research Production Center for Water Management, Land Reclamation and Ecology “Soyuzvodproject” Moscow, Russia

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International Advisory Board Chair: Richard H. McCuen The Ben Dyer Professor of Civil & Environmental Engineering Department of Civil & Environmental Engineering University of Maryland Maryland Andras Bardossy Lehrstuhl für Hydrologie und Geohydrologie Institute für Wasser- und Umweltsystemmodellierung Universität Stuttgart Stuttgart, Germany Ronny Berndtsson Department of Water Resources Engineering & Center for Middle Eastern Studies Lund University Lund, Sweden Corrado Corradini Department of Civil and Environmental Engineering University of Perugia Perugia, Italy James F. Cruise Department of Civil and Environmental Engineering University of Alabama Huntsville, Alabama João Pedroso de Lima Department of Civil Engineering University of Coimbra Coimbra, Portugal M. Fiorentino Department of Environmental Engineering and Physics University of Basilicata Potenza, Italy R. S. Govindaraju Delon and Elizabeth Hampton Hall of Civil Engineering Purdue University West Lafayette, Indiana Nilgun B. Harmancioglu Dokuz Eylul University Water Resources Management Research & Application Center (SUMER) Izmir, Turkey Sharad K. Jain Water Resources Systems Division National Institute of Hydrology Roorkee, India M. Levent Kavvas Hydrologic Research Laboratory Department of Civil and Environmental Engineering University of California Davis, California Lev Kuchment Laboratory of Hydrological Cycle Water Problems Institute of the Russian Academy of Sciences, Moscow, Russia Venkat Laxmi Department of Earth and Ocean Sciences University of South Carolina Columbia, South Carolina Upmanu Lall Department of Earth and Environmental Engineering and Department of Civil Engineering and Engineering Mechanics Columbia Water Center International Research Institute for Climate and Society Columbia University New York, New York

P. E. O’Connell School of Civil Engineering and Geosciences Newcastle University New Castle upon Tyne, United Kingdom C. S. P. Ojha Department of Civil Engineering Indian Institute of Technology Roorkee Roorkee, India Umed S. Panu Department of Civil Engineering Lakehead University Thunder Bay, Canada Fernando Falco Pruski Agricultural Engineering Department Federal University of Viçosa Viçosa, Brazil Liliang Ren Professor, State Key Laboratory of Hydrology, Water Resources and Hydraulic Engineering Hohai University Nanjing, China  Dan Rosbjerg Department of Environmental Engineering Technical University of Denmark Kongens Lyngby, Denmark Jose D. Salas Professor Emeritus, Department of Civil and Environmental Engineering Colorado State University Fort Collins, Colorado Sergio E. Serrano Department of Civil & Environmental Engineering Temple University Philadelphia, Pennsylvania Ashish Sharma School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia Witold Strupczewski Water Resources Department Institute of Geophysics Polish Academy of Sciences Warsaw, Poland Ezio Todini Retired Professor, Dipartimento di Scienze Biologiche, Geologiche e Ambientali University of Bologna Bologna, Italy Dawen Yang Department of Hydraulic Engineering Tsinghua University Beijing, China Qiang Zhang Department of Water Resources and Environment Sun Yat-sen University Guangzhou, China

liii

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Practitioner Advisory Board Chair: R. D. Singh National Institute of Hydrology Roorkee, India

Aaron R. Byrd Research Hydraulic Engineer & Branch Chief Hydrologic Systems Branch Coastal and Hydraulics Laboratory Engineer Research Development Center U.S. Army Corps of Engineers Vicksburg, Mississippi Anthony S. Donigian, Jr. Principal Consultant AQUA TERRA Consultants Mountain View, California Donald K. Frevert Hydraulic Engineer and Team Leader, Retired Technical Services Center Bureau of Reclamation Lakewood, Colorado Randall (Randy) W. Gentry Argonne National Laboratory Environmental Science Division Argonne, Illinois

Mohamed M. Hantush Research Hydrologist National Risk Management Research Laboratory U.S. Environmental Protection Agency Cincinnati, Ohio T. Moramarco National Research Council, Institute for Hydrogeological Protection, Perugia, Italy John W. Mueller State Conservation Engineer USDA-NRCS Temple, Texas Biswajit Mukhopadhyay National Practice Leader | Water Resources North American Infrastructure Jacobs Engineering Group, Inc. Dallas, Texas Dallas, Texas J. Obeysekara Hydrologic & Environmental Systems Modeling South Florida Water Management District

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PART

1

FUNDAMENTALS

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Chapter

1

The Hydrologic Cycle BY

VIJAY P. SINGH

ABSTRACT

Hydrology can be called as the study of the hydrologic cycle. There are several biogeochemical cycles in nature that are interactive, but the dominant among these cycles is probably the hydrologic cycle. This chapter discusses the hydrologic cycle and its association with carbon and nitrogen cycles. It also discusses the impact of climate change on the hydrologic cycle. Before discussing the hydrologic cycle, it will be useful to discuss the characteristics of water at a more fundamental level that impact the behavior of water at large scales. 1.1  CHARACTERISTICS OF WATER

Water is the most common substance on earth and covers more than 70% of its surface. It is vital for life on earth and for that matter anywhere in the universe. Water is a simple compound where each water molecule is composed of two hydrogen (H) atoms and one oxygen (O) atom. This atomic structure causes the water molecules to have unique electrochemical properties. Each hydrogen atom shares its single electron with the oxygen atom, and thus these atoms are bonded together as H-O-H. If we consider two water molecules, then it is seen that there occurs an attraction between a hydrogen atom of one water molecule and the oxygen atom of another molecule. The bonding between these molecules has low energy and is often referred to as hydrogen bonding (Brady and Weil, 2008). This bonding is responsible for polymerization of water as well as water’s high boiling point, specific heat, and viscosity in comparison with other hydrogen-containing compounds. It is the structure of water molecule that determines its ability to influence soil processes as well as its existence as a liquid at temperatures observed on Earth, and makes it a powerful solvent. The bulk water, what is usually discussed in hydrology, is the collection of water molecules. The arrangement of these molecules, that constitutes the bulk water, is still being debated. The retention and movement of water in soils are partly caused by two forces: cohesion and adhesion. Cohesion is the attraction of water molecules for each other, and adhesion is the attraction of water molecules for solid surfaces. In soil water, adhesion is also called adsorption. In the soil, the water molecules are attached to solid surfaces by virtue of adhesion. These water molecules are attached to other water molecules by virtue of cohesion, and these are attached to other molecules farther away from the solids, and so on. Thus, through forces of cohesion and adhesion, the water is retained and it moves through the soil. The forces of cohesion and adhesion are also responsible for another property of water, called surface tension, commonly observed at the liquid-air interfaces. Surface tension together with adhesion causes capillarity (the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity). It is known that electrons and protons are the elementary units of charge and they have opposite charges, that is, one has a positive charge, while the other has a negative charge. Electrons and protons play a central role in determining how water behaves. Since the oxygen atom has a charge of –2 and each hydrogen atom has a plus charge, the water molecule is reduced to have no charge or is neutral, since the negative and positive units of charge cancel each other. When protons latch on to water molecules, the result is the formation

of hydronium ions that are highly mobile and can wreak havoc. A positively charged water molecule is a hydronium ion. If we look at the structure of the water molecule, then it is found that the hydrogen atoms are tied to the oxygen atom in a V-shaped arrangement at an angle of 105° resulting in an asymmetrical configuration. This also means that the shared electrons are closer to oxygen than to hydrogen and the charges are not evenly divided. This phenomenon is characterized as polarity. Thus, there are two sides of the configuration, one side on which hydrogen atoms are located and the other opposite side. The hydrogen side tends to be electropositive and the opposite side electronegative. When water changes from liquid to solid or vapor or vice versa, the water molecules arrange themselves in distinctly different patterns during the phase change. When water becomes ice, the molecular arrangement of ice results in an increase in volume and a decrease in density. The pattern of molecules is highly organized and is in a rigid geometric form. When water freezes to ice, there occurs an expansion of molecules, causing ice to float on water. In the case of liquid water, water molecules organize themselves in small groups of joint particles. It is this property that permits water to move and flow. The molecules of water vapor tend not to form bonds among themselves and are in a state of high energy, causing molecules to be moving. The polarity characteristic is the cause for the attraction of water molecules to electrostatically charged ions and colloidal surfaces. This is how hydration occurs. For example, cations, such as Na+, K+, and Ca2+, get hydrated because of their attraction to the negative or oxygen end of water molecules. Likewise, negatively charged clay surfaces attract water through the positive or hydrogen end of molecules. Salt gets dissolved in water because its ionic components have more attraction for water molecules than for each other. Water is found to have three phases: solid (as snow and ice), liquid, and vapor. It is found in the liquid phase and the solid phase on the land surface and beneath, in the liquid phase only in the oceans and seas, and in the vapor phase in the atmosphere. Recently, another phase, which is beyond these three phases, has been discovered (Pollack, 2013). This is the fourth phase and is referred to the “exclusion zone” (EZ). The EZ forms next to the many submerged materials and is unexpectedly a large zone of water. It derives its name from the fact that it excludes practically everything. It contains a lot of charge and its characteristics are different from those of the bulk water. There are many water-related mysteries or phenomena in nature that we observe but we do not understand them well, because they comprise the EZ that we do not quite understand. In his masterpiece book, Pollack (2013) details such phenomena and the social behavior of water. Examples of such mysteries include water serving as a glue when building sand castles from wet sand, tsunami waves traveling very long distances before petering out, slipperiness of ice, swelling upon bruising or breakage, freezing warm water, flow of water upward from plant roots through narrow columns, cracking of concrete by upwelling tree roots, spreading of droplets on surfaces, walking on water, formation of isolated clouds, floating of ice, consistency of yogurt, migration of microspheres away from the center in a beaker of water, the bridge made up of water, and floating water droplets, among others. These phenomena entail crowds of water molecules, not water at the molecular level. Nevertheless, they defy easy explanation and show that we know little about 1-3

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1-4    The Hydrologic Cycle

the interaction of water molecules with other water molecules or water’s social behavior. Surprisingly, we understand little about Earth’s most common substance—water. Water has some unique properties that no other liquid on Earth seems to enjoy. It has high specific heat, conducts heat more easily, has a neural pH, is a universal solvent, has high surface tension, exists in all three phases, occupies larger volume upon freezing, and has maximum density at 4°C. 1.2  DEFINITION OF HYDROLOGY

Hydrology deals with bulk water primarily in liquid, gaseous, and solid phases. This means that hydrology incorporates the study of snow and glaciers. It deals with both quality and quantity that vary in space and time, so hydrology deals with how water is distributed. Water occurs over the land surface, in soil, and in geological formations. Water is dynamic, so hydrology deals with the movement of water (surface and subsurface). Water can be stored in lakes and reservoirs as well as in subterranean environment, so hydrology deals with the storage of water. The occurrence, distribution, movement, and storage of water quality and quantity are not always entirely deterministic, so hydrology encompasses their statistical characteristics. Thus, hydrology can be defined as the science that deals with the space, time and frequency characteristics of the occurrence, distribution, movement, and storage of the quantity and quality of the waters of the Earth. Frequently, hydrology is partitioned into surface-water hydrology, vadose zone hydrology, groundwater hydrology, and snow and glacier hydrology. In all of these partitions, both water quantity and quality are dealt with. To emphasize quality aspects, water quality hydrology is sometimes considered a separate branch. Likewise, other partitions of hydrology are agricultural hydrology, forest hydrology, wetland hydrology, environmental hydrology, ecosystems hydrology, atmospheric hydrology, arid lands hydrology, coastal hydrology, urban hydrology, and geohydrology. These partitions point to the broad scope and the interdisciplinary nature of hydrology. Also, these branches are not entirely insulated from each other. Hydrology should be viewed in a broader context, for water is sometimes on the surface and at other times, it is below the surface but then again it reappears on the surface. Depending upon the methods of analysis and synthesis, hydrology is also divided into different branches, such as mathematical hydrology, numerical hydrology, digital hydrology, systems hydrology, parametric hydrology, empirical hydrology, statistical hydrology, and stochastic hydrology. Based on the emphasis of scientific concepts, hydrology can be divided into physical hydrology, chemical hydrology, and biological hydrology.

1.3  HYDROLOGIC CYCLE

There are two systems through and between which the water moves: Earth and atmosphere. Earth can be divided into oceans and seas, and the land part. Oceans and seas form what is called the hydrosphere. The land part, also called land sphere, is divided into the land surface part and subsurface part that includes unsaturated (soils) and saturated parts (geologic formations). The land surface also includes water bodies, such as lakes and reservoirs, wetlands, canals and rivers, channels, and lagoons. The unsaturated part is referred to as pedosphere and the saturated part as lithosphere. The characteristics of water movement, distribution, occurrence, and storage are different in these spheres, and to understand these characteristics, this partitioning is essential. The interconnectivity of these spheres is essential for water moving through the hydrologic cycle. On the continents, the water evaporates from the hydrosphere, pedosphere, and lithosphere, and is transpired by vegetation and plants into the atmosphere. The water vapor moves in the atmosphere, and under suitable conditions, it condenses and precipitates over the land surface and over oceans and seas. The liquid part of the precipitated water (or melted snow and ice) runs off in part over the land surface and in part into the pedosphere from where water either transpires/evaporates back to the atmosphere or percolates to groundwater (or lithosphere). Evaporation takes place from open surfaces (rivers, lakes, wetlands, etc.) as well. The water that is not reevaporated back to the atmosphere continues to the oceans, where a net evaporation takes place. Conversely, a net precipitation occurs over the continents. This endless circulation of water from the hydrosphere and land sphere to the atmosphere and back is called the hydrologic cycle. It has no beginning and no end. The hydrologic cycle can be viewed as a natural machine, a constantly running, distillation and pumping system. The primary source of energy for the operation of this machine is the Sun that supplies heat energy. In Fig. 1.1, the spatial and temporal averages of these energy components are illustrated. Of particular importance for hydrology is the loss of heat energy from the Earth’s surface. The 29% share of heat loss can be further subdivided into 22% loss as latent heat and 7% loss as sensible heat. It should be emphasized that there are huge variations from the average picture both in time (diurnal, seasonal, and yearly) and in space. Together with the force of gravity, this energy keeps the water moving as evaporation and transpiration from the earth to the atmosphere, as condensation and precipitation from the atmosphere to the earth, and as streamflow and groundwater from the Earth to the oceans. Thus, the hydrologic cycle encompasses three major systems with the hydrosphere as

Shortwave solar radiation

Space

(342

Longwave radiation and heat transfer

30% reflected and scattered

W/m2)

70% radiated

100% 26% reflected and scattered

Earth’s atmosphere

26% absorbed

65% radiated 109% absorbed

5% lost to space

4% reflected

Earth’s surface

96% radiated back down

Greenhouse effect

29% lost as latent and sensible heat

Back radiation 47% absorbed

Figure 1.1  The global energy balance.

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Schematic Representation of the Hydrologic Cycle     1-5 

Storage in ice and snow

Moisture over land Condensation Precipitation on land Evaporation from land

Surface runoff

Precipitation on ocean

Evapotranspiration Freshwater storage Lake

Infiltration

Evaporation Sub

sur

fac e

flow

Evaporation from ocean

Lake

Groundwater flow

Surface outflow

Groundwater outflow Ocean

Groundwater storage

Ocean storage Figure 1.2  The hydrologic cycle [Source: http://water.usgs.gov].

the major source of water, the atmosphere as the deliverer of water, and the land sphere as user of water. It may be noted that water is transported, temporarily stored, and may change state in each sphere. Consider, for example, the atmosphere where water occurs as vapor flow, stored as vapor storage in the atmosphere, and condenses and precipitates under change of state from vapor to either that of liquid or solid. Much more water stays in storage for longer periods of time than in movement through the hydrologic cycle.

land surface, replenishing soil moisture, and percolating down to recharge groundwater. Part of the infiltrated water includes subsurface runoff or interflow, and groundwater runoff or baseflow. Part of rainfall and snowmelt runs off over the land surface, joins streamflow, and eventually joins the sea. Streamflow includes surface runoff as well as subsurface runoff. It may be noted that these different components have different time scales and their significance is different at different scales.

1.4  COMPONENTS OF THE HYDROLOGIC CYCLE

1.5  SCHEMATIC REPRESENTATION OF THE HYDROLOGIC CYCLE

The major components of the hydrologic cycle are precipitation, evaporation, infiltration, groundwater, and streamflow. Precipitation includes all forms of water that falls from the atmosphere, including rainfall, snowfall, hail, sleet, drizzle, dew, and fog. Rainfall and snowfall constitute the main forms of precipitation. Snow is a form of stored water that remains where it falls until melting occurs. Each year, approximately 505,000 km3 (121,000 mi3) of water falls as precipitation of which about 78% [approximately 398,000 km3 (95,000 mi3)] falls over the oceans. The precipitation falling as rain is about 107,000 km3 (26,000 mi3) and as snow is about 1000 km3 (240 mi3). Precipitation is partitioned into four parts: (1) interception, (2) evaporation, (3) infiltration, and (3) runoff. Part of precipitation is intercepted by trees, buildings, and other abstract objects. Most of this portion is evaporated to the atmosphere. A small portion of snow also evaporates. Oceans contribute about 86% of global evaporation that reduces oceanic temperature through evaporative cooling. Part of rainfall fills surficial depressions, forming small ponds, where some water infiltrates and some evaporates. Another part of rainfall infiltrates the

The hydrologic cycle can be illustrated as shown in Fig. 1.2, depicting the various components and their principal directions of flow. This, however, does not provide any information on the relative significance of these components which varies with the space (watershed, land system, and global) and time scales. The cycle is truly an endless circulation at the global scale, but not necessarily so at smaller scales. At the global scale, the atmosphere, the hydrosphere, and the land sphere are connected and interactive, as shown in Fig. 1.3. The major components of the cycle at this spatial scale are precipitation, evaporation and transpiration, and streamflow. The time scale is also much larger and hence more transient components, such as infiltration, surface runoff, interception, etc., become embedded in other components. The global scale hydrologic cycle is a closed system and it is normally neither modeled nor used for water resources planning and management at this scale. The general circulation models are, however, employed for climate impact studies. Of greater significance is the hydrologic cycle at the watershed scale which can be quite small or very large. At this spatial scale, all components of the

Precipitation Atmospheric system

Precipitation ET

River flow Earth system

Tidal flow, sea rise, GW

Ocean system

Evaporation Figure 1.3  A global schematic of the hydrologic cycle [Source: Singh, 1992].

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1-6    The Hydrologic Cycle Precipitation

Global

Global weather systems

ET 10,000 Km

Exfiltration

1000 Km

Infiltration

Subsurface system Percolation (recharge)

Surface runoff

100 Km Subsurface runoff

Upward moisture movement

Aquifer system

Stream flow

Local

Soil formation

Development of major river rasins

Drainage network formation Soil erosion Shallow ground water Nutrient circulation cycles

Mesoscale weather Mesoscale systems soil (floods) moisture variation

10 Km 1 Km

Ground water runoff

Runoff cycle

Space

Land system

Global CO2 variations

Thunder storm

Time

2

Second Minute

4

6 Day

8

10

Year Century

Figure 1.4  A schematic of the hydrologic cycle of the earth system [Source: Singh, 1992].

12

14

Log/ 16 second

One million One billion years years

Figure 1.5  Spatial and temporal scales of hydrologic processes and cycle [Source: National Researach Council, 1991].

hydrologic cycle play their role, but the significance of their role depends on the time scale. For example, at small time scales, evapotranspiration may not be significant, but infiltration plays a dominant role. The reverse happens at the large time scale. If the watershed is forested or highly urbanized, then interception may be significant. At a large watershed scale, the hydrologic cycle is illustrated in Fig. 1.4. 1.6  SCALES IN HYDROLOGIC CYCLE

The significance of the components of the hydrologic cycle is closely connected with their space and time scales. Further, mathematical relationships describing the different components depend on the scale. Therefore, these scales play a critical role when simulating the cycle or the components thereof. 1.6.1  Time Scales

The components of the hydrologic cycle have vastly different time scales of movement and residence times. The residence time of a reservoir is the average time the water will spend in the reservoir (i.e., the storage divided by the flux through the reservoir), and thus is also a measure of the age of water of that reservoir. For example, the rate at which surface runoff moves is very high as compared with that of groundwater. Likewise, the residence time of surface water is a small fraction of that of groundwater. Groundwater may reside for over 10,000 years below the earth’s surface before leaving, whereas water spends about 2–6 months in rivers, about 1–2 months in the soil, and only a little over a week in the atmosphere. These times play a critical role in numerical simulation of the hydrologic cycle. The time step should be sufficiently small so that the variations in the component processes are captured in sufficient detail without putting undue burden on data collection and computational efforts. Shiklomanov (1999) has presented the estimated periods of renewal of water resources on the earth, as shown in Table 1.1. Table 1.1  Periods of Water Resources Renewal on the Earth Water of hydrosphere and land sphere

Period of renewal

World Ocean

2,500 years

Groundwater

1,400 years

Polar ice

9,700 years

Mountain glaciers

1,600 years

Ground ice of the permafrost zone

10,000 years

Lakes

17 years

Bogs

5 years

Soil moisture

1 year

Channel network

16 days

Atmospheric moisture

8 days

Biological water

Several hours

[Source: Shiklomanov, 1999]

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The spatial and temporal scales of some hydrologic processes and cycles are shown in Fig. 1.5. 1.6.2  Spatial Scales

Spatial scales that are important for hydrologic investigations are hillslope, catchment, basin, regional, continental, and global. At the global scale, the atmosphere, the hydrosphere, and the land and lithosphere are strongly connected through the processes of precipitation, evaporation, and runoff. Global fluxes and circulation patterns are investigated at this scale and are important for assessing the impact of climate change on the hydrologic cycle. Figure 1.2 shows a global view of the hydrologic cycle. The catchment scale has a very broad range, ranging from a few square kilometers to tens of thousands of kilometers. For small catchments, the processes of precipitation, runoff, infiltration, soil moisture, and runoff are most important. As catchment becomes large, groundwater in most cases also plays an important role. 1.7  IMPACT OF CLIMATE CHANGE ON THE HYDROLOGIC CYCLE

It is now widely accepted that as a result of anthropogenic influences, there is global warming and the climate is changing. The increased emission of greenhouse gases is considered to be the principal cause of increase in Earth’s temperature. This is supported by measured temperature data series as well as paleoclimatological records. The implications of climatic change for the hydrologic cycle are many. The increase in temperature would cause higher evapotranspiration; changes in patterns, timing, intensity, and distribution of precipitation; melting of polar ice caps and recession of glaciers; more frequently occurring droughts; increased frequency of flooding; higher melting of polar ice and glaciers; sea water level rise; inundation of islands of low elevations as well as cities adjacent to seas; changes in vegetation dynamics; ecosystem health; and so on. These changes are already being witnessed. Nevertheless, it is not understood with acceptable degree of certainty as to what, where, and how much changes are or will occur. On the other hand, an increase in temperature may mean greater precipitation, some of which may occur in the form of snow at the poles, leading to an additional accumulation of ice. Since the three spheres constitute a continuum, changes in the land sphere and lithosphere would cause a change in the climate. For example, changes in vegetation mean changes in evapotranspiration, soil moisture, albedo and radiation balance; burning of fossil fuels; increased use of water day-to-day needs; increased irrigation; increased industrial activities; and large-scale water transfer between basins would contribute to climate change. Climate change manifests itself in changed patterns of spatial and temporal variability in the components of the hydrologic cycle. 1.8  INFLUENCE OF HUMAN ACTIVITIES AND LAND USE CHANGES ON HYDROLOGIC CYCLE

There are a multitude of changes, minor or major, caused in watersheds by human activities. These changes influence virtually all components of the hydrologic cycle. The watershed changes can be either point or nonpoint

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Relation between Hydrologic cycle and Carbon and Nitrogen Cycles     1-7 

changes. Examples of point changes are structural changes, such as dam construction, channel improvement, and detention storage, and these changes affect watershed response in terms of evaporation, seepage, residence time, etc. Examples of nonpoint land use changes, that affect catchment response, include forestry, agriculture, mining, and urbanization. The seriousness of hydrologic consequences depends on the magnitude of watershed changes. There is growing need to quantify the impact of man-made changes on the hydrologic cycle in order to anticipate and minimize the potential environmental detriment and to satisfy water resources requirements of the society. Agricultural operations exercise a significant influence on the hydrologic response of watersheds. Agriculture alters land cover, influencing evapotranspiration; changes upper soil layers, altering infiltration, soil moisture, leaching and evaporation; irrigating land, causing recharge and changing water table; applying fertilizers, causing soil and water pollution; and tilling the soil, causing soil erosion by wind and water. These changes lead to changes in evapotranspiration, overland flow, channel flow, and infiltration. Urbanization transforms forested and agricultural land into urban areas where houses, roads, schools, buildings, shopping malls, parks, parking lots, sewers, etc., are constructed. Urbanized lands are dominated by paved or impervious surfaces that reduce infiltration and evapotranspiration and increase storm water. The hydrologic effects of urbanization include increases in the volume and peak of direct runoff for a given rainfall; reduction in time of travel resulting in lower lag time and lower time of concentration; diminution of baseflow; reduction in infiltration; increase in the withdrawal of groundwater; increase in pollution of rivers and aquifers; endangering the ecology; and increase in temperature of urban areas, causing a change in microclimate. The principal forest activities are afforestation (planting trees), deforestation (cutting trees), and management. Forest management includes road construction, erosion control and water management, electrification, and chemical sprays. The immediate effect of these activities is the change in vegetal cover. Deforestation virtually eliminates the interception of precipitation. Forest litter removal changes infiltration capacity of the soil and has a pronounced effect on raindrop impact and the resulting soil erosion. With the loss of vegetative cover, evapotranspiration decreases. These changes increase direct runoff, reduce surface roughness, and decrease recharge to groundwater. Typical structural changes include a dam, a weir, levees, spurs, dykes, dredging, channel improvement works, etc. A dam reservoir, in general, reduces the

peak of outflow from the reservoir and the volume of flow downstream may be considerably reduced if the reservoir water is diverted elsewhere. Channel improvements include straightening of the channel, removal of vegetation, lining of the channel, maintenance of bends, or increasing slope. These changes may translate into decreased channel roughness increasing flow velocity and hence peak discharge. Depending upon the bed material, infiltration through the bed and banks also modifies flow characteristics. 1.9  RELATION BETWEEN HYDROLOGIC CYCLE AND CARBON AND NITROGEN CYCLES

The hydrologic cycle is a biogeochemical cycle. The movement of water over and below the Earth is fundamental to the cycling of biogeochemicals. This suggests that the hydrologic cycle not only impacts other biogeochemical cycles, such as the carbon cycle and the nitrogen cycle, but is also impacted by them in a feedback system. Consider, for example, runoff and streamflow. Eroded sediment and phosphorus are carried by surface runoff to streams and rivers and further on to water bodies. Phosphorus, applied to agricultural lands through fertilizer in excess of what plants consume, is carried by surface runoff to lakes and reservoirs, and that causes eutrophication. Likewise, nitrogen is transported by surface runoff and groundwater and streamflow to water bodies. As an example, the water has become hypoxic in the Gulf of Mexico as a result of nitrates from fertilizer being transported by runoff from agricultural lands to rivers and then by the Mississippi. Transport of eroded sediment and dissolved salts causes salinity of the oceans and seas. 1.9.1  Carbon Cycle

Carbon is ubiquitous in all living things as well as is part of nonliving things, such as ocean, air, and earth. Fossil fuels, such as coal and oil, are made of carbon. By joining oxygen, it forms carbon dioxide that is present in the atmosphere. The movement of carbon back and forth among biosphere, atmosphere, hydrosphere, and geosphere defines the carbon cycle, as shown in Fig. 1.6. It is one of the biogeochemical cycles and consists of various sinks and stores of carbon and processes by which sources and sinks exchange carbon. The carbon cycle is a closed system, suggesting a fixed amount of carbon in the world. The processes in the carbon cycle are interactive. Fossil fuel burning and land use change are two of the important processes impacting the

Sunlight Auto and factory emissions

CO2 cycle

Photosynthesis Plant respiration

Organic carbon

Decay organisms

Animal respiration

Dead organisms and waste products

Fossils and fossil fuels

Root respiration

Ocean uptake

Figure 1.6  Carbon cycle [Source: https://eo.ucar.edu].

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1-8    The Hydrologic Cycle

Figure 1.7  Nitrogen cycle [Source: http://cnx.org and USGS].

balance of carbon. Oceans contain the greatest amount of actively cycled carbon in the world. Only the lithosphere stores more carbon than oceans. The geologic component of the carbon cycle operates more slowly than other parts of the global carbon cycle. Carbon leaves the geosphere in several ways, for example, during the metamorphosis of carbonate rocks. In the atmosphere, carbon exists primarily in gaseous form as carbon dioxide (CO2) and methane. Both these gases absorb and retain heat in the atmosphere, and are partly responsible for the greenhouse effect. In comparison with carbon dioxide, methane produces a large greenhouse effect per volume, but it exists in much lower concentration and is more short-lived. This explains why carbon dioxide is a more important greenhouse gas than methane. Carbon dioxide traps heat in the atmosphere, and were it not for it and other greenhouse gases, the earth would be a frozen planet. Because of so much more emission of carbon dioxide to the atmosphere, we are experiencing global warming. Plants employ carbon dioxide and sunlight to make their own food through the process of photosynthesis, and carbon becomes part of plants. On burning fossil fuels, most of the carbon enters the atmosphere as carbon dioxide. Living beings release CO2 back to the atmosphere during respiration. Large exchange of CO2 also takes place between the hydrosphere and the atmosphere. The dissolved CO2 in the oceans is used by marine biota for photosynthesis, thus carbon dioxide is entering the oceanic biosphere as well. Also, from the atmosphere carbon dioxide directly dissolves into water bodies (oceans, lakes, reservoirs, etc.) as well as dissolves in precipitation as raindrops. When dissolved in water, it reacts with water molecules and forms carbonic acid that contributes to the ocean acidity. Carbon leaves the terrestrial biosphere in different ways and at different time scales. For example, combustion releases organic carbon rapidly into the atmosphere. Carbon stored in soil can remain there for up to thousands of years before being washed into rivers by erosion or released into the atmosphere through soil respiration. 1.9.2  Nitrogen Cycle

Nitrogen (N2) is a colorless, odorless, nontoxic gas, and is essential for all known forms of life on Earth. Being close to 78%, it is the largest constituent of the atmosphere, but this form is relatively nonreactive and unusable by plants. Nitrogen in plants is used in chlorophyll molecules, which are essential for photosynthesis and plant growth. Natural fixation converts gaseous

01_Singh_ch01_p1.1-1.10.indd 8

nitrogen into compounds, such as nitrate or ammonia, which can be used by plants. This “fixed” nitrogen (or reactive nitrogen) is a limiting factor for plant growth in both managed and wild environments. The nitrogen cycle, as shown in Fig. 1.7, is the process by which nitrogen is transformed to various chemical forms through biological and physical processes. Important processes in the nitrogen cycle include fixation, ammonification, nitrification, and denitrification. The nitrogen cycle constitutes an important component of the ecosystem. The nitrogen cycle has hydrologic significance because the availability of nitrogen affects the rate of key ecosystem processes, including primary production and decomposition. When plant and animal wastes decompose, they add nitrogen to the soil. Bacteria in the soil convert those forms of nitrogen into forms that plants can use. Plants use the nitrogen in the soil to grow. People and animals eat plants; then animal and plant residues return nitrogen to the soil again, completing the cycle. Human activities, such as fossil fuel combustion, use of artificial nitrogen fertilizers, and release of nitrogen in wastewater, significantly impact the global nitrogen cycle. Nitrogen enters the nitrogen cycle from different sources: (1) air through several unique types of microorganisms that can convert N2 gas to inorganic forms usable by plants; (2) manure and decaying of plant materials; (3) application of commercial nitrogen fertilizers; and (4) as inorganic nitrogen from the atmosphere and factories. It leaves the cycle in four ways: (1) denitrification, (2) bacteria change nitrate in the soil to atmospheric nitrogen, (3) volatilization, and (4) turning urea fertilizers and manures on the soil surface into gases. Nitrogen reenters the cycle through one of the aforementioned processes or through other processes. As shown in Fig. 1.7, nitrogen changes from organic matter in the soil, to bacteria, to plants, and back to the organic matter. Nitrogen is present in the environment in a variety of chemical forms: organic nitrogen, ammonium (NH4+), nitrite (NO2−), nitrate (NO3−), nitrous oxide (N2O), nitric oxide (NO), or inorganic nitrogen gas (N2). Figure 1.7 shows how these processes fit together to form the nitrogen cycle. The main component of the nitrogen cycle starts with the element nitrogen in the air. Upon interaction with oxygen, several compounds, such as nitric oxide (NO) and nitrogen dioxide (NO2) are formed. Eventually nitrogen dioxide may react with water in rain to form nitric acid (HNO3), which may be utilized by plants as a nutrient. Nitrogen in the air becomes part of the biological matter through bacterial action and algae through a process called nitrogen fixation. Legume plants, such as clover, alfalfa, and soybeans, form

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REFERENCES    1-9 

nodules on the roots, where nitrogen-fixing bacteria take nitrogen from the air and convert it into ammonia (NH3). The main source of nitrogen in soils is from organic matter, which largely arises from plant and animal residues. The conversion of ammonia to nitrate is performed primarily by soil-living bacteria and other nitrifying bacteria. Onsite sewage facilities, such as septic tanks, release large amounts of nitrogen into the environment by discharging through drain fields into the ground. Ammonia (NH3) is highly toxic to fish, and the level of ammonia discharged from wastewater-treatment facilities must be closely monitored. Nitrogen and hydrogen react under great pressure and temperature in the presence of a catalyst to make ammonia. The reaction of ammonia and nitric acid produces ammonium nitrate, which may be used as a fertilizer. Animal wastes when decomposed also return to the Earth as nitrates. Nitrate is a concern for water quality when it is converted to the nitrate (NO3–) form because nitrate is very mobile and easily moves with water in the soil. High levels of nitrates can be toxic to newborns, causing anoxia or internal suffocation. Denitrification is the reduction of nitrates back into the largely inert N2 gas, completing the nitrogen cycle. This process is performed by some bacterial species in anaerobic conditions. Nitrous oxide gas (N2O) is a side product of this reaction. It is also produced as a result of agricultural fertilization, biomass burning, cattle and feedlots, and industrial sources. Nitrous oxide is also a greenhouse gas and is currently the third largest contributor to global warming, after carbon dioxide and methane. Although not as abundant in the atmosphere as carbon dioxide, for an equivalent mass, it is nearly 300 times more potent in its ability to warm the planet. In the atmosphere, ammonia (NH3) acts as an aerosol, decreases air quality and clings to water droplets, eventually resulting in nitric acid (HNO3) that produces acid rain. Ammonia and nitrous oxides actively alter atmospheric chemistry. They are precursors of tropospheric ozone production, which contributes to smog and acid rain, damages plants, and increases nitrogen inputs to ecosystems.

01_Singh_ch01_p1.1-1.10.indd 9

1.10 CONCLUSION

The hydrologic cycle is receiving a lot of attention these days from various quarters of the society, partly because of climate change; frequently occurring natural disasters, such as floods and droughts; increasing demand for water; and the growing recognition of water-energy-food nexus. Even laymen are now talking about the specter of climate change and the hydrologic cycle. Fortunately, in primary and junior high schools, the hydrologic cycle is being taught and this is causing an interest in water issues and climate change at the grass root level. The hydrologic cycle is fundamental to other cycles and it is important that the cycle is managed properly. The fate of human civilization and economic prosperity will depend on how well water is managed. Ancient civilizations understood this dependence well. It is now time to learn from them and begin a concerted effort to respect nature and its fundamental elements, especially air, water, and soil. REFERENCES

Brady, N. C. and R. R. Weil, The Nature and Properties of Soils, Pearson/ Prentice Hall, Columbus, OH, 2008. PhysicalGeography.net, Introduction to the Hydrosphere, http:www. physicalgeography.net/fundamentals/8n.html. Accessed on April 20, 2015. National Research Council. Opportunities in the Hydrologic Sciences, National Academy of Sciences, National Academy Press, Washington, D.C., 1991. Pollack, G. H., The Fourth Phase of Water: Beyond Solid, Liquid and Vapor, Ebner & Sons Publishers, Seattle, Washington, D.C., 2013, p. 357. Shiklomanov, I. A., World Water Resources: Modern Assessment and Outlook for the 21st Century (Prepared in the framework of IHP, UNESCO), State Hydrology Institute, St. Petersburg, Russia, 1999. Singh, V. P., Elementary Hydrology, Prentice Hall, Englewood Cliffs, New Jersey, 1992.

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Chapter

2

Watersheds, River Basins, and Land Use BY

DAWEN YANG, A.W. JAYAWARDENA, AND ZHENTAO CONG

ABSTRACT

This Chapter describes the components, characteristics, hydrological processes including the characteristics that impact them, river basins and their management, and land use, in the context of the basin scale hydrological cycle which may be considered as at the lower end of the hierarchically nested hydrological cycle. It further describes the basic components of a watershed and its delineation, topography, geology, soil type, hillslope, stream network, wetlands, artificial canals and waterways including how they impact the hydrological processes. The Chapter ends with a brief description of river basin management including the water policies in a few selected countries/ regions, and the effect of land use on the basin scale hydrological cycle including the link between land use, runoff and water quality. 2.1 INTRODUCTION

Hydrological cycle refers to the continuous circulation of earth’s water resources in solid, liquid, and vapor phases, linking the atmosphere, land, and the oceans, and powered by the energy of the sun. It can be viewed as a nested system at different scales. At the top of the hierarchy is the global hydrological cycle, in which there is no water added or lost and is a closed system. There are no inputs or outputs. At the next level is the regional or continental scale hydrological cycle, which is an open system, which can be nested into the global hydrological cycle. It may include the atmosphere, land surface, and/or the oceans. At the lower end is the basin scale hydrological cycle, which is a local open system with inputs and outputs and is of primary importance to hydrologists and water professionals. It can be defined as the hydrological cycle in a river basin bounded by its watershed and its outlet which may sometimes be the ocean. This chapter describes the components, characteristics, hydrological processes including the characteristics that affect them, river basins and their management, and land use, in the context of the basin scale hydrological cycle. The name “watershed” has its origin in the early nineteenth century from the combination of “water” and “shed” in the sense “ridge of high ground” suggested by German Wasserscheide. The Oxford English Dictionary defines watershed as an area or ridge of land that separates waters flowing to different rivers, basins or seas. Merriam Webster Dictionary defines it as a region or area bounded peripherally by a divide and draining ultimately to a particular watercourse or a body of water. United States Environmental Protection Agency defines it as the area of land where all of the water that is under it or drains off of it goes into the same place. Collins English Dictionary defines it as the dividing line between two adjacent river systems, such as a ridge. Watershed also has another definition in common usage. Oxford Dictionary defines this second usage as an event or period marking a turning point in a situation. Vocabulary.com defines it as an event marking a unique or important historical change of course or one on which important developments depended. Collins English Dictionary defines it as an important period or factor that serves as a dividing line. Similar definitions of both usages can be found in other dictionaries too. In the context of hydrology, it is the first definition that matters. There are,

however, different terminologies used in different countries and regions that convey the same meaning. For example, “watershed” and “drainage basin” are used synonymously in North America. In the United Kingdom, and in countries that follow the UK terminology the term “catchment area” is used. A second usage of the term “catchment area” is to mean an area covered/served by an institution such as a school or a hospital. A watershed (or a drainage basin or a catchment area) is the basic unit within which the processes of the basin scale take place and are of interest to hydrologists and water engineers. It is an open system in contrast to the global hydrological cycle which is a closed system. Inputs to the watershed include precipitation (rainfall, snowfall, and other minor forms) and solar energy for evaporation. Outputs include runoff into the seas or to other water bodies/ watersheds, infiltration into the subsoil, which may percolate further into deeper subsoils and finally to aquifers, evaporation, and transpiration from vegetation (evapotranspiration) and moisture replenishment. Water storage in the watershed may include interception storage, depression storage, soil moisture storage, water absorbed through roots by osmosis and stored in vegetation, water stored in natural or artificial water bodies, groundwater storage, and channel storage. Processes of water movement in a watershed include precipitation (from atmosphere to land surface), stemflow (intercepted water flowing along the stems of vegetation), infiltration (from land surface to subsoil), surface runoff (along the land surface), soil moisture movement (in the partially saturated soil zone), percolation (gravity flow from subsoil to deeper soil layers and finally to aquifers), and interflow or throughflow (flow below the surface in the downslope direction). A drainage basin is usually exoreic, meaning that the water finds its way finally to one of the major oceans. However, there are some endorheic drainage basins, particularly in arid areas without any rivers where the water does not find its way to an ocean and ends up underground, or in inland lakes within the basin or returned back to the atmosphere by evaporation. Examples of basins that do not drain into an ocean include those that drain into Caspian sea, Aral Sea, and other inland lakes. In the context of groundwater flow, the watershed below the land surface may not always coincide with that above. There can be watersheds within watersheds in a nested manner and arranged hierarchically. All land is part of one watershed or another. 2.2  COMPONENTS OF WATERSHEDS

In general, the components of a watershed include hill slopes, a stream network, water bodies, wetlands, swamps, bogs, artificial canals and waterways, hydraulic structures, floodplains, riparian zones, gullies and rills, vegetation, soils, etc. Underlying a surface watershed is a subsurface groundwater storage and seepage system, the boundaries of which may not coincide with the surface boundaries. The subsurface may include aquifers which may be unconfined or confined. Underlying the aquifers, or separating confined and unconfined aquifers lies the impermeable bedrock forming the base of a watershed. Some of the key components are described in more detail in the following sections. 2-1

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2-2     Watersheds, River Basins, and Land Use 2.2.1  Hill Slope

The simplest—and somewhat ideal—watershed has the shape of an inverted two-sided roof top. Water falling on the two sides, which may have equal or unequal areas drain into a stream which connects the two hill slopes. Natural watersheds can be of different sizes and shapes. Contributions to watershed runoff come mainly from hill slopes. They may be in the form of overland flow which consists of precipitated water in excess of the infiltration capacity which for modeling purposes is assumed to flow down the hill slope in the form of a thin sheet, and throughflow, which for modeling purposes is often assumed to flow beneath the surface of the hill slope in a saturated form. The latter type of flow exists in watersheds with highly porous surface soil layers or with subsoils consisting of soil pipes in which the saturated and partially saturated flows through the subsoil take place. Hill slope hydrology is an important area of interest to hydrologists as the first stage of water movement after precipitation has taken place along hill slopes. In modeling hill slope hydrology, the kinematic wave approximation to the St. Venant’s equations is normally made since the slope terms (physical slope as well as the friction slope) in the momentum equation are at least an order of magnitude larger than the other terms. The kinematic wave approximation has been first applied to flood flows (Lighthill and Whitham, 1955), and subsequently used in several other catchment hydrological studies (e.g., Woolhiser and Ligget, 1967; White and Jayawardena, 1975; Jayawardena and White, 1977, 1979). An advantage in using the kinematic wave equations is that a downstream boundary condition is not required for solving the constitutive equations. The flow is upstream driven. 2.2.2  Stream Network

The stream network in a watershed transports water and sediments downstream to a main stream, river, or to another water body. The denser the stream network, the faster the surface waters flow down to the main stream. Some streams in some watersheds are perennial whereas some are only ephemeral and become dry after the rain stops. Stream networks are usually characterized by their Horton-Strahler numbers (Horton, 1945; Strahler, 1952) in which each segment of a stream is treated as a node in a tree with the first order at the upstream highest level. For example, most upstream segments of the stream will be order-1. When two order-1 segments join together, they form an order-2 stream. When two order-2 segments join together, they form an order-3 stream. However, if an order-2 stream intersects an order-1 stream, it still retains order-2 status. That is, a lower order stream joining a higher order stream does not change the order of the higher order stream. The lowest order stream (order-1) has no tributaries. The highest order recorded is in Amazon River with order-12 at the mouth. Most streams are of orders 1–3. A typical stream network for a catchment in China is shown in Fig 2.1. Associated with Horton-Strahler number is another index called the bifurcation ratio which is the ratio of the number of streams of any order to the number of streams of the next higher order. A lower bifurcation ratio indicates fewer streams whereas a higher one

will have more streams implying a higher risk of flooding in a low bifurcation stream network. Typical values range from 2 to 4. Another characteristic is the fractal dimension of an individual stream that is a measure of its irregularity as seen by its meandering. In the case of a stream network, the fractal dimension is a measure of the network’s ability to fill space (or a surface area in this case). The more branches and more sinuosity the network has, the greater is its ability to fill such space. A network that completely fills a surface like space has a fractal dimension of two, whereas normal networks which are not completely spacefilling, would have fractal dimensions less than two, and which vary from place to place over the surface. The geometrical pattern of a stream network can be viewed as a fractal with a fractal dimension, D, between 1 and 2. For irregularly shaped surface areas, the perimeter (length)–area relationship is (Mandelbrot, 1982)

( P(ε))1/D α (A(ε))1/2 (2.1)

where P(ε) is the perimeter of the irregular shape measured with a measuring stick of length ε, A(ε) is the area measured in units of ε2, and α is a symbol to denote the proportionality relationship. This relationship can be extended to stream networks too. Hack (1957) has obtained the following relationship for rivers in Virginia and Maryland in USA:

P = 1.4 A D (2.2)

where P is measured along the longest stream, A is measured from topographical maps, and the fractal dimension D = 1.2. Other researchers (e.g., Rosso et al., 1991) have found slightly different values for D ranging from 1.036 to 1.29. Equation (2.2) establishes a link between fractal dimensions and Horton’s law of stream numbers.1 It can be shown (Xie, 1993, p. 46) that Du



A2 P = P0   (2.3)  A0 

where

Du =

ln(rb ) (2.4) ln(rL )

and rb and rL, respectively, are the bifurcation and length ratios of streams defined as:

rb =

Ni (2.5a) N i +1



rL =

Pi (2.5b) Pi−1

In Eqs. (2.5a) and (2.5b), Ni is the number of streams of order i, and Ni+1 is the number of streams of order i+1, Pi is the average length of streams of order i, Pi–1 is the average length of streams of order i - 1, and P0, A0 are the average length and area of the smallest streams. The bifurcation and length ratios rb and rL can respectively be obtained as the gradients of the plots of Ni vs. Ni+1 and Pi vs. Pi –1. The overall bifurcation ratio may also be obtained by averaging the bifurcation ratios of different orders. However, it is to be noted that rivers and streams are not exactly self-similar. 2.2.3  Water Bodies

Water bodies include lakes, reservoirs, detention ponds, polders, glaciers, icecaps, lagoons, moats, fjords, bays, swamps, wetlands, marshes, rivers, waterways, canals, and other areas which collect and store water. A water body can be as small as a puddle and as large as an ocean, and can be natural or artificial. They have a significant role to play in the basin scale hydrological cycle by way of increasing the time of travel of water from origin to destination. Water bodies may have different functions such as providing for drinking water, irrigation, recreation, hydropower generation, floodwater detention, aquatic life, etc., with different water quality requirements for different functions.

Figure 2.1  Sketch of a typical drainage basin of the Wuli River in Northern China. (The numbers in each stream segment refer to the Horton-Strahler numbers as explained in Sec. 2.2.2.)

02_Singh_ch02_p2.1-2.8.indd 2

1 Horton’s law of stream numbers states that there exists a geometric relationship between the number of streams of a given order Ni and the corresponding order, i. The parameter of this geometric relationship is the Bifurcation Ratio,  rb. Horton’s law of stream lengths states that there exists a geometric relationship between the average length of streams of a given order and the corresponding order, i. The parameter of this relationship is the Length Ratio, rL. Horton’s law of stream areas states that there exists a geometric relationship between the average areas drained by streams of a given order and the corresponding order i. The parameter of this relationship is the area ratio, RA

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COMPONENTS OF WATERSHEDS    2-3  2.2.4  Wetlands, Swamps, and Bogs

Under the  Ramsar international wetland conservation treaty, wetlands are defined as areas of marsh, fen, peatland or water, whether natural or artificial, permanent or temporary, with water that is static or flowing, fresh, brackish or salt, including areas of marine water the depth of which at low tide does not exceed 6 m. They may incorporate riparian and coastal zones adjacent to the wetlands, and  islands  or bodies of marine water deeper than 6 m at  low tide lying within the wetlands. They are lands where the soil is saturated with water, which determines the type of plant and animal communities that can live in the soil. They generally include marshes, swamps, bogs, and similar areas. Wetlands play important roles in the environment. They purify the water through their biofilters; they assist in flood control by detaining the floodwaters temporarily; they support shoreline stability; they are biologically diverse ecosystems that support many types of plants and animals. Plants may be completely submerged, floating, or partially submerged (emergent). Fish are the most abundant biotic species, but other amphibians such as frogs, reptiles, alligators, and crocodiles also make wetlands their home. Wetlands occur in almost all countries. The largest may be in the Amazon River Basin. Wetlands can be tidal or non-tidal. There are also artificial (man-made) wetlands, but they take a long time (of the order of 10-100 years) to attain the same vegetation composition as in natural wetlands. Wetlands are known to be carbon sinks thereby mitigating the effect of greenhouse gases. However, they are sources of nitrous oxide from the soils through denitrification.and nitrification processes. Nitrous oxide is a more potent greenhouse gas than carbon dioxide.   Forested wetlands are generally known as swamps. The largest swamps are found in the Amazon, Mississippi, and Congo River basins. A bog (sometimes called peatland) occurs where the ground surface water is acidic and is a collection of dead plant material. They are carbon sinks, if left undisturbed. 2.2.5  Artificial Canals and Waterways

In addition to the natural stream network, many watersheds have artificial canals and waterways. They may run in the same direction as the natural waterway or may cut across a watershed and connect to another watershed. A waterway is used to transport goods and people or for connecting places such as cities. Examples include the waterways in cities such as Amsterdam, Bangkok, Suzhou, and Venice. Aqueducts, or conveyancing canals, on the other hand are used to transport water from a source to a destination. They transport water for domestic use, irrigation and hydropower generation. The longest, and still existing, is perhaps the “Grand Canal” (also known as the Beijing-Hangzhou Grand Canal) of China. It passes through the city of Tianjin and the provinces of Hebei, Shandong, Jiangsu, and Zhejiang for a distance of about 1800 km. The oldest parts of the canal date back to the fifth century BC, although the various sections were eventually combined during the Sui dynasty by emperor Yang Guang (581–618 AD). It has been reconstructed several times and is now (since June 2014) a UNESCO World Heritage site. A more recent Mega project, also in China, is the South to North Water Transfer Project aimed at transferring water from the Yangtze River to Beijing, Tianjin, and other northern cities of China via two routes known as the middle and the eastern routes. The eastern route, which has a length of 1156 km with a flow capacity of 150 m3/s, generally follows the “Grand Canal,” and will divert water from the lower reach of Yangtze River to the eastern Huang-Huai-Hai Plain and terminates in Tianjin. The middle route, which has a length of 1267 km with a flow capacity of 130 m3/s, will divert water from the Danjiangkou Reservoir on the Han River, a tributary of Yangtze River, to Beijing, and Tianjin. This project, which commenced in 2002, and part of which is already completed is expected to be fully operational in the year 2015 at a cost of about 350 billion RMB (approximately US$ 57 billion). There have been irrigation canals built from ancient times. They originate from artificial reservoirs or natural lakes, or from rivers across which hydraulic structures such as low dams and weirs are built to divert the water flow. In almost all irrigation canals, the flow of water is by gravity, but in some arid regions, where the source is groundwater, water has to be pumped to the surface at a high energy cost. Irrigation started with the dawn of civilization as it was necessary for human beings to ensure food security throughout the year. The earliest recorded attempts to harness the waters of natural rivers may have been in Egypt where cuts across embankments of the River Nile were made to divert water to irrigable areas circa 6000 BC, in Mesopotamia (present Iraq and Iran), where the waters of the Tigris and Euphrates rivers were utilized circa 4000 BC, in India where irrigation canals were constructed circa 2600 BC, in China where the Zhibo channel was constructed circa 613 BC in the pre-Qin dynasty period (Xu, 2006), and the Dujiangyan irrigation system in Sichuan Province constructed circa 256 BC that still supplies water for irrigation. Nilometers, invented circa 3500 BC, were used to measure the rise and fall of the river stage in the Nile, which helped to understand the timing of flooding and in turn the timing of irrigation. The “Egyptian bucket wheel,” called

02_Singh_ch02_p2.1-2.8.indd 3

“Noria,” was invented circa 750 BC to lift water from a flowing river by clay buckets arranged at the circumference of a water wheel. This device enabled water to be lifted to a higher elevation using the energy of the flowing water. An underground canal system known as “qanats” (in Arabic) was used in Mesopotamia circa 550 BC. Such systems can be found even at the present time in the Xinjiang Province of China where the snow melt water from the Tianshan mountains are conveyed to the “grape valley” area near Turpan and in the city of Kumul via a network of underground canals. Ancient irrigation works in Sri Lanka which date back to circa 300 BC during the reign of King Pandukabaya are considered to be some of the most complex irrigation systems of the ancient world. They include but are not limited to many artificial reservoirs (known as “tanks” in local terminology), canals, and sluices, a few of which still remain. Other engineering marvels of the time include the construction of “Yoda Ela” (meaning giant canal), which maintained a gradient of approximately 10–20 cm/km for a distance of about 87 km during the reign of King Dathusena in fifth century AD to transfer excess waters from an upstream reservoir known as “Kala Wewa” in the Polonnaruwa district to a downstream reservoir known as “Tissa Wewa” in the Anuradhapura district, “Minipe Yoda Ela,” another giant canal to divert water from the longest river in Sri Lanka to provide water for irrigation on the left bank, also circa fifth century AD, which has now been extended and expanded and in operation, and a giant reservoir known as “Parakrama Samudraya” (meaning Parakrama sea; Parakrama is the name of the king who built it) which still provides water for irrigation and built during the reign of King Parakramabahu I, circa twelfth century, AD. An important feature of irrigation canals in Sri Lanka is that they have been, and are still being, constructed along topographical contours with only one embankment thereby receiving the overland runoff from the other side. The Romans built aqueducts to carry water from snowmelt in the Alps to cities in the valleys below. The first aqueduct, the “Aqua Appia” was built in 312 BC and was followed by many others throughout the Roman Empire, some below ground surface and some elevated. Over a period of 500 years, 11 aqueducts were built, the last one being “Aqua Alexandrina” built in 226 AD. Roman aqueducts consisted of infiltration galleries, steep chutes or drop shafts, settling tanks, tunnels, covered trenches, bridges to support the aqueduct, siphons, and a distribution system at the destination. Prehistoric developments of irrigation canals also have taken place in Mexico circa 500 BC. Canals have also been constructed to connect seas. Two of the well-known sea-level waterways are the Suez canal in Egypt that connects the Mediterranean Sea and the Red Sea opened in November 1869 after 10 years of construction, and the Panama canal managed by the Panama Canal Authority that connects the Atlantic Ocean and the Pacific Ocean opened in August 1914 after more than 30 years since construction began due to technical and health issues. These two canals cut the respective sea voyages by about 7000 km and 12,875 km. Design of Alluvial Canals Most, if not all, artificial canals for irrigation are constructed on natural formations and are therefore prone to erosion and deposition and should be designed to maintain channel stability. A “stable channel,” is defined as an unlined channel for carrying water, the banks and beds of which are not scoured by the moving water and in which objectionable deposits of sediments do not occur. The widely used methods of design include the method of maximum permissible velocity, the regime method, and the tractive force method. Design formulae for the first method, which aims at the maximum velocity which will not cause erosion or deposition, are based on measured velocities in stable channels in various places and under various flow conditions. The second method is empirical and based on observations made in the field, with notable field data from India, Egypt, and Pakistan. The design approach is to relate the depth of flow, bed width, friction slope, and velocity empirically to the discharge and soil particle size. The third method, which requires a relationship between roughness coefficient and grain size and has some theoretical basis, was developed by the United States Bureau of Reclamation. Significant contributions to this topic can be found in the studies by Kennedy (1895), Lacey (1930), Shields (1936), and Lane (1955) among others. 2.2.6  Hydraulic Structures

There are many types of hydraulic structures built from ancient times to serve many functions. They include dams, embankments, spillways, sluices, gates, locks, weirs, flumes, siphons, culverts, and storm surge barriers among others. The main function of a hydraulic structure is to change the natural flow in a water body by diverting, controlling, or completely stopping. In some instances, a hydraulic structure may be built to measure the flow rate. Dams, which store water behind the upstream face, may be classified according to several criteria such as the purpose, material used to build the dam and structural design. The purposes of a storage dam may be for flood mitigation,

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2-4     Watersheds, River Basins, and Land Use Table 2.1  Ten Tallest Dams in The World Name of dam

Height of dam (m)

Type

Country

River

Jinping-I

305

Concrete arch

China

Yalong

Nurek

304

Earthfill embankment

Tajikistan

Vakhsh

Xiaowan

292

Concrete arch

China

Lancang Jiang

Xiluodu

285.5

Concrete arch

China

Jinsha Jiang

Grande Dixence

285

Concrete arch

Switzerland

Dixence

Inguri

271.5

Concrete arch

Georgia

Inguri

Vajont (disused)

261.6

Concrete arch

Italy

Vajont

Manuel Moreno Torres

261

Earthfill embankment

Mexico

Grijalva

Nuozhadu

261

Embankment

China

Lancang Jiang

Tehri

260.5

Rockfill and earthfill

India

Bhagirathi

Mauvoisin

250

Concrete arch

Switzerland

Val de Bagnes

[Source: http://en.wikipedia.org/wiki/List_of_tallest_dams_in_the_world]

irrigation, hydro power generation, water supply, fisheries, and recreation. A detention dam, which temporarily stores water during high flows, can also act as a storage dam. A diversion dam, on the other hand, diverts the flow in a river or canal, to a different area for purposes such as irrigation or water supply. They are generally low in height. A dam (or part of it) may also be of an overflow type or nonoverflow type. Structurally, a dam may be a gravity type, or arch type, or buttress type, or embankment type. A gravity dam resists the external forces by its weight, whereas an arch dam resists the external forces by the arch action. The dead weight of an arch dam is much less than that of a gravity dam and requires less material to construct, but needs very strong abutments. Buttress dams are “mini” gravity dams with inclined upstream faces supported by struts on the downstream side. Dams are constructed using concrete, rock, earth, and sometimes their combinations. Embankmenttype dams are constructed using soil, gravel, and other natural material and generally not as high as concrete dams. They resist the external forces by the shear strength of the fill material, and, may be earthfill or rockfill. Components of a dam include a spillway, a control structure to control and measure the outflow, a discharge channel, and energy dissipaters. Table 2.1 gives a list of the 10 tallest dams in the world until 2015. Storm surge barriers are a relatively new type of hydraulic structures designed and constructed across estuaries to prevent the storm surges from the sea penetrating into the river under extreme weather conditions. They are usually gated structures that allow the passage of ships under normal conditions. Among them are the world’s largest, the Oosterschelde, located between the islands Schouwen-Duiveland and Noors-Bevelanand, and the world’s largest movable structure, the Maeslantkering, across the New Waterway (Nieuwe Waterweg, a canal that connects to River Rhine forming the artificial mouth of the River Rhine) that connects Rotterdam with North Sea, both in the Netherlands, the world’s second largest movable structure, the Thames Barrier, located downstream of central London to protect London from North Sea flooding, and the Inner Harbour Navigation Canal Lake Borgne Surge Barrier constructed near the confluence of and across the Gulf Intracoastal Waterway and the Mississippi River Gulf Outlet near New Orleans in United States. The design of hydraulic structures involves two stages: firstly, the hydraulic design to meet the flow requirement criteria and the second stage, the structural design to resist the external forces. Design guidelines, procedures as well as examples can be found in most hydraulic engineering text books. 2.3 

DELINEATION OF A WATERSHED

2.3.1  Use of Contour Maps

For many years, delineation of a watershed was done manually using topographical contour maps. The procedure involves demarcating a curve starting from a specified outlet of the watershed and following it upstream at right angles to the contour lines. It can be done by drawing arrows representing the flow directions drawn perpendicular to each contour and in the direction of the steepest gradient. The location of a catchment divide is taken to be where flow directions diverge or where the arrows point in opposite directions. 2.3.2  Use of Digital Elevation Models

A digital elevation model (DEM) gives a three-dimensional representation of a terrain surface. It can be represented as a RASTER2 or as a vector-based A RASTER is a grid of squares which is a commonly used dataset in Geographical Information Systems. 2

02_Singh_ch02_p2.1-2.8.indd 4

triangular irregular network,3 the latter being usually measured whereas the former is computed. Data for determining a DEM can be acquired by direct land surveying or remote sensing. Remote sensing employs aerial surveys by flying objects, Interferometric Synthetic Aperture Radar, with two passes of a radar satellite, Shuttle Radar Topography Mission (SRTM) with a single pass with satellites equipped with two antennas, Advanced Spaceborne Thermal Emission and Reflection Radiometer  (ASTER), etc. They may also be obtained by interpolation between digital contour maps obtained by direct land surveying. Publicly available DEM’s of the entire world include GTOPO30 and ASTER, but their resolutions and quality in some areas vary. Higher quality and higher resolution DEM’s are available commercially. With the availability of DEM’s, watershed delineation can be done using several packages of Geographical Information Systems (GIS) software. Examples of software packages include ArcGIS distributed by Environmental Systems Research Institute (ESRI), Geographical Resources Analysis Support System (GRASS), Topographic PArameteriZation (TOPAZ) tools, and River Tools from River Systems, Inc. (RSI). In the widely used ArcGIS environment, the “Watershed” tool contained in the “Hydrology toolset” of the “Spatial Analysis Toolbox” can be used to demarcate the catchment boundary and the catchment area for a specified outlet point. However, the use of DEM’s for hydrological analysis has a problem with depressions. A group of raster cells completely surrounded by cells of higher elevation may represent a real depression or an imaginary depression arising from interpolation errors during DEM generation. It is important to arrive at a depressionless DEM before it can be used for any hydrological analysis. Algorithms for removing spurious depressions (Jenson and Domingue, 1988) are embedded into GIS software such as ArcView. 2.4  WATERSHED HYDROLOGICAL PROCESSES

The primary hydrological process in a watershed is the capture of precipitation which may be in the form of rain, snow or other forms, and the release of part of the water gathered into a network of streams downslope to a watershed outlet and to store the remainder either above or below the ground surface. Part of the precipitation intercepted by the watershed surface may also be returned back into the atmosphere by direct evaporation and evapotranspiration. In addition to the water-related functions, a watershed is also the home to many types of biotic species including humans, plants, animals, and microbes. There are designated watersheds for drinking water supply purposes where no developments and/or settlements are permitted as well as nondesignated watersheds where other types of development activities can take place. Other development activities may include agriculture, grazing for cattle, and other domesticated animal species, industry, and residential. When there are water bodies in a watershed, they may also be used for recreation purposes and for fisheries. 2.5  CHARACTERISTICS OF A WATERSHED THAT IMPACT ON HYDROLOGICAL PROCESSES 2.5.1 Topography

Topography, which is formed over millions of years of geomorphological processes, plays an important role in water movement in watersheds. Steep 3 A TIN is a vector-based representation of a surface made up of irregularly distributed nodes and lines with three-dimensional coordinates (x, y, and z) that are arranged in a network of nonoverlapping triangles.

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CHARACTERISTICS OF A WATERSHED THAT IMPACT ON HYDROLOGICAL PROCESSES     2-5 

catchments have a faster time of concentration4 leading to a relatively quick response to rainfall. In the catchment scale, topography may be quantified by an average slope of the overland flow plane and an average slope of the main stream. These characteristics have been used to define certain empirical relationships for estimating the time of concentration and other process parameters. Topography also determines the flow patterns and pathways, and storages, which influences the residence time of water within the catchment and has important influences on the flow response transient as well as on the quality of water. The contact time with surface and subsurface material has direct control on chemical and biochemical processes that determine the quality of water. Topography also determines the type of vegetation that can adapt to the watershed landscape. In steep catchments, the likelihood of soil erosion and transport of the eroded sediments to the mainstream is quite high leading to a new dimension in catchment hydrology. Sometimes, the topography of a catchment is often represented by a few integrated parameters for purposes of modeling catchment processes. Examples of such parameters include the overland slope, the mainstream slope and the topographical index. The topographical index is a measure of the extent of flow accumulation at a specified point on the topographical surface. It was first defined in the context of TOPMODEL (Bevan and Kirkby, 1979) as topographical wetness index as:

λ = ln

α (2.6) tan β

where λ is the topographical wetness index, α is the local upslope area draining through a certain point per unit contour length, and tan β is the local slope in radians. It is used to describe the spatial distribution of soil moisture. Topographical index controls flow accumulation, soil moisture, evapotranspiration, chemical and biological composition in soil, etc., and is used in modeling vegetation, spatial distribution of soil moisture, mapping soil organic matter, mapping slope failure susceptibility, and many more. A similar index, known as downslope index has also been suggested to quantify topographic controls on hydrology (Hjerdt et al., 2004) as:

tan αd =

d (2.7) Ld

where Ld is the horizontal distance to the point with an elevation d meters below the elevation of the starting cell, following the steepest-direction flow path. 2.5.2 Geomorphology

The natural shape of a terrain is determined by its geomorphology, which can be considered as the science aimed at understanding the processes that shape the earth’s crust over a range of time scales. The near-static shape of the earth has been formed over millions of years by the physical and chemical processes taking place on or near the earth’s surface. The more dynamic shapes nested within the overall large static-scale shape take place over shorter time scales and are caused by the actions of wind, rain, snow, ice, climate, and human activities. There can also be abrupt changes caused by geologic activities such as earthquakes, volcanic eruptions, landslides, and tsunami. Changes in small-scale features such as formation of drainage channels, rills, gullies, and permanent river channels take place over shorter time scales. The main processes that control the short-term changes are weathering, erosion, and deposition. Sediment production by weathering and erosion usually takes place in the higher altitudes, the transport in the middle altitudes, and the deposition in the lower altitudes. For short-term hydrological analysis excluding sediment dynamics, a catchment may be considered as geomorphologically in steady state. Fluvial geomorphology refers to the study of the formation of rivers and how they respond to human and climate induced changes in the watershed. It is an important area of concern from the point of view of river management to reduce flood vulnerability and damages while maintaining a healthy aquatic habitat. Channel changes may have implications for water supply, navigation, aquatic habitat, and protection of property. Human-induced changes include construction of dams and reservoirs, sand mining in the river bed, while a common natural disturbance is a flood. The response to such changes includes bed and bank erosion, bed aggradations, and changes in the channel cross section. Degradation of the channel takes place downstream of a dam due to reduction of sediment load, or due to increase in stream power (product of discharge and slope) or decrease in sediment yield as a result of conservation measures. Aggradations can occur due to increase of upstream sediment supply due to, for example, construction activities, increase of downstream bed level due to, for example, sea level rises, and due to basin wide increase in sediment supply. When exposed to such changes, the sediment equilibrium is disturbed, but over a period of time, rivers attain new 4 Time of concentration is the time taken by a water particle to travel from the furthest point in the catchment to its outlet.

02_Singh_ch02_p2.1-2.8.indd 5

state of sediment equilibrium, a condition when the overall erosion is balanced by the overall deposition. This condition is quite common when a dam is constructed across a sediment-carrying river. Attempts to incorporate geomorphological features into hydrograph computations have been introduced by defining a geomorphologically-based unit hydrograph (Rodrigues-Itube, 1979; Gupta et al, 1980) which has subsequently been critically reviewed (Shamseldin and Nash, 1998), but applied in several situations in unguaged areas where the geometric properties of the catchment have been used to parameterize the physical processes (Yang et al., 1998; Yang et al., 2002). 2.5.3 Geology

The underlying geology of a watershed determines the mechanisms of subsurface flow of water. When the surface layers are highly porous, much of the precipitation goes into the sub-soil by infiltration. In rocky, sandy, or gravel geologic formations, the infiltrated water in turn may percolate downward to the water table or to deeper aquifers to reappear with a time delay as base flow.5 The chemical composition of geology also determines the quality of ground water and subsequently surface water. In the process of the subterranean water flowing through geological formations, many types of minerals necessary for human health get dissolved. The water extracted from such formations is commercially marketed as spring waters by many vendors. Geology also determines the movement and storage of water in aquifers. Groundwater accounts for about 0.5% of earth’s fresh water and is a major source for municipal and irrigation use. Depending upon the geology, aquifers may be confined or unconfined. Confined aquifers are generally deep and are sandwiched between two impermeable layers of rock, whereas unconfined aquifers are shallow and confined from below only. In confined aquifers, water is held under pressure, which in some cases can be high enough to flow out to the ground surface as a jet if a hole (well) is drilled through the confining layer, whereas in unconfined aquifers, the phreatic surface is at atmospheric pressure. Unconfined aquifers are prone to pollution from the surface, whereas confined aquifers are relatively free from surface pollution. 2.5.4  Soil Type

Soils can be classified in different ways. Engineers classify soils according to their engineering properties; soil scientists classify them according to their morphology, behavior, and genesis. The widely used engineering classification is that used by the United Soil Classification System (USCS) which has three categories: coarse-grained soils (e.g., sands and gravels); fine-grained soils (e.g., silts and clays); and highly organic soils (peat). These categories may be further subdivided by taking into account properties such as color, water content, strength, etc., which are not specified in the coarse classification. Subclassifications include clay (fine particle size), silty clay, silty clay loam, silt loam, silt, clay loam, loam, sandy clay, sandy clay loam, sandy loam, loamy sand, and sand (coarse particle size) that are represented in a triangle. The soil science classification is more suited to land-use applications. US Department of Agriculture classifies according the grain size in the form of a soil texture triangle. They may also be classified as residual soils or transported soils depending on their formation, which may be either by weathering rocks or transported by wind, water, or gravity, from a different location. In addition to these classifications, there are also region specific classifications. Soil type in a watershed determines the type of vegetation it can support. Soil is a non-renewable resource in the watershed, which requires careful management to support high-quality vegetation that includes agricultural crops. Soils carry water, nutrients, and minerals necessary for plant growth, which is an indicator of a watershed health. Soils and plants have a symbiotic relationship. Plants modify and develop soils by creating pore spaces that increase the capacity to store more water, add organic matter when leaves die and decompose, protect the surface from erosion, while soils act as the foundation for all types of plants. 2.5.5  Shape, Size, and Slope

The shape of the watershed contributes to the speed with which the precipitated water reaches the outlet. The size determines the quantity of runoff generated by the watershed. The widely used rational method for estimating the peak discharge relates it directly to the catchment area (Q = CIA, Q-peak discharge, I-rainfall intensity, A-catchment area, C-runoff coefficient). The size, shape, and slope also determine the hydrograph6 shape. For example, increase in the catchment size tends to increase the base length7 of the hydrograph; shape of main stream and valleys tends to affect the velocity and hence 5 Baseflow is that part of streamflow which comes from deep subsurface flow and delayed shallow subsurface flow. It is also sometimes referred to as dry weather flow. 6 A hydrograph is a graph that displays the flow rate as a function of time. 7 Base length is the length along the time axis between the beginning of the hydrograph and the end of the hydrograph.

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2-6     Watersheds, River Basins, and Land Use

the time to peak; in a large catchment, depending upon where the high-intensity occurs, the base length of the hydrograph may increase or decrease. If the high-intensity rainfall occurs near the outlet, the time base will be short. In comparison, if the high-intensity rainfall occurs at the far end of the catchment, the time base would be relatively longer. If the watershed hillslopes are steep, the overland flow will be swift, whereas in relatively flat watersheds, more water will infiltrate into the subsoil and delay the runoff process. The same applies to the slopes of the stream network. 2.6  RIVER BASIN

A river basin is the land area drained by a river and its tributaries. The basic difference between a river basin and a watershed is that a river basin has a well-defined network of rivers most of which are perennial, which a watershed may not have. For example, in arid zones, drainage in a watershed may take place via overland flow and/or ephemeral drainage channels, which become dry once the precipitation has ceased. In a river basin, the river network is formed by geomorphological actions over long periods but could be altered by human actions. The flow in the main river and tributaries take place perennially albeit at a reduced rate during dry weather. In ancient times, most civilizations started along river banks, as they provided the basic essentials of life, namely, water and food. Examples include the Indus Valley Civilization (also known as Harappan culture) circa 6000 BC along the River Indus (India and Pakistan), Ancient Nile Valley Civilization along the River Nile in Egypt circa 5000 BC and the Yellow River Valley civilization in China circa 1500 BC, all of which were considered as hydraulic empires. Significant hydraulic civilizations also existed in ancient Somalia, Sri Lanka, Mesopotamia, Mexico, and Peru. A river basin is a land area for human habitation as well as the home for many other biotic species. In the modern times, it is also a place for many types of economic activities such as agriculture and aquaculture, industry, and tourism. Vegetation forms an important and essential component of a river basin as a link that provides exchange of water from the land to the atmosphere. The vegetation canopy intercepts the rain thereby reducing the kinetic energy of rainfall which in turn reduces soil erosion. Plant roots make the underlying soil more porous thereby increasing the capacity to store more soil water, and transport water and minerals from the soil to the body of the plant. Dead leaves from vegetation when decomposed provide nourishment to the soil. Before decomposition, they also provide some protection to the soil surface. Plants also contribute to reduction of greenhouse gases by their ability to absorb carbon dioxide and release oxygen during the photosynthesis process. The greatest impact on a river basin is made by human population. Humans have a higher demand for water from the river basin. They alter the natural river basin by actions, such as deforestation, building dams, urbanization and other infrastructural development, migration of people and the accompanying increase of waste generation, all of which give rise to negative impacts on the natural health of the river basin. There is a danger that such actions also increase the risk of flooding the lowland areas of a river basin. Most human activities in river basins alter and accelerate the natural hydrological cycle thereby threatening the sustainability of many river basins in the world today. 2.7  RIVER BASIN MANAGEMENT

Earth’s water resources are limited and the per capita share of water is continuously decreasing as a result of increasing population. Approximately 97.5% of earth’s water resources are contained in the oceans which is not suitable for human consumption except via an expensive process of desalination. The fresh water contained in ice caps and glaciers, inland lakes, soil water, ground water, atmospheric water, and river and stream storage constitute the remaining 2.5% of which only about 0.5% is accessible. Although renewable and abundant in nature, fresh water has spatial and temporal variability. This temporal and spatial variability and the changing life styles of human population lead to water stress8 and water scarcity. To ensure sustainability of the Earth system while providing water, food, and energy security to all inhabitants of the earth it is imperative to have fair and equitable water management policies. These were forged into development goals during several summits of world leaders. These include the U.N. Millennium Summit of world leaders held at the UN headquarters from September 6–8, 2000, during 8 Water stress index is the fraction of the total annual runoff available for human use. The widely used indicators of water stress, water scarcity, and water shortage are the Falkenmark indices (Falkenmark, 1989). Based on water usage in multiple countries, Falkenmark assigns the following threshold values: > 1700 m3 per capita per year, no stress; 1000–1700 m3 per capita per year, stress; 500–1000 m3 per capita per year, scarcity; < 500 m3 per capita per year, shortage.

02_Singh_ch02_p2.1-2.8.indd 6

which the millennium development goals were established, the Rio Summit (Earth Summit), held in Rio de Janeiro, June 3–14, 1992, as an inter governmental summit under the auspices of the United Nations Conference on Environment and Development (UNCED) and the aptly named commemorative summit Rio+20 held in Rio de Janeiro, from June 20–22, 2012. The focus of Rio+20 Summit was to ensure sustainability of the Earth system while providing water, food, and energy security to all inhabitants of the earth. The goal was to forge a global agreement by establishing a set of sustainable development goals acceptable to all parties. The water policies and water laws enacted in the recent times for better management in a few selected countries are summarized in the following sections. 2.7.1  Water Policy in Japan

In Japan, the National Government is regulating the management of rivers and associated water resources through the River Law. According to the law, each river administrator (national or regional government) is in charge of comprehensive river management, which include flood management, river works, maintenance of infrastructure (e.g., dams and dikes), authorization of water (resources) rights, and conservation of the river environment (including water quality, regulation of river area use, etc.). Ground water is controlled by local governments. Water supply is controlled by municipal governments and the private sector. Large rivers (109 in total) come under the river administration while other rivers are managed by local governments. Japan had a river administrative system for over 100 years. The first modern river administrative system started after the enactment of the “old river law” in 1896, designed primarily for flood control. The “new river law” was enacted in 1964 for the purpose of flood control and water use. Under this law, integrated river system management and water-use regulations were introduced. Subsequently, to meet the changing social and economic needs, the river law was amended in 1997 with the objective of establishing a comprehensive river administrative system for flood control, water use, and the conservation of the environment. Water reuse is practiced in industry and for recreational purposes. Desalination by reverse osmosis is done in a small way in Fukuoka and Okinawa. Interbasin water transfers also take place for optimum utilization. The water sector in Japan is administered under five ministries: Ministry of Land, Infrastructure, and Transport; Ministry of Health, Labour, and Welfare; Ministry of Agriculture, Forestry, and Fisheries; Ministry of Economy, Trade, and Industry; and Ministry of the Environment. In rare situations, where conflicts occur, they are settled in an amicable manner in the traditional Japanese way. There is also an initiative by the Cabinet Office to strengthen the coordination of functions of related ministries. Japan has 14 water related legislative enactments of which River Law is one. 2.7.2  Water Policy in the European Union

There have been three water-management directives in the European Union (EU). The Urban Waste Water Treatment Directive (91/271/EEC) of May 21, 1991 is concerned about discharges of municipal and some industrial wastewaters. The Drinking Water Directive (98/83/EC) of November 3, 1998 concerns potable water quality and the Water Framework Directive (2000/60/EC) of October 23, 2000 concerns water resources management. Recently, the water environment in the EU is under great pressure from economic activities, urban and demographic developments, and climate change and has put forward a “Blueprint to Safeguard Europe’s Water Resources,” which outlines actions that concentrate on better implementation of current water legislation, integration of water policy objectives into other policies, and filling the gaps in particular as regards water quantity and efficiency with a time frame target set to 2050. 2.7.3  Water Policy in the United States

In United States, conflicts continue to arise in constructing a framework of policies due to the contrasting climates of the Eastern and Western United States. The difficulty lies in a lack of understanding of integrating policies in both sides of the country, while still maintaining a fair amount of regional control. California state has been the first to attempt to integrate both types of the traditional policies. Major water policies are under the federal government, for example, the Clean Water Act (CWA) of 1948. Water policy, management, and pollution usually tend to fall under the jurisdiction of local and state governments. Federal government gives grants to local and state governments to construct facilities for sewage treatment. In the Western United Sates, the “Prior appropriation doctrine” in which “first in time, first in rights” principle together with the relative powers of the two parties is practiced. Under this doctrine, the more powerful the party who diverts water, the more rights he has. Landowners can use the rivers as they see fit, as long as they do not disturb the natural flow. In the Eastern United States, the “Riparian rights doctrine” in which landowners whose property adjoins a body of water have the right to make reasonable use of it as it flows through or over their property, is practiced.

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LAND USE    2-7  2.7.4  Water Policy in India

India has more than 18% of the world’s population, but has only 4% of world’s renewable fresh water resources and only 2.4% of world’s land area. Indian National Water Policy has been enacted and adopted in 1987, reviewed and updated in 2002 and 2012. A new Draft National Water Framework Bill was introduced in April 2013. It covers uses of water, adaptation to climate change, enhancing water available for use, demand management and water use, efficiency, water pricing, conservation of river corridors, water bodies and infrastructure, project planning and implementation, management of floods and droughts, water supply and sanitation, institutional arrangements, transboundary rivers, database and information systems for sharing information, and research and training needs. 2.7.5  Water Policy in China

China with a population more than 1.3 billion has an annual per capita fresh water resources availability of 2100 m3, which is about a third of the world average. However, due to the monsoon climate and complex geographical conditions, temporal and spatial distribution of water resources is highly uneven resulting in difficulties in food production and economic development in some parts of the country. For example, north China has about 70% of farmland, but only 16% of water resources compared with south China, which has only about 30% of farmland but with 84% water resources. The maximum annual precipitation in southeast China is over 2000 mm, and the minimum annual precipitation in northwest China is less than 50 mm. Recently, China has embarked on implementing a strict management system to optimally utilize her water resources in the face of conflicting demands in the economic sector and the protection of eco-environment. China has a centralized system of water management. The Ministry of Water Resources formulates general policies, and develops strategies for water administration, and enacts related laws and regulations. The Ministry of Water Resources has established seven river basin commissions, namely, Changjiang (Yangtze River), Yellow River, Huai River, Hai River, Songliao River, Pearl River, and Lake Tai. The Water Law, enforced in July 1998 and revised in 2002, is the most fundamental legislation concerning water administration in China. This law organizes activities concerning the general development, use, and conservation of water resources, river development and flood-control works, etc., as well as coordination of socioeconomic activities and related agencies concerned with water. The Water Law covers general rules for the development and utilization of water resources, the maintenance of water supplies, the river basin and water use facilities, water resources management, water disputes and the supervision and inspection of law enforcement, legal responsibility, and supplementary provisions such as flood control, prevention, and control of water pollution. 2.7.6  River Commissions/River Authorities (International and National)

Many trans-boundary rivers experience conflicts when it comes to sharing the resources of the river. They may arise as a result of differences in the social and economic levels of the riparian countries, or be due to unilateral actions taken usually by upstream countries. For example, if an upstream country decides to build a dam for hydropower generation or for other purposes, the natural flow of the river is disturbed and controlled which may adversely affect the downstream countries. International river commissions and/or agreements have been established in regions with trans-boundary rivers to avoid or minimize such conflicts and to share and assist in the management of the resources in an optimal and equitable manner. Examples include the Mekong River Commission (MRC), established on April 5, 1995 with the participation of Cambodia, Lao PDR, Thailand, and Vietnam, to which China and Myanmar joined as “dialogue partners” in 1996; Organization of the Amazon Cooperation Treaty (OACT), established on July 3, 1978 with the participation of Brazil, Peru, Bolivia, Colombia, Ecuador, Venezuela, Guyana, Suriname, and French Guiana; the Danube Commission, initially established in 1948 by seven riparian countries but subsequently expanded to include Austria, Bulgaria, Croatia, Germany, Moldova, Slovakia, Romania, Russia, Ukraine and Serbia; the Central Commission for Navigation on the Rhine (CCNR), formally constituted in 1815 with the participation of Germany, Belgium, France, The Netherlands, and Switzerland; the International Sava River basin Commission (ISRBC), established on December 29, 2004 with the participation of Bosnia and Herzegovina, Federal Republic of Yugoslavia, Serbia, Republic of Croatia and Republic of Slovenia; and the Mosel Commission established on December 21, 1962 with the participation of France, Germany, and Luxembourg. The latter four commissions operate within the European Commission. Similar bilateral treaties have been signed by India and Bangladesh with the establishment of the Joint Rivers Commission, which came into being on November 24, 1972 to share the waters of the 57 trans-boundary rivers of

02_Singh_ch02_p2.1-2.8.indd 7

Bangladesh (54 with India and 3 with Myanmar) including the three mighty rivers the Ganges, the Brahmaputra, and the Meghna. The earliest national river authority may perhaps be the Tennessee Valley Authority (TVA), which is a federally owned corporation in the United States established in May 1933 to develop the Tennessee Valley which had been seriously affected by the great depression. Similar national authorities or commissions include the Yangtze River Water Resources Commission, the Yellow River Conservancy Commission, the Pearl River Water Resources Commission, Huai River Water Resources Commission, the Hai River Water Resources Commission, the Songliao River Water Resources Commission, and the Taihu Basin Authority in China. 2.8  MAJOR RIVER BASINS IN THE WORLD

The ranking of a river can be based on the value of a wide range of ranking parameters such as length, average or maximum discharge, annual volume of flow to ocean, catchment area, depth of flow, width, sediment yield, population in the river basin, and many other characteristics. However, in general, the size and importance of rivers are ranked in terms of their length or discharge. The ten largest rivers in the world by discharge at the mouth are Amazon (Brazil), Congo (Congo), Yangtze (China), Brahmaputra (Bangladesh), Ganges (India), Yenisei (Russia), Mississippi (USA), Orinoco (Venezuela), Lena (Russia), and Parana (Argentina). In Asia, the major river basins include the Yangtze, the Brahmaputra, the Mekong, and the Indus; in Africa the Nile and the Congo; in Europe, the Danube and the Rhine; in South America, the Amazon, the Orinoco and the Parana; in North America, the Mississippi, the Colorado, the Columbia, and the St Lawrence. Table 2.2 gives a list of the 10 longest rivers in the world. Depending of the criterion used, the rankings in Table 2.2 may change. It is also to be noted that the data available sometimes differ from source to source. 2.9  LAND USE

Land use in a watershed governs several processes in the hydrological cycle as well as the productivity of the land. Classification of land use is not unique but, based on the vegetation type, nine traditional classes can be identified. They are forests, land put to nonagricultural uses, barren and uncultivable land, permanent pastures and other grazing lands, miscellaneous tree crops and groves not included in the net area sown, culturable waste, fallow land other than current fallows, current fallows, and net area sown. A much simpler classification based on the services provided by the land considers four categories, namely, land underlying buildings and structures, land under cultivation, recreational land and associated surface water, and other land and associated surface water. 2.9.1  Types of Land Use

The main function of land from the point of view of food production and security is agriculture. According to Food and Agricultural Organization (FAO) approximately 30% of the world land area is used for agriculture and another 30% for forests (http://www.fao.org/docrep/018/i3107e/i3107e04.pdf). The percentage of land area taken for residential purposes is relatively small although the percentage of urban areas is on the increase as a result of migration of people for better opportunities. With the demand for increased food production to feed the increasing population the percentage of land used for agriculture is likely to increase in the future. Since land is a finite and nonrenewable resource, there is always competition among various users. For example, in many countries agricultural and forestland are being converted to urban areas thereby threatening the sustainability of the environment and related ecosystems. Table 2.2  Ten Longest Rivers of the World Ranking

Name

Country/region

Length (km)

Area (106 km2)

Average discharge (m3/s)

1

Nile

Africa

6,648

3.35

5,100

2

Amazon

South America

6,500

7.05

175,000

3

Yangtze

China

6,280

1.80

30,440

4

MississippiMissouri

The United States

6,020

3.22

16,792

5

Yenisei

Russia

5,539

2.60

19,830

6

Yellow

China

5,400

0.75

1,775

7

Ob-Irtysh

Russia

5,410

2.98

12,300

8

Congo

Congo

4,700

3.48

41,300

9

Amur

Asia

4,440

1.86

8,600

10

Lena

Russia

4,400

2.49

17,000

(Source: Water Conservancy Encyclopedia China, Vols. 1–4)

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2-8     Watersheds, River Basins, and Land Use 2.9.2  Effect of Land Use on the Basin Scale Hydrological Cycle

Land use practices have major impacts on water (quantity and quality), soil, nutrients, and other bio-species. Deforestation tends to increase runoff and sediment generation by erosion. Forest covers act as carbon sinks thereby contributing to the reduction of global warming. They also tend to make the underlying subsoil more porous thereby allowing more infiltration to take place which in turn reduces overland flow. Conversion of forest land to urban areas has negative impacts on the sustainability of the land-water ecosystem. Urban areas also have low infiltration capacities thereby reducing the water available in the subsoil that supports plant growth. Inappropriate land use can alter the basin scale hydrological cycle which in some situations may lead to irreversible consequences. Land use affects evapotranspiration, initiation of surface runoff, and washout of nutrients from soil, among other hydrological processes. Changes in land use in a catchment may be caused by a number of factors, such as urbanization, agricultural development, land cover changes by afforestation and deforestation, and construction of water infrastructures, such as dams, reservoirs, and water import/export facilities. Such changes have direct influences on runoff generation both temporally and spatially. They can also affect the partitioning of precipitation between evapotranspiration and runoff. For example, it has been reported that an annual runoff reduction of more than 50% has taken place in Loess Plateau (Zhang et al., 2008) and in the mountainous catchments in Haihe River (Xu et al., 2014) in China as a result of soil conservation measures such as terracing, afforestation, and construction of sediment traps. 2.9.3  Land Use and Runoff

Land use has a direct relationship with runoff production. In vegetated catchments, the rate and quantity of overland runoff production is much less than those of bare catchments. Erosion and sediment generation in bare catchments are higher leading to land degradation and accumulation of sediments downstream. Urbanization which increases the impervious surface areas in catchments leads to a decrease in infiltration which in turn increases surface runoff. An increase in impervious surface areas associated with urbanization also slows down evaporative cooling, allowing surface temperatures to rise to levels higher than in rural areas. Urbanization in flood plain areas increases the risk of flooding due to increased peak discharge and volume, and decreased time to peak. Urban watersheds, on average convert about 90% of the storm rainfall to runoff (runoff coefficient ≈ 0.9), whereas nonurban forested watersheds the conversion rate is about 25% (runoff coefficient ≈ 0.25). Catchment runoff also increases after deforestation and, as expected, decreases after afforestation. Water infrastructure development such as construction of dams, reservoirs, and diversions from/and to such structures can also affect the runoff. The effects of land use change on river flows are more evident in arid climates, where the low flows are more sensitive to land use changes. 2.9.4  Land Use and Water Quality

Higher concentration of population resulting from urbanization also substantially degrades water quality, especially when there is no wastewater treatment. The resulting degradation of inland and coastal waters impairs water supplies, causes oxygen depletion and fish kills, increases harmful algal blooms, and contributes to the spread of waterborne diseases. It is important to note that land use changes are aimed at increasing economic productivity, but the price that has to be paid is the impairment to ecosystems and biodiversity, with potentially serious and unquantifiable costs. Intensive agriculture using various types of fertilizes to increase productivity has increased food production but at the same time caused extensive environmental damage. Intensive agriculture increases erosion and sediment yield, and leaches nutrients and agricultural chemicals to groundwater, streams, and rivers. Agriculture has become the largest source of excess nitrogen and phosphorus in waterways and coastal zones. Increased fertilizer use has led to degradation of water quality in many regions. 2.10 CONCLUSION

In this early chapter of the book, an attempt has been made to highlight and describe the basic components of a watershed including their main characteristics, basin scale hydrological processes and the characteristics that impact them, river basins, and their management, including water policies in some selected countries/regions, and land use and their relationship to other hydrological processes. The Chapter is introductory in its presentation since much of the related details will appear in subsequent chapters. Watersheds, or river basins, have been the cradles of civilization since ancient times. What appears to be the challenge for human population is to ensure the continuity and sustainability of the ecosystems in watersheds, which in recent times have been threatened by human activities. A balanced approach in which optimal utilization of the resources of the watershed, the river system, and land productivity are sought while at the same time ensuring the sustainability of the entire system appears to be the way forward.

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The authors acknowledge the sources of the information reproduced in the two Tables in this chapter. REFERENCES

Bevan, K. J. and M. J. Kirkby, “A physically based, variable contributing area model of basin hydrology,” Hydrological Sciences Bulletin, 24: 43–69, 1979. Falkenmark, M., “The massive water scarcity threatening Africa—why isn’t it being addressed,” Ambio, 18 (2): 112–118, 1989. Gupta V. K., E. Waymire, and C. T. Wang, “A representation of an instantaneous unit hydrograph from geomorphology,” Water Resources Research, 16: 855–862, 1980. Hack, J. T., “Studies of longitudinal stream profiles in Virginia and Maryland,” U.S. Geological Survey Professional Paper 294-B, U.S. Government Printing Office, Washington, D.C., pp. 45–97, 1957. Hjerdt, K. N., J. J. McDonnell, J. Seibert, and A. Rodhe, “A new topographic index to quantify downslope controls on local drainage,” Water Resources Research, 40 (w05602): 1–6, 2004. Horton, R. E., “Erosional development of streams and their drainage basins: hydro-physical approach to quantitative morphology,” Geological Society of America Bulletin, 56 (3): 275–370, 1945. Jayawardena, A. W. and J. K. White, “A finite element distributed catchment model, I—Analytical basis,” Journal of Hydrology, 34 (3–4): 269–286, 1977. Jayawardena, A. W. and J. K. White, “A finite element distributed catchment model, II—Application to real catchments,” Journal of Hydrology, 42 (3–4): 231–249, 1979. Jenson, S. K. and J. O. Domingue, “Extracting topographic structure from digital elevation data for geographic information systems analysis,” Photogrammetric Engineering Remote Sensing, 54: 1593–1600, 1988. Kennedy, R. G., “The prevention of silting in irrigation canals,” Proceedings of the Institution of Civil Engineers, London, 1895, Vol. 119, pp. 281–290. Lacey, G., “Stable channels in alluvium,” Proceedings of the Institution of Civil Engineers, London, 1930, Vol. 229, pp. 259–384. Lane, E. W., “Design of stable channels,” Transactions, ASCE, 20 (2776): 1234–1279, 1955. Lighthill, M. H. and Whitham, G. B., “On kinematic waves, I. Flood movement in long rivers,” Proceedings of the Royal Society, London, Series A, 1955, Vol. 229, pp. 281–316. Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman, New York, pp. 188–189, 1983. Rodriguez-Iturbe, I. and J. B. Valdes, “The geomorphologic structure of the hydrologic response,” Water Resources Research, 15: 1409–1420, 1979. Rosso, R., B. Bacchi, and P. La Barbera, “Fractal relation of mainstream length to catchment area in river networks,” Water Resources Research, 27 (3): 381–388, 1991. Shamseldin A. Y. and J. E. Nash, “The geomorphological unit hydrograph—a critical review,” Hydrology and Earth System Sciences, 2 (1): 1–8, 1998. Shields, A., “Anwendung der ahnlichkeitsmechanik und der turbulenzforschung auf die geschiebebewegung,” Mitteilungen der Preussischen Versuchsanstalt fur Wasserbau und Schiffbau, No. 26, Germany, Berlin, 1936 (English Translation by W.P. Ott and J.C. van Uchelen, California Institute of Technology, Pasadena, CA, 1936, p. 36). Strahler, A. N., “Hypsometric (area-altitude) analysis of erosional topology, Geological Society of America Bulletin 63 (11): 1117–1142, 1952. White, J. K. and A. W. Jayawardena, “Discussion of ‘A finite element approach to watershed hydrology’ by C. Taylor et al.,” Journal of Hydrology, 27: 357–358, 1975. Woolhiser, D. A. and J. A. Ligget, “Unsteady one dimensional flow over a plane; the rising hydrograph,” Water Resources Research, 3 (3): 753–771, 1967. Xie, H., Fractals in rock mechanics, Geomechanics Research Series 1, A. A. Balkema, Rotterdam, The Netherlands, 1993. Xu, X., D. Yang, H. Yang, and H. Lei, “Attribution analysis based on the Budyko hypothesis for detecting the dominant cause of runoff decline in Haihe basin,” Journal of Hydrology, 510: 530–540, 2014. Xu, Q. Water Conservancy Encyclopedia China [M]. China Waterpower Press, Beijing, 2006. Yang, D., S. Herath, and K. Musiake, “Development of a geomorphologybased hydrological model for large catchments.” Annual Journal of Hydraulic Engineering, JSCE, 42: 169–174, 1998. Yang, D., S. Herath, and K. Musiake, “A Hillslope-based hydrological model using catchment area and width functions.” Hydrological Sciences Journal, 47 (1): 49–65, 2002. Zhang, X., L. Zhang, J. Zhao, P. Rustomji, and P. Hairsine, “Responses of streamflow to changes in climate and land use/cover in the Loess Plateau, China.” Water Resources Research, 44: W00A07, 2008.

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Chapter

3

Water Balance BY

C. PRAKASH KHEDUN AND VIJAY P. SINGH

ABSTRACT

The water balance is an expression of the law of conservation of mass, formulated as the continuity equation. The formulation requires that the balance is done for a system of specified geometry at a specified time scale. In hydrology, the system can be a watershed, a channel or a segment thereof, a lake, a plant root zone, or an aquifer. The system can be extended to a region, continent, or even the globe, but it becomes difficult to accurately compute the different components of the water balance for such a large system because of the lack of data, especially at small time scales, and our less than complete understanding of the interactions between components. The time scale, depending on the spatial size of the system, can be hourly, daily, weekly, monthly, seasonal, or yearly. The objective of this chapter is to present rudimentary aspects of water balance and its computation. 3.1 INTRODUCTION

Computation of water balance of a hydrologic system entails computing the individual components of the hydrologic cycle and their interactions. The role of water in each process or component of the cycle is complex. As water moves in the hydrological cycle, it not only interacts with other processes within the cycle, but also dynamically interacts with other components of the atmosphere, biosphere, cryosphere, lithosphere, and pedosphere. The amount of water in each state—ice, liquid, and vapor—varies over time. For example, on a seasonal scale, ice cover and rain-to-snow ratio changes with fluctuations in temperature. Furthermore, changes, whether natural or anthropogenic, in any one of the components of the hydrological cycle or the Earth’s system, affects the balance of water in the hydrological cycle. Although Chap. 1 deals with the hydrologic cycle, it is appropriate here to revisit different components from the water balance perspective. After a brief description of the hydrological cycle, its components, and hydrologic fluxes, a discussion on the amount of water on the Earth highlighting the spatial distribution of water is given. Empirical equations for the computation of water budget is given next, followed by a presentation of different water balance models, ranging from the simple bucket model to more complex land surface models. Finally, the influence of natural and anthropogenic effects on the water balance is discussed. 3.2  HYDROLOGIC FLUXES

The hydrologic cycle and its various components are shown in Fig. 3.1. The main components of the hydrological cycle are precipitation, interception, evapotranspiration (evaporation from free water surfaces and transpiration from plants), depression storage, infiltration, surface runoff, interflow, and groundwater flow. The interchange of water between open water bodies and land via the atmosphere not only ensures the replenishment of water in lakes and rivers, but also prevents deterioration in water quality. The rate of renewal varies spatially and temporally, depending on the climate. This renewal, or

turnover, time is defined as τ o = S / O , where S is the storage capacity of the water body and O is the drainage rate. The residence time, which is the time that a water molecule remains in a reservoir, can also be defined. Since every water molecule has a different residence time, the average residence time is ∞ most commonly used. It is defined as τr = ∫ τr ⋅ f (τr ) ⋅d τr , where f (·) is a 0 function and τ r is the residence time of one molecule (Mussy and Higy, 2011). This continuous renewal allows a larger volume of water to be withdrawn than is actually available on land at a point in time. For example, the total volume of water available for consumption is about 2000 km3, while the total volume of water withdrawn is 3800 km3/year—only 10% of the maximum available renewable freshwater resources [Fig. 3.2 (Oki and Kanae, 2006)]. Evaporation from the oceans represents the bulk of the moisture in the atmosphere, while terrestrial evapotranspiration contributes only about 20%. Further, transpiration contributes between 80 and 90% of the total water flux from land, recycling 62,000 ± 8000 km3 of water per year (Jasechko et al., 2013). 3.3  WATER ON THE EARTH

Over 70% of the Earth is covered with water. Unfortunately, most of this water is in the oceans; about 97% of the world’s water is saltwater and is undrinkable or unusable without some form of treatment to remove the salt. Desalination, however, is an extremely expensive and energy intensive process. Thus, the water in the ocean is inaccessible for most countries or communities around the world. Only about 3% of the water on Earth is fresh. The freshwater reserves include: rivers and lakes, marshes, the vadoze zone, groundwater aquifers, glaciers and other permanently snow-covered regions, permafrost, biological entities, and the atmosphere (Table 3.1). Not all of this water is readily accessible or exploitable—68.7% is locked in ice caps and glaciers and 30.1% is in fresh groundwater, leaving a very small portion (1.3%) in surface and other freshwater sources for human consumption. 3.3.1  Earth’s Water Balance

On the continental scale, the water balance of the major rivers and self-contained hydrologic systems gives an estimate of renewable water resources available for human consumption (Table 3.2). Water on Earth is not uniformly distributed. Asia, followed by South America and Africa, receives the largest volume of precipitation, while Antarctica receives the least. Evaporation in Antarctica is nil, while on all the other continents, it is greater than half of the total volume of precipitation. The total evaporation from land as a whole is equal to about 60% of the total precipitation. Discharge from rivers in Asia and South America combined is more than 50% of Earth’s total river discharge (Table 3.3). In Africa, one of the largest continents by area, river flow is only 10% of the world’s total flow. This equates to a mean discharge of 4.8 l/s/km2—the least among all continents. In contrast, the mean discharge for South America is 21 l/s/km2, while that for Asia, Europe, and North America is about 10 l/s/km2 for each.

3-1

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Precipitation Evapotranspiration

Channel precipitation Interception

Surface runoff

Depression storage

Ground surface

Infiltration

Soil moisture

Interflow

Percolation

Capillary rise

Groundwater storage

Groundwater flow

Streamflow Underground flow into or out of the area Figure 3.1  Components of the hydrologic cycle [Source: US EPA, 2015].

Figure 3.2  Global hydrological fluxes (1000 km3/year) and storages (1000 km3) [Source: Oki and Kanae, 2006]. 2

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Water on THE Earth    3-3  Table 3.1  Water Reserves on Earth (Source: Shiklomanov, 1993) Volume (103 km3)

Distribution area (103 km2)

Percentage of global reserves

Layer (m)

of total water

of fresh water

World ocean

361,300

1,338,000

3,700

96.5

Ground water

134,800

23,400

174

1.7

–

10,530

78

0.76

30.1

Fresh water Soil moisture

16.5

0.2

–

0.001

0.05

Glaciers and permanent snow cover

16,227

24,064.1

1,463

1.74

68.7

Antarctic

13,980

21,600

1,546

1.56

61.7

Greenland

1,802

2,340

1,298

0.17

6.68

Arctic islands

226

83.5

369

0.006

0.24

Mountainous regions

224

40.6

181

0.003

0.12 0.86

Ground ice/permafrost

21,000

300

14

0.022

Water reserves in lakes

2,0548.7

176.4

85.7

0.013

–

1,236.4

91

73.6

0.007

0.26

Fresh Saline

822.3

Swamp water

85.4

2,682.6

0.006

–

11.47

103.8 4.28

0.0008

0.03 0.006

River flows

148,800

2.12

0.014

0.0002

Biological water

510,000

1.12

0.002

0.0001

0.003

Atmospheric water

510,000

0.025

0.001

0.04

12.9

Total water reserves

510,000

1,385,9844.61

2,718

Total fresh water reserves

148,800

35,029.21

235

100

–

2.53

100

Table 3.2  Water Balance at the Continental Scale (Source: Shiklomanov, 1993) Precipitation Continent

(mm)

Evaporation

(km ) 3

(mm)

Runoff

(km ) 3

(mm)

(km3)

Europe

790

8,290

507

5,320

283

2,970

Asia

740

32,200

416

18,100

324

14,100

Africa

740

22,300

587

17,700

153

4,600

North America

756

18,300

418

10,100

339

8,180

South America

1,600

28,400

910

16,200

685

12,200

Australia and Oceania

791

7,080

511

4,570

280

2,510

Antarctica

165

2,310

0

0

165

2,310

Land as a whole

800

119,000

485

72,000

315

47,000

Areas of external runoff

924

110,000

529

63,000

395

47,000a

Areas of internal runoff

300

9,000

300

9,000

34

1,000b

Including underground water not drained by rivers. Lost in the region through evaporation.

a

b

Table 3.3  River Runoff for Each Continent (Source: Shiklomanov, 1993) Annual river runoff Continent

(mm)

(km3)

Portion of total runoff (%)

Area (103 km2)

Specific discharge (l/s/km2)

Europe

306

3,210

7

10,500

9.7

Asiaa

332

14,410

31

43,475

10.5

Africab

151

4,570

10

30,120

4.8

North Americac

339

8,200

17

24,200

10.7

South America

661

11,760

25

17,800

21.0

45

348

1

7,683

1,610

2,040

4

1,267

Australiad Oceania

1.44 51.1

Antarctica

160

2,230

5

13,977

5.1

Total land area

314

46,770

100

149,000

10.0

Asia includes Japan, the Philippines, and Indonesia. Africa includes Madagascar. North and Central America. d Australia includes Tasmania. a

b c

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3-4    Water Balance 3.3.2  Per Capita Water Availability

3.4  WATER BALANCE MODELING

Water balance is based on the principle of conservation of mass, where the change in storage ( ∆S ) is the difference between the inflow and outflow of water, expressed as:

∆S = P − R − E − G (3.1)

where P is the precipitation, R is the runoff, E is the evaporation, and G is the deep groundwater storage. The water balance, or water budget, can be computed for part of the hydrologic cycle (soil profile, terrestrial, or atmosphere component), or over a defined spatial dimension (root zone, catchment area, continental, or global). The water budget can also be computed for any temporal scale. 3.4.1  Water Balance Models

The hydrologic cycle is governed by both local processes and remote oceanatmosphere interactions. On the local scale the land surface exerts a nonnegligible influence on the atmospheric boundary layer through momentum, energy, and water fluxes. This may occur over large spatial and temporal scales, and influences both the weather and the global climate system. This interaction can be broadly classified into two types: that affecting the physical climate system and that impacting the biogeochemical cycles. The physical system involves the exchanges of radiation, sensible heat, latent heat, and momentum, and has an immediate impact on the wind regime, precipitation, skin temperature, and soil moisture. The biogeochemical cycles involve the exchanges of gases, which affect the radiative transfer characteristics, and consequently, the energy budget of the planet (Sellers, 1991). These processes affect the movement of water, its interactions within the hydrological cycle, and its interactions with other components of the atmosphere, biosphere, cryosphere, lithosphere, and pedosphere. These processes are complex. Understanding and modeling the role of water in these processes helps in the development of more precise hydrological models. A large number of water balance models have been reported in the literature (Singh, 1995; Singh and Frevert, 2002a, b, 2006; Singh and Woolhiser, 2002). In this section, some of the different water balance models, ranging from the simple one-parameter model to more complex ones that include most known hydrological processes, are presented for purposes of illustrating the rudiments for computing water balance. Simple Bucket Model The “bucket” (or leaky bucket) hydrologic model is the simplest water balance model and the fundamental building block of most rainfall-runoff models (Fig. 3.3). In this model, a single-layer soil sample is simulated as a

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Precipitation

Evaporation Runoff

Field capacity

Total annual precipitation or river discharge gives a limited idea of the amount of water available for human consumption, especially given that the world’s population is not uniformly distributed. Per capita water availability depends on both the amount of renewable water available and the total population dependent on it. On a continental scale, Australia and Oceania, even though have the lowest percentage runoff, have the highest per capita water availability. Asia, home to two-third of the world’s population, has the highest annual river runoff by volume, but has the least per capita water availability (3920 m3/ person/year), followed by Europe (4230 m3/person/year), and Africa (5720 m3/ person/year) (Shiklomanov, 1998). At the country scale, water availability depends on a number of factors, including: surface area, climate regime, geographic and physical conditions, geomorphology, and population. Two countries on the same continent may have very different per capita water availability. Brazil and Peru, for example, both located in South America have distinctly different renewable per capita water availability: 28,124 and 54,023 m3/inhabitant/year, respectively; the latter being one of the highest in the world (FAO, 2015). Iceland (515,152 m3/inhabitant/year), Guyana (301,250 m3/inhabitant/year), and Suriname (183,673 m3/ inhabitant/year) have the highest total internal renewable water resources per capita, mainly due to their small population (330,000, 800,000, and 539,000, respectively). Canada, with a population of over 35 million, has a total internal renewable water resource per capita of 81,007 m3/inhabitant/year. On the other side of the spectrum, Kuwait (negligible), Bahrain (3 m3/inhabitant/year), and the United Arab Emirates (16.05 m3/inhabitant/year) have the lowest total internal renewable water resource per capita, mainly due to their geographic location. China and India, two of the most populous countries, have only 1986 and 1155 m3/inhabitant/year, respectively. Of the 181 countries included in the FAO’s AQUASTAT database, 39% have less than 1700 m3/inhabitant/year; 27% have less than 1000 m3/inhabitant/year; and 16% have less than 500 m3/inhabitant/year, where 1700, 1000, and 500 m3/inhabitant/year are the United Nations thresholds for water stressed, scarce, and absolute scarcity, respectively. The number of people living in each of these regions is increasing, hence putting more pressure on the limited water resources available.

Dry soil

Water depth

Wet soil

Figure 3.3  Manabe’s simple bucket hydrologic model.

bucket. When it rains, if precipitation is greater than evaporation, the moisture capacity of the soil sample increases up to the point where it cannot hold any more water and any excess water thus runs off. This simplification is not totally accurate; it has been established that runoff occurs if the rate of precipitation exceeds the rate of infiltration irrespective of the degree of soil saturation (Loaiciga et al., 1996). Budyko (1956) used the simple bucket model to calculate Earth’s energy balance and Manabe (1969) was the first to include this hydrologic parameterization scheme within a general circulation model (GCM). The simple bucket model ignores vegetation and other important hydrologic processes; subsequent models have been developed to capture more complex and realistic hydrologic processes. Bucket with a Bottom Hole Model Kobayashi et al. (2001) proposed the bucket with a bottom hole model (BBH); a simple bucket model with a hole at the bottom that allows both downward and upward movement of water across the bottom surface. BBH is an active surface soil layer of depth D (about 10 cm thick), where the water content can go below the field capacity and an underlying soil moisture reservoir. The change in W, the equivalent depth of water in the soil layer, is given by:

∆W = W (t + 1) − W (t ) = Pr (t ) − E(t ) − Gd (t ) − Rs (t ) (3.2)

where t is the time in days, Pr is the daily precipitation, E is the daily evapotranspiration, Gd is the daily gravity drainage (positive) or capillary rise (negative), and Rs is the daily surface runoff. E  =  M ⋅ E p, where M = min [W /σWmax , 1], E p is the daily potential evapotranspiration (PET), Wmax is the total waterholding capacity (porosity  ×   D ), and s is the resistance of ground cover to W − a  evapotranspiration. Gd = exp   − c, where a is nearly equal to or less than  b  WFD, the field capacity, b is the soil moisture recession parameter, and c is the daily potential capillary rising. When W equals bucket capacity WBC = ηWmax, where η (≤ 1) is the moisture retaining capacity of the soil, and Pr > E + Gd ,

Rs = Pr − E − Gd (3.3)

runoff is still possible even if W < WBC, when Pr > E + Gd + (WBC − W )

Rs = Pr − E − Gd − (WBC − W ) (3.4)

A modified version of the model (BBH-B) was later proposed to capture bioprocesses. It is a two-layer model with the intent of modeling interception losses and macropore flow in the root zone (Kobayashi et al., 2006). Simple Water Balance Model Schaake et al. (1996) developed a “simple water balance” (SWB) model which is intermediate between the one-parameter simple bucket model and the more complex hydrological models. The main aim of the model was to improve the representation of runoff, with respect to the simple bucket model, without including every complex physical process involved. The model parameters include both a physical representation and a statistical averaging of the processes. SWB can be applied to the catchment scale or to represent runoff in the coupled atmospheric-hydrological models (e.g., the NOAH land surface model). It can help in refining our understanding of the level of complexity required in realistically modeling land surface processes and to estimate model parameters.

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Water Balance Modeling    3-5 

Thornthwaite’s Water Balance Model Thornthwaite (1931) produced climatic maps of North America and later that of the Earth (Thornthwaite, 1933) using evaporation data to compute a moisture index. He utilized two concepts: precipitation effectiveness and temperature efficiency, defined as the accumulated sum of monthly precipitation to evaporation ratio and the accumulated sum of temperature to evaporation ratio, respectively, in producing those maps. Later, Thornthwaite (1948) proposed a revised climate classification scheme and introduced the term potential evapotranspiration (PET) (Brutsaert, 2005). He recognized that the moisture index should not be based on actual evaporation and transpiration from the soil but on PET. Potential evaporation was derived empirically from mean temperature. The method was further modified by Thornthwaite and Mather (1955) to accommodate a wider range of soils and vegetation types. Air temperature is assumed to be correlated with the integrated effects of net radiation and other controls of parameters affecting evapotranspiration. It is used as an index of the energy available for evapotranspiration, shared between heating of the atmosphere and evapotranspiration (Calvo, 1986). PET is computed over a 12-h (amount of daylight) and a 30-day month:

a

 10T  PET = 1.6  a  (3.5)  I 

where Ta is the mean monthly air temperature (°C), and I is the heat index for the station, which is the sum of 12 monthly heat indices, given by:

and Mather (1957). It was written in APL and later translated in other languages. McCabe and Markstrom (2007) developed a monthly water balance model, based on Thornthwaite’s method, driven by a graphical user interface. The model can be used for water balance assessment, research, or as a classroom instruction tool. The main components of the model are: snow accumulation, direct runoff, snowmelt, evapotranspiration and soil-moisture storage, and runoff (Fig. 3.4). The inputs to the model are monthly temperature (T, in °C), monthly total precipitation (P, in mm), and the latitude of the location of interest. The latter determines the day length, a variable that determines PET. The hydrological partitioning and steps in the computations are summarized as follows. Snow Accumulation  The model first determines the amount of monthly precipitation (P) that is rain ( Prain ) or snow (in mm). Mean monthly temperature (T) determines whether precipitation is snow, rain, or a mix of both. When the mean monthly temperature is below Tsnow, all precipitation is considered snow and if T is greater than Train, all precipitation is considered to be rain. When T is between Tsnow and Train, the amount of precipitation that is snow is given by:

12

T  I = ∑ ai  5  i =1

1.514

(3.6)

and a  = 0.000000675 I 3 − 0.0000771I 2 + 0.01792 I + 0.49239 . The values for a and I can be obtained from tables in Thornthwaite and Mather (1957). The information required to compute the water balance at a site includes: latitude, mean monthly air temperature, mean monthly precipitation, necessary conversion and computational tables, and information on water holding capacity of the depth of soil for which the balance is to be computed. Black (2007) notes that the latter method to compute water balance is useful for “description, classification, management, and research,” but given that the method is based on monthly data, precipitation falling at the end of the month may not be allocated to the appropriate month. Several computer programs have been written to compute water balance using the Thornthwaite method. Black (1981) was probably the first to write a computer program to implement the method described in Thornthwaite

 T −T  Psnow = P ×  rain  (3.7) Train − Tsnow 



and Prain = P − Psnow . The value for Train has been empirically found to be 3.3°C, while that for Tsnow varies with elevation. For elevations less than 1000 m, Tsnow = −10°C and for elevations more than 1000 m, Tsnow = −1°C . Direct Runoff  Direct runoff (DRO) is the fraction of rainfall that does not infiltrate, for example, from impervious surfaces, and runoff from rainfall excess after infiltration. The percentage of precipitation that becomes DRO has been empirically found to be about 5%. The amount of precipitation after DRO is Premain = Prain − DRO . Snow Melt  The fraction of snow storage (snowstor) that melts in a month (SMF) is given by:  T − Tsnow  SMF =   × meltmax (3.8) Train − Tsnow 



where the meltmax is the maximum melt rate set to 0.5. If SMF is greater than meltmax, then SMF = meltmax. The snow water equivalent of the amount of snow (SM) that is melted in a month is given as SM = Psnow × SMF. The total liquid water input to the soil ( Ptotal ) is the sum of SM and Premain.

Temperature (T)

Precipitation (P)

Potential evapotranspiration (PET) Snow (Psnow)

Actual evapotranspiration (AET)

Snow storage (snostor)

Snow melt (SM)

Rain (Prain) Direct runoff (DRO)

Surplus runoff (RO) Soil-moisture storage capacity (STC)

Soil-moisture storage (ST)

Figure 3.4  Components of the Thornthwaite monthly water balance (Source: McCabe and Markstrom, 2007).

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3-6    Water Balance Evapotranspiration and Soil-Moisture Storage  Actual evapotranspiration (AET) is a function of PET, Ptotal , soil-moisture storage (ST), and soil moisture withdrawal (STW). Monthly PET is calculated through the Hamon equation (Hamon, 1961)

PETHamon = 13.97 × d × D 2 × Wt (3.9)



where PETHamon is PET in mm, d is the number of days in a month, and D is 4.95 × e0.062×T the mean monthly hours daylight units in 12 h, and Wt = ,a 100 3 saturated water vapor density term, in g/m . When Ptotal for a month is less than PET, AET is equal to Ptotal plus the amount of moisture that can be withdrawn from storage in the soil. STW decreases linearly with decreasing ST, such that as the soil becomes drier, it becomes more difficult to remove water from the soil and less is available for AET:

  ST   STW = STi−1 − abs ( Ptotal − PET ) ×  i−1   (3.10)  STC   

where STi−1 is the soil moisture storage for the previous month and STC is the soil moisture storage capacity, estimated to be 150 mm. If Ptotal + STW < PET, then water deficit is calculated as PET − AET; if Ptotal > PET, then AET = PET and water in excess of PET replenishes ST. When ST > STC, the excess water is deemed surplus (S) and eventually runs off. Runoff Generation  Runoff (RO) is generated from S at a specified rate of 0.5, which determines the fraction of surplus that becomes runoff for a month and the excess is carried over to the next month. The total monthly runoff RO total = DRO + RO. Steenhuis and Van Der Molen (1986) extended the Thornthwaite–Mather procedure to use daily input values and account for delay caused by percolation through the unsaturated zone in calculating recharge from soil moisture balance. A computer program to compute soil water balance with spatial and temporal variations in groundwater recharge in a geographic information system setting has been developed by Westenbroek et al. (2010). Soil and Water Assessment Tool The Soil and Water Assessment Tool (SWAT; Arnold et al., 1998; http://swat .tamu.edu) is a modeling effort of the USDA Agricultural Research Service. It has been in continuous development for over 30 years to meet the growing and diverse needs of the hydrological, environmental, and water resources management community. SWAT is a physically based continuous-time model that operates on a daily time step and is capable of predicting the impact of water management on sediment and other nonpoint source pollutions, and agricultural yields in ungagged watersheds. The main components of the model are: weather, hydrology, soil temperature and properties, plant growth, nutrients, pesticides, bacteria and pathogens, and land management. The hydrologic modeling is carried out at the watershed or subwatershed scale. A basin is either subdivided into smaller subbasins or can be subdivided down to hydrologic response units (HRUs), characterized by their prevailing land use, soil type, and management characteristics. The model requires daily precipitation, minimum and maximum temperature, solar radiation, relative humidity, and wind speed as inputs. The average air temperature determines if precipitation should be in the form of snow or not. Evapotranspiration can be computed via either Penman–Monteith (Monteith, 1965), Priestly–Taylor (Priestley and Taylor, 1972), or Hargreaves (Hargreaves et al., 1985) equations. Hydrologic simulations and water balance at the HRU scale includes canopy interception of precipitation, partitioning of precipitation, snowmelt, and irrigation water between surface runoff and infiltration, redistribution of water within the soil profile, evapotranspiration, lateral subsurface flow from the soil profile, and return flow from shallow aquifers. A number of routines are available in SWAT to simulate crop yields, biomass outputs, nutrients and pesticide applications, and conservation and water-management practices. Flow from all HRUs are aggregated to the subbasin level and routed using either the variable-rate storage method or the Muskingum method. Sediment, nutrient, pesticide, and bacterial loadings or concentrations are also aggregated at the HRU level and routed through channels, reservoirs, and other water bodies within the basin. SWAT has been widely adopted across the world. Among others, it is being used in total maximum daily load (TMDL) analyses and to support conservation efforts in the United States (U.S.). A complete review of the SWAT model and its applications is available in Gassman et al. (2007). Variable Infiltration Capacity Model The variable infiltration capacity model (VIC: Liang et al., 1994; http:// www.hydro.washington.edu/Lettenmaier/Models/VIC/index.shtml) is a

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semidistributed macroscale hydrologic model that can solve both the water and energy balance. It can be run in either the water balance mode or the water-and-energy balance mode; the former being less computationally intensive. VIC was developed for incorporation in GCMs and has undergone several updates and modifications over the last 20 years. It has been used in studies ranging from water resources management, including the effect of climate change, to land-atmosphere interaction. VIC operates at the grid scale and subgrid variations are captured statistically. The main features of VIC is its ability to incorporate subgrid heterogeneity in land surface vegetation classes, subgrid variability in soil moisture storage capacity, drainage from the lower soil moisture zone as a nonlinear recession, and topography, thus enabling more realistic hydrology in mountainous regions by allowing for orographic precipitation and temperature lapse rates. Within the grid-based VIC, vegetation characteristics [e.g., leaf area index (LAI), albedo, minimum stomatal resistance, relative fraction of roots in each soil layer, etc.] are assigned for each vegetation tile. Evapotranspiration is obtained through the Penman–Monteith equation. Total AET include evaporation from the canopy layer, transpiration from each vegetation tile, and evaporation from bare soil, weighted by their respective surface cover area fractions. For each land cover type, there is one canopy layer. It intercepts rainfall according to the Biosphere-atmosphere transfer scheme parameterization as a function of LAI [Dickinson et al., 1986]. There can be multiple soil layers for each land cover type. Soil characteristics (e.g., texture, hydraulic conductivity, etc.) are constant within each grid, even though vegetation characteristics may vary. In a three-layer representation, the two top layers are designed to represent the dynamic response of soil to the infiltrated rainfall; if the middle layer is wetter, diffusion is permitted from the latter to the upper layer. The third layer receives moisture from the middle layer through gravity drainage following the Brooks–Corey relationship for unsaturated hydraulic conductivity (Brooks and Corey, 1964). Runoff from the bottom layer is modeled according to drainage description in the Arno model (Franchini and Pacciani, 1991). For each time step, soil moisture distribution, infiltration, drainage between soil layers, surface runoff, and subsurface runoff are calculated for each land cover tile. For each grid cell, total heat fluxes (latent heat, sensible heat, and ground heat) effective surface temperature, and total surface and subsurface runoffs are obtained by aggregating over the land cover tiles weighted by their fractional coverage. Each grid is modeled independently without any horizontal water flow and runoff is obtained for each grid cell. Runoff is not uniform across the cell. A stand-alone routing model collects and transport the gridbased surface runoff and base flow toward the outlet of each grid. Water in the channel is no longer included in the water budget accounting procedure. Once runoff exits a cell, a channel routing, based on the Saint–Venant equation, simulated drainage and discharge at the basin outlet. A detailed description of the model, including a historical overview, complete set of algorithms, calibration, and validation procedures, is available in Gao et al., (2009). Noah The community Noah [the acronym comes from the first letter of the following four organizations: the National Centers for Environmental Protection, the Oregon State University (Department of Atmospheric Sciences), the Air Force, and the Hydrologic Research Lab—NWS (now Office of Hydrologic Development) land surface model (Chen et al., 1996; Koren et al., 1999; http:// www.ral.ucar.edu/research/land/technology/lsm.php) is a stand-alone onedimensional column model. It can simulate soil moisture (both liquid and frozen), soil temperature, skin temperature, snowpack depth, snowpack water equivalent, canopy water content, and the water and energy flux terms of the surface water and energy balance (Mitchell, 2005). The model has a snow layer and a canopy layer. The snow layer simulates snow accumulation, sublimation, melting, and heat exchange at snowatmosphere and snow-soil interfaces. Precipitation is deemed snow if the temperature of the lowest atmospheric layer is less than 0°C. The surface skin temperature is determined by applying a single linearized surface energy balance equation (Mahrt and Ek, 1984). The ground heat flux is governed by the diffusion equation for soil temperature. The governing equations for the physical processes of the soil-vegetation-snowpack medium are solved through a finite difference spatial discretization and the Crank–Nicolson time integration scheme. The total evaporation, in the absence of snow, is the sum of evaporation of precipitation that has been intercepted by the plant canopy, transpiration from the roots and canopy of vegetation, and direct evaporation from the topmost soil layer. Potential evaporation is calculated using Penman-based energy balance and includes a stability-dependent aerodynamic resistance after Mahrt and Ek (1984). The wet canopy evaporation follows Noilhan and Planton (1989) and Jacquemin and Noilhan (1990) and evaporation from bare soil follows Mahfouf and Noilhan (1991).

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Natural and Anthropogenic Effects on the Water Balance     3-7 

Figure 3.5  Effect of ENSO around the world. [Source: National Oceanic and Atmospheric Administration (2012).]

The soil profile extends to a depth of 2 m, divided into four layers (0–0.1, 0.1–0.4, 0.4–1, and 1–2 m) from the ground surface to the bottom. The root zone is confined to the upper 1 m, while the lower layer acts as a reservoir with gravitational free-drainage at the bottom (Chen and Dudhia, 2001). The volumetric soil moisture content is determined using the diffusive form of Richard’s equation, which is derived from Darcy’s law under the assumption of rigid, isotropic, homogeneous, and 1D vertical flow. Surface runoff is the excess after infiltration (Schaake et al., 1996). Noah has been tested and validated in both coupled and uncoupled modes (Mitchell, 2005). A complete description of the model physics and computational algorithm is available in Chen and Dudhia (2001) and Grunmann (2005). 3.5  NATURAL AND ANTHROPOGENIC EFFECTS ON THE WATER BALANCE

The hydrological cycle and each of the components of the water balance is sensitive to local changes happening within the watershed and to changes in the climate. These changes are noticeable at various spatial and temporal scales. The most immediate changes are noticed when land use land cover is affected. Natural variations in the climate and changes due to anthropogenic contribution of greenhouse gases and aerosols in the atmosphere are felt on most hydro-meteorological variables and affect the water balance. 3.5.1  Natural: Climate Variability

The most common drivers of climate variability include large-scale circulation patterns, sunspots, and volcanic eruptions. The former is the most important variable and can have dramatic influence on the climate, weather, and other hydro-meteorological variables. The lag and duration between the onset of these phenomena and their impacts locally can range from a few months to years, and at times, even decades. A number of climate variability patterns have been identified. Some exert negligible influence on the hydrometeorological cycle, while others can drastically affect water balance. The extent and severity with which they influence local conditions is often a function of the geographical distance between the watershed of interest and the poles of the climate teleconnection patterns. The most important large-scale circulation pattern is the El Niño-Southern Oscillation (ENSO). ENSO is a coupled ocean-atmosphere phenomenon related to sea surface temperature anomalies in the central and eastern equatorial Pacific and the associated sea-level pressure difference, the Southern Oscillation (SO). ENSO has a recurrence pattern of 3–6 years and every event normally lasts for around a year. El Niño events are often, but not always, followed by La Niña events, also referred to as the cold phase of ENSO.

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The effect of ENSO is neither distributed uniformly across the globe, not even across a single continent, nor in time during its period of occurrence (Lyon, 2004). As ENSO cycles through its different phases (El Niño, La Niña, and neutral), its impacts vary both spatially and temporally (Fig. 3.5). The two hydrological variables that are most affected during ENSO are precipitation and temperature. Some regions may experience above-average rainfall, leading to devastating floods, while others may endure extended periods of precipitation deficit leading to excruciating droughts, the drying of biomass, and forest fires. ENSO affects water balance mostly in the southeastern part of Africa and Madagascar and the eastern part along the equator. Dry and warm conditions are felt during El Niño from December through February in the southeastern part, while the eastern part benefits from higher precipitation. The opposite pattern is noted during La Niña. The influence of ENSO is visible in the seasonal vegetation dynamics as highlighted in the normalized difference vegetation index (NDVI) (Philippon et al., 2014). The hydrology of Asia and Australia can be seriously affected during ENSO. El Niño causes abnormally warm and dry conditions during December through February over the region extending from India to northern Australia. Dry conditions extend further south affecting hydrological conditions in the eastern part of Australia, leading to an increase in the risk of bush fires (Fuller and Murphy, 2006; Williams and Karoly, 1999). The Indian summer monsoon, which lasts from June to September and accounts for up to 80% of the rainfall in the region, is negatively correlated with ENSO—the region receives less than average rainfall during El Niño events (Kirtman and Shukla, 2000). Delays in the onset of the monsoon and changes in the amount and pattern of precipitation affects soil moisture and may lead to agricultural drought, and has a strong impact on several major rivers [e.g., the Huang He (Yellow) and Yangtze Rivers in China, the transboundary Mekong, GangesBrahmaputra-Megna, Indus, etc.] fed from the Himalayan glaciers (Thompson et al., 2000). Nearly 1.5 billion people in the most populous region in the world depends on those rivers for sustenance. Given the geographical distance separating the central Pacific Ocean and Europe, the influence of ENSO is hardly noticeable on the hydrology of the European regions (Fraedrich, 1990, 1994). El Niño (La Niña) episodes have been associated with enhanced cyclonic (anticyclonic) activities. El Niño has also been associated with more variable winter conditions, which affect snowfall, and ultimately streamflow in snow fed rivers. Correlation between ENSO and mass of glaciers in the European Alps is almost negligible at the yearly timescale (Durand et al., 2009), but weak correlations may be detected on a multidecadal scale (Efthymiadis et al., 2007). ENSO alters the path of the jet stream, influencing weather and storm tracks on the North American continent. A dip in the position of the jet

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3-8    Water Balance

stream, during El Niño, results in warmer winters in western Canada and southern Alaska and northeastern U.S., while the southern U.S. experiences cooler and wetter conditions. California experiences above average precipitation, which can often lead to flash floods (Mo and Higgins, 1998; Schonher and Nicholson, 1989). Influx of excess moisture during El Niño also increases snowfall in the Sierra Nevada mountain range (Kunkel and Angel, 1999). The southern part of Mexico, on the other hand, experiences warmer and drier summers. During La Niña, the dip in the jet stream shifts west of its normal position toward the Central Pacific resulting in cooler winters in eastern Canada, while the weather conditions in southern U.S. are warm and dry. Water availability in some sections of the Rio Grande basin, which extends from the Rockies in Colorado to the Gulf of Mexico, can increase by over 300% and decrease by nearly 100% during the warm and cold phases of ENSO, respectively (Khedun et al., 2012). In South America, El Niño has been associated with severe drought in Mexico and most of Brazil, while Argentina, Paraguay, and Uruguay have experienced increased precipitation. The coastal areas of Ecuador, northern Peru, and southern Chile also benefit from above-average precipitation, while snowfall in the mountainous Andes decreases, causing the glaciers to retreat. During La Niña, however, the region can receive up to four times more precipitation (Haylock et al., 2006). Extreme precipitation events have also been noted during ENSO (Grimm and Tedeschi, 2009). ENSO is not the only climate variability pattern to influence water balance. Other climate teleconnection patterns, such as the Pacific Decadal Oscillation (PDO), the North Atlantic Oscillation, the Atlantic Multidecadal Oscillation, the Quasi Biennial Oscillation, the El Niño Modoki, etc., have been found to influence the hydrological cycle, either alone, or in conjunction with ENSO. In the U.S., for example, PDO has been found to modulate the effect of ENSO, where El Niño (La Niña) during the positive (negative) phase of the PDO, may lead to stronger climate responses, than when they are evolving in opposite phases (Gershunov and Barnett, 1998; Khedun et al., 2014). 3.5.2  Anthropogenic Influences

Earth’s surface has been subject to considerable anthropogenic influences, especially over the last century. Modification of the natural state of the watershed upsets local level water balance, and as these alterations expand, the country and continental scale hydrological patterns and water balance are affected. Changes in the climate as well affect the spatiotemporal hydrometeorological variables and thus the water balance. Land Use Land Cover Changes The relationship between climate, land cover, and the hydrological cycle is complex. Climate determines vegetation type and density of cover, which, in turn, influences the hydrology of the watershed—both the above- and subsurface components of the hydrological cycle are affected by land cover characteristics. Vegetation cover intercepts and stores a fraction of precipitation; part of which eventually reaches the ground through stemflow and throughfall, while some evaporates back into the atmosphere. A higher proportion of rainfall is intercepted during light- or short-duration events compared to high intensity or long duration ones. Other vegetation characteristics, such as leaf area, leaf density, and orientation also determine interception storage. Part of the water that reaches the ground and does percolate deep into groundwater storage is available for evaporation. Some evaporates directly from the soil surface, while others transpire through the plant cover. Evapotranspiration is influenced by plant type, growth stage and growth cycle, and evaporation demand of the atmosphere, which is controlled by temperature, humidity, and wind speed. The complex nature of the interaction between precipitation with vegetation and its impact on local hydrology is extensively discussed in the literature (e.g., Crockford and Richardson, 2000; Llorens and Domingo, 2007; Martinez-Meza and Whitford, 1996). Disruption of the vegetation cover, for example, when forests are replaced with agriculture, urbanization, etc., directly impacts the hydrology and water balance of the watershed. Infiltration capacity of the soil and evapotranspiration are reduced, and surface runoff and streamflow increase when forests are cleared, and the reverse trend is observed when cleared areas are reforested. Urbanization further increases the area of impermeable surfaces, again inhibiting the infiltration of precipitation and reduces evapotranspiration leading to greater runoff and streamflow. The magnitude of these changes depends on the watershed characteristics, the prevailing climate, and spatial area impacted by land use land cover changes. Results from a study based in the upper Midwest United States showed that the conversion of forest to cropland resulted in a decrease in evapotranspiration by 2.5% and an increase in runoff by 20% and baseflow by 4%, while urbanization caused a decrease in evapotranspiration and baseflow by 70 and 38%, respectively while surface runoff increased by 1200% (Mishra et al., 2010).

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Changes in the vegetation type or crop cycle also affect the hydrology of a watershed by altering evapotranspiration regime. The conversion of rangeland to cropland, for example, has been found to enhance recharge (Scanlon et al., 2005). A decrease in evapotranspiration and increase in streamflow and baseflow was observed when cropping systems changed from perennial to annual (Schilling et al., 2008). Infrastructure related to urbanization, such as the construction of roads, form barriers, modify drainage patterns, and disrupt the movement of water within the basin and thus affect water balance. Other anthropogenic factors, such as water abstraction from surface and groundwater sources, diversions and interbasin water transfers, and discharge of effluent, also affect the watershed scale water balance. Dams alter flow regimes and affect both the upstream and downstream hydrology of the watershed—marked impacts on flow, sediment, aquatic and riparian ecology, and the environment are visible especially downstream, almost immediately after the construction of a dam, and changes in the channel geomorphology becomes apparent in the long term (Lu and Siew, 2006; Mathias Kondolf and Batalla, 2005; Singer, 2007). Around the world, between 4510 and 5330 km3 of water, equivalent to 11–13% of total annual river runoff (~41,000 km3/year), is impounded behind major dams (Gornitz, 2000). Groundwater Abstraction The component of precipitation that drains through the soil matrix and does not evaporate or transpire back to the atmosphere through plants becomes groundwater. This water is stored in aquifers. Aquifers are classified as unconfined, confined, and fossil aquifers. Unconfined aquifers do not have a confining upper layer and their upper boundary is the water table. They are often shallow and are recharged by water percolating through the unsaturated zone directly above the aquifer and are thus prone to changes in climatic factors, especially precipitation and temperature. Confined aquifers are found between layers of impermeable material both above and below the aquifer. They are saturated with water and are under pressure. These aquifers are recharged by precipitation in the area where the aquifer crops out and has similar characteristics as an unconfined aquifer. Fossil aquifers have groundwater that has remained trapped, as a result of geologic activities, in the rock formations for thousands or millions of years. Unlike confined and unconfined aquifers, they are not replenished and are thus a nonrenewable resource. Examples of fossil aquifers include the Ogallala underlying the U.S. Great Plains and the Nubian Sandstone Aquifer, one of the largest known fossil aquifers, located on the Eastern end of the Sahara Desert. Surface and ground water balance are interlinked. Water balance models can be used to estimate the contribution of groundwater to surface water or vice versa. Groundwater and surface water interaction comprises natural exchange fluxes with rivers, fluxes due to groundwater abstraction, aquifer recharge, and changes in evapotranspiration (Rassam et al., 2012). Groundwater abstraction can be from both replenishable and nonreplenishable aquifers; it is being mined to meet domestic, industrial, and agricultural uses. Globally, groundwater contributes 43% (545 km3/year) of the total consumptive irrigation water use (Siebert et al., 2010) and has played a key role in containing the food crisis. In several agricultural areas around the world, groundwater has become the primary source of irrigation. Satellite measurements have revealed dramatic loss of groundwater resources in northern India (Rodell et al., 2009), western U.S. (Famiglietti et al., 2011), and elsewhere (Famiglietti, 2014). Groundwater depletion is even more apparent during extended droughts (Long et al., 2013). Therefore, groundwater, especially fossil water, is an important component of the hydrological cycle and any water balance exercise. It can be deemed an input to the water balance equation, just like precipitation, and accounted as a flux of water vapor, through evaporation and evapotranspiration, into the atmosphere. Recent studies (DeAngelis et al., 2010; Puma and Cook, 2010) have shown that irrigation can significantly increase downwind precipitation. Groundwater also, subsequently, increases the volume and rate of streamflows to the ocean and has been found to be partly responsible for sea-level rise (Gornitz, 2000; Konikow, 2015; Pokhrel et al., 2012). Climate Change Earth’s climate is constantly changing. Evidence from tree rings and other paleo-climatological records indicate that changes in the climate are also reflected in the hydrological system at both spatial and temporal scales (Gray et al., 2004; Gray and McCabe, 2010). Global warming, due to anthropogenic emissions of greenhouse gases, a result of industrialization, is having a discernible influence on the physical and biological systems across the globe. An increase in the amount of energy on Earth’s surface results in an intensification of the hydrological cycle, which disrupts every component of the water balance.

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REFERENCES    3-9 

A warmer planet implies that the atmosphere has a higher water holding capacity, as given by the Clausius–Clapeyron equation

des Lv (T ) es = (3.11) RvT 2 dT

where es is the saturation vapor pressure, T is the temperature, Lv is the latent heat of vaporization, and Rv is the gas constant for water vapor (Shelton, 2009). A higher atmospheric concentration of water vapor, a greenhouse gas, results in the capture of more of the outgoing radiation from the Earth’s surface, thus leading to even further warming. Warmer temperatures enhances evapotranspiration, which results in lower soil moisture for plants and a decrease in groundwater recharge and reduced baseflow, and therefore, less water in the streams, rivers, lakes, and wetlands. It also affects snow accumulation and enhances the melting of land-based glaciers resulting in changes in the timing and volume of runoff and streamflow in snow and glacier-fed catchments. On the other hand, an increase in evaporation from large water bodies such as lakes and oceans implies more water in the atmosphere and hence an increase in global precipitation. Higher precipitation results in an increase in the soil moisture and recharge component of the water balance equation, where the increase in precipitation is greater than that in evapotranspiration. This complex interaction between the climate and the hydrological cycle, and the redistribution of water in the water balance equation, has been explored through GCMs. The Intergovernmental Panel on Climate Change (IPCC; Cisneros et al., 2014) states that as the climate changes, precipitation across the globe is projected to increase. However, this increase will not be equally distributed across the planet. Watersheds located in high latitudes and equatorial Pacific are likely to experience an increase in mean annual precipitation, while those located in the mid-latitude and subtropical dry regions are likely to suffer a decrease in mean precipitation. On the other hand, mean precipitation in catchments located in several mid-latitude wet regions is likely to increase. Soil moisture in dry regions is likely to further decrease, hence increasing the frequency of meteorological and agricultural droughts, leading to less surface and groundwater. Intensification of the hydrological cycle also implies changes in the frequency, intensity, spatial extent, duration, and timing of extreme precipitation events. Changes at both the lower tail (e.g., reduction in precipitation leading to droughts) and the upper tail (e.g., high-intensity rainfall resulting in floods) of range of observed values can be expected. These changes may be in the following three ways: (i) a shift in the mean which will result in less lowmagnitude events and more high-magnitude events; (ii) an increase in standard deviation and thus variability which equates with more low- and high-magnitude events; and (iii) a change in the shape of the distribution, where low magnitude events remain almost constant while high magnitude events increases (IPCC, 2012). 3.6 CONCLUSION

Hydrological processes are complex. Each process plays an important role in the water balance—both at the spatial and temporal scales. Natural factors, such as climate variability patterns, and anthropogenic factors, such as land use land cover and climate change, influence the hydrological cycle and hence affect, to varying degrees, the water balance components. A wide range of models, ranging from the one-parameter simple bucket model to more complex ones, aimed at realistically representing each process, has been developed. These models help in our understanding of the role of water and its interactions within the hydrological cycle and with other physico-chemical and biological processes in other elemental cycles. Insights into the movement of water and its vulnerabilities can help in the assessment and mitigation of the impacts of natural and anthropogenic factors on water availability. REFERENCES

Arnold, J. G., R. Srinivasan, R. S. Muttiah, and J. R. Williams, “Large area hydrologic modeling and assessment. Part I: model development,” Journal of the American Water Resources Association, 34 (1), 73–89, 1998, doi:10.1111/ j.1752-1688.1998.tb05961.x. Black, P. E., The Thornthwaite Water Budget in APL, SUNY College of Environmental Science and Forestry, Syracuse, New York, 1981. Black, P. E., “Revisiting the Thornthwaite and Mather water balance,” Journal of the American Water Resources Association, 43 (6), 1604–1605, 2007, doi:10.1111/j.1752-1688.2007.00132.x. Brooks, R. H. and A. T. Corey, Hydraulic Properties of Porous MediaRep., Colorado State University, Fort Collins, Colorado, 1964.

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PART

2

DATA COLLECTION AND PROCESSING

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Chapter

4

Hydrometeors and Quantitative Precipitation Estimation BY

BEKKI WARD HARJO 

ABSTRACT

Hydrometeorological data are collected using radars, gages, satellites, and human observations. Quantitative precipitation estimates are developed from multisensor analysis and are sometimes supplemented with climatological data. Automated algorithms can help identify and estimate hydrometeors, but human-performed quality control is still needed to ensure a high-quality final estimate.

that it covers a limited area, so precipitation occurs between gaging sites. Thus gages may not always sample rainfall maxima and minima, which are crucial to a good hydrometeorological analysis. 4.2.2  Remotely Sensed Measurements

4.1 INTRODUCTION

Ground-based radars can provide estimation of the phase, quantity, and elevation of hydrometeors in the atmosphere. Satellites can provide visual images, thermal imagery, and platforms for radiometers to help determine the quantity and phase of hydrometeors and also show the extent of snow and ice after an event has occurred.

4.1.1  Precipitation Phases

4.2.3  Ground-Based Indirect Measurements

Hydrometeors are various forms of water in the atmosphere. Hydrometeors in liquid form include rain, dew, fog, drizzle, and cloud droplets. Solid hydrometeors include snow, hail, ice crystals, and graupel. Mixed-phase hydrometeors include wet snow and water-covered hail. Most hydrometeors are large enough to fall as precipitation, typically formed when ice crystals in the clouds grow big enough to overcome updrafts in the clouds and begin falling as snow. A much smaller percentage of precipitation forms as liquid drops and grows by the process of collision and coalescence when ambient cloudlevel temperatures are above freezing. 4.1.2  Impacts at Surface of Hydrometeor Types and Temperatures

The impacts at the earth’s surface can be significantly different based on the temperature of the hydrometeor and the temperature of the air near the earth’s surface, especially when the temperature is near freezing. A difference of 1°C or 2°C in temperature of the air, the hydrometeor, or the  ground surface can mean the difference between an ice storm (with ice buildup on trees and power lines) and a very cold rain. It is possible for rain to remain supercooled liquid when it falls through the atmosphere through slightly below freezing air temperatures, but once the droplets hit the earth or another surface, if the temperature of the surface is below freezing, the water droplets will quickly freeze and cause a buildup of ice on the surface. 4.2  TYPES OF HYDROMETEOROLOGICAL DATA 4.2.1  Ground-Based Direct Measurements

Ground-based measurements provide a ground truth when analyzing radarbased precipitation data at specific points. Direct measurement instruments include rain gages, snow pillows, and hail pads, which all measure the quantity of hydrometeors once they have fallen to the ground. Snow pillows measure the weight of the fallen snow and convert it to a liquid equivalent. Hail pads show the approximate size and representative quantity of hail that fell at a location. Rain gages at the Earth’s surface can provide point measurements of the amount of precipitation that has fallen. However, rain gage density is such

Indirect measurements of hydrometeorological data are used to re-create the historical estimates of rain and snow. Tree rings can be used as a proxy of annual precipitation in many locations. Striations in glaciers can also be used as a surrogate measurement of long-term annual precipitation. 4.3  REMOTE SENSING OF PRECIPITATION 4.3.1  Doppler Radars—Single Polarization

Real-time hydrometeorological data are collected with a network of Doppler radars. These Doppler radars utilize radar pulses to estimate the size, type, location, and movement of hydrometeors in the atmosphere. Doppler radar is the preferred sensor for determining the timing and location of precipitation. But as will be discussed later in this chapter, the amount of estimated precipitation can at times be in considerable error. Doppler radars have different operational modes, including clear air and precipitation modes. Different radar wavelengths are used worldwide. Longer wavelengths are beneficial to obtaining adequate radar coverage in convective precipitation and severe weather. Shorter wavelengths can provide uniform radar coverage for widespread precipitation patterns over a smaller range than radars with longer wavelengths due to attenuation issues. The measurement that radar makes of the strength of the radar pulse after returning from hitting hydrometeors or other objects is called reflectivity. The reflectivity radar product (Z) is the most important single polarization radar product conveying hydrometeorological information. Relationships between the reflectivity and the rainfall rates have long been established. Areas of extremely high reflectivity indicate that heavy rainfall or hail is falling, while snow typically shows as bands of low reflectivity. The power of returned radar pulses is used to calculate a reflectivity factor (Z) using the Probert–Jones equation. The Z factor is then used in various equations (often referred to as Z–R power relationships) to calculate a rainfall rate, R. The strength of the return signal to the radar is determined by the quantity and size of hydrometeors within the radar beam. Different Z–R relationships can be used based on the meteorological conditions. The equation Z = 300 R1.4 is used as the default equation over much of the United States, 4-3

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4-4     Hydrometeors and Quantitative Precipitation Estimation

but the equation Z = 250 R1.2 is used for tropical rain processes. Large numbers of Z–R relationships have been experimentally tested over the years and additional research is still being conducted to determine appropriate Z–R relationship for different locations, climates, and atmospheric conditions. There are automated algorithms that can estimate the precipitation based on the radar products alone or in combination with automated rain gages. Cumulative rainfall products can also be automatically produced in a suite of radar products, typically on a 1-h, 3-h, or 24-h, or storm total basis. A single relationship between the reflectivity and gage data is calculated and can be applied to the entire field of the radar, if the radar operator so chooses to employ it. This is used to mitigate the shortcomings in radar precipitation estimates due to atmospheric condition or hydrometeor type. However, it is important to note that these unbiased, fully automated products may contain inaccurate precipitation rates because of radar artifacts or limitations in the radar data, such as incomplete radar beam filling, blocked beams, overshooting beams, range degradation, melting layer, hail, or a nonstandard meteorological atmosphere. 4.3.2  Doppler Radars—Dual Polarization

Doppler radars with dual polarization send out radar pulses in both the horizontal and vertical directions. This technology can help distinguish the types of hydrometeors and other objects found in the radar beam by measuring their length and width. Based on the reflectivity, differential reflectivity (the measurement of the strength and frequency shift of both the horizontal and vertical radar pulse returns), velocity, correlation coefficient (the measurement of how similarly the horizontally and vertically polarized pulses are behaving within a pulse volume), and specific differential phase (KDP) (the phase shift of both the horizontal and vertical radar pulse returns) products, the object detected is classified into one of 10 different classifications: Hail, Graupel, Big Drops, Heavy Rain, Rain, Wet Snow, Dry Snow, Ice Crystals, Ground Clutter, and Biological. Being able to quickly identify and classify the types of hydrometeors makes dual-polarization radar a more valuable tool over the single-phase Doppler radar technology. The difference between the horizontally polarized return power and the vertically polarized return power helps define the shape of the hydrometeors. Figure 4.1 shows how dual-polarization technology is able to differentiate between different types of hydrometeors based upon the distinctive length and width ratios of the radar returns. There are several radar products available with dual polarization Doppler radar that convey hydrometeorological information. Of the dual polarization fields now available, the KDP product is one of the most useful for determining areas of heavy rain. The phase of the radar pulse shifts as it passes through hydrometeors and back again. The magnitude of this shift is a good indicator of the size and density of the precipitation. As with the single polarization radar, there are automated algorithms that can estimate the instantaneous precipitation rate, or the cumulative precipitation based solely on radar products, or using a multisensor analysis with automated rain gages. 4.3.3  Phased Array Radars

Phased array radars are now being used to provide information about current weather conditions, especially about storm structure during severe weather. In the future, it is expected that phased array radar will be able to better provide information about hydrometeors as well. Phased array radars are different from the traditional Doppler radars in that the radar pulses can be electronically steered to sample certain areas of the atmosphere, instead of the entire radar unit physically moving in a defined rotational pattern of differing preset elevation angles.

Figure 4.1  Dual polarization technology helps classify types of hydrometeors.

04_Singh_ch04_p4.1-4.8.indd 4

4.3.4 Satellites

With little ability to obtain gage data or coverage from land-based radars, satellite information is particularly valuable over the oceans and countries such as Mexico with limited surface-based sensors. Recent technological advances in satellites allow for the use of multiple radiometers using multiple algorithms to estimate the quantity of hydrometeors. Satellites estimate precipitation based on thermal imagery and microwave radiometers. As the cloud height increases, the temperature of the hydrometeors in the clouds decreases. The temperature differential within the cloud formation can help survey precipitation formation and severe weather patterns. In general, colder cloud tops are correlated with heavier precipitation rates, and warmer cloud tops with lighter precipitation rates. Early satellite programs focused on rainfall measurements on the tropics as these latitudes can be easier to obtain adequate coverage from orbit. Current satellite programs seek to expand the coverage over land. However, there are current limitations with satellite measurements at higher latitudes and over land. Higher latitude measurements by geostationary satellites are hindered by a high angle of the Earth’s surface from the satellite. Satellite observations over land are affected by increased albedo in snow-covered areas and localized phenomena such as mountains that alter cloud formation. Due to these limitations, satellite precipitation estimates are generally less skillful than those derived from radar and have larger areal displacement and quantitative errors. However, satellite-derived precipitation estimates are useful in multisensor precipitation algorithms, especially in areas lacking radar coverage or ground-based rain gage observations. 4.4  HYDROMETEOROLOGICAL DATA PROCESSING 4.4.1  Gridded Multisensor Precipitation Estimates

In an effort to mitigate radar deficiencies, a gridded quantitative precipitation estimate, using multiple sensors, is performed. For a hydrometeorological analysis of large areas, a mosaic of radar coverage is used in combination with rain gages, snow pillows, and other information. In areas without radar coverage, climatological data are generally used to estimate the precipitation that fell, augmented by available gage data and satellite estimates. Radar precipitation estimates are raised or lowered based on the relationship between the radar and gage readings. Both field and local biases are calculated and either can be applied. In order to minimize accumulative errors, the quality-control step of using additional 24-h manual precipitation observations to calibrate the hourly data from the past day is performed. A daily postanalysis is performed to adjust the hourly precipitation estimates from the previous day. The postanalysis works by time distributing the 24-h accumulation into individual 1-h amounts based on the estimated precipitation expected to have fallen in each individual hour. The multisensor optimal estimation procedure is then re-run for each of the individual hours making up the 24-h total, but including the newly time distributed gage observations. When multiple sensors are used to process hydrometeorological data, various methodologies can be used to estimate the contribution of each input. A grid is developed by optimally combining data from radars and gages. A mosaic of radar coverage is typically developed from lowest level available radar coverage. However, it is also possible to distance-weight the radar information or to utilize the average or maximum value of overlapping radar coverage to obtain a more accurate estimate in some weather conditions. The influence of individual gages is also generally distance weighted, unless there is local knowledge to the contrary. In some areas that tend to experience more uniform precipitation patterns and have few gages, a single radar bias correction is developed from the available gages compared to the radar field. This relationship, known as a field bias, then is applied to the entire radar field to obtain precipitation estimates. This radar bias is generally less reliable than localized biases. Human input to quantitative precipitation estimates continues to be important even as technology advances. Unlike humans, algorithms cannot make subjective decisions about current meteorological conditions or gage behavior and have no familiarity with the region’s topography, meteorological processes, and areas of insufficient radar coverage. An example of a 24-h gridded multisensor quantitative precipitation estimate (QPE) incorporating a mosaic of radar coverage, rain gages, snow pillows, precipitation climatology, human observations, and other information is shown in Fig. 4.2. The QPE in the top part of Fig. 4.2 is based only on a mosaic of raw radar average reflectivity. The circular rings show the radar coverage available from each Doppler radar in the area. The QPE in the bottom section of Fig. 4.2 includes the radar reflectivity, observed hourly gage data and the biases calculated from these data, daily human observations, precipitation climatology, and input from the hydrometeorological forecaster’s local knowledge.

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Hydrometeorological Data Quality Assurance and Control     4-5 

Figure 4.2  24-Hour radar only and multisensor quantitative precipitation estimates.

4.5  HYDROMETEOROLOGICAL DATA QUALITY ASSURANCE AND CONTROL

It is important that hydrometeorological data go through appropriate data quality control. While the spatial and temporal resolution of radar-based products is superior to other sensors, there are several limitations of Doppler radar data, ranging from physical limitations to artificial data

04_Singh_ch04_p4.1-4.8.indd 5

artifacts. There are a multitude of items in addition to hydrometeors within a radar beam that can return signal back to the radar unit. And as discussed earlier, satellite hydrometeorological data also have limitations as they do not produce the same accuracy of information as the landbased Doppler radars used in conjunction with gages and climatological information.

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4-6     Hydrometeors and Quantitative Precipitation Estimation 4.5.1  Nonprecipitation Radar Echoes

4.5.7  Cone of Silence

Anthropogenic items can appear on radar as ground clutter. While the majority of Doppler radars have been built near airports or other open places, buildings and structures built near radars can block radar beams. Moving man-made items, such as wind turbines and traffic on elevated roadways, can also return radar signals. Natural obstructions such as mountains can also obstruct radar beams. Even far away trees can block radar beams in very flat regions. There are algorithms in place within the processing of the radar data, which mitigate the effects of ground clutter and other nonmeteorological targets. Biological matter can be clearly seen on the radar at times. Birds can be picked up on the radar leaving their roosts in the morning and bats can be seen leaving theirs at night. Insects and small birds in gust fronts make the gust front show as a clearly delineated line across the radar image. Chaff is a radar countermeasure that consists of small strips of aluminum, metalized glass fiber, or plastic fiber released into the atmosphere during military maneuvers. The chaff shows up on radars as a streak of high reflectivity. Smoke plumes from fires and ash from volcanic eruptions show up on radars as the smoke and ash are blown away by the wind.

A Doppler radar typically sends out radar beams from angles of 0.5° to 19.5°. Because all the beams are transmitted radially and at preselected angles based on the volumetric scan type chosen for operations, the radar is prevented from transmitting beams overhead. Because of the fixed angles of operability, there will be a cone-shaped area centered over the radar where the radar will not be able to see, although returns from hydrometers can still be obtained at low elevations near the radar.

4.5.2  Radar Artifacts

Certain atmospheric conditions can cause the radar beam to be anomalously bent back into the ground causing significant ground clutter to show up in radar data. In addition, spurious spikes and radar artifacts occur in Doppler radar data frequently. A three-body scatter spike is an anomalous radar artifact that can occur in the radar image down radial of areas with large or wet hail. Side lobe errors can occur when the returned power signal to the radar unit is quite high. During the radar volume scans that occur during sunrise, a spurious radar spike will occur to the east of the radar location toward the rising sun. The same phenomenon occurs again at sunset with a spurious radar spike occurring to the west toward the setting sun. These spikes occur because the sun also produces radiation at the same wavelength of the radar. 4.5.3  Beam Blockage

The radar beam can be physically blocked by mountains, buildings, water towers, or other items. It can also be blocked by intense precipitation, especially by significant quantities of large hail. There are times that radar beams are blocked by areas of high reflectivity in the atmosphere, such as severe storms. Storms with intense hail or rain can essentially block or attenuate radar beams from traveling through the storm’s core or lowering the power or the return signal from returning back to the radar unit. 4.5.4  Range Degradation and Earth Curvature

As a radar beam travels, the beam samples higher altitudes due to the Earth’s curvature. Storm systems with low clouds, particularly winter weather events like snow, can be undersampled and not show up on radar when they are not located near the radar. When radar beams travel farther than the operating range of the radar, range folding can occur when the returns are confused with those returning from within the operating range. Atmospheric conditions can also cause changes in the typical range of a Doppler radar. Subrefraction occurs when the beam bends less than the curvature of the Earth, resulting in a longer effective radar range, but possibly underestimating cloud heights. Super refraction happens when the radar beam bends back toward the Earth’s surface more than the curvature of the Earth. Low-level inversions where the temperature is cooler at the Earth’s surface than at higher elevations in the atmosphere can produce super refraction, especially near dawn and dusk. Ducting is extreme super refraction where the radar beam bends back toward the Earth’s surface instead of increasing with elevation. Super refraction tends to show an increase in radar returns from ground clutter. 4.5.5  Nonuniform Beam Filling

In general operations, no radar beam is filled with the exact same size and type of hydrometeor. Water droplets, hail, and snow exist in various sizes and shapes. Only the average information of all the information in each area is returned to the radar. If large wet hail is located in the same area as small raindrops, the average reflectivity for the entire area will be returned back to the radar. This would raise the reflectivity of the entire area, showing up as a heavier rain. 4.5.6  Bright Banding

The elevation when the snow begins to melt, as it is falling, is detected as an area of higher reflectivity and the radar algorithm is programmed to estimate this as heavy rain. This melting level can produce a concentric band or circle of higher reflectivity known as the “bright band” in the radar products.

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4.5.8 Virga

Virga is falling precipitation that evaporates or sublimates before reaching the ground surface. Hydrometeors in virga produce radar returns, but the rain or snow does not reach the ground. Virga is common in regions with high temperatures and low humidity. 4.5.9 Incorrect Z–R Relationships and Algorithm Errors

Different types of storms have different meteorology patterns that require different Z-R relationships. Using an incorrect Z–R relationship can have a detrimental impact on precipitation estimates. Weather with low reflectivity, such as light snow, often has radar returns too low to be considered hydrometeors by the algorithm, even though precipitation is falling and reaching the ground. The high reflectivity of hail can cause the radar algorithm to calculate a much higher rainfall estimate because of the presence of hail. 4.5.10  Data Errors

Sampling errors are common in rain and snow measurements. Rainfall gages may introduce errors such as tipping buckets not keeping pace with heavy rainfall. If automated rain gages are not heated, they may report zero precipitation during freezing events. Automated gages may also introduce errors such as dew being measured as rainfall or by continuing to drip rain after the rainfall events have ended because of a partially clogged opening. Completely clogged gages report zero precipitation during events. Rain gages require routine maintenance because they can be totally or partially clogged with bird nests, spider webs, leaves, or other debris. Snow pillows can have measurement errors due to the mechanical nature of the gages. Because snow pillows measure the weight of the snow, changes in barometric pressure and temperature can cause the gage to measure erroneous positive or negative values. The siting of gages to keep appropriate distances from trees and buildings can prevent some gage errors. However, strong winds may prevent rain or snow from falling into rain gages or snow pillows even in open areas. Blowing snow can also alter snowpack estimates. The consistency of hydrometeorological data is dependent upon the training and skill of the human observer. Human observations can differ for a variety of reasons, such as differences in eyesight acuity or the introduction of numerical rounding into observations. Snowfall observations can be particularly inconsistent. Observations of snow are reported for inches of depth of fresh fallen snow or for older more densely packed snow. Snow depths also vary from place to place due to drifting. In order to mitigate differences due to the consolidation of the older snow pack and localized drifting, human observations of the water equivalent of the snow that fell over the past 24 hours are reported. However, differences occur between observers due to their methods and equipment when melting the snow to determine the snow water equivalent. Some of the liquid water will evaporate if the snow is melted too fast. 4.6  QUANTITATIVE PRECIPITATION ESTIMATE, DATA USE, ARCHIVING, AND ACCESSIBILITY

QPE data are used in weather and climate forecasts, including flash flood warnings, fire weather forecasting, and droughts. QPE data are crucial in impact-based weather forecasts, such as forecasted snow, ice, or freezing rains during rush hour traffic. Some of the usages of QPE data in engineering include hydrologic and hydraulic modeling, flood studies, water resources planning, and development activity. There are multiple planning studies that use QPE data, such as precipitation recurrence intervals, climate stationarity studies, probable maximum precipitation (PMP), and probable maximum flood (PMF) studies. Uses of these planning studies include application in building codes, development standards for public works, and coastal surge studies. QPE information is used in reservoir planning and yield studies, while short-term QPE data are used in the operations of reservoirs, locks and dams, and hydropower facilities. Many facets of society use quantitative precipitation estimates, including farmers, ranchers, landscapers, outdoor enthusiasts, and many corporations. Corporations use them to correlate sales to weather patterns. Kayakers, skiers,

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Quantitative Precipitation Estimate, Data Use, Archiving, and Accessibility     4-7 

hunters, and others use them to plan their outdoor activities. Quantitative precipitation estimates are used in drought monitoring and irrigation planning to maximize crop yields. Estimations of the anticipated streamflow resulting from the melt of a mountain snowpack can be estimated in the early spring to assist farmers and reservoir controllers in calculating the quantity of water that will be available for irrigation later during the year from downstream reservoirs.

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QPE data are not useful unless they are available. The period of available data should be as long as possible as the length of the data archive will affect the usefulness of many studies and applications. It is also important that data be archived in a format that is readily accessible to others, such as Standard Hydrometeorological Exchange Format (SHEF), network Common Data Form (netCDF), or WaterXML. As more GIS applications are developed, it is also important that the QPE data be GIS compatible.

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Chapter

5

Streamflow Data BY

ROBERT R. HOLMES, JR.

ABSTRACT

The importance of streamflow data to the world’s economy, environmental health, and public safety continues to grow as the population increases. The collection of streamflow data is often an involved and complicated process. The quality of streamflow data hinges on such things as site selection, instrumentation selection, streamgage maintenance and quality assurance, proper discharge measurement techniques, and the development and continued verification of the streamflow rating. This chapter serves only as an overview of the streamflow data collection process as proper treatment of considerations, techniques, and quality assurance cannot be addressed adequately in the space limitations of this chapter. Readers with the need for the detailed information on the streamflow data collection process are referred to the many references noted in this chapter. 5.1 STREAMFLOW

Streamflow is the volumetric discharge, expressed in volume per unit time (typically cubic feet per second, ft3/s, or cubic meters per second, m3/s), that occurs in a stream or channel and varies in time and space. Knowledge of the quantity and temporal and spatial distribution of streamflow is crucial to the science of hydrology, emergency management, and associated fields such as water resources infrastructure design, water resources management, and environmental protection.

Figure 5.1  Streamgage mounted on the downstream pier of a bridge over the Gasconade River at Rich Fountain, Missouri, USA, with a hydrographer making a discharge measurement using an ADCP mounted on a tethered boat during a flood in August, 2013.

5.2  TYPES OF STREAMFLOW DATA

Streamflow data encompass both water–surface elevation (stage) and volumetric discharge (discharge), being collected at various intervals depending on the need. Often streamflow data are collected through means of permanent or semipermanent facilities installed on or near the stream, which are termed streamgages (Fig. 5.1), although other names such as gauges, streamflow monitoring stations, and river gages are used to refer to these installations. Continuous streamflow data are collected at a frequency sufficient to fully characterize the streamflow hydrograph. Typical frequency of continuous streamflow data at streamgages ranges from 1 to 5 min for small streams (drainage area typically less than 50 km2), which are “flashy” (Fig. 5.2a) to 60  min for large rivers (drainage area typically greater than 250,000 km2), which rise and fall more gradually (Fig. 5.2b).

This chapter is restricted to discussing only those streamflow data derived from simple stage-discharge or index-velocity ratings. The chapter on streamflow ratings (see Chap. 6), as well as Rantz (1982), describes more involved rating methods that utilize water–surface slope and rate of stage change for the ratings. Internationally, procedures for collecting stage and discharge data can be found in selected publications by the International Organization for Standardization (ISO) and the World Meteorological Organization (WMO). In many parts of the world and certainly in the United States, the methods described in reports by the U.S. Geological Survey (USGS) have been adopted. The streamflow rating is developed by correlating discretely measured discharge (discharge measurements) with concurrently collected stream data. In essence, the autonomously collected stream data act as a surrogate for discharge by means of the streamflow rating.

5.3  STREAMGAGE OPERATION

5.3.1  Selection of a Streamgage Site

The purpose of a streamgage is to obtain a continuous record of discharge. Currently (2015), there is no widely-accepted, economically-viable method to directly determine discharge continuously, although research to develop methods for this purpose is ongoing (Brakenridge et al., 2012; Fulton and Ostrowski, 2008; Costa et al., 2006; Bjerklie et al., 2005). Rather, continuous time series of discharge are typically computed by applying a streamflow rating relation (streamflow rating) to autonomously collected stream data, such as stage, slope of water surface, rate of stage change, or stream index velocity.

Selection of the exact location of the streamgage along a reach should maximize the quality of the streamflow ratings following existing guidelines (Rantz, 1982; Levesque and Oberg, 2012; World Meteorological Organization, 2010a; Sauer and Turnipseed, 2010; Carter and Davidian, 1968). Site selection involves three major factors. First and foremost, the streamgage has to be located such that it can collect reliable and accurate stream data. Second, the site must permit the making of discrete discharge measurements (discharge measurements) in a safe and efficient manner. 5-1

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5-2    Streamflow Data

25

(a) Boneyard data

Discharge (m3/s)

20

15

10

5

0 5/29/2008 0:00

25,000

5/31/2008 0:00

6/2/2008 0:00

5/29/2008 6/4/2008 0:00 0:00 Date and time

6/6/2008 0:00

6/10/2008 0:00

(b) St. Louis data

Discharge (m3/s)

20,000

15,000

10,000

5000

0 1/31/2008 0:00

3/21/2008 0:00

5/10/2008 0:00

6/29/2008 0:00

8/18/2008 0:00

Date and time Figure 5.2  Hydrograph for (a) Boneyard Creek near Urbana, Illinois, USA (drainage area 11.6 km2) and (b) Mississippi River at St. Louis, Missouri, USA (drainage area 1,810,000 km2) (Data from U.S. Geological Survey National Water Information System, http://waterdata.usgs.gov/nwis).

Third, the site must be well suited to the strategy chosen for developing and maintaining the streamflow rating. Reliability of stream data means that the data are complete and its delivery uninterrupted as near real-time data have proven crucial to efficient water resources management. Ideally, the streamgage should be located in relatively calm pools or downstream from protective features to minimize the chances of damage or failure of the structure, sensors, data logger, telemetry, and associated accessories. Care should be taken to elevate moisture-sensitive equipment above the potential maximum flood height. Accuracy of stream data requires that the sensors be located so that the sensor is reporting the true ambient value for that parameter. For example, in the collection of stage, locations for sensor placement should be avoided where the water–surface elevation is artificially influenced by localized features, such as bridge piers, inducing water–surface elevations different than the true ambient stream elevation. Additionally, accurate stage data depend on a stable mounting platform for the sensor apparatus. The location of the discharge measurements should provide hydraulic conditions that enable an accurate and safe measurement across the spectrum of hydrologic conditions, including flood. The section where the discharge measurement is made must be close enough in proximity to the streamgage

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that the concurrency between the autonomously collected stream data and the discharge measurement is well established. Otherwise, the discharge measurements will require adjustment for time of travel (Rantz, 1982). The strategy for developing the streamflow rating has major implications for the location of the streamgage. A simple rating is used where the relation between stage and discharge is single-valued or univariate, meaning the rating has only one value of discharge for a particular stage (Fig. 5.3a). For simple ratings, the relation is controlled by features of the stream (Fig.5.3b). These features are known as “controls” (see Chap. 6 on rating controls) and the accuracy and stability of the simple rating are directly related to the shape and stability, respectively, of the control. Locating the streamgage where the controls are well-established is important, particularly the low-water (section) control as that provides stability for the simple rating at low-water. With the exception of steep mountain streams, the streamgage should be located upstream from the most stable low-water control that can be found during reconnaissance. Controls with cross-section shapes resembling a narrow “V-shape” will promote increased sensitivity to small changes in stage and result in improved accuracy of streamflow estimates. In some cases, it may be desirable to construct an artificial low-water control to increase stability and ease the determination of a low-water rating.

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Streamgage Operation    5-3 

(a) Rating curve k

Overban

Stage (m or ft)

trol con

el nn 1 Cha 2

Discharge (m3/s or ft3/s) (b) Stream cross section Overbank

Channel control Section control

Streamgage section control

(c) Stream profile Streamgage and associated water-level sensor Water surface for low flow when section control effective

Water surface for medium flow when channel control effective

Water surface for high flow when channel control involves both main channel and overbank

Flow age amg Stre ction se rol cont

Figure 5.3  (a) Streamgage simple rating curve with effective controls identified along with the appropriate slope of the rating for those controls, (b) stream cross section view showing water levels associated with controls, and (c) reach of river showing streamgage in relation to associated controls at various stages.

5.3.2  Methods of Stage Measurement

Where the channel slope is subject to backwater or highly unsteady flow conditions, the water–surface slope also may need to be incorporated into the rating. Backwater conditions often arise due to the presence of a downstream dam or, perhaps, from variable inflow from a contributing stream. Passage of a significant flood wave along a low-slope channel may also induce slope effects. In some cases, an auxiliary gage must be established to measure the slope along the reach between it and the primary streamgage, and appropriate site and logistical evaluations for the auxiliary gage must be included in the site evaluation process. The fall between the two should exceed 0.5 ft, the slope should be uniform along the entire reach, and the reach should be as far upstream from the backwater as possible. Unfortunately, the need for an auxiliary station or for incorporation of other treatments for slope effects may not be evident until several high-flow measurements documenting the passage of flood waves are obtained and plotted against concurrent stage to reveal the presence of a loop-rating. Rantz (1982) provides additional details.

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A fundamental part of any streamgage is stage data, the collection of which is governed by guidelines and standards to ensure quality data (Sauer and Turnipseed, 2010; World Meteorological Organization, 2010a). The stage of a stream or river is the height of the water–surface elevation above a particular local datum, often known as the gage datum. The gage datum is often established such that a zero value is below the channel bed to ensure nonnegative values at all stages. Gage datum is often surveyed into known geodetic datum to relate stage to known elevations of adjoining physical features and enable comparison of stages between streamgages. Stage is measured in feet or meters, often determined and reported to an accuracy of the nearest one hundredth of a foot or one thousandth of a meter (Sauer and Turnipseed, 2010). Stage can be determined either by nonrecording manual observation or through automated sensors attached to recording data loggers. Manual observation systems include vertical graduated plates (staff gages) attached

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5-4    Streamflow Data

to fixed mounts in the water or weighted tapes or wires in which the observation of stage is made by lowering the weight to the water surface from a known elevation and subtracting the length of the wire or tape. The traditional automated stage sensor over the years has been the float and counterweight in a stilling well, but new technology has provided an array of options including submersible and nonsubmersible pressure transducers, optical sensors, and radio detection and ranging (radar) sensors (Sauer and Turnipseed, 2010). Selection of the sensor must be determined based on economics and site conditions. For example, a float and counterweight in a stilling well sensor provide excellent data; however, the combination of construction cost and sedimentation of the well may mean that a noncontact (sensor is mounted above the water) radar stage sensor may provide a better alternative when all factors are considered. Conversely, regular large magnitude winds or surface ice at a site might induce enough errors in stage measurement to influence the choice of a submersible pressure sensor over a radar stage sensor. Often the data loggers for automated stage sensor systems are attached to telemetry, which transmit the data in near real-time to centralized computer systems. Options for telemetry include satellite, telephone, and radio. Telemetered data provide a number of advantages including timely detection and correction of streamgage problems, enhancement of flood warning and forecasting, and operation of water resources projects. 5.3.3  Methods of Discharge Measurement

Discharge measurements are made using a variety of methods following standard protocols (Turnipseed and Sauer, 2010; World Meteorological Organization, 2010a; Herschy, 2008; Rantz, 1982; Benson and Dalrymple, 1967) and are grouped into two broad categories: direct and indirect methods. Direct measurements of discharge use instruments, sensors, or devices to “directly” measure the discharge on site when the streamflow of interest is occurring. Indirect methods are used primarily to measure the peak discharge from a flood event after the flood has passed and the stage has receded. Indirect methods utilize forensic surveys of both the remnant high water marks and the channel geometry in concert with the estimates of the boundary roughness to estimate the peak discharge from the principles of conservation of energy, momentum, and mass. A variety of indirect measurement techniques exist that can be employed based on the stream hydraulics and presence or absence of structures (Benson and Dalrymple, 1967; Matthai, 1967; Bodhaine, 1968; Dalrymple and Benson, 1967; Hulsing, 1967). Most of the discharge measurements made at streamgages are direct measurements based on the velocity–area method, whereby the cross section of the stream is discretized into subsections (Fig. 5.4) and the total discharge is

b1 b2 1

b3

b4

b5

the sum of the products of the area and mean velocity of each subsection (International Organization for Standardization, 2007a) n

Q = ∑(ai vi ) (5.1)



i =1

where Q is the total discharge, ai is the area of subsection i, and vi is the average velocity of subsection I. The midsection velocity–area method to determine discharge (Fig. 5.4) is a commonly used cross section discretization method in the United States (Turnipseed and Sauer, 2010). With the advent of acoustic Doppler current profilers (ADCP) technology, efficient discretization of the channel cross section into small subsections (both horizontal and vertical discretizations) has increased the number of data points collected in each measurement and reduced the measurement time. Thousands of velocity and depth observations (Fig. 5.5) are made as the ADCP travels approximately perpendicular to streamflow (Mueller et al., 2009) with the ADCP mounted rigidly either inside a tethered boat or near the bow of a manned boat (Fig. 5.5). Instruments to determine velocity include mechanical current meters (both vertical axis and horizontal axis-type meters), electromagnetic meters, acoustic Doppler velocity meters (ADVMs), and optical strobe velocity meters (Turnipseed and Sauer, 2010; Bureau of Reclamation, 2001). Mechanical meters have been used around for over 100 years. The predominant mechanical meter used in the United States today is the AA Price mechanical current meter, which was first introduced in the nineteenth century (Frazier, 1974); however, the ADVMs have largely supplanted the use of mechanical meters by many water resources agencies around the world. 5.3.4  Methods of Index Velocity Measurement

Continuous measurement of velocity, termed index velocity, is used at selected sites to derive streamflow by the index–velocity method (see Chap. 6). Four types of metering systems have typically been used to measure index velocity (Rantz, 1982): standard mechanical current meters, deflection meters, electromagnetic meters, and acoustic meters. Over time, the acoustic velocity meter (AVM) and ADVM have become the predominant tools for index velocity methods (Levesque and Oberg, 2012). The AVM consists of two sensors positioned at an angle to the flow and utilize the acoustic travel time technique (Laenen, 1985). ADVMs, first introduced in 1997, consist of a single unit utilizing the Doppler principal (Morlock et al., 2002). In the United States, the relatively low-cost but robust ADVM has become more predominant than the AVM (Levesque and Oberg, 2012). The ADVM is mounted in an up-looking orientation to ensonify a vertical volume of water, or it is mounted in a side-looking orientation (Fig. 5.6) to ensonify a horizontal volume of water.

b6

Water surface

d2

bn–1 bn dn–1

d3 2

d4

n

(n–1)

d5 3

d6

4

Streambed

5 6

Explanation Boundary of subsection

1, 2, 3, .......... n Subsection number b1, b2, b3 ..... bn Width of subsection d1, d2, d3 ..... dn Depth of subsection Figure 5.4  For the midsection velocity–area method, the total discharge is determined by the sum of the products of the area and mean velocity of each subsection.

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Streamgage Operation    5-5 

ADCP mounted rigidly near bow of a manned boat

0

Depth (m)

20

19

28

37

0

250

500 Lateral distance (m)

750

0.004

1.051 2.098 3.145 Velocity magnitude (m/s)

4.192

1000

Figure 5.5  Photo of data collection with associated data output from a manned-boat ADCP measurement for the Mississippi River below Cairo, Illinois, USA, on May 1, 2011.

Bridge pier Horizontally ensonified volume Vertically ensonified volume

Figure 5.6  Diagram of the ensonified volume from a horizontally- and vertically-mounted ADVM.

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5-6    Streamflow Data 5.4  QUALITY ASSURANCE OF STREAMGAGE DATA

5.5  DERIVED STREAMFLOW UNCERTAINTY

The quality of streamgage data is directly dependent on (1) regular field inspections and maintenance of streamgages, (2) collection of quality discharge measurements through time and throughout the range of stage, and (3) proper construction, application, and verification of streamflow ratings. High-quality stage data are obtained by adhering to standard techniques and quality assurance measures (Sauer and Turnipseed, 2010; World Meteorological Organization, 2010a; Herschy, 2008). At each streamgage, an independent nonrecording reference gage should be established, and its elevation should be related to the local gage datum. This datum should be verified by regular differential surveys and tied to multiple stable reference marks away from the immediate vicinity of the streamgage (Kenney, 2010). Regular field visits to the streamgage, augmented by intermittent high-streamflow visits, should be made to verify the recording gage against the reference gage and ensure the working condition of the equipment. Corrections to the principle gage of as little as 0.003 m are made if the low-water control is a permanent weir (Kennedy, 1983). Accuracy of streamflow data begins with high-quality stage data. The following are stage-accuracy standards of the USGS as given by Sauer and Turnipseed (2010):

Uncertainty in computed (derived) streamflow is a product of the uncertainty from three primary sources: streamgage data (stage, index velocity, and slope), discharge measurements, and streamflow rating. These three sources have their many sources of uncertainty. For example, discharge measurement sources of uncertainty include the uncertainty in measurement of crosssectional area, uncertainty in measurement of water velocity, uncertainty in computation assumptions, and random and systematic errors (Turnipseed and Sauer, 2010). Each of these in turn arise from other sources of uncertainty; for example, the uncertainty in cross sectional area arises from uncertainty in width and depth measurements. Clearly, uncertainty has a cascading effect. Although much work has been done to quantify the uncertainty of discharge measurements (Cohn et al., 2013; International Organization for Standardization, 2007b; Oberg and Mueller, 2007; Herschy, 1999; Pelletier, 1989) and the uncertainty of streamflow ratings used to derive continuous streamflow data (Morlot and others, 2014; Schmidt, 2002; Clarke, 1999; Masson and others, 1987), no quantitative standard for determination of uncertainty of derived streamflow has yet to be agreed upon. The USGS assigns a qualitative assessment of the general accuracy of the derived streamflow based on the amount of missing stage data, the quality of the stage data, the hydrographer-assigned accuracy rating of discharge measurements, and the stability of the streamflow rating. An accuracy assignment of “excellent” means that 95% of the daily mean values of streamflow are correct within 5%; “good” is within 10%; “fair” is with within 15%; and “poor” is less than “fair” (Kennedy, 1983).

“..the overall accuracy of stage data ….is either 0.01 feet [3.05 mm] or 0.2 percent of the effective stage, whichever is greater. For example, the required accuracy would be 0.06 ft [18.3 mm] at an effective stage of 30 ft [9.14 m], 0.02 ft [6 mm] at 10 ft [3.05 m], and 0.01 ft [3.048 mm] at all effective stages less than 7.5 ft [2.29 m]. Effective stage is defined as the height of the water surface above the orifice, intake, or other point of exposure of the sensor in the water body.”

The International Organization for Standardization (2008) does not give a standard for stage data, but rather outlines three performance classes of water level measurement devices based on their nominal uncertainty. Nominal uncertainty of less than ±0.1%, less than ±0.3%, and less than ±1% of the range of stage correspond to performance classes 1, 2, and 3, respectively. The discharge measurements are crucial to establishing the streamflow rating. Discharge measurements should be made both through time and across the range of stage using proper protocols and techniques (Turnipseed and Sauer, 2010; World Meteorological Organization, 2010a; International Organization for Standardization, 2007a; Herschy, 2008). These techniques include a thorough review of the measurement notes and associated electronic files and records to ensure that prevailing hydraulic conditions are consistent with the assumptions on which the measurement computations are based. Once the streamflow rating is established, continued discharge measurements are important to either verify or adjust the streamflow rating. The hydrographer’s judgment will determine the required density and frequency of discharge measurements based on the stability of the streamflow rating (Kennedy, 1983, 1984). Proper development and maintenance of the streamflow rating using standard protocols and techniques (World Meteorological Organization, 2010b; Kennedy, 1983) are crucial to the derivation of a continuous time series of discharge. Development of streamflow ratings is addressed in more detail in Chap. 6.

5.5.1  Streamflow Data Dissemination and Archival

Dissemination and archival of real-time and historic data are important for management, emergency response, and design for water and environmental resources. Real-time streamflow data are often needed for timely operational decisions by water resources and emergency managers. Data are often telemetered by way of satellite radio, telephone, or terrestrial radio to central locations to enable real-time decisions. These data are often concurrently broadcast and made available via the World Wide Web (Table 5.1). Historic data are important to enable contextual understanding of streamflow probabilities, rainfall-runoff characteristics, and stream hydraulics. These data are sometimes also available on the World Wide Web (Table 5.1). Although certain countries of the world have collected streamflow data for decades, their data have not always been made widely available. Web portals such as the World Hydrological Cycle Observing System (WHYCOS) and Global Runoff Data Centre (GRDC) both operated by the World Meteorological Organization (Table 5.1) exist to promote free exchange of streamflow data around the world and particularly in those countries who do not have centralized capability to serve their data. Value-added synthesis of streamflow data such as USGS WaterWatch (Fig. 5.7) provides powerful mapbased hydrologic situational awareness.

Table 5.1  Sources of Streamflow Data and Assessments Region Canada

Agency/Organization Environment Canada Water Office

United Kingdom Environment Agency

Type of information/Data

Data age

Web location of streamflow data

Stage and Discharge

Real Time and Historic http://wateroffice.ec.gc.ca/index_e.html

Stage

Real Time

http://apps.environment-agency.gov.uk/ river-and-sea-levels/default.aspx

United States

U.S. Army Corps of Engineers River Gages Network Stage

Real Time and Historic http://rivergages.com

United States

U.S. Bureau of Reclamation Water Operations

Reservoir Releases and Volumes

Real Time

http://www.usbr.gov/main/water/

United States

U.S. Geological Survey National Water Information System

Stage and Discharge

Real time and Historic

http://water.usgs.gov

United States

U.S. Geological Survey WaterWatch

Assessment Maps

Real Time and Historic http://waterwatch.usgs.gov/

World

World Meteorological Organization GRDC

Stage and Discharge

Historic

World

World Meteorological Organization WHYCOS

Stage and Discharge

Real Time and Historic http://www.whycos.org/whycos/

05_Singh_ch05_p5.1-5.8.indd 6

http://www.bafg.de/GRDC/EN/Home/ homepage_node.html

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REFERENCES    5-7 

Explanation—Percentile classes

Low

90

Normal

Above normal

Much above normal

High

Figure 5.7  WaterWatch assessment of hydrologic condition across regions of the United States on May 16, 2014 (http://waterwatch.usgs.gov/).

REFERENCES

Benson, M. A. and T. Dalrymple, General field and office procedures for indirect discharge measurements, U.S. Geological Survey Techniques of WaterResources Investigations Book 3, Reston, Virginia, 1967, Chap. A1, p. 12. Bjerklie, D. M., D. Moller, L. C. Smith, and S. L. Dingman, “Estimating discharge in rivers using remotely sensed hydraulic information,” Journal of Hydrology, 309: 191–209, 2005. Bodhain, G. L., Measurement of peak discharge at culverts by indirect methods, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1968, Chap. A3, p. 60. Brakenridge, G. R., S. Cohen, A. J. Kettner, T. De Groeve, S. V. Nghiem, J. P. M. Syvitski, and B. M. Fekerte, “Calibration of satellite measurements of river discharge using a global hydrology model,” Journal of Hydrology, 475: 123–136, 2012. Bureau of Reclamation, Water Measurement Manual, Water Resources Technical Publication, Washington, D.C., 2001, http://www.usbr.gov/pmts/ hydraulics_lab/pubs/wmm/. Carter, R. W. and J. Davidian, General procedure for gaging streams, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1968, Chap. A6, p. 13. Clarke, R. T., “Uncertainty in the estimation of mean annual flood due to rating-curve indefinition,” Journal of Hydrology, 222: 1–4, 1999. Cohn, T. A., J. E. Kiang, and R. R. Mason, Jr., “Estimating discharge measurement uncertainty using the interpolated variance estimator,” ASCE Journal of Hydraulic Engineering, 139: 502–510, 2013. Costa, J. E., R. T. Cheng, F. P. Haeni, N. Melcher, K. R. Spicer, E. Hayes, W. Plant, et al., “Use of radars to monitor stream discharge by noncontact methods,” Water Resources Research, 42: 1–14, 2006.

05_Singh_ch05_p5.1-5.8.indd 7

Dalrymple, T. and M. A. Benson, Measurement of Peak Discharge by the Slope-Area Method, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1967, Chap. A2, p. 12. Frazier, A. H, Water Current Meters in the Smithsonian Studies in History and Technology, No. 28, Smithsonian Institution Press, Washington, D.C., 1974, p. 95. Fulton, J. and J. Ostrowski, “Measuring real-time streamflow using emerging technologies: radar, hydroacoustics, and the probability concept,” Journal of Hydrology, 357: 1–10, 2008. Herschy, R. W., Streamflow Measurement, 3rd ed., Taylor and Francis Group, United Kingdom, 2008, p. 507. Herschy, R. W., Uncertainties in hydrometric measurements, Hydrometry, 2nd ed. John Wiley, New York, 1999, pp. 355–370. Hulsing, H., Measurement of peak discharge at dams by indirect method, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1967, Chap. A5, p. 29. Kennedy, E. J. Discharge ratings at gaging stations, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1984, Chap. A10, p. 59. Kennedy, E. J., Computation of continuous records of streamflow, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1983, Chap. A13, p. 53. Kenney, T. A., Levels at gaging stations, U.S.Geological Survey Techniques and Methods Book 3, Reston, Virginia, 2010, Chap. A19, p. 60. International Organization for Standardization, Hydrometry—Water Level Measuring Devices, ISO 4373:2008, Geneva, Switzerland, 2008, p. 20. International Organization for Standardization, Hydrometry—Measurement of Liquid Flow in Open Channels Using Current Meters or Floats, ISO 748:2007, Geneva, Switzerland, 2007a, p. 58.

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5-8    Streamflow Data

International Organization for Standardization, Hydrometry—VelocityArea Method Using Current Meters—Collection and Processing of Data for Determination of Uncertainties in Flow Measurement, ISO 1088, Geneva, Switzerland, 2007b. International Organization for Standardization, Measurement of Liquid Flow in Open Channel—Water Level Measuring Devices, ISO 4373–1979, Geneva, Switzerland, 1979, p. 18. Laenen, A., Acoustic velocity meter systems, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1985, Chap. A17, p. 38. Levesque, V. A. and K. A. Oberg, Computing discharge using the index velocity method, U.S. Geological Survey Techniques and Methods Book 3, Reston, Virginia, 2012, Chap. A23, p. 148. Masson, J., M. Ghio, C. Lallement, C. Parsy, and J. Philippe, Debitmetrie: Precision des Stations de Jaugeage, La Houille Blanche, France, 1987, pp. 333–338. Matthai, H. F., Measurement of peak discharge at width contractions by indirect methods, U.S. Geological Survey Techniques of Water-Resources Investigations Book 3, Reston, Virginia, 1967, Chap. A4, p. 44. Morlock, S. E., H. T. Nguyen, and J. H. Ross, “Feasibility of acoustic Doppler velocity meters for production of discharge records from U.S. Geological Survey streamflow-gaging stations,” U.S. Geological Survey Water Resources Investigations Report 01-4157, Reston, Virginia, 2002, p. 56. Morlot, T., C. Perret, A. C. Favre, and J. Jalbert, “Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: quality and station management indicators,” Journal of Hydrology, 517: pp. 173–186, 2014.

05_Singh_ch05_p5.1-5.8.indd 8

Mueller, D. S., C. R. Wagner, M. S. Rehmel, K. A. Oberg, and F. Rainviulle, Measuring discharge with acoustic Doppler current profilers from a moving boat, U.S. Geological Survey Techniques and Methods Book 3, Reston, Virginia, 2009, Chap. A22, p. 95. Oberg, K. A. and D. S. Mueller, “Validation of streamflow measurements made with acoustic Doppler current profilers,” ASCE Journal of Hydraulic Engineering, 133: 1421–1432, 2007. Pelletier, P. M., “Uncertainties in streamflow measurement under winter ice conditions, a case study: The Red River at Emerson, Manitoba, Canada,” Water Resources Research, 25 (8): 1857–1868, 1989. Rantz, S. E., Measurement and computation of streamflow, U.S. Geological Survey Water-Supply Paper 2175, 1982, p. 631. Sauer V. B. and D. P. Turnipseed, Stage measurement at gaging stations, U. S. Geological Survey Techniques and Methods Book 3, Reston, Virginia, 2010, Chap. A7, p. 87. Schmidt, A. R., Analysis of stage-discharge relations for open-channel flows and their associated uncertainties, Ph.D. Thesis, University of Illinois, Urbana-Champaign, 2002, p. 329. Turnipseed, D. P. and V. B. Sauer, Discharge measurements at gaging stations, U.S. Geological Survey Techniques and Methods Book 3, Reston, Virginia, 2010, Chap. A8, p. 87. World Meteorological Organization, Field work, Manual on Stream Gauging, World Meteorological Organization Number 1044-V1, 2010a, Vol. 1, p. 252. World Meteorological Organization, Computation of discharge, Manual on Stream Gauging, World Meteorological Organization Number 1044-V2, 2010b, Vol. 2, p. 198.

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Chapter

6

Streamflow Ratings BY

ROBERT R. HOLMES, JR.

ABSTRACT

6.2  RATING CONTROLS

Autonomous direct determination of a continuous time series of streamflow is not economically feasible at present (2014). As such, surrogates are used to derive a continuous time series of streamflow. The derivation process entails developing a streamflow rating, which can range from a simple, single-valued relation between stage and streamflow to a fully dynamic one-dimensional model based on hydraulics of the flow.

For a simple rating, it is assumed that the relation between stage and discharge is controlled by a section (section control) or reach of channel (channel control); both of these features are referred to as station control (Kennedy, 1984). Corbett (1943) explains this as follows: “In order to have a definite and enduring stage-discharge relation, the channel must have certain physical features capable of regulating or stabilizing flow past the gage to such an extent that for a given stage of the water surface, the discharge will always be the same.” A station control essentially “controls” the relation between the stage and discharge and prevents hydraulic disturbances from translating past the control (and affecting the stage-discharge relation). Depending on the state of flow (subcritical or supercritical flow), the control will either be downstream or upstream of the streamgage location. The flow type is determined by the Froude number (F) as:

6.1 INTRODUCTION

In the absence of discrete measurements of streamflow (discharge measurements), streamflow typically is determined through surrogate measures of other variables such as stage, water-surface slope, rate of water surface rise, or index velocity at a streamgage. Streamflow is derived from the surrogate variables using what is termed a “rating.” The rating model is developed and calibrated utilizing discharge measurements, collected onsite by field staff using standard methods (Turnipseed and Sauer, 2010; International Standards Organization, 2007). Types of ratings used are broken into two broad categories: simple stagestreamflow ratings (simple rating) and complex ratings. A simple rating relates discharge to stage, assuming a unique univariate (single-value) relation between stage and discharge (Fig. 6.1). Simple ratings do not work well for streamgages on low-gradient streams, streams with variable backwater (Fig. 6.2), streams with large amounts of channel or overbank storage, or streams with highly unsteady flow. A complex rating relates discharge to stage and other variables because of the lack of a unique, univariate relation between stage and discharge. Complex rating methods range from simply adding a second independent variable to the process of computing streamflow to computer models that solve conservation-of-momentum and conservation-of-mass partial differential equations. Traditionally, the simplest form of rating model (stage streamflow) is tried first, with progressively more complex models developed if the simpler forms fail to provide an accurate relation. Rating development and verification constitute an ongoing process throughout the life of a streamgage, requiring systematic analysis and periodic evaluation to refine (or reconstruct) the rating in response to changes in stream hydraulics. This analysis and evaluation must be undergirded by a thorough understanding of both the science of river hydraulics and the many assumptions made for the various rating methods. Those sites producing the best rating-derived streamflow data are the result of a systematic process that is put in place to regularly analyze available rating methods in light of new data, keeping in mind that alterations and corrections made on the basis of new data must accurately reflect the underlying stream hydraulics—and therefore are rational and defensible.



F=

V = gD

Q 3

gA T

(6.1)

where V is the mean velocity, g is the acceleration of gravity, D is the hydraulic depth, Q is the discharge, A is the cross-sectional area, and T is the top width. Flow is subcritical when F is less than 1 and supercritical when F is greater than 1. The critical state of flow is when F equals 1. For subcritical flow, the control will be downstream of the streamgage and that control will not allow any effects downstream of the control (due to changing streamflow or channel conditions) to translate upstream through the control. For supercritical flow (Froude number greater than 1), the control will be upstream from the streamgage and would not allow any effects upstream of the control (due to changing streamflow or channel conditions) to translate downstream through the control. Thus, depending on the state of flow, the stage-discharge rating relation at the streamgage will neither be impacted by hydraulic changes that occur upstream (subcritical flow) or downstream (supercritical flow) from the control, nor will effects translate in the other direction through the control feature. For the remainder of this chapter, the discussion will focus on the more common condition of subcritical flow (control downstream). For section control, a single section of the channel is the controlling feature and, strictly defined, is configured (e.g., by constriction, bump in channel bottom, gravel/rock riffle, artificial weir/flume, or some combination) such that the flow passes through the critical-flow state. At the state of critical flow, F equals 1 and from Eq. (6.1) the stage (which is directly related to depth of flow) is only dependent on the discharge and the channel geometry. This aspect is very useful for ratings, as it implies a unique relation between stage and discharge (assuming unchanging channel geometry). The equation for a

6-1

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6-2    Streamflow Ratings

U.S. Geological Survey streamgage 07015720, Bourbeuse River near High Gate, Missouri 102.10 82.10 62.10 52.10 42.10 32.10

Explanation Simple rating Discrete measurement of streamflow

Stage, in ft

22.10 12.10 10.10 8.10 7.10 6.10 5.10 4.10 3.10 2.90 2.70 2.60 2.50 2.40 2.30 2.20 0.00499995 0.01

0.1

1

10

100

Streamflow, in ft3/s

1000

10,000

77,800

Data from U.S. Geological Survey, 2014

Figure 6.1  Example of a simple rating for the Bourbeuse River near High Gate, Missouri, USA (U.S. Geological Survey streamgage 07015720).

U.S. Geological Survey streamgage 05586100, Illinois River at Valley City, Illinois 28.00

Explanation

26.00

Simple rating

24.00

Discrete measurement of streamflow

22.00 20.00

Stage, in ft

18.00 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.60 2000

10,000

30,000

50,000

70,000

Streamflow, in ft3/s

90,000

110,000 118,700

Data from U.S. Geological Survey, 2014

Figure 6.2  Example of a site for which the simple rating shown will not determine streamflow accurately, Illinois River at Valley City, Illinois, USA (U.S. Geological Survey streamgage 05586100).

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Rating Controls    6-3 

straight-line rating (plotted in logarithmic space) that characterizes flow over an artificial or natural section control can be expressed (Rantz, 1982) as:

Q = P(G − e )N (6.2)

where Q is streamflow, P and N are constants depending on the geometry and configuration of the natural section and its approach section, G is the stage, and e is the stage (or gage height) of zero flow (GZF). GZF corresponds to the stage when streamflow just ceases. The exponent N ranges from 1.5 for rectangular sharp-crested weirs to 3.3 for Columbus-type controls (Rantz, 1982). For natural control sections, N is almost always equal to or greater than 2. The values of P are also highly variable, being influenced largely by the shape and configuration of the artificial or natural control section. For medium-to-high streamflow, the influence of the section control is lost (submerged) and channel control is typically in effect. Channel control consists of all the physical features of the downstream channel (such as geometry, bed slope, expansion/contraction, sinuosity, and bed/bank roughness) that determine the stage of the river for a particular discharge (Rantz, 1982). Channel control can be conceptualized and formulated using the flowresistance relation for open channels; flow resistance in an open channel can be described by the Manning equation (Yen, 1992): k 2 1 V = R 3 S f 2 (6.3) n



where the constant k is 1.49 for English units and 1 for SI units, n is the Manning resistance coefficient, R is the hydraulic radius, and Sf is the friction slope. The hydraulic radius is defined as:

R=

A (6.4) P

where P is the wetted perimeter. It should be noted that Chezy’s1 equation is equally appropriate for describing the flow resistance (French, 1985), because Manning’s equation has its roots in the Chezy equation (Yen, 1992). Because the streamflow is the product of the mean velocity and the crosssectional area, A, we can express. Eq. (6.2) in terms of streamflow as: 1 2 k Q = AR 3 S f 2 (6.5) n



For steady, uniform flow the friction slope (Sf) is equal to the bed slope (S0) (Chow, 1959) and Eq. (6.5) becomes

2 1 k Q = AR 3 S0 2 (6.6) n Equation (6.6) represents the assumption of steady, uniform flow and undergirds the simple stage-discharge relation for channel control; Eq. (6.6) demonstrates that for a channel reach of unchanging geometry, roughness, and bed slope, a unique relation between stage and streamflow occurs. For a rectangular channel, Eq. (6.6) becomes



5 1 k (6.7) Q = BD 3 S0 2 n where B is channel width and D is depth of flow. Steady, uniform flow rarely exists in natural streams and rivers because of the dynamic and spatial variability of open-channel flow (unsteadiness, convective acceleration, and differential hydrostatic pressure forces). When no abrupt changes occur in the flow depth (and thus velocity), this flow is called gradually varied flow (Chow, 1959). As such, the controls on the stagestreamflow rating are complex. For channel control in simple ratings, the assumption is made that the bed slope (S0) can be substituted for the friction slope (Sf) in Eq. (6.6). This assumption needs further examination. When flow is predominantly one dimensional, the friction slope can be expressed through a rearrangement of the one-dimensional unsteady equation of motion (Henderson, 1966) as:





S f = S0 −

∂ y v ∂v 1 ∂v (6.8) − − ∂ x g ∂ x g ∂t

where y is flow depth, x is distance in the longitudinal direction of flow, v is the velocity in the longitudinal direction, and t is time. Eq. (6.8) readily shows the effects of differential hydrostatic pressure force (∂ y ⁄ ∂ x ), convective acceleration (∂v ⁄ ∂ x ), and unsteadiness (∂v ⁄ ∂t ) on the friction slope. Additionally, the conservation of mass in one-dimensional flow with no sidechannel inflow can be stated as (Henderson, 1966): ∂y ∂Q +T = 0 (6.9) ∂x ∂t



with all variables in Eq. (6.9) having been previously defined. The components to control complexity in Eq. (6.8), caused by the nonuniformity and unsteadiness of the flow, combined with Eq. (6.9), produce a hysteresis effect in the stage-discharge relation as shown in Fig. (6.3). This hysteresis is the so-called loop rating (Kennedy, 1984), or loop-rating curve (Henderson, 1966).

988 986 984 982

Stage, in ft

980 978 976 974 972 970 968

0

1000

2000

3000

4000

5000

6000

7000

Streamflow, in ft3/s Figure 6.3  Example of a loop rating based on an unsteady flow routed by the unsteady flow option of the one-dimensional hydraulic model HEC-RAS (U.S. Army Corps of Engineers, 2010) through a nonuniform rectangular channel.

1

Chezy’s equation V = C RS f where C is Chezy’s roughness coefficient.

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6-4    Streamflow Ratings

Equations (6.8) and (6.9) conceptually capture all one-dimensional causes of hysteresis, including those phenomena that are dominated by conservation of mass concepts as demonstrated in Fig. 6.4 (Kennedy, 1984) which shows the loop effects when the hysteresis is tied to channel storage during the rise and fall of a hydrograph or when streams with large floodplains transition into and out of overbank flows. Although all natural streams have hysteresis (Fig. 6.3), a complex rating is not required in all cases because sometimes the degree of hysteresis is small and well within the uncertainty of discharge measurements made by the hydrographer. Unsteady flows caused by rainfall events move as a “wave” down the river. In those cases where the streamflow can be accurately determined from a simple rating, the wave is said to move as a “kinematic wave” (Henderson, 1966)—Sf is approximated by S0 and conditions meet the steady-uniform flow assumption. If the wave does not fit the kinematic-wave model, or is subject to other factors (such as backwater) that preclude use of a simple rating, Eqs. (6.8) and (6.9) conceptually capture the complexity of the flow dynamics and provide the basis for rating development.

26

6.3  SIMPLE RATINGS

Ratings are developed by first plotting the discharge measurements. For many sites, a simple rating will suffice, with segments of the simple rating represented by a power relation (Rantz, 1982) illustrated in Fig. 6.5 and shown as Eq. (6.2). Because the independent variable stage is plotted on the y-axis (this is standard practice), the slope term N is defined as the run over the rise instead of the traditional rise over the run. In the United States, particularly in the U.S. Geological Survey, simple ratings are often developed using logarithmic plots to take advantage of the ability to linearize Eq. (6.2) as follows: Log Q = Log P + N Log(G − e ) (6.10) The value of e usually corresponds to the GZF (for section control) or effective GZF (for channel control). The term e is a value that, when subtracted from stage in the curvilinear relation defined by the stage-discharge measurements, will cause the logarithmic rating curve to plot as a straight line; a straight-line relation helps when ratings need to be extended (or interpolated) in unmeasured ranges of stage. The value e is also called the scale offset. For

(a) Rating loops from channel storage

24 Constant-stage rating

22

Bankfull stage

20

Stage, in ft

18 16

Overbank rise

14 12

Within-banks rise

10 8 6 4 2 26

(b) Rating loops from return of overbank flow Recession from high overbank rise

24 22 20 18 Stage, in ft

Bankfull stage

Recession from minor overbank rise

16 14

All rising stages or recessions from within-banks rises

12 10 8 6 4 2

0

10

20 30 Streamflow, in thousands of ft3/s

40

50

Figure 6.4  Typical loop ratings from single-storm events (adapted from Kennedy, 1984).

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Simple Ratings    6-5 

of the floodplain, greatly increasing the conveyance capacity of the stream. Thus, for each increment of stage rise, a greater amount of water is conveyed opposed to streamflow confined within the banks of the main channel.

b= 1.7

2

(G–e), in ft

1

6.3.1  Curve Shaping and Rating Extension for Simple Ratings

P

1.5

Equation of curve shown is Q = 125(G–e)1.7 1 100

200 Q, in ft3/s

300

400

Figure 6.5  Theoretical simple rating plotted as a straight line on log-log grid (adapted from Kennedy, 1984).

regular-shaped section controls, the value of e will be the actual GZF. For channel controls and irregular-shaped section controls, the value of e will be the effective GZF, which usually is greater than the actual GZF (Rantz, 1982). Although logarithmic plotting is predominant, there are benefits to plotting the low-flow part of the rating in arithmetic rectangular coordinates. This allows for more detailed analysis at small changes of stage and enables zero discharge to be plotted (which cannot be done in logarithmic space). Simple ratings often have multiple linear segments, in which each segment represents a different control feature. For example, the lower part of the rating may have section control and the upper part have channel control; each segment often will have a different effective GZF (e). Kennedy (1984) provides much discussion about the use of scale offsets and their selection. Experience with rating characteristics has shown that the term b typically will have a value of less than 2 for channel control and greater than 2 for section control (Kennedy, 1984). The lower value of b for channel control is to be expected when one contrasts the exponent on the depth of flow in Eq. (6.7) with the exponent on depth in Eq. (6.2). Figure 6.1 demonstrates the higher slope for the lower stages where section control is in effect and the transition to channel control above approximately 7 ft stage where the slope is less than 2. Above approximately 16 ft stage, the curve bends back to the right. This bend back to the right is not the result of the return of the simple rating to section control, but rather it stems from the stage rising above the elevation

Discharge measurements over a range of stage form the basis for construction of accurate rating curves. Given discharge measurements have associated errors, and adequate discharge measurements are sometimes not available to define the entire range of the rating, hydraulic theory helps guide accurate construction of the complete rating curve. Channel surveys that provide detailed cross-section geometry (horizontal positions and associated elevations) enable proper geometrical characterization of the streamgage reach. Figure 6.6 shows surveyed cross sections of three different types of channel cross-section/control configurations and associated shape of the simple rating. Figure 6.6a has no section control and a channel cross section that has no major break in geometry. The lack of section control and uniform cross-section geometry result in a single-segment linear rating of constant slope. Figure 6.6b shows no section control and a major break in the channel cross section, where the water exceeds the bankfull stage and flows through the wide floodplain on both sides of the main channel. Channel control exists at all stages up to the bankfull condition of the main channel (shown by the lower segment of the rating), and then the rating bends over to the right as the stage exceeds the bankfull condition. Knowledge of the bankfull stage allows for proper shaping of the rating—the rating bends to the right at the bankfull stage, as the water conveyance increases substantially above this point. Figure 6.6c shows an effective section control inside the main channel, with a wide floodplain existing on both sides once the stage exceeds bankfull conditions. This results in a three-segment rating curve as shown to the right in Fig. 6.6c. Additionally, for those sites that have section control at lower stages, determination of GZF provides an anchor point on the lower end of the rating. Determination of GZF is done in the field, either by direct survey (using precision surveying equipment) or by determining the maximum depth of flow over the section control and subtracting this value from the observed stage. When an artificial section control is present, the stage-streamflow relation (for the section control portion of the rating) can be computed using a weir equation. Calibration of an appropriate weir equation is done by utilizing discharge measurements made during section control conditions. Slope-conveyance and step-backwater analyses are two hydraulic methods that can be used to provide quantitative estimates of the relation between

(a)

Rating shape

Channel shape

1

G–e

No flood plain (channel control)

1) or contraction (a < 1) 15-1

15_Singh_ch15_p15.1-15.8.indd 1

8/22/16 11:45 AM

15-2    Harmonic Analysis and Wavelets

(a)

Annual runoff m3/s

1000 800 600 400 200 0

1820

1840

1860

1880

1900

1920

1940

1960

1980

2000

1940

1960

1980

2000

Years (b)

Period (years)

4 8 16 32

1820

1840

1860

1880

1900

1920

Years

Global wavelet spectrum

(c)

3.5

x 104

3 2.5 2 1.5 1 0.5 0 128

64

32

16 Period (years)

8

4

Figure 15.1  Morlet wavelet analysis of the mean annual runoff of the Vannern Gota river. From top to bottom: (a) temporal fluctuations of annual runoff, (b) two-dimensional Morlet wavelet spectrum, (c) global Morlet wavelet spectrum. In the third plot, the dashed line correspond to the 90% statistical confidence interval from red noise.

factor of the wavelet function y(t), corresponding to different scales of observation. The parameter t can be interpreted as a temporal translation or shift of the function y(t), which allows for a study of the signal x(t) locally around the time t (Fig. 15.1). A frequency interpretation of Eq. 15.5 is also of interest. Effectively, using the Parseval theorem, the wavelet coefficients of a continuous time-signal x(t) can also be obtained via the relation:

C x (a , τ ) =

1 +∞ 1 +∞ ˆ (aω )e iωτ dω = xˆ(ω ) aψ xˆ(ω )Ψ a ,τ (ω )dω (15.6) ∫ −∞ 2π 2π ∫−∞

This formulation indicates that the wavelet coefficients correspond to the filtering of xˆ [Fourier transform of x(t)] by a set of band-pass filters Ya,t related to the Fourier transform of the wavelet function y(t) [denoted by ˆ (ω )] via the relation: ψ

ˆ (aω )e iωτ (15.7) Ψ a ,τ (ω ) = a ψ

From this point of view, the parameter a can be physically interpreted as a dilation or contraction of the filter (corresponding to different nonoverlapping frequency intervals), and the parameter t can be interpreted as a phase shift.

15_Singh_ch15_p15.1-15.8.indd 2

The wavelet function must fulfill some strict mathematical conditions (Heil and Walnut, 1989; Jawerth and Sweldens, 1994). For example, the time-scale localization property requires that wavelet functions be characterized, in both time and frequency domains, by compact supports or, at least, by a sufficiently fast decay. Mathematically, these properties can be formalized by two conditions called “admissibility conditions.” As mentioned earlier, one of these conditions is a rapid decrease of y(t) around the origin of time, implying that: +∞

∫−∞ ψ(t )dt = 0 (15.8)



In addition to a normalization of the wavelet function (the wavelet function must satisfy ψ = 1, where ⋅ is the classical L2-norm), more restrictive conditions must be imposed on the wavelet function in order to obtain a reconstruction formula. More precisely, the continuous-time signal x(t) can be retrieved from its wavelet decomposition coefficients Cx(a,t) only if the wavelet function has N vanishing moments, that is,

+∞ k

∫−∞ t

ψ (t )dt = 0, k = 1,, N − 1 (15.9)

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SIGNAL ENERGY REPARTITION IN THE WAVELET FRAME    15-3 

the condition (15.11), is then said to have regularity of order N. These conditions are called “regularity conditions.” There is a very large choice in wavelet functions. The simplest wavelet function is the Haar function, which is used for wavelet spectrum estimation in this chapter (presented later). It is defined by: 1 for 0 < x < 1 / 2  ψ ( x ) = −1 for 1 / 2 ≤ x < 1 (15.10) 0 elsewhere 



Then, several other wavelets such as Daubechies or Symlets (Daubechies, 1990, 1992) are also of practical use when dealing with hydrological signals. Conditioned by these restrictions, there exists a reconstruction formula, which allows the synthesis of the continuous time signal x(t) through its wavelet coefficients CX(a,t): x (t ) =



1 Kψ

+∞

+∞

∫0 ∫−∞ Cx (a, τ)ψ a ,τ (t )

dadτ (15.11) a2

Moreover, another property of the continuous wavelet transform is its ability to conserve energy between time domain and time-scale domain, that is, +∞

∫−∞ x



2

(t )dt =

1 Kψ

+∞

+∞

∫0 ∫−∞ Cx (a, τ)

2

dadτ (15.12) a2

One disadvantage of these nonorthogonal wavelets is the variance repartition of the signal in the continuous wavelet framework is characterized by a redundancy of information among the wavelet coefficients. This redundancy is mainly due to two properties. First, the Morlet wavelet and all other mother wavelet used in continuous wavelet transform do not strictly have a fully compact support although it decays to zero. Second, redundancy also comes from continuous shift in time and scale of the wavelet during the continuous transform. Then, the correlation between coefficients appears as intrinsic to the wavelet-kernel and not to the analyzed signal. The wavelet coefficient interpretation is then strongly dependent on the chosen projection basis, which must be adapted to the specific problem at hand. For example, the Mexican Hat wavelet allows for a good temporal resolution but a poor frequency resolution, whereas the Morlet wavelet allows for a good frequency resolution although with a less precise temporal resolution (Labat, 2000). 15.3  DISCRETE TIME WAVELET TRANSFORM AND MULTIRESOLUTION ANALYSIS

The continuous wavelet analysis is essentially a signal analysis tool but because of the redundancy issue, it cannot be considered as an energyrepartition tool. Therefore, discrete time wavelet transform is now presented. For practical applications, the scientist or engineer does not have at his or her disposal a continuous-time signal process but rather only a discrete-time signal, which is denoted herein by x(i). Then, the time-scale domain needs to be discretized and one can choose the general form ({a0-j, k.t0.a0-j}, (k,j) ∈ Z2), where a0 (a0 > 1) and t0 are constants. The discretized version of Eq. (15.5) defines the discrete wavelet transform coefficients, as follows:

Cx ( j , k ) = ∫

+∞

x (t )ψ ∗ j ,k (t )dt  with  ψ j ,k (t ) = a0 j 2 ψ (a0 jt − kτ0 ) (15.13)

−∞

where ∗ corresponds to the complex conjugate. The integer j is the scale factor (analogous to the parameter a) and the integer kt0 is the translation factor (analogous to the parameter t). As in the continuous-time wavelet transform, the reconstruction formula can be applied with appropriate choice of the coefficients a0 and t0. This allows for representation of a discrete-time signal process in terms of series expansion and leads ultimately to the concept of multiresolution analysis. Once the time-scale domain has been discretized, some particular values for a0 and t0, corresponding to an octave representation (Daubechies, 1990, 1992) of the scales, are chosen, namely: a0 = 2 and t0 = 1. The continuous grid is then replaced by a discrete dyadic grid of the form: ({2-j, k.2j}, (k,j)∈Z2). The orthogonal discrete wavelet transform coefficients Cj,k are then defined by the convolution product:

C xj ,k = ∫

15_Singh_ch15_p15.1-15.8.indd 3

+∞

−∞

x (t )ψ ∗ j ,k (t )dt  with  ψ j ,k (t ) = 2 j 2 ψ (2 j t − k ) (15.14)

In such an orthonormal basis, all wavelets y are orthonormal to both their translates and their dilates. Therefore, it is possible to construct a complete basis under which all square integrable signals x(i) can be expanded as a linear combination of translates and dilates of orthonormal wavelets: +∞ +∞

x (i ) = ∑



∑ C xj ,k (i )ψ j ,k (i ) (15.15)

j =0 k =−∞

Equation (15.15) is the discrete-time formulation of signal synthesis; it can also be interpreted as successive approximations (in a mean square sense) of the analyzed discrete signal x(i) by the sequence {xn(i)}n=1,..,N defined by: n−1 +∞

xn (i ) = ∑



∑ C xj ,k (i )ψ j ,k (i ) (15.16)

j =0 k =−∞

This constitutes the conceptual basis of multiresolution analysis. Multiresolution wavelet analysis (Mallat, 1989) allows the decomposition of a function or signal in a progression of successive “approximations” and “smooths,” corresponding to different scales j. The difference between the actual signal and its approximation of order n is called the “residual.” Intuitively, the approximation is relatively smooth and the detail, being composed of high frequency components, is rougher. Note that the detail corresponds to the difference between two successive levels of approximation j and j + 1. To sum up, the multiresolution analysis allows an orthogonal decomposition of a sampled hydrologic signal in terms of approximations and details of increasing order of resolution. Because of the orthogonality of the decomposition, it should be noted that the multiresolution induces a more efficient and easier interpretation of energy distribution on the different scales of the decomposition. 15.4  SIGNAL ENERGY REPARTITION IN THE WAVELET FRAME

The repartition of energy in the signal allows for the determination of the scales, which concentrate the essential dynamics of a signal. Therefore, a determination of the temporal variations of the distribution of energy across the scales is an important application of the wavelet transform. Previously, the Fourier transform allowed for the determination of the distribution of energy across frequencies w using the Power spectrum density PXX(w) defined for a signal x(t) by:

PXX (ω ) =

2

1 T x (t )2 exp(−i 2πωt )dt (15.17) T ∫0

The companion concept of wavelet energy was defined first by Hudgins (1993) and then by Brunet and Collineau (1995). Other results concerning wavelet variance estimators and applications can be found in Percival (1995) or Percival and Mofjeld (1997). This concept is based on the conservation of energy between the time domain and the time-scale domain, that is,

+∞

∫−∞ x

2

(t )dt =

1 Kψ

+∞

+∞

∫0 ∫−∞ Cx (a, τ)

2

dadτ (15.18) a2

Therefore, the total signal variance can be distributed in the wavelet domain and the wavelet spectrum WX(a,t) of a continuous-time signal x(t), which can be defined, by analogy with the Fourier analysis, as the modulus of its wavelet coefficients (Liu, 1995):

2

WX (a, τ ) = C X (a, τ )C ∗ X (a, τ ) = C X (a, τ ) (15.19)

This wavelet spectrum can also be averaged in time or in scale, but at the cost of a loss of information (Torrence and Compo, 1998). On the one hand, time averaging gives the variance signal distribution between the different scales, referred to as the global averaged wavelet power spectrum. On the other hand, scale averaging allows the temporal identification of a particular component (i.e., the temporal variability at a given scale or range of scales) of the signal. The determination of the characteristic periods of oscillation is achieved using the global wavelet spectrum. Statistical confidence limits of the peaks identified in the wavelet power spectrum have been discussed by Torrence and Compo (1998), Ge (2007), Liu et al. (2007), and Schaefli et al. (2007). Two

8/22/16 11:45 AM

15-4    Harmonic Analysis and Wavelets

limits are classically considered in order to restrict analysis to statistically significant-only fluctuations: • Only periodicity different from red noise is taken into consideration and 95% and 90% confidence intervals are generally depicted in the figures, following Torrence and Campo (1998) calculations. • Edge effects [and then limitations of the integration time interval in Eq.  (15.19)], lead to the presence of a “cone of influence.” This corresponds to the region where variance cannot be estimated properly because of edge effects. That leads to an under-estimation of variance as a result of zero-padding of the original time series (Torrence and Compo, 1998). Due to the edge effects leading to the presence of a cone of influence, if T is the total length of the series, only periods smaller than T/(2√2) are considered as significant. According to Meneveau (1991), the concept of wavelet variance can be also defined by using the orthogonal multiresolution approach. The total energy (variance) TX(m) captured by the detail Dm at scale m of a signal x(i) of length 2M can be estimated as: Tx (m) =



1

2 M −m

2 M −m

∑ ( Dmx , j )

2

(15.20)

j =1

Note that these concepts were first implemented to study intermittent turbulent signal above the forest canopy (Katul et al., 1994; Turner et al., 1994; Katul and Parlange, 1995). One of the main advantages of the discrete orthogonal wavelet transform is its ability to concentrate the energy of the signal in a limited number of coefficients. 15.5  WAVELET ANALYSIS OF THE TIME–SCALE RELATIONSHIP BETWEEN TWO SIGNALS

In this section, we focus on the introduction of wavelet indicators to investigate the relationship between two given signals (Labat et al., 2000, 2002), for example, rainfall rates/discharges or discharge/climate circulation indices. Classical cross-correlation makes it possible to quantify the degree of similarity between two signals, denoted as. The cross-correlation RXY between two stochastic processes x(t) and y(t) is defined as: RXY (t1 − t 2 ) =

C XY (t1 − t 2 ) with CXY(t1 – t2)=E[x(t – t1)y(t – t2)] (15.21) C XX (0) ⋅ CYY (0)

that indicates that cross-correlation analysis provides a quantitative measure of the relatedness of two signals, shifting in time with respect to each other. It allows for the identification of common components occurring at the same delay. However, if the signals include nonstationary components, the crosscorrelation becomes invalid. The same conclusion arises if both signals are characterized by highly variable processes occurring over a wide range of scales. The continuous wavelet cross-correlation is introduced, which overcomes the limitations of classical correlations. Considering two correlated processes x(t) and y(t), the wavelet crosscorrelation function, denoted as WCXY(a,u), for a given scale a and a given time delay u can be defined as:

WC XY (a,u) = E[WXX (a, τ ) ⋅WYY (a, τ + u)] (15.22)

At this point, some authors (see later) recommend distinguishing the real part noted as RWCXY (a,t) from the imaginary part IWCXY (a,t) of the wavelet cross-correlation, arguing that the real part alone is sufficient to quantify the strength of correlation between the two signals at a given scale a. The wavelet cross-correlation is then defined by:

WR XY (a,u) =

RWC XY (a,u) (15.23) RWC XX (a,0) ⋅ RWCYY (a,0)

Sello and Bellazzini (2000) propose to consider only the real part of the wavelet transform and define a wavelet local correlation coefficient WLCC(a,u) based * on the wavelet cross-spectrum, WXY (a,u) = WXX (a,u) ⋅WYY (a,u), as:

Real(WXY (a,u)) WLCC(a,u) = (15.24) WXX (a,u) ⋅ WYY (a,u)

15_Singh_ch15_p15.1-15.8.indd 4

However, real-part or imaginary-part wavelet analysis does not provide a clearly separate information. As in the Fourier analysis, only modulus and arguments are considered here as interpretative. This conclusion is confirmed by Mizuno-Matsumoto et al. (2001, 2002), who observed, in a preliminary analysis, no significant differences between the definitions. Wavelet crosscorrelation is thus defined as: 2



WR XY (a, τ ) =

RWC XY (a, τ ) + IWC XY (a, τ )

2

WC XX (a,0) ⋅ WCYY (a,0)

(15.25)

where RWCXY and IWCXY are, respectively, the real and the imaginary part of the cross-wavelet correlation function. In conclusion, the introduction of wavelet cross-correlation allows for a rapid identification of the degree of correlation between processes at a given scale and the determination of the delay between these processes. 15.6  WAVELET CROSS SPECTRUM AND COHERENCE

By analogy with the Fourier cross-spectrum, Liu (1995) first defines the wavelet cross-spectrum WXY(a,t) between two signals x(t) and y(t) by:

WXY (a, τ ) = C X (a, τ )C ∗Y (a, τ ) (15.26)

where CX(a,t) and C∗Y(a,t) are, respectively, the wavelet coefficient of the continuous-time signal x(t) and the conjugate of the wavelet coefficient of y(t). The averaging techniques applied to the wavelet spectrum are also used here for expressing the cross-covariance of the signals x and y and its distribution through scales. However, Maraun and Kurths (2004), among others, demonstrated that wavelet cross-spectrum appears to be nonsuitable to interpret interrelation between two processes (ENSO and NAO fluctuations in their example) and recommend the introduction and use of wavelet coherence analysis. Effectively, cross wavelet analyses only indicate the presence of common periodicities between two signals but no causal relationship is put in evidence. The notion of coherence in signal processing consists, from a general point of view, of a measure of the correlation between two signals or between two representations of these signals. To overcome the problems inherent to nonstationary signals, it has been proposed to introduce the wavelet coherence (Torrence and Webster, 1999; Lachaux et al., 2002). The main issue is that an application of a similar formula to Eq. (15.25) leads to a coherence always equal to 1, as pointed out by Labat et al. (1999a) and Labat (2005). Attempts that have been made to avoid this problem essentially differ on two points: the fundamental definition of wavelet coherence and the estimation of the wavelet cross-spectrum. Torrence and Webster (1999) propose to determine wavelet coherence using a smoothed estimate of the wavelet spectrum and to define a smooth wavelet spectrum and cross spectrum, denoted as SWXX(a,t) and SWXY(a,t), respectively: t +δ /2



SWXX (a, τ ) = ∫



SWXY (a, τ ) = ∫

t −δ /2

* WXX (a, τ )WXX (a, τ )dadτ (15.27)

t +δ /2

t −δ /2

* WXX (a, τ )WYY (a, τ )dadτ (15.28)

The scalar d represents the size of the two-dimensional filter (Lachaux et al., 2002). It constitutes an important parameter of the wavelet coherence and must be adequately determined for an acceptable estimation of the wavelet coherence. We therefore recommend to the reader to properly apply the Torrence and Webster (1999) indications. The wavelet coherence can then be defined by analogy with Fourier coherence as:



WC(a, τ ) =

SWXY (a, τ )  SWXX (a, τ ) ⋅ SWYY (a, τ )   

(15.29)

Schwarz inequality still ensures that WC takes a value between 0 and 1. Statistical confidence limits of wavelet coherency based on red-noise background are discussed in Grinsted et al. (2004), Maraun and Kurths (2004), and Schaefli et al. (2007).

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REFERENCES    15-5  15.7  APPLICATIONS OF WAVELET TRANSFORMS IN HYDROLOGY AND EARTH SCIENCES

Wavelet analysis has become more and more popular in the Earth sciences, especially in order to identify coherent structures in time series. In fact, there is a good chance that wavelets will be used by scientists from all different fields dealing with time series that clearly exhibit transient processes. This popularity is possible largely because of the existence of free wavelet programs (especially in MATLAB) made available through the Web [see, for example, www.amara.com/current/wavelet.html, atoc.colorado.edu/research/wavelets, or Uvi wave toolbox (Gonzales et al., 1996) for univariate analyses, and www.pol.ac.uk/home/research/waveletcoherency/ for wavelet bivariate analyses). Table 15.1 provides a brief review of wavelet applications in hydrology. In hydrology, wavelets can, at the fundamental level, be used to detect transient or periodic processes in relevant time series. But as mentioned earlier, continuous wavelet transform is also well suited for identification of interannual variability in hydrologic and meteorologic time series. Some applications of wavelets in hydrology also concern much longer periods, ranging from a century to several million years. Multiresolution concepts can also be used as a stochastic tool to generate synthetic time series and recent developments in wavelet hydrologic modeling are also based on the introduction of artificial

neural networks (ANNs) coupled to wavelet models, the so called hybrid wavelet-ANN models and derivatives. 15.8 PERSPECTIVES

In conclusion, wavelet transforms appear as a powerful tool to identify transient processes in hydrological time series whereas multiresolution analyses for now appear as restricted to isolate a given component corresponding to a given scale. Wavelet cross-correlation and wavelet coherence are also now being extensively used to provide more detailed information about the time–scale relationship between two signals. In these ways, wavelet analysis provides a better investigating tool than classical Fourier analysis, and thus deserves far more attention. However, it must be kept in mind that wavelets cannot be considered as the “Grail,” since the interpretation of the results from the wavelet analysis must always rely on physical processes. Wavelet research in hydrology should be dedicated to the development of wavelet-based models coupled to neural networks or Volterra expansion taking into account the intrinsic multiscale and nonlinear nature of many physical relationships. Wavelets will become a more complete and better simulation tool for the study of multiscale properties of hydrological and climatological processes.

Table 1.1  Studies on the Applications of Wavelets in Hydrology Climatology

Baliunas et al. (1997), Benner (1999), Higuchi et al. (1999), Lucero and Rodriguez (1999), Compagnucci et al. (2000), Janicot and Sultan (2000), Jury and Melice (2000), Kulkarni (2000), Saco and Kumar (2000), Jung et al. (2002), Lucero and Rozas (2002), Tyson et al. (2002), Min et al. (2003), Zheng et al. (2003), Anctil and Coulibaly (2004), Coulibaly and Burn (2004), Coulibaly (2006), Partal and Küçük (2006), Su et al. (2008), Sujino et al. (2004), Franco Villoria et al. (2012), Labat et al. (2012), Carey et al. (2013), de Souza et al. (2010), Dong et al. (2014), Dieppois et al. (2013), Bourrel et al. (2014), Zhang et al. (2014), Küçük et al. (2009), Li et al. (2009), Gobena et al. (2009), Liu et al. (2014), Briciu and Mihaila (2014), Xu et al. (2014), Restrepo et al. (2014), Chevalier et al. (2013), Hierro et al. (2013), Assani et al. (2008), Masséi et al. (2011), Nalley et al. (2012), Xu et al. (2013), Zhu and Meng (2010), Li et al. (2012), Rivera et al. (2013)

Precipitation fields and micrometeorology

Kumar and Foufoula-Georgiou (1993), Takeuchi et al. (1994), Turner et al. (1994), Katul and Parlange (1995), Kumar (1996), Venugopal and Foufoula-Georgiou (1996), Katul et al. (1998, 2001), Szilagyi et al. (1999), Furon et al. (2008)

Hydrology

Dohan and Whitfield (1997), Fraedrich et al. (1997), Whitfield and Dohan (1997), Smith et al. (1998), Jury and Melice (2000), Labat et al. (2000, 2001, 2004, 2005), Gaucherel (2002), Lafrenière and Sharp (2003), Lark et al. (2003), Labat (2005, 2006, 2008), Schaefli et al. (2007), Kang and Lin (2007), Kuss et al. (2014), Slimani et al. (2009), Dong et al. (2014), Zhang et al. (2013), Lauzon et al. (2004), Parent et al. (2006)

Hydrologic modeling

Anctil and Tapé (2004), Bayazit and Aksoy (2001), Bayazit et al. (2001), Chou and Wang (2002), Aksoy et al. (2004a, b), Sujono et al. (2004), Cannas et al. (2006), Chou (2007), Lane (2007), Adamkowsi (2008a, 2008b), Kisi (2008, 2009), Partal and Cigizoglu (2008), Unal (2004), Kim and Waldes (2003), Sahay and Sehgal (2014), Shoaib et al. (2014), Nourani et al. (2009a, 2009b, 2014a, 2014b), Goyal (2013, 2014), Sehgal et al. (2014), Sahay and Srivastava (2014), Tiwari and Chatterjee (2010), Wang et al. (2009), Hu et al. (2008), Salerno and Tartari (2009), Rao and Krishna (2009), Coulibaly and Baldwin (2008), Xu et al. (2014), Pramanik et al. (2011), Danandeh et al. (2013), Karthikeyan and Nagesh Dumar (2013), Yu et al. (2013), Rathinasamy et al. (2013), Sang et al. (2009)

REFERENCES

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Unal, N. E., H. Aksoy, and T. Akar, “Annual and monthly rainfall data generation schemes,” Stochastic Environmental Research and Risk Assessment, 18 (4): 245–257, 2004. Venugopal, V. and E. Foufoula-Georgiou, “Energy decomposition of rainfall in the time-frequency-scale domain using wavelet packets,” Journal of Hydrology, 187: 3–27, 1996. Walker, J. S., “Fourier analysis and wavelet analysis,” Notices of The American Mathematical Society, 44 (6): 658–670, 1997. Wang, W., J. Jin, and Y. Li, “Prediction of inflow at three gorges dam in Yangtze River with wavelet network model,” Water Resources Management, 23 (13): 2791–2803, 2009. Whitfield, P. H. and K. Dohan, “Identification and characterization of water quality transients using wavelet analysis. II Application to electronic water quality data,” Water Science and Technology, 36 (5): 337–348, 1997. Xu, J., Y. Chen, and W. Li, “The nonlinear hydro-climatic process: a case study of the Tarim Headwaters, NW China, Water Resources Research in Northwest China, Springer, The Netherlands, 2014, pp. 289–310. Xu, J., Y. Chen, W. Li, Q. Nie, Y. Hong, and Y. Yang, “The nonlinear hydroclimatic process in the Yarkand River, northwestern China,” Stochastic Environmental Research and Risk Assessment, 27 (2): 389–399, 2013. Xu, J., Y. Chen, W. Li, Q. Nie, C. Song, and C. Wei, “Integrating wavelet analysis and BPANN to simulate the annual runoff with regional climate change: a case Study of Yarkand River, Northwest China,” Water Resources Management, 28 (9): 2523–2537, 2014. Yu, S. P., J. S. Yang, and G. M. Liu, “A novel discussion on two long-term forecast mechanisms for hydro-meteorological signals using hybrid waveletNN model,” Journal of Hydrology, 497: 189–197, 2013. Zhang, D., H. Hong, Q. Zhang, and X. Li, “Attribution of the changes in annual streamflow in the Yangtze River Basin over the past 146 years,” Theoretical and Applied Climatology, 119(1): 323–332, 2015. Zhang, Y. Y., D. Y. Zhong, and B. S. Wu, “Multiple temporal scale relationships of bankfull discharge with streamflow and sediment transport in the Yellow River in China,” International Journal of Sediment Research, 28 (4): 496–510, 2013. Zheng, D., X. Ding, Y. Zhou, and Y. Chen, “Earth rotation and ENSO events: combined excitation of interannual LOD variations by multiscale atmospheric oscillations,” Global and Planetary Change, 36: 89–97, 2003. Zhu, L. and J. Meng, “Study on rainfall variations in the middle part of Inner Mongolia, China during the past 43 years,” Environmental Earth Sciences, 60 (8): 1661–1671, 2010.

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Chapter

16

Outlier Analysis and Infilling of Missing Records in Hydrologic Data BY

UMED S. PANU AND WAIWAH NG

ABSTRACT

Hydrologic data analyses invariably suffer from challenges arising from the existence of outliers and missing records in datasets. In absence of adequate understanding and/or interpretation of such datasets usually lead to biased results in hydrologic analyses. Fundamental definitions and germane discussion necessary for understanding the concepts of outliers and missing records as well as their handling methods in hydrologic analyses are offered. The significance of outliers and missing records in hydrologic datasets is espoused in view of analysis, forecasting, and synthesis of hydrologic data. Specific topics including suggestions for improvement of existing methods, and future research directions pertaining to handling of outliers and missing records are discoursed. This chapter endeavors to provide succinct yet fundamental background information concerning outliers and missing records in hydrologic datasets. 16.1 INTRODUCTION

In geophysical sciences, such as hydrology, observations are regularly recorded to capture changes in historical processes under a specific space interval or time period. Hydrologic data may contain extreme observations (outliers) due to subtle changes experienced by the process of a system, or due to the impact of external factors. Such outliers are indeed true observations. In contrast, outliers in dataset can be caused by instrumental errors (e.g., faulty equipment, malfunction of instrument, and improper exposure of instrument), or human errors (e.g., misreading the instrument, wrong entry of data in records/computer, and error in computation). In hydrology, the concern for outliers commonly arises in context of regional homogeneity assessment and frequency-analysis studies. Without a careful investigation of possible causes and/or overlooking the importance of outliers, practitioners may incorrectly include or omit outliers in their analysis. Consequently, the over/ underestimation of the impact of outliers may lead to over conservative or under design in water resources systems. At times, the datasets may be riddled with missing records. Missing records are common to hydrologic datasets for reasons such as loss of records, malfunction of instruments, temporary termination of the data collection at a specific location or period because of relatively less interest to the survey or data-collection agencies, or lack of financial, human, or apparatus resources to continue measurements. Under such different circumstances, some practical and rational approaches are required to handle missing records. The presence of missing records in a dataset presents unique challenges to the analyst because these scenarios lead to less-reliable estimates of parameters, which bias the results of analysis. Methods for handling of missing records are

required, and mainly governed by the data characteristics in terms of their occurrence in space and time. This chapter deals with the basic definitions and presents discussion necessary for the understanding of the concepts of outliers and missing records, as well as their handling methods in hydrologic analysis. The definitions, types, causes, identification/treatment methods of outliers as well as the infilling procedures of missing records are presented. Specific topics including how to improve existing methods and approaches, and future research directions are discussed in subsequent sections. 16.1.1  Characteristics of Hydrologic Data

Measurement and reporting of hydrologic data on various hydrologic processes can be classified into four types (Yevjevich, 1972): (1) data recorded in time; (2) data recorded in space; (3) data recorded in time and space; and (4) data concurrently recorded in time and/or space on two or more variables. Most of historically collected data on hydrologic processes, specifically on streamflows, belong to the first type. Streamflows and other hydrologic phenomena are observed in discrete time and thus time becomes an indexing variable. The resulting temporal data can be referred to as univariate time series with single variable, or multivariate time series with multiple variables. Hereafter, the temporal hydrologic data is referred to as hydrologic data. Hydrologic data, such as streamflow and precipitation records, typically comprises of data without negative values. The existence of lower bound of zero and occurrences of high outliers cause positive skewness in the distribution of hydrologic data. On the other hand, rivers or streams originating in lakes and swamps cause negative skewness because of prolonged periods of low flows. The lower limit of zero and presence of skewness invariably make the hydrologic data to be non-normally distributed and their distributions with heavy tails (Klemes, 2000). In addition, hydrologic data may be regarded composed of deterministic and stochastic components. The deterministic component may be comprised of cyclicity and/or some kind of monotonic trend. The stochastic component could be pure random or may display some persistence. The existing techniques in hydrologic literature do not explicitly consider groups of similar data values and are rather based on the consideration of individual data values where both cyclicity and trend effects are removed by applying prespecified, sometimes arbitrary, procedures. In contrast, techniques, such as pattern recognition, explicitly consider similar values in the form of groups (or patterns) where cyclicity is considered as an integral part of the pattern itself and hence no need arises for its isolation. In a seminal paper, Unny et al. (1981) respecified a “deterministic trend” as a “known trend,” which in any case, must be removed before analysis.

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16-2     Outlier Analysis and Infilling of Missing Records in Hydrologic Data 16.1.2  Outliers and Missing Records and Their Significance

Outliers in a dataset in essence are undesirable (or unwanted) values similar to weed plants in a cultivated agricultural field. For example, maize plants, although physically exist in a wheat field (i.e., statistically considered as a part of the population), are indeed considered by the farmer as undesirable or unwanted (i.e., weed plants). In other words, outlier (e.g., maize plants) may be subjectively determined based on the interest of the user. Outliers (i.e., extreme high and/or low observations) being true observations on the phenomena and/or hydrologic process and yet may be considered as observations which do not conform to the remainder of data. A key question for an analyst is whether an observation is an outlier or not, and should such an observation be discarded from the dataset simply because it appears to be inconsistent with rest of the data. In the design and management of water resources systems, it is important to be aware of anomalies (i.e., outliers) in data as they can induce bias in the estimation of parameters, and may consequently lead to improper water resource management policies or infrastructure design. When the interior data of a distribution are in error, these data are called as inliers. Inliers can arise due to systematic or processing error. Statistical analysis can be seriously affected by inliers because inliers are difficult to distinguish from good data values. Sometimes, they are even difficult to find and correct. Relevant detail pertaining to methods of handling inliers in the context of flood frequency analysis is found elsewhere (Singh and Nakashima, 1981). In addition to outliers, the existence of missing records is also one of the common problems observed in hydrologic data. This problem can affect the fitting of statistical models and thus, leads to increment of uncertainty in model outputs (Little and Rubin, 1987). Assessment of incomplete datasets invariably raises the level of complexity in statistical analyses. The varying nature of missing observations under different circumstances requires adequate, practical, and rational treatment because improper handling of missing observations may significantly distort the results of analyses. 16.2  CONCEPTS AND METHODS FOR OUTLIER ANALYSIS

In some fields, outlier detection is also known as event detection, novelty detection, anomaly detection, deviant discovery, change point detection, fault detection, intrusion detection, or misuse detection. Presently, majority of studies are devoted to three types of outliers, namely: contextual outliers (i.e., anomalous in specific contextual attributes, e.g., longitude/latitude for the spatial data, and time for temporal data), point outliers (i.e., individually anomalous), and collective outliers (i.e., a collection of anomalies). 16.2.1  DEFINITION AND TYPES OF OUTLIERS

An outlier can broadly be conceptualized (Aggarwal, 2013) as comprising of a regular component, a noise component, and an anomaly component as depicted as follows. Strong or weak outlier Regular data

Noise Inconsistency

Grade of OUTLYINGNESS increases from left to right Figure 16.1  A Pictorial representation of conceptual aspects of an outlier.

Despite an intuitive understanding on the meaning of outlier by many analysts, there exist varied interpretations. Barnett and Lewis (1978) stated that an outlier is “an observation” (or subset of observations), which appears to be inconsistent with the remainder of that set of data. Similarly but more precisely, Hawkins (1980) defined an outlier as “an observation which deviates so much from other observations as to arouse suspicions that it was generated by a different mechanism.” Alternatively, Beckman and Cook (1983) defined an outlier as “a collective to refer to either a contaminant or a discordant observation.” In turn, Iglewicz and Hoaglin (1993) provided interpretation of a discordant observation as “any observation that appears surprising or discrepant to the analyst” and likewise a contaminant observation as “any observation that is not a realization from the target distribution.” An outlier is generally recognized either as an error or a true observation. The terms such as “mixing” and “slippage” are also used to explain the outlier phenomena (Hawkins, 1980). Tabachnick and Fidell (1996) even visualized

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four types of outliers caused by their nature of occurrence: incorrect data entry, computer coding failure, data contamination, and true observation(s). Knowledge on the causes of outliers can help in the determination of a suitable treatment. Confirmed erroneous observation(s) can simply be removed. As for the treatment of true extreme observation(s), the user needs to perform statistical outlier tests to assess whether or not the extremal observation(s) likely belong to the same population. The user will then decide the treatment of the identified outlier(s) as to be “removed,” “retained,” or “revised” (Grubbs, 1969). Usually, such a decision-making process may involve investigation of outliers using other variables, risk-assessment considerations, and the experience of the user. 16.2.2  Approaches for Outlier Analysis

Outlier analysis can be viewed as comprising of two components, namely, the outlier detection and the outlier treatment. The statistical purpose of identifying suspicious observation(s) is to prevent data contamination, to recognize mixed groups, to avoid distortion of the presumed underlying probability distribution, and to enhance the quality of hydrologic data. Outlier detection and their treatment constitute a broad field encompassing a wide array of applications. Outlier analysis in different data fields typically requires specific types of techniques such as in time series data, temporal outlier analysis examines anomalies in the behavior of the data across time. The data type clearly governs the selection of the method/technique to be used for outlier analysis. A key aspect of any outlier detection technique is the nature of hydrologic data. Outlier analysis in temporal data may be categorized (Gupta et al., 2014) in a wide variety of ways representing different facets of the analysis. Outlier analysis in various aspects of hydrologic data can also be visualized to discover rare and interesting occurrences similar to traditional techniques, but the hydrologic data is likely to offer new challenges. Outliers can be detected in a variety of ways such as through concentration of residuals, Studentized t-test distribution, cluster analysis (Mahalonobis distance, Mds), etc.; however, there exists some commonalties among the various competing outlier detection methods. The process of outlier detection can be divided into four basic processing units. (1) An observation by itself is natural and boundless that cannot be discriminated as outlier without suitable specification of a defining boundary. (2) Adjustment of a dataset may be carried out to enhance the tractability of a dataset. (3) A proper presumed type of distribution or pattern can isolate a target group from other groups in the dataset. (4) Depending upon the interest of the user, the data excluded outside the boundary can be considered as outliers. The manner in which the sample dataset is arranged and bounded will directly affect the outcome. Without clearly specifying the scope and purpose of an outlier analysis, the results of such an analysis will vary. The treatment of outlier(s) is highly dependent on the interest of the user. 16.2.3  Methods for Outliers Detection

In streamflow analyses, outliers are referred to extreme high/low flows in historical records. Such extremal observations can highly deviate from the rest of the observations (Hu, 1987). Outliers in a dataset may bias the estimates of missing observations and simply discarding outliers could lead to underestimation of the true missing observations. Thus, the selection and use of an appropriate outlier detection approach is not only critical but also crucial in the design and operation of water resource systems. Classical outlier detection is confined to pure statistical considerations with disregard to the nature of the extremal observation(s) and to the need of users. However, two major categories of outlier detection analysis are based on the premises of accommodation (i.e., outlier inclusion with/without revision) and discordance (i.e., outlier exclusion). In a particular application, it is important to evaluate the available options of the selected method to meet the detection sensitivity for a specific case. For example, in flood frequency analyses, a conservative outlier detection approach is suggested because omissions of outlier(s) can result in serious consequences due to under or over estimation of the design flow. This can be achieved by using more sensitive outlier detection methods. Technically, the selection of a suitable outlier detection method involves different considerations, such as parametric versus nonparametric, sensitive versus insensitive, one-tail or two-tails, time and efficiency, type of application, number of outliers, number of variables, sample size, selection of probability distribution, determination of the level of significance, outlier treatment, and risk assessment. Least square regression is an especially useful method for multivariate outlier detection. Autoregressive Integrated Moving Average (ARIMA) is useful for detection of outlier(s) in time series datasets.

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Concepts and Methods for Handling Missing Records     16-3 

The U.S. Water Resources Council (US-WRC) method is one of the most commonly used hydrologic techniques in North America for outlier detection and outlier deletion (McCuen, 2004). This method is based on the principles of hypothesis testing with an underlying assumption of normal distribution, but applied to the log Pearson Type III (LP3) distributed data (Grubbs and Beck, 1972). Studies on the hydrologic outlier analysis by Kottegoda (1984) compared different statistical outlier methods in maximum flow data series, whereas Singh (1980), Hu (1987), Spencer and McCuen (1996), and Panu et al. (2000) primarily examined the application of the US-WRS method in hydrologic data series. Pegram (1997) examined detection and identification of extremal observation(s) in rainfall data and Singh (1980) in groundwater data series. Other than US-WRC method, the US-EPA (U.S. Environment Protection Agency) also employs an outlier analysis approach from the Scout multivariate analysis package (Singh, 1993), which includes the nonparametric outlier approach (e.g., Mds) for extremal data detection. For flood frequency analysis in Canada, the Consolidated Frequency Analysis (CFA) package (Pilon et al., 1985) is utilized for the outlier testing process. In water resources engineering, the nonstatistical considerations (e.g., the intent of outlier analysis and the interest of the user) are usually more critical than the statistical procedures (outlier detection). To cope with the needs of different users, statisticians have developed numerous methods for outlier(s) detection as briefly described below. • Box-plot: The box-plot serves in revealing the center of the data, their spread and distribution, and the presence of outliers. A box-plot displays the three quartiles as well as the minimum, the median, and the maximum of the data. Quantiles are values taken at intervals from the inverse of the cumulative distribution function of a random variable. It includes a rectangular box that shows the interquartile range with the lower edge at the first quartile, and the upper edge at the third quartile. A line is drawn through the box at the second quartile (which is the 50th percentile or the median). A line at either end of the box extends to the extreme values which are observations between 0 and 1.5 times that of the interquartile range from the edge of the box. Observations lying between 1.5 and 3 times the interquartile range from the edge of the box are called outliers. Observations that are beyond three times the interquartile range from the edge of the box are called extreme outliers (Montgomery and Runger, 1994). Further, the box-plot method is simple to use and easy interpret. The disadvantage of the method is its reliance on the interquartile range, which may not have a strong relationship to the nature of the dataset. • Mahalonobis Distance: The Mds of a point is a measure of the distance of the point from the mean point of a cluster. It is used to find outliers in multivariate data and, unlike an absolute distance measurement, it takes into account the “shape” of the cluster. The Mds method is very sensitive to intervariable changes in the sample dataset. The Scout is a multivariate analysis package of the US-EPA (Singh, 1993) and can be used for computation of the Mds. One of the major functions of this package is outlier(s) detection and utilizes both the classical and the robust approaches to outlier analysis. • Chauvenet’s criterion: This criterion, based on the normal distribution, advises the rejection of an extreme observation if the probability of occurrence of such a deviant observation from the mean of the n measurements is less than ½ n. Obviously, the choice of ½ n is arbitrary, but is reasonable and can be validated (Taylor, 1984). Nevertheless, some analysts disagree with such a statement. Analysts at the U.S. National Bureau of Standards (Natrella, 1963) point out that such a criterion easily rejects when the sample size, n, is too small. • Extreme studentized: The extreme studentized method is similar to the Chauvenet’s criterion. Instead of using normal error integral distribution, it uses t-distribution. And it also assumes that extreme observations in either direction are considered rejectable. Population mean is unknown and standard deviation is assumed to be the sample standard deviation. The test procedure is to: (1) choose α (the probability [or risk]), a user is willing to accept in rejecting an observation that really belongs into the group. (2) Look up the q1 − α (n, ∞) value from the Standard Studentized Deviate Table with n being the number of observations in the sample, and v being the degrees of freedom. (3) Compute w = q1 − α * σ and (4) if (Xn − X1) > w, reject the observation that is a suspect; otherwise, retain it (Natrella, 1963). The use of the t-distribution, instead of the normal distribution, makes it more realistic in reflecting the use of the sample distribution rather than population distribution. This is especially true for cases where the sample size is less than 30. • Dixon method: The objective of the Dixon method (Dixon, 1951) is to investigate the significance of the difference between a suspicious extremal observation and other observations in the sample. The population that is being

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sampled is assumed to be normal. For this test procedure, consider a sample of size n, where the sample is arranged with the suspect value in front and followed by its nearest neighbor and then remainder values are arranged in ascending (or descending) order. The order is determined based on whether the suspect value is the largest or the smallest. The null hypothesis that the outlier belongs to the sample is rejected if the test statistic value exceeds the tabulated critical value (Kanji, 1993). The advantage of the Dixon method is that it does not consider the mean and standard deviation, which in some cases cannot be accurately obtained. This method is good for a small sample size, but its constraint is that the sample size should be greater than 3 and less than 25. For sample sizes larger than 25, the Dixon method may not be appropriate. • The US-WRC method: The US-WRC method is the most popular method used for flood frequency analysis in North America. The method is based on Grubb’s test and normality assumption. The test procedure for a sample size (n) is to obtain the value of deviate (Ko) at 10% significance level from the Table for Outlier Test Deviates (Ko) [Appendix 4 of Bulletin 17B (U. S. Interagency Advisory Committee, 1982)]. Compute the mean (Ỹ) and standard deviation (Sy) of the logarithms of observations including the extremal observation(s) being considered, but excluding zero-flood observations, peak flows below the gage base, and outliers (extremal observations) previously detected and identified. Compute the value of the detection criterion for high outliers (Yoh) as Yoh = Ỹ + Ko Sy. Compare the logarithm of the extremal observation(s) being considered (Yh) with the criterion (Yoh); if Yh > Yoh then the extremal observation can be considered as a high outlier (McCuen, 2004). • Consolidated Frequency Analysis method: In the CFA analysis package (Pilon et al., 1985), the identification of outliers is conducted based on the methods suggested by Grubbs and Beck (1972). Only high outliers are considered for evaluation because of their importance in flood frequency analysis. 16.2.4  Methods for Outliers Treatment

Often, a hydrologic data analyst is faced with a crucial question whether an observation is an outlier or not, and should such an observation be discarded from the dataset simply because it appears to be inconsistent with rest of the hydrologic data. The treatment of outliers is still under debate because of the user specific requirements and the varied nature of hydrological data sets. The principal argument of handling outliers has been presented by Rousseeuw and Leroy (1987), who proposed the use of robust regression methods, such as the least median of squares to cope with outliers in the dataset. On the other hand, Dixon (1951) has proposed the outlier removal approach. Irrespectively, there are suggestions that whenever outlier(s) is eliminated from the results, one should at least report outlier(s) along with results of the statistical analysis. Another remedial suggestion pertaining to outlier(s) treatment is to enlarge the sample size to aid in better assessing the nature of extremal observation(s). Moreover, an adequate decision on a specific treatment should be based on the results of analysis obtained from cases of the “with” and “without” outlier(s) in the dataset. In some cases, the removal or preservation of outliers in the dataset does not affect the design outcome. The International Organization for Standardization also recommends that outlier(s) should not be excluded purely based on statistical evidence, and further investigations for discrepancies are required. However, it is not always possible to know the exact cause of such discrepancies in individual data values. Basically, three approaches for the outlier treatment are available for consideration, such as outlier exclusion, outlier revision, and outlier inclusion. The exclusion of outliers is particularly selected when the data are suspect to contain erroneous observations. In some cases, outliers can be excluded because the observation exceeds the limit of measuring equipment. However, most of the time, the nature of observed outliers is still uncertain. Frequently, engineering practitioners tend to arbitrarily adopt the treatment of excluding outliers that can enhance the fitness to a specific model (e.g., linear regression and normal distribution). Some scientists or engineers may suggest a conservative treatment concept that data should not be excluded unless a reasonable doubt does exist. In that case, modification can be an approach to reduce the impact of extreme data and to facilitate the analysis. However, the degree of modification is hard to justify. The inclusion approach is to adopt only those statistical methods that are robust enough to manage the extreme data without being unduly biased by individual outliers. Such an approach is getting acceptance in hydrological analyses (e.g., L-moment and nonparametric methods). 16.3  CONCEPTS AND METHODS FOR HANDLING MISSING RECORDS

Conceptual understanding of a missing record satisfying the varying nature of its interpretations by various users/investigators is challenging because the significance and complexity of a missing record varies according to the

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16-4     Outlier Analysis and Infilling of Missing Records in Hydrologic Data

time-scale or space-scale usage of a data series. In hydrologic data, missing records have a common occurrence and can have a significant impact on the conclusions that can be inferred from the data. The existence of missing records is not uncommon whenever and wherever a hydrologic time series (or a set of observations) is recorded. Missing records in such time series represent a great challenge for statistical analysis as a complete dataset is normally required. Understanding the occurrence and nature of missing records, and statistical properties of the observed data are essential for the infilling of missing records. 16.3.1  Definition and Types of Missing Records

A missing observation or missing data (or record) in a particular spatial or temporal location for a variable signifies the nonexistence of a value either in numerical or any other format. The categorization of missing records into three basic types by Little and Rubin (1987) is well accepted in the literature, namely: (1) Missing Completely at random (MCAR): this is the highest level of randomness. It occurs when the probability of an instance having a missing value for an attribute does not depend on either the known values or the missing values. At this level of randomness, any missing records treatment method can be applied without risk of introducing bias in the database. (2) Missing at random (MAR): the probability of an instance having a missing value for an attribute depends on the known values and not on the value of the missing observation itself. That is, the missing observation exists due to some external influences, which can be explained by other measured variables in the study. The pattern of missing records is considered to be traceable or predictable. (3) Non-ignorable (NI): when the probability of an instance having a missing value for an attribute depends on the value of that attribute, then the pattern of missing observations is nonrandom and is not predictable from other variables in the database. Missing observations can only be explained by the very variable(s) for which the observation is missing. In fact, the MCAR assumption is not easy to meet. Most of the time, MAR is the only assumption being met in practical cases. Even such a widely accepted assumption may not always be true because missing values on a number of variables and the assessment of dependence on other variables may be considerably complex. The purpose of understanding the cause and/ or nature of the missing records is to facilitate the selection of a suitable method and ensure compliance of the selected method with the underlying assumptions. 16.3.2  Approaches for Handling Missing Records

There exist three primary approaches for handling missing records such as: (1) Omitting Missing Records: Procedures based on complete records are under this category. When some variables are not recorded, such an approach simply discards incomplete records and only analyzes the complete records. This is a relatively easy approach to carry out and deemed satisfactory with small amounts of missing records. However, it is not usually efficient because it is not able to fully utilize the available information and thus can lead to serious biases. (2) Infilling Missing Records: The missing records are routinely infilled and the resultant completed data are analyzed by standard methods. (3) Accommodating Missing Records: Based on a predefined model, a broad class of parameters estimation procedures is developed using the dataset containing missing records (without infilling). This approach is flexible and avoids the use of ad hoc methods. In hydrology, many studies concerning handling of missing records in streamflow or precipitation data rely on the association between predictor and predictant variables. Basically, missing records can be estimated based on assumptions related to probability distribution, spatial dependence, temporal dependence, or other forms of dependence. For instance, missing records can be estimated using the neighboring observations within a time series itself (Elshorbagy et al., 2000b) or the spatial observations from nearby sites (Ng et al., 2009). For infilling purposes, it is normally preferable to use the neighboring observations exhibiting higher correlation with missing records. For infilling missing records in daily streamflow data, the temporal dependence normally plays a significant role compared to the probability distribution of the dataset. It is because the behavior of streamflow evolves continuously in a time series and the strength of embedding autocorrelation is more apparent. Therefore, the missing records of one site can be estimated through correlation relationships of the univariate time series itself (i.e., through an autocorrelation model) or from corresponding data of a nearby stream(s), or of a gauging station located either upstream or downstream on the same river. A typical approach for infilling missing records may involve an estimation based on linear regression models, which may perform reasonably well in

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cases of highly correlated data with small gaps. However, in cases where estimation of missing records exceeds the allowable sequential correlation regime (or the dataset is seemingly random-like), then traditional estimation approaches may encounter difficulties for infilling missing records. Thus, overreliance on temporal correlations in the estimation process may not be appropriate. For infilling of missing records in daily precipitation data, the probability distribution and spatial dependence among nearby sites are normally more relevant than the temporal dependence of precipitation series. It is because the temporal dependence normally suffers from the adverse effect of temporal intermittence, which generally is more apparent than the adverse impact induced by the spatial intermittence. Therefore, the missing records of one site can be estimated through the correlative relationships expressed in parameters of a multivariate normal distribution model. The missing records can also be infilled or estimated through a conditional sampling obtained from the distribution model. Most streamflow infilling and/or of records extension procedures are based on ordinary least-squares (OLS) regression equations that are used to estimate individual missing values at stations with short-term records. OLS regression, unfortunately, results in extended records with smaller variances than the historical records. Techniques other than OLS regression that can be used to estimate missing values and that tend to preserve the variance of the historical records include regression plus noise (Matalas and Jacobs, 1964) and two alternatives to regression described by Hirsch (1982), which are referred to as MOVE-l and MOVE-2 (Maintenance of Variance Extension, Types 1 and 2). Alley and Burns (1983) developed a procedure capable of selecting any of the above mentioned procedures for infilling of missing streamflow records, which is routinely utilized by the United States Geological Survey (Parrett et al., 1989 and Vining et al., 1996.) The procedure selects the best base station from several base stations and also has an option that allows for consideration of both cyclic (monthly) and noncyclic (annual) extension equations for each individual prediction. With the cyclic (monthly) equation, missing data are estimated from streamflow data for the selected month; with the noncyclic (annual) equation, missing data are estimated from all concurrent streamflow data for the period of record. Specifically, the missing data for a particular month at a site of interest are estimated using both the cyclic (monthly) equation and the noncyclic (annual) equation, and a standard error of prediction is computed for both estimates. The equation that has the smallest standard error of prediction then is used to estimate the missing data. Similar methods including the kriging interpolation method have been utilized in Canada, among others by Ontario Ministry of Natural Resources (2008) for addressing the issues of missing records in climatic data. Some of the most commonly used methods for handling missing records are presented as follows. 16.3.3  Methods for Handling of Missing Records

There exist numerous methods for handling of missing records which may vary from ad hoc methods to involving linear or nonlinear regression approach. Such methods can be broadly grouped into two categories. The first category involves methods based on assumptions of normal probability distribution or temporal dependence in a univeriate time series. The methods in the second category are developed based on assumptions of spatial dependence among observations at multiple sites using multivariate time series. Some of the most commonly employed methods are briefly described as follows. Methods Based on the Assumption of Normal Probability Distribution or Temporal Dependence (Category-I) The following methods are commonly used for infilling of missing records in datasets exhibiting strong temporal dependence (e.g., daily streamflows data), or datasets complying the underlying assumption of normal distribution (e.g., annual maximum peak flow data). These methods are applicable to different observations of hydrologic variables as long as the nature and characteristics of variables obeying the underlying assumption. Additional detail on these classical statistical methods can be found elsewhere (Little and Rubin, 1987). • List-wise or case-wise data-deletion method: This method omits the entire missing record from the analysis. Such an operation may result in reduction of the sample size for analysis, but such an approach is generally acceptable if the dataset is large enough and the number of missing records is relatively small. • Pair-wise data-deletion method: This method computes the statistics based on the available pair-wise data, such as that used in cases of bivariate correlation or covariance. Similar concerns as described for the list-wise deletion are also applicable in this method. • Mean substitution method: This method substitutes a mean, mode, or median value computed from available cases to infill missing records on the

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Discussion and Concluding Remarks on methods for outliers and infilling of missing records      16-5 

remaining cases. This method is simple, but may underestimate the variance of the selected variable. Also, in a trendy time series, this method may miss a number of valuable coherent signals around the missing records. • Last value/observation carried forward method: This is a simple method as its name implies, but may only be reasonable when the data set is consistent and stable. • Hot deck imputation method: This method identifies the case that is most similar to that with a missing record and substitutes the most similar known case value for the missing record. • Linear model: An equation is developed based on the available data for a given variable. An improvement of this method may involve uncertainty in the imputed record. The model can be static or dynamic (e.g., ARIMA, exponential moving average, OLS model, etc.). This method is acceptable in cases of trendy series or in the existence of a significantly clear temporal pattern. However, with cases of noisy data, infilling may be difficult and a nonlinear model may be required. • Nonlinear model: Mathematical models where the exponent is not equal to 1 [e.g., artificial neural networks (ANNs), etc.] are used. • Expectation Maximization (EM) algorithm: The EM algorithm is an iterative procedure with two discrete steps. In the expectation (E) step, one computes the expected values of the log-likelihood of the available data. In the maximization (M) step, one substitutes the expected values for the missing records obtained from the E step, and then maximizes the likelihood function as if no data were missing when obtaining the new parametric estimates. The procedure iterates through these two steps until convergence occurs. • Data-segmentation method: This method groups data into segments based on assigned common patterns or criteria and then substitutes the missing records value using statistical information (e.g., group mean) of the segment. • Multiple imputation approach: Instead of the single imputation approach, the multiple imputations approach is gaining increasing attention and acceptance because the results of such analyses are further extendable to other statistical analyses. One advantage of the multiple imputations approach is its ability to generate more than one dataset for imputation. Another advantage is that it does not underestimate the variability of the infilled records, which is a common problem with the single imputation approach. • k-Nearest Neighbor method: It is one of the simplest methods to infill missing records through selecting k number of nearest neighbors (i.e., candidates) for the predictor. One of the randomly drawn candidates becomes the observation to infill missing record(s). The nearest neighbor can be selected either geometrically or by taking sites with highest correlation to the target site. Methods Based on the Assumption of Spatial Dependence (Category-II) The following methods are commonly used for infilling of missing records in datasets exhibiting strong spatial dependence (e.g., precipitation data at multiple sites within a homogenous region). Additional detail on the methods can be found elsewhere (ASCE, 1996; Teegavarapu, 2012). • Thiessen method: The Thiessen method (1911) is for approximating the distribution area around precipitation gages for the purpose of distributing average precipitation depths over an area. A Thiessen polygon network is manually constructed using perpendicular bisectors to lines between gages and carefully removing overlapping bisectors until an even spatial distribution is obtained. Computerized versions of this method are available (Diskin, 1969; Shih and Hamrick, 1975; Croley and Hartmann, 1985). • Isohyetal method: The Isohyetal method is still a useful and fairly accurate technique for determining the spatial distribution of precipitation (France, 1985). An isohyetal map is constructed by plotting the stations and their appropriate precipitation values on a topographic map of the area. A user judgement decides on the placement of isohyetal contours affected by topography within the area. Estimated average precipitation is computed by the weighted average for the subject area. Hamlin (1983) demonstrated that the number of stations and the skill of the analyst make a difference in the resolution of isohyetal maps. • Linear and multiple regression methods: These methods offer a straightforward and useful technique in extrapolating precipitation records of an area with incomplete records. Regression relationships between two or more stations are developed with the existing data. Their relationships are then used to infill missing records. Salas (1993) provided an excellent discussion on criteria for improving estimators of parameters in details. • Polynomial interpolation method: This method (Tabios and Salas, 1985) utilizes a polynomial function fitted to the stations. A least squares statistical treatment resolves the weighting of a station based on coordinates of the station.

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• Kriging technique: Kriging is a spatial interpolation technique that arrives at interpolated surfaces with minimum error. The technique is used regularly in geotechnical analysis and also utilized in hydrologic analysis, among other by Yates et al. (1986), Dingman et al. (1988), and Seo et al. (1990). The advantage of kriging over OLS techniques is that the estimated values at observation sites are equal to the actual measurement (Gebhardt et al., 1988). In kriging technique, the precipitation at a site is considered as a function of a predictable trend and a random component. Kriging may treat the trend as a constant (simple kriging), or may describe the trend as a polynomial (universal kriging). A difference in variance of measurements between rain gages is related to the distance between rain gages, called a semivariogram. The direction (of the distance) is assumed not to affect rain gage values. The above mentioned categories are a typical classification of different missing records handling methods/techniques. Some of the methods stated earlier can fall into more than one category. The selection of a method should still be determined in view of the nature of missing records and the type of data structure. For example, the selection of a method may remain inconclusive for estimating missing daily precipitation due to varied correlations arising from considerations of neighboring observations, the period and the length of the dataset with missing records, the distance between sites, and the impact of temporal/spatial intermittence. 16.4  DISCUSSION AND CONCLUDING REMARKS ON METHODS FOR OUTLIERS AND INFILLING OF MISSING RECORDS

There exists numerous methods for the consideration; however, the selection of a proper method to handle the problem of outliers or missing records would be a challenge as many criteria may have to be considered. For instance, the selection of an infilling method depends on criteria, such as the record length (observed and missing), the presumed type of missing data, the structure of the observed data (any spatial and temporal dependence established), and the intent of the use (i.e., effectiveness, efficiency, robustness, accuracy, precision, and parsimoniousness). As this chapter intends to serve as a comprehensive material for the target readers of the handbook, a broad discussion relevant to foregoing considerations is presented in this section; which explores the relevant topics related to outliers and missing records along with the outstanding problems concerning the existing methods. 16.4.1  Outlier Detection and Treatment

The existing methods for outlier detection and treatment as described earlier perform reasonably in handling outlier problems. However, these methods do have limitations in their application because either the dataset does not meet the underlying assumptions of the method or the method itself has inherent problems, such as conceptualization, mathematical or computational shortcomings. To overcome such shortcomings, many investigators endeavor to explore the use of alternative methods. In the case of univariate hydrologic data, outlier detection and treatment have received reasonable attention. However, with multivariate hydrologic data, this has not been the case, despite the fact that multivariate analysis is well established in statistics. Recently, a pioneering attempt in this direction has been reported by Chebana and Ouarda (2011). It was found that outliers have a serious effect on the modeling of hydrologic data in multivariate settings. The identification of outliers has always been a difficult task to undertake due to the existence of a few consistent identification procedures and guidelines for application to water resources as well as many other domains related to the environment and natural resources. There is a need to identify procedures for outliers especially due to the availability of increasing amount of data in a variety of forms. 16.4.2  Infilling of Missing Records

The commonly used techniques for the estimation of missing data in hydrology, such as streamflows, are based on regression analysis, time series analysis, ANNs, and interpolation techniques. Despite progress in the development of methods/techniques for infilling of missing records, some issues remain largely unsettled. Numerous techniques of data infilling can be primarily classified into two classes: statistical methods and computer intelligence methods. Regardless of the common usage of statistical methods in handling missing records, there has been a slow, but steady shift toward computational intelligence methods even though these methods will take time to attain a higher accuracy. Without detailed knowledge of the processes under investigation, these types of techniques develop functional relationships between

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16-6     Outlier Analysis and Infilling of Missing Records in Hydrologic Data

input and output data by finding an optimum set of network parameters through the training process. ANNs have become one of the most promising techniques for infilling of missing records (Panu et al., 2000). Invariably, the ANN-based techniques have been shown to be superior compared to the traditional statistical methods for infilling missing records. 16.4.3 Limitations of Existing Methods for Outliers and Infilling of Missing Records

A review of limitations of various methods utilized in handling of outliers and missing records is summarized succinctly in general terms as follows. • Univariate analysis: In many instances, it is understood that more than one independent variable is correlated with the dependent variable and consequently effect and/or contribute to resulting value(s) of the dependent variable. Univariate analysis may be simple to implement, but it may underestimate the complexity of a system through the omission of impacts of all contributing factors. • Linear regression approach:  Linear regression approaches used for the outlier analysis or handling missing records have proven to be a very simple but valuable tool for modeling and thus are widely used in all disciplines. However, such an approach may be relatively inflexible and restrictive in modeling data with high variability both in time and space. • Parametric approach: Many of the methods related to outlier detection and infilling of missing records are based on the parametric approach. This approach does not effectively deal with the variability in the data, and hence the assumption corresponding to a parametric approach is subjectively selected requiring justification and validity. • Single-valued approach: The single-valued approaches ignore information contained in and among groups of data. A few published reports (Panu et al., 1978; Unny et al., 1981) are available in hydrologic literature that discuss and describe the use of concepts of extracting information from data groups. • Ad hoc analysis: Ad hoc analysis may be able to address a single and specific problem, but its usefulness may be limited because it lacks diversity in application. A general and robust analysis method is therefore required. 16.4.4  Handling Outliers or Infilling Missing Records First?

Which operation should come first, the outlier analysis or handling missing records? The answer to this dilemma is still inconclusive because the result of each operation may or may not affect the operation of the other. However, four potential scenarios may occur: (1) the results of these two operations (i.e., outlier detection and infilling of missing records) do not affect each other, proceeding with either of the operations first is not a concern; (2) the existence of missing records does not affect the outlier analysis, but not the other way around, the outlier analysis may proceed first; (3) the outlier does not affect the handling of missing records, but not the other way around, the missing records may proceed first; and (4) the results of these two operations may affect each other, thus performing whichever operation first may bias the result of analysis. One of the methods to manage the problem is through alternatively conducting each operation iteratively until a stable result is obtained. 16.5 FUTURE RESEARCH DIRECTIONS

Toward addressing some of the outstanding problems mentioned earlier in Sec. 16.4.3, several potential future research directions are envisioned. Since the subject matter related to missing data and outliers are rather diverse, the future research directions are presented through a broad discussion of relevant issues and concerns. • Multivariate analysis: Multivariate techniques allow researchers to investigate relationships between variables in an overarching way and to quantify the relationship between variables. They can control association between variables by using cross tabulation, partial correlation and multiple regressions, and introduce other variables (e.g., output of a radar or a regional climate model) to determine the links between the independent and dependent variables or to specify the conditions under which the association occurs. This will enhance our understanding and interpretations of the relationship among data and would allow for more powerful and robust tests of significance compared to univariate analysis. • Use of nonlinear regression and non-normal distribution: To address the need for flexibility in analysis, the nonlinear regression approach may be required. The advantages of using a nonlinear approach utilizing ANN for handling missing streamflow records have been demonstrated among others by Panu et al. (2000) and Khalil et al. (2001). Also, Bardossy and Pegram (2013) proposed a method for estimating missing records using a non-normal distribution based on copulas. • Nonparametric approach: This approach is gaining interest through the k-Nearest neighbor for infilling of missing records (Elshorbagy et al., 2000;

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Ng et al., 2007; and others). Another nonparametric approach based on the Generalized Regression Neural Network (Ng et al., 2007) has also been used for infilling of missing records with reasonable success. • Grouping approach: Proper grouping may help generalize relationships within grouped elements and also help to specialize relationships among grouped elements. Grouping may not be able to specialize the accuracy within groups, but the generalization of the infilling process may help develop more robust mechanisms for handling data sets containing outliers. In general, grouping appears everywhere. Some group formations may simply be arbitrary and hence not justifiable either hydrologically or physically. Other group formations may be physically valid, such as grouping into four seasons. Therefore, rational and systematic data organization (e.g., clustering, partition, and data segmentation) is necessary for meaningful and successful data infilling analyses. The concept of Grouping has been presented by Panu et al. (1978). Some studies in relation to this concept have also been adopted for the simulation of precipitation approach using the dictionary formulations (Ng and Panu, 2010). For example, one of the grouping approaches utilizes the concepts of chaos theory in the development of infilling technique for the missing records in streamflow data (Elshorbagy et al., 2002; Ng et al., 2007; and others). Recently, the utility and benefits of considering groups in hydrologic datasets for infilling of missing records have been specified by Harvey et al. (2012). • Accommodation analysis: Accommodation analysis refers to methods of handling outliers or missing records that require neither detection of outliers nor infilling of individual missing records. The methods under this category are considered robust enough to handle problematic data directly without the need of a single or specific address to the problematic observation or sequence of observations. For example, some studies utilize L-moments to handle data with outliers without detection. Also, users can directly estimate parameters of a distribution using likelihood estimation without the need for infilling the missing records. • Implication of climate change in outlier analysis or infilling of missing records: Global climate change directly affects regional precipitation patterns that subsequently change streamflows, and there is a good correspondence between local precipitation/streamflow patterns and atmospheric circulation (Stehlik et al., 2002). It is envisioned that additional considerations relative to outlier analysis or handling of missing records are required due to global climate change. For example, since more extreme events are envisioned to occur in the future due to climate change, some outlier detection methods may become oversensitive in identifying outliers, if more frequent extreme events are anticipated to occur in the future. For infilling of missing data, multiple imputation techniques may become more attractive due to their increased ability to handle the higher uncertainty anticipated to occur in hydrologic data under global climate change. In addition, the development of suitable techniques involving the downscaling approach of the variable of interest at specific time scale is likely to be advantageous in infilling of missing records. • Data mining, machine learning, and computer intensive computations: With the advancement of computing technology over the past decades, hydrological analyses have also gone through a significant change. Improved computing hardware and data collection techniques have produced a large data base available for analysis. Data mining plays an important role in using and interpreting such data. Many machine learning algorithms have also been developed; for example, ANN and Genetic Algorithm techniques are gaining interest among many researchers, scientists, and practitioners in handling missing records and outlier-related problems. As computing becomes faster and more efficient, some studies that would not have been feasible earlier, have now become possible due to the availability of high capacity operational systems. REFERENCES

Aggarwal, C. C., Outlier Analysis, Springer, New York, 2013. Alley, W. M. and A. W. Burns, “Mixed-station extension of monthly streamflow records: American Society of Civil Engineers,” Journal of Hydraulic Engineering, 109 (10): 1272–1284, 1983. ASCE Task Committee on Hydrology Handbook of Management Group D, Hydrology Handbook (ASCE Manuals on Engineering Practice No. 28, 2nd ed.), ASCE, New York, NY, USA, 1996. Bardossy, A. and G. G. Pegram, “Infilling missing hydrological data-methods and consequences,” AGU Fall Meeting Abstracts, San Francisco, California, USA , 2013, p. 1. Barnett, V. and T. Lewis, Outliers in Statistical Data, John Wiley & Sons, New York, 1978. Beckman, R. J. and R. D. Cook, “Outliers with discussion,” Technometrics, 25(2): 161–163, 1983.

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REFERENCES    16-7 

Box, G. E. P. and G. M. Jenkin, Time Series Analysis: Forecasting and Control (Revised edition in 1986), Campbell, San Francisco, CA, 1976. Chebana, F. and T. B. M. J. Ouarda, “Depth-based multivariate descriptive statistics with hydrological applications,” Journal of Geophysical Research, 116: D10120, doi: 10.1029/2010JD015338, 2011. Croley, T. E. II and H. C. Hartmann, “Resolving Thiessen polygon,” Journal of Hydrology, 76(1): 363–379, 1985. Dingman, S. L., D. M. Seely-Reynolds, and R. C. Reynolds, “Application of kriging to estimating mean annual precipitation in a region of orographic influence,” JAWRA, 24(2): 329–339, 1988. Diskin, M. H., “Thiessen coefficients by a Monte Carlo procedure,” Journal of Hydrology, 8(3): 323–335, 1969. Dixon, W. J., “Ratios involving extreme values,” Annals of Mathematical Statistics, 22(1): 68–78, 1951. Elshorbagy, A. A., U. S. Panu, and S. P. Simonovic, “Group-based estimation of missing hydrological data: I. Approach and general methodology,” Hydrological Sciences Journal, 45(6): 849–866, 2000. Elshorbagy, A. A., U. S. Panu, and S. P. Simonovic, “Group-based estimation of missing hydrological data: II. Application to streamflows,” Hydrological Sciences Journal, 45(6): 867–880, 2000a. Elshorbagy, A. A., S. P. Simonovic, and U. S. Panu, “Estimation of missing streamflow data using principles of chaos theory,” Journal of Hydrology, 255(1): 123–133, 2002b. Gandin, L. S., “Objective analysis of meteorological fields (Ed. Hardin, R.),” Jerusalem: Israel Program for Scientific Translations, 242, 1965. Reproduced by the Royal Meteorological Society, UK, Quarterly Journal, 92(393): 447, 1966. Gebhardt, K., C. Bohn, and N. Gordon, “Estimating daily precipitation in the northern Great Basin,” Saval Ranch Research and Evaluation Project Final Report, Bureau of Land Management, Reno, NV, 1988, 4 (25), pp. 4–25. Grubbs, F. E., “Procedures for detecting outlying observations in samples,” Technometrics, 11: 1–21, 1969. Grubbs, F. E. and G. Beck, “Extension of sample sizes and percentage points for significance tests of outlying observations,” Technometrics, 14(4): 847–854, 1972. Gupta, M., J. Gao, C. Aggarwal, and J. Han, “Outlier detection for temporal data: a survey,” IEEE Trans. on Knowledge and Data Engineering, 25(1): 1–20, 2014. Hamlin, M. J., “The significance of rainfall in the study of hydrological processes at basin scale,” Journal of Hydrology, 65(1–3): 73–94, 1983. Hawkins, D. M., Identification of Outliers, Chapman and Hall, London, 1980. Harvey, C. L., H. Dixon, and J. Hannaford, “An appraisal of the performance of data-infilling methods for application to daily mean river flow records in the UK,” Hydrology Research, 43(5): 618–636, 2012. Hirsch, R. M., “A comparison of four streamflow record extension techniques,” Water Resources Research, 18(4): 1081–1088, 1982. Hu, S., “Problems with outlier test methods in flood frequency analysis,” Journal of Hydrology, 96(1): 375–383, 1987. Iglewicz, B. and D. Hoaglin, “How to detect and handle outliers,” American for Quality, Milwaukee, WA, 1993, Vol. 16. Kanji, G. K., 100 Statistical Tests, SAGE, London, 1993. Khalil, M., U. S. Panu, and W. C. Lennox, “Groups and neural networks based streamflow data infilling procedures,” Journal of Hydrology, 241 (3): 153–176, 2001. Klemeš, V., “Tall tales about tails of hydrological distributions,” Journal of Hydrologic Engineering, 5 (3): 227–231, 2000. Kottegoda, N. T., “Investigation of outliers in annual maximum flow series,” Journal of Hydrology, 72 (1–2): 105–137, 1984. Little, R. J. A. and D. B. Rubin, Statistical Analysis with Missing Data, Wiley, New York, 1987. Matalas, N. C. and B. A. Jacobs, “A correlation procedure for augmenting hydrologic data, U.S. Geological Survey Professional Paper 434-E, 1964, p. 7, USGS, Reston , Virginia, USA. McCuen, R. H., Hydrologic Analysis and Design, 3rd ed., Prentice Hall, Upper Saddle River, New Jersey, 2004. Montgomery, D. C. and G. C. Runger, Applied Statistics and Probability for Engineers, Wiley, New York, 1994. Muthén, B., D. Kaplan, and M. Hollis, “On structural equation modeling with data that are not missing completely at random,” Psychometrika, 52 (3): 431–462, 1987. Natrella, M. G., Experimental Statistics, National Bureau of Standards Handbook 91, U.S. Government Printing Office, Washington, D.C., 1963. Ng, W. W., U. S. Panu, and W. C. Lennox, “Chaos based analytical techniques for daily extreme hydrological observations,” Journal of Hydrology, 342 (1): 17–41, 2007.

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Ng, W. W., U. S. Panu, and W. C. Lennox, “Comparative studies in problems of missing extreme daily streamflow records,” Journal of Hydrologic Engineering, 14 (1): 91–100, 2009. Ng, W. W. and U. S. Panu, “Comparisons of traditional and novel stochastic models for the generation of daily precipitation occurrences,” Journal of Hydrology, 380 (1): 222–236, 2010. Ontario Ministry of Natural Resources, Climate Data Gap Filling Project, Government of Ontario, Toronto, 2008, p. 37. Panu, U. S., T. E. Unny, and R. K. Ragade, “A feature prediction model in synthetic hydrology based on concepts of pattern recognition,” Water Resources Research, 14 (2): 335–344, 1978. Panu, U. S., M. Khalil, and A. Elshorbagy, “Streamflow data infilling techniques based on concepts of groups and neural networks,” Artificial Neural Networks in Hydrology, Springer, The Netherlands, 2000, Vol. 36, pp. 235–258. Parrett, C., D. R. Johnson, and J. A. Hull, “Estimates of monthly streamflow characteristics at selected sites in the upper Missouri river basin, Montana: base period water year 1937–1986,” U.S. Geological Survey, Water-Resources Investigations Report 89–4082, 1989, p. 109, USGS, Reston, Virginia, USA. Pegram, G., “Patching rainfall data using regression methods. 3. Grouping, patching and outlier detection,” Journal of Hydrology, 198 (1–4): 319–334, 1997. Pilon, P. J., R. Condie, and K. D. Harvey, Consolidated Frequency Analysis Package—CFA—User Manual. Water Resources Branch, Inland Waters Directorate, Environment Canada, Ottawa, Ontario, Canada, 1985. Rousseeuw, P. J. and A. Leroy, Robust Regression and Outlier Detection, Wiley, New York, 1987. Salas, J. D., “Analysis and modeling of hydrologic time series,” Handbook of Hydrology, edited by D. R. Maidment, McGraw-Hill, New York, 1993, pp. 19.1–19.72. Seo, D. J., W. F. Krajewski, and D. S. Bowles, “Stochastic interpolation of rainfall data from rain gages and radar using cokriging: 1. Design of experiments,” Water Resources Research, 26 (3): 469–477, 1990. Shih, S. F. and R. L. Hamrick, “Modified Monte Carlo technique to compute Thiessen coefficients,” Journal of Hydrology, 27 (3–4): 339–356, 1975. Singh, A., “Omnibus robust procedures for assessment of multivariate normality and detection of multivariate outliers,” Multivariate Environmental Statistics, North-Holland, Amsterdam, 1993. Singh, D. P., Flood frequency modeling and outliers. Organic Geochemistry, ASCE, New York. 1980. Singh, K. P. and M. Nakashima, A new methodology for flood frequency analysis with objective detection and modification of outliers/inliers, Illinois Institute of Natural Resources, State Water Survey Division, SWS Report # 272, 1981, Urbana-Champaign, Illinois, USA. Spencer, C. S. and R. McCuen, “Detection of outliers in Pearson type III data,” Journal of Hydrologic Engineering, 1: 2–10, 1996. Stehlik, I., J. J. Schneller, and K. Bachmann, “Immigration and in situ glacial survival of the low-alpine Erinus alpinus (Scrophulariaceae),” Biological Journal of the Linnean Society, 77: 87–103, 2002. Tabachnick, B. G. and L. S. Fidell, Using Multivariate Statistics, Harper Collins College Publishers, New York, 1996. Tabios, G. Q. and J. D. Salas, “A comparative analysis of techniques for spatial interpolation of precipitation,” Water Resources Bulletin, 21(3): 365–380, 1985. Taylor, S. J., “Estimating the variances of autocorrelations calculated from financial time series,” Applied Statistics, 33 (3): 300–308, 1984. Teegavarapu, R. S. V., Floods in a Changing Climate: Extreme Precipitation,” Cambridge University Press, Cambridge, UK, 2012. Thiessen, A. H., “Precipitation for large areas,” Monthly Weather Review, 39 (7): 1082–1084, 1911. U.S. Interagency Advisory Committee on Water Data, Guidelines for Determining Flood Flow Frequency, Bulletin 17-B of the Hydrology Subcommittee: Reston, Virginia, U.S. Geological Survey, Office of Water Data Coordination, 1982, p. 183. Unny, T. E., U. S. Panu, C. D. McInnes, and A. K. C. Wong, “Pattern analysis and synthesis of time dependent hydrologic data,” Advances in Hydrosciences, Academic Press, New York, Vol. 12, 1981, pp. 222–244. Vining, K. C., D. R. Johnson, and C. Parrett, “Synthesis of monthly natural flows for selected sites in the Musselshell River Basin, Montana, base period 1929–1989,” U.S. Geological Survey, Water-Resources Investigations Report 96–4094, 1996, p. 43, USGS, Reston, Virginia, USA. World Meteorological Organization (WMO), Report No. 3: The Beaufort scale of wind force, Reports on Marine Science Affairs, WMO, Geneva, 1970. Yevjevich, V., Probability and Statistics in Hydrology, Water Resources Publication, Fort Collins, CO, 1972.

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Chapter

17

Linear and Nonlinear Regression BY

KUMER PIAL DAS

ABSTRACT

17.1.2  Assumptions of Linear Regression

Regression analysis is one of the most commonly used statistical methods of analyzing hydrologic data. This analysis is used to detect a relation between the values of two or more variables, of which at least one of them is independent in nature. The objective is to test whether there is any statistically significant relation exist among dependent variable and independent variable(s). For example, regression analysis can be used for detecting relations between crop growth and water table depth or runoff and rainfall data. This chapter describes, with example, simple linear regression model, bivariate regression model, multiple regression model, and nonlinear regression model. It also provides example related to hydrology and the steps of using MS Excel for analyzing hydrologic data.

Let us take a look at the following assumptions of linear regression: • A linear relationship exists between dependent and independent variables. There are a number of ways to check whether a linear relationship exists between two variables. However, creating a scatterplot is the most convenient way of checking linearity. Steps of creating a scatterplot using Microsoft Excel is straightforward and we can create a linear regression line (trend line) as well. When the chart window is highlighted, a Chart menu appears. From the Chart menu, we can add a trend line to the chart. If the relationship displayed in your scatterplot is not linear, an execution of linear regression is not recommended. Alternative approaches are: nonlinear regression or polynomial regression. • Both dependent and independent variables should be measured at the continuous level (i.e., they are either interval or ratio variables). • Statistical independence of the errors means that all errors are independent of one another. • Homoscedasticity of the errors can be interpreted as variances of the errors constant over the range of x (e.g., in the case of time series data, it would be constant over time), versus the predictions and any independent variable. If the variance changes over a range of x, which is known as heteroscedastic, we need to transform the response variable and possibly the predictor variables to achieve a linear and homoscedastic relationship. • Approximate normality of the error distribution. In other words, we assume that εi ∼ N (0,σ 2 ). Two common methods of checking this assumption are: (a) a histogram (with a superimposed normal curve) or (b) a normal P–P plot. It should be noted that the most common model used in hydrology is based on the aforementioned assumptions of a simple linear regression model.

17.1  LINEAR AND NONLINEAR REGRESSION

Regression is perhaps the most popular form of statistical analysis. It can be used for two purposes: explanation and prediction. The use of regression analysis (both linear and nonlinear) in hydrology is obvious and evident because in numerous occasions, we wish to investigate how the changes in one or more variable(s) affect each other. These variables may be linked by an exact straight line relationship or by a nonlinear relationship. This chapter briefly discusses these relationships that are used in hydrology. 17.1.1  Simple Linear Regression Model

In simple linear regression model, there is only one predictor variable and the regression function is linear. The model is concerned with relationships of the form: Response variable = Model function + Random error In mathematical terms we write, yi = β0 + β1xi + εi , i = 1, 2,…, n  (17.1) where 1. yi and xi are the values of the response or dependent variable (y) and predictor or independent variable (x) in the ith trial, respectively. 2. β0 and β1 are the unknown parameters and usually known as regression coefficients. In particular, β0 is the y-intercept (the point where the regression line crosses the y-axis)   and β1 is the slope of the line (the change in outcome variable y with a unit change in the predictor variable x ). 3. Finally, it is assumed that   εi is a random error term which is independent of xi .  Moreover, E(εi ) = 0, and  V (εi) =  σ 2 . The standard deviation of εi is constant over the whole range of variation of xi , this property is called “homoscedasticity.” Also, E(εiε j ) = 0 (for all i , j ; i ≠ j , i , j = 1, 2,..., n) which states that the random errors are uncorrelated. Since the model (17.1) is linear in regard to the parameters β0 and β1 , it is called a simple linear regression model. Unless a model is specifically called nonlinear, it can be considered linear in its parameters, and the word linear is usually omitted. The value of the highest power of a predictor variable in the model is called the order of the model. Note that the model (17.1) is linear in the independent variable x too and that is why it is also called a first-order linear model.

17.1.3  Correlation Coefficient

Correlation measures the dependability of the relationship, it is a measure of how well one variable can predict the other and it is important to know the correlation coefficient before we start fitting the regression model. The Pearson (or product moment) correlation coefficient is denoted by r and it is a measure of linear association. The value of r does not depend on which of the two variables under study is labeled x and y . It is also independent of the units in which x and y are measured. The value of r is always in between -1 and +1 [r = 1 can happen if and only if all (xi , yi ) pairs lie on a straight line with positive slope and r = −1 is the case when this happens with negative slope]. Datasets that follow some nonlinear monotonic function will exactly have r < 1. The square of the sample correlation coefficient provides the value of the coefficient of determination (r 2 ) that would result from fitting the simple linear regression model. The correlation coefficient r can form the basis of a statistical test of independence where the null hypothesis is that yi ’s are not dependent from the xi ’s and are identically and independently distribr n−2 uted normal random variables. The test statistics t is defined as = . The 1− r 2 null hypothesis is rejected if t > tα ,n−2. 2

17-1

17_Singh_ch17_p17.1-17.10.indd 1

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17-2     Linear and Nonlinear Regression

to be in standard form because we get b0 and b1 only on the left side of the following normal equations:

17.1.4  Fitting the Regression Line

The n pairs of measurements (xi , yi ), i = 1, 2,…, n can be plotted as points in the Cartesian plane. This will result in a scatter diagram. A generic scatter plot is displayed in Fig. 17.1. The parameters β0 and β1 are somewhat in the nature of population parameters which can never be known exactly and must be estimated by their corresponding statistics denoted by b0 and b1. One widely used method known as ordinary least squares (OLS) is a statistical technique which attempts to find the function which most closely approximates the data (a “best fit”). In other words, it is an approach to fitting a model to the observed data. In technical terms, the OLS is used to fit a straight line through a set of data points, so that the sum of the squared vertical distances (called residuals) from the actual data points is minimized. From Eq. (17.1) the sum of squares of deviation from the true line, also known as the error sum of squares, can be written as: n

n

i =1

i =1

Q ( β0 , β1 ) = SSε = ∑εi 2 = ∑( yi − β0 − β1xi )2 

n (b0 ) + nX (b1 ) = nY , and nX (b0 ) + ∑ Xi2 (b1 ) = ∑ XiYi  By solving the two equations in Eq. (17.3) for b0 and b1, we get b0 = Y − b1 X  where n

2



(17.4)

Here, SSXY is called the corrected sums of cross product and SSXX is called the corrected sum of squares. Given the estimators b0 and b1, the fitted model is

(17.2)

yˆ = b0 + b1x = y − b1x + b1x = y + b1 ( x − x ) 

(17.5)

The aforementioned estimation procedure is called regressing y upon x and Eq. (17.5) is called the fitted regression line or regression equation. Since equality of all xi   is not a possibility, it can be claimed that there will be a unique solution with respect to (b0 , b1 ). Example 1 (Rainfall and Runoff) Consider a data set of rainfall and runoff information obtained from Pee Dee River, South Carolina, USA. A sample of annual data are shown as follows. We will construct a scatter plot for these data and find the fitted regression line for them.

∂Q(β0 , β1 ) ∂Q(β0 , β1 ) = 0 and = 0 ∂β 0 ∂β1 bb bb 0 1

0 1

∑ i=1XiYi − nXY = ∑ i=1( Xi − X )(Yi − Y ) = SSXY n n SSXX ∑ i=1Xi2 − nX ∑ i=1( Xi − X )2 n

b1 =

The method begins with assumed values for b0 and b1 and suppose that the relation between the independent and dependent variable is given by y = b0 + b1xi; some b0 ’s and b1’s will be better fits than others. Let yˆi = b0 + b1xi   be the value of the y  estimated by the regression equation when x has the value xi ; then the difference between the actual value (yi ) and estimated value ( yˆi ) is called the residual or the error denoted by ei  (Fig. 17.1). Mathematically, ei = yi − yˆi = yi − b0 − b1xi . As expected, different b0 ’s and b1’s will cause each ei to have a different value. The OLS method requires that we choose our estimates b0  and b1,  so that when substituted for β0  and β1 in Eq. (17.2), we get the least possible value of SSε . We can determine b0 and b1  by using calculus. More specifically, we differentiate Eq. (17.2) twice, first with respect to β0  and then with respect to β1   and set those results equal to zero. In other words,

(17.3)

Table 17.1  Eighteen Observations of Rainfall and Runoff Data

These two partial derivatives produce the following two equations: n

−2∑( yi − b0 − b1xi ) = 0 

Year 1966 1967 1968 1969 1970 1971

i =1 n

−2∑ xi ( yi − b0 − b1xi ) = 0  i=1

After some algebraic development, we get the following set of equations popularly known as normal equations for estimating β0  and β1 . They are said

Rainfall Runoff (X) (in) (Y) (cfs) 57.81 50.89 45.14 28.28 57.99 73.06

668 692 760 792 505 2010

Year 1972 1973 1974 1975 1976 1977

Rainfall Runoff (X) (in) (Y) (cfs) 46.76 54.46 46.82 54.12 53.35 46.5

915 988 624 1740 541 770

Year 1978 1979 1980 1981 1982 1983

Rainfall Runoff (X) (in) (Y) (cfs) 36.56 56.49 37.3 54.43 65.2 55.35

776 1170 1160 510 1360 984

(xn, yn) (xn, y^n)

(xi, yi) ei = yi – y^i

y

(xi, y^i)

y^ = b0 + b1x b0 = y + b1x ∑(x – x–)(yi – y–) b1 = i ∑(xi – x–)2

(x3, y3) (x1, y^1) (x1, y1) x1

x2

x3

xi

xn x

Figure 17.1  Scatter plot of n pairs of measurements ( xi , yi ), i = 1, 2,…, n .

17_Singh_ch17_p17.1-17.10.indd 2

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Linear and Nonlinear Regression     17-3  17.1.5  Creating a Scatter Diagram with Excel

A scatter diagram of the aforementioned dataset is displayed in Fig. 17.2. Scatterplot [runoff against rainfall (1966–1983)] 2500

The left term of the aforementioned equation is the total variation in the response Y , the first term in the right side is the variation in mean response, and the second term is the residual value. In other words, the aforementioned equation quantifies the basic regression line concepts, Data = Fit + Residual. If we square both sides of this and sum from i = 1,2,..., n, we obtain

2000

n

n

n

i=1

i=1

i=1

∑( yi − y )2 = ∑( yˆi − y )2 + ∑( yi − yˆi )2



(17.7)

Runoff (cfs)

n

It can be shown that the cross-product term, CPT = 2∑( yˆi − y )( yi − yˆi )

1500

i=1

becomes zero. The aforementioned equation may also be written as SST = SSM + SSE , where, as before, SS stands for sum of squares and T, M, and E are notations for total, model, and error (residual), respectively. For simple linear regression, the mean square model (MSM ) is defined as MSM =

1000 500 0

n

0

10

20

30

40 50 Rainfall (in)

60

70

80

2 SSM ∑i=1( yˆi − y ) = , where degrees of freedom, DFM = 1, because in the DFM 1 model, there is one explanatory variable, x . Similarly, the corresponding mean n

From the data, we have, n = 18, x = 51.14, y = 942.5,  Sxx = ∑( xi − x )2 = 104.09,

2 SSE ∑i=1( yi − yˆi ) = which is nothing DFE n−2 but the estimate of the variance about the population regression line (σ 2 ). Note that, each sum of squares is associated with a number known as its DF which indicates how many independent pieces of information involving the n independent numbers y1 ,  y 2 ,...,  yn are needed to compile the sum of squares.

and Sxy = ∑( xi − x )( yi − y ) = 1797.96.

For example, the SS about the mean,

Figure 17.2  Scatter plot for rainfall and runoff data (Example 1).

square error (MSE ) is defined as MSE =

The data described in Example 1 will be used to find the fitted regression line. 18

i=1

18

i =1

SXY 1797.96 = = 17.27 and b0 = y − b1 x = 942.5SXX 104.09 (17.27)*51.14 = 59.31 which provides the fitted regression line as: Thus, from Eq. (17.4) b1 =

i=1

pieces and so the sum of squares has (n − 1) DF. So, corresponding to Eq. (6), we can show the split of DF as n − 1 = 1 + (n − 2) 

yˆ = 59.31 + 17.27 x 



We can tabulate for each of the 18 values xi , at which a yi   observation is available, the fitted value yˆi , and the residual yi − yˆi as in Table 17.2. As expected from the definition of the least squares, the residuals sum to zero. However, in practice, the sum may not be exactly zero due to rounding. 17.1.6  Analysis of Variance

ANOVA calculations are displayed in an analysis of variance table, which has the format for simple linear regression according to Table 17.3. Table 17.3  ANOVA Table for Simple Linear Regression

Analysis of variance (ANOVA) consists of calculations that provide information about levels of variability within a regression model. It also forms a basis for tests of significance. Consider the following identity:

Source

DF

Sum of squares

yi − y = ( yˆi − y ) + ( yi − yˆi ) 

Model

1

∑( yˆi − y )2

(17.6)

n

i=1

Table 17.2  Observations, Fitted Values, Residuals, and Sum of Squares (Example 1) Obs no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

yi 668 692 760 792 505 2,010 915 988 624 1,740 541 770 776 1,170 1,160 510 1,360 984

Sum

17_Singh_ch17_p17.1-17.10.indd 3

yˆi 1,058 938 839 548 1,061 1,321 867 1,000 868 994 981 862 691 1,035 703 999 1,185 1,015

yi − yˆi

( yi − yˆi )2

yˆi − y

( yˆi − y )2

-390 -246 -79 244 -556 689 48 -12 -244 746 -440 -92 85 135 457 -489 175 -31

152,100 60,516 6,241 59,536 309,136 474,721 2,304 144 59,536 556,516 193,600 8,464 7,225 18,225 208,849 239,121 30,625 961

115 -4 -104 -395 118 379 -76 57 -75 51 38 -80 -252 92 -239 57 243 73

13,225 16 10,816 156,025 13,924 143,641 5,776 3,249 5,625 2,601 1,444 6,400 63,504 8,464 57,121 3,249 59,049 5,329

2,387,820

n

∑( yi − y )2 needs (n −1) independent

559,458

Error

n−2

n −1

F

n

MSM =

2 SSM ∑i=1( yˆi − y ) = 1 DFM

MSE =

2 SSE ∑i=1( yi − yˆi ) = DFE n−2

MSM /MSE

n

n

∑( yi − yˆi )2 i=1

Total

Mean square

n

∑( yi − y )2 i=1

SST DFT

The F column provides a statistic for testing the alternative hypothesis that β1 ≠ 0 against the null hypothesis that β1 = 0. We compare the ratio F = MSM / MSE with the 100(1 − α )% point of the tabulated F (1, n − 2) distribution in order to determine whether β1 can be considered nonzero on the basis of the data. So, for a 0.05-level test, we would reject H 0 if F exceeds the 95th percentile of F (1, n − 2). In other words, when the numerator (MSM ) is large relative to the denominator (MSE), then the ratio is large and there is evidence against the null hypothesis. 17.1.7  Inferences Related to Linear Regression

Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data. Thus, often we want to make inferences about β1 , the slope of the population regression model. For example, we may test:       H 0 : β1 = 0  (X and Y are not linearly related) H1 : β1 ≠ 0  (X and Y are linearly related)

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17-4     Linear and Nonlinear Regression

βˆ1 s(βˆ1 ) which follows the t distribution with (n − 2) DF. Alternatively, we can test whether x and y   are positively linearly related (β1 > 0) or x and y are negatively linearly related (β1 < 0). Even though we have discussed two different tests (a t-test and an F-test) for the test of H 0 : β1 = 0  versus H1 : β1 ≠ 0,  we can show that these two tests are equivalent due to the mathematical fact that F (1, v ) = {t (v )}2 . For example, using the data mentioned in Example 1, we can test whether rainfall is linearly related to runoff by testing the aforementioned hypothesis. We have used the following steps in MS Excel 2013 to analyze the data in Example 1: 1. Go to the Data tab and click Data Analysis (if you do not have Data Analysis button, you may need to load the Analysis ToolPak Add-In). 2. Select Regression and click OK. 3. In the resulting dialogue box, select the Y Range (A1: A 20). This is the predictor variable. 4. Similarly, select the X Range (B1:B20). These are the explanatory variables. 5. In the same dialogue box, click the Confidence Level box and enter the desired level of confidence in the corresponding cell to the right. 6. Select an Output Range. 7. Click Residuals. 8. Click OK. The resulting output obtained from Excel has been displayed in Table 17.4. For example, using the data mentioned in Example 1, we can test whether rainfall is linearly related to runoff by testing the aforementioned hypothesis. We have used the following steps in MS Excel 2013 to analyze the data in Example 1: 1. Go to the Data tab and click Data Analysis (if you do not have Data Analysis button, you may need to load the Analysis ToolPak Add-In) 2. Select Regression and click OK. 3. In the resulting dialogue box, select the Y Range (A1: A20). This is the predictor variable. 4. Similarly, select the X Range (B1:B20). These are the explanatory variables. 5. In the same dialogue box, click the Confidence Level box and enter the desired level of confidence in the corresponding cell to the right. 6. Select an Output Range. 7. Click Residuals. 8. Click OK. The resulting output obtained from Excel has been displayed in Table 17.4. The test statistic for the aforementioned hypothesis is defined as t =

Table 17.4  ANOVA Table for Example 1 (Rainfall and Runoff Data) Summary output

17.2  MEASURES FOR GOODNESS OF FIT

Sxy By using the definition of correlation coefficient (r = ), the regression σ x .σ y equation can also be rewritten from Eq. (17.5) as: yˆ − y = r

σy σx

(x − x ) 

(17.8)

By squaring the aforementioned expression and taking the expected value of the squares, the error variance can be written as: σ ε2 = σ y2 (1 − r 2 )  17.2.1  Inference on Regression Coefficients

In order to place confidence intervals on β0  and β1  and to test hypothesis concerning them, it is essential to know the variance of b0 and b1 which will be denoted by σ b20 and σ b21 ,  respectively. Considering the normality assumption of random error [εi ~ N (0,σ 2 )],  for all i independent, the inference for the slope (β1 ) can be obtained using Eq. (17.4): n

b1 =

where ki =

n

∑i=1(xi − x )( yi − y ) = ∑i=1(xi − x ) yi = n k y ∑i i n n ∑i=1(xi − x )2 ∑i=1(xi − x )2 i=1

( xi − x )

n

∑ j=1(x j − x )2



and yi ~ N (β0 + β1xi ,σ 2 ).

    σ2 σ2 2 b ~ N β ,  It follows that 1 which  1  . Thus, σ b1 = n n  ∑ ( x j − x )2  ∑ j=1(x j − x )2 j=1   indicates that SE (b1 ) =

s n

∑ j=1(x j − x )2

.

b1 is distributed as a t distriSE(b1 ) b1 − β1 bution with (n − 2) DF. In other words, ~ tn−2 . The confidence limits of SE(b1 ) β1 can be estimated from If the model is correct, then the quantity

Lβ1 = b1 − tn−2 (1 − α 2) SE (b1 ) 

Regression statistics Multiple R 0.435591 R square 0.18973985 Adjusted R square 0.139098 Standard error 386.25 Observations 18

U β1 = b1 + tn−2 (1 − α 2) SE (b1 )  Test of the hypothesis concerning β1 can be made by noting that b1 − β1 ~ tn−2 . Thus, H 0 : β1 = 0 versus H1 : β1 ≠ 0 is tested by computing the test SE(b1 ) b statistic t = 1 under H 0 . H 0 is rejected if t > tn−2 (1 − α 2). A typical SE(b1 ) P-value graph is displayed in Fig. 17.3. The inference for the intercept (β0 ) can be obtained in a similar fashion. In this situation, we start with the following equation:

ANOVA SS

Source

df

Regression Residual Total

1 16 17

Coefficient

SE

t-stat

Intercept X variable 1

59.19 17.27

465.33 8.92

MS

F

Significance F

3.74

0.070

P value

Lower 95%

Upper 95%

0.1272 1.93

0.900367 0.07

-927.26 -1.64

558,996.8 558,996.8 2,387,126 149,195.4 2,946,123

n

1045.65 36.19

Thus, we estimate with 95% confidence that the runoff in Pee Dee River, South Carolina, USA for each additional increase of 1% rainfall will increase by an amount somewhere between -1.64 and 36.19 cfs. Note that the experimental range of predictor variable X is varied from 28.28 and 73.06 in. Our estimate is best when the value of rainfall is within the experimental range. In general, more rainfall should produce more runoff. But the lack of significance and apparent nonlinearity of the hydrologic response of the basin can be observed from a low R 2 and higher P value. The model is explaining about

17_Singh_ch17_p17.1-17.10.indd 4

19% of the variation. This dataset indicates that the correlation between rainfall and runoff may depend on other characteristics. It is our understanding that there are a few other characteristics in this example that may influence the relationship between rainfall and runoff.

b0 = y − b1x = ∑ i=1

n

n

yi − ∑ ki xyi = ∑ci yi  n i=1 i=1

1 (x − x ) x and yi ~ N (β0 + β1xi ,σ 2 ) which leads us to where ci = − n i n ∑ ( x j − x )2 j=1

   2    b0 ~ N  β0 , σ 2  1 + n x  n 2  x − x ( )  ∑ j=1 j  

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Multiple Linear Regression    17-5 

represents a hyperplane in the (k + 1)-dimensional space of ( x1 ,   x 2 ,…, x k ,   y ). The regression coefficient (β j ). represents the expected rate of change in the response variable as x j changes when the remaining predictor variables x1 ,   x 2 ,…, x j −1 ,  x j+1 ,…, x k are kept fixed. Since β j is the expected change in y as x j changes by one unit while all other independent variables remain fixed, the regression coefficients β ’s are also referred to as partial regression coefficients.

0.30

tn–2

0.25

Density

17.3.2  Estimation of Regression Coefficients

0.20

The OLS method that we have used to fit simple linear regression can be extended to fit the multiple linear regression model. The least-squares method proceeds by minimizing the sum of squared deviations of the observed yi   (i = 1, 2,…,  n) from true means obtained from Eq. (17.9) E(yi) = β0 + β1xi1 + β 2 xi 2 + ..... + β k xik , that is by minimizing

0.15

0.10

n

Q(β0 , β1 ,…, β k ) = ∑( yi − β0 − β1xi1 − β 2 xi 2 − ..... − β k xik )2 

(17.10)

i =1

0.05 –3

–2

–1

0 tobs

1

2

3

over choices of (β0 , β1 ,…, β k ). As before we use calculus to minimize the aforementioned Eq. (17.9). We take the partial derivatives of Eq. (17.10) with respect to β0 , β1 ,…, and   β k and equating them to zero. We will have the following (k + 1) equations:  ∂Q n  = −2 ∑ yi − βˆ0 − βˆ1xi1 − βˆ2 xi 2 − ..... − βˆk xik = 0                        ∂β 0 i=1  n  ∂Q ˆ ˆ 2 ˆ ˆ  ∂β = −2 ∑ yi xi1 − β0 xi1 − β1xi1 − β 2 xi1xi 2 − ..... − β k xi1xik = 0 i 1 = 1    ……………  n  ∂Q = −2 ∑ y x − βˆ x − βˆ x x − βˆ x x − ..... − βˆ x 2 = 0 i ik 0 ik 1 i1 ik 2 ik i 2 k ik  ∂β i=1 k   (17.11)

(

P value = Area in the two tails Figure 17.3  A typical P-value graph.

  x2 1  Thus, σ b20 = σ 2  + n , which indicates that SE(b 0)  = n ∑ ( x j − x )2    j=1    x2 . s 1 + n n 2 ( ) − x x ∑   j=1 j The basis for confidence interval and hypothesis tests can be defined as b0 − β0 ~ tn−2 . The confidence limits of β1 can be estimated from SE(b0 )

)

(

)

(

)

The aforementioned set of equations can also be converted into standard form, referred to as normal equations, as follows:                        n

n

n

n

i=1

i=1

Lβ0 = b0 − tn−2 (1 − α 2) SE (b0 ) 

nβˆ0 + βˆ1 ∑xi1 + βˆ2 ∑xi 2 +…+ βˆk ∑xik = ∑yi                               

U β0 = b0 + tn−2 (1 − α 2) SE (b0 ) 

βˆ0 ∑xi1 + βˆ1 ∑xi12 + βˆ2 ∑xi1xi 2 +…+ βˆk ∑xi1xik = ∑xi1 yi

i=1



Thus, H 0 : β0 = 0 versus H1 : β0 ≠ 0 is tested by computing the test statistic b0 t= under H 0 . H 0 is rejected if t > tn−2 (1 − α 2). The significance of the SE(b0 ) overall regression equation can be evaluated by testing the above hypothesis (H 0 : β0 = 0 ) too. If the hypothesis is accepted, then yˆ = y or the regression line does not explain a significant amount of the variation in y. In this situation, we recommend using y as an estimator of y regardless of the value of x . 17.3  MULTIPLE LINEAR REGRESSION 17.3.1 Introduction

Quite often we wish to model the response variable as a function of several other quantities in the same equation because it is reasonable to have a response variable being dependent on more than one quantity. For example, the peak rate of runoff from watersheds in a given area may be dependent on the rainfall, watershed area, and slope of the mainstream. Multiple regression is used to model situations like these where the objective is to build a model that relates a dependent variable to more than one independent variable. Multiple regression analysis is often of practical value because it can yield important insights into data dynamics. The general multiple linear regression model is an extension of model discussed in Sec. 17.1: yi = β0 + β1x1 + β 2 x 2 + ,..., + β k x k + ε , 

(17.9)

where, y is a dependent variable, x1 , x 2 ,..., x k   are independent variables and β1 , β 2 ,..., β k are unknown parameters. This model is linear in the parameters β j . As before, ε is a random error incurred when observing y at ( x1 ,  x 2 ,…, x k ). Also, E(ε ) = 0 and  V (ε ) = σ 2 . The multiple linear regression model (17.9)

17_Singh_ch17_p17.1-17.10.indd 5

i=1

n

n

n

n

n

i=1

i=1

i=1

i=1

i=1

n

n

n

n

n

i=1

i=1

i=1

i=1

i=1

(17.12)

……………

βˆ0 ∑xik + βˆ1 ∑xi1xik + βˆ2 ∑xik xi 2 +…+ βˆk ∑xik 2 = ∑xik yi

Solving the above system of (k + 1) equations for the (k + 1) unknowns βˆ0 , βˆ1 ,…, and βˆk , we obtain the least square estimators for β0 , β1 ,…, and   β k . 17.3.3  An Example Related to Hydrology

Example 2 Consider a similar example discussed in the simple regression model, where monthly runoff is likely to be dependent on the rainfall in the same month and in the previous months. Consider the following set of hypothetical data. To analyze these data using Excel, we proceed exactly as we did in the simple linear regression example. The only change here is to include more than one column in the Input X Range. Note, however, that the regressors must to be in contiguous columns. Moreover, it should be noted that Excel restricts the number of regressors. Excel calculates the values of β0 , β1 , …, β k   to make the predictions from Eq. (17.9) as accurate (in the sense of minimizing the sum of squared errors) as possible. 17.3.4  ANOVA for Multiple Regression

The ANOVA calculations for multiple regression are nearly identical to the calculations for simple linear regression, except that the DF are different,

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17-6     Linear and Nonlinear Regression

significant which is difficult to justify physically. But again note that this is a trial dataset. In a situation like this, stepwise regression, which is a semiautomated process of building a model by successively adding or removing variables based solely on the t-statistic of their estimated coefficients can be used. The regression coefficients table is very informative. The column Coefficients gives the least squares estimates of β j and the column SE gives the standard errors of the least square estimates b j of β j . The column t-stat gives the computed t-statistic H 0 : β j = 0 against H a : β j ≠ 0 . As expected, the Column P value gives the P value for test of H 0 : β j = 0 versus H a : β j ≠ 0 . For this example, this equals the P{ t > t − stat}, where t is a t-distributed random variable with 12 DF and t-stat is the computed value of the t-statistic given in the previous column. It should also be noted that this P value is for a twosided test, and it must be adjusted in case of a one sided test. Columns Lower 95% and Upper 95% provide a 95% confidence interval for β j . In summary, the regression output provides us a fitted model for the aforementioned example as follows:

Table 17.5  Fifteen Observations of Rainfall and Runoff Data

Month (i)

Runoff month (i) (y) (mm)

Rainfall in month (i) (x1) (mm)

Rainfall in month (i–1) (x2) (mm)

Rainfall in month (i–2) (x3) (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

948 940 940 945 941 957 935 952 943 935 945 941 935 950 945

680 662 588 557 584 852 527 574 689 514 591 459 312 600 745

452 504 422 528 547 515 512 545 325 495 542 410 478 548 565

562 247 258 323 228 525 260 615 315 254 654 412 528 605 445

y = 912.5 + 0.03x1 + 0.003x 2 + 0.023x 3  17.4  NONLINEAR REGRESSION 17.4.1 Introduction

since the number of explanatory variables is more than one. In the multiple regression case, the F column provides a statistic for testing the alternative hypothesis that at least one of the parameters, β j ≠ 0 for j =1, 2,..., k against the MSM with MSE the 100(1 − α )% point of the tabulated F (k , n − k − 1) distribution. When the numerator (MSM ) is large relative to the denominator (MSE), then the ratio is large and there is evidence against the null hypothesis. It must be noted that the F-test in multiple regression does not indicate which of the parameters β j for j =1, 2,..., k is not equal to zero, rather it indicates only that at least one of the parameters is linearly related to the response variable. SS The ratio M is an important quantity and is denoted by R 2 . It is known SST as the squared multiple correlation coefficient. As before, a measure of quality of the regression is the F statistic. Formally, this F statistic tests the following null and alternative hypothesis. H0 : b1 = 0, b2 = 0, . . . , bk = 0 H1 : At least one of β1 , β 2 ,...,  β k is not zero. It must be noted that β0 is not involved in this test. The fraction of the variation in Y explained by the regression is denoted by R 2  and is defined as SS R 2 = Model . SSTotal Understanding and interpreting the regression output is very important. R2 is of greatest interest of all. R 2 = 0.796569375 means that 79.7% of the variation of runoff ( yi ) around its mean ( y ) is explained by the regressors x1 , x 2 , and x 3 . The standard error (SE) refers to the estimated standard deviation of the error term. It is sometimes called the SE of the regression. The column labeled F gives the overall F-test of H 0 : β1 = 0, β 2 = 0,...,  β k = 0 versus H1 : At least one of β1 , β 2 ,...,  β k is not zero. The column labeled Significance F has the associated P-value. Since 0.000403 < 0.05, we reject the null hypothesis at significance level 0.05. It should be noted that Excel can be used to calculate the P value as well. For example, FINV(0.000403,3,11) = 14.36. The hypothetical dataset provides a good illustration with high R 2 and significant regression. However, the dataset also indicates that β 2 is not null hypothesis that β1 = β 2 = ... = β k = 0. We compare the ratio F =

The linear least squares method fits a straight line or a flat plane to a set of data points. However, it is possible that the true relationship we want to model is curved rather than flat. In this situation, we use nonlinear regression where we discuss models which are nonlinear in the parameters. There are many examples in hydrology where nonlinear regression is used. For example, daily evaporation can be computed by nonlinear regression with temperature, solar radiation, wind velocity, relative humidity, and sunshine hours. In this section, we present some commonly used nonlinear regression models and discuss point and interval estimation for unknown parameters. We also illustrate how nonlinear regression functions can be transformed to linear regression model. 17.4.2  Some Commonly Used Families of Nonlinear Regression in Hydrology

While a linear equation has one basic form, nonlinear equations can take many different forms and that is why nonlinear regression covers many different forms. A few popular nonlinear equation forms are: 1. An exponential growth (decay) equation is defined as Yi = β0e β1xi + εi , where β1 is the growth (decay) rate and εi is the error term. The equation states that the response variable is growing (or shrinking) at a constant rate of β1 . 2. A constant-elasticity equation can be defined as Yi = β0 xiβ1 + εi . This is primarily used for demand curves equation, where x  is some continuous variable that is always larger than 0, β0  determines the scale, and β1  is the elasticity of y with respect to x. As before, εi is the error term and has a mean of 1 and is always larger than 0. The equation states that the elasticity of  y with respect to x is constant, β1 . 3. Sigmoidal curves (S-shaped curves) arise in various applications, including engineering, signal detection theory, and physiological or pharmacological studies. Various types of growth data often correspond to sigmoidal curve. For example, the weekly water/air temperature relationship can be described by a continuous S-shaped curve. Some of the nonlinear regression functions that have been used in such situations include the Gompertz model, Logistic regression model, Morgan–Mercer–Flodin model, Weibull-type functions, − ( β1 +β2 xi ) and Richard’s model. The Gompertz model is defined as: Yi = β0e −e + εi. As it can be seen from the mathematical definition, the Gompertz model is a three-parameter model with the parameters β1 , β 2 ,  and β3 . The logistic regresβ0 sion model has three parameters too and can be defined as: Yi = + εi . 1 + e −( β1 +β2 xi )

Table 17.6  ANOVA Table for Multiple Linear Regression Source Model

DF

k

Sum of squares n

∑( yˆi − y )2

Mean square n

MSM =

2 SSM ∑i=1( yˆi − y ) = 1 DFM

MSE =

n ˆ 2 SSE ∑ i=1(Yi − Yi ) = DFE n − k −1

i=1

Error

n − k −1

n

∑( yi − yˆi )2 i =1

n

Total

17_Singh_ch17_p17.1-17.10.indd 6

n −1

∑( yi − y )2 i=1

F MSM /MSE

SST DFT

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Nonlinear Regression    17-7  Table 17.7  ANOVA Table for Example 2 (Rainfall and Runoff Data) Summary output Regression statistics Multiple R R square Adjusted R square Standard error Observations ANOVA

0.892507353 0.796569375 0.741088295 3.268703567 15

Source

df

SS

MS

F

Significance F

Regression Residual Total

3 11 14

460.20 117.53 577.73

153.40 10.68

14.36

0.000403

Coefficient

SE

t-stat

P value

Lower 95%

Upper 95%

912.5 0.03 0.003 0.023

7.53 0.00 0.01 0.00

121.1 4.64 0.27 4.03

1.52 × 10-18 0.00 0.79 0.0019

895.91 0.02 -0.03 0.01

929.08 0.05 0.03 0.03

Intercept X variable 1 X variable 2 X variable 3

This logistic regression model can be used to flood susceptible mapping and risk area delineation (Pradhan, 2009). The Weibull model can be defined as: Yi = β0e

 X β2 {− i  }  β1 

transformations of the response variable, the predictor variables, the parameters, or any combination of these such that the transformed function is linear in the unknown parameters. In addition, transformations are applied to correct problems of non-normality or unequal variances. As expected, transformations need to apply to the original data prior to performing regression and it must be assumed that the error in the transformed equation has the desired properties (normal distribution with mean 0). Many transformation techniques are available; however, only a relatively small number of them are regularly used: (a)  Usually transformations are chosen from the “power family” of transformations, where each value is replaced by x p , with p being an integer or half-integer. For example, p = −0.5 indicates a reciprocal square root transformation and the data values must all be positive. (b)  Log transformation is another common approach. Exponential growth or decay equation discussed earlier can be transformed by logarithmic transformations. Antilogarithm technique is also used. (c)  Some other transformation techniques include: polynomial regression and Box-Cox transformations. Polynomial regression model can be approximated by the forward selection procedure or backward elimination approach. Box-Cox technique allows one to find an optimum transformation of the response variable using maximum likelihood methods.

+ εi . Finally, the Richard’s model has the following definition

β0 + εi . [1 + e −( β1+β2 xi ) ]β3 4. The segmented polynomial model has some use too. It can be defined as:  β + β x + β 2 xi2 + εi ,                     xi ≤ α Yi =  0 1 i   γ 0 + γ1xi + εi ,                                    xi > α

Yi =

Numerous other useful families of nonlinear regression functions exist. We will consider intrinsically the following form of the nonlinear models: Yi = f ( xi′ ; β ) + εi 

(17.13)

where f ( xi′ ; β ) is a nonlinear function relating E(Y ) to the predictor variables, xi . Moreover, xi is a k ×1 vector of independent variables, β is a p ×1 vector of parameters, and εi′ s are identically and independently distributed random variables with mean 0 and variance σ 2 .

17.4.5  Example 3 (A Trial Dataset)

To illustrate the fitting and analysis of a nonlinear regression model, we will use a simple example, where the model has only two parameters and the sample size is reasonably small. Say we would like to develop a regression model for predicting annual maximum series (Y), measured in gallon of a desired variable based on geographical distances (X), measured in ft. Observed annual maximum series and geographical distances expected to follow an exponential decay model (Kjeldsen and Jones, 2009). Consider the following dataset (Table 17.8) to explore the relationship between X and Y. Our objective is to (1) describing this relationship mathematically and (2)

17.4.3 Assumptions

The assumptions for nonlinear regression is almost identical to that for linear regression with an exception that in this case, the regression function is a nonlinear function of the unknown parameters. 17.4.4  Transforming to Linearity

Sometimes, it is possible to transform a nonlinear regression function Yi to a linear function. Appropriate transformation techniques have to be used, so that

Table 17.8  Thirty-Six Hypothetical Data of Distances and Annual Maxima

17_Singh_ch17_p17.1-17.10.indd 7

Catchment (i)

Distances (X) (ft)

Annual maximum (Y) (gallon)

1 2 3 4 5 6 7 8 9 10 11 12

10 14 15 20 35 50 70 100 20 120 180 250

1325 1256 1254 1200 1025 800 758 854 1525 652 650 556

Catchment (i)

Distances (X) (ft)

Annual maximum (Y) (gallon)

13 14 15 16 17 18 19 20 21 22 23 24

300 300 330 380 450 520 530 600 650 700 752 760

468 1250 425 389 400 354 300 258 241 201 25 156

Catchment (i)

Distances (X) (ft)

Annual maximum (Y) (gallon)

25 26 27 28 29 30 31 32 33 34 35 36

821 842 900 924 980 1021 1054 1124 1230 1254 1289 1321

124 114 78 69 60 52 45 40 41 38 40 35

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17-8     Linear and Nonlinear Regression Table 17.9  ANOVA Table for Example-3 (Distances vs. Annual Maxima)

Scatterplot of Example 3

Annual maximum

1800 1600

Summary output

1400

Regression statistics Multiple R R square Adjusted R square Standard error Observations

1200 1000 800 600 400

0.377835964 36

ANOVA

200 0

0.959133908 0.919937854 0.917583085

0

200

400

600

800

1000

1200

1400

Distances Figure 17.4  Scatter plot of distances and annual maxima (Example 3).

creating a model from this dataset. A scatter plot of the data is shown in Fig. 17.4. As it can observed from the scatterplot, the relationship between the predictor variable and the response variable is exponential in this example and should follow a two-parameter nonlinear exponential model (Yi = β0e β1xi + εi ) discussed in (1). Since a model is available, our next step would be to estimate the regression parameters β0 and β1 . The aforementioned scatterplot clearly indicates a curvilinear trend to the data, to be more specific the above plot can be fitted to an exponential function. It is possible to linearize the relationship between X and Y by using the log transformation of the data on the y-axis. In other words, we take the base of the natural logarithms, log e ( yi ). The log-transformed data are displayed in Fig. 17.5. A linearized trend can be observed from this logtransformed scatterplot, which indicates that a simple linear regression model can be used to describe the relationship between distances and annual maximum.

Source

df

SS

MS

F

Significance F

Regression Residual Total

1 34 35

55.77 4.85 60.63

55.77 0.14

390.674

0

Coefficient

SE

t-stat

P-value

Lower 95%

Upper 95%

7.10 -0.0029

0.10 0.00

69.12 -19.76

0 0

6.89 0.00

7.31 0.00

Intercept X variable 1

17.4.6  Parameter Estimation

Method of least squares is widely used in nonlinear regression too. Geometric illustration and linear approximation can also be used to estimate parameters in nonlinear regression model. The least square estimates of β defined in Eq. (17.13) is denoted by βˆ and is the set of parameters that minimizes the sum of squared residuals (errors): n

( )}

{

SSE (βˆ ) = ∑ (Yi − f xi′ ; βˆ



i=1

2



(17.14)

SSE and it is an unbiased ( n − p) estimate of σ 2 . As in linear regression, our objective is to obtain the normal equations. To obtain those equations, we consider the partial derivatives of SSE (βˆ ) with respect to each βˆ j , and set them equal to zero. This gives us a system of p equations. Each normal equation is given by

Log annual maximum

Mean squared error (MSE) is defined as MSE =

8 7 6 5 4 3 2 1 0

Scatterplot of log-transformed data (Example 3)



∂SSE (βˆ ) = −2∑ (Yi − f ( xi′ ; βˆ ) ∂βˆ n

j

0

200

400

600 800 Distances

1000

1200

1400

Figure 17.5  Scatter plot of log-transformed data for distances and annual maxima (Example 3).

i=1

n

∑i=1( Xi − X )(Yi − Y ) = −0.0029, n ∑i=1( Xi − X )2

α = Y − Xβ = 5.5 − (0.0029) ∗ 553.2 = 7.10.

So, Yˆ = α + β X = 7.10-0.0029X. So, our regression equation can be written as log e (Yˆ ) = 7.10-0.0029X which can be written as Yˆ = e7.10e−0.0029 X or Yˆ = 1214.38e−0.0029 X . We can calculate the ANOVA table (Table 17.9) and resulting signifi-

∑ y = 197.91, ∑(x − x )2 =  ∑( x − x )( y − y ) = −19229, ∑( y − yˆ )2 = 4.85,

cance with the following quantities: ∑ x =19916 , 6,629,868,

∑( y − y )

2

and ∑ ( yˆ − y )2 = 55.75.

17_Singh_ch17_p17.1-17.10.indd 8

= 60.63,



j



(17.15)

n

n

∑{(Yi − βˆ0eβ x }eβ x  = 0, and ∑{(Yi − βˆ0eβ x }βˆ0eβ x (xi ) = 0. . Even for ˆ

ˆ

1 i

i=1

β=

ˆ 

i

These partial derivatives are functions of the parameters and an explicit solution for βˆ cannot be obtained. For example, consider the exponential ∂Y growth model discussed earlier: Yi = β0e β1 xi + εi. The derivatives are: i = e β1xi ∂β 0 ∂Yi β1 xi = β0e ( xi ). From Eq. (17.14), the normal equations are: and ∂ β1 1 i

We follow the steps in linear regression examples (Examples 1 and 2) to model the relationship. We have the following information: n = 36 , X = 553.2, and Y = 5.5 [note that this is the mean of transformed Y = log e ( yi )].



}∂ f ∂(xβˆ′; β ) = 0 

{

ˆ

1 i

ˆ

1 i

i=1

this straightforward example, there is no analytic solution for βˆ . The estimation of parameters usually requires the use of iterative methods (e.g., Gauss–Newton algorithm and Levenberg–Marquardt algorithm) and most of the commonly used statistical software packages provide routines for calculating βˆ1 , βˆ2 ,…, βˆ p. 17.4.7  Example 4 (Weekly Water/Air Temperature)

It has been documented on many occasions [(e.g., in (Maidment, 1993)] that the relationship between weekly water and air temperature can be well explained by a continuous S-shaped curve. Mohseni et al. (Gupta and Guttman, 2013) proposed the following equation to estimate stream temperature, Ts :

Ts = µ +

α −µ  1 + e y (β − Ts )

(17.16)

8/22/16 12:04 PM

REFERENCES    17-9 

where, µ is a parameter used to generate nonzero minimum temperature, α is the estimated maximum stream temperature, γ is a measure of the steepest slope of the function, and β is the air temperature at the point of inflection. The slope (γ ) is a function of the slope tanθ at the point of inflection and can 4 tanθ . The logistic function Eq. (17.16) will produce an α −µ S-shaped curve and can be explained using the nonlinear technique. The least square method discussed earlier will be used to estimate the four parameters (α , β ,γ , and µ ). Similar argument discussed in Eq. (17.14) will lead us to the following: be estimated as γ =



∑εi2 = ∑{Tobs − µ − 1 + eγ (β −T ) } n

i=1

n

i=1

2

α −µ

s



(17.17)

Note that, α and µ are the linear coefficients and thus can be estimated using the normal equations directly. However, the other two parameters (β and γ ) must be estimated using a numerical approximation method (e.g., Gauss–Newton algorithm).

17_Singh_ch17_p17.1-17.10.indd 9

REFERENCES

Gupta, B. C. and I. Guttman, Statistics and Probability with Applications for Engineers and Scientists, Wiley, Hoboken, NJ, 2013. Haan, C. T., Statistical Methods in Hydrology, The Iowa State University Press, Ames, IO, 1977. Kjeldsen, T. and A. J. David, “An exploratory analysis of error components in hydrological regression modeling,” Water Resources Research, 45: 1–13, 2009. Maidment, D. R., Handbook of Hydrology, McGraw-Hill, New York, 1993. Mohseni, O., G. H. Stefan, and R. E. Troy, “A nonlinear regression model for weekly steam temperatures,” Water Resources Research, 34 (10): 2685–2692, 1998. Pradhan, B., “Flood susceptible mapping and risk area delineation using logistic regression, GIS and remote sensing,” Journal of Spatial Hydrology, 9 (2): 1–18, 2009.

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Chapter

18

Time Series Analysis and Models BY

ÓLI G. B. SVEINSSON AND JOSE D. SALAS

ABSTRACT

18.2  PROPERTIES OF HYDROLOGICAL TIME SERIES

Analysis and modeling of time series of hydroclimatic processes have been widely used for planning and management of water resources systems. Time series models are capable of modeling complex dependence structure across multiple sites and multiple time scales, and may be useful for generating synthetic time series, forecasting, evaluating operational rules for water resources systems, sizing reservoirs, detecting trends and shifts, filling in missing data, and extending short records. This chapter includes a wide range of models and modeling techniques for representing hydrological processes, such as precipitation and streamflows, in continuous and discrete time scales. The emphasis for continuous models has been around the concept of point processes while that for discrete time has been based on short memory and long memory models such as AR, ARMA, and FARMA including their discrete counterparts as well as periodic and product models. In addition, univariate and multivariate shifting mean (SM) models are included so as to represent complex shifting mechanisms of the underlying hydroclimatic processes that may arise from the effect of low frequency components of the atmospheric and oceanic systems. Furthermore, the chapter discusses disaggregation and nonparametric approaches and some basic issues pertaining to uncertainty and nonstationarity. Time series analysis and modeling of hydrologic data can be used for evaluation of impacts and risks that can arise from changes due to natural variability, human intervention, and climate change in hydrologic time series.

Hydroclimatic series are generally defined at discrete time intervals, such as annual and seasonal (e.g., monthly, weekly, daily, and hourly). Seasonal data are usually affected by the annual hydrologic cycle (e.g., monthly data show a 12-month periodic structure). In addition, hourly data may exhibit distinct diurnal cycle due to Earth’s rotation. Other cycles due to sunspot activity or large-scale atmospheric forcing, such as El-Niño-Southern Oscillation may also affect the variability of hydroclimatic data. Time series are said to be stationary if the statistical properties of the time series do not change with time, that is, the probability distribution of the process is the same at all times. Conversely, if any statistical property depends on time then the process is nonstationary with regards to that statistical property. For example, monthly hydrologic time series, which generally have periodic mean and periodic covariance are nonstationary. Statistical analysis of hydroclimatic time series generally includes estimating basic sample statistics, such as mean, variance, skewness, autocorrelation function (ACF), partial autocorrelation function (PACF), and spectrum. Other statistics, such as storage-drought- and flood-related statistics, may be of particular interest for analyses of streamflow series (Salas, 1993; Sveinsson, 2014; Yevjevich, 1972). Basic unbiased sample statistics of time series yt of length N, such as the mean, variance, and skewness coefficient are calculated, respectively, as:

18.1 INTRODUCTION



The field of stochastic hydrology has been developed in the early 1950s and 1960s by several hydrologists, such as Hurst, Matalas, Thomas, Fiering, and Yevjevich, who developed the basic concepts, models, and applications thereof and inspired the work and contributions by many others in the 1970s, 1980s, and 1990s, which are included in books and chapters such as Bras and Rodriguez-Iturbe (1985), Hipel and McLeod (1994), Loucks et al. (1981), Salas et al. (1980), and Salas (1993). Many of the statistical procedures and time-series models are available in common software packages, such as Matlab, R, SAS, SPSS, specialized software for classical time series analysis (e.g., Brockwell and Davis, 2002), and time-series analysis of hydrologic data, such as SPIGOT (Grygier and Stedinger, 1990) and SAMS (Salas et al., 2006; Sveinsson et al., 2009). Time-series models may be used for generating (simulating) and forecasting hydrologic processes such as streamflows that may occur in the future. Synthetically generated series enable estimating the probability distribution of key decision parameters, such as reservoir storage, planning and testing operating rules, estimating future power output of hydroelectric systems, and evaluating the performance of an irrigation system. Forecasting data may be used for operation and management of water resources systems. This chapter includes a variety of models and methods developed in literature. Some of them have been used for modeling complex water systems, such as the Colorado River, the Great Lakes and St. Lawrence River, and the Nile River systems.

N

N ∑ ( yt − y )3 1 N 1 N 2 2 1 (18.1) y = ∑ yt ; s = ∑ ( yt − y ) ; g = ( N −t =1)( N t =1 N − 1 t =1 N − 2)s 3

where y is a measure of location, s is a measure of spread, and g is a measure of shape of the distributional properties of the data. Hydrologic series are generally autocorrelated although the series of maximum annual values may be uncorrelated. The lag-h autocorrelation function ACF(h) is ρh = Ch /C0, where Ch is the lag-h autocovariance function ACVF(h), Ch = E[( yt +h − µ )( yt − µ )]. The corresponding sample estimate rh is simply the cross-correlation between consecutive values of the series at lag-h, that is,

rh =

ch 1 N −h , with ch = ∑ ( yt +h − y )( yt − y ) c0 N t =1

(18.2)

where ch is the lag-h sample autocovariance function and c0 is the maximum likelihood estimate of the sample variance. The autocorrelation of annual streamflow series usually arises from storage effects of the soil, groundwater, and lake storage. The correlogram is a plot of rh versus h, where a fast-decaying correlogram is an indication of a short memory process, whereas a slowdecaying correlogram may suggest a long-memory process. The PACF is also a measure of serial dependence and is obtained by repeatedly fitting an autoregressive model of order p to the time series with PACF(p) being equivalent to the autoregressive parameter fp in the fitted AR(p) model. 18-1

18_Singh_ch18_p18.1-18.12.indd 1

8/22/16 12:06 PM

18-2     Time Series Analysis and Models

Approximate confidence intervals for both the ACF and PACF of a white noise process at the α significance level are ± z1–α/2/N0.5 (Brockwell and Davis, 2002), where z1–α/2 is the 1–α/2 quantile of the standard N(0,1) distribution, for example, approximate 95% confidence intervals are given by ±1.96/N0.5. Both ACF and PACF are used to select the order of the autoregressive-moving average [ARMA] (p, q) model (Sec. 18.5.1). For multiple time series, the sample lag-h cross-correlation coefficient across sites i and j is rhij = chij / c0ii c0jj , N −h

1 ∑ ( yt(+i )h − y (i ) )( yt( j ) − y ( j ) ) , h = …, –2, –1, 0, 1, 2, …., is the lag-h N t =1 cross-covariance estimator and c0ii is the maximum likelihood estimator of the variance at site i. Similarly, for seasonal time series yv,t, where ν = 1,...,N is the year, t = 1,...,w is the season, and w is the number of seasons per year, the unbiased seasonal sample statistics are calculated for each season t in the same way as in Eq. (18.1) earlier. Likewise, the sample lag-h ACF of a seasonal time series is the correlation between yv,t and yv,t–h, that is, where chij =

rh ,τ =

ch ,τ c0,τ c0,τ −h

, with ch ,τ =

1 N ∑ ( yν ,τ − yτ )( yν ,τ −h − yτ −h ) (18.3) N ν =1

Note that the lag-1 season-to-season correlation coefficient r1,1 represents the correlation between season 1 of the current year with season w of the previous year. If the lag-h season-to-season correlations are markedly different throughout the year, for example, different dependence structure during the spring freshet and during the winter, then it is an indication that a model with periodic parameters should be used to preserve the season-to-season correlations. Note that for preserving both annual and seasonal statistics of the sample data, disaggregation models must be applied (Sec. 18.8). Spectral analysis of the time series may be used to detect cyclic components. The spectral density function can be estimated in various ways by both parametric and nonparametric methods (e.g., Brockwell and Davis, 1991, 2002; Jenkins and Watts, 1968; Percival and Walden, 1993; Salas et al., 1980) and most software packages today include one or more methods for estimation of the spectrum. Similarly, for slowly decaying correlograms (longmemory processes), the Hurst slope (Koutsoyiannis, 2002; Salas, 1993) should be estimated. 18.3  TIME-SERIES MODELING

A variety of models are available for modeling univariate and multivariate series of annual and seasonal time scales. The selected model aims at reproducing important statistics of the process under consideration. Most parametric time series models assume that the underlying process is normally distributed and stationary in the mean and variance. Depending on the time series under consideration, the first step prior to model fitting may involve the following analysis (Brockwell and Davis, 2002; Salas, 1993; Sveinsson, 2014): Test for normality Transform to normal if necessary

Test for trends Remove trends if necessary

Is seasonality present in the mean and variance? Remove if necessary by seasonal standardization

The foregoing choices whether or not to transform to normal, remove trends, and seasonality, depend on the properties of the selected model as we will see later. Generally, the typical steps involved in time series modeling are: (1) model identification, (2) parameter estimation, and (3) model testing and validation. The sample correlation of the time series (ACF and PACF) is often used to get an idea of the type of model that may be fitted to the given series, and the model with the minimum residual variance is often selected as the best. This does not penalize for the number of parameters and a common practice is using the Akaike Information Criterion (AIC) and the Schwarz Information Criterion (SIC) for selecting the best model which penalize for the number of parameters used in the model (Brockwell and Davis, 2002; Shumway and Stoffer, 2000). All the sample information on lack of fit is contained in the residuals. In most time-series models, the residuals are assumed to be normally distributed with mean zero. Thus, the frequency distribution of the residuals should resemble as normally distributed. In addition, the residuals should be uncorrelated and also independent of the explanatory variables used in the model. Non-normal residuals may indicate lack of transformation of the data, while

18_Singh_ch18_p18.1-18.12.indd 2

correlated residuals may suggest that a higher order model is needed. Furthermore, synthetically simulated series from the model of the same length as the historical series should be capable of approximately reproducing the historical statistical properties of the original time series. A number of alternative time-series models are presented and discussed in the following sections. They include univariate models, such as ARMA and periodic ARMA (PARMA) (periodic ARMA), fractional ARMA and PARMA for modeling long-memory series, gamma autoregressive (GAR) models for gamma distributed data as well as periodic gamma autoregressive (PGAR), shifting mean (SM) models capable of modeling processes exhibiting shifts or jumps in the mean level, Markov Chain and discrete ARMA and PARMA for modeling discrete processes, and product models for modeling intermittent processes. Multivariate versions of the ARMA, PARMA, and SM models are also included. In addition, disaggregation models for modeling jointly the annual and seasonal process of both single and multiple time series are presented as well as nonparametric models as an alternative to parametric models. Furthermore, the use of exogenous parameters in models for short-term forecasting can improve forecasting performance over models based on past observations of the forecasted process. Some of these models are briefly reviewed. 18.4  MODELING OF CONTINUOUS TIME PROCESSES

Alternative models are available for modeling hydrological processes at various time scales such as annual, monthly (seasonal), and daily. In principle, if one can develop a model that can adequately represent the underlying process in continuous time, then one should be able to derive the model at any level of aggregation such as a day or month. For illustration, we will consider modeling rainfall in which rainfall occurrence in continuous time follows a Poisson process (Le Cam, 1961). Let the number of storms N(t) arriving to a given point in a time interval (0,t) be Poisson distributed with parameter lt (l = storm arrival rate). Thus, if n storms arrive in the interval (0,t) at times t1,...,tn, the number of storms in any time interval T is also Poisson with parameter lT. Assume further that the rainfall amount R associated with a storm arrival is white noise, that is, rainfall amounts r1,...,rn are associated with storms occurring at t1,...,tn, and N(t) and R are independent. Such rainfall generating process has been called Poisson white noise (PWN). The cumulative rainfall in the interval (0,t), denoted Z (t ), is a compound Poisson process and the cumulative rainfall over successive nonoverlapping time intervals T, is given by Yi = Z (iT ) − Z (iT − T ), i = 1,2,... . The statistical properties of Yi, assuming that Z(t) is generated by a PWN model, has been widely studied (e.g., Cadavid et al., 1992; Eagleson, 1978). For example, its ACF is equal to zero for all lags greater than zero, which contradicts actual observations. In PWN, the rainfall is assumed to occur instantaneously with zero duration, which is unrealistic. Instead, the Poisson rectangular pulse (PRP) model (Rodriguez-Iturbe et al., 1984) assumes that rainfall occurs with a finite random duration D and a random intensity I. The pair of intensities and durations (i1,d1),...,(in,dn) correspond to n storms occurring at times t1,...,tn in the interval (0,t). The storms may overlap; consequently, the process Yi becomes autocorrelated. Although the PRP model is better conceptualized than PWN, it is still limited when applied to rainfall data. Alternative models based on the concept of clusters have been suggested (Le Cam, 1961; Neyman and Scott, 1958). The Neyman–Scott cluster process can be thought as a two-level mechanism for rainfall generation, storm generation governed by a Poisson process, and associated with each storm, a number of precipitation bursts that are Poisson (or geometric) distributed occurs. Then, if an instantaneous random depth R describes the rainfall burst, the resulting process is known as Neyman–Scott white noise, while if the rainfall burst is a rectangular pulse the process is known as Neyman–Scott rectangular pulse. Extensive developments, applications, and reviews on such models have been made (e.g., Burlando and Rosso, 1991; Cadavid et al., 1992; Cowpertwait and O’Connell, 1997; Entekhabi et al., 1989; Foufoula-Georgiou and Guttorp, 1986, 1987; Kavvas and Delleur, 1981; Obeysekera et al., 1987; Rodriguez-Iturbe et al., 1984, 1987a, and 1988). In addition, temporal rainfall models based on Cox processes (e.g., Smith and Karr, 1983), renewal processes (e.g., Buishand, 1977; Foufoula-Georgiou and Lettenmaier, 1987), Barlett–Lewis processes (e.g., Gyasi-Agyei and Willgoose, 1997; RodriguezIturbe et al., 1987a), and space-time multidimensional rainfall models have been developed (e.g., Islam et al., 1988; Krajewski and Smith, 1989; Smith and Krajewski, 1987; Wilks, 1998). 18.5  UNIVARIATE MODELING

Univariate time series models are used for modeling hydroclimatic processes, such as streamflow at a single site. There are various types of models available

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Univariate Modeling    18-3 

depending on the characteristics of the underlying time series. In this section, we give an overview of commonly used univariate models, some of them have been specifically developed for streamflow, while some others for precipitation. In general, models intended for shorter time scales, such as hourly or daily, are more complex than models for larger time scales. 18.5.1  ARMA and FARMA Models

The stationary ARMA models (Bras and Rodriguez-Iturbe, 1985; Brockwell and Davis, 2002; Box et al., 1994; Hipel and McLeod, 1994; Salas, 1993) is likely the most commonly used for modeling streamflow and other hydroclimatic series. Physical justification of ARMA models for rainfall-runoff process (Sec. 18.11) have been suggested (e.g., Salas and Smith, 1981). The ARMA(p,q) model of autoregressive order p and moving average order q is defined as: p

q

i =1

j =1

Yt = µY + ∑ φi (Yt −i −µY ) + ε t − ∑θ j ε t − j

(18.4)

where Yt is normally distributed with mean µY and variance s2(Y), εt is an independent normally distributed noise with mean zero and variance σ 2 (ε ) and independent of past observations, and {f1,…,fp} and {q1,…, qq} are  the autoregressive and moving average parameters, respectively. Simpler models such as AR(1) Yt = µY + φ1 (Yt −1 − µY ) + ε t and ARMA(1,1) Yt = µY + φ1 (Yt −1 − µY ) + ε t − θ1ε t −1 are often adequate, but higher order ARMA models may be necessary depending on the particular case. The characteristics of ACF and PACF of ARMA(p,q) models are summarized in Table 18.1. ARMA models are flexible and can accommodate features of alternative models, such as fractional Gaussian noise (Mandelbrot and Wallis, 1969), broken line (Mejia et al., 1972), and SM (Sveinsson et al., 2003). For details on parameter estimation of ARMA models [e.g., method of moments (MOM), least squares, and maximum likelihood], the interested reader may refer to Box et al. (1994) and Brockwell and Davis (2002). Fractional ARMA models, that is, FARMA(p, d, q), are direct generalization of ARMA and ARIMA models (Brockwell and Davis, 1991; Hipel and McLeod, 1994; Hosking, 1981). FARMA models are stationary with fractional differencing (1–B)d = 1 – dB – 0.5 d(1 – d) B2 – … , where B is the backward shift operator BnYt = Yt–n. Note that –0.5 < d < 0.5 is applied instead of an integer d as is used for the nonstationary ARIMA model. Then, the fractionally differenced time series is fitted by an ARMA model as: q

φ ( B)(1 − B)d (Yt − µY ) = ε t − ∑θ j ε t − j



(18.5)

j =1

where f(B) = 1 – f1B – f2B2 – … – fpBp is the autoregressive operator. Also note that due to the fractional differencing operator, the FARMA model has an infinite autoregressive part. An interesting feature of FARMA process is its long memory, where the ACF converges to zero with increasing lag-h at a much slower rate than that for the traditional ARMA process. The FARMA model has been used for modeling monthly and daily flows (Montanari et al., 1997, 2000), and the FARMA(0, d, 0) is similar to fractional Gaussian noise, which has been used in studies of the long-term dependence in hydroclimatic series (Kousoyiannis, 2002). An overview of fitting FARMA models is given by Brockwell and Davis (1991) and Hipel and McLeod (1994). Unlike ARIMA models, where d is fixed prior to estimating the parameters of the differenced series, all parameters in the FARMA model must be estimated simultaneously. Note that for practical applications 0 < d < 0.5. 18.5.2  GAR Model

The GAR(1) model is similar to the AR(1) model except that the underlying process is gamma distributed instead of normal. The GAR(1) model can be expressed as (Lawrence and Lewis, 1981):

Zt = φ Zt −1 + ε t (18.6)

where et is an independent noise term, f is the autoregression coefficient, and Zt is a three-parameter gamma distributed variable with marginal density function given by: fZ (z ) =

AR(1)

AR(p)

MA(q)

ARMA(p,q)

ACF

Decays geometrically

Tails off

Zero for lag > q

Tails off

PACF

Zero for lag > 1

Zero for lag > p

Tails off

Tails off

18_Singh_ch18_p18.1-18.12.indd 3

(18.7)

in which l, a, and b are the location, scale, and shape parameters, respectively. The noise term, εt, can be obtained by (Lawrence, 1982):

 η=0  Uj ε = λ (1 − φ ) + η , where  M  η = ∑ j =1Y j φ 

if if

M = 0 (18.8) M >0

where M is Poisson distributed with mean – b ln(f), Uj, j =1,2,.... are independent identically distributed (iid) random variables with uniform (0,1) distribution, and, Yj, j = 1,2,.... are iid exponential random variables with mean (1/a). The GAR(1) model has four parameters, {f, l, a, b}. It may be shown that the population moments of Zt are functions of the model parameters as: µ=λ+

β 2 β , σ2 = 2, γ = , ρ1 = φ α α β

(18.9)

where m, s2, g, and r1 are the mean, variance, skewness coefficient, and the lag-1 autocorrelation coefficient, respectively. Estimation of the GAR(1) model parameters can be made by the MOM using the results obtained by Matalas (1967), Wallis and O’Connell (1972), and the simulation studies conducted by Fernandez and Salas (1990). For more details on the estimation method and applications thereof, the interested reader is referred to Fernandez and Salas (1990). 18.5.3  Shifting Mean Models

The shifting mean (SM) model (Sveinsson and Salas, 2006; Sveinsson et al., 2003, 2005, 2009) is characterized by sudden shifts or jumps in the mean and can be used to model high and low frequency shifts of hydroclimatic processes (Sveinsson et al., 2003). The underlying process is assumed to be characterized by multiple stationary states, which only differ from each other by having different means that vary around the long-term mean. Thus, the process is stationary and autocorrelated, where autocorrelation arises from the sudden shifting pattern in the mean. The SM model is defined as: t



Xt = Yt + Zt , where Zt = ∑ Mi I( Si −1 ,Si ] (t ) (18.10) i =1

where {Xt} is a sequence of random variables representing the hydrological process of interest; {Yt} is a sequence of iid Normal (mY, σ Y2 ) variables that are independent of the mean level sequence {Zt}, {Mt} is iid Normal 2 (0, σ M = σ Z2 ), and I is the indicator function with Si = N1 + N 2 ++ N i , S0 = 0 , in which Ni is the length the process remains in mean level i (a discrete stationary delayed renewal sequence with Ni iid positive geometric with parameter p). Thus, the average length of each state of the process is 1/p. SM models have been used for modeling the Pacific Decadal Oscillation (PDO) index and the annual flows of the Niger River (Sveinsson et al., 2003), the quartermonthly annual maximum outflows of Lake Ontario, and the 7-day annual low flows of the Paraná River, Argentina (Sveinsson et al., 2005), where Yt was modeled by skewed distributions. SM model with AR(1) persistence (Sveinsson, 2002; Sveinsson and Salas, 2006), and Bayesian version of the SM model has been proposed (Fortin et al., 2004), and it has been shown that the SM models belongs to the class of hidden Markov-chain models (Fortin et al., 2004; Thyer and Kuczera, 2000). Parameters are estimated by the MOM as:

Table 18.1  Properties of the ACF and PACF for ARMA(p,q) Models

α β ( z − λ )β −1 exp [−α ( z − λ )] Γ(β )

ρˆ ( X ) ρˆ ( X ) 2 2 ; µˆY = µˆ X ; σˆ M pˆ = 1 − 2 = σˆ X2 1 (18.11) ; σˆ Y2 = σˆ X2 − σˆ M ρˆ1 ( X ) (1 − pˆ )

where parameters are feasible if ρˆ1 ( X ) > ρˆ 2 ( X ) > ρˆ12 ( X ) . To reduce the effects of sample variability of the estimated ACF and avoid unfeasibility of the SM model parameters, Sveinsson et al. (2005, 2009) suggested fitting the exact σ 2 (1 − p)h form of the model ACF ρh ( X ) = M2 , h = 1,2, to the sample ACF 2 σ Y +σ M 2 using least squares, that is, rh = a bh with a = σ M / σ X2 and b = 1 – p, where 0 < {a, b} < 1.

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18-4     Time Series Analysis and Models 18.5.4  Markov-Chain Models

We have seen in Sec. 18.4 that the models and properties of cumulative processes, such as rainfall over successive nonoverlapping time periods, that is, discrete time rainfall, can be derived from continuous time models. However, one may formulate models directly at discrete time scales, such as hourly and daily. In these cases, the theory of Markov chains has been widely used for modeling not only rainfall, but also other hydrologic processes such as streamflow, soil moisture, and water storage in reservoirs (e.g., Buishand, 1977; Chang et al., 1984; Feyerherm and Bark, 1964; Katz, 1977a, b; Katz and Parlange, 1995; Richardson and Wright, 1984; Roldan and Woolhiser, 1982). Consider that a discrete time process X(t) = j, j=1,...,r (where j denotes a state and r is the number of states) is represented by a first-order Markov chain (a simple Markov chain). For instance, in modeling daily rainfall, one may consider only two states, j=1 for a dry day (no rain) and j=2 for a wet day. A simple Markov chain is defined by a square transition probability matrix P(t), with elements pij(t)=P[X(t)=j|X(t–1)=i] for all i,j pairs. The marginal distribution of the chain being at any state j at time t is denoted by qj(t)=P[X(t)=j], j=1,...,r, and qj(0) is the distribution of the initial states. The Markov chain is homogeneous or stationary if P(t) does not depend on time and in this case, the notations P, pij, and qj are used. Common estimation methods include the MOM and maximum likelihood and estimation of the n-step transition probability pij(n ) is available in literature (e.g., Guttorp, 1995; Wilks, 1995). To test whether a simple Markov chain is an adequate model for the process under consideration, one can check some of the underlying assumptions of the model and whether it is able to reproduce some relevant properties of the rainfall process. For example, one can compare the n-step transition probability pij(n ) with that obtained from the data, pˆij(n ). Likewise, the AIC has been helpful in selecting the order of Markov chains (e.g., Katz, 1981). Although in some cases, simple Markov chains may be adequate for representing the occurrence of rainfall, often more complex models may be necessary. For instance, in modeling daily rainfall throughout the year, the parameters of the Markov chain may vary with time and the estimates of the transition probabilities pij can be fitted with trigonometric series to smooth out sample variations (e.g., Roldan and Woolhiser, 1982). Higher order Markov chains may be necessary as well. Chin (1977) analyzed daily precipitation records for many stations across the United States and concluded that for the majority of stations Second- and third-order models were preferred for the winter months, while a first-order model was better for the summer months. In addition, for daily rainfall, Roldan and Woolhiser (1982) used maximum likelihood for estimating Fourier series coefficients for alternating renewal processes and Markov chains while Katz and Parlange (1995) and Martinez and Salas (2004) used mixed models with periodic Markov chains for hourly rainfall. 18.5.5  DAR and DARMA Models

Markov chains have been widely used for modeling hydrologic time series that are temporally correlated. However, in many cases geophysical data may not be Markovian. Chang et al. (1984) showed that simple Markov chains do not properly describe some sequences of daily rainfall, and alternative models such as DARMA (discrete autoregressive moving average) were needed. DARMA models are the discrete counterparts of ARMA models and they are more suitable for modeling persistence characteristics with longer memory than simple Markov chains. DARMA(p,q) models (p = autoregressive order and q = moving average order) were originally developed by Jacobs and Lewis (1978). Low-order DARMA models have been used for modeling precipitation and streamflow (e.g., Buishand, 1978; Chang et al., 1984; Chebaane et al., 1992, 1995; Chung and Salas, 2000; Cancelliere and Salas, 2010). Some relevant definitions and properties regarding DAR(1) and DARMA(1,1) models are briefly reviewed as follows. The DAR(1) model may be defined as (Jacobs and Lewis, 1978):

Xi = Vi Xi−1 + (1 − Vi )Yi , i = 1, 2,, (18.12)

where Vi is an independent Bernoulli (0,1) process with parameter P[Vi = 1] = λ , 0 ≤ λ ≤ 1 and Yi is another independent Bernoulli (0,1) process with P[Yi = 1] = π 1 . Model (18.12) implies that Xi = Xi−1 with probability λ and Xi =Yi with probability 1 − λ . Its ACF is given by ρk ( X ) = λ k , k ≥1. It may be shown that the DAR(1) process is a first-order Markov chain with one-step transition probabilities given by:  λ + (1 − λ )π j , if i = j  , i , j = 0, 1 (18.13) p(i , j ) = P( X k+1 = j | X k = i ) =  if i ≠ j  (1 − λ )π j ,

18_Singh_ch18_p18.1-18.12.indd 4

Likewise, the DARMA(1,1) model may be expressed as (Jacobs and Lewis, 1978):

Xi = U iYi + (1 − U i )Zi−1 , i = 1, 2, (18.14)

in which Ui is an iid Bernoulli (0,1) process with parameter P[U i = 1] = β , 0 ≤ β ≤ 1, so that Xi = Yi with probability β , and Xi = Zi−1 with probability 1 − β , Yi is another iid Bernoulli variable with parameter p1 (as aforementioned), Zi–1 is a DAR(1) process with parameters l and p1 as in Eq. (18.12), and the variables {Xi}, {Zi}, and {Yi} are stationary and have the same probability distribution p0 and p1. The ACF of Xi is ρk ( X ) = cλ k−1 , k ≥1, where c = (1 − β )(λ + β − 2λβ ). It is a slower decaying correlogram (longer memory) than that of the DAR(1). The run length distributions, estimation procedures, and applications of the binary DAR(1) and DARMA(1,1) processes can be found in literature (e.g., Buishand, 1978; Chang et al., 1984; and Chung and Salas, 2000). 18.5.6  Product Models

The AR, ARMA, FARMA, SM, and GAR models discussed in the previous sections are useful for modeling hydrologic processes, such as sreamflow in perennial rivers, but they are impractical for intermittent streamflows in ephemeral streams and precipitation. Intermittent processes can be modeled as the product (Jacobs and Lewis, 1978): Yt = Xt Zt (18.15)



where Yt is a nonnegative intermittent hydrologic variable, Xt is a discrete autocorrelated variable, Zt is a positive-valued continuous autocorrelated variable, and Xt and Zt are assumed to be mutually uncorrelated. For example, the variable Xt may be represented by a dependent (1,0) Bernoulli process and Zt by an AR(1). Then the product Yt is intermittent and autoregressive. Product models have been applied for modeling short-term rainfall (e.g., Buishand, 1977; Chang et al., 1984) and intermittent streamflows (Chebaane et al., 1995; Salas and Chebaane, 1990). Further elaboration on product models considering periodic processes will be illustrated in Sec. 18.6.5. 18.6  UNIVARIATE PERIODIC MODELING

For seasonal hydrologic time series (e.g., monthly series) seasonal statistics such as the mean and variance may be reproduced by a stationary model, such as ARMA by removing seasonality in the underlying seasonal series, that is, subtracting the seasonal mean and dividing by the seasonal standard deviation. This procedure assumes that season-to-season correlations are the same throughout the year. However, hydrologic time series, such as monthly streamflows, generally exhibit different month-to-month correlations. Periodic models have been suggested in literature for modeling such periodic dependence structure (e.g., Hipel and McLeod, 1994; Loucks et al., 1981; Salas, 1993; Salas et al., 1980). Note that despite preserving the seasonal covariance, annual statistics may not be preserved. Sections 18.6.3 and 18.8 later further refer to this issue. 18.6.1  PARMA and PFARMA Models

PARMA models have been suggested in literature for modeling periodic dependence structure. For w seasons, a PARMA(p, q) model consists of w individual ARMA(p, q) models, where the dependence is across seasons instead of years. A PARMA(p, q) model may be expressed as (e.g., Hipel and McLeod, 1994; Salas, 1993) p

q

i =1

j =1

Yν ,τ = µτ + ∑ φi ,τ (Yν ,τ −i − µτ −i ) + εν ,τ − ∑θ j ,τ εν ,τ − j

(18.16)

where Yν ,τ represents the hydrologic process for year n and season t and µτ and σ τ2 (Y ) are the seasonal mean and seasonal variance, respectively. The noise term εn,t is uncorrelated normally distributed with mean zero and variance σ τ2 (ε ), {f1,t,…,fp,t} are periodic autoregressive parameters, and {q1,t,…, qq,t} are periodic moving average parameters. In many practical applications, low-order models, such as PAR(1), PAR(2), and PARMA(1,1) may be adequate, although residuals should always be tested to ensure adequate model fit. For details on parameter estimation and applications, the interested reader is referred to the references mentioned earlier. Fractional PARMA, that is, FPARMA(p,d,q) (Hipel and McLeod, 1994), is developed in a similar way as FARMA (Sec. 18.5.1). For each season of the year, a separate FARMA model is fitted as for the PARMA model. Hui and Li (1988) derived maximum likelihood estimation for the special case FPARMA(p,d,0).

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Univariate Periodic Modeling     18-5  18.6.2  PGAR Model

Stochastic modeling of skewed hydrologic series has always been a challenge because most models available in literature assume that the underlying process is normally distributed, thereby requiring data transformations to normality prior to applying the models. While this procedure has been a standard practice, it is always useful having an alternative where no transformation to normality is necessary. Section 18.5.2 presented the GAR model, which is stationary autoregressive with a gamma marginal distribution. While such a model is useful for some hydrologic series such as annual streamflow, its application to processes such as weekly rainfall is impractical. An alternative model for periodic hydrological processes at the seasonal time scales (e.g., monthly) is the PGAR (Fernandez and Salas, 1986). Let the underlying periodic process denoted by Qn, t be autoregressive having a gamma marginal distribution with parameter set (at, bt, lt), respectively, the scale, shape, and location parameters. Then, Zν ,τ = Qν ,τ − λτ is gamma with two parameters at and bt . There are two types of periodic gamma models, an additive model Zν ,τ = φτ Zν ,τ −1 + εν ,τ , where Zn, t is defined for season t (t =1,…, w) of year n (n =1,…,N) in which w is the number of seasons per year and N is the number of years considered, Zν ,0 = Zν −1,ω , ft is a periodic autoregressive parameter, and en, t is a random noise that has a particular distribution such that Zn, t is gamma. The model requires that bt increases from one season to the next or remains constant throughout the year. The second type of periodic gamma model is a product Zν ,τ = Zνθτ,τ −1 Wν ,τ , where qt is a periodic autoregressive exponent and Wn, t is a random noise, which also has a particular distribution such that Zn, t is gamma. The product model requires that the shape parameter bt decreases from one season to the next or remains constant. In each case, details of the noises en, t and Wn, t can be found in Fernandez and Salas (1986). In order to accommodate a varying shape parameter bt across the year, a mixture model was suggested as: Zν ,τ = φτ Zν ,τ −1 + Zνθτ,τ −1 Eν ,τ (18.17) in which Eν ,τ = εν ,τ if βτ ≥ βτ −1 or Eν ,τ = Wν ,τ if βτ < βτ −1 and the parameters 1/2

 α  β  ft and ft are given by φτ = ρ1,τ  τ   τ  if βτ ≥ βτ −1 otherwise ft = 0  α τ −1   βτ −1   β 1/2 and θτ = ρ1,τ  τ  if bt < bt–1, otherwise θτ = 0, where ρ1,τ is the correla βτ −1  tion between Zn, t and Zn, t–1. For further details on the PGAR model, the interested reader is referred to Fernandez and Salas (1986). The model has been tested extensively and its performance compares favorably respect to the traditional approach based on normal transformation and using normalbased models. However, the PGAR model is limited to a lag-1 correlation and is only available for single sites. Applications of the PGAR model to weekly and monthly streamflows are available (e.g., Fernandez and Salas, 1986; Chebaane et al., 1992). 18.6.3  XPARMA Model

A desirable objective in simulating hydrological processes is the preservation of statistics at multiple time scales (e.g., monthly and annual). For example, for generating seasonal flows two approaches have been followed in literature. The first has been generating seasonal flows directly using a periodic model, while the second has been generating annual flows in a first step, then obtain the seasonal flows by temporal disaggregation (Sec. 18.8). Low-order PARMA models generally do not capture the annual flow properties when annual flows are obtained from seasonal flows (Obeysekera and Salas, 1986). While disaggregation approach has been useful from the practical standpoint, one may not be satisfied from the theoretical standpoint, since it is only logical that if a model that is applied directly to seasonal flows fails to reproduce the statistics at the aggregated level (annual), it must be because the model was not right in the first place. The XPARMA model below offers an alternative. The stationary ARMA(p,q) process may be expressed as φ ( B)Yt = θ ( B)ε t in which Yt is stationary with mean zero, et = normal noise with mean zero and variance σ 2 (ε ), φ ( B) = 1 − (φ1B1 + ... + φ p B p ) and θ ( B) = 1 − (θ1B1 + ...+ θ q Bq ) with B j Zt = Zt − j . Likewise, the PARMA(p,q) process may be expressed as φτ ( B)Yν ,τ =θτ ( B)εν ,τ where Yν ,τ = periodic correlated process with mean zero, εν ,τ = normal noise with mean zero and variance σ τ2 (ε ), ν is the year and t is the season, φτ ( B) = 1 − (φ1,τ B1 + ... + φ p ,τ B p ), and qt = (B) = 1– (θ1,τ B1 + ... + θ q ,τ Bq ) with B j Zν ,τ = Zν ,τ − j for j ≤ t, otherwise B j zν ,τ = zν −1,ω +τ − j . Box et al. (1994) suggested a multiplicative model that could be useful for seasonal series. Let Yt represent the underlying seasonal series after removing

18_Singh_ch18_p18.1-18.12.indd 5

the seasonal mean. We will fit an ARMA(P,Q)ω model to Yt as Φ( Bω )Yt =Θ( Bω )ξt in which xt is correlated normal with mean zero and variance σ 2 (ξ ) , Φ( Bω ) = 1 − (Φ1B1ω + ...+ Φ P B Pω ), and Θ( Bω ) = 1 − (Θ1B1ω + ...+ ΘQ BQω ) . For instance, for P=2 and Q=0, the model is written as Yt = Φ1 Yt −ω + Φ 2Yt −2ω + ξt , which is like an ARMA(2,0), but defined over blocks of ω seasons (years). Recall that t is time in seasons. In addition, the residual xt is fitted as φ ( B)ξt =θ ( B)ε t . Then, combining the two aforementioned models above gives Φ( Bω )φ ( B)Yt =Θ( Bω )θ ( B) ε t

(18.18)

which is the stationary multiplicative ARMA(p,q)x(P,Q)ω process. The drawback with the foregoing multiplicative model for seasonal processes such as streamflows is that there is no provision for reproducing periodic covariance structure. A model that incorporates such periodic features is the multiplicative PARMA (XPARMA) which may be written as (Salas and Abdelmohsen, 1992): Φτ ( Bω )φτ ( B)Yν ,τ = Θτ ( Bω )θτ ( B) εν ,τ

(18.19)

where Yn,t = periodic series after removing the seasonal mean, en,t = normal uncorrelated with mean zero and variance σ τ2 (ε ), Ft (Bw) = 1–(F1,t B1w +...+ Fp,t BPw), and Qt (Bw) = 1–(Q1,t B1w +...+ QQ,t BQw). For instance, the PARMA(1,1) x(1,1)ω is a multiplicative process which is written as Yn,t  = F1,t Yn–1,t + φ1,τ Yν ,τ −1 − Φ1,τ φ1,τ Yν −1,τ −1 + εν ,τ − Θ1,τ εν −1,τ − θ1,τ εν ,τ −1 + Θ1,τ θ1,τ εν −1,τ −1. Monthly flows of the Nile River at Aswan (1871–1989) were used for comparing the PAR(1), PARMA(1,1), and XPARMA(1,1)x(1,1)12 models. The model parameters were estimated by least squares based on the logarithmic transformed data. For each model, samples of the same length as the historical record were generated which were used for comparing the monthly and annual statistics versus those obtained from the historical record. The Nile River flows show significant year-to-year dependence for monthly flows, significant month-to-month correlations as well as significant dependence structure for annual flows. The results showed that generally all models reproduced the month-to-month correlations quite well. However, the yearly correlations for each month and the annual correlations were not reproduced by the PAR(1) and PARMA(1,1) models. On the other hand, the correlations derived from the XPARMA(1,1)x(1,1)12 model resembled closely those from the historical record. Likewise the storage- and drought-related statistics were better reproduced by the multiplicative model (Salas and Abdelmohsen, 1992). 18.6.4  PDAR and PDARMA Models

Just as PAR and PARMA models are useful for modeling continuous valued hydrological process at the seasonal time scale, PDAR and PDARMA models are useful for discrete valued processes, such as the occurrence of hourly rainfall or the occurrence of seasonal streamflows in dry regions. For illustration, we assume modeling the occurrence of hourly rainfall in places where there is evidence of the daily cycle. The occurrence of hourly rainfall may be represented by Xν ,τ = 1 if rainfall occurs during the hour t of day v and Xν ,τ = 0 otherwise. Such binarydependent variable may be expressed mathematically by a periodic discrete autoregressive model of order one, PDAR(1), as: Xν ,τ = Vν ,τ Xν ,τ −1 + (1 − Vν ,τ )Wν ,τ ; ν =1,2,...; τ =1,2,...,ω

(18.20)

such that Xν ,0 = Xν −1,ω , where ω is the number of hours. In addition, Vν ,τ and Wν ,τ are independent Bernoulli with P[Wν ,τ = 1] = π 1,τ , 0 ≤ π 1,τ ≤ 1, P[Wν ,τ = 0] = π 0,τ = 1 − π 1,τ , and P[Vν ,τ = 1] = λτ , 0 ≤ λτ ≤ 1. Therefore, Xν ,τ = Xν ,τ −1 with probability λτ and Xν ,τ = Wν ,τ with probability 1 – λτ . Furthermore, it may be shown that PDAR(1) is equivalent to a periodic simple Markov chain with transition probability matrix 0 0

Pτ =

1

 λτ + (1 − λτ )π 0,τ    (1 − λ )π τ 0,τ 

1 (1 − λτ )π 1,τ

λτ + (1 − λτ )π 1,τ

    

(18.21)

A number of properties that are useful for parameter estimation, simulation, and forecasting can be determined (Chebaane et al., 1992, 1995) such as: (a) the expected value E[ Xν ,τ ] = µτ ( X ), that is, the probability of rainfall during the hour t, P[ Xν ,τ = 1], τ =1,...,ω ; (b) the variance and the lag-k autocovariance; (c) the n-step periodic transition probability matrix Pτ(n ); and (d) the probabilities of the durations of dry and wet periods (i.e., the sequence of nonrainy and rainy hours, respectively).

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18-6     Time Series Analysis and Models

Katz and Parlange (1995) suggested that in order to improve the estimation of upscale statistics, for example, daily statistics, one could improve the modeling of the occurrence process Xν ,τ . For example, low-order periodic DARMA (PDARMA) models, such as the PDARMA(1,1) model, can be formulated (Chebaane et al., 1992) as: Xν ,τ = Uν ,τ Wν ,τ + (1 − Uν ,τ ) Zν ,τ −1

(18.22a)

Zν ,τ = Vν ,τ Zν ,τ −1 + (1 − Vν ,τ )Wν ,τ

(18.22b)

where Uν ,τ and Vν ,τ are independent Bernoulli processes with probabilities P[Uν ,τ = 1] = β1,τ , 0 ≤ βτ ≤ 1 and P[Vν ,τ = 1] = λτ , 0 ≤ λτ ≤ 1, and Wν ,τ as defined earlier. The parameters of PDAR and PDARMA models can be estimated by the MOM and maximum likelihood (Chebaane et al., 1992; Katz and Parlange, 1995). Applications of the PDAR(1) model for modeling the occurrence of hourly rainfall have been made (e.g., Katz and Parlange, 1995; Martinez and Salas, 2004). Also Salas and Chebaane (1990) and Chebaane et al. (1992) used the PDAR(1) model for modeling the occurrence of monthly streamflows in arid regions. 18.6.5  Product Periodic Models

In Sec 18.5.6, we have considered product models for stationary processes. In this section, we consider periodic (seasonal) processes, such as hourly rainfall in some regions and monthly streamflows in dry regions. For the ease of explanation, let Yν ,τ be a positive intermittent variable that represents hourly precipitation (or any time interval smaller than a day such as 4 h) in which n = day, t = time interval (h) within the day. The variable Yν ,τ can be expressed as the product: Yν ,τ = Xν ,τ Zν ,τ ; ν = 1,2,...; τ =1,2,...,ω

(18.23)

where w = number of time intervals in the day (e.g., w = 24 for hourly precipitation), Xν ,τ is a binary (0,1) periodic-dependent variable that represents the occurrence process, and Zν ,τ is a periodic-dependent continuous variable representing the precipitation amount. In addition, Xν ,τ and Zν ,τ are assumed to be mutually independent. Alternatively, Eq. (18.23) can be expressed as Yν ,τ = Zν ,τ if Xν ,τ = 1 and Yν ,τ = 0 if Xν ,τ = 0 . The model is assumed to be valid for a given season or month of the year. Modeling of the discrete periodic process Xν ,τ has been described in Sec. 18.6.4. Therefore, the remaining part of this section addresses briefly modeling of the continuous variable Zν ,τ . The variable Zν ,τ represents the amount of rainfall during the rainy hours, that is, Zν ,τ > 0 if Xν ,τ = 1. The variable Zν ,τ is generally skewed and autocorrelated, and may be modeled in several ways depending on the data at hand. For example, depending whether or not the hour-to-hour correlations vary with the hour a model with constant or periodic parameters will be selected. In addition, short-term precipitation may be highly skewed. For example, the skewness coefficient of July daily rainfall for Denver Airport is about 5 based on 1949–1990 record. Thus, one will need to include this characteristic in the model. Two possibilities are: Gaussian models that require a previous step of transformation and non-Gaussian models. They are further described as follows. Several alternatives are available for modeling Zν ,τ , such as low-order ARMA and PARMA models after transformation. For example, suppose a power transformation is used to transform Z into a normal variable N as Nν ,τ = [Zν ,τ ]cτ , where ct is an exponent that may vary with the hour, and a PAR(1) model is fitted to represent the variable N as:

Nν ,τ = µτ ( N ) +

σ τ (N ) φτ  Nν ,τ −1 − µτ −1 ( N ) + σ τ ( N ) εν ,τ σ τ −1 ( N ) 

(18.24)

in which e is a standard normal random variate. The parameters of this model are µτ ( N ) , σ τ2 ( N ) , and φτ . Using the power transformation one could find a relationship between the moments of N and Z. For illustration if ct =1/2 one can find that µτ ( Z ) = µτ2 ( N ) + σ τ2 ( N )

(18.25a)

σ τ2 ( Z ) = 2σ τ2 ( N )[σ τ2 ( N ) + 2 µτ2 ( N )] (18.25b) Katz and Parlange (1995) used the PAR(1) model for July hourly rainfall of Denver Airport except that the transformation parameter was assumed to be a constant value, that is, ct = 1/8 and the autoregression coefficient ft was also assumed a constant value throughout the day. The parameters µτ ( N ) and σ τ2 ( N ) were estimated from the transformed data and used the moment relationships as a way of verifying the estimates of µτ ( Z ) and σ τ2 ( Z ) . Alternatively, Martinez and Salas (2004) applied an estimation procedure in

18_Singh_ch18_p18.1-18.12.indd 6

which µτ ( Z ) and σ τ2 ( Z ) were estimated from the conditional hourly rainfall data and used the moment Eqs. (18.25a) and (18.25b) to estimate the parameters µτ ( N ) and σ τ2 ( N ). The autoregressive parameter ft was determined by the MOM. Clearly using an autocorrelated non-Gaussian model directly in terms of the underlying precipitation variable Z would be desirable. Two alternatives that assume that Z is gamma distributed are the GAR model and the PGAR model as described in Secs. 18.5.2 and 18.6.2. 18.7  MULTIVARIATE MODELING

Multivariate models are used for modeling temporal and spatial dependence structure at multiple lags. Analysis and modeling of multiple time series is more involved than for a single time series. The basic principles are similar except that vector and matrix notations are used for model definition, properties, and parameter estimation. Full multivariate models generally have many parameters, but simpler contemporaneous models in which the parameter matrices are diagonal are practical. Both models can preserve complex temporal dependence structure for each site, but only lag-0 correlations across sites are preserved in contemporaneous models, which is usually considered adequate for hydroclimatic series, while higher lagged correlations can be preserved in full multivariate models. 18.7.1  Multivariate ARMA and PARMA Models

Properties of multivariate ARMA and PARMA models have been widely studied (e.g., Bernardo and Smith, 1994; Box et al., 1994; Brockwell and Davis, 1991, 2002; Hipel and McLeod, 1994; Shumway and Stoffer, 2000). Simpler contemporaneous ARMA and PARMA models are often used for hydroclimatic data at multiple sites (Hipel and McLeod, 1994; Salas, 1993; Salas et al., 1980). The full multivariate ARMA (MARMA) and the contemporaneous ARMA (CARMA) models for n sites are expressed as: p

q

i =1

j =1

Yt = ∑ Φi Yt −i + ε t − ∑ Θ j ε t − j

(18.26)

where Yt is a n × 1 column vector of normally distributed zero mean variables (i.e., it is assumed that the mean of the data have been subtracted prior to modeling). The CARMA model is more parsimonious than MAR. For example, the number of parameters for MAR(2) and CARMA(2,0) models are 2n2 + 0.5(n2 + n) and 2n + 0.5(n2 + n), respectively. Similarly, the full multivariate MPARMA and contemporaneous PARMA with periodic parameters, that is, CPARMA, can be expressed as: p

q

i =1

j =1

Yν ,τ = ∑ Φi ,τ Yν ,τ −i + εν ,τ − ∑ Θ j ,τ εν ,τ − j (18.27) where Yν ,τ is a n × 1 column vector of normally distributed zero mean variables for year n and season t. The autoregressive and moving average parameter matrices are full n × n matrices for the MARMA and MPARMA models, but they are diagonal for the CARMA and CPARMA models. The n × 1 noise vector {εt} is iid MVN (0,G) and vector {εν ,τ } ~ iid MVN ( 0 , Gτ ) for the stationary and periodic models, respectively. For all models, G and Gt are full n × n matrices, enabling the preservation of the lag-0 correlations across sites. In practice low-order models, such as MAR(1), MAR(2), and MARMA(1,1), and their periodic counterparts (Hipel and McLeod, 1994; Matalas, 1967; Salas and Pegram, 1979; Salas et al., 1980) have been used for modeling streamflows at multiple sites. The moment equations of the MAR(p) model p

p

i =1

i =1

are C 0 = G + ∑Φi CTi and C h = ∑Φi C h−i , h ≥ 1, where C h = E[Yt YtT−h ] is

the

lag-h population cross-covariance matrix. For the MAR(1) model, the paramˆ = c c −1 and G ˆ = c − c c −1c T , where c is the lag-h eters are estimated by Φ 1 1 0 0 1 0 1 h sample cross-covariance matrix. Similarly, for the MPAR(1) model, the −1 T ˆ = c c −1 ˆ parameters are estimated by Φ 1,τ 1,τ 0,τ −1 and Gτ = c 0,τ − c 1,τ c 0,τ −1c 1,τ . Since the CARMA and CPARMA models have diagonal parameter matrices, they can be decoupled into univariate ARMA and PARMA models and the parameters estimated independently for each site using the corresponding univariate model-estimation procedures. This allows for identification of the best univariate model for each site. After having estimated the diagonal parameter matrices what remains is estimation of the noise variance-covariance matrix G or Gt. In either case, the process is the same and it is illustrated here for the CARMA model, which requires that the individual ARMA models are causal. Causality implies that CARMA(p, q) can be written as an infinite mov∞ ∞ ing average model, for example, Yt = ∑ Ψ j ε t − j with C 0 = ∑ Ψ j GΨ Tj , where p

j =0

j =0

Ψ 0 = I and Ψ j = −Θ j + ∑Φi Ψ Tj −i . Since for contemporaneous models, Ψ j i =1

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Nonparametric Models    18-7 

are diagonal then the ith row and jth column element of G is estimated from ∞ Gˆ ij = c0ij ∑ ψkiiψkjj . k =0

18.7.2  Multivariate SM Model

Multiple time series of different variables and different hydrologic regimes may require mixing of models. For example, shifts in time series of one hydrologic variable may not be present in the series of another variable because of differences in their geographic locations. Sveinsson (2002) and Sveinsson and Salas (2006) developed a multivariate contemporaneous shifting mean (CSM) model for simulating the annual net basin supplies of the Great Lakes. Since not all of the Great Lakes data showed shifting patterns, the CSM model was mixed with a CARMA model for simulating jointly the entire Great Lakes system. The CSM-CARMA model is capable of preserving the spatial lag-0 correlations between different sites and other important statistical characteristics of the individual sites. In addition, for sites modeled by the CSM model certain dependence structure in both time and space is preserved through the mean level process Zt. The CARMA portion of the model is capable of preserving more complex time dependence structure for each individual site depending on the AR and MA order of the model. In Sveinsson and Salas (2006), the Yt component in the CSM model was also contemporaneous AR(1). The mixed CSM-CARMA model for n sites can be written as:  X (1) t     (n1 )  Xt  (n +1) 1  Xt    (n )  Xt

  Y (1) t        (n1 )   Yt  =  (n1 +1)   Yt      (n ) Y   t

  (1)   Zt        Zt(n1 )  +    0        0

        

(18.28)

where n = n1+ n2, the first n1 sites follow a CSM model and the remaining n2 sites a CARMA(p,q) model. For the CSM model, the mean level sequence t

Zt = ∑ Mi I (Si −1 ,Si ] (t ) is defined in the same way as for the univariate SM i =1

model (Sec. 18.5.3) and {Yt } ~ iid MVN (µ Y , C Y (0)) . If the shifts in the means are caused by changes in natural processes, such as low-frequency variability of the climate system and the mean levels are correlated in space at lag-0, then {Mi } ~ iid MVN[0 , C M (0)] and a necessary and sufficient condition for {Zt} to be stationary in the covariance C Z (h) = (1 − p )h C M (0), h = 0,1, , is that N1 , N 2 , is a common sequence for all sites, that is, the time series of the hydrologic variables within a geographic region would all exhibit shifts at the same time periods (note that if {Mi} are assumed uncorrelated in space the stationarity condition where N1 , N 2 , is a common sequence is not necessary). The parameter-estimation procedure for the CSM model is similar as for other contemporaneous models. That is, first the CSM is uncoupled into univariate SM models and the parameters at each site estimated using the univariate SM model procedures (Sec. 18.5.3). Then the common parameter of the positive geometric distribution, p, for all sites is taken as the average of the at site sample estimates [or as a weighted average if the sample records are of different lengths as in Sveinsson and Salas (2006)]. Subsequently, the parameters of each univariate SM model are reestimated using the common pˆ. Lastly, the nondiagonal elements of CM(0) and CY(0) ˆ (0) = (1 − pˆ )−1 C ˆ (1) and made symmetric by replacing are estimated from C M X ij (0) and cˆMji (0) with their respective averages, the off diagonal elements cˆM ˆ (0) = C ˆ (0) − C ˆ (0) . then C Y X M Estimation of the CARMA part of the model (if n2>0) is made by fitting the best ARMA model for each site using the parameter estimation procedure for the multivariate CARMA model (Sec. 18.7.1). For estimation of the variancecovariance matrix of the noise (G) of the CARMA component Yt, the procedure of the CARMA model is applied, where each of the elements of Yt corresponding to the CSM component is considered as an ARMA(0,0) model. The upper left n1 × n1 part of the n × n estimated G matrix is replaced by the estimate of CY(0). 18.8  DISAGGREGATION MODELS

Disaggregation models are capable of reproducing statistics at different aggregation levels, such as annual and seasonal. Disaggregation in space and time (downscaling) is important for modeling complex hydroclimatic processes, whether it is downscaling precipitation to fine time scales or disaggregating flows and precipitation in a complex stream network system (Hipel and McLeod, 1994; Santos and Salas, 1992; Salas, 1993; Sveinsson et al., 2009) such

18_Singh_ch18_p18.1-18.12.indd 7

as for the Great Lakes system in North America (Fagherazzi et al., 2005) or the Nile River system in Africa (Salas et al., 1995). Software packages such as SAMS (Salas et al., 2006; Sveinsson et al., 2009) and SPIGOT (Grygier and Stedinger, 1990) have been developed for applications of alternative disaggregation modeling techniques. In general, annual flow series at key sites or regions can be disaggregated in stages, for preserving the temporal (annual and seasonal) and spatial statistics. The disaggregation models introduced by Valencia and Schaake (1973) and Mejia and Rousselle (1976) are often used in practice. The Mejia– Rousselle model for spatial disaggregation of annual or seasonal data from n key sites to m subsites can be respectively represented as: Yν = A Xν + B εν + C Yν −1

(18.29)

Yν ,τ = Aτ Xν ,τ + Bτ εν ,τ + Cτ Yν ,τ −1

(18.30)

where X is a vector of flows at key sites, which is modeled by say an ARMA or CARMA model, Y is a vector of flows at subsites, and A, B, and C are full m × n, m × m, and m × m parameter matrices, respectively. The independent noise vector εv is iid MVN(0, G = BBT), and εv,t is iid MVN (0, Gt = BtBtT), where B (or Bt) is the Cholesky decomposition of G (or Gt). This model preserves the lag-1 correlation coefficient in space and time between all subsites through the matrices C and Ct . The parameters are estimated by the MOM (Hipel and McLeod, 1994; Salas, 1993; Sveinsson et al., 2009). The model can be repeatedly applied for further spatial disaggregation from subsites to subsequent sites as is common when modeling complex streamflow networks. The disaggregation aforementioned schemes were initially proposed for temporal disaggregation of annual flows into seasonal flows, requiring many parameters for temporal disaggregation. Alternatively, condensed models (Lane) (Lane, 1979, 1981) and contemporaneous models (Spigot) (Grygier and Stedinger, 1988, 1990) with periodic parameters for temporal disaggregation have been suggested reducing the number of parameters required significantly (Sveinsson, 2014). The Lane and Spigot models for disaggregating annual Yν to seasonal Yν ,τ data (year ν, season t) for n sites can be respectively represented as: Yν ,τ = Aτ Yν + Bτ εν ,τ + Cτ Yν ,τ −1

Yν ,τ = Aτ Yν + Bτ εν ,τ + Cτ Yν ,τ −1 + Dτ εν ,τ

(18.31) (18.32)

where Lane’s model in Eq. (18.31) has full n × n periodic parameter matrices and Spigot model in Eq. (18.32) has diagonal contemporaneous periodic matrices except for matrix B which is full. Both models preserve the lag-1 season-to-season correlations between all subsites through the matrix C and the lag-0 correlations in space through matrix B. These models can be repeatedly applied in a stage-wise manner down to finer time scales (Santos and Salas, 1992), further reducing the number of parameters in each disaggregation step. Lane’s model requires adjustments of seasonal values to ensure additivity up to the aggregated level while Spigot model includes a transformation dependent term [last term in Eq. (18.32)] ensuring approximate additivity of the model in the original domain. For annual data, only spatial disaggregation models are needed, while for seasonal data both spatial and temporal disaggregation models are used. The foregoing models can be applied in different order for seasonal data, for example, “Spatial-Temporal Disaggregation” first disaggregating annual data in space from key sites to subsites to subsequent sites, etc., and then applying temporal disaggregation for different groups of sites, or “Temporal-Spatial Disaggregation” first disaggregating key sites annual data down to seasonal and then disaggregating the seasonal data in space from key sites to subsites, etc. When using disaggregation models for data generation for data requiring normalizing transformation or when various elements of the parameter matrices are simplified in condensed or contemporaneous models, then summability may be lost and adjustments may be needed to ensure additivity constraints in the original domain (Grygier and Stedinger, 1988; Sveinsson et al., 2009). 18.9  NONPARAMETRIC MODELS

Nonparametric models are an attractive alternative to parametric models. They do not require the data to be transformed to normal and they can capture features, such as bimodality and nonlinear dependence structure that is difficult to capture with traditional parametric models (Lall, 1995), although it has also been suggested for hydrologic data to log-transform the data prior to kernel density estimation (Rajagopalan et al., 1997, 2010) or use skewed gamma kernels (Salas and Lee, 2010). The simplest nonparametric models are pure resampling or bootstrapping models, such as the simple bootstrap based

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18-8     Time Series Analysis and Models

on resampling with replacement of the historical time series. To account for autocorrelation block bootstrapping can be used (Vogel and Shallcross, 1996). Nonparametric models based on conditional kernel density estimation have been used for monthly streamflow simulation and disaggregation of annual to monthly (Lee et al., 2010; Sharma et al., 1997; Tarboton et al., 1998). The k-nearest neighbor resampling models have been used for resampling monthly streamflows, disaggregating annual to monthly flows, and simulating rainfall and other climatic variables (Lall and Sharma, 1996; Lee and Ouarda, 2011; Prairie et al., 2006, 2007, and 2008; Rajagopalan and Lall, 1999; Salas and Lee, 2010; Yakowitz and Karlsson, 1987). 18.10  STOCHASTIC SIMULATION, FORECASTING, AND UNCERTAINTY

Stochastic simulation and forecasting and methods to deal with parameter uncertainty have been discussed in literature (e.g., Lettenmaier and Wood, 1993; Salas, 1993; Valdés et al., 2002; Sveinsson, 2014; Salas et al., 2014). Timeseries models enable generating (simulating) equally likely synthetic series that may be used for planning and management of water resources systems. Random generators of univariate and multivariate standard normal variables are available in many software packages. Exact generation procedures are available for simple models (e.g., Salas and Abdelmohsen, 1993), while approximations are generally used for complex correlated models that depend on past observations and residuals’ initial assumptions. In practice, a short warm-up period is often used to remove the effects of initial assumptions, which is equivalent to generating a longer time series that is needed and discarding the values of the warm-up period. The length and the number of the simulated series may vary. For comparing model and historical statistics, the length of the simulated series should be the same as the length of the historical record, while for operational and planning studies the length of the simulated series should match the planning horizon. The number of generated series (e.g., 1000) will provide the information to estimate the probability distribution or the uncertainty of the decision parameter. Forecasting can be done based on the models presented in this chapter simply by starting simulations at the end of the historical records conditioned on the most recent historical observations with all noises initialized to zero. Adaptive forecasting or Kalman filtering techniques (Lettenmaier and Wood, 1993) can be used allowing the model parameters to be updated at each time step prior to making the next step-ahead forecast. Forecast performance of different model alternatives is often measured in terms of mean-absoluteforecast error and root-mean-square-forecast error (Lettenmaier and Wood, 1993). Forecasting models including exogenous information usually perform better than models based only on past observations of the underlying process. For example, for streamflow forecasting the exogenous variables may be basin precipitation, climatic indices such as El Niño-Southern Oscillations (ENSO) and PDO or other measures of atmospheric and oceanic circulation influencing the local climate (e.g., Sveinsson et al., 2008b; Salas et al., 2011). If there are many exogenous variables available alternative models could be fitted and the resulting forecasts combined by averaging or weighting (Sveinsson et al., 2008a). Furthermore, the parameter space may be reduced using principal components on the exogenous variables. Popular models including exogenous variables are transfer function noise (TFN) models and simpler alternatives, such as ARMAX and PARMAX, which are extensions of the ARMA and PARMA models (Box et al., 1994; Brockwell and Davis, 2002; Hipel and McLeod, 1994). A TFN model and ARMAX model with one exogenous variable can be presented respectively as: Yt − µY =

ω ( B) b θ ( B) B ( Xt − µ X ) + εt δ ( B) φ ( B)

(18.33)

Yt − µY =

ω ( B) b θ ( B) B ( Xt − µ X ) + εt φ ( B) φ ( B)

(18.34)

with Yt being the dependent variable, Xt being the exogenous variable leading the dependent variable by lag-b, ω(B), δ(B), θ(B), and f(B) are polynomials in B and εt is the white noise term. When Yt is weekly autocorrelated, it is not uncommon to have the order p = 0 and q = 0 for the autoregressive operator f(B) and the moving average operator θ(B), respectively, and thus the only input being the exogenous variable. The main difference between the two models relates to the estimation procedure, although both models can be expressed as traditional regression models. In the ARMAX model, typically both series are prewhitened using appropriate ARMA models and then causal relationships between the residual series are analyzed at different lags. In the TFN model, the input series Xt is prewhitened and the output series Yt is then filtered with the same model. The impulse response function ω(B)/δ(B)

18_Singh_ch18_p18.1-18.12.indd 8

is then directly proportional to the cross-correlation coefficients at different lags of the residual series. In practice, the selected model(s) should be physically consistent and be included among those with the lowest AIC and SIC. For details on alternative estimation procedures, refer to Box et al. (1994), Brockwell and Davis (2002), and Hipel and McLeod (1994). Uncertainty of model parameters is a well-known issue, but not always taken into account to estimate predictive distribution or confidence limits of decision variables or forecasts. For some models, such as ARMA and PARMA, the sample distributions of the parameters are defined (e.g., Brockwell and Davis, 2002; Hipel and McLeod, 1994), Also, since most parametric modeling schemes are linear in the parameters and assume that the data are normally distributed, the inclusion of parameter uncertainty can be achieved by adopting the theory of parameter inferences from the classical univariate and multivariate regression model (Johnson and Wichern, 2002). In addition, Bayesian modeling (Bernardo and Smith, 1994) offers a framework to take into account the uncertainty in the model parameters. When multiple time series are synthetically generated from a parametric model for analysis of the distribution of certain design related statistics, it can be argued that parameter uncertainty is being somewhat implicitly embedded in the generation process, since for each synthetic series the parameters could be reestimated and they certainly would be different than the original parameter set. For complex models, repeated one-step ahead prediction can be used for estimating the prediction interval. Furthermore, if non-normality of the residuals is of concern bootstrapping of the residuals with replacement and reestimation of the model parameters can be used as an alternative for estimating the prediction interval (Sveinsson et al., 2008a). 18.11  CONCEPTUAL STOCHASTIC MODELING

In previous sections, we presented stochastic models, such as ARMA and FARMA that may be useful for simulating hydrological processes, such as streamflows. However, these models are also used in several fields other than hydrology. In this section, we briefly illustrate a conceptual framework which may justify applying the foregoing models for streamflow processes. For this purpose, we follow the concepts outlined in Fiering (1967) and Salas and Smith (1981), where the modeling framework is considered at the annual time scale. The key input in the simple basin system is precipitation. Some studies in literature suggest that precipitation is white or near white noise, say short memory Markovian (e.g., Blender et al., 2006; Potter, 1979); while others indicate that in some regions precipitation exhibits long-term variability and shifting patterns similar to those observed in some atmospheric/oceanic indices (e.g., Enfield et al., 2001) and Koutsoyiannis and Langousis (2011) suggested that precipitation has long memory. In the following conceptual watershed both short- and long-memory precipitation are assumed. Let us define a basin, where Xt is average annual precipitation, Gt is the groundwater storage, and Qt is the annual streamflow (t is in years). The following assumptions are made (Fig. 18.1): annual infiltrated precipitation, It = aXt ; annual evapotranspiration, Et = bXt ; and annual surface runoff Rt = (1 − a − b)Xt , in which 0 0

(21.15a)

1−

b( x − c ) >0 a

1

 (t − µ )2   x −µ exp  − dt = Φ   σ  2σ 2  

F(x ) = ∫

σ >0

Φ(): standard Normal distribution function (c.d.f.)

µ > 0; λ > 0

 λ  x   2λ   λ  x   F(x ) = Φ   − 1  + exp  µ  Φ  − x  µ + 1  (21.13b)  x  µ   

µ > 0; λ > 0 a ∈R µ ∈R

−∞

1 σ (2π ) 2

(21.12b)

  2λ   λ  x −a  λ  x −a  − 1  + exp   Φ  − + 1  (21.14b) F(x ) = Φ      µ  x − a  µ  x −a  µ  ln x − µ  F(x ) = Φ   σ 

(21.15b)

σ >0

σ x = exp(2µ + σ 2 )(exp(σ 2 ) −1)

a,  µ , ∈R

 ln( x − a) − µ  (21.16b) F(x ) = Φ    σ

(21.15d)

(21.15c) exp(−

(ln( x − a) − µ )2 ) (21.16a) 2σ 2

σ >0

1  x f ( x ) = exp  −  ,  x > 0 (21.17a)  a a 1  b( x − c )  f (x ) =  1 −  a a 

x

µ ∈R

1

Inverse Gaussian Three-Parameter

c.d.f.

1−b b

1    b( x − c )  b  exp  −  1 − (21.18a)    a    

1  (x − c)  (x − c)  f ( x ) = exp  − − exp  −  (21.19a)  a a a    x −c ≥0

a>0 a > 0; b, c ∈R a>0 c ∈R

 x F ( x ) = 1 − exp  −  (21.17b)  a   b( x − c )  1  b F ( x ) = exp  −  1 −   (21.18b) a    

  x −c F ( x ) = exp  − exp  −    a 

(21.19b)

(Continued)

21_Singh_ch21_p21.1-21.12.indd 3

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21-4    Frequency Distributions Table 21.3  Commonly Applied Continuous Probability Distributions in Hydrology (Continued) Distribution

p.d.f. f (x ) =

Log-EV I

Parameters

1  ln x − c  ln x − c   exp  − − exp  −  (21.20a)  ax a a   

ln x − c ≥ 0

c.d.f.   ln x − c   F ( x ) = exp  − exp  −    a  

(21.20b)

        c ∈R − β −1

Extreme Value Type II (EVII)

f (x ) =

β  x −c   a a 

Extreme Value Type III (EVIII)

f (x ) =

β  x −c −  a a 

Weibull

a x f (x ) =   b b

Gamma

f (x ) =

a −1

(21.21a)

a, β > 0; x >c

  x − c  −β  F ( x ) = exp  −      a  

(21.21b)

  x −c β  exp  −  −   (21.22a) a    

a > 0;  c ∈R ,  β > 0

  x −cβ  F ( x ) = exp  −  −   a    

(21.22b)

  x − c  −β  exp  −      a  

β −1

x ≤c

Generalized Gamma f ( x ) =

a>0

  x a exp  −    ,  x > 0 (21.23a)   b 

1  x   aΓ (b )  a 

b −1

c  x    b  a  aΓ    c

 x exp  −  ,  x > 0 (21.24a)  a

b −1

c

  x  exp  −    (21.25a)   a 

a, b > 0 a, b > 0

a, b, c > 0

  x a F ( x ) = exp  −      b 

(21.23b)

x

F(x ) =

1 x a b −1 y exp(− y )dy ; y = (21.24b) Γ(b) ∫0 a

b x c Γ  ,     c  a  F(x ) = 1 −  b Γ   c

(21.25b)

  Γ(,): incomplete Gamma function. b −1

Pearson Type III

f (x ) =

1  x −c   aΓ(b)  a 

 x −c exp  −  ,  x ∈[c , +∞ ) (21.26a)  a 

Log-Pearson Type III

f (x ) =

1  ln x − c    axΓ(b)  a 

b −1

 ln x − c  exp  −  (21.27a)  a 

      x ∈[exp(c ),  +∞ )

Beta

f (x ) =

Logistic

f (x ) =

Three-Parameter Log-Logistic

f (x ) =

Γ(a + b) a−1 x (1 − x )b−1,  x ∈(0, 1) (21.28a) Γ(a)Γ(b)

  x −b a  1 + exp  −    a  

2

1

; x ∈(−∞ , +∞ ) (21.29a)

2

1 1  −  +1  b( x − c )  b   b( x − c )  b + a1+ 1+ 1        a a  

(21.30a)

Generalized Two-Parameter Pareto

21_Singh_ch21_p21.1-21.12.indd 4

a, b > 0 c ∈R a, b > 0

x −c

F(x ) =

x −c 1 a y b −1 exp(− y )dy ;  y = (21.26b) Γ(b) ∫0 a

F(x ) =

1 Γ(b) ∫0

F(x ) =

Γ(a + b) x a−1 x (1 − x )b−1 dx (21.28b) Γ(a)Γ(b) ∫0

ln x − c a

y b−1 exp(− y )dy ;  y =

ln x − c (21.27b) a

b ∈R ,  a > 0

F(x ) =

a > 0,

F(x ) =

b ≥ 1,

1  x −b 1 + exp  −   a 

(21.29b)

1  b( x − c )  1+ 1+   a 

−

1 b

(21.30b)

c ∈R a−1 exp(−(1 − b) y ) (1 + exp(− y ))2

(21.31a)

  b( x − c )   −b −1 ln  1 −  ,  b ≠ 0   a  y=  x − c , b = 0  a  Two-Parameter Pareto

a, b > 0 c ∈R



 x −b exp  −   a 

x ≥c Generalized Logistic f ( x ) =



f ( x ) = bab x −b−1 ,  x > a    f (x ) =    

1

−1

(21.32a)

1  bx  b bx  1 −  ,  x > 0, < 1, b ≠ 0 a a a (21.33a) 1 x exp(− ), b = 0 a a

a > 0,

b, c ∈R

a, b > 0

a > 0, b ∈R

F(x ) =

1

1

 b( x − c ) b 1+ 1−   a 

F(x ) = ∫

x

−∞

, b ≠ 0

 ( x − c )(1 − b)  exp  −   a x −c   a  exp  −  + 1   a  

 x F(x ) = 1 −    a

−b

2

dx , b = 0

1 (21.31b) 2

(21.32b)

1     bx   b 1 1 − −     a   , b ≠ 0  (21.33b) F(x ) =    x 1 − exp − ,  = 0 b   a  

8/22/16 12:22 PM

Classification of Continuous Frequency Distributions    21-5  Table 21.3  Commonly Applied Continuous Probability Distributions in Hydrology (Continued) Distribution

Generalized ThreeParameter Pareto

Chi-Square

p.d.f.    f (x ) =    

1

Parameters

−1

1  b( x − c )  b 1−  , b ≠ 0 a a 

(21.34a)

1  x −c exp  −  , b = 0  a a 

n

 x  −1 f (x ) = n exp  −  x 2 , x > 0 (21.35a)  2 n   22 Γ    2 1

a>0 b, c ∈R

n :  degree of freedom

c.d.f. 1    b( x − c )  b 1 1 − −   a  (21.34b) F(x ) =    x −c  1 − exp  − a  

F(x ) =

n Γ x    2 2

n Γ    2 x

(21.35b)

 n

  −1  n Γ x   = ∫ 2 t  2  exp(−t )dt = γ  2 0 2

Chi

f ( y) =

y n−1

n −1 n 22 Γ  

  2

 y2  exp  −  ,  y = x ,  X ~ χ 2 (n) (21.36a)  2

−1

2 1   x − a  1 +    (21.37a)  b πb  

Cauchy

f (x ) =

Burr Type XII

f ( x ) = kcx c −1 (1 + x c )−( k +1) ,  x ≥ 0 (21.38a) M j = E( X j ) = k

j  j   Γ  k −  Γ  + 1  c  c  Γ(k + 1)

Burr Type XII (Three-Parameter)

Burr Type III

f ( x ) = kcx − c −1 (1 + x − c )− c −1 ,  x > 0 (21.40a) j  j  Γ  k +  Γ 1−   c  c M j = E(Y ) = k Γ(k + 1)

f ( x ) = k exp(− x )(1 + exp(− x ))−( k +1) ,  x ≥ 0 (21.41a)

Lomax

k x f (x ) =  1 +  a a

Kappa (Two-Parameter)

f (x ) =

,  x ≥ 0 (21.42a)

2a   x 2 x   ( a − 1)  b  + a  b  a   1

a  x a x  a +      b  

x a2c    b

Kappa (Three-Parameter)

f (x ) =

Halphen-A

f A (x ) =

+1

a ∈R ,  b > 0

1 1  x − a F ( x ) = + arctan   b  2 π

k, c > 0

F ( x ) = 1 − (1 + x c )− k (21.38b)

a,  k , c > 0

 c −k  1− 1+ x    a   (21.39b) F(x ) =  −k   x c   1 −  1 +    a   

(21.37b)

k, c > 0

k

 xc  F(x ) =  (21.40b)  1 + x c 

(21.40c)

Burr Type II

− ( k +1)



 n  n F ( y ) = Γy 2   / Γ   (21.36b)  2  2 2

(21.38c)

 c −1 c − ( k +1)  ckx  1 + x      a  a   f (x ) =  ; x ≥ 0 (21.39a) − ( k +1)  ckx c −1   x  c    1 +  a   c   a 

j

n : degree of freedom

n x   ,  : lower incomplete Gamma function. 2 2

k>0 k>0

F ( x ) = (1 + exp(− x ))− k (21.41b)  x F(x ) = 1 −  1 +   a

,  x ≥ 0

(21.43a)

 x  b 

−k

(21.42b)

a

a, b > 0

F(x ) =

a, b, c > 0

ac ac  x   x  F(x ) =    a +     b   b 

1

  x a a  a +  b    

(21.43b)

ac

ac

 x  x  a +      b 

2

,  x ≥ 0

(21.44a)

1   x m  x n−1 exp  −a  +   ,  x > 0 (21.45)   m x  2mn Kn (2a)

Kn () : modified Bessel function of second kind

−1

(21.44b)

m > 0, a > 0 n ∈R

of other n. Halphen-B

fB (x ) =

  x  2 ax  1 x 2n−1 exp  −   +     m2n efn (a) m   m

x > 0 ; efn () : exponential factorial function

(21.46)

m > 0, n > 0 a ∈R (Continued)

21_Singh_ch21_p21.1-21.12.indd 5

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21-6    Frequency Distributions Table 21.3  Commonly Applied Continuous Probability Distributions in Hydrology (Continued) Distribution

p.d.f. f IB ( x ) =

Halphen-IB

x >0

Parameters 2

  m  am  1 x −2n−1 exp  −   + m−2n efn (a) x    x

(21.47)



1

f (x ) =

Tsallis

(1 − (1 − m)ax )1−m

∫0

,  x > 0 (21.48a)

1

∞

(1 − (1 − m)at )1−m dt

1

 m-exponential expm ( x ) = (1 − (1 − m)x )1−m ∞

  f ( x ) = expm (−ax )  ∫ expm (−ax )dx   0 

(21.48b) (21.48c)

−1

∞  f ( x ) = f m ( x )  ∫ f m (t )dt  0 

Kousoyiannis

f ( x ) = (1 + k(a0 + a1 x c1 ))

Wakeby

x = −a(1 − F )b + c(1 − F )− d + a0 (21.51)

Sinepower

f ( x ) = mnx n−1 cos( x n )(sin( x n ))m−1 , x ≥ 0 y − y min x= ym − y min

(21.49)

−1−

x=

a>0

−1

Escort

Extended Sinepower f ( x ) =

c.d.f.

m > 0, n > 0 a ∈R

1 k

m: Tsallis entropy index

x c2 −1 (21.50)



1 − F0 mny n−1 cos( y n )(sin( y n ))m zm − z min



a, b, c , d > 0; a0 ∈R

(21.52a)

m, n > 0

F ( x ) =  sin( x n ) (21.52b)

(21.53a)

m, n > 0

F ( x ) = F0 + (1 − F0 )(sin( y n ))m

z −z z − z min x ,  xm = m min ,  y = z max − z min z max − z min xm

m

(21.53b)

F0 : CDF at z = z min

(21.54a) General distribution f ( x ) = (1 − F0 )abx a−1 (1 − x a )b−1 ,  x ∈[0,1] for double bounded z − z min x= variable z max − z min a Boughton K = A+ , a = 1.009 A2.04 (21.55) T   −A ln ln    T − 1 T  : recurrence interval determined from the plotting position formula; K: frequency factor for the selected recurrence interval as: − − log Q = q+ KS , q = mean(log q)

a, b ≥ 0

F ( x ) = F0 + (1 − F0 )(1 − (1 − x a ))b

(21.54b)

F0 : CDF at z = z min

S = std(log q) Compound distribution ExponentialGamma

f ( x|b) = b exp(−bx )  x ≥ 0, b > 0 : Exponential

(21.56a)

∞

f ( x ) = ∫ f ( x|b) f (b)d b = 0

∞

f ( x ) = ∫ f ( x|b) f (b)d b = 0

21_Singh_ch21_p21.1-21.12.indd 6

 xa  F(x ) = 1 −  1 +  c  

−n

a, c , n > 0

 xa  F(x ) = 1 −  1 +  c  

−1

f (x ) =

c  x   dB(a, b)  d 

ac −1

(21.56b)

(21.57b)

ancn x a−1 ,x ≥0 ( x a + c )n+1

Weibull-Exponential f ( x|b) = abx a−1 exp(−bx a ),  x ≥ 0, a, b > 0 f (b) = c exp(−cb), c > 0

Generalized Beta of second kind

 x F(x ) = 1 −  1 +   a

akb k −1 f (b) = exp(−ab), a, k > 0 : Gamma Γ( k ) − k +1 ∞ k x ( ) f ( x ) = ∫ f ( x|b) f (b)d b =  1 +  0   a a

Compound Weibull f ( x | b) = abx a−1 exp(−bx a ),  x ≥ 0, a, b > 0  (Weibull) (21.57a) (Weibull-Gamma) cnbn−1 f (b) = exp(−cb), c > 0, n > 0;(Gamma) Γ(n)

−k

a, k > 0

(21.58a)

a, c > 0

(21.58b)

acx a−1 ( x a + c )2

  x c  1 +  d    

B(,): complete Beta Function.



− ( a +b )



(21.59a)

a, b, c , d > 0

F(x ) =

B y (a , b )

  x  −c  ;  y =  1 +    B(a, b)   d 

−1



(21.59b)

  By (,): incomplete Beta function

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Continuous Frequency Distributions   21-7  21.4.3  Log-Normal Distributions

21.4.8  Extreme Value Type III (EVIII) Distribution

The log-normal probability distribution arises when the causative factors are multiplicative. Since Chow (1951, 1954) proposed a log-probability law in hydrology, this distribution has been widely used for frequency analysis of hydrological data. Aitchison and Brown (1957) gave a full account of the log-normal distribution. Let Y = log X, 0 < x < ∞. If Y follows a normal distribution, X follows a log-normal distribution. The two-parameter PDF and CDF of X are given as Eqs. (21.15a)–(21.15b), where m and σ are the mean and standard deviation of Y. The parameters m and σ can be related to the mean and standard deviation of original random variable X given as Eqs. (21.15c)–(21.15d). If z = (ln x − µ )/σ , the variable Z follows the standardized normal distribution. The three-parameter log-normal distribution is frequently used in frequency analysis of extreme floods, seasonal flow volumes, flow duration curves, rainfall intensity-duration, soil water retention, pore radius, and pore capillary pressure (e.g., Stedinger, 1980). The PDF and CDF of the threeparameter log-normal distribution are given as Eqs. (21.16a)–(21.16b). Again, the standardized variable Z following the standard normal distribution is obtained as z = [ln(x − a) − my]/σy .

The EV III distribution has an upper bound. The EV III distribution belongs to the Weibull family also known as inverse Weibull distribution. The EV III distribution may be obtained if the shape parameter b > 0. Let b = 1/β , its PDF and CDF are given as Eqs. (21.22a)–(21.22b). In EV III distribution, parameters a,  β , c are the scale, shape, and location parameters, respectively. The EV III distribution has been used for frequency analysis of low flow extremes. Figure 21.1 shows the GEV density function plots for different values of shape parameter b to represent the EVI (Gumbel), EVII, and EVIII distributions.

0.35 0.3

21.4.5  Generalized Extreme Value Distribution

The generalized extreme value (GEV) distribution, introduced by Jenkinson (1955) and recommended by Natural Environment Research Council (1975) of Great Britain, is widely used for flood frequency analysis, particularly in western Europe. Its PDF and CDF are given as Eqs. (21.18a)–(21.18b) in which parameters a, b , c are the scale, shape, and location parameters, respectively. The range of X is controlled by shape parameter, b; for b > 0, it is bounded as −∞ < x ≤ c + (a/b); for b < 0, it bounded as c + (a/b) ≤ x < ∞. Depending on the value of shape parameter b, the GEV distribution specializes into the extreme value type I (or Gumbel) distribution for b = 0, the extreme value type II (or Fréchet) distribution for b < 0, the extreme value type III distribution for b > 0, the reverse Raleigh distribution for b = 2, and the reverse exponential distribution for b = 1. EV I, EV II, and EV III distributions are further discussed in what follows. 21.4.6 Extreme Value Type I (EVI) and Log-EVI Distributions

The extreme value type I (EVI) distribution, also known as the Gumbel distribution (Gumbel, 1941, 1942), is one of the most popular distributions for frequency analysis of extremes of floods, droughts, and rainfall and meteorological data. By limiting the shape parameter b → 0, the GEV distribution converges to EV I (or Gumbel) distribution. Its PDF and the CDF are given as Eqs. (21.19a)– (21.19b), respectively. In the EV I distribution, a is the scale parameter and c is the location parameter. Additionally, the EV I distribution has a constant skewness coefficient (about 1.14) and constant kurtosis (about 5.4). If the return x −c period T is defined as T = 1/(1 − F) and let y = then the reduced variate a T − 1    yT can be defined as yT = − ln  − ln  . .   T    Let Y = lnX. If Y has an EV I distribution then X has a log-EV I distribution with PDF and CDF given as Eqs. (21.20a)–(21.20b). The log-EV I distribution has been used for frequency analysis of instantaneous flood peaks (Reich, 1970). The greatest attraction of EVI is that it is expressed in closed form. 21.4.7  Extreme Value Type II (EVII) Distribution

The EVII distribution, known as Fréchet distribution, may be obtained if the shape parameter b  30 the gamma distribution tends to approach a normal distribution with zero mean and unit variance, where the variate u is defined through the following transformation: χ2 2 9υ u =[( )1 3 + − 1]( )1 2 . υ 9υ 2 21.4.11  Generalized Gamma Distribution

Using the Boltzmann–Gibbs–Shannon (BGS) entropy, Papalexiou and Koutsoyiannis (2012) derived the three-parameter generalized gamma distribution and tested it on 11,519 daily rainfall records from across the globe. The PDF and CDF of this distribution can be expressed as Eqs. (21.25a)–(21.25b), respectively, where a is the scale parameter, and b, c are the shape parameters. The generalized gamma distribution is a flexible distribution that includes exponential, two-parameter gamma, Weibull, chi-squared, and other distributions as special cases. Figure 21.2 plots the generalized gamma distribution and its special cases. 21.4.12  Pearson Type III and Log-Pearson Type III Distributions

The Pearson type III (PIII) distribution is popular in flood and extreme rainfall frequency analysis and can be considered as a three-parameter gamma

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21-8    Frequency Distributions

The three-parameter log-logistic distribution (LLD) has been used for frequency analysis of precipitation and streamflow data. Its PDF and CDF can be written as Eqs. (21.30a)–(21.30b) where a, b, c are the scale, shape, and location parameters.

Generalized Gamma and its special distributions

0.35

Generalized Gamma: a = 4, b = 3, c = 2 Gamma: a = 4, b = 3 Exponential: a = 4 Weibull: a = 4, b = 3

0.3

21.4.15  Generalized Logistic distribution

The generalized logistic (GLO) distribution has been recommended as the standard for flood frequency analysis in the United Kingdom (IH, 1999; Kjeldsen and Jones, 2004). The GLO distribution is a three-parameter distribution proposed by Hosking and Wallis (1997). Its PDF and CDF can be written as Eqs. (21.31a)–(21.31b) in which a, b , c are the scale, shape, and location parameters, respectively. The range of random variable X is dependent on a a these parameters as: x ∈( −∞ , c +  for b > 0, x ∈ c + , +∞ ) for b < 0, and b  b x ∈(−∞ , +∞ ) for b = 0. The event with T-year return period can be expressed a in the closed-form as: xT = c + (1 − (T − 1)− b ). b

0.25

pdf

0.2 0.15 0.1 0.05

21.4.16  Pareto Distributions

0

0

5

10 x

15

20

Figure 21.2  Plots of generalized gamma probability density function and its special cases.

distribution. Its PDF and CDF can be expressed as Eqs. (21.26a)–(21.26b), where a, b, and c are the scale, shape, and location parameters, respectively. The log-Pearson type III (LPIII) distribution is one of the most frequently used distributions for frequency analyses of hydrologic extremes and was recommended by Water Resources Council (1967) of the United States. If Y = lnX follows the PIII distribution, X follows the LPIII distribution with PDF and CDF expressed as Eqs. (21.27a)–(21.27b), where a, b, and c are the scale, shape, and location parameters, respectively. Figure 21.3 shows an example of the density functions of the PIII and LPIII distributions.

21.4.17  Chi-Square Distributions

21.4.13  Beta Distribution

The beta distribution is used in reliability and safety analysis of civil engineering systems where variables or parameters are generally bounded and skewed random quantities. This distribution is a Pearson type I distribution. Its PDF and CDF can be expressed as Eqs. (21.28a)–(21.28b), where a, b are the shape parameters. Singh (1996) found the solution of kinematic converging and diverging overland flows in terms of the beta distribution. 21.4.14  Logistic and Log-Logistic Distributions

The logistic distribution has been applied to frequency analysis of floods. Its PDF and CDF can be expressed as Eqs. (21.29a)–(21.29b), where a is the scale parameter, and b is the location parameter. If Y = [ln(X − b)/a)]/a has a logistic distribution then X has a log-logistic distribution. The log-logistic distribution (LLD) is similar in shape as the log-normal distribution and has been used for frequency analysis of survival data and precipitation data. Pearson type distribution

0.04

PIII(4,6,2) LPIII(2,1.2,2.07)

0.035 0.03

pdf

0.025 0.02

For a Chi-square random variable with n degrees of freedom, the PDF and the CDF can be expressed as Eqs. (21.35a)–(21.35b). Lancaster (1969) gave a full account of this distribution. The standard Chi-square distribution with n degrees of freedom is the same as the gamma distribution with shape parameter n/2 and scale parameter 2. The mode of the distribution is at 0 if n ≤ 2, and at n − 2 for n > 2. If y = x , where X has a chi-square distribution with n degrees of freedom, Y has a chi-distribution with n degrees of freedom with PDF and CDF as Eqs. (21.36a)–(21.36b). It may be interesting to note that Eq. (21.36) becomes the half normal distribution for n = 1, the Raleigh distribution for n = 2, and the Maxwell–Boltzmann distribution in continuous domain for n = 3. Originally the MB distribution was derived as a discrete distribution. The mode of Chi-square distribution occurs at 0 if n ≤ 1 and at n − 1 if n  > 1. 21.4.18  Cauchy Distribution

The Cauchy distribution has PDF and CDF given as Eqs. (21.37a)–(21.37b) in which a and b are the location and scale parameters, respectively. Cauchy distribution has no mean, variance, or higher moments. It is interesting to note that the distribution is symmetrical about x = a, has median as a, and has upper and lower quartiles as a ± b. The PDF exhibits points of inflexion at a ± b 3 . At these points the values of CDF are 0.273 and 0.727, whereas the corresponding values are 0.159 and 0.841 for the normal distribution. The Cauchy distribution has heavier tails than the Gumbel distribution and can be used for frequency analysis of extremes. Nathan and McMahon (1990) state that although the data may fit it, it is unsuitable for low flow analysis. 21.4.19  Burr Distributions

Three distributions of the Burr family (Burr, 1942) that have been more widely used in hydrology and hydrometeorology are types II, III, and XII, because their forms and their inverses are expressed in simple closed forms. Tadikamalla (1980) showed that several distributions are special cases of the Burr type XII distribution.

0.015 0.01 0.005 0

The Pareto distribution, introduced by Pickands (1975), is a heavy tailed distribution and is applied to model peak over threshold flood values. A full account of the Pareto distributions was given by Arnold (1983). Three forms of the Pareto distribution have been used: (1) two-parameter, (2) generalized two-parameter, and (3) three-parameter generalized. The PDF and CDF of two-parameter Pareto distribution can be expressed as Eqs. (21.32a)–(21.32b), where a and b are the scale and shape parameters, respectively. This is a special form of the Pearson type VI distribution. The two-parameter generalized Pareto distribution has a PDF and CDF given as Eqs. (21.33a)–(21.33b), where a, b are the scale and shape parameters, respectively. If the shape parameter b = 0, this distribution reduces to the exponential distribution. The three-parameter generalized Pareto distribution has its PDF and CDF expressed as Eqs. (21.34a)–(21.34b), where a, b , c are the scale, shape, and location parameters, respectively. Again, if b = 0, this distribution reduces to the exponential distribution with a location parameter.

21.4.20 Burr Type XII Distribution

20

40

60

80

100 x

120

140

160

Figure 21.3  Plots of PT III and LPT III probability density functions.

21_Singh_ch21_p21.1-21.12.indd 8

180

200

The PDF and CDF of this distribution can be expressed as Eqs. (21.38a)– (21.38b) in which k , c are the shape parameters. The PDF is unimodal at x = [(c − 1)/(kc + 1)1 c ], provided c >1, and L-shaped if c ≤ 1. The Burr type XII is a two-parameter distribution but can be extended to (i) a three-parameter

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Conclusion   21-9 

distribution by incorporating the scale parameter, and (ii) a four-parameter distribution by incorporating both the location and scale parameters. A scale parameter a can be incorporated in Eq. (21.38) in two ways given as Eqs. (21.39a)–(21.39b). To express the PDF and CDF for the four-parameter Burr XII distribution, we can simply substitute x in Eq. (22.39) as x − m, given µ ∈R , x ≥ µ . It is interesting that the j-th power moment of X following the two-parameter Burr type XII distribution can be computed using Eq. (21.38c). 21.4.21 Burr Type III Distribution

Applying the transformation Y = 1/X to the Burr Type XII distribution, the Burr type III distribution can be expressed as Eqs. (21.40a)–(21.40b) with the j-th power moment of Y given as Eq. (21.40c). Tadikamalla (1980) showed that as k → ∞, the distribution function reduces to F(x) = exp(−x −c) which can be considered as the distribution of a reciprocal Weibull variate. In other words, if X has the Weibull distribution, Y = 1/X has the aforementioned reciprocal Weibull distribution. 21.4.22 Burr Type II Distribution

If X = c log y, Eqs. (21.40a) and (21.40b) lead to the PDF and CDF for the Burr type II distribution which can be expressed as Eqs. (21.41a)–(21.41b). 21.4.23  Lomax Distribution

For analyzing business data, Lomax (1954) employed a distribution whose PDF and CDF can be written as Eqs. (21.42a)–(21.42b) in which a is a scale parameter and k is a shape parameter. The Lomax distribution is a special case of the three-parameter Burr type XII distribution with c = 1. Lomax (1954) f (x ) k = defined the failure rate function as Z = . 1 − F ( x ) a( x + a) 21.4.24  Kappa Distributions

For analyzing precipitation data, Mielke (1973) used the two-parameter kappa distribution with PDF and CDF, respectively, as Eqs. (21.43a)–(21.43b) where a, b are the shape and scale parameters, respectively. The three-parameter kappa distribution can be written for its PDF and CDF as Eqs. (21.44a)– (21.44b) where a, c are the shape parameters and b is the scale parameter. Hosking (1994) discussed the four-parameter kappa distribution that is a reprameterization of the three-parameter kappa distribution with an additional location parameter. Singh and Deng (2003) estimated its parameters using entropy. It specializes into the three parameter kappa distribution plus Pareto, GEV, Gumbel, logistic, and uniform distributions, and is related to the Burr type III and type XII distributions. 21.4.25  Halphen Distributions

Perreault et al. (1999a, b) described the Halphen distribution types A, B, and IB (Halphen, 1941, 1955). Following their work, these three distributions have PDFs as Eqs. (21.45)–(21.47). The gamma and inverse gamma distributions can be shown to be limiting cases of the Halphen distributions. These distributions can be applied to a variety of hydrologic variables. Perreault et al. (1999a, b) discussed mathematical properties for these distributions. El Adlouni et al. (2010) compared the performances of Halphen and GEV distributions for flood frequency analysis. They showed Halphen distributions are good candidates to model the extreme value variables. 21.4.26  Tsallis Distribution

Drainage basin characteristics, such as width function, exhibit power law or multifractal scaling. For frequency analysis of such characteristics, the Tsallis distribution that is derived from the Tsallis entropy may be appropriate (Keylock, 2005). The Tsallis distribution can be expressed as Eq. (21.48a) in which m > 0 is the entropy index, and a is a parameter. Equation (21.48a) is related to the exponential distribution. Using the definition of m-exponential given as Eq. (21.48b), Eq. (21.48a) can be expressed as Eq. (21.48c). For m → 1, Eq. (21.48) will converge to the exponential distribution.

would lead to several exponential and power distributions, including beta prime, gamma, Weibull, and Pareto. 21.4.29  Wakeby Distribution

Houghton (1978) introduced a five-parameter distribution, called Wakeby, for modeling flood flows. In inverse form, the Wakeby distribution is expressed as Eq. (21.51) in which F is the uniform variate (0, 1), a, b, c, and d are the shape parameters, and a0 is a location parameter. This distribution has found significant use for modeling hydrologic extremes (Griffiths, 1989; Rahman et al., 2015). 21.4.30  Sinepower Distributions

Kumaraswamy (1976, 1978, 1980) introduced sinpower distributions for modeling bounded data. Let Y be a random variable with lower bound ymin and mode ymode. Using a transformation, the sinepower distribution has PDF and CDF as Eqs. (21.52a)–(21.52b). It may be noted that x is expressed in radians, and (xmin ) = 0 when x = xmin = 0; f (x) = 0 when x = 0 and x = xmax, since xnmax = p /2 and cos(xn) = 0. Kumaraswamy (1976) showed  that n = ln(p/2)/ ln(xmax),

(

)

n  n − 1 tan x mode  n and m = sec 2 x mode −  . Kumaraswamy (1978) extended the n n x mode    sinepower distribution by using a transformation, whose PDF and CDF are expressed as Eqs. (21.53a)–(21.53b). Kumaraswamy (1980) presented a more general distribution for double bounded variables with PDF and CDF as Eqs. (21.54a)–(21.54b) using a transformation.

(

)

21.4.31  Boughton Distribution

Boughton (1980) empirically proposed a three-parameter distribution expressed by Eq. (21.55) for modeling flood flows from 78 watersheds from eastern Australia. He also showed a relation between frequency factor and recurrence interval. 21.4.32  Compound Distributions

Some aforementioned distributions are examples of compound distributions. Dubey (1970) has shown that the Lomax distribution is analogous to the exponential-gamma distribution given as Eqs. (21.56a)–(21.56b) which is the same as Eqs. (21.42a)–(21.42b) for the Lomax distribution. 21.4.33  Compound Weibull Distribution

The compound Weibull distribution is obtained by considering the scale parameter in the Weibull distribution as gamma distributed (Dubey, 1966) expressed as Eqs. (21.57a)–(21.57b) which is the same as the Burr type XII distribution. 21.4.34  Weibull Exponential Distribution

The Weibull-exponential distribution can be obtained by considering the scale parameter b in Eq. (21.58a) as exponentially distributed, that is, n = 1. Then, the Weibull exponential distribution can be expressed as Eqs. (21.58a)– (21.58b) which is the same as the Burr type XII distribution. 21.4.35  Generalized Beta of the Second Kind Distribution

Papalexiou and Koutsoyiannis (2012) derived a four-parameter beta of second kind (GB2) distribution with the use of the BGS entropy, and applied it to 11,519 daily rainfall records from across the globe. The PDF and CDF of this distribution can be expressed as Eqs. (21.59a)–(21.59b), where d is the scale parameter and a, b, c are the shape parameters. The GB2 is a flexible distribution that nests Beta Prime, lognormal, Weibull, gamma, generalized gamma, log-logistic, Burr types III and XII, Lomax, and other distributions. Figure 21.4 plots the generalized beta distribution of second kind and some of its special case.

21.4.27  Escort Distribution

The distribution is given as Eq. (21.49). Keylock (2005) used this distribution for analyzing the recurrence interval of extreme floods. 21.4.28  Kousoyiannis Distribution

Using Tsallis entropy, Koutsoyiannis (2005a, b) proposed a four-parameter distribution that can be expressed as Eq. (21.50), where a is a scale parameter, and c1, c2, and k are shape parameters. Parameter a0 is determined by substituting Eq. (21.50) into the total probability equation so is not considered as a parameter. Here, k = (1 − m)/m where m is the Tsallis entropy index. With appropriate transformations, Koutsoyiannis (2005a) showed that Eq. (21.50)

21_Singh_ch21_p21.1-21.12.indd 9

21.5  CONCLUSION

A wide variety of frequency distributions, ranging from one-parameter to five-parameter, are available for hydrological analyses. Some of the distributions are quite general and lead to other distributions as special cases. Literature survey shows that for most hydrological frequency analyses two- or three- or exceptionally four-parameter distributions suffice. All of these distributions are derived without reference to hydrology. It would be desirable to investigate if some of the distributions could be derived using hydrological laws and then estimate their parameters using measurable hydrologic characteristics. This might further help unify some of these distributions.

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21-10    Frequency Distributions

Generalized Beta distribution of second kind 1

Generalized B2 (0.5,4,2,2) Generalized Gamma (b to inf, d = b*b1/a) Beta 2 (5,4,1,1)(c = d = 1) BR12 (1,4,0.8,2)(a = 1) BR3 (3,1,0.1,2)(b = 1) Log−Logistic (1,1,1,3)(a = b = c = 1)

0.9 0.8 0.7

pdf

0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8

10

x Figure 21.4  Plots of generalized beta distribution of second kind and some of its special cases. REFERENCES

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Chapter

22

Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation BY

EZIO TODINI AND DANIELA BIONDI

ABSTRACT

This chapter discusses the nature of the different sources of uncertainty and their effect on the parameters estimates. Techniques aimed at estimating the most appropriate parameter values are also introduced, from the simple trial and error and the least squares approaches to more sophisticated Bayesian techniques involving assumptions on the prior distributions and the definition of appropriate likelihood functions. Models are simplified representations of complex reality, observations are affected by direct measurement or indirect estimation error, while scaling from the infinitesimal to coarser finite scales introduces representation errors, etc. Therefore, model reproductions of natural phenomena are hardly perfect and parameters are thus introduced as the knobs, which allow tuning models thus improving their adequacy to observed realizations. Chapter 22 discusses parameters nature, which can vary from purely stochastic and uncertain to deterministic, linked to physically meaningful quantities, but still distorted by the presence of model and measurement errors. In all cases the presence of uncertainties and errors in models, data, initial and boundary conditions requires the use of appropriate techniques to estimate the expected value of parameters conditional to the chosen models and the observations (deterministic estimation) or, in a Bayesian perspective, the posterior density of parameters (probabilistic estimation). Calibration and data assimilation techniques are introduced as means to adjust, update or improve parameter estimated values. Assessment of simulation uncertainty, namely the uncertainty of the model output for any given known observation, is also presented together with the sensitivity analysis of models on the errors of different nature. 22.1  INTRODUCTION

Hydrological models are simplified representations of reality, combining prior knowledge assumptions to information extracted from observations. From the very simple hydrological models, with no prior assumptions made, to the more complex ones based on physically meaningful structural assumptions, they all require calibration. Models are a combination of structure, describing the links among state variables, and parameters. Parameters, according to the modeler conceptualization, have different meanings, ranging from simple model “tuning” knobs to physically meaningful quantities. Whichever the meaning, calibration is needed in order to match the model output (simulation/prediction) to the observations. Traditional calibration

allows either to estimate the parameters value if no assumptions on their nature are made or to adjust their value in order to compensate for the unavoidable simplifications and errors in model and data, if the parameters are assumed to have physical meaning. More recently, under the Bayesian influence, calibration has been extended from the original scope of finding a specific “optimal” value for the parameters to the derivation of a parameter density (the “posterior” parameter density) incorporating all the uncertainties descending from the model assumptions, the required simplifications, and the data errors. Under this approach, parameter estimation is then a way for assessing and reducing model prediction uncertainty. One important issue is to bear in mind the difference between calibrating a model and estimating the “true” values of parameters (if they exist). Calibrating, means tuning the parameter values in order to match the model output with observations, while estimating the parameters aiming at finding their true, albeit unknown value, means finding values that not necessarily will improve the model performances, due to the not necessarily correctness of the model, the eventual nonlinear links between parameters and model predictions and measurement errors. Therefore, an unbiased optimal parameter estimator will not necessarily lead to a calibrated model. Calibration of parameters is sometimes confused with identification, which falls beyond the scope of this chapter. While calibration refers exclusively to parameters, identification also involves adjusting model structure. Frequently, data-driven models are based on the linear/nonlinear combination of simple elementary structures (such as the linear channel and linear reservoir in Nash cascade (Nash, 1958, 1960), the elementary nonlinear reservoir in the Tank model (Sugawara, 1967, 1995) and the nonlinear generic building block in the hydrological models proposed by Fenicia et al. (2011) or the “perceptrons” (Rosemblatt, 1958; Minsky and Papert, 1969) in the Artificial Neural Networks models. Identification, which also involves calibration for any selected hypothetic structure, is then needed to find a parsimonious combination of the elementary models producing the most likely fit with observations. 22.2  PARAMETER UNCERTAINTY

This section aims at clarifying the concept of uncertainty in parameter estimates, which depends on several factors including the prior assumptions on the nature of the model (regression type, linear/nonlinear, structural, physically based, etc.), of parameters (undefined as in regression models or physically meaningful as, for instance, the roughness coefficient in hydraulic 22-1

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22-2     Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation

models), as well as the errors in observations and the estimation approach [moments, least squares, maximum likelihood(ML), multicriteria, etc.] used in the parameter calibration phase.

Ot + ∆t = Ot e−k∆t + (1 − e−k∆t )I + e t + ∆t (22.3)

22.2.1 The Nature of Models and Parameters

Ot + ∆t = I + a (Ot − I ) + e t + ∆t (22.4)

The nature of a model depends largely on the assumptions made by the modeler on the basis of his prior knowledge. A wide range of modeling possibilities exist from the very simple linear regression models, only requiring a weak hypothesis on the probability distribution of errors, to the structural physically based models, where prior hypotheses are made on the links and interactions among state variables and parameters. To better understand, let us start from the simple linear regression model, which can be viewed as the estimation of the correlation coefficient among standardized Normal variables. When applying the linear regression model, no additional prior assumptions are needed on causative relations between predictand, which is the variable of interest, and the predictor, the explanatory variable used to infer its value. It is well known in statistics that correlation does not imply cause and effect. The most well-known example commonly used to show this concept is the storks and babies correlation estimated by Dr. Gustav Fischer on annual counts made 1930–1936 in Oldenburg Germany (Fischer, 1936, 1938, 1940), reproduced in Fig. 22.1, which resulted in R = 0.94. 80 75

Babies

70 65 60 y = 0.1506x + 36.231 R2 = 0.88374

55 50 130

150

170

190

210

230

250

Storks Figure 22.1  Regression between annual number of babies born and annual number of storks seen in Oldenburg in the years 1930–1936.

In the example, the number of babies born, the predictand, is well predicted on the basis of the number of storks, the predictor, although no cause and effect can be implied. Causative relations (namely, cause and effect, the basic assumption in most of hydrological and hydraulic models), always descend from the axiomatic hypothesis of a continuum (a continuum of mass, energy, relation, etc.). To clarify this point, let us apply the linear regression model to a more physically based (and therefore, causative) example of a linear reservoir. In this case, we “a priori” assume a continuum of mass reflected into the mass balance equation:

dV = I − O (22.1) dt

in which the change in time of the storage V is linked to I, the inflow, and O, the outflow. It is evident from Eq. (22.1) that our prior assumption on a continuum, makes the inflow causative on the outflow. Moreover, let us assume the outflow O to be proportional to the stored volume, namely, O = kV, through a parameter k > 0, which is a first-order approximation of a hydraulic correct momentum-balance equation. If I is constant in time, the solution to Eq. (22.1), an ordinary differential equation, is Ot + ∆t = Ot e−k∆t + (1 − e−k∆t)I (22.2) If one wants to estimate parameter k from observations, measurement and model-schematization errors must also be taken into account, which can be done by adding an error term et + ∆t to the previous equation, leading to:

22_Singh_ch22_p22.1-22.20.indd 2

which can be transformed into a simple linear regression model by setting a = e−k∆t: It is evident that in order to preserve the causative nature of Eq. (22.1), the value of a must be greater than zero and smaller than 1, because 0 ≤ e − k∆t ≤ 1;  ∀k > 0,  ∆t ≥ 0. Nonetheless, when measurement errors are large, and due to the linearity assumption on the link between outflow and stored volume, the linear regression estimated value for a may even result negative (α < 0) or larger than 1 (α > 1), because the range of existence of α in the linear regression, if not explicitly imposed, lies between −∞ and +∞, thus loosing the physical (at least in terms of mass balance) nature of our model. Therefore, a way to impose the physicality on the parameter is to perform a “constrained” linear regression, by introducing a number of constraints representing our prior assumptions in the estimation phase. In the case of the example, it is sufficient to add the constraint 0 < α < 1 to the minimization of the sum of squared errors (essentially the simple linear regression), by solving the following quadratic programming problem:

N 2 minα ∑ ti =1 [(Oti +∆t − I ) − α (Oti − I )] (22.5)  s .t .                       0 < α < 1   

where N is the number of observed data. Following the same concept, when estimating the Unit Hydrograph ordinates, Natale and Todini (1976a, b) showed that physically meaningful constraints, aimed at imposing mass balance between inflow and outflow (the sum of all the ordinates of the Unit Hydrograph, must equal 1) and that water flows downstream (all the ordinates must be non-negative), reduce the range of existence parameters, thus leading to sharper parameter estimates. Needless to say that introducing constraints in the estimation requires a certain care otherwise when the imposed constraints are incorrect, the estimation variance will still be reduced at the cost of increasingly higher biases. The aforementioned concepts can be broaden by introducing in the estimation phase all the available prior knowledge, either in the form of prior probability densities assumptions, as advocated by Bayesian Inference, or by introducing physically meaningful constraints in order to obtain sharper parameter estimates. The case of the Muskingum model can be used to illustrate this point. The Muskingum routing model was developed by McCarthy (1938, 1940) as a linear model with constant parameters, which fixed value had to be estimated from input and output observations. Similarly to the linear reservoir case presented earlier, the model was physically based in terms of mass balance, while approximated in terms of the momentum balance. Cunge (1969) found a way of linking the Muskingum original parameters to physical quantities (namely, discharge, water stage, celerity, surface width, friction slope, time, and space integration steps) and, at the same time, to better represent the momentum balance equation leading to a parabolic flood routing model. In order to estimate the previously mentioned physical quantities, the Muskingu–Cunge model requires more information than the original one, such as the cross section geometry and the friction coefficient, but, although calibration of the friction coefficient (the unique physically meaningful parameter) is still needed, the model performs better and where calibration is not available can still be run from literature-based friction coefficient values. Summarizing, the physical nature and the structure of a model depends on the amount of information the modeler has introduced in terms of prior knowledge of the phenomena under study. Similarly, with the increasing physical meaning of model equations, also parameters may be assumed to retain physical meaning, which allows to derive first guess estimates from available information and to reduce their domain of existence, potentially leading to sharper parameter or parameter probability densities estimates. 22.2.2  Sources of Uncertainty

There are several sources of uncertainty affecting the parameter estimates. The first source of uncertainty is due to the model itself because a model is always a schematization and a simplification of a more complex reality. The used schematizations and simplifications never allow reproducing reality at 100% and the discrepancy will resort into differences (eM ) between observed and modeled quantities, affecting the estimated parameter values. A second source of uncertainty is due to the input (e I ) and output (e O ) observation errors. Sometimes, these errors may be negligible, but most of the times, they are large strongly influencing the parameter estimates. Given the tendency of estimating parameters on the basis of deviations of model output from the observations, input errors and output errors may have substantially different effects on parameter estimates.

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Parameter estimation    22-3 

Errors in initial and boundary conditions assumptions (ε IC ; ε BC ) also affect many structural models, and in particular, the nonlinear chaotic models (such as most of the meteorological and climatological models) resorting into large uncertainties affecting the determination of the parameter values. In addition to all these sources of uncertainty converging into the parameter-estimation process, the final parameter estimate values will be unavoidably affected by the selected estimation procedure (ε ES ). 22.2.3 Parameter Estimation: A Way for Assessing and Reducing Uncertainty

As stated in the introduction, model parameters can either be viewed as unknown quantities of interest to be determined from observations, or as tuning knobs essentially aimed at calibrating the model. In both cases, given that we do not know their true value (if it exists) parameters must be viewed as uncertain variables strongly affected by all the errors described earlier.

ϑ {ε M , ε I , ε O , ε IC , ε BC , ε ES } (22.6)

If the scope is to find the true albeit unknown value of parameters, the estimation must aim at deriving unbiased minimum variance estimates of the parameters, starting from assumptions on the probability distribution of errors and, possibly, on the description of a prior probability density of the parameter values. On the contrary, if the scope is to calibrate a model, the estimation must aim at deriving unbiased minimum variance estimates of the model output, and not necessarily of the parameters still starting from assumptions on the probability distribution of errors and, following a Bayesian approach, on the description of a prior probability density of the parameter values. Finally, given that the scope for building and using models is to improve available knowledge allowing taking better decisions, a third, more rewarding objective of calibration, known in the literature as Bayesian Inference, is to find the full posterior density of parameters, not limiting the process to the estimation of their expected value, or sometimes their modal one. In this approach, regardless to their potential physical nature, parameters are considered as the final receptors of all the uncertainty and the scope is to improve the description of their prior probability density using the observations. The resulting sharper posterior probability density is then used, by means of the model, to compute predictive uncertainty as well as expected benefits and costs affecting decisions, as described in Chap. 26. 22.3  PARAMETER ESTIMATION

This section describes the standard techniques commonly used for estimating parameters of hydrological models, such as the maximum likelihood, the least squares, and the more recent multicriteria approaches. All these approaches aim at finding either an expected value or a modal value for the parameters. In addition, Bayesian approaches will be described, which aim at inferring the full posterior probability density of parameters to be used in the assessment of predictive uncertainty and in the estimation of the expected utility in decision making (see Chap. 26). No mention is made of other approaches, such as moments and cumulants (Dooge, 1973), etc., which were used in the pre- and early computer era to estimate parameters of linear hydrological models, such as the Muskingum or the Nash cascade. Today, moments, cumulants, probability weighted moments, and L-moments are essentially used to directly estimate parameters of probability distribution functions. The main properties required in an estimator are: consistency, unbiasedness, efficiency, robustness, and minimum variance. • Consistency: An estimator is consistent if the estimator converges to the true value almost surely as the number of observations approaches infinity. • Unbiasedness: An estimator is unbiased if its expected value is equal to the true value. • Efficiency: An estimator is efficient if it produces the smallest error covariance matrix among all unbiased estimators, it is also regarded optimally using the information in the measurements. A well-known efficiency criterion is the Cramér–Rao (Rao, 1945; Cramér, 1946) lower bound. • Robustness: An estimator is robust if it is insensitive to the gross measurement errors and the uncertainties of the model. • Minimal variance: Variance reduction is the central issue of various Monte Carlo approximation methods, most improvement techniques are variance-reduction oriented. 22.3.1  The Least-Squares Estimator

The introduction of the least squares (LS) approach is generally attributed to Carl Friederich Gauss (1809), who used it in the determination of planetary orbits.

22_Singh_ch22_p22.1-22.20.indd 3

In its simplest formulation, LS allows estimating the parameters of a simple or a complex model by minimizing the sum of squares of deviations between observed and model predicted values. The Ordinary Least Squares Given a vector of observations Y ≡ [ y1 ,  y 2 , …,  yn ] (water levels, discharges, etc.) with n, the number of observations, using a model M ( X , ϑ ), a function of known and/or observed quantities X and of a set of unknown (or at least uncertain) parameter values ϑ, it is possible to simulate the observations by  ≡ [ y ,  y , …,  y ]. producing a corresponding vector of forecasts, namely,   Y 1 2 n  is a product of the model M ( X ,ϑ ), it will be different Given that  Y for any k k  given realization ϑ  of the set of parameters, namely,  Y(ϑ  ). The least-square approach allows to estimate the model parameters by minimizing the follow (ϑ), namely: ing function of the residuals d(ϑ) = Y −  Y

{

}

 (ϑ )] = min 1 ∑n [ y − y (ϑ )]2 (22.7) ϑ LS = minϑ J[ε (ϑ )] = minϑ J[Y − Y i =1 i ϑ i n In principle, to just estimate a set of parameters, one does not need to make assumptions when using the LS approach. Nonetheless, if one wants to be able of evaluating the properties of the estimator or drawing confidence limits around the predicted values, a number of assumptions on the nature of the residuals are required. The classical assumption, which makes the least squares identical to the simple linear regression, is that d are assumed to be weakly stationary (namely, stationary up to the second-order moment), independent and Normally distributed random variables. Under this assumption, the least squares approach, called the ordinary least squares (OLS) approach, is derived as the minimum of the trace of the covariance assuming a multivariate homoscedastic Normal distribution

{

}

ϑ OLS = min trace  E ( ε − µ ε )( ε − µ ε )T = min trace Σ ϑ

{

ϑ

(22.8)

}

with le the mean of d and ¬ =E ( ε − µ ε )( ε − µ ε )T = σ ε2I, its covariance matrix, a diagonal matrix with constant variances under homoscedasticity assumptions, which leads to:

T

(ε − µε ) (ε − µε ) ϑ OLS = min trace  Σ = min  = min σ ε2 (22.9) ϑ ϑ ϑ n

The Generalized Least Squares Frequently, the assumptions of weak stationarity and independence do not hold, which implies the need for extending OLS to the heteroscedastic case, where ¬ is a full symmetrical matrix. In theory, minimizing the trace of the correlation matrix, can do it: ( ε − µ ε )T Σ ε−1 ( ε − µ ε ) (22.10) ϑ ϑ n In practice, this approach, which is called the generalized least squares (GLS) approach, cannot be directly solved because, while trace Σ ε = nσ ε2 can be estimated under the assumption of ergodicity, this is not the case for the full matrix. To solve the GLS problem, special approaches have been devised, such as the feasible generalized least squares (FGLS). In the first stage, one solves the OLS problem. Using the OLS-estimated parameters, ϑ  , one can  (ϑ ) from which one can iteratively produce a set of residuals ε (ϑ ) =   Y − Y obtain an approximated estimate of ¬. Please note that the GLS approach corresponds to maximizing the natural logarithm of probability of d under the assumption of a multi-Normal distribution.

{

}

min trace  E ( ε − µ ε )Σ ε−1 ( ε − µ ε )T = min  



1 − ( ε − µ ε )T Σε−1 ( ε − µ ε )

e 2 ϑ GLS = max log = min( ε − µ ε )T Σ ε−1 ( ε − µ ε ) ϑ ϑ (2π )n/2 || Σ ε ||1/2

(22.11)

apart from constants terms that are not relevant in maximization. The Constrained Least Squares As described in Sec. 22.2.1, there are several cases for which, unacceptable parameter estimates can result, as for instance, when the numerosity of the sample is rather small and at the same time, the variance of errors is large. It is therefore necessary to reduce the domain of existence of parameter values by introducing constraints describing, in the estimation phase, physically meaningful ranges or regularity conditions. An example of the constrained linear systems (CLS) approach is given for the estimation of unit hydrograph ordinates, which may result unstable, oscillating, negative, and generally not preserving the mass balance condition (namely, that their sum must equal 1), when OLS or GLS approaches are taken. Several researchers dealt with the problem, as for instance, Eagleson

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22-4     Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation

et al. (1966) who introduced non-negativity constraints by means of linear programming. Few years later, Natale and Todini (1976a,b), followed by Bree (1978) and Bruen and Dooge (1984), introduced non-negativity constraints, mass balance constraints, and regularity conditions in the estimation of unit hydrograph ordinates. A quadratic programming approach was effectively used to solve the constrained least squares problem and the reduction of estimation variance was shown both analytically and numerically by Natale and Todini (1976a, b). As a concluding remark, linear or nonlinear constrained minimization approaches should be used whenever the numerosity of the available data sample is insufficient to allow for reliable parameter estimates and/or the variance of the difference between observed and computed values is quite large. 22.3.2  The Maximum Likelihood Estimator

The maximum likelihood estimator (MLE) was introduced by Fisher (1922) as an alternative approach to the method of least squares. MLE allows estimating parameters of a probability distribution given a sample of observations. Given a sample of n-independent observations ( x1 ,  x 2 , …,  xn ), extracted from a population Ω x with probability density function f x ( x ϑ ), where ϑ are the relevant parameters, the probability density of the sample can be computed as: n

f ( x1 ,  x 2 , …,  xn ϑ ) = f x ( x1 ϑ ) × f x ( x1 ϑ ) ×…  ×   f x ( xn ϑ ) = ∏ i =1 f x ( xi ϑ ) (22.12)

Equation (22.12) shows that the probability of observing a specific sample of length n can be estimated conditionally on the “knowledge” of the parameters ϑ, namely, on the assumption that ϑ values are known. The Likelihood function was introduced by looking at f ( x1 ,  x 2 , …,  xn ϑ ) the other way round, namely, by interpreting Eq. (22.12) as the likelihood of observing a specific value for the parameters once the value of the sample is known. In other words:

(ϑ ; x1 ,  x 2 , …,  xn ) = f ( x1 ,  x 2 , …,  xn ϑ ) (22.13)

The MLE of the parameter values is then obtained by maximizing the likelihood of observing them conditionally upon the observed sample.

ϑ MLE = max ϑ (ϑ ; x1 ,  x 2 , …,  xn ) (22.14)



Please note that formally, under a Bayesian inferential approach, Eq. (22.13) is not fully correct. A proper relation should in fact be derived from the Bayes theorem as: '(ϑ ; x1 ,  x 2 , …,  xn ) = f (ϑ x1 ,  x 2 , …,  xn ) =

f ( x1 ,  x 2 , …,  xn ϑ ) f (ϑ ) (22.15) f ( x1 ,  x 2 , …,  xn )

In Eq. (22.15), while the denominator is independent from the parameters and can be neglected in view of the maximization, this does not apply to the density of parameters f (ϑ), which unfortunately is unknown. In practice, the validity of Eq. (22.13) is introduced by assuming f (ϑ) as a uniform probability density (in other words, we assume a “noninformative prior” on the parameters), therefore, independent from the value of ϑ, and the Likelihood can be viewed as proportional to the density of the parameters given the sample:

(ϑ ; x1 ,  x 2 , …,  xn )  ∝ f (ϑ x1 ,  x 2 , …,  xn ) (22.16)

Given the multiplicative nature of the Likelihood, it is common practice to minimize its natural logarithm (the Log-Likelihood):

ϑ MLE = max ϑ log L(ϑ ; x1 ,  x 2 , …,  xn ) = max Σni=1 log  f x ( xi ϑ ) (22.17) ϑ

Please note that, when the MLE is not directly applied to the estimation of parameters of a probability distribution, but rather to estimate the parameters of a model, such as a hydrological or a hydraulic model, one must convert the observations into variables for which a probability distribution can be assumed. For instance, one can proceed at defining a vector of residuals  (ϑ ), namely, the differences between the observed quantities of ε (ϑ ) = Y − Y interest and the modeled ones, similarly to what described in Sec. 22.3.1 for the LS estimator. If the observations are not independent, the Likelihood can still be defined pending the availability of an expression for the multivariate joint density of the sample. For instance, if the sample is distributed according to a multi-Normal density, then the MLE becomes

1 − ( ε − µ ε )T Σε−1 ( ε − µ ε )

e ϑ MLE = max log 2 n/2 = min( ε − µ ε )T Σ ε−1 ( ε − µ ε )  (22.18) ϑ ϑ (2π ) || Σ ε ||1/2

which, as can be noticed, coincides with the GLS estimator.

22_Singh_ch22_p22.1-22.20.indd 4

Under a set of regularity conditions (Newey and McFadden, 1994) and provided that the Likelihood function is a continuous function twice differentiable in the domain of existence of the observations, which, in the case of hydrological models, as previously mentioned, are the residuals d(ϑ), MLE shows interesting properties. It can be proven that, under the aforementioned conditions, the MLE is a consistent, efficient, and asymptotically Normal estimator. Consistency means that the estimated value will tend to the true value when the numerosity of the sample tends to infinity; efficiency is the property for which the covariance of the MLE, which can be estimated by using Fisher Information or Fisher Information Matrix in the multivariate parameter case (Lehmann and Casella, 1998), reaches the Cramér–Rao lower bound (Rao, 1945; Cramér, 1946) when the sample size tends to infinity; asymptotic normality implies that the MLE tends to the Gaussian distribution with mean the true value and covariance matrix equal to the inverse of the Fisher Information Matrix (Lehmann and Casella, 1998). 22.3.3  The Multicriteria Approaches

As discussed in the previous sections dealing with least square or ML approaches, parameter estimation is essentially based on the minimization or maximization of a convex function of departures of the modeled values from the observed ones, with the objective of best reproducing observations using the model. Sometimes, when dealing with lumped or semidistributed hydrological models, the objective of parameter estimation requires reflecting other issues, such as the timing of the peak flow; improved performances in reproducing low, medium, or high flows; better preservation of mass balance; etc. (Gupta et al., 1998). This has motivated multicriteria calibration approaches in which multiple sets of observations and/or multiple evaluation criteria are employed (Gupta et al., 1998; Legates and McCabe, 1999; Madsen, 2000; Yapo et al., 1998). Moreover, in the case of fully distributed physically based models, measurements are not limited to the flow at the gauged sections; several additional state variables measured in space and time are also available (Refsgaard, 1997), such as snowpack elevation and water equivalent, soil moisture, groundwater table elevation, areas at saturation, etc. Assimilation of all these additional data may be appropriate to better reflect the internal physical nature of the state variables. Accordingly, the new calibration criteria are either introduced into the objective function assigning them individual weights and aggregated into a single measure to be maximized (or minimized) using single-objective optimization procedure or compromised by searching for Pareto-optimal, namely boundary nondominated, solutions (Pareto, 1896–1897, 1906) through evolutionary multiobjective optimization (EMO) algorithms. Application of the single-objective optimization can be found in Madsen (2000), who used the aggregation (or scalarization) approach together with the SCE-UA single-objective optimization algorithm developed by Duan et al. (1992). Several applications of Pareto-optimal compromise solutions can be found in the literature using the NSGA-II (Nondominated Sorted Genetic Algorithm-II due to Deb et al., 2002), the Multiobjective Complex Evolution (MOCOM-UA due to Yapo et al., 1998), the Multiobjective Shuffled Complex Evolution Metropolis (MOSCEM) algorithm (Vrugt et al., 2003), and the AMALGAM (A Multi-ALgorithm Genetically Adaptive Multiobjective) method (Vrugt and Robinson, 2007). In the latter case, a set of Pareto-optimal solutions is initially found, while the most satisfactory solution is successively selected either subjectively or by maximizing an appropriate (subjective) utility function, not necessarily an economic benefit/cost function, but rather a function expressing the decision maker’s perception of benefits, costs, or risk (see Chap. 26 for a more detailed definition). Several examples of application of multicriteria approaches in calibration and validation of hydrological models can be found in the literature (Duan et al., 1992; Seibert, 2000; Beldring, 2002; Boyle et al., 2003; Vrugt et al., 2003), while an extensive list of multicriteria and Pareto-optimal applications can be found in Efstratiadis and Koutsoyiannis (2010). 22.3.4  The Bayesian Inferential Approaches

In order to introduce the Bayesian Inferential approaches, it is worthwhile quoting Draper and Krnjajić (2013): “In the Bayesian modeling paradigm for inference, prediction and decision-making, there are three fundamental ingredients: ϑ, something unknown or only partially known to You (a generic person wishing to reason sensibly in the face of uncertainty; D, an information (data) source that You judge to be relevant to decreasing Your uncertainty about ϑ; and B” (please note that B may correspond to a hydrological model and/or to) “a set of propositions (true-false statements) summarizing Your background assumptions and judgments about relevant aspects of ϑ (e.g., that ϑ ≥ 0 if ϑ represents the mean remission time for a specified set of patients with a given disease) and D (e.g., that the data set arose as the result of a randomized controlled trial with the following design: ...).

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Data assimilation    22-5 

Decision Theory (Berger, 1985; Bernardo and Smith, 1994; De Groot, 2004), and as discussed earlier by Draper and Krnjajić (2013) decisions should be taken on the basis of an expected utility E{U ( y )}, which can be a combination of expected benefits and expected losses. U(y) can be estimated only if the probability density function of the conditioning variable y is known. For instance, y can be the water level to be compared to a threshold value above which losses are no more null. Since both in planning and management y is generally unknown, the expected value of the utility function can be defined as:

In this framework the physicist R. T. Cox (1946, 1961) (also see Jaynes, 2003) began with a conditional plausibility operator p(A\B) acting on propositions A and B and calibration results for Bayesian model specification developed from this a full probability calculus in which, for example, propositions such as ϑ ≥ t for real-valued ϑ bring the usual machinery of cumulative distribution functions and densities to bear on the process of uncertainty quantification. Cox proved that—under a reasonable set of axioms involving internal logical consistency and representation of degrees of plausibility by real numbers Your uncertainty quantification, if You wish to be rational, should be based (for inference and prediction) on the conditional probability distributions p(ϑ|B) and p(D|ϑ, B); for decision-making, it had previously been shown by Ramsey (1926) that the only other relevant ingredients are a set A of possible actions a and a utility function U(a, ϑ*), expressing (in real-valued terms) the gain that would result if You chose action a and the unknown ϑ actually took the value ϑ*. In this formulation p(ϑ|B)—usually called Your prior distribution, although temporal considerations need not arise in specifying it—quantifies all of Your information about ϑ external to D, and p(D|ϑ, B)—typically referred to as Your sampling distribution, even though it is not necessary in this approach to consider other data sets that might have been observed but were not—quantifies Your predictive uncertainty about D given ϑ, before D has arrived. Note that Cox’s use of the phrase logical consistency has nothing to do with the repeated-sampling idea of asymptotic consistency ….”





• ( inference) p(ϑ D , B) = c   p(ϑ B) p(D ϑ , B), in which c is a positive normalizing constant and p(ϑ D , B)—usually called Your posterior distribution, although again temporal considerations are not central to the formulation—quantifies the totality of Your information about ϑ, both internal and external to D; ∗ ∗ • (prediction) p(D D , B) = ∫ p(D ϑ , D , B) p(ϑ D , B)dϑ , where D* is Θ

D , B) = arg max a∈A E(ϑ D ,B ) {U (a,ϑ )} = arg max a∈A ∫ U (a,ϑ ) p(ϑ D , B)dϑ Θ

(22.19) in other words, a* is the action that maximizes expected utility, where the expectation is over Your posterior uncertainty about ϑ given D.”

The described Bayesian approach shows that if one wants to improve the effectiveness of prediction as well as the reliability of decisions, it is not sufficient to estimate a specific “optimal value” ϑ* for the parameters in the inferential problem, but it is rather necessary to assess the full parameter probability distribution (or density) in the form of the “posterior” probability density of parameters p(ϑ D , B) given a set of propositions B (usually your model and/or physical constraints) and a set of observations D. The “posterior” probability density can then be used to estimate the expected predictive density as well as the expected utility function value to be optimized (see Chap. 26). 22.3.5 Parameter Estimation vs. Posterior Parameter Densities Estimation

Why one should spend larger amount of computational efforts in order to estimate the full posterior density of parameters f ′′(ϑ D ,  B) (with B a model and/or a set of propositions and D the set of observations used to assess the posterior density), instead of estimating a single “optimal” set ϑ* is a major aspects to be clarified. To do so, let us consider that even in the case where perfect values for ϑ exist, due to a correct physical interpretation of the phenomenon under study, all the errors floating around (see Sec. 22.2) will unavoidably distort their estimated value ϑ . This is the classical nature of the “inverse problem,” where even small perturbations in the observations can generate large errors in the estimated parameter values. Moreover, not necessarily perfect values for ϑ exist, due to the inaccuracies in the description of phenomena or the coarseness of the model used, which will make the optimal parameter values vary randomly or according to specific situations. In other words, if not the parameters, at least the parameter estimates must be viewed as random variables, which best description is uniquely provided by the full probability density f ′′(ϑ D ,  B). Bearing in mind that the most common scope for using models, both in the planning and in the management phases, is to improve decisions, following

22_Singh_ch22_p22.1-22.20.indd 5

U ( y ) f ( y )d y (22.20)

E[ f { y | y (ϑ )}] = ∫

Ωϑ

f [ y | y (ϑ ) ] f "(ϑ | D , B)d ϑ (22.21)

with Ωϑ the domain of existence of the parameters and the explicit dependence on D and B has been omitted for the sake of clarity. Note that the integral appearing in Eq. (22.21) is usually evaluated via Monte Carlo sampling (see Chap. 23), to overcome the analytical complexity. Using the expected predictive density of Eq. (22.21), one can then reliably estimate the expected value of the utility function E{U ( y )} to be used within the decision-making framework:

future data” (please note with reference to Chap. 27: not the model forecasted quantity but rather the real unknown one) “and Q is the space over which You acknowledge Your uncertainty about ϑ. Often D provides no additional information about D* if ϑ is known, in which case this equation simplifies to p(D ∗ D , B) = ∫ p(D ∗ ϑ , B) p(ϑ D , B)dϑ and involves only Θ previously-defined quantities; and • (decision) The optimal action a* is given by = (a∗

+∞

0

where f (y) is the probability density expressing our incomplete knowledge (in other words, our uncertainty) on the value that will actually occur. In order to assess the value of y, we make a model providing an estimate for y, namely, ŷ, which hopefully will be less uncertain than y, although still not fully known. Following the Bayesian approach ŷ can be expressed as a function of the parameters, namely, ŷ (ϑ) and, as described in Sec. 22.3.4 a posterior density for the parameters can be assessed. Moreover, following what is described in Chap. 26, the conditional density of y given the model predicted value ŷ, namely, f (y|ŷ (ϑ)) can be derived using one of the available predictive uncertainty postprocessors (Chap. 26, Sec. 26.6) and used to estimate an expected predictive density:

“It then follows from Cox’s and Ramsey’s results in this framework that the three basic statistical activities of inference, prediction and decision-making are each governed by a single equation (familiar in Bayesian work), as follows:

a∗

E {U ( y )} = ∫



+∞

E{U ( y )} = ∫ U ( y )E [ f { y | y (ϑ )}]d y (22.22) 0

The alternative approach where an optimal set of parameters ϑ* is estimated instead of the full posterior parameter density, namely:

+∞

E{U ( y )} = ∫ U ( y ) f [ y | y (ϑ ∗ )]d y (22.23) 0

leads to a different result, because, unavoidably:

E[ f { y | y(ϑ )}] = ∫

Ωϑ

f [ y | y (ϑ )] f ′′(ϑ | D , B)d ϑ ≠   f [ y | y (ϑ )] (22.24)

This is what hydrologists tend to do in practice, and, as shown by Liu et al. (2005), due to the integral nature of Eq. (22.23), it can often be used in decision-making, particularly when using structural and physics-based models, provided one can show that the difference between the two expressions is not large, namely,

E[ f { y | y (ϑ )}] = ∫

Ωϑ

f [ y | y (ϑ )] f ′′(ϑ D ,  B)d ϑ ≈   f [ y | y (ϑ ∗ )] (22.25)

22.3.6  Choice of the Most Appropriate Parameter Estimation Approach

It is rather difficult to provide guidelines on which is the most appropriate parameter estimation tool to be used. This will depend at the same time on the availability of information on the quantitative description of the phenomenon to be represented (structural equations, time-independent and timedependent data describing the spatial domain, the initial and boundary conditions, the forcing input functions, etc.) as well as the quality of results one would like to obtain. Simple approaches with few available input-output data should restrict themselves to OLS approaches. If long series of input-output data are available, one can apply GLS and ML approaches. Multicriteria approaches can be used with more complex structural models when the approximations in the internal model links can distort parameter estimates. Nonetheless, whenever important decisions are linked to the modeling exercise, it is essential that one resorts to the Bayesian approaches in order to derive a representation of the full density of parameters and not only an expected or a modal value. 22.4  DATA ASSIMILATION

A part of the challenge for every modeling/forecasting problem is the combination of the knowledge embodied in a model and of the information that can be gained from independent observations to find the best representation of

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22-6     Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation

the dynamic behavior of a system. Both model predictions and observations are imperfect and affected by uncertainty, but an appropriate merging has proved to synergistically provide more accurate results, that cannot be obtained when used individually (McLaughlin, 2002; Liu and Gupta, 2007; Reichle, 2008; Liu et al., 2012). Charney et al. (1969) first suggested combining current and past data in a meteorological model, using the model’s equations to provide continuity in time and dynamic coupling amongst the meteorological fields. From this approach, several techniques, known as data assimilation (DA), have developed in meteorology, hydrology, and other fields. In the broader sense, DA is the process with which the information content of observations (data) is transferred into other data records, into model input state variables, parameter estimates, or model output. DA is also highly related to calibration, but, when dealing with models, it is generally applied to already “calibrated” models in order to update state estimates or to adjust the model performance in accordance to the latest measured quantities. For instance, one may refer to DA when infilling data record gaps or when setting up the input state variables of a model, such as the snow cover or the soil moisture content, based on recent point or spatial measurements. DA is also directly used to update model output to match the latest observations or to improve model parameter estimates. In the case of record gap infilling, the DA problem consists in using the information content of the surrounding measurement records to estimate the expected value of the missing observations, possibly using approaches leading to minimum variance estimates. DA techniques were firstly developed and used operationally in meteorology (Daley, 1991) and oceanography (Bennet, 1992). The application of DA in hydrology is more recent, although in dramatic growth under the motivation in understanding and possibly reducing the uncertainty involved in hydrologic modeling, but mostly because of the increasing availability of satellite observation at higher spatiotemporal resolutions. Hydrological applications range from addressing the assimilation of traditional in situ observations, such as stage, discharge, rainfall, soil moisture, and snowpack measurements (e.g., Todini et al., 1976; Todini and Wallis, 1978; Kitanidis and Bras,1980a, b; Walker and Houser, 2001; Young, 2002; Seo et al., 2003, 2009; Vrugt et al., 2005; Clark et al., 2008) to incorporating remotely sensed retrievals of various quantities, such as soil moisture, land surface temperature, snow water equivalent, and/or snow cover area or extent, surface water level (Entekhabi et al., 1994; Houser et al., 1998; Castelli, 1999; Galantowicz et al., 1999; Walker et al., 2001; Boni et al., 2001; Brocca et al., 2010; Reichle et al., 2010; Rodell and Houser, 2004; Giustarini et al., 2011; De Lannoy et al., 2012, among others). Although, most DA applications were meant for updating model dynamic states, stochastic DA techniques have been also developed and applied to joint-state-parameter estimation (Todini, 1978a, b; Thiemann et al., 2001; Moradkhani et al., 2005a, b; Labarre et al., 2006), by coupling state DA with parameter calibration techniques (Vrugt et al., 2005) either using a technique known as “state augmentation,” that is, essentially extending state vector with the model parameters (e.g., Gelb, 1974) or by adding a second Kalman Filter (Todini, 1978a, b; Labarre et al., 2006) or an Ensemble Kalman Filter (Moradkhani et al., 2005a) in the space of parameters. Anyway, the focus of DA is largely on model state updating, referring to the problem of assimilating observations that can be distributed in space and in time. There are two basic approaches to DA, variational and sequential techniques (Bouttier and Courtier, 1999), which essentially differ in their numerical cost, their optimality, and their suitability for real-time data assimilation. 22.4.1 State-Space Formulation

To illustrate the basic concept of data assimilation, let us consider a generic, typically nonlinear, dynamic state-space formulation of a stochastic model, that predicts the system state vector xt at time t as a function of the system state vector estimate at previous time steps (for the sake of clarity, the forcing, or control, vector ut, and model parameters ϑ are not included).

x t = M( x t −1 ) + w t (22.26)

where M() is a linear or nonlinear model operator and wt a model error vector. Bearing in mind that there is not always an immediate relationship between the state variables and the observations, the following equation provides a general map that relates states to observations zt:

z t = H( x t ) + v t (22.27)

where H() is a linear or nonlinear observation operator, and vt the observation error.

22_Singh_ch22_p22.1-22.20.indd 6

In most of DA techniques, to render the problem mathematically treatable, these equations are linearized introducing the tangent linear operators Mt and Ht:

x t = Mt x t −1 + w t (22.28)



z t = Ht x t + v t (22.29)

Moreover, assumptions on the statistical properties of the errors terms are also made: many approximations consider terms wt, and vt as zero-mean white noise (namely, uncorrelated in time) sequences with Gaussian probability distribution, characterized by covariance matrices Qt and Rt respectively, and null cross covariances. 22.4.2  Variational Data Assimilation

The aim of variational data assimilation (VDA) techniques is to find the best fit, in the least-squares sense, between selected output quantities of the model and their observed equivalent, by minimizing an objective function or cost function J. The cost function J quantifies an aggregate of errors (assumed independents and additive at different time), each being weighted by the corresponding error covariance matrix. Considering a time-independent problem, the objective function J can be written as:   J ( x ) = 1 / 2( x − x b )T B −1 ( x − x b ) + 1 / 2(z − z )T R −1 (z − z ) (22.30)



where xb refers to a “background” estimate of the state vector, for example, the result from a previous assimilation analysis, characterized by a background error covariance matrix B. In practice, xb usually corresponds to the “first guess,” that is, the point used to initiate the minimization procedure. VDA can thus be considered simply as an optimization problem, where the state vector x is “calibrated,” that is, the variable with respect to which the cost function is minimized. When the vector of model-predicted observations zˆ is calculated through a linear observation operator H, J is a quadratic functional of x with a unique minimum and provides a solution which is equivalent to the BLUE (best linear unbiased estimator) result (see Sec. 22.4.3). Depending on the spatial dimension of the state variable, VDA methods can be one-dimensional (1D-Var), two-dimensional (2D-Var), and threedimensional (3D-Var). Introducing the time dimension into the assimilation scheme provides the four-dimensional variational assimilation (4D-Var, 3D in space + 1D in time). Figure 22.2 shows a schematic representation of 3D-Var and 4D-Var. The optimization objective function can be reformulated to handle observations at several different times within a prescribed assimilation window, as follows:   J ( x ) = 1 / 2( x − x b )T B −1 ( x − x b ) + 1 / 2 Σ N (z − z t )T R −1 (z − z t ) (22.31) o

o

o

o

o

t =0

t

t

t

where the subscript refers to the time, varying from the initial time to N the number of observation time points within the assimilation interval. The 4D-Var is a generalization of 3D-Var, where the control variable is the state vector xo at the beginning of each assimilation window that is calibrated to allow also for the best fit between the model trajectory and the observations over the assimilation interval. In this sense, the 4D-Var is a dynamic observer method, well suited for smoothing problems. It is worth mentioning that in the 4D-Var, the model equation is imposed as a strong constraint in the minimization of the objective function, that is, the sequence of model states xt must be a solution of the model equation, which is considered to be perfect. A straightforward and elegant way to incorporate the model constraint in the cost function is to introduce Lagrangian multipliers. In application involving high-dimensional and nonlinear models, minimizing the objective function in Eq. (22.31) is not analytically feasible, and advanced numerical (i.e., variational) methods are required. The solution is sought iteratively, by performing several evaluations of the cost function and of its gradient ∇J in order to approach the minimum using suitable descent methods (steepest descent, conjugate gradient, or quasi-Newton method). Nevertheless, to obtain a treatable minimization problem, especially because of the calculation of the second term of J, simplifications and approximations are needed, like linearizing the state and observation equations. With these premises, the adjoint model technique is typically used as it permits efficient calculation of the gradient of the cost function and is proved to be large-scale efficient (e.g., Huang and Yang, 1996). KF and 4D-Var are valid and produce equivalent results at the end of the interval if model operator M and observation operator H are linear (namely, M and H), whereas the covariance propagation in 4D-Var is implicit and only applies within the assimilation interval.

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Data assimilation    22-7 

Statistical Correction Statistical correction can be considered as a derivative of the direct insertion technique. The approach assumes that the model pattern is correct but contains a nonuniform bias: the predicted states are thus scaled and shifted to match mean and variance of the observations. State

State

Adding another term to the objective function in Eq. (22.31) that takes into account errors in model formulation, yields the so-called weak-constraint 4D-Var: this method is numerically very costly and barely used. Examples of application of VDA techniques in hydrology are given by Castelli et al. (1999), Reichle et al. (2001), and Seo et al. (2003, 2009).

Xb

Xb0

X

X0 Observation Background Analysis

Updated prediction Previous prediction

Assimilation window

Time

Time

Figure 22.2  Schematic representation of different assimilation approaches. 3D-Var and other sequential DA techniques (left); 4D-Var (right). 22.4.3 Sequential Data Assimilation

The sequential techniques (or direct observer) approach the DA problem by sequentially updating the model “background” states, the prior knowledge about the system states, using the additional information brought by a new observation z. The improved estimate of the state vector xa, which is called the “analysis,” is obtained through a correction applied to the background states, which depends on the difference between the observations and the model predicted observations (known as the innovation) multiplied by a weighting factor Kt:

x ta = x bt + K t (z t − zˆ t ) (22.32)

The subscript t refers to the time of the update. The fundamental difference between the various assimilation methods is the choice of the gain matrix Kt, which represents the relative uncertainty in the observations and in model estimates, and is a number between 0 and 1 in the scalar situation. With the aim to look for an unbiased estimate of x ta with minimal variance, that is, a BLUE, and assuming a linear observation operator, that can be achieved choosing a proper Kt from the solution to the variational optimization problem posed in Eq. (22.30), that is, such that objective function J is a minimum. This has been shown analytically (see, for instance, Jazwinski, 1970; Gelb, 1974) to produce:

K t = PtbHtT (Ht PtbHtT + Rt )−1   (22.33)

where Ptb

is the background error covariance matrix of x bt . It can also be shown that choosing the optimal least squares gain, the analysis error covariance Pta matrix simplifies into

Pta = (I − K t Ht )Ptb (22.34)

where I is the identity matrix. Please note that Eqs. (22.32)–(22.34) are similar to the basic equations of the KF approach to data assimilation developed in the 1960s for optimal systems control, which will be detailed in Sec. 22.4.4 together with its variants. Other sequential methods can be assimilated to Eq. (22.32) by choosing a proper gain (Walker and Houser, 2005), such as direct insertion, statistical correction, successive correction, optimal interpolation/statistical interpolation, and nudging. While these approaches, briefly described as follows, are computationally efficient and easy to implement, the updates do not always account for observation uncertainty and do not propagate the state covariance matrix explicitly. Houser et al. (1998) provide an example of application of several of these alternative assimilation approaches. Direct Insertion It is one of the earliest and most simplistic approaches to data assimilation. As the name suggests, the forecast model states are directly replaced with the observations, whenever one is available, by assuming explicitly that the model has no useful information and that the observations are perfect.

22_Singh_ch22_p22.1-22.20.indd 7

Successive Correction The method consists of successively adjusting a background estimate for an individual model state by nearby observations. The analysis at each time step is found through a series of iterations that use weight W to smooth observations into the model states (Bratseth, 1986) by modifying the states at all grid points within a user-defined influence radius for each observation. The Cressman function is one of the most commonly used weighting system to account for the distance between the model grid point and the observation. At each time step t, the approach is usually applied consecutively to each observation s as follows:

x ts+1 = x ts + Wts (z t −   z t )

where z t  is evaluated through the observation operator considering the state estimate at the sth iteration and the analysis x ta is the value obtained at the end of the iterations. Optimal Interpolation The method, also referred to as statistical interpolation, approximates the “optimal” solution of Eq. (22.32) through an algebraic simplification of the computation of the gain K in Eq. (22.33). The analysis equation can be regarded as a list of scalar equation, one per model state in the vector x, and the analysis increment for each of them is given by the corresponding line of K times the innovation vector. The fundamental hypothesis in optimal interpolation is that for each model variable only a few observations are important in determining the analysis increment, so that background and observation error covariance submatrices restricted to selected observations are considered in the gain matrix. The algorithm also requires the background error covariance matrix to be specified, that is often approximated with a fixed structure for all time steps and relies on the design of empirical autocorrelation functions given only by distance (Lorenc, 1981). Nudging Nudging (or Newtonian relaxation) has its roots in control theory and fundamentally consists of adding to the model equation a feedback term to gradually “nudge” the model state toward the observations. The correction term is assumed proportional to the difference between the observation and the equivalent quantity computed by the model, multiplied by a gain matrix accounting for the uncertainties of both modeled and observed variables. The model solution can be nudged toward either individual observations; in this case, it is referred to as observation nudging, or gridded analysis obtained from observation using other assimilation schemes (Stauffer and Seaman,1990), leading to the so-called analysis nudging. Nudging differs from the previously introduced sequential DA techniques in the fact that through the numerical model the time dimension is included and thus it can be considered as a continuous form of four-dimensional data assimilation methods, that is, those that use information from more than one

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22-8     Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation

time like 4D-Var. Nudging allows the model to adjust gradually in a realistic manner over multiple time steps to changes introduced by DA. 22.4.4  The Kalman Filter

The typical sequential data assimilation technique is based on the KF (Kalman, 1960; Kalman and Bucy, 1961). Given its importance, this section provides the basic elements of the KF, while a more extensive derivation can be found in Jazwinski (1970) or Gelb (1974). Before introducing the KF, it is worthwhile explaining the domain of application of a “filter.” A filter is essentially a data assimilation tool. In systems engineering, there is a clear distinction between “filter,” “predictor,” and “smoother.” A filter aims at estimating the value of a quantity at time t0 conditionally to available observations up to time t0; a predictor aims at estimating the value of a quantity at time t > t0 conditionally to available observations up to time t0; a smoother aims at estimating a quantity at time t < t0 conditionally to available observations up to time t0. The KF (Kalman, 1960; Kalman and Bucy, 1961) is the recursive extension of the Wiener Filter (Wiener, 1949) to linear, or locally linearized in time, stationary as well as nonstationary processes. The original formulation of the KF is in continuous time, but in hydrological applications it has always been applied in its discretized form. In discrete form, the KF is build around two vectors: the state vector xt , which contains the n variables that uniquely characterizes the system under study and the measurement vector zt , which contains the m available measurements. In its simplest discrete formulation, two equations describe the system. The first one represents the system dynamics through the evolution in time of the state vector, while the second one represents the measurement system by describing which of the state variables are being directly or indirectly measured. x t = Φt ,t −1   x t −1 + Γ t   w t   known as the “model” or “system” equation (22.35) and z t = Ht   x t + v t   known as the “measurement” equation

(22.36)

where x t   [1, n] is the state vector, namely, the vector containing all the n sufficient state variables able to fully represent the system’s dynamics; Φt ,t −1 [n, n] is the state transition matrix transforming the state vector from its state at time (t − 1) to its state at time t, which may vary at each step in time; w t  [1,  p ≤ n  ] is an unknown random Gaussian time independent process, with mean w t and covariance matrix Qt , used to represent the model error, while matrix Γt [n,  p] is an appropriate matrix to relate the dimensions; z t  [1, m ≤ n] is the measurement vector, namely, the vector containing the m observations and matrix Ht [n, m] is an appropriate matrix to relate the dimensions; vt [1, m] is the measurement error, represented as a random Gaussian time independent process with mean v t and covariance matrix Rt, which is also assumed as being independent from the model error wt . For the sake of simplicity, following the original derivation by Kalman (1960) the “control” term has been omitted from the model equation, while a measurement error term has been added. The KF aims at finding x t t the unbiased minimum variance estimate of the unknown state xt together with its error covariance matrix Pt t conditional to the knowledge of an unbiased a priori state estimate x t t −1 and its covariance matrix Pt t −1 (which fully represent the stochastic process due to the hypothesis of Gaussian errors) and the latest noise-corrupted measurements zt together with its measurement error statistics (again mean and covariance are sufficient due to the Gaussian hypothesis). This estimate is obtained using the following equations, a neat derivation of which can be found in Gelb (1974). At each step in time, the estimates of the state and of the error covariance are extrapolated from the previous step: x t t −1 = Φ   x + Γ w t   state extrapolation (22.37) t ,t −1

t −1 t −1

t

Pt t −1 = Φt ,t −1   Pt −1 t −1 ΦtT,t −1 + Γ t  Qt   Γ tT   error covariance extrapolation (22.38) Then the following quantities are estimated: υt = z t − Ht   x t t −1 − v t   known as the “Innovation”

(22.39)

K t = Pt t −1HTt (Ht   Pt t −1  HTt + Rt )−1   known as the “Kalman Gain”

(22.40)

Finally the a priori estimates can be updated to include the latest measurements: x t t = x t t −1 + K t υt      state update

(22.41)

Pt t = (I − K t  Ht  )  Pt t −1   error covariance update

(22.42)

22_Singh_ch22_p22.1-22.20.indd 8

With respect to the Wiener filter, the advantage of the KF lays in the fact that optimality (in terms of unbiasedness and minimum variance) is recursively imposed at each step in time as opposed to the batch form required by the Wiener filter. This also allows to locally linearizing in time non linear dynamic processes to produce what is known as the extended Kalman filter (EKF) (Jazwinski, 1970). The basic problem in applying the KF is that the optimality conditions only hold when the state transition matrix Φt ,t −1, together with the model and measurement error statistics ( w t , Qt , v t ,  Rt ) are fully known. Mehra (1970) provided a solution to the problem of estimating the unknown error statistics by imposing the time independence of the innovation, a condition associated with the KF optimality. The estimation of the state transition matrix parameters (generally known as the “hyperparameters”) showed to be a far more complex problem. Several approaches can be found in the literature for solving the nonlinear estimation problem originated by the simultaneous estimation of both state and parameter values, ranging from developing the KF in the parameter space (Mayne, 1965; Sage and Husa, 1969) to the use of the EKF on state vector enlarged with the parameters (Jazwinski, 1970) or from the use of ML Cooper and Wood (1982a, b) (Gupta and Mehra, 1974; Wood and O’Connell, 1985) to the method of moments (Wojcik, 1993) and to full Bayesian approaches (Mantovan et al., 1999). Following the instrumental variables (IV) approach (Kendall and Stuart, 1967; Young, 1974; Young and Whitehead 1977), Todini (1978a, b) realized that the posterior state estimate x t t is the best possible IV, being totally independent from measurement noise and a minimum variance estimator of the true unknown state due to optimality of the KF. Accordingly, he developed the mutually interactive state parameter (MISP) estimation technique by using two mutually conditional KFs: one in the space of the state conditional to the previous step parameter estimates and one in the space of the parameters conditional to the previous and the last updated state estimates (Todini, 1978a, b). MISP, was recently found by Mantovan et al. (1999) very close to ML, but much less demanding in computer time, while a full Bayesian approach, requiring the use of the Gibbs sampler (Gelfand and Smith, 1990; Carter and Khon, 1994), to produce the posterior distributions, had to be abandoned due to its exaggerated computer time requirements. MISP was also recognized as the most efficient unbiased estimator in other fields, such as the automatic speech recognition (Labarre et al., 2006). Several examples of application of KF, in hydraulic and hydrological applications can be found in Chiu (1978), while specific applications to rainfall runoff models can be found in Hino (1973), Szollosy-Nagy (1976) for on-line estimation of linear transfer function models, in Todini and Wallis (1978) for on-line state and parameter estimation of a threshold-type multiple input single output ARX model and in Georgakakos (1986a, b) for the on-line update of the Sacramento model parameters using an EKF. Before concluding this section, it is worthwhile pointing out that the KF, being a “filter” is not a “predictor.” Therefore, it is an excellent tool for assimilating observations and calibrating state and parameter values, but rapidly becomes very poor when used for “forecasting” at future times, since observations are no more available. Therefore, the KF cannot be directly used for assessing what will be defined as “forecasting uncertainty,” unless the problem is reversed and the model forecast, which is the only available information at future times, is now taken as a “pseudomeasurement” of a future unknown realization. The Extended Kalman Filter The KF provides optimal estimates of the state for linear systems when both the model and the measurement systems are affected by independent Gaussian noise. When dealing with nonlinear dynamical systems, application of the KF implies local linearization Smith et al. (1962), McElhoe (1966) are among the first ones who studied techniques to apply the standard linear KF methodology to a linearization of the true nonlinear system. This approach, called EKF, is suboptimal and can easily lead to divergence. The EKF became the algorithm of choice in numerous nonlinear estimation, and machine learning applications, including estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification. To overcome the EKF limitations, new developments have appeared which aim at overcoming the need for linearizing nonlinear models and the need for the Gaussian assumption of model and measurement errors. New approaches appeared such as the ensemble Kalman filter (EnKF) (Evensen, 1994, 2003) and its simplified version known as the unscented Kalman filter (UKF) (Julier and Uhlmann , 1997; Wan and van der Merwe, 2001) and finally the particle filter (PF) (Gordon et al., 1993), which is the closest to a full Bayesian approach, together with its simplified version, the unscented particle filter (UPF) (van der Merwe et al., 2000).

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Data assimilation    22-9 

Interesting overviews of all these development can be found in Chen (2003) and Feng et al. (2011). The Ensemble Kalman Filter The EnKF, introduced by Evensen (1994, 2003), is an interesting approach aimed at overcoming the linearization problem, which produces suboptimal estimates when dealing with complex nonlinear models. EnKF is presently used for assimilating data in large nonlinear ocean and atmospheric models as well as in hydrological models. The simple and innovative approach of the EnKF is to modify the state and covariance extrapolation [Eqs. (22.37) and (22.38)]. In the EnKF, one starts by extracting an ensemble of m possible states at time t0 , namely x it0 t0, with ∀  i = 1, …, m, from a prior distribution, usually a Gaussian distribution with mean µ x t and covariance matrix Px t . 0 0 The extrapolation of each individual member is then performed using the model M instead of the state transition matrix, as in the standard KF: x it t −1 = M(x it −1 t −1 ) (22.43)



The state and the error covariance are then directly estimated from the extrapolated members: x t t −1 = µ x t t −1 + Γ t  w t   state extrapolation

(22.44)

Pt t −1 = Px t t −1 + Γ t  Qt  Γ tT   error covariance extrapolation

(22.45)

1 m 1 m where µ x t t −1 = ∑x it t −1 and Px t t −1 = ∑ i =1 ( x it t −1 − µ x t t −1 )2, while the other m i =1 m KF equations remain unchanged. After the first step from t0 to t0 +1, there is no more need of generating a new ensemble because it is possible to update each individual member of the ensemble:

x it t = x it t −1 + K t (z t − Ht   x it t −1 − v t ) (22.46)

EnKF has been extensively used in several domains due to its simplicity. The only requirement is to generate a sufficiently numerous sample to reliably allow to estimate the mean and the covariance matrix (generally ≤ 100). Also, the EnKF needs hyper parameters, which in this case are the model M parameters, to be estimated. In that respect, an interesting extension of the MISP approach to EnKF was recently introduced by Mordakhani et al. (2005). The Unscented Kalman Filter The UKF (Julier and Uhlmann, 1997; Wan and van der Merwe, 2001) is a variant of the EnKF, aimed at reducing the numerosity of ensemble members to be generated, which may prove useful when the model simulation cost is very high. Instead of estimating the extrapolated mean and covariance matrix from a numerous randomly chosen ensemble, as in the EnKF, a small but representative “deterministic” sample is used in UKF to calculate mean and covariance terms. Essentially, ( 2n + 1), sigma points (n being the state vector dimension), are chosen based on a square-root decomposition of the prior covariance. These sigma points are propagated through the true nonlinearity, without approximation, and then a weighted mean and covariance are estimated. UKF has the advantage of reducing the number of required simulations for estimating the extrapolated mean and covariance matrix, but, as opposed to the EnKF suffers from the asymmetry of the error probability distribution and should be used for symmetrical distributions if not Gaussian. The Particle Filter The PF introduced by Gordon et al. (1993) was developed on the basis of Smith and Gelfand (1992) sampling-resampling approach and falls in the category of the sequential Monte Carlo (SMC) algorithms aimed at correctly describing the posterior density of the state vector given the model and the measurements. The required posterior density function is represented by a set of random samples with associated weights, which are then used in the estimation phase. In practice, a sample of the state is generated from a prior distribution and the prediction phase, similar to that of the EnKF, implies routing this sample through the model. The update phase of the PF implies resampling and weights estimation (Smith and Gelfand, 1992), for instance, using the principle of the “important sampling” (Doucet, 1998). As the number of samples becomes very large, the SMC representation approaches the usual functional description of the posterior density function, and the SMC filter approaches the optimal Bayesian estimate. The advantages of the PF lie in the fact it allows estimating the full posterior probability density in cases of nonlinear systems with non-Gaussian error

22_Singh_ch22_p22.1-22.20.indd 9

probability distributions. Nonetheless, there are several disadvantages that limit the use of the PF. The large number of samples, much larger than that required by the EnKF, limit the PF to low dimensionality problems. In addition in the PF, there is a danger of degeneration of posterior densities, which requires frequent careful resamplings. An example of application of the PF in hydrology can be found in Moradkhani et al. (2005). Unscented Particle Filter In van der Merwe et al. (2000), the unscented transformation was used to approximate the appropriate “proposal distribution” of the PF, which resulted in the so-called UPF, allowing for a noticeable reduction of computational efforts. The UPF couples the PF with the UKF (Mohammadi and Asif, 2011; Zhao, 2014). In the UPF, the optimal proposal distribution function is approximated as a Gaussian distribution whose statistics (mean and error covariance matrix) are computed using the UKF. Presently, applications of the UPF and its variants are mostly limited to object tracking (Rui and Chen, 2001). 22.4.5  Choice of the Most Appropriate DA Approach

The identification of the most suitable assimilation technique for a given variable is still an open issue in DA for hydrological applications. A number of questions and issues concerning hydrological DA, and particularly land surface DA, are open areas of research mainly related to the development: of bias correction schemes (prior to data assimilation) in order to reduce systematic bias between observations and model outputs; of techniques facing the spatial resolution problem; of improved representation of model and observational errors; of data assimilation algorithms that can efficiently account for the typical nonlinear nature of hydrological models; of optimization of data assimilation computational efficiency for use in large operational hydrological applications; of multivariate hydrological assimilation methods to exploit observations from multiple sensors with complementary information (Houser et al., 2010). The key factors in determining the choice of the most appropriate assimilation techniques depend on the suitability for the prescribed application, on the computational feasibility, and on the optimality of the solution. Filtering problems, like real-time assimilation, where the data is ingested sequentially as it becomes available, can be solved using sequential approaches or with variational methods if a new smoothing problem is defined at each measurement time. This is usually the framework for forecasting problems, where the observations up to the current time are used to update the initial conditions. Smoothing techniques are suited for analyzing historic data (reanalysis), so that having observations for an entire study interval, they allows to include information also from later observations to improve the state estimation at a given time. Sequential DA methods that considers approximations to the optimal filter equations or alternative solving methodologies, like direct insertion or methods based on the old Creissman analysis scheme, are computationally efficient and relatively easy to implement, but the updates often account for observational accuracy in an empirical way, and information on estimation uncertainty is limited. Furthermore, the common assumption that the observations are more accurate than model forecasts, can lead to discontinuities in the time series of the updated state that can cause possible model shocks. Moreover, these approaches are ineffective in data sparse regions (Houser et al., 2010). On the other hand, methods based on probabilistic interpretation in defining error statistics, such as the variational and other sequential methods, although under different simplifying assumptions are able in theory to represent the uncertainty in the background, in the observations and in the resulting analysis. In this perspective, operational 4D-Var method, which has been widely used in weather forecast (traditionally viewed as an initial value problem; by contrast, typical land surface models are dissipative in nature), assimilates uncertain data into a deterministic model under the assumption that the model is perfect (strong constraint). In weak-constraint 4D-Var, a term in the penalty function involving model errors is included. Through the 4D-Var, being a smoothing algorithm, there is a larger potential for dynamically based balanced analyses and performs satisfactorily when data are sparse. Nevertheless, three major drawbacks of 4D-Var must be mentioned. First, its numerical cost is high compared to approximate versions of the KF or ensemble-based methods (e.g., EnKF and PF). Second, its formalism does not allow to determine the analysis error directly. Finally, the definition of the adjoint model, is time consuming and may be difficult for a system (e.g., hydrological models) which exhibits nonlinearities and threshold processes (Lahoz and Schneider, 2014). The classic KF provides the optimal state estimate for linear systems and provides information on estimation uncertainty through a dynamic updating

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22-10     Calibration, Parameter Estimation, Uncertainty, Data Assimilation, Sensitivity Analysis, and Validation

of the forecast (background) error covariance through time: on the other hand, it is of limited use in hydrological applications where physical models are often nonlinear. For nonlinear dynamics the EKF can be used, but neither the optimality of the analysis nor the equivalence with 4D-Var holds, and if the nonlinearities are strong, it is prone to become unstable. The computational cost (larger than that of 4D-Var even with small models) of propagating error information result in making both KF and EKF impractical for high dimensional systems. Ensemble based-data assimilation methods, like EnKF and PF approaches and their variants, although can be computationally demanding depending on the size of the analyzed ensemble, are becoming attractive for near realtime applications as they do not require the derivation of a tangent linear operator or adjoint equations, and offer great flexibility with respect to the form of model error that can be included. 22.5  SENSITIVITY ANALYSIS

Sensitivity analysis (SA) is the study of how the variation in the output of a model can be apportioned, qualitatively or quantitatively, to different sources of variation (Saltelli et al., 2000), including model parameters, forcing, initial conditions or boundary conditions. An extensive analysis of the relation between model input factors and output variables represents an essential screening tool that can be helpful for manifold purposes: identifying potential deficiencies in model structure; providing guidance for model simplification; identifying model inputs having a significant influence on model responses; finding critical regions in the space of inputs. A related practice is uncertainty analysis, which has a greater focus on assessing  how the data errors propagate through the model and affect the output (see Chap. 26). In this section, although often referring to a generic “input” or “factor,” the focus is mainly on analyzing the sensitivities of model outputs to model parameters: in this perspective, one of the main goal of SA, is to rank, qualitatively or quantitatively, the various parameters based on their relative importance in influencing the model response, with the aim of providing information to achieve an appropriate and parsimonious model structure as well as the minimum number of parameters to facilitate calibration. Referring to the degree to which a parameter affects model output, the terms “influential,” “important,” “effective,” or “correlated” are almost interchangeably used in literature, while it should be pointed out that only the model responses could be properly defined as “sensitive” or “insensitive” with respect to the parameter variation. Any SA approach can be considered as having two components: (1) a strategy for sampling the model parameters space and (2) a numerical or graphical measure to quantify the impacts of sampled parameters on the model output of interest. The implementation of these two components varies widely and several studies have reviewed and classified existing SA methods (Iman and Helton, 1988; Hamby, 1994; Melching, 1995; Frey and Patil, 2002; Cacuci, 2003; Oakley and O’Hagan, 2004; Saltelli et al., 2000, 2004, 2008). Many techniques have been applied also to hydrological models (e.g., Carpenter et al., 2001; Sieber and Uhlenbrook, 2005; van Griensven et al., 2006; Pappenberger et al., 2006; Tang et al., 2007) even if the increasing complexity of distributed models poses several challenges in terms of their computational demands and their potential overparameterization. Throughout this section, the generalized model functional form is represented by y = f ( x ), where it has been assumed a single model output  y, and x is the vector of independent variables xi with i=1,…n potentially including model parameters, forcing, initial conditions, etc. It is worth noting that SA can be accomplished considering either a particular model output y   or a generic error function describing the degree of model fit to the observations. SA techniques can be classified as local, screening and global methodologies, depending on how the parameters are perturbed. A synthetic description of the main approaches is provided in the following section. 22.5.1 Local Sensitivity Analysis

Local methods focus on the study of model response behavior for small changes in input factors, generally considered individually, around a specified value in the parameter space, called “nominal value,” which can be taken from the literature, or assumed as the calibrated or the mean value of the parameters. The major drawbacks of these methods are their inability to fully explore the input space and to account for parameter interactions, making them prone to underestimating actual model sensitivities (Tang et al., 2007). Moreover, the results are function of selection of the nominal value.

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Differential Analysis Differential analysis evaluates local sensitivity at the nominal value of the parameter space by taking the partial derivative of y with respect to an input factor  xi , while all parameters are held constant to their nominal value. The derivative values are themselves the metrics of sensitivity: Sj =



∂y ∂ xi

(22.47) x nom

where the subscript x nom indicates that the derivative is taken at the nominal value in the space of the input. Derivatives are usually normalized by the reference value at which the derivative is calculated, thus removing the effects of units and allowing the comparison of sensitivity indices related to parameters with different units of measure. Inherent to this calculation are the assumptions that the higher order partials are negligible and there is no correlation between input parameters. The calculation of model derivatives is a delicate task. A straightforward and frequently used way for approximating the sensitivities is realized by applying finite difference schemes, described in literature as the brute force or the indirect method (as it does not require manipulation of the underlying model equations and it is applied in a blackbox manner); other methods for calculating the local sensitivities are the direct method, Greens function method, and the polynomial approximation method. A powerful and promising strategy for reliable and accurate derivatives evaluation is the implementation of algorithmic differentiation (Hwang et al., 1987; Griewank, 2000), which is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. 22.5.2 Screening Techniques

Screening techniques are SA methods that also make use of local sensitivity measures to compute parameter sensitivities, but calculated at a larger number of points in the parameter space. One-at-a-time (OAT) designs (Daniel, 1973) and Morris’ design (Morris, 1991), briefly described in the following, are among the most popular screening methods. Other screening SA methods are the factorial, the fractional factorial and the iterated fractional factorial designs, and sequential bifurcation (Box and Draper, 1987). One at a Time Conceptually, one of the simplest and most common approaches is that of perturbing repeatedly model inputs OAT, holding others at their nominal values (Daniel, 1973). A sensitivity ranking can be obtained quantifying the difference (or the percentage change) in the model output due to the change in the input variable. Each input factor, for instance, can be individually perturbed by a factor of its standard deviation (Downing et al., 1985) or varied from its minimum to its maximum value. Morris’ Design The experimental design proposed by Morris (1991) is composed of individually randomized OAT experiments, in which the impact of changing one factor at a time is evaluated in turn, exploring several regions of the parameter space. The method relies on a local sensitivity measure, called elementary effect of xi (EE i), due to a perturbation ∆, and expressed as: f ( x1 ,…, xi + ∆ ,…, xn ) − f ( x1 ,…, xn ) (22.48) ∆ The sensitivity measures proposed by Morris for each factor xi are the mean and the standard deviations of the elementary effects computed at different randomly sampled points in a parameter space normalized to the unit n-dimensional hypercube.

EE i ( x1 ,…, xn , ∆ ) =

22.5.3 Global Sensitivity Analysis

Global SA methods attempt to explore the full parameter space within predefined feasible parameter ranges and distributions, and to address the effect on model output resulting from considering all input parameters simultaneously and from their interactions. The available approaches to perform global SA, can be grouped, not without overlapping, into random sampling methods, variance-based methods and response surface techniques. Random Sampling Methods These approaches consist in sampling at random the vector of input parameters and propagating their effect through the model to explore the relative mapping of the output functions. The parameters are sampled from a prior parameters distribution using one of the several available strategies, such as random Monte Carlo sampling, importance sampling or Latin hypercube sampling (McKay et al., 1979).

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Validation techniques    22-11 

The sensitivity measures inferred from sampling SA methods can be evaluated in several ways: scatter plots, regression and correlation-based methods, and sensitivity tests involving segmented input distributions.

Vij = V [E ( y | xi , xj)] − Vi − Vj (22.50)

Scatter Plots Parameter sensitivity can be assessed qualitatively by plotting the output variable against individual input variables. Scatter plots are useful for quick visual assessment of the influence of individual inputs on model response: the pattern of the plot can directly reveal linear or monotonic relations, the existence of thresholds, or other complex dependencies. Other widely used graphical tools to analyze relations between inputs and outputs are cobweb plots.

The Vi terms are called first-order terms, or main effect of xi , and they represent the reduction of V that would be expected if the true value of xi is known: it is worth noting that the Vi estimate is a more general sensitivity measure than a regression coefficient, as it also works for nonlinear models. In the same way, the second-order terms Vij accounts for the interaction effect between the parameters xi and xj. In this framework, the sensitivity of the output variance to each individual parameter or parameter combinations is then assessed based on the Sobol’s sensitivity indices:

Regression- and Correlation-Based Methods The regression-based SA relies on determining a multiple regression between the model output and the model parameters as explanatory variables. In some way, this relationship is used to replace a highly complex model with a simplified “response surface.” Regression analysis typically involves fitting a linear relationship between inputs and the output which in its simplest form corresponds to the first order polynomials. The fitted regression coefficients (RCs), representing the expected change in the output y per unit change in xi when all the remaining independent variables are constant, provide an absolute sensitivity measure of model parameters. A standardization process is often necessary to remove the effects of units to allow comparisons and rankings. The standardized regression coefficients (SRCs) are obtained by the ratio of parameter and output standard deviations. If the inputs are independent, the SRCs give the fractional contribution of each factor to the variance of y. Whenever a nonlinear but monotonic relationship between input and output exists, a rank transformation (Iman and Conover, 1979) should be applied to get a linear relationship: rank regression coefficient and standardized rank regression coefficient are the correspondent qualitative sensitivity measures. A drawback of rank-based sensitivity measures is that the properties of the model y can be distorted by the rank transformation (Saltelli and Sobol’, 1995). Correlation coefficients can also be used as a measure of sensitivity. The Pearson correlation coefficient, or the corresponding Spearman correlation coefficient based upon rank-transformed values, can be used to characterize the degree of linear relationship between the output values and sampled values of individual inputs. A possible way to account for correlations among other input variables relies in the calculation of partial correlation coefficients or partial rank correlation coefficient. Sensitivity Tests Involving Segmented Input Distributions (Monte Carlo Filtering) Alternative SA techniques involve some form of dividing or segmenting input parameters into two or more empirical distributions based on an associated partitioning of the output distribution according to some criterion (Hamby, 1994). Either graphical methods (e.g., marginal cumulative distribution function plots) or statistical tests, such as Kolmogorov–Smirnov and Cramer–von Mises t tests, test and Mann–Whitney test, F test and squaredranks tests, are used to compare the input distributions, or some of their characteristics (e.g., means or variances), created by segmentation. If the distribution of a certain parameter, xi , is proved to be different in the subsamples, xi is considered to be an influential parameter. Another kind of approach involving segmented input distribution, often applied in hydrology, is the generalized SA, or regional SA (RSA), by Spear and Hornberger (1980), which inspired the generalized likelihood uncertainty estimation procedure by Beven and Binley (1992). The RSA approach partitions the MC-generated parameter sets into two groups “behavioral/non-behavioral,” according to model performance or behavior. Variance-Based Methods Variance-based methods are a particular form of sampling-based SA techniques allowing a full exploration of the input space, accounting for interactions and nonlinear responses. The approach considers that each parameter is responsible for a fraction of the output variance and takes this fraction as the reference sensitivity measure. For independent input factors, the total variance of a model output y can be decomposed into a series of contributions that result from individual parameters as well as parameter interactions: V(y) = ∑i Vi + ∑i  1, is finite if and only if K (q ) < (q − 1). These are also the conditions for nondegeneracy and finiteness of the q-moment of ε n. The order q∗ > 1 above which the moments of Z and ε n diverge is found from the condition K (q∗ ) = (q∗ − 1). For an extension of these results, see Guivarc’h (1990). Scaling of the dressed moments of order q < q* follows immediately from Eq. (28.10), which gives

E[(ε n+1 )q ] = m K ( q ) E[(ε n )q ] (28.11)

One can further show that if the spectral density of the cascade S(ω ) exists, then S(ω ) ∝| ω |−1+ K (2) (28.12)



Since K(2) > 0, the exponent in Eq. (28.12) is larger than −1. Comparison with Eq. (28.8) shows that stationary multifractal measures have a slowerdecaying spectral density than H-sssi processes. This slower power-law decay of S(ω ) is often taken as first evidence of multifractality. Distributions of Z and ε n The distribution of the dressed densities ε n depends on the distributions of W and Z. In turn, the distribution of Z depends on the distribution of W and the multiplicity m. The latter dependence is embodied in the fundamental identity d

Z=



1 m ∑Wi Zi (28.13) m i =1

where the variables Wi and Zi are independent copies of W and Z, respectively. Based on Eq. (28.13), the distribution of Z can be found iteratively as follows (Veneziano and Furcolo, 2003): one starts by setting Z(0) = 1 and then updates the distribution of Z( j ) at steps j = 1, 2,… using

Z( j ) =

1 m ∑Wi Z( j−1)i (28.14) m i =1 d

where the variables Z( j −1)i are independent copies of Z( j−1). As j → ∞, Z( j ) → Z . An important feature of Z and ε n is that when the order of moment diver∗ gence q* is finite the distributions have algebraic upper tails P[Z > z ] ~ z − q and − q∗ P[ε n > ε ] ~ ε . However, for ε n the power-law behavior effectively starts at values of e that increase rapidly as the resolution level n increases and at high resolutions the power-law tail may not be observable, even in large samples. In the high-resolution limit n → ∞, the power-law tail becomes unimportant and the distribution of ε n is attracted to one of three types: lognormal, log-Levy, or a third type with a lognormal or log-Levy distribution of ε n+ = (ε n | ε n > 0) (Veneziano and Furcolo, 2003). Extremes Many applications require knowledge of the high quantiles of en or the distribution of the maximum of en over all the tiles of a cascade at level n. Several such results can be derived, in approximation, from a property of the random variables Ar in Eq. (28.5). Specifically, let g be a given positive number. It follows from Cramer’s large deviation theorem (Cramér, 1938; Stroock, 1999) that as r → ∞

P[ Ar > r γ ] ~ r − C (γ ) (28.15)

where ~ denotes equality up to a factor that varies slowly with r at infinity and C(g ) is the Legendre transform of K(q),

C(γ ) = max{qγ − K (q )} (28.16) q

Since the bare densities en ; b have the same distribution as Ar for r = mn, Eq. (28.16) has direct implications on the upper tail of those densities when n is high. What happens for the dressed densities en = en ; b Z is less obvious because of the effect of Z. The problem has been studied by Schertzer and Lovejoy (1987) and Veneziano (2002), where the following is proven. Define a “dressed moment scaling function” K d (q ) as

log E[ε rq ]  K (q ), = r →∞ log r ∞ ,

K d (q ) = lim

q < q∗ q ≥ q∗

(28.17)

where q* is the order of moment divergence and let Cd (γ ) be the Legendre transform of K d (q ),

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28-4    Scaling and Fractals



γ ≤γ ∗

C(γ ), Cd (γ ) = max{qγ − K d (q )} =  ∗ ∗ ∗ q C(γ ) + q (γ − γ ), ∗

γ >γ ∗

(28.18)

∗

with C(γ ) in Eq. (28.16) and γ = K ′(q ) the slope of K(q) at q*. Then, as n → ∞, P[ε n > mnγ ] ~ m− nCd (γ ) (28.19)



∗

Note that for γ > γ , Eqs. (28.18) and (28.19) give P[ε n > mnγ ] ~ m− nq γ , ∗ which by putting ε = mnγ becomes P[ε n > ε ] ~ ε − q . This shows that when the * order of moment divergence q is finite, the dressed densities have an algebraic upper tail, as previously noted. Equation (28.19) is fundamental to the estimation of the extremes of stationary multifractal measures. Since there are mn cascade tiles at resolution level n, the product mn P[ε n > ε ] is the expected number of exceedances of level e by en in a single cascade realization. One may take the reciprocal of this quantity, ∗

Tn (ε ) =



1 (28.20) mn P[ε n > ε ]

as the return period of e expressed in units of cascade realizations. Solving Eq. (28.20) for e gives the return period values e(n, T). Using Eqs. (28.19) and (28.20), one can show that (Hubert et al., 1998; Veneziano and Furcolo, 2002) mnγ 1 T 1/q1 , ε (n,T ) ~  n 1/q* m T ,



n → ∞ , T finite n finite, T → ∞

(28.21)

where g1 is such that C(γ 1 ) = 1 and q1 = C '(γ 1 ) is the moment order associated with g1; see Fig. 28.1. This result has been used to explain the approximate scaling of the intensity-duration-frequency curves with return period T and temporal resolution mn.

28.4.2  Estimation of Universal Parameters

Stationary universal multifractal measures are widely used to model geophysical fields including atmospheric turbulence and temperature, cloud reflectivity, topographic surface, ocean surface radiance, rainfall and river flow (eg., Schertzer and Lovejoy, 1987; Lovejoy and Schertzer, 1990; Lavallee et al., 1991; Schmitt et al., 1992; Olsson et al., 1993; Schmitt et al., 1993; Tessier et al., 1993; Deidda et al., 2004, 2006; Lilley et al., 2006; Lovejoy and Schertzer, 2006; De Montera et al., 2009; Serinaldi, 2010; Sun and Barros, 2010; Veneziano and Lepore, 2012). The amplitude scaling factors Ar have (maximally negatively skewed) log-Levy distribution and in the stationary case the moment scaling function is given by

K (q) K (q) (q–1)

1 q1

(0, 0)

q*

q

g1 1 –1 Figure 28.1  Illustration of the moment-scaling function K(q) and the parameters q1, q*, and g1 in Eq. (28.21). 28.4  INFERENCE OF SCALING FOR STATIONARY MULTIFRACTAL MEASURES

Methods to estimate the scaling parameters from data depend on the type of scale invariance considered. Here we focus on stationary multifractal measures X(dt), whose scaling is characterized by the random factor Ar in Eq. (28.5) or equivalently by the moment scaling function K (q ) = log r ( E[ Arq ]). First, we consider the general problem of K(q) estimation and then remark on specialized methods for universal multifractal measures. 28.4.1  Estimation of K(q)

Since the factors Ar are not directly accessible, one typically finds K(q) from the scaling of moments of variables X (h), such as average densities and wavelet coefficients. The range of finite (and hence usable) moments depends on which specific variable is analyzed and never exceeds the range of finite moments of Ar . For example, if Ar has lognormal distribution then all its moments are finite, but the average densities and wavelet coefficients have finite moments only over a finite range of q.

28_Singh_ch28_p28.1-28.6.indd 4

The standard method to estimate K(q), say for a stationary multifractal measure in the unit interval, is to partition that interval into sub-intervals of length 2− n, where n = 1, 2, … is the resolution level, and for each n calculate the empirical moments of en, the average measure density at level n. After these initial operations, the function K(q) is estimated in two steps (Harris et al., 1997). In the first step, one calculates the empirical moments µˆnq = 〈ε nq 〉 and for each q finds Kˆ (q ) as the slope of the linear regression of log 2 ( µˆnq ) against n over a range [nmin , nmax ] of resolution levels. We refer to Kˆ (q ) as a nonparametric estimator of the moment scaling function. In the second step one fits a parametric model Kˆ par (q ) to Kˆ (q ) over a range [qmin , qmax ] of moment orders. As the resolution level nmax → ∞, Kˆ (q ) becomes linear with positive slope g1 for q > q1 where g1 and q1 are illustrated in Fig. 28.1 (Ossiander and Waymire, 2000, 2002). In practice nmax is finite and Kˆ (q ) displays gradual straightening as q increases, with an asymptotic slope γˆ1 that varies from sample to sample. Using numerical simulation, Harris et al. (1997) studied the bias and variance of Kˆ (q ) for different q and nmax . The bias and variance of the parametric estimator Kˆ par (q ) depend in addition on the range of moment orders used in the second step. Following the approach of Harris et al. (1997), Furcolo and Veneziano (2008) assessed the performance of both nonparametric and parametric estimators and suggested alternatives to reduce their mean square error.

 C1 (qα − q ),  K (q ) =  α − 1 C1q ln(q ),

for α ≠ 1 for α = 1

(28.22)

where 0 < a ≤ 2 is the index of stability of the Levy distribution and C1 is a positive constant that controls the dispersion of ln(Ae ) (Schertzer and Lovejoy, 1987). For a = 2, ln(Ae ) has normal distribution with mean value m = −C1 and variance σ 2 = 2C1 (this corresponds to the sub-class of lognormal multifractal measures), whereas for a → 0 the distribution of Ae develops a mass e−C1 at eC1 and a mass (1 − e−C1) at zero (this limit corresponds to socalled beta multifractal measures). A popular way to estimate a and C1 is the double trace moment (DTM) method of Lavallée et al. (1991), which involves certain sequences of exponentiation, normalization, and averaging operations. Veneziano and Furcolo (1999) observed that the original DTM method uses scaling relations that hold only in approximation and produces biased estimates of the parameters (especially C1). They also suggested an unbiased variant of the method. 28.5  PROCESSES WITH LIMITED SCALE INVARIANCE

Many natural phenomena display systematic deviations from scale invariance or are scale invariant in some limited sense. In some cases, the mechanism that generates a process is scale invariant, but the process itself is not. For example, this is the case when a multiplicative cascade construction is terminated at a finite resolution. Similarly, processes may have different scaling properties in different resolution ranges, meaning that the generating mechanism has distinct scale-invariance properties in different ranges, whereas the process itself is not scaling. In yet other cases, a process may become scaleinvariant after a simple transformation. One example is exponentiated fractional Brownian motion. Exponentiation destroys scale invariance, but the fact that the log of the process is scale invariant may still be of interest. In what follows, we describe four different deviations from multifractal scaling that have been noted in rainfall, turbulence, and hydraulic conductivity. Bounded Cascades The generator W of a stationary multifractal cascade has the same distribution at all levels n, whereas bounded cascade models (Menabde, 1998; Menabde and Sivapalan, 2000) allow the distribution to depend on n. The variation with n is typically assumed to be systematic. For example, in a bounded variant of the lognormal cascade one often assumes that ln(Wn ) has normal distribution with mean value −0.5c nσ 2 and variance c nσ 2 , where σ 2 > 0 and 0 < c < 1 are

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References    28-5 

given constants. The result is essentially exponentiated fractional Brownian motion exp{BH (t )} with H = −0.5logm (c ) and m the multiplicity of the cascade. While the process exp{BH (t )} does not scale, in the small-scale limit its increments have the same H-ss property as those of BH(t). In hydrology, bounded cascades have been proposed as models of rainfall (Menabde et al., 1997; Menabde and Sivapalan, 2000) and exponentiated fBm has been used to represent the hydraulic conductivity of natural aquifers (Molz and Bowman, 1993). Extended Self-Similarity Extended self-similarity (ESS) is a departure from multifractality originally introduced to better describe turbulent velocity fields (e.g., Benzi et al., 1993). If ∆v(τ ) is the longitudinal velocity increment at distance t in the mean flow direction, then a standard multifractal assumption is that for any r > 1 a random variable Ar exists such that d

∆v(τ ) = Ar ∆v(rτ )



⇒ E[| ∆v(τ )|q ] ∝ τ − K (q )

(28.23)

where K (q ) = log r E[ Arq ]. Equation (28.23) holds reasonably well at high Reynolds numbers, but breaks down at small scales where viscosity becomes significant. To extend validity to the latter regime, Benzi et al. (1993) suggested the weaker condition

E[| ∆v(τ )|q1 ] ∝ E[| ∆v(τ )|q2 ]K (q1 )/ K (q2 ) (28.24)

One can show (Veneziano and Langousis, 2010) that there are similarities between the ESS property in Eq. (28.24) and the bounded cascades mentioned earlier. Dependence of the Generator Wn on the State at Level n In studies of rainfall, Olsson (1998), Guntner et al. (2001), Veneziano et al. (2006), and others found it necessary to generalize the cascade construction by making the distribution of Wn depend not only on n but also on the bare rainfall intensity in the parent tile at level n − 1. Note that in this extension, Wn depends on the bare rainfall intensity at level n − 1 (dependence on the dressed intensity at level n − 1 is present also in standard multifractal cascades). Segmented Multifractality A special case of dependence of W on the scale level n is when the cascade generator W is distributed like Wi for ni < n ≤ ni +1, where 1 = n1 < n2 < . . . < ns = ∞. We refer to this property as segmented multifractality. Measures with segmented multifractality are actually not multifractal at any scale, but if a range [ni , ni+1 ] is wide, the measure displays approximate multifractality inside that range. For example, if it exists, the spectral density S(w) displays approximate power-law behavior in the frequency range associated with [ni , ni+1 ]. The existence of distinct scaling regimes in rainfall time series has long been recognized; see for example Fraedrich and Larnder (1993). 28.6 CONCLUSION

This review of scale invariant methods in hydrology is necessarily concise and the reference list is definitely incomplete. Indeed, the literature on fractal/ multifractal applications, even restricted to hydrology, is vast and growing at a rapid pace. However, the evolution of the underpinning concepts and methods, which are emphasized here, is more gradual. This should hopefully make the present review relevant for years to come. Our aim has been to present a unified picture of scale invariance, emphasizing the conceptually simple (yet far reaching) difference between self-similarity and multifractality and the commonality of how self-similar and multifractal processes can be generated. In our own work, we have found it beneficial to think of multifractality as a scale invariance condition, as presented here. There are areas in which further understanding and development are needed. One is the robust testing and inference of scale invariance: one too often concludes that an object is scale invariant from the observed approximate scaling of the moments. This may be deceptive as there are many nonscaling processes whose moments approximately scale. The problem is especially critical for the extremes of a process, which are controlled by high-order moments that are difficult to estimate from data. It would be desirable to go beyond the use of moments and develop inference techniques that use marginal distributions at various scales. At a minimum, after a scale-invariant model has been fitted to data, one should verify that the model accurately reproduces the observed marginal distributions at multiple scales.

28_Singh_ch28_p28.1-28.6.indd 5

In many cases nature displays departures from exact scale invariance. As scale-invariant modeling and analysis are becoming well established, the next frontier should include a more systematic understanding and treatment of deviations from scale invariance. This would lead to models that, while in general more complicated than those based on self-similarity and multifractality, have wider applicability and better represent observed phenomena. ACKNOWLEDGMENT

The authors gratefully acknowledge the useful comments by an anonymous reviewer. REFERENCES

Barnsley, M. F., Fractals Everywhere, 2nd ed., Academic Press, Boston, 1988, pp. 205–207. Benzi, R., S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, “Extended self-similarity in turbulent flows,” Physical Review E, 48, R29–R32, 1993. Beran, J., Statistics for Long-Memory Processes, Vol. 61, CRC Press, Boca Raton, FL, 1994. Cramér, H., “Sur un nouveau théoreme-limite de la théorie des probabilities,” Actualités scientifiques et industrielles, 736: 5–23, 1938. De Montera, L., L. Barthès, C. Mallet, and P. Golé, “The effect of rain-no rain intermittency on the estimation of the universal multifractals model parameters,” Journal of Hydrometeorology, 10, 493–506, 2009. Deidda, R., M. G. Badas, and E. Piga, “Space-time scaling in high-intensity Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA-COARE) storms,” Water Resources Research, 40, 2004. doi:10.1029/2003WR002574. Deidda, R., M. G. Badas, and E. Piga, “Space–time multifractality of remotely sensed rainfall fields,” Journal of Hydrology, 322: 2–13, 2006. Falconer, K. J., “Random fractals,” Mathematical Proceedings of the Cambridge, 100: 559–582, 1986. Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley & Sons, NewYork, NJ, 1996. Fournier, A., D. Fussell, and L. Carpenter, “Computer rendering of stochastic models,” Communications of the ACM, 25: 371–384, 1982. Fraedrich, K. and C. Larnder, “Scaling regimes of composite rainfall timeseries,” Tellus A, 45A: 289–298, 1993. Frisch, U., and G. Parisi. “Fully developed turbulence and intermittency,” Turbulence and predictability in geophysical fluid dynamics and climate dynamics, edited by Ghil, M., R. Benzi and G. Parisi, North-Holland Publishing Co., Amsterdamn, 1985, pp. 71–88. Furcolo, P. and D. Veneziano, “Improved moment-scaling estimation for stationary multifractal measures,” Symposium on New Statistical Tools in Hydrology, Capri, Italy, October 13–14, 2008. Graf, S., “Statistically self-similar fractals,” Probability Theory and Related Fields, 74, 357–392, 1987. Guivarc’h, Y., “Sur une extension de la notion de loi semi-stable,” Annales de l’I.H.P. Probabilités et Statistiques, 26 (2): 261–285, 1990. Guntner, A., J. Olsson, A. Calver, and B. Gannon, “Cascade-based disaggregation of continuous rainfall time series: the influence of climate,” Hydrology and Earth System Sciences, 5: 145–164, 2001. Gupta, V. K. and E. C. Waymire, “A statistical analysis of mesoscale rainfall as a random cascade,” Journal of Applied Meteorology, 32: 251–267, 1993. Gupta, V. K., O. J. Mesa, and D. R. Dawdy, “Multiscaling theory of flood peaks: regional quantile analysis,” Water Resources Research, 30: 3405–3421, 1994. Hack, J. T., Studies of longitudinal river profiles in Virginia and Maryland, US Geological Survey Professional Paper, 1957, p. 294. Halford, D., “A general mechanical model for |f|αspectral density random noise with special reference to flicker noise 1/|f|,” Proceedings of the IEEE, 56: 251–258, 1968. Harris, D., A. Seed, M. Menabde, and G. Austin, “Factors affecting multiscaling analysis of rainfall time series,” Nonlinear Processes in Geophhysics, 4: 137–156, 1997. Horton, R. E., “Erosional development of streams and their drainage basins—hydrophysical approach to quantitative morphology,” Geological Society of American Bulletin, 56: 275–370, 1945. Hubert, P., H. Bendjoudi, D. Schertzer, S. Lovejoy, A multifractal explanation for rainfall intensity-duration-frequency curves, Proceedings of the Workshop “Heavy Rains and Flash Floods,” Publ., 2049, 21–28, Natl. Res. Counc., Washington, D.C., 1998.

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28-6    Scaling and Fractals

Hurst, H. E., “Long-term storage capacity of reservoirs,” The American Society of Civil Engineers, 116: 770–799, 1951. Hutchinson, J. E., “Fractals and self similarity,” The Indiana University Mathematics Journal, 30: 713–747, 1981. Kahane, J. P. and J. Peyriere, Sur certaines martingales de Benoit Mandelbrot, Advances in Mathematics, 22: 131–145, 1976. Lamperti, J., “Semi-stable stochastic processes,” The American Mathematical Society, 104: 62–78, 1962. Lavallée,D., S. Lovejoy, D. Schertzer. Universal multifractal theory and observations of land and ocean surfaces, and of clouds, SPIE proceedings 1558, San Diego, pp. 60–75, 1991. Lavallée, D., D. Schertzer, and S. Lovejoy, On the determination of the codimension function, Non-Linear Variability in Geophysics, Springer, Netherlands, 1991, pp. 99–109. Le Cam, L., “A stochastic description of precipitation,” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, 1961, pp. 165–186. Lee, J. S., D. Veneziano, and H. H. Einstein, Hierarchical fracture trace model, Rock Mechanics: Contributions and Challenges, Balkema, Rotterdam, 1990, pp. 261–268. Lilley, M., S. Lovejoy, N. Desaulniers-Soucy, and D. Schertzer, “Multifractal large number of drops limit in rain,” Journal of Hydrology, 328: 20–37, 2006. Lovejoy, S. and B. B. Mandelbrot, “Fractal properties of rain, and a fractal model,” Tellus A, 37: 209–232, 1985. Lovejoy, S. and D. Schertzer, “Fractals, raindrops and resolution dependence of rain measurements,” Journal of Applied Meteorology, 29: 1167–1170, 1990. Lovejoy, S. and D. Schertzer, “Multifractals and rain,” In New Uncertainty Concepts in Hydrology and Water Resources, edited by Z. W. Kundzewicz, 61–103, 1995. Retrieved from http://www.cambridge.org/US/academic/subjects/ earth-and-environmental-science/hydrology-hydrogeology-and-waterresources/new-uncertainty-concepts-hydrology-and-water-resources Lovejoy, S. and D. Schertzer, “Multifractals, cloud radiances and rain,” Journal of Hydrology, 322: 59–88, 2006. Mandelbrot, B. B., “Intermittent turbulence in self-similar cascades; divergence of high moments and dimension of the carrier,” Journal of Fluid Mechanics, 62: 331–358, 1974. Mandelbrot, B. B., The Fractal Geometry of Nature (updated and augmented edition), W. H. Freeman, New York, NY, 1983, p. 468. Mandelbrot, B. B. and T. Vicsek, “Directed recursion models for fractal growth,” Journal of Physics A: Mathematical and General, 22: L377, 1989. Mauldin, R. D. and S. C. Williams, “On the hausdorff dimension of some graphs,” The American Mathematical Society, 298: 793–803, 1986. Menabde, M., “Bounded lognormal cascades as quasi-multiaffine random processes,” Nonlinear Processes in Geophysics, 5: 63–67, 1998. Menabde, M. and M. Sivapalan, “Modeling of rainfall time series and extremes using bounded random cascades and Levy-stable distributions,” Water Resources Research, 36: 3293–3300, 2000. Menabde, M., D. Harris, A. Seed, G. Austin, and D. Stow, “Multiscaling properties of rainfall and bounded random cascades,” Water Resources Research, 33: 2823–2830, 1997. Molz, F. J. and G. K. Bowman, “A fractal-based stochastic interpolation scheme in subsurface hydrology,” Water Resources Research, 29: 3769–3774, 1993. Olsson, J., “Evaluation of a scaling cascade model for temporal rainfall disaggregation,” Hydrology and Earth System Sciences, 2: 19–30, 1998. Olsson, J., J. Niemczynowicz, and R. Berndtsson, “Fractal analysis of highresolution rainfall time-series,” Journal of Geophysical Research: Atmospheres, 98: 23265–23274, 1993. Ossiander, M. and E. C. Waymire, “Statistical estimation for multiplicative cascades,” Annals of Statistics, 28 (6): 1533–1560, 2000. Ossiander, M. and E. C. Waymire, “On estimation theory for multiplicative cascades,” Sankhyā: The Indian Journal of Statistics, Series A, 64 (2): 323–343, 2002. Over, T. M. and V. K. Gupta, “A space-time theory of mesoscale rainfall using random cascades,” Journal of Geophysical Research: Atmospheres, 101: 26319–26331, 1996.

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Rodríguez-Iturbe, I. and A. Rinaldo, Fractal River Basins: Chance and Selforganization, Cambridge University Press, Cambridge, UK, 1997. Samoradnitsky, G. and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, CRC Press, Boca Raton, FL, 1994. Schertzer, D. and S. Lovejoy, “Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes,” Journal of Geophysical Research: Atmospheres, 92: 9693–9714, 1987. Schmitt, F., D. Lavallee, D. Schertzer, and S. Lovejoy, “Empirical determination of universal multifractal exponents in turbulent velocity-fields,” Physical Review Letters, 68: 305–308, 1992. Schmitt, F., D. Schertzer, S. Lovejoy, and Y. Brunet, “Estimation of universal multifractal indices for atmospheric turbulent velocity fields,” FractalsComplex Geometry Patterns and Scaling in Nature and Society, 1: 568–575, 1993. Serinaldi, F., “Multifractality, imperfect scaling and hydrological properties of rainfall time series simulated by continuous universal multifractal and discrete random cascade models,” Nonlinear Processes in Geophysics, 17: 697–714, 2010. Shreve, R. L., “Statistical law of stream numbers,” Journal of Geology, 74: 17–37, 1966. Strahler, A. N., “Quantitative analysis of watershed geomorphology,” Transactions of the American Geophysical Union, 38: 913–920, 1957. Stroock, D. W., Probability Theory: An Analytic View, Cambridge University Press, Cambridge, UK, 1999. Sun, X. and A. P. Barros, “An evaluation of the statistics of rainfall extremes in rain gauge observations, and satellite-based and reanalysis products using universal multifractals,” Journal of Hydrometeorology, 11: 388–404, 2010. Tessier, Y., S. Lovejoy, and D. Schertzer, “Universal multifractals—theory and observations for rain and clouds,” Journal of Applied Meteorology, 32: 223–250, 1993. Tricot, C., Curves and Fractal Dimension, Springer-Verlag, New York, NY, 1995. Veneziano, D., “Basic properties and characterization of stochastically selfsimilar processes in Rd,” Fractals, 7: 59–78, 1999. Veneziano, D., “Large deviations of multifractal measures,” FractalsComplex Geometry Patterns and Scaling in Nature and Society, 10: 117–129, 2002. Veneziano, D. and P. Furcolo, “A modified double trace moment method of multifractal analysis,” Fractals-Complex Geometry Patterns and Scaling in Nature and Society, 7: 181–195, 1999. Veneziano, D. and P. Furcolo, “Multifractality of rainfall and scaling of intensity-duration-frequency curves,” Water Resources Research, 38 (12): 1306, 2002. doi:10.1029/2001WR000372 Veneziano, D. and A. K. Essiam, “Flow through porous media with multifractal hydraulic conductivity,” Water Resources Research, 39: 1166, 2003. doi:10.1029/2001WR001018, 6 Veneziano, D. and P. Furcolo, “Marginal distribution of stationary multifractal measures and their Haar wavelet coefficients,” Fractals-Complex Geometry Patterns and Scaling in Nature and Society, 11: 253–270, 2003. Veneziano, D. and A. Langousis, Scaling and fractals in hydrology, Advances in Data-Based Approaches for Hydrologic Modeling and Forecasting, edited by B. Sivakumar and R. Berndtsson, World Scientific, 2010. Retrieved from http://www.worldscientific.com/doi/abs/10.1142/9789814307987_0004 Veneziano, D. and C. Lepore, “The scaling of temporal rainfall,” Water Resources Research, 48: 2012. doi:10.1029/2012WR012105 Veneziano, D., P. Furcolo, and V. Iacobellis, “Imperfect scaling of time and space-time rainfall,” Journal of Hydrology, 322: 105–119, 2006. Voss, R. F., Fractals in nature: from characterization to simulation, The Science of Fractal Images, Springer-Verlag, New York, NY, 1988, pp. 21–70. Willgoose, G., R. L. Bras, and I. Rodriguez-Iturbe, “A coupled channel network growth and hillslope evolution model: 1. Theory,” Water Resources Research, 27: 1671–1684, 1991. Yaglom, A. M., Correlation Theory of Stationary and Related Random Functions, Springer, New York, NY, 1987. Zahle, U., “Self-similar random measures .1. Notion, carrying hausdorff dimension, and hyberbolic distribution,” Probability Theory and Related Fields, 80: 79–100, 1988.

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Chapter

29

Nonlinear Dynamics and Chaos BY

BELLIE SIVAKUMAR

ABSTRACT

The inherent nonlinear nature of hydrologic systems has been known for several decades. Nonlinearity in hydrologic systems is evident in many different ways and at almost all spatial and temporal scales. “Chaos” is a special form of nonlinearity and generally refers to situations where simple nonlinear deterministic systems with sensitive dependence on initial conditions leading to complex and random-looking behavior. Since the fundamental properties of chaotic systems (i.e., nonlinear interdependence, hidden determinism and order, and sensitivity to initial conditions) are highly relevant in hydrology, numerous studies, since the 1980s, have applied the concepts of chaos theory to study hydrologic systems. The application areas and problems include rainfall, river flow, rainfall-runoff, lake volume, sediment transport, groundwater flow and contaminant transport, modeling, prediction, noise reduction, scaling, disaggregation, missing data estimation, reconstruction of system equations, parameter estimation, catchment classification, and others. This chapter provides an overview of chaos theory and its applications in hydrology. It includes: (1) a brief account of the history of the development of chaos theory; (2) a description of some of the most important methods for chaos identification and prediction; (3) a discussion on the potential issues in the applications of chaos theory methods to hydrologic (and other real) systems; and (4) a brief review of the applications of chaos theory in hydrology. 29.1  INTRODUCTION

The inherent nonlinear nature of hydrologic systems has been known for several decades now (e.g., Minshall, 1960; Izzard, 1966; Amorocho, 1967; Dooge, 1967; Singh, 1979). Nonlinearity in hydrologic systems is evident in many different ways and at almost all spatial and temporal scales. The hydrologic cycle itself is an example of a system exhibiting nonlinear behavior, with almost all of the individual components exhibiting nonlinear behavior as well. The climatic inputs and landscape characteristics are changing in a highly nonlinear fashion, and so are the outputs, often in unknown ways. The rainfall-runoff process is nonlinear, almost regardless of the basin area, land use, rainfall intensity, and other influencing factors; see Singh (1988) for a comprehensive review of earlier black-box and conceptual models of nonlinear rainfall-runoff process. Although the nonlinear nature of hydrologic systems had been recognized at least as early as the 1960s, much of the early research during 1960s–1980s, including development and application of time series models, essentially resorted to linear (deterministic and stochastic) approaches (e.g., Thomas and Fiering, 1962; Crawford and Linsley, 1966; Freeze and Harlan, 1969; Yevjevich, 1972; Valencia and Schaake, 1973; Fleming, 1975; Salas and Smith, 1981; Bras and Rodriguez-Iturbe, 1985), which continue to be prevalent in hydrology. At least two factors contributed to this situation: (1) the assumption that linear approaches are generally sufficient in view of simplicity and relative to other uncertainties; and (2) lack of computational power to develop the (perhaps more complex) nonlinear mathematical models.

However, significant developments in computational power and major advances in measurement technology and mathematical concepts over the past three decades have facilitated formulation of nonlinear approaches as viable alternatives for complex hydrologic systems. This subsequently led to applications of nonlinear approaches to study the nonlinear and related properties of hydrologic systems. These applications started to emerge during 1980s–1990s, and have skyrocketed since then. The nonlinear approaches that are popular in hydrology include, among others, nonlinear stochastic methods, data-based mechanistic models, artificial neural networks, support vector machines, evolutionary computing, fuzzy logic, wavelets, entropybased techniques, and chaos theory (e.g., Rodriguez-Iturbe et al., 1989; Young and Beven, 1994; Kumar and Foufoula-Georgiou, 1997; Singh, 1997; ASCE Task Committee, 2000; Sivakumar, 2000; Dibike et al., 2001; Kavvas, 2003; Gupta et al., 2007; Şen, 2009); see Sivakumar and Berndtsson (2010) and Jayawardena (2014) for further details on the applications of many of these nonlinear approaches in hydrology. This chapter focuses on nonlinear deterministic dynamics, popularly known as chaos theory. In the nonlinear science literature, the term “chaos” is normally used to refer to situations where complex and random-looking behavior arises from simple nonlinear deterministic systems with sensitive dependence on initial conditions (Lorenz, 1963); the converse also applies. The three fundamental properties inherent in this definition: (1) nonlinear interdependence; (2) hidden determinism and order; and (3) sensitivity to initial conditions are highly relevant in hydrologic systems and processes. For example: (1) nonlinear interactions and interdependence are dominant among the components and mechanisms in the hydrologic cycle; (2) determinism and order are prevalent in daily temperature, annual river flow, and many other hydrologic processes at one or more scales; and (3) runoff is highly sensitive to the time at which rainfall occurs and solute transport in surface and sub-surface waters is highly sensitive to the time at which contaminants are released. The first property represents the “general” nature of hydrologic phenomena, whereas the second and third represent their “deterministic” and “stochastic” natures, respectively. Further, despite their complexity and random-looking behavior, hydrologic phenomena may also be governed by only a very few degrees of freedom (e.g., runoff in a well-developed urban catchment depends essentially on rainfall), another fundamental idea of chaos theory (Sivakumar, 2004a). The finding that “complex and random-looking” behavior is not necessarily the outcomes of complex systems but can also be from simple nonlinear deterministic systems with sensitivity to initial conditions has far reaching implications in hydrologic modeling, since most outputs from such systems (e.g., time series of rainfall, river flow, water quality) are typically “complex and randomlooking.” One crucial implication of this finding is the need, first of all, to identify the dynamic nature of the given system towards selecting an appropriate modeling approach, as opposed to simply resorting to a particular approach based on certain preconceived notion. In view of this, and particularly facilitated by the development of new nonlinear time series methods, applications of nonlinear dynamics and chaos concepts in hydrology started to 29-1

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29-2      Nonlinear Dynamics and Chaos

emerge in the 1980s (e.g., Hense, 1987; Rodriguez-Iturbe et al., 1989), and have seen an enormous growth since then (e.g., Wilcox et al., 1991; Berndtsson et al., 1994; Jayawardena and Lai, 1994; Puente and Obregon, 1996; Porporato and Ridolfi, 1997; Sivakumar et al., 1999a, 2001a, 2001b, 2014; Lambrakis et al., 2000; Sivakumar, 2002, 2003; Regonda et al., 2004; Dodov and FoufoulaGeorgiou, 2005; Hossain and Sivakumar, 2006; Dhanya and Nagesh Kumar, 2010, 2011; Kyoung et al., 2011; Sivakumar and Singh, 2012). Comprehensive accounts of such applications can be found in Sivakumar (2000, 2004a, 2009). The application areas and problems include rainfall, river flow, rainfall-runoff, lake volume, sediment transport, groundwater contaminant transport, modeling, prediction, noise reduction, scaling, disaggregation, missing data estimation, reconstruction of system equations, parameter estimation, and catchment classification, among others. The outcomes of such studies are encouraging, as they generally reveal the adequacy of simpler models to represent hydrologic systems and the possibility of more accurate short-term predictions. Despite this, there continue to be criticisms and skepticisms on such studies and outcomes, essentially on the basis of potential limitations in the applications of chaos methods for hydrologic time series, such as issues associated with data size, data noise, zeros, and others (e.g., Schertzer et al., 2002; Koutsoyiannis, 2006). While such criticisms and skepticisms cannot be dismissed altogether and indeed need careful consideration, there are also concerns about potential “falsealarms” arising from such criticisms; see Sivakumar et al. (2002a) and Sivakumar (2005) for details. Amid all of this, what is clear is that chaos studies and their outcomes certainly provide different perspectives and new avenues to study hydrologic systems. In fact, arguments as to the potential of chaos theory to serve as a bridge between our traditional and dominant deterministic and stochastic theories have also been put forward (Sivakumar, 2004a, 2011c). This chapter aims to provide an overview of chaos theory and its applications in hydrology. The rest of the chapter is organized as follows. Section 29.2 presents briefly the history of the development of chaos theory. Section 29.3 describes some of the most important methods for chaos identification and prediction, especially those that have found applications in hydrology. Section 29.4 discusses some of the potential issues in the applications of chaos methods to real systems. Section 29.5 reviews the applications of chaos theory in hydrology. Section 29.6 offers some final thoughts toward the future. 29.2  CHAOS THEORY: A BRIEF HISTORY

The roots of chaos theory date back to the studies, in about 1900, of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction: the “three-body problem.” Poincaré found that there can be orbits that are non-periodic, and yet not forever increasing nor approaching a fixed point. However, chaos theory remained in the background for almost half a century. The invention of high-speed computers in the 1950s changed this situation, as computers allowed better experimentation with equations, especially the process of repeated iteration of mathematical formulas to study nonlinear dynamic systems. Such experiments led to Edward Lorenz’s discovery, in 1963, of chaotic motion on a “strange attractor” (Lorenz, 1963). Lorenz studied a simplified model of convection rolls in the atmosphere to obtain insight into the unpredictability of the weather. He found that the solutions to his equations never settled down to equilibrium or to a periodic state, but instead continued to oscillate in an irregular, aperiodic fashion. Furthermore, when the simulations were started from two slightly different initial conditions, the resulting behaviors became totally different. The implication was that the system was inherently unpredictable, as tiny errors in measuring the current state of the atmosphere could amplify rapidly. However, Lorenz also showed that there was structure (in the chaos), as the solutions to his equations, when plotted in three dimensions, fell onto a butterfly-shaped set of points. The 1970s witnessed the main developments in chaos theory. Ruelle and Takens (1971) proposed a new theory for the onset of turbulence in fluids, based on abstract consideration about “strange attractors.” May (1976) found examples of chaos in iterated mappings arising in population biology, and stressed on the pedagogical importance of studying simple nonlinear systems, to counterbalance the often misleading linear intuition fostered by traditional education. Study of other simple nonlinear models, such as the Henon map (Henon, 1976) and the Rössler system (Rössler, 1976), also revealed the hidden beauty of chaos. Feigenbaum (1978) discovered that there are certain universal laws governing the transition from regular to chaotic behavior, and established a link between chaos and phase transitions. The study of chaos then moved to the laboratory, with ingenious experiments set up and chaotic behavior studied in fluids, mechanical oscillators, and semiconductors (e.g., Swinney and Gollub, 1978; Linsay, 1981).

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The positive outcomes from these laboratory experiments encouraged search for chaos outside the “controlled” space—in nature. To this end, advances in computational power and measurement technology facilitated development, in the 1980s, of a new set of mathematical techniques for chaos identification and prediction based on time series, with concepts of data reconstruction, dimensionality, entropy, and predictability (e.g., Packard et al., 1980; Takens, 1981; Grassberger and Procaccia, 1983a, b; Wolf et al., 1985; Farmer and Sidorowich, 1987). These techniques have, since then, been employed for identification and prediction of chaos in many real systems, including atmospheric, biologic, ecologic, economic, engineering, environmental, financial, political, and social systems. 29.3  CHAOS CONCEPTS AND IDENTIFICATION METHODS

In the context of nonlinear dynamics and chaos, a wide variety of methods have been developed for reconstruction of a time series, estimation of the invariants for chaos identification, and prediction. These methods include phase space reconstruction, correlation dimension method, Kolmogorov entropy method, Lyapunov exponent method, false nearest neighbor algorithm, nonlinear local approximation prediction method, close returns plot, surrogate data method (for detection of nonlinearity), and 0-1 test, among others. Some of the more popular methods are briefly described here. To put the methods in a proper context, a simple demonstration of their utility for identification of chaos (or distinguishing between stochastic and chaotic time series) is also presented. Due to space constraints, however, results for only two of the methods are presented. 29.3.1 Phase-Space Reconstruction

Phase space is a useful concept for representing the evolution of a dynamic system (e.g., Packard et al., 1980). It is essentially a graph, whose coordinates represent the variables necessary to completely describe the state of the system at any moment. The trajectories of the phase space diagram describe the evolution of the system from some initial state and, hence, represent the history of the system. The “region of attraction” of these trajectories in the phase space provides at least important qualitative information on the nature of the system dynamic properties. Phase space can be reconstructed based on a single-variable data series or multivariable data series (in the latter case, it is called state space). The idea behind this concept is that a nonlinear system is characterized by selfinteractions, and that data of a single (or multi) variable should carry sufficient information about the entire, and often large-dimensional, system. Many methods are available for phase space reconstruction from an available time series, including the Whitney’s embedding theorem, Takens’ delay embedding theorem, fractal delay embedding prevalence theorem, and filtered delay embedding (such as singular value decomposition or singular spectrum analysis or principal component analysis); see Sauer et al. (1991) for a comprehensive account. However, the Takens’ delay embedding theorem is the most widely used one in chaos identification studies, and is described here. Given a single variable series, Xi, where i = 1, 2, ..., N, a multi-dimensional phase space can be reconstructed according to the Takens’ delay embedding theorem as follows (Takens, 1981):

Yj = (Xj, Xj+t , Xj+2t ,..., Xj+(m–1) t ) (29.1)

where j = 1, 2,..., N – 1(m – 1)t ; m is the dimension of the vector Yj, called embedding dimension; and t is an appropriate delay time. A correct phase space reconstruction in a dimension m generally allows interpretation of the system dynamics (if the variable chosen to represent the system is appropriate) in the form of an m-dimensional map fT, given by

Yj+T = fT (Yj) (29.2)

where Yj and Yj+T are vectors of dimension m, describing the state of the system at times j (current state) and j + T (future state), respectively. With Eq. (29.2), the task is basically to find an appropriate expression for fT (e.g., FT) to predict the future. The utility of phase space diagram for system identification is demonstrated here through its reconstruction for two artificially generated series (Fig. 29.1a and b) that look very much alike (both “complex” and “random”) but nevertheless are the outcomes of systems (equations) possessing significantly different dynamic characteristics. The first set (Fig. 29.1a) is the outcome of a pseudo random number generation function Xi = rand( )

(29.3)

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CHAOS CONCEPTS AND IDENTIFICATION METHODS     29-3 

which yields independent and identically distributed numbers (between 0 and 1). The second set (Fig. 29.1b) is the outcome of a fully deterministic two-dimensional map (Henon, 1976): Xi+1 = a – Xi2 + bYi;

Yi+1 = X (29.4)

which yields irregular solutions for many choices of a and b, but for a = 1.4 and b = 0.3, a typical sequence of Xi is chaotic. Figure 29.1c and d presents the phase space plots for these two series. These diagrams correspond to reconstruction in two dimensions (m = 2) with delay time t = 1, that is the projection of the attractor on the plane {Xi, Xi+1}. For the first set, the points (of trajectories) are scattered all over the phase space (i.e., absence of an attractor), a clear indication of a “complex” and “random” nature of the underlying dynamics and potentially of a high-dimensional (and possibly random) system. On the other hand, the projection for the second set yields a very clear attractor (in a well-defined region), indicating a “simple” and “deterministic” (yet non-repeating) nature of the underlying dynamics and potentially of a low-dimensional (and possibly chaotic) system. 29.3.2 Correlation Dimension Method

The dimension of a time series is, in a way, a representation of the number of dominant variables present in the evolution of the corresponding dynamic system. Correlation dimension is a measure of the extent to which a data point affects the position of the other points of the attractor in phase space. The correlation dimension method uses the correlation integral (or function) for determining the dimension of the attractor and, hence, for distinguishing between low-dimensional and high-dimensional systems.

Many algorithms have been formulated for the estimation of the correlation dimension of a time series, but the Grassberger-Procaccia algorithm (Grassberger and Procaccia, 1983a) has been the most popular. The algorithm uses the concept of phase space reconstruction for representing the dynamics of the system from an available single-variable time series (Eq. 29.1). For an m-dimensional phase space, the correlation function, C(r), is given by C(r ) = lim



r →0 N →∞

where a is a constant and n is the correlation exponent or the slope of the Log C(r) versus Log r plot. The slope is generally estimated by a least square fit of a straight line over a certain range of r (scaling regime) or through estimation of local slopes between r values. The distinction between low-dimensional (and perhaps deterministic) and high-dimensional (and perhaps stochastic) systems can be made using the n versus m plot. If n saturates after a certain m and the saturation value is low, then the system is generally considered to exhibit low-dimensional deterministic 1.35

Value, X

Value, X

0.81

0.60 0.40 0.20 0

200

400

600

800

–0.27

–1.35

1000

Time

200

400 600 Time

800

1000

1.35 0.81

0.60

0.27

Xi+1

0.80

0.40 0.20

–0.27 –0.81

0.00 0.00

0.20

0.40

0.60

0.80

–1.35 –1.35

1.00

Xi

(c)

–0.81

–0.27

0.27

0.81

1.35

Xi

(d)

10.0

1.4 Correlation exponent

Correlation exponent

0

(b)

1.00

Xi+1

0.27

–0.81

(a)

(e)

i, j (1≤i < j ≤ N )

C(r ) ≈ α r ν (29.6)



0.80

8.0 6.0 4.0 2.0 0.0

∑ H (r − | Yi − Yj |) (29.5)

where H is the Heaviside step function, with H(u) = 1 for u > 0, and H(u) = 0 for u ≤ 0, where u = r – ||Yi – Yj||, r is the vector norm (radius of sphere) centered on Yi or Yj. If the time series is characterized by an attractor, then C(r) and r are related according to

1.00

0.00

2

N →∞ N ( N − 1)

0

2

4 6 8 Embedding dimension

10

1.2 1.0 0.8 0.6

0 (f)

2

8 4 6 Embedding dimension

10

Figure 29.1  Random versus chaotic data: (a) and (b) time series; (c) and (d) phase space; and (e) and ( f  ) correlation dimension.

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29-4      Nonlinear Dynamics and Chaos

dynamics. The saturation value of n is defined as the correlation dimension (d) of the attractor, and the nearest integer above this value is generally an indication of the number of variables dominantly governing the dynamics. On the other hand, if n increases without bound with increase in m, the system under investigation is generally considered to exhibit high-dimensional stochastic behavior. To demonstrate the utility of the dimension concept, Fig. 29.1e presents the correlation dimension results for the first set (Fig. 29.1a), whereas those for the second set (Fig. 29.1b) are shown in Fig. 29.1f. In each case, embedding dimensions from 1 to 10 are used for phase space reconstruction. It is clear that the first set is the outcome of an infinite-dimensional system (i.e., absence of saturation in dimension), whereas the second set is the outcome of a lowdimensional system (with a correlation dimension value of 1.22). 29.3.3 Kolmogorov Entropy Method

The Kolmogorov entropy, or K entropy, is the mean rate of information created by the system. It is important in characterizing the average predictability of a system of which it represents the sum of the positive Lyapunov exponents (see Sec. 29.3.4 for details of Lyapunov exponent). The Kolmogorov entropy quantifies the average amount of new information on the system dynamics brought by the measurement of a new value of the time series. In this sense, it measures the rate of information produced by the system. The value of K = 0 for periodic or quasi-periodic (i.e., completely predictable) time series; K = ∞ for white noise (i.e., unpredictable by definition); and 0 < K < ∞ for chaotic system. It is important to note that it is very difficult to directly estimate K entropy for a time series representing a system with greater than two degrees of freedom. Therefore, in the study of chaos, the only alternative is to estimate an approximation of the K entropy. This approximation is oftentimes the lower bound of the K entropy, which is the K2 entropy. Grassberger and Procaccia (1983b) proposed the first algorithm for estimation of Kolmogorov entropy in the context of chaos analysis. Their algorithm uses the correlation sum or correlation integral, and the K2 entropy is given as follows: and

1  K 2 (m) = lim   {log[Cm (r )] − log[Cm+1 (r )]} (29.7) r →0  τ 



K 2 = lim[K 2 (m)]  (29.8) m→∞

where t is the delay time, Cm(r) is the value of C(r) when the embedding dimension is m, Cm+1(r) is the value of C(r) when embedding dimension of phase space is m + 1. Again, the choice of t and m is key to calculation of entropy. In practice, we need to consider dimension of embedding phase space as well as t which has better simulating effect. The K2 entropy and Kolmogorov entropy are thought to have the same qualitative behavior. For chaotic systems with noise, a variation to the above, known as the modified correlation entropy, has also been proposed (Jayawardena et al., 2010). 29.3.4 Lyapunov Exponent Method

Lyapunov exponents are the average exponential rates of divergence (expansion) or convergence (contraction) of nearby orbits in the phase space. Since nearby orbits correspond to nearly identical states, exponential orbital divergence means that systems whose initial differences that may not be possible to resolve will soon behave quite differently; in other words, predictive ability is rapidly lost. Any system containing at least one positive Lyapunov exponent is considered to exhibit chaotic behavior, with the magnitude of the exponent reflecting the timescale on which system dynamics become unpredictable. A negative Lyapunov exponent indicates that the orbit is stable and periodic. A zero Lyapunov exponent is an indication of a marginally or neutrally stable orbit, which often occurs near a point of bifurcation. An infinite Lyapunov exponent value is an indication of a stochastic system. Many algorithms have been formulated for calculation of the Lyapunov exponents from a time series (e.g., Wolf et al., 1985; Eckmann et al., 1986; Rosenstein et al., 1993; Kantz, 1994). Among these, the algorithm by Wolf et al. (1985) has been most widely used. The algorithm tracks a pair of arbitrarily close points over a trajectory to estimate the accumulated error per timestep. The points are separated in time by at least one orbit on the attractor. The trajectory is defined by the fiducial and test trajectories. They are tracked for a fixed time period or until the distance between the two components of the trajectory exceeds some specific value. In sequence, another test point near the fiducial trajectory is selected and the estimation proceeds. The end product is that the stretching and squeezing are averaged. The computation of a Lyapunov exponent proposed by Wolf et al. (1985) can be explained as follows. Let us assume an initial data point, Y1, and its neighbor, Z1, that are L1 units apart. Over ∆t, a series of timesteps from 1 to k,

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the two points Y and Z evolve until their distance, L¢1, is greater than some arbitrarily small e. The Y value at k becomes Y2 and a new nearest neighbor, Z2, is selected. This procedure continues until the fiducial trajectory reaches the end of the time series. The replacement of the old point by its substitute point and the replacement of the error direction by a new directional vector constitutes a renormalization of errors along the trajectory. The largest Lyapunov exponent, λ1, is then given by

λ1 =

M L′j 1  ∑ log 2 (29.9) N∆t j =1 Lj

where M is the number of replacement steps (where some arbitrarily small e is exceeded) and N is the total number of timesteps that the fiducial trajectory progressed. 29.3.5 False Nearest Neighbor Algorithm

The false nearest neighbor (FNN) algorithm (Kennel et al., 1992) provides information on the minimum embedding dimension of the phase space required for representing the system dynamics. It examines, in dimension m, the nearest neighbor YjNN of every vector Yj, as it behaves in dimension m + 1. If the vector YjNN is a true neighbor of Yj, then it comes to the neighborhood of Yj through dynamical origins. On the other hand, if the vector YjNN moves far away from vector Yj as the dimension is increased, then it is declared a “false nearest neighbor” as it arrived in the neighborhood of Yj in dimension m by projection from a distant part of the attractor. When the percentage of these false nearest neighbors drops to zero, the geometric structure of the attractor has been unfolded and the orbits of the system are now distinct and do not cross (or overlap). A key step in the false nearest neighbor algorithm is to determine how to decide upon increasing the embedding dimension that a nearest neighbor is false. Two criteria are generally used (e.g., Sangoyomi et al., 1996): • If Rm+1( j) ≥ 2RA, the jth vector has a false nearest neighbor, where Rm+1(j) is the distance to the nearest neighbor of the jth vector (i.e., YjNN) in an embedding of dimension (m + 1), and RA is the standard deviation of the time series Xi, i = 1, 2, …, N. • If [Rm+1( j) – Rm( j)] > eRm( j), the jth vector has a false nearest neighbor, where e is a threshold factor (generally between 10 and 50), and the distance Rm+1( j) is computed to the same neighbor that was identified with embedding m, but with the (m + 1)th coordinate (i.e., Xj-mt appended to the jth vector and to its nearest neighbor with embedding m). The first criterion is needed because with a finite dataset, such as hydrologic series, under repeated embedding, one may stretch out the points such that they are far apart and yet cannot move any farther apart upon increasing the dimension. The second criterion checks whether the nearest neighbors have moved far apart on increasing the dimension. The appropriate threshold e is generally selected through experimentation. 29.3.6 Nonlinear Local Approximation Prediction Method

An important purpose of identification of chaos in a time series is to attempt more reliable short-term predictions than those possible with other methods (e.g., stochastic methods), since the deterministic nature of chaotic systems allows reliable short-term predictions, although their sensitive dependence on initial conditions precludes long-term ones. An early method for prediction of chaotic time series was proposed by Farmer and Sidorowich (1987). Since then, there have been many advances (e.g., Casdagli, 1989, 1992; Sugihara and May, 1990), not only for prediction but also for chaos identification using the prediction results. All these methods are essentially based on “local approximation.” The first step in chaos prediction method is phase space reconstruction of a time series according to Eq. (29.1) to reliably represent the underlying dynamics in the form of an m-dimensional map fT (Eq. 29.2). The problem then is to find an appropriate expression for fT (e.g., FT ) to predict the future. There are several possible approaches for determining FT . One promising approach is the “local approximation method” (Farmer and Sidorowich, 1987), which uses only nearby states to make prediction. The basic idea in the local approximation method is to break up the domain FT into local neighborhoods and fit parameters in each neighborhood separately. In this way, the underlying system dynamics are represented step by step locally in the phase space. To predict Xj+T based on Yj (an m-dimensional vector) and past history, k nearest neighbors of Yj are found on the basis of the minimum values of ||Yj – Yj′||, with j¢ < j. If only one such neighbor is considered, the prediction of Xj+T would be X ′j +T . For k number of neighbors, the prediction of Xj+T could be taken as an average of the k values of X ′j +T . The value of k is determined by

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ISSUES IN CHAOS IDENTIFICATION AND PREDICTION      29-5 

trial and error. It is also relevant to note that k = 1 for m = 1 is a very limited case of prediction, and is similar to the “method of analogues,” originally proposed by Lorenz (1969). In general, however, both k and m are varied to find out the optimum predictions. The prediction accuracy can be evaluated using a variety of measures, such as correlation coefficient, root mean square error, and coefficient of efficiency. In addition, direct time series plots and scatter diagrams can also be used to choose the best prediction results among a large combination of results achieved with different embedding dimensions (m) and number of neighbors (k), and also different delay times. While the local approximation method primarily serves as a prediction tool, the prediction results can also be used to distinguish between deterministic and stochastic systems, i.e., an inverse way to identify chaos. In general, a high prediction accuracy from the local approximation method may be considered as an indication of a deterministic system, whereas a low prediction accuracy is expected if the dynamics are stochastic. However, this is not a very effective way, since the predictions are strongly influenced by the two main parameters involved (m and k) as well as the lead time (T) considered. It should be noted that while the parameter m may be fixed a priori based on reliable available information (e.g., results from correlation dimension and false nearest neighbor methods), there is no reliable way to fix the parameter k in advance. Furthermore, even if m is reliably known a priori, it would still be helpful to make the predictions for different m values, so that the optimum m can be compared and cross-verified against the dimension obtained from other methods. With results from the nonlinear local approximation prediction method at hand, the following guidelines may be adopted to distinguish between stochastic and chaotic systems: • Embedding dimension (m): If the time series exhibits deterministic chaos, then the prediction accuracy would increase to its best with the increase in the embedding dimension up to a certain point (low value of m), called the optimal embedding dimension (mopt), and would remain close to its best for embedding dimensions higher than mopt. For stochastic time series, there would be no increase in the prediction accuracy with an increase in the embedding dimension and the accuracy would remain the same for any value of m (e.g., Casdagli, 1989). • Neighbors (k): Smaller number of neighbors would give the best predictions if the system dynamics are deterministic, whereas for stochastic systems the best predictions are achieved when the number of neighbors is large. This approach is also called the deterministic versus stochastic algorithm (Casdagli, 1992). The idea behind this approach is that since small k represents local models, it would be more appropriate for deterministic systems, while global models (large k) would be more appropriate for stochastic systems. However, if the best prediction is obtained using neither deterministic nor stochastic models but intermediate models (i.e., intermediate number of neighbors), then such a condition can be taken as an indication of chaotic behavior with some amount of noise in the data or chaos of moderate dimension (e.g., Casdagli, 1992). • Lead time (T): For a given embedding dimension and for a given number of neighbors, predictions in deterministic systems deteriorate considerably faster than in stochastic systems when the lead time is increased. This is due to the sensitivity of deterministic chaotic systems to initial conditions (Sugihara and May, 1990). 29.4  ISSUES IN CHAOS IDENTIFICATION AND PREDICTION

The reliability of the aforementioned for chaos identification and prediction in real-time series, such as hydrologic time series, has been under considerable debate. This is mainly due to some of the basic assumptions in the development of the above methods and, therefore, their potential limitations when applied to real data. For instance, almost all of the above methods have been developed with an inherent assumption that the time series is infinite and noise-free, but real time series are always finite and often contaminated by noise. Much of the criticism on chaos analysis, however, has been directed at the correlation dimension method. Part of the criticisms has been due to a number of studies carried out on data requirements for dimension estimation, but the fact that the correlation dimension method has been the most widely used method for chaos detection has also contributed to such criticisms. In what follows, some of the important issues associated with chaos methods are discussed. 29.4.1 Delay Time

Almost all of the chaos identification and prediction methods involve the reconstruction of the time series in phase space. An appropriately constructed

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phase space is crucial for reliable invariant estimates and predictions. The delay embedding techniques, such as those of Takens’ embedding theorem (Takens, 1981), widely used for phase space reconstruction involve the use of a delay time, t. An appropriate delay time, t, for phase space reconstruction is necessary, because only an optimum t gives the best separation of neighboring trajectories within the minimum embedding phase space, whereas an inappropriate t may lead to unreliable outcomes (e.g., underestimation or overestimation of invariants). For example, if t is too small, then there is little new information contained in each subsequent datum and the reconstructed attractor is compressed along the identity line. This situation is termed as redundance (Casdagli et al., 1991), and the result of which is an inaccurate estimation of the invariants, such as an underestimation of the correlation dimension (e.g., Havstad and Ehlers, 1989). On the other hand, if t is too large, and the dynamics happen to be chaotic, then all relevant information for the phase space reconstruction is lost, since neighboring trajectories diverge, and averaging in time and/or space is no longer useful (Sangoyomi et al., 1996). This situation is termed as irrelevance (Casdagli et al., 1991), and this may result in an inaccurate estimation of the invariants, such as an overestimation of the correlation dimension. In view of the importance of t, several methods and guidelines have been proposed in the literature for the selection of an appropriate t. These approaches are based on series correlation (e.g., autocorrelation, mutual information, high-order correlations), phase space extension (e.g., fill factor, wavering product, average displacement), and multiple autocorrelation and non-bias multiple autocorrelation (e.g., Holzfuss and Mayer-Kress, 1986; Fraser and Swinney, 1986; Albano et al., 1991; Rosenstein et al., 1994; Judd and Mees, 1998). Two of the more widely used methods are briefly discussed here: the autocorrelation function method and the mutual information method. The autocorrelation function method is the most commonly used method for delay time selection, for at least two reasons: (1) its computation is simple; and (2) it is one of the most fundamental and standard statistical tools in any time series analysis. Within the autocorrelation function method, there are several guidelines for the selection of t. For instance, Holzfuss and Mayer-Kress (1986) recommended using a value of lag time (or index lag) at which the autocorrelation function first crosses the zero line. Schuster (1988) suggested the use of the lag time at which the autocorrelation function attains 0.5, while Tsonis and Elsner (1988) suggested the selection of the lag time at which the autocorrelation function crosses 0.1. Despite its widespread use, the appropriateness of the autocorrelation function method for the selection of t has been seriously questioned. For example, Fraser and Swinney (1986) pointed out that the autocorrelation function method measures only the linear dependence between successive points and, thus, may not be appropriate for nonlinear dynamics. They suggested the use of the local minimum of the mutual information, which measures the general dependence, not just the linear dependence, between successive points. They reasoned that if t is chosen to coincide with the first minimum of the mutual information, then the recovered state vector would consist of components that possess minimal mutual information between them, i.e., the successive values in the time series are statistically independent but (also) without any redundancy. For a discrete time series, with Xi and Xi-t as successive values, for instance, the mutual information function, I(t), is computed according to

 P( X ,  X )  i i −τ I (τ ) = ∑ P( Xi , Xi−τ )log 2   (29.10)  P ( Xi ) P( Xi−τ )  i ,i −τ

where P(Xi) and P(Xi–t) are the individual probabilities of Xi and Xi–t, respectively, and P(Xi, Xi–t) is the joint probability density. The mutual information method is a more comprehensive method of determining proper delay time values (e.g., Tsonis, 1992). However, the method has the disadvantage of requiring a large number of data, unless the dimension is small, and is computationally cumbersome. For some attractors, it may not really matter which method is used for the selection of t. For example, when applied to the Rössler system (Rössler, 1976), the autocorrelation function, the mutual information, and the correlation integral methods all provide a value of t approximately equal to one-fourth of the mean orbital period (Tsonis, 1992). However, for some other attractors, the estimation of t might depend strongly on the approach employed. An obvious way to have more confidence in the selection of t may be to use different methods and check the consistency of the resulting t values. A reliable alternative to address the issue of delay time selection is to try to fix the delay time window tw = t(m – 1), rather than just the delay time itself, since the delay time window is the one that is of actual interest at the end to represent the dynamics. An early attempt in this regard was made by

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29-6      Nonlinear Dynamics and Chaos

Martinerie et al. (1992). Comparing the delay time window and delay times estimated using the autocorrelation function and mutual information methods, Martinerie et al. (1992) did not observe a consistent agreement between them. This is because, tw is basically the optimal time for representing the independence of the data, whereas autocorrelation function and mutual information methods determine only the first local optimal times in their estimation of t. Kugiumtzis (1996) put emphasis on the relation between tw and dynamics of the underlying chaotic system and suggested to set tw > tp, the mean orbital period, with tp approximated from the oscillations of the time series. Kim et al. (1998), through analysis of time series generated from the Lorenz system, the Rabinovich-Fabrikant system, and the three-torus, showed that with an increase in the embedding dimension, the correlation dimension converges more rapidly for the case of tw held fixed than for the case of t held fixed. Kim et al. (1999) subsequently developed a new technique to estimate both t and tw. This technique, called the C–C method, uses the Brock–Dechert–Scheinkman (BDS) statistic (Brock et al., 1996), which has its base on the correlation integral for testing nonlinearity in a time series. In the absence of clear-cut guidelines on t selection, a practical approach is to experiment with different t values to ascertain its effect, for example, on the estimation of invariants. Such an exercise is particularly fruitful for synthetic time series, since the dynamic properties (e.g., invariants) of such series are known a priori. One has to be careful, however, in adopting this approach for real time series, since their dynamic properties are not known, and determination of which is indeed the task at hand. Nevertheless, the exercise can offer some important clues. 29.4.2 Data Size

The issue of data size (or length) has perhaps attracted the fiercest criticism on studies employing chaos theory-based methods to real time series. A common criticism is that the correlation dimension, arguably the most widely used indicator of chaos, is underestimated when the data size is small. The basis of this criticism lies with a basic assumption of the method that the time series is infinite. The recognition that the effects of data size may be different for different chaos identification and prediction methods has led to studies to estimate the minimum data requirement for different methods. However, most of such studies have been directed at the correlation dimension method (e.g., Smith, 1988; Havstad and Ehlers, 1989; Nerenberg and Essex, 1990; Ramsey and Yuan, 1990; Lorenz, 1991; Tsonis et al., 1993; Sivakumar et al., 2002c; Sivakumar, 2005), many of them attempting to link the minimum data size (Nmin) to the embedding dimension (m) or correlation dimension (d). Some examples are presented here. The study by Smith (1988) was the first to address the minimum data size for correlation dimension estimation in terms of embedding dimension. Smith (1988) concluded that the minimum data size was equal to 42m, where m is the smallest integer above the dimension of the attractor. Nerenberg and Essex (1990) demonstrated that the procedure by Smith (1988) was flawed and that the data requirements might not be so extreme. They suggested that the minimum number of points required for the dimension estimate is Nmin ~ 102+0.4m. Havstad and Ehlers (1989) used a variant of the nearest neighbor dimension algorithm to compute the dimension of the time series generated from the Mackey–Glass equation (Mackey and Glass, 1977), whose actual dimension is 7.5. Using a dataset of as small as 200 points, Havstad and Ehlers (1989) reported an underestimation of the dimension by about 11%. Ramsey and Yuan (1990) concluded that for small sample sizes, dimension could be estimated with upward bias for chaotic systems and with downward bias for random noise as the embedding dimension is increased. They proved that, due to these bias effects, a correlation dimension estimate of 0.214 could imply an actual correlation dimension value of as high as 1.68. Lorenz (1991) argued that, while underestimation of correlation dimension for small sample sizes may occur, different [climatic] variables yield different estimates and that a suitably selected variable could yield a fairly accurate estimate of dimension even if the number of points were not large. A number of studies have followed and/or supported one or more of the above-mentioned guidelines as to the potential underestimation of the correlation dimension when the data size is small (e.g., Tsonis et al., 1993; Wang and Gan, 1998; Schertzer et al., 2002). However, several counter-arguments to these guidelines have also been made (e.g., Sivakumar et al., 2002a, c; Sivakumar, 2005); see below for some details. As of now, a clear-cut guideline on the minimum data size for the correlation dimension estimation continues to be elusive. In the absence of clear-cut guidelines, a practical way to address the minimum data size requirement is by decreasing (or increasing) the length of the time series step-by-step and estimating the correlation dimension. The length of data below which significant changes are observed can be taken as the minimum data size required. While this procedure may have some drawbacks

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when it comes to real time series, as the properties (e.g., correlation dimension) are not known a priori, it is nevertheless still useful, if sufficient caution is exercised in its implementation. This approach has been adopted by many studies, including in hydrology (e.g., Jayawardena and Lai, 1994; Wang and Gan, 1998; Sivakumar et al., 1999a; Sivakumar, 2005). 29.4.3 Data Noise

Another important assumption in the development of chaos identification and prediction methods is that the data are noise-free. However, all real data are contaminated by noise. This means, we usually observe a noisy time series Xi, i = 1, 2, …, N, which is composed of a clean signal Yi from a deterministic system and some amount of noise ni, according to

Xi =  Yi +  ni (29.11)

There are two types of noise: measurement noise and dynamical noise. Measurement noise refers to the corruption of observations by errors, which are external and independent of the dynamics, and may be caused by, for example, the measuring device. Dynamical noise, in contrast to measurement noise, is a feedback process wherein the system is perturbed by a small random amount at each timestep. Noise affects the performance of many techniques of identification, modeling, prediction, and control of deterministic systems (e.g., Schreiber and Kantz, 1996; Kantz and Schreiber, 1997; Sivakumar et al., 1999b). The severity of the influence of noise depends largely on the level and the nature of noise. In general, most dynamical measures of determinism are reasonably robust to small levels of noise, but as the noise level approaches a few percent, estimates can become quite unreliable (Schreiber and Kantz, 1996; Kantz and Schreiber, 1997). Therefore, the estimation of the level of noise in data is an important first step to understand its possible effects. Several attempts have been made to determine the level of noise in a time series (e.g., Schreiber, 1993a; Schouten et al., 1994; Heald and Stark, 2000; Jayawardena et al., 2008; Xu et al., 2012), and have made different assumptions about the nature of noise, even when using somewhat similar approaches. If it is found that the level of noise is only moderate, and there are hints that there is a strong deterministic component in the signal, then one can attempt the second step of separating the deterministic signal from the noise. While this two-step procedure is the most appropriate approach to assess the effects of noise and their mitigation (see Sivakumar et al., 1999b), it is often difficult to implement this for real data, since determination of the level of noise for such data is a very complicated process. Therefore, many studies have employed only the second step, i.e., noise reduction. A large number of nonlinear noise reduction methods have been proposed in the literature (e.g., Kostelich and Yorke, 1988; Schreiber and Grassberger, 1991; Schreiber, 1993b; Luo et al., 2005; Chelidze, 2014). The different noise reduction methods differ in the way the dynamics are approximated, how the trajectory is adjusted, and how the approximation and the adjustment steps are linked to each other; see Kostelich and Schreiber (1993) for an early survey. It has been reported that most of these methods reduce noise by a similar amount and their performances do not differ much (Kantz and Schreiber, 1997). However, noise reduction algorithms are generally chosen on the basis of their robustness, ease of use and implementation, and the computing resources needed. 29.4.4  Presence of Zeros

Most of the issues associated with chaos identification and prediction methods are common to almost all fields. For instance, delay time and embedding dimension for phase space reconstruction are inherent issues in the application of chaos methods to any real time series. Similarly, smaller data size and presence of noise are common problems in dealing with observed time series, regardless of the field of study. These are the reasons why these issues have received considerable attention in the analysis of chaos in real time series. However, there are also other problems that have not received the necessary attention in chaos analysis, although their effects can be as serious as those of the above issues. An important reason for this is that such issues are essentially encountered in just one or a few fields. For instance, the issue of the presence of a large number of zeros in time series is perhaps unique to the field of hydrology and water resources, as is the case with rainfall (and streamflow) observations, especially at high temporal resolutions. One possible influence of the presence of a large number of zeros is that the reconstructed hyper-surface in the phase space will tend to a point and may result in an underestimation of the correlation dimension (e.g., Tsonis et al., 1994; Sivakumar, 2001a). It is not yet clear how to deal with the issue of a large number of zeros in a time series. It is even questionable whether zero values should be eliminated from the time series in chaos analysis for more reliable results, since such values are also often indicative of, and important to understand, how the

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FINAL REMARKS    29-7 

dynamics of the system evolve; there can, however, be exceptions depending upon the task at hand (e.g., Sivakumar et al., 2001b). In fact, removal of the zeros can have serious effects and result in unrealistic outcomes. What is needed, therefore, is a careful understanding of the potential influences of the inclusion or exclusion of the zero values with respect to the particular task at hand. On the other hand, there may be alternative means to address the problem of zeros in chaos analysis, such as the verification of the results from one method with that of another. 29.5  HYDROLOGIC APPLICATIONS

Applications of the ideas of nonlinear dynamic and chaos theories in hydrology have been one of the most exciting areas of hydrologic research during the past two decades or so. Extensive details of such applications and the associated issues are reported in Sivakumar (2000, 2004a, 2009); see also Phillips (2003, 2006) for some details of nonlinearity, complexity, and chaos in geomorphology. Here, only a brief overview of chaos studies in hydrology is presented. The first ever study on chaos theory application in a hydrologic context was probably the one conducted by Hense (1987) on rainfall time series, although chaos in rainfall time series had already been investigated previously in the context of climate and weather (e.g., Fraedrich, 1986). It is fair to say, however, that it was the study by Rodriguez-Iturbe et al. (1989), which also investigated the presence of chaos in rainfall time series, that attracted many researchers about chaos theory in hydrology and was a significant motivation to many other studies that followed during the subsequent period. Studies during the early 1990s focused mainly on the investigation and prediction of chaos in rainfall, river flow, and lake volume time series in a purely single-variable data reconstruction sense (e.g., Sharifi et al., 1990; Wilcox et al., 1991; Berndtsson et al., 1994; Jayawardena and Lai, 1994; Abarbanel and Lall, 1996; Puente and Obregon, 1996; Sangoyomi et al., 1996; Porporato and Ridolfi, 1996). These early studies employed various chaos identification and prediction methods, including correlation dimension method, Lyapunov exponent method, Kolmogorov entropy method, nonlinear prediction method, and Poincáre maps. Some of these studies and several others that followed also addressed important methodological and data issues, including minimum data size requirement for correlation dimension estimation (e.g., Tsonis et al., 1993; Kim et al., 1998; Sivakumar et al., 1998, 1999a), effects of data noise on chaos identification and prediction (e.g., Berndtsson et al., 1994; Tsonis et al., 1994; Porporato and Ridolfi, 1997; Sivakumar et al., 1999b, c), effects of the presence of zeros on chaos identification and prediction (e.g., Tsonis et al., 1994; Koutsoyiannis and Pachakis, 1996; Wang and Gan, 1998; Sivakumar et al., 1999a), delay time/delay window selection for phase space reconstruction (e.g., Sangoyomi et al., 1996; Sivakumar et al., 1998, 1999a; Kim et al., 1998, 1999; Pasternack, 1999), stochastic processes possibly leading to positive chaos identification (e.g., Wang and Gan, 1998; Pasternack, 1999; Sivakumar et al., 1999a), and others (e.g., Liu et al., 1998; Krasovskaia et al., 1999). At the very beginning of the twenty-first century, Sivakumar (2000) published the first ever review of chaos theory applications in hydrology, with particular emphasis on addressing the important issues in chaos theory applications in hydrology and also interpreting the reported results. The review significantly helped allay many of the earlier fears and misgivings about chaos studies in hydrology and their outcomes and completely changed the course of chaos theory in hydrology. It also led to further advances in theory and application, with study of other hydrologic processes and associated problems. The processes studied include: rainfall-runoff (e.g., Sivakumar et al., 2000, 2001a; Dodov and Foufoula-Georgiou, 2005), sediment transport (e.g., Sivakumar, 2002; Sivakumar and Jayawardena, 2002, 2003; Sivakumar and Wallender, 2004, 2005), soil nutrient cycles (e.g., Manzoni et al., 2004), and subsurface flow and solute transport (e.g., Faybishenko, 2002; Sivakumar et al., 2005a; Hossain and Sivakumar, 2006). The hydrologic problems studied include: scaling and data aggregation/disaggregation (e.g., Sivakumar, 2001a, b; Sivakumar et al., 2001b, 2004; Regonda et al., 2004; Sivakumar and Wallender, 2004; Salas et al., 2005; Gaume et al., 2006) including the development of a new chaotic approach for disaggregation of hydrologic data (Sivakumar et al., 2001b), missing data estimation (e.g., Elshorbagy et al., 2002a), reconstruction of system equations (e.g., Zhou et al., 2002), regional hydrology and river flow regimes (Sivakumar, 2003), parameter estimation (Hossain et al., 2004), and model integration (e.g., Sivakumar, 2004b). A number of studies also addressed the issues of data size, noise, zeros, selection of optimal parameters, and others (e.g., Jayawardena and Gurung, 2000; Sivakumar, 2001a, 2005; Islam and Sivakumar, 2002; Elshorbagy et al., 2002b; Jayawardena et al., 2002; Phoon et al., 2002; Schertzer et al., 2002; Sivakumar et al., 2002a, c, 2006; Khan et al., 2005; Salas et al., 2005; Koutsoyiannis, 2006). Some studies also compared hydrologic predictions based on chaos methods with those based on

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other techniques, such as stochastic methods and artificial neural networks (e.g., Jayawardena and Gurung, 2000; Lambrakis et al., 2000; Sivakumar et al., 2002b, c; Laio et al., 2003). Attempts were also made to analyze hydrologic time series based on multi-variable phase space reconstruction (e.g., Porporato and Ridolfi, 2001; Jin et al., 2005; Sivakumar et al., 2005b), rather than reconstruction based on just a single variable. There were also attempts to use other data analysis methods for prediction of hydrologic time series that exhibit chaotic behavior (e.g., Karunasinghe and Liong, 2006). The significant inroads made in the application of chaos theory in hydrology over two decades led to an increasing realization in more recent years on the opportunities and possibilities for studying large-scale problems. It is fair to say that this realization has come at a critical juncture, as we are (and will be in the future) facing some tremendous global-scale challenges in hydrology and water resources, such as assessment of impacts of global climate change, development of a generic modeling framework, study of interactions between hydrologic systems and other Earth and socio-economic systems, and study of issues related to transboundary waters (river basins as well as aquifers), among others. Details of these challenges are available in, for example, Paola et al. (2006), Wagener et al. (2010), Sivakumar (2011a, b), and Sivakumar and Singh (2012); see also Sivakumar and Singh (2015) for compilation of some of the grand challenges in hydrology. During the last few years, some initial attempts have been made to apply the ideas of chaos theory to study these global-scale hydrologic challenges, especially for catchment classification and global climate change. For instance, Sivakumar et al. (2007) explored the utility of phase space reconstruction for assessing the complexity of hydrologic systems and thus their classification, through analysis of several real river-related hydrologic series observed around the world. Sivakumar and Singh (2012) offered a more comprehensive scientific background and discussion for proposing system complexity as a basis for catchment classification framework and nonlinear dynamic concepts as a suitable methodology for assessing system complexity. They also presented a classification of 117 catchments in the western United States, based on the application of correlation dimension method to monthly streamflow data. Kyoung et al. (2011) examined the dynamic characteristics of rainfall under conditions of climate change, through analysis of observed and global climate model (GCM)simulated (present and future) monthly rainfall in the Korean Peninsula. Based on the application of the correlation dimension and close returns plot, among others, they reported that the nature of rainfall dynamics falls more on the chaotic dynamic spectrum than on the linear stochastic spectrum and also that future GCM-simulated rainfall exhibits stronger nonlinearity and chaos compared to the present rainfall, thus emphasizing the need for a chaotic dynamic-based framework for downscaling GCM outputs. While studies on these global-scale hydrologic challenges using chaos theory and related concepts have been gaining momentum, other chaos studies in hydrology have and continue to grow as well. These include studies on rainfall (e.g., Jothiprakash and Fathima, 2013), regional streamflow variability and classification (e.g., Vignesh et al., 2015), sediment transport (e.g., Sivakumar and Chen, 2007), arsenic contamination in groundwater (e.g., Hill et al., 2008), ensemble prediction of chaotic hydrologic time series (including multivariate prediction with climate inputs) (e.g., Dhanya and Nagesh Kumar, 2010, 2011), river stage (Khatibi et al., 2012), and others (e.g., Sivakumar, 2007; Kim et al., 2009), including advances in methodologies (e.g., Jayawardena et al., 2008). A more recent account of chaos studies in hydrology is presented in Sivakumar (2009), which also reiterates the need for a middle-ground approach in hydrology and the role chaos theory can play in its formulation. Further philosophical and pragmatic arguments to this end are made by Sivakumar (2011c). 29.6  FINAL REMARKS

Since its introduction in hydrology in the late 1980s, chaos theory has and continues to be a fascinating topic in hydrologic research. As a result, there has been a tremendous growth both in our understanding of the concepts of chaos theory and in their application areas. We have come a long way, starting from simple (and perhaps sometimes blind) applications of chaos methods looking for something (or anything) to more-refined applications for betterdefined purposes at local scales and now with an aim to tackle global-scale hydrologic challenges. Along the way, we have encountered several difficulties, identified some potential problems, and resolved at least a few of them. There is also an increasing recognition that chaos theory, with its underpinning concepts of nonlinear interdependence, hidden determinism and order, and sensitivity to initial conditions, can encompass all the key ideas of our currently dominant, yet extreme, deterministic and stochastic views. Consequently, it can offer a more balanced and realistic approach for hydrologic modeling, than either the deterministic or the stochastic approach can

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on their own. With issues around water resources starting to take a prominent stage and as the need for a better, and more balanced, approach becoming clear, now is indeed an exciting time for research on chaos in hydrology. ACKNOWLEDGMENT

This work is supported by the Australian Research Council (ARC) Future Fellowship grant (FT110100328). Bellie Sivakumar acknowledges the financial support from ARC through this Future Fellowship grant. REFERENCES

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29-10      Nonlinear Dynamics and Chaos

Singh, V. P., “The use of entropy in hydrology and water resources,” Hydrological Processes, 11: 587–626, 1997. Sivakumar, B., “Chaos theory in hydrology: important issues and interpretations,” Journal of Hydrology, 227 (1–4): 1–20, 2000. Sivakumar, B., “Rainfall dynamics at different temporal scales: a chaotic perspective,” Hydrology and Earth System Sciences, 5 (4): 645–651, 2001a. Sivakumar, B., “Is a chaotic multi-fractal approach for rainfall possible?” Hydrological Processes, 15 (6): 943–955, 2001b. Sivakumar, B., “A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers,” Journal of Hydrology, 258: 149–162, 2002. Sivakumar, B., “Forecasting monthly streamflow dynamics in the western United States: a nonlinear dynamical approach,” Environmental Modelling & Software, 18 (8–9): 721–728, 2003. Sivakumar, B., “Chaos theory in geophysics: past, present and future,” Chaos, Solitons & Fractals, 19 (2): 441–462, 2004a. Sivakumar, B., “Dominant processes concept in hydrology: moving forward,” Hydrological Processes, 18 (12): 2349–2353, 2004b. Sivakumar, B., “Correlation dimension estimation of hydrologic series and data size requirement: myth and reality,” Hydrological Sciences Journal, 50 (4): 591–604, 2005. Sivakumar, B., “Nonlinear determinism in river flow: prediction as a possible indicator,” Earth Surface Processes and Landforms, 32 (7): 969–979, 2007. Sivakumar, B., “Nonlinear dynamics and chaos in hydrologic systems: latest developments and a look forward,” Stochastic Environmental Research and Risk Assessment, 23: 1027–1036, 2009. Sivakumar, B., “Global climate change and its impacts on water resources planning and management: assessment and challenges,” Stochastic Environmental Research Risk Assessment, 25 (4): 583–600, 2011a. Sivakumar, B., “Water crisis: from conflict to cooperation—an overview,” Hydrological Sciences Journal, 56 (4): 531–552, 2011b. Sivakumar, B., Chaos theory for modeling environmental systems: philosophy and pragmatism, System Identification, Environmental Modelling, and Control System Design, edited by L. Wang and H. Garnier, Springer-Verlag, London, 2011c, pp. 533–555. Sivakumar, B. and R. Berndtsson, Advances in Data-Based Approaches for Hydrologic Modeling and Forecasting, World Scientific, Singapore, 2010. Sivakumar, B. and J. Chen, “Suspended sediment load transport in the Mississippi River basin at St. Louis: temporal scaling and nonlinear determinism,” Earth Surface Processes and Landforms, 32 (2): 269–280, 2007. Sivakumar, B. and A. W. Jayawardena, “An investigation of the presence of low-dimensional chaotic behavior in the sediment transport phenomenon,” Hydrological Sciences Journal, 47 (3): 405–416, 2002. Sivakumar, B. and A. W. Jayawardena, “Sediment transport phenomenon in rivers: an alternative perspective,” Environmental Modelling & Software, 18 (8–9): 831–838, 2003. Sivakumar, B. and V. P. Singh, “Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework,” Hydrology and Earth System Sciences, 16: 4119–4131, 2012. Sivakumar, B. and V. P. Singh, “Special issue on grand challenges in hydrology,” Journal of Hydrologic Engineering, ASCE, 20 (1): A2014001, 2015. Sivakumar, B. and W. W. Wallender, “Deriving high-resolution sediment load data using a nonlinear deterministic approach,” Water Resources Research, 40: W05403, 2004, doi: 10.1029/2004WR003152. Sivakumar, B. and W. W. Wallender, “Predictability of river flow and sediment transport in the Mississippi River basin: a nonlinear deterministic approach,” Earth Surface Processes and Landforms, 30: 665–677, 2005. Sivakumar, B., S. Y. Liong, and C. Y. Liaw, “Evidence of chaotic behavior in Singapore rainfall,” Journal of the American Water Resources Association, 34 (2): 301–310, 1998. Sivakumar, B., S. Y. Liong, C. Y. Liaw, and K. K. Phoon, “Singapore rainfall behavior: chaotic?” Journal of Hydrological Engineering, ASCE, 4 (1): 38–48, 1999a. Sivakumar, B., K. K. Phoon, S. Y. Liong, and C. Y. Liaw, “A systematic approach to noise reduction in chaotic hydrological time series,” Journal of Hydrology, 219 (3–4): 103–135, 1999b. Sivakumar, B., K. K. Phoon, S. Y. Liong, and C. Y. Liaw, “Comment on ‘Nonlinear analysis of river flow time sequences’ by Amilcare Porporato and Luca Ridolfi,” Water Resources Research, 35 (3): 895–897, 1999c. Sivakumar, B., R. Berndtsson, J. Olsson, K. Jinno, and A. Kawamura, “Dynamics of monthly rainfall-runoff process at the Göta basin: a search for chaos,” Hydrology and Earth System Sciences, 4 (3): 407–417, 2000. Sivakumar, B., R. Berndtsson, J. Olsson, and K. Jinno, “Evidence of chaos in the rainfall-runoff process,” Hydrological Sciences Journal, 46 (1): 131–145, 2001a.

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Sivakumar, B., S. Sorooshian, H. V. Gupta, and X. Gao, “A chaotic approach to rainfall disaggregation,” Water Resources Research, 37 (1): 61–72, 2001b. Sivakumar, B., R. Berndtsson, J. Olsson, and Jinno, “Reply to ‘which chaos in the rainfall-runoff process?’ by Schertzer et al.” Hydrological Sciences Journal, 47 (1): 149–158, 2002a. Sivakumar, B., A. W. Jayawardena, and T. M. G. H. Fernando, “River flow forecasting: use of phase-space reconstruction and artificial neural networks approaches,” Journal of Hydrology, 265 (1–4): 225–245, 2002b. Sivakumar, B., M. Persson, R. Berndtsson, and C. B. Uvo, “Is correlation dimension a reliable indicator of low-dimensional chaos in short hydrological time series?” Water Resources Research, 38 (2): 2002c, doi: 10.1029/ 2001WR000333 Sivakumar, B., W. W. Wallender, C. E. Puente, and M. N. Islam, “Streamflow disaggregation: a nonlinear deterministic approach,” Nonlinear Processes in Geophysics, 11: 383–392, 2004. Sivakumar, B., T. Harter, and H. Zhang, “Solute transport in a heterogeneous aquifer: a search for nonlinear deterministic dynamics,” Nonlinear Processes in Geophysics, 12: 211–218, 2005a. Sivakumar, B., R. Berndtsson, M. Persson, and C. B. Uvo, “A multi-variable time series phase-space reconstruction approach to investigation of chaos in hydrological processes,” International Journal of Civil & Environmental Engineering, 1 (1): 35–51, 2005b. Sivakumar, B., W. W. Wallender, Horwath, J. P. Mitchell, S. E. Prentice, and B. A. Joyce, “Nonlinear analysis of rainfall dynamics in California’s Sacramento Valley,” Hydrological Processes, 20 (8): 1723–1736, 2006. Sivakumar, B., A. W. Jayawardena, and W. K. Li, “Hydrologic complexity and classification: a simple data reconstruction approach,” Hydrological Processes, 21 (20): 2713–2728, 2007. Sivakumar, B., F. M. Woldemeskel, and C. E. Puente, “Nonlinear analysis of rainfall variability in Australia,” Stochastic Environmental Research Risk Assessment, 28 (1): 17–27, 2014. Smith, L. A., “Intrinsic limits on dimension calculations,” Physics Letters A, 133 (6): 283–288, 1988. Sugihara, G. and R. M. May, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series,” Nature, 344: 734–741, 1990. Sun, J., Y. Zhao, J. Zhang, X, Luo, and M. Small, “Reducing coloured noise for chaotic time series in the local phase space,” Physical Review E, 76: 026211, 2007. Swinney, H. L. and J. P. Gollub, “Hydrodynamic instabilities and the transition to turbulence,” Physics Today, 31 (8): 41–49, 1978. Şen, Z., Fuzzy Logic and Hydrologic Modeling, CRC Press, Boca Raton, FL, 2009. Takens, F., Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898, edited by D. A. Rand and L. S. Young, Springer, Berlin, 1981, pp. 366–381. Thomas, Jr., H. A. and M. B. Fiering, Mathematical synthesis of streamflow sequences for the analysis of river basins by simulation, Design of Water Resource-Systems: New Techniques for Relating Economic Objectives, Engineering Analysis, and Governmental Planning, edited by A. Mass, M. M. Hufschmidt, R. Dorfman, H. A. Thomas, Jr., S. A. Marglin, and G. M. Fair, Harvard University Press, Cambridge, MA, 1962, pp. 459–493. Tsonis, A. A., Chaos: From Theory to Applications, Plenum Press, New York, 1992. Tsonis, A. A. and J. B. Elsner, “The weather attractor over short timescales,” Nature, 333: 545–547, 1988. Tsonis, A. A., J. B. Elsner, and K. P. Georgakakos, “Estimating the dimension of weather and climate attractors: important issues about the procedure and interpretation,” Journal of the Atmospheric Sciences, 50: 2549–2555, 1993. Tsonis, A. A., G. N. Triantafyllou, J. B. Elsner, J. J. Holdzkom II, and A. D. Kirwan, Jr., “An investigation on the ability of nonlinear methods to infer dynamics from observables,” Bulletin of the American Meteorological Society, 75: 1623–1633, 1994. Valencia, D. R. and J. L. Schaake “Disaggregation processes in stochastic hydrology,” Water Resources Research, 9 (3): 211–219, 1973. Vignesh, R., V. Jothiprakash, and B. Sivakumar, “Streamflow variability and classification using false neighbor neighbor method,” Journal of Hydrology, 531: 706–715, 2015. Wagener, T., M. Sivapalan, P. A. Troch, B. L. McGlynn, C. J. Harman, H. V. Gupta, P. Kumar, et al., “The future of hydrology: an evolving science for a changing world,” Water Resources Research, 46: W05301, 2010, doi:10.1029/ 2009WR008906. Wang, Q. and T. Y. Gan, “Biases of correlation dimension estimates of streamflow data in the Canadian prairies,” Water Resources Research, 34 (9): 2329–2339, 1998.

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Wilcox, B. P., M. S. Seyfried, and T. M. Matison, “Searching for chaotic dynamics in snowmelt runoff,” Water Resources Research, 27 (6): 1005–1010, 1991. Wolf, A., J. B. Swift, H. L. Swinney, and A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, 16: 285–317, 1985. Xu, P., W. K. Li, and A. W. Jayawardena, “Noise level estimation for a chaotic time series,” International Journal of Bifurcation and Chaos, 22 (3): 1250052, 2012.

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Chapter

30

Copula Modeling in Hydrologic Frequency Analysis BY

CHRISTIAN GENEST AND FATEH CHEBANA

ABSTRACT

This chapter provides an introduction to the copula approach to multivariate data modeling and surveys some of its uses in hydrologic frequency analysis (HFA). The various steps involved in the construction and validation of a copula model are outlined. Flood peak and volume data from the Fraser River at Hope (British Columbia, Canada) are used to illustrate the methodology, and resources for carrying out the analysis are provided. Possible extensions of these techniques to regional and nonstationary frequency analysis, as well as to spatial contexts, are briefly discussed. 30.1  INTRODUCTION

Floods, droughts, and extreme precipitation can have serious social and economic consequences, including widespread material destruction and loss of human lives. It is crucial, therefore, to rely on appropriate models in order to predict the frequency and magnitude of such events. Hydrologic frequency analysis is a set of statistical techniques designed for this purpose. Hydrologic phenomena involve many interrelated variables, for example, flood peak, volume, and duration. It is natural, therefore, to adopt a multivariate framework for their analysis, and it is in fact often essential to do so for accurate risk assessment. In contrast, univariate HFA can provide only a limited evaluation of extreme events and their probability of occurrence, because treating each event characteristic in isolation ignores possibly crucial relations between the variables at play. This chapter describes the use of copula models in the context of multivariate HFA. The merits of this approach in statistics are now well known (Genest and Favre, 2007; Joe, 2014) and references therein. In hydrology, copula modeling techniques have been used to study events such as floods, storms, droughts, sediments, rainfalls, and sea storms (Grimaldi and Serinaldi, 2006; Kao and Govindaraju, 2007; Chebana and Ouarda, 2011; Vandenberghe et al., 2011; Requena et al., 2013; Volpi and Fiori, 2014). Copulas are also useful to study a single feature, for example, flood peak, at multiple sites; for an illustration, see El Adlouni and Ouarda (2008). Copula models are introduced in Sec.  30.2, where their advantages are highlighted. An overview of inference techniques for these models is then given in Sec. 30.3, and their use for multivariate quantile and return period estimation is described in Sec. 30.4. The methodology is illustrated in Sec. 30.5 using flood peak and volume data from the Fraser River at Hope (British Columbia, Canada). Extensions are outlined in Sec.  30.6, and specialized resources are listed in Sec. 30.7. Although copula modeling in hydrology only developed in the last decade or so, the literature on the subject is already substantial. Only selected references are provided here. The readers can consult the website of the International Commission on Statistical Hydrology for an up-to-date list of publications related to copulas and their applications in HFA and other hydrologic contexts, from the construction of drought

severity-duration-frequency curves and copula-based dissimilarity measures for streamflow prediction in ungaged catchments to models for temporal sampling errors in satellite-derived rainfall estimates and for checking dam spillway adequacy. 30.2  DESCRIPTION OF COPULA MODELS

A copula is a multivariate distribution function whose margins are uniform on the interval (0, 1). Given a random vector X = (X1, …, Xd) with joint distribution function H and margins F1,…,  Fd, Sklar (1959) showed that there always exists a copula C such that, for all x1, …, xd ∈ (−∞, +∞), Pr( X1   ≤ x1 , …,  X d   ≤   x d ) = H ( x1 , …,  x d ) = C{F1 ( x1 ), …, Fd ( x d )}. (30.1) If the marginal distribution functions F1, …, Fd are continuous, then C is unique and can be retrieved from H by setting, for all u1, …, ud ∈(0, 1), C(u1 , …, ud ) = H {F1−1 (u1 ), …, Fd −1 (ud )}

(30.2)

where, for each j ∈{1, …, d }, Fj–1 denotes the generalized inverse of Fj. In other words, C is the distribution function of the vector (U1, …, Ud) = (F1(X1), …, Fd (Xd )). When the distribution H of X is unknown, Eq. (30.1) remains valid but C and F1, …, Fd are undetermined. A copula model for X then consists of assuming that C ∈C, F1 ∈ F1, …, Fd ∈Fd , (30.3) where C is a class of copulas indexed by a (possibly vector-valued) parameter q while F1, …, Fd are classes of univariate distribution functions indexed by parameters h1, …, hd, respectively. The main advantage of this approach is that each variable can be modeled separately. Once suitable parametric classes of univariate distributions (e.g., Gamma, Normal, or Student t) have been selected for each variable, they can be “glued” together with an appropriate choice of copula. Classical multivariate models typically offer much less flexibility because the margins are often all of the same type; for example, all margins of a multivariate Normal distribution are Normal. When they use copula models, analysts face no restriction (other than practicality) in the choice of C and F1, …, Fd in Eq. (30.1). This advantage is of particular interest in hydrology, where variables associated with a given event (e.g., flood peak and volume) may vary widely in nature, scale, and distribution. Copula models eliminate the need to transform the variables in the (often illusory) hope of making them fit into a classical framework. This is also convenient because for interpretation purposes, practitioners tend to prefer models that involve the original variables. Another distinct advantage of copula modeling is its reliance on the common tools of univariate HFA. Before proceeding with the construction of a univariate model, an analyst would first carry out an exploratory 30-1

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30-2    Copula Modeling in Hydrologic Frequency Analysis

analysis aimed at identifying outliers and checking the basic assumptions of stationarity, homogeneity, and serial independence of the time series. The same preliminary steps are prerequisite to copula modeling. As the procedures required to accomplish this task and select suitable univariate parametric models are already well documented in the HFA literature (Rao and Hamed, 2000), it is taken for granted henceforth that Assumption: The data at hand constitute a random sample from a joint distribution H with continuous margins F1, …, Fd. This implies, in particular, that the underlying dependence structure is represented by a unique copula C that does not change over time. The success of the multivariate modeling exercise then depends on the choice of a suitable copula C within some parametric class C. The basic procedure to be followed is described in Sec. 30.3. To fix ideas, some of the simple classes of copulas commonly used to model low-dimensional hydrologic data are listed below. For additional examples and book-length treatments of the mathematical properties of copulas, see Joe (2015) or Nelsen (2006). 30.2.1 Archimedean Copulas

A copula C is said to be Archimedean if it can be written in the form C(u1 , …, ud ) = ψ {ψ −1 (u1 )  +    +   ψ −1 (ud )}

(30.4)

in terms of a generator y: [0, ∞) → [0, 1] such that y(0) = 1, y(t) → 0 as t → ∞ and y is d-monotone in the sense of McNeil and NeŠlehová (2010). For instance, the generator of the Clayton copula with parameter q  ∈  (0,  ∞) is defined, for all t ∈ [0, ∞) by y(t) = (1+qt)−1/q. Many other examples, including the Frank and Gumbel parametric families, can be found, for example, in Genest and MacKay (1986), Nelsen (2006), Salvadori et al. (2007). Archimedean copulas are symmetric in their arguments. Thus if a vector (U1,  …, Ud) is distributed as per Eq. (30.4), every subset of its components inherits the same dependence structure. 30.2.2 Extreme-Value Copulas

A copula is said to be extreme-value (EV) if, for all t > 0 and u1, …, ud ∈ (0, 1), Ct(u11/t, …, ud1/t) = C(u1, …, ud) (30.5) This property is characteristic of the dependence structure in multivariate extreme-value distributions; it is particularly attractive for data that are block maxima or that can be construed as such, for example, yearly maxima of daily flow measurements (Salvadori and De Michele, 2011). In dimension d = 2, EV copulas can be expressed simply in terms of a convex map A: [0, 1] → [0, 1] such that, for all t ∈ [0, 1], A(t) ≥ max(t, 1 – t). To be specific, one has, for all u1, u2 ∈ (0, 1), C(u1 ,u2 ) = exp[ln(u1u2 )A{ln(u2 ) / ln(u1u2 )}]

(30.6)

Examples of parametric families of EV copulas include the Galambos, Gumbel, Hüsler–Reiß, Marshall–Olkin, and Student t v – EV (Joe, 2014; Kotz and Nadarajah, 2000; Salvadori and De Michele, 2010; Tawn, 1988).

Preliminary steps

Although EV copulas are not necessarily symmetric in their arguments, the EV dependence structure is inherited by all subsets of a vector whose copula satisfies property (30.5). Members of the Gumbel family are the only copulas that are both EV and Archimedean (Genest and Rivest, 1989). 30.2.3 Meta-Elliptical Copulas

Suppose that a random vector is of the form X = RAT, where R is a strictly positive continuous random variable, A is a d × d matrix of constants, and T is a random vector that is both independent of R and uniformly distributed on the unit sphere Td = {(t1 , , td ): t12   + +  td2   =  1}. The vector X then has an elliptical distribution with mean 0 and dispersion matrix S = AAT, where T denotes matrix transposition; the copula associated with the distribution of X is said to be meta-elliptical (Genest et al., 2007). For instance when R2 is chi-square with d degrees of freedom, X has a multivariate Normal distribution; its underlying copula, which is parameterized by S, is called the Gaussian copula. The multivariate Student t copula with v degrees of freedom also belongs to this class. Similar to EV copulas, meta-elliptical copulas can be asymmetric in their arguments but all subsets of a vector whose copula is meta-elliptical will inherit the same type of dependence structure (e.g., if X has a multivariate Student t copula with v degrees of freedom, then so do all subsets of d ≥ 2 components of X). 30.2.4 More Elaborate Copula Structures

Many parametric copula families available in the literature are such that all subsets of variables share a common type of dependence structure. When this property is too limitative, one can resort to asymmetric extensions, for example, hierarchical Archimedean copulas (Joe, 2014) or Liouville copulas (McNeil and NeŠlehová, 2010). The state-of-the-art in this regard is vine copula models, built from a tree-like structure of bivariate copulas consisting of conditional distributions beyond the first level of the hierarchy (Kurowicka and Joe, 2011). In such models, various subsets of variables can have widely different dependence structures. However, this added flexibility comes at the cost of more complex inference procedures than those that will be reviewed below. 30.3  OVERVIEW OF MODEL SELECTION

Given a random sample from some unknown multivariate distribution H, how can one select a suitable parametric copula family to model its dependence, estimate its parameters, and validate this choice? A brief practical guide is provided below, based largely on (Genest and Favre, 2007). It is assumed henceforth that the margins F1, …, Fd are continuous to ensure that the unknown copula C in Eq.  (30.1) is unique. A discussion of the discontinuous case can be found in Genest and NeŠlehová (2007). Figure 30.1 summarizes the key issues to be considered. The items in the chart are not ordered because the various steps are interrelated and one may have to go back and forth between them. A rough order is provided below and illustrated in the data analysis of Sec. 30.4. The preliminary steps are those needed to ensure that the assumption stated in Sec. 30.2 holds. As the required procedures are not specific to copula modeling, the reader can refer, for example, to Chebana and Ouarda (2011)

Preprocessing of the series at hand to ensure that the data form a random sample from a distribution with continuous margins (in particular, stationarity, homogeneity, and serial independence must hold, at least approximately)

Modeling steps

Multivariate Fx,y Margins Fx and Fy

Copula C (u, v)

Refer to U-HFA for each variable Graphics

Dependence measures

Goodness of testing Inference steps

Tail dependence

Parameter estimation

Selection criteria

Evaluation of return periods and multivariate quantile

Figure 30.1  Main steps of HFA and procedure for copula fitting and selection.

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Overview of model selection    30-3 

or Chebana et al. (2013) for descriptions in the context of hydrologic flood data. In what follows, the (possibly de-trended or otherwise preprocessed) data are denoted X1 = (X11, …, X1d), …, Xn = (Xn1, …, Xnd) and are considered to form (at least approximately) a random sample from distribution H with margins F1, …, Fd and copula C. 30.3.1 Visualizing Data from Copulas

If the margins F1, …, Fd were known, in view of Eq. (30.2), the observations could be transformed into a random sample from C by setting U1 = (F1(X11), …, Fd (X1d)), …, Un = (F1(Xn1), …, Fd(Xnd)). The vectors U1, …, Un could then be plotted if d = 2 (or two components at the time when d > 2) to get a sense of the dependence embodied in C. In practice, however, the margins are unknown and must be estimated. As this process involves uncertainty and is prone to error, inference about C is best based on conservative estimates of these margins defined, for all j ∈{1, …, d} and x ∈ (−∞, ∞), by Fnj ( x ) =

1 n ∑1( Xij ≤ x ) (30.7) n + 1 i=1

where 1(A) is the indicator of the event A, that is, 1(A) = 1 when the event A occurs and 1(A)  =  0 otherwise. One may then construct a pseudo-sample from C by letting, for each i ∈ {1, …, n}, Ûi = (Ûi1, …, Ûid) = (Fn1(Xi1), … Fnd(Xid)) (30.8) Note that, for each i ∈ {1, …, n} and j ∈ {1, …, d}, (n + 1) × Fnj(Xij) is the rank of Xij among X1j,  …,  Xnj. Thus if the points Û1,  …,  Ûn could be visualized in d-dimensional space, each of their marginal distributions would be found to be exactly uniform on the set {1/(n + 1), …, n/(n + 1)} because ties do not occur (almost surely) when F1,  …,  Fd are continuous. Statistical theory further guarantees that the joint pattern described by Û1,  …,  Ûn contains all the information needed to identify C. In formal terms, consider the empirical distribution of the pseudo-sample Û1, …, Ûn defined, for all u1, …, ud ∈ (0, 1), by

is an important clue. Among other characteristics that should be taken into account are: the presence of symmetry or asymmetry, the strength of dependence, signs of extreme-value behavior, etc. Various indices can be computed to help with this task. 30.3.2 Numerical Summaries

There exist several measures of dependence, asymmetry, and tail behavior whose population values depend on the underlying copula alone, and whose sample values can be computed from the vectors of ranks derived from the observations. Prominent among them are Spearman’s rho (r) and Kendall’s tau (t). In dimension d  =  2, Spearman’s rho is simply Pearson’s correlation coefficient, but computed from the ranks of the observations. Kendall’s tau is defined theoretically as the difference between the probability of concordance (positive slope) and the probability of discordance (negative slope) between two observations chosen at random. Its sample version is defined in a similar way. In families of bivariate copulas indexed by a real-valued parameter q, there is typically a one-to-one correspondence between q, r, and t; in the Clayton model, for instance, t = q/(q + 2) for all q ∈ (0, ∞) and similarly for r, though the latter cannot be expressed in closed form as a function of q. Some copula families have restricted ranges for the values of r or t, for example, t ∈ [−2/9, 2/9] in the Farlie–Gumbel–Morgenstern (FGM) bivariate copula family; such models may thus be judged inadequate from the start in certain contexts. Multivariate definitions are available for many coefficients, including Spearman’s rho and Kendall’s tau; see, for example, Genest et al. (2011b), Joe (1993), and Schmid and Schmidt (2007). Several other rank-based/copula-based coefficients are possible; see, for example, Genest and Nešlehová (2012b) and Nelsen (2006). Among them, Joe’s lower and upper tail dependence coefficients (Joe, 1993) are particularly important for hydrologic applications. In the bivariate case, they are given respectively by λ l = lim C(q , q ) q , λu = 2 − lim{1 − C(q , q )} / (1 − q ) q↓0

q↑1

(30.11)

1 Cˆn (u1 ,...,ud ) = ∑1(Uˆ i1 ≤ u1 ,...,Uˆ id ≤ ud ) (30.9) n i=1

whenever the limits exist. Rank-based estimates are defined analogously; for instance,

This so-called empirical copula is a consistent estimator of C, that is, it converges [in distribution for all choices of u1,  …,  ud ∈ (0,  1)] to C as the sample size n grows indefinitely; see, for example, Segers (2012). A plot of the vectors Û1, …, Ûn (in dimension d = 2, or else looking at pairs of components) provides many clues as to the type of dependence embodied in C; it can also assist in selecting a parametric form for C. This procedure called a rank-plot can be used, for example, to check whether all pairs share the same type of dependence. Alternatively, one could transform the pseudo-observations to the Normal scale by setting, for each i ∈ {1, …, n}, Ôi = (F-1(Ûi1), … F-1(Ûid)), where F denotes the N(0, 1) distribution. One could then plot Ô1, …, Ôn, called a rankit plot. In large samples, rank-plots and rankit plots become too dense and hence ineffective. They can then be replaced, for example, by rank-based estimates of the copula density (Genest et al., 2009b). In high-dimensional problems, however, rank-based plots of the various pairs of variables shed only partial light into the overall dependence structure. Additional insight can be gained by looking at one-dimensional projections of the multivariate data, for example, by setting, for all i ∈ {1, …, n},

λun = 2 − (n / k )Cˆn (1 − k / n,…,1 − k / n)

n

ˆ = Cˆ (Uˆ ,…,Uˆ ) W i n i1 id

(30.10)

ˆ is the proportion of points among Û , …, Û whose compoIn words, W i 1 n nents are all smaller than those of Ûi. The empirical distribution function Kˆ n ˆ ,, W ˆ is an estimate of the distribution K of the associated with the set W 1 n variable W = C(U1, …, Ud) = H(X1, …, Xd); the latter is unknown but can be computed for any hypothesized choice of C or H. For example, in dimension d = 2, the assumption that the variables X1, X2 (or equivalently U1, U2) are independent implies that, for all w ∈ (0, 1), K(w) = w – w ln(w). One could then check whether independence is reasonable by comparing visually Kˆ n and K or by drawing a Q-Q plot. This can be a useful complement to a rankplot, even when dealing with only two variables, but the point is that this procedure can be used whatever the number of variables because the pseudoˆ ,, W ˆ are always real-valued and hence have a univariate observations W 1 n distribution; see Genest and Boies (2003) for details. These and other graphical tools provide insight into the dependence structure of the data and help to select suitable parametric copula families. As mentioned earlier, the presence or not of similar patterns between different pairs

(30.12)

is associated to lu for an appropriate choice of k; more sophisticated approaches are preferable, however; see, for example, Schmidt and Stadtmüller (2006). Tail dependence coefficients aim at detecting and quantifying the presence of extreme dependence between X1 and X2. For instance when C is a bivariate extreme-value copula with dependence function A, as per Eq. (30.6), one has ll = 0 and lu = 2 – 2A(0.5); other examples of calculation are given, for example, in Ghoudi et al. (1998). Computing ll and lu can sometimes reveal important differences between otherwise similarly looking dependence structures. For instance, both coefficients vanish for the Gaussian copula but they are strictly positive for the Student t copula. As a result, models based on the Gaussian copula can sometimes lead to an underestimation of joint extreme risks (McNeil et al., 2005). Alternative multivariate extreme coefficients are considered in Frahm (2006). 30.3.3 Model Fitting

Suppose that a copula model (30.3) is being considered for a d-variate distribution H. Assume that following an exploratory data analysis, parametric families F1, …, Fd and C have been selected for the margins and the copula, respectively. Further assume that densities exist for all members of these families and denote them by lower-case letters. A natural way to estimate the parameters is to maximize the joint loglikelihood, viz. n

n

d

(θ, η1 ,…, ηd ) = ∑ ln[cθ {F1,η1 ( Xi1 ),…, Fd ,ηd ( Xid )}] + ∑∑ ln{ f j ,η j ( Xij )} (30.13) i =1

i =1 j =1

In principle, this expression can be maximized, but this can be tedious and time-consuming, especially in high dimensions. To circumvent this problem, Joe (2005) examined the merits of proceeding in two steps. In the first step, estimates of the marginal distribution parameters h1, …, hd are obtained. For each j ∈ {1, …, d}, this is done by maximizing n

( η j ) = ∑ ln{f j ,η j ( Xij )}

(30.14)

i =1

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30-4    Copula Modeling in Hydrologic Frequency Analysis

In the second step, the resulting estimates of the marginal parameters are plugged into the first summand of Eq. (30.13) to obtain a log-likelihood function for the dependence parameter q, viz. n

(θ) = ∑ln[cθ {F1,ηˆ1 ( Xi1 ),…, Fd ,ηˆ d ( Xid )}] i =1

(30.15)

As shown by Joe (2005), this technique typically yields estimates of q that are consistent and asymptotically Normal, but somewhat less efficient than those derived from the full maximization of Eq. (30.13), except in the neighborhood of independence. However, the estimator of q can be severely biased if the marginal models are specified incorrectly (Kim et al., 2007). A safer, and yet conceptually simple, solution is to base the estimate of q on the likelihood function derived from the pseudo-observation Û1, …, Ûn, viz. n

n

i =1

i =1

(θ) = ∑ln[cθ {Fn1 ( Xi1 ),…, Fnd ( Xid )}] = ∑ln{cθ (Uˆ i1 ,…,Uˆ id )}

(30.16)

which is rank-based. The consistency and asymptotic Normality of this estimator are established under mild regularity conditions in Genest et al. (1995), where a consistent estimate of the asymptotic variance is also proposed. The maximum pseudo-likelihood estimator is generally less efficient than the full maximum likelihood estimator when the margins are properly specified, but much preferable than either of the two previous methods in case of marginal misspecification (Kim et al., 2007). Several commonly used bivariate copula families are indexed by a realvalued parameter q that can be expressed as a simple algebraic function of Spearman’s rho or Kendall’s tau, say q = g(r) or q = h(t). In a moment-based method, a quick-and-dirty estimate qn of q is then given either by qn = g(rn) or qn = h(tn), where rn and tn are the classical estimators of r and t, respectively. Both solutions are acceptable; they also provide reliable starting values for numerical maximization of the pseudo-likelihood, given that rn and tn are rank-based, consistent, and asymptotically Normal. Inversion of Spearman’s or Kendall’s tau can also be used to estimate dependence parameters in multivariate copula families. In d-variate meta-elliptical copula models, for example, the (i, j)th element sij of the correlation matrix S can be estimated via Kendall’s tau through the relation qij = (2/p) arcsin(sij); see, for example, Genest et al. (2007). The above moment-based estimators are generally less efficient than maximum pseudo-likelihood estimators. In addition, they sometimes need to be adjusted to ensure that they remain within the admissible range of values for q. In the bivariate FGM copula family, for instance, one has r  =  q/3 and t = 2q/9, so that qn = 3rn and qn = 9tn /2, both of which can fall outside the interval [−1, 1]. Similarly, the correlation matrix S of a d-variate meta-elliptical copula obtained by inversion of Kendall’s tau may fail to be positive semidefinite. Alternative approaches are being developed; for example, a method based on multivariate L-moments was recently proposed in Brahimi et al. (2015) to account for various hydrologic constraints.

30.3.5  Procedures for Extreme-Values Copulas

Given their importance in modeling catastrophic events, extreme-value copulas have been the object of a great deal of attention in recent years. Let Ed denotes the class of d-variate extreme-value copulas. To see whether an extreme-value copula applies to a given situation, one should first test the hypothesis HE: C  ∈  Ed. This is called a test of extremeness. The oldest and simplest procedure of this type, in the bivariate case, consists of setting Iij = 1(Xj1 ≤ Xi1, Xj2 ≤ Xi2) for all i, j ∈ {1, …, n} and computing SnGKR = −1 +

n n 8 9 ∑Iij − n(n − 1)(n − 2) ∑ Iij I kj (30.19) n(n − 1) i≠ j i≠ j ≠k

This statistic is asymptotically Normal with zero mean under HE. To carry out the test, the variance of SnGKR must be estimated, either by the jackknife or using approximations given in Ben Ghorbal et al. (2009) for its finite-sample and asymptotic variance under HE. This test performs very well against a wide range of alternatives; see Genest and Nešlehová (2012a) and Kojadinovic and Yan (2010) for additional tests of extremeness. When there is no evidence against HE, one may seek an adequate parametric extreme-value copula family for the data. To validate a particular class A of Pickands dependence functions, one may then wish to test HA: A ∈ A using a GoF test. A natural way to accomplish this is to compare the best representative Aθn from the class A to a model-free, nonparametric estimator An. In the bivariate case, the simplest such rank-based estimator is defined, for all t ∈ [0, 1], by  1 n AnP (t ) = 1  ∑ ξˆ i (1 − t ) (30.20)   n i=1 where, for each i ∈ {1, …, n}, ξˆ i (t ) = min{− ln(Uˆ i1 ) t , − ln(Uˆ i 2 ) (1 − t )}

30.3.4 Model Validation

Before a copula model is used, for example, for prediction, it is important to check whether the selected copula family C is appropriate. This can be accomplished by testing the hypothesis H0: C ∈ C. Goodness-of-fit (GoF) tests for copula models are typically based on a comparison between the empirical copula Cˆn, which stands for the pseudo-sample, and the “best representative” Cθn from the parametric class C. Any convenient rank-based estimator qn of q can be used, provided that it is consistent. For moderate sample sizes, an informal graphical procedure consists of comparing the rank-plot to a random sample of equal size from Cθn . The simplest formal test (Genest et al., 2009c) is based on a measure of discrepancy between the best representative Cθn under H0 and the empirical copula Cˆn , viz. n

2 Sn = ∑{Cˆn (Uˆ i1 ,…,Uˆ id ) − Cθn (Uˆ i1 ,…,Uˆ id )} (30.17) i =1

This test is consistent for any copula family. An alternative test, which exhibits good power when dealing with Archimedean copulas, is based on a comparison between the empirical Kendall distribution and its parametric estimate under H0, viz. n

T = {Kˆ (W ∑ n ˆi ) − K θn (Wˆi )}2 (30.18) n i =1

30_Singh_ch30_p30.1-30.10.indd 4

Both Sn and Tn have unwieldy limiting distributions that depend on the unknown value of the dependence parameter q under H0. For this reason, one must resort to resampling techniques to compute the p-values associated to these test statistics. Two commonly used procedures are the parametric bootstrap (Genest and Rémillard, 2008) and the multiplier method (Kojadinovic et al., 2011). In addition to being a function of sample size, the performance of all GoF tests for copulas typically depends on the discrepancy between the true underlying copula and the hypothesized one. For extensive power studies and general recommendations, see, for example, Berg (2009) and Genest et al. (2009c). More elaborate GoF tests have been proposed based, for example, on the Rosenblatt transformation or on smooth nonparametric estimates of the copula density. Tests tailored for specific dependence structures such as Clayton or Gaussian copulas are also available; see, for example, Berg and Quessy (2009) for general moment-based GoF test statistics. Tests of the more general hypothesis that C belongs to some specific nonparametric class of copulas are also beginning to emerge; the case of extreme-value copulas considered below provides an example.

(30.21)

This estimator is a rank-based version of the original estimator of A due to Pickands (1981). In practice, however, it is generally preferable to use another rank-based estimator Aˆ nCFG , called the CFG estimator after Capéraà, Fougères, and Genest (1997); the latter is based on a geometric mean (with offset) rather than a harmonic mean as in Eq. (30.20). As shown in Genest and Segers (2009), both estimators are consistent and asymptotically Normal at every fixed t ∈ (0, 1). For multivariate extensions of the CFG estimator in the known-margin case, see Gudendorf and Segers (2011) and Zhang et al. (2008). In the absence of evidence against HE, one can proceed with testing the hypothesis HA that the unknown Pickands dependence function belongs to some parametric family A. This idea is explored in Genest et al. (2011a) in dimension d = 2, where tests are based on the statistics 1

1

GKNY −1 = ∫ { Aθn (t ) − Aˆ nCFG (t )}2 dt and TnGKNY−2 = ∫ { Aθn (t ) − Aˆ nP (t )}2 dt Tn 0 0 (30.22)

As the limiting null distributions of these test statistics depend on the unknown parameter value q, one must resort to a parametric bootstrap or the multiplier method to compute the associated p-values. From the simulation study in Genest et al. (2011a), it appears that the statistic TnGKNY−1 typically leads to a more powerful test than the statistic TnGKNY−2 . Moreover, the

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An illustration: The Fraser River at Hope     30-5 

procedure based on TnGKNY−1 outperforms the general-purpose blanket test based on the statistic Tn =

∫

[0,1]2

{Cˆn (u1 ,u2 ) − Cθn (u1 ,u2 )}2 dCˆn (u1 ,u2 ) (30.23)

as well as other similar procedures reviewed and studied in Genest et al. (2009c). 30.3.6  Selection Criteria (Adapted AIC and BIC for Copula)

Once GoF tests have been applied, one can rely on selection criteria to help choose the best copula family by ranking them. Criteria such as the AICcop and the BICcop can be used, viz. n R S AICcop = − 2∑ ln c  i , i ; θˆ   + 2kcop , BICcop = AICcop − {2 − ln(n)}kcop  n + 1 n +1    i =1 (30.24)

where c ( .,.; θ ) is the copula density function and kcop is the number of parameters of C. The selected copula is the one with the smallest criterion value. See, for example, De Michele et al. (2013) and Dias and Embrechts (2010) for illustrations in hydrology and finance, respectively. 30.4  MULTIVARIATE QUANTILE AND RETURN PERIOD

Suppose that a multivariate distribution H with representation (30.1) has been selected. This section describes how this model can then be used to estimate the quantiles and return periods that are commonly used, for instance, as criteria in the design of hydraulic structures and water supply systems. In univariate HFA, the return period or recurrence interval of an event of the form {X > x} or {X  x} is defined by TX = µ / {1  −  Pr( X   ≥ x )}, where m is the mean inter-arrival time between two consecutive occurrences of X. By analogy, the multivariate version is thus given by T=

1 1 = (30.25) p Pr(critical event)

Various definitions of the multivariate return period have been derived from this notion. In the bivariate case, for instance, the event P could take the forms Π1 = { X1 ≥ x1 ∩ X 2   ≥ x 2 },     Π2 = { X1 ≥ x1 ∪ X 2 ≥ x 2 } Π3 = { X1 ≥ x1 | X 2 ≥   x 2 },       Π4 = { X1 ≥ x1 | X 2   ≤   x 2 } Π5 = { X1 ≥ x1 | X 2   =   x 2 },       Π6 = {H ( X1 ,  X 2 )  ≥  t }



(30.26)

for some choices of x1, x2, and t. Events P1 to P5 are all related to the probabilities of exceedance. Events of the type P6 are called Kendall return periods. The choice between the multiple definitions depends on how the system (for example, a hydraulic device) responds to a specific forcing. This mechanism has a unique probabilistic description that results in a particular type of probability p (univariate, multivariate, conditional, etc.), which in turn corresponds to a unique choice of T according to the reciprocal transformation 1/T = p. The probability of each of the above events can be expressed in terms of  the  corresponding copula, for example, Pr(Π2 ) = 1– H ( x1 ,  x 2 ) = 1– C{F1 ( x1 ),  F2 ( x 2 )}. Definitions are also available for partial duration series.

30_Singh_ch30_p30.1-30.10.indd 5

Furthermore, relationships between univariate return periods and the joint return period can be derived. The reader is referred to Salvadori et al. (2007) for more details about these concepts. Comparing different multivariate return periods means comparing the probabilities describing different sets of events corresponding to different mechanisms of failure, only one of which describes the response of the system at hand (Serinaldi, 2015). Therefore, conclusions about supposed overestimation or underestimation are illogical and misleading because every univariate, multivariate, and conditional T and p correctly describes its own event set. However, differences exist among the above multivariate return periods, not only in terms of the selected design events, but also regarding their specific statistical meaning, as pointed out in Requena et al. (2013) and Volpi and Fiori (2014), respectively. The reader is referred to Requena et al. (2015) for a recent summary. Mathematical relations can be found, for example, T2 ≤ min(TX1, TX2) ≤ max(TX1, TX2) ≤ T1 ≤ T3 (30.27) In the statistical literature, a number of multivariate extensions of the notion of quantile are available. The version often used in hydrology involves the values of x1 and x2 such that H(x1, x2) = Pr(X1 ≤ x1, X2 ≤ x2) = p for a given p in (0,1). This version is presented in Belzunce et al. (2007) and adapted to the hydrologic context in Chebana and Ouarda (2011). Given that there are typically infinitely many combinations (x1, x2) for which H(x1,  x2)  =  p, the corresponding quantile is a curve. More precisely, for example, when considering the event { X1   ≤ x1 ,  X 2   ≤ x 2 }, the quantile curve can be expressed as follows:

{

}

QX ,Y ( p) = ( x , y ) ∈ 2 such that x = FX−1 (u), y = FY−1 (v ); u, v ∈[0,1]: C(u, v ) = p (30.28)

This definition is simple, intuitive, interpretable, and probability-based. More properties and applications of multivariate quantiles in hydrology can be found in Chebana and Ouarda (2011) and Salvadori et al. (2011b). 30.5  AN ILLUSTRATION: THE FRASER RIVER AT HOPE

Yearly measurements of peak flow (Q) and volume (V) for the Fraser River at Hope (British Columbia, Canada) are available from 1913 to 2011 without any missing values, so the sample size is n = 99. Both series are stationary, homogeneous, and exhibit no serial dependence. The following analysis focuses on the study of the pair (Q, V) for this station. A rank-plot and a K-plot of the pairs (Q, V) are given in Fig. 30.2. They clearly reveal the presence of positive dependence between the two variables. This is confirmed by the values of Spearman’s rho (rn = 0.60) and Kendall’s tau (tn = 0.42), which are both significantly different from 0, as easily verified with tests of independence based on these two statistics. The issue then arises as to how best to model this dependence. For illustration purposes, six copula families were considered: the Clayton, Frank, Galambos, Gumbel, Hüsler–Reiß, and Plackett families of copulas. All of them can achieve the observed values of r and t; in contrast, the FGM copula family cannot model such a strong level of dependence and could be discarded offhand. Figure 30.3 shows scatter plots of simulated samples of size N = 500 from each of these copulas once their parameters have been estimated by inversion of Kendall’s tau. It is clear from Fig. 30.3 that even for a fixed level of dependence (as measured here by Kendall’s tau), different copulas express different types of dependence between the variables: the Clayton copula exhibits lowertail dependence (accumulation of points in the lower left corner of the plot), the three extreme-value copulas exhibit upper-tail dependence, and the remaining two copulas have neither. A visual comparison of the plots gives only partial clues as to which of these models provides the best fit of the data (represented in the rank-plot). Yet the choice of copula is crucial for predictive purposes, as illustrated in Fig. 30.4, where the scatter plots of the synthetic pairs were converted to the original units, namely, a lognormal distribution with mean 9.04 and standard deviation 0.19 for peak flow (Q), and a GEV distribution with parameters x = −0.22, m = 41,916.23, and b = 8,046.58 for volume (V). As a first step in the selection of a copula structure, GoF tests can be applied. This is done in Table 30.1 using the statistic Sn, whose p-value was estimated using R = 1000 parametric bootstrap samples. The only model that can be rejected this way is the Clayton family (p-value < 5%); all the others appear to be consistent with the dataset at the 5% level. Models that are not rejected should not be ranked in order of merit according to their p-value, however. Adapted AIC and BIC criteria should be used instead to discriminate between them. Among the models listed in Table 30.1, the Frank copula

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30-6    Copula Modeling in Hydrologic Frequency Analysis

Rank plot between Q and V 1.0

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Q Figure 30.2  Rank-plot and K-plot for the pair (Q, V).

looks like the best choice, given that it has the lowest criteria values. This is not the end of the story, however, because selecting this specific copula family would be an implicit admission that there is no upper- or lower-tail dependence in the data. Given that joint extremes are of particular interest in HFA, one should always pay attention to tail dependence. Given that the null hypothesis of extremeness cannot be rejected [for example, p-value = 0.17 for the original test of Ghoudi et al. (1998)], a nonparametric estimate of the uppertail dependence coefficient based on the CFG estimator is given by λˆ UCFG = 0.4725 . This value should be compared to the parametric estimates given in Table 30.2 for each of the copulas exhibiting upper-tail dependence. From Table 30.2, it looks as though either one of the Galambos, Gumbel, and Hüsler–Reiß extreme-value copulas provides an adequate representation of the tail-dependence. In view of the AIC and BIC, there also appears to be

Rank plot

little to choose between them. In what follows, the Gumbel copula is used for convenience; the same choice was made, for example, in a recent study of Spanish floods (Requena et al., 2013). Contours associated to the fitted Gumbel copula corresponding to the events {U   ≥ u, V   ≥ v} and {U ≤ u, V ≤ v} with the same probability of occurrence are shown in Fig. 30.5. Figure 30.6 shows what happens to the left plot once the margins are factored in. As mentioned earlier, infinitely many pairs (Q, V) are associated to a given return period in the bivariate case. For comparison purposes, various return periods are given in Table 30.3. One can see that the inequalities displayed in Eq. (30.27) are satisfied here for each copula. This result suggests that ignoring the upper-tail dependence (which would have occurred if the Frank copula had been preferred to the Gumbel copula) in joint extreme event modeling can lead to underestimation of the risk and becomes problematic for high return periods.

Clayton copula

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Figure 30.3  Scatter plots of simulated series from different copulas with N = 500.

30_Singh_ch30_p30.1-30.10.indd 6

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Extensions    30-7  Clayton copula

Frank copula

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Figure 30.4  Scatter plots of simulated series from different copulas and margins with N = 500.

Table 30.1  Parameter Estimates Based on the Maximum Pseudo-Likelihood Approach, GoF Test Based on the Statistic Sn, and Criteria Results for the Fraser River Data at Hope Copula

Parameter

Sn

p-value

AIC

BIC

Clayton

0.907

0.0914

0.0005

−26.22965

−23.63453

Frank

4.494

0.0228

0.2602

−40.34380

−37.74868

Plackett

6.739

0.0255

0.2133

−39.42036

−36.82524

Galambos

0.915

0.0351

0.0804

−37.29895

−34.70383

Gumbel

1.632

0.0353

0.0794

−36.46249

−33.86737

Hüsler–Reiß

1.363

0.0364

0.0894

−37.50378

−34.90866

30.6  EXTENSIONS 30.6.1 Multivariate Regional HFA

Regional HFA aims to assess hydrologic risks in sites where hydrologic information is unavailable or insufficient to perform an analysis. Similar information from neighboring gaged sites can then be used to derive estimates at targeted, ungaged sites. Regional HFA consists of two main steps: (i) delineation of homogeneous regions and (ii) regional estimation. In the multivariate context, step (i) was treated in Chebana and Ouarda (2007) based on multivariate L-moments; step (ii) was the object of Chebana et al. (2009) by extending the index-flood model. Applications to floods may be found, for example, Table 30.2  Parametric Estimates of the Tail-Dependence Coefficient Copula

λUC ( θ ) −1 θ

λˆ UC

Galambos

2

Gumbel

2 − 21 θ

0.4706775

Hüsler–Reiß

2 – 2F(1/q)

0.4631934

0.4686745

F is the univariate standard Normal distribution

30_Singh_ch30_p30.1-30.10.indd 7

in Ben Aissia et al. (2015) and Chebana et al. (2009); for an application to droughts, see Sadri and Burn (2011). 30.6.2 Nonstationary Multivariate HFA

Proper assessment of risk factors requires that the statistical inference be valid during the projected life span of the structure. However, such assumptions are sometimes unrealistic. For instance, statistically significant trends have been identified in extreme values of different hydro-climatological series in different parts of the world. The reality of non-stationary hydro-meteorological extremes needs to be properly addressed as probability distributions with constant parameters may no longer be valid under non-stationary conditions. In the multivariate case, Ben Aissia et al. (2014) studied the evolution of the dependence between flood variables through time for a long simulated climate change series; non-stationarity was showed to be present, and that it could affect the marginal parameters as well as those of the copula. This topic was recently considered in a hydrologic context by Bender et al. (2014) and Ouarda and Chebana (2014). 30.6.3  Spatial Copulas

The motivation for using a copula to describe the dependence structure of spatial data was first discussed by Bárdossy (2006). The objective was to develop a more general approach for characterizing spatial patterns that traditional methods fail to capture. Building a copula that allows the formulation of the spatial structure according to the separating distance can be a difficult task and not all copulas are suited for spatial analysis. The Gaussian copula is one special case that insures continuity with existing methods. Alternatively, Bárdossy (2006) and Bárdossy and Li (2008) proposed the ν -transformed copulas that, in contrast to the Gaussian copula, account for radial asymmetry, a property observed with variables having different interrelation between the lower and upper quantiles of the marginal distribution. Recently, vine copulas have also been proposed as a tool to model various form of spatial dependence Gräler and Pebesma (2011). For predictive purposes, Bárdossy and Li (2008) showed that copulas provide a practical way of writing a predictive distribution at a new location. Adopting a copula framework for spatial modeling is a way of generalizing traditional kriging techniques such as indicator kriging, trans-Gaussian

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30-8    Copula Modeling in Hydrologic Frequency Analysis

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0.4

0.4

0.3

0.1

0.2

0.3

0.3 2 0.

0.4

0.5

0.4

0.1

0.5

0.2

0.6

0.6

0.1

0.1

0.5

0.6

0.1

0.7

0.7

0.7

U2

0.8

0.8

0.6

0.4

0.2

0.8

0.5

0.3

0.1

0.9

0.9

0.7

0.8

0.9

U1

Simulated pairs Observed data Contour

Simulated pairs Observed data Contours

Figure 30.5  Contours for the events {U ≤ u, V ≤ v} (left) and {U   ≥ u, V   ≥ v} (right) having the same probability based on a Gumbel copula fit of the Fraser River data at Hope.

80,000 70,579

0.999 (T=1000) 0.995

65,488

0.99 (T=100)

60,000

0.95 0.9 (T=10)

Volume

56,627 0.8 0.6

40,000

0.3

0.5

0.7 0.4

0.2

5000

10,000

15,177

20,000

13,143

10,794

0.1

15,000

20,000

Peak flow Figure 30.6  Representation, on the original scales, of the contours for the events {U ≤ u, V ≤ v} having the same probability based on a Gumbel copula fit of the Fraser River data at Hope.

Table 30.3  Joint Return Periods (TX∨,Y , TX∧,Y , and ρ∨t ) with Frank and Gumbel Copulas Corresponding to the Combinations of the Univariate Quantiles Return periods T = TX1 = TX2

qT (m3/s)

vT (m3)

Gumbel

ρt∨

TX∧,Y

TX∨,Y

Univariate quantiles

Frank

Gumbel

Frank

Gumbel

Frank

10

10,794

66,383

7

5

18

56

15

20

100

13,143

109,216

67

51

196

4,838

158

1,182

1000

15,177

181,030

670

501

1,970

476,381

1586

110,703

kriging, and normal-rank kriging Kazianka and Pilz (2010), as well as providing additional flexibility. In the context of physiographical kriging, Durocher et  al. (2016) showed that adopting the spatial copula framework to predict flood quantiles at ungaged location allows to reduce considerably the bias created by the failure of the normality assumption and leads to improved predictive power. 30.7  RESOURCES AND FURTHER SPECIFIC REFERENCES

Most of the procedures described here are available in the R Project for Statistical Computing, for example, through the packages copula, envir, and

30_Singh_ch30_p30.1-30.10.indd 8

qrmlib. For a historical overview of copula modeling in statistics and related fields, see Genest et al. (2009a). For further reading on copulas and multivariate extremes, including threshold models, see, for example, McNeil et al. (2005). For a current list of publications related to copulas and their applications in hydrology, see www.stahy.org. In terms of specific references, for a short overview about multivariate HFA, see Chebana (2012). More details about copula modeling and additional developments can be found in Genest and Nešlehová (2012b), and more specifically about extremes in Genest and Nešlehová (2012a). A number of recent papers dealt with extending the return period notion to the multivariate setting, such as Salvadori et al. (2011a,b), Gräler et al. (2013), Requena et al. (2013).

8/22/16 1:53 PM

REFERENCES    30-9 

Multivariate quantiles were adapted to the hydrologic context by Chebana and Ouarda (2011). All the above is related to at-site HFA. Regional estimation at ungaged sites was introduced in Chebana and Ouarda (2009); for recent applications, see, for example, Grimaldi and Serinaldi (2006), Sadri and Burn (2011), Requena et al. (2016). In the presence of non-stationarity in one or more of the multivariate distribution components (margins or copula), Bender et al. (2014) and Ouarda and Chebana (2014) proposed models to take into account this trend. ACKNOWLEDGMENTS

Funding in partial support of this work was provided by the Canada Research Chairs Program, the Natural Sciences and Engineering Research Council of Canada (RGPIN/39476-2011), the Canadian Statistical Sciences Institute, and the Fonds de recherche du Québec - Nature et technologies (2015-PR183236). REFERENCES

Bárdossy, A., “Copula-based geostatistical models for groundwater quality parameters,” Water Resources Research, 42 (11): 2006. Bárdossy, A. and J. Li, “Geostatistical interpolation using copulas,” Water Resources Research, 44 (7): 2008. Belzunce, F., A. Castaño, A. Olvera-Cervantes, and A. Suárez-Llorens, “Quantile curves and dependence structure for bivariate distributions,” Computational Statistics & Data Analysis, 51 (10): 5112–5129, 2007. Ben Aissia, M-A., F. Chebana, T. B. M. J. Ouarda, P. Bruneau, and M. Barbet, “Bivariate index-flood model: case study in Québec, Canada,” Hydrological Sciences Journal, 60 (2): 247–268, 2015. Ben Aissia, M-A., F. Chebana, T. B. M. J. Ouarda, L. Roy, P. Bruneau, and M. Barbet, “Dependence evolution of hydrological characteristics, applied to floods in a climate change context in Québec,” Journal of Hydrology, 519 (A): 148–163, 2014. Bender, J., T. Wahl, and J. Jensen, “Multivariate design in the presence of non-stationarity,” Journal of Hydrology, 514: 123–130, 2014. Ben Ghorbal, N., C. Genest, and J. Nešlehová, “On the Ghoudi, Khoudraji, and Rivest test for extreme‐value dependence,” The Canadian Journal of Statistics, 37 (4): 534–552, 2009. Berg, D., “Copula goodness-of-fit testing: an overview and power comparison,” The European Journal of Finance, 15 (7–8): 675–701, 2009. Berg, D. and J-F. Quessy, “Local power analyses of goodness-of-fit tests for copulas,” Scandinavian Journal of Statistics, 36 (3): 389–412, 2009. Brahimi, B., F. Chebana, and A. Necir, “Copula representation of bivariate L-moments: a new estimation method for multiparameter two-dimensional copula models,” Statistics, 49 (3): 497–521, 2015. Capéraà, P., A-L. Fougères, and C. Genest, “A nonparametric estimation procedure for bivariate extreme value copulas,” Biometrika, 84 (3): 567–577, 1997. Chebana, F., “Multivariate analysis of hydrological variables,” Encyclopedia of Environmetrics, 2nd ed., Wiley, Chichester, 2012. Chebana, F. and T. B .M. J. Ouarda, “Multivariate L-moment homogeneity test,” Water Resources Research, 43 (8): 2007. Chebana, F. and T. B. M. J. Ouarda, “Index flood-based multivariate regional frequency analysis,” Water Resources Research, 45 (10): 2009. Chebana, F. and T. B. M. J. Ouarda, “Multivariate quantiles in hydrological frequency analysis,” Environmetrics, 22 (1): 63–78, 2011. Chebana, F., T. B. M. J. Ouarda, P. Bruneau, M. Barbet, S. El Adlouni, and M. Latraverse, “Multivariate homogeneity testing in a northern case study in the province of Québec, Canada,” Hydrological Processes, 23 (12): 1690–1700, 2009. Chebana, F., T. B. M. J. Ouarda, and T. C. Duong, “Testing for multivariate trends in hydrologic frequency analysis,” Journal of Hydrology, 486: 519–530, 2013. De Michele, C., G. Salvadori, R. Vezzoli, and S. Pecora, "Multivariate assessment of droughts: frequency analysis and dynamic return period," Water Resources Research, 49 (10): 6985–6994, 2013. Dias, A. and P. Embrechts, “Modeling exchange rate dependence dynamics at different time horizons,” Journal of International Money and Finance, 29 (8): 1687–1705, 2010. Durocher, M., T. B. M. J. Ouarda, and F. Chebana, “On the prediction of extreme flood quantiles at ungauged locations with spatial copula,” Journal of Hydrology, 533:523–532, 2016. El Adlouni, S. and T. B. M. J. Ouarda, “Study of the joint law flow-level by copulas: case of the Châteauguay river,” Canadian Journal of Civil Engineering, 35 (10): 1128–1137, 2008. Frahm, G., “On the extremal dependence coefficient of multivariate distributions,” Statistics & Probability Letters, 76 (14): 1470–1481, 2006.

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Genest, C. and J-C. Boies, “Detecting dependence with Kendall plots,” The American Statistician, 57 (4): 275–284, 2003. Genest, C. and A-C. Favre, “Everything you always wanted to know about copula modeling but were afraid to ask,” Journal of Hydrologic Engineering, 12 (4): 347–368, 2007. Genest, C., A-C. Favre, J. Béliveau, and C. Jacques, “Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data,” Water Resources Research, 43 (9): 2007. Genest, C., M. Gendron, and M. Bourdeau-Brien, “The advent of copulas in finance,” The European Journal of Finance, 15 (7–8): 609–618, 2009a. Genest, C., K. Ghoudi, and L-P. Rivest, “A semiparametric estimation procedure of dependence parameters in multivariate families of distributions,” Biometrika, 82 (3): 543–552, 1995. Genest, C., I. Kojadinovic, J. Nešlehová, and J. Yan, “A goodness-of-fit test for bivariate extreme-value copulas,” Bernoulli, 17 (1): 253–275, 2011a. Genest, C. and R. J. MacKay, “Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données,” The Canadian Journal of Statistics, 14 (2): 145–159, 1986. Genest, C., E. Masiello, and K. Tribouley, “Estimating copula densities through wavelets,” Insurance: Mathematics & Economics, 44 (2): 170–181, 2009b. Genest, C. and J. Nešlehová, “A primer on copulas for count data,” ASTIN Bulletin, 37 (2): 475–515, 2007. Genest, C. and J. Nešlehová, “Copula modeling for extremes,” Encyclopedia of Environmetrics, 2nd ed., Wiley, Chichester, 2012a, pp. 530–541. Genest, C. and J. Nešlehová, “Copulas and copula models,” Encyclopedia of Environmetrics, 2nd ed., Wiley, Chichester, 2012b, pp. 541–553. Genest, C., J. Nešlehová, and N. Ben Ghorbal, “Estimators based on Kendall’s tau in multivariate copula models,” Australian and New Zealand Journal of Statistics, 53 (2): 157–177, 2011b. Genest, C. and B. Rémillard, “Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models,” Annales de l’Institut Henri-Poincaré - Probabilités et Statistiques, 44 (6): 1096–1127, 2008. Genest, C., B. Rémillard, and D. Beaudoin, “Goodness-of-fit tests for copulas: a review and a power study,” Insurance: Mathematics & Economics, 44 (2): 199–213, 2009c. Genest, C. and L-P. Rivest, “A characterization of Gumbel’s family of extreme value distributions,” Statistics & Probability Letters, 8 (3): 207–211, 1989. Genest, C. and J. Segers, “Rank-based inference for bivariate extremevalue copulas,” The Annals of Statistics, 37 (5B): 2990–3022, 2009. Ghoudi, K., A. Khoudraji, and L-P. Rivest, “Propriétés statistiques des copules de valeurs extrêmes,” The Canadian Journal of Statistics, 26 (1): 187–197, 1998. Gräler, B. and E. Pebesma, “The pair-copula construction for spatial data: a new approach to model spatial dependency,” Procedia Environmental Sciences, 7: 206–211, 2011. Gräler, B., M. J. van den Berg, S. Vandenberghe, A. Petroselli, S. Grimaldi, B. De Baets, and N. E. C. Verhoest, “Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation,” Hydrology and Earth System Sciences, 17 (4): 1281–1296, 2013. Grimaldi, S. and F. Serinaldi, “Design hyetograph analysis with 3-copula function,” Hydrological Sciences Journal, 51 (2): 223–238, 2006. Gudendorf, G. and J. Segers, “Nonparametric estimation of an extremevalue copula in arbitrary dimensions,” Journal of Multivariate Analysis, 102 (1): 37–47, 2011. Joe, H., “Multivariate dependence measures and data analysis,” Computational Statistics & Data Analysis, 16 (3): 279–297, 1993. Joe, H., “Asymptotic efficiency of the two-stage estimation method for copula-based models,” Journal of Multivariate Analysis, 94 (2): 401–419, 2005. Joe, H., Dependence Modeling With Copulas, CRC Press, Boca Raton, FL, 2015. Kao, S-C. and R. S. Govindaraju, “A bivariate frequency analysis of extreme rainfall with implications for design,” Journal of Geophysical Research: Atmospheres, 112 (D13), 2007. Kazianka, H. and J. Pilz, “Copula-based geostatistical modeling of continuous and discrete data including covariates,” Stochastic Environmental Research and Risk Assessment, 24 (5): 661–673, 2010. Kim, G., M. J. Silvapulle, and P. Silvapulle, “Comparison of semiparametric and parametric methods for estimating copulas,” Computational Statistics & Data Analysis, 51 (6): 2836–2850, 2007. Kojadinovic, I. and J. Yan, “Nonparametric rank-based tests of bivariate extreme-value dependence,” Journal of Multivariate Analysis, 101 (9): 2234– 2249, 2010.

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30-10    Copula Modeling in Hydrologic Frequency Analysis

Kojadinovic, I., J. Yan, and M. Holmes, “Fast large-sample goodness-of-fit tests for copulas,” Statistica Sinica, 21 (2): 841–871, 2011. Kotz, S. and S. Nadarajah, Extreme Value Distributions: Theory and Applications, Imperial College Press, London, 2000. Kurowicka, D. and H. Joe, Dependence Modeling: Handbook on Vine Copulae, World Scientific, Singapore, 2011. McNeil, A. J., R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, Princeton, New Jersey, 2005. McNeil, A. J. and J. Nešlehová, “From Archimedean to Liouville copulas,” Journal of Multivariate Analysis, 101 (8): 1772–1790, 2010. Nelsen, R. B., An Introduction to Copulas, Springer, New York, 2006. Ouarda, T. B. M. J. and F. Chebana, “Multivariate non-stationarity hydrological frequency analysis,” Submitted, 2014. Pickands III, J., “Multivariate extreme value distributions,” Proceedings of the 43rd Session of the International Statistical Institute, Vol. 49, 1981, pp. 859–878, 894–902. Rao, A. R. and K. H. Hamed, Flood Frequency Analysis, CRC Press, Boca Raton, FL, 2000. Requena, A. I., F. Chebana, and L. Mediero, “A complete procedure for multivariate index-flood model application,” Journal of Hydrology, 535: 559– 580, 2016. Requena, A. I., L. Mediero, and L. Garrote, “A bivariate return period based on copulas for hydrologic dam design: accounting for reservoir routing in risk estimation,” Hydrology and Earth System Sciences, 17 (8): 3023–3038, 2013. Sadri, S. and D. H. Burn, “A fuzzy C-means approach for regionalization using a bivariate homogeneity and discordancy approach,” Journal of Hydrology, 401 (3–4): 231–239, 2011. Salvadori, G. and C. De Michele, “Multivariate multiparameter extreme value models and return periods: a copula approach,” Water Resources Research, 46 (10): 2010.

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Salvadori, G. and C. De Michele, “Estimating strategies for multiparameter multivariate extreme value copulas,” Hydrology and Earth System Sciences, 15 (1): 141–150, 2011. Salvadori, G., C. De Michele, and F. Durante, “Multivariate design via copulas,” Hydrology and Earth System Sciences Discussions, 8 (3): 5523–5558, 2011a. Salvadori, G., C. De Michele, and F. Durante, “On the return period and design in a multivariate framework,” Hydrology and Earth System Sciences, 15 (11): 3293–3305, 2011b. Salvadori, G., C. De Michele, N. T. Kottegoda, and R. Rosso, Extremes in Nature: An Approach Using Copulas, Springer, Dordrecht, 2007. Schmid, F. and R. Schmidt, “Multivariate extensions of Spearman’s rho and related statistics,” Statistics & Probability Letters, 77 (4): 407–416, 2007. Schmidt, R. and U. Stadtmüller, “Non-parametric estimation of tail dependence,” Scandinavian Journal of Statistics, 33 (2): 307–335, 2006. Segers, J., “Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions,” Bernoulli, 18 (3): 764–782, 2012. Serinaldi, F., “Dismissing return periods!” Stochastic Environmental Research and Risk Assessment, 29 (4): 1179–1189, 2015. Sklar, A., “Fonctions de répartition à n dimensions et leurs marges,” Publications de l’Institut de statistique de l’Université de Paris, 8: 229–231, 1959. Tawn, J. A., “Bivariate extreme value theory: models and estimation,” Biometrika, 75 (3): 397–415, 1988. Vandenberghe, S., N. E. C. Verhoest, C. Onof, and B. De Baets, “A comparative copula-based bivariate frequency analysis of observed and simulated storm events: a case study on Bartlett-Lewis modeled rainfall,” Water Resources Research, 47 (7): 2011. Volpi, E. and A. Fiori, “Hydraulic structures subject to bivariate hydrological loads: return period, design, and risk assessment,” Water Resources Research, 50 (2): 885–897, 2014. Zhang, D., M. T. Wells, and L. Peng, “Nonparametric estimation of the dependence function for a multivariate extreme value distribution,” Journal of Multivariate Analysis, 99 (4): 577–588, 2008.

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Chapter

31

Entropy Theory BY

VIJAY P. SINGH

ABSTRACT

31.2.1  Real Domain

Since the development of mathematical foundations of informational entropy by Shannon in 1948, there has been a proliferation of concepts related to entropy and their application in a multitude of areas, including hydrology. Now there is a well-developed entropy theory. The theory is so versatile that there is hardly any area where its application is not found. Over the past half a century, applications of entropy have been multitudinous. The objective of this chapter is to describe the entropy theory and show how it is applied in hydrology.

There are different types of entropy (Kapur, 1989; Singh, 2013), such as Shannon, Tsallis, exponential, epsilon entropy, algorithmic, Hartley, Renyi, and Kapur. Here, we define only two types that are more commonly used: Shannon and Tsallis, and the first will be detailed in the remainder of the chapter. Consider a discrete random variable X that takes on values x1, x2,…, xN occurring with probabilities p1, p2,…, pN, respectively. The Shannon (1948) entropy is defined as the weighted average information, and is given as :

31.1 ORIGIN



The concept of entropy originated in statistical physics in the nineteenth century and is related to the second law of thermodynamics. Classical thermodynamics is concerned with only macroscopic states of matter characterized by observable properties, such as temperature, pressure, volume, etc. In 1850, Rudolf Clausius (Papalexiou and Koutsoyiannis, 2012) defined entropy in terms of these macroscopic quantities and this is what is known as thermodynamic entropy (Templeman, 1989). A macroscopic state may be made up of various combinations of microscopic states. In order to infer on the nature of macroscopic state of a thermodynamic system, analyzing microscopic states of such a system, in 1872 Ludwig Boltzmann (Tamburo, 1992) formulated a new definition of entropy as a measure of disorder in the system, thus constituting a basis for statistical thermodynamics. J. Willard Gibbs (cited in Papalexiou and Koutsoyiannis, 2012) further advanced the statistical concept of entropy. Adopting this definition, Shannon (1948) developed the mathematical foundation of what is now known as informational entropy theory. Koutsoyiannis (2013; 2014) has given a historical account of the entropy concept. 31.2 DEFINITION

Entropy has been interpreted in many different ways. Often, it is considered as a measure of uncertainty, information, or surprise. When an event of very low probability occurs, we are surprised because we did not anticipate its occurrence. The occurrence of this event is more uncertain than an event of higher probability. Clearly, more information will be needed to predict lowprobability events than high-probability events. Information reduces uncertainty. If p(x) is the probability of occurrence of an event x, there are different ways by which the uncertainty or information associated with the event can be expressed. Likewise, if there are, say, n events, then entropy provides a measure of uncertainty or information associated with these events. Often, a random variable is defined in real domain, that is, in space or time, but for certain problems, such as spectral analysis, a random variable may be characterized by frequency. Then, entropy is defined a little differently. In the real domain, it has been defined in different ways and hence the information measure has been expressed differently.

N

H = − k ∑ pi log pi (31.1) i =1

where H is Shannon entropy, and k is a parameter whose value depends on the base of the logarithm used. Depending on the base of the logarithm, the unit of measurement of entropy changes. For example, the unit of entropy is bit for base 2, Napier or nat for base e, and decibel or dogit for base 10. In general, k can be taken as unity. The information gained from the occurrence of any event x with probability p(x) is –log p(x). Equation (31.1) satisfies a number of desiderata, such as continuity, symmetry, additivity, expansibility, recursivity, and others (Shannon and Weaver, 1949). If X were a deterministic variable, the probability that it will take on a certain value is one, and the probabilities of all other alternative values will be zero. Then, Eq. (31.1) shows that H(x) = 0 which can be viewed as the lower limit of the values the entropy function may assume. This corresponds to the absolute certainty; that is, there is no uncertainty and the system is completely ordered. On the other hand, when all xi’s are equally likely; that is, the variable is uniformly distributed (pi = 1/N, i = 1, 2,…, N), then Eq. (31.1) yields H ( X ) = H max ( X ) = log N (31.2)



This shows that the entropy function attains a maximum, and Eq. (31.2) thus defines the upper limit. If X is a continuous variable with probability density function (PDF), f(x), then the Shannon entropy can be written as:

∞

∞

0

0

H ( X ) = − ∫ f ( x )log f ( x )d x = − ∫ log[ f ( x )]d F ( x ) = E[− log f ( x )] (31.3)

Here F(x) is the cumulative (probability) distribution function (CDF) of X, E[·] is the expectation of [·], and H(X) is the entropy of variable X. The Tsallis (1988) entropy can be expressed as: N



Hm = k

1 − ∑ pi m i =1

m −1

=

k N ∑[ pi − pim ] (31.4) m − 1 i=1

31-1

31_Singh_ch31_p31.1-31.8.indd 1

8/22/16 1:54 PM

31-2    Entropy Theory

where Hm is the Tsallis entropy and k is often taken as unity. For m → 1, the Tsallis entropy reduces to the Shannon entropy. Exponent m can be positive or negative, characterizes the degree of nonlinearity, and is often referred to as nonextensivity index or Tsallis entropy index. Tsallis (2002) noted that superextensivity, extensivity, and subextensivity occur when m < 1, m = 1, or m > 1, respectively. For m ≥ 0, m < 1 corresponds to the rare events and m>1 corresponds to frequent events (Tsallis et al., 1998). The information gain from the occurrence of any event i can be expressed as: ∆Ii =



N

1 (1 − pi m−1 ), m −1

∑ pi = 1 (31.5) i =1

where ΔIi is the gain in information from an event i which occurs with probability pi. For continuous variable X, the Tsallis entropy can be expressed as: Hm ( X ) = Hm ( f ) =

∞

∞

1 1 {1 − ∫ [ f ( x )]m }d x (31.6) { f ( x ) − [ f ( x )]m }d x = m − 1 ∫0 m −1 0

If one guesses a PDF of X, referred to as a prior distribution {qi} and finds its a posterior distribution denoted as {pi}, then one defines relative entropy or cross-entropy (Kullback and Leibler, 1951) as: p D = ∑ pi log  i  (31.7) q  N



i =1

31.2.2  Frequency or Spectral Power Domain

Entropy is also defined in the frequency or spectral power domain for doing spectral analysis wherein the random variable is frequency or spectral power. Let frequency f be considered a random variable, and the normalized spectral density P(f) be considered its PDF. Burg (1975) defined entropy as: H( f ) =



N

M

i =1

j =1

∑ pi = 1, pi ≥ 0; ∑ q j = 1, q j ≥ 0 (31.12)

Note that each value of a random variable represents an event. The joint probability of xi and yj can be denoted as p(xi, yj) = pij; that is, p( xi , y j ) is the joint probability of X = xi and Y = yj; N is the number of values X takes on; and M is the number of values Y takes on. Then, the joint entropy of X and Y can be defined as:

N M

H ( X ,Y ) = − ∑ ∑ p( xi , y j )log p( xi , y j ) (31.13) i =1 j =1

i

From now onward only the Shannon entropy is discussed in this chapter.



Clearly H(X) depends on the probability distribution of X and the same for Y. H(X, Y) in Eq. (31.11) indicates the total amount of uncertainty that X and Y entail or the total amount of information they convey. Consider, for example, the flood process characterized by flood peak (X) and flood volume (Y) if they are independent. A practical implication is that more observations on X and Y would reduce the uncertainty more than more observations of only X or Y. Let X take on values x1, x2, x3,…, xN with corresponding probabilities P = {p1, p2, p3,…, pN}, and let Y take on values y1, y2, y3,…, yM with corresponding probabilities Q={q1, q2, q3,…, qM}, such that

W

∫ log[P( f )]d f (31.8)

Then for the case of the flood process, the uncertainty would be represented by Eq. (31.13). Equation (31.13) can be generalized to any number of variables. If p(xi, yj), i = 1, 2, 3,…, N; j = 1, 2,…, M, are the joint probabilities; and p(xi| yj) and p(yj | xi) are conditional probabilities, then the conditional entropies H ( X Y ) and H (Y X ) can be expressed mathematically as:

N M

H ( X Y ) = − ∑ ∑ p( xi , y j )log p( xi y j ) (31.14a) i =1 j =1

−W

where P( f ) is the power spectrum as a function of frequency related to the autocovariance function, and W is the Nyquist frequency or band width. Equation (31.8) is often known as the Burg entropy. The entropy of the power spectrum density can also be defined in the Shannon sense as:

H( f ) = −

W

∫ P( f )ln[P( f )]d f (31.9)

−W

This definition is called the configurational entropy, which is an extension of the Burg entropy (Frieden, 1972; Gull and Daniell, 1978). The relative entropy of the spectral density can also be defined as:

H ( P , q) =

W

 P( f ) 

∫ P( f )ln  q( f )  d f

(31.10)

−W

where q( f ) is the prior spectral density function and P( f ) is the posterior spectral density function. These three definitions are primarily used for forecasting. Configurational entropy can be considered as a special case of minimum relative entropy (MRE), when prior information is given as a uniform distribution.



i, j

Equation (31.14a) or (31.14b) can be easily generalized to any number of variables. Now consider the conditional entropy for two variables denoted as H ( X Y ). The conditional entropy H(X|Y) is a measure of the amount of uncertainty remaining in X even with the knowledge of Y; the same amount of information can be gained by observing X. The amount of reduction in uncertainty in X equals the amount of information gained by observing Y. When X and Y are dependent, as may frequently be the case, say, for example, flood peak and flood volume, then their joint (or total) entropy equals the sum of marginal entropy of the first variable (say X) and the entropy of the second variable (say Y) conditioned on the first variable (X) (i.e., the uncertainty remaining in Y when a certain amount of information it conveys is already present in X). H(X, Y) = H(X) + H(Y|X) (31.15) or H(X, Y) = H(Y) + H(X|Y) (31.16) The conditional entropy can be expressed as

31.3  FORMS OF ENTROPY

There are different types of entropy or measures of information: marginal entropy, approximation entropy, sample entropy, conditional entropy, joint entropy, transinformation, and interaction information. The marginal entropy is the entropy of a single variable and is defined by Eq. (31.2) if the variable is continuous, or Eq. (31.1) if the variable is discrete. Other types of entropies are defined when more than one variable is considered. For two random variables X and Y that are not necessarily independent, the joint entropy H ( X ,Y ) is the total information content contained in both X and Y; that is, it is the sum of marginal entropy of one of the variables and the uncertainty that remains in the other variable when a certain amount of information that it can convey is already present in the first variable. This leads to the definition of the joint entropy of two independent variables as:

31_Singh_ch31_p31.1-31.8.indd 2

H(X, Y) = H(X) + H(Y) (31.11)

H (Y X ) = − ∑ p( xi , y j )log p( y j xi ) (31.14b)



N M

H ( X Y ) = − ∑ ∑ p( xi , y j )log p( xi y j ) = H ( X ,Y ) − H (Y )

(31.17)

i =1 j =1



N M

H (Y X ) = − ∑ ∑ p( xi , y j )log p( y j xi ) = H ( X ,Y ) − H ( X ) (31.18) i =1 j =1

Transinformation represents the amount of information common to both X and Y or repeated in both or shared between X and Y, and is denoted as T(X, Y). Transinformation is also referred to as mutual information and is a measure of the dependence between X and Y and is always non-negative. It is a measure of the amount of information random variable X contains about random variable Y; that is, if something is known about X, transinformation indicates the amount of information known about Y and vice versa.

8/22/16 1:54 PM

Total Correlation    31-3 

It equals the difference between the sum of two marginal entropies and the total entropy:

Now the Shannon entropy can be written as:

T(X, Y) = H(X) + H(Y) — H(X, Y) (31.19)



−∞

When X and Y are independent, T(X, Y) = 0. The information transmitted from variable X to variable Y is represented by the mutual information T(X, Y) and is given (Lathi, 1969) as: T ( X ,Y ) = ∑ ∑ p( xi , y j )log



i

j

p( xi y j ) p( xi )

Let w = (ay + b)1/c . Then, Eq. (31.26) can be simplified as:

T ( X ,Y ) = ∑ ∑ p( xi , y j )log i

j

p( xi , y j ) p( xi ) p( y j )

(31.21)

Although transinformation indicates the dependence of two variables, its upper bound varies from site to site (it varies from 0 to marginal entropy H). Therefore, Yang and Burn (1984) normalized T by dividing by the joint entropy as: H T ( H − H Lost ) = DIT = = 1 − Lost (31.22) H H H



The ratio of T by H is called directional information transfer (DIT) index. Mogheir and Singh (2002) called it as Information Transfer Index (ITI). The physical meaning of DIT is the fraction of information transferred from one site to another. DIT varies from zero to unity when T varies from zero to H. The zero value of DIT corresponds to the case where sites are independent and therefore no information is transmitted. A value of unity for DIT corresponds to the case where sites are fully dependent and no information is lost. Any other value of DIT between zero and one corresponds to a case between fully dependent and fully independent. DIT is not symmetrical, since DITXY = T/H(X) will not, in general, be equal to DITYX = T/H(Y). DITXY describes the fractional information inferred by station X about station Y, whereas DITYX describes the fractional information inferred by station Y about station X. 31.5  ENTROPY UNDER TRANSFORMATION OF VARIABLES

Let there be two random variables X and Y with PDF f X ( x ) and fY ( y ) and CDF FX ( x ) = P( X ≤ x ) and FY ( y ) = P(Y ≤ y ). Let Y = ( X − b)/ a or X = aY + b. Then, one can write FY ( y ) = P( X ≤ ay + b ) = FX (ay + b ) and f Y ( y ) = af X (ay + b) (31.23) Following Saeb (2012), the Shannon entropy can be written as: H (Y ) = − ∫ fY ( y )log[ fY ( y )]d y

 a H (Y ) = − log   + (c − 1)E[log(− X )] + H ( X ) (31.28)  c

31.6  INFORMATIONAL CORRELATION COEFFICIENT

The coefficient of determination r2 is a measure of the amount of variance that can be explained by the regression relationship between random variables X and Y, and 1–r2 represents the amount of unexplained variance. Here r is the classical Pearson correlation coefficient under the assumption that X and Y are normally distributed and linearly correlated, and it is often used as a measure of transferred information by regression. In a similar manner, the informational correlation coefficient R measures the mutual dependence between these two random variables and is a measure of transferable information but does not assume any type of distributional relationship between them. It is a dimensionless quantity and is expressed in terms of transinformation as:

R = 1 − exp(−2T0 ) (31.29)

where T0 is the transinformation or mutual information representing the upper limit of transferable information between two variables X and Y. When X and Y are normally distributed and linearly correlated, R reduces to the classical Pearson correlation coefficient. The coefficient of nontransferred information t1 measures the percentage of information left in Y after transfer to X, and can be expressed as: t1 =



T0 − T1 (31.30) T0

where T1 is the transinformation computed for the relationship between X and Y. For example, if the relationship is described by the bivariate normal distribution, T1 is given as: 1 T1 = − ln(1 − r 2 ) (31.31) 2

Otherwise, T1 is the transinformation between X and Y and T0 will be the marginal entropy of X. Quantity t1 basically describes the relative portion, T0  – T1, of the untransferred information. Likewise, 1 – t1 expresses the percentage of transferred information. Both R and t1 may be used to test the validity of the assumed relationship between two random variables X and Y.

−∞ ∞

= − ∫ af X (ay + b)log[af X (ay + b)]d y −∞ ∞

= − ∫ af X (ay + b)[log(a) + log f X (ay + b)]dy −∞

31.7  TOTAL CORRELATION

∞

∫





∞

= − log a −

∞

a H (Y ) = − ∫ f X (w )[log x 1−c f X (w )]d w = log  a  + (c − 1)E[log X ] + H ( X )  c c 0 (31.27)

Similarly, if X is negative, that is, X ≤ 0 , and Y = [(− X )c − b]/ a, then following the same method as previously, the entropy of Y can be expressed as:

31.4  DIRECTIONAL INFORMATION TRANSFER INDEX



∞ a a = − ∫ (ay + b)(1−c )/c f X [(ay + b )1/c ]log[ (ay + b )1/c f X ((ay + b)1/c )]d y (31.26) c c − b /a

(31.20)

or

∞

H (Y ) = − ∫ fY ( y )log[ fY ( y )]d y

f X (w )log f X (w )dw

−∞

= − log a + H ( X ) or H ( X ) = log a + H (Y ) (31.24)             

The total correlation can be considered as a measure of general dependence, including both linear and nonlinear relationship, among multiple variables. The total correlation for N variables ( X1 , X 2 ,, X N ) can be defined as (McGill, 1954; Watanabe, 1960) as:

Likewise, let Y = ( X c − b)/ a , where X is a positive random variable, that is, x ≥ 0. Then, X = (aY + b)1/c , y > −b / a. In a similar fashion as mentioned earlier



a FY ( y ) = P[ X ≤ (ay + b)1/c ] = FX [(ay + b)1/c ] , f Y ( y ) = (ay + b)(1−c )/c f X [(ay + b )1/c ] c

Equation (31.32) shows that the total correlation is always positive, because the sum of marginal entropies of the N variables will be greater than their joint entropy. It is symmetric with respect to its arguments. If these N

(31.25)

31_Singh_ch31_p31.1-31.8.indd 3

N

C( X1 , X 2 ,..., X N ) = ∑ H ( Xi ) − H ( X1 , X 2 ,..., X N )

(31.32)

i =1

8/22/16 1:54 PM

31-4    Entropy Theory

variables as well as their combinations are independent, C will be zero. If N = 2, Eq. (32.32) will reduce to the usual transinformation T. The total correlation can be computed without resorting to the computation of multivariate entropy by recalling the grouping property of total correlation and accordingly a systematic grouping of bivariate entropies (Kraskov et al., 2003). 31.8  THEORY OF ENTROPY

The entropy theory can be composed of three main components: (1) defining entropy, (2) entropy optimizing, and (3) concentration theorem for testing the entropy-based probability distribution. The first component is already discussed and remaining two are now briefly discussed.

distribution shape is gathered from empirical data, (4) constraints should be such that they are more or less preserved in the future, and (5) specify constraints in terms of the laws of mathematical physics-mass conservation, momentum conservation, and energy conservation-or constitutive laws, if possible. It is convenient to express the information in terms of moments, such as mean, variance, covariance, cross-covariance, or linear combinations of these statistics. The constraints encode relevant information. If observations are available, then, for simplicity of analysis, let the information on the random variable, Xε (a, b), be expressed in terms of constraints Cr, r = 0, 1, 2,…, n, defined as: b

C0 = ∫ f ( x )d x = 1 (31.33)



a

31.8.1  Entropy Maximizing

Jaynes (1957) formulated the principle of maximum entropy (POME) that states that the minimally prejudiced assignment of probabilities is that which maximizes entropy subject to the given information; that is, POME takes into account all of the given information and at the same time avoids consideration of any information that is not given. POME reasons that for given information the best possible distribution that fits the data would be the one with the maximum entropy, since this contains the most reliable assignment of probabilities. The information usually included in POME is specified as some statistics, including, for example, mean, variance, covariance, crossvariance, etc., or linear combinations of these statistics. Since the POMEbased distribution is favored over those with less entropy among those which satisfy the given constraints, according to the Shannon entropy as an information measure, entropy defines a kind of measure on the space of probability distributions. Intuitively, distributions of higher entropy represent more disorder, are smoother, more probable, less predictable, or assume less. The POME-based distribution is maximally noncommittal with regard to missing information and does not require invocation of ergodic hypotheses. Constraints encode relevant information. POME leads to the distribution that is most conservative and hence most uninformative. If a distribution with lower entropy were chosen, it would mean that we would be assuming information that was not available, while a distribution with higher entropy would violate the known constraints. The maximum entropy leads to a probability distribution of particular macrostate occurring among all possible arrangements (or microstates) of the events under consideration. The Shannon entropy is maximum when the probability distribution of the random variable is that one which is as close to the a priori distribution as possible. This is referred to as the principle of minimum cross entropy (POMCE) which minimizes the Bayesian entropy (Kullback and Leibler, 1951). This is equivalent to maximizing the Shannon entropy. 31.8.2  Concentration Theorem

The concentration theorem, formulated by Jaynes (1958; 1968), has two aspects. First, it shows the POME-based probability distribution best represents our knowledge about the state of the system by showing the spread of lower entropies around the maximum entropy value. Second, POME is the preferred method to obtain this distribution. The basis for these two aspects is contained in the Shannon inequality and the relation between entropy and the Chi-square test. 31.9  METHODOLOGY FOR APPLICATION

The methodology for application of the entropy theory involves the following steps: (1) specification of constraints, (2) maximization of entropy which can be accomplished using the method of Lagrange multipliers, (3) determination of the entropy-based PDF, (4) determination of the Lagrange multipliers in terms of constraints, (5) determination of entropy in terms of constraints, and (6) derivation of the desired relation. These parts are now briefly discussed for a continuous variable case having a PDF as f(x) and CDF as F(x). 31.9.1  Choice and Specification of Constraints

Entropy maximizing shows that there is a unique correspondence between a probability distribution and the constraints that lead to it. Therefore, choosing appropriate constraints is fundamental to the entropy formalism. The rationale for choosing different types of constraints in hydrology is discussed by Papalexiou and Koutsoyiannis (2012). Constraints encode the information or summarize the knowledge that can be garnered from empirical observations or theoretical considerations. There can be a large number of constraints that can perhaps summarize the information on the random variable. Therefore, the following considerations, when choosing appropriate constraints, are (1) constraints should be simple, (2) constraints should be as few as absolutely needed, (3) an idea about the

31_Singh_ch31_p31.1-31.8.indd 4

b

Cr = ∫ g r ( x ) f ( x )d x = g r ( x ), r = 1,2,..., n (31.34)



a

where gr(x), r =1, 2,…, n, represent some functions of x, n denotes the number of constraints, and g r ( x ) is the expectation of gr(x), with g0(x) = 1. Equation (31.34) states that the PDF must satisfy the total probability. Other constraints, defined by Eq. (31.34), have physical meaning. For example, if r = 1 and g1(x) = x, Eq. (31.34) would correspond to the mean x ; likewise, for r = 2 2 and g 2 ( x ) = ( x − x ) , it would denote the variance of X. For many hydrologic engineering problems, more than two or three constraints are not needed. 31.9.2  Maximization of Entropy

In order to obtain the least-biased f(x), the entropy given by Eq. (31.1) is maximized, subject to Eq. (31.34) and (31.34), and one simple way to achieve the maximization is the use of the method of Lagrange multipliers. To that end, the Lagrangean function L can be constructed as: b

b

n

b

a

a

r =1

a

L = − ∫ f ( x )ln f ( x )d x − (λ0 − 1)[ ∫ f ( x )d x − C0 ] − ∑ λr [ ∫ f ( x ) g r ( x )d x − Cr ]

(31.35)

where λ1 , λ2 ,..., λn are the Lagrange multipliers. In order to obtain f(x) which maximizes L, one may recall the Euler-Lagrange calculus of variation, and therefore one differentiates L with respect to f(x) (noting X as parameter and f as variable), equates the derivative to zero, and obtains

n ∂L = 0 ⇒ −[1 + ln f ( x )] − (λ0 − 1) − ∑ λr g r ( x ) = 0 (31.36) ∂f r =1

31.9.3  Probability Distribution

Equation (31.36) leads to the PDF of X in terms of the given constraints: n

f ( x ) = exp[− λ0 − ∑ λr g r ( x )]



(31.37)

r =1

where the Lagrange multipliers λr, r = 0,1, 2,…, n, can be determined with the use of Eqs. (31.34) and (31.34). Equation (31.37) is also written as:

f (x ) =

n 1 exp[− ∑ λr g r ( x )] (31.38) Z (λ1 , λ2 ,..., λn ) r =1

where Z = exp(λ0 ) is called the partition function. Integration of Eq. (31.37) leads to the cumulative distribution function or simply probability distribution of X, F(x), as:

x

n

a

r =1

F ( x ) = ∫ exp[− λ0 − ∑ λr g r ( x )]d x (31.39)

Substitution of Eq. (31.37) in Eq. (31.1) results in the maximum entropy of f(x) or X as:

n

n

n

r =1

r =1

r =1

H = λ0 + ∑ λr g r ( x ) = λ0 + ∑ λ1E[ g r ( x )] =λ 0 + ∑ λr Cr (31.40)

where E[g(x)] is the expectation of g(x). Equation (31.40) shows that the entropy of the probability distribution of X depends only on the specified constraints, since the Lagrange multipliers themselves can be expressed in terms of the specified constraints. Equations (31.1), (31.34), (31.34), (31.37), and (31.40) constitute the building blocks for application of the entropy theory. Entropy maximization results in the probability distribution that is most

8/22/16 1:55 PM

Methodology for Application    31-5 

conservative and hence most informative. If a distribution with lower entropy were chosen, it would mean the assumption of information which is not available. On the other hand, a distribution with higher entropy would violate the known constraints. Thus, it can be stated that the maximum entropy leads to a probability distribution of a particular macro-state among all possible configurations of the states (or events) under consideration. 31.9.4  Determination of Lagrange Multipliers

Equation (31.37) is the POME-based probability distribution containing Lagrange multipliers λ0, λ1, λ2,…, λn as parameters which can be determined by inserting Eq. (31.37) in Eqs. (31.34) and (31.34):

b

n

a

r =1

exp(λ0 ) = ∫ exp[− ∑ λr g r ( x )]dx , r = 1,2,..., n (31.41) b

n

a

r =1

Cr = ∫ g r ( x )exp[− ∑ λr g r ( x )]dx , r = 1,2,..., n

n

(31.42)

a

r =1

λ0 = ln ∫ exp[− ∑ λr g r ( x )]dx , r = 1,2,..., n (31.43)



Equation (31.43) shows that λ 0 is a function of λ1 , λ2 ,..., λn and expresses the partition function Z as: b

m

a

r =1

Z (λ1 , λ2 ,..., λm ) = ∫ exp[− ∑ λr g r ( x )]dx (31.44)



Differentiating Eq. (31.43) with respect to λr, one gets b



n

λr g r ( x )]dx ∫ g r ( x )exp[−∑ r =1

∂λ0 =− a ∂λr

b

n

λr g r ( x )]dx ∫ exp[−∑ r =1

, r = 1,2,..., n (31.45)

Multiplying the numerator as well as the denominator of Eq. (31.45) by exp(–λ0), one obtains b



n

λr g r ( x )]dx ∫ g r ( x )exp[− λ0 − ∑ r =1 b

n

∫ exp[− λ0 − ∑ λr g r (x )]dx a

, r = 1,2,..., n (31.46)

r =1

b

n

a

r =1

Cr = ∫ g r ( x )exp[− λ0 − ∑ λr g r ( x )]dx , r = 1,2,..., n (31.48)

Therefore, Eqs. (31.47) and (31.48) yield



∂λ0 = −C r , r = 1, 2,..., n (31.49) ∂λr

λ0 = λ0 (λ1 , λ2 ,.., λn ) (31.50)

Equation (31.50) can be differentiated with respect to λr, r = 1, 2,…, n, and each derivative can be equated to the corresponding derivative in Eq. (31.49). This would lead to a system of n–1 equations with n–1 unknowns, whose solution would lead to the expression of Lagrange multipliers in terms of constraints.

31_Singh_ch31_p31.1-31.8.indd 5

0

∞

∞

∞

0

0

0

L = − ∫ ( x ) In f ( x )dx − (λ0 − 1)[ ∫ f ( x )d x − 1] − λ1[ ∫ x f ( x )d x − k] (31.53)

Taking the derivative of L with respect to f(x) and equating it to 0, one obtains

∂L = 0 ⇒ −[1 + In f ( x )] − (λ0 − 1) − λ1x = 0 (31.54) ∂ f (x )

Therefore,

f(x) = exp(–λ0 – λ1x) (31.55)

Equation (31.55) is the POME-based distribution with λ0 and λ1 as parameters. Substituting Eq. (31.55) in Eq. (31.34), one obtains

∞

∫ exp (− λ0 − λ1x ) dx = λ1 exp (λ0 ) = 1 (31.56) 0

f ( x ) = λ1 exp (− λ1x ) (31.57)





∞

1

∫ λ1x exp (− λ1x ) dx = k or λ1 = k

(31.58)

0

Thus Eq. (31.57) becomes 1 x f ( x ) = exp(− ), k = x (31.59) k k



which is the exponential distribution. 31.9.6  Entropy Minimization

For a continuous random variable X in the range 0 and infinity, let q(x) be the prior PDF and p(x) be the posterior PDF. Note q(x) is a prior estimate of p(x). The objective is to determine p(x) subject to specified constraints and given the prior PDF. Then the relative entropy or cross-entropy of p(x) relative to q(x) is expressed by Eq. (31.58) where both the prior and the posterior PDFs satisfy: ∞

∫ q(x )dx = 1



0

Likewise, Eq. (31.43) can be written analytically as:

∫ x f (x )dx = E[x] = x = k (31.52)

The least-biased f(x) is determined by maximizing Eq. (31.1), subject to Eqs. (31.33) and (31.52). To that end, the Lagrangian L is constructed as:

b

n ∂λ0 = − ∫ g r ( x )exp[− λ0 − ∑ λr g r ( x )]dx , r = 1,2,..., n (31.47) ∂λr r =1 a

Note that substitution of Eq. (31.47) in Eq. (31.34) yields

The maximum entropy is a function of the Lagrange multipliers and constraints. Because the Lagrange multipliers can be expressed in terms of constraints, the entropy can be expressed in terms of constraints alone. It can be shown that H(X) is a concave function of constraints. To illustrate, consider a random variable X varying over a semi-infinite Interval (0, ∞) and having a PDF f(x). From observations, the expected value of X, E(x), is known. The objective is to derive the PDF f(x) of X. In this case the constraint equation is given as:

Inserting Eq. (31.57) in Eq. (31.34), one gets

The denominator in Eq. (31.46) equals unity by virtue of Eq. (31.33). Therefore, Eq. (31.46) becomes

r =1

Substituting Eq. (31.55) in Eq. (31.51), one gets

a

∂λ0 =− a ∂λr

n

H ( X ) = λ0 + ∑ λr g r ( x )] (31.51)





Equation (31.41) can be written for the zeroth Lagrange multiplier as: b

Substitution of Eq. (31.37) into Eq. (31.1) yields the maximum entropy:

∞

and

31.9.5  Maximum Entropy

(31.60)

∞

∫ p(x )dx = 1



(31.61)

0

In order to derive p(x) by applying POMCE, the following constraints can be specified:

∞

∫ g r ( x ) p( x )dx = g r = Cr , r = 1,2..., m (31.62) 0

8/22/16 1:55 PM

31-6    Entropy Theory

The problem is to determine p(x), subject to the information given by Eqs. (31.60) through (31.62). Equation (31.58) does not lead to a unique p(x) but does restrict the permissible densities that could be plausible. To obtain p(x) uniquely, D(p, q) is minimized, subject to Eqs. (31.60)–(31.62), and minimization can be achieved using the method of Lagrange multipliers where the Lagrangian function can be expressed as: ∞

L = ∫ p( x )In 0

m

∞

r =1

0

p( x ) dx + (λ0 − 1)[ ∫ p( x )d x − 1] q( x ) 0

where λr , r = 1,2,..., m, are the Lagrange multipliers. Minimizing the variation in L with respect to p(x) or differentiating Eq. (31.63) with respect to p(x) while recalling the calculus of variation and equating the derivative to zero, the following is obtained:



p( x ) =

which can be expressed as:

m

p( x )



p( x ) =

0

∞

x − λ2 exp (− x / k )exp (− λ0 − λ1x )dx = 1 (31.76) k 0



∫

∞

x 1−λ2 exp (− x / k )exp (− λ0 − λ1x )dx = x (31.77) k 0



∫

∞

x 1−λ2 In x exp (− x / k )exp (− λ0 − λ1x )dx = In x k 0



∫

Since δp is arbitrary, terms inside the parentheses must vanish. Therefore, In

m p( x ) + 1 + (λ0 − 1) + ∑ λr g r ( x ) = 0 (31.65) q( x ) r =1

p( x ) = q( x )exp[− λ0 − ∑ λr g r ( x )]



(31.66)

r =1

The Lagrange multipliers are determined using Eqs. (31.60) through (31.62). Let exp(− λ0 ) = C . Then Eq. (31.63) can be written as: m

p( x ) = q( x )C exp[− ∑ λ r g r ( x )] (31.67)



r =1

which can be written in a product or multiplicative form: m

p( x ) = Cq( x ) Π exp[− λr g r ( x )] (31.68)



r =1

Equation (31.68) is also conveniently written as

p( x ) =

m 1 q( x )exp[− ∑ λr g r ( x )] (31.69) Z (λ0 ) r =1

where Z is called the partition function obtained by substituting Eq. (31.69) in Eq. (31.61):

∞

m

0

r =1

Z = exp(λ0 ) = ∫ q( x )exp[− ∑ λr g r ( x )]dx (31.70)

The Lagrange multipliers are determined using Eq. (31.70) with known values of constraints as:

−

∂In Z 1 ∂Z =− = g r ( x ) = Cr (31.71) ∂λr Z ∂λr

Equation (31.71) does not lend itself to an analytical solution except for simple cases, but numerical solution is not difficult. Equation (31.71) shows that specific forms of p(x) depend on the specification of q(x) and gr(x). Here two simple cases of the prior q(x) are dealt with. The prior PDF is exponential, given as:

1 q( x ) = exp(− x / k ) (31.72) k

and the constraints are mean and mean log specified as:

∞

∫ In xp(x )dx = In x (31.73) 0

31_Singh_ch31_p31.1-31.8.indd 6

(31.78)

Equations (31.76)–(31.78) can be solved numerically for the Lagrange multipliers. 31.9.7  Entropy Maximizing in Frequency Domain

Equation (31.65) leads to the posterior PDF p(x): m

(31.75) x − λ2 exp (− x / k )exp (− λ0 − λ1x ) k

Substitution of Eq. (31.76) in Eqs. (31.60), (31.51), and (31.74) yield, respectively,

λr g r ( x )}dx = 0 (31.64) ∫ δ p{ln[ q(x ) ] + 1 + (λ0 − 1) + ∑ r =1



1 exp (− x / k )exp (− λ0 − λ1x − λ2 In x ) (31.74) k

∞

(31.63) + ∑ λr [ ∫ g r ( x ) p( x )d x − Cr ]

∞

Equation (31.67) in light of Eqs. (31.60), (31.52) and (31.73) becomes

Let a time series (e.g., streamflow) y over time t be denoted as y(t) and let yt (y1, y2,…, N) define a set of observations generated sequentially in time, t = 1,…, N, where N is the total number of observations. Then, the time series can be expressed using the Fourier series as: q

yt = α 0 + ∑ α k cos(2π f kt ) + βk sin(2π f kt ) (31.79)



k =1

where f k is the kth frequency defined as f k = k / N , and αk and βk are the Fourier coefficients defined as: N N α0 = y ; α k = 2 ∑ yt cos(2π f kt ); β = 2 ∑ y sin(2π f t ) (31.80) k t k N t =1 N t =1

where y is the mean of y. The coefficients can be computed by the least square method. If yt is normalized, α0 = 0. Frequencies f k = k / N are called harmonics of the fundamental frequency 1/N. The spectral power, xk, at a given frequency f k , is defined as:

xk =

N (α k 2 + βk 2 ) (31.81) 2

For each frequency f k, there is a one corresponding spectral power. The power spectrum can be defined as:

G( f ) =

N (a f 2 + b f 2 ) 2

(31.82)

where f is the frequency, and af and bf are Fourier coefficients defined in Eq. (31.80) with f instead of f k. The spectral density of the time series is defined by dividing the power spectrum by the variance of observed series values, σ 2 , as: P( f ) = G( f ) / σ 2



(31.83)

The spectral density defined as above is equivalent to P( f ) =



∞

∑ y(t )e−2πtif

t =−∞

2

2

= Y ( f ) (31.84)

where i = −1 , and Y(f) is the Fourier transform of y(t). The power spectrum can be expressed in terms of autocovariance function, Rk, as:

G( f ) =

N −1

∑

k=− N +1

Rk exp(−i 2π fk ) (31.85)

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Hydrologic Modeling Using Entropy theory    31-7 

and the spectral density in terms of autocorrelation function as: N −1

∑

P( f ) =



k =− N +1

ρk exp(−i 2πfk ) (31.86)

where ρk is the autocorrelation of lag k, and i = −1 . One can also express the autocovariance and autocorrelation functions in terms of power spectrum and spectral density, respectively, by taking the inverse Fourier transform of Eqs. (31.85) and (31.86) as:

W

∫

Rn =

G( f )ei 2π fn∆t df (31.87)

The second integral on the left side of Eq. (31.93) becomes the cepstrum of autocorrelation defined in Eq. (31.92). Now the right side of Eq. (31.94) can be cast as: W



−W

W

∫

ρn =

W

∫ G( f )d f

∫ P( f )d f are equivalent in the case n = 0, and the

−W

−W

integral of G(f) equals R(0) and the integral of P(f) equals 1. The entropy theory can be applied to derive the least-biased spectral density which is needed for forecasting. The spectral density can be derived using the Burg entropy, the configurational entropy, or relative entropy. To that end, one can use either frequency f or spectral power as a random variable. The procedure remains the same in either case and is illustrated by maximizing the configurational entropy defined by Eq. (31.9), subject to constraints defined by Eq. (31.88). Entropy maximizing can be done using the method of Lagrange multipliers in which the Lagrangian function can be formulated as:

L( f ) = −

W

W

N

λn[ ∫ P( f )exp(i 2π fn∆t )d f − ρn ] ∫ P( f )ln[P( f )]d f − n∑ =− N

−W

−W

(31.89)

where λn, n = 0, 1, 2,…, N, are the Lagrange multipliers. Taking the partial derivative of Eq. (31.80) with respect to P(f) and equating the derivative to zero, one obtains W

N ∂ L( f ) = 0 = − ∫ {ln[P( f )] +1 + ∑ λn exp(i 2π fn∆t )}df (31.90) ∂ P( f ) n=− N −W



W

s =− N

−W

∫e

i 2π f (n− s ) ∆t

df =

N

∑ λsδ n−s (31.97)

s =− N

N

δ n + e(n) = − ∑ λsδ n−s (31.98)

where e(n) is the cepstrum of the autocorrelation and σn is the delta function. Equation (31.98) can be expanded as a set of N linear equations:

W

and

N

)ei 2π fn∆t d f = − ∑ λs

s =− N

P( f )ei 2π fn∆t df (31.88)

−W

Thus,

i 2π fn∆t

Thus, summing Eqs. (31.95) and (31.97) and Equating to Eq. (31.96), the result of integration of (31.92) is:

−W

and

N

λne ∫ (−n∑ =− N

λ0 = −1 − e(0)

λ1 = −e(1) 

(31.99)

λ N = − e( N ) Equation (31.94) shows that the Lagrange multipliers can be determined from the values of cepstrum which entail the spectral density that is obtained from Eq. (31.91). For finite length of data, Nadeu (1992) developed a simple method for computing cepstrum based on the use of the causal part of autocorrelation, where ρ(n) is used only for 0 < n ≤ N instead of − N ≤ n ≤ N . Thus, cepstrum can be estimated by the following recursive relation:

n−1 k e(n) = 2[ρ(n) − ∑ e(k ) ρ(n − k )], n > 0 k=1 n

(31.100)

In order to compute e(k), one needs autocorrelation from lag 0 to k. Thus, for given N lag autocorrelations, the cepstrum of autocorrelation can be computed up to lag N, and beyond this lag, the cepstrum is defined as 0. Then, the calculated cepstrum from lag 0 to T can be used to compute the Lagrange multipliers using Eq. (31.99).

Equation (31.90) yields the least-biased P(f): N

∑ λnei 2π fn∆t ) (31.91)

P( f ) = exp(−1 −



n=− N

The next step is to compute the Lagrange multipliers which can be efficiently done by cepstrum which, by definition, is the inverse Fourier transform of the logarithm of the spectrum: e(n) =



π

1 2 πnif ∫ log P( f ) e d f (31.92) 2π −π

It is a measure of the rate of change in the spectrum bands. Thus, taking the logarithmic transform of Eq. (31.92) for computing the Lagrange multipliers, one obtains 1 + log[P( f )] = −



N

∑ λnei 2π fn∆t

n=− N

(31.93)

Taking the inverse Fourier transform of Eq. (31.93), one gets

W

∫ {1 + log[P( f )]}e

i 2π fn∆t

df =

−W

W

N

λne ∫ (−n∑ =− N

i 2π fn∆t

)ei 2π fn∆t d f (31.94)

−W

where the first part of the left side of Eq. (31.94) can be denoted as:

W

∫e

i 2π fn∆t

df =

−W

W

∫ cos(2π fn∆t )d f =

−W

sin(π n) (31.95) πn

Thus, when n = 0, Eq. (31.95) is equivalent to 1; otherwise it equals 0. Therefore, it can be written using the delta function as:

W

∫e

−W

31_Singh_ch31_p31.1-31.8.indd 7

i 2π fnt

 1, n = 0 d f = δn =   0, n ≠ 0

(31.96)

31.10  HYDROLOGIC MODELING USING ENTROPY THEORY

From a hydrologic point of view, applications of the entropy can be distinguished into three classes: (1) statistical or empirical, (2) physical, and (3) mixed. Applications in the first category essentially entail analysis of data with little physics. Here the focus is on determining the probability distribution of the random variable under consideration, subject to specified constraints. Examples of such applications include derivation of probability distributions, frequency analysis, parameter estimation, network evaluation and design, flow forecasting, reliability assessment of water distribution systems, spatial analysis, grain-size distribution, system complexity analysis, regionalization of watersheds, and clustering of data. Information is specified in the form of constraints and the determination is based on POME or POMCE and entropy is maximized. Applications in this category are common in water engineering. In the second category, the focus is on deriving a functional relation in time or space between a dependent variable and an independent variable. Here three aspects are involved. First, the entropy theory is applied to determine the probability distribution of the random variable under consideration. Second, the cumulative probability distribution relating the flux and concentration is hypothesized. Third, the first and second aspects are connected to obtain the relation in time or space as the case may be. Examples in this category include rainfall-runoff modeling, infiltration, soil moisture movement, velocity distribution, hydraulic geometry, channel cross-section, sediment concentration and discharge, sediment yield, river bed profile, flow duration curve, and rating curve. The third category is a mixture of the first two categories wherein analysis is partly empirical and partly physical. Examples include geomorphic relations for elevation, slope, and fall; and reliability of water distribution systems. Singh (1997; 2011a; 2011b) has reviewed entropy applications and numerous examples of applications are described in Singh (2013; 2014; 2015).

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31-8    Entropy Theory 31.11 CONCLUSION

The entropy theory is a versatile tool for modeling in hydrology. It can serve as a connecting link when modeling seemingly disparate hydrologic processes. In other words, it can be a unifying theory for hydrologic science and engineering. A great advantage of the theory is that it provides a means for incorporating the known information into the derivation of probability distributions and to estimate the distribution parameters in terms of the known information. In this manner, it obviates the need for using the usual parameter estimation techniques. Also, it brings the model and the modeler closer. REFERENCES

Burg, J. P., Maximum entropy spectral analysis, Unpublished Ph.D. thesis, Stanford University, Palo Alto, CA, 1975, p. 123. Frieden, B. R., “Restoring with maximum likelihood and maximum entropy,” Journal of the Optical Society of America, 62 (4): 511–517, 1972. Gull, S. F. and G. J. Daniell, “Image-reconstruction from incomplete and noisy data,” Nature, 272 (5655): 686–690, 1978. Jaynes, E. T., “Information theory and statistical mechanics,” I. Physical Review, 106: 620–630, 1957. Jaynes, E. T., “Probability Theory in Science and Engineering,” Colloquium Lectures in Pure and Applied Science, No. 4, Socony Mobil Oil Company, Dallas, TX, 1958. Jaynes, E. T., “Prior probabilities,” IEEE Transactions on System Science and Cybernetics, 70: 939–952, 1968. Kapur, J. N., Maximum Entropy Models in Science and Engineering, Wiley Eastern, New Delhi, India, 1989. Kapur, J. N. and H. K. Kesavan, Entropy Optimisation Principles with Applications, Academic Press, San Diego, p. 408. Koutsoyiannis, D., “Physics of uncertainty, the Gibbs paradox and indistinguishable particles,” Studies in History and Philosophy of Modern Physics, 44: 480–489, 2013. Koutsoyiannis, D., “Entropy: from thermodynamics to hydrology,” Entropy, 16: 1287–1314, 2014, doi: 10.3390/e16031287. Kraskov, A., H. Stogbauer, and P. Grassberger, “Hierarchical clustering based on mutual information,” 2003, arXiv preprint q-bio. QM/0311039. Kullback, S. and R. A. Leibler, “On information and sufficiency,” Annals of Mathematical Statistics, 22: 79–86, 1951. Lathi, B. P., An Introduction to Random Signals and Communication Theory, International Textbook Company, Scanton, PA, 1969. McGill, W. J., “Multivariate information transmission,” Psychometrica, 19 (2): 97–116, 1954.

31_Singh_ch31_p31.1-31.8.indd 8

Mogheir, Y. and V. P. Singh, “Application of information theory to groundwater quality monitoring networks,” Water Resources Management, 16 (1): 37–49, 2002. Nadeu, C., “Finite length cepstrum modeling—a simple spectrum estimation technique,” Signal Process, 26 (1): 49–59, 1992. Papalexiou, S. M. and D. Koutsoyiannis, “Entropy based derivation of probability distributions: a case study to daily rainfall,” Advances in Water Resources, 25: 51–57, 2012 Saeb, A., On Extreme Value Theory and Information Theory, Unpublished Ph.D. dissertation, University of Mysore, Mysore, India, 2013. Shannon, C. E., “A mathematical theory of communications, I and II,” Bell System Technical Journal, 27: 379–443, 1948. Shannon, C. E. and W. Weaver, The Mathematical Theory of Communication, University of Illinois Press, Urbana, IL, 1949. Singh, V. P., “The use of entropy in hydrology and water resources,” Hydrological Processes, 11: 587–626, 1997. Singh, V. P., “Hydrologic synthesis using entropy theory: review,” Journal of Hydrologic Engineering, 16 (5): 421–433, 2011a. Singh, V. P., “Entropy theory for earth science modeling,” Indian Geological Congress Journal, 2 (2): 5–40, 2010, 2011b. Singh, V. P., Entropy Theory in Environmental and Water Engineering, Wiley-Blackwell, Sussex, U.K., 2013 Singh, V. P., Entropy Theory in Hydraulic Engineering, ASCE Press, Reston, VA, 2014. Singh, V. P., Entropy Theory in Hydrologic Science and Engineering, McGraw-Hill Education, New York, 2015. Tamburo, A. M., “Random walk between order and disorder,” Entropy and Energy Dissipation in Water Resources, edited by V. P. Singh and M. Fiorentino, Kluwer Academic, Dordrecht, The Netherlands, 1992, pp. 131–136. Templeman, A. B., Entropy and Civil Engineering Optimization, NATO/ASI on Optimization and Decision Support Systems in Civil Engineering, Edinburg, U.K., 1989, p. 17. Tsallis, C., “Possible generalization of Boltzmann-Gibbs statistics,” Journal of Statistical Physics, 32 (½): 479–487, 1988. Tsallis, C., “Entropic nonextensivity: a possible measure of complexity,” Chaos, Solitons and Fractals, 12: 371–391, 2002. Tsallis, C., R. S. Mendrs, and A. R. Plastino, “The role of constraints within generalized nonextensive statistics,” Physica A, 261: 534–554, 1998. Watanabe, S., “Information theoretical analysis of multivariate correlation,” IBM Journal of Research and Development, 6 (1): 66–82, 1960. Yang, Y. and D. H. Burn, “An entropy approach to data collection network design,” Journal of Hydrology, 157: 307–324, 1984.

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Chapter

32

Entropy Production Extremum Principles BY

ROBERT K. NIVEN AND HISASHI OZAWA

ABSTRACT

We review a number of extremum principles of nonequilibrium thermodynamics based on dissipation, power consumption or entropy production, and their significance to hydrology. The connections between these and various allied methods are reviewed and mapped using the rigorous framework supplied by maximum entropy analysis. Two conjugate maximum and minimum entropy production principles, originating from Paltridge, Gaggioli, and others, are found to be of considerable importance and warrant further investigations in hydrology. 32.1 INTRODUCTION

For several decades, the maximum entropy production (MaxEP) principle of Paltridge (1975, 1978) has been applied as a heuristic tool to infer the stationary state of a variety of far-from-equilibrium dissipative systems, including many fluid and heat flow systems of interest to hydrologists. Examples include the atmospheric, oceanic, and mantle circulation systems on the Earth and other planets (e.g., Paltridge, 1975, 1978; Vanyo and Paltridge, 1981; Ozawa and Ohmura, 1997; Lorenz et al., 2001; Shimokawa and Ozawa, 2001, 2002; Ozawa et al., 2003; Kleidon and Lorenz, 2005); turbulent shear and heat convection systems (Ozawa et al., 2001, 2003); global planetary cycles and the biosphere (Kleidon, 2004, 2009a-b, 2010a-b; Kleidon and Lorenz, 2005); vegetation spatial distribution (Kleidon et al., 2007; del Jesus et al., 2012); ecosystem selection (Meysman and Bruers, 2007); photosynthesis and plant optimization (Dewar et al., 2006; Dewar, 2010); fluvial geomorphology (Paik and Kumar, 2010; Beven, 2015); particle sedimentation (Chung and Vaidya, 2008, 2011); jumpwise colloidal processes (Kozvon et al., 2002); crystal growth (Martyushev and Axelrod, 2003); tropical cyclones (Ozawa and Shimokawa, 2015); plasma dynamics (Yoshida and Mahajan, 2008; Kawazura and Yoshida, 2010, 2012); and earthquake dynamics (Main and Naylor, 2008). Several reviews and monographs have been published (Ozawa et al., 2003; Kleidon and Lorenz, 2005; Martyushev and Seleznev, 2006; Kleidon, 2010a; Dewar et al., 2013). Numerous studies also reveal the existence of a “conjugate Paltridge” minimum entropy production (MinEP) principle (e.g., Paulus and Gaggioli, 2004; Martyushev, 2007; Niven, 2010b; Kawazura and Yoshida, 2010, 2012), quite different to the MinEP principle of Prigogine (1967). Of course, the application of variational methods to nonequilibrium systems has a long pedigree, with a variety of minimum or maximum dissipation, power or entropy production principles proposed by Helmholtz, Rayleigh; Onsager, and Machlup (1953); Prigogine (1967), and in turbulent fluid mechanics (upper-bound theory) (e.g., Malkus, 1956, 2003; Busse, 1970; Howard, 1972), finite-time thermodynamics (e.g., Salamon and Berry, 1983; Nulton et al., 1985), and engineering design (Bejan, 1996). One apparent advantage of the

Paltridge MaxEP heuristic (and its conjugate)—as applied by many researchers—is that the user can omit most details of the dynamics. If correct, this would suggest the operation of a new variational principle of science, consistent with conservation laws but which acts outside the framework of the four laws of equilibrium thermodynamics. The aim of this chapter is to impart a more rigorous theoretical foundation and understanding of minimum and maximum entropy production (MinEP and MaxEP) principles, and to review their current and possible future applications in hydrology. In Sec. 32.2, we examine the entropy production concept for both macroscopic and infinitesimal dissipative systems, and then review and attempt to classify the many variational methods based on dissipation, power consumption, or entropy production. From this review, we draw out the significance of the MaxEP principle of Paltridge for flow systems subject to particular constraints, as well as its conjugate MinEP principle for flow systems subject to different constraints. In Sec. 32.3, we apply the maximum entropy method formulated in Chap. 31 to the analysis of flow systems, both locally and globally, in each case leading to an extremum principle based on a thermodynamic-like “flux potential.” Under different conditions, this furnishes subsidiary MinEP and MaxEP principles, in the same fashion that the minimum free energy principle of thermodynamics gives rise to subsidiary minimum and maximum internal energy (or enthalpy) principles. This analysis provides a theoretical justification of the two conjugate Paltridge principles. In Sec. 32.4, we review the existing (relatively sparse) literature on applications in hydrology, leading in Sec. 32.5 to our conclusions and recommendations for further research. 32.2  BACKGROUND AND REVIEW 32.2.1  Entropy Production

From the second law of thermodynamics, the thermodynamic entropy is not conserved but rather is preserved; that is, once created, it cannot be destroyed. For a macroscopic open system subject to external flows and internal entropy-generating processes, the rate of increase in entropy is measured by the total (thermodynamic) entropy production, as defined by (Jaumann, 1911; de Groot and Mazur, 1984; Niven and Noack, 2014):

σ =

∂S in + FSout ,tot − FS ,tot ≥ 0 ∂t

(32.1)

where S is the thermodynamic entropy within the system, t is time, and FSin,tot and FSin,tot are, respectively, the flow rates of thermodynamic entropy out of and into the system. For a fluid flow system represented by a control volume (CV) enclosed by its control surface (CS), this becomes (de Groot and Mazur, 1984; Niven and Noack, 2013):

32-1

32_Singh_ch32_p32.1-32.8.indd 1

8/22/16 1:55 PM

32-2     Entropy Production Extremum Principles

σ = ∫∫∫



∂ ρs dV +  ∫∫ ( jS + ρsv ) ⋅ n dA ≥ 0 ∂t CS

32.2.2  Entropy Production Extremum Principles

(32.2)

where Q is the volumetric flow rate, PL is the loss in piezometric pressure, H L is the head loss, and g is the gravitational acceleration. The sign of (32.7) indicates spontaneous flow in the direction of decreasing pressure or head. For open channel flow, H L is the negative change in total head H L = −∆H = −∆( z + y + U 2 /2 g ) , where z is the channel base elevation, y is the water surface elevation, and U is the mean cross-sectional velocity; furthermore, for uniform flow H L = SL ≈ Sx, where S is the bed slope, L is the downstream length, and x is the downstream horizontal distance. Whether microscopic or macroscopic, for nonradiative systems (32.4) and (32.5) or (32.6) we can refer to pairs of conjugate fluxes and thermodynamic forces, which combine to give individual components of the entropy production. If a pair of extremum principles are identical in all respects but lead to opposing extrema (minimum or maximum), depending on whether they are constrained by flux(es) or thermodynamic force(s), they can be termed conjugate extremum principles.

A schematic diagram is given in Fig. 32.1 of several variational methods proposed in equilibrium and nonequilibrium thermodynamics. In equilibrium thermodynamics, the purpose of an extremum principle is to predict the equilibrium position of the system. Analogously, for a nonequilibrium system, a useful extremum principle should facilitate the prediction of the stationary state, usually termed the steady-state flow2. Some principles have also been proposed for transient flows. As evident from Fig. 32.1, a tremendous assortment of variational methods—each denoted by a small roman letter— have been proposed. The validity of and borderlands between many of these methods have not been adequately charted; indeed, the authors do not even proclaim that the present representation is correct or complete. Furthermore, some methods are of relatively narrow applicability (indeed, some may not be valid at all). Briefly summarizing the essential points: 1. All of equilibrium thermodynamics and many branches of nonequilibrium thermodynamics are underpinned by the maximum entropy (MaxEnt) method (a) developed by Boltzmann (1877) and Planck (1901), rewritten in generic form by Jaynes (1957, 2003). This method is discussed in detail in Chap. 31. For example, applying MaxEnt to the contents of a thermodynamic system (b) gives the four laws of thermodynamics (c) and, for an isolated system, can be applied directly (max S ) to infer its equilibrium state (d). For an open thermodynamic system, MaxEnt leads to a thermodynamic potential Φeq (analogous to a Planck potential or free energy/temperature)—representing the (negative) entropy of the universe—which is minimized (min Φeq ) at equilibrium (e). Both isolated and open equilibrium formulations have strong connections to empirical thermodynamics (e.g., Clausius 1876; Gyftopoulos and Beretta, 2005) (f), finite-time thermodynamics (e.g., Salamon and Berry, 1983; Nulton et al., 1985) (g), and probabilistic dynamics based on the Liouville, Hamiltonian, Fokker-Planck, or Master equations (e.g., Lanczos, 1970; Risken, 1996) (h). However, while the framework of equilibrium thermodynamics indicates the direction of spontaneous change of a system—from which the total global or local entropy production (32.1)–(32.3) must be positive—it makes no statement concerning the rate of change. Many prominent researchers have been confused on this point. For this reason, any justification of an entropy production extremum principle must lie outside the established framework of equilibrium thermodynamics. 2. The conservation of mass, momentum, energy, and charge gives rise to the conservation laws of fluid mechanics, thermodynamics, and chemical reactions (i), including the continuity, Navier-Stokes, energy and mass action equations (e.g., de Groot and Mazur, 1984; White, 2006). These can be used to directly model steady-state and time-variant flows (j). However, these analyses can involve considerable complexity, due to the nonlinear Reynolds stress terms and chemical reaction kinetics, motivating the present search for direct variational methods. 3. A spectral representation of turbulent flow, in conjunction with dimensional arguments, yields the Kolmogorov (1941) equation and related analyses of the turbulent energy cascade (k). Furthermore, the MaxEnt analysis of spectral decompositions of flow systems (l), e.g., by Galerkin proper orthogonal decomposition, can also be used to infer spectral coefficients (modal amplitudes) in certain classes of flows (e.g., Noack and Niven, 2012, 2013). 4. As will be discussed, the application of MaxEnt to an open, flow-controlled system (m) gives a thermodynamic-like potential Φst which is minimized at steady state (min Φst ; Niven, 2009, 2010a) (n). This behavior is analogous to that of the thermodynamic potential of an open equilibrium system (e). Related analyses give steady-state analogs of the four laws of thermodynamics (Niven, 2009) (o), as well as the Fluctuation Theorem (p) for the ratio of probabilities of forward and backward fluxes of the same magnitude (e.g., Evans et al., 1993). The MaxEnt analysis of hybrid systems constrained by both thermodynamic contents and flow rates gives the field of Extended Irreversible Thermodynamics (e.g., Jou et al., 1993) (x). 5. An important pair of variational principles consist of the MaxEP principle of Paltridge (1975, 1978) (q) and its conjugate MinEP principle (e.g., Paulus and Gaggioli, 2004; Martyushev, 2007; Niven, 2010b; Kawazura and Yoshida, 2010, 2012) (r). The former is commonly allied with a MaxEP orthogonality principle developed by Ziegler (1977) (u), often applied to mechanical systems. These methods select the observed steady state from a set of possible steady-state flows; they are therefore of considerable utility and, as discussed, have been widely applied. Examples include the state of a heat convection system based on Rayleigh

Individual components of the entropy production can be negative, for example heat transfer and chemical reaction, provided they are coupled to be non-negative in total.

2 The term “steady-state” is somewhat misleading, since it refers only to the mean flow and not its fluctuations. A steady-state flow need not be steady in time, only in the mean.

CV

where ρ is the fluid density, s is the specific thermodynamic entropy (per unit mass of fluid), jS is the nonfluid entropy flux (per unit area of the control surface), v is the fluid velocity (whence ρsv is the fluid-borne entropy flux), n is the unit normal pointing out of the control surface, “⋅” is the vector scalar product, and dV and dA are infinitesimal volume and area elements, respectively. Using σ = ∫∫∫ σˆ dV to define the local entropy production σˆ , expressed CV per unit volume of fluid, we obtain from (32.2) using Gauss’ theorem: Ds ∂ ρs + ∇ ⋅ ( jS + ρ sv ) = ρ + ∇ ⋅ jS ≥ 0 σˆ = (32.3) Dt ∂t where D/Dt is the substantial derivative. Whether local or global, from the second law of thermodynamics the total entropy production must be nonnegative.1 By a standard analysis of a nonradiative system at local thermodynamic equilibrium, subject to flows of heat, fluid, chemical species, and with chemical reactions, it can be shown that the non-fluid entropy flux and entropy production are given by (de Groot and Mazur, 1984):

1 µ jS =   jQ − ∑  c  jc T T c



(32.4)

τ : ∇v 1 µ ˆ  G  − ∑ ξd ∆  d  (32.5) σˆ = jQ ⋅∇   − ∑ jc ⋅∇  c  − T T c T  T d



where T is the absolute temperature, jQ is the heat flux, jc and µc are respectively the flux and chemical potential of the cth chemical species, τ is the stress tensor (here positive in compression), ξˆd and G d are, respectively, the rate and molar Gibbs free energy of the dth chemical reaction, and “:” is the tensor scalar product.

Equations (32.4) and (32.5) can be summarized respectively by jS = ∑ jr λr and r σˆ = ∑ jr ⋅ fr , where jr ∈ jQ , jc , τ , ξd are generalized fluxes, λr ∈{1/ T , − µc / T } r −1 are intensive variables, and fr ∈{∇T , −∇( µc / T ), −∇v / T , −∆(G d / T )} are

{

}

thermodynamic forces conjugate to jr. Note that the fluid velocity (volumetric flux) v does not appear as a flux in (32.5), but is connected to these relations via (32.3) or by the specific Gibbs-Duhem equality (de Groot and Mazur, 1984). For systems with a potential field or interactions with electromagnetic radiation, more complicated (not necessarily bilinear) relations are required (de Groot and Mazur, 1984; Niven and Noack, 2014). Relations (32.4) and (32.5) are often extended to larger (macroscopic) systems with uniform flow rates and gradients:

σ = ∫∫∫ σˆ dV = ∫∫∫ ∑ jr ⋅ fr dV



CV

CV

r

=

uniform flows

− ∑ Fr ∆λr (32.6) r

where Fr ∈{ FQ , Fc } are generalized flow rates and ∆λr are their conjugate intensive variable differences. For example, the total steady-state entropy production in an incompressible fluid flow with only turbulent dissipation—typical of hydrological flows—can be calculated by (Adeyinka and Naterer, 2004):

σ bulk = − ∫∫∫



CV

1

32_Singh_ch32_p32.1-32.8.indd 2

τ : ∇v P ρ gH L dV = Q L = Q T T T

(32.7)

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BACKGROUND AND REVIEW    32-3 

(ab) Steepest entropy ascent (Beretta)

Extended (x) irreversible thermodynamics (b) MaxEnt on contents (a) MaxEnt (Boltzmann, Jaynes)

(c) Zeroth,first, second and third laws of thermodynamics

?

(d)

?

(f) Empirical thermodynamics

Isolated systems max Ssys

Fluctuation (p) theorem (steady state)

(n)

Turbulence cascade (Kolmogorov)

?

(y)

d

(l)

(u)

?

Ziegler orthog.

(j)

? (i) Conservation laws (mass, momentum, energy, charge)

(r) MinEP (ac) design (Bejan)

? MinEP steady state (e.g., Paulus/Gaggioli, Yoshida, Niven)

MaxEP steady state (Paltridge)

?

Finite-time (z) steady-state MinEP limit

Global

st

(q)

Upper bound theory (Malkus, etc.)

Time-variant flows

Local

Open (flow) systems (variable , )

MaxEnt on modes (e.g., Fourier, Galerkin) (k)

?

Open systems Suniv = eq

univ

?

Quantum thermodynamics (Beretta) (aa) Finite-time (g) thermodynamics

(e)

(m) MaxEnt on fluxes

(o) Steady-state analogues of four laws of thermodynamics

Prob. dynamics (h) (Liouville, Hamitonian, Fokker–Planck and Master eqs; ? Fisher information)

(t) Linear regime (Onsager)

?

?

?

(s) Onsager “min dissip”

(w)

Min/max power (networks) ?

Prigogine MinEP

(v)

Figure 32.1  Schematic diagram of relationships between extremum and other allied principles in equilibrium and nonequilibrium systems.

number (Ozawa et al., 2001, 2003), of a turbulent flow system based on Reynolds number (Niven, 2010b), or of a chemical degrading system based on chemical or biological processes (Meysman and Bruers, 2007). Furthermore, the type of extremum (MinEP or MaxEP) appears to depend on the choice of constraint from a given conjugate pair. For example: • For flow in series or parallel pipes, the choice of laminar or turbulent flow regime is consistent with MaxEP for a fixed flow rate, but with MinEP for a fixed head loss (Paulus and Gaggioli, 2004; Martyushev, 2007; Niven, 2010b). • For a zonal flow heat transfer system, the choice of convection regime is consistent with MaxEP for a flux-driven system, but with MinEP for a temperature-driven system (Kawazura and Yoshida, 2010, 2012). • For a heat convection system with parallel connections such as a Bénard cell, the choice of convection regime is consistent with MinEP for a flux-driven system, but with MaxEP for a temperature-driven system (Kawazura and Yoshida, 2012). The full spectrum of this MaxEP/MinEP inversion for different flow systems has not been properly delineated. Apart from the MaxEnt analysis based on fluxes given herein (see Sec. 32.3), few adherents of the Paltridge MaxEP principle have paid much attention to its observed conjugate MinEP form in the literature. 6. A so-called “minimum dissipation” principle was developed by Onsager and Machlup (1953), building on work by Helmholtz and Rayleigh, involving maximizing the entropy production less a dissipation function (s). This method applies only in the linear transport regime (t), i.e., in which the fluxes are linear functions of the forces jr = ∑ Lrk fk , where Lrk k

is the phenomenological coefficient between the rth and kth processes (Onsager, 1931). 7. A well-known theorem was given by Prigogine and coworkers (e.g., Prigogine, 1967), involving MinEP with respect to certain fluxes or forces, constrained by their conjugate parameters (v). The theorem can be derived in the linear transport regime (t), and selects the stationary state from the set of transient states of the system. However, since the

32_Singh_ch32_p32.1-32.8.indd 3

stationary state can be calculated by other methods, it is not very useful (Jaynes, 1980); indeed, there does not appear to be any application of Prigogine’s method in any branch of engineering. Unfortunately, there is widespread confusion between Prigogine’s and other MinEP principles in the literature, and also an incorrect view that Prigogine’s principle “contradicts” other principles, such as of Paltridge. 8. For decades, a MinEP or minimum power principle has been applied to the analysis of flow networks subject to fixed flow rates, primarily electrical circuits (e.g., Jeans, 1925) and more recently fluid flow networks (Paulus and Gaggioli 2004) (w). A conjugate MaxEP or maximum power principle also exists under conjugate constraints of fixed potential differences (e.g., Županović et al., 2004), often confused with the Paltridge principle (see Niven, 2010b). However, these two principles are of the same character, and can be proven to be valid only in networks with a linear or power-law relationship between flow rates and potential differences, in the power-low case with a common power exponent (Niven, 2010b). 9. In the field of “upper bound theory” within turbulent fluid mechanics, the steady state is identified by maximizing a functional of the total, mean, or turbulent dissipation under various constraints (e.g., Malkus, 1956, 2003; Busse, 1970; Howard, 1972) (y). While enjoying some success, such methods have been criticized for their seemingly ad hoc choice of extremum functional and lack of theoretical justification. 10. A minimum entropy or MinEP principle of rather different character has been derived in finite-time thermodynamics, as a fundamental bound to the “cost” of moving a thermodynamic system between two equilibrium states at a specified (finite) rate (e.g., Salamon and Berry, 1983; Nulton et al., 1985) (g). The same theoretical framework has also been applied to derive the MinEP bound for moving a flow system between two steady states at a finite rate (Niven and Andresen, 2009) (z). 11. A quantum formulation of thermodynamics is given by some authors (e.g., Beretta et al., 1984) (aa). Using this formulation, or more direct approaches, a “steepest entropy ascent” principle has been formulated to predict the transient state of an isolated thermodynamic system (Beretta, 2006) (ab).

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32-4     Entropy Production Extremum Principles

12. A quite different MinEP principle involves minimizing the entropy production of an engineered system to obtain the most efficient design (e.g., Bejan, 1996) (ac). This principle serves a different purpose to the predictive principles examined previously. 32.3 MAXIMUM ENTROPY ANALYSIS

We now embark on MaxEnt analyses of a nonequilibrium flow system, respectively for an infinitesimal (local) or macroscopic control volume. 32.3.1  Analysis of an Infinitesimal Fluid Element

Consider an open infinitesimal fluid element subject to instantaneous fluxes of heat, chemical species, and momentum, as well as instantaneous rates of various chemical reactions. The uncertainty in the system is expressed by the joint probability of these quantities. Departing from previous treatments (Niven 2009, 2010a), we here adopt the continuum assumption for each flux or rate, giving the joint probability density function (pdf) defined by



 jQ ≤ ϒ Q ≤ jQ + djQ   jc ≤ ϒ c ≤ jc + djc , ∀c p( X )dX = Prob.   τ ≤ ϒ τ ≤ τ + dτ  ˆ ˆ ˆ  ξd ≤ ϒ d ≤ ξd + dξd , ∀d

ˆ where X = { jQ ,{ jc }, τ ,{ξd }}, dX = djQ ∏ djc dτ c

      

(32.8)

ˆ

∏ dξd , and ϒ x is the random d

variable of quantity x. The relative entropy, here termed the local flux entropy, then is p( X ) H = − ∫ d X p( X )ln ( X ) (32.9) q Ω X

where Ω X represents the domain of all variables. Adopting the global expectation notation 〈gr 〉 =



∫ d X p( X ) g r (X )

ΩX



(32.10)

for any function g r , we see that maximizing (32.9) subject to normalization (〈1〉 = 1) and specified values of each mean flux and reaction rate {〈 jQ 〉, 〈 jc 〉, 〈 τ 〉, ˆ 〈ξd 〉 } gives

p* ( X ) =

ˆ q  exp  −ζ Q ⋅ jQ − ∑ζ c ⋅ jc − ζ τ : τ − ∑ζ d ⋅ ξd  (32.11)   Z c d

where Z is the partition function and ζ x is the Lagrangian multiplier for quantity x. From (32.9) and (32.11) the maximum entropy is

ˆ H* = ln Z + ζ Q ⋅ 〈 jQ 〉 + ∑ζ c ⋅ 〈 jc 〉 + ζ τ : 〈 τ 〉 + ∑ζ d ⋅ 〈ξd 〉 (32.12) d

c

Comparing the latter to the local entropy production (32.5), we see that each Lagrangian multiplier must be proportional to the mean gradient or potential difference conjugate to its flux or rate (Niven, 2009):  1 µ ∇v G ˆ  H* = ln Z − κ −1 〈∇ 〉⋅ 〈 jQ 〉 − ∑ 〈∇ c 〉⋅ 〈 jc 〉 − 〈 〉 : 〈 τ 〉 − ∑ 〈∆ d 〉⋅ 〈ξd 〉 T T T  T  c d −1  ˆ    = ln Z − κ σ (32.13) where κ is a physical constant (of units J K−1 s−1 m−3), and σˆ  = ∑ 〈 jr 〉⋅〈fr 〉 is r

the local entropy production in the mean. Strictly, the latter is not the same as the total entropy production, since by the Reynolds decomposition

〈σˆ 〉 = ∑ 〈 jr ⋅ fr 〉 = ∑ 〈 jr 〉⋅〈fr 〉 + ∑ 〈 jr ' ⋅ fr '〉 = σˆ  + ∑ 〈 jr ' ⋅ fr '〉 (32.14) r

r

r

32_Singh_ch32_p32.1-32.8.indd 4

Φst = − ln Z = − H* − κ −1 σˆ 

32.3.2  Analysis of a (Macroscopic) Control Volume

We now examine an open macroscopic control volume, in which instantaneous fluxes of fluid, heat, and chemical species cross its control surface. From (32.2) at the steady state, it is not necessary to account separately for dissipation or internal chemical reactions, since their entropy production must be manifested as heat or material fluxes through the boundary (Jaynes, 1980). It is, however, necessary to account for the entropy carried by the fluid velocity v through the boundary, since this is transported into or out ˆ = { j ,{ j }, v }, the total uncertainty is now expressed of the system. Writing X Q c by the joint probability of the fluxes, each as a function of position x on the boundary:  jQ ( x ) ≤ ϒ Q ( x ) ≤ jQ ( x ) + djQ ( x )    ˆ ( x ), x ) dX ˆ dx = Prob.  jc ( x ) ≤ ϒ c ( x ) ≤ jc ( x ) + djc ( x ), ∀c  p( X  v ( x ) ≤ ϒ ( x ) ≤ v ( x ) + dv ( x )    (32.16) v     x ≤ ϒ x ≤ x + dx   The control surface flux entropy can then be defined by integration around the boundary: HCV = − ∫ d x CS

∫

Ω Xˆ ( x )

ˆ ˆ ˆ p( X ˆ , x )ln p( X , x ) ˆ p( X ˆ , x )ln p( X , x ) = − d A dX dX  ∫∫ ∫ ˆ ˆ q( X , x ) q( X , x ) Ωˆ CS X(x )

(32.17) where Ω Xˆ ( x ) is the domain of all fluxes at each position on the control surface. By an analogous argument to the previous section, maximizing (32.17) subject to normalization and constraints on the mean value of each flux around the boundary {〈 jQ ( x )〉,〈 jc ( x )〉,〈 v ( x )〉}, the inferred pdf and maximum entropy are ˆ , x ) = q exp  −η ( x ) ⋅ j ( x ) − ∑η ( x ) ⋅ j ( x ) − η ( x ) ⋅ v ( x )    (32.18) p* ( X Q c c v  Q  ZCS   c

(

* HCS = ln ZCS +  ∫∫ dA ηQ (x ) ⋅〈 jQ (x )〉 + ∑ηc (x ) ⋅〈 jc (x )〉 + ηv (x ) ⋅〈 v (x )〉 CS

c

)

(32.19)

r

where the last term is the mean-fluctuating component (Niven and Noack, 2014). Equation (32.13) can be further rearranged in terms of a local flux potential:

We see that for an open system, the differential dΦst accounts (in a negative sense) for the interplay between the change of (flux) entropy within the system, dH* , and the transfer of (thermodynamic) entropy (in the mean) from the system to the environment, d σˆ  . From this we conclude that (1) the flux and thermodynamic entropy concepts, while distinct, are explicitly connected through (32.15); and (2) we can interpret Φst = − ln Z as a thermodynamiclike potential which should be minimized at the stationary state of the system. The flux potential Φst therefore governs the state of the system. Furthermore, depending on the states accessible to the system, the interplay between changes in H* and σˆ  to minimize Φst can be manifested in three outcomes: 1. The system could increase both H*and σˆ , to give ∆H > 0 and ∆ σˆ  / κ > 0 2. The system could increase H* and decreaseσˆ , provided ∆H ≥| ∆ σˆ  / κ |≥ 0 3. The system could decrease H* and increase σˆ , provided ∆ σˆ  / κ ≥| ∆H |≥ 0 These outcomes are analogous to what is observed in chemical thermodynamics, in the competition between (so-called) “entropic” and “enthalpic” processes (Atkins, 1982). Finally, we see that if one is unaware of the flux entropy H* concept, the first and third outcomes above might be identified heuristically as “MaxEP processes,” while the middle outcome would be identified as a “MinEP process.” We also note these processes are formulated in the same manner as the Paltridge MaxEP or conjugate MinEP form, based on products of mean fluxes and mean thermodynamic forces, rather than the total entropy production (Niven and Noack, 2014). We therefore establish a connection between the MaxEnt analysis of an infinitesimal fluid element and the two Paltridge extremum principles, united by a thermodynamic-like potential which is minimized at the steady-state flow.

(32.15)

where ηr are the Lagrangian multipliers and ZCS is the global partition function. Comparing (32.19) to (32.2) and (32.4), we recognize each multiplier as the mean conjugate intensive variable or entropy density in the direction of the unit normal, i.e., ηr ( x ) = n 〈λr ( x )〉 for jr ∈{ jQ , jc } and η v ( x ) = n 〈 ρ( x )s( x )〉. From (32.19) and the steady-state form of (32.2)

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REFERENCES    32-5 

 1 µ (x ) * = ln ZCS − κ −1  HCS ∫∫ dA n T (x ) ⋅ 〈 jQ (x )〉 − ∑ n Tc(x ) ⋅  CS c

(32.20)

 〈 jc ( x )〉+ n 〈 ρ ( x )s( x )〉⋅ 〈 v ( x )〉   −1 = ln ZCS − κ ∫∫ dAω ( x )

 CS

 where ω ( x ) is the mean half-boundary entropy production per unit area at position x, due to the absolute fluxes out of the enclosed volume through the boundary [for more details see Niven and Noack (2014)]. Rearrangement gives the control surface flux potential:

 * ΦCS = − ln ZCS = − HCS − κ −1  ∫∫ dA ω (x ) CS

(32.21)

As with (32.15), this again expresses the (negative) interplay between changes of (control surface flux) entropy within the system and the transfer of (thermodynamic) entropy from the system to the environment. We again recover three possibilities, of which two involve apparent maximization of an entropy production term, while one involves its apparent minimization. Once again, we obtain a connection between the MaxEnt analysis of a (macro­scopic) control volume and the two conjugate Paltridge extremum principles, united by minimization of a thermodynamic-like potential at the steady-state flow. 32.4  REVIEW OF APPLICATIONS IN HYDROLOGY AND HYDRAULICS

Applications of the Paltridge MaxEP or MinEP principles to hydrology and hydraulics only commenced recently. Kleidon and Schymanski (2008), Kleidon et al., (2009), and Westhoff and Zehe (2013) developed detailed MaxEP models for hydrological water balance, while Porada et al., (2011) and Zehe et al., (2013) have applied MaxEP to hillslope drainage. Zehe et al., (2010) proposed a maximum dissipation model for soil infiltration. Westoff et al., (2014) and Wang et al., (2015) found the empirical relations for the partitioning of precipitation between runoff and evaporation—and subsequent runoff and soil wetting—to be consistent with MaxEP. Paik and Kumar (2010) and Beven (2015) examined the possibility of an optimality principle for landscape evolution, while Hergarten et al., (2014) proposed a minimum dissipation principle for surface drainage networks. Quijano and Lin (2014) review a variety of optimality principles for the critical zone (rock-soil-waterair-organism interface). Other analyses based directly on MaxEnt are examined in Chap. 31. As mentioned, MaxEP may be important in controlling spatial vegetation patterns (Kleidon et al., 2007; del Jesus et al., 2012), while the optimization of the photosynthesis cycle and various whole-plant processes is consistent with MaxEP (e.g., Dewar et al., 2006; Dewar, 2010). Further research is needed on the connection between these phenomena and the MaxEnt analyses of Sec. 32.3. As discussed, analyses of simple series or parallel pipe networks show that the choice of laminar or turbulent flow is consistent with MaxEP if flowconstrained or MinEP if head-constrained (Niven, 2010b). Chung and Vaidya (2011) and Vaidya (2013) found the settling of various particles at low Reynolds number, including elongated particles, dual spheres or a sphere near a wall, to be consistent with MaxEP. Shimokawa and Ozawa (2010) and Ozawa and Shimokawa (2015) found that tropical cyclones evolve to a highEP steady state, and may tend to move along trajectories with higher rates of entropy production. Finally, as discussed, there is now a sizeable literature on MaxEP as a driving force for the Earth’s climate system (e.g., Paltridge, 1975, 1978; Ozawa and Ohmura 1997; Lorenz et al., 2001; Shimokawa and Ozawa, 2001, 2002; Ozawa et al., 2003; Kleidon and Lorenz, 2005), which could have many implications for global atmospheric and oceanic circulation patterns, water and energy cycles, and species distributions under climate change (Kleidon, 2004, 2009a-b, 2010a-b; Kleidon et al., 2007). 32.5 CONCLUSION

This chapter presents a review of extremum principles for nonequilibrium systems, with emphasis on the MaxEP principle of Paltridge (1975, 1978) and its conjugate MinEP form. A crude “treasure map” of such principles, containing some hints on the known and unknown borders and connections between different formulations, is provided in Fig. 32.1. The MaxEnt method of Jaynes (1957) is then applied to local and global nonequilibrium systems, in each case giving a flux potential which is minimized to determine the steady state.

32_Singh_ch32_p32.1-32.8.indd 5

This furnishes subsidiary (pseudo-) MinEP and MaxEP principles, consistent with the two conjugate Paltridge principles. The relatively sparse literature on applications in hydrology and hydraulics is then reviewed. This chapter indicates the need for further substantial, rigorous research on the theoretical development of nonequilibrium extremum principles, and their application throughout hydrology and hydraulics. REFERENCES

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Chapter

33

Data-Based Mechanistic Modeling BY

PETER C. YOUNG

ABSTRACT

There are numerous approaches to the mathematical modelling of hydrological systems. Many of these are based on an approach that the philosopher Karl Popper (1959) termed the “hypothetico–deductive” scientific method. Here, the model, or theory in more general terms, forms a hypothesis that is tested against data, usually obtained from experiments, often within a laboratory where experiments can be carefully planned. Unfortunately, it is not easy to plan experiments in hydrology and the modeller is forced to rely on data collected from the hydrological system during its “normal operation,” where the planning of experiments is difficult or impossible. This chapter describes the data-based mechanistic (DBM) approach to modelling that follows an alternative “hypothetic–inductive” approach where the model structure is first inferred from the data with the minimum of a priori assumptions and only then interpreted in physically meaningful terms that may relate to preexisting hypotheses or may provide new insight into the nature of the hydrological system. 33.1 INTRODUCTION

Data-based mechanistic (DBM) modeling is a method theory that has been developed over many years and exploits a largely inductive approach to scientific investigation and data-based modeling. The inductive approach to science and mathematical modeling has a long history; indeed, it was the approach preferred by Newton (1643–1727) who wrote in his magnum opus, the Principia Mathematica (Newton, 1687), Hypotheses non fingo, or “I frame no hypotheses.” But the inductive approach was not Newton’s invention; it had been discussed by Francis Bacon (1561–1626) and, almost contemporaneously with Newton, by Robert Boyle (1627–1691). During the twentieth century, scientific philosophers, such as Karl Popper (1959) and Thomas Kuhn (1962), looked at the philosophy of science in the context of the scientific research that they observed at the time; research that was dominated by enormous advances that occurred in physics during the latter part of the nineteenth century and the first half of the twentieth century. Popper, in particular, was a proponent of what he termed the hypothetico– deductive method. Here, the model, or theory in more general terms, forms a hypothesis that is tested against data, usually obtained from experiments, often within a laboratory where experiments could be carefully planned. And, in Popper’s view, the aim is not to prove the hypothesis, but rather to attempt its “falsification” and consider it to be “conditionally valid” until falsified. Whether Kuhn subscribed to the hypothetico–deductive concept is not clear. Rather, he viewed science from a paradigmatic standpoint in which most ordinary science worked within and embroidered research according to defined paradigms, while the more fundamental achievements of science were those that questioned or even overturned these current paradigms (as Einstein’s theories of relativity and the development of quantum mechanics radically changed the Newtonian view of the World). In this regard, the

hypothetico–deductive approach to scientific research used by ordinary scientists often tends to be too constrained by current paradigms: hypotheses are made within the current paradigm and do not often seek to question it. So, in a modeling context, which is better: the inductive approach that concentrates on inference from experimental or monitored data, followed by its interpretation in physical terms; or the hypothetico–deductive approach that relies on the creation of a prior hypotheses that normally have a direct physical interpretation and are then tested against such data? The answer is, of course, that they are not mutually exclusive methodologies and should be combined in a constructive way to yield an array of models that satisfy different objectives. My own predilection is to concentrate on an inductive approach whenever the availability of suitable data allows for this. Such an approach can often yield a useful, physically meaningful model rather quickly, without being overly constrained by existing hypotheses; and, as we see later in Sec. 33.5, it may also be an aid in the falsification of hypothetico–deductive models. But suitable data are not always available and hypothetico–deductive simulation models provide an obvious alternative in this data-scarce situation that occurs so often in the natural sciences. Moreover, in making inferences about the model structure, inductive analysis is normally guided by existing or new hypotheses concerning the physical interpretation of parsimonious model structures. In other words, inductive and hypothetico–deductive modelings are synergistic activities, the relative contributions of which will depend on the system being modeled and the information of all types, not only timeseries data, that are available to the scientist and modeler. With above caveats in mind, this chapter first outlines the main stages of the DBM approach to the modeling of dynamic systems from observational time series data. DBM modeling has been applied to the investigation of dynamic processes in a wide variety of systems from engineering, through economics to ecology. The DBM modeling philosophy recognizes that, in contrast to human-designed and constructed dynamic systems, the nature of more evolutionary systems, particularly at the holistic or macro-level (e.g., global climate, river catchment, macro-economy), is still not well understood. Reductionist approaches to modeling such systems, based on the aggregation of hypothetico–deductive models at the micro level, or the application of micro-scale laws at the macro level, often result in large models that suffer from identifiability problems [sometimes termed equifinality in the hydrological literature (e.g., Beven, 1993), following von Bertalanffy (1968)], and are not fully identifiable from the available data. In other words, the information content in the data is not sufficient to allow for the unambiguous estimation of all the model parameters. By contrast, in DBM modeling the model is identified and estimated from the data in a minimally parameterized or parsimonious form that is, therefore, clearly identifiable. Of course, this identifiable model may not answer all questions about the behavior of the dynamic system; rather it represents only that part of the system behavior that is identifiable from the available data. In the present world of science, this kind of inductive modeling is now more the 33-1

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33-2    Data-Based Mechanistic Modeling

domain of the statistician and time series analyst, as well as those engineers like myself who prefer to use efficiently parameterized, data-based models whose physical interpretation is inferred from the data-based model, taking a balanced but questioning account of any existing physical interpretations and models. However, the DBM approach is rather different to that of the conventional statistician and time series analyst in that it attempts to build a bridge between the hypothetico–deductive and inductive modeler. For example, most time series models are formulated as black-box, discrete-time, difference equations that do not have a very transparent connection with physical, biological, and ecological dynamic processes of the kind that occur in nature. Whereas DBM modeling encourages the identification and estimation of continuous-time, differential equation descriptions that normally have an immediate physical interpretation, with parameters that have defined physical units (see, e.g., Garnier and Wang, 2008; Garnier and Young, 2014). In this way, the best aspects of both approaches are melded together into a unified procedure that tries to reach the right balance between parsimony, identifiability, and realistic descriptive ability. Of course, as we shall see in later practical examples, it sometimes makes sense to use discrete-time models since these fit better into the context of the modeling problem being considered. It will be argued that the DBM approach is often a more appropriate method of scientific inference in research on natural systems, where the natural laws at the macro-level, as used in reductionist modeling, are normally untestable by planned experimentation, which is often difficult or impossible in the broad gamut of the natural sciences. In such applications, DBM modeling tends to question existing paradigms if they are not compatible with the measured data. In this manner, it not only helps to avoid the possibility of false hypotheses and overly parameterized, poorly identifiable models, but also provides a compelling reason for exploiting the powerful and relatively novel tools of statistical inference that have been developed to service the requirements of DBM modeling. These DBM modeling tools are conveniently collected together as computational routines (m-files) in the freely available CAPTAIN Toolbox for MatlabTM (website details in Conclusion). These are used to generate the modeling results for the examples presented in this chapter, as well as other hydrological examples in my book Recursive Estimation and Time Series Analysis: an Introduction for the Student and Practitioner (Young, 2011) and the many references cited therein. This latter book also provides a comprehensive and detailed description of the identification and estimation methods used for DBM modeling in this chapter. As a result, only the main aspects of the DBM modeling methodology are presented here, particularly as they relate to the data-based modeling of hydrological systems. 33.2  THE MAIN STAGES OF DBM MODELING

Although the term “data-based mechanistic modeling” was first used in Young and Lees (1993), the basic concepts of this DBM approach to modeling dynamic systems have been developed over many years. For example, they were first applied seriously within a hydrological context in the early 1970s, with application to the modeling of water quality and flow in rivers (Young and Beck, 1974; Young, 1974) and set within a more general framework shortly thereafter (Young, 1978). Since then, they have been applied to many different systems in diverse areas of application (see, e.g., Young, 1998; Lees, 2000; Young, 2006; Young and Ratto, 2009; and the prior references therein). In its latest form, the DBM modeling strategy consists of the 10 major methodological stages that are outlined as follows and considered in greater detail in later sections. While all of these procedures can be used in the process of DBM model synthesis, they may not all be required in any specific application; rather, they are possible stages and associated methodological tools to be used at the discretion of the modeler. 1. M  odeling objectives: The important first stage in any modeling exercise is to define the modeling objectives and to consider the types of model that are most appropriate to meeting these objectives. Since the concept of DBM modeling requires adequate data if it is to be completely successful, this stage also includes considerations of scale and the likely data availability at this scale, particularly as they relate to the defined modeling objectives. Of course, the objectives of modeling are often specific to the system being studied and it is difficult to generalize. In all cases, however, the objectives normally include the enhancement of our understanding of the system, most often in terms of its inherent dynamic behavior. But they may well also include the use of the model for forecasting the future behavior of the system and/or the design of control and management systems that will ensure desired system response. Provided the necessary data are available, the DBM model is in an ideal form for performing such tasks since it seeks to identify the dominant

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modes (see stage 5) of dynamic behavior that are so important in forecasting and control system design. Moreover, the parsimonious nature of the DBM model facilitates such design because the most effective methods normally require an efficiently parameterized model. 2. Inductive DBM modeling: If experimental time series data are available, the inductive stage of model synthesis can begin. Here, an appropriate model structure is identified by a process of statistical inference applied directly to these real time-series data. This is normally based on a generic class of dynamic models. Although discrete-time models could be utilized, linear, stochastic models described by continuous-time transfer functions (i.e., lumped parameter ordinary differential equations) are often preferred since these have advantages when used in modeling physical systems (see, e.g., Sec. 33.3.1 and the examples given in Garnier and Young, 2014). If adequate time series data are available at the start of the study, this analysis will constitute the first and most important stage of inductive DBM modeling. The result of the analysis will be a functioning stochastic-dynamic model of the system based, at this stage, entirely on inductive reasoning and avoiding any prejudicial incorporation of existing hypothetico–deductive modeling results. However, based on the philosophy of DBM modeling, the model should be credible to those working in the discipline or disciplines for whom the model is intended (here, hydrology). This credibility will, of course, depend on the discipline, but, in general terms, it will mean that the model can be interpreted in “physically meaningful” terms, where this phrase should be construed widely to include not only physical but also other factors that affect the hydrological system. 3.  Hypothetico–Inductive (HI-DBM) modeling: Normally, the DBM modeler will be well acquainted with other models and modeling methods that have used previously to meet the objectives defined in stage 1. Following the initial, purely inductive DBM analysis in stage 2, therefore, it is necessary to consider these in relation to the DBM results. Such existing models can take a range of different forms, from the large computer simulation models, as discussed later in stages 4 through 6, to relatively small conceptual models. In hydrology, for example, these have been termed “hybrid metric-conceptual” models (Wheater et al., 1993). Whatever their form, however, these models need to be evaluated carefully in relation to the results obtained at the inductive DBM modeling stage 2. For example, smaller conceptual models may have elements that can immediately and usefully replace any equivalent elements in the inductively identified DBM model that are less satisfactory in physically meaningful terms and require further elucidation in this regard. An example of such hypothetico–inductive DBM (HI-DBM) analysis is outlined in Sec. 33.5. In the case of large, high-order simulation models, this HI-DBM analysis can lead on from any emulation modeling that has been carried out, as discussed later in stage 6. Here, the dynamic behavior of the large model is emulated by a lower-order model that represents its dominant modes of dynamic behavior, and the HI-DBM model may involve the consideration of conceptual elements identified by such emulation. 4. Stochastic simulation modeling: In the initial phases of modeling, it may be that real observational data will be scarce, so that major modeling effort is often centered on simulation modeling, normally based on largely deterministic concepts, such as the conservation laws (mass, energy momentum, etc.). One approach in this situation is to convert these deterministic simulation equations into a stochastic form by assuming that the associated parameters and inputs are inherently uncertain and can only be characterized in some suitable stochastic form, such as a probability distribution function (pdf ) for the parameters and a continuous or discrete time-series model for the inputs. The subsequent stochastic analysis normally uses Monte Carlo Simulation (MCS) analysis to perform sensitivity analysis (see, e.g., Ratto et al., 2007, and the prior references therein) and identify the most important parameters which lead to the simulated model behavior. Such analysis can be useful for DBM and HI-DBM modeling because it identifies those aspects of the large model that appear most important in defining its dominant dynamic behavior. This approach will not be discussed in this chapter, but a typical example of the methodology is described in Parkinson and Young (1998). 5. Dominant mode analysis (DMA): While sensitivity analysis can be useful in defining important behavioral mechanisms in a large, highorder model, DMA has proven to be more important as a precursor to DBM modeling because it utilizes the same statistical modeling methodology and tools employed in stages 2 and 3. Here, these tools are applied to time-series data obtained from planned experimentation, not on the system itself but on the high-order computer simulation model of

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Linear DBM Models    33-3 

the system that, in effect, becomes a surrogate for the real system. In particular, the analysis yields low-order approximations to the highorder simulation model that are often able to explain its dynamic response characteristics to a remarkably accurate degree with more than 99.99% of the large model output variance explained by the reduced order model output. Typical examples are the dominant mode emulation of large climate simulation models in Young et al. (1996) and Young and Parkinson (2002). Such a reduced-order model is referred to as nominal emulation model (NEM) because it emulates the dynamic behavior of the large simulation model for a nominal set of its parameter values. 6. Dynamic emulation modeling: A more complete understanding of the links between a high-order simulation model and its reduced-order representation obtained in stage 5 can often be obtained by performing multiple DMA analysis over a user-specified range of simulation model parameter values. The mapping between the large and reduced-order model parameters or responses then yields a full dynamic emulation (or meta) model (DEM) that can replace the simulation model over a wide range of parameter values. This approach to high-order model emulation is based primarily on the previous success of DMA in emulating large climate simulation models in the nominal sense mentioned above. It was introduced in Young and Ratto (2009), while Young and Ratto (2011) describe in detail two methods of emulation: stand-alone parameter mapping and response mapping, with application to the emulation of the Nash-Cascade hydrological model and a large economic model. A comprehensive discussion on emulation modeling of all types is available in Castelletti et al. (2012). 7. Hypothetico–deductive and inductive model reconciliation: If emulation modeling has been carried out prior to the acquisition of data, then, if at all possible, the DBM model obtained at the inductive DBM modeling stage 2 should be reconciled with the dynamic emulation version of the simulation model. Otherwise, if time-series data are available at the start of the study and a purely inductive DBM model is available, any available and relevant emulation model that merits such investigation should be considered at this stage and reconciled with the DBM model. Although such reconciliation will depend on the nature of the application being considered, the DBM model obtained from the real data should have strong similarities with the reduced-order dynamic emulation model. If this is not the case, the differences need to be investigated, with the aim of linking the reduced-order model with the high-order simulation model via the parametric mapping of the dynamic emulation model. An example of such analysis is described in Chap. 12 of Young (2011). 8. Conditional validation: An important stage of DBM model synthesis should always be an attempt at model validation (see, e.g., Young, 2001a). The word “attempt” is important since validation is a complex process and even its definition is controversial. Some academics (e.g., Konikow and Bredehoeft, 1992), within a ground-water context, and Oreskes et al. (1994) in relation to the whole of the earth sciences, question even the possibility of validating models. However, one specific, quantitative aspect of validation is widely accepted, namely “predictive validation” or “cross-validation,” in which the predictive potential of the model is evaluated on data other than that used in the identification and estimation stages of the analysis. When validated in this narrow sense, it can be assumed that this conditionally valid model represents the best theory of behavior currently available that has not yet been falsified in a Popperian sense. Statistical evaluation of the model, by exploring whether the statistical diagnostics are satisfactory (e.g., no significant autocorrelation in the residuals or cross correlation between the residuals and input variables; no evidence of un-modeled nonlinearity, etc.), is always possible and can engender greater confidence in the conditional validity of the model. Of course, the DBM modeler has to be careful when considering such diagnostics. Residual autocorrelation will almost always be present in models obtained from the analysis of real data. Consequently, scientific judgment is necessary to decide whether it is acceptably described by a purely stochastic process (aleatory error), or whether it is caused in part by model deficiency that needs further attention (epistemic error), see, e.g., Helton and Burmaster (1996). 9. Continuing data assimilation: Model validation is not the end of the modeling process. As additional data are received, they should be used to evaluate further the model’s ability to meet its objectives. Then, if possible, both the model parameters and structure can be modified if they are inadequate in any way. This process, often referred to as data assimilation, can be achieved in a variety of ways. Since most data assimilation methods attempt to mimic the Kalman filter (KF), however, it is likely to involve recursive updating of the model parameter and state estimates in

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some manner, as well as the use of the model in a forecasting (predictive) sense; see, e.g., Young (2010b). The exact form of these DA procedures will depend to some extent on the nature of the application and the associated DBM model, but they can range from relatively simple, computationally efficient methods based around techniques such as the KF, to computationally intensive numerical Bayesian procedures that exploit Monte Carlo methods, such as the ensemble Kalman filter (EKF), particle filter (PF), and unscented Kalman filter (UKF). This process of data assimilation is often made simpler in the DBM case because the estimation methods used in DBM modeling are all inherently recursive in form (Young, 2011) and so can be used directly to process monitored data in real-time, as it is acquired (Young, 2011, 2002; Romanowicz et al., 2006). 10. Application: The ultimate utility of a DBM model is how well it attains its objectives. Normally these objectives are dominated by its ability to perform defined practical tasks. In addition to the advancement of our understanding of the system being modeled, the most obvious practical applications in hydrology are adaptive flow/level forecasting and control, as well as model-based flood warning and management systems. Normally, data assimilation includes forecasting rather naturally because the updating of the model parameters and state variables includes both the interpolation and extrapolation of state variables that are important to the achievement of the modeling objectives. Moreover, it is interesting to note that forecasting is either an explicit or implicit aspect of many advanced control systems, such as those based on model predictive control (MPC), so that DBM models are in an ideal form for utilization in the design of such control systems within a hydrological context. As regards the implementation of the above stages in DBM modeling, it is important to note that the DBM approach is generic, so that any suitable stochastic-dynamic model type and associated optimization procedure can be utilized when it is applied in practice to observational time-series data. However, the models and statistical modeling methods outlined in the subsequent sections have been developed specifically with the DBM type of modeling in mind and are implemented in the CAPTAIN Toolbox. 33.3 LINEAR DBM MODELS

Linear models are obviously useful for modeling linearly behaving hydrological systems. However, they can also be used for piece-wise linear systems, where the behavior is linear under similar environmental conditions. For example, in the case of rainfall-flow data, they can describe single storm events that occur under the same soil-moisture and flow conditions. And following from this, they can form a significant component in a nonlinear rainfall-flow model where the nonlinearity acts on the input rainfall, producing the effective rainfall that provides an input to the underlying, predominantly linear system. We shall see later, in Sec. 33.4, that such a nonlinear system is a special example of a state-dependent parameter model. A typical, linear, continuous-time, DBM model is normally presented in a useful transfer function (TF) form, but, to start with, these will be considered also in the equivalent differential equation form that is more familiar to a wider audience. In the case where a single output variable x(t) (e.g., flow, level pollution concentration, etc.) is related to nu input variables ui(t), i = 1, 2, . . . , nu, the simplest model takes the following general form [see, e.g., Chap. 8 in Young (2011)]: nu d mi ui (t − τ i ) d n−1x (t ) d n x (t ) + a1 + . . . + an x (t ) = ∑ bi 0 + . . . + bimi u(t − τ i ) n n−1 dt mi dt dt i =1 (33.1)

where τi , i = 1, 2, . . . , nu are pure time delays to allow for any such delays in the system dynamics, while aj , j = 1, 2, . . . , n and bij , i = 1, 2, . . . , nu; j = 0, 1, 2, . . . , mi are normally constant coefficients. A model such as this can be transformed straightforwardly into an equivalent state-space form. More importantly in the present DBM modeling context, however, it can also be written as a TF model in the equivalent common denominator operator form:

nu

x (t ) = ∑ i =1

bi 0s mi + bi1s mi −1 +  + bimi n

s + a1s

n−1

+  + an

nu

ui (t − τ i ) = ∑ i =1

Bi (s ) ui (t − τ i ) (33.2) A(s )

where sr is the derivative operator, i.e., sr = dr/dtr and {Bi(s); A(s)}, i = 1, 2,. . ., nu are the appropriately defined polynomials in this operator. The model can be extended easily to additive TFs with different denominators, i.e., where A(s) in (33.2) is replaced by Ai(s). And a multi-output system can be defined as an

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33-4    Data-Based Mechanistic Modeling

amalgamation of single output models such as this. Given that state space (SS) models are so popular in recent hydrological systems analysis, the reader might ask why such a multivariable TF model is favored in DBM modeling. This is discussed in Young (2013), but the main reasons are (1) the uniqueness and parametric efficiency of the TF model which, for example, has only n + m parameters in the SISO case compared with n2 + n for the general SS alternative; and (2) the fact that it can be converted easily to any SS model whose state variables best suite the nature of the hydrological system being modeled. The generic model used for the analysis reported in the present paper is the multi-input, single-output (MISO) hybrid Box-Jenkins model, which is simply a stochastic version of the TF model (33.2) with different denominators and an additional, serially correlated or “colored” noise component, denoted at the sampling instants by ξ(k), k = 1, 2, . . . , N, where N is the total sample size and the uniform sampling interval is denoted by ∆t time units.1 This model is composed of the following three component equations: nu

Deterministic Model: x (t ) = ∑



i =1

    Noise Model: ξ (k ) =



Bi (s ) ui (t − τ i ) (33.3a) Ai (s )

D( z −1 ) e(k ); C( z −1 )

where the polynomials Ai (s) and Bi (s) are defined as follows: Ai (s) = sni + ai1sni −1 + ai2 sni −2 + . . . + aini ;  i = 1, 2, . . . , nu Bi (s) = bi0smi + bi1smi −1 + bi2 smi −2 + . . . + bimi ;  i = 1, 2, . . . , nu

(33.4)

and the i subscript is omitted in the single-input, single-output (SISO) case. Note that this is a hybrid (continuous-discrete) model because the main, underlying deterministic model in Eq. (33.3a) is represented by a continuoustime transfer function with output and input variables x(t) and ui(t), i = 1, 2, . . . , nu, having a time argument t; and x(k) is the sampled measure of x(t) at the sampling instants k = 1, 2, . . . , N, with an argument k. The y(k) observations are then x(k) contaminated by the colored noise ξ(k), as shown in Eq. (33.3c). This colored noise, which represents the effect of all stochastic influences on the system, including measurement and modeling errors, as well as unmeasured or diffuse inputs, is assumed in Eq. (33.3c) to be generated by passing the zero mean, serially uncorrelated sequence of random variables (white noise) e(k), with variance σ2, through an autoregressive moving average (ARMA) filter (see Young, 2011), which converts the white noise into the serially correlated, colored sequence of random variables ξ(k). The associated polynomials C (z−1) and D(z−1) are defined as follows: C (z−1) = 1 + c1 z−1 + c2 z−2 + . . . + cp z−p

D(z−1) = 1 + d1 z−1 + d2 z−2 + . . . + dq z−q

(33.5)

where z−r is the backward shift operator, i.e., z−r ξ(k) = ξ(k − r). The completely discrete-time equivalent of the model 33.3 can be obtained by discretization at the specified sampling interval of ∆T time units (see, e.g., App. B in Young, 2011) and takes the form: nu

 System TF Model: x (k ) = ∑ i =1

Bi ( z −1 ) u(k − δ i ) (33.6a) Ai ( z −1 )

D( z −1 ) e(k ); C( z −1 )

e(k ) = N (0, σ 2 ) (33.6b)

   Output Observation : y (k ) = x (k ) + ξ (k )

(33.6c)

  Noise TF Model:   ξ (k ) =

where δi are the time delays measured in sampling intervals and the main TF polynomials are defined as follows: 1 Nonuniform sampling is possible in the case of continuous-time models (see, e.g., Garnier and Young, 2014), but this is not considered here.

33_Singh_ch33_p33.1-33.12.indd 4

i = 1, 2, . . . , nu

Bi (z ) = bi0 + bi1 z + bi2 z + . . . + bimi z

i = 1, 2, . . . , nu

−1

−1

 ;

−mi

−2

(33.7) For convenience, the parametric nomenclature of the main transfer function model in Eq. (33.6a) is made the same as in (33.3a). However, it is important to note that these parameters are quite different since, in the completely discrete-time model (33.6a), they are dependent on the sampling interval ∆t. In other words, for a unique continuous-time model such as Eq. (33.3), there are an infinite number of discrete-time models (33.6). In each case, the discrete-time model can be obtained from the continuous-time model by the continuous to discrete model transform, which is available in Matlab as the routine c2d. This transformation is dependent on an assumption about the behavior of the input variable u(t) between its sampled values u(k), where the most common assumption is the zero-order hold (zoh), in which the input is assumed to remain constant over the sampling interval. This unique nature of the continuous-time model reveals another advantage of this model form: once its structure has been identified and the associated parameters have been estimated, it can be converted to a discrete-time model at any sampling interval that best suits the modeling objectives.

e(k ) = N (0, σ 2 ) (33.3b)

Observation Equation: y (k ) = x (k ) + ξ (k ) (33.3c)



Ai (z−1) = 1 + ai1 z−1 + ai2 z−2 + . . . + aini z−ni ;

The full model structure for both of the above models is defined by [n1 . . . nnu m1 . . . mnu τ1 . . . τnu p q]

(33.8)

although, in the later practical examples, it is sometimes convenient to refer only to the deterministic part of the model, which defines the structure of the main hydrological relationship between the input and the output; i.e., p and q are omitted, while the subscript 1, 2, . . . , nu is also omitted in the SISO case, where the structure is simply [n m τ]. Identification of these model structure parameters, as well as the estimation of the associated polynomial parameters that characterize the model, is a well-known statistical problem. In DBM modeling, it is normally solved by maximum likelihood optimization using the refined instrumental variable (RIV) methods for estimating continuoustime (RIVC) and discrete-time (RIVD) models of this type (see Young, 2011, 2015, and the prior references therein). These are available in the CAPTAIN Toolbox as the rivcbjid and rivbjid routines, for model structure identification and parameter estimation; and the rivcbj and rivbj routines for subsequent final estimation of the parameters in the identified structure, together with their associated statistical uncertainty bounds. The most common measure of how well the model explains the data is the simulation coefficient of determination, R2T , which is directly equivalent to the Nash-Sutcliffe efficiency (Nash and Sutcliffe, 1970) used in hydrology, and is defined as follows:

{

R2T = 1 − σ ξ2 /σ

2 y

}

(33.9)

where σ y2 is the mean square value of the flow y(k) about its mean value; and σ ξ2 is similarly the mean square value of the model simulation error ξˆ(k) = y(k) − xˆ(k), where xˆ(k) is the noise-free, deterministic output of the model, i.e., model-generated value of the sampled output x(k) in the models (33.3) and (33.6). There are numerous model structure (order) identification criteria, but only the most common of these (AIC, BIC, and YIC: see pages 176–179 in Young, 2011) are available in rivcbjid and rivbjid. Here, the AIC and BIC are based on a cost function that balances the variance of the residual errors, ê(k) [or ξˆ(k) if C (z−1 ) = D(z−1 ) = 1.0], with the number of parameters in the model, while the YIC is based on the properties of the instrumental product matrix (IPM) used in instrumental variable algorithms such as RIV estimation (see Wellstead, 1978; Young et al., 1980). Model order identification can also be aided by recursive parameter estimation, which is available as an option in the CAPTAIN rivcbj and rivbj routines. In particular, it can help in diagnosing whether there are any signs of either time variation (see Sec. 33.3), poor identifiability, or the presence of epistemic error, where the recursive estimates of the nominally constant parameters tend to vary, with large uncertainty bounds and no clear convergence (see, e.g., Young, 2010c or 2011). 33.3.1  A Solute Transport and Dispersion Example

The DBM approach in this context has been described in a number of papers (see, e.g., Beer and Young, 1983; Wallis et al., 1989; Blazkova et al., 2012), where the resulting model has been termed the aggregated dead zone (ADZ) model for solute concentration changes in a reach of a river or channel system. In the case of solute transport through soils (Beven and Young, 1988),

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Linear DBM Models    33-5 

b0 s + b1 b0 s + b1 x (t ) = u(t − 42) = 4 u(t − 42); (s + α 1 )3 (s + α 2 ) s + a1s 3 + a2 s 2 + a3s + a4

5 95% confidence bounds Model output Tracer data

4.5 4 Concentration (mg/L)

this name has been modified to aggregated mixing zone (AMZ), and for more general applications (Young and Lees, 1993), the active mixing volume (AMV). The ADZ model is compared with the distributed parameter advection dispersion equation (ADE) model in Young and Wallis (1993). The term “dead zone” is somewhat misleading because these are the zones where most mixing occurs, but it derives from earlier literature on the subject (see Valentine and Wood, 1977). Research on solute transport and dispersion is aided by the ability to conduct simple planned experiments using conservative tracer materials, such as the fluorescent red dye, Rhodamine WT (see, e.g., Wallis et al., 1989), or potassium bromide (KBr), as used in the tracer experiments of Martinez and Wise (2003), who kindly provided me with the resultant data set utilized below. These consist of the tracer concentration changes monitored at various locations in a wetland area of Florida, following an impulsive or gulp injection u(k) of the tracer at the upstream, input location. This study, which was designed to investigate the effects of domestic wastewater passing through the system, used a sampling interval of ∆T = 2 hours. With such an impulse input of tracer material, one might expect there to be some initial mixing behavior before the concentration changes y(k) become relatively smooth at any of the downstream output sites. This needs to be noted when interpreting the nature of the estimated DBM model (see the following text). The standard DBM analysis of such tracer data is to use them to identify and estimate a TF model, in either continuous or discrete-time form, based on the measurements u(k) and y(k). Discrete-time DBM modeling is described in Young (2001b), where a [4 2 22 0 0] model is identified for the TF between u(k) and y(k), with the latter measured at one of the output sites and the model estimated in a constrained form to ensure four physically interpretable real eigenvalues in the estimated TF model. However, it is clear that a continuous-time DBM model makes more sense since most solute transport and dispersion models are formulated in continuous-time, differential, or partial differential equation (PDE) terms that relate directly to mass conservation (see later). Such a continuous-time analysis is described in Young (2011), but here we will re-consider the data is a slightly different manner using hybrid continuous-time modeling with the rivcbjid/rivcbj routines in CAPTAIN. As in the previous analysis of these data, the roots (poles) of this estimated continuous-time TF denominator polynomial are complex so, using the same arguments as those used in the above references, it is constrained to have real poles (see Special TF Optimization in CAPTAIN in the “Technical Matters” section of http://captaintoolbox.co.uk/Captain_Toolbox.html/Captain_ Toolbox.html). The basic TF model structure is identified well, with a fourth order denominator and first-order numerator, as in the previous analysis. However, there is some uncertainty in the pure time delay, which was identified in Young (2001b) and Young (2011) as 22 and 20 sampling intervals, respectively. In order to see how selecting a different time delay affects the continuous-time modeling results, therefore, let us consider a [4 2 21 20 0] structure, in which the additive colored noise process is identified as an AR(20) process: i.e.,

3.5 3 2.5 2 1.5 1 0.5 0 –0.5

0

100

200

300

400 500 600 Time (hours)

700

800

900

Figure 33.1  The output of the DBM aggregated dead zone model compared with the tracer data (note: impulsive input of 186.3 applied after 82 hours).

differential equation model representing the changes in the mass of solute M (t), based on simple conservation considerations:

dM (t ) d {Vc2 (t )} = = Q(t − τ )c1 (t ) − Q(t )c2 + R(t ) dt dt

(33.12)

where pure time delay τ is introduced to account for advection; R(t) accounts for any solute gains or losses within the reach; c1(t) and c2(t) are the concentrations of the solute at the input and output of the reach, respectively, and V is the AMV, i.e., the volume of water in the reach that is estimated to be effective in the transport and dispersion of the solute. In general, because of the inefficient mixing in the reach as a whole, V < Va, where Va is the actual volume of water in the reach. Note that, for simplicity, the concentration within the AMV is assumed to be the same as that measured at the output of the reach c2(t). Thus, if we let x(t) = c2(t), u(t) = c1(t), and R(t) = 0 (no inflow within the reach), the equation can be rearranged to the form

Q dx (t ) Q = u(t − τ ) − x (t ) (33.13) V dt V

or, in TF terms,

x (t ) =

b0 u(t − τ ) (33.14) s + a1

where a1, = b0 = Q/V, so that the steady state gain G = b0/a1 = 1.0 (fully conser vative with no solute loss over the reach) and the time constant, here the residence time, is T = V/Q time units, while the travel time is defined as τ + T. If (33.10) e(k ) = N (0,σ 2 ) R(t) ≠ 0, then G < 1.0 or > 1.0, representing loss or addition of solute, respectively, over the reach. This model is a single-input, first-order example of the general continuous-time model (33.3a). From a DBM perspective, however, it is important to note that, as in this example, a single ADZ model may not be where the reader can evaluate the values of a1 to a4 in terms of α1 and α2. Note able to explain all of the concentration changes in the reach so a higher-order here that the AR(20) noise model indicates considerable serial correlation in model, consisting of several first-order ADZ elements, such as Eq. (33.14), in ξ(k), although with very low variance (σ2 = 0.007, see below), and that the series or parallel, may be identified statistically from the tracer data, dependtime delay of 21 samples becomes a temporal time delay of 42 hours because ing on the length and nature of the reach, as well as the type of input u(t). In of the ∆t = 2-hour sampling interval. The parameter estimates are as follows, this more general case, the total travel time is the sum of the pure time delay with the adjusted (see as follows) standard errors shown in parentheses: τ and the time constants of the ADZ elements. This is, of course, the type of higher-order model identified in Eq. (33.10) for the Martinez-Wise data. αˆ1 = 0.0801(0.0013); αˆ2 = 0.0101(0.00037); σˆ2 = 0.007 Returning to this identified model, the autocorrelation function of the estimated residuals e(k) shows that they are reasonably white, with low bˆ0 = 2.040 × 10−4 (1.0213 × 10−5 ); bˆ1 = 5.0929 × 10−6 (1.998 × 10−7) variance, although they are somewhat heteroscedastic (i.e., their variance (33.11) changes over time, a quite common characteristic of hydrological systems). In order to allow for this changing variance, it has been estimated using time The tracer data are explained well by this model, as shown in Fig. 33.1, with variable parameter estimation; see Sec. 33.4 and Young and Pedregal (1996). R2T = 0.997% (i.e., 99.7% of the output variance explained by the model). This is used to adjust the parametric error covariance matrices that define the From the DBM modeling standpoint, it is now necessary to interpret the standard errors on the parameters, as well as the MCS analysis used to infer model (33.10) in physically meaningful terms. Assuming that there is a the uncertainties in the estimates of the derived time constants (T1 = 1/α1 and constant flow rate Q m3/s out of a reach in a river system, the simplest continT2 = 1/α2 ), the steady-state gain, G = b1 /a4, and the 95% confidence bounds uous-time ADZ model is the following lumped parameter, first-order

y(k ) = x (k ) + ξ (k ) 1 ξ (k ) = e( k ) 1 + c1z −1 +  + c20 z −20

33_Singh_ch33_p33.1-33.12.indd 5

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33-6    Data-Based Mechanistic Modeling

on the model response in Fig. 33.1. This MCS analysis, based on 10,000 random realizations, yields the sample distributions for the three parameters shown in Fig. 33.2 and the associated estimates of the parameters are as follows: Tˆ1 = 12.49(0.196); Tˆ2 = 99.34(3.65); Gˆ = 0.987(0.084) (33.15) which compare well with the estimates obtained in the previous analysis (Young, 2011). Note how, in statistical terms, Gˆ = 0.987 ± 0.084 is insignificantly different from unity, so that no tracer appears to have been lost between input and output, and the varying width of the 95% confidence bounds in Fig. 33.1 is a reflection of the changing variance in the residuals e(k). The sharp widening around 115 to 125 hours, for example, indicates the uncertainty in the estimation of the advective time delay arising from the lumped approximation of this delay, when the physical process is naturally more distributed, or possibly some variation in the time delay over the test period. However, this limitation of the ADZ model has only a minor effect on the overall dynamics of the estimated model. Finally, how can we interpret the higher-order nature of the model (33.10)? Such an interpretation is likely to be somewhat subjective, but I feel that part of the model is accounting for the initial nonuniform mixing following the initial impulsive input of the tracer. This illustrates the well-known experimental conclusion that it is best to base solute transport and dispersion modeling on an input measured someway down the system from the injection point, where mixing has stabilized. A DBM analysis of this kind, again using the Martinez-Wise data, is described in Young (2011), where a model based on inputs downstream of the injection site is identified, in which all of the TF elements are now first order. 33.4  TIME VARIABLE AND STATE-DEPENDENT PARAMETER MODELS

Although many processes in hydrology are nonlinear, the range of application of linear models in hydrology can be extended by considering time variable parameter (TVP) models, whose parameters and structure may change over time. This leads to the idea of adaptive linear models whose dynamics can change in response to the updating of the model parameters using recursive

Slow path time constant, T2

Steady-state gain, G

1

1

0.8

0.8

0.8

0.6

0.4

0.2

Normalized frequency

1

Normalized frequency

Normalized frequency

Quick path time constant, T1

parameter estimation (Young, 2011) within a data assimilation context. The RIV class of estimation algorithms have the advantage that they are based on “pseudo-linear regression” (Solo, 1978) and can, therefore, exploit the optimal IV version of the best known and simplest of recursive algorithms, the recursive linear least squares (RLS) algorithm, thus allowing for the estimation of time-variable parameters. Remarkably, RLS was developed originally by Gauss sometime before 1826 (see App. A of Young, 2011) and can be interpreted as a simple example of the KF (see, e.g., Young, 2010b). There is insufficient space available in this chapter to discuss recursive estimation in detail, but the above references provide a full description of its development and application. If the time variations in the parameters of an estimated TVP model are found to be related to the variations of measured variables from the system under study, they can be assumed to be state dependent. Such state-dependent parameter (SDP) models (Priestley, 1988; Young, 1993, 2000, 2001c; Ratto et al., 2007) have figured prominently in the development of nonlinear DBM models for rainfall-flow models from early contributions in Young (1993) and Young and Beven (1994) to recent publications, such as Romanowicz et al. (2006), McIntyre et al. (2011), Young (2013), and Leedal et al. (2013). The complete SDP transfer function model is reviewed in Young (2013), where it is then applied to the well-known data from the humid Leaf River basin, as outlined in Sec. 33.5. Here, for simplicity and to illustrate how the methodology is widely applicable, let us consider the analysis of daily data from another, quite different semi-arid catchment, analysis that any reader who has access to Matlab and the CAPTAIN Toolbox can run through using the command-line demonstration example provided in the CAPTAIN package. This rfdemo example is illuminating since it shows how both TVP and SDP modeling can be exploited in this rainfall-flow context. It concerns the DBM modeling of daily rainfall-flow data from the Canning, an ephemeral river in Western Australia (for further details, see Young et al., 1997; Young, 2008). The discrete-time model is identified and estimated on the basis of the data from 1985.2 to 1987.1, and it is then validated on two sets of data from 1977 to 1978.5 and 1979.4 to 1980. For those readers who are not able to use the demonstration example, however, let us briefly review the results obtained. First, the analysis shows that although a linear model is unable to explain the data in a satisfactory manner, a first-order TVP model estimated using

0.6

0.4

0.2

0 11.5

12

12.5

13

13.5

Residence time (hours)

0

0.6

0.4

0.2

90

100

110

120

0 0.6

Residence time (hours)

0.8

1

1.2

1.4

Conservativity measure

Figure 33.2  MCS estimated sample distributions of the three derived parameters T1 , T2, and G.

33_Singh_ch33_p33.1-33.12.indd 6

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HypotheticO–Inductive DBM Modeling    33-7 

the dtfm routine in CAPTAIN reveals that the main identified parameter variation is in the numerator b0 parameter, with little estimated change in the denominator a1 parameter. The resulting TVP model provides a large improvement in modeling ability, with R2T = 0.963. Moreover, the TVP estimates of the numerator parameter correlate very well with the measured flow. It is not surprising, therefore, that if flow is considered as the variable on which the numerator parameter is dependent, non-parametric SDP estimation, using the sdp routine in CAPTAIN, also produces a quite good first-order SDP model, with R2T = 0.910. Final identification of the model structure using the rivbj routine suggests a [2 3 0 0 0] model and the resulting estimated model has an R2T = 0.958. This model validates well on the periods noted above, with R2T = 0.954 and 0.928, respectively. The use of flow as the SDP variable may seem strange at first sight, but, not surprisingly, it appears to be acting as a good surrogate measure of the soil moisture. Moreover, the estimated TF model can be decomposed by the residuez routine in Matlab to a parallel pathway model, where the instantaneous effect accounts for a small 5.84% of the flow; the quick pathway, probably representing the surface flow contributions to the flow changes, has a time constant of 2.57 days (contributing 52.53% of the flow), while the slow pathway, probably accounting mainly for the subsurface and groundwater effects (and contributing 41.63% of the flow), has a time constant of 18.7 days. These provide a reasonable physical interpretation of the model in hydrological terms and, therefore, satisfy the DBM modeling requirements in this regard. Note that, in the demonstration example, the simplified refined instrumental variable (SRIV) is used for final model estimation because the noise ξ(k) is heteroscedastic and its modeling by an ARMA process is questionable, so p and q are set to zero. However, the SRIV algorithm produces consistent estimates with low variance, so the results are reliable. For completeness, however, full RIV identification suggests a [2 3 0 6 0] model with an AR(6) noise model. This model has an R2T = 0.953 and the residual variance estimate is σ2 = 0.0031, i.e., standard deviation of 0.056 m3/s. The main reason for estimating such a discrete-time, fully stochastic model is that it makes flow forecasting based on the state space, KF form of the model, with the state variables defined by the outputs of the quick and slow flow pathways, quite straightforward (see, e.g., Young, 2002; Romanowicz et al., 2006; Young, 2011; and the prior references cited therein). In order to remove the need for forecasting rainfall, which is always a daunting task, this requires that there is a pure time delay between rainfall and flow and, in this example, the model does not have such a delay. Fortunately, however, a false pure time delay of 1 day can be introduced into the model, yielding an identified [2 3 1 8 0] structure that leads to a reasonable 1-day-ahead forecasting error standard deviation of 0.060 m3/s.

33.5 HYPOTHETICO–INDUCTIVE DBM MODELING

Hypothetic–inductive DBM (HI-DBM) modeling cannot be generalized because it depends on the particular application and the available hypothetical elements that can be evaluated in the context of the identified and estimated DBM model. However, a recent, comprehensive example (Young, 2013) illustrates well the nature of the HI-DBM modeling approach. Here, the HI-DBM analysis is applied to daily mean areal rainfall (mm/d), potential evapo transpiration (PET) in mm/d, and streamflow (m3/d) data from the well-known, humid, Leaf River basin (1944 km2 ) located north of Collins, Mississippi, over the 40 water-years from October 1948 to September 1988. The associated HyMOD conceptual model (e.g., Vrugt et al., 2009, and the prior references therein) provides the hypothetical element in the form of the probability distributed model (PDM) hypothesis; namely, a procedure for soil moisture accounting and determining the value of runoff production according to a probability-distributed storage capacity model that is driven by the rainfall, r(k), and PET, Ev(k), inputs (see Moore, 2007). As in the case of the most rainfall-flow models, such as the Canning River model considered in the previous section, HyMOD is of a Hammerstein form, with the PDM element converting the measured rainfall into an effective rainfall measure. However, the model is unusual because this effective rainfall provides the input to a fourth-order linear dynamic system, rather than the more normal second-order system (as in the Canning River example). This fourth-order system consists of a third-order Nash Cascade (Nash, 1958), i.e., a serial connection of three identical, quick-flow storage tanks, each modeled as a first-order discrete-time system, in parallel with a single slow-flow tank, again modeled as a first-order discrete-time system. There are two versions of the HyMOD model (see Young, 2013) and Fig. 33.3a compares the optimized responses of these models over a typical section of the Leaf River data. Version 2 explains the data best, with R2T = 0.814. This atypical fourth-order structure of the HyMod model arises from the somewhat unusual response characteristics of the Leaf River and is reflected also in the initial inductive DBM modeling analysis, which identifies a Hammerstein model form, with a power-law nonlinearity, similar to that identified for the Canning River in Sec. 33.4, in series with a number of possible linear model structures which explain the flow data better than HyMOD. One of these, like HyMod, is a fourth-order [4 4 0 1 5] model, with its eigenvalues constrained to be real. However, the best identified linear model element is a simple, unconstrained, first-order [1 7 0 1 5] TF model, with an R2T = 0.871. This is an interesting result because it shows that the fast response is complete in 7 days and so can be modeled very well by the seven coefficients in the TF numerator polynomial, with the very low-amplitude slow response modeled by the first-order TF denominator polynomial, with an

30 HYMOD 95% uncertainty bounds HYMOD model version 1 Measured flow HYMOD model version 2 HYMOD model version 2 (Vrugt)

25

95% uncertainty bounds Version 1 HI-DBM model Measured flow Version 1 HI-DBM ex-post predictions

30

25

Flow (mm)

Flow (mm)

20

15

10

15

10

5

0

20

5

1949.3

1949.35

1949.4 Date

1949.45

0

1949.3

1949.35

1949.4 Date

1949.45

Figure 33.3  The simulated outputs of selected optimized HyMOD and DBM models compared with the flow data y(k) over a typical short segment of the Leaf River data: (a) left, HYMOD; (b) right, HI-DBM.

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33-8    Data-Based Mechanistic Modeling

associated time constant of 31.9 days, within a standard error range of between 28.4 and 36.2 days. Moreover, the response does not contain the slower, second-order exponential recession that normally characterizes well the response of most rainfall-flow hydrographs and is usually modeled by a second-order process, with quick and slow dynamic modes, as in the Canning River example and most other previously identified DBM models of rainfallflow behavior. It is interesting to note that the unit hydrograph (unit impulse response) of the [1 7 0 1 5] model is redolent of higher-order, serial connections of firstorder dynamic processes and is probably the reason why the Nash Cascade of three quick-flow stores had to be used in the HyMOD model, rather than the more usual, single store. In fact, if the number of quick-flow tanks in the model is increased to 10 (with no increase in the number of estimated parameters since the tanks are identical), then the estimated model explains the flow data quite well, with R2T = 0.860 compared with the standard HyMOD model’s R2T = 0.814. Moreover all of the parameter estimates seem well defined, without any signs of poor identifiability, suggesting that the previously utilized HyMOD model structure for the Leaf River data might be best modified to this alternative, higher order form. This illustrates how the initial, simple, and computationally efficient DBM modeling can be useful in a diagnostic role, quickly revealing unusual dynamic model structures, such as the [1 7 0 1 5] DBM model and the 10-tank HyMOD model, models that are capable of efficiently characterizing the Leaf River dynamics and yet might be unlikely products of conceptual model formulation. The HI-DBM model is obtained by replacing the effective rainfall nonlinearity in the DBM model by the conceptual PDM effective rainfall mechanism in the HyMOD model. As such, this change not only introduces a conceptual nonlinear mechanism that has a clearer physical interpretation than the SDP powerlaw nonlinearity, but it also converts the DBM model from one that is limited to forecasting applications (see Young, 2013) into a stochastic simulation model that has a much wider application potential. Based on these considerations and the initial DBM modeling results, the general HI-DBM model has the structure shown in Fig. 33.4. Although, in the DBM modeling analysis, the constrained [4 4 0] TF model does not yield as good estimation and validation results as the unconstrained [1 7 0] model, it is, perhaps, more appealing to hydrologists because it has a structure that is similar to that used in the HyMOD model. The main difference between the structure of the TF model in this figure and the linear part of the HyMOD model is that the Nash-Cascade of three, identical, first-order stores in HyMOD is modified by the addition of a simple, three-parameter, moving average prefilter PF(z−1) = 0.073 + 0.499z−1 + 0.344z−2 in the quick-flow routing equation. Since this prefilter polynomial operates only on the PDM-generated effective rainfall input ue (k), it is simply modifying ue (k) to better explain the flow data. And the fact that the model then explains these data significantly better than the standard HyMOD model suggests that this is one other area where the HyMOD model hypothesis might be reinvestigated.

Stochastic input

A dynamic emulation model (DEM) is a special model form designed to overcome some of the identifiability issues associated with high-order and complex hydrological simulation models and produce a low-order model that emulates the behavior of the simulation model as closely as possible over a specified range of its parameter values (see, e.g., Castelletti et al., 2012). One approach to DEM synthesis that exploits DBM modeling methodology and DMA (see stage 5 in Sec. 33.2) for model order reduction is described in Young and Ratto (2009) and shown diagrammatically in Fig. 33.5. This uses non-parametric or parametric nonlinear function estimation to identify and map the multivariate relationship between the high-order model parameters and those of the low-order model. The latter could involve a suitable known nonlinear function, as in the later practical example; a function identified by SDP-type estimation, as in Young and Ratto (2009); or, in more complex cases, smoothing spline ANOVA models (see, e.g., Gu, 2002). Normally, the emulation model obtained in this manner is in the form of either a constant parameter linear transfer function relationship; or a state-dependent parameter, nonlinear transfer function model (SISO, MISO, or MIMO); and it is estimated from discrete-time, sampled data in either continuous or discretetime, using the CAPTAIN tools or other equivalent estimation procedures. Examples of DBM emulation modeling applied to hydrological models are reported in Young and Ratto (2009), Young (2010a), and Young and Ratto (2011). Another typical example (Young et al., 2009) is based on an HEC-RAS model representation of the River Severn formed from 89 cross-sectional nodes between the Montford and Buildwas locations, which includes the town of Shrewsbury. HEC-RAS solves the dynamic Saint-Venant partial differential equation (PDE) model using an implicit, Preissman-type finite difference method. The main computational engine is based on the UNET computational scheme originally developed by Barkau et al. (1989) and this is run in an unsteady flow mode and forced with an upstream boundary condition defined by 15-minute observations of flow at Montford between December and March 2002. The DEM analysis was carried out by David Leedal at Lancaster and the six nodes through Shrewsbury were selected as the sites for the DEM output, using data from the HEC-RAS simulated water surface level for all the crosssection nodes. These nodes (references cs: 82, 79, 77, 73, 70, and 68) represent sites that may be useful for a subsequent real-time flood forecasting system. The purpose of the research is then to develop initial, low-order, DBM submodels to emulate the relationship between the upstream boundary condition as an input time series and the water surface level at each of the chosen HECRAS nodes as output series. The HEC-RAS simulated data with a nominal set of parameter values is shown in Fig. 33.6. In this case, given the finite difference nature of the simulation model, it makes sense to consider discrete-time models. Consequently, the initial DBM analysis of the data in Fig. 33.6 was carried out using the CAPTAIN rivbjid routine, which identifies the following first order, nonlinear SDP model:

ARMA(1,5) noise model

e(k)

D(z –1) C(z –1)

α = 0.876

PET

33.6  DBM EMULATION MODELING OF HIGH-ORDER SIMULATION MODELS

Nash-Cascade of 3 quick-flow tanks each time constant Tq = 0.98 days

Prefilter PF(z –1)

Ev(k) Rain

r(k)

xq(k)

Flow

87.6%

PDM rainfall nonlinearity F r(k),Ev(k), qs

{

x(k)

}

Effective rain ue(k) = F r (k), Ev(k), qs

{

1– a = 0.124

}

Slow-flow tank Time constant Ts = 25.4 days

+

y(k)

12.4%

xs(k)

Figure 33.4  Block diagram of the HI-DBM model for the Leaf River (from Young, 2013).

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DBM Emulation Modeling of High-Order Simulation Models     33-9 

High order model: (e.g., 17th order and circa 180 parameters)

High dynamic order model estimated parameters

Reduced order model: (e.g., 3rd order and 27 parameters)

Parameterized state-dependent parameter regression (SDR) relationships

. . . . . .

. . . . .

Reduced dynamic order model parameters

Figure 33.5  The process of dynamic emulation model synthesis: note that the order and number of parameters referred to for the high and low order models relate to a typical example (from Young and Ratto, 2009).



y j (k ) =

b j ,0

1 + a j ,1z −1

f j ( y j (k − 1)) u (k − δ ) + ξ j (k ); j = 1  6 (33.16)

where the slightly modified nomenclature aj, 1 and bj, 0 is used to differentiate this single-input, multiple output (SIMO) model from the MISO models used heretofore. The input transformation function fj (yj (k − 1)) is the SDP for cross-section j, with the parameter dependent on the measured yj (k − 1), which acts as a time-variable gain on the input data, as in the rainfall-flow

models of Secs. 33.4 and 33.5. The nonlinear SDP functions in each case were parameterized and estimated using the adaptive neuro-fuzzy inference system (ANFIS) algorithm from the Matlab Fuzzy Systems Toolbox package (see, e.g., Vernieuwe et al., 2005) and are plotted in Fig. 33.7a as a function of y(k–1) (denoted in the figure by yk–1). Table 33.1 displays the SRIV estimated TF parameter values âj, 1, ˆbj, 0, and δ for each cross-section. It also shows the time constants derived from the aj,l parameters, which demonstrate that the

5.5 5 4.5 4

Level (m)

3.5 3 2.5 2 1.5 1 0.5 0

0

200 input

cs 82

400 cs 79

600 Time (hours) cs 77

800 cs 73

1000 cs 70

1200 cs 68

Figure 33.6  The data used for nominal emulation model identification and estimation. The bold line is the upstream boundary level for the HEC-RAS simulation, the thin lines are levels at six cross-sections representing the River Severn through Shrewsbury (this and subsequent Fig. 33.7 are from Young et al., 2009).

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33-10    Data-Based Mechanistic Modeling

cs 82

cs 79

0.6

0.6

0.5

0.5

0.5

–0.8

cs 77

TFa parameter

0.4

0.4

0

2

1

cs 73

0.95

0.3

0.3 0

0.4

2

0

cs 70

1

0.95

0.9

0.75 0

yk–1

4 SDP

(a)

cs 68

0.8 0

ANFIS fit

–0.95

1.5 1

0.4 0

5

yk–1

–0.9

–1 0.5

0.6

0.85

0.8

2

0.8

0.9

0.85

1

–0.85

1

State−dependent parameter, Cj,k

0.7

yk–1

5

ANFIS fit conf. int.

Inbank mannings multiplier

(b)

1.4

1 0.6

Parametric fit

Outbank mannings multiplier Data

Figure 33.7  The estimated nonlinear functions of the HEC-RAS emulation model: (a) SDP input nonlinearities; (b) the emulation mapping surface for the â70,1 DBM emulation model parameter.

âj,1 = exp(− (pj,1 α p1j , 2 + pj,3 α p j , 4 )) + pj,5 (33.17) 2 p , 4

p ,2

where α 1j and α 2j are the HEC-RAS parameters (in-bank and out-ofbank scaling factors, respectively) and pj,i, where j indicates the cross-section identities from Fig. 33.7a, are the five mapping parameters, i = 1, 2,..., 5 optimized to minimize the sum of squares of the errors between âj,1 and the

mapped estimate from Eq. (33.17). Figure 33.7b compares the resulting estimated surface defined by Eq. (33.17) for â70,1 with the estimates of this parameter obtained from the MCS realizations. Despite its very simple lumped parameter approximation to the distributed parameter HEC-RAS model, the resulting DEM is quite successful, with a coefficient of determination in a validation run, based on a new Monte Carlo set generated using randomly selected values of the HEC-RAS model parameters, of R2T = 0.989 even though, for this illustrative exercise, no attempt was made to include the parameters of the input nonlinearity into the DEM analysis. In more general terms, the DBM emulation of PDE models such as this has also produced promising results. For example, another more recent study, similar to the previous (Tych and Young, 2012), has considered the emulation of the popular OTIS model, which simulates the distributed parameter transient storage (TS) model for solute transport and dispersion used by Martinez and Wise in their wetland area study (see Sec. 33.3.1). Figure 33.8 from the preceding reference shows a typical validation run for this DEM. Also, I have developed an emulation model for a semidistributed ADZ (SDADZ) model (see Sec. 12.5 of Chap. 12 in Young, 2011), which is a simpler alternative to the OTIS model, and compared the results with the discrete-time DBM model of the Martinez and Wise data. Finally, it is important to recall, from stages 5 through 7 of the DBM modeling strategy in Sec. 33.2, that the DEM model is not in any way an alternative to DBM modeling; it simply exploits DBM modeling methodology to evaluate how an existing large computer simulation model can be represented by a much lower-dimensional DBM-type of model in order to consider how this can be reconciled with a DBM model identified and estimated using real data from the same system.

0.9

1 Tracer concentration

identified first-order TF model structure provides a reasonable mechanistic interpretation, with the time constant increasing with distance from the input, ranging from 0.65 hours at cross-section 82, to 6.57 hours at crosssection 68. Also, as required, the emulation model explains the data well, with R2T > 0.99 for all cross-sections, so that the noise terms ξj (k), which represent the modeling error since there is no measurement noise on the data, have a very low variance, accounting for less than 1% of the simulated output variance. More importantly, the explanation is quite similar in the case of the validation exercises, where a completely new input series is used to drive the HEC-RAS model. The results of the preceding analysis show that the nominal model can accurately reproduce the level-to-level relationship of the HEC-RAS model for the chosen cross-sections when the HEC-RAS model parameters are set at their nominal values. This good performance can be attributed to the rich range of dynamics that can be reproduced by the surprisingly simple nonlinear DBM model form. However, it is now necessary to map the relationship between the DBM model parameters and whichever HEC-RAS model parameters are considered most important in the proposed application. In order to identify and estimate these mapping characteristics, a Monte Carlo ensemble of 1000 HEC-RAS simulation realizations was generated by introducing variation into the Manning’s coefficients. In particular, the full set of 89 nominal coefficients were multiplied by a factor sampled from a uniform probability distribution between 0.5 and 1.5. These scaling factors were sampled independently (i.e., with zero covariance) and, in each of the 1000 realizations, the simulated levels generated by the HEC-RAS model were collected at each of the selected cross-sections. For simplicity, the input SDP nonlinearities obtained in the nominal emulation model estimation were maintained for the full DEM identification, so that the only DEM parameters that had to be reestimated, at each realization, were the two, linear TF model parameters. The ensemble of these parameter estimates was then used in the mapping analysis. For example, in the case of the estimated âj,1 parameter, the resulting relationship with the two HEC-RAS parameters was identified in the following nonlinear form for all of the submodels:

0.8

0.5 10

11

Table 33.1  SRIV Estimation Results from rivbj Location

82

79

77

73

70

68

âj,1

–0.213

–0.330

–0.436

–0.681

–0.764

–0.859

^

bj,O

0.7708

0.673

0.557

0.311

0.224

0.141

δ

6

6

6

6

6

6

T (h)

0.65

0.90

1.21

2.60

3.72

6.57

33_Singh_ch33_p33.1-33.12.indd 10

0

5

10

15

20

25

Time in days Figure 33.8  A typical validation result showing the OTIS and emulation model outputs (concentration in mg/l) for a randomly selected point in the full model parameter space. The shaded area shows the 95% uncertainty band and the full model simulation line is practically indistinguishable from the emulator output.

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REFERENCE    33-11  33.7 CONCLUSION

This chapter has outlined the data-based mechanistic (DBM) approach to modeling stochastic dynamic systems and, in particular, shown how it provides a useful methodological framework for the inductive modeling of hydrological systems from observational time-series data. DBM modeling is a general method theory consisting of several analytical stages of model structure identification and associated parameter estimation. These can also include the assimilation of hypothetical conceptual elements using a hypothetico–inductive DBM (HI-DBM) modeling strategy, or the development of a low-order dynamic emulation model (DEM) for a large, high-order simulation model. In this manner, HI-DBM and DEM synthesis can provide a bridge between hypothetico–deductive (and normally large), conceptual computer simulation models and the parsimonious (parametrically efficient, low-order) DBM models. Although it is possible to employ any statistically acceptable methods of time-series model identification and estimation for DBM modeling, the various statistical tools available in the CAPTAIN Toolbox2 for Matlab have been designed specifically to service the DBM modeling requirements, thus providing a convenient set of routines for any reader who has access to the Matlab software environment. Finally, other recent applications that have used DBM modeling methods successfully are McIntyre and Marshall (2010), Ockenden and Chappell (2011), Wagener and McIntyre (2012), Vaughan and McIntyre (2012), Jones and Chappell (2014), and Smith et al. (2014).

REFERENCES

Barkau, R., M. Johnson, and M. Jackson, “UNET: a model of unsteady flow through a full network of open channels,” Proceedings of the 1989 National Conference on Hydraulic Engineering, American Society of Civil Engineers, New York, 1989. Beer, T. and P. C. Young, “Longitudinal dispersion in natural streams,” American Society of Civil Engineers, Journal of Environmental Engineering, 109: 1049–1067, 1983. Beven, K. J., “Prophecy, reality and uncertainty in distributed hydrological modelling,” Advances in Water Resources, 16: 41–51, 1993. Beven, K. J. and P. C. Young, “An aggregated mixing zone model of solute transport through porous media,” Journal of Contaminant Hydrology, 3: 129–143, 1988. Blazkova, S. D., K. J. Beven, and P. J. Smith, “Transport and dispersion in large rivers: application of the aggregated dead zone model,” edited by L. Wang and H. Garnier, System Identification, Environmetric Modelling and Control, Springer-Verlag, London, 2012, pp. 367–382. Castelletti, A., S. Galelli, M. Ratto, R. Soncini-Sessa, and P. Young, “A general framework for dynamic emulation modelling in environmental problems,” Environmental Modelling & Software, 34: 5–18, 2012. Garnier, H. and P. C. Young, “The advantages of directly identifying continuous-time transfer function models in practical applications,” International Journal of Control, 87 (7): 1319–1338, 2014. Garnier, H. and L. Wang, editors. Identification of Continuous-Time models from Sampled data. Springer, London, 2008. Gu, C., Smoothing Spline ANOVA Models, Springer-Verlag, Berlin, 2002. Helton, J. C. and D. E. Burmaster, “Guest editorial: treatment of aleatory and epistemic uncertainty in performance assessments for complex systems,” Reliability Engineering and Systems Safety, 54: 91–94, 1996. Jones, T. D. and N. A. Chappell, “Streamflow and hydrogen ion interrelationships identified using data-based mechanistic modelling of high frequency observations through contiguous storms,” Hydrology Research, 45 (6): 868–892, 2014. Konikow, L. and J. Bredehoeft, “Ground water models cannot be validated,” Advances in Water Resources, 15: 75–83, 1992. Kuhn, T., The Structure of Scientific Revolutions, University of Chicago, Chicago, 1962. Leedal, D., A. H. Weerts, P. J. Smith, and K. J. Beven, “Application of databased mechanistic modelling for flood forecasting at multiple locations in the Eden catchment in the National Flood Forecasting System (England and Wales),” Hydrology and Earth System Sciences Journal, 17: 177–185, 2013. Lees, M. J., “Data-based mechanistic modelling and forecasting of hydrological systems,” Journal of Hydroinformatics, 2: 15–34, 2000. 2 This can be downloaded free via a link from my website http://captaintoolbox.co.uk/ Captain_Toolbox.html, which also provides information on CAPTAIN routines and DBM modeling; or directly from http://www.lancaster.ac.uk/staff/taylorcj/tdc/download.php

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Martinez, C. J. and W. R. Wise, “Analysis of constructed treatment wetland hydraulics with the transient storage model OTIS,” Ecological Engineering, 20 (3): 211–222, 2003. McIntyre, N. and M. Marshall, “Identification of rural land management signals in runoff response,” Hydrological Processes, 24 (24): 3521–3534, 2010. McIntyre, N., P. C. Young, B. Orellana, M. Marshall, B. Reynolds, and H. Wheater, “Identification of nonlinearity in rainfall-flow response using databased mechanistic modelling,” Water Resources Research, 47: W03515, 2011, doi:10.1029/2010WR009851. Moore, R. J., “The PDM rainfall-runoff model,” Hydrology and Earth System Sciences Journal, 11: 483–499, 2007. Nash, J. E., “The form of the instantaneous hydrograph,” Proceedings of IASH General Assembly of Toronto, IAHS Publ. No. 42, Vol. 3, 1958, pp. 114–118. Nash, J. E. and J. V. Sutcliffe, “River flow forecasting through conceptual models: discussion of principles,” Journal of Hydrology, 10: 282–290, 1970. Newton, I., Mathematical principles of natural philosophy, Philosophiae Naturalis Principia Mathematica, London, 1687. Ockenden, M. C. and N. A. Chappell, “Identification of the dominant runoff pathways from data-based mechanistic modelling of nested catchments in temperate UK,” Journal of Hydrology, 402 (1–2): 71–79, 2011. Oreskes, N., K. Shrader-Frechette, and K. Belitz, “Verification, validation, and confirmation of numerical models in the earth sciences,” Science, 263: 641–646, 1994. Parkinson, S. and P. C. Young, “Uncertainty and sensitivity in global carbon cycle modelling,” Climate Research, 9: 157–174, 1998. Popper, K., The Logic of Scientific Discovery, Hutchinson, London, 1959. Priestley, M. B., Nonlinear and Nonstationary Time Series Analysis, Academic Press, London, 1988. Ratto, M., A. Pagano, and P. Young, “State dependent parameter metamodelling and sensitivity analysis,” Computer Physics Communications, 177: 863–876, 2007. Romanowicz, R. J., P. C. Young, and K. J. Beven, “Data assimilation and adaptive forecasting of water levels in the River Severn catchment,” Water Resources Research, 42: W06407, 2006, doi:10.1029/2005WR004373. Smith, P. J., K. J. Beven, D. Leedal, A. H. Weerts, and P. C. Young, “Testing probabilistic adaptive real-time flood forecasting models,” Journal of Flood Risk Management, 7 (3): 265–279, 2014. Solo, V., Time series recursions and stochastic approximation, Ph.D. thesis, Australian National University, Canberra, Australia, 1978. Tych, W. and P. Young, “A Matlab software framework for dynamic model emulation,” Environmental Modelling and Software, 34: 19–29, 2012. Valentine, E. M. and I. R. Wood, “Longitudinal dispersion with dead zones,” Journal of the Hydraulics Division, ASCE, 103 (9): 975–990, 1977. Vaughan, M. and N. McIntyre, “An assessment of DBM flood forecasting models,” Proceedings of the Institution of Civil Engineers-Water Management, 165 (2): 105–120, 2012. Vernieuwe, H., O. Georgieva, B. de Baets, V. Pauwels, N. Verhoest, and F. P. D. Troch, “Comparison of data-driven Takagi-Sugeno models of rainfall-discharge dynamics,” Journal of Hydrology, 302: 173–186, 2005. von Bertalanffy, L., General System Theory: Foundations, Development, Applications, George Braziller (revised edition 1976), New York, 1968. Vrugt, J. A., C. J. F. ter Braak, H. V. Gupta, and B. A. Robinson, “Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modelling,” Stochastic Environmental Research and Risk Assessment, 23: 1011–1026, 2009, doi: 10.1007/s00477–008–0274–y. Wagener, T. and N. McIntyre, “Hydrological catchment classification using a data-based mechanistic strategy system identification,” edited by L. Wang and H. Garnier, System Identification, Environmental Modelling, and Control System Design, Springer-Verlag, London, UK, 2012, pp. 483–500. Wallis, S. G., P. C. Young, and K. J. Beven, “Experimental investigation of the aggregated dead zone model for longitudinal solute transport in stream channels,” Proceedings of the Institution of Civil Engineers, Part 2, 87: 1–22, 1989. Wellstead, P. E., “An instrumental product moment test for model order estimation,” Automatica, 14: 89–91, 1978. Wheater, H. S., A. J. Jakeman, and K. J. Beven, “Progress and directions in rainfall-runoff modelling,” edited by A. J. Jakeman, M. B. Beck, and M. J. McAleer, Modelling Change in Environmental Systems, Wiley, Chichester, Chap. 5, 1993, pp. 101–132. Young, P. C., “Recursive approaches to time-series analysis,” Bulletin of Institute of Mathematics and Its Applications, 10: 209–224, 1974. Young, P. C., “A general theory of modeling for badly defined dynamic systems,” edited by G. C. Vansteenkiste, Modeling, identification and Control in Environmental Systems, North Holland, Amsterdam, 1978, pp. 103–135.

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33-12    Data-Based Mechanistic Modeling

Young, P. C., “Time variable and state dependent modelling of nonstationary and nonlinear time series,” edited by T. Subba Rao, Developments in Time Series Analysis, Chapman and Hall, London, 1993, pp. 374–413. Young, P. C., “Data-based mechanistic modeling of environmental, ecological, economic and engineering systems,” Environmental Modelling & Software, 13: 105–122, 1998. Young, P. C., “Stochastic, dynamic modelling and signal processing: time variable and state dependent parameter estimation,” edited by W. J. Fitzgerald, A. Walden, R. Smith, and P. C. Young, Nonlinear and Nonstationary Signal Processing, Cambridge University Press, Cambridge, 2000, pp. 74–114. Young, P. C., “Data-based mechanistic modelling and validation of rainfallflow processes,” edited by M. G. Anderson and P. D. Bates, Model Validation: Perspectives in Hydrological Science, John Wiley, Chichester, 2001a, pp. 117–161. Young, P. C., “Data-based mechanistic modelling of environmental systems,” Proceedings, IFAC Workshop on Environmental Systems, First Plenary Session Keynote Paper, Yokohama, Japan, 2001b. Young, P. C., “The identification and estimation of nonlinear stochastic systems,” edited by A. I. Mees, Nonlinear Dynamics and Statistics, Birkhauser, Boston, 2001c, pp. 127–166. Young, P. C., “Advances in real-time flood forecasting.” Philosophical Transactions of the Royal Society: Physical and Engineering Sciences, 360 (9): 1433–1450, 2002. Young, P. C., “The data-based mechanistic approach to the modelling, forecasting and control of environmental systems,” Annual Reviews in Control, 30: 169–182, 2006. Young, P. C., “Real-time flow forecasting,” edited by H. Wheater, S. Sarooshian, and K. D. Sharma, Hydrological modelling in arid and semi-arid areas, Cambridge University Press, Cambridge, U.K., 2008, pp. 113–138. Young, P. C., “The data-based mechanistic approach to the emulation of large dynamic hydrological process models,” edited by C. Walshe, Proceedings British Hydrological Society 3rd International Symposium, British Hydrological Society, London, 2010a, ISBN 1 903741 17 3. Young, P. C., “Gauss, Kalman and advances in recursive parameter estimation,” Journal of Forecasting (special issue celebrating 50 years of the Kalman Filter), 30: 104–146, 2010b. Young, P. C., “Real-time updating in flood forecasting and warning,” edited by G. J. Pender and H. Faulkner, Flood Risk Science and Management, WileyBlackwell, Oxford, U.K., 2010c, pp. 163–195.

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Young, P. C., Recursive Estimation and Time-Series Analysis: An Introduction for the Student and Practitioner, Springer-Verlag, Berlin, 2011. Young, P. C., “Hypothetico-inductive data-based mechanistic modeling of hydrological systems,” Water Resources Research, 49 (2): 915–935, 2013. Young, P. C., “Refined instrumental variable estimation: maximum likelihood optimization of a unified Box-Jenkins model,” Automatica, 52: 35–46, 2015. Young, P. C. and M. B. Beck, “The modelling and control of water quality in a river system,” Automatica, 10: 455–468, 1974. Young, P. C. and K. J. Beven, “Data-based mechanistic (DBM) modelling and the rainfall-flow nonlinearity,” Environmetrics, 5: 335–363, 1994. Young, P. C., A. J. Jakeman, and R. McMurtrie, “An instrumental variable method for model order identification,” Automatica, 16: 281–296, 1980. Young, P. C., A. J. Jakeman, and D. A. Post, “Recent advances in the databased modelling and analysis of hydrological systems,” Water Science and Technology, 36: 99–116, 1997. Young, P. C., D. Leedal, K. J. Beven, and C. Szczypta, “Reduced order emulation of distributed hydraulic models,” Proceedings 15th IFAC Symposium on System Identification SYSID09, St. Malo, France, 2009. Young, P. C. and M. J. Lees, “The active mixing volume: a new concept in modelling environmental systems,” edited by V. Barnett and K. Turkman, Statistics for the Environment, John Wiley, Chichester, 1993, pp. 3–43. Young, P. C. and S. Parkinson, “Simplicity out of complexity,” edited by M. B. Beck, Environmental Foresight and Models: A Manifesto, Elsevier, Oxford, 2002, pp. 251–294. Young, P. C., S. Parkinson, and M. J. Lees, “Simplicity out of complexity: Occam’s razor revisited,” Journal of Applied Statistics, 23: 165–210, 1996. Young, P. C. and D. J. Pedregal, “Recursive fixed interval smoothing and the evaluation of LIDAR measurements,” Environmetrics, 7: 417–427, 1996. Young, P. C. and M. Ratto, “A unified approach to environmental systems modelling,” Stochastic Environmental Research and Risk Assessment, 23: 1037–1057, 2009. Young, P. C. and M. Ratto, “Statistical emulation of large linear dynamic models,” Technometrics, 53: 29–43, 2011. Young, P. C. and S. G. Wallis, “Solute transport and dispersion in channels,” edited by K. J. Beven and M. J. Kirkby, Channel Network Hydrology, John Wiley, Chichester, 1993, pp. 129–173.

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Chapter

34

Decomposition Methods BY

SERGIO E. SERRANO

ABSTRACT

Adomian’s decomposition method (ADM) constitutes a useful alternative to model linear, and especially nonlinear, equations in surface, subsurface, and contaminant hydrology. ADM offers the advantages of both analytical and numerical procedures, while minimizing many of their disadvantages. It exhibits the simplicity, stability, and spatial and temporal continuity of analytical solutions, in addition to the ability to handle irregularly shaped domains, and nonlinearity typical of numerical solutions. The most important feature in ADM is its simplicity; many problems that were considered tractable by complex numerical methods only, are now easily approached with ADM. This chapter presents a simple introduction to the features of ADM with some application examples to the modeling of various phenomena in hydrology. These include nonlinear kinematic and dynamic flood waves, infiltration under variable rainfall, nonlinear soil moisture propagation, groundwater flow in irregularly shaped aquifers, groundwater flow in heterogeneous aquifers subject to pumping and recharge, stream-aquifer interaction, propagation of nonlinear hydraulic transients in aquifers, contaminant transport in soils and aquifers subject to nonlinear reactions, and stochastic analysis in hydrology without small perturbation restrictions. 34.1  INTRODUCTION: ADOMIAN’S DECOMPOSITIONS METHOD

The fundamental building blocks of hydrologic models are usually made of partial differential equations governing fundamental laws in a watershed, a soil, or an aquifer. Traditionally, these equations must be solved analytically or numerically. Classical analytical solutions (e.g., Fourier series and Laplace transform) are simple to program, are usually stable, and offer a spatially and temporally continuous description of hydrologic variables (e.g., Steward et al., 2009; Read, 2007; Read and Volker, 1993; Chapman and Dressler, 1984; Kirkham, 1957; and Philip, 1957). However, classical analytical solutions require regular domain shapes (e.g., simple one-dimensional or rectangular geometries), and cannot handle nonlinearities. On the other hand, numerical solutions (e.g., finite differences, finite elements, and finite volume) reduce the differential equations to a simpler system of algebraic equations, can manage irregular domain shapes, and may deal with numerical approximations of nonlinearities. However, numerical solutions yield the state variables at discrete nodes only (i.e., they require a grid), are difficult to program (i.e., they require specialized computer software), and often have difficulties with instabilities and round-off errors. An alternative method, Adomian’s decomposition method (ADM; Adomian, 1994, 1991, 1986, 1983), offers the simplicity, stability, and spatial continuity of analytical solutions, in addition to the ability to handle system nonlinearities and irregular geometries typical of numerical solutions. Many studies have reported new solutions to a wide class of equations (ordinary, partial, differential, integral, integrodifferential, linear, nonlinear, deterministic, or stochastic) in a variety of fields of mathematical physics, science, and engineering (see, e.g., Rach, Wazwaz, and Duan, 2013; Duan and Rach, 2011;

Rach, 2008; Wazwaz, 2000; Adomian, 1994, 1991). For nonlinear equations in particular, decomposition is one of the few systematic solution procedures available. Other analytical methods to solve nonlinear equations have been proposed, such as the homotopy perturbation method (e.g., He, 2006), and the variational iterations method (e.g., He, 1999), with few published applications in hydrology as yet. ADM consists in deriving an infinite series that in many cases converge to an exact solution. The convergence of decomposition series has been rigorously established in the mathematical community (Gabet, 1994, 1993, 1992; Abbaoui and Cherruault, 1994; Cherruault, 1989; Cherruault et al., 1992), and in the hydrologic community (Serrano, 2003c, 1998). In many complex linear and nonlinear problems, an exact closed form solution is difficult to obtain. However, in these cases the usual fast convergence rate of decomposition series provides the modeler with a sufficiently accurate approximate solution. A convergent decomposition series made of the first few terms usually provides an effective model in practical applications. With the concepts of partial decomposition and of double decomposition (Adomian, 1994, 1991), the process of obtaining an approximate solution was further simplified. Also, recent contributions suggest that the choice of the initial term greatly influences the rate of convergence and the complexity in the calculation of individual terms, especially for nonlinear equations (Wazwaz and Gorguiz, 2004; Wazwaz, 2000). Thus, as long as the initial term in a decomposition series, usually the forcing function or the initial condition, is described in analytic form, a partial decomposition procedure may offer a simplified approximate solution to many modeling problems. In hydrology, several fundamental works using decomposition have been published on groundwater flow (Moutsopoulos, 2007; Serrano, 1995; Serrano and Unny, 1987); analytical steady and transient groundwater modeling in irregularly shaped aquifers (Serrano, 2013, 2012; Patel and Serrano, 2011; Tiaiff and Serrano, 2014); fracture flow (Moutsopoulos, 2009); contaminant transport (Serrano, 1988; Adomian and Serrano, 1998; Serrano and Adomian, 1996); special problems involving non-Fickian and scale-dependent contaminant transport (Serrano, 1997b, 1996, 1995b); hydraulics of wells in heterogeneous aquifers (Serrano, 1997a); stream-aquifer interaction (Serrano and Workman, 2008, 1998; Serrano et al., 2007; Srivastava et al., 2006); modeling in heterogeneous aquifers (Srivastava and Serrano, 2007; Serrano, 1995a); nonlinear moving boundaries in unconfined aquifers (Serrano, 2003a); infiltration in unsaturated and hysteretic soils (Serrano, 2004, 2003b, 2001, 1998, 1990a); catchment hydrology and nonlinear flood propagation (Serrano, 2006; Sarino and Serrano, 1990); nonlinear reactive contaminant transport (Serrano, 2003c); and sediment transport in alluvial streams (Tayfur and Singh, 2011). Several fundamental problems that were considered tractable with numerical methods only have been easily approached with decomposition. Besides simplicity, an analytical solution offers a continuous spatio temporal distribution in heads, gradients, velocities and fluxes, thus reducing instability. Combination of analytical decomposition with numerical methods offers an ideal modeling scenario that exhibits the advantages of both analytical and numerical procedures (Serrano, 1992).

34-1

34_Singh_ch34_p34.1-34.6.indd 1

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34-2    Decomposition Methods 34.2  REGIONAL FLOW IN AN UNCONFINED AQUIFER

This section illustrates the basis of ADM with a simple problem. For detailed introduction, practical examples in hydrology, and computer programs, see Serrano (2011, 2010). Consider the two-dimensional regional groundwater flow in an unconfined aquifer with Dupuit assumptions. The governing equation is given by



Rg ∂2 h ∂2 h , 2 + 2 =− ∂y ∂x T

0 ≤ x ≤ lx , 0 ≤ y ≤ l y (34.1)

where h(x, y) is the hydraulic head; x and y are the planar coordinates with respect to an origin; the aquifer dimensions are lx = 860 m and ly = 2000 m, respectively; the aquifer transmissivity is T = 700 m2/month, and the mean recharge from rainfall is Rg = 10 mm/month. Equation (34.1) is subject to a mixed set of boundary conditions given by

h(0, y ) = f1 ( y ),

∂h (lx , y ) = 0, h( x ,0) = f 2 ( x ), h( x , l y ) = f3 ( x ) (34.2) ∂x

where f1( y) = 241 – 0.001y, f2(x) = Cx2 + Ax + f1(0), f3(x) = Ex2 + Bx + f1(ly), C = –Rg /(2T), A = –2Clx, B = –2Elx, and E = –Rg /T. The functions f1, f2, and f3 have been derived from field measurements of the stage of the boundary rivers. Now define the operators Lx = ∂2/∂x2 and Ly = ∂2/∂y2. The inverse −1 operators L−1 x and L y are the corresponding twofold indefinite integrals with respect to x and y, respectively. Thus, Eq. (34.1) reduces to Lx h + L y h = −



Rg T

(34.3)

Two partial decomposition expansions are possible: The x-partial solution and the y-partial solution. The x-partial solution, hx, results from operating with L−1 x on Eq. (34.3) and rearranging. hx = − L−x1



Rg T



Rg

− L−x1L y (hx 0 + hx1 + hx 2 + ) (34.5)

T

The choice of hx0 often determines the level of difficulty in calculating subsequent terms and the rate of convergence (Adomian, 1994; Wazwaz, 2000). A simple choice is to set hx0 as equal to the first term in the right side of Eq. (34.5). Thus, the first approximation to the solutions is



hx 0 = − L−x1

Rg T

= k1 ( y ) + k2 ( y )x −

Rg x 2 2T

(34.6)

where the integration “constants,” k1 and k2, must be found from the x boundary conditions in Eq. (34.2). Hence, Eq. (34.6) becomes hx 0 = f1 ( y ) +



R g lx x T

−

Rg x 2 2T

(34.7)

Equation (34.7) satisfies the governing Eq. (34.1) and the x boundary conditions in Eq. (34.2), but not necessarily the ones in the y direction. Now, to obtain the y-partial solution to Eq. (34.3), hy, operate with L−1 y on Eq. (34.3) and rearrange hy = − L−y1



Rg T

− L−y1Lx hy (34.8)

Expanding hy in the right side as an infinite series hy = hy0 + hy1 + hy2 +…, Eq. (34.8) becomes

hy = − L−y1

Rg T

− L−y1Lx (hy 0 + hy1 + hy 2 + ) (34.9)

Taking hy0 as the first term in the right side of Eq. (34.9) one obtains the first approximation



hy 0 = − L−y1

34_Singh_ch34_p34.1-34.6.indd 2

Rg T



= k3 ( x ) + k4 ( x ) y −

Rg y 2T

2

(34.10)

 f ( x ) − f 2 ( x ) Rg l y  Rg y 2 (34.11) hy 0 = f 2 ( x ) +  3 + y− ly 2T  2T 

The y-partial solution satisfies the differential Eq. (34.1) and the y boundary conditions in Eq. (34.2), but not necessarily those in the x direction. We now have two partial solutions to Eq. (34.1): the x-partial solution (34.7), and the y-partial solution (34.11). Since both are solutions to h, a combination of the two partial solutions yields



 hx 0 ( x , y ) + hy 0 ( x , y )  h0 ( x , y ) =   (34.12) 2  

where h0 is the first combined term. Higher-order terms are derived similarly. The ith order term in the x-partial solution, hxi, may be derived from Eq. (34.5) as:

hxi = k4i+1 ( y ) + k4i+2 ( y )x − L−x1L y hi−1 (34.13)

where hi – 1 is the previous combined term in the decomposition series, and k4i + 1 k4i + 2 are such that homogeneous x boundary conditions in Eq. (34.2) are satisfied. Similarly, the ith-order term in the y-partial solution, hyi, may be derived from Eq. (34.9)

hyi = k4i+3 ( x ) + k4i+4 ( x ) y − L−y1Lx hi−1 (34.14)

where hi – 1 is the previous combined term in the decomposition series, and k4i + 3 and k4i + 4 are such that homogeneous y boundary conditions in Eq. (34.2) are satisfied. Similarly to Eq. (34.12), the ith combined term is given by

− L−x1L y hx (34.4)

Expanding hx in the right side as an infinite series hx = hx0 + hx1 + hx2 +… , Eq. (34.4) becomes hx = − L−x1

where the integration “constants,” k3 and k4, are found from the y boundary conditions (34.2)



 hxi ( x , y ) + hyi ( x , y )  hi ( x , y ) =   (34.15) 2  

Lastly, approximate the final solution with N terms, h ≈ h0 + h1 + …+ hN , where each term in the series is a combination of two partial solutions, one in x and one in y. The aforementioned procedure may be extended to threedimensional transient problems. In such case, each combined term in the decomposition series has four components, one in each spatial coordinate, and one in the temporal coordinate. Due the high rate of convergence of decomposition solutions, the hydrologist often finds that few terms in the above iteration might be reasonably accurate in many practical applications. In this example, the first three terms at the center of the aquifer x = lx/2, y = ly/2 are h0 = 248.52 m, h1 = 5.46 m, and h2 = 2.13 m. It is easy to program the aforementioned solution in any standard mathematics software, such as Maple. Figure 34.1 shows the regional groundwater flow distribution. Patel and Serrano (2011) showed that the maximum relative error of a four-term decomposition solution with respect to the exact analytical solution of a similar problem is less than 2%. Serrano (2012) also showed similar results when comparing a transient regional groundwater flow with respect to its exact analytical solution. Consideration of pumping wells at known coordinates in the aquifer is an easy extension to the aforementioned description. Similarly, aquifer heterogeneity, nonlinearity in the differential equation, and irregular aquifer geometries can be incorporated into the analysis (Tiaif and Serrano, 2014, 2013; Patel and Serrano, 2011). For practical examples with detailed programs, see Serrano (2010). 34.3  PROPAGATION OF NONLINEAR KINEMATIC FLOOD WAVES IN RIVERS

This section illustrates the application of ADM series, double decomposition, and successive approximation to the solution of nonlinear hydrologic problems. Consider the following nonlinear kinematic flood routing equation that combines the continuity and momentum equations, when the acceleration and the pressure terms in the latter are neglected:

∂Q  ∂Q  = QL , Q(0, t ) = QI (t ), Q( x ,0) = QI (0) (34.16) + αβQ β −1   ∂T  ∂x

where Q(x, t) is the flow rate (m3/s); QL is the lateral flow into the channel per unit length (m3/s); x is the distance from a streamflow station with a known hydrograph; t is time (hour); a is a constant with dimensions (m2 – 3bsb); QI(t) is the inflow hydrograph at t = 0; and b is a dimensionless constant. Using the

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Propagation of Nonlinear Kinematic Flood Waves in Rivers     34-3 

100

ADM Finite elements

90 80 70 247 Q (m3/s)

248 245

h (m)

244

60 50 40

243 30

242 241 240 2000

1500

1000 y (m)

500

0

800 700 600 500 400 300 x (m) 200 100

0

Figure 34.1  Regional groundwater head distribution with contours every 0.5 m, beginning at 239.5 m.

method of characteristics, an implicit solution to Eq. (34.16) was obtained by Serrano (2006) Q( x , t ) = QI (t − F (Q − QL x )x ) + QL x ,



F (Q ) = αβQ β −1 (34.17)

To approximate an explicit solution define the decomposition series Q = ∑ i∞=0 Qi , and use the concept of double decomposition to expand the initial term Q0 = ∑ ∞j=0 Q0 j in the right side of Eq. (34.17) (Serrano, 2006; Adomian, 1994). Thus, the first term in Eq. (34.17) is   ∞   Q0 = Ql  t − F  ∑ Q0 j − QL x  x  + QL x (34.18)  j =0   



Now expand F = ∑ k∞=1 Fk , where Fk is calculated using one of the many algorithms for the Adomian polynomials (Duan and Rach, 2011; Rach, 2008; Wazwaz, 2000). Using the traditional Adomian polynomials about an initial term f0 = Q00 − QL x = QI (t ) − QL x (Adomian, 1994), the first few terms are

20 10 0

10

20

30 Time (h)

40

50

60

Figure 34.2  Analytical versus numerical hydrograph at x = 75 km. Finite elements data kindly provided by Szymkiewicz (2010).

As stated, the convergence rate of ADM series is so high that only a few terms are needed to assure an accurate solution. Thus, if, for instance, Q02 is a reasonable approximation, then Q ≈ Q0 ≈ QI (t − x[F0 ( f0 ) + F1 ( f0 ) + F2 ( f0 )]) + QL x , f0 = Q02 ( x , t ) − QL x (34.21) As an illustration, assume that a = 5/3, b = 3/5, QL = 0, and an inflow hydrograph given by t  t  (1− tmax ) (34.22) e QI (t ) = q0 + (qmax − q0 )  t max 



where q0 = 5 m3/s, qmax = 100 m3/s, and tmax = 4 h. Figure 34.2 shows a comparison of a three-term analytical nonlinear hydrograph, Eqs. (34.19)– (34.21), at x = 75 km and the numerical hydrograph calculated by the modified finite element method according to Szymkiewicz (2010). The time to peak calculated by the two methods is in excellent agreement. The analytical solution seems to better preserve the peak magnitude, in agreement with kinematic wave theory. There appears to be some minor differences in the recession limb, possibly because of numerical dissipation. However, the calculation of the analytical hydrograph is much simpler than the numerical one. The analytical solution lends itself easily to extended simulations in rivers. Figure 34.3 shows observed and predicted nonlinear hydrographs at the

F0 ( f0 ) = F0 ( f0 ) F1 ( f0 ) = F0 ( f0 )

d F ( f0 ) d f0

F2 ( f0 ) = F1 ( f0 )

d F ( f0 ) F0 ( f0 )2 d 2 F ( f0 ) + d f0 2! d f02

F3 ( f0 ) = F2 ( f0 )

d 2 F ( f ) F 3 ( f ) d 3 F0 ( f0 ) d F ( f0 ) + F1 ( f0 )F2 ( f0 ) 0 2 0 + 1 0 d f0 3! d f03 d f0

1000 Q (m3/s)

(34.19)





Observed at Philadelphia Nonlinear

1200

800 600 400

Combine Eqs. (34.18) and (34.19) to successively approximate Q0: Q00 = QI (t − xF0 ( f0 )) + QL x ,

f0 = QI (t ) − QL x

Q01 = QI (t − x[F0 ( f0 ) + F1 ( f0 )]) + QL x ,

f0 = Q00 ( x , t ) − QL x

Q02 = QI (t − x[F0 ( f0 ) + F1 ( f0 ) + F2 ( f0 )]) + QL x ,

f0 = Q01 ( x , t ) − QL x



34_Singh_ch34_p34.1-34.6.indd 3

200

(34.20)

100

150

200 250 Time (days)

300

Figure 34.3  Observed and predicted nonlinear flow rates at the Schuylkill River, September 2004–May 2005.

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34-4    Decomposition Methods

Philadelphia station in the Schuylkill River (Southeast Pennsylvania), given an inflow hydrograph at the Norristown station (Serrano, 2006). The simulations included variable lateral flow that included groundwater flow and variable effective precipitation. The standard deviation of the absolute error between observed and predicted was only 13.21 m3/s.

0.50

∂θ  ∂θ ∂  D(θ )  = 0, 0 < x < ∞ , 0 < t , θ (0, t ) = θb , θ (∞ , t ) = θi , θ ( x ,0) = θi − ∂x  ∂t ∂ x 

Parlange (1971)

0.45

Observed ADM

0.40

34.4  NONLINEAR INFILTRATION IN UNSATURATED SOILS

0.35 q

This section presents a combination of ADM with a successive approximation (Serrano, 2004) that yields a simple approximate solution to a highly nonlinear equation, when traditional numerical solutions present numerous accuracy, complexity, and instability problems. Consider the horizontal infiltration equation in a semi-infinite homogeneous soil with a constant boundary condition maintained on one end:

Philip (1955)

0.30 0.25 0.20 0.15

(34.23)

where q = soil volumetric water content; x = horizontal distance (m); t = time (h); qb = 0.458 is the water content at the left boundary; qi = 0.086 is the initial α water content; D(θ ) = c1e λθ is the soil-water diffusivity (m2/h); c1 = 1 m2/h; l = 500; and a = 11 (Serrano, 1998). Expanding the nonlinear diffusivity as the series D(θ ) = ∑ i∞=0 Di , Eq. (34.23) becomes ∂θ ∂  ∞  ∂θ  −  ∑ Di  = 0 (34.24) ∂t ∂ x  i=0  ∂ x 



Similar to Eq. (34.19), the Adomian polynomials about the first term, q0, are given by (Adomian, 1994) D0 = D(θb )



D1 = θ1

dD(θ 0 ) dθ 0

D2 = θ 2

dD(θ 0 ) θ 02 d 2D(θ 0 ) + dθ 0 2! dθ 02

D3 = θ 3

dD(θ 0 ) d 2D(θ 0 ) θ13 d 3D(θ 0 ) + θ1θ 2 + dθ 0 dθ 02 3! dθ 03

(34.25)



Now recursively calculate each Di, which is used to approximate the next qi. Calculations end when the water content distribution reaches a desired accuracy. From Eq. (34.25), use the left boundary condition, q ≈ qb, as an initial estimate of the water content. With the approximation D(q) ≈ D0 = D(q b), Eq. (34.24) reduces to the classical heat flow equation with constant coefficient whose solution is (Zauderer, 1983)  x  θ 0 = θi + (θb − θi )erfc   (34.26)  4 D0t 



where erfc( ) denotes the “complementary error function.” With q ≈ q0, an improved estimate of the water content, calculate   x θ1 = θi + (θb − θi )erfc   (34.27)  4 D(θ 0 )t 



Now from Eq. (34.25), calculate D1 and obtain an improved diffusivity D(q) ≈ D0 + D1. The improved



  x θ 2′ = θi + (θb − θi )erfc   (34.28)  4(D0 + D1 )t 

solution to Eq. (34.24) becomes With θ ≈ θ 2′ , an improved estimate of the water content, calculate



34_Singh_ch34_p34.1-34.6.indd 4

  x θ 2 ≈ θi + (θb − θi )erfc   (34.29) 4( ( θ ) D t ′ 2  

0.10 0 0

0.05 0.10

0.15 0.20 0.25 0.30 0.35 0.40 Distance, x (m)

Figure 34.4  Water content versus distance at t = 1 h according to ADM, two numerical solutions, and laboratory observation.

This process may be continued: from Eq. (34.25), calculate D2, obtain an improved diffusivity D(q) ≈ D0 + D1 + D2, then similar to Eq. (34.28) derive θ 3′ , then similar to Eq. (34.29) derive q3. Many studies report that ADM series converges fast and only a few terms are needed. Thus, q ≈ q3 may be a good approximation to the water content. Figure 34.4 shows profiles of the water content versus distance profiles at t = 1 h according to four sources: ADM, the classical numerical solution of Philip (1955), the numerical solution of Parlange (1971), and laboratory observations (Serrano, 2004). The ADM solution appears to better predict the position of the wetting front, and the subsequent shape of the tail, than Philip’s (1955) or Parlange’s (1971) solutions. The decomposition solution is simpler; it provides a continuous spatio-temporal description, and it does not exhibit the stability and discretization restrictions of numerical solutions. Extensions of ADM to physically based vertical infiltration and distribution were derived by Serrano (2004). Simple physically based ADM models of infiltration in watersheds subject to variable rainfall have been proposed (Serrano, 2010, 2004). Explicit ADM solutions to the Green and Ampt equation were derived by Serrano (2003b, 2001). 34.5  SUMMARY AND CONCLUSION

The basic features of hydrologic modeling with ADM have been presented via three simple examples: regional groundwater flow in aquifers, the propagation of nonlinear kinematic flood waves in rivers, and nonlinear infiltration in unsaturated soils. ADM offers the advantages of both analytical and numerical procedures, while minimizing many of their disadvantages. It exhibits the simplicity, stability, and spatial and temporal continuity of analytical solutions, in addition to the ability to handle irregularly shaped domains, and nonlinearity typical of numerical solutions. ADM may be applied to a wide class of problems in hydrology, including, infiltration under variable rainfall, groundwater flow in irregularly shaped aquifers, groundwater flow in heterogeneous aquifers subject to pumping and recharge, stream-aquifer interaction, propagation of nonlinear hydraulic transients in aquifers, contaminant transport in soils and aquifers subject to nonlinear reactions, and stochastic analysis in hydrology without small perturbation restrictions. Expanded references to such work, including practical computer programs, have been included. The most important feature in ADM is its simplicity; many problems that were considered tractable by complex numerical methods only, are now easily approached with ADM. The method has proved invaluable to hydrologic analysis and preliminary design under scarce data. Future work should be devoted to the application of ADM to unresolved problems in nonlinear hydrology. As new algorithms for the Adomian polynomials continue to appear in the research literature, the possibilities to solved increasingly complex problems are promising.

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References    34-5  REFERENCES

Abbaoui, K. and Y. Cherruault, “Convergence of Adomian’s method applied to differential equations,” Computers & Mathematics with Applications, 28 (5): 103–109, 1994. Adomian, G., Solving Frontier Problems in Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. Adomian, G., “A review of the decomposition method and some recent results for nonlinear equations,” Computers & Mathematics with Applications, 21 (5): 101–127, 1991. Adomian, G., Non-Linear Stochastic Operator Equations, Academic Press, New York, 1986. Adomian, G., Stochastic Systems, Academic Press, New York, 1983. Adomian, G. and S. E. Serrano, “Stochastic contaminant transport equation in porous media,” Applied Mathematics Letters, 1191: 53–55, 1998. Chapman, T. G. and R. F. Dressler, “Unsteady shallow groundwater flow over a curved impermeable boundary,” Water Resources Research, 20: 1427– 1434, 1984. Cherruault, Y., “Convergence of Adomian’s method,” Kybernetes, 18 (2): 31–38, 1989. Cherruault, Y., G. Saccomardi, and B. Some, “New results for convergence of Adomian’s method applied to integral equations,” Mathematical and Computer Modelling, 16 (2): 85–93, 1992. Duan, J.-S. and R. Rach, “A new modification of the Adomian decomposition method for solving boundary value problems for higher-order nonlinear differential equations,”Applied Mathematics and Computation, 218: 4090– 4118, 2011. Gabet, L., “The decomposition method and distributions,” Computers and Mathematics with Applications, 27 (3): 41–49, 1994. Gabet, L., “The decomposition method and linear partial differential equations,” Mathematical and Computer Modelling, 17 (6): 11–22, 1993. Gabet, L., Equissed’uneThéorieDécompositionnelleet Application aux Equations aux DérivéesPartialles, Dissertation, Ecole Centrale de Paris, France, 1992. He, J-H.,“Homotopy perturbation method for solving boundary value problems,” Physics Letters A, 350: 87–88, 2006. He, J-H.,“Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, 34: 699–708, 1999. Kirkham, D., Theory of land drainage, Drainage of Agricultural Lands, AgronomyMonograph7, ASA, CSSA, and SSSA, Madison, WI, 1957. Moutsopoulos, K. N., “Exact and approximate analytical solutions for unsteady fully developed turbulent flow in porous media and fractures for time dependent boundary conditions,” Journal of Hydrology, 369 (1–2): 78–89, 2009. Moutsopoulos, K. N., “One-dimensional unsteady inertial flow in phreatic aquifers, induced by a sudden change of the boundary head,” Transport in Porous Media, 70: 97–125, 2007. Parlange, J. Y., “Theory of water movement in soils: I. One-dimensional absorption,” Soil Science, 111 (2): 134–137, 1971. Patel, A. and S. E. Serrano, “Decomposition solution of multidimensional groundwater equations,” Journal of Hydrology, 397: 202–2109, 2011. Philip, J. R., “The theory of infiltration: 1. The infiltration equation and its solution,” Soil Science, 83: 345–358, 1957. Philip, J. R., “Numerical solution of equations of the diffusion type with diffusivity concentration-dependent,” Transactions of the Faraday Society, 51 (7): 391, 1995. Rach, R., “A new definition of the Adomian polynomials,” Kybernetes, 37 (7): 910–955, 2008. Read, W. W., “An analytical series method for Laplacian problems with mixed boundary conditions,” Journal of Computational and Applied Mathematics, 209: 22–32, 2007. Read, W. W. and R. E. Volker, “Series solutions for steady seepage through hillsides with arbitrary flow boundaries,” Water Resources Research, 29: 2871–2880, 1993. Sarino, and Serrano, S. E., “Development of the instantaneous unit hydrograph using stochastic differential equations,” Stochastic Hydrology and Hydraulics, 4 (2): 151–160, 1990. Tiaiff, S. and Serrano, S. E., “Regional groundwater flow in the Louisville aquifer: new analytical solution in irregular domains,” Ground Water, 53: 550–557, 2014, doi:10.1111/gwat.12242. Serrano, S. E., “A simple approach to groundwater modeling with decomposition,” Hydrological Sciences Journal, 58 (1): 1–9, 2003, doi:10.1080/02626 667.2012.745938.

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Serrano, S. E., “New approaches to the propagation of nonlinear transients in porous media,” Transport in Porous Media, 93 (2): 331–346, 2012. Serrano, S. E., Engineering Uncertainty and Risk Analysis, 2nd ed., A Balanced Approach to Probability, Statistics, Stochastic Models and Stochastic Differential Equations, HydroScience Inc., Ambler, PA, 2011. Serrano, S. E., Hydrology for Engineers, Geologists, and Environmental Professionals, 2nd ed., An Integrated Treatment of Surface, Subsurface, and Contaminant Hydrology, HydroScience Inc., Ambler, PA, 2010. Serrano, S. E., “Development and verification of an analytical solution for forecasting nonlinear kinematic flood waves,” Journal of Hydrologic Engineering, ASCE, 11 (4): 347–353, 2006. Serrano, S. E., “Modeling infiltration with approximate solutions to Richard’s equation.” Journal of Hydrologic Engineering, ASCE, 9 (5): 421–432, 2004. Serrano, S. E., “Modeling groundwater flow under a transient non-linear free surface,” Journal of Hydrologic Engineering, ASCE, 8 (3): 123–132, 2003a. Serrano, S. E., “Improved decomposition solution to Green and Ampt equation,” Journal of Hydrologic Engineering, ASCE, 8 (3): 158–160, 2003b. Serrano, S. E., “Propagation of nonlinear reactive contaminants in porous media,” Water Resources Research, 39 (8): 1228–1242, 2003c. Serrano, S. E., “An explicit solution to the Green and Ampt infiltration equation,” Journal of Hydrologic Engineering, ASCE, 6 (4): 336–340, 2001. Serrano, S. E., “Analytical decomposition of the non-linear infiltration equation,” Water Resources Research, 34 (3): 397–407, 1998. Serrano, S. E., “The Theis solution in heterogeneous aquifers,” Ground Water, 35 (3): 463–467, 1997a. Serrano, S. E., “Non-Fickian transport in heterogeneous saturated porous media,” Journal of Engineering Mechanics, ASCE, 123 (1): 70–76, 1997b. Serrano, S. E., “Hydrologic theory of dispersion in heterogeneous aquifers,” Journal of Hydrologic Engineering, ASCE, 1 (4): 144–151, 1996. Serrano, S. E., “Analytical solutions of the nonlinear groundwater flow equation in unconfined aquifers and the effect of heterogeneity,” Water Resources Research, 31 (11): 2733–2742, 1995a. Serrano, S. E., “Forecasting scale-dependent dispersion from spills in heterogeneous aquifers,” Journal of Hydrology, 169: 151–169, 1995b. Serrano. S. E., “Semi-analytical methods in stochastic groundwater transport,” Applied Mathematical Modelling, 16: 181–191, 1992. Serrano, S. E., “Modeling infiltration in hysteretic,” Soil and Advanced Water Resources, 13 (1): 12–23, 1990a. Serrano, S. E., “Stochastic differential equation models of erratic infiltration,” Water Resources Research, 26 (4): 703–711, 1990b. Serrano, S. E., “General solution to random advective-dispersive equation in porous media. Part II: Stochasticity in the parameters,” Stochastic Hydrology and Hydraulics, 2 (2): 99–112, 1988. Serrano, S. E. and S. R. Workman, “Experimental verification of models of nonlinear stream aquifer transients,” Journal of Hydrologic Engineering, ASCE, 13 (12): 1119–1124, 2008. Serrano, S. E., S. R. Workman, K. Srivastava, and B. Miller-Van Cleave, “Models of nonlinear stream aquifer transients,” Journal of Hydrology, 336 (1–2): 199–205, 2007. Serrano, S. E. and S. R. Workman, “Modeling transient stream/aquifer interaction with the nonlinear Boussinesq equation and its analytical solution,” Journal of Hydrology, 206: 245–255, 1998. Serrano, S. E. and G. Adomian, “New contributions to the solution of transport equations in porous media,” Mathematical and Computer Modelling, 24 (4): 15–25, 1996. Serrano, S. E. and T. E. Unny, “Semi group solutions of the unsteady groundwater flow equation with stochastic parameters,” Stochastic Hydrology and Hydraulics, 1 (4): 281–296, 1987. Srivastava, K. and S. E. Serrano, “Uncertainty analysis of linear and nonlinear groundwater flow in a heterogeneous aquifer,” Journal of Hydrologic Engineering, ASCE, 12 (3): 306–318, 2007, doi:10.1061/(ASCE)10840699(2007)12:3(306). Srivastava, K., S. E. Serrano, and S. R. Workman, “Stochastic modeling of stream-aquifer interaction with the nonlinear Boussinesq equation,” Journal of Hydrology, 328: 538–547, 2006, doi:10.1016/j.jhydrol.2005.12.035. Steward, D. R., X. Yang, and S. Chacon, “Groundwater response to changing water-use practices in sloping aquifers using convolution of transient response functions,” Water Resources Research, 45: 2009, doi:10.1029/ 2007WR006775. Szymkiewicz, R., Numerical modeling in open channel hydraulics, Water Science and Technology Library, Springer, New York, 2010.

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34-6    Decomposition Methods

Tayfur, G. and V. P. Singh, “Simulating transient sediment waves in aggraded alluvial channels by double-decomposition method,” Journal of Hydrologic Engineering, ASCE, 16 (4): 362–370, 2011. Workman, S. R., S. E. Serrano, and K. Liberty, “Development and application of analytical model of stream/aquifer interaction,” Journal of Hydrology, 200: 149–63, 1997. Wazwaz, A-M.,“A new algorithm for calculating Adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, 111: 53–69, 2000.

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Wazwaz, A.M. and A. Gorguis A., “Exact solutions for heat like and wave like equations with variable coefficients,” Applied Mathematics and Computation, 149, 15–29, 2004. Zauderer, E., Partial Differential Equations of Applied Mathematics. John Wiley, New York, 1983.

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Chapter

35

Network Theory BY

BELLIE SIVAKUMAR, FITSUM M. WOLDEMESKEL, KOREN FANG, AND VIJAY P. SINGH

ABSTRACT

Connections are everywhere in hydrology. Unraveling the nature and extent of connections in hydrologic systems, as well as their interactions with other Earth systems, has always been a great challenge. Numerous scientific concepts and mathematical methods have been proposed and applied for studying the various connections in hydrology. Despite such efforts and progress, an adequate knowledge of such connections continues to be elusive. A key reason for this situation is the absence of a strong scientific theory that is suitable for studying all types of connections encountered in hydrology. To this end, modern developments in network theory, and complex networks in particular, seem to provide new avenues. Applications of the concepts of complex networks have recently started to emerge and are gaining significant momentum at the current time. This chapter provides an overview of network theory, especially the theory of complex networks, and its applications in hydrology. The chapter: (1) reviews the basic concepts of a network and the history of the development of network theory (including graph theory, topology, random graph theory, and complex networks); (2) describes different types of networks (regular, random, small-world, and scale-free) and different measures of their properties (e.g., centrality, clustering, adjacency, distance, and community structure) that have been widely studied; and (3) a brief review of their applications in hydrology, including studies on rainfall monitoring networks, streamflow monitoring networks, river networks, and virtual water trade networks. 35.1 INTRODUCTION

Connections are everywhere in hydrologic systems. The hydrologic cycle or water cycle is perhaps the best example of connections, with every component of the hydrologic cycle connected to every other component. Depending upon the place they occupy in the hydrologic cycle and the spatial and temporal scales of interest, the components can be connected directly or indirectly, and strongly or weakly. Similar to the hydrologic cycle, connections exist within each component as well (e.g., spatial and temporal connections in rainfall). Unraveling the nature and extent of connections in hydrologic systems, and also their interactions with other Earth systems (e.g., climatic, geomorphic, ecologic, and environmental), has always been a major challenge in hydrology. Until now, a plethora of scientific concepts and mathematical methods have been proposed and applied for studying the connections associated with hydrologic systems. Such concepts and methods are based on, among others, time, distance, correlation, variability, scale, patterns, information, and many other properties/measures as well as their combinations and variants; see, for example, Gupta et al. (1986), Salas et al. (1995), Grayson and Blöschl (2000), Yang et al. (2004), Mishra and Coulibaly (2009), Li et al. (2012), Sivakumar and Singh (2012), and Niu (2013) for some details. Despite such efforts and progress, our understanding of connections in hydrologic systems remains largely inadequate. Several major factors contribute to this situation, such as the inherent complexity of the hydrologic cycle and catchment systems, natural and anthropogenic influences, and data and computational constraints.

However, a key reason is the absence of a strong scientific theory that is suitable for studying all types of connections encountered in hydrology. This deficiency has increasingly been realized in recent times, with many calls for a generic theory or a broader framework (e.g., Dooge, 1986; Paola et al., 2006; Sivakumar, 2008; Young and Ratto, 2009). A host of yet new challenges, such as impacts of global climate change (e.g., Sivakumar, 2011a), issues associated with water planning, management, and conflicts (e.g., Sivakumar, 2011b), predictions in ungaged basins (e.g., Hrachowitz et al., 2013), and interactions between hydrology and society (e.g., Montanari et al., 2013), puts further emphasis on the importance of a generic theory for hydrology. In the context of connections, network theory or graph theory can offer useful ideas. The concept of networks originated in the mid-eighteenth century (Euler, 1741), and witnessed significant developments over the next three centuries, including topology (Listing, 1848), trees (Cayley, 1857), and random graph theory (Erdös and Rényi, 1959, 1960); see Harary (1969), Bondy and Murty (1976), and Bollobás (1998) for extensive details of these developments and their applications. However, developments since the late 1990s, including small-world networks (Watts and Strogatz, 1998), scale-free networks (Barabási and Albert, 1999), network motifs (Milo et al., 2002), and community structure (Girvan and Newman, 2002), put under the broad umbrella of the science of complex networks, have offered a whole new dimension and various avenues for studying connections encountered in natural and social systems; see, for example, Watts (1999), Barabási (2002), and Estrada (2012) for extensive details of these concepts and their applications in various fields. Applications of such complex networks-based concepts in hydrology and water resources are just starting to emerge, including for rainfall monitoring networks (e.g., Malik et al., 2012; Boers et al., 2013; Scarsoglio et al., 2013; Sivakumar and Woldemeskel, 2015; Jha et al., 2015), streamflow monitoring networks (e.g., Sivakumar and Woldemeskel, 2014; Halverson and Fleming, 2014; Fang et al., 2015), river networks (Rinaldo et al., 2006; Zaliapin et al., 2010; Czuba and Foufoula-Georgiou, 2014, 2015), and virtual water networks (e.g., Suweis et al., 2011; Konar et al., 2011; Carr et al., 2012; Dalin et al., 2012; D’Odorico et al., 2012; Tamea et al., 2013). The outcomes of these studies are certainly encouraging, as they offer important information about the connections in these networks and also suggest the potential of the network-based concepts for studying other hydrologic networks. Indeed, in light of the recent calls for a broader framework in hydrology, Sivakumar (2015) has also made an argument in favor of network theory as a suitable tool to study all types of connections in hydrology and, hence, to serve as a generic theory. The purpose of this chapter is to provide an overview of network theory, especially in the context of complex networks, and its applications in hydrology. The rest of the chapter is organized as follows. Section 35.2 reviews the basic concept of a network and the history of the development of network theory. Section 35.3 presents a brief description of the different types of networks, and Sec. 35.4 describes the different measures for identification of their properties. Section 35.5 presents a review of the applications of the ideas of network theory in hydrology. Section 35.6 offers some closing remarks. 35-1

35_Singh_ch35_p35.1-35.10.indd 1

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35-2    Network Theory 35.2  NETWORK THEORY: CONCEPT AND HISTORY

35.2.3 Topology

35.2.1  Concept of a Network

Topology is the study of geometric properties and topological spaces, through analysis of such concepts as space, dimension, and transformation. The basic ideas of topology go back to the works of Gottfried Wilhelm von Leibniz in the seventeenth century, with suggestions establishing the geometry of positions as a field without any measurements or calculations. However, the term “topology” was first introduced only in the nineteenth century, as topologie (in German) by Johann Benedict Listing (Listing, 1848). Significant developments in topology were made during late nineteenth century and early twentieth century, with important contributions in the areas of set theory by Georg Cantor (Cantor, 1874), homotopy and homology (now part of algebraic topology) by Henri Poincaré (Poincaré, 1895), and on topological space by Felix Hausdorff (Hausdorff, 1919). Topology had become a major branch of mathematics by the middle of the last century. Topology has many sub-fields, including: • general topology, which deals with the basic set-theoretic definitions and constructions used in topology; • algebraic topology, which uses tools from abstract algebra to study topological spaces; • differential topology, which deals with differentiable functions on differentiable manifolds; and • geometric topology, which not only focuses on low-dimensional manifolds (i.e., dimensions 2, 3, and 4) and their interaction with geometry, but also includes higher-dimensional ones. Extensive details on the concepts of topology and their applications are available in Bourbaki (1966), Mendelson (1990), and Adams and Franzosa (2007), among others. In hydrology, the concept of topology has been extensively discussed with reference to river/channel networks; see, for example, Scheidegger (1967), Shreve (1967), Smart (1970), Coffman and Turner (1971), Kirkby (1976), and Mark and Goodchild (1982) for some earlier studies.

A network or a graph is a set of points connected together by a set of lines, as shown in Fig. 35.1. The points are referred to as vertices or nodes and the lines are referred to as edges or links; here, we use the terms nodes for points and links for lines. Therefore, mathematically, a network can be represented as G = {P, E}, where P is a set of N nodes (P1, P2,…,PN) and E is a set of n links. Figure 35.1 shows a network with N = 7 nodes and n = 8 links. In this network, P = {1, 2, 3, 4, 5, 6, 7} and the set of links is E = {{1, 7},{2, 3},{2, 5},{2, 7}, {3,7 },{4, 7},{5, 6},{6, 7}}. Figure 35.1a, consisting of a set of identical nodes connected by identical links, is perhaps the simplest form of network. This kind of network, however, is rarely seen in nature and society, since natural and social networks are often far more complex. There are many ways in which networks may be more complex. For instance, a network may: (1) have more than one different type of node and/or link (Fig. 35.1b); (2) contain nodes and links with a variety of properties associated with them, such as different weights for different nodes and links depending on the strength of nodes and connections (Fig. 35.1c); (3) have links that can be directed (pointing in only one direction Fig. 35.1d), with either cyclic (i.e., containing closed loops of links) or acyclic form; (4) have multilinks (i.e., repeated links between the same pair of nodes), self-links (i.e., links connecting a node to itself), and hyperlinks (i.e., links connecting more than two nodes together); and (5) bipartite (containing nodes of two distinct types, with links running only between unlike types). 35.2.2  Origin of Network Theory (Graph Theory)

The origin of the concept of networks can be traced back to the works of the Swiss mathematician Leonhard Euler, during the first half of the eighteenth century, on one of the most famous problems in mathematics, the Seven Bridges of Königsberg (Euler, 1741). The problem studied by Euler is as follows. The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges, thus making it a network (for transportation). The problem Euler attempted to solve was to find a route through the city that would cross each bridge once and only once (including complete crossing of each bridge at once). While Euler proved that the problem had no solution, he also found that the choice of route inside each land mass is irrelevant and that the only important feature of a route is the sequence of bridges crossed. These findings allowed Euler to reformulate the problem in abstract terms, eliminating all features except the list of land masses and the bridges connecting them. This reformulation laid the foundations of what would become popularly known as graph theory. The graph theory has historically been a branch of the broader field of discrete mathematics, but is now a branch of combinatorics. Since this initial work by Euler, graph theory has witnessed a number of important theoretical developments, including topology (Listing, 1848), trees (Cayley, 1857), random graph theory (Erdös and Rényi, 1959), and complex networks (Watts and Strogatz, 1998). A brief description of these developments is presented next. Node 1 6

Link

2

7 5

3 4

In graph theory, a tree (Cayley, 1857) is an undirected connected graph without cycles. Therefore, a tree has to be a simple graph, since self-loops and parallel links both form cycles. A graph G with n nodes is a tree, when it satisfies the following equivalent conditions: (1) the graph is connected without cycles; (2) the graph has n – 1 links and no cycle; (3) the graph is connected and has n – 1 links; (4) there exists a unique path between any two nodes of the graph; and (5) the graph becomes a nontree if a link is removed (i.e., the graph becomes disconnected) or added (i.e., the graph creates cycle). There are different types of trees, including: • rooted tree—one node serves as a starting point and all the branches (i.e., links) fan out from this; • binary tree—a rooted tree in which there are at most two descending branches at any node; • polytree—a directed acyclic graph whose underlying undirected graph is a tree; and •  spanning tree—a partial graph G′ = {P, E′} of a connected graph G = {P, E} is a spanning tree of G if G′ is a tree. There is at least one spanning tree for any connected graph. A spanning tree of the graph that optimizes the weight or cost (compared to all spanning trees of the graph) is called an optimal spanning tree. The concept of tree is highly relevant in hydrology, especially in river networks; see, for example, Horton (1945), Strahler (1957), Tokunaga (1978), Rodriguez-Iturbe and Rinaldo (1997), Rinaldo et al. (2006), and Zaliapin et al. (2010) for details. 35.2.5  Random Graph Theory

(a)

(b)

(c)

(d)

Figure 35.1  Networks: (a) an undirected network with only a single type of node and a single type of link; (b) a network with a number of discrete node and link types; (c) a network with varying node and link weights; and (d) a directed network in which each node has a direction.

35_Singh_ch35_p35.1-35.10.indd 2

35.2.4 Trees

A random graph is, as the name suggests, a network of nodes connected by links in a purely random fashion. The roots of random graph theory can be traced back to the work of Solomonoff and Rapoport (1951). However, the model introduced by Erdös and Rényi (1959, 1960) is the best-known version today. This model—which is now known as the Erdös–Rényi model—is also sometimes known as the “classical” random graph. Therefore, assuming a network consisting of N nodes, the n links that connect these N nodes are chosen randomly from all the N(N – 1)/2 possible links. Therefore, there are a total of C[nN ( N −1)/2] graphs with N nodes and n links, forming a probability space in which every realization is equiprobable. Considering that every pair of nodes is connected with probability p, the total number of links is a random variable with the expectation value E(n) = p[N(N – 1)/2]. If G0 is a graph with nodes P1,P2, …,PN and n links, then the probability of obtaining such a graph by the graph construction process is P(G0) = pn(1 – p)(N – 1)/2 – n. As random graph theory basically examines the properties of the probability space associated with graphs with N nodes as N → ∞ , many properties of

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NETWORK TYPES    35-3 

random graphs can be determined in a probabilistic sense. The classical random graph model of Erdös and Rényi (1959, 1960) assumed that almost every graph has a property Q if the probability of having Q approaches 1 as N → ∞ . The primary purpose of random graph theory is to determine the connection probability p at which a particular property of a graph will most likely arise. Erdös and Rényi (1959, 1960) discovered that many important properties of random graphs appear quite suddenly. In other words, at a given probability p, either almost every graph has some property Q (e.g., every pair of nodes is connected by a path of consecutive links) or almost no graph does. The transition from a property’s being very unlikely to its being very likely is usually swift. For many such properties, there is a critical probability pc(N). If p(N) grows more slowly than pc(N) as N → ∞, then almost every graph with connection probability p(N) fails to have Q. If p(N) grows somewhat faster than pc(N), then almost every graph has the property Q. Therefore, the probability that a graph with N nodes and connection probability p = p(N) has property Q satisfies



(a)

(b)

(c)

(d)

 p( N )  → 0  0    if      pc ( N )  lim PN , p (Q ) =    (35.1) N →∞  1     if      p( N )  → ∞  pc ( N ) 

The random graph theory addressed a number of important questions that are relevant for studying real complex systems, such as: (1) Is a typical graph connected? (2) Does it contain a triangle of connected nodes? and (3) How does its diameter depend on its size? Consequently, the theory gained considerable attention and applications in many different fields until the end of the last century, including in natural and physical sciences, engineering and technology, economics, and social sciences; see, for example, Harary (1969), Bondy and Murty (1976), and Bollobás (1998) for extensive details. The random graph theory has found numerous applications in hydrology, especially in river/channel networks; see, for instance, Shreve (1966, 1967, 1969), Kirkby (1976), Smart and Werner (1976), Smart (1978), Moon (1980), and Werner (1982) for some earlier studies. 35.2.6  Complex Networks

While random graph theory has and continues to be applied for studying a broad range of networks in nature and society, including in hydrology, it also has certain important deficiencies. For instance, the assumption that all networks are wired randomly together makes the application of this theory to real networks questionable, since order and determinism are inherent in real systems and networks. Indeed, advances in some other areas of complex systems science, which revealed nonlinear deterministic dynamics, selforganization, and scale invariance as inherent properties of complex systems (e.g., Lorenz, 1963; Mandelbrot, 1983; Bak, 1996), also led to reconsideration of the assumption of random connections in complex networks. In addition, significant changes in technology, economy, and society during the last few decades resulted in real networks that are highly irregular, complex, large, and dynamically evolving, which necessitated the development of new methods and measures for studying network properties. All these led to a renewed and fresh perspective of the study of complex networks in the late 1990s (Watts and Strogatz, 1998; Barabási and Albert, 1999), under the new science of networks. Such studies also led to new discoveries about complex networks, such as small-world networks (Watts and Strogatz, 1998), scale-free networks (Barabási and Albert, 1999), network motifs (Milo et al., 2002), as well as other notable advances, such as a new method for identifying community structure (Girvan and Newman, 2002). Since then, the science of complex networks has found applications in many different fields, including natural and physical sciences, social sciences, medical sciences, economics, and engineering and technology (e.g., Albert et al., 1999; Bouchaud and Mézard, 2000; Newman, 2001a; Liljeros et al., 2001; Montoya and Solé, 2002; Tsonis and Roebber, 2004; Miguens and Mendes, 2008; Suweis et al., 2011; Sivakumar and Woldemeskel, 2014). Further details of complex network theory and its applications can be found in, for example, Watts (1999), Barabási (2002), and Estrada (2012). The concepts of complex networks, the measures to identify their properties, and their applications in hydrology are the focus of the rest of this chapter.

Figure 35.2  Network types: (a) regular network; (b) random network; (c) smallworld network; and (d) scale-free network.

shows, for example, four different types of networks: a completely regular network (Fig. 35.2a), a completely random network (Fig. 35.2b), a smallworld network (Fig. 35.2c), and a scale-free network (Fig. 35.2d). The last two networks belong to random networks as well, but have different properties when compared to classical random networks. They are also generally called complex networks. A brief description of these types of networks is presented here. 35.3.1  Regular Networks

Regular or ordered networks (Fig. 35.2a) are simple types of networks. Regular networks have a fixed number of nodes, with each node having the same number of links connecting it in a specific way to a number of neighboring nodes. A simple example of a regular network is a 3D Cartesian grid. If each node is linked to all other nodes in the network, then the network is a fully connected one. Regular networks display a wide range of properties, because there are many ways to construct them while keeping the degree uniform across all nodes. In general, however, regular networks are highly clustered, with clustering coefficient C (a measure of local density—see Sec. 35.4.2 for details) close to 1.0 and, therefore, such networks are stable. However, they also have long average path lengths L (a global measure of separation—see Sec. 35.4.4 for details), and so often have a high degree of inefficiency. In the context of complex networks, stability means that the removal of any randomly chosen node will have little effect on the network as a whole, while efficiency means that information may easily be propagated across the network because the average path length is small. 35.3.2  Random Networks

In random networks or classical random networks (e.g., Erdös and Rényi, 1959, 1960), the nodes are connected at random (Fig. 35.2b). In this case, the degree distribution pk (a measure of network spread—see Sec. 35.4.3 for details) is a Poisson distribution. The problem with these networks is that they are not very stable, and removal of a number of nodes at random may fracture the network to noncommunicating parts. As the name suggests, classical random networks have no patterns. They have a small clustering coefficient (often close to zero) and a small average path length. These mean that random networks tend to be unstable but efficient. Some other important details about the random networks have already been presented in Sec. 35.2.5 (random graph theory).

35.3  NETWORK TYPES

35.3.3  Small-World Networks

The recent concepts of complex networks and the new measures to identify their properties can be better explained by comparing such networks with the previously known types (i.e., regular and random). To this end, Fig. 35.2

While the concepts of regular networks and random networks have strong theoretical underpinnings, they nevertheless have only limited application in real networks encountered in nature and society. This is because, most

35_Singh_ch35_p35.1-35.10.indd 3

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35-4    Network Theory

real-world networks are neither completely ordered nor completely random, but rather exhibit important properties of both (Watts and Strogatz, 1998). Indeed, developments in some other areas of complex systems science also emphasize the influence of both order and randomness in real systems, as revealed by, for example, nonlinear dynamic and chaos theories (e.g., Lorenz, 1963; May, 1976). According to Watts and Strogatz (1998), some properties of real networks can be embodied by simple mathematical models that interpolate between order and randomness. They represented order by a uniform one-dimensional lattice (where each node is connected to its k nearest neighbors on the lattice) and characterized randomness by a tunable parameter p (which specifies the fraction of randomly rewired links). They showed that these properties can be quantified with simple statistics, such as the clustering coefficient C (see Sec. 35.4.2) and the average shortest path length L (see Sec. 35.4.4). In their model, both C and L can be measured as a function of p. When p = 0 (completely ordered), the network is large [L(0) ∼ N/2k] and highly clustered [C(0) ∼ ¾], as mentioned earlier (Sec. 35.3.1). On the other hand, when p = 1 (completely random), the network is small [L(1) ∼ ln(N)/ln(k)] and poorly clustered [C(1) ∼ k/N], suggesting that path lengths are short only when clustering is low, as highlighted earlier (Sec. 35.3.2). The model by Watts and Strogatz (1998) indeed exhibits a broad region of p values in which C(p) is high relative to its random limit C(l), yet L(p) is as small as possible. Watts and Strogatz (1998) coined the term small-world networks to refer to networks in this class, after similar concepts as manifestation of small worlds; see Karinthy (1929) and Milgram (1967). The small-world network concept of Watts and Strogatz (1998) highlights the fact that most networks, despite their often large size, have a relatively short path between any two nodes (see Fig. 35.2c). Therefore, small-world networks are one form of random networks, but different from classical random networks: they are stable (unlike classical random networks) and efficient (like classical random networks). 35.3.4  Scale-Free Networks

The random graph of Erdös and Rényi (1959, 1960) and the small-world network of Watts and Strogatz (1998) display Poissonian degree distributions. However, there are real-world networks that display a very different degree distribution, which is characterized by a few nodes of high degree and a large proportion of nodes with relatively low degree (Barabási and Albert, 1999). Although such networks can display a wide range of “fat-tail” degree distributions, the easiest way of conceptualizing such topological characteristics is to consider a model in which P(k) ∼ k–γ, where γ is the degree exponent. In other words, a model in which the probability of finding a node with degree k decreases as a power-law of its degree. Barabási and Albert (1999) were the first ones to address the origin of the power-law degree distribution observed in networks. They argued that the scale-free nature of real networks is rooted in two generic mechanisms shared by many real networks as follows: 1. The random graph and small-world network assume that the network starts with a fixed number N of nodes that are then randomly connected or rewired, without modifying N. However, real-world networks generally describe open systems that grow by the continuous addition of new nodes. The networks start with only a few nodes and the number of nodes increases throughout the lifetime of the networks by the subsequent addition of new nodes (e.g., World Wide Web). 2. The random graph and small-world network assume that the probability that two nodes are connected is independent of the nodes’ degree, that is, new links are always placed randomly. However, most real networks exhibit preferential attachment, in such a way that the likelihood of connecting to a node basically depends on the node’s degree. These two mechanisms inspired the introduction of the Barabási–Albert model (Barabási and Albert, 1999) and led, for the first time, to a network with a power-law degree distribution. The dynamic properties of the scale-free model can be addressed using various analytic approaches. Barabási and Albert (1999) proposed the continuum theory, which focuses on the dynamics of node degrees. Dorogovtsev et al. (2000) introduced the master-equation approach, while Krapivsky et al. (2000) proposed the rate-equation approach. Finally, from an overall perspective, the goal of random graph and smallworld network is to construct a graph with correct topological features. However, the modeling of scale-free networks puts the emphasis on capturing the network dynamics. The underlying assumption behind evolving or dynamic networks is that if the processes that assembled the networks are captured correctly, then their topology can also be obtained correctly. Therefore, dynamics takes the driving role, and topology is only a by-product of this modeling philosophy.

35_Singh_ch35_p35.1-35.10.indd 4

35.4  NETWORK MEASURES

Over the past two decades or so, a large number of measures have been developed to study the properties of complex networks. The measures include centrality, clustering, adjacency, distance, community structure, bipartivity, fragments (or subgraphs), communicability, and global invariants, among others; see, for example, Estrada (2012) for details. Such measures not only identify/quantify different properties of networks, but also often offer crossverification, and possible confirmation, of results. For some measures, there are also different definitions, submeasures, and corresponding methods, as appropriate. In what follows, a brief description of some of these network measures is presented: centrality (degree centrality), clustering (clustering coefficient), adjacency (degree distribution), distance (average shortest path length), and community structure. 35.4.1  Centrality (Degree Centrality)

Centrality is one of the most basic and intuitive measures of a network, as it identifies the “most important” or “dominant” nodes. The concept of centrality goes back to the studies of Bavelas (1948) and Leavitt (1951) for communication networks; see Freeman (1979) for an early comprehensive review. However, Jeong et al. (2001) and Newman (2001b) were among the first to use the concept in the context of complex networks. A number of centrality-based measures have been proposed in the network literature, such as degree centrality, centrality beyond nearest neighbors (e.g., Katz centrality, eigenvector centrality, subgraph centrality, pagerank centrality, and vibrational centrality), closeness centrality, betweenness centrality, and information centrality; see, for example, Estrada (2012) for details. Among these, the degree centrality has been one of the most widely used measures. The idea behind the use of degree centrality as a network measure is that it identifies whether a given node, say i in a network, is more central or more influential than another node in the network. The degree centrality of node i in a network of N nodes is defined as the number of first neighbors (or simply neighbors) of node i divided by the total number of possible neighbors (N – 1) in the network. Let us consider a selected node i in a network of N nodes, having ki links which connect it to ki other nodes. For illustration, Fig. 35.3 presents a network consisting of nine nodes (i.e., N = 9), with the node i having four links (i.e., with four other nodes) (see Fig. 35.3, left panel). In this case, the four nodes corresponding to the four links are the neighbors of node  i, which are identified based on some conditions (e.g., correlation between node i and other nodes in the network), while the total number of possible neighbors for node i is eight (i.e., N – 1). The procedure is repeated for each and every node of the network. 35.4.2  Clustering (Clustering Coefficient)

One of the most fundamental properties of networks is their tendency to cluster. The concept of clustering has its origin in sociology, under the name fraction of transitive triples (Wasserman and Faust, 1994), but Watts and Strogatz (1998) were the first to use this concept in the context of complex networks. The tendency of a network to cluster is quantified by the clustering coefficient. There exist several definitions of clustering coefficient; see Watts and Strogatz (1998), Barrat and Weigt (2000), and Newman (2001a) for details. However, the clustering coefficient method proposed by Watts and Strogatz (1998), which measures the local density, is very widely used. A brief description of its calculation is presented here. Let us consider first a selected node i in the network, having ki links which connect it to ki other nodes (Fig. 35.3, left panel), as mentioned earlier. If the neighbors of the original node i were part of a cluster, there would be ki(ki – 1)/2 links between them. As shown in Fig. 35.3 (right), with four neighbors of the node i part of the cluster, there are 4(4 – 1)/2 = 6 links in the

Neighbors = 4

i

Actual connections = 3

i

Figure 35.3  Connections in networks and calculation of clustering coefficient: nearest neighbors and actual connections.

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NETWORK MEASURES    35-5 

cluster of node i. The clustering coefficient of node i is then given by the ratio between the number Ei of links that actually exist between these ki nodes (shown as solid lines on Fig. 35.3, right) and the total number ki(ki – 1)/2 (i.e., all lines on Fig. 35.3, right),

Ci =  

2 Ei (35.2) ki (ki − 1)

The procedure is repeated for each and every node of the network. The average of the clustering coefficients Ci’s of all the individual nodes is the clustering coefficient of the whole network C. The clustering coefficient of the individual nodes and of the entire network can be used to obtain important information about the type of network, grouping (or classification) of nodes, and identification of dominant nodes (e.g., super nodes), among others. For instance, as mentioned earlier, a high clustering coefficient (close to 1.0) indicates a regular network (Fig. 35.2a), while a very low clustering coefficient (close to 0), with C = p, indicates a random network (Fig. 35.2b). The clustering coefficient of a small-world network (Fig. 35.2c) and a scale-free network (Fig. 35.2d) is generally not only smaller than that of the regular network, but also considerably larger than that of a comparable random network (i.e., having the same number of nodes and links). 35.4.3  Adjacency (Degree Distribution)

In a network, several structural properties are related to adjacency relationships between nodes. There are different ways to measure the adjacency relations in a network, including node adjacency, degree distribution, degree– degree correlation, and link adjacency. Among these, the degree distribution is a particularly useful measure (e.g., Barabási and Albert, 1999), especially for the identification of the type of network, and is widely used in network studies. A brief description is as follows. In a network, different nodes may have different number of links. The number of links (k) of a node is called as node degree. The degree is an important characteristic of a node, as it allows one to derive many measurements for the network. The spread in the node degrees is characterized by a distribution function p(k), which expresses the fraction of nodes in a network with degree k. This distribution, called degree distribution, is often a reliable indicator of the type of network. In a random graph, since the links are placed randomly, the majority of nodes have approximately the same degree, and close to the average degree 〈k〉 of the network. Therefore, the degree distribution of a completely random graph is a Poisson distribution with a peak at P(〈k〉), and is given by:

p( k ) =  

e−k k k (35.3) k!

Similarly, depending upon the properties of networks, degree distribution can be Gaussian, given by:

 ( k −   k )2   2 σ k2 

− 1 p( k ) =   e  √ 2πσ k

(35.4)

exponential, given by:

p( k ) ~ e − k/k (35.5)

power-law or scale-free, given by:

p( k ) ~  k − γ (35.6)

or other. Among these distributions, the power-law or scale-free distribution has attracted the most attention in the literature on complex networks, as such has been found in a large number of natural and social networks (e.g., Barabási and Albert, 1999; Kim et al., 2004; Keller, 2005; Clauset et al., 2010). The fractal or scale-free nature of numerous natural and social systems and their ability also to self-organize themselves, already well-documented in the literature (e.g., Mandelbrot, 1983; Bak, 1996; Barnsley, 2012), give both credence and motivation to further advance research on scale-free networks. Although it is true that some scale-free networks display an exponential tail, the functional form of P(k) still deviates significantly from the Poisson distribution expected for a random graph.

35_Singh_ch35_p35.1-35.10.indd 5

35.4.4  Distance (Average Shortest Path Length)

Several distance-based metrics are used for studying the topology of networks, including the average shortest path length, resistance length, and generalized network length. The average path length (or distance) is considered as one of the most robust measures of network topology, along with clustering coefficient and degree distribution. In a network, the shortest path length of a node pair i and j is the number of links on the shortest path connecting the node pair. If the node pair is unconnected, then the value of the shortest path length is set to infinite. The average path length (L) of a network with N nodes is the average over all nodes of the shortest path between every combination of node pairs, and is given by L=



1 ∑ dij (35.7) N ( N − 1)

where dij is the distance between pair i and j. This definition for average shortest path length, however, diverges if there are unconnected nodes in the network, since the distance between such nodes is set to infinite; see also da F. Costa et al. (2007). Consideration of only the connected node pairs avoids this divergence problem, but such also introduces a distortion for networks with many unconnected pairs of nodes. The consequence of this is a small value of the average path length, which is expected only for networks with a high number of connections. A closely related measurement is the global efficiency (E), proposed by Latora and Marchiori (2001):

E=

1 1 ∑ (35.8) N ( N − 1) dij

where the sum takes all pairs of nodes into account. The global efficiency quantifies the efficiency of the network in sending information between nodes, with the assumption that the efficiency for sending information between two nodes i and j is proportional to the reciprocal of their distance. The reciprocal of the global efficiency is the harmonic mean of the geodesic distances [see Eq. (35.7)], given by

h = 

1 (35.9) E

The fact that the harmonic mean eliminates the divergence problem otherwise encountered in the average shortest path length makes is a more appropriate measurement for networks with more than one connected component. The average shortest path length offers important information about the type of network. For instance, as mentioned earlier, regular networks, with their high clustering (i.e., stable), have long average path lengths (i.e., inefficient). Random networks have short average path lengths (i.e., efficient) but have low clustering (i.e., unstable). Small-world networks have short path lengths and have high clustering, and so are both stable and efficient. 35.4.5  Community Structure

In many complex networks, nodes cluster together into distinct groups, each of which is more densely linked together when compared to the rest of the network. The properties of these groups are more or less independent of the properties of individual nodes and of the network as a while. These groups are known as communities and this kind of network structure is known as community structure. Identification of communities in networks, especially large networks, is particularly useful, since nodes belonging to the same community are more likely to share properties and dynamics. Further, the number and characteristics of the existing communities provide subsidies for identifying the type of a network and in understanding its dynamic evolution and organization. Community identification is highly relevant and useful in hydrology, including in the contexts of catchment classification and data interpolation/ extrapolation. Despite the importance of the concept of community in networks, there is no consensus about its definition, and there exist several different formal definitions. The strictest definition is a clique, a subset where all nodes are neighbors. In other words, the shortest path between any node pair within a clique is 1. By relaxing this requirement, the definition of n-cliques is found, where the maximum shortest path between any node pair is n. An alternative set of community definitions is based upon the relative frequency of links. One definition in this category is a LS set, in which nodes have more links to nodes within the set than to other nodes. Radicchi et al. (2004) provided a simple and intuitive definition of a community based on the comparison of

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35-6    Network Theory

the link density among nodes. As per this definition, a community is defined as a set of nodes which has more internal links than external links. A number of methods have been developed for community detection in networks, based on the different definitions of a community and on the need for faster approximations in large networks. Some of these methods rely on the modularity, Q, which quantifies the quality or strength of a community (see later for details). The community detection methods include edge betweenness centrality, greedy optimization, multilevel modularity optimization, leading eigenvector, label propagation, walktrap, info map, optimal, and several others. The first six methods are briefly described here, as they are more widely used. 1. Edge betweenness centrality: It is a measure of how central a particular link is in a network. It measures the number of shortest paths which pass through the link. The links with the highest centrality are thought to act as bridges joining the communities together. Removing these links will split the network into more densely connected communities. As the algorithm (Newman and Girvan, 2004) removes links, a dendrogram is constructed for later analysis to determine the communities. 2. Greedy algorithm: The greedy algorithm, formulated by Clauset et al. (2004), attempts to optimize the modularity (i.e., quality or strength of a community—see later for further details) directly by first placing each node into its own community. The change in property Q from joining any two communities together is then calculated. The community pairs that will produce the largest change in Q are merged together and this is repeated until a maximum value of Q is obtained. 3. Multilevel modularity optimization: This algorithm was proposed by Blondel et al. (2008) and utilizes a similar greedy approach with each node first assigned to its own community. The nodes are then reassigned one by one into a community that will provide the highest increase in modularity. When no more nodes can be reassigned, then each community is considered a node on its own and the process is repeated. This process continues until there is only one node left or if the modularity cannot be increased further. 4. Leading eigenvector method: This method (Newman, 2006) takes advantage of the algebraic properties from the matrix representations of networks in order to optimize the modularity. It defines the modularity matrix as:   B = A − P   (35.10)



where A is the adjacency matrix of the network and P is the probability matrix of links existing according to the configuration model. The eigenvector of the modularity matrix for the largest positive eigenvalue is calculated and nodes are then separated based on the sign of the corresponding element in the eigenvector. 5. Label propagation method: This method (Raghavan et al., 2007) first assigns each node with its own unique label. Each node will then adopt the label which the majority of its neighbors have in an iterative process. Densely connected nodes will reach a consensus on a unique label for the group. When the labels cannot propagate any further, the process ends and nodes with the same label are grouped as one community. 6. Walktrap method: This algorithm, proposed by Pons and Latapy (2006), uses short random walks to define a distance between nodes and communities. Nodes that are within the same community should have short distances between them, while nodes from different communities would have long distances. Similar to the algorithms mentioned earlier, the initial step separates each node into its own community. Communities are then joined on the basis of minimizing the mean of the squared distances between each node and its community. This iterates until a single community consisting of all the nodes is formed. The dendrogram formed during this process is then used to decide on the final community partitions. Given that there are many methods for the identification of communities, with each providing differing results, there is a need to evaluate the quality (or strength) of the identified communities as well as the similarities between identification methods. One of the most popular methods for quantifying the quality of a community division is modularity, which is defined as:

Q = ∑(eii − ai2 ) (35.11) i

where ai is the fraction of links in the network which connect to community i and eii is the fraction of links that exist within the community. A high modularity community will have dense intercommunity connections while

35_Singh_ch35_p35.1-35.10.indd 6

the intracommunity connections are sparse. Other methods include the Silhouette index, performance, and Davies-Bouldin Validation Index. Still other methods to compare whether the communities identified by one algorithm is similar to those found by another algorithm exist. The normalized mutual information (NMI) measure (Danon et al., 2005) is one such method. The NMI utilizes a confusion matrix where the rows represent the communities identified in one method and the columns represent the communities from another method. The measure of similarity between methods is given by   N ij N  −2∑ ∀i ∑ ∀jN ij log   ∑ iN ij ∑ jN ij  NMI( A, B ) =  (35.12)  ∑ N ij   ∑ N ij  j i ∑ ∀i∑ jNij log  N  + ∑ ∀j∑ iNij log  N      where i corresponds to the communities found with identification method A, j corresponds to the communities found with identification method B, N is the total number of nodes in the network, and Nij is the number of nodes that appear in both community i and community j. If the identified communities are identical, then the NMI will have the maximum value of 1, while a value of 0 means that the identified communities are completely independent of each other. 35.5  APPLICATIONS IN HYDROLOGY

Successful applications of the concepts of complex networks in various fields have inspired applications of such concepts in hydrology and water resources as well. Studies in this direction are still at a very early stage, and the applications have thus far been limited to rainfall monitoring networks, streamflow monitoring networks, river networks, and virtual water networks (e.g., Rinaldo et al., 2006; Zaliapin et al., 2010; Suweis et al., 2011; D’Odorico et al., 2012; Scarsoglio et al., 2013; Sivakumar and Woldemeskel, 2014, 2015). Sivakumar (2015) discussed the general relevance of the concepts of complex networks in hydrology and presented three specific examples (the hydrologic cycle, streamflow monitoring network, and parameter estimation in hydrologic models) as to the treatment of hydrologic systems as complex networks. In light of the generality of the concept and early promising outcomes, Sivakumar (2015) also argued that network theory can serve as a generic theory for hydrology. In what follows, a brief account of the applications of network theory in hydrology is presented. For details, the interested reader is directed to the respective studies. 35.5.1  Rainfall Monitoring Networks

To our knowledge, the study by Malik et al. (2012) was the first one to apply the concepts of network theory to rainfall. Malik et al. (2012) examined the spatial and temporal characteristics of extreme (summer) monsoonal rainfall over South Asia, by analyzing daily gridded rainfall data over the period 1951–2007 from the APHRODITE (Asian Rainfall Highly Resolved Observational Data Integration Towards the Evaluation of Water Resources) project. They applied a host of network-based methods, including degree centrality, degree distribution, clustering coefficient, and closeness centrality. Following up on the study by Malik et al. (2012), Boers et al. (2013) applied similar network concepts to investigate the spatial characteristics of extreme rainfall synchronicity of the South American Monsoon System (SAMS), through analysis of a 15-year long (January 1998–December 2012) gridded daily rainfall data with a spatial resolution of 0.25° × 0.25°, obtained from the Tropical Rainfall Measuring Mission (TRMM) 3B42 V7 satellite product. Scarsoglio et al. (2013) applied the complex networks concepts to examine the spatial dynamics of annual precipitation around the globe. They analyzed a 70-year long (January 1941–December 2010) gridded precipitation data from the Global Precipitation Climatology Center (GPCC) Database using a number of methods, including degree centrality, clustering coefficient, degree distribution, and shortest path length. Sivakumar and Woldemeskel (2015) examined the spatial connections in rainfall in Australia using concepts of complex networks. They employed the clustering coefficient method and the degree distribution method to monthly rainfall observed over a period of 68 years (1940–2007) at 230 raingage stations across Australia. Their study was the first study that employed complex networks concepts to ground-measured rainfall data, as opposed to the above studies that used gridded rainfall data. They also considered the influence of rainfall correlation threshold on network properties, by carrying out the analysis for several different thresholds. They identified regions/stations of

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APPLICATIONS IN HYDROLOGY    35-7 

high and low connectivities, which have important implications for optimal raingage density and locations. Their results also suggested that the rainfall monitoring network is not a classical random network, but more likely an exponentially truncated power-law network. Following up on the study by Sivakumar and Woldemeskel (2015), Jha et al. (2015) attempted to offer hydrologic explanation for the outcomes of network-based methods. For this purpose, they applied the clustering coefficient method to two different raingage networks in Australia (57 stations from Western Australia and 45 stations from the Sydney region) and interpreted the results in terms of topographic properties of raingage stations (latitude, longitude, and elevation) and characteristics of rainfall data (mean, standard deviation, and coefficient of variation). 35.5.2  Streamflow Monitoring Networks

The study by Sivakumar and Woldemeskel (2014) was the first ever that applied the concepts of complex networks to streamflow monitoring networks. The study employed the degree centrality method and the clustering coefficient method to examine spatial connections in monthly streamflow observed over a period of 52 years (1951–2002) at 639 streamflow gaging stations in the contiguous United States. The study also investigated the influence of streamflow correlation threshold on the aforementioned properties of the network. One of the important outcomes of the study is that even nearest stations can have significantly different properties as part of a network and even distant stations can have similar properties. This has important implications for interpolation and extrapolation of data, including for predictions in ungaged basins. The study also revealed that the streamflow monitoring network is not a classical random graph, but of some other nature, which has implications for model development, including identification of the nature and complexity of the model. The outcomes of the study also led to a series of other studies, including the role of hydrologic regions on network properties (i.e., “regional” vs. “global” network), optimum number of gaging stations, and catchment classification using community structure, details of which will be reported elsewhere. Halverson and Fleming (2015) applied the concepts of complex networks to a network of 127 streamflow monitoring stations in Canada. In addition to the investigation of whether regional streamflow hydrology might be quantitatively represented as a formal network, their study aimed at assessing whether the results from the network-based methods might offer important information as to the optimal design of streamflow monitoring systems. They employed a host of network-based methods, including clustering coefficient, degree distribution, average shortest path length, betweenness, and community structure. The community structure analysis identified 10 separate communities, each of which defined by the combination of its median seasonal flow regime and geographic proximity to other communities, with important implications for catchment classification. Based on such results, they proposed that an idealized sampling network should sample high-betweenness stations as well as small-membership communities which are, by definition, rare or undersampled relative to other communities, while at the same time retaining some degree of redundancy to maintain network robustness. Fang et al. (2015) introduced the concept of complex networks and community structure to classify catchments in large-scale river basins, considering the Mississippi River basin as a representative basin. They applied six community detection methods: edge betweenness, greedy algorithm, multilevel modularity optimization, leading eigenvector, label propagation, and walktrap. They examined the influence of correlation threshold on classification. The consistency among the methods in classifying catchments was assessed using the NMI index. Their results suggested the role of geography and river branching on classification and also a notable influence of correlation threshold on the number and size of communities identified. They also reported high degree of consistency in the performance among the six methods, except for the leading eigenvector method at lower thresholds. Detailed investigations as to the specific role of river branching on classification are currently underway. 35.5.3  River Networks and Processes

The relevance of the concepts of networks to the structure and dynamics of rivers has been known for many decades now, especially from the point of view of topology, trees, and random networks (e.g., Horton, 1945; Strahler, 1957; Shreve 1966, 1967; Scheidegger, 1967; Kirkby, 1976; Tokunaga, 1978; Werner, 1982; Rodriguez-Iturbe and Rinaldo, 1997). It is no surprise, therefore, that the first applications of complex networks concepts in hydrology were to study the river networks. Among the new network concepts, scalefree networks and network motifs (subgraphs) are readily applicable to river networks, since river networks exhibit scale-invariance, self-organization, and

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many related properties (e.g., Mandelbrot, 1983; Rodriguez-Iturbe and Rinaldo, 1997; Peckham and Gupta, 1999). Rinaldo et al. (2006) introduced the concepts of complex networks in hydrology, through a review of theoretical and observational developments on the form and function of natural networks in different contexts in different fields and their relevance in hydrology. They discussed the properties and dynamic origin of the scale-invariant structure of river patterns and its relation to optimal network selection, and argued that purely random or deterministic constructs are unsuitable for a proper description of river networks and other natural network forms. They reported, through application of degree distribution, clustering coefficient, and average path length methods, the emergence of nontrivial phase transitions with increasing links-tonodes ratios and that different features like scale-free or small-world networks are obtained for particular cases; see also Colizza et al. (2004). They concluded that the emergence of the structural properties in river (and other natural) networks may not necessarily be due to embedded rules for growth, but may rather reflect the interplay of dynamic mechanisms with an evolutionary selective process. Rinaldo et al. (2014) discuss further on the statics, dynamics, and complexity of the evolution and optimal selection of river networks. Zaliapin et al. (2010) applied the concepts of network theory to study environmental transport problems in river networks. Specifically, they studied the dynamic topology of directed trees. They described the static geometric structure of a drainage network by a tree, referred to as the static tree, and introduced an associated dynamic tree that describes the transport along the static tree. Applying connectivity concepts, such as hierarchical aggregation and clustering, they showed that dynamic trees are also self-similar just as their corresponding static trees, but that their properties differ systematically from those of the corresponding static trees. They also reported an unexpected phase transition in the dynamics of river networks (one from California and two from Italy) and demonstrated universal features of this transition, with interpretation in a hydrologic context. Czuba and Foufoula-Georgiou (2014) proposed a simplified networkbased predictive framework of sedimentological response in a basin, incorporating network topology, channel characteristics, and transport-process dynamics to perform a nonlinear process-based scaling of the river-network width function to a time-response function. They developed the processscaling formulation for transport of mud, sand, and gravel, using simplifying assumptions including neglecting long-term storage. They applied the methodology to the Minnesota River basin in the United States. Their results indicated, among others, that the network topology and sedimenttransport dynamics combined to produce a double peaked response function for sand, suggesting that there exists a resonant frequency of sediment supply that could lead to an unexpected downstream amplification of sedimentological response. Subsequently, Czuba and Foufoula-Georgiou (2015) extended this framework to understand the internal dynamics of the basin for sediment transport, that is, how sediment is organized and where sediment accumulates due to the combined effects of river network structure (topology and associated geometry) and transport dynamics (accounting for slopes, channel morphology, bed shear stress, grain size, etc.). Specifically, they developed a dynamic connectivity framework and applied it to understand sand transport in the Greater Blue Earth River Network in the Minnesota River basin. 35.5.4  Virtual Water Trade Networks

Virtual water trade networks have been the most studied networks, thus far, in the context of complex networks in hydrology and water resources. Suweis et al. (2011) applied the complex network theory to examine the connections in virtual water trade networks. They used a simple model to describe the topological and weighted properties of the virtual water trade network between world nations for staple food products, using degree distribution, average nearest neighbor degree, and clustering coefficient methods. Assuming the gross domestic product and yearly rainfall on agricultural areas in each country as sole drivers, they identified and quantified the high degree of globalization of water trade and showed that only a small group of nations play a key role in the connectivity of the network and in the global redistribution of virtual water. Konar et al. (2011) applied the concepts of complex networks to quantify the topology of the virtual water network associated with international food trade (58 commodities from five major crops and three livestock types). They analyzed both directed and weighted properties of the networks. They reported that the number of trade connections follows an exponential distribution, except for the case of import trade relationships, while the volume of water that each nation trades compares well with a stretched exponential distribution, indicating high heterogeneity of flows between nations. They also reported that there is a power-law relationship

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35-8    Network Theory

between the volume of water traded and the number of trade connections of each nation. Carr et al. (2012) employed the degree distribution method to the virtual water network associated with international food trade over the period 1986–2008. They found that the total flow over the period had more than doubled, and the number of links had increased by 92%. The network grew fast by establishing new links during 1990–2001 and then by increasing the fluxes through existing links during 2002–2008. The virtual water network was found to have become more homogeneous but most of the flow was concentrated in a few links and hubs, while several countries exhibited only few, and mostly weak, connections. Dalin et al. (2012) used the concepts of complex networks, including the degree distribution method, to study the virtual water network associated with international food trade during 1986– 2007 and linked it to trade policies, socioeconomic circumstances, and agricultural efficiency. They found that both regional and national virtual water trade patterns significantly changed, with Asia increasing its virtual water imports by more than 170%, switching from North America to South America as its main partner, whereas North America oriented to a growing intraregional trade. D’Odorico et al. (2012) studied the community structure of the virtual water network. They employed a method based on the maximization of the modularity (Newman and Girvan, 2004) to the virtual water network over the period 1986–2008. Identifying an increase in the ratio between virtual water flows within communities and the total global trade of virtual water, they reported the existence of well-defined clusters of virtual water transfers. However, geographic proximity was found to only partly explain the community structure of virtual water trade. Tamea et al. (2013) introduced the average distance traveled by virtual water as a measure for the analysis of globalization in the virtual water trade. They visualized the fluxes between a country and its importing/exporting partners with a geographical representation shaping the trade network as a virtual river/delta. Using Italy as a country of interest, they found that food trade has a steadily growing importance compared to domestic production, with a major component represented by plant-based products, and luxury products taking an increasingly larger share. Replicating the analysis to 10 other countries, which triggered similar investigations on different socioeconomic actualities, they showed that integrating the local (country) perspective and the global perspective can shed new light on the structure and dynamics of virtual water trade network. Several other studies have also examined the structure and dynamics of virtual water trade networks, either in the context of complex networks or in other contexts that support the results reported by the above studies. Examples of such studies are those by D’Odorico et al. (2010), Konar and Caylor (2013), Konar et al. (2013), O’Bannon et al. (2014), Dalin et al. (2014), and Tamea et al. (2014). 35.6 CONCLUSION

A fundamental requirement in studying the dynamics of hydrologic systems and phenomena is the identification of the nature and extent of connections in space and time. Research over the past century on hydrologic connections, through numerous scientific concepts and mathematical techniques, has indeed resulted in significant progress. Nevertheless, our understanding of such connections still remains far from adequate. To this end, the concepts of network theory, and especially the recent concepts of complex networks, offer a whole different dimension and new avenues, for their holistic perspective of systems and problems encountered in hydrology and water resources. For instance, small-world, scale-free, community structure, and several other new concepts are highly relevant for hydrologic systems, whose structure and dynamics are a combination of order and randomness. As the aforementioned review, albeit brief, suggests, the relevance of the concepts of complex networks in hydrology as well as the potential areas for their applications are obvious; see also Sivakumar (2015). These new concepts help obtain important information toward addressing a wide range of problems in hydrology. A few important ones among such problems are: interpolation and extrapolation of data, development of a catchment classification framework, and formulation of an integrated framework for water planning and management (taking into account also other Earth systems and socioeconomic systems). As such problems are at the core of many of the most important recent and current initiatives and activities in hydrology and water resources, such as predictions in ungaged basins, downscaling of outputs from global climate models for hydrologic analysis, and study of human-water interactions, network concepts can go a long way in hydrology. Indeed, as Sivakumar (2015) argued, network theory is a suitable tool for studying all types of connections and, hence, can serve as

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a generic theory for hydrology. The future of network theory in hydrology looks bright! ACKNOWLEDGMENTS

This work is supported by the Australian Research Council (ARC) Future Fellowship grant (FT110100328). Bellie Sivakumar acknowledges the financial support from ARC through this Future Fellowship grant. We thank Dawen Yang and Jun Niu for their review comments and suggestions on an earlier version of the manuscript. REFERENCES

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35-10    Network Theory

Salas, J. D., J. W. Delleur, V. Yevjevich, and W. L. Lane, Applied Modeling of Hydrologic Time Series, Water Resources Publications, Littleton, CO, 1995. Scarsoglio, S., F. Laio, and L. Ridolfi, “Climate dynamics: a network-based approach for the analysis of global precipitation,” PloS One, 8 (8): e71129, 2013, doi:10.1371/journal.pone.0071129. Scheidegger, A. E., “On the topology of river nets,” Water Resources Research, 3 (1): 103–106, 1967. Shreve, R. L., “Statistical law of stream numbers,” Journal of Geology, 74: 17–37, 1966. Shreve, R. L., “Infinite topologically random channel networks,” Journal of Geology, 75: 178–186, 1967. Shreve, R. L., “Stream lengths and basin areas in topologically random channel networks,” Journal of Geology, 77: 397–414, 1969. Sivakumar, B., “Dominant processes concept, model simplification and classification framework in catchment hydrology,” Stochastic Environmental Research and Risk Assessment, 22 (6): 737–748, 2008. Sivakumar, B., “Global climate change and its impacts on water resources planning and management: assessment and challenges,” Stochastic Environmental Research and Risk Assessment, 25 (4): 583–600, 2011a. Sivakumar, B., “Water crisis: from conflict to cooperation—an overview,” Hydrological Sciences Journal, 56 (4): 531–552, 2011b. Sivakumar, B. “Networks: a generic theory for hydrology?” Stochastic Environmental Research and Risk Assessment, 29: 761–771, 2015. Sivakumar, B. and V. P. Singh, “Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework,” Hydrology and Earth System Sciences, 16: 4119–4131, 2012. Sivakumar, B. and Woldemeskel, F. M., “Complex networks for streamflow dynamics,” Hydrology and Earth System Sciences, 18: 4565–4578, 2014. Sivakumar, B. and F. M. Woldemeskel, “A network-based analysis of spatial rainfall connections,” Environmental Modelling and Software, 69: 55–62, 2015. Smart, J. S., “Use of topologic information in processing data for channel networks,” Water Resources Research, 6 (3): 932–936, 1970. Smart, J. S., “The analysis of drainage network composition,” Earth Surface Processes and Landforms, 3 (2): 129–170, 1978. Smart, J. S. and C. Werner, “Applications of the random model of drainage

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composition,” Earth Surface Processes and Landforms, 1: 219–233, 1976. Solomonoff, R. and A. Rapoport. “Connectivity of random nets,” Bulletin of Mathematical Biophysics, 13: 107–117, 1951. Strahler, A. N., “Quantitative analysis of watershed geomorphology,” Eos Transactions American Geophysical Union, 38: 913–920, 1957. Suweis, S., M. Konar, C. Dalin, N. Hanasaki, A. Rinaldo, and I. RodriguezIturbe, “Structure and controls of the global virtual water trade network,” Geophysical Research Letters, 38: L10403, 2011, doi:10.1029/2011GL046837. Tamea, S., P. Allamano, J. Carr, P. Claps, F. Laio, and L. Ridolfi.” Local and global perspectives on the virtual water trade,” Hydrology and Earth System Sciences, 17: 1205–1215, 2013. Tamea, S., J. A. Carr, F. Laio, and L. Ridolfi, “Drivers of the virtual water trade,” Water Resources Research, 50: 17–28, 2014. Tokunaga, E., “Consideration on the composition of drainage networks and their evolution,” Geographical Reports of Tokyo Metropolitan University, 13: 1–27, 1978. Tsonis, A. A. and P. J. Roebber, “The architecture of the climate network,” Physics A, 333: 497–504, 2004. Wasserman, S. and K. Faust, Social Network Analysis, Cambridge University Press, Cambridge, 1994. Watts, D. J., Small Worlds: The Dynamics of Networks Between Order and Randomness, Princeton University Press, Princeton, 1999. Watts, D. J. and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature, 393: 440–442, 1998. Werner, C., “Analysis of length distribution of drainage basin parameter,” Water Resources Research, 18 (4): 997–1005, 1982. Yang, D., C. Li, H. Hu, Z. Lei, S. Yang, T. Kusuda, T. Koike, and K. Musiake, “Analysis of water resources variability in the Yellow River of China during the last half century using historical data,” Water Resources Research, 40: W06502, 2004, doi:10.1029/2003WR002763. Young, P. C. and M. Ratto, “A unified approach to environmental systems modeling,” Stochastic Environmental Research and Risk Assessment, 23: 1037–1057, 2009. Zaliapin, I., F. Foufoula‐Georgiou, and M. Ghil, “Transport on river networks: a dynamic tree approach,” Journal of Geophysical Research, 115: F00A15, 2010, doi:10.1029/2009JF001281.

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Chapter

36

Hydroeconomic Analysis BY

PETER BAUER-GOTTWEIN, NIELS RIEGELS, MANUEL PULIDO-VELAZQUEZ, JULIEN J. HAROU, XIMING CAI, CLAUS DAVIDSEN, AND SILVIO J. PEREIRA-CARDENAL

ABSTRACT

Hydroeconomic analysis and modeling provides a consistent and quantitative framework to assess the links between water resources systems and economic activities related to water use, simultaneously modeling water supply and water demand. It supports water managers and decision makers in assessing trade-offs between different water uses, different geographic regions, and various economic sectors and between the present and the future. Hydroeconomic analysis provides consistent economic performance criteria for infrastructure development and institutional reform in water policies and management organizations. This chapter presents an introduction to hydroeconomic analysis and modeling, and reviews the state of the art in the field. We review available economic water-valuation techniques and summarize the main types of decision problems encountered in hydroeconomic analysis. Popular solution strategies for hydroeconomic problems are introduced. Typical problems and issues in real-world applications of hydroeconomic analysis and modeling are discussed. The chapter ends with a discussion of challenges, limitations, and future research directions in this field. 36.1 INTRODUCTION

Around the globe, natural resources and ecosystems are coming under increasing pressure due to human development (Steffen et al., 2007). In many parts of the world, demands for freshwater and pollution loads to receiving water bodies are increasing (Postel et al., 1996; Rockstrom et al., 2009). Moreover, future renewable freshwater supplies are uncertain due to the stochastic nature of climate/weather and the insufficiently understood effects of anthropogenic climate change (Oki and Kanae, 2006). In this context, water resources management over all spatial and temporal scales has had to reconsider the demand fulfillment paradigm and to consider whether society might be better served by redistributing water among sectors, for example, from irrigated agriculture to aquatic ecosystems (Booker et al., 2012). Hydroeconomic analysis provides a systematic and quantitative framework to assess the links between water resources systems and economic activities and reveal economic trade-offs between different water users (Harou et al., 2009; Maass et al., 1962). Complex real-world water resources systems are mapped into simplified conceptual models that represent the topology and connectivity of the flow network as well as the main water-storage options in the system. Water uses are classified into a number of sectors and use locations, and are represented by economically respectful demands—true value-sensitive functions rather than historical use amounts. Hydroeconomic analysis and modeling has been carried out over a range of spatial scales from single reservoir, irrigation district, and sub-basin, to basin-scale, regional, and even global models. Hydrologic and economic submodels have been coupled in two principal ways: simulation and optimization approaches. While early studies traditionally focused on costs, new era hydroeconomic studies generally use the net present value (NPV), that is, the sum of discounted benefits net of costs, to express economic performance of projects and operational strategies. In the simulation approach, NPV of observed or modeled water allocations is quantified. Simulation approaches are useful when the focus is on the evaluation of water resource interventions such as new policies or infrastructure developments or combinations of these. For instance, the economic implications of widely used decision rules in water resources management, such as the rule of

prior appropriation, can be evaluated using a simulation approach. In the optimization approach, water is allocated to maximize net economic value, or, alternatively, to minimize the total costs of water allocation and water-supply curtailment, occurring at all use locations in the system. The management problem is cast into an optimization framework where the objective is to maximize net value and the constraints reflect various physical and institutional characteristics of the system as well as environmental flow requirements. The applications of hydroeconomic analysis are numerous and diverse. One class of applications involves planning problems. Often, the economic effects of planned changes (e.g., new hydraulic infrastructure, institutional reform) have to be assessed a priori. Hydroeconomic analysis and modeling can quantify differences in costs and benefits between the with and withoutproject scenarios to analyze if a certain water project is deemed economically acceptable (cost-benefit analysis; e.g., Griffin, 1998). One example is the design of a new surface water reservoir in a multireservoir system for water supply. In order to assess the benefits that the reservoir would generate (to be compared with the costs of the project), we need to estimate how this additional storage capacity would change water deliveries to the competing uses during all periods. The policy may involve new operating rules for all reservoirs in the systems, which need to be defined during the design process. A hydroeconomic model with value-sensitive demand functions can be used to assess the welfare changes in the system for developing the cost-benefit analysis and for finding an economically efficient design. Another application of hydroeconomic modeling and analysis is decision support in water resources management. Operators are confronted with trade-offs in water allocation in the spatial dimension, in terms of intersectoral competition at a certain time, and in terms of intertemporal decisions affecting current and future benefits/costs. Economic efficiency in water management requires considering opportunity costs in water-allocation decisions. Hydroeconomic analysis tools can reveal those opportunity costs, which can be used to inform water-allocation decisions and in the design of efficient economic instruments, such as water pricing (Pulido-Velazquez et al., 2013a). Dynamic optimization algorithms can support the development of operating rules and scheduling decisions and reveal intertemporal trade-offs to identify the long-term optimal management. Hydroeconomic analysis has also been used to guide and inform reallocation policies. For instance, in Europe, the EU Water Framework Directive required reallocation of water to ecosystems. Reallocation can be achieved using economic tools such as flexible water pricing (e.g., Riegels et al., 2013) and water markets (Erfani et al., 2014). The performance of these tools can be partially assessed a priori using hydroeconomic modeling and analysis. This chapter starts with an overview of available economic water-valuation methods. We proceed with a review of typical decision problems in hydroeconomic analysis and modeling, and of common solution strategies for these problems. Subsequently, we discuss applications and implementation of hydroeconomic models for informing real-world problems. We discuss limitations of current methodologies and future research directions and provide conclusions. 36.2  ESTIMATING THE ECONOMIC VALUE OF WATER

A key task in hydroeconomic analysis is the determination of the water users’ water demand function, especially the relationship between the economic 36-1

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36-2    Hydroeconomic Analysis

value of water and the amount of water delivered. A broad range of valuation approaches have been presented (e.g., Booker et al., 2012, provide a recent and comprehensive overview). Applicability and suitability depend on the specific water services being valued as well as on the purpose of the valuation exercise and the data/information availability (Young, 2005). Microeconomic theory differentiates two types of resource users: producers and consumers (e.g., Varian, 2010). Producers use water to produce a product and their objective is to maximize the profits obtained from selling the product on the market. Consumers use water to derive utility and their objective is to maximize the utility they obtain from consuming a mix of water and other goods, subject to a budget constraint (Griffin, 2006). In hydroeconomic analysis, the challenges are to determine the relationship between the amount of water delivered and the amount of product produced or utility obtained (the production and utility functions) in order to estimate the marginal value of water. Numerical efforts have been put on the development of functional relationships between water use output (e.g., crop yield, industrial product, profit, etc.) and water and other inputs (McKinney et al., 1999), and demand functions, which connect water demand by sector or by individual water user and their willingness to pay. The estimation of economic value is particularly challenging and controversial for ecosystem water uses. Challenges exist with the valuation of more comprehensive ecosystem services especially biodiversity and water quality, which is the concern of environmental economics (de Groot et al., 2002). Furthermore, the lack of quantified relationships between ecological indicators and hydrologic variables often limits the incorporation of ecological benefits into hydroeconomic analysis (e.g., Yang et al., 2008). 36.3  WATER DEMAND FUNCTIONS

If production or utility functions are known for a given water user, the water demand function can be derived by solving the profit maximization problem for the producer and the constrained utility maximization problem for the consumer (Griffin, 2006). Most common production functions show decreasing returns to scale, that is, the marginal benefit decreases with increasing water allocation. At the point of operation that is optimal from the perspective of the individual user, the marginal net benefit (= the first derivative of the benefit with respect to water allocation) is zero. Producer and consumer surplus are defined as the area between the user’s willingness to pay (WTP) function and the cost function. At water prices higher than the choke price, water users are assumed to switch to alternative supplies (e.g., bottled water). Surplus is considered as the standard measure of welfare in microeconomic theory. Figure 36.1 illustrates these basic concepts of water resources economics. Conventional water resources management models use a centralized approach to explore the system-wide optimal solution, assuming that (1) there is a topdown management process in which all watershed stakeholders completely obey a “super brain,” (2) complete information exchange exists among the stakeholders in different locations and production sectors, and (3) perfect economic efficiency is realized, that is, the marginal welfare of natural/raw/source water use is identical for all agents. These assumptions are usually not fulfilled

Water price Choke price Consumers’ surplus + Market value = Gross benefits

A Demand function Water demand B Figure 36.1  Water demand function for a water user, based on Harou et al. (2009). As the water price increases, the users demand for water decreases. For a situation with a water price equal to A (e.g., $/m3), the water demand (e.g., m3) will be B.

36_Singh_ch36_p36.1-36.10.indd 2

in real-world watershed management situations. Decentralized modeling approaches may be partially free of those assumptions (Wallace et al., 2003; Yang et al., 2009). For example, Yang et al. (2009) present a decentralized optimization method that depicts a watershed as a multiagent system, which is expected to support stakeholder-driven, bottom-up watershed management. This method is applied to the Yellow River Basin for water allocation by incorporating a more realistic institutional setting of the basin (Yang et al., 2012). Other examples in this modeling direction include Barreteau et al. (2004), Berger et al. (2006), and Ng et al. (2011). Britz et al. (2013) presented the MOPEC method (multiple optimization problems with equilibrium constraints) to solve multiuser allocation problems under the assumption that each user maximizes her individual welfare, while users are connected to each other through flow network topology and market equilibria. 36.3.1  Estimating Water Demand Functions

Most water uses can be defined as agriculture, industry, power generation, recreation, navigation, domestic, or ecosystem uses. In real-world applications, there is typically insufficient information to parameterize water production and utility functions for different uses. Different methods and approaches have been proposed for the estimation of water demand functions as well as point estimates of the marginal economic value of water, using varying levels of complexity and data requirements. Obtaining reliable data at the appropriate spatial aggregation level is an important challenge in hydroeconomic analysis. Water can be either an intermediate (production) or a final good (consumption). When water is an input to a production process (e.g., irrigation, hydropower generation, commercial, or industrial uses), water demand is derived from the demand for the final output and depends on the production process. When water is a final good (e.g., residential or recreational water uses), water provides direct utility to consumers; this utility translates into a certain WTP for it, influenced by the preferences of consumers and their budget constraints. Different economic theories (of consumer’s and producer’s demands) (Hanemann, 1999; Young, 2005) are applicable to each case, affecting the selection of the valuation method. For intermediate goods (derived demand, e.g., agriculture and industry), a widely used method for making point estimates of marginal economic values is residual imputation (RI; Young, 2005). The basic idea in RI is to subtract all non–water-related costs from the observed gross benefit and to attribute the residual value to the water. The challenge with RI is how to deal with non–water-production factors that contribute to the production but are not easily quantifiable, such as, for instance, entrepreneurial skill and experience. If such factors are ignored in RI, the resulting value of water will be inflated. Estimates of marginal values of water using the RI approach also depend on the time horizon of the planning problem. For short-term planning problems, marginal value may be higher because producers do not have the ability to shift sunk investments in machinery or other physical capital (so-called “sunk costs”) to other production processes. In seasonal irrigation planning, even costs of seeds and fertilizers may be considered sunk costs if marginal value is being estimated, for example, in the middle of a crop growth cycle. In the long term, however, all inputs can be reallocated to other uses and should be subtracted from the residual value used to estimate marginal value of water. Econometric and especially mathematical programming approaches have been widely applied to define demand functions for water in irrigation (Booker et al., 2012; Young, 2005). Irrigation water demands are usually simulated in hydroeconomic models using exogenous linear or quadratic equations, relating water application to economic benefits (e.g., PulidoVelazquez et al., 2007a; Pulido-Velázquez et al., 2006). In other cases, crop yield functions are explicitly included in a river basin hydroeconomic model (e.g. Cai et al., 2003b) so that the marginal values are obtained from multiplying marginal physical productivities by crop prices. Howitt (1995) introduced a calibration method for estimating agricultural water demand functions, positive mathematical programming (PMP), which has since become popular in hydroeconomic modeling studies. The central idea is that observed allocations of land and water represent profit-maximizing behavior by farmers. Shadow prices on observed land and water use are used to parameterize production functions, with the assumption that these shadow prices incorporate unobserved information, such as soil characteristics, consideration of risk and uncertainty, and farmer expertise. The approach can be implemented using cross-sectional surveys of land use, water use, crop yields, prices, and input costs, making it suitable for use with datasets that are commonly available in hydroeconomic studies. The PMP approach estimates a production function, not a demand function, although it can be easily used to estimate a water demand function by solving the profit maximization problem at different water price increments.

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Considerations in the design of hydroeconomic analysis studies     36-3 

For determining water demand for final uses (e.g., urban demands), econometric approaches are dominant (Dalhuisen, 2003; Renzetti, 2002). For urban demands, a simplified but popular method used to estimate water demand functions is the point expansion method. In this approach a pair of observed water price and water use is taken as one point on the demand function. Extrapolation to different allocations or different user prices is possible if assumptions are adopted about the slope of the demand function or the elasticity of the demand (Griffin, 2006). This method is applicable for any sector where a price-quantity pair can be determined for anchoring the demand function. When estimating water demand functions or WTP, it is important to differentiate between “at site” and the “at source” value of water. For example, if a demand function is estimated from observed consumer water use using the point expansion method, this reflects demand for water that has already been abstracted from natural waters, treated, and delivered to the point of use (i.e., the at-site value). However, in an investigation of trade-offs between water uses in a river basin, the value of water at the point of abstraction is the relevant value for comparison to competing uses (the at-source value). The atsource value should include any producer-side benefits to the supplying water utility (Young, 2005). However, the nonprofit status of most utilities implies that producer-side benefits are often transferred to consumers in some fashion (especially by underpricing). The estimation of economic water demand functions for ecosystem uses is a new area for research and professional practice. In this context, the emerging “ecosystems services” paradigm can be useful for defining what are generally understood as “ecosystem water uses” (for an overview of the ecosystem services concept, see Braat and de Groot, 2012). The ecosystem services paradigm argues that ecosystems provide a number of benefits to humans and that the best way to ensure efficient conservation of ecosystems is to measure the contributions of ecosystems to human welfare. A number of methods have been applied to estimating ecosystem water use values. These methods can be broadly grouped into two categories: revealed preference methods and stated preference methods. Revealed preference methods estimate ecosystem values from observations of market transactions involving related private goods, such as prices for homes located near ecosystem resources, or costs associated with travel to visit natural areas. Stated preference methods estimate ecosystem values by asking people what they would be willing to pay for proposed or hypothetical changes in ecosystem service provision. A useful overview of the uses of stated and revealed preference methods in water resource economics is provided in Young (2005). Estimating the value of ecosystem water uses is only one of two steps needed to estimate a water demand function. It is also necessary to develop a relationship between water supply and the associated level of ecosystem service provision. For example, if river restoration activities are under consideration in order to provide recreational and aesthetic benefits, it may be necessary to estimate a relationship between the amount of flow in the river and the amount of benefits provided. Estimation of this relationship may be complicated by the fact that, in the case of river flows, the ecosystem service level depends on many features of the flow regime, including the timing of flows and the amount of variability in the flow regime (for a discussion of the hydrological regime, see Arthington et al., 2006). A number of operational approaches to link measures of the river flow regime to levels of ecosystem service provision have been proposed (e.g., Poff et al., 2010; Smakhtin et al., 2004). Because estimation of ecosystem values and modeling of ecosystem service provision are complicated and uncertain, almost all hydroeconomic studies represent demand for ecosystem water uses as hard constraints (for an  example of the use of an economic demand function for recreational ecosystem service in a hydroeconomic optimization study, see Ward and Pulido-Velazquez, 2008). Because of generalized lack of data, scaling problems and methodological issues, economic values of water are always uncertain. At the same time, water values are among the most sensitive parameters in hydroeconomic analysis and modeling (Riegels et al., 2011). Rigorous uncertainty analysis and analysis of robustness of results is presently not standard practice in hydroeconomic analysis and modeling. Some hydroeconomic optimization studies include the uncertainty of economic performance into the objective function, penalizing highly variable performance with a risk aversion term (e.g., BlancoGutiérrez et al., 2013). In most hydroeconomic modeling studies, economic effects are quantified as surplus changes based on prespecified water demand functions. In reality, the perturbations generated by water management decisions will propagate further into the economy and may change output prices, labor costs, and other parameters used in the derivation of the demand functions. For instance, if hydropower has a significant share of total power supply in a given system, then the scheduling decisions of the hydropower operators can

36_Singh_ch36_p36.1-36.10.indd 3

change the power price on the market (Pereira-Cardenal et al., 2014). For system-scale applications, it is important to internalize output prices and other exogenous variables to achieve a more realistic simulation of the effects occurring throughout the entire economy. Several attempts to couple water resources models with general equilibrium models of the economy have been reported (e.g., Luckmann et al., 2014). 36.4  CONSIDERATIONS IN THE DESIGN OF HYDROECONOMIC ANALYSIS STUDIES

A hydroeconomic analysis study generally (although not always) combines a hydrologic or river basin model with some form of economic analysis. In some cases, economic water demand functions are integrated directly into the hydrologic model and optimization is used to allocate water across space and time in order to maximize water values or other criteria. Typical water sources include surface water, groundwater, reclaimed water, desalinated water, or transfer water from other basins. Each water source is characterized by its availability, storage capability, quality, and pumping and conveyance cost. The allocation problem is often complicated further by the presence of multiple and contradicting management objectives. While economic efficiency can be an important objective in water resources management, other objectives, such as reliability, equity, and sustainability may also be important. 36.4.1  Multiple User Allocation

Figure 36.2 illustrates a representative multiuser allocation problem. The figure shows three water use locations, each with water demand and each producing net benefit (NB), aligned along a river reach with supply Q and an in-stream flow requirement R at the downstream end of the reach. If the supply Q is insufficient to meet all three demands and the in-stream flow requirement, then the diversions x must be curtailed to be less than the demands (or the amount of water allocated to the in-stream flow requirement must be reduced). Hydroeconomic analysis is often concerned with the question of what these diversions or flow reductions should be. If the net benefits at the use locations are linear functions of the water allocations and if only one single time period is considered, this optimization problem becomes a simple linear program: max(WV1 ⋅ x1 + WV2 ⋅ x 2 + WV3 ⋅ x 3 ) subject to x1 + x1 + x 3 ≤ Q − R xi ≤ Di where xi are the water allocations to the three users, WVi are the water values at the three use locations, Di are the demands, Q is the available river flow, and R is an in-stream water requirement. In real-world applications, of course, river network topology and connectivity are typically much more complex. Moreover, as discussed earlier, uses are not always rival and return flows generated by one user can be reused further downstream in the system (Griffin, 2006). The economic theory and modeling of multiple users connected by their withdrawals, return flows, and instream flow demands is presented and discussed in Griffin and Hsu (1993). Net benefits involve a marginal value of water that increases with lower deliveries, which is better represented by nonlinear economic demand functions. Water-quality requirements by various users and reactive transport processes in the river network may introduce additional constraints into the allocation problem. Hydrologic models are mostly built for simulation experiments, while economic models are often of the optimization type. Compartment models couple the simulation and optimization models usually through genetic algorithms (GA; Nicklow et al., 2010), while holistic models combine physical

R Q

X1 NB1

X2

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NB3

Figure 36.2  Basic river basin allocation problem. Q is the inflow to the system, xi are the water allocations to the individual users, NBi are the net benefits obtained by the users, and R is an instream flow requirement.

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36-4    Hydroeconomic Analysis

relationships, economic constraints, and objectives in a so-called simulationembedded optimization model. Such a model is often complicated due to nonlinearity and discontinuity involved in the simulation module. Both types of models meet computational difficulties when they are applied to large water resources systems. A hybrid GA and linear programming (LP) approach was developed to find approximate global solutions or feasible solutions for large hydroeconomic models formulated as nonlinear programming models with high nonlinearity and nonconvexity (Cai et al., 2001a); a “pieceby-piece” approach (Cai et al., 2001b) was used to solve models that can be naturally decomposed into components (e.g., hydrologic and economic components). Network-flow optimization was used to solve other large hydroeconomic models (e.g., Draper et al., 2003). 36.4.2  Intertemporal Allocation

When allocation decisions today have an impact on future water availability, the allocation problem must be handled using a dynamic framework. Many river system allocation problems are characterized by intertemporal tradeoffs because of the possibility of storing water for future use, including reservoir and groundwater storage (Fig. 36.3, Gisser and Sanchez, 1980; Loucks and van Beek, 2005). In the case of surface reservoirs, hedging operating rules are usually designed to minimize the risk of high losses due to large water deficits in severe future drought periods (Bayazit and Unal, 1990; You and Cai, 2008a; You and Cai, 2008b). In finance and economics, the term hedging is used to describe an action or instrument that reduces the exposure of an individual or organization to any significant losses/gains due to uncertain future developments. In irrigation agriculture, water is allocated over the course of the entire growing season, while harvest and associated economic benefits materialize at the end of the growing season. Such “delayed yields” in irrigated agriculture (Tilmant et al., 2008), storage in water supply systems, time delays in network routing and water quality effects can also contribute to intertemporal trade-offs. One important issue is how to discount costs and benefits occurring in the future in order to trade them off against costs and benefits in the present. A large body of literature discusses discounting and the choice of appropriate discount rates in environmental studies (e.g., Heal, 1998). Hydroeconomic analysis studies that aim to provide advice about intertemporal allocation in real-world operations, such as reservoir operating rules, should address the role of uncertainty. Future water availability is uncertain, particularly in the case of surface water resources, and hydroeconomic analysis of intertemporal trade-offs should consider the value of hedging against uncertainty. Hydroeconomic studies to support real-world operations are often formulated as stochastic dynamic optimization problems. A starting point for this type of problems is stochastic dynamic programming (SDP; Stedinger et al., 1984, Loucks and van Beek, 2005), although applications are limited by the well-known curse of dimensionality in complex water resources systems. A widely used variant of SDP known as the water value method, determines and archives the shadow prices for any combination of time and system state (Stage and Larsson, 1961, Wolfgang et al., 2009). The water values can subsequently be used as decision rules in a more highly resolved simulation model (e.g., Wolfgang et al., 2009; Davidsen et al., 2015a). The SDP method has many advantages: it can handle discontinuous and nonlinear functions; it provides the optimal solution from any given stage, and can handle a large amount of stages. However, computing and memory requirements increase exponentially with the number of states, which limits the amount of states that can be included in the problem. Hence, only three or four reservoirs can be optimized at a time (Labadie, 2004). Several extensions to DP have been proposed to overcome this challenge

(Rani and Moreira, 2009). One of the most promising methods is stochastic dual dynamic programming (SDDP; Pereira and Pinto, 1991), which has been successfully applied to optimize multi-purpose reservoir systems (e.g., Goor et al., 2011; Tilmant and Kelman, 2007). An alternative to dynamic programming methods are heuristic optimization techniques, which do not guarantee the optimal solution, but provide a feasible solution often close to the optimal one. A popular class of such methods are evolutionary algorithms, which have been used extensively in water resource management (Cai et al., 2001a; Nicklow et al., 2010). 36.4.3  Conjunctive Use of Multiple Water Resources

In many allocation problems, managers have access to more than one water source to satisfy demands. Sources may include surface water, groundwater, reclaimed water, desalinated water, and transfer water from other basins (e.g., Luckmann et al., 2014). Often, water can also be supplied from multiple reservoirs which are connected and managed jointly as a multireservoir system (Labadie, 2004). In systems with important groundwater components, the model should include surface and ground water systems, and stream-aquifer interaction. Otherwise, significant economic opportunities, opportunity costs, and externalities may be ignored (e.g., Pulido-Velazquez et al., 2008). Flexible management of conjunctive use facilities under flexible water allocation can generate substantial economic benefits in water resource systems operation, which can be assessed through hydroeconomic models (e.g., Harou and Lund, 2008; Pulido-Velazquez et al., 2004, 2007b). If different water sources have to be managed jointly, managers have to consider differences in fixed and variable costs between the various water sources. Such costs may depend on the state of the system (e.g., depth to the groundwater table; Riegels et al., 2013), on the rate of abstraction (e.g., regional and local drawdown in groundwater pumping) and on the water quality (e.g., desalination; Karagiannis and Soldatos, 2008). Often, cost functions are complex and introduce nonlinearity and nonconvexity into the optimization problems, which requires more advanced and CPU-intensive optimization algorithms (e.g., Pulido-Velázquez et al., 2006, Davidsen et al., 2016). 36.4.4  Conflicting Objectives and Pareto Fronts

In hydroeconomic optimization, managers and analysts often encounter multiple and contradicting management objectives (Xevi and Khan, 2005). Contradicting objectives may be related to competing users which can be geographically defined (transboundary river systems, Fig. 36.4) or defined in terms of water use (e.g., irrigation agriculture vs. aquatic ecosystems). A traditional hydroeconomic approach is to value all benefits and costs so they can be aggregated into one commensurate monetized objective function for an optimization model. However, sometimes, objectives cannot be easily expressed in monetary units and the requirement to monetize can weaken the analysis or underappreciate stakeholder interests. Such criteria may include reliability metrics, measures of the spatial and sectorial distribution of impacts, and sustainability metrics. The inherently multiobjective nature of most water resources management problems has led to further development of multiobjective optimization approaches (Maier et al., 2014; Nicklow et al., 2010; Reed et al., 2013). Such approaches do not provide one single best solution, but a whole set of Pareto-optimal or noninferior solutions, the so-called Pareto frontier. A Pareto-optimal policy is characterized by the fact that there is no alternative policy that is better in terms of at least one objective and not worse in terms of all other objectives. A selection between two Pareto-optimal policies will always have to involve higher-level information, subjective preferences, and/ or political choice.

Country A

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q

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g1

Groundwater aquifer

Figure 36.3  River basin allocation problem with surface and groundwater storage. Q is the surface water inflow to the system, q is the groundwater recharge, xi are the surface water allocations to the individual users, gi the groundwater allocations to the individual users, NBi are the net benefits obtained by the users, and R is an instream flow requirement.

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R Q

X3

Country B

q

X2

X3 NB2

NB3

g2

g3

Groundwater aquifer

Figure 36.4  Transboundary river basin allocation problem. Q is the surface water inflow to the system, q is the groundwater recharge, xi are the surface water allocations to the individual users, gi are the groundwater allocations to the individual users, NBi are the net benefits obtained by the users, and R is an instream flow requirement.

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Applications and implementation of hydroeconomic analysis for management and decision support      36-5 

Country A

Country B R

Q

X1

Water treatment q

r1 NB1 g1

X2

r2

X3

r3

NB2

NB3

g2

g3

Groundwater aquifer

Figure 36.5  Transboundary river basin allocation problem with water quality decision variables. Q is the surface water inflow to the system, q is the groundwater recharge, xi are the surface water allocations to the individual users, gi the groundwater allocations to the individual users, ri the polluted return flows to the river, NBi are the net benefits obtained by the users, and R is an instream flow requirement. Pollution can be removed before or after use at each of the use locations.

36.4.5  Water Quality

Another challenge for the hydroeconomic modeling and analysis community is joint management of water quantity and quality. Different water users and ecosystems have different water quality requirements. Water-quality management can operate with both pretreatment (water is purified prior to delivery to the users) and posttreatment (water is purified prior to discharge into the environment) as illustrated in Fig. 36.5. The optimal policy will depend on complex interactions of different water quality processes (advective and dispersive transport, reactions, sedimentation, etc.), water flow in the river (dilution), available treatment technologies and water-quality requirements for human use and ecosystems. Hydroeconomic models have commonly targeted pure water quality problems (e.g., Cools et al., 2011; Hasler et al., 2014), while salinization problems (e.g., Cai et al., 2003a) have dominated the joint waterquantity and water-quality-management studies. There are only a few hydroeconomic modeling and optimization studies on other basin-scale water-quality problems [e.g., biological oxygen demand, nutrients and heavy metals (e.g., de Moraes et al., 2010; Davidson et al., 2015b)]. One class of approaches focuses on stream water quality (e.g., Cardwell and Ellis, 1993; Karamouz et al., 2010). Peña-Haro et al. (2009) presented a hydroeconomic modeling framework for optimal management of groundwater nitrate pollution from agriculture. Ejaz and Peralta (1995) used a response matrix optimization-simulation approach to maximize waste loads from a sewage treatment plant while complying with water-quality constraints (dissolved oxygen, TDS, chlorophyll-a, and nutrients). Other studies include also upstream reservoir releases in the decision problem (e.g., de Azevedo et al., 2000; Hayes et al., 1998). Inclusion of a reservoir in the water-quality problem (illustrated in Fig. 36.5) couples the water-quality decisions in time, that is, pollutants remain or slowly degrade in the reservoir. Castelletti et al. (2014) used a reinforcement learning algorithm coupled with a hydrodynamic-ecological model to find Pareto-optimal operation policies for a reservoir equipped with a multipurpose selective withdrawal system (SWS) under both water quantity and quality constraints. The water-quality constraints were formulated as multiple operating objectives to avoid algal bloom, sedimentation, and excessive water temperature. Ahmadi et al. (2012) used a fuzzy multiobjective GA-based approach to maximize upstream agricultural production and mitigate unemployment impacts of land use changes while complying with selected water quality and quantity constraints. Davidsen et al. (2015b) used the water value method to find optional water allocation and water-treatment at basin scale water downstream water quality restraints. A hybrid GA-LP implementation (Cai et al., 2001) was used to handle nonlinearity and nonconvexity in the objective function. A GA-based approach was also used in the context of a simplified SDP framework to resolve water quantity and quality conflicts summarized in a Nash bargaining setup (Karamouz et al., 2008; Kerachian and Karamouz, 2007).

36.5  APPLICATIONS AND IMPLEMENTATION OF HYDROECONOMIC ANALYSIS FOR MANAGEMENT AND DECISION SUPPORT

Implementation of hydroeconomic models faces many difficulties in terms of model formulation, temporal and spatial scale issues, model calibration, solution, and result interpretation as well as extensive data requirements in multiple areas (Cai, 2008). In general, there are two approaches to combining hydrologic and economic components—“compartment modeling” and

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“holistic modeling” (Braat and van Lierop, 1987). The compartment approach treats hydrologic and economic components as separate submodels, which interact by data exchange. In holistic modeling, the submodels are combined into a single consistent model, which is typically solved in its entirety, and information between components is transferred endogenously (Cai et al., 2003a; Draper et al., 2003). Holistic models depict the coupled human-natural interrelationships and mimic the impact of driving forces and feedbacks from the environment (Cai et al., 2006). Data requirements for hydroeconomic analysis refer to the type of data and the level of aggregation over spatial and temporal scales in order to develop either a holistic model or a compartment model. The divergent numerical scales of these items and uncertainties will raise difficulties in judging the reliability of model outputs. Especially in calculus-based nonlinear programming (NLP) models, water resources and economic data need to use appropriate units so that their numerical magnitudes appear similar (Cai et al., 2006). The correlations between some economic variables and hydrologic ones can make uncertainty analysis with a hydroeconomic model complex. Both lumped and distributed models are applied to water resources and economic modeling. Hydroeconomic modeling requires an appropriate matching of the spatial aggregation with both components. The spatial aggregation of water resources modeling needs to facilitate economic analysis, while, at the same time, it has to be effective enough in generating impacts to hydrologic system operation and water allocation. The ideal match-up would provide effective information transfer between the two components, while economizing on the complexity of the modeling framework and reducing aggregation error. One typical challenge is that the boundaries of the economic systems of a specific resources problem may not a priori be the same as those of the hydrologic systems. Hydrologic simulation usually adopts a top-down structure with the spatial scale starting from the river basin and going down to the crop land. Water balance is examined crossing multiple scales, for example, at the basin level deciding water allocation among various demand sites, at the demand site level determining water allocation to crop lands; and finally at the crop land level determining irrigation schedule and crop acreage. Meanwhile economic modeling usually adopts a “bottom-up” structure, for example, with the spatial scale starting from crop lands and going up to the whole basin; and profit from irrigation water use is calculated at the crop land scale, and is aggregated to the demand site and the basin to optimize basin-level water use profit. The top-down and bottom-up structures should be coupled to allow interactions between the two processes crossing all spatial levels (Cai, 2008). Economic analysis generally uses larger time intervals (seasonal or annual) and a longer time horizon (e.g., for long-term forecasts) than hydrologic models. This is because some production activities (for instance, crop production) are associated with certain time periods. Moreover, economic damage caused by some environmental consequences of production activities, for example, crop damage caused by soil salinity accumulation (Cai et al., 2003a), need a longterm projection. Other economic activities, such as water leasing usually involve relatively short time intervals, such as months. For hydrologic simulation, the time interval should be small enough to reflect real-world processes and capture the transition change of physical systems, which will affect economic costs and benefits. The time horizon should be long to reflect the regional hydroclimatic cycle and economic and environmental consequences resulting from flow regulation and water and land uses. Special model formulation is often needed to match the temporal scales particularly in holistic hydroeconomic models (see examples in Cai, 2008; Foster et al., 2014). Most applications of hydroeconomic modeling and analysis fall into three broad categories: planning problems, operational problems, and assessments of water policies. 36.5.1  Planning Problems

Hydroeconomic models can be used to evaluate the impact of future changes in supply and demand as can occur due to infrastructure investment or implementation of demand management (water conservation) schemes. Hydroeconomic models can inform cost-benefit analyses (Griffin, 1998) valuing the sectoral economic impacts of new assets such as reservoirs, transfers, or other hydraulic infrastructure. Whereas traditionally cost-benefit analysis is used to evaluate single assets, hydroeconomic modeling is a systems approach (also referred to as “integrated assessment”) seeking to evaluate how changes propagate through spatially distributed engineered-natural water systems. With/without studies can quantify the benefits and/or opportunity costs to different sectors of one or more interventions. For example, Pulido-Velázquez et al. (2006) and Harou and Lund (2008) quantify the opportunity costs of infrastructure bottlenecks using shadow values (appropriate for water systems with water markets).

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36-6    Hydroeconomic Analysis

Capacity-expansion problems are a special class of planning problems consisting of phasing investments over time. In capacity-expansion problems, we need to select, size, and schedule options to meet the predicted demands at maximum benefit/minimum cost over a multiyear period (O’Laoghaire and Himmelblau, 1974). For example, a least-cost deterministic capacity expansion optimization model was formulated by Padula et al. (2013) for sizing and scheduling of water supply and demand options in a large regional water supply system in South East England. 36.5.2  Operational Problems

Operation of an existing water resources system requires managers to take day-to-day decisions. Decisions today typically have an impact on what is possible in the future and operational management must therefore balance present and future performance. The most important operational problem in water resources management is reservoir scheduling. Many studies address optimal management of single or multipurpose reservoirs and reservoir systems (see review by Labadie, 2004). Typical objectives in reservoir management include water demand fulfillment, hydropower production, flood protection, and recreational value (Loucks and van Beek, 2005). A hydroeconomic modeling and optimization approach reveals the shadow price of stored water in the reservoir as a function of time, present inflow, and reservoir level (e.g., Davidsen et al., 2015a; Stage and Larsson, 1961). Shadow prices can subsequently be used as decision rules in operational management. In the power systems literature, many studies have addressed hydropower scheduling applications (e.g., Wolfgang et al., 2009). From the power systems perspective, hydropower is a unique power source because of its low variable cost, its dispatch flexibility, and low ramp on/ramp off costs. These characteristics of hydropower become increasingly important with the increased penetration of highly intermittent renewable power sources in most power systems. Groundwater management also involves operational real-time decisions, for instance, to minimize groundwater pumping cost, to minimize energy footprint of delivered groundwater, and to minimize the risk of pollution (e.g., Ahlfeld and Laverty, 2011; Bauer-Gottwein et al., 2016; Fowler et al., 2008; Hansen et al., 2012; Mayer et al., 2002). 36.5.3  Assessment of Water Policies

New water policies typically change the way in which water is allocated or controlled. Examples include water rights or water licensing changes, water markets, water pricing, and water allocation. Hydroeconomic models help assess new water policies by quantifying how policies lead to economic outcomes at the system-wide level. An important problem in hydroeconomic analysis is to quantitatively specify policy instruments that provide incentives to steer water- allocation patterns into a more desirable direction. Changing water pricing schemes is one approach to reallocating water resources (Pulido-Velazquez et al., 2013a; Riegels et al., 2013). Hydroeconomic modeling and analysis can inform innovative pricing policies by providing information on the opportunity cost of water at any point in space and time (Pulido-Velazquez et al., 2013a). Some countries have introduced markets for water or mechanisms for water right exchange through markets. If one assumes the economic value of water is the exclusive driver to water exchanges, hydroeconomic models can be used to estimate the benefits and water allocations that would occur if a water market functioned without transaction costs. Examples of water markets simulated with hydroeconomic models include Chile (Bauer, 1997; Cai et al., 2006), California (Draper et al., 2003; Jenkins et al., 2004), Australia, and the United Kingdom (Erfani et al., 2014). Although the market mechanism ensures efficient allocation of water under ideal conditions, real-world water markets are often dysfunctional and inactive because of high transaction costs and market failures due to the public good character of water (Rosegrant and Binswanger, 1994). Real-world water markets also tend to be limited because quantified and enforced water rights are often absent or transferability of such water rights is illegal. Recent advances in hydroeconomic optimization of flow path models (Cheng et al., 2009; Erfani et al., 2013) enable tracking of individual transactions of water in the optimal allocation. This approach allows for a detailed and realistic representation of transaction costs in models of water markets (Erfani et al., 2014). 36.6  DISCUSSION OF CHALLENGES, LIMITATIONS, AND FUTURE DIRECTIONS

Most hydroeconomic analysis and modeling work present to date has been undertaken with the ambition to inform decision-making. In contrast to other types of hydrologic and water resources analysis, hydroeconomic modeling offers the opportunity to simulate and test the impact of economic policies on

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water resources management. Several studies have had a considerable impact on the water-management debate through the simulation of different assumptions about the economic use of water (e.g., Cai et al., 2003b; Draper et al., 2003). However, few, if any, hydroeconomic models formulate, test, and make decisions about economic policy. Reasons include: the perception that water is essential for many of its uses and therefore not appropriate for economic analysis; lack of conceptual and/or technical clarity about how to simulate economic policies in a hydroeconomic modeling framework; and, as with other types of hydrologic and water resources planning models, the perception that model results may be too uncertain to formulate serious policy conclusions. One source of uncertainty is coefficients and parameters in the hydrologic as well as the economic compartment of hydroeconomic models that are unobserved and cannot be verified empirically. Prompted by challenges associated with model uncertainty, a literature has emerged describing a variety of systematic approaches to support decision making in planning conditions characterized by scientific uncertainty. Two approaches that have found applications in water resources planning are the Info-Gap approach (introduced by Ben-Haim, 2004; recent water-management applications include Hine and Hall, 2010, and Riegels et al., 2011) and Robust Decision Making (introduced by Lempert et al., 2006; applications include Groves and Lempert, 2007, and Lempert and Groves, 2010). The general goal of the approaches is to use ensemble computer simulation to characterize a range of plausible uncertain futures and use this information to design water resource plans that perform reasonably well across this range of future conditions. Matrosov et al. (2013) provide a recent comparison of the Info-Gap and Robust Decision Making approaches. Although hydroeconomic optimization models have the capacity to simulate economic policies directly, the level of aggregation and simplification used in some optimization schemes also limits the uptake of these models for real-world policy making. In many hydroeconomic models, network topology is simplified, storage facilities are aggregated, process descriptions are lumped, and the system representation in the model leaves out aspects of the physical system. If model results contain aggregated variables, such as aggregated fluxes and storages, decision makers and managers may have difficulty understanding simulated quantities. In a deterministic optimization approach, where the water balance is solved through a solution of a single optimization problem that assumes perfect foresight of future water availability, the network representation can be more complex; however, model credibility suffers from unrealistic operations resulting from the perfect foresight assumption. Stochastic optimization offers the possibility to simulate more realistic operations, but typically requires a highly simplified system representation because of computational challenges. Other optimization techniques (e.g., genetic algorithms, SDDP) together with the increasing availability of computational resources may help combine a detailed system representation with realistic operations in optimization models. Combined simulation-optimization approaches, where a simplified optimization model is used to formulate policies and operating rules that are then tested in a detailed simulation model (e.g., Hurford et al., 2014; Pulido-Velazquez et al., 2013b) are another approach to reconciling optimization with the need for realistic detail, and offer the advantage of using simulation models that are trusted by decision makers (for a related argument for coupled optimization-simulation, see Labadie, 2004). Unlike in other areas of modeling, it is not straightforward to develop baseline simulations representing observed conditions in hydroeconomic modeling. This is not always necessary when models are used, for example, to identify system constraints limiting the efficient allocation of water (e.g., Draper et al., 2003; Pulido-Velázquez et al., 2006). Calibrating hydroeconomic models is challenging because of the large number of parameters involved from different components and also because of interdependence among parameters to be calibrated. In addition, time series of economic data may not be available to the same extent as hydrological variables, such as streamflow and rainfall. Changes in hydrologic and economic parameters affect not only hydrologic model outputs, but also the economic outputs, and vice versa. The PMP methodology described earlier appears to have some promise as a method for calibrating hydroeconomic models. The method was used, for example, by Medellín-Azuara et al. (2012) to simulate farmer’s adoption of irrigation technology. Cai and Wang (2006), adapted the PMP method by using a numerical approach based on a hybrid genetic algorithm (GA; Goldberg, 1989) and nonlinear programming, and developed a calibrated baseline scenario of a hydroeconomic model for the Maipo River Basin in Chile. Another problem for the application of hydroeconomic models in practical decision-making is that, given their complexity and specificity, the models often have to be adapted for a certain system. Usually there is a lack of generic decision-support system (DSS) tools that could facilitate the process of implementation and application of the model. DSSs often involve

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References    36-7 

capabilities of computer assisted graphical design, geographically referenced databases, and interactive and user-friendly graphical interfaces and tools for input management, results display, and analysis. Some examples of DSS which incorporate hydroeconomic analysis capabilities are CALVIN (Draper et al., 2003; Jenkins et al., 2004) or AQUATOOL (Andreu et al., 1996; PulidoVelazquez et al., 2007b, 2013c). On the other hand, model platforms allow model developers to link a generic user interface and data-management system to external model codes. One example is HydroPlatform, an open-source model platform that facilitates data introduction and management, allowing data transfer to external water-management network flow models (Harou et al., 2010). SIMGAMS and OPTIGAMS are generic tools for hydroeconomic modelling that link HydroPlatform to a generic hydroeconomic code for simulation and optimization in GAMS, respectively (Lopez-Nicolas and Pulido-Velazquez, 2014; Pulido-Velazquez et al., 2013c), while SDP_GAMS is a generic tool oriented for stochastic programming models (Macian-Sorribes and Pulido-Velazquez, 2014). When used for operational support or to support ongoing planning processes (e.g., adaptive management), a static representation of economic behavior may be inappropriate. Factors influencing economic behavior are usually changing, for example, because of changes in the physical environment, new legislation, or changes in risk perception. Like other fields of science that deal with complex and dynamic systems (e.g., climate science), operational hydroeconomic modeling can benefit from methods to update system state and model parameters, including economic parameters. Data-assimilation methods that combine model predictions with observations have had a significant impact in other fields of modeling, particularly in data-rich environments. Maneta and Howitt (2014) presented an approach to update agricultural production functions in real-time using data-assimilation techniques, an approach that could be used, for example, with remotely sensed agricultural land use. Hydroeconomic modeling methodologies have been used in the simulation of coupled hydrohuman systems to understand the complexity of the systems, using agent-based modeling approaches. In some applications, agent-based modeling is used to simulate rational economic behavior, but with a decentralized approach that allows system behavior to emerge from the interactions of individual, rational economic agents (e.g., Zhao et al., 2013). Agent-based modeling has also been proposed as a way to relax assumptions about the behavior of economic agents and replace them to some extent by other behavioral theories such as bounded rationality (Armstrong and Huck, 2010). Instead of postulating profit or utility maximizing behavior, economic agents are assumed to behave according to a set of rules which can, in some cases, evolve as agents learn about the environment and the behavior of other agents (e.g., Ng et al., 2011; Pahl-Wostl, 2002). In some cases, for example, when modeling markets, rules can be added to economic drivers to improve upon the realism of agent-behaviors in models (Erfani et al., 2014). Developing integrated system-scale approaches to management of food, energy, and water resources under climate change, the so-called “waterenergy-food nexus” is one of today’s key challenges in environmental engineering. Joint management of water, energy, and agricultural resources has been receiving increasing attention from researchers, operators, and decision makers because recent technological and institutional developments have tightened the coupling links between these resource systems (e.g., Olsson, 2012): (1) with increasing penetration of highly intermittent renewable energy sources (solar and wind), the importance of hydropower as balancing and storage facility in the power market has been increasing. (2) It is becoming more and more difficult to meet cooling water requirements from thermal power plants, given tighter environmental regulation and climatic extremes. (3) The advent of biofuels has resulted in direct competition for scarce water between the food and energy sectors. (4) Irrigation agriculture is responsible for an increasing share of food production to meet growing global food demand. (5) Freshwater ecosystems are a significant water user and are among the most vulnerable and endangered ecosystems on the planet. Hydroeconomic modeling concepts provide a consistent and quantitative framework to inform the water-energy-food nexus debate and to provide decision support for joint management of multiple resource systems (PereiraCardenal et al., 2015, 2014) and the trade-offs involved (Hurford and Harou, 2014). 36.7 CONCLUSION

The main advantage of hydroeconomic modeling is its ability to track the full temporal and spatial distribution of economic benefits that result from water use and costs (losses) that result from water scarcity. No other approach has similar capabilities. By representing spatially distributed economic water demands throughout the water resources network, such models quantify the

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marginal value of water throughout the system, and therefore, would allow us to assess the economic gains or losses coming from different water policies, new infrastructures, new operating rules, or water allocation decisions to cope with water scarcity. Despite the significant potential of hydroeconomic analysis, its practical application for real decision-making processes has so far been quite limited. To this end, uncertainties inherent in hydroeconomic models must be analyzed, quantified, and reported, and the models should be fused with observations to obtain reliable and sharp predictions. Decentralized hydroeconomic modeling approaches may enable analysts to relax some of the restrictive assumptions about human behavior inherent in microeconomic theory, allow more modeling flexibility, and yield more realistic results. Multicriteria approaches will increasingly enable hydroeconomic analysis to include important benefits that are usually hard to monetize, such as supply reliability, and resilience and ecological value. Finally, water resources systems will increasingly have to be managed jointly with other resources systems (food, power, etc.) to overcome the challenges of population growth, economic development, and global environmental change. REFERENCES

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36-8    Hydroeconomic Analysis

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PART

4

HYDROLOGIC PROCESSES AND MODELING

37_Singh_ch37_p37.1-37.10.indd 1

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Chapter

37

Weather and Climate BY

MICHAEL ANDERSON

ABSTRACT

Weather and climate play a foundational role in hydrologic engineering. Weather reflects the short-term conditions of the atmosphere while climate reflects the long-term statistical distribution of atmospheric conditions driven by variability in the atmospheric composition, ocean, land surface and solar input conditions. Knowledge of weather and climate and the physical processes that drive them can improve the development and use of hydrologic engineering tools and practice. Such knowledge will be critical in the coming decades as impacts from climate change materialize. This chapter starts with a description of the intersection of weather and climate with hydrologic engineering. Elements of weather and monitoring of weather that impact hydrologic engineering is presented next. Climate concepts follow including a discussion of the role of variability. The chapter closes with some thoughts on the impacts of climate change on hydrologic engineering. 37.1 INTRODUCTION

Weather and climate play a foundational role in hydrologic engineering. Weather is the short-term conditions of the atmosphere (i.e., daily) whereas climate is the long-term average conditions of the atmosphere (i.e., decadal). Weather systems leading to flood and drought, the seasonal cycles of weather over the course of a year, and annual to decadal variability all impact an area’s hydrology. Knowledge of weather and climate and the physical processes that drive them can improve the development and use of hydrologic engineering tools and practice. This knowledge will be critical in the coming decades as impacts from climate change appear. Figure 37.1 illustrates the hydrologic cycle that is foundational to hydrologic engineering. Precipitation falls on the landscape either as liquid (rain) or solid (snow, sleet, hail, etc.). Water, as rain or melting snow and ice, can infiltrate into the ground or runoff via overland and channel flow. Channel flow moves water from the land surface to terminal reservoirs, lakes, seas, and oceans. Evaporation and transpiration from plants return water to the atmosphere where it is moved by the winds and condenses into clouds leading back to precipitation. Texts like Bedient and Huber (1988) and Brutsaert (2005) cover introductory concepts to the hydrologic cycle as it relates to hydrologic engineering. Weather and climate are critical elements to understanding the role of the hydrologic cycle in hydrologic engineering. In general, the relation between the earth’s atmosphere and solar radiation drives the current climate processes that set the stage for day-to-day weather variability that drives the hydrologic cycle. Climate processes establish the spatial and temporal variability of weather and lead to changes in the statistical properties of weather variables (i.e., temperature, precipitation, wind, atmospheric pressure, etc.) that impact hydrologic engineering. Climate change impacts can modulate the relationships between different flow paths in the hydrologic cycle changing the timing and magnitude of runoff, recharge, evapotranspiration, and snow accumulation and melt. In this chapter the introductory concepts of weather, climate, and the hydrologic cycle that are central to civil engineering methods are presented. The chapter starts with a description of the intersection of hydrologic engineering with weather and climate. Next, a general overview of weather

concepts will be presented. An essential part of weather includes the monitoring methods that provide the data inputs for hydrologic engineering applications, which are covered highlighting important considerations when using such data including gridded products. Climate concepts are covered next, including variability across space and time scales. Some of the fundamental hydrologic engineering applications are based on assumptions of a stationary climate. However, these assumptions are being challenged under an increasing awareness of a changing climate. With that in mind, climate change impacts on hydrology and hydrologic engineering will be introduced to close out the chapter. 37.2  HYDROLOGIC ENGINEERING AND INTERSECTION WITH WEATHER AND CLIMATE

Hydrologic engineering methods for planning and design of storm-water runoff management, reservoir sizing, forecasting of stream flow for water supply, or flood management all rely on observations and statistics of weather variables like precipitation and temperature. Engineering tools, such as the rational method, univariate and multivariate regression, and rainfallrunoff or watershed models all make use of weather data and climate concepts. Underlying the methods used to create these tools are assumptions about the processes that drive weather and climate for a given location over time. A better understanding of weather and climate principles will improve an engineer’s judgment on when to use which tool for the job at hand. In this section, selected engineering applications are used to demonstrate how weather and climate underlie those applications. Foundational documents for these methods are referenced. The section starts with planning applications, moves to operational applications such as snowmelt forecasting, and ends with a look at watershed modeling. 37.2.1 Planning

Engineering planning utilizes design concepts to estimate the size of a structure like a spillway or storm-water detention basin. These concepts are used to establish thresholds for flood protection that go into project development and planning documents. Concepts of how much rain or runoff, how long the event lasts, and how frequently one can expect events of that magnitude have been developed based on statistics of observed quantities. Modifications are made when new observations exceed past events. One of the most widely used engineering applications using these concepts is the Rational Method. The method developed in 1889 by Kuichling relates the magnitude of runoff (peak discharge, Q) to the intensity of rainfall i over the area A that the rain falls upon with runoff coefficient c, which is a function of soil type, drainage basin slope, and ground cover, as shown in Eq. (37.1).

Q = ciA (37.1)

In this method, weather and climate are tied to the estimate of the intensity, i. Federal documents such as the NOAA Atlas 2 (1973) and updates like NOAA Atlas 14 (2006) provide estimates of intensities over different durations for different return periods for different locations. For a given location, tables of intensities for different durations are compiled. Different volume numbers are used to organize the data for different regions or states. As an example, California data can be found in NOAA Atlas 14 Volume 6 (2011). 37-3

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37-4    Weather and Climate

Evapotranspiration Reservoir Water treatment plant

Runoff Stream

Diversion canal

River

Municipal/ industrial use

Treatment plant

Wildlife refuge Irrigated agriculture

Unconfined aquifer Confined aquifer

Agricultural supply well

Injection well

Recharge basin

Groundwater table

Municipal/ industrial supply well

Shallow monitoring well Deep monitoring well

Aquitard

Figure 37.1  Schematic of the hydrologic cycle [Source: California Department of Water Resources].

To develop such tables of intensity-duration-frequency data, historical rainfall totals over different durations are analyzed and fit to probability distributions which relate the magnitude to the return period. Statistical methods such as L-Moments (Hosking and Wallis, 2005) or empirical fitting functions (Shorack and Wellner, 1986) are used for the distribution fitting. The observed rainfall peaks are evaluated to make sure that they meet the assumptions of independence and that they come from a homogeneous population to satisfy the requirements of the statistical theory used to develop the method. Another important engineering application that uses statistical methods to fit observational data is the Bulletin 17B (USGS, 1982) method for sizing flood peaks and volumes. The Bulletin 17B process was started to provide a nationwide standard (log-Pearson Type III) to sizing flood peaks and volumes for flood management projects. In this application, the flood peak or volume for a given return period Q is described in Eq. (37.2):

logQ = logX + KSlogX (37.2)

where X is the mean of the annual peak flow time series, S is the standard deviation of the time series of the log of the annual peak flows, and K is a parameter based on the skew and return period. Tables of the coefficient K can be found in the Bulletin 17B documentation. In this application, the mean, standard deviation, and skew–the first three moments of the statistical distribution of observed values–are used to estimate the size flood for a given return period. Federal programs like the National Flood Insurance Program (https://www.fema.gov/media-library/ assets/documents/101759) and the Army Corps of Engineers Levee Certification Program (http://www.usace.army.mil/Missions/CivilWorks/ LeveeSafetyProgram.aspx) use the threshold of a 1% exceedance or 100-year return period event at the time of publication. The foundational assumptions for this method are that the largest annual flood peaks for a given location are independent events that come from the same statistical distribution. This assumption is necessary to satisfy the sample statistics theory used to develop the method. It is further assumed that the longer the data record, the closer the observed statistical properties approach the real distribution. Such concepts presume that the same weather types generate the flood peaks each year and that the statistical properties of the

37_Singh_ch37_p37.1-37.10.indd 4

flood events do not change in time. Efforts are made to sort through historical records to sort out different flood peaks associated with rainfall or snowmelt. If spatial independence can be established a space for time exchange is used to extend short records. The assumption of stationarity of hydrologic variables has recently been questioned as evidence of anthropogenic climate change mounts. Nonstationarity has also arisen from changes to the landscape such as forest clearing or from channel changes upstream for different water management purposes. Workshops on addressing nonstationarity issues have been held (Colorado Water Institute, 2010) and it remains an active area of research. 37.2.2 Forecasting

Hydrologic forecasting can be either event driven estimates of how much runoff is generated from observed or predicted precipitation or how much runoff is expected over the spring from a melting snow pack. Here we will focus on the seasonal snow pack melt forecasts used to estimate water volumes used for water supply in the Western United States. For watersheds in 11 western states, snowmelt forecasts are provided by the Natural Resources Conservation Service at the National Water and Climate Center (http://www. wcc.nrcs.usda.gov/). In California, the forecasts are provided by the Department of Water Resources as part of the California Cooperative Snow Surveys (http://www.water.ca.gov/floodmgmt/hafoo/hb/sss/). In the mountainous areas of the western United States, snow typically accumulates from December through March. From April through July, the accumulated snow melts as the sun angle moves higher in the sky and temperatures warm (Lundquist and Flint, 2006). April 1 is the historical average date of maximum accumulation. Some years have heavy snow packs that build into May and have melt that extends past July. On the other end of the spectrum dry years tend to have smaller snow packs that peak earlier and melt out before the end of July. An average snowpack trace for three regions in California is shown in Fig. 37.2. Also shown in Fig. 37.2 are the historical maximum and minimum traces of each region illustrating the wide range of possible outcomes. Statistical methods have been used that relate runoff volumes with select observations of the amount of water in the snow in the watershed and expected precipitation over the forecast period. As long as the runoff in the forecast period stays within the historical bounds of observations, these

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Weather     37-5 

% April 1 average

250

California snow water content–percent of April 1 average North

200 150 100 50 0

% April 1 average

250

Central

200 150 100 50 0

% April 1 average

250

Soutn

200 150 100 50 0

Dec

Jan Average

Feb

Mar

Apr

1976–1977 (min)

May

Jun

Jul

1982–1983 (max)

Figure 37.2  Regional plots of snow accumulation and melt for California [Source: California Data Exchange Center].

methods tend to work fairly well. Another alternative is to use watershed models to simulate the physical processes of the watershed (see, e.g., Kavvas et al., 2013). Watershed modeling in hydrologic engineering is described briefly in the next section. 37.2.3  Watershed Modeling

Watershed models are collections of mathematical representations of hydrologic processes to simulate the movement of water through a watershed. The level of complexity of the models varies from simple, lumped parameter models to detailed, spatially distributed models using some variation of mathematical representations of the different physical processes. A review of the history of watershed models can be found in Singh and Frevert (2002a 2002b). The level of complexity of the model often corresponds with the level of detail required of the input data. Parameterized relationships to represent different watershed processes may not require much input data, but need to be calibrated and verified against observed runoff data. These calibrations “fit” the model to the rainfall and runoff data. The underlying relationships being simulated are presumed to be constant through time and under different climate conditions, such as droughts or floods. Lack of performance by these models is usually tied to the violation of the aforementioned assumption. More detailed distributed models require detailed data at many points in the watershed. Variables, such as maximum and minimum temperature, precipitation, wind, and solar or net radiation, are used in the computation of the equations to follow water through the watershed. For many watersheds, such detailed data are not available. This is where derived data products, such as gridded data fields, have been very helpful. Gridded datasets are regular-spaced values of a variable like temperature or precipitation that are derived from point observations. Interpolation schemes are used to translate point observations to all grid spaces in the domain. An example of this process is the Precipitation-Elevation Regressions on Indipendent Stopes Model (PRISM) method developed at Oregon State University (Daley et al., 1994). Gridded datasets are models themselves and are discussed further in the next section which looks at weather. 37.3 WEATHER

Weather drives the atmospheric component of the hydrologic cycle. It is driven by solar energy from the sun to the atmosphere, land surface, and

37_Singh_ch37_p37.1-37.10.indd 5

oceans and the subsequent thermodynamic balancing of the atmosphere. There are several components of the atmosphere that help set the stage for weather events and are described in the following paragraphs. For more information on weather and the atmosphere, see Carlson (1991), Gill (1982), Holton (1992), and Pedlosky (1987). The atmosphere is made up of a series of layers defined by physical characteristics. The weather that impacts the hydrologic cycle occurs in the lowest layer of the atmosphere known as the troposphere. It extends from the surface to about 12 km. In this layer, temperature decreases with altitude. The rate of temperature decrease with altitude is known as the lapse rate. At the top of the troposphere is the tropopause which is a layer of atmosphere with little to no change in temperature with altitude. It separates the troposphere from the stratosphere and tends to cap the vertical extent of moisture-bearing weather systems. See Holton (1992) for more information on atmospheric structure. In general, more heat through solar radiation is input to the atmosphere in the tropics, from the equator to 23° N/S latitude. The warmer temperatures and vast areas of ocean in the tropics also create high concentrations of water vapor. In the atmosphere, heat (in latent and sensible forms) and water vapor are moved from the tropics toward the polar regions, latitude greater than 66°, through the middle latitudes, the region from 23° latitude to 66° latitude by wind patterns called the atmospheric global circulation. These regions have distinct characteristics to their weather patterns and the resulting temperature and precipitation that drive hydrology. Differences in solar input are also affected by the rotation of the earth, which results in day and night and tilt of the earth which results in the seasonal cycle. One important component of the atmospheric global circulation for hydrology is the jet stream. The jet stream is a narrow band of fast moving air near the top of the troposphere that moves eastward across the middle latitudes. There are areas of high pressure and low pressure that form during the evolution of the atmosphere as the earth spins and tilts and heat is transferred from equator to poles. These areas of high and low pressure form and dissipate over the course of days to weeks. The space and time scale of these pressure regions is called the synoptic scale. Synoptic scale evolution of the atmosphere helps mix the air between equator and poles and in the vertical. An observer on the ground would note different weather conditions as the synoptic systems evolve and pass overhead. More information on atmospheric dynamics can be  found in Holton (1992) and information on global circulation of the atmosphere can be found in Grotjahn (1993), and Cushman-Roisin (1994).

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37-6    Weather and Climate

The areas of high and low pressure have different characteristics that set the stage for the weather that is observed on the ground. Areas of high pressure are associated with sinking air. Sinking air is warming and is associated with warmer temperatures, less relatively humid air, and no precipitation. The areas of sinking motion tend to be large with little pressure differences resulting in little horizontal wind. Air rotates clockwise around high-pressure centers. A weather observer underneath a high-pressure system would note sunny skies and warm temperatures. Areas of low pressure are where air is rising and cooling. Air rotates counterclockwise around low-pressure systems. Rising, air cools and thus can hold less water. As the air cools and has less capacity to hold the water vapor, it was carrying condensation (the transition in water phase from gas to liquid) occurs and clouds form. If the temperatures drop below freezing (0°C) then the condensed droplets can freeze and become ice particles if the right nuclei are present. The relationship between temperature and the amount of water vapor the atmosphere can hold is defined by the Clausius Clapyron equation (see Haltiner and Williams, 1980). The saturation vapor pressure (mbar) over a water surface as a function of temperature (°C) (Gill, 1982) can be approximated as: Log10 ew(t) = (0.7859+0.03477t)/(1+0.00412t) (37.3) In this equation, t is temperature (°C), and ew is water vapor (mbar). As can be seen from Eq. (37.3), the relationship between temperature and the amount of water vapor is highly nonlinear. This equation describes how warmer tropical air will hold much more water vapor than colder, polar air. The transition from warmer temperatures to colder temperatures from equator to pole and their mixing in the middle latitudes through the synoptic scale pressure systems are where most precipitation form and fall to the ground. These transition zones between the high-and low-pressure systems that evolve in the middle latitudes are anything but gradual and smooth. Areas of strong gradients in pressure and temperature can form which are called frontal zones. As the pressure gradient increases, the winds increase in speed along the pressure gradient due to the spinning of the earth. This effect, the Coriolis Effect, enables the gradients to persist and possibly intensify. The frontal zone also has a strong temperature gradient. Frontal zones are described as a warm front, cold front, or stationary front depending on the relative temperature of the advancing front. Each has a different vertical structure and relationship to the location of the low pressure. See Carlson (1991) for more details. The winds along the foot of frontal zones, especially cold fronts, can be quite intense and if supplied by a source of water vapor typically from the tropics can form narrow features called atmospheric rivers (Zhu and Newell, 1998). These narrow bands of intense atmospheric water vapor are responsible for 90% of the equator to pole transport of moisture. Figure 37.3 shows an image of water vapor concentration and atmospheric rivers over oceans from satellite data. The bands are only hundreds of kilometers wide, but can extend thousands of kilometers from the tropics and across the middle latitudes. When atmospheric rivers are entrained into middle latitude low-pressure systems, heavy precipitation occurs ahead of the passing of the cold front as the intense moisture, which is located in the lowest 5 km of the atmosphere, interacts with the dynamics of the front and the land surface. Further information

50N 40N

(b) 7 Nov 2006 a.m. composite Atmospheric river

625 mm in 72 h

30N 20N 10N Eq Figure 37.3  Microwave imagery of water vapor over the Pacific Ocean with atmospheric river annotated.

on atmospheric rivers and their relation to extreme precipitation events can be found in Ralph et al. (2010) and Ralph and Dettinger (2012). As noted earlier, the land surface plays an important role in precipitation processes. Winds encountering mountains are forced up and over them. The forcing of the air by the topography causes increased convergence of water vapor along with cooling. The combined effects of rising and cooling during a frontal precipitation event result in enhancement of precipitation. This is called orographic precipitation and is why mountainous areas tend to receive higher amounts of rain and snow and is described in detail by Rhea (1978). Another important process that can happen along weather fronts is called convection. Warm, moist air is lighter than colder, dryer air. Because of this difference in weight, warmer moister air will want to rise and the colder, dryer air will want to sink. This buoyant motion is called convection and can result in strong vertical motions in the atmosphere. As the warm, moist air rises buoyantly it cools rapidly and water vapor condenses releasing heat that further accelerates the rising and further condensation. The rapid vertical motions can result in the development of thunderstorms which can drop intense precipitation and hail. The winds of synoptic systems help to move thunderstorms across the landscape. The scale of individual cells of convection is on the order of tens of kilometers. Sometimes multiple thunderstorms form along a frontal boundary and multiple thunderstorms are moved over an observer on the ground. This is called training and can result in extended periods of heavy precipitation and ultimately flooding. Carlson (1991) provides more information on convective processes. Convective precipitation is also associated with tropical systems known as hurricanes, typhoons, or cyclones. These tropical weather systems form when tropical convection becomes self-sustaining at large scales and organized to start rotating around a low pressure center. The low pressure center of a hurricane is called the eye of the hurricane. As a hurricane develops, the pressure at the center can drop, which increases the strength of the winds rotating around the eye. Warm ocean waters help support convection providing the hurricane with vast amounts of water vapor that condenses in the thunderstorms generating more energy for the storm. Landfalling hurricanes result in very strong winds and heavy precipitation that can cause significant damage. Figure 37.4 shows that hurricanes produce rainfall totals over 3 days over the

Figure 37.4  Three-day accumulation ranges of precipitation associated with atmospheric rivers and hurricanes over the Continental United States [Source: From Ralph and Dettinger, 2012].

37_Singh_ch37_p37.1-37.10.indd 6

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Climate     37-7 

Gulf and Atlantic states equal to the precipitation of landfalling atmospheric rivers on the Pacific Coast (Ralph and Dettinger, 2012). These are the most intense precipitation processes in the United States.

estimation of values. While work continues to generate improved values, the products are still widely accepted and used. These gridded products have also helped in the advancement of the description of a location’s climate.

37.4  OBSERVING WEATHER

37.5 CLIMATE

As noted earlier, an observer on the ground watches weather change as synoptic systems, fronts, thunderstorms, and hurricanes pass overhead. The observer records the impact of these events by measuring atmospheric variables, like pressure, temperature, amount and form of precipitation, and wind speed and direction. Other observations, like solar and terrestrial radiation, atmospheric water vapor content, snow water equivalent, and soil moisture help inform the hydrologic response. There are many different ways in which weather observations are gathered. The National Weather Service uses trained volunteers to record daily temperature, maximum and minimum temperature, snowfall, and snow depth through their Cooperative Observer network. Grass-roots weather observers, such as the Community Collaborative Rain, Hail and Snow (CoCoRaHS) group (http://www.cocorahs.org) or Weather Underground (http://www. wunderground.com/), use volunteer observers to record weather data that can be useful for hydrologic engineering projects. Care must be taken to review the data for quality control and completeness. In remote areas, weather instruments on towers are used. Data are relayed by satellite communications to provide near–real-time information for hydrologic forecasting. The data can also be used to drive hydrologic models for other engineering activities. While the data gathered in this manner is very important, it is a point measurement in a large and variable landscape. To address this issue, new instrumentation using radar and satellite technology have been invented. Radar technology has been successfully applied to determine spatial rain fields and the intensity and form of precipitation. The National Weather Service has deployed a network of radar systems across the United States for this purpose (see Anagnostou et al., 2004; Baeck and Smith, 1998; Austin, 1987). In the near future, aircraft measurements including drones will be used for weather observations (Painter et al., 2015). Vertically pointing radar has been used to identify the altitude at which falling precipitation changes from snow to rain to help determine freezing elevation. This metric has been identified as an important factor in determining the amount of the watershed producing direct runoff from a weather event which impacts the size and volume of the hydrograph. California has deployed a picket fence of these radar units down the length of the state to monitoring the freezing elevation (White et al., 2013). Data is also being gathered from satellites. The Geostationary Operational Environmental Satellite is perhaps the most common source of satellite data tracking clouds, cloud heights, and temperatures in the visible and infrared spectrums respectively. Other satellites, which carry the Special Sensor Microwave/Imager provide a more direct estimate total water vapor, but only work with water vapor at the lower boundary of the atmosphere. Other data are needed to track this information over land. It was the application of some of these satellite data led to the discovery of atmospheric river phenomena. The visualization of water vapor movement over the globe from satellites showed how the abundant moisture of the tropics is drawn into the middle latitudes in narrow filaments that while hundreds of kilometers wide can be thousands of kilometers in length. Such work has helped advance and inform flood forecasting efforts and improved our understanding of how the atmosphere works. It is important to note that the point scale measurements gathered from the ground are vital to the operation of the radar and satellite observing systems. Data from the point-scale measurements are used to calibrate and validate the data and image-processing routines used to transform satellite and radar energy readings into the desired hydrometeorological observations. Radar and satellite products can also be challenged by rapidly varying terrain, such as mountains. Mountains can block or distort the signals which impact the data-transformation routines and resultant data. Used together, suites of observations from stations, radar, satellites, and aircraft, can provide a wealth of valuable information for hydrologic engineering applications. While this information is valuable, its uneven coverage over the landscape can lead to challenges in hydrologic simulation. To that end, effort has been undertaken to create gridded interpolations of the observed data fields. One of the most prominent gridded datasets is the PRISM product. PRISM was developed at Oregon State University in 1994 (Daley et al, 1994). Currently, PRISM maps are available at 4 km and 800 m for annual, monthly, and daily values. Such data can be very helpful for distributed hydrologic models as noted earlier. It should be noted that gridded data products are not perfect. Lack of data in key locations, like mountain tops can lead to errors in the

Climate can be thought of as the expected state of the atmosphere at a given time. In summer, a location may be expected to be warm and dry or warm with the chance of thunderstorms. In winter, expectations may be for colder weather and rain or snow. The description of these expectations can be done using sampling statistics methods. Each weather observation is a sample that helps to inform the statistical moments (mean, variance, and skew) of the distributions of weather variables that describe a location’s climate. Historically temperature and precipitation have been the primary variables used to describe the climate of a location. The time scale such as daily, monthly, seasonal, annual, decadal, or longer can be used to characterize the observed statistical distributions that represent a location’s climate. The National Climatic Data Center of the National Oceanographic and Atmospheric Administration uses a 30-year window to construct the statistical measures of climate for a location that are referred to as the 30-year normal values. They are updated once per decade and account for new methodologies in computing such statistics accommodating issues, like missing or bad data, data from multiple sources with differing types of observation errors (see Arguez et al., 2012). These values are often the values used in Hydrologic Engineering planning and design. Using the same statistical methods with different 30-year time periods, one can evaluate how the climate distributions are evolving in time. The change in these distributions through time due to the evolution of the state of the atmosphere, land surface, and ocean systems is called climate variability. Humaninduced changes to the atmosphere and land surface that modulate the statistical distributions of temperature and precipitation are referred to as anthropogenic climate change (IPCC, 1995). Figure 37.5 shows a depiction of the year-to-year variability in precipitation for the continental United States. The values depicted are the coefficient of variation, the standard deviation divided by the mean. This value provides a sense of how far each annual precipitation totals typically deviates from their long-term mean values. Low values imply that any one year’s total is not that different from the average value. In this case, the mean value may be a reasonable representation of the annual precipitation. Larger values imply that each year’s accumulated precipitation can be quite different from the mean and that the mean is not necessarily a reasonable depiction of what transpires in any given year. In such cases, it is advantageous to know about the variance in the data and its possible sources, such as the Pacific Decadal Oscillation, Atlantic Multi-decadal Oscillation, or El Nino Southern Oscillation. These “oscillations” are identified recurring changes in the state of the ocean-atmosphere-land surface system. For example, the El Nino Southern Oscillation refers to a coupled ocean-atmosphere phenomena in the tropical Pacific (see Philander, 1989). During its warm phase or El Nino, the eastern tropical Pacific sea-surface temperatures are warmer than normal and sea level pressure tends to be high over Tahiti and low over Darwin, Australia.

37_Singh_ch37_p37.1-37.10.indd 7

Fraction 0.00 0.15 0.30 0.45 0.60 0.75 0.90 Figure 37.5  Interannual variability of precipitation depicted through the coefficient of variation (mean divided by standard deviation) of National Weather Service Cooperative Observer sites [Source: M. Dettinger 2011].

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37-8    Weather and Climate

The northern hemisphere jet stream and storm tracks tend to be displaced to the south during winter and in some instances split into two paths impacting winter precipitation patterns. Temperatures over the United States tend to be warmer than average. In its cold phase, or La Nina, the eastern tropical Pacific is cooler than normal and the pressure relationship between Darwin and Tahiti flips. The wintertime jet stream over the northern hemisphere develops a looping pattern with a high pressure system located somewhere over the Pacific Basin. The looping jet stream brings colder air into North America. If the high-pressure system is located over the eastern Pacific or the west coast of North America, drought conditions can set in as storms are diverted up and over the high-pressure system away from much of the west coast. Such expectations have been used to try to inform hydrologic engineering applications (e.g., Dracup and Kahya, 1994; Piechota and Dracup, 1996). Variability can also occur over longer periods. Figure 37.6 depicts an annual time series of precipitation with an 11-year running average of precipitation for the state of California. These smoothed time series damp out the large year-to-year variability and indicate a fluctuation in precipitation outcomes on the order of a decade. This same fluctuation also shows up in runoff records both in California and the Colorado River Basins. While recognizable in the smoothed observations, associations to specific processes in the climate system have yet to be made.

Such fluctuations can be difficult to analyze with periods of records that extend only for 100 years or so. To address this challenge, reconstruction of hydrologic time series has been made using paleo-proxies, such as tree-rings, lake sediments, and chemical isotope signatures (see Bradley, 1985 or Cronin, 2010). Examples of such reconstructions include the works of Woodhouse et al. (2006) for the Colorado River, and Meko et al. (2014) for the Sacramento, San Joaquin, and Klamath Rivers. Along with variability over different time scales in time series of precipitation and temperature data sets is the appearance of trends. These trends are long term, often monotonic changes in the distribution indicating a response to changes in the atmosphere-ocean-land surface system. In urbanized areas, the elevated night-time temperatures associated with the heat retention of the built environment is known as the urban heat island (Kim, 1992). Such changes have significant influence on the populations of those urbanized areas for health, energy use, etc. This has been documented in EPA (http:// epa.gov/heatislands/resources/pdf/BasicsCompendium.pdf). 37.6  CLIMATE CHANGE

Another type of change that has garnered increasing concern and attention relates to changes in Earth’s radiation balance associated with the burning of

California statewide precipitation Oct–Sep 40

Inches

30

20

10

0 1900

1910

1920

1930

Solid line denotes 11-year running mean

1940

1950

1960

1970

1980

1990

Year

Linear trend 1895–present

+ 2.45 ± 3.09 in.

(+ 10 ± 13%) per 100 year

Linear trend 1949–present

– 2.06 ± 8.75 in.

(– 8 ± 38%) per 100 year

Linear trend 1975–present

– 8.99 ± 21.46 in.

(– 39 ± 93%) per 100 year

Wettest year Driest year Oct–Sep

2014

2000

Western regional climate center

40.44 in. ( 176%) in 1983

Mean 22.90 in.

9.23 in. ( 40%) in 1924

Stdev 6.53 in.

12.05 in. ( 52%)

2010

Rank 3 of 119

Figure 37.6  Time series of precipitation with 11 year moving average from the California Climate Tracker (http://www.wrcc.dri.edu/monitor/cal-mon/index.html).

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Climate Change    37-9 

Climate change effects on water resources Total precipitation may increase or decrease Increased air temperature Less snowpack More precipitation as rain than snow due to higher temperatures Earlier runoff from snow melt

Increased evaporative demand on lakes and landscapes

Changes in timing and amount of river flows

Changes in water resource system operations

Sea level rise Figure 37.7  Schematic of expected hydrologic impacts from climate change.

fossil fuels that began with the industrial revolution. The burning of fossil fuels has increased the concentration of carbon dioxide and other greenhouse gases in the atmosphere which is effective in trapping earth’s heat in the atmosphere. As a result, temperatures have risen over the twentieth century and look to continue to do so through the twenty-first century. These changes in temperature will impact the way weather events develop and evolve in space and time. As a consequence of those changes, significant hydrologic changes are expected which will profoundly impact hydrologic engineering. A summary of expected hydrologic impacts due to climate change are illustrated in Fig. 37.7. Higher temperatures will lead to more precipitation falling as rain than snow which will change the timing and magnitude of watershed runoff as well as increase evaporative demand on the landscape. The relationships between infiltration and groundwater recharge, and surface and subsurface flow may change too as snow pack accumulation and melt patterns change, vegetation changes, and soil moisture amounts and timing change. Inflows into reservoirs will change which will impact water resources management in basins served by those reservoirs. Reservoir operations rules derived from past hydrologic relationships may not be as effective. At the downstream end, sea level rise will impact coastal infrastructure, increase seawater intrusion into coastal aquifers, and may cause backwater effects during floods. Much of the work documenting observed change and anticipating future change has been done in the past 25 years. The Intergovernmental Panel on Climate Change (IPCC) has issued a series of reports beginning in 1990 with updates approximately every 5 years (http://www.ipcc.ch/index.htm). These reports are the collaborative efforts of scientists from many nations. Using

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these efforts national and state-level efforts have been undertaken to attempt to document and plan for a changing climate and its associated impacts. Examples include the United States third national assessment competed in 2014 (USGCRP, 2014) and the California Department of Water Resources report in (2006). Efforts will continue as more impacts manifest themselves. Hydrologic Engineering will need to respond to the challenge by developing new approaches to designing infrastructure that will be resilient to climate change impacts and work across a range of possible outcomes. 37.6.1  Concluding Remarks

Weather and climate play important roles as drivers of hydrologic engineering design and applications. Climate sets the stage for weather events to unfold across the landscape and is driven by energy balances in the land-ocean-atmosphere system. Weather is the time evolution of the atmosphere across the earth. Precipitation, temperature, wind, solar radiation, and relative humidity are weather variables observed to provide input into hydrologic tools and applications. Knowledge of how the processes work that control the magnitude and variation of these variables enables the engineer to accurately judge the right method for the job at hand. Understanding how climate modulates the weather and evolves in response to changes in the ocean-atmosphere-land system can provide insight into the conditions hydrologic systems will face and how the hydrologic system responds. This can be beneficial for planning for and managing extreme events like floods and droughts as well as for identifying important thresholds for different resource management objectives. Anticipating changes in hydrologic conditions in response to climate change enables adaptive strategies to be developed for a more robust and resilient civilization.

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37-10    Weather and Climate REFERENCES

Anagnostou, E. N., M. N. Anagnostou, W. F. Krajewski, A. Kruger, and B. J. Miriovsky, “High-resolution rainfall estimation from X-band polarimetric radar measurements,” Journal of Hydrometeorology, 5: 110–128, 2004. Arguez, A., I. Durre, S. Applequist, R. S. Vose, M. F. Squires, X. Yin, R. R. Heim, Jr., et al., “NOAA’s 1981–2010 U.S. Climate Normals: An Overview,” Bulletin of the American Meteorological Society, 93: 1687–1697, 2012. Austin, P. M., “Relation between measured radar reflectivity and surface rainfall,” Monthly Weather Review, 115: 1053–1071, 1987. Baeck, M. L. and J. A. Smith, “Rainfall estimation for the WSR-88D for heavy rainfall events,” Weather and Forecasting, 13: 416–436, 1998. Bedient, P. B. and W. C. Huber, Hydrology and Floodplain Analysis, Addison Wesley, Reading, MA, 1988. Bradley, Raymond S., Quaternary Paleoclimatology: Methods of Paleoclimatic Reconstruction, Allen & Unwin, Boston, 1985, ISBN 0-04-551067-9. Brutsaert, W., Hydrology—An Introduction, Cambridge University Press, Cambridge, UK, 2005, ISBN-13: 978-0521824798 California Department of Water Resources, “Progress on incorporating climate change into management of California’s water resources,” Technical Memorandum Report, 2006. Carlson, T., Mid-latitude Weather Systems, Routledge, London, UK, 1991. Colorado Water Institute, Workshop on Nonstationarity, Hydrologic Frequency Analysis, and Water Management, January 13–15, 2010 Boulder, CO. Colorado Water Institute Information Series No. 109, 2010. Cronin, Thomas N., Paleoclimates: Understanding Climate Change Past and Present, Columbia University Press, New York, 2010, ISBN 978-0-23114494-0. Cushman-Roisin, B., Introduction to Geophysical Fluid Dynamics, Prentice Hall, New York, NY, 1994. Daly, C., R. P. Neilson, and D. L. Phillips, “A statistical-topographic model for mapping climatological precipitation over mountainous terrain,” Journal of Applied Meteorology, 33: 140–158, 1994. Dracup, J. A. and E. Kahya, “The relationships between U.S. streamflow and La Niña events,” Water Resources Research, 30 (7): 2133–2141, 1994. Gill, A., “Atmosphere-ocean dynamics,” International Geophysics Series, Vol. 30, Academic Press, London, UK, 1982. Grotjahn, R., Global Atmospheric Circulations Observations and Theories, Oxford Press, Oxford, UK, 1993. Haltiner, G. and R. Williams, Numerical Prediction and Dynamic Meteorology, 2nd ed., John Wiley & Sons, New York, NY, 1980. Holton, J. R., An Introduction to Dynamic Meteorology, 3rd ed., Academic Press, London, UK, 1992. Hosking, J. R. M. and J. R. Wallis,  Regional Frequency Analysis: An Approach Based on L-moments, Cambridge University Press, Cambridge, UK, 2005, p. 3, ISBN 0521019400. IPCC, Climate Change 1995 The Science of Climate Change Contribution of Working Group I to the Second Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, 1995. Interagency Advisory Committee on Water Data, March 1982. Bulletin 17B, “Guidelines for Determining Flood Flow Frequency,” Published by the U.S. Department of the Interior, Geologic Survey. Kavvas, M. L., S. Kure, Z. Chen, N. Ohara, and S. Jang, “WEHY-HCM for modeling interactive atmospheric-hydrologic processes at watershed scale. I: Model Description,” Journal of Hydrologic Engineering, 18 (10): 1262–1271, 2013. Kim, H. H., “Urban heat island,” International Journal of Remote Sensing, 13 (12): 2319–2336, 1992.

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Kuichling, E., “The relation between the rainfall and the discharge of sewers in populous districts,” Transactions, American Society of Civil Engineers, 20: 1–56, 1889. Lundquist, J. and A. Flint, “Onset of snowmelt and streamflow in 2004 in the western United States: how shading may affect spring streamflow timing in a warmer world,” Journal of Hydrometeorology, Vol. 7: 1199–1216, 2006. Meko, David M., Connie A. Woodhouse, and Ramzi Touchan, Klamath/ San Joaquin/Sacramento Hydroclimatic Reconstructions from Tree Rings. Report to California Department of Water Resources, 2014, 117 pages. NOAA Atlas 2, Precipitation Frequency Atlas of the United States, 1973. NOAA Atlas 14, Precipitation Frequency Atlas of the United States, 2006. NOAA Atlas 14, Precipitation Frequency Atlas of the United States Volume 6 California, 2011, Vol. 6. Painter, T. H., D. F. Berisford, J. W. Boardman, K. J. Bormann, J. S. Deems, F. Gehrke, P. Kirchner, et al., “The airborne snow observatory: fusion of imaging spectrometer and scanning LiDAR for studies of mountain snow cover,” Remote Sensing of Environment, 2015, in review. Pedlosky, J., Geophysical Fluid Dynamics, Springer-Verlag, New York, NY, 1987. Philander, S., El Niño, La Niña and the Southern Oscillation, Academic Press, London, UK, 1989. Piechota, T. and J. Dracup, “Drought and regional hydrologic variation in the United States: Associations with El Niño-Southern Oscillation,” Water Resources Research, 32 (5): 1359–1373, 1996. Ralph, F. M., E. Sukovich, D. Reynolds, M. Dettinger, S. Weagle, W. Clark, and P. J. Neiman, “Assessment of extreme quantitative precipitation forecasts and development of regional extreme event thresholds using data from HMT2006 and COOP observers,” Journal of Hydrometeorology, 11: 1286–1304, 2010. Ralph, F. M. and M. Dettinger, “Historical and national perspectives on extreme west coast precipitation associated with atmospheric rivers during December 2010,” Bulletin of the American Meteorological Society, 93: 783–790, 2012. Rhea, O., Orographic precipitation model for hydrometeorological use. Atmospheric Paper 278, Colorado State University, Fort Collins, CO, 198 pages, 1978. Singh, V. P. and D. K. Frevert (Eds.), Mathematical Models of Small Watershed Hydrology and Applications. 950 pp., Water Resources Publications, Highlands Ranch, Colorado, 2002a. Singh, V. P. and D. K. Frevert (Eds.) Mathematical Models of Large Watershed Hydrology. 891 pp., Water Resources Publications, Highlands Ranch, Colorado, 2002b. Shorack, G. R. and J. A. Wellner, Empirical Processes with Applications to Statistics. New York: John Wiley & Sons, 1986, ISBN 0-471-86725-X. USGS, Guidelines for determining flood flow frequency, Bulletin 17B of the Hydrology Subcommittee Interagency Advisory Committee on Water Data, US Department of the Interior, 1982. USGCRP, Climate Change Impacts in the United States. US National Climate Change Assessment, 2014. White, A. B., M. L. Anderson, M. D. Dettinger, F. M. Ralph, A. Hinojosa, D. R. Cayan, R. K. Hartman, et al., “A 21st century California observing network for monitoring extreme weather events,” Journal of Atmospheric and Oceanic Technology, Vol. 30: 1585–1603, 2013. Woodhouse C. A., S. T. Gray, and D. M. Meko, “Updated streamflow reconstructions for the Upper Colorado River basin,” Water Resources Research, 42: W05415, 2006, doi:10.1029/2005WR004455. Zhu, Y. and R. E. Newell, “A proposed algorithm for moisture fluxes from atmospheric rivers,” Monthly Weather Review, 126: 725–735, 1998.

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Chapter

38

Hydroclimatology: Global Warming and Climate Change BY

JOHN W. NIELSEN-GAMMON

ABSTRACT

Many factors cause the global and local climate to change, leading to ongoing and future impacts to many aspects of the hydrologic cycle such as peak intensity and spatial distribution of precipitation. Climate models are able to simulate many of the observed changes, particularly at larger scales, and are an important tool for estimating future changes according to projections of future greenhouse gas concentrations and other change agents. Using model projections for hydrologic applications requires applying appropriate downscaling techniques to convert the model-simulated variables to hydrologic information specific to a particular information need at a particular location.

38.1  INTRODUCTION: THE AMBIGUITY OF CLIMATE

Climate can be succinctly defined as the statistical properties of the weather. Climate change, in turn, is defined by the American Meteorological Society (AMS, 2015) as “any systematic change in the long-term statistics of climate elements (such as temperature, pressure, or winds) sustained over several decades or longer.” These definitions leave plenty of room for ambiguity. Climate statistics incorporate the variability associated with such phenomena as El Niño, which is a coupled ocean-atmosphere phenomenon with a period ranging from less than 2 years up to 8 years (Wittenberg, 2009). Some slower modes of variability, such as the Atlantic Multidecadal Oscillation, have characteristic periods greater than 50 years (Gray et al., 2004), and the weather variations associated with them may thus persist for several decades. This would technically meet the preceding definition of climate change. Major volcanic eruptions spew tiny solid and aqueous particles, collectively known as aerosols, into the stratosphere, where they reduce incoming solar radiation for several years. A single volcanic eruption is not generally said to cause climate change, but a period of enhanced volcanic activity appears to have helped bring about the Little Ice Age. If the Little Ice Age can be regarded as an example of climate change, it is difficult to see where the transition from ordinary climate variability to climate change occurs. An engineer designing a weather-sensitive project such as a flood control structure typically has a closely related definition of climate: the expected statistics of weather throughout the design lifetime of the project. As long as climate change is not taking place, such statistics may be obtained through historical observations and related information. Even with climate change, the change may be slow enough that the historical observations may still provide a suitable estimate of future statistics. The conventional period established by the World Meteorological Organization for calculating climatic averages is 30 years (Arguez and Vose, 2011). If the climate is changing, the weather observations taken 30 years ago are less relevant to present-day and future climate than observations from last

year. The Climate Prediction Center has developed the concept of optimal climate normals (Huang et al., 1996), whereby the averaging interval is selected to be the interval (usually 10–15 years) whose average conditions provides the most accurate prediction of future average conditions. If the rate of change is large and the natural variability is small, the optimal averaging interval will tend to be small too. Temperature typically has a systematic trend that is fairly large compared to interannual variability, so short averaging periods work best for temperature. Precipitation has relatively large natural variability and thus requires longer averaging times. Shortening the averaging time is a luxury not available to one who wishes to estimate the probability of rare events such as major floods. Such events are so rare that data from as many years as possible are needed to accurately estimate the magnitude of extreme events. This is one important reason that rare event probabilities are particularly subject to inaccuracies in the case of climate change. Another important reason is that rare event probabilities can change dramatically if there is merely a slight shift in the central value of the probability distribution (Fig. 38.1). We can now define climate change from an engineering perspective: a change in the statistical properties of the weather such that perfect past observations would be insufficiently representative of statistical properties over a design lifetime. The general statistical term for such a situation is nonstationarity. Climate change is presently proceeding at a sufficiently rapid pace that, for many engineering applications, stationarity is no longer an appropriate assumption (Milly et al., 2008). The remainder of this chapter assumes that nonstationarity is a concern and discusses the physical processes that are causing present-day climate change, observations of climate change impacts already present in the hydrologic cycle, and the use of global climate model (GCM) output for estimating possible future climate changes. 38.2  NATURAL AND HUMAN INFLUENCES ON PRESENT-DAY CLIMATE

The knowledge that the climate is presently changing at an historically rapid rate comes not just from comparisons between recent climate observations and reconstructions of past climate, but also from observations and calculations of the magnitude of processes driving imbalances in the climate system. The combined rate and magnitude of warming is apparently larger than has happened for several millennia, and the magnitude of those processes driving climate change are collectively much larger than any such drivers over the past several millennia as well (Masson-Delmotte et al., 2013). Global-scale changes in climate are most directly produced by changes in the radiative balance of the climate system. The Earth receives electromagnetic radiation from the Sun and emits electromagnetic radiation into space. The majority of the received radiation is absorbed by the Earth’s surface and the vast majority of emitted radiation escaping to space is emitted by greenhouse 38-1

38_Singh_ch38_p38.1-38.8.indd 1

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38-2     Hydroclimatology: Global Warming and Climate Change 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –4 –0.1

–3

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0

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–0.2 –0.3 –0.4 Original PDF

PDF with new mean

Change in low extremes (*10)

Change in high extremes (*10)

Figure 38.1  The change in the frequency of extreme events (plus or minus two standard deviations) associated with a change in the mean. Note that the probability change is asymmetric, with roughly twice as large an increase in high extreme events as the decrease in low extreme events.

gases and clouds; the energy loop is closed by net transfer energy from the surface to the atmosphere by primarily nonradiative processes. Any overall difference between what is absorbed and what is emitted will cause the climate system to gain or lose energy. Over multiple years, this balance is so exact that the present global warming is estimated to be caused by an energy imbalance smaller than one part in 300 (Trenberth et al., 2014b). The fundamental reason for this stability is a local equilibrium in the climate system: Any gain in energy will cause the temperature of the atmosphere to rise, thereby increasing the rate at which energy is emitted to space. Likewise, a drop in temperature produces a decrease in emitted energy. Any energy imbalance causes a response in the climate system that reduces that imbalance. Past long-term climate changes all seem to have involved changes in one or more of the following factors affecting the planetary energy balance: atmospheric gaseous composition, snow and ice cover, solar intensity, volcanic activity, vegetation distribution, and atmospheric dust. For example, during the last glacial maximum about 21,000 years ago, when much of North America and Europe was covered by thick ice sheets, snow and ice cover and atmospheric gaseous composition were both having a strong cooling effect compared to present-day climate, while vegetation distribution and atmospheric dust were having a weak cooling effect (though the actual size of the dust effect is difficult to pin down) (Schmittner et al., 2011). Although multiple factors are usually at work, the root cause of a climate change is typically a single process or event. For example, the recovery from the glaciation 21,000 years ago is thought to have been triggered by orbitcaused changes in the relative amount of sunlight received in summer versus winter, which caused an increase in summertime melting of the Northern Hemisphere ice sheets. Because of their sheer size after nearly 100,000 years of accumulation, it appears that they pushed the continents down far enough that melting could take place rapidly at relatively low elevations (Abe-Ouchi et al., 2013). Atmospheric carbon dioxide then began rising through some combination of outgassing due to warmer temperatures and changes in the ocean circulation that brought carbon dioxide–rich waters to the surface. As rainfall patterns changed, vegetation became more widespread and atmospheric dust decreased. By the time the last of the continental North American ice sheet melted, this particular climate change had been complete. Of those four factors, the three that have changed substantially in the past few hundred years are atmospheric composition, vegetation, and suspended particles. All three changes are in this instance due to humanity’s activities rather than orbital triggering of ice sheet changes. The specific causal mechanisms are fossil fuel burning (affecting composition and suspended particles) and agriculture (affecting all three). The largest present-day imbalance by far is caused by changes in atmospheric composition, specifically increases in the concentrations of carbon dioxide, methane, chlorofluorocarbons, nitrous oxide, ozone, and other compounds (Myhre et al., 2013). These increases have a warming effect because they decrease the amount of energy emitted to space. Next in present-day importance is the increase in suspended particles, which by itself would have a cooling effect due to a reduction in incoming solar radiation. Some suspended particles absorb solar radiation and have their own warming effect.

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Land use changes appear to have had a small net cooling effect. Some warming was contributed by an increase in solar intensity, though intensity seems to have peaked around the middle of the twentieth century and has declined since the 1970s. Of these energy imbalances, the easiest to quantify are imbalances due to atmospheric gaseous composition changes. Most difficult are imbalances due to suspended particles, because these particles may strongly affect the radiative properties of clouds and top-down global observations of cloud properties are only available for the past few decades. Because gas and suspended particle concentrations have increased in tandem, it is nearly impossible to separate out their influences through statistical analysis of changes in past temperatures. Likewise, because the direct and indirect radiative impacts of suspended particles is only known to within a factor of 10, the total energy imbalance driving global warming is not known very accurately (Trenberth et al., 2014b). This makes estimating the magnitude of future climate changes from observations of recent changes difficult. Other methods of estimating climate sensitivity have their own limitations. Any GCM can be driven by specified processes that create an energy imbalance and the size of the simulated climate response can be determined from the model simulation. At present, different climate models exhibit different climate sensitivities, and there is no guarantee that the actual climate sensitivity lies within the range of sensitivities of the different climate models. The differences in sensitivity are primarily due to the effects of simulated clouds, which for a number of reasons are particularly challenging for GCMs (Boucher et al., 2013). Climate sensitivity may also be estimated from past climate changes, but both the size of the energy imbalance and the size of the resulting climate change must be estimated from the limited geological evidence available. The overall climate change response to an energy imbalance involves changes to the temperature, humidity, cloud cover, snow cover, atmospheric and ocean circulations, and other aspects of the climate system. Atmospheric changes happen rapidly, within hours to weeks of a sudden energy balance change. The response of the ocean is more gradual, and the atmosphere continues to change in response to ocean surface changes. Other changes are even slower, such as the increase in energy absorption caused by the retreat of the major ice sheets of Greenland and Antarctica. These changes are called feedbacks, and their individual impacts can also be separately quantified, though for our purposes it is more useful to think of the climate system as an integrated whole with a complex set of time-dependent responses on the way toward a new energy equilibrium. Many of the important time-dependent responses involve changes to the hydrological cycle, and such changes are discussed next. 38.3  IMPACTS OF CLIMATE CHANGE ON THE HYDROLOGICAL CYCLE IN THE TWENTIETH AND TWENTY-FIRST CENTURIES 38.3.1 

Evapotranspiration

The rate of evaporation from the oceans into the atmosphere depends primarily on the wind speed and on the saturation vapor pressure deficit, which is the difference between the actual partial pressure of water vapor and the partial pressure of water vapor at 100% relative humidity at the actual temperature. According to basic thermodynamics, an increase in surface temperature would lead directly to an increase in saturation vapor pressure and hence an increase in saturation vapor pressure deficit, which in turn would imply an increase in the evaporation rate, all else being equal. GCMs predict an increase in evaporation rate over the oceans averaging a bit less than 2% per degree of warming (Held and Soden, 2006), which is much smaller than the 7% increase in saturation vapor pressure per degree. As is discussed in the following text, models tend to increase tropospheric humidity at 7% per degree. This difference implies an overall reduction in wind speed. Although wind speed variations can be highly localized and noisy, a comprehensive survey of papers documenting wind speed changes finds that globally averaged surface wind speeds seem to have decreased (McVicar et al., 2012). Over land, much of the flux of water into the atmosphere is via transpiration. In addition to the same constraints on water vapor fluxes as over the ocean, transpiration is also affected by changes in plant cover and the partly temperature-dependent regulation of transpiration by plants. Atmospheric increases in carbon dioxide additionally have two primary direct but competing effects on transpiration. First, stomata need not open as much to intake the same amount of carbon dioxide, thereby allowing plants to lose less water vapor through transpiration. Second, the resulting increased plant growth would create larger plants, which in turn would transpire more. Direct measurements of evapotranspiration over land are sparse and quite spatially vari-

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Global Climate Models    38-3 

able, but overall they indicate an increase in evapotranspiration (Hartmann et al., 2013). The average rate of change in evapotranspiration is not known with sufficient precision to compare against the evapotranspiration increases simulated by climate models, but observations and climate models are consistent with respect to the spatial pattern of evapotranspiration changes. Over water, the various specific humidity data sets are too inconsistent with each other to permit a statement on humidity trends there (Kent et al., 2014). In the free troposphere (between about 1 km and 12–15 km above ground level), and at ground level over land, average humidity seems to be increasing at something close to the 7% per degree warming change in saturation water vapor content (Hartmann et al., 2013; Willett et al., 2010). Climate models consistently predict that relative humidity at the surface and troposphere ought to increase at about 7% per degree as well (Collins et al., 2013; Willett et al., 2010). Wind patterns in the atmosphere can alter the distribution of water vapor within the atmosphere, and are closely related to the production of precipitation. In the tropics, the primary large-scale circulation features are the Hadley cells, which feature upward motion in convective regions near the equator and downward motion over the subtropics. During the past several decades, the Hadley cells have become taller and the poleward edges of the Hadley cells have migrated toward the poles (Hartmann et al., 2013), implying an expansion of the Hadley cells and associated expansion of areas of low relative humidity from the subtropics toward the midlatitudes. Climate models likewise predict an expansion of the Hadley cells, but the observed expansion has been greater than that predicted by model simulations. The discrepancy has been attributed to a specific form of multi-decadal natural variability in the Pacific Ocean, which alters the pole-to-equator temperature gradient and thus affects the extent of the Hadley cells (Allen et al., 2014; Adam et al., 2014). Over land, the surface relative humidity has stayed roughly constant. Climate models predict a decrease, which can be understood as caused by the enhanced warming over land due to radiative changes (Sherwood et al., 2010) and, particularly in the subtropics, by the increased desiccation of air as it is carried to higher altitudes within the Hadley cell before subsiding back down to the surface (Lau and Kim, 2015). 38.3.2 Precipitation

Precipitation on a global and annual basis is constrained to equal evaporation, since the atmosphere only stores about 1 or 2 weeks’ worth of precipitation. Thus, if one is known precisely, so is the other. Unfortunately, neither is known precisely, due to the difficulty of measuring evaporation over land and of measuring precipitation over water. New satellite-based precipitation measurements will help close the budget, but it will take decades before the data are extensive enough to pin down the globally averaged trend. Model simulations of precipitation resemble the observed global pattern in most areas (Flato et al., 2013). Notable exceptions are a tendency to simulate too little precipitation in the western tropical Pacific and to simulate too much precipitation over the tropical oceans south of the equator. The general consensus from the various global rainfall data sets is that rainfall seems to be increasing globally on average (Hartmann et al., 2013). There are particular places in the Northern Hemisphere, such as the central United States, where there is a clear positive trend. Together with the 2% increase in evaporation per degree of warming in climate models is a 2% increase in precipitation per degree of warming (Collins et al., 2013). These trends are consistent with observations, but the range of estimated observed precipitation increases is so large that the observations do not constrain the models. The dependence of observed rainfall trends on latitude seems to be simulated fairly well by models. Both observations and models indicate a decrease of precipitation in the subtropics (except for summertime) and an increase at higher latitudes that is largest in wintertime. Although this pattern is frequently generalized as “wetter places get wetter and drier places get drier,” there are many places where that generality does not hold. In the models, the subtropical change is associated with the changes in Hadley cell discussed earlier, while the change at higher latitudes is due to the increased moisture in a warmer world. Regional variations in trends are in many areas considerably different in observations compared to models. Multidecadal natural variability explains some, but not all, of the differences (Gu and Adler, 2015). Because of warmer temperatures, some of what used to fall as snow is now falling as rain. This has implications, for example, for the snow pack in the western mountains of North America in places where the rain-snow line has a strong influence on snow accumulation, and observed changes in springtime mountain snowpack are consistent with warming temperatures. The overall Northern Hemisphere snow cover extent is decreasing at a statistically significant rate in springtime; trends in fall and winter are for the most part indistinguishable from zero (Hernández-Henríquez et al., 2015). Springtime

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declines are particularly important for humans (implying a longer dry season) and for the climate system (the change in reflectivity of the Earth’s surface near the summer solstice affects Earth’s radiation balance). Changes in extreme precipitation can be distinct from those in mean precipitation. Because extreme precipitation, by its nature, is more erratic, observed trends are less spatially coherent than trends in mean precipitation. However, increasing temperatures directly imply an increase in the maximum water vapor in the atmosphere and hence the maximum amount of rainfall, so an increase in extreme precipitation is a robust expectation from rising temperatures. Globally, there are more places where heavy precipitation is increasing than decreasing. Across much of the United States, for example, 1- and 2-day extreme rainfall events are becoming much more frequent (DeGaetano, 2009; Janssen et al., 2014; Walsh et al., 2014). GCMs tend to underestimate this particular increase (Janssen et al., 2014). Overall, models have wide spread but are reasonably centered around the average observed rate of increase in extreme 1-day rainfall in the Northern Hemisphere midlatitudes, but they tend to underestimate the rate of increase in extreme rainfall in the tropics (Asadieh and Krakauer, 2015). 38.3.3 

Land Surface Hydrology

In the absence of a global soil moisture network or long-term satellite-based estimates of soil moisture, model-simulated changes in soil moisture are mostly unverified. The most prominent decreases in soil moisture are projected to occur in the Mediterranean, southwestern North America, and southern Africa (Collins et al., 2013). All three of these changes are consistent with the expansion of the Hadley cell and resulting changes in precipitation and humidity. Drought seems to be increasing globally, but with large regional variations. The lack of confidence regarding globally averaged drought trends is caused in part by the divergent trends in global precipitation analyses and in part by challenges with estimating potential evapotranspiration in a world with a changing climate and changing atmospheric composition (Trenberth et al., 2014a). Global observed streamflow trends are mixed, and a comprehensive estimate of streamflow trends is difficult because many major rivers are ungauged (Hartmann et al., 2013). A modeled increase in streamflow can be inferred, even for those models without a hydrologic routing component, by the simulated increase in precipitation minus evaporation over land (Collins et al., 2013). There is no detected trend in flood frequency globally, which is consistent with the lack of a streamflow trend but inconsistent with the observed increase in extreme rainfall. Presumably, getting from extreme rainfall to flooding involves mediation from flood control measures and better local water management. Substantially decreased streamflow is simulated for the Mediterranean, southern Africa, and the Middle East. Despite the decline in simulated soil moisture and earlier findings, southwestern North America is not projected to have a substantial decline in streamflow in the latest model simulations (Collins et al., 2013). There is a substantial observed and simulated increase in streamflow at high latitudes, with contributions from increased precipitation and melting of semipermanent ice in glaciers and ice sheets. This is one of the main components of global sea level rise, along with thermal expansion. Melting of ice from Greenland and Antarctica is an increasing component of sea level rise, and there is an additional contribution from aquifer depletion (Church et al., 2013). 38.3.4 

A Comment on Extremes

The preceding discussion of droughts, floods, and extreme rainfall should make it clear that it is difficult to generalize about extreme weather event trends. For example, the general statement “global warming causes more droughts and floods” is quite misleading. While droughts are increasing on average worldwide, the increase is by no means uniform, nor is the rate of increase known. Meanwhile, floods are not increasing on a global average basis, and extreme rainfall increases are only guaranteed to lead to flooding increases if societal or infrastructure changes are not at the same time reducing the flooding potential. Many actions taken by society, either intentionally or inadvertently, affect flood risk, and one shouldn’t generalize to an overall consequence when only one contributing factor is being considered. Even if floods increase in the future, the area of overlap, where both droughts and floods are increasing, will certainly be less than half of the globe. 38.4  GLOBAL CLIMATE MODELS

The preceding section compared observed trends in the water cycle to global climate model (GCM) output. This section provides some perspective on climate model performance by discussing the components of a climate model and the sources of discrepancies between model output and reality.

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38-4     Hydroclimatology: Global Warming and Climate Change

A state-of-the-art GCM is actually a set of models that exchange information as they run. A minimal GCM will include an atmospheric dynamical model, an ocean model, and a land surface model. Common additional components include a snow model, a sea ice model, a hydrologic model, a vegetation model, and an atmospheric chemistry model. GCMs are typically designed so that some component models are optional and can be replaced by steady-state assumptions or simplistic approximations. The atmospheric dynamical and ocean model both solve a set of equations known as the Navier-Stokes equations or an approximate version of them. These partial differential equations govern fluid flow and represent Newton’s second law of motion, the first law of thermodynamics, conservation of mass, and an equation of state, which for the atmosphere is the Ideal Gas Law. The equations describe how mass, momentum, and energy vary smoothly throughout the fluid, with the fluid continuously creating and responding to changes in forces or internal energy. The equations are exact if the atmosphere and ocean behave as an ideal fluid. To make numerical integration possible, the fluid state is approximated by a set of numbers that collectively describe variations in the state of the fluid (wind, temperature, pressure, moisture, or salinity) that are larger than a certain spatial scale. This is done by dividing the atmosphere into a handful of levels (typically 25–40) and either creating a horizontal grid or using a truncated expansion of continuous functions such as spherical harmonics. The neglect of finer-scale processes creates errors, in part because energy is constantly cascading to finer scales—there is no clean scale separation between phenomena that can be resolved and phenomena that cannot. Other errors are introduced when the equations are integrated forward over a finite time interval. Various techniques are available for numerical integration in time and derivative calculation in space, but tradeoffs must be made among accuracy, desirable properties (such as conservation of energy or conservation of mass), and speed. Ultimately, the accuracy of the simulation is limited by the available computer power. Configured in this way, the GCM is not rigged or preprogrammed to include specific large-scale phenomena such as cold fronts, El Niño, the Hadley cells, the jet streams, the Gulf Stream, and so forth, except that some of them will exist in the starting conditions. Instead, the model continuously creates and regenerates these and other weather and climate features as emergent phenomena resulting from the laws of physics and thermodynamics acting on a quasi-continuous fluid. Climate models are validated by how well they reproduce the climate of the past and present. Some processes occurring at smaller scales have important, fundamental influences on the larger-scale motions and thus cannot be completely neglected. To include the influences of these processes, modelers rely on relationships between the local larger-scale conditions and the smaller-scale processes, in effect predicting or estimating the processes that the model cannot simulate directly based on what the model can simulate directly. These relationships, programmed into a climate model, are called parameterizations. The most important parameterizations from a hydrological perspective are those involving clouds and precipitation. Clouds are too small in scale to be simulated directly by climate models. Climate models must infer the presence and properties of clouds within a grid cell from the grid cell’s average values of water vapor, temperature, and vertical motion. In many current versions of climate models, the properties of aerosol particles that can serve as cloud condensation nuclei are also simulated and are taken into account by the cloud parameterization. The cloud parameterization also generates precipitation if the clouds and the condensation rate within them are determined to be large enough. Condensation occurs if there is on average upward motion at a particular level in the grid box and at least a portion of that level within the grid box is determined to be at or above saturation with respect to water vapor. The cloud parameterization produces only part of the precipitation simulated by a climate model. The remainder, including almost all of the precipitation in the tropics, is simulated by the convective parameterization. The convective parameterization represents the effect of moist convection, which visually takes the form of cumulus or cumulonimbus clouds, including thunderstorms. The effects of moist convection include a vertical redistribution of heat, moisture, and, in some parameterizations, momentum; the possible generation of precipitation; and the possible generation of hydrometeors that are tracked at the grid scale and will be dealt with by the cloud parameterization. Convective parameterizations can be quite complicated. In general, most of them include two interacting components: the trigger function, which determines whether subgrid-scale convection is present, and the cloud model, which determines the consequences of subgrid-scale convection. The trigger function is based on a calculation of the amount of instability or on the rate at which instability is being generated, and it may also include an assessment of the extent of inhibition to convection due to such features as lower-troposphere temperature inversions. The cloud model may be as simple as an

38_Singh_ch38_p38.1-38.8.indd 4

asymptotic relaxation to a prescribed vertical distribution of heat and moisture or as complicated as a one-dimensional column model of convection complete with calculated entrainment/detrainment and updraft and downdraft strengths at every level. It has been a difficult challenge to create convective parameterizations that perform well in both the tropics and higher latitudes. Tropical convection is typically very deep and develops from weak instability. Midlatitude convection is often much stronger because instability can increase substantially above a capping inversion before the onset of convection. It is also common for models to have a specialized convective parameterization for shallow convection, the type of convection that typically forms at the top of the boundary layer and produces little if any precipitation. Unfortunately, the convective parameterization is a critically important component of a climate model, because the general circulation of the atmosphere is driven by the atmospheric convection and heat redistribution that takes place in the tropics in response to solar heating. Models have difficulty reproducing even such seemingly simple characteristics of convection as the local time of day at which convection is most likely to occur. The different models have different strengths and weaknesses, but they also share some common strengths and weaknesses. The various GCMs cannot be assumed to span the range of reality, because reality does not include smallscale processes that are entirely predictable, as is assumed by most parameterizations. Even model ensembles in which the various parameters controlling the parameterizations are varied throughout their plausible numerical ranges cannot by their nature encompass reality. On the whole, then, climate models produce simulations of weather and climate on imaginary planets that are very much like our own. 38.5  WORKING WITH CLIMATE MODEL PROJECTIONS

There are many sources of difficulty and inaccuracy in estimating future climate states. For example, for estimating future flood frequency in England, Kay et al. (2009) list human emissions, GCM structure, internal variability, regional climate model structure, downscaling, hydrologic model structure, and hydrologic model parameters. The previous section discussed uncertainties caused by GCM structure. This section considers uncertainties associated with human emissions, regional climate model structure, and/or downscaling. 38.5.1 Projections

For forecasts multiple decades into the future, neither initial atmosphere nor initial ocean conditions are relevant. The primary source of predictability of future climate states arises from knowledge of future changes in atmospheric radiative forcing. A true climate model forecast would consist of a GCM driven by a forecast of radiative forcing changes. Comprehensive forecasts of radiative forcing changes do not exist. Volcanic eruptions are unpredictable, and solar variability is presently unpredictable more than a few years in advance. Changes in the atmospheric composition caused by human activities are believed to be much more predictable over the near term, in part because predicting human emissions is much like predicting the course of a supertanker: because it takes so long to institute a significant change, continuation of the present evolution for quite some time is likely. By the end of the twenty-first century, though, a very wide range of emissions and subsequent atmospheric conditions is possible. Estimates of possible ways that climate-relevant human activities might change over time are called scenarios. In general, scenarios are based on assumptions of population, global wealth, commerce, and technology. So-called “business as usual” scenarios, such as the SRES (Special Report on Emission Scenarios; Nakicenovic and Swart, 2000) scenarios, assume that there will be no intentional efforts to reduce emission for climate change mitigation purposes, though emissions reductions due to air quality regulations or technological improvements are included in the scenarios. Other scenarios assume emissions are reduced as a result of policy decisions to limit human influence on climate. As of 2015, the two most recent coordinated sets of GCM runs for the next century are CMIP3 and CMIP5. (CMIP stands for Coupled Model Intercomparison Project.) The six SRES scenarios in CMIP3 were all business-as-usual scenarios: A1fi, A2, A1B, A1T, B2, and B1 (Nakicenovic and Swart, 2000). They differed in their assumptions regarding demographics, international cooperation, and so forth. Three of these scenarios have emissions peaking and then declining by the end of the twenty-first century through a combination of reduced population growth and reduced carbon intensity. The IPCC report putting forth these scenarios advised that they were all to be considered as equally likely.

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Working with Climate Model Projections    38-5 

The CMIP5 group took a different approach (van Vuuren et al., 2011). Instead of specifying scenarios and calculating emissions, they specified several sets of postulated emissions (called representative concentration pathways, or RCPs) and then had scenarios created that were consistent with those emissions. The RCPs were designed to span a wide range of possibilities. The RCP with the greatest greenhouse gas concentrations by the end of the twenty-first century, RCP 8.5, is virtually identical to the higher SRES A1fi business-as-usual scenario; most SRES scenarios and other business-as-usual scenarios have lower emissions than RCP 6.0 and RCP 8.5 through the middle of the twenty-first century and finish the century with net emissions somewhere between RCP 6.0 and RCP 8.5. At the low end, RCP 2.6 envisions such aggressive action to combat greenhouse gas emissions that by the last few decades of the twenty-first century there is net removal of carbon dioxide from the atmosphere. The final RCP, 4.5, lies in between RCP 2.6 and RCP 6.0 and includes fairly strong greenhouse gas emission reductions. There is no single correct scenario to use. Or rather, it is not known which scenario, if any, will turn out to be the correct one. One way to deal with this uncertainty is to use model output generated from several different scenarios. Another approach is to use model output generated from one of the high-end emissions scenarios (such as A1FI or RCP 8.5) and present the results in terms of the amount of change per degree of global warming by a certain year (e.g., Collins et al., 2013; Swain and Hayhoe, 2015). To be confident with this approach, it is useful to determine the extent to which the local hydrological parameters of interest change linearly with global mean surface temperature and are independent of the rate of change of surface temperature. 38.5.2  Bias Adjustment

Ideally, a GCM will simulate climatic conditions exactly like those that are observed. In practice, this is impossible, partly because natural variability prevents the true long-term average climate conditions from ever being perfectly known, but primarily because the models themselves are imperfect. Some aspects of the climate system can be reproduced very accurately by global climate models. Other aspects give climate models greater difficulty. Global precipitation distributions and amounts fall into the latter category. Suppose a GCM’s present-day simulation is 200 mm of precipitation, its future simulation is 150 mm and present-day observations show 100 mm. Even though the model’s future climate is wetter than the present-day climate we observe, should we interpret the model projection as indicating that precipitation will decrease? If so, by how much? Bias adjustment says that the model’s precipitation decreased by 25%, from 200 to 150 mm, so that we should apply a 25% adjustment to the observed climatic conditions, yielding a projected precipitation amount of 75 mm? The latter approach, applying a multiplicative bias adjustment, is the most common one. In general, one is not interested merely in the future mean conditions, but also (or exclusively) in the future variability or extremes. Thus the problem can be stated: given an observed present-day probability distribution, a simulated present-day probability distribution, and a projected probability distribution, how should one use the knowledge of the difference between the observed and simulated present-day probability distribution to improve the accuracy of the projected probability distribution? There are a number of possible approaches. One common one is called empirical quantile mapping: Assume that the first percentile of the observed distribution corresponds to the first percentile of the simulated present-day distribution and in turn to the first percentile of the simulated future distribution, and apply the same assumption to all percentiles. Then the bias between the observed and simulated present-day percentiles may be computed and applied as an adjustment to the future percentile values. The quantile approach seems fine in principle, but it easy to create pathological cases. For example, consider observed precipitation ranging from 40 to 60 mm, simulated present-day precipitation ranging from 20 to 80 mm, and simulated future precipitation ranging from 40 to 90 mm. At first glance, these seem like plausible numbers rather than pathological ones. At the high end, the present-day observations are 75% of the simulated value, so the high end in the future climate ought to be reduced to 90 × 0.75 = 67.5 mm. Meanwhile, at the low end, the present-day observations are 200% of the simulated value, so the low end in the future climate ought to be increased to 40 × 2 = 80 mm. The low end of the adjusted precipitation distribution is greater than the high end of the adjusted precipitation distribution. An alternative common approach is to separately consider changes in the mean and standard deviation, but that too can lead to inconsistencies. 38.5.3 

Statistical Downscaling

The preceding discussion has assumed that one is interested in the future state of some variable that is both observable and appears in the model output. Generally this will not be the case; it is rare to want to work with average daily

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or monthly precipitation over an externally defined 5000-km2 grid area. Instead, what will be of interest will be precipitation at a single gauge location, or at a particular spatial resolution, or averaged over a particular drainage basin, over a range of accumulation intervals that may include intervals shorter than the gaps between model output. The conversion of climate model output from a coarse spatial and temporal scale to a fine spatial and temporal scale is called downscaling. There are two types of downscaling: statistical and dynamical. Statistical downscaling is based on empirical relationships between modeled quantities and the quantities of interest. Statistical downscaling is only possible if a direct relationship exists between the desired model output variable and the actual output variables. Such a relationship can be established using observations from the climate record; the task becomes much harder if such an observational relationship is not readily available due to lack of appropriate observations. If observations exist, the statistical downscaling problem becomes similar to the bias correction problem. We know the probability distribution of the desired variable, the present-day simulated distribution of a corresponding model output variable, and the future simulated distribution of the model output variable. As with bias correction, the task becomes one of computing a transfer function between the present-day model output variable distribution and the observed distribution of the desired variable, and applying that transfer function to the simulated future output variable distribution. Since the process is similar, the concerns are similar. An important additional concern also arises: the stability of the transfer function between the observed variable and the model variable. One needs the two variables to be so closely related that a certain value of the model variable would be expected to correspond to the same value of the observed value in the present-day climate and in a climate that has undergone all the changes associated with global warming. An example of two variables that would seem to satisfy this condition is observed gauge precipitation and model-simulated grid-box average precipitation. But if the increased moisture content of a warmer atmosphere leads to greater concentration of rainfall in smaller areas, the mapping of peak grid-box average precipitation onto peak rain gauge precipitation will change over time. Rather than relying on mapping of a single simulated variable to the observed variable of interest, one can use information from multiple simulated variables. Taken to its logical extreme, this leads to an approach known as weather typing, in which the observed weather pattern or patterns most closely corresponding to the model-simulated situation is identified and the corresponding desired input variables are drawn from the observations for that day or days. The literature on statistical downscaling and bias adjustment is rapidly expanding as new techniques are developed and tested; a sampling of papers relevant to hydrologic downscaling is listed here. Wilby et al. (2004) provides an early but reasonably comprehensive guide to addressing many issues related to statistical downscaling, including those mentioned previously. Maraun et al. (2010) provides a more up-to-date and detailed overview of  downscaling issues specifically as applied to precipitation variables. Teutschbein et al. (2011) provide a more detailed comparison of three downscaling techniques as applied to hydrological impacts in a specific Scandinavian catchment. Ho et al. (2012) compare the bias adjustment paradigm described previously with a different but equally valid approach known as change factors and shows that large differences can result in the downscaled output. Estrada et al. (2013) caution against the danger that the statistical assumptions inherent in the techniques applied to statistical downscaling may not commonly hold. 38.5.4 

Dynamical Downscaling

An alternative approach to obtaining the needed information from large-scale climate simulations is dynamical downscaling. Dynamical downscaling consists of using the output from the large-scale climate simulation to drive a meteorological (or coupled) model over a limited area at much higher spatial resolution, with output available at much higher temporal resolution. The challenge of running a limited-area model can be daunting and usually requires considerable computing power and expertise to avoid detrimental impacts from phenomena such as model climate mismatch. Fortunately for the nonatmospheric scientist, several sets of model output from dynamical downscaling are available online from various sources, so a new model run may not be necessary. Dynamical downscaling provides a way forward when observations are not available with which to apply statistical downscaling techniques. Even so, just as the GCM will generally not accurately simulate the real climate, neither will the limited-area model. Also, if one wishes to produce point data, a statistical downscaling step must be applied to the limited-area model output. So it is common for dynamical downscaling to include bias adjustment and statistical downscaling components too.

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38-6     Hydroclimatology: Global Warming and Climate Change 38.6 CONCLUSION

Hydrologists attempting to deal with future conditions are presently stuck in a gray area. On the one hand, it is no longer possible in certain circumstances to assume stationarity and rely exclusively on historical observations for estimating future conditions. On the other hand, techniques for estimating future conditions under a changing climate are still in their infancy, and it is not clear whether all the important issues involved with such estimations have yet been identified, let alone resolved into best practices. There do at least exist standard approaches, such as those utilized by the National Climate Assessments to identify policy-relevant current and future climate changes on a subnational scale (Melillo et al., 2014). While it is tempting to treat downscaled climate information available online as coming from a black box, it is essential to be aware of the assumptions, tradeoffs, and sources of error inherent to each downscaled data set and to understand how those issues might apply to the particular question and location at hand. REFERENCES

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Chapter

39

Spatial and Temporal Estimation and Analysis of Precipitation BY

RAMESH S. V. TEEGAVARAPU

ABSTRACT

Spatial and temporal estimation and analysis of precipitation are critical and essential tasks for hydrologic modeling and water resources management. This chapter presents a comprehensive overview of methods for estimation of mean areal precipitation (MAP) and missing data. MAP estimation methods ranging from conceptually simple methods using rain gauge observations to computationally intensive methods that use multi-sensor observations are presented and discussed. Methods to obtain serially complete precipitation datasets when missing data exist are needed for many hydrologic, climate variability and change studies. Deterministic and stochastic spatial interpolation methods and their variants for estimation of missing precipitation data are described and reviewed. Recently developed variants of spatial interpolation methods that benefit from advanced data-driven, data mining, and optimization techniques are also discussed. The chapter presents issues and challenges related to the estimation of MAP and filling of missing precipitation data, and points to new methods that can overcome the limitations of some of the existing methods. 39.1 INTRODUCTION

The estimation of mean rainfall or mean areal precipitation (MAP) over a region or watershed and the development of gap-free serially complete precipitation data are two critical tasks for hydrologic modeling and water resources management. The accuracy of MAP estimation depends on the rain gauge network or precipitation estimates from any other sources (e.g., radar and satellite) used. While this chapter deals with quantification of variability of precipitation in space and time based on observations, instruments used to measure precipitation, measurement errors and uncertainty in measurement process that are relevant to estimation are not discussed. Readers are advised to refer to works of WMO (1983), Linacre (1992), Strangeways (2007), and Sene (2010) for an exhaustive discussion on precipitation measurement devices, networks, radar-based precipitation estimation, and advances in hydrometeorology. Deterministic weighting and stochastic interpolation methods have been used for the creation of rainfall surfaces (fields) or gridded precipitation data and also estimating missing rainfall data at points in space. Some of these methods are also used in bias correction procedures for radar data (Teegavarapu, 2013b). Methods for MAP estimation and missing data are discussed in several sections of this chapter. A summary of issues and considerations given for selection of MAP and missing data estimation methods are also provided. 39.2  ESTIMATES OF MEAN AREAL PRECIPITATION

MAP, also referred to as equivalent uniform depth of rainfall over a region or areal average rainfall, is required in many hydrologic modeling studies as a main precipitation input to lumped hydrologic simulation models or

rain-runoff (RR) models. Several conceptually simple methods with low computational complexity to methods that require enormous computational effort are available in literature (WMO, 1994; ASCE, 1996; Dingman, 2011). A few commonly used methods are discussed briefly in the next sections. 39.2.1  Gauge Mean Method

The gauge mean method (GMM) is simplest method for estimating mean precipitation depth over an area of interest (i.e., watershed or basin) using an arithmetic average of all the observations at the gauges for a given time interval. All the gauges that are inside an area of interest are used for calculation of mean precipitation. The GMM can also be applied by considering rain gauges (or stations) outside the region of interest provided the observations recorded by these gauges are representative of precipitation inside the region. The method as an arithmetic average of all the available observations is objective. However, applicability of this method for estimation of MAP is recommended when the region has: (1) uniform precipitation with little spatial variability of rainfall; (2) precipitation patterns that are not influenced by any topographic or physiographic features; and (3) the gauges are uniformly distributed across the region. 39.2.2  Thiessen Polygon-Based Weighting Method

The Thiessen (or Voronoi) polygon method (Thiessen, 1911; Brassel and Reif, 1979; Boots, 1986) provides an area-weighted average of all precipitation observations based on development of polygons with an assumption that any point in the polygon will have similar rainfall characteristics to those recorded at a rain gauge located in that polygon. Thiessen polygons are constructed by connecting gauges by straight lines and drawing perpendicular bisections that meet to form irregular polygons. If polygon areas extend beyond the boundary of the region, only partial areas within the region are considered. This method is an objective approach for estimation of MAP. The polygons need to be recreated every time the gauging network changes. Also, the method does not include or benefit from physiographic information relevant to the region that influences the spatial variability of the precipitation. A graphical illustration of Thiessen polygons created based on five rain gauges in a watershed is shown in Fig. 39.1. The MAP (qA) is estimated used using Eqs. (39.1) and (39.2). The variables wj, qj, and Aj represent the weight, precipitation amount, and polygon area associated with rain gauge j, respectively. The number of rain gauges is referred by variable ng. ng



θ A = ∑w jθ j                (39.1) j =1

−1



ng   w j = A j ∑A j     ∀j (39.2)  j =1 

39-1

39_Singh_ch39_p39.1-39.10.indd 1

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39-2     Spatial and Temporal Estimation and Analysis of Precipitation

2 3

5

1 Centroid

4

Figure 39.2  Illustration of centroid distance calculations for estimation of MAP. Figure 39.1  Location of rain gauges and Thiessen polygons. 39.2.3  Isohyetal Method

39.2.7  Line Method

The isohyetal method or eyeball method (Linsley et al., 1949; France, 1985; Dingman, 2011) requires the creation of isohyets (i.e., lines or contours of equal rainfall amounts) and calculation of area in-between the isohyets. Isohyets are created using available rain gauge measures at different points in a region by linear interpolation between any two points and adopting some form of smoothing or surface fitting method to generate smooth curves. MAP is based on area-weighted average precipitation value calculated based on the isohyet values.

Line method for calculation of mean average precipitation was introduced by Goel and Aldabagh (1979). In this method, lines are drawn between any two points with precipitation measurements as shown in Fig. 39.3. The lines are not allowed to cross other lines. Once the lines are drawn, their lengths are measured and are used in a weighted average estimation of MAP. Gauges lying outside the region can be used for the line method. The MAP is estimated using Eq. (39.5) and the variable Li,j in Eq. (39.5) is the length of line from point i to point j (i.e., distance between gauges).

The inverse centroidal distance method belongs to a class of weighting techniques for estimation of mean precipitation over an area. Initially, the centroid of the basin or watershed or any region of interest is found. Distances of the rain gauges from the centroid are calculated and the inverses of these distances (dc−, jf ) raised to an exponent (f ) are used as weights (wc,j) in a distance weighted method along with precipitation measures at these gauges. The MAP is estimated using Eqs. (39.3) and (39.4) and is illustrated in Fig. 39.2. ng





(39.3) θ A = ∑wc , jθ j                j =1

ng   wc , j = dc−, jf ∑dc−, jf   j=1 

∑ i=1∑ j=1Li , j  (0.5 *[θi + θ j ]) (39.5) ng ng ∑ i=1∑ j=1Li , j ng

39.2.4  Inverse Centroidal Distance Method

θA =

ng

39.2.8  Triangle Method

The triangle method, as the name suggests, involves creation of triangles using the rain gauges (or control points) as vertices. The triangle method is discussed in detail by Sumner (1988) and Shuttleworth (2012). The method involves development of triangles connecting any three points and making sure that they are equilateral as possible. Once triangles are developed the area covered by each triangle is calculated and the average of

−1

(39.4)

2

39.2.5  Multiquadratic Surface Method

The multiquadratic method (Shaw and Lynn, 1972; Shaw et al., 2011) uses a three-dimensional description of the rainfall surface. The coefficients used to define the surface are estimated using available equations using different values of precipitation amounts at different locations in space. Once the coefficients are estimated, the equation can be used to obtain volume of rainfall by integration over the area of the catchment or basin and finally dividing the volume of rainfall by catchment area. More details of this method can be found in work by Shaw et al. (2011).

3 5

1

39.2.6  Percent Normal method

The percent normal method (WMO, 1994) is beneficial in regions where the physiography influences the spatial variability of rainfall at different temporal resolutions. If average precipitation for a storm event over a region is required, then the storm precipitation is expressed as a percentage of mean annual or mean seasonal precipitation, and iso-percental maps are used for preparing isohyetal maps (WMO, 1994).

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4

Figure 39.3  Illustration of line method for estimation of MAP.

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Estimates of Mean Areal Precipitation     39-3 

precipitation value for each triangle is obtained using the average of the three gauges. The MAP is calculated by obtaining the weighted average using the product of mean precipitation value and the area of each triangle and divide the product by total area from all the triangles. Sumner (1988) suggests that gauges that are outside the region of interest can also be used in this method. 39.2.9  Two-Axis and Hyposometric Methods

The two-axis method, often referred to as Bethlahmy’s two-axis method, is another way of obtaining mean annual precipitation for a region. The method is developed by and described in detail by Dingman (2011) and WMO (1994). The hyposometric method is a surface fitting method and is particularly suitable for mountainous regions and requires four curves, and they are: (1) area elevation; (2) station precipitation; (3) station elevation; and (4) area precipitation. The procedures for construction of curves, use of these curves, and estimation of MAP are available from WMO (1994). 39.2.10  Kriging Method

Variance-dependent stochastic interpolation techniques based on geostatistical approaches (Isaaks and Srivastava, 1989), such as kriging are also often used for estimation of MAP. Cokriging of radar and rain gauge data has been employed by Krajewski (1987) to estimate MAP. Seo et al. (1990a; 1990b) and Seo (1996) described the use of cokriging and indicator kriging for interpolating rainfall data. Chua and Bras (1982) report an application of kriging to estimate MAP for mountainous regions considering drift and spatial dependency of precipitation influenced by orographic effects. Goovaerts (2000) report the use of simple kriging with varying local means, kriging with an external drift; and colocated kriging for incorporating a digital elevation model into the spatial prediction of rainfall. Phillips et al. (1992) evaluated the performance of three geostatistical methods for obtaining mean annual precipitation estimates on a regular grid of points in mountainous terrain. The methods evaluated include: (1) kriging; (2) kriging elevation-detrended data; and (3) cokriging with elevation as an auxiliary variable. The last two methods provided better estimates compared to kriging. Rogelis and Werner (2013) investigated the use of ordinary kriging (OK), universal kriging, and kriging with external drift with individual and pooled variograms for realtime rainfall field estimation in areas with complex topography. In their study, they found that interpolators using pooled variograms provided results comparable to those obtained when the interpolators were applied to the storms individually. 39.2.11  Other MAP Estimation Methods

This section provides a brief survey of MAP estimation methods and their applications that consider and use basin-specific characteristics, physiographic features, and orographic effects, and merged data from different measurement and estimation sources of precipitation (i.e., multisensor estimates). Objective analysis (OA) proposed by Gandin (1965) can be used for interpolation of precipitation data for MAP. OA uses an optimal interpolation analysis scheme to obtain best possible estimate of a meteorological field (e.g., precipitation in this context) at a regular network of grid points using observations at sites available in space by minimizing the mean square interpolation error for a large ensemble of analysis situations (Bergman, 1978). A few other methods listed by Dingman (2011), such as polynomial, Lagrange polynomial and spline surface, optimal interpolation via kriging, and empirical orthogonal functions (EOFs) can also be used for estimation of MAP. Application of bicubic spline to estimate MAP was discussed by Shaw and Lynn (1972). Finite element method with altitude corrections and approach using product of distances enclosing or radiating from a rain gauge as weights was developed and evaluated for MAP estimation by Hutchinson and Walley (1972) and Lal and Al-Mashidani (1978), respectively. Sugawara (1992) discusses the limitations of Thiessen polygon-based weights and reports a method wherein weights are calculated based on observed discharge. Variants of regression models (Daly et al., 1994, 2002) that use elevation, topography and proximity to coastal area, and distances as independent variables are also available for estimation of MAP. Descriptions and application of other methods suitable for desert environments and varying topography can be found in works of Sen and Eljadid (2000) and Dawdy and Langbein (1960), respectively. Sen (1988) and Akin (1971) provided different weighting methods for estimation of MAP using polygons defined by rain gauge networks. Trend surface analyses using linear, quadratic, and cubic functions for estimation of MAP were reported by Mandeville and Rodda (1973). They conclude that trend surface methods provide better estimates of MAP compared to those from Thiessen polygon or isohyetal methods. Abtew et al. (1993) evaluated Thiessen polygon, inverse-distance, multiquadratic, polynomial, optimal and point interpolation, ordinary and universal kriging for point, and areal

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estimation of rainfall in South Florida. They conclude that multiquadric, kriging, and optimal interpolation are best three methods among those evaluated. Inverse-distance weighting method with optimal parameters (exponent and search radius) was used to estimation of spatial rainfall distribution by Chen and Liu (2012). A similar exercise was reported by Noori et al. (2014) for spatial interpolation of rainfall. Mair and Fares (2011) used Thiessen polygon, inverse distance weighting (IDW), linear regression, OK, and simple kriging with varying local means to estimate MAP in mountainous regions. Use of artificial neural networks (ANNs) for evaluation of spatial distribution of precipitation was reported by Tsangaratos et al. (2014) and for MAP by Bryan and Adams (2002). Gridded precipitation data developed using rain gauge data (referred to MAPO) with the help of spatial interpolation methods and radar-based data (referred to as MAPX) are used by the National Weather Service (NWS) of the United States (NOAA, 2015) for estimation of MAP. The reader is advised to refer to comprehensive reviews of methods for estimation of MAP in mountainous and non-mountainous provided by Anderson (2002) and Jones (1983). Lebel et al. (1987) used scaled estimation error variance computed from a scaled climatological variogram model of the rainfall field as a criterion to evaluation three MAP estimation methods. Distancebased methods are not only used for estimation of MAP, but also for understanding spatial variability of precipitation extremes (Teegavarapu et al., 2013). Radar- (Raghavan, 2003; Sene, 2010; Teegavarapu 2013b, 2015a) and satellite-based [e.g., Tropical Rainfall Monitoring Mission (TRMM), Teegavarapu, 2013b] and Global Precipitation Measurement (GPM), Sene (2010) precipitation data available at a prespecified spatial resolution are now available for calculation of MAP. Details of radar-based precipitation estimation and bias correction procedures are discussed exhaustively by Teegavarapu (2013b) and Sene (2010). Transformations of precipitation data from one grid size to another using spatial interpolation methods (Teegavarapu et al., 2012a) are essential in many situations to estimate MAP. Grimes et al. (1999) estimated MAP using optimal merging of the estimates provided by satellite information and estimates obtained from rain gauges. Sokol (2003) reported the use of rain gauge and bias-corrected radar-based precipitation estimates for estimation of MAP. Several other spatial interpolation methods such as natural neighbors, nearest neighbors, triangular irregular networks (TINs), trend surface models, and thin plate splines (TPS) can be used for estimation of MAP. An exhaustive review of several interpolation methods along with discussions about the assumptions need, complexities encountered, and computational requirements for accurate surface generation is provided by Burrough and McDonnell (1998) and Li and Heap (2008). Table 39.1 provides main features and limitations of a few select MAP estimation methods. The methods and references listed and the features and limitations documented should not be considered complete. 39.2.12  Summary of Issues for MAP Estimation

Advancements in radar and satellite-based meteorological measurements (Sene, 2010) provide two alternative sources to rain gauge-based precipitation measurements for accurate estimation of MAP. Radar-based precipitation estimates, even while becoming more reliable and accurate, require appropriate bias corrections before use (Teegavarapu et al., 2015a). Moulin et al. (2009) indicate that major source of input uncertainty in hydrologic models comes from the lack of representativeness of a discrete set of gauges of a network and from the necessity to interpolate the rain rates between these points. A number of factors need to be considered when selecting suitable methods for MAP estimates and they include: (1) rain gauge density; (2) spatial location of the gauges including inter rain gauge distances; (3) availability of optimal network of gauges for characterizing spatial and temporal variability of precipitation in a region; (4) topographic and physiographic variations of the region including existence of separable homogeneous rain areas; (5) utility of rain gauge(s) that are outside of the domain of interest; (6) long-term temporal rain gauge-based datasets; (7) accuracy of rainfall measurements from rain gauges considering random and systematic errors; (8) availability of radar or satellite-based precipitation datasets at a reasonable spatial resolution (i.e., tessellation) considering the watershed size; (9) availability of bias corrected radar and satellite-based precipitation data; (10) validated and quality assured and quality controlled (QAQC) rainfall data that is error, anomaly, and gap free; and (11) the spatial interpolation method used to generate rainfall fields or surfaces. While radar and satellite estimated precipitation data are ideally suited for MAP estimation, ground-based reliable rain gauge network of sufficient density and with spatially uniform placement of gauges in a region is critical as the data from rain gauges are used for bias corrections of radar (Teegavarapu et al., 2015a) and satellite data. Exhaustive comparison of different methods to select the best method for estimation of MAP is recommended in several studies (e.g., Tabios and Salas, 1985; Singh and Chowdhury, 1986; Teegavarapu, 2012). Improved monitoring network designs based on

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39-4     Spatial and Temporal Estimation and Analysis of Precipitation

Table 39.1  Main Features and Limitations of Different MAP Estimation Methods MAP estimation method

Features of the method

Limitations/issues

Reference

Arithmetic average

Conceptually simple

No consideration for topography or other features

McCuen (2004)

Bicubic spline

Objective and simple method

Computationally intensive

Shaw and Lynn (1972)

Centroidal distance and variants

Simple, objective

Rain gauge network dependent

Shuttleworth (2012)

EOF

Conceptually sound

Computationally intensive

Dingman (2011)

Finite element

Conceptually sound

Computationally intensive

Hutchinson and Walley (1972)

Gridded precipitation (radar)

Objective, dependent on good quality data

Needs bias corrected data, good spatial resolution required

Sokol (2013), Teegavarapu (2013b), NOAA (2015)

Gridded precipitation (rain gauge)

Spatially interpolated data

Underestimation of higher-end extremes

Teegavarapu et al. (2013), Goly and Teegavarapu (2014), NOAA (2015)

Hyposometric

Conceptually sound

Inverse-distance weighting

Simple and objective

No consideration for topography or other features

Chen and Liu (2012), Mair and Fares (2011)

Isohyetal

Objective and simple method

Requires meteorological expertise in creation of isohyets

Linsley et al. (1949), France (1985)

Kriging

Most widely used, objective, provides uncertainty in estimates

Selection of variogram, anisotropy issues

Chua and Bras (1982), Krajewski (1987), Goovaerts (2000), Rogelis and Werner (2013)

Line

Simple and objective

No consideration for topography or other features

Goel and Aldabagh (1979)

Multiquadratic surface

Surface generation method

Requires good rain gage network density

Shaw and Lynn (1972), Shaw et al. (2011)

Percent normal

Simple and objective

Needs long-term data

WMO (1994)

Polygon weighting

Simple and objective

No consideration for topography or other features

Akin (1971), Sen (1988)

Regression

Fixed functional form

Rigid functional form, validation of assumptions about residuals

Daly et al. (1994, 2002)

Satellite data

Spatial variation is captured well

Needs bias corrected data, good spatial resolution required

Grimes et al. (1999) Ding man (2011)

Thiessen polygon

Objective and widely used

Rain gauge network dependent, no physiographic information considered

Thiessen (1911), Brassel and Reif (1979)

Triangle

Simple and objective

No consideration for topography or other features

Sumner (1988) and Shuttleworth (2012)

Two axis

Conceptually simple

No consideration for topography or other features

WMO (1994)

WMO (1994), Dingman (2011)

geostatistical approaches (Teegavarapu et al., 2015b) information-theoretic methods (Xing et al., 2013), entropy and multiobjective-based techniques (Mogheir et al., 2013) and fuzzy theory and multiple criteria analysis methods (Chang and Lin, 2014) can help in the MAP estimation. Ly et al. (2013) provided a survey commonly used spatial interpolation methods for areal rainfall estimation and discuss advantages and limitations of different methods. A summary of conceptually simple techniques used methods for MAP estimation was provided by Rainbird (1967). 39.3  MISSING PRECIPITATION DATA ESTIMATION METHODS

Existence of serially complete, continuous precipitation data from rain gauge measurements is practically impossible due to a number of reasons. Instrument malfunctions, systematic and random errors, transcription errors, and extreme weather events sometimes disturb the measurement process and also introduce gaps in precipitation measurements. Understanding the mechanisms by which missing data occur is crucial for development of missing data estimation methods. A few mechanisms are defined and they are missing at random (MAR), missing completely at random (MCAR), and missing not at random (MNAR) by Little and Rubin (2002). In many missing precipitation data-estimation studies, gaps can be attributed to be as MCAR suggesting that number and temporal occurrence of gaps (missing) in precipitation data a site (i.e., rain gauge) are not dependent on the data at the site or any other sites. Estimation of missing precipitation data can be carried using temporal and spatial interpolation methods. The former relies on the persistence with strong serial autocorrelation and the latter depends on reliable time consistent observations at different rain gauges. Chronological pairing (CP) of data is needed to estimate missing values using spatial interpolation. At low temporal resolutions (e.g., 15 min or 1 h), rainfall observations often show strong persistence allowing for infilling a missing observation in a time interval using observation from the immediate previous time interval (referred to as temporal interpolation). This way of infilling

39_Singh_ch39_p39.1-39.10.indd 4

the data is referred to also as last observation carry forward (LOCF) procedure (Buuren, 2012). A high serial autocorrelation value at lag one is essential for success of this method. If several missing data exist in a series of continuous time intervals, a constant or a mean value can be used as values in these time intervals. This process is referred as the baseline observation carry forward (BOCF) approach (Buuren, 2012). BOCF method is also referred to as single imputation. Serially complete precipitation data without any gaps is needed for climate variability and change studies (Teegavarapu, 2013a, 2013b) and therefore spatial or temporal interpolation becomes essential to fill the gaps. 39.3.1  Conceptually Simple or Naïve Estimation Methods

These methods are conceptually simple and use rainfall information in space and time from nearby rain gauges to estimate missing data at base station (i.e., site or gauge at which missing data exist). Arithmetic average referred to as gauge mean estimator (GME), single best estimator (SBE), and climatological mean estimator (CME) use an average value of observations from the nearby rain gauges, observations from one rain gauge, and mean of historical observations available for that specific temporal slice at the rain gauge which is known to be having the missing data, respectively. The optimal number of rain gauges in GME and most representative rain gauge in SBE can be obtained by optimization or by use of any proximity metric that is Euclidean or predefined distance metric or using correlation between base station and any other station. A special case of using only three nearby gauges for estimation was referred to as three-station average method by Paulhus and Kohler (1952). These methods are often used as benchmark methods against whom more advanced spatial interpolation methods are compared and evaluated. 39.3.2  Weighting Methods

Traditional weighting and data-driven methods generally are used for estimating missing precipitation. Inverse distance (Wei and McGuinness, 1973; Simanton and Osborn, 1980) is one such method that is widely used in

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New Methods for Missing Data Estimation      39-5 

hydrological applications. The inverse distance weighting method (IDWM) is the most commonly used approach for estimating missing data in the hydrological and geographical sciences. In the United States especially in the operational hydrology literature, IDWM is often referred to as the NWS method. The IDWM is routinely used for estimating missing rainfall data (ASCE, 1996). In the field of quantitative geography, IDWM and inverse exponential weighting methods are commonly used for spatial interpolation (O’Sullivan and Unwin, 2003). Several variants of IDWM have been developed. Hodgson (1989) modified IDWM to include a learned search approach that reduces the number of distance calculations. To incorporate topographical aspects, Shepard (1968) proposed a modified IDWM that is referred to as a barrier method. The quadrant method (McCuen, 2004) uses the nearest gauge to the base gauge in each quadrant for estimation of missing precipitation data in weighting scheme similar to IDWM. Nearest gauge in each quadrant is defined by Euclidean distance. Normal ratio method (Paulhus and Kohler, 1952; Tung, 1983; McCuen, 2004; Teegavarapu and Chandramouli, 2005) is another conceptually simple method for estimation of missing data that relies on the assumption related to average annual precipitation values at sites. 39.3.3  Surface Interpolation Methods

Variance-dependent surface interpolation methods, belonging to the general family of kriging, have been applied to hydrological interpolation problems (Grayson and Bloschl, 2001; Vieux, 2001). These interpolation schemes are based on the principle of minimizing the estimate of variance at points where measurements are unavailable. Kriging in various forms has been used to estimate missing precipitation data at stations (Teegavarapu and Chandramouli, 2005) as well as to interpolate precipitation from point measurements (Vieux, 2001; Dingman, 2011). Ashraf et al. (1997) compared interpolation methods (kriging, inverse distance, and cokriging) to estimate missing precipitation values. They indicated that kriging provided the smallest root mean square error (RMSE). Application of kriging for areal mapping of precipitation was reported by Karnieli (1990). Regression and time series analysis methods (Salas, 1993) belong to data-driven approaches. Global interpolation methods that use trend surface analysis and regression (Chang, 2006) provide several advantages compared to deterministic weighting techniques. Trend surface methods use polynomial equations of spatial coordinates to approximate points with known values. Regression models using elevation information (Lloyd, 2009), locally weighted polynomials (Regonda et al., 2006), thin splines (Xia et al., 1999), and local and global trend surface models (Teegavarapu, 2012) are available for estimation of missing precipitation data. Guillermo et al. (1985) evaluated polynomial interpolation by leastsquares and Lagrange approach, inverse distance, multiquadric interpolation, optimal interpolation, and kriging techniques for estimation of missing data. They indicated kriging and optimal interpolation techniques are superior to the other techniques. 39.4  LIMITATIONS OF ESTIMATION METHODS

Teegavarapu and Chandramouli (2005) reported several limitations and advantages of using deterministic and stochastic spatial interpolation techniques to estimate missing precipitation data at a base station using data at all other stations. They indicated that all interpolation techniques will fail to provide accurate estimates of missing precipitation data in two situations: (1) when precipitation is measured at all or a few other stations, but no precipitation actually occurred at the base station, and (2) when precipitation is measured at the base station but no precipitation is measured or occurred at all other stations. In case 1, all spatial interpolation techniques will produce a positive estimate whereas in reality zero precipitation is recorded at the base station. It is impossible to estimate missing precipitation data in the second case since point observations are used to estimate the missing value at the base station using spatial interpolation algorithms alone. All interpolation techniques produce a zero estimate for situations encountered in case (2). Data from other sources (e.g., radar or satellite-based estimates) can be used in these situations to estimate the missing values. However, the reliability of radar-based precipitation measurements is always a contentious issue (Teegavarapu, 2008) considering several radar-based estimation errors (Sene, 2010). Limitations of spatial interpolation methods have been reported in recent studies. Vieux (2001) pointed out several limitations of the IDWM, with a major one being the “tent pole effect” that leads to greater estimates closer to the point of interest. Grayson and Bloschl (2000) list several limitations of Thiessen polygons and IDWMs. They have suggested that these methods should not be recommended for spatial interpolation considering their limitations. However, they recommend thin splines and kriging for interpolating hydrologic variables. The Thiessen polygon approach has the major limitation of not providing a continuous field of estimates when used

39_Singh_ch39_p39.1-39.10.indd 5

for spatial interpolation (O’Unwin and Sullivan, 2003). Brimicombe (2003) indicated that the main point of contention in applying IDWM to spatial interpolation is selecting the number of relevant observation points used for the spatial interpolation. Many spatial interpolation methods for missing data estimation lack: (1) a procedure for objective selection of optimal number of neighbors (or gauges) and neighborhood size (in reference to local or global interpolation); (2) optimal weights in distance or correlation based or other weighting schemes; (3) the ability to preserve site-specific statistics of precipitation data; (4) ability to provide uncertainty in the estimates; (5) mechanism to define rain or no-rain conditions at the site of interest or correct the estimates based on these conditions; (6) ability to preserve all the precipitation characteristics [e.g., dry and wet spells and extremes, transitions (e.g., wet to dry), autocorrelation, and several others]; and (7) the ability to preserve spatial variance and regional statistics. 39.5  NEW METHODS FOR MISSING DATA ESTIMATION

Distance weighting methods often used for estimating missing rainfall records are revisited by Teegavarapu and Chandramouli (2005). Conceptual revisions are incorporated into their methods and the revised methods are tested for estimation of missing rainfall values. The revisions addressed two main issues relating to the definition of distance used in the calculations and selection process of the nearby gauges. 39.5.1  Variants of Weighting, Data-Driven, and Copula-Based Methods

Teegavarapu and Chandramouli (2005) developed and evaluated several variants of inverse distance method and data-driven methods for estimation of missing precipitation records at a site. These methods include coefficient of correlation weighing method (CCWM), artificial neural network estimation method, modified Thiessen polygon method, nearest neighbor weighting method, and its revised version and kriging estimation method (KEM). All the methods are data intensive and sensitive, with KEM being computationally most intensive of all the methods. Use of conceptually acceptable surrogate measures for distances and improvised weighting factors improved the estimates from these variants. CCWM has proven to be a better weighting method compared to other interpolation methods in several studies (e.g., Kim et al., 2008; Westerberg et al., 2009). The kriging method is considered a reliable interpolation technique (O’Sullivan and Unwin, 2003), but is plagued by several limitations. These include selecting the appropriate semivariogram, assignment of arbitrary values to sill and nugget parameters and distance intervals, observation value-insensitive variance estimates, and the computational burden to interpolate the surfaces. Universal functional approximators such as ANNs are used for fitting a semivariogram model using the raw data in OK to estimate missing precipitation data by Teegavarapu (2007). The use of ANN eliminates the need for the predefined authorized semivariogram models to capture the spatial variation of data, and the trial-and-error process involved in estimation of semivariogram parameters. Use of modular neural networks for estimation of missing precipitation is reported by Kajornrit et al. (2012). Kuligowski and Barros (1998) report improved estimates of missing data using ANN compared to those from distance-weighted and linear regression models at a number of rain gauge sites. Similar conclusion about improvements in estimates was reported by Teegavarapu and Chandramouli (2005). Association rule mining (ARM)-based spatial interpolation (Teegavarapu, 2009) is used to improve the precipitation estimates provided by traditional and improved deterministic and stochastic spatial interpolation techniques. The ARM methodology besides offering insights into the spatiotemporal precipitation data patterns and the associations among observations, also helped in addressing one major ubiquitous limitation of all spatial interpolation techniques in accurately estimating missing precipitation records. Considerable improvements in the estimates were achieved when ARM was used in conjunction with the interpolation techniques. Copulabased methods for infilling are found to be superior compared to a number of interpolation methods [nearest neighbor, linear regression, fuzzy rules, OK, multiple linear regression (MLR), and MLR using the expectation maximization (EM) algorithm] in a study by Bardossy and Pegram (2014). Landot et al. (2008) used a copula-based method for infilling long-term gaps in precipitation data in Florida with the help of support vector machine (SVM). However, the copula-based method reported by Landot et al. (2008) was found to be inferior to CCWM and linear optimization method in a study by Aly et al. (2009). Presti et al. (2010) investigated simple substitution, parametric regression, ranked regression, and rank-invariant method of linear and polynomial regression analysis (Theil method). They concluded that Theil method and simple substitution method provided better results among all those compared.

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39-6     Spatial and Temporal Estimation and Analysis of Precipitation 39.5.2 

Optimal Spatial Interpolation Methods

Range and cluster-based optimization methods in space and time were developed by Teegavarapu (2012). Optimization models using mixed integer linear and nonlinear programming formulations are developed for estimation of missing precipitation data at a gauge. The formulations use binary variables for selection of rain gauges that participate in the spatial interpolation process and also in the process of selection of the optimum cluster of rain gauges for estimation of missing data. Several variants of these mathematical programming formulations are proposed by Teegavarapu (2012) to improve the estimates of missing precipitation data. The variants include: (1) IDWM with optimal number of gauges; (2) optimal quadrant method; (3) IDWM with optimal parameters and number of gauges; and (4) linear weighting method with optimal clusters of rain gauges. These variants investigated in their study provided improved estimates of precipitation data compared to those obtained from different traditional distance-based weighting and kriging methods. A nonlinear mathematical programming model using binary variables is proposed and investigated by Teegavarapu and Pathak (2008) and Teegavarapu et al. (2010) to infill missing precipitation records using radarbased rainfall estimates. The model identifies the cluster of radar values that can be used for infilling the rain gauge records. The application of the model also revealed several interesting insights, which suggest that optimal selection of radar-based precipitation values using weights in the grid surrounding a rain gauge is essential for the infilling process. Spatial and temporal variability of weights for different clusters are evident. Linear programming method using topographical information was discussed by Yoo (2013). 39.5.3  Nearest-Neighbor and Proximity Metric-Based Methods

Similarity of two observation datasets based on real or categorical attribute values can be evaluated using proximity metrics based on numerical taxonomy concepts (Ling, 2010). These metrics provide a measure of proximity between any two observed datasets and, in turn, can be used to assign weights in the interpolation schemes to estimate missing data. Ten different binary and real-valued distance metrics were used as proximity measures by Teegavarapu (2014b) to obtain optimal weights in spatial interpolation methods for the estimation of missing data. In a recent study, Pappas et al. (2014) provided a conceptually simple method using nearest neighbors (observations in time at one site) for estimation of missing data. 39.5.4  Corrections of Spatially Interpolated Estimates

Corrections of spatial interpolation-based precipitation estimates considering rain or no-rain conditions at a nearest rain gauge selected by a proximity metric or correlation can be used. Corrections are suggested by Teegavarapu (2009) using data mining-based association rules. Improved estimates can be obtained compared to use of methods with no corrections. However, overestimation of precipitation magnitudes in many instances is common. The SVM-based approach was used to select rain gauges for correction by Aly et al. (2009). The single best estimator (SBE) method and its optimal variant are developed by Teegavarapu (2011) and Teegavarapu (2014b), respectively, for postcorrections of missing data estimates at a gauge. 39.5.5  Site and Regional Statistics Preserving Methods

Preserving site-specific statistics along with accurate estimation of missing data is essential when data gaps at a gauge are filled. Postcorrection methods using quantile–quantile (Q–Q) and equiratio quantile matching and single best estimation methods were suggested by Teegavarapu (2014a). The methods preserved site statistics and also provide improved estimates of several WMObased extreme precipitation indices that are closer to observed series than those obtained without these corrections. Teegavarapu (2013b) and El-Sharif and Teegavarapu (2012) proposed and evaluated several multiobjective optimization formulations for preserving site-specific statistics and correlations among different sites and spatial variability of precipitation data evaluated using a variogram. 39.5.6  Evaluation Measures and Selection of Interpolation Methods

The spatial interpolation methods can be evaluated using error measures and performance measures such as: RMSE, mean absolute error MAE, and goodness-of-fit measure criterion, coefficient of correlation (ρ), or determination (R2), based on observed and estimated rainfall values at the base station. Factors such as intuitive reasonableness of model or approach, conceptual accuracy, and simplicity of the model are considered to make an objective assessment, and finally selection of the models or approaches used for estimation of missing data (Teegavarapu and Chandramouli, 2005).

39_Singh_ch39_p39.1-39.10.indd 6

User-specified weights can be attached to performance metrics or measures as numerical values (Teegavarapu, 2012) or fuzzy membership functionbased values (Teegavarapu and Elshorbagy, 2005) to quantify the importance of one measure over the other. In addition to these performance measures, a bias plot or analysis of bias is also recommended as a means of understanding the structure of errors, model-specific bias, and distribution of errors over a specific range of values or time. The methods can be qualified using the criteria of accuracy and precision, which are related to the bias and variance of the residuals (Kanevski and Maignan, 2004). Small variations in rainfall intensity can introduce significant changes in the runoff values generated from distributed rainfall-runoff (RR) models (Vieux, 2001). Any improvement in the rainfall magnitude estimation, however minute it may be, can be considered significant, as rainfall is a crucial input that governs the response of hydrologic systems and the results of continuous simulation models. Similar arguments were made by Xu and Vandewiele (1994) to suggest that errors in precipitation values may lead to significant effects on the model performance and also on parameters. Two error measures (RMSE and absolute error) are generally used to assess and compare the performance of interpolation techniques. However, these global error assessment measures may not provide a complete assessment of methods as they are average measures calculated for a specific period of time. The temporal distribution and magnitude of the precipitation affects the outputs of RR models, which use precipitation as an input. One of the best ways to evaluate spatial interpolation methods is to use the estimated precipitation values in a hydrologic simulation model and assess the impact of the major input (i.e., rainfall) on the estimation of one or several hydrological variables. A recent study by Teegavarapu (2014a) provides a number of performance measures, error measures, and statistical hypothesis test to evaluate different spatial interpolation methods. The measures included summary statistics, quantile–quantile (Q–Q) plots, statistical hypothesis tests (e.g., two sample Kolmogorov– Smirnov and Chi-square tests) to check the similarity of probability distributions, differences in two state first-order Markov chain transition probabilities for dry–wet, dry–dry, wet–dry, and wet–wet conditions, autocorrelations at several temporal lags and deviations in extreme precipitation indices (WMO, 2009; Teegavarapu et al., 2012b) derived based on observed and filled precipitation datasets. Contingency measures (i.e., concordance, error rate, sensitivity, and specificity) and several forecast-verification measures and skill scores are used by Teegavarapu (2014a) for comparison of different spatial interpolation methods. Finally, the homogeneity of the filled datasets should be tested using one of the tests: Pettitt’s (Pettitt, 1979), Buishand’s (Buishand, 1982), or Alexandersson’s standard normal homogeneity test (SNHT) (Alexandersson, 1986), or Von Neumann ratio test (Von Neumann, 1941). Table 39.2 provides a list of missing data estimation methods their main features and limitations. The list and discussion of features and limitations provided is not considered to be exhaustive. 39.6  SUMMARY OF ISSUES FOR MISSING PRECIPITATION DATA ESTIMATION

An exhaustive survey of the literature indicates several methods ranging from conceptually simple to computationally intensive are available that can be used for estimation of missing precipitation data. Selection and use of one or more methods for estimation is an exhaustive task and will depend on several constraints and is also dictated by numerous objectives related to any specific study. The focus, in general will be on the selection of those methods that are conceptually sound, field tested, and robust. The geographical region of study, ease of method development by modelers, transferability, and utility of a part or complete computational code from a similar model available elsewhere, practical applicability, data availability, and time constraints are a few important factors that are considered in the selection process. Also, the objectives and goals set forth by study and agency needs will primarily direct the identification and finally the selection process. Computationally efficient algorithms are required for implementation of improved estimation methods based on auxiliary data other than precipitation datasets. Surrogate measures of correlation among observations are required for applying weighting techniques. Use of correlation coefficients (Teegavarapu and Chandramouli, 2005), statistical distance (Ahrens, 2006), and proximity metrics (Teegavarapu, 2013) are few recommendations for development of conceptually simple weighting methods. Universal functional approximation methods, such as (ANNs), are viable techniques for estimation of missing precipitation. However, overfitting, lack of generalization, unrealistic numeric results, and replicability and repeatability are some of the main concerns of these techniques. Reliable data-driven, artificial intelligence, or statistical techniques are required for prediction of ‘‘rain’’ or ‘‘no-rain’’ conditions. These techniques precede the application of spatial interpolation techniques for estimating

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Conclusion    39-7 

Table 39.2  Main Features and Limitations of Different Missing Precipitation Data Estimation Methods Missing data-estimation method

Features of the method

Limitations/issues

Reference

ANN based

Data driven, function approximation

Generalization and issues with unrealistic estimates

Kuligowski and Barros (1998), Teegavarapu and Chandramouli (2005), Kajornrit et al. (2012)

ARM based

Data- and rule-driven method

Long-term data is required

Teegavarapu (2009)

Climatological mean estimator

Conceptually simple

Last resort method, not accurate

Aly et al. (2011)

Correlation coefficient weighting

Improvised inverse distance method

Data driven, needs long-term data

Teegavarapu and Chandramouli (2005), Kim et al. (2008), Westerberg et al. (2009)

CME

Conceptually simple, benchmark method

Underestimation of variance

Aly et al. (2011), Teegavarapu (2012)

Inverse-distance weighting

Conceptually simple, widely used

Network density dependent, no meteorological and topographical influences considered

Tung (1983), ASCE (1996)

K-means cluster-based optimization

Provides ability to select clusters in space

Subjective parameters (e.g., number of clusters)

Teegavarapu (2014b)

K-nearest neighbor-based optimization

Provides ability to select clusters in time

Subjective parameters (e.g., number of clusters)

Teegavarapu (2014b)

Kriging

Widely used, provides estimation uncertainty

Selection of variogram models, anisotropy considerations

Ashraf et al. (1997), Vieux (2001), Dingman (2011)

Local and global polynomial

Surface fitting

Variability of rainfall is not captured if rain gauge density is not sufficient

Regonda et al. (2006), Teegavarapu (2012)

Normal ratio

Simple, objective

Long-term data is required

Paulhus and Kohler (1952), Teegavarapu and McCuen (2004), Chandramouli (2005)

Optimal spatial interpolation and variants

Objective, efficient method

Needs optimization solvers, nonglobal optimal solutions

Yoo (2013), Teegavarapu (2014b)

Optimal spatial interpolation—neighbor selection

Objective, efficient method, provides optimal selection of neighbors

Computational issues with optimization solvers

Teegavarapu (2012), Teegavarapu (2014b)

Proximity-metric–based optimization

Objective method with weights estimated based on proximity metrics

Relies heavily on data length

Teegavarapu (2009)

Quadrant

Simple, objective

Proximity of gauges with similar data characteristics important, availability of long-term data

McCuen (2004)

SBE

Conceptually simple, benchmark method

Needs a highly correlated station

Teegavarapu (2009)

Spatial interpolation—with post corrections

Objective method, improvement over other methods in preserving site specific statistics

Bias may be introduced due to corrections

Teegavarapu (2014a)

Thin plate splines (TPS)

Assumption of smooth surface

Variability of rainfall not captured based on the assumption

Xia et al. (1999), Chang (2006)

Time series models

Autoregressive models, objective

Strong serial autocorrelation at few lags is needed

Dingman (2011)

Universal function approximation–based kriging

Modified version of kriging

Function approximation of semi variogram, generalization issues

Teegavarapu (2009)

missing precipitation records. Rain-radar relationships need to be characterized via functional forms to effectively utilize them in estimating missing precipitation. Optimum clustering of gauges for efficient and robust estimation of missing precipitation records can be identified. Mathematical programming formulations with binary variables can be used for this purpose for determination of optimal number of sites (i.e., neighborhood size) and clusters. Spatial and temporal variability of rainfall processes, and local and regional climatology are essential factors that need to be considered when estimating missing data at a site. In a few data-sparse situations, naïve methods described in this chapter are the only way out. In general, filling missing precipitation data more than 5% of the entire dataset leads to underestimation of higher-end extremes and an increase in bias in the estimation of precipitation extremes (Teegavarapu, 2014a). Multiple imputation (infilling) methods (Buuren, 2012) based on bootstrap sampling and optimal interpolation (Teegavarapu, 2013b) are useful in quantifying the uncertainties associated with estimates provided by spatial interpolation methods. Software focused on estimation of missing data is rarely generic. Variance deflation, over- and underestimation of lower- and higher-end extremes, respectively, and alteration of statistical distributions are inevitable consequences of spatial interpolation methods (Goly and Teegavarapu, 2014; Teegavarapu, 2014a) often used for generation of gridded datasets. Point and spatially complete gridded precipitation datasets are equally valuable for analysis of precipitation extremes and characteristics. Justifiable inferences about variability in precipitation characteristics and extremes can be drawn when both the datasets are used independently (Goly and Teegavarapu, 2014).

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A comprehensive survey of interpolation methods for environmental data applications is provided by Li and Heap (2008), and Akkala et al. (2010) and kriging-based methods is provided by USEPA (2004). Practical application of all spatial interpolation methods require time consistent precipitation measurements at all the rain gauges that are used for development of parameter values (e.g., weights, exponents, etc.) in these methods. If one or more gauges are missing data, they need to be filled first before interpolation can be used for estimation. The filling process can be carried out using a conceptually simple estimation method, temporal interpolation, or data from sources other than rain gauge if available. If one or more rain gauges have substantial amount of missing data, then parameters of the interpolation method need to be re-estimated using the available rain gauges with serially complete precipitation records. If the amount of data missing is negligible and missingness can be explained as MCAR, listwise and pairwise deletion of data methods common in social sciences arena (Enders, 2010) can be used. However, if missing data are not estimated or gaps in data are not filled, meaningful conclusions about the data cannot be made for any hydrologic or water resources modeling studies. 39.7 CONCLUSION

MAP and missing data estimation are the two topics of focus in this chapter. This chapter provides a brief summary of available methods. Different conceptually simple methods available for MAP along with computationally demanding approaches are discussed. Increasing availability of radar- and

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39-8     Spatial and Temporal Estimation and Analysis of Precipitation

satellite-based precipitation datasets is leading to improved estimates of MAP when compared with those obtained from rain gauges. Combined use of rain gauge and radar or satellite precipitation datasets can also lead to improvements in estimates. Availability of good quality and bias-corrected radar or satellite-based precipitation data is essential especially in those regions with very low rain gauge density. Missing precipitation data at a site or multiple sites is a ubiquitous problem that can be solved by using data and regionspecific traditional spatial interpolation, geostatistical, or surface generation methods. Recent developments in estimation of missing data methods have also been documented in this chapter. Methods that preserve site and regional summary statistics, probability distributions, reduce under- and overestimation of higher- and lower-end extremes, maintain various time series-specific properties, and correctly estimate the dry and wet states are required. Variants of spatial interpolation methods to incorporate corrections to spatially interpolated precipitation estimates, optimal neighborhood and number of sites selections, and local as opposed global interpolations are proven to have potential for improving missing precipitation data estimates. REFERENCES

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References    39-9 

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39-10     Spatial and Temporal Estimation and Analysis of Precipitation

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Wei, T. C. and J. L. McGuinness, “Reciprocal distance squared method, a Computer Technique for Estimating Area Precipitation, Technical Report ARS-Nc-8.” U.S. Agricultural Research Service, North Central Region, OH, 1973. Westerberg, I., A. Walther, J-L. Guerrero, Z. Coello, S. Halldin, C-Y. Xu, D. Chen, et al., “Precipitation data in a mountainous catchment in Honduras: quality assessment and spatiotemporal characteristics,” Theoretical and Applied Climatology, 101 (3–4): 381–296, 2009. WMO, World Meteorological Organization: Guide to Meteorological Instruments and Methods of Observation, 6th edition, Part I, WMO-No. 8, Geneva, Switzerland, Chap. 6, 1983. WMO, Guide to Hydrological Practices, WMO-No. 168, World Meteorological Organization, Geneva, Switzerland, 1994. WMO, Guidelines on Analysis of Extremes in a Changing Climate in Support of Informed Decisions for Adaptation, World Meteorological Organization, Geneva, Switzerland, 2009. Xia, Y., P. Fabian, A. Stohl, and M. Winterhalter, “Forest climatology: estimation of missing values for Bavaria, Germany,” Agricultural and Forest Meteorology, 96: 131–144, 1999. Xing, T., Z. Xuesong, and J. Taylor, “Designing heterogeneous sensor networks for estimating and predicting path travel time dynamics: an information-theoretic modeling approach,” Transportation Research Part B: Methodological, 57: 66–90, 2013. Xu, C-Y. and G. L. Vandewiele, “Sensitivity of monthly rainfall runoff models to input errors and data length,” Hydrological Sciences Journal, 39 (2): 157–176, 1994. Yoo, J., “Linear programming method considering topographical factors used for estimating missing precipitation,” Journal of Hydrologic Engineering, 18 (5): 542–551, 2013.

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Chapter

40

Snow Distribution and Snowpack Characteristics BY

MING-KO WOO AND PHILIP MARSH

ABSTRACT

In cold regions, snow is not only hydrologically important, but it also exerts influences on ecosystems and human activities. The distribution of snow is controlled by the location and amount of snowfall, redistribution after deposition, and losses through sublimation and melt. Snowcover exhibits variations at several spatial scales, from patterns that vary globally and regionally to unevenness at local and micro-scales. Once snowfall is deposited on the ground, snowpack properties undergo modifications through snow metamorphism that alters the ice crystals, water content, and density of the pack. Snow properties affect the movement of liquid water through the snowpack. Of equal importance are the optical and thermal properties of the snow; the former influences the albedo therefore the energy balance and the latter affects insulation of the underlying ground. Both the distribution and characteristics of snow undergo changes during the cold season. Snow condition in the melt period plays a major role in runoff generation. 40.1 INTRODUCTION

Snow is hydrologically and climatically important in cold regions, and it also influences terrestrial ecosystems and human activities. Physical properties of snow such as its reflectivity or albedo, and its low temperatures and insulating capabilities, have feedback on the climate. The amount of snow and the pattern of the snowcover, as well as the internal condition of the snowpack, are important considerations for melt and for runoff generation that delivers water for use but creates flood hazards. 40.2  PROCESSES CONTROLLING SNOW DISTRIBUTION 40.2.1 Snowfall

Latitude, altitude, and proximity to large water bodies affect the amount of precipitation. For winter precipitation, atmospheric moisture content and the tracks of storms are also major considerations. The Arctic, with a relatively dry atmosphere, has lower precipitation than the subarctic and temperate regions where active winter cyclones bring about frequent and intense precipitation events. In addition to latitudinal differences in precipitation, topography gives rise to notable contrasts in snowfall pattern. As onshore wind encounters a topographic barrier such as a mountain range, the moistureladen air is forced to rise and cooling of the rising air yields high orographic precipitation on the windward slope, usually increasing with elevation. The western side of Norwegian mountains, for example, receives 2000 to 3000 mm of precipitation from the onshore westerlies, whereas on the leeward side such as around Oslo, annual precipitation drops to around 1000 mm (Killingtveit, 1994). Another situation is where winter air flows over a large body of open water, such as the Great Lakes of North America. Moisture is picked up by the air during its passage over the water body, but as the air encounters the cold land downwind of the water body, the air is chilled and its moisture-holding

capacity is reduced to precipitate snow. This gives rise to lake-effect snowfall. The same effect can develop when warm air flows over an open sea, and deposits snow on the cold land downwind. In cold regions including the cold temperate zone, the Arctic and the Antarctic, snowfall tends to be larger in areas of high precipitation. Partition of precipitation into rainfall and snowfall is predicated largely upon air temperature, but both forms of precipitation can coexist around 0°C. In general, a larger portion of annual precipitation falls as snow at high latitudes and at high altitudes compared with the lower latitudes and lower elevations. Examples of mean monthly precipitation at low-elevation stations located in different climatic regions of Canada demonstrate the effects of latitude and continentality on the proportion of snowfall relative to rainfall (Fig. 42.1). Arctic regions where the winters are long (lasting for over 7 months or longer) but where precipitation is often of low intensity, the annual snowfall is small. Coastal belts of the temperate region, such as Chile, Norway, and coastal mountains of western North America, snowfall is much higher than the continental interior, such as the prairies. Large snowfall nourishes mountain glaciers and ice caps, like those of British Columbia and Alaska (Meier, 1990). 40.2.2  Snow Redistribution

Once fallen, some of the snow undergoes redistribution. In vegetated areas, precipitation is intercepted by plants. The stand closure of the vegetation plays a role as it limits the amount of snow that can reach the ground as throughfall (Pomeroy and Gray, 1995). Coniferous and deciduous trees before their leaves have fallen are notably effective in intercepting snowfall. For the latter, the weight of snow can break the leaf-clad branches, but the branches of the conifers are more pliant and can bend without breaking under heavy snow. In open areas including the tundra, grasslands, and open fields, wind is the principal agent that redistributes snow. Topography and vegetation play important roles (Essery and Pomeroy, 2004; Sturm et al., 2001). Snow is lifted from its place of rest and entrained with the wind flow. The snow moves by creeping close to the surface, by saltation in which snow particles skip along the surface, or by suspension in the air. These processes transfer the snow from exposed sites to locations sheltered from the wind. Large amounts of snow accumulated on upper hillslopes are subject to avalanches in which the snow mass moves rapidly down the slopes (Schaerer, 1981). When the snow is saturated with rain-water or meltwater, the mixture of snow and water, known as slush, can flow quickly down the slopes or along valleys (Gude and Scherer, 1995). Sublimation from the snowpack occurs when ice crystals are converted to water vapor, which is then lost to the atmosphere. This process is favored by an atmosphere of low humidity and by high wind speed, and is further enhanced during blowing snow events. In vegetated areas, the snow intercepted by leaves and branches is also subject to sublimation. An experiment that continuously weighed a cut tree in the boreal forest, with intercepted snow on this tree, showed that about one-third of the intercepted snow could be sublimated in the winter (Pomeroy et al., 1998). 40-1

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40-2     Snow Distribution and Snowpack Characteristics

Figure 40.1  Partitioning of mean monthly precipitation into rainfall and snowfall for stations representative of different geographical settings in Canada.

40.3  SPATIAL PATTERNS OF SNOW AT VARIOUS SCALES

The distribution of snow on the ground is highly uneven and the unevenness increases as the melt season progresses. Depending on the duration that the snow occupies the ground, the snowcover may be considered as ephemeral, persistent, or prolonged. Areas with ephemeral snowcover, including the warm and cool temperate zones, are subject to frequent above-freezing winter temperatures, or have multiple episodes of rain events, both of which lead to the removal of snow only days or weeks after its deposition. Most cold temperate regions have a persistent snowcover that lasts for several months. At higher latitudes, the snowcover is prolonged for over 6 months every year and some snow has acquired semipermanent status or is incorporated into glaciers and ice caps. Within a climatic region, spatial variation in snow pattern reflects the influence of storm tracks, topography, and proximity to water bodies. Windward and leeward slopes have different amount of snow. Differential snowmelt rates further the variation in snowcover on slopes of different elevations and orientations due to temperature, radiation, and wind. In extensive open areas with sparse vegetation, such as the Arctic and open fields of the prairies and steppes, terrain enables local acceleration or reduction in wind flow, leading to the development of typical snow patterns indicative of snow scour and deposition (Rees et al., 2013). On a local scale (102–103m in length scale), minor snow features in the forms of ripples, dunes, and sastrugi are commonly superimposed on the broad snowfields. One special snow phenomenon is created by lake-effect snowfall in the Great Lakes area where the wind picks up moisture from the open water surface of the lakes and deposits snow on the downwind sides of the lakes (Phillips and McCulloch, 1972). As the winds are predominantly from the west during the winter, repeated snowfall events accumulate much snow to form snowbelts with snowcovers that last longer than their nearby areas. A difference in vegetation affects the snowcover distribution. Marsh et al. (2003), for example, found that end-of-winter snow water equivalent (SWE) near the treeline in the Mackenzie Valley, NWT, was 98 mm on the tundra, 141 mm for the shrubs, and 155 mm for the woodland. In a forest environ-

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ment, interception and sublimation loss leave the forest floors with less snow than open areas. The amount of snow on the forest floor is uneven due to throughfall, sublimation, and shedding of the intercepted canopy snow. When melt begins, shadows cut down short-wave radiation, but tree trunks, being warmer than the snow, emits long-wave radiation to melt the snow preferentially around the trees to form tree-wells, which are bowl-shaped depressions in the snow surrounding the trunks (Sturm 1992; Woo and Steer 1986). Snowmelt rate is lower in the forest than the open areas that are more exposed to high radiation and wind. The snow then lingers in the forest even after the open sites become snow-free. Logging and forest fire not only modify the timing and the rate of snowmelt, but the presence of wood debris and subsequent growth of grasses and brushes also alter the snow accumulation pattern, as does the plantation of trees in the disturbed woodlands. Urban areas exhibit snow patterns distinct from their surrounding countryside. Buildings and other structures alter the wind direction and speed to affect snowfall and snow deposition. Snow removal from roofs and streets create bare surfaces during the winter. Urban heat island effect and heat advection from bare surfaces render the cities and towns clear of snow much before their rural neighbors. From the air, cities present a mixed pattern of snow on rooftops, open grounds, and linear features where the snow plows along highways. 40.4  SNOWPACK CHARACTERISTICS 40.4.1  Snowpack Metamorphism

Falling snowflakes occur in many shapes and sizes, including stellars, columns, plates, and grauples, for example (Fierz et al., 2009), but after a few days or even hours, they undergo rapid metamorphic changes that can occur extremely quickly since the snow is typically close to the freezing point. The resulting snowpack (referring to the accumulation of snow at a specific site) consists of snow grains (or particles) with impurities and pore spaces filled with air, water vapor, and liquid water when the snow is wet. Snowpacks are typically layered as the result of individual storms (either snowfall or blowing snow deposition) (Colbeck, 1991). Understanding snow stratigraphy, and

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Discussion and Conclusion    40-3 

how it evolves in time, provides key information required for understanding snow hydrology, avalanche formation, and heat and moisture exchange between the underlying soil, snowpack, and the atmosphere. An international snow on the ground classification has been provided by Fierz et al. (2009), a regional classification by Sturm et al. (1995), and observational methods by Pielmeier and Schneebeli (2003). Rapid metamorphic changes described by Colbeck (1997) include changes in grain size, grain shape, snow density, and specific surface area (SSA), defined as the surface area per unit mass (Proksch et al., 2015) and degree of sintering between particles. Each change significantly impacts the optical, thermal, and hydrologic properties of the snowpack. For example, optical properties are strongly related to changes in snow grain size, mechanical properties to the bonds between grains, and thermal properties to microstructure (Kaempfer and Schneebeli, 2007). In snowpacks with a temperature below 0°C and with little liquid water, metamorphism depends on whether the snowpack is isothermal or has a temperature gradient, while in snowpacks with a temperature of 0°C, metamorphism depends on the liquid water content. With equitemperature metamorphism (ETM) the snow is isothermal at a scale that includes a significant portion of the snowpack, but at the grain scale there are small temperature and vapor pressure gradients related to particle size and curvature around and between snow particles. These gradients lead to sublimation of ice from the convex portions of the particles and condensation on the concave areas, with large grains growing at the expense of small grains and producing well-rounded snow particles (Colbeck, 1982). Deposition also occurs at the concave connections where two particles touch (referred to as sintering) to result in grain bonding (Kaempfer and Schneebeli, 2007). Temperature gradient metamorphism (TGM) occurs when large gradients of greater than 10 to 20°C m–1 exists, often where the base of the snowpack is warmer than the snow surface, but also when temperature of the snow surface layer is reduced by radiative heating and cooling. Moisture is transferred along this temperature-induced vapor pressure gradient, with sublimation and condensation from one snow particle to the next, described by Yosida (1955) as “hand-to-hand transport,” and the resulting recrystalization produces kinetic growth crystal forms (Colbeck, 1997). Pinzer et al. (2012) demonstrated that the rapid flux of vapor can lead to a turnover of up to 60% of the total snow mass per day. A typical kinetic growth form, referred to as depth hoar, is in form of faceted snow crystals (Fierz et al., 2009) that typically have large grain size, low density, and low strength. Collapse of depth hoar layers under loading (storm or human) is a common avalanche trigger, while the low thermal conductivity of these layers retards heat transfer through the snow and decouples atmospheric conditions from soil freezing (Sturm and Benson, 1997). In snow at 0°C and with liquid water fully or partially filling the pore spaces, melt-freeze metamorphism (MFM) and grain growth rate depend on water content. In typical freely draining snowpacks with a low liquid water content, the air in the pores occupies continuous paths throughout the snowpack and liquid saturation is below 14% of the pore volume. Grain growth is much faster than during ETM and the resulting snow is characterized by clusters of rounded grains (Colbeck, 1982). With higher saturation, the liquid is continuous throughout the pores and air only occurs in isolated bubbles. Grain growth is much higher than in nonsaturated snow (Colbeck, 1982; Marsh, 1987). Snow grains develop into a true equilibrium form, with rounded and individual grains. In such “slush,” the snow has very low strength and is cohesionless. 40.4.2  Optical Properties of Snow

Snowmelt, a key driver of streamflow in cold regions, is controlled by turbulent fluxes of heat and absorption of shortwave radiation, and in many cases the shortwave energy balance dominates melt (Male and Gray, 1982). The actual absorption of shortwave radiation is controlled by various factors (Warren, 1982). Snow albedo is low (less than 0.4) in the near infrared (IR) (i.e., wavelengths > 0.7 μm) due to the strong absorption by ice. As a result, near-IR radiation plays an important role in snowmelt. Albedo in the visible wavelengths (0.3 to 0.7 μm) ranges from near 0.4 to 1.0. Snow albedo decreases with increasing grain size, ranging from 0.9 for a radius of 0.1 mm to 0.68 at 2.5 mm. As snow grains enlarge during melting, albedo decreases and melt rate increases. Snow albedo is also very sensitive to impurities in the snow, especially for strongly absorbent impurities, including soot, dust and volcanic ash, with the greatest effect for snow with large grain size (Warren and Wiscombe, 1980). Especially important is black carbon (BC) (Doherty et al., 2010) produced by incomplete combustion by diesel engines, coal burning, forest fires, agricultural fires, and residential wood burning. Hadley and Kirchstetter (2012) noted that BC of only 10 to 100 parts per billion by mass reduces albedo by 1 to 5%, sufficient to increase melt and have a positive radiative climate forcing. Albedo, especially in the near-IR (Warren, 1982), is higher as the sun approaches the horizon, and this effect is

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increased in the visible when trace amounts of impurities are present in the snow. As clouds absorb radiation in similar ways to snow, they increase both the fraction of shortwave radiation reaching the snow surface as well as the diffuse radiation, leading to an increase the snow albedo. 40.4.3  Thermal Properties of Snow

Snow is an excellent insulator, and hence strongly controls the snowpack temperature gradient. Sturm et al. (1997) reviewed the many attempts to estimate snow thermal conductivity and presented data that showed a strong relationship of conductivity for rounded grains or wind-blown fragments with snow density. Sturm and Johnson (1992) noted that snow texture (size, shape, bonding, and spatial arrangement of grains) also impacts thermal conductivity, but such observations are difficult to make, and it is believed that for faceted grains (i.e., depth hoar) there is little connection between density and thermal conductivity. To demonstrate such changes, Sturm et al. (2002) suggested a timeline of changes in thermal conductivity, where after a snowfall, the thermal conductivity would increase as the snow hardens, then decrease with increasing depth hoar formation, and finally if the snow is wetted by melting and then refreezes, the conductivity would again increase. Calonne et al. (2011) used both experimental observations and modeling to study various snow properties, one of the results being the finding of anisotropy, with the vertical thermal conductivity of facetted grains being 1.5 times higher than the horizontal values, and the opposite occurring for rounded grains. 40.4.4  Liquid Water Movement Through Snow

Melting snowpack consists of a complex mixture of rounded and sintered grains, or faceted particles, of low density and low thermal conductivity. When the snow is melting and freely draining, water saturation is typically less than 7% for coarse grained snow and 7 to 15% for finer grained snow (Marsh, 2005). When melt stops and a snowpack drains, the water saturation declines to the irreducible value, between 2 and 7% depending on snow properties. As snow grains grow in size when wetted, the irreducible water saturation decreases with time since wetting. The permeability of melting snowpacks is related to both grain size and density, with a commonly used parameterization provided by Shimizu (1970). When meltwater enters into a dry snowpack, a wetting front forms that divides the snowpack into a zone with an upper layer of wet snow at 0oC and a lower zone with no liquid water (Marsh and Woo, 1984). However, the wetting front quickly becomes heterogeneous and preferential flow paths form (Marsh and Woo, 1984; Schneebeli, 1995; Waldner et al., 2004). Such preferential flow paths play an important role in moving meltwater quickly through the snowpack, but modeling of these features remains a challenge. The interaction of flow within these flow paths and stratigraphic boundaries in cold snowpacks results in the formation of large ice lenses, and in sufficiently cold snowpacks water also freezes in the vertical flow paths to develop ice columns (Marsh and Woo, 1984). Lateral flow of water, especially through deep snowcovers (Eriksson et al., 2013) also plays an important role in the movement of meltwater to the stream channel. 40.5  DISCUSSION AND CONCLUSION

Both the spatial distribution and characteristics of snowcovers are altered as snowfall is deposited and as the snowpack undergoes accumulation, blowing and metamorphic processes. Sturm et al. (1995) presented a seasonal snow classification for snowcovers that evolve over the winter and melt periods in major environments across the globe. With the warmest snow, ephemeral snowcovers are typically less than 0.5 m deep, often formed of a single snowfall, with melt metamorphism beginning soon after deposition, and melt from both the surface and base leading to complete melting of the snow over a short period of time. A snowcover may form again after the next snowfall. In cooler maritime climates, a deeper (up to 3 m) snowcover with multiple layers deposited by individual storms is typical. At warm temperatures, TGM is rare while ETM occurs over much of the winter. However, winter melt events are common and MFM dominates, producing ice layers and large melt-rounded snow grains. A typically protracted spring melt season results in gradual removal of the snow. Under increasingly cold conditions, such as those of the prairies and tundra, winter melt events become infrequent and total snowfall is usually low. The snow is normally shallow (less than 1 m) and consists of a small numbers of layers. Substantial blowing snow produces wind slabs, sastrugi, and deep snowdrifts. Although ETM occurs, large temperature gradients usually lead to dominance of TGM. Both the magnitude and duration of the temperature gradients and TGM increase in the tundra regions, causing a large portion the snowcover to be composed of depth hoar by the end of winter. Taiga snowcovers are also dominated by low snowfalls, but low winds

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40-4     Snow Distribution and Snowpack Characteristics

Bare ground Snow cover Water/Ice Cloud

Sep. 30, 2010

Oct. 16, 2010

Nov. 17, 2010 Elevation (m) 2700 30

May 1, 2011

0

May 17, 2011

50 100

200 Km

Figure 40.2  Seasonal change in snow distribution in Liard basin in northern Canada during the winter of 2010–11. [Source: MODIS Snow Cover 8-Day L3 Global 500m

Grid (MOD10A2), obtained from the NSIDC.]

limit the frequency of blowing events. Under cold temperatures, large temperature gradients and TGM prevail, ending with snow that is of low density and dominated by depth hoar. Due to their shallow depth, prairie, tundra, and taiga snowcovers often disappear quickly during spring melt. In contrast, alpine and mountain areas have highly variable snowcovers depending on orientation, steepness, and elevation. Consequently, the snowpack varies from being cold and deep, commonly with many layers and a base of depth hoar, to being shallow and ephemeral. The melt period is similarly variable, ranging from protracted melt at high elevations with deep snow, to short and early melt in low elevations where the snow is relatively shallow and warm. Snow distribution pattern evolves continually from the time of snowcover formation to the time when snow disappears, as is illustrated by the meltseason snowcover of Liard River basin (area 275,000 km2) in northwestern Canada, where large-scale topography exerts prominent control. Western Liard basin lies in the Western Cordillera, reaching more than 3000 m and dissected by many headwater tributaries that unite and drain eastward. The eastern section of the basins is part of the Interior Plains, rising from 170 to 800 m at the foothills. Figure 40.2 shows the changing snowcover pattern in the winter of 2010–11. Topographic effect is demonstrated by late buildup but early disappearance of snow on lowlands in the east, branching out into the valleys that extend into the western mountain chains. High mountains remained snow-clad in mid-May. Within the highland zone, the southern mountains lost their snow earlier than those in the northern end of the basin, suggesting the influence of latitude. The changing mosaic of snow distribution and continuous alteration of snow properties significantly affect snowmelt but are themselves modified by melt events. These considerations prominently control melt runoff generation, which is the topic for another chapter of this handbook. ACKNOWLEDGMENT

We thank Dr. Laura Brown for preparing the figures.

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REFERENCES

Calonne, N., F. Flin, S. Morin, B. Lesaffre, S. R. Du Roscoat, and C. Geindreau, “Numerical and experimental investigations of the effective thermal conductivity of snow,” Geophysical Research Letters, 38: 1–6, 2011, doi:10.1029/2011GL049234. Colbeck, S. C., A review of sintering in seasonal snow, CRREL Rep., 97–10, Cold Regions Research and Engineering Laboratory, U.S. Army Corps of Engineers, 1997, p. 12. Colbeck, S. C., “An overview of seasonal snow metamorphism,” Reviews of Geophysics, 20 (1): 45, 1982, doi:10.1029/RG020i001p00045. Colbeck, S., “The layered character of snow covers,” Reviews of Geophysics, 29 (1): 81–96, 1991. Doherty, S. J., S. G. Warren, T. C. Grenfell, A. D. Clarke, and R. E. Brandt, “Light-absorbing impurities in Arctic snow,” Atmospheric Chemistry and Physics, 10 (23): 11647–11680, 2010, doi:10.5194/acp-10-11647-2010. Eiriksson, D., M. Whitson, C. H. Luce, H. P. Marshall, J. Bradford, S. G. Benner, T. Black, et al., “An evaluation of the hydrologic relevance of lateral flow in snow at hillslope and catchment scales,” Hydrological Processes, 27 (5): 640–654, 2013, doi:10.1002/hyp.9666. Essery, R. and J. Pomeroy, “Vegetation and topographic control of windblown snow distributions in distributed and aggregated simulations for an Arctic tundra basin,” J. Hydrometeorology, 5 (5): 735–744, 2004. Fierz, C., R. L. Armstrong, Y. Durand, P. Etchevers, E. Greene, D. M. McClung, K. Nishimura, et al., The international classification for seasonal snow on the ground, IHP-VII Technical Documents in Hydrology No. 83, IACS Contribution No. 1, UNESCO-IHP, Paris, 2009. Gude, M. and D. Scherer, “Snowmelt and slush torrents–preliminary report from a field campaign in Kärkevagge, Swedish Lappland,” Geografiska Annaler, 77A: 199–206, 1995. Hadley, O. L. and T. W. Kirchstetter, “Black-carbon reduction of snow albedo,” Nature Climate Change, 5 (March), 2012, doi:10.1038/ NCLIMATE1433.

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Killingtveit, Å., Hydrology in northern Norway, Northern Hydrology: International Perspectives, edited by T. D. Prowse, C. S. L. Ommanney, and L. E. Watson, National Hydrology Research Institute Science Report No. 3, Saskatoon, Canada, 1994, pp. 81–107. Kaempfer, T. U. and M. Schneebeli, “Observation of isothermal metamorphism of new snow and interpretation as a sintering process,” Journal of Geophysical Research: Atmospheres, 112 (D24): 24101 2007, doi:10.1029/2007J D009047. Male, D. H. and R. J. Granger, “Snow surface energy exchange,” Water Resources Research, 17 (3): 609–627, 1981. Marsh, P., “Grain growth in a wet arctic snow cover,” Cold Regions Science and Technology Journal, 14 (1): 23–31, 1987, doi:10.1016/0165-232X (87)90041-3. Marsh, P., “Water flow through snow and firn,” Encyclopedia of Hydrological Sciences, Vol. 4, Part 14, John Wiley & Sons, Chichester, UK, 2005, pp. 97–123. Marsh, P., C. Onclin, M. Russell, and S. Pohl, “Effects of shrubs on snow processes in the vicinity of the Arctic treeline in NW Canada,” Proceedings of the 14th Northern Research Basins Symposium and Workshop, Kangerlussuaq, Greenland, 2003, pp. 113–118. Marsh, P. and M. K. Woo, “Wetting front advance and freezing of meltwater within a snow cover 1,” Observations in the Canadian Arctic, Water Resources Research, 20: 1853–1864, 1984. Meier, M. F., Snow and ice, The Geology of North America, edited by M. G. Wolman and H. C. Riggs, Geological Society of America, Boulder, CO, Vol. O-1: Surface Water Hydrology, 1990, pp. 131–158. Phillips, D. W. and J. A. W. McCulloch, The Climate of the Great Lakes Basin, Atmospheric Environment Service, Toronto, 1972, p. 40. Pielmeier, C. and M. Schneebeli, “Developments in the stratigraphy  of  snow,” Surveys in Geophysics, 24: 389–416, 2003, doi:10.1023/B:GEOP.0000006073.25155.b0. Pinzer, B. R., M. Schneebeli, and T. U. Kaempfer, “Vapor flux and recrystallization during dry snow metamorphism under a steady temperature gradient as observed by time-lapse micro-tomography,” Cryosphere, 6: 1141–1155, 2012, doi:10.5194/tc-6-1141-2012. Pomeroy, J. W. and D. M. Gray, Snow Cover: Accumulation, Relocation and Management, National Hydrology Research Institute Science Report No. 7, Saskatoon, Canada, 1995, p. 144. Pomeroy, J. W., J. Parvianinen, N. Hedstrom, and D. M. Gray, “Coupled modeling of forest snow interception and sublimation,” Hydrological Processes, 12: 2317–2337, 1998. Proksch, M., H. Lowe, and M. Schneebeli, “Density, specific surface area, and correlation length of snow measured by high-resolution penetrometry,”

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Journal of Geophysical Research: Earth Surface, 120: 346, 2015, doi: 10.1002/2014JF003266. Rees, A., M. English, C. Derksen, P. Toose, and A. Silis, “Observations of late winter Canadian tundra snow cover properties,” Hydrological Processes, 28 (12): 3962–3977, 2013, doi: 10.1002/hyp.9931. Schaerer, P. A., Avalanches, Handbook of Snow: Principles, Processes, Management and Use, edited by D. M. Gray and D. H. Male, Pergamum Press, Toronto, 1981, pp. 475–518. Schneebeli, M., “Development and stability of preferential flow paths in a layered snowpack,” IAHS Publications, 228 (228): 89–95, 1995. Shimizu, H., “Air permeability of deposited snow,” Contributions No. 1053, Institute of Low Temperature Science, Hokkaido, Japan, pp. 1–32, 1970. Sturm, M. and C. Benson, “Vapor transport, grain growth, and depth-hoar development in the subarctic snow,” Journal of Glaciology, 43 (143): 42–59, 1997. Sturm, M., “Snow distribution and heat flow in the taiga,” Arctic Alpine Research, 24: 145–152, 1992. Sturm, M., “Thermal conductivity and heat transfer through the snow on the ice of the Beaufort Sea,” Journal of Geophysical Research, 107 (C10), 2002, doi:10.1029/2000JC000409. Sturm, M., J. Holmgren, and G. E. Liston, “A seasonal snow cover classification system for local to global applications,” Journal of Climate, 8: 1261–1283, 1995. Sturm, M. and J. B. Johnson, “Thermal conductivity measurements of depth hoar,” Journal of Geophysical Research, 97 (Figure 2): 2129, 1992, doi:10.1029/91JB02685. Sturm, M., J. McFadden, G. Liston, S. Chapin, C. Racine, and J. Holmgren, “Snow-shrub interactions in Arctic tundra: a hypothesis with climatic implications,” Journal of Climate, 14: 336–344, 2001. Waldner, P. A., M. Schneebeli, U. Schultze-Zimmermann, and H. Flühler, “Effect of snow structure on water flow and solute transport,” Hydrological Processes, 18 (April 2002): 1271–1290, 2004, doi:10.1002/hyp.1401. Warren, S. G., “Optical properties of snow,” Reviews of Geophysics and Space Physics, 20 (1): 67–89, 1982. Warren, S. G. and W. J. Wiscombe, “A model for the spectral albedo of snow. II: Snow containing atmospheric aerosols,” Journal of Atmospheric Sciences, 37: 2734–2745, 1980. Woo, M. K. and P. Steer, “Monte Carlo simulation of snow depth in a forest,” Water Resources Research, 22: 864–868, 1986. Yosida, Z. and Colleagues, “Physical studies on deposited snow 1: Thermal properties,” Institute of Low Temperature Science, 7: 19–74, 1955.

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Chapter

41

Time-Space Modeling of Precipitation BY

M.L. KAVVAS, N. OHARA, K. ISHIDA, AND A. ERCAN

ABSTRACT

For the water resources planning and management studies during the last century, the necessary long-term simulations of the precipitation fields over a target region were performed by stochastic precipitation models. After the necessary advances in computational resources and modeling technologies were achieved toward the end of the 20th century, in the 21st century it has been possible to simulate the precipitation fields at a target region also by deterministic numerical atmospheric models at sufficiently fine spatial and temporal resolutions. This chapter reports on the existing approaches to the stochastic and deterministic numerical modeling of precipitation, as well as the remote sensing approaches. Modeling of time-space precipitation is reported with respect to three purposes: the water resources planning, the design of hydraulic structures, and the real-time management of water resources systems. Numerical modeling approaches to regional-scale or watershed-scale time-space precipitation to simulate historical, real-time, and future conditions are presented. A recently developed methodology for the physically-based Probable Maximum Precipitation (PhysPMP) estimation, or more technically Maximum Precipitation estimation (MPE), is described. MPE approach, which does not require the assumption of stationarity, can account for the physical features of precipitation modeling, including the orographic effect and the blocking effect of mountains, that cannot be properly handled by the traditional PMP approach. 41.1 INTRODUCTION

In the planning and long-term management of the water resources of a geographical region, the engineer needs to consider a wide range of possibilities for the water conditions during the future planning/operational horizon at the target water resources system. This requires the projection of the uncertain but possible hydrologic conditions (streamflows, soil water, evapotranspiration, groundwater recharge rates, and levels, etc.) during the planning/ operational horizon at the target region. However, since the hydroclimate conditions during the future are uncertain, one needs to consider a wide range of future hydrologic water supply and water demand conditions. Typically, the future hydrologic conditions are simulated by numerical hydrologic models of the target region with the projected atmospheric inputs. Precipitation is the fundamental one among these inputs, since it provides the essential water input to the hydrologic model of the target region. Since a typical planning/operational horizon is in the order of a century, it is necessary to simulate the hydrologic conditions, and, thereby, the necessary precipitation inputs, at fine time increments over the horizon of a century for each individual projection that represents one possible future realization. In order to develop sound policies for the target water resources system, it is necessary to simulate a sufficiently large ensemble of future hydrologic projections to be able to develop sound statistical inferences for the outcomes of possible policies for this system. As such, it is desirable to have at

least 10 × 100 year projections of the precipitation sequences, ideally at hourly increments, at either some specified locations or over the whole of the target region. Until recently, it was not practical to perform simulations of the precipitation fields at a target region at sufficiently fine spatial (less than 10-km grid) and temporal (less than or equal to 1 h) resolutions by the deterministic numerical atmospheric models. Therefore, for the water resource planning/management studies during the last century, the necessary longterm simulations of the precipitation fields over a target region were performed by stochastic precipitation models. In the case of the design of ordinary hydraulic structures, it is necessary to estimate the design flood for a specified return period which is typically taken to be 100 years. In order to develop such an estimate, it is then necessary to estimate the flood peak discharge that would correspond to the 100-year return period. However, only in few locations around the world flow records that exceed 100 years exist. In most locations, it is necessary to extend the flow record-based empirical flood frequency curve to 100-year or longer return periods either by means of a fitted theoretical frequency distribution or by performing the simulation of an ensemble of flow projections (described earlier) from which one can then estimate the 100-year or longer return period flood hydrographs that can then be used for the design of the hydraulic structures. Within this framework, it is again necessary to perform longterm simulations of the precipitation fields as the necessary input to the target region’s hydrologic model’s simulations. Due to the aforementioned problems with the numerical atmospheric models, during the last century simulations of the precipitation fields for hydrologic design were again performed by stochastic methods. Within the aforementioned framework, first the existing approaches to the stochastic modeling of precipitation will be discussed. Since starting around the end of the twentieth century and the beginning of the twenty-first century, the necessary advances in computational resources and modeling technologies for the simulation of the precipitation processes by deterministic numerical atmospheric models for long time periods at fine time-space resolutions has been achieved, the deterministic numerical atmospheric modeling approaches to the simulation of the precipitation fields will then be reported. 41.2  STOCHASTIC MODELING OF PRECIPITATION

Precipitation is a complicated process which is the outcome of the interaction of dynamical, thermodynamical, and microphysical atmospheric processes occurring in continuous time and space. However, the precipitation process, as observed at a gauging station or at a group of stations at hourly or daily increments at a study region, emerges as a temporal stochastic process with uncertain times for the occurrence of wet periods, uncertain dry periods, and uncertain precipitation amounts. Within this context, first the approaches to the modeling of precipitation as a temporal stochastic process will be discussed. Then the more comprehensive, yet more complicated models of timespace precipitation will be reported. 41-1

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41-2     Time-Space Modeling of Precipitation 41.2.1  Stochastic Modeling of Precipitation in Time at Fixed Spatial Locations

As observed at a gauging station, the precipitation occurrence process can be modeled as a two-state process, consisting of a wet and a dry state. Once a wet state occurs, this state can be marked by a random precipitation quantity for the considered time increment (daily or hourly). This two-state precipitation occurrence process can be modeled conveniently as a two-state Markov chain with a two-by-two transition probability matrix that describes the dry-to-dry, dry-to-wet, wet-to-dry, and wet-to-wet state transition probabilities when moving from one time increment to the next. Once the state probabilities of being in a wet state and in a dry state at the time origin of the process are defined, it is straightforward to estimate the probability of being in a specified state (wet or dry) at any specified time of the process. Assuming that the outcome of the process (wet or dry) at any given day is only dependent on the state of the process in the previous day, the Markov chain model is a practical model that was applied by many researchers during the twentieth century to the modeling of precipitation occurrences (Gabriel and Neumann, 1957; Wiser, 1965; Feyerherm and Bark, 1967; Smith and Schreiber, 1973; Roldan and Woolhiser, 1982; Woolhiser and Osborne, 1985). It is also possible to generalize the two-state Markov chain for wet and dry state occurrences to a multistate Markov chain where the precipitation amounts can be classified into a number of interval classes during a wet time increment. This way one ends up with an n by n transition probability matrix, where n becomes the number of precipitation depth intervals plus one for the dry state. Various researchers applied the multistate Markov chain model to describe the occurrences and amounts during a precipitation process at discrete time increments (Hopkins and Robillard, 1964; Haan et al., 1976; Srikanthan and McMahon, 1983). Also, higher-order Markov chain models for winter time daily rainfall over continental United States were developed by Chin (1977). While being easy to apply, the Markov chain models suffer from having a short memory, being unable to account for the observed persistence during long wet periods due, for example, to the passage of a synoptic extratropical cyclonic system, and also during long dry periods. A promising generalization to the Markov chain models is the Markov renewal process (MRP) model of FoufoulaGeorgiou and Lettenmaier (1986) for daily rainfall occurrences. The fundamental advantage of the MRP model over a Markov chain is that on a given day the probability of being on a wet state (rainy day) or a dry state does not depend only on the previous day being wet or dry, but on the number of days since a wet day. It has a Markovian renewal structure in that the length of the interval in between the occurrences of a particular state has a probability distribution (taken as a geometric distribution) determined by Markovian transition probabilities among wet and dry states. Within this framework, the MRP can account for the effect of different rainfall mechanisms. Since, as observed at a gauging station, the precipitation process can be interpreted as a wet-dry two-state process, it is possible to also model the process as an alternating renewal process of dry and wet periods that alternate randomly in time (Grace and Eagleson, 1966; Todorovic and Yevjevich, 1969; Galloy et al., 1981; Roldan and Woolhiser, 1982). The basic assumption of the alternating renewal process model is the independence of the wet and dry period durations. As such, different probability distributions for the wet and dry period durations can be assigned. If the process is considered in discrete time increments, a negative binomial distribution for both the wet and the dry period durations (Galley et al., 1981), or a truncated geometric distribution for wet period durations and a truncated negative binomial distribution for dry period durations (Roldan and Woolhiser, 1982) can be taken. Furthermore, given that the process is in a wet period, a precipitation depth needs to be assigned to that wet period independently. Aside from the restrictive independence assumption among the sequential dry and wet period durations, another serious issue with the alternating renewal process model is how to disaggregate the total precipitation depth during a wet period among the individual wet days. A simple solution to this problem is to consider either a sequence of independent rain depths for each of the individual time increments of a wet period, or to consider a multistate Markov chain in order to account for the autocorrelation among the precipitation depths of the sequential wet days during a wet period. As a way to relax the independence assumption in the wet and dry period durations of the alternating renewal process model, and to expand the short memory structure of the Markov chain model for the wet–dry day occurrences, discrete autoregressive moving average models (DARMA, Jacobs and Lewis, 1978) were also applied to daily rainfall sequences (Buishand, 1978; Chang et al., 1984). The appeal of DARMA models is their flexible structure in modeling discrete-state time series with zero values, accommodating various memory lengths in a parsimonious manner. Hence, they can be useful in modeling droughts with long-term memories in a parsimonious way. However, while the precipitation process is seasonal, DARMA models are

41_Singh_ch41_p41-1-41-14.indd 2

stationary. Hence, they need to be applied to predefined time periods within which the stationarity of the precipitation process can be assumed. Precipitation being basically a continuous time-space process, a different approach can also be taken to modeling precipitation in time by means of continuous-time point process stochastic models. Point process stochastic models consider the occurrence locations of precipitation events in time or in time space as random points. After modeling, the occurrence of precipitation event origin locations, then a mark, either as a random value, or a random rectangular pulse or a random signal can be attached to each origin location in order to describe the precipitation amounts as accumulated values at desired time increments. The simplest of such point process stochastic models of precipitation is the Poisson process model. The Poisson process has the unique property that the number of event occurrences in nonoverlapping time-space intervals are mutually independent with Poisson probability distribution. The time intervals between consecutive events are also independent with exponential distribution. The Poisson process can be described by a single parameter function, the occurrence rate function, that may be either a constant value, or a function of time, depending on whether the studied process is considered as stationary or nonstationary in time. Accordingly, the Poisson process was employed by various researchers for modeling precipitation fields (Todorovic and Yevjevich, 1969; Gupta and Duckstein, 1975; Eagleson, 1978; Rodriguez-Iturbe et al., 1984, 1987; Valdes et al., 1985; Foufoula-Georgiou and Guttorp, 1987, among others). In order to model precipitation depths at various time intervals, Rodriguez-Iturbe et al. (1984) marked the Poisson occurrence time locations of the precipitation events either by an independent random depth value [Poisson white noise model (PWNM)] or by a rectangular pulse with random independent precipitation rate and independent duration [Poisson rectangular pulse model (PRPM)]. In their application of these models to hourly and daily precipitation sequences in the United States and Venezuela, they observed that the parameter estimates in these models vary with the time increment sizes. They also observed that these models cannot preserve the correlation structure of the hourly precipitation sequences. Foufoula-Georgiou and Guttorp (1987) also reported the inadequacy of the PWNM in modeling short time increment rainfall. Based on their studies of the extratropical cyclonic precipitation systems in the United States, Austin and Houze (1972) and Hobbs and Locatelli (1978) observed that precipitation is organized in distinct regions within the synoptic scale (>104 km2) cyclones with respect to temporal and spatial scales as follows: (a) synoptic areas which have lifetimes of one to several days and horizontal spatial extents > 104 km2, (b) large mesoscale areas (LMSAs), also called “rainbands,” which last several hours and have spatial horizontal extents in the range 103–104 km2; (c) small mesoscale areas (SMSAs) or precipitation cores which last from several minutes to an hour and have horizontal spatial extents in the 100–800 km2 range; and (d) rain cells which last from several minutes to an hour and have horizontal spatial extents in the order of 10 km2. They observed that rainbands are located within synoptic areas of the cyclonic system while the rain cores and cells are mainly located within rainbands. Within this framework, the extratropical cyclonic systems demonstrate a clustered precipitation field structure both in time and space. This cluster structure of the precipitation systems are also typical for the severe precipitation-producing mesoscale convective systems (MCSs; Parker and Johnson, 2004; Trapp, 2013) and for tropical cyclones (TCs; Lonfat et al., 2004). Based on a statistical analysis of rainfall sequences in Indiana, the United States, Kavvas and Delleur (1976, 1981) detected the clustering structure of the precipitation sequences in time by means of the precipitation occurrence counts spectrum and counts variance-time functions. Hence, they developed a Poisson cluster point process model of Neyman–Scott-type (NSPCM) (Neyman and Scott, 1958) for the precipitation occurrence process in time by means of its various moments. The NSPCM is a two-level Poisson cluster model, where in the primary level, there is a Poisson process for the arrivals of rainfall generating mechanisms (RGMs), representing cyclonic fronts or convective systems. At the secondary level of NSPCM, to each RGM of the primary level process there corresponds a random number (geometrically distributed) of individual rain cells that are independently located (with exponential distribution) with respect to the RGM arrival time, forming a cluster around that RGM. Rodriguez-Iturbe et al. (1984) expanded the NSPCM by marking each of the individual rain cell occurrences with an independent random rain depth and deriving the moment of the rainfall depths for various time intervals [Neyman–Scott white noise marked Poisson cluster process (NSWNPCP)]. Their application of the NSWNPCP model to hourly and daily rainfall depths in Colorado and Venezuela showed that NSWNPCP is superior to Poisson occurrence process-based marked rainfall depth models. However, a separate application by Valdes et al. (1985) showed that NSWNPCP could not model the statistics of extreme precipitation satisfactorily. In order to overcome the shortcomings of NSWNPCP, Rodriguez-Iturbe

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Deterministic numerical modeling of time-space precipitation     41-3 

et al. (1987a, 1987b) introduced the Bartlett–Lewis process for rain cell arrivals with each of the rain cells then marked by a rectangular pulse with an independent rain rate and independent rain duration [Bartlett–Lewis rectangular pulse model (BLRPM)]. Two types of probability distributions were selected for rectangular rain pulses in order to differentiate among the stratiform rain and the convective rain. Application of BLRPM to Colorado rainfall at various intervals yielded satisfactory results for rain depths at various time intervals. Also, the statistics of extremes were deemed to be modeled satisfactorily. It may be noted that Bartlett–Lewis process is again a Poisson cluster process with both the primary level and the secondary level arrival processes taken as Poisson processes. The BLRPM was later generalized by Hanaish et al. (2011) to rectangular pulses where the rain intensity distribution is conditional on the random rain cell duration. The Bartlett–Lewis two level Poisson cluster model was also extended by Cowpertwait et al. (2007) to a three-level Poisson cluster process of instantaneous rain pulse arrivals, accounting for both the stratiform and convective rainfall. Yet another type of Poisson model, the doubly stochastic Poisson process (DSPP) was introduced by Smith and Karr (1983) to the modeling of daily rainfall with application to Potomac river basin. In this model, the occurrence rate becomes random with a zero state and a positive state. Given that the process is in the positive state, it takes on values according to a Markov chain. The aforementioned Poisson cluster models have rich structures to account for the correlation structure of the precipitation occurrences and statistics of the extremes. However, they all suffer from parameter estimation problems since the rain gauge observations they use at hourly or daily intervals provide only the aggregated products (precipitation depths) of a fundamentally continuous-time process, as first observed by Foufoula-Georgiou and Guttorp (1987). Also, they have many parameters whose estimation are sensitive to the size of the time increments being considered. One way to avoid these problems is to utilize concurrent weather radar and rain gauge data for the modeling of sequential precipitation depths at a selected number of ground locations. In this direction, Kavvas and Herd (1985) developed a filtered Poisson cluster process model (FPCPM), where the time-varying precipitation depths at any selected time interval can be obtained from the trajectories that are drawn by the radar-detected rain fields that pass over the selected ground location. The basic advantage of this modeling approach is that its parameters can be estimated directly from the radar-observed rain field arrivals, and the time-varying rainfall intensities can be constructed by the simultaneous use of the radar reflectivity fields and rain gauge observations. The model can also accommodate the nonstationary characteristics of the temporal precipitation process by means of its nonstationary Poisson cluster process structure. Since December 2001 Stage-IV multisensor precipitation analyses at as fine as one hour intervals by U.S. National Weather Service (NWS) (Lin and Mitchell, 2005) became available. Stage-IV multisensor analyses combine rain gauge data with radar-estimated rainfall, and are quality controlled by the U.S. NWS’ river forecast centers. Stage-IV data are ideally suited to the application of FPCPM for the identification of the arrival times of the rain fields, and for the observation of the precipitation depths as they evolve in time. 41.2.2  Stochastic Modeling of Precipitation in Time and Space

Building upon the reported conceptualizations of the cyclonic precipitation fields by Austin and Houze (1972) and Hobbs and Locatelli (1978), and exploiting the property of a time-space Poisson process that the number of events in nonoverlapping time-space boxes are mutually independent, several researchers developed various forms of the Poisson cluster random field model for precipitation in time and space (Waymire et al., 1984; Smith and Karr, 1985; Valdes et al., 1985; Rodriquez-Iturbe et al., 1986; Kavvas et al., 1987). Among these studies, the ones that also account for the observed kinematic behavior of the rain fields shall be discussed here. The first Poisson cluster random field model for precipitation in time space was by Waymire et  al. (1984). In this time-space filtered three-level Poisson cluster process model for ground rainfall in time space, which was developed in terms of the mean and covariance functions of the time-space ground rainfall intensity, the rainband arrivals in time, described by a stationary Poisson process, make up the primary level. At the secondary level, small mesoscale rain areas (SMSAs or rain cores) are born according to a spatially stationary planar Poisson process at the arrival time of each rainband. Once the time-space birth location of an SMSA is realized, a time × two dimensional-space stationary stochastic process is taken for the births of rain cells within the disk of the SMSA. Each cell is also assigned a random lifetime. As such, the births of rain cells is modeled as a time-space stationary Neyman–Scott process. A common constant velocity is taken for all cells. Then the rain intensity contributions of all born cells are superimposed in time and space to compose the ground rainfall field. Based on their simulation of the time-space rain

41_Singh_ch41_p41-1-41-14.indd 3

fields by Waymire et al.’s (1984) model, Valdes et al. (1985) contended that the model can simulate the clustering, kinematic, and life evolution of rain cells. Kavvas et al. (1987, 1988) also developed a kinematic, time space Poisson cluster random field model for ground rainfall in time space. This time-space nonstationary, kinematic two-level filtered Poisson cluster model was developed in terms of its characteristic function and mean function of time-space rain intensity. At the primary level of this model, cyclone centers are born by a time-space nonstationary Poisson process in order to mimic the preferred birth locations of extratropical cyclone centers over the United States. Once born, each of the cyclone centers move as a nonhomogeneous Markov process. At the secondary level of the model, rain cores and rain cells are born according to another time-space nonstationary Poisson process with respect to each cyclone center time-space location at preferred time-space locations. Each born rain core/cell assumes a random circular area, random rain intensity, random life length, and random velocity. The ground rain fields are formed from the time-space superimposition of the circular rain core/cell areas. The simulated rain fields by Saquib et al. (1988) resembled the radarobserved characteristics of the rainbands. While, the Poisson cluster-type stochastic precipitation field models advanced the realism in the description of precipitation fields in time and space, they suffered from parameter estimation and validation problems (Georgakakos and Kavvas, 1987; Smith and Krajewski, 1987). Since at the time of their development routine concurrent rain gauge-radar rainfall data (the aforementioned Stage-IV data) were not available, the calibration of these continuous time-space models could only be performed indirectly in terms of fitting certain rainfall moments of these models to ground rain gauge observations. No rigorous validation of these models could be performed. However, with the current availability of StageIV multisensor precipitation data (Lin and Mitchell, 2005), it may be possible to calibrate and validate Poisson cluster-type precipitation random field models in time and space. A paradigm shift in time-space precipitation modeling has happened in mid-1980s by the recognition that the precipitation fields are scaling (Lovejoy, 1982; Lovejoy and Mandelbrot, 1985; Lovejoy and Schertzer, 1985; Waymire, 1985; Schertzer and Lovejoy, 1987, etc.). Scaling transformations stretch or contract processes or geometric objects/fields in time and/or space. A process or an object is self-similar if it is invariant under a subset of scaling transformations. Within the framework of a physical process, self-similarity is the case where the solution of the process at a specified time-space scale can be related to the solution of the process at another time-space scale. Meanwhile, a fractal may be thought as a geometric object/field that is self-similar at various scales. About three decades ago, Mandelbrot (1983) introduced the fractal geometry of nature, the degree of irregularity, or fragmentation of which is identical at all geometric scales. Meanwhile, if the studied geometric field is self-similar with different scale factors at different scales, then that field becomes a multifractal, and needs to be modeled accordingly (Schertzer and Lovejoy, 1987). In mid-1980s Lovejoy and Mandelbrot (1985) proposed a fractal model for stochastic precipitation fields that scale with the same scale factor at all scales. Later on, starting with Schertzer and Lovejoy (1987), the stochastic precipitation fields were modeled as multifractal cascades by various researchers (Gupta and Waymire, 1990, 1993; Hubert et al., 1993; Over and Gupta, 1994; Burlando and Rosso, 1996; Veneziano et al., 1996; Menabde et al., 1997; Deidda, 2000; Lovejoy and Schertzer, 2005; Lovejoy and de Lima, 2015, etc.). Based on various observations of the precipitation fields in time and space (e.g., Lovejoy and Schertzer, 2005; Lovejoy and de Lima, 2015), the fractional-integrated flux model (FIF) which was originally developed in Schertzer and Lovejoy (1987), emerged as a viable model for the time-space stochastic precipitation fields that does preserve the multifractal characteristics of precipitation at various scales (Lovejoy and Schertzer, 2005; Lovejoy and de Lima, 2015). The fundamental advantage of the multifractal models over the point process models in modeling stochastic time-space precipitation fields is that they can describe these fields with very few parameters. For example, the FIF model can describe the evolution of precipitation fields at multiple scales with three parameters. Its details are given in Schertzer and Lovejoy (1987). 41.3  DETERMINISTIC NUMERICAL MODELING OF TIME-SPACE PRECIPITATION

While significant advances in the stochastic modeling of precipitation fields have been achieved in the last 50 years, there are still some fundamental issues to be resolved in modeling precipitation with respect to its short-term (with several hours to several days lead time) forecasting in time and space over a target region, and the estimation of maximum precipitation at the drainage basin of a designed large hydraulics structure toward the estimation of maximum flood at the site of the structure.

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41-4     Time-Space Modeling of Precipitation

In the Americas and elsewhere around the globe, one of the mesoscale atmospheric systems that produce the heaviest precipitation are the MCSs (Schumacher and Johnson, 2005). The precipitation in MCSs is organized in distinct convective and stratiform regions with distinct intensity textures, locations and dynamics (Parker and Johnson, 2004; Schumacher and Johnson, 2005; Trapp, 2013). Another very important mesoscale atmospheric system that renders extreme precipitation is a TC. The TCs that form hurricanes in the United States and typhoons in Asia again have their precipitation fields organized in distinct patterns in intensity and location with distinctly defined dynamics that play important roles in the time-space distribution of precipitation over a region (Marks Jr., 1985; Powell, 1990a, 1990b; Lonfat et al., 2004; Lin et al., 2010). In order to be able to estimate the maximum precipitation over a target watershed due to MCSs or to TCs, or to be able to forecast the precipitation in short term due to these systems over a target region, it is necessary to use deterministic numerical atmospheric models. Time-space precipitation can be described by a numerical atmospheric model, which simulates atmospheric processes to predict weather or climate based on initial conditions and/or boundary conditions, depending upon time horizons. A numerical atmospheric model numerically solves the governing equations of the atmosphere, including Navier–Stokes equations and heat and mass conservation equations for air, water, and various atmospheric substances. These equations are nonlinear partial differential equations for which exact solutions are unobtainable, and even numerical methods often suffer from instability due to the nonlinearity. Furthermore, to handle subgrid variability within a computational grid cell, the numerical atmospheric models at present require parameterizations for solar radiation, clouds and precipitation, heat exchange, soil, vegetation, surface water, and the effects of terrain. The complexity level of the model, which determines the computational requirements, depends on the scale of the process typically characterized by a grid resolution. A general circulation model (GCM) is a class of numerical atmospheric models for the general circulation of the planetary atmosphere. Nowadays, some GCMs have a component to model the ocean, and solve the general circulation of the planetary atmosphere and ocean together. When specifying a model, a GCM that models the atmosphere, the land, and the ocean interactively is called an atmospheric-oceanic GCM (AOGCM), whereas a GCM that models only the atmosphere and land interactively is called an atmospheric GCM (AGCM), while a GCM that models only the ocean is called an oceanic GCM (OGCM). A GCM typically solves a set of nonlinear equations called primitive equations that consist of momentum equations, thermal energy equations, and various mass conservation equations under the hydrostatic approximation. GCMs are used to integrate the dynamics of the physical components of the earth’s climate system including atmosphere, ocean, land, and sea ice. Since they cover the whole earth as their modeling domain, the GCMs are typically run under initial conditions for long-term climate projections in the order of many years, or for medium term forecasts typically with seasonal or 6-month horizons, or for short-term weather forecasts with a typical horizon of several days to 2 weeks. As such, in mathematical terms they solve initial value problems. For short-term forecasting of weather at global scale, a class of numerical atmospheric models, called the global weather forecasting system (GWFS), are utilized. The main purpose of a GWFS is the real-time weather forecasting. Since the results of real-time weather forecasting strongly depend on the initial state, GWFS typically uses a data assimilation system (DAS) to fit the initial state to observation data, such as surface observations, balloon data, wind profiler data, aircraft reports, buoy observations, radar observations, and satellite observations. Over the last several decades, the effectiveness of the GWFSs has been significantly improved even in terms of rainfall (e.g., Hamill et al., 2013). There are several major meteorological centers operationally running GWFSs, including the environmental modeling center of the U.S. National Weather Service, a member of the U.S. National Centers for Environmental Prediction (NCEP), the European Center for Medium Range Weather Forecasting (ECMWF), the United Kingdom Met Office, the Canadian Meteorological Center, the Japanese Meteorological Agency (JMA), the U.S. Navy, the Commonwealth Scientific and Industrial Research Organization, and few others. The GWFS modeling has become increasingly essential for daily weather forecasting, aviation traffic controls as well as hydrological analyses (e.g., Lynch, 2008). A mesoscale or a regional atmospheric model is a limited-area model with fundamentally the same representations of atmospheric processes as those in the GCMs or the GWFSs. The regional climate model (RCM) and the numerical weather prediction (NWP) model belong to this class of atmospheric models. A RCM is used for long-term climate assessment, and a NWP model is used for short-term weather simulation. Thus, the purposes of an RCM and

41_Singh_ch41_p41-1-41-14.indd 4

NWP are different, but a regional atmospheric model is frequently used both as a RCM as well as in NWP in practice. Unlike currently operational GCMs and some GWFSs, the mesoscale models solve the nonhydrostatic equations, which tend to be more computationally intensive, to capture the details of vertical atmospheric dynamics. While the application of a global-scale model, such as a GCM or a GWFS, essentially solves an initial value problem, a regional atmospheric model application deals with an initial and boundary value problem, as it needs external boundary conditions typically supplied from a global model or a larger-scale regional model. A regional atmospheric model can be used for atmospheric simulation with fine spatial resolution in a limited area. Sometimes simulation at a fine resolution is required to obtain detailed information on precipitation fields. For example, if a study region or watershed is in a mountainous area, a fine resolution is required to account for the effect of topography (e.g., Kanamitsu and Kanamaru, 2007; Maraun et al., 2010; Soares et al., 2012). The application of a hydrologic model frequently focuses on a geographical region or a watershed. Such an application would require less than 10-km spatial grid resolution (e.g., Boé et al., 2007; Lin et al., 2015). A global model can literally cover the whole globe and does not require external boundary conditions from another model simulation, but requires very substantial computational resources in order to be able to simulate the atmospheric phenomena at fine spatial resolution (such as less than or equal to 10 km). Thus, a regional atmospheric model is useful for simulating precipitation at a fine resolution that is necessary for a regional-scale or a watershed-scale hydrologic analysis and modeling. A regional atmospheric model requires external boundary conditions from a global model or a larger-scale regional model within which it is nested. Most regional atmospheric models are built with domain nesting functionality. Such regional atmospheric models can gradually downscale the coarse spatial resolution atmospheric data, supplied by a global model, into a fine spatial resolution over the target model region through a sequence of nestings of regional models with gradually decreasing domains and spatial grid resolutions. Such a process is called dynamical downscaling. Many meteorological centers and agencies provide atmospheric data simulated by a global model that can be used by a regional atmospheric model. Such simulated global atmospheric data can be divided into three categories; reanalysis data, forecasting data, and future climate change projections. Reanalysis data are atmospheric simulations by a global atmospheric model that are fitted to observation data by a DAS, such as the three-dimensional variational data assimilation system (3D-VAR), the four-dimensional variational data assimilation system (4D-VAR), the ensemble Kalman filter (EnKF; Evensen, 1994) etc. Therefore, reanalysis data, after being dynamically downscaled by a regional atmospheric model over the target study region, are frequently used as inputs to hydrologic models in order to reconstruct historical hydrological conditions over the target region. Table 42.1 summarizes the currently available reanalysis data. With respect to short-term forecasting, a GWFS provides global weather forecasts at coarse grid resolution (~0.5o) over a target study region with horizon from several days to several weeks ahead in real time. Such real-time weather forecast data are provided every 6 h or every 12 h. These global forecasts at coarse resolution are then dynamically downscaled by a regional atmospheric model at fine spatial resolution (< 6 km) and at hourly intervals as input to a hydrologic model for its hydrologic forecasts in real-time over the target region. Nowadays, there is another type of forecasting system to predict climate instead of weather. It is called seasonal forecasting system, and its forecasting period is more than several months. Such global coarse resolution forecasts with seasonal to 6-month time horizons, when downscaled by a regional atmospheric model at fine grid resolution ( 1) of the radiative term yielding an estimate of E0 that is primarily a function of radiation; the PenPan equation (Rotstayn et al., 2006), which uses meteorological and radiative data to synthesize pan evaporation observations, with uses in streamflow simulation and forecasting and climatological studies and predictions. Reference ET In no small part due to the uncertainty about the definition of Ep (see Sec. 42.3.6), and to meet the ET-estimation needs of agricultural users, the concept of reference ET was formulated (Doorenbos and Pruitt, 1977). In this concept, the prevailing meteorologic and radiative conditions are used to estimate an ET rate for a heavily specified, well-watered surface: this estimate is called reference ET (ET0). To estimate actual ET from a cropped surface (ETc), ET0 is then multiplied by coefficients that account for variations from the reference conditions due to such factors as crop type, phenology, salinity, and stress. Most current estimation of ET0 relies on a specific parameterization of the Penman–Monteith (PM) model (Monteith, 1965), itself an offshoot of Penman’s (1948) seminal approach. The general PM formulation parameterizes the moisture availability by two resistances to the diffusion of

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Relevant Concepts and Methods    42-3 

water vapor from within the leaves of a vegetative canopy out to the dynamic boundary layer. 42.1.5  Satellite Remote Sensing of ET

Optical and thermal satellite imagery of earth’s surface is commonly used to estimate ET from field to landscape scales, and is especially useful for water rights, water resource decisions, and global water budget accounting. Remote sensing of ET has allowed scientists and water managers to investigate and make management decisions based on figures that incorporate the spatial and temporal variability of ET. Advances in ET estimation that incorporate remotely sensed imagery with local weather station or gridded weather data allow for these important spatial and temporal ET characteristics to be estimated. For a review of recent remote sensing techniques for estimating ET, see Kalma et al. (2008) and Gonzalez-Dugo et al. (2009). Recent case studies and highlights on the uses and benefits of field scale remote sensing applications for water resource decision making can be found in Serbina and Miller (2014). 42.1.6  Emerging and Scientific Methodologies

Advances in instrumentation, data collection, and processing techniques have permitted previously research-oriented ET-estimation technologies, such as eddy correlation (EC) and scintillometry techniques, to gain traction as operational methodologies. Eddy Correlation The eddy correlation (EC) technique for estimating ET has become common among researchers and practitioners alike over the last decade. The EC technique is considered one of the most accurate and direct approaches to study micrometeorology, land surface energy balance, land and atmospheric interactions and feedbacks, remote sensing models of ET, and vegetation physiological and ecological processes. Practitioners commonly rely on the EC technique for vegetation monitoring, water budget estimation, and precision agriculture. The technique estimates vertical fluxes of heat, water vapor, and other gases by measuring instantaneous deviations in vertical wind speed and the flux of interest. The development of eddy flux networks over the last decade, such as Ameriflux, CarboEurope, FLUXNET, ICOS, iLEAPS, NEON, and OzFlux, has led to more unification of data-collection and processing steps. However, differences in experimental purpose, design, and station instrumentation make EC-processing steps difficult to standardize for operational applications. Scintillometry Scintillometry is an emerging technique for indirectly estimating integrated ET over large areas through estimation of H and available energy (Qn). The technique relies on measuring the transmission and scattering of electromagnetic radiation through the turbulent atmosphere. H is estimated by measuring atmospheric scintillations caused by turbulence-induced refractive index fluctuations of the air, in combination with measurements of site-specific meteorological conditions and application of Monin-Obukhov similarity theory (MOST). Scintillometer systems consist of an optical or radio transmitter, receiver, data logger, and signal-processing unit. Optical transmitters and receivers are most commonly used in ET studies. Path-averaged H estimates can be made over path lengths of hundreds to a few thousand meters. Common research applications include estimating ET and H over large areas, validating airborne and satellite remote sensing estimates of ET and H, estimating water budgets, and validating mesoscale numerical models. Costs of scintillometry systems are currently equivalent to those of well-equipped EC and available energy measurement systems. Scintillometry is well suited for long-term measurement due to the system being fairly insensitive to environmental conditions such as rain, snow, and dust, and requires little power and maintenance, making the system ideal for measurements in remote areas. 42.2  RELEVANT CONCEPTS AND METHODS 42.2.1  Constraints and Drivers of ET and E0

ET is subject to three physical constraints: the hydrologic limit is the availability of water to evaporate and/or transpire from the soil, plants, open water, and ice; the radiative limit is the availability of energy to drive the evaporative process; and the advective limit is the ability of the atmosphere to absorb and bear away moisture. Hydrologic Limit The hydrologic limit can be described as:

42_Singh_ch42_p42.1-42.18.indd 3

 ∂Θ  ET ≤ −  (42.3)  ∂t  max

where ∂Θ / ∂t is the time rate of change of moisture avaialability in units of mass flux, the negative sign is due to ∂Θ/ ∂t itself being negative; and the subscript “max” indicates the fastest rate at which the soil can exfiltrate water to the surface for evaporation. Crucially, moisture availability is seldom knowable at scales useful to practitioners, and this has led to the development of the more tractable concept of evaporative demand (E0), which is the maximal ET possible subject to only the remaining radiative and advective limits. The idealized flux of E0 has become central to hydrologic science and practice. It is primarily used in the estimation of ET, which may be derived from E0 by adding a further constraint derived from some parameterization of Θ. While ET has long been scaled from E0 using a linear function of Θ, feedbacks between ET and E0 have also been recognized for some time (Bouchet, 1963), and they have been coupled in a nonlinear fashion in what is now known as the Complementary Relationship (CR; see Sec. 42.2.3). Radiative Limit The net available energy (Qn) is the energy available for turbulent heat transfers to the dynamic boundary layer as either latent heat lET or sensible heat H and it is derived from the instantaneous balance of the energy storages and fluxes shown in Fig. 42.1 and described as follows:

∂W = (1 − α )Rd + Ld − Lu − λ ET − H − G − C − Ad (42.4) ∂t

where all terms are in flux units (W m−2, multiply by 86,400/l for mm day−1). On the LHS, ∂W/ ∂t is the time rate of change of heat storage in the evaporating layer, positive for increasing energy storage. On the RHS, positive fluxes are into the evaporating surface, negative out: α is the surface albedo (-); Rd is the downward shortwave radiation incident at the surface; Ld and Lu are the longwave radiation fluxes inward to and outward from the surface, respectively; λ ET is the latent heat flux (the energy equivalent of ET); H is the sensible heat flux; G is the net heat flux conducted from the evaporating surface into the soil (or ground heat flux) below; C is the energy absorbed by biochemical processes in vegetation in the control volume; and Ad is heat gained by advection to the control volume. Although Eq. (42.4) is defined explicitly for water evaporating from the liquid state, its broad concept still applies for evaporation from snow and ice—that is, sublimation—although some of the terms change their relative importance and, indeed, sign. The phase change of solid to vapor requires a further energetic input, so the latent heat of fusion lf must be added to the latent heat of vaporization l, noting that l exceeds lf by a factor of about 7.5. The conduction of H with the ground below the snowpack G is usually negative (i.e., heat travels to the snowpack from the ground), Ad accounts for heat advected from precipitation (Prcp) onto the snowpack, and is always positive. Lu usually exceeds Ld, while ET and H are mostly negative. The thermodynamics of the snowpack itself vary both seasonally from the accumulation phase through the snowmelt phase (itself consisting of warming, ripening, and finally liquid runoff phases) and diurnally with an internal freeze/thaw cycle. The change in energy storage term ∂W / ∂t dominates across seasonal time-scales, and ET may become a minor flux at short time-scales during snowmelt periods when, in fact, the opposite flux—condensation of moisture from the atmosphere onto the snowpack—often occurs (Hood et al., 1999); in fact, over a year, condensation and evaporation are of the same order of magnitude, leaving snowmelt the largest water balance term. It is for this reason and the complications pertaining to its measurement and modeling that we do not further examine sublimation in this chapter: interested readers are encourage to seek out Dingman (2015). As the net available energy (Qn) for lET and H is dominated by radiative terms it is also commonly referred to as the “radiative driver”; it may be better understood by rearranging Eq. 42.4 as follows:

λ ET + H = Qn = (1 − α) Rd + Ld − Lu − G − C − Ad −

∂W (42.4a) ∂t

which elements of Eq. 42.4a appear in the estimation of Qn depends on the surface assumptions, time step, and data availability: C, Ad, and ∂W/∂t are generally ignored, as is G on daily or annual time steps. For a water body, evaporation lE is expressed as a residual of the energy balance of the entire water body as a control volume from:

λ E = Rn + Ln + Qv + Qb − Qw − H − Qx

(42.5)

where lE is the latent heat flux (for open-water evaporation, we no longer discuss lET, as there is no transpiration); Rn is the net shortwave radiation [or (1 – α)Rd from Eq. (42.4)]; Ln is the net longwave radiation [or Ld – Lu from Eq. (42.4)]; Qv is the net advected energy to the water body from surface and

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42-4     Evapotranspiration and Evaporative Demand

Figure 42.1  Instantaneous energy balance at an evaporating surfaces. All fluxes and heat-storage changes are in flux units (W m−2). The grey box represents the control volume to and from which all fluxes pass and within which all heat-storage changes are considered.

groundwater inflows and outflows and direct precipitation (Prcp); Qb is the energy exchange from bottom sediments to the water body; Qw is the energy advected away from the body of water to the atmosphere by the evaporated water; H is the sensible heat flux convected and conducted from the water body; and Qx is the change in stored energy in the water body [∂W /∂t in Eq. (42.4)] (all terms are W m−2). The flux Qw occurs over land but is rarely estimated because it is small, and the temperature of the evaporating water is usually not known. For open-water applications Qw can be estimated based on the difference between the water surface temperature and a base temperature [see Eq. (42.29)]. Complete knowledge of Qn requires extensive instrumentation not generally available on an operational basis: much parameterization is therefore required, and this often relies on, or is mixed with, observations of surrogate variables (Morton 1983a; Allen et al., 1998; Brutsaert, 2005). Advective Limit The advective limit describes the ability of the dynamic boundary layer to absorb and bear away moisture, and is described following Dalton’s (1802) general form of the mass transfer equation for evaporation from water:

E = M(esat − ea ) (42.6)

where E is the evaporation rate; esat and ea are the saturation vapor pressure at the temperature of the water surface and the actual vapor pressure of the air, respectively; and M is a mass transfer coefficient that is a function of wind speed, atmospheric stability, surface roughness, thermally induced turbulence, barometric pressure, and the density and viscosity of the air. Expressing M as a “vapor transfer function” or “wind function” f  (Uz) of wind speed Uz (m s−1) at height z (m) above the surface that describes the variation of vapor transfer of the air as a function of Uz gives us a generic expression of the advective limit to ET from all surfaces:

λ ET ≤ f(U z )(esat − ea ) (42.7)



where λET is in W m ; esat and ea (Pa) are estimated from the Clausius– Clapeyron equation at T and dewpoint temperature (Tdew), respectively (both in °C); and f(Uz) is in W m–2 Pa–1. [Note the different temperatures at which esat is estimated in Eqs. (42.6) and (42.7): in Eq. (42.6), esat is estimated at the water surface temperature, whereas in Eq. (42.7), it is at the air temperature, commonly measured at a 2-m height.] The RHS of Eq. (42.7) is also known as the drying power of the air (EA). The accurate parameterization of the advective component is essential to any physically based ET or E0 model (Hobbins et al., 2001a, 2001b; Sugita et al., 2001). f(Uz) has many functional forms depending on site-specific factors and experimental design; a complete expression for f(Uz) under neutral conditions—that is, a stable atmospheric boundary layer—is given by Brutsaert and Stricker (1979), Eq. (42.17). However, due to the complexities resulting from the effects of atmospheric instability acting over short time periods and onerous data requirements, such theoretical formulations are generally dropped in favor of empirical relationships, such as Penman’s (1948) simple, empirical, linear U2 function (m s−1). The relationship between ET, E0, and their meteorological and radiative drivers may be summarized thus: λ ET ≤ λ E0 = g (Qn ,U z ,T , ea ) (42.8) –2

42_Singh_ch42_p42.1-42.18.indd 4

where the function g(…) represents a Penman-like combination of the advective and radiative drivers expressed in terms of the input variables. ET is then always constrained to the lowest of the hydrologic limit [Eq. (42.3)] and the function g(…) in Eq. (42.8). Uses of E0 There are three traditional uses of E0, and three general types: Epan; Ep; and ET0. The primary use of E0 is as a starting point in the estimation of ET, generally through some parameterization of Θ that depends on the specific use for the required ET estimate: traditionally, LSMs scale ET from Ep as a function of a model state that tracks soil moisture, or, where vegetation is present, the transpiration (the dominant term in ET) is parameterized with a bulk stomatal resistance that reflects water availability; remote sensing models commonly scale ET from Ep or ET0 based on modeled surface energy balances or vegetation indices. A second use is in irrigation scheduling and consumptive use estimation in agriculture, wherein ET0 estimates are scaled by applying coefficients specific to crops, their phenology, soil stress, etc. (see Sec. 42.2.4). The third, more recent use of E0 has developed within the climatological community, which has recently focused on its long-term trends (particularly in Epan, due to its long-standing and widespread global data record), and as a diagnostic indicator of the dryness of the lower atmosphere and its causes (see Sec. 42.3.2). 42.2.2  Climatologic Estimators of ET

Budyko (1974) realized that ET couples the water and energy cycles, thus examining ET in the context of its water and energy limits (see Sec. 42.2.1) would permit an ecologically useful analysis of the variation of large-scale, long-term hydroclimatology. This is best expressed graphically (see Fig. 42.2), by displaying the relation between two indices representing hydroclimatic behavior across temporal scales long enough to allow steady-state conditions to prevail and spatial scales large enough for influences of climate on the landsurface/climate interactions to fully develop. The dryness index Φ (on the x-axis) is the ratio of E0 [where E0 is generally estimated from a radiationbased parameterization, commonly the Priestley–Taylor equation; Priestley and Taylor (1972)] and Prcp; the evaporative index ε (on the y-axis) is the ratio of ET to Prcp. He proposed a relation between the two indices through an empirical relationship now known as the “Budyko curve,” as follows:

 1 ε B = Φtanh   (1 − cosh Φ + sinh Φ) (42.9)  Φ

Figure 42.2 shows the relation between Φ and ε in theory and as applied to 229 Australian basins from Donohue et al. (2010). The horizontal extent represents the hydroclimatic spectrum: water-limited to the right; energy-limited to the left. In water-limited (arid-tending) areas, ET is limited by water availability: it is asymptotic to, but generally cannot exceed, Prcp. In energy-limited (humid-tending) areas, ET is asymptotic to, but cannot exceed, E0. The Budyko curve represents the variation in the behavior of ET between these two limits, and so can be used to define long-term, large-scale ET in regions where E0 and Prcp—both generally more available than ET—are known. Not

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Relevant Concepts and Methods    42-5 

Figure 42.2  The Budyko (1974) concept in theory and practice. Solid straight lines denote the water and energy limits on ET (as labeled), to which the behavior of ET is asymptotic. The curve shows Budyko’s idealized relation between Φ and ε. The vertical arrows represent the ET and Runoff portions of Prcp, respectively. The circles represent the behavior as observed in 229 Australian basins by Donohue et al. (2010).

only can ET be estimated (lower vertical arrow in Fig. 42.2) in an ungaged region, but so too can Runoff (i.e., Runoff = E0 - ε. Prcp), as the angled dashed line represents the totality of the hydrologic cycle (recall the large-scale, longterm constraints in which the paradigm operates). While Budyko (1974) was able to close water budgets to within 10%, Fig. 42.2 also indicates scatter of basins’ behavior around the curve (there are other curves developed). This so-called “Budyko scatter” (δ) is the difference between the evaporative index predicted by the Budyko curve (εB) and that observed, or δ = ε - εB = (ET - ETB)/Prcp, where ETB is the Budyko-predicted ET. The absolute value of δ increases at lower time and space scales that increasingly contravene the central assumption of long-term regionality. Attempts to increase the ET-predictive power of these relations (i.e., to decrease δ) at smaller time and space scales involve characterizing the basin surface, including land cover, soil, and topography; but vegetation dynamics and particularly storage effects must also be accounted for (Milly, 1994; Donohue et al., 2010). The Budyko (1974) framework remains a useful paradigm for hydrologists: for example, Szilagyi and Jozsa (2009a) used meteorological and radiative data in the combined contexts of the CR (Sec. 42.2.3) and Budyko’s coupled water/energy balance to derive various ecosystem characteristics— mean effective relative soil moisture and maximum soil water storage available to plants—useful for predicting ecosystem responses to climate change. 42.2.3  Complementarity of ET and E0

The first considerations of feedbacks across the land surface-atmosphere interface linking both the supply of and demand for evaporated moisture from the land surface led Bouchet (1963) to hypothesize a complementarity between variations in regional-scale ET and those in point-scale E0. While his CR hypothesis still lacks a rigorous proof, it has nevertheless been observed across many hydroclimates and regions (e.g., Hobbins et al., 2004) and models predicated upon it have proved useful in estimating regional-scale ET in areas where moisture data are scarce (e.g., Morton, 1983a; Hobbins et al., 2001a; 2001b). Briefly, Bouchet (1963) proposed that, considering a specified energy budget, the passage of air masses over land surfaces modifies them according to the proportion of net radiative energy transferred back to the atmosphere as latent heat (lET). Decreases in ET due to reductions in water availability lead to an energy release as an increase in sensible heat (H in Fig. 42.1) and hence to temperature increases and humidity decreases of the dynamic boundary layer, which lead to an increase in the drying power of the air (EA) through an increase of vapor pressure deficit (VPD, or esat – ea) and ultimately to an increase in E0. Less-important effects relate to changes in albedo (which determines the shortwave radiation balance) and the surface temperature (which affects the longwave radiation balance). The CR is indicated in Fig.  42.3, where the Ew term is the theoretical ET rate for a large-scale wet environment. As moisture availability increases, the opposite occurs: less of

42_Singh_ch42_p42.1-42.18.indd 5

the energy available goes to H, leading to smaller increases in T, VPD, or EA, and thus smaller increases in E0 above Ew. At the limit when moisture is completely available (i.e., Θ = 1 in Fig. 42.3), ET is at its maximum, and converges with E0 on Ew. Required for the relationship are a regional-scale homogeneity of land surface and water availability and a time-scale sufficient for complete development of the aforementioned feedbacks between surface moisture and the dynamic boundary layer: Brutsaert and Stricker (1979) state that scales on the order of thee days and hundreds of meters are sufficient. The original hypothesis (Bouchet, 1963) for the CR held that:

ET + E0 = kEw (42.10)

where the value of the dimensionless constant k is 2. This value implied that all the energy released due to moisture limitations is taken up by E0, leading to a direct, symmetrical feedback between the energy released at the surface (q2, from ET decreasing) and the increase in the demand for latent heat flux (q1, leading to E0 increasing)—that is, q1/l = q2/l in Fig. 42.3. However, recent work has uncovered different feedback strengths depending on the particular E0 (i.e., ET0, Ep, and Epan) and Ew used, leading to discussion of asymmetrical (q1/l > q2/l) complementarity (Szilagyi, 2007; Huntington et al., 2011). A generic expression for the CR that permits both symmetry and asymmetry can be given as:

bET + E0 = (1 + b)Ew (42.11)

where b is the scaling factor (dimensionless), taking a value of 1 for symmetry, and exceeding 1 for asymmetry (see Fig. 42.3). The CR hypothesis has led to a family of ET models that break with the traditional conception of E0: that is, one that ignored feedbacks and interactions across the land surface-atmosphere interface and held E0 to be independent of ET. In this traditional paradigm, ET is generally modeled either as a simple, often empirical function of E0 derived from pan evaporation (Epan) observed at nearby weather stations, or by ground-based models that tend to rely on gross assumptions regarding processes in the soil, vegetative, and atmospheric systems. Instead, CR models rely on these surface-atmosphere feedbacks and thus bypass the poorly understood dynamics within each component, and that incur minimal data requirements as to the nature of the land surface that bedevils traditional ET estimation. They can be applied across unmetered basins at a regional spatial scale and so can be of particular use to water managers. Widely used examples of such models are the WREVAP suite of models for areal and lake evaporation (Morton et al., 1985) and the Advection-Aridity approach (Brutsaert and Stricker, 1979). In the field of drought-monitoring, attention is turning to the use of the CR to uncover ET-related drought dynamics that may be more easily detected as a signal in E0 (see Sec. 42.3.3). Further, as we show in Sec. 42.3.2, consideration of the CR

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42-6     Evapotranspiration and Evaporative Demand

ET and E0 rates (normalized by Ew)

6

5

4

Increasing b

3 E0 2

0

Ew

q1/λ

1

q2/λ

0

0.1

ET 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Notional moisture availability, Θ (saturation fraction)

0.9

1

Figure 42.3  The CR hypothesis showing ET and E0 converging on Ew at high moisture availability, diverging at low moisture availability. The depth equivalent of the energy released by decreasing moisture availability limiting ET below Ew is indicated as q2/l, and that released to the overpassing air as q1l. b varies from 1 (for symmetrical relationship, indicated by solid grey line for the resulting E0) to 5 (b from 2 to 5 indicated by dashed grey lines for resulting E0).

resolves the long-standing “evaporation paradox,” wherein Epan is observed to decline around the globe in the face of a global ET rate that must be rising to supply the increasing global Prcp and cloudiness (Brutsaert and Parlange, 1998). In an analysis of long-term trends in basin-budget derived ET (and therefore independent of the CR) in 655 basins representing 19.4% of the continental US (CONUS) by area, Hobbins (2004) found that the CR hypothesis explained trend directions in 92% of the basins, compared to only 57% basins explained by the traditional paradigm. Finally, McMahon et al. (2013) provides an excellent literature survey of the development and current state of the science and application of the CR in the estimation of multiday ET and lake evaporation and in studies of hydroclimatic change. 42.2.4 Reference ET Concept

The most popular method to estimate ET from cropped surfaces (crop ET, ETc) is via the concept of reference evapotranspiration (ET0). This concept obviates direct knowledge of moisture availability at operational scales, asking instead “assuming well-defined, ideal surface conditions, what would be the moisture supply from the land surface to the atmosphere?” Answering this question provides ET0, the theoretical ET for an idealized, well-watered reference surface—an upper limit constrained only by the advective and radiative drivers. Additional assumptions as to the prevailing soil- and vegetationmoisture conditions and the vegetation mix and phenology are then applied to ET0 to account for the divergence from the well-defined reference conditions and thereby scale ET0 to ETc. This divergence may arise due to a variety of factors: crop factors (e.g., type, variety, and development stage), management factors (e.g., management of disease, pests, and soil), and environmental factors (e.g., salinity and water stress). The ETc so derived is primarily used in irrigation scheduling and in water demand models. ETc may be met by water stored in the soil profile, Prcp and, when ETc exceeds available soil water, by irrigation scheduled to meet the deficit, so that ET reaches ETc. Beyond these traditional uses, ET0 (the starting estimate for ETc) is now used in combination with remotely sensed vegetation indices and land-surface temperature (LST) to estimate actual ET from crops and natural landscapes. In this section, we summarize the main points of the concept of ET0 and its use to derive ETc so as to raise the issues that need to be addressed by users. We do not attempt to detail the procedures, as this is well covered in the seminal material—particularly the U.N. Food and Agriculture Organization FAO-56 standard (FAO-56; Allen et al., 1998) and the American Society of Civil Engineers (ASCE) Environmental Water Resources Institute (EWRI) Standardized Reference ET Equation report (ASCE05; Allen et al., 2005). The intent of these reports is to resolve uncertainties arising from the variety of ET0 model inputs and applications and to codify approaches to ET0 estimation

42_Singh_ch42_p42.1-42.18.indd 6

internationally (FAO-56) and for the United States (ASCE05); here we cite their specific sections where applicable. In these approaches, ET0 is derived using the PM formulation for ambient meteorological and radiative conditions either measured over the cropped surface in question, at a nearby agrometeorological station, or, increasingly, derived from meteorological reanalyses. ET0 assumes a reference surface: a healthy, well-fertilized, extensive crop of either short grass (0.12 m high) or alfalfa (0.5 m high), completely shading the ground, actively growing under well-watered conditions, of specific albedo (0.23), and with a stomatal resistance rs of 70 s m−1 and aerodynamic resistance ra of 208/U2 s m−1 (U2 is 2-m wind speed). It further assumes no advected energy from water or heat storage effects. This physically based formulation retains the Penman (1948)based combination form but includes in its advective driver a parameterization of the fine-scale diffusive characteristics of the plants, replacing Penman’s (1948) linear approximation of the complex vapor transfer process by a “big leaf ” approximation. In this approximation, bulk resistances parameterize moisture availability and vegetation roughness: water vapor overcomes first the rs to diffusion from the leaf stomates into the canopy, thereby capturing moisture availability, and then overcomes the ra from the canopy into the overpassing air, thereby capturing vegetation roughness. Many smaller terms drop away from the radiative driver Qn (see Fig. 42.1), leaving Rn, Ln, and G, which are variously observed, ignored at various time steps (e.g., G at a daily time step), or parameterized as functions of other variables [e.g., Ln as a function of T, ea, and extraterrestrial solar radiation, itself a function of solar and orbital geometry as described in Eq. (4.4.4) in Shuttleworth (1992)]. The ASCE05 approach standardizes both ET0 estimation and the transferability of crop coefficients for agriculture and landscape use. Its general expression is C γ n 86400 0.408∆ e −e T ( Rn + Ln − G) + ET0 = U 2 sat 3 a 106 10 ∆ + γ (1 + CdU 2 ) ∆ + γ (1 + CdU 2 ) (42.12) where ET0 is the standardized reference ET (mm day–1) for 0.12-m grass (in which case it is commonly abbreviated as ETos) or 0.5-m alfalfa (commonly ETrs) crop surfaces. The 0.408 coefficient (m2 mm MJ−1) represents the inverse of the latent heat of vaporization; Δ and γ are in the same units (commonly Pa K–1); the 86,400/106 term converts Rn, Ln, and G from W m–2 to MJ m−2 day−1; the 1/103 term converts the last term (in ea and esat ) from Pa to kPa. Values of the “numerator constant” Cn (here in K mm s3 Mg−1 day−1 for

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Relevant Concepts and Methods    42-7 

daily time steps) and the “denominator constant” Cd (s m−1) specific to time step and the choice of grass or alfalfa, as well as the derivation of all other variables, are detailed in Allen et al. (2005). The FAO-56 equation, which yields identical results on a daily time step, is shown as Eq. (6) in Allen et al. (1998). As we discuss in Secs. 42.3.4 and 42.3.6, simpler formulations for ET0 (e.g., Blaney–Criddle and Hargreaves) are no longer recommended for use. To estimate ETc from ET0 various user-selected coefficients that vary from ~0 to 1.4 are applied to account for the difference between the idealized reference condition (i.e., ETos or ETrs) and field crops under “standard conditions” (i.e., free from environmental or water stresses) must be accounted for. These differences may arise due to variations in field crop height (which affects ra and vapor transfer), canopy-soil albedo (which affects Qn), the stomatal resistance rs, and evaporation from soil; they are accounted for using the “single crop coefficient” approach, as follows: ETc = K c ET0 (42.13)



where Kc is the crop coefficient (dimensionless), which varies with crop type, its location, and development stage (see Fig. 42.4). The calibration of Kc values for particular crop types, regions, and development stages is the subject of a vast literature, but representative values are summarized in Chapter 6 of the the FAO-56 report (Allen et al., 1998) and in an earlier FAO report known as FAO-24 (Doorenbos and Pruitt, 1977). For standard conditions but when the proportion of evaporation from bare soil may be significant, such as for capturing specific wetting events (irrigation or Prcp), the “dual crop coefficient” approach is used:

ETc = ( K cb + K e )ET0 (42.14)

where Kcb is the basal crop coefficient, which accounts for transpiration from the crop; and Ke is the soil evaporation coefficient (both Kcb and Kc are dimensionless). The estimation of ETc using a dual crop coefficient approach is covered in detail in Chap. 7 of the FAO-56 report (Allen et al., 1998). For crops grown under nonstandard conditions—whether due to soilwater stress or salinity, nontypical or nonpristine vegetation types, manageKc 1.4

Kcmid

1.2

ET24i = ET0 Fi × ET0_ 24i (42.16)



where ET0_24i is the 24-h ET0 for day i; and ET0Fi is the fraction of ET0 for the ith day. Time integration of ET over an extended time period is estimated from: m ET = ∑ ET0 Fi × ET0_ 24i (42.17) i =n where n and m are the first and last days of the desired time period, respectively. In irrigated and natural, well-watered environments (i.e., riparian and wetland areas) the use of a PM -based ET0 such as ETrs is recommended for time integration due to the ability of ET0 to capture the potential effects of advection on ET. For vegetated surfaces where no net transfer of advected energy exists, such as deserts and rangelands, an evaporative fraction (EF) approach is recommended for estimating ET24 and time integration similar to that employed in the Surface Energy Balance for Land (SEBAL, Bastiaanssen et al., 1998). In this case, the instantaneous EF is defined as: λ ETinst EFinst = (42.18) ( Rn + Ln − G )inst where lETinst and (Rn + Ln - G)inst are estimated through the satellite energy balance model; and 24-h ET (ET24i) is estimated as:

1 0.8 0.6

thermal radiance and optical reflectance, incoming solar radiation, vegetation indices, and other surface parameters (Allen et al., 2007; Anderson et al., 2012). Net radiation is estimated from incoming longwave and shortwave radiation, while satellite-derived estimates of surface temperature, emissivity, and surface albedo are used to calculate outgoing longwave and shortwave radiation. Ground heat flux is commonly estimated using satellite-derived net radiation, surface temperature, and vegetation indices. In the widely used Mapping EvapoTranspiration at high Resolution using Internalized Calibration (METRIC) model, sensible heat flux is estimated with LST and the Calibration using Inverse Modeling at Extreme Conditions (CIMEC) procedure (Allen et al., 2007). Once Rn, Ln, G, and H are estimated, lET is calculated as a residual of the energy balance (Eq. 42.4). The instantaneous ET rate at satellite overpass time (ETinst) is calculated by dividing lET by l. ETinst is then used to compute ET0F per satellite image pixel, where estimates of ET0 at the time of the satellite overpass are derived from a local weather station or gridded weather dataset. The ET0F is analogous to an instantaneous crop coefficient. ET for the 24-h period (ET24) for day i is typically estimated as:

Kcini

0.4

Kcend

0.2 0

Time (days) Initial

Crop development

Mid season

Late season

Figure 42.4  The development of Kc through the growing season for a notional crop, showing crop growth stages. [Source: http://www.fao.org/nr/water/cropinfo_sugarcane. html]

ment practices, or simply during the non-growing period—further coefficients apply to account for deviations from the standard, well-watered conditions. For example, under soil water-stressed conditions, Kc in Eq. (42.13) and Kcb in Eq. (42.14) are multiplied by the water stress coefficient Ks, which ranges from 0 for zero water availability to 1 for fully irrigated conditions. ETc-estimation procedures for nonstandard conditions are detailed in Part C of the FAO-56 report (Allen et al., 1998). 42.2.5  Satellite Remote Sensing of ET

Satellite remote sensing is commonly used to estimate ET from agricultural areas by scaling grass or alfalfa standardized reference ET0 as:

ET = ET0 F × ET0 (42.15)

where ET0 is the reference ET; and ET0F is the fraction of the respective ET0. ET0F is typically estimated from remotely sensed vegetation indices or a LSTbased surface energy balance (SEB) model. Most SEB models estimate required variables from a series of calculations involving weather data,

42_Singh_ch42_p42.1-42.18.indd 7

ET24i = EFinst ( Rn + Ln − G)24i (42.19)

Daily time series of per-pixel ET0F and EF are commonly estimated between image-acquisition dates through linear or spline interpolation of ET0Finst and EFinst values. Time-interpolated ET0Finst and EFinst values are multiplied by daily ET0 or empirical 24-h average Rn + Ln and G estimates (Brunt, 1932; de Bruin, 1987) to estimate per-pixel daily ET time series. Empirical models for estimating ET and ET0F are commonly based on satellite-derived vegetation indices. Heilman et al. (1982) proposed a linear relationship between a vegetation index and ET0F, which has been supported by various other studies (Tasumi et al., 2005; Calera-Belmonte et al., 2005). If no local calibration data exists, Allen et al. (2011a) suggest that ET0F can be generally estimated as: ETrs F = 1.25 NDVI (42.20) or ETos F = 1.25 NDVI + 0.2 (42.21)



where ETrsF and ETosF are the relative fractions of alfalfa and grass reference ET, respectively. NDVI is the normalized difference vegetation index, defined as:

NDVI = ( ρNIR − ρRed ) ( ρNIR + ρRed ) (42.22)

where ρ is the at-surface or at-satellite (i.e., top of atmosphere) reflectance; NIR is near-infrared waveband from ~0.7 to 1.3 μm; and Red is the visible waveband from ~0.60 to 0.70 μm. Due to differences in satellite wave bandwidths and the use of at-surface or at-satellite reflectances for computing NDVI, Eqs. (42.20) and (42.21) are general relationships and should be used with caution, especially in areas of low vegetation cover, or where bare soil evaporation is a significant component of ET. 42.2.6  Eddy Correlation

Up to tens of meters above the land or water surface, vertical transport is mostly driven by turbulence. Turbulent eddies near the surface cause deviations about the mean values of vertical wind w’, specific humidity q’, and

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42-8     Evapotranspiration and Evaporative Demand

temperature θ’. On average, there are positive correlations between high vertical wind and humidity deviations, negative correlations between low vertical wind and humidity deviations. By exploiting these correlations, estimates of ET can be made through direct measurements of the deviations in vertical wind speed and specific humidity at frequencies of 5–20 Hz, and computing the cross-correlation between q’ and w’ for averaging periods that usually range from 15 min to 1 h depending on surface conditions and the surrounding environment. The integrated product of the means of instantaneous deviations q’ and w’ results in the expression of ET from advective vapor transport by turbulence (Swinbank, 1951) as: ET = ρa q′w ′ (42.23) where ra is the density of air; q’ and w’ are, respectively, deviations of specific humidity and vertical wind speed about their mean values; and the overbar denotes averages taken over the time period considered. Sensible heat flux may also be computed following the same theory as: H = ρac p θ ′w ′ (42.24) where cp is the specific heat of air. Measurements of w’, q’, and θ’ require specialized instruments but are widely available and used in practice, such as a three-dimensional sonic anemometer, high-frequency hygrometers and gas analyzers, and fine wire thermocouples to measure w’, q’, and θ’, respectively. Numerous corrections are required to account for temperature and humidity effects on air density and eddy structure, instrument separation, frequency response, coordinate rotation, and hygrometer type (Fuehrer and Friehe, 2002). Net radiometers, ground heat flux plates, and soil moisture and temperature sensors are typically collocated with EC instrumentation to evaluate the land-surface energy imbalance (see Sec. 42.3.5). For details on the EC theory, see Foken (2008a) and Brutsaert (2005), and for EC networks, such as FLUXNET and EUROFLUX, see Baldocchi et al. (2001) and Aubinet et al. (2000). For detailed discussions on measurement accuracy, and recommended reporting requirements and documentation, see Allen et al. (2011a; 2011b). 42.2.7 Scintillometry

The scintillometry technique estimates H by relating scattered electromagnetic radiation due to atmospheric turbulence to the structure parameter of the refractive index of air, Cn2 (Wang et al. 1978). Scintillometry-estimated H (Hsc) is obtained from: H sc = ρac pT *u* (42.25) where T * is the temperature scale; and u* is the friction velocity. T * can be derived relying on MOST relationships and the structure parameter of temperature following Wyngaard et al. (1971). Several variants for estimating T * are commonly used depending on the study purpose and design. For more detailed background and recent developments in scintillometry see Moene et al. (2009). Recent novel applications of scintillometry include the following: estimation of open-water evaporation (McGloin et al., 2014; McJannet et al., 2011); comparison to EC measurements of H and ET (Hoedjes et al., 2007; Liu et al., 2013); validation of remotely sensed ET (Marx et al., 2008); validation of catchment-scale hydrologic modeling (Samain et al., 2011); and development of a scintillometer network designed to support remote sensing, hydrologic, and meteorological models (Kleissl et al., 2009). The main advantage of scintillometry is that it can be used to estimate aggregated H fluxes over different scales, and can therefore be used to estimate aggregated ET if reliable available energy fluxes are acquired over the same scales. 42.2.8  Water Balance Estimates of ET

Mass balance ET may be estimated for a basin or a water body by taking advantage of the law of conservation of mass across a naturally defined control volume. “Basin-budget” or “water balance” ET is estimated as a residual of the water budget—that is, the expression of all fluxes and state-changes (ET, Prcp, groundwater recharge, surface and subsurface runoff, diversions, and storage changes) estimated or observed—across the spatial extent of a basin or control volume of a water body (see Sec. 42.2.9) at a tractable time step (time and space scales for which fluxes and storage changes are quantifiable). For a basin, the expression of the natural water balance that yields the estimate of ET (shown in Fig. 42.5 as the sum of E from water surfaces and ET from vegetation and bare soil, as well as the sublimation from snowpack that is not shown) is as follows: ET = Prcp - Runoff - GW + D - ΔS (42.26) where all fluxes and storage changes are in units of time-rate changes of depth (e.g., mm day-1) and in the directions specified. Prcp is spatially averaged over the basin. Runoff is the streamflow recorded by streamgage, and is assumed to repre-

42_Singh_ch42_p42.1-42.18.indd 8

sent the sum of surface and subsurface runoff fluxes from all areas within the basin upstream of the gage (surface-water outflow less surface-water inflow in Fig. 42.5). Streamflow is generally assumed to be one of the most accurate hydrologic measurements, but any differences between the stream gage contributing area and the basin boundaries must be accounted for. GW is the surface water transferred to regional  groundwater and is unrecorded at the streamgage; it is generally ignored (and difficult to quantify), but can be a significant unmeasured quantity depending on the system (groundwater outflow less groundwater inflow in Fig. 42.5). D is the net diversion of water into the basin from pipelines and canals; depending on the basin, this may be zero or it may contribute a large portion of the water balance (imported water less exported water in Fig. 42.5). ΔS is the net basin-wide increase in water storage within the basin across the time step. This storage may be in lakes (or reservoirs), rivers, snowpack, and both the saturated and unsaturated soil zones (depth below and height above the water table in Fig. 42.5, respectively), and it may be a function of the hydrologic, climatic, geologic, and pedologic characteristics of the basin. This term is generally ignored at the annual time step, but will generally be significant at subannual time scales. The concept of water years (in the United States, they are defined as starting October 1 of the previous year) is used to minimize the effects of interannual ΔS. Evaporation from a water body E can be estimated through the water balance, as follows: E = Prcp + SWin + GWin - SWout - GWout - B - ΔS (42.27) where Prcp is precipitation onto the water surface; SWin and GWin are surface and groundwater inflows, respectively; SWout and GWout are surface and groundwater outflows, respectively; B is bank storage; and ∆S here is the change in water body storage.
 The uncertainty in water balance E and ET estimates is a summative function of the uncertainties in the input variables, so the fewer input variables the better. The process is simpler for basins that are minimally disturbed, that is, with minimal interyear, in-basin storage and no trans-basin water transfers [e.g., the basins of the Hydro-Climatic Data Network of Slack and Landwehr (1992)]. Then further assuming no net groundwater losses (i.e., GW = 0) and that the gage is sited at the natural outlet, it is possible to reduce the water balance down to ET = Prcp – Runoff. The primary drawback of the water budget approach is the time step required to reduce uncertainties to an acceptable level and that, in the case of basin budgets, the ET so derived is a lumped value across the spatial extent of the basin. Often the primary use of the mass balance approach is to provide a physical ET (or E) estimate against which other estimation procedures may be calibrated (such as for groundwater recharge). 42.2.9  Open-Water Evaporation

Open-water evaporation E is one of the most difficult surface energy fluxes to quantify and is rarely directly measured in the natural environment. Common indirect techniques include pan evaporation measurements combined with application of pan coefficients, water budget, energy budget, mass transfer, and a combination of energy and mass-transfer techniques. The EC technique is considered the most accurate direct approach, given ideal environmental conditions, water body physical setting, and experimental design (Brutsaert, 2005). The water budget technique is considered the most accurate indirect approach, especially in arid environments, where evaporation is a relatively large water budget component (Morton, 1994). Primary factors governing E include net radiation, heat storage, air temperature, water surface “skin” temperature, humidity, wind speed, stability of the atmosphere, advection of water and heat into and out of the water body, and salinity. Additionally, the aerodynamics of the water surface, turbidity of the water, and inflow and outflow rates control the rate of transfer between energy balance variables. All of these factors are important to consider when deciding which technique is most appropriate for application. In most cases, the lack of adequate hydrologic and meteorological data limits the application of physically robust techniques for estimating open-water evaporation. Pan Evaporation The most common operational technique for estimating E is the pan evaporation approach: E = Kp Epan (42.28) where Kp is a pan coefficient (dimensionless); and Epan is measured pan evaporation, typically from a standardized pan such as the U.S. Weather Bureau Class A pan (this is the standard in the United States; China has a micropan standard). Epan measurements reflect the combined effects of net radiation, air and water temperature, humidity, and wind speed on a point-scale reference open water surface. Kp adjusts Epan to more accurately represent the physical conditions of the water body of interest and to account for the fact that the pan has limited heat storage potential and is exposed to the air on its sides and bottom, allowing for heat exchange and thus affecting the energy balance. Annual Kp values

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Relevant Concepts and Methods    42-9 

Figure 42.5  Components of a water balance for part of a watershed, indicating inputs, outputs, and storages. [Source: Healy et al. (2007). Used with permission.]

typically range from 0.65 to 0.85, while monthly Kp values can range from 0.3 to 1.7 depending on water body characteristics, such as depth, turbidity, and potential for heat storage. In addition to considering potential effects of heat storage for the water body, appropriate Kp values depend on pan type, ground cover, evaporative conditions within the fetch of the pan, presence of aquatic plants, and salinity of the water body. While Epan measurements are still widely used, they are very susceptible to microclimatic conditions surrounding the pan (including freezing, which limits their operations in time and space) and subject to the care of station maintenance (Thom et al., 1981). Further, some limitations apply as to the appropriate minimum time scale of observations (Roderick et al., 2009a). Ideally, the pan technique relies on locally calibrated monthly and annual Kp values. For a review of considerations and techniques for selecting Kp values, see Grayson et al. (1996) and Alvarez et al. (2007). Water Budget Evaporation can be expressed as a residual of the water budget volume or depth per unit time following the continuity equation for a generalized water body as shown in Sec. 42.2.8. Estimating open-water evaporation as a residual of the water budget is sometimes considered the most accurate approach for large water bodies in arid environments due to the fact evaporation is a large component of the water budget in such cases, and errors in estimated terms are small and insignificant relative to E. Energy Budget This approach has been most widely used in research, and it is likely the most data-intensive approach for estimating evaporation due to the need to consider the entire water body as a control volume rather than as just the surface in the case of a land surface energy balance. This requirement brings about

42_Singh_ch42_p42.1-42.18.indd 9

many issues including the need to estimate net advected energy to the water body, energy advected from evaporating water, energy exchange from bottom sediments, and–most difficult to estimate–the change in heat storage. Solar radiation penetrating and absorbed by the water body affects the magnitude and timing of E. Energy storage depends on climate and physical characteristics of the water body, such as volume, depth, geometry, clarity, and surrounding environment. For shallow and turbid water bodies, energy storage effects on E are small compared to deep, clear water bodies. Stored energy can be partitioned into E, heating of the air, emission of longwave radiation, or advection of heat in the discharging water. Most energy storage occurs during spring and summer months and is not readily available for E until conditions are such that it is transferred to the surface by conduction or convection, at which point the water surface temperature may exceed the air temperature, thereby causing a large portion of energy to be partitioned from stored heat into sensible heat or longwave emission rather than E, reducing the total evaporation from the water body. The change in energy storage is typically estimated through repeated thermal profile measurements of the water column; smaller energy budget periods require more frequent thermal profile measurements. For water bodies with large depth fluctuations and thermal stratification, heat-storage estimates can be obtained using discrete layer volumes and respective average temperatures to compute the integrated energy storage changes for each layer (Saur and Anderson, 1956; Moreo and Swancar, 2013). Since both H and lET are unknown in the energy budget equation they are expressed in the Bowen ratio [Bowen, 1926; Eq. (42.1) in Sec. 42.1.4], where Ts (the water surface “skin” temperature, °C) is preferably measured with a thermal infrared sensor, and T is measured at a reference height usually 1–3 m above the water surface [Eq. (42.4)]. b is also commonly estimated over water at two reference heights above the water surface with automatic sensor

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42-10     Evapotranspiration and Evaporative Demand

exchange mechanisms, which reduces sensor bias and drift effects on estimated T and e gradients. Substituting H = blET and Qw into the energy balance equation (Eq. 42.5) yields the commonly used EBBR expression of open-water evaporation: Rn + Ln + Qv − Qx (42.29) E= ρw [λ (1 + β ) + c w (Ts − Tb )] where E is the evaporation rate (m s−1); Rn, Ln, Qv, and Qx are in W m–2; cw is the volumetric heat capacity of water (4186 J kg−1 °C−1 at 15 °C); ρ is the density of water (1000 kg m−3); Ts is the water surface temperature (°C); and Tb is a base temperature usually assumed to be 0°C for convenience. Extended discussions on the EBBR for open water can be found in Winter et al. (2003) and Brutsaert (2005). There is some debate as to whether the EBBR technique for estimating E is useful operationally or for secular analysis: some view it as one of the more accurate techniques for long-term monitoring; others point to many limitations regarding the practicality of accurately estimating the most sensitive variables (Morton, 1994). More accurate instrumentation and advanced automated measurement techniques have become widely used in the last few decades, making research applications of the EBBR especially useful for understanding evaporation processes, developing semiempirical approaches, and comparing and combining energy fluxes with those obtained by other techniques such as EC and aerodynamic mass transfer techniques (Allen and Tasumi, 2005; Moreo and Swancar, 2013). Aerodynamic Mass Transfer The aerodynamic mass-transfer technique is one of the oldest and most common techniques for operational estimation of E due to its simplicity and relative accuracy. A detailed evaluation and sensitivity analysis of commonly used aerodynamic mass transfer equations is given by Singh and Xu (1997a; 1997b). More recent applications of the aerodynamic mass-transfer approach for estimating evaporation rely on turbulent transport theory for estimating vapor flux where

λ ET= λρaCEu1 (qs* − q2 ) (42.30)

and CE is the Dalton number, a bulk transfer coefficient of water vapor; qs* is the specific humidity at the water surface temperature (kg kg−1); q2 is the specific humidity of air at reference level z2 (kg kg−1); and u1 is wind speed at reference level z1 (m s−1). The coefficient CE is only constant for neutral atmospheric conditions and constant surface roughness. CE for neutral conditions can be expressed for a water surface following Brutsaert (2005) from:

CE =

k2 , (42.31) ln( z 2 / z 0m )ln( z1 / z 0h )

where k is the von Kármán constant (0.41); and z2 and z1 are measurement heights of wind speed and humidity (m), respectively; and z0m and z0h are roughness lengths of momentum and humidity (m), respectively. For lakes of medium size with fetches of 1–10 km CE values of ranging from 0.0011 to 0.0013 can be used as a first approximation (Brutsaert, 2005; Tasumi, 2005). For stable and unstable conditions, stability correction factors must be considered. Allen and Tasumi (2005) used Bowen ratio exchange arm measurements of q at two different heights in Eq. (42.30), and results compared favorably to EC estimates of E. A common challenge in applying the aerodynamic mass transfer approach is obtaining accurate weather data for q and u representative of water surface conditions. An advantage of the approach is that the required variables are fairly easily obtainable using floating rafts or buoys, and can be used in an automated, near-real-time fashion to compute operational estimates of E. Combination Methods Combination energy-aerodynamic mass transfer methods are commonly used for land applications (e.g., see Sec. 42.2.4) and are typically based on Penman (1948; 1956) and Penman-Monteith (Monteith, 1965) formulations. When applied to water bodies, the same limitations of the energy balance approach apply to the combination approach in that estimates of Rn, Ln, Qx, and Qv are required, as are representative open-water estimates of es, ea, and Uz. Available energy is often used for open-water evaporation estimates through the use of the Priestley–Taylor equation. Because of the absence of an aerodynamic term in the Priestley–Taylor equation (Priestley and Taylor, 1972) it is considered to represent equilibrium evaporation (i.e., Ew), something that rarely occurs from water bodies surrounded by water-limited environments. For this reason, the use of the Priestley–Taylor equation has

42_Singh_ch42_p42.1-42.18.indd 10

only limited application unless calibrated. Morton (1983b) developed a numerical energy-aerodynamic approach by iteratively solving energy balance and vapor transfer equations to estimate ambient Ep, which is used to obtain an estimate of the equilibrium wet environment surface temperature at which to estimate the slope of the saturation vapor pressure curve (∆) and Priestley–Taylor potential evaporation for a spatially extensive wet surface (i.e., Ew). The relationship between Ep and E is generally complementary (discussed in Sec. 42.2.3); hence, the model is called the Complementary Relationship Lake Evaporation (CRLE) model. The modified Priestley– Taylor equation (Morton, 1983b) used in the CRLE is

E = b1 + b2

∆w ( Rw − Qx ) (42.32) ∆w + γ

where Rw is the net radiation for a water surface; b1 and b2 are calibration constants [Morton (1983b) suggests 13 W m−2 and 1.12, respectively]; and Δw is the wet-environment slope of the saturation vapor pressure curve (kPa °C−1) computed with the wet-environment surface temperature. The energy storage term (Qx) is solved for using an approach outlined by Morton (1983b) and Morton (1986) that simulates the hypothetical, linear, heat storage by lagging absorbed shortwave radiation. Further details on the combination equation are in Brutsaert (2005). For background and development of the CRLE, see Morton (1983b; 1986), with recent applications by Jones et al. (2001) and Huntington et al. (2015). 42.3  OUTSTANDING PROBLEMS AND DIRECTIONS FOR FUTURE WORK

In this section, we examine ongoing research themes and outline directions for future work, including advances in the data availability for ET and E0 and the science of their estimation, with the aim of closing the gap between the well-established science and its practice in operational settings that so often plagues the study and application of ET and E0. In the following sections, we examine an often-obscured practice still rooted in the past (T-based parameterizations of E0), and one that is being dragged forwards into the science (representation of ET and E0 in drought monitoring). 42.3.1 Reanalyses

In the hydrological context, reanalyses are spatially and temporally comprehensive datasets of meteorological, radiative, land-surface, and atmosphere fluxes and states describing the earth-atmosphere system. They are used in long-term (generally several decades), large-scale (continental to global) analyses of weather and climate. Observations of fluxes and states at the earth’s surface and various levels in the atmosphere are objectively combined in a numerical model to synthesize the overall state of the earth-atmosphere system (at several levels from the surface to above the stratosphere). Reanalyses are used to characterize and compare current and past climate, to assist in predictions and near-real-time monitoring (e.g., as in drought) across sectors (drought monitoring, agriculture, water resource management, energy, and insurance) in climatology, and in providing climate services. A significant example of a reanalysis dataset is the North American Land Data Assimilation System (NLDAS; Mitchell et al., 2004), which is commonly used—for example, to verify ET-based drought monitoring tools by Anderson et al. (2013). Figure 42.6 demonstrates a reanalysis dataset (the CRU CL 2.0 Global Climate Dataset) using as examples maps of global mean annual ET and evaporative demand (using ET0). The two fluxes are plotted to the same range so as to demonstrate how ET and ET0 vary and interrelate across the water- and energy-limited spectrum. The spatial pattern of ET reflects the regional availabilities of both water and energy: ET is lowest in dry regions, both hot and cold, and is highest in warm, wet regions. It follows energy availability (mostly latitudinal) when water is available, and water availability otherwise. The spatial pattern of ET0, which assumes ample water availability, reflects both energy availability and its complementarity with ET in water-limited regions. As an example of this interrelation, note that in the water-limited Saharan, Arabian, and Australian deserts, ET is very low whereas ET0 (the water that would evaporate given such dry atmospheric conditions) is very high; contrast that with the energy-limited areas (e.g., Indonesia and the basins of Amazon and Congo rivers), where the availability of water is high and ET and ET0 rates converge, with ET higher than its global mean and ET0 lower than at similar latitudes. 42.3.2  Trends in ET and E0

Understanding the impacts of climate change on ET and E0 is crucial for the operational implications for future regional water budgets and, from a

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Outstanding Problems and Directions for Future Work     42-11 

Figure 42.6  Mean annual (top) ET and (bottom) ET0 across the globe (1961–1990), as determined from the CRU CL 2.0 Global Climate Dataset (University of East Anglia, United Kingdom) reanalysis dataset. ET is estimated using a simple 1-dimensional water balance model, ET0 is estimated using the Penman Monteith formulation of the FAO-56. [Source: FAO. Used with permission.]

research perspective, to understand changing E0 as indicative of lower atmosphere changes and to provide physical evidence-based global climate model (GCM) constraints. Here we summarize ET and E0 trends, identify their drivers, and outline primary research questions. Trends in ET Observations of hydrologic storages and fluxes (including Prcp and ET) and simulations by GCMs (Huntington, 2006) all point to an ongoing and future intensification of the global hydrologic cycle under warming [e.g., Jung et al. (2010) note an increase in globally averaged ET of +0.71±0.1 mm year−2 from 1982 to 1997], while atmospheric water vapor’s residence time (the atmospheric reservoir) is increasing as atmospheric circulation weakens (Bosilovich et al., 2005; Held and Soden, 2006). Prcp affects ET rates through its direct effects on water availability: in water-limited hydroclimates, ET will vary directly with Prcp, whereas in energy-limited hydroclimates the effects of Prcp on ET will be through indirect effects such as changes in cloudiness, albedo, and vegetation effects. Long-term global mean Prcp and ET must balance, but there are spatial distinctions: oceanic Prcp and ET are both increasing; but terrestrial Prcp (mainly in the tropics) and ET are both (mostly) declining. Terrestrially, wet regions are getting wetter and dry regions are getting drier, while an increasingly variable ET–Prcp residual increases the intensity of both floods and droughts. There are also temporal nuances in the hydrologic intensification: Jung et al. (2010) note no hydrologic intensification after 1997, and suggest that soil moisture limitation in the southern hemisphere drives global ET to decline (though nonsignificantly) since then, with other possible mechanisms, such as stomatal closure under increased CO2, land-use change, and decreasing Uz (stilling, see later). Other change drivers include Θ, seasonality, land cover changes, and consumptive use. Few influential studies estimate global terrestrial ET [interested readers should turn to Wang and Dickinson’s (2012) comprehensive review of observation, modeling, and variabilities], thus many vital research questions remain. If a limit to the energy- and T-driven acceleration of the global hydrologic cycle has indeed been reached, will this lead to productivity and terrestrial carbon sink declines and so to a growing surface energy budget increasingly partitioned in favor of H, thereby increasing land-surface warming and intensifying regional land-atmosphere feedbacks? As global Runoff has increased more

42_Singh_ch42_p42.1-42.18.indd 11

than Prcp [from increased ocean evaporation and in places, on land, increased Prcp and/or decreased ET (Labat et al. 2004)], what is suppressing terrestrial ET? An exception to these declining terrestrial Prcp and ET is North America. Observations over CONUS show that Prcp is increasing faster than Runoff; the implied acceleration in ET is supported by observations of basin-budgetderived ET over the last half of the twentieth century (Milly and Dunne 2001; Brutsaert, 2006; Huntington, 2006) and by ET modeling using CR models (Szilagyi, 2001; Hobbins, 2004) and LSMs. Trends vary with model- and observation-type, period, and region, but generally do not exceed 2 mm year−2. Trends in E0 Long-standing, worldwide observations of Epan yield the most widely distributed set of E0 trend analyses and have revealed phenomena central both to E0 and to its drivers. Epan trend studies are notable for their consensus: Roderick et al. (2009a; 2009b) and Fu et al. (2009) summarize studies showing declining Epan almost globally over the last 30–50 years, with a conservatively estimated overall trend of about -2 mm year−2, equivalent to a total decline in radiative forcing of about 4.8 W m−2 over a 30-year period, or an order of magnitude higher than the 0.02 W m−2 year−1 trend in the top-of-atmosphere (TOA) imbalance (Hansen et al., 2005). Drivers of trends in ET and E0 ET and E0 trends are driven by trends in their hydroclimatic drivers and relate to each other within the context of the CR (see Sec. 42.2.3). Both ET and E0 are subject therefore to trends in T, humidity, Uz, and radiation, with ET further subject to water availability trends. Of all driver trends, those in T are the most easily estimated: all else equal, rising T raises E0 through an increase in VPD, which increases ET if water is available. But all else is seldom equal and the effects of T-trends are often overshadowed by other drivers’ trends (e.g., Roderick et al., 2007). Indeed, relying on secular analyses of T alone has led to oversimplistic, incorrect conclusions about E0 and ET trends and to discussion of an “Evaporation Paradox,” which has also been cited in observations that were once difficult to explain: ET and E0 trending in opposite directions in some regions, but parallel in others; decreasing Epan in the face of increasing Prcp (e.g., Cong et al., 2009), cloudiness, GCM-derived ET (e.g., Brutsaert and Parlange, 1998), and T (e.g., Roderick and Farquhar, 2002).

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42-12     Evapotranspiration and Evaporative Demand Table 42.1  Directions of Long-Term Trends in Annual ET, Qn, and EA for 42 Years across 655 Hydrologically Undisturbed Basins across CONUS. Possible Combinations Represent Whether the Combined Trends in Qn and EA Explain the ET Trend Direction Within the Context of (CR) the CR Hypothesis, or (Trad) a Traditional Relationship Between ET and E0, Respectively: � = Explained,  = Not Explained. [Source: Hobbins (2004)] Physically possible combinations?

Trend directions % of basins in combination

EA

Qn

ET

CR complementary relationship

Trad traditional paradigm

0.3

+

+

+





2.8

-

+

+





7.6

+

-

+

×



43.2

-

-

+



×

0

+

+

-



×

0.5

-

+

-

×



2.4

+

-

-





43.2

-

-

-





This paradox has been resolved as the interrelations of E0 and ET and of their drivers have become better understood. Brutsaert and Parlange (1998) propose the CR to explain opposite ET and Epan trends in water-limited environments. Roderick and Farquhar (2002) further explain that, in non–waterlimiting environments, Epan and ET may decline together due to decreasing Qn (dimming, see later). Hobbins et al. (2004) show that both relations apply: (i) ET and Epan are complementary under constant radiative component (Qn) due to variations in the advective component (EA); and (ii) under declining (or increasing) Qn, both Epan and ET decline (or increase) together without contravening their complementarity. In examining whether the directions of trends in independently derived ET (i.e., through a water budget) are physically resolvable with those of its components Qn and EA, Hobbins (2004) found that the CR explained the directions of over 90% of ET trends from 655 basins ( in the CR column in Table 42.1), including 43% whose ET trend directions would have been described as paradoxical under the traditional paradigm ( in the Trad column in Table 42.1), wherein decreasing EA and decreasing Qn together result in increasing ET. Global Dimming Changes in the radiation budget result from changes in both the longwave flux [increasing concentrations of greenhouse gasses such as methane, CO2, and water vapor contribute to a TOA radiative imbalance of 0.9 ± 0.5 W m−2 (Trenberth and Fasullo, 2010)] and the shortwave flux (due to solar variability and changing transmission and reflection due to atmospheric dust, aerosols, clouds, and humidity). Stanhill and Cohen (2001) found evidence of “global dimming” across the second half of the twentieth century: globally averaged Rd fell by 2.7% decade−1 for a total (in 2000) of 20 W m−2. They conclude that aerosols and cloud cover and their indirect interactions are the most likely causes of this dimming; however, they are the most difficult to quantify. A reverse of the dimming—or rebrightening—has been observed in many regions (Pinker et al., 2005; Wild et al., 2005). This turn-around may be due to economic factors and/or clean-air legislation. Wild et al. (2005) anticipate a stronger greenhouse signal being observed from a brighter atmosphere, as the aerosol direct and indirect dimming effects are alleviated. Brightening would have the same effect on surface warming as greenhouse gas-forcing, but would double the impact on the hydrologic cycle (Romanou et al., 2007). The main research questions provoked by dimming point to the need for resolution of the issue of aerosol effects on the hydrologic cycle: what is the direction of the aerosol indirect effect on Rd? Are the dimming and rebrightening phenomena global or regional? What are the effects of rebrightening due to anthropogenic aerosol mitigation: will the hydrologic cycle further intensify as clean-air legislation and energy production reduce regional aerosol loads such as in India and China? What will be their effects on the crucial Asian monsoon? Beyond minimizing observational uncertainties, we also need improved parameterizations of aerosol dynamics and effects in GCMs, particularly of aerosol-cloud interactions.

42_Singh_ch42_p42.1-42.18.indd 12

Stilling Globally, observational studies—for example, in Australia, China, Italy, the United States, Canada, and the Tibetan Plateau [see Roderick et al. (2009b) for a bibliography]—show declining land-surface Uz, or “stilling.” Decreasing Uz leads to decreasing eddy diffusivity and decreasing turbulence, which, all else equal, decrease E0 and reflect a complementary ET increase. Possible mechanisms for stilling are: the oasis effect, whereby irrigation affects local wind circulation in both magnitude and direction; irrigation development resulting in increased surface roughness over large distances; Uz instrumentation affected by local shielding, either by crop growth or by trees planted specifically to reduce Uz over crops; climatological factors acting on a regional or supra-regional scale unrelated to local ET changes. However, the ubiquity of observations of declining Uz argues against stilling being either solely an observational quality problem or an isolated, regional one. Neither reanalyses nor gradients derived from daily Pa surfaces have proved useful in attributing Uz trends to large atmospheric circulation changes, suggesting that daily Uz is not dominated by large-scale atmospheric patterns. Clearly, more research into identifying and quantifying the drivers of declining Uz and its effects on E0 and ET is necessary. 42.3.3  ET, E0, and Drought Monitoring and Forecasting

Evaporative dynamics are generally not well represented in current operational drought monitoring, due to ET being poorly parameterized, whether independently of E0 or derived from E0 by LSMs. Of the direct use of E0 itself as an indicator of the state of the lower atmosphere, there is no representation. When E0 is used, it is as a driver of ET and is mostly derived from T (see Sec. 42.3.4) for reasons of convenience and operational inertia. The state of the lower atmosphere in drought monitors is instead represented simply by Prcp and T; typical of this operational underparameterization of drought characteristics is the Palmer Drought Severity Index (PDSI; Palmer, 1965) that is used widely across the world and underpins much of the U.S. Drought Monitor (USDM), the foremost drought-monitoring tool across CONUS. Many authors have described the shortcomings of the E0 parameterization in the PDSI (e.g., Hobbins et al., 2008; Sheffield et al., 2012). Basically, it relies on an oversimplified relationship between E0 and T that cannot capture physical dynamics in energy-limited areas, or areas where advection is crucial to E0 variability (e.g., the desert Southwest CONUS). In long-term drought-trend analyses it forces more droughtiness than is observed in reanalyses (simply, due to rising T forcing rising E0, whereas E0 has been shown to be falling in general). There is a move in drought monitoring to include ET-based products, which would represent a significant narrowing of the science/operations gap. These are based on a physical understanding of ET processes and recent remote-sensing products (Sec. 42.2.5). While there is as yet no inclusion of E0-based products, the current near-real-time availability of land-based, remotely sensed, and reanalyses-based driving data (see Sec. 42.3.1) is provoking their development. One example is the Evaporative Demand Drought Index (EDDI; Hobbins et al., 2016; McEvoy et al., 2016a), which is showing promise as a leading indicator of agricultural drought across many parts of CONUS at time frames pertaining to both flash (i.e., fast-developing) and extended droughts. Such E0-based indices must be physically based (EDDI is based on ASCE05 ET0). ET-based metrics directly measure a hydrologic flux (ET should decrease in sustained droughts, and may in fact increase at the onset of flash droughts in the period between increasing Qn and/or EA and water availability becoming limiting), but are often only available with considerable latency due to the data-QA/QC procedures necessary to incorporate remotely sensed drivers (e.g., to eliminate cloud effects) and interpolation between satellite overpasses (in the case of nongeostationary satellites). E0-based measures, while they do not directly measure a hydrologic flux (recall that E0 is an idealized flux) nevertheless offer some advantages over ET-based metrics: they are ground-based, and so do not require R/S data streams; they have limited latency (NLDAS data are currently available after ~100 h); and their dynamics respond positively to both flash droughts and sustained droughts. E0 rises both in response to drivers of flash drought (increased T, Rd, or Uz, or decreased ea) and via the CR in response to drivers of sustained drought (decreased Prcp), while the response of ET to drought depends on drought type: ET (like E0) rises in response to drivers of flash drought, but falls (opposite to E0) in response to sustained drought. There is also a potential role for near-real-time E0 estimates in drought monitors (e.g., the U.S. Drought Monitor) as physically based inputs to LSMs and drought models (e.g., PDSI), to replace the T-based E0 estimators currently in vogue (see Sec. 42.3.4). E0 and ET-based measures may act as a complement to each other in drought indices: this is clearly an avenue ripe for further research. As forecast drivers become available at the short term (e.g., weather elements from numerical weather predictors used by weather forecasters, such as the National Weather Service), and at the mid-term (e.g., the Climate

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Outstanding Problems and Directions for Future Work     42-13 

Forecast System version 2, CFSv2), forecasting of E0 becomes possible (McEvoy et al., 2016b). This is a highly desired outcome, with calls from many sectors and at many time scales; for example, irrigation scheduling and drought monitoring at the shortest time scale, or drought forecasting and water resource planning at seasonal time scales. Such a product is the Forecast Reference ET product of the U.S. National Weather Service, which forecasts ET0 at 1- to 7-day lead times, driven by meteorological and radiative forcing fields that are already operationally forecast by numerical weather predictors. 42.3.4  Warning on Temperature-Based E0 Parameterizations

A persistent, significant problem in the operational estimation of E0 (and through LSMs, ET) is the continued use of outdated E0 parameterizations that are typically based on T alone (e.g., Thornthwaite, Hargreaves and Samani, Hamon, and Blaney–Criddle). The motivation for using such T-based parameterizations is that they are simple to use, they appear intuitive, and their data requirements are minimal (often T alone). Their underlying assumption is that there is a strong relationship between the radiative budget Qn (or the component of Qn that is most commonly measured, Rd) and either 2-m T or the diurnal temperature range (DTR = Tmax - Tmin). This encourages the use of T (or DTR) as a sole input for parameterization of Rd (or Qn). In estimating E0, representing the effects of its other physical drivers—humidity and wind speed—is assumed not to be as important in estimating E0 as is capturing the effects of the T and Rd drivers, so one or both of these minor drivers is ignored. Let us call this the “T-Rd assumption.” The problem is that these T-based parameterizations are being used outside of the geography and, particularly, time-frame appropriate to their underlying assumptions. For instance, Thornthwaite (1948) derived his approach to estimating E0 as a taxonomic aid in classifying climates around the globe: he needed a simple technique for which the data were available at his required global extent and climatologic time frame. However, due to its simplicity, his T-based E0-estimation approach and its descendants are now used in dynamic modeling of E0 (a particularly egregious example of which is to estimate ET from the simple hydrology bucket model that underpins the PDSI, see Sec. 42.3.3)—a purpose for which it was never intended and for which it is ill-suited (Hobbins et al., 2008). Such T-based parameterizations

are often deeply obscured within LSMs (such as the PDSI) to such a degree that users of such LSMs may be completely oblivious to their presence and, thus, that they are relying on poor models. However, not only has this underlying assumption been shown not to hold, but its effects in estimating both trends and short-term variability of E0 are dangerous. Hobbins et al. (2012) showed that across much of CONUS, T and Rd are, in fact, negatively correlated across much of the year (and positively correlated at other times and in other regions), giving the lie to the central assumption itself. Regarding E0-trends, Roderick et al. (2007) showed that trends in observed Epan in Australia are almost completely a function of (declining) U2, not in T or Rd, while Hobbins et al. (2008) showed (also in Australia) that using T-based E0 in the PDSI in secular drought analyses results in almost uniformly increasing PDSI trends (increasing droughtiness), contrary to both observations and the results when a more-realistic, physically based E0 is used. Also note that each driver’s relative contributions to the variability of both Epan and ET0 varieties of E0 across CONUS is dynamic, with different drivers dominating regionally at different ties of the year and at different time scales (Hobbins et al., 2012; Hobbins, 2016). Figure 42.7 illustrates the dangers of estimating long-term E0 from T-based parameterizations. While the T-based Ep from the PDSI inhabits much the same T-response space as Ep from observations of Epan (see Fig. 42.7a), it is constrained to a much-reduced variability and exhibits physically unrealistic T-responses in that (i) it is zero below 0°C, and (ii) it declines above 38°C (100°F; not shown). When used in trends analysis, the almost uniformly upwards trends in T (see Fig. 42.7b) result in similarly upward trends in Ep from PDSI (see Fig. 42.7c), in marked contrast to observed trends, which are, at the majority of stations, downwards (see Fig. 42.7d). Regarding the operational estimation of E0, our advice here is that the drivers of evaporation from hydrologic models must be carefully examined: particularly the parameterization of E0. A priori, we should assume that the best E0 parameterizations are physically based and are driven by observations (or predictions) of all drivers; if T-based E0 estimators are used, their use should be supported within the context of the purpose to which the E0 is being put, and for the particular region, season, and time scale. A variability analysis (e.g., Hobbins, 2016) or a trends analysis (e.g., Roderick et al., 2007) will assist in answering these questions. Clearly, however, in many places,

(b) (a)

(c)

(d)

Figure 42.7  Comparing the effects of a T-based versus a physically based parameterization of E0 at 27 Australian and eight New Zealand stations, 1975–2004: (a) shows the T-response of monthly E0 (mm mon-1 on the y-axis) estimated by the T-based Ep from the PDSI [grey triangles; see Hobbins et al. (2008); Supporting Information] and contemporaneously observed Epan (dark circles) to mean monthly T (°C, on the x-axis); (b), (c), and (d) show station trends in T (°C a−1), Ep from the PDSI (mm a-1); and Ep trends from Epan observations using U.S. class-A pans (mm a-2), respectively. [Source: Hobbins et al. (2008). Used with permission.]

42_Singh_ch42_p42.1-42.18.indd 13

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42-14     Evapotranspiration and Evaporative Demand

T-based E0 estimators should be avoided at all time scales, and they are not recommended everywhere for any season, or for all seasons in most places. 42.3.5  Energy Imbalance Problem

Many land surface energy balance experiments using EC techniques show that available energy [see Eq. (42.4a)] is larger than the sum of turbulent fluxes lET and H by 10–20% (Foken, 2008b), leading to an energy balance residual Res, defined as:

Res = Rn + Ln − G − λ ET − H (42.33)

Energy imbalance has been a significant topic of research and discussion among the micrometeorological community over the last decade (Foken 2008b, Leuning et al., 2012). There are many reasons for energy imbalance, but the most obvious is the disparity in measurement height and horizontal scales (i.e., footprint) between available energy and turbulent flux sensors. Footprints for measured Rn, Ln, and G are ~10 m and 0.1–1 m, respectively, while the footprint for lET and H is ~100  m at measurement heights of 2–10 m (Foken, 2008b). Errors in measured Rn, Ln, and incorrect estimates of G and heat storage due to phase-lag errors associated with heat storage in soils, air, and biomass are emerging as significant contributors to energy imbalance (Liebethal, 2006; Higgins, 2012; Leuning et al., 2012). While measurement errors clearly contribute to energy imbalance they alone cannot solve the closure problem, nor can the many processing steps and corrections required to solve the closure problem. Other sources of imbalance are flux divergence due to terrain and vegetation heterogeneities, interaction of scales, and low-frequency mesoscale eddies not being captured by the instrumentation (Foken, 2008b). Energy imbalance corrections are commonly applied by researchers and practitioners even though sources of errors or the energy balance terms in question are not fully understood. Historically, turbulent fluxes have been targeted through increasing lET by Res (Stannard et al., 1994), or distributing Res across lET and H according to b (Twine et al., 2000). More recently, Allen (2008) and Higgins (2012) used least squares linear regression to identify and correct terms that describe large variances in Res. These studies found that G and soil heat storage accounted for the largest variance. This technique is attractive due to the statistical identification of systematic bias and coefficients that are relatively unique due to the lack of correlation among the peaks and timing of energy balance terms. Methodological uncertainties and measurement errors are inevitable, but uncertainties and errors can be avoided with intelligent experimental design and general guidelines. Tools and guidelines for measurement, assessment, and reporting of flux measurements are summarized in detail by Foken and Wichura (1996), Allen (2008), and Allen et al. (2011a; 2011b). 42.3.6 Potential and Reference ET

Confusion regarding the concepts of potential ET (Ep) and reference ET (ET0) stems from misunderstandings of their basic assumptions; when applied in dry climates, the impacts of these assumptions can be important to understand and consider (Sellers, 1964). As discussed under Penman and other Combination-Based Equations, Penman (1948) described the concept of Ep as the ET from a surface sufficiently moist where ET is constrained only by the remaining radiative and advective limits. The concept of ET0 is also traced to Penman (1948), who moved beyond the definition of Ep by further specifying a cropped surface—usually grass or alfalfa. Standardizing the term “reference ET” to reduce confusion on the assumed “surface of known properties” was a primary motivation of the ASCE-EWRI Evapotranspiration Task Committee that developed Eq. (42.12) for estimating ETsz (Allen et al., 2005). Penman’s assumption of a sufficiently moist surface stems from the fact that Ts and ea are commonly unknown in defining the T and ea gradients over the surface. For a wet surface, these two quantities are related by the Clausius–Clapeyron equation, and Penman used the finite difference form of this equation to eliminate Ts-T from the energy balance (as Ts is generally unknown). Penman also assumed zero G, and that Ln was a function of T and ea over the surface. Based on these assumptions, it is obvious that the use of data collected over insufficiently moist surfaces directly violates both concepts, unless explicitly stated that the estimates represent the ambient Ep or ET0 from a hypothetical wet surface small in size. Larger wet environments invoke surface-atmosphere interactions that condition the near surface boundary layer, resulting in localscale complementarity, as discussed in Sec. 42.2.3. Even field-scale feedbacks have been well documented in irrigated areas surrounded by water-limited regions (Allen et al., 1983; Temesgen et al., 1999; Szilagyi and Schepers, 2014). Despite this common knowledge, practitioners and researchers alike routinely and erroneously apply ET0 equations to estimate well-watered ETc using arid and nonconditioned weather data.

42_Singh_ch42_p42.1-42.18.indd 14

Because constants and empirical relationships inherent in Ep and ET0 equations assume a well-watered environment, and the fact that Kc ratios were developed under well-watered conditions, it is therefore required that weather data conditioned by the well-watered environment be used to compute ET0 for estimating ETc. This requirement has been the motivation for developing irrigated environment weather station networks, such as AgriMet, CIMIS, NICE Net, AZMET, to more accurately estimate ET0 and provide benchmark datasets for correcting water-limited weather data. In efforts to reflect the weather that would exist over a well-watered surface, various adjustment techniques have been developed and applied to weather collected in water-limited regions. Temesgen et al. (1999) developed and proposed empirical T and humidity adjustments using agricultural weather data. Abatzoglou (2011) showed that bias correction of gridded weather data can improve well-watered ET0 estimates, but that biases still exist. More recently, a physically based strategy using theoretical blending height-profiling procedures was developed and applied to water-limited weather station and ambient gridded weather data. Results were shown to better reflect weather of a well-watered reference environment (Allen et al., 2013a). For practical applications, it is suggested that weather data collected from ET0 networks be used for estimating ETsz and ETc. If no representative agrometeorological data exist in the area of interest, gridded weather data can be used provided some bias corrections are applied (e.g., Temesgen et al., 1999; Abatzoglou, 2011). Monthly spatially distributed climatologies of dewpoint depression (i.e., daily average dew point temperature minus daily Tmin) and Uz derived from agricultural weather data can also be applied to arid weather data to better reflect agricultural conditions (Huntington and Allen, 2010; Huntington et al., 2015). Developing agricultural representative weather data for operational ETc estimation continues to be an outstanding problem within the ET community. However, recent and future advancements will hopefully soon be well established and used for operational and practical applications. 42.3.7  Remote Sensing of ET

Developing accurate ET maps through the use of satellite or airborne imagery that are useful for water management remains a challenge. While estimating ‘snapshot’ ETinst or anomalies of ETinst operationally is fairly common, given freely available satellite imagery, such as Landsat and MODIS, developing field-scale seasonal and annual ET maps that are physically based, fairly accurate, and usable for water rights and management still requires expert knowledge, human oversight, and research codes. The main challenges are accounting for Prcp and ET from wetting events prior to or after image dates, detection and treatment of clouded areas of the image, estimating representative daily ET0 and EF to account for daily variability of weather between image dates, and the fusion of data from multiple satellite sensors and gridded weather datasets to increase the frequency of useable images, spatial resolution, and representation of local weather conditions. Consideration of these issues significantly improves the quality and accuracy of resulting seasonal and annual ET estimates. In arid regions, where annual ET from natural vegetation is nearly equivalent to annual Prcp, an empirical single-image approach has proven useful for estimating annual ET and ET from groundwater (ETg) as it avoids the difficulties and associated errors with time integration between image dates (Groeneveld et al., 2007; Beamer et al., 2013). Application of remotely sensed surface energy balance approaches for estimating ETinst and seasonal ET in arid regions remains a challenge because values of Qn and H are typically large, commonly leading to ET errors that exceed ET itself—this is a particular problem in estimating annual ETg (annual ET - Prcp) for phreatophyte shrub areas. For a review of outstanding issues related to ET mapping using optical and thermal satellite imagery, see Gowda et al. (2008) and Irmak et al. (2012). Some recent advances in satellitebased remote sensing of ET include consideration of hydrologic states and fluxes between image dates to improve monthly and seasonal ET (Kjaersgaard et al., 2011; Pereira et al., 2015), automated calibration of the CIMIC process (Allen et al., 2013b; Morton et al., 2013), use of the CR with remotely sensed surface temperature (Szilagyi and Jozsa, 2009b), development of representative ET0 (Allen et al., 2013a) and wind fields in mountainous terrain (Allen et al., 2013c), integration of soil-vegetation-atmosphere transfer models (Kustas and Anderson, 2009), and fusion of data from multiple satellite sensors to map field-scale ET (Cammalleri et al., 2013). Future directions for practical applications of remotely sensed ET for water management include developing frameworks to integrate recent advancements into operational workflows, and testing the sensitivity of different automated calibration and data-fusion techniques. Using cloud computing, parallel processing, and data-fusion techniques on systems such as the Google Earth Engine cloud computing and environmental monitoring platform shows exceptional promise for integrating recent advances for efficient, practical, and operational remotely sensed ET

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REFERENCES    42-15 

applications (Allen et al., 2014). Future field-scale research and applications that rely on reflectance and surface energy balance-based modeling rests on the availability of Landsat-like paired optical and thermal infrared data that are at least at the 100-m resolution, with at least 4- to 16-day return intervals (Miller et al., 2013; Serbina and Miller, 2014). ACKNOWLEDGMENTS

M. H. was supported by the NOAA-National Integrated Drought Information System (NIDIS) and the Famine Early Warning System Network (FEWS NET). J. H. was partially funded by the U.S. Geological Survey Landsat Science Team, a Nevada NASA EPSCoR Space Grant, and U.S. Bureau of Reclamation WaterSMART and Science and Technology grants for improved irrigation demands and open-water evaporation monitoring. REFERENCES

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Chapter

43

Rainfall Interception, Detention, and Depression Storage BY

DARRYL E. CARLYLE-MOSES

ABSTRACT

Rainfall depth is an important consideration for estimating the amount of water available for hydrologic processes such as infiltration, transpiration, and runoff generation; nevertheless, not all rain is available to take part in the terrestrial portion of the hydrologic cycle. Interception of rain by vegetation canopies and the forest floor layer, as well as storage of water in surface depressions, represent avenues by which rain may evaporate directly back to the atmosphere. Canopy interception loss Ic denotes the portion of rain that strikes vegetation canopies and is stored upon that vegetation and subsequently evaporated, while forest floor interception loss If is the fraction of rain that is retained by and evaporated from the litter-layer, bryophytes, lichens, and/or grasses that compose the understory. Rain may also be temporarily stored directly on the soil surface or on artificial surfaces such as concrete and asphalt, constituting detention or depression storage SD. The goal of this chapter is to provide an overview of interception loss, both from vegetation canopies and from the forest floor, as well as detention and depression storage, and to highlight the quantitative importance of these hydrologic components, the physical processes involved, and the current measurement techniques and modeling approaches applied. 43.1  CANOPY INTERCEPTION LOSS

Tree canopies and their associated water storage capacities coupled with evaporation during and after a storm event can substantially diminish the quantity of rain that reaches the ground. For example, Wallace and McJannet (2006) found Ic accounted for 25% of the rain that fell at a lowland tropical rainforest in Australia over a 3-year period, while Licata et al. (2011) estimated annual Ic over 2 consecutive years to range from 35 to 44% of rainfall in stands of cordilleran cypress (Austrocedrus chilensis) in Argentina. Annual and season-long Ic can exceed 50% of rainfall (e.g., McMinn, 1960) and often represents an important and even dominant component of evapotranspiration from natural forests and plantations (Benyon and Doody, 2014). At the storm-event scale, the percentage of rain partitioned into Ic typically decreases asymptotically with rainfall depth (Carlyle-Moses and Gash, 2011). For trees and forests with complete canopy cover, Ic may be equated with the total depth of rain during small events that do not saturate the canopy (i.e., no rain reaches the ground). However, with progressively larger rain events differing areas within the canopy saturate with additional rain input becoming canopy-drip throughfall or flow along branches and down the tree boles as stemflow. For relatively large events that entirely saturate the canopy Ic should not be considered negligible. For example, the fraction of rainfall partitioned into Ic for a cedar (Juniperus flaccida)—white oak (Quercus glaucoides)—ash (Fraxinus cuspida) forest of northeastern Mexico remained quasi-constant at approximately 15% of rainfall for events ranging in depth from ≈10 to 120 mm (Carlyle-Moses and Gash, 2011).

Forest and tree characteristics influence Ic by controlling above-ground water storage capacities and regulating during- and post-event evaporation from that storage. Canopy cover fraction is an important control on the quantitative importance of Ic as it represents the effective area from which rain can be stored upon and evaporated (Gash et al., 1995), while leaf area index (LAI) is considered to be an important factor influencing the canopy storage capacity (Deguchi et al., 2006). However, differing foliage characteristics, including leaf hydrophobicity (Holder, 2013) and morphology (Licata et al., 2011), may result in dissimilar storage and, by extension, Ic under similar canopy cover and LAI conditions. In addition, the proportion of cover that is comprised of branches and boles or epiphytes as opposed to leaf foliage should also be considered. Wood has a much greater storage capacity per unit surface area than foliage (Herwitz, 1985), while epiphytes may have large storage capacities and retain water over long periods that may bridge events (Pypker et al., 2011; Allen et al., 2014). Interception loss typically increases with forest age, with positive relationships between Ic and stand basal area, diameter at breast height, and tree height being reported; however, stand density may not serve as a good indicator of the Ic efficiency of a stand since forests self-thin with age (see Llorens and Domingo, 2007; Carlyle-Moses and Gash, 2011). Although much variability exists among and within forest types, coniferous forests typically partition a greater amount of annual rainfall as Ic than their broad-leaved deciduous and broad-leaved evergreen counterparts. Miralles et al. (2010), using multisatellite observations coupled with the climate prediction center morphing technique (CMORPH) rainfall product and the Gash analytical interception model (see as follows), derived global estimates of annual Ic. Canopy interception loss of rainfall for coniferous forests was found to average 22%, while for deciduous and broad-leaved evergreen forests Ic averaged 19 and 13% of annual rainfall, respectively. However, as Miralles et al. (2010) note, differences in Ic among the forest types may have more to do with the rainfall regime of a region than the characteristics of the forest. For example, coniferous forests are commonly found in areas subjected to long-duration synoptic rainfall, while tropical broad-leaved forests are exposed to short-duration, high-intensity rainfalls and large annual rain depths (Miralles et al., 2010). In addition, forests with widely differing features, but located in the same region, may have very similar Ic (see Krämer and Hölscher, 2009), highlighting the importance of an area’s rainfall regime in evaluating the relative importance of this component of evapotranspiration. During the dormant-season temperate, deciduous forests may be efficient at partitioning rainfall into Ic despite having no leaf cover. Intuitively, the relatively large storage capacities associated with the wood component of the canopy and the increased exposure of this retained water to wind in the absence of leaves may result in relatively large Ic, especially when considered on a per unit of cover basis. Deguchi et al. (2006), in a Japanese deciduous forest, found that Ic over a 3-year period decreased from 17.6% of rainfall

43-1

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43-2     Rainfall Interception, Detention, and Depression Storage

under growing-season conditions to 14.3% under dormant conditions; however, the decrease in the Ic efficiency of the stand (24.2%) was much lower than the 41.1% decrease in LAI. As aforementioned, Ic is composed of during-event evaporation E [L T-1] and the evaporation of water held in storage at the end of an event Ce [L]:

I c = Et + Ce (43.1)

where t [T] is the storm duration. The relationship between event Ic [L] and rainfall depth P [L] is often expressed as a linear equation (e.g., Helvey and Patric, 1965) of the form:

I c = aP + b (43.2)

Estimates of E and Ce may be obtained from Eq. (43.2). Using rain events that are assumed to be large enough to satisfy the canopy storage capacity, the slope a represents the ratio of mean E from the saturated canopy to the mean intensity of rain falling on that saturated canopy, R [L T-1], that is, a = E / R . If R is known (see methods described by Valente et al., 1997), E  may be derived as (Gash, 1979):

E = aR (43.3)

The intercept b of Eq. (43.2) represents a first approximation of Ce as well as its storage capacity S [L], since Ce = S under saturation conditions (Gash, 1979). However, as a consequence of evaporation and drainage prior to the canopy reaching saturation, b may not be a good indicator of Ce or S (see Gash, 1979). Instead, other regression-based methods, such as the use of envelope curves (see Valente et al., 1997), are recommended. Miralles et al. (2010) suggest, based on reviews of field studies from around the globe, that the E and S, scaled to the fraction of canopy cover, Ec and Sc, have a fairly narrow range, averaging 0.3 ± 0.1 mm h-1 and 1.2 ± 0.4 mm, respectively, across a wide array of forest types. The evaporation of intercepted rainfall differs from transpiration in that E is only depended on aerodynamic conductance, whereas transpiration is a function of both aerodynamic conductance and canopy conductance. This has important implications in that E, and thus Ic may occur during periods when the stomata of leaves are closed (e.g., at night for many species) or during the dormant period for deciduous trees and forests (Helvey and Patric, 1965). Although E is often simulated using the Penman-Monteith equation, Carlyle-Moses and Gash (2011) suggest that this approach should be abandoned. The evaporation of rainfall stored on vegetation canopies is often greater than that of transpiration under the same environmental conditions and as such Ic needs to be modeled separately. In addition, it has long been recognized that net radiation, the energy source in the Penman-Monteith approach, plays a minor role in the evaporation of intercepted rainfall from forest and tree canopies (Stewart, 1977). Assuming a mean Ec rate of 0.3 ± 0.1 mm h-1 from forest canopies (Miralles et al., 2010), the average supply of energy required to sustain Ec is 206 ± 69 W m-2. The energy involved in the Ic process is considered to be advected sensible heat with the evaporation from and the associated cooling of wetted canopies creating a sensible heat sink (Stewart 1977; Carlyle-Moses and Gash, 2011). Sources of sensible heat involved in the Ic process include the relatively warmer understory, surrounding vegetation not wetted by the rain event, and oceans (Stewart, 1977; Carlyle-Moses and Gash, 2011). Asadian and Weiler (2009) suggest that one reason Ic percentages from urban trees may be so great is due to the enhanced sensible heat generated by the urban heat island effect. For some vegetation the source of advected energy may vary seasonally. Ringgaard et al. (2014), for example, conclude that the sensible heat source for E in a Norway spruce (Picea abies) plantation in Denmark is at the regional scale (North Sea) over the winter and at the local scale during the summer. With net radiation playing a minor role and canopy conductance not applicable under saturated canopy conditions, Pereira et al. (2009) suggest that E be derived using a diffusion equation with canopy surface temperature equated with the wet-bulb temperature of the air: ρaC p λE = g [VPD ] (43.4) γ b where l [L2 T-2] is the latent heat of vaporization, ra [M L-3] is the density of air, cp is the heat capacity of air [L2 T-2 Θ-1], γ represents the psychrometric constant [M L-1 T-2 Θ-1], g b [L T-1] is the aerodynamic conductance for water vapor integrated over the distance between the foliage surface and the adjacent air, and VPD [M L-1 T-2 ] is the vapor pressure deficit between the saturated surface and the overlying air. Canopy interception loss is almost always measured indirectly by taking the difference between precipitation incident on the canopy, either measured above the canopy or in a near-by clearing, and the understory precipitation inputs of throughfall and stemflow. Obtaining an accurate estimate of Ic is

43_Singh_ch43_p43.1-43.4.indd 2

complicated by the spatial variability exhibited by understory precipitation. Evaluations of sampling and scaling methodologies of understory precipitation are provided by Hanchi and Rapp (1997), Ritter and Regalado (2014), and Zimmermann and Zimmermann (2014). Several physically based Ic simulation models exist (Muzylo et al., 2009); however, the Gash model (Gash, 1979) and reformulations of that model (Gash et al., 1995; Valente et al., 1997) are the most commonly used (e.g., Jian et al., 2014). The reformulated Gash model is an analytical model that ­combines the conceptual framework of the sparse numerical Rutter model with the simplicity of empirical equations (Valente et al., 1997). The model considers Ic in three distinct phases: (1) the wetting-up of the canopy phase, (2) the saturated canopy phase, and (3) the canopy drying phase. The parameters required for the model include Sc, E c, and R . In addition, estimates of the ratio of the mean evaporation rate from the trunks and the canopy e, the trunk storage capacity St [L], and the trunk partitioning coefficient pd—which represents the fraction of total drainage under saturated conditions that flows down the trunks as stemflow—are required to run the model. Methods of deriving the various parameter values are provided by Valente et al. (1997). Reformulated Gash model components of rainfall partitioning by a vegetation canopy, including Ic, throughfall and stemflow, and their formulation are provided in Table 43.1. Table 43.1  Components of Rainfall Partitioning by Forests Canopies in the Sparse Gash Model Canopy partitioning component

Formulation

Rainfall thresholds Depth of rainfall required to ­saturate canopy, P ′

−

Depth of rainfall required to ­saturate tree boles P ′′

R

(1 − ε ) Ec

 (1 − ε ) Ec  Sc ln1 −  R  

R St + P′ R − (1 − ε ) Ec pd c

Interception loss components Interception loss from the canopy for m events of insufficient depth to saturate the canopy, P < P ′

m

c∑P, j j =1

Interception loss from the canopy for n events of sufficient depth to saturate the canopy, P ≥ P ′

 (1 − ε ) Ec n c nP′ + ∑ P , j − P′ R  j =1

Interception loss from the trunks for q events that saturate the trunks, P ≥ P ′′

qSt

Interception loss from trunks for n – q events that do not saturate the canopy, P′′ > P ≥ P ′

 (1 − ε ) Ec  n pd c 1 − ∑ P , j − P′ R  j=1 



)

(



)

(

Understory precipitation components Throughfall

m+n

(1 − c ) ∑ P , j + j =1

(1 − p )c 1 − (1 −Rε ) Ec  ∑  1 − (1 −Rε ) Ec  n

d

Stemflow

j =1

 (1 − ε ) Ec  q pd c 1 − ∑ P , j − P′ − qSt R  j=1 

(



)

Definition of terms: R [L T-1] mean rate of rainfall on a saturated canopy; Ec [L T-1] mean evaporation rate from a saturated canopy on a per unit canopy basis; Sc [L] canopy storage capacity per unit area of canopy; St [L] the storage capacity of the tree boles; ε (dimensionless) the ratio between the evaporation rate from the tree boles and the canopy; c (dimensionless) the canopy cover fraction; and pd (dimensionless) the trunk partitioning coefficient. (Source: Valente et al., 1997.)

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References    43-3  43.2 FOREST FLOOR INTERCEPTION LOSS

Not all throughfall and stemflow inputs reach the soil surface. Below the forest or tree canopy the litter layer, mosses, and grasses intercept and store a proportion of understory precipitation. This forest floor interception loss If may be appreciable. Price et al. (1997), for example, found that the feather moss (Pleurozium schreberi) carpet below the canopy of a black spruce (Picea mariana) stand in northern Manitoba stored and subsequently evaporated 23% of the summer throughfall input, while Gerrits et al. (2010) found that If for a beech (Fagus sylvatica) forest in Luxemborg accounted for 10 to 35% of rainfall. Forest floor interception loss may be defined as follows (Gerrits and Savenje, 2011):

If =

dS f dt

+ E f = Pu − F (43.5)

where Sf [L] and Ef [L T-1] are water storage by and evaporation from the forest floor, respectively, Pu [L] represents understory precipitation (throughfall + stemflow) input, F [L] is the total infiltration into the ground, and t is time. Evaporation rates from forest floors may be considerably lower than those of wetted overstorey canopies due to a decrease in wind speed and the associated aerodynamic conductance below the canopy. Additionally, net radiation, which may be an important source of energy for Ef, is often much less for the forest floor layer compared to the overstorey canopy (Gerrits and Savenje, 2011). Although Ef may be relatively small, If may be appreciable, especially where the thickness (or mass) of that forest floor is ample and there is enough time between rain events to allow for sufficient drying. Gerrits and Savenje (2011) state that the frequency of rain events is the dominant factor influencing the magnitude of If. The importance of If may also vary by season, especially for deciduous forests where the thickness of the litter layer increases during leaf fall and, as a consequence of the reduction in LAI, short-wave radiation more readily penetrates to the forest floor (Gerrits and Savenje, 2011). Additionally, If for a given rainfall may differ before and after the dormant season since snow cover compresses and flattens leaf litter reducing its storage capacity (Gerrits et al., 2010). Measurement techniques concerning If include lab methods, which focus on determining Sf using sample submersion or artificial rainfall, and field methods, which typically use lysimeters. Gerrits et al. (2010) developed a method that allows for both Ef and Sf to be evaluated separately in the field. Two aluminum basins, with one mounted on top of the other, are measured continuously using strain gauges with the upper permeable basin containing forest floor material and the lower basin acting as a reservoir for the drainage from the upper basin. A simple water balance approach is applied [see Eq. (43.5)] in deriving the components of If. Gerrits et al. (2010) provide a modified version of the original Rutter model (Rutter et al., 1971) that includes a forest floor interception reservoir. 43.3  DETENTION AND DEPRESSION STORAGE

In addition to canopy and forest floor interception, a portion of rainfall may be held in storage on the surface of soil (or other media) where it is also subjected to evaporation. Although detention storage and depression storage have been differentiated by some (e.g., Viessman and Lewis, 2002), here the two terms are considered synonymous with each other (Ponce, 1989; Mishra and Singh, 2010) and the term “depth of ponding” has also been used to describe this surface storage component (e.g., Dingman, 2008). Depression storage SD is the amount of rain that is retained on the surface in microdepressions, ditches, and other terrain irregularities where water is allowed to collect and pond. In response to rainfall, SD is present to some degree for all impervious surfaces, including concrete, asphalt, and highly compacted soils, with the depth of SD being dependent on surface roughness and the local slope gradient (Borselli and Torri, 2010). Depression storage may develop on pervious surfaces, such as soils, if either the rainfall rate exceeds the infiltration capacity of the surface (saturation from above) or if infiltration ceases as a result of the surface pores becoming saturated (saturation from below). Thus, for soils and other pervious surfaces, the development and depth of SD are dependent not only on surface roughness and slope but also on the characteristics of the surface, including its infiltration capacity and hydraulic conductivity (Dingman, 2008). Water held in SD either evaporates or infiltrates into the soil while rainfall that exceeds the SD capacity of the surface may contribute to surface runoff. The importance of SD varies as a function of time and thus the duration of rainfall (Mishra and Singh, 2010). At the beginning of a rain event SD may play an important role. However, as the storm event progresses maximum SD is satisfied with any additional rainfall contributing to runoff and the relative

43_Singh_ch43_p43.1-43.4.indd 3

importance of SD diminishes (Mishra and Singh, 2010). Following Linsley et al. (1949), the depth of water held in surface depressions at any given time can be approximated as

(

)

VS = SD−Max 1 − e − kPe (43.6)

where VS [L] is the equivalent depth of storage at some time of interest, SD-Max [L] is the maximum depression storage of the surface, Pe [L] is the effective rainfall depth, and k [L-1] is a constant. The constant k may be found by assuming that for small values of Pe all effective rainfall is used to satisfy SD–Max (dVS/dPe = 1), giving k = 1/SD–Max (Viessman and Lewis, 2002). The maximum depression storage is an important property for runoff assessment in response to rain events. Singh (1989) provides reference values of SD–Max for large impervious and pervious areas. Several empirical models have been developed relating SD–Max to an index of surface roughness (e.g., Onstad, 1984). Borselli and Torri (2010) suggest an empirical model of the form:

SD−Max = c + de kX e − gSL (43.7)

where c is the minimum depression storage [L]; X is a generic roughness index; SL [L L-1, %] is the slope gradient expressed as a percentage; and d, k, and g are empirical coefficients. Borselli and Torri (2010) provide the coefficient values associated with Eq. (43.7) for a variety of roughness indices. 43.4 SUMMARY

Important pathways of rainfall occur before infiltration, or surface runoff can take place and should not be considered negligible even for large rainfall events. This is especially true for canopy interception loss, whose quantitative importance is owed, in part, to relatively high during-event evaporation rates that are sustained by advected sensible heat. Forest floor interception loss and depression storage, where the bulk of evaporation occurs after the storm has ceased, may not be inconsequential, depending on not only surface characteristics but also the rainfall regime of the area. REFERENCES

Allen, S. T., J. R. Brooks, R. F. Keim, B. J. Bond, and J. J. McDonnell, “The role of pre-event canopy storage in throughfall and stemflow by using isotopic tracers,” Ecohydrology, 7: 858–868, 2014. Asadian, Y. and M. Weiler, “A new approach in measuring rainfall interception by urban trees in coastal British Columbia,” Water Quality Resources Journal of Canada, 44: 16–25, 2009. Benyon, R. G. and T. M. Doody, “Comparison of interception, forest floor evaporation and transpiration in Pinus radiata and Eucalyptus globulus plantations,” Hydrological Processes (in press), 2014, doi: 10.1002/hyp.10237. Borselli, L. and D. Torri, “Soil roughness, slope and surface storage relationship for impervious areas,” Journal of Hydrology, 393: 389–400, 2010. Carlyle-Moses, D. E. and J. H. C. Gash, “Rainfall interception loss by forest canopies,” Forest Hydrology and Biogeochemistry: Synthesis of Past Research and Future Directions, Ecological Series 216, edited by D. F. Levia, D. E. Carlyle-Moses, and T. Tanaka, Springer-Verlag, Heidelberg, Germany, 2011, pp. 407–423. Deguchi, A., S. Hattori, and H-T. Park “The influence of seasonal changes in canopy structure on interception loss: application of the revised Gash model,” Journal of Hydrology, 318: 80–102, 2006. Dingman, S. L., Physical Hydrology, 2nd ed., Waveland Press, Long Grove, IL, 2008, p. 656. Gash, J. H. C., “An analytical model of rainfall interception by forests,” Quarterly Journal of the Royal Meteorological Society, 105: 43–55, 1979. Gash, J. H. C., C. R. Lloyd, and G. Lachaud, “Estimating sparse forest rainfall interception with an analytical model,” Journal of Hydrology, 170: 79–86, 1995. Gerrits, A. M. J., L. Pfister, and H. H. G. Savenije, “Spatial and temporal variability of canopy and forest floor interception in a beech forest,” Hydrological Processes, 24: 3011–3025, 2010. Gerrits A. M. J. and H. H. G. Savenije, Forest Floor Interception. Forest Hydrology and Biogeochemistry: Synthesis of Past Research and Future Directions, Ecological Series 216, edited by D. F. Levia, D. E. Carlyle-Moses, and T. Tanaka, Springer-Verlag, Heidelberg, Germany, 445–454, 2011. Hanchi A. and M. Rapp, “Stemflow determination in forest stands,” Forest Ecology and Management, 97: 231–235, 1997. Helvey J. D. and J. H. Patric, “Canopy and litter interception of rainfall by hardwoods of eastern United States,” Water Resources Research, 1: 193–206, 1965. Herwitz S. R., “Interception storage capacities of tropical rainforest canopy trees,” Journal of Hydrology, 77: 237–252, 1985.

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43-4     Rainfall Interception, Detention, and Depression Storage

Holder C. D., “Effects of leaf hydrophobicity and water droplet retention on canopy storage capacity,” Ecohydrology, 6: 483–490, 2013. Jian S., C. Zhao, S. Fang, and K. Yu, “Soil water content and water balance simulation of Caragana korshinskii Kom. In the semiarid Chinease Loess Plateau,” Journal of Hydrology and Hydromechanics, 62: 89–96, 2014. Krämer I. and D. Hölscher, “Rainfall partitioning along a tree diversity gradient in a deciduous old-growth forest in Central Germany,” Ecohydrology, 2: 102–114, 2009. Licata J. A., T. G. Pypker, M. Weigandt, M. H. Unsworth, J. E. Gyenge, M. E. Fernandez, T. M. Schlichter, et al., “Decreased rainfall interception balances increased transpiration in exotic ponderosa pine plantations compared with native cypress stands in Patagonia, Argentina,” Ecohydrology, 4: 83–93, 2011. Linsley Jr., R. K., M. A. Kohler, and J. L. H. Paulhus, Hydrology for Engineers (3rd ed., McGraw-Hill College Division, New York, 1949, p. 508. Llorens P. and F. Domingo, “Rainfall partitioning by vegetation under Mediterranean conditions. A review of studies in Europe,” Journal of Hydrology, 335: 37–54, 2007. McMinn R. G., Water Relations and Forest Distribution in the Douglas-fir Region on Vancouver Island, Publication 1091 of the Canadian Department of Agriculture, Science Service, Forest Biology Division, Ottawa, ON, 1960, p. 71. Miralles D. G., J. H. Gash, T. R. H. Holmes, R. A. M. de Jeu, and A. J. Dolman “Global canopy interception from satellite observations,” Journal of Geophysical Research: Atmosphere, 115: D16122, 2010. Mishra S. K. and V. Singh, “Soil Conservation Service Curve Number (SCSCN) Methodology,” Water Science and Technology Library, Springer, The Netherlands, Vol. 42, 2010, p. 516. Muzylo A., P. Llorens, F. Valente, J. J. Keizer, F. Domingo, and J. H. C. Gash, “A review of rainfall interception modelling,” Journal of Hydrology, 370: 191– 206, 2009. Onstad C. A., “Depressional storage on tilled soil surfaces,” Transaction of the American Society of Agricultural Engineers, 27: 729–732, 1984. Pereira F. L., J. H. C. Gash, J. S. David, and F. Valente, “Evaporation of intercepted rainfall from isolated evergreen oak trees: do the crowns behave as wet bulbs?” Agricultural and Forest Meteorology, 149: 667–679, 2009.

43_Singh_ch43_p43.1-43.4.indd 4

Ponce V. M., Engineering Hydrology: Principles and Practices, Prentice Hall, Englewood Cliffs, NJ, 1989, p. 640. Price A. G., K. Dunham, T. Carleton, and L. Band, “Variability of water fluxes through the black spruce (Picea mariana) canopy and feather moss (Pleurozium schreberi) carpet in the boreal forest of northern Manitoba,” Journal of Hydrology, 196: 310–323, 1997. Pypker T. G., D. F. Levia, J. Stailens, and J. T. Van Stan II, Canopy structure in relation to hydrological and biogeochemical fluxes, Forest Hydrology and Biogeochemistry: Synthesis of Past Research and Future Directions, Ecological Series 216, edited by D. F. Levia, D. E. Carlyle-Moses, and T. Tanaka, SpringerVerlag, Heidelberg, Germany, 2011, pp. 371–388. Ringgaard R., M. Herbst, and T. Friborg, “Partitioning forest evapotranspiration: interception evaporation and impact of canopy structure, local and regional advection,” Journal of Hydrology, 517: 677–690, 2014. Ritter A. and C. M. Regalado, “Roving revisited, towards an optimum throughfall sampling design,” Hydrological Processes, 28: 123–133, 2014. Rutter A. J., K. A. Kershaw, P. C. Robins, and A. J. Morton, “A predictive model of rainfall interception in forests I. Derivation of the model from observations in a plantation of Corsican pine,” Agricultural and Forest Meteorology, 9: 367–384, 1971. Singh V. P., Hydrologic Systems. Watershed Modelling, Prentice Hall, Inglewood Cliffs, NJ, Vol. II, 1989, p. 448. Stewart, J. B., “Evaporation from a wet canopy of a pine forest,” Water Resources Research, 13: 915–921, 1977. Valente, F., J. S. David, and J. H. C. Gash, “Modelling interception loss for two sparse eucalypt and pine forests in central Portugal using reformulated Rutter and Gash analytical models,” Journal of Hydrology, 190: 141–162, 1997. Viessman Jr., W. and G. L. Lewis, Introduction to Hydrology, 5th ed., Prentice Hall–Pearson Education, Upper Saddle River, NJ, 2002, p. 612. Wallace J. and D. McJannet, “On interception modelling of a lowland coastal rainforest in northern Queensland, Australia,” Journal of Hydrology, 329: 477–488, 2006. Zimmermann A. and B. Zimmermann, “Requirements for throughfall monitoring: the roles of temporal scale and canopy complexity,” Agricultural and Forest Meteorology, 189–190: 125–139, 2014.

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Chapter

44

Watershed Geomorphological Characteristics BY

FAITH A. FITZPATRICK

ABSTRACT

The ­descriptive and conceptual models (Davis, 1899) became process based (Gilbert, 1877; Horton, 1945), which then ultimately developed into dynamic quantitative geomorphology (Strahler, 1952a; 1957). Schumm and Lichty’s (1965) framework for time, space, and causality encompasses how watershed geomorphological characteristics fit into a range of temporal scales (Table 44.1). For example, if the time span of interest is a few hundred years (between geologic scales and typical engineering scales), watershed runoff and sediment yield (as affected by geologic setting, climate, vegetation, and relief) influence drainage network morphology, hillslope morphology, channel hydrology, and hydrology at a point along a stream (water and sediment discharge). At engineering scales, the drainage network morphology, hillslope morphology, and channel morphology influence water and sediment discharge at a particular point along a stream. At the time of the second edition of the Handbook (1964), another treatise by Leopold, Wolman, and Miller (1964) was published. Leopold et al. (1964) is still considered a key reference book for fluvial geomorphologists, especially for its treatment of “The Drainage Basin as a Geomorphologic Unit” (Chap. 5), which, combined with Strahler’s and Horton’s work forms the basis of much of the watershed measurements that are still done today. A quote from Leopold, Wolman, and Miller (1964) gives that sense of interconnection of watersheds with the streams that drain them as well as interconnection of geomorphological principles with river and hydraulic engineering: “Each denudational agent, depending upon its density, gradient, and mass at a particular place, is capable of applying a given stress on the materials available. A certain amount of work may be performed by the application of this stress, and the results of this work are the landforms that we see developed in various parts of the world.” (p. 4)

This chapter describes commonly used geomorphological characteristics that are useful for analyzing watershed-scale hydrology and sediment dynamics. It includes calculations and measurements for stream network features and areal basin characteristics that cover a range of spatial and temporal scales and dimensions of watersheds. Construction and application of longitudinal profiles are described in terms of understanding the three-dimensional development of stream networks. A brief discussion of outstanding problems and directions for future work, particularly as they relate to water-resources management, is provided. Notations with preferred units are given. 44.1  INTRODUCTION AND LITERATURE REVIEW

This chapter represents an update to Strahler’s description of watershed geomorphological characteristics in “Quantitative Geomorphology of Drainage Basins and Channel Networks” (Strahler, 1964). Watershed geomorphological characteristics are formed by the downhill movement of water and how the land is eroded into different forms or morphology because of this movement. Strahler’s career encompassed a time period during which the methods of quantitative geomorphology emerged, and advanced fluvial geomorphology from “purely qualitative and deductive” to “rigorous quantitative science capable of providing hydrologists with numerical data of practical value” (Strahler, 1964, pp. 4–40). Strahler’s quantitative geomorphology had its roots in Horton’s (1932; 1945) quantification of geomorphological characteristics of drainage basins, stream networks, and erosion processes over time. Previous to the quantitative approach, landscape evolution models were more ­qualitative or descriptive in nature, such as William Morris Davis’s analysis ­framework and classification of cycles of erosion (Davis, 1899).

Table 44.1  Factors that Influence Watersheds and Stream Network Characteristics at Three Temporal Scales of Decreasing Duration Cyclic (geologic, millions of years)

Graded (centuries to millennia)

Steady (days to decades)

Time

Drainage basin variables

I

NA

NA

Initial relief

I

NA

NA

Geology (lithology, structure)

I

I

I

Climate

I

I

I

Vegetation (type and density)

D

I

I

Relief or volume of system above base level

D

I

I

Hydrology (runoff and sediment yield per unit drainage area)

D

I

I

Drainage network morphology

D

D

I

Hillslope morphology

D

D

I

Channel morphology

D

D

I

Hydrology (discharge of water and sediment from system)

D

D

D

I, independent; D, dependent; NA, not applicable. [Source: Schumm and Lichty, 1965]

44_Singh_ch44_p44.1-44.12.indd 1

44-1

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44-2    Watershed Geomorphological Characteristics

A holistic fluvial systems approach is now popular that interconnects multiple spatial and temporal scales of quantitative and qualitative descriptors and physical mechanistic processes (e.g., see Sear et al., 2003; Brierley and Fryirs, 2005; Rosgen, 2006). This systems approach has a backbone of classical geomorphological concepts including geomorphological work, equilibrium, thresholds, feedbacks, and relaxation and response time associated with watershed changes, sediment dynamics, and disturbance mechanisms (Mackin, 1948; Chorley, 1957, 1962; Schumm, 1956, 1973; Schumm and Lichty, 1965; and Bull, 1991). Currently, we are at a state of integration of scales of geomorphologic processes, knowing that we cannot treat watersheds and stream networks as simply as the laws of gravitational and molecular stresses and chemical processes (Strahler, 1952a). In addition, geomorphological principles help to link hydraulic and chemical-based processes with the larger spatial and temporal variations that shape the development of watersheds and fluvial landforms from perturbations in climate, tectonics, base level, and humans (Church, 2002; Brierley and Fryirs, 2005). Watershed geomorphological characteristics are measured for multiple reasons that support theoretical, empirical, and statistical analyses. An example of theoretical analyses—formulation and testing of specific hypotheses based on established principles—is Schumm’s (1973) paper on geomorphological thresholds for gully formation and river meandering. Empirical applications involve collection of field data to form mathematical relations between watershed variables and stream characteristics, such as Emmett’s (1975) classic work on the relations of stream network characteristics to channel characteristics for the Upper Salmon River Area, Idaho,” Watershed geomorphological characteristics have been used successfully to help describe watershed hydrology and sediment delivery across varying physiography (Langbein and Schumm, 1958) and also have been used in models of surface water sediment and nutrient modeling (Robertson et al., 2009). Other applications using watershed characteristics include studies of channel and hillslope instability, effects of dams and dam removals on fluvial systems, flood and debris flow hazards, wetland classification (Brinson et al., 1995), groundwater susceptibility (http://www.dnr.state.mn.us/whaf/about/ scores/geomorphology/gw_contamination.html), ecology (Gordon et al., 1992), habitat studies (Fitzpatrick et al., 1998), and climate- or vegetationchange effects on river meandering, incision, and sedimentation (Knox et al., 1974; Knox, 1977; Fitzpatrick et al., 1999; Fitzpatrick and Peppler, 2010). 44.2  WATERSHEDS AND DRAINAGE NETWORKS

Quantitative measurements and calculations of watershed geomorphological characteristics described in this section include watershed boundary ­delineations, stream network characteristics, watershed areal characteristics,

drainage patterns, and three-dimensional development and evolution of stream networks. Watershed geomorphological characteristics are completely dependent on data from maps or remote sensing techniques, thus it is of utmost importance to document the spatial resolution, scale, and source of all map data. Comparative studies cannot be done successfully unless the same method of measurement, map source, and scale are used. 44.2.1  Watershed Boundary Delineation

A watershed is defined as the area of land that drains water, sediment, and dissolved materials toward a common outlet. The term watershed carries the same meaning as “drainage basin” and “catchment,” and usage of one term over another is determined by regional and disciplinary preferences. The bounds of a watershed are determined by a drainage divide, also known as an interfluve. Topographic-derived drainage boundaries are delineated by drawing lines connecting the highest points of elevation along a land surface that separate diverging flow paths. Usually, this looks like a ridge of high ground that falls off either side into a series of ravines and valleys. The delineation can be done by hand with maps of elevation contour lines or automatically with a geographic information system (GIS) and digital elevation model (DEM) data. Historically, watershed areas were measured with the use of planimeters (http://persweb.wabash.edu/facstaff/footer/Planimeter/ planimeter.htm). In the United States, methods were standardized by the U.S. Geological Survey (USGS) to measure the area of watersheds upstream of streamgaging stations. An example of drainage divide lines drawn on a 1:24,000 USGS topographic quadrangle map is provided in Fig. 44.1. Before GIS, each USGS Water Science Center maintained an official set of paper copies of these quadrangle maps with drainage boundaries. Topographic divides are easiest to delineate in well-developed, geologically old watersheds. In geologically young landscapes, wetlands and ­interconnected lakes may form watershed divides making it difficult to determine the flow direction. In urban areas, storm sewers often transcend topographic boundaries and additional data on storm-sewer networks are needed. In arid areas, water is often artificially routed through pipes or canals across topographic divides into adjacent watersheds for storage or supply for agricultural and industrial uses and drinking water supplies. Some watersheds contain depressions or closed basins with no surface water outlet, such as kettle ponds in glaciated terrain or playas in the arid western United States. These areas are referred to as noncontributing areas by the USGS, meaning they do not directly contribute to surface water drainage. In sandy areas, infiltration rates may exceed any known precipitation rates, leaving no trace of surficial drainage because all precipitation either evaporates or infiltrates to groundwater aquifers. Watershed boundaries are not just only delineated for surface-water drainages, but can also be defined along groundwater divides, which may be different

Figure 44.1  Example of U.S. Geological Survey traditional method of delineating drainage divides on 1:24,000-scale topographic maps. Delineations with crosshatching are noncontributing areas.

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Watersheds and Drainage Networks    44-3 

Table 44.2  Hydrologic Unit Codes and Levels of Classification for U.S. Watersheds HUC code number of digits

Description

2

Region (first level)

4

Subregion (second level)

6

Basin (third level)

8

Subbasin (fourth level)

10

Watershed (fifth level)

12

Subwatershed (sixth level)

to trace the stream in upstream or downstream directions. The blue lines originally were drawn with 5- to 10-ft contour intervals of elevations as a base map and are a simplification of all the possible drainage lines that could be drawn from contour crenulations, especially for ephemeral streams. A stream network generated in a GIS from airborne light detection and ranging (LiDAR)-based 1-ft contours may be denser, as shown in Fig. 46.2, and helpful for studies of small watersheds; however, additional time and resources are required for field verification and establishing continuous flow networks. The NHD remains the preferred dataset in the United States for many agencies on which regional stream network geomorphological measurements are made because of its wide availability and consistent national coverage, consistent flow networks, and comprehensive history of field verification and updates.

[Source: http://nhd.usgs.gov/wbd_facts.html]

than surface-water divides because they depend on the subsurface horizontal and vertical extent of the aquifer and groundwater gradients (Fitzpatrick et al., 2015). It is not uncommon for the surface water and groundwater divides to be different in areas with nonhomogeneous rock types and glacial deposits. Topographic drainage divides form the framework of the hydrologic unit code (HUC) system in the United States and the Caribbean, which uses a national hierarchical system of stream categorization into six levels as shown in Table 44.2 (Seaber et al., 1987). The newest and most up-to-date delineations for the United States are included in the watershed boundary dataset (WBD). The WBD contains boundaries for the 8-, 10-, and 12-digit HUCS and is available at http://www.nrcs.usda.gov/wps/portal/nrcs/main/national/ water/watersheds/dataset/. 44.2.2  Stream Network Characteristics

This section describes four commonly used stream network measurements of stream order, bifurcation ratio, drainage density, and overland flow. A brief overview of drainage patterns is included. The first step before measuring geomorphological characteristics of stream networks is mapping the flow patterns of water, or the science of hydrography. If not already available, the stream network is usually mapped during the delineation of drainage divides. It is important to know the source and resolution of the line work representing the stream network, along with the assumptions that went into their mapping. In the United States, stream networks are commonly derived from blue lines on USGS 7.5-min 1:24,000-scale topographic quadrangle maps. The quadrangle maps show perennial (solid blue lines) as well as ephemeral (dashed blue lines) streams. These streams form the basis for the national hydrography dataset (NHD) (http://nhd.usgs.gov/; Simley and Carswell, 2009). Each line segment in the NHD contains networking codes that allow a user

Stream Order A hierarchical system for describing the spatial position of a stream in a stream network was introduced in the United States by Horton (1945) and later modified by Strahler (1952b; 1957; Fig. 44.3). The Laws of Horton’s stream orders summarize mathematical empirical relations that describe how stream networks are arranged and related to hydrologic characteristics (Horton, 1945; Melton, 1959). The mathematical relations reflect the watershed size; geographic relations among the number, length, and size of channels; stream branching; and physiography of the landscape drained by the stream network. These empirical relations were especially useful before the use of GIS for measuring the number of streams and stream lengths. Perennial, intermittent, and ephemeral streams can be assigned a stream order number. In Strahler’s stream ordering system (Fig. 44.3a), the smallest tributaries are given an order number of 1. Where two streams of order 1 join, a stream of order 2 is formed. Where two streams of order 2 join a stream of order 3 is formed, and so on. If a stream of order 1 joins a stream of order 2, the downstream segment stays at order 2. The order of the stream at the outlet is the “Principle Order.” Thus, in the example shown in Fig. 44.3, the principle order is 3. Once the ordering of a stream network is defined using the Strahler method, each of the orders can be summarized for the watershed in terms of the number of stream segments in each order No (Table 44.3). As the order increases, the number of stream segments, No, decreases as shown in Fig. 44.4. The number of segments in the principle stream order is always equal to 1. Conversely, mean stream length and drainage area increase in an orderly fashion that can be explained by a power function. Another ordering system uses the Shreve method (1967) and is typically used to describe the relation of a stream to the next waterbody downstream. This measurement is important in the aquatic ecology fields for explaining longitudinal connection among organisms whose habitat requirements differ

(a)

(c)

(b)

(d)

Figure 44.2  Examples of detailed stream networks delineated from 1-ft LiDAR overlain with elevation contours from different map sources for a 1.5-km reach of the Bark River, Wis. (A) 1:24,000-scale topographic map, where the stream was represented by a single blue line with no tributaries. (B) Contours from a 30-m DEM that are too coarse to show a valley. (C) Contours from a 10-m DEM show a valley. (D) LiDAR based 1-ft contours show multiple tributaries and valleys.

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44-4    Watershed Geomorphological Characteristics

Strahler

1

1

1 1

(a)

1

1 2

1

2

1

2

1

(b)

1 1 2

1

(a)

3 Rb = 5

Shreve

1

1

1

1 3

3

2

1

Shreve number = 12

4

Q

1

5 9 1 1

(b)

2

11

Table 44.3  Strahler Stream Order and Number of Stream Segments Shown in Example Stream Network in Fig 44.3a Stream order

Number of stream segments, No

Bifurcation ratio, Rb

1

12

3

2

4

4

3

1

over life stages. Each stream segment is counted and all the segments are summed at a confluence (Fig. 44.3b). A downstream link number for a stream segment is the Shreve order calculated for the next downstream stream segment. In this example, the tributary with a Strahler stream order of 1 has a downstream link number of 12, indicating that it drains to a much larger river.

Number of streams

50

10 5

Mean stream length (km)

Bifurcation Ratio The bifurcation ratio, Rb, is the ratio of the number of stream segments of any order to the number of stream segments of the highest order (Horton, 1945; Strahler, 1957). Rb = N u /N u+1

Nu = 615e–1.33u RB = 3.78

It is usually calculated for the second highest stream order or the mean value of all orders measured. The number is usually between 2 and 4, and indicates the peakedness of runoff. Low ratios are associated with flashy hydrographs with high flood peaks, as shown in Fig. 44.5 for a small watershed. When the log of the number of streams is plotted against order, the relation is usually described by a linear regression equation. Drainage Density Drainage density reflects runoff characteristics; for example, dense stream networks have flashy high peak flows (Horton, 1945; Leopold et al., 1964) because the reduced friction in closely spaced channels produce faster flows compared to rough terrain and vegetation encountered by overland flow in less dense networks. The density of a stream network (D) is the quotient of the cumulative length of streams (L) and the total drainage area (A): D = ∑L/ A Drainage density has units of length per unit area such as km/km2. Drainage density is a result of a watershed’s geology, climate, time, and human impacts (Table 44.4). Two watersheds may have similar drainage densities but still have different bifurcation ratios, which will further define the hydrologic characteristics (Leopold et al., 1964). Because the calculation involves stream length, it can be quite tedious to calculate by hand, especially for large watersheds, but relatively easy to do in a GIS.

– Lu = 0.21e0.97u RL = 2.64 10 5

1 0.5

1 1 (a)

t

Figure 44.5  Bifurcation ratio examples. Stream networks with (A) high bifurcation ratios have low flood peaks compared to (B) stream networks with low bifurcation ratios.

12

Figure 44.3  Stream ordering examples: (A) Strahler (modified from Strahler, 1954, 1957) and (B) Shreve (modified from Shreve, 1967).

100

2 3 4 Stream order

Nu = Number of streams RB = Bifurcation ratio

A

Rb = 17

1

Mean drainage area (km2)

2

1

B

2

1 1

5

– Au = 0.18e1.48u RA = 4.39

50

10 5

1 1

(b)

100

2 3 4 Stream order

– Lu = Average length of streams of order u RL = Stream length ratio

5

1 (c)

2 3 4 Stream order

5

– Au = Average catchment area of streams of order u RA = Catchment area ratio

Figure 44.4  Horton’s Laws of stream networks for (A) number of streams, (B) mean stream length, and (C) mean drainage area (from Horton, 1945).

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Watersheds and Drainage Networks    44-5 

Table 44.4  Driving Variable Effects on Drainage Density Watershed characteristic

Low density

High density

Infiltration

High

Low

Geology

Resistive

Erodible

Age

Young

Old

Climate

Humid

Arid Large

Flood size

Small

Sediment production

Low

High

Relief

Gentle

Steep

Leopold et al. (1964) and Hack (1957) found in calculations of many stream lengths and drainage areas that total stream length can be estimated from drainage area by the relation (in English units of square miles for area):

∑ L = 1.4 A∧ 0.6 This equation was more commonly used before GIS was widely available because it saved time by not having to measure stream lengths by hand. In general, there is about 1.3 km of stream length for every square km of drainage area. Emmett’s (1975) analyses for tributaries of the East Fork Salmon River yielded a similar relation as shown in Fig. 44.6a.

100

Stream length, in miles

10

L = 1.5 DA0.55

1 Explanation Compilation by stream order Compilation by drainage area

0.1 0.1

1

(a)

100

10 Drainage area, in square miles

1000

10,000

Discharge: cubic feet/second

Bankful discharge QB = 28.3 DA0.69

1000

Average annual discharge

100

10 (b)

1

10

100 Drainage area: square miles

1000

10,000

Figure 44.6  Relations among drainage area and (A) stream length (from Emmett, 1975) and (B) discharge for tributaries to East Fork Salmon River (from Emmett, 1975; Leopold, 1994).

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44-6    Watershed Geomorphological Characteristics

Length of Overland Flow Once the total drainage area and cumulative length of streams have been computed, the average length of overland flow in a watershed can be calculated (Horton, 1945). The length of overland flow is used in surface-water modeling to help predict the quantity of water needed to exceed a threshold of erosion that begins rill and channel development. It is also related to the constant of channel maintenance (C), the reciprocal of drainage density which has units of length (km), which is the average distance between channels. One-half of channel maintenance length is the length of overland flow. C = A/∑L

Drainage Patterns Drainage patterns of a watershed reflect rock type, geologic structure and tectonics, and fracturing patterns (Howard, 1967). Some common drainage patterns are shown in Fig. 44.9 and originate from Howard’s (1967) detailed overview. Common terms used to describe drainage patterns include dendritic (tree-like), rectangular or trellis, radial, and parallel (Horton, 1945). If the geologic setting is uniformly resistant to erosion, the land surface is gently sloped, or surficial deposits are horizontally layered, a dendritic pattern develops, similar to the branching of a tree. Parallel patterns develop on steeply sloping surfaces or in areas with parallel elongate landforms, where long tributaries are oriented parallel to, and eventually join, the parent stream from opposite sides. Trellis drainages develop on land surfaces with nonuniform geology, alternation of resistant and less-resistant rocks, dipping, folded or faulted rocks, or exposed lake floors or sea beds with beach ridges. Rectan­ gular patterns indicate repeated jointing and fracturing at right angles. A radial pattern would be found on a volcano or dome with uniform geology, whereas an annular pattern would develop on a dome with concentric rings of usually layered sedimentary bedrock units with differing resistance to erosion. Multibasinal patterns develop on glacial deposits with hummocky surfaces, limestone solution or karst landscapes, and permafrost. Contorted drainages represent multiple tectonic events that altered the course of already developed drainage patterns, metamorphic rocks, dikes, and veins.

climate (Leopold et al., 1964). Like other measurements, map resolution affects the accuracy of a drainage area calculation, but the most important factor is the correct identification of the drainage divide, and the knowledge of other factors that may affect the contributing area of a watershed, such as diversions and withdrawals for irrigation and drinking water supplies; and storm sewer networks in urban areas. The presence of these features is important because many watershed geomorphologic measurements are normalized by drainage area before comparisons are done among streams. Basin Length and Shape There are many ways to describe the areal shapes and dimensions of watersheds. For basin length, upon which many descriptors rely, one of the most common methods is to measure the distance along a line centered through the watershed from its mouth to where the main channel meets its drainage divide (Schumm, 1956; Gardiner, 1975; Harvey and Eash, 1996; Fitzpatrick et al., 1998; Fig. 44.7). Sometimes a straight line from the mouth to the highest point in the basin is used, but this method may not be as accurate because, depending on the orientation of the basin, this line may fall outside of the actual drainage basin, and is usually shorter than the actual basin length. Basin shape has many descriptors including form, circularity, elongation, and compactness. These features, among others equally important, help to describe the geologic setting and resultant watershed hydrology and sediment regime (Horton, 1932; Morisawa, 1958; Strahler, 1964). Basin shape was described by Horton (1932; 1945) as usually a pear-shaped ovoid. Horton thought that the predominance of this pear-shape form was evidence that drainage basins are formed by sheet erosion processes acting on an inclined surface (Strahler, 1964). Horton defined basin form (Rf) as the quotient of the drainage area (A) divided by the basin length squared (L2b ): R f = A/L2b The units of basin shape are dimensionless, but usually the area is in km2 and the basin length is in km. This measurement is the same as the elongation ratio (Re; Fryirs and Brierley, 2013): Re = A0.5 / Lb

44.2.3  Watershed Areal Characteristics

Watershed areal characteristics are major drivers in forming stream network patterns, watershed hydrology, longitudinal sediment dynamics, and ­ultimately slope and channel morphology over a range of time scales (Leopold et al., 1964; Schumm and Lichty, 1965; Fryirs and Brierley, 2013). These characteristics are easily calculated in a GIS with high-resolution DEMs. Traditionally, characteristics were calculated by hand from 1:24,000-scale USGS quadrangle maps. Drainage Area The drainage area forms one of the most important features of a watershed, as it provides a scaling factor for many other characteristics related to hydrology and sediment (Leopold et al., 1964; http://streamstatsags.cr.usgs.gov/ss_defs/ basin_char_defs.aspx). Streamflow characteristics, including base flow, ­bankfull flows, and flood frequencies are empirically related to drainage area (Fig. 44.8b), usually by physiographic and climatic regions (Reis, 2006) (http://water.usgs.gov/osw/streamstats/ssinfo1.html). For example, streamflow can be estimated using drainage area as follows: Q = aAb where Q is discharge or streamflow in units of volume per unit time, A is drainage area, and coefficient a and exponent b relate to the physiography and

A ratio of 1.0 is a round basin. A ratio closer to 0.6 is an elongated basin, and the more elongate a basin the slower the runoff and less flashy the floods compared to a more round basin. Elongated basins tend to have relatively high bifurcation ratios as well (Fig. 44.5). Another basin shape measurement, the circularity ratio (Rc), compares a basin’s drainage area (A) to the area of a circle (Ac) with the same circumference (Miller, 1953; Fig. 44.8): Rc = A/Ac Using a circle as a standard reference for drainage basin, however, is a lessfavorable method because of the sharply defined point at the mouth (Strahler, 1964). In general, watersheds with low ratios of about 0.4 are more elongate and shapes are likely controlled by geologic setting and structure. More round basins have ratios of 0.6–0.7 (Fryirs and Brierley, 2013). The compactness ratio (Cc) is related to Rc in that it compares a basin’s perimeter to the circumference of a circle of equal area (Reddy, 2005): Cc = P / (4π A) Highest altitude

Longest from mouth, following main streamline

Most direct length from mouth to highest point Figure 44.7  Basin length measurements and definitions.

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Watersheds and Drainage Networks    44-7 

C2

C1

Basin Relief and Slope Basin relief (H) is the measure of the difference in altitude between the highest point along a drainage divide and the altitude at the river mouth. Figure 44.7 shows how the highest point in a watershed along a drainage divide might be slightly offset from the extension of the main streamline used to measure basin length. The basin relief ratio (Rh) is a dimensionless value of the quotient of basin relief divided by basin length (Lb), which provides a measure of how fast the elevation drops across a watershed: Rh = H/Lb

C1 = C2 Figure 44.8  Comparison of a common pear-shaped watershed with a circle with the same circumference (C) (adapted from Fryirs and Brierley, 2013).

The relief ratio is a measure of the steepness of a watershed and is an indicator of the potential erosion of the landscape (Strahler, 1964). Schumm (1956) used the same basin length measurement used for basin shape ratio—the overall longest distance along the basin length from the mouth to the drainage divide following the principal drainage line. Because the measurements

(a) Dendritic

(b) Parallel

(c) Trellis

(d) Rectangular

(e) Radial

(f) Annular

(g) Multi-basinal

(h) Contorted

Figure 44.9  Examples of drainage patterns (from Howard, 1967). AAPG © 1967. Reprinted by permission of the AAPG whose permission is required for further use.

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44-8    Watershed Geomorphological Characteristics

included in the basin relief ratio make up the sides of a right triangle, it can also be referred to as basin slope. As part of estimating streamflow statistics at ungaged streams in the United States using a National Streamflow Statistics Program (Reis, 2006; Turnipseed and Ries, 2007), channel slope (S) is calculated by dividing the elevation difference by stream length between points that are 10 and 85% of the total stream length. 44.2.4  Drainage Development and Three-Dimensional Evolution of Watersheds

The descriptions of watershed geomorphological characteristics in the aforementioned sections have focused on two-dimensional, spatially related characteristics, and how they might differ in terms of the major driving factors—climate, geology and tectonics, human disturbance, and time. This section expands on the temporal and spatial scales for describing drainage development and evolution. Drainage Development and Evolution Stream networks develop where erosive forces of water overcome resistant forces (friction and gravitational) of particles (Horton, 1945; Leopold et al., 1964). The fundamental requirement for channel initiation is that precipitation exceeds infiltration and produces overland flow. Rills form almost ­immediately when overland flow erodes soil on bare hillslopes leaving small, mostly ­temporary, intermittent channels that are no more than a few centimeters wide and deep. When overland flow further reaches a depth that fills and overtops R Hg

rills, water starts to flow slightly in a lateral direction that eventually erodes the divides between rills and forms a master rill that is lower in elevation than surrounding rills. With additional erosion from overland flow, the divides between the rills break down, and relatively smooth surfaces are formed between rills or channels in a process called “cross grading” (Horton, 1945). Permanent and more continuous channels form and deepen as flows continue to erode in upstream and lateral directions and capture smaller channels. Glock (1931) described the development of stream networks as a conceptual model of initiation, growth, extension, and elaboration, followed by a stage of integration marked as a partial simplification of the network as time progresses (Clifford, 2011). This process describes the branches and tree-like framework of most drainage systems, with many small branches connected to fewer large branches. Horton’s (1945) laws on stream orders reflect this process. Leopold et al. (1964) noted the random nature of cross grading and micropiracy, and that most landscapes have inconsistencies in erodibility and topography, which adds variability to the regular geometric patterns proposed by Horton. These same principles that reflect the conservation of energy and the effects of gravity are responsible for the relation between slope and stream order, where larger stream orders have decreasing slopes (Leopold et al., 1964). The Longitudinal Profile Longitudinal profiles offer vertical views of stream network development and evolution (Fig. 44.10) and help to describe the continuum of streamflow and

Reference level Elevation of stream head

O

Convex up

Elevation

Yo Concave up

A2

A1

M

Local base level

Yc

Sea level

N

(a)

500 450

Bad river

Marengo river

Sandy glacial till, poorly developed drainage network, no valley

Altitute, in meters

400 350 Bedrock outcrop 300 Sandy post-glacial shorelines, entrenched valley

250 200 150 (b)

Clay plain, entrenched/alluvial valley 120

100

80 60 40 River kilometer, from mouth

20

0

Figure 44.10  Features and examples of longitudinal profiles. (A) Characteristics of longitudinal profiles. (B) Longitudinal profile of the Bad River and its tributary the Marengo River. The streams are steep in the upper part of their profiles but have poorly developed networks with common wetlands that slow sediment production and reduce streamflow flashiness. Most of the sediment in the watershed is generated from 60-m high eroding valley sides from river kilometer 55 to 65. From river kilometer 55 to the mouth, the river intersects highly erodible clayey glacial lacustrine deposits.

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Watersheds and Drainage Networks    44-9 

Original channel New channel Drainage divide Tip of new channel Tip of original channel

Ero

Aggradation

Stage I premodified

Stage IV degradation and widening

n

sio

Aggradation/new alluvium

Stage II channelized

Stage V aggradation and widening

Stage III degradation

Stage VI quasi-equilibrium

Aggradation/new alluvium Degradation/erosion Stage VI

Aggraded

Stage V

Stage IV

Stage III

Stages I, II

Aggradation

Figure 44.11  Spatial and temporal changes in stream networks and effects on longitudinal profiles and erosion and deposition: (A) headward expansion of a stream network from land clearing and erosion (modified from Strahler, 1958), (B) temporal changes due to headward knickpoint migration along channelized streams (from Simon and Hupp, 1986).

sediment dynamics (erosion, transfer, and storage) from headwaters to the mouth of a watershed (Schumm, 1977). The profile consists of a graph ­showing elevation changes along a length of a stream. The elevation of its mouth (M)—at a lake, ocean, or the valley bottom of a larger river—is called its local base level, which may be at various altitudes above sea level (Z) and may vary over time. Similar to Horton’s mathematical relations and Strahler’s stream orders, characteristics of discharge, slope, and sediment characteristics progress along a predictable longitudinal continuum that is useful for understanding aquatic ecology/physical habitat connections (Allan and Castillo, 2007) as well as river engineering and management (Thorne et al., 1998). Streams generally develop smooth concave up profiles that reflect the overall longitudinal energy expenditure of a river system (Hack, 1957). Streams with smooth concave up profiles are considered graded or in equilibrium— the channel slope has adjusted to be able to efficiently transport the stream-

44_Singh_ch44_p44.1-44.12.indd 9

flow and sediment load provided by the watershed. Equilibrium profiles are expected to have consistent increases in discharge and channel size with decreases in slope and bed material size (Fryirs and Brierley, 2013). Gilbert (1877) theorized that upward concavity could be explained by increasing discharge, and as discharge increases and channel cross-sectional area increases, along with relative decreases in frictional losses, the channel is able to transport more bedload on a lesser slope. Steep long profiles result in flashier hydrographs than profiles with low relief. Variations in lithology, geologic structure, and tectonic uplift can disrupt the profile (Hack, 1957; Radoane et al., 2003). Convex sections of the longitudinal profile with steep slopes are usually where the stream intersects erosion resistant bedrock. These may be local sources of sediment or geologic controls on sediment supply. Knickpoints are discontinuities in the longitudinal profile, the most obvious being falls over bedrock outcroppings. Tectonic uplift,

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44-10    Watershed Geomorphological Characteristics

tributary junctions, and changes in base level can also cause profile irregularities. Dams, culverts, and other manmade structures function as artificial base level and grade controls. Young (generally less than 15,000 years old) tributaries draining into the Great Lakes in the United States have breaks in their somewhat convex longitudinal slopes coincident with ancient glacial lake shorelines (Fitzpatrick et al., 1999; Fitzpatrick et al., 2006; Fig.  44.10). Underlying stratigraphy not present on the land surface may be affecting sediment characteristics, channel slopes, and profile shapes, even in alluvial sections. For the Bad River example in Fig. 44.10, the lower knickpoint at river kilometer 35 is a set of falls over a bedrock ridge of Precambrian sandstone within thick glaciolacustrine clay deposits. The shape of longitudinal profiles can be described quantitatively in multiple fashions. Regression equations can be constructed that are based on simple linear, logarithmic, exponential, and power functions (Strahler, 1964; Radoane et al., 2003). Another method that is useful for comparing longitudinal profiles on a per unit basis (important for comparing streams of different sizes), is to determine the concavity ratio (Ca) of two measured areas on a longitudinal profile graphic: Ca = A1 /A 2 where A1 is the numerically integrated area between the curve of the profile and a straight line connecting the endpoints of the curve, and A2 is the area of a right triangle with the hypotenuse as the same straight line (Radoane et al., 2003). The Ca ranges from 0 to 1.0. If the profile is close to a straight line the value is close to 0 and if the profiles are more L-shaped the Ca is closer to 1.0. In general, the older the river the more concave its longitudinal profile and closer the Ca to 1.0. Accelerated Erosion and Drainage Evolution Over the time span of human alterations, the drainage network density can change from having a dependent to independent relation with watershed hydrology and downstream streamflow and sediment characteristics and move from steady state to unstable conditions (Table 44.1). One prominent example of this phenomenon is the denudation of vegetative cover for watersheds that have been subject to intensive agricultural uses (Strahler, 1958). Agriculture-associated excess runoff from erodible soils and unconsolidated deposits leads to gullying, headcutting, and permanent headward expansion of drainage networks (Lowdermilk, 1932; Happ et al., 1940; Strahler, 1958; Costa, 1975; Knox, 1977; Knox, 2006). Even if the vegetative cover was returned to its predisturbance condition, the expanded drainage network remains. Knickpoints can continue to migrate gradually and episodically in an upstream direction for decades, centuries, and millennia. Accelerated erosion associated with headward expansion leads to accelerated aggradation in main stem valley bottoms (Fig. 44.11a) (Strahler, 1958). This phenomenon of a continuum of aggradation along a longitudinal profile is also observed from headward incision caused by channelization (Fig. 44.11b) (Simon and Hupp, 1986). In the first example, aggradation is caused by simultaneous watershed-wide soil and gully erosion. In the second example in Fig. 44.11 for channelization, aggradation proceeds over time from downstream to upstream following upstream progression of erosion. Methods for measuring erosion and aggradation along longitudinal profiles are described in detail in Fitzpatrick (2014), and require field measurements and coring. 44.3  OUTSTANDING PROBLEMS AND DIRECTIONS FOR FUTURE WORK

The use of watershed geomorphological characteristics for hydrologic analyses continues to grow and expand as high-resolution DEM and stream network data become more readily available at a global scale. Watershed geomorphological characteristics are almost ubiquitously measured and calculated from paper or digital maps, making it a necessity to keep track of spatial scale, resolution, and source of the original data. Linking land conservation, management, and planning to stream hydrology, sediment dynamics, and contaminant transport can easily cover large regions and river basins through virtually unlimited computing power for geospatial analyses in a GIS. However, with large computing power and massive digital datasets comes a potential disconnect between the user and the limitations of the source data and standards used for mapping. This is probably the largest outstanding problem for watershed geomorphology—standard documentation and assembly of metadata associated with large geospatial data sets. Technical standard methods for calculating metrics consistently are still needed (Brookes, 1995). The Open Geospatial Consortium, an international multidisciplinary consortium of private companies, government agencies, and academia, is working toward a consensus-based process and development of standards to make

44_Singh_ch44_p44.1-44.12.indd 10

geospatial data more accessible and usable, especially for hazards and human safety issues (https://daac.ornl.gov/spatial_data_access.shtml). In the United States, the widely available and georeferenced NHD digital hydrography data are based on blue lines (dashed and solid) drawn by cartographers on the nationally available USGS 7.5-min 1:24,000-scale quadrangle maps. Leopold (1994) summarized the problems inherent in a hydrography dataset based on “….blue lines that do not reflect any statistical characteristics of streamflow occurrence… Rather, the choice of what is to be shown as an interrupted blue line is based on consistent portrayal...” Hydrology and geomorphology students and young practitioners that use the U.S. digital hydrography may not have seen or held a paper copy of a 7.5-min quadrangle map, let alone thought about why a streamline exists in one swale and not another. Even so, as mentioned earlier, the NHD remains the preferred dataset on which regional stream network geomorphological measurements are made by many agencies in the United States because of its consistent flow network and thorough metadata, even though more detailed networks can be quickly generated from remotely sensed DEM with airborne LiDAR sensors. Like all geodetic data, however, it is important to document the accuracy not only of the LiDAR data, but also the generated DEM data (Hodgson and Bresnahan, 2005). Directions for future work for application of watershed geomorphological characteristics lie in watershed and longitudinal connections of hydrology and sediment dynamics for river and floodplain restoration (Brookes, 1995). Emerging environmental issues need interdisciplinary research that includes a combination of hydrology, geomorphology, and ecology, including priority areas of landscape ecology and landscape evolution modeling, ecological restoration, and climate change adaption (Renschler et al., 2007). Research needs include understanding problems of erosion and sedimentation, and identifying and fingerprinting sources of fine-sediment-related contaminants and turbidity. Additional research is needed in applying watershed geomorphological principles to ecological integrity studies, land conservation management, flood planning and management, fisheries and habitat enhancement, and integration of aesthetic and recreational opportunities. Watershed geomorphological characteristics are important in assessments of drinking water sources, flood reservoir lifespans, and navigation planning (Binnie Black and Veatch, 2001). Ultimately, human alterations to water generation and supply can interfere with the application of these measurement. This is especially important in semiarid or mountainous regions, where interbasin transfers, diversions, and withdrawals are common. Watershed responses to changes in sediment dynamics occur along a much longer time continuum than hydrologic changes because of legacy effects on the storage and transport of sediment from upstream to downstream (Church, 2002). Common factors that affect sediment dynamics may positively interact or counteract each other, making it important to understand historical as well as current perturbations, such as dam building, agricultural drainage, ditching, or mining (Table 44.5). Understanding time scales of change and channel evolution in terms of a continuum of sediment transport from headwaters to mouth areas is important for management considerations that seek to reduce sediment loads by controlling upland erosion (Sear et al., 2003). This is especially pertinent when it comes to fine sediment quantity and quality, and understanding stream network sources of sediment, such as how increased runoff not only causes soil erosion but irreversible stream network extension, gully erosion, and landslides (Owens et al., 2005). In conclusion, defining spatial, longitudinal, and vertical continuums in a watershed is an important first step to defining any hydrologically related problem or issue, no matter how local or widespread. Watershed geomorphological characteristics will help give perspective, depth, and understanding to complex hydrologic, water quality, and ecological studies.

Table 44.5  Common Watershed-Scale Perturbations Increased sediment supply Climate change (increase in rainfall)

Decreased sediment supply Climate change (decrease in rainfall)

Upland drainage

Dams/river regulation

Deforestation

Reduced cropping and grazing

Mining spoil additions

Cessation of mining

Urban development (construction phase)

Revegetation, paving over bare ground

Agricultural drainage

Sediment management

[Source: Sear et al., 2003]

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References    44-11  REFERENCES

Allan, J. D. and M. M. Castillo, Stream Ecology, Structure and Function of Running waters, 2nd ed., Springer, Dordrecht, The Netherlands, 2007. Binnie Black and Veatch, Fluvial Design Guide: R&D Technical Report W109 (W5-027 Version 11/01), Surrey, Redhill, 2001, https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/290794/str-w109a-e-e.pdf. Brierley, G. J. and K. A. Fryirs, Geomorphology and River Management: Applications of the River Styles Framework, Blackwell Publications, Oxford, 2005. Brinson, M. M., F. R. Hauer, L. C. Lee, W. L. Nutter, R. D. Rheinhardt, R. D. Smith, and D. Whigham, “A guidebook for application of hydrogeomorphologic assessments to riverine wetlands,” Technical Report WRP-DE-11, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. NTIS No. AD A308 365, 1995. Brookes, A., “Challenges and objectives for geomorphology in U.K. river management,” Earth Surface Processes and Landforms, 20: 593–610, 1995. W. B. Bull, Geomorphologic Responses to Climate Change, Oxford University Press, New York, 1991, p. 326. Chorley, R. J., Illustrating the Laws of Morphometry, Geological Magazine, Vol. 94, 1957, pp. 140–150. Chorley, R. J., Geomorphology and General Systems Theory, U.S. Geological Survey Professional Paper 500-B, Washington, DC, 1962. Church, M., “Geomorphologic thresholds in riverine landscapes,” Freshwater Biology, 47: 541–557, 2002. Clifford, N., “Rivers and drainage basins,” edited by J. A. Agnew, and D. N. Livingstone, The SAGE Handbook of Geographical Knowledge, SAGE, London, 2011, pp. 502–527. Costa, J. E., “Effects of agriculture on erosion and sedimentation in the piedmont province, Maryland,” Geological Society of America Bulletin, 86: 1281–1286, 1975. Davis, W. M., “The geographic cycle,” Geographical Journal, 14: 481–504, 1899. Emmett, W., “The channels and waters of the upper Salmon River area, Idaho,” U.S. Geological Survey Professional Paper 870-A, 1975. Fitzpatrick, F. A., “A geologic approach to field methods in fluvial geomorphology,” edited by M. J. Thornbush, C. D. Allen, and F. A. Fitzpatrick, Geomorphological Fieldwork: Developments in Earth Surface Processes, Vol. 18, Chap. 5.2, 2014, pp. 201–230. Fitzpatrick, F. A. and M. P. Peppler, “Relation of urbanization to stream geomorphologic and habitat characteristics in nine metropolitan areas of the United States,” U.S. Geological Survey Scientific Investigations Report 2010–5056, 2010, p. 29. Fitzpatrick, F. A., J. C. Knox, and H. E. Whitman, “Effects of historical landcover changes on flooding and sedimentation, North Fish Creek, Wisconsin,” U.S. Geological Survey Water-Resources Investigations Report 99–4083, 1999, p. 12. Fitzpatrick, F. A., M. C. Peppler, M. M. DePhilip, and K. E. Lee, “Geomorphologic characteristics and classification of Duluth-area streams, Minnesota,” U.S. Geological Survey Scientific Investigations Report 2006–5029, 2006, accessed May 15, 2014, http://pubs.usgs.gov/sir/2006/5029/. Fitzpatrick, F. A., M. C. Peppler, D. A. Saad, D. M. Pratt, and B. N. Lenz, “Geomorphologic, flood, and groundwater-flow characteristics of Bayfield Peninsula streams, Wisconsin, and implications for brook-trout habitat,” U.S. Geological Survey Scientific Investigations Report 2014–5007, 2015. Fitzpatrick, F. A., I. R. Waite, P. J. D’Arconte, M. R. Meador, M. A. Maupin, and M. E. Gurtz, “Revised methods for characterizing stream habitat in the National Water-Quality Assessment Program,” U.S. Geological Survey Water-Resources Investigations Report 98–4052, 1998, p. 67. Fryirs, K. A. and G. J. Brierley, “Geomorphologic Analysis of River Systems,” An Approach to Reading the Landscape, John Wiley & Sons, Chichester, 2013. Gardiner, V., “Drainage basin morphometry,” Technical Bulletin 14, British Geomorphological Research Group, Norwich, 1975, p. 48. Gilbert, G. K., “Report on the geology of the Henry Mountains,” U.S. Geo­graphical and Geological Survey of the Rocky Mountain Region, Washington, DC, 1877. Glock, W. S., “The development of drainage systems: a synoptic view,” Geographical Review, 21: 475–482, 1931. Gordon, N. D., T. A. McMahon, and B. L. Finlayson, Stream Hydrology, an Introduction For Ecologists, John Wiley & Sons, Chichester, 1992. Hack, J., “Studies of longitudinal stream profiles in Virginia and Maryland,” U.S. Geological Survey Professional Paper, 294–B: 59 P, 1957. Happ, S. C., G. Rittenhouse, and G. C. Dobson, “Some principles of accelerated stream and valley sedimentation,” Technical Bulletin 695, U.S. Department of Agriculture, Washington, DC, 1940. Harvey, C. A. and D. A. Eash, “Description, instructions, and verification for Basinsoft, a computer program to quantify drainage-basin characteristics,” U.S. Geological Survey Water-Resources Investigations Report 95–4287, 1996, http://pubs.usgs.gov/wri/1995/4287/report.pdf.

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Horton, R. E., “Drainage basin characteristics,” Transactions of the American Geophysical Union, 13: 350–361, 1932. Horton, R. E., “Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphology,” Geological Society of America Bulletin, 56: 275–370, 1945. Howard, A. D., “Drainage analyses in geologic interpretation: a summation,” The American Association of Petroleum Geologists Bulletin, 51 (11): 2246–2259, 1967. Knox, J. C., “Human impacts on Wisconsin stream channels,” Annals of the Association of American Geographers, 67 (3): 323–342, 1977. Knox, J. C., “Floodplain sedimentation in the Upper Mississippi Valley— Natural versus human accelerated,” Geomorphology, 79: 286–310, 2006. Knox, J. C., P. J. Bartlein, and W. C. Johnson, “Environmental assessment of sediment sources and sedimentation distributions for the Lake LaFarge watershed and impoundment,” Environmental Analysis of the Kickapoo River Impoundment: Institute for Environmental Studies, Report 28, University of Wisconsin, Madison, WI, 1974, pp. 77–116. Langbein, W. B. and S. A. Schumm, “Yield of sediment in relation to mean annual precipitation,” American Geophysical Union Transactions, 39: 1076–1084, 1958. Leopold, L. B., A View of the River, Harvard University Press, Cambridge, MA, 1994. Leopold, L. B., M. G. Wolman, and J. P. Miller, Fluvial Processes in Geomorphology, Dover Publications, New York, 1964. Lowdermilk, W. C., “Acceleration of erosion above geologic norms,” American Geophysical Union, 15th Transactions, 2: 505–509, 1932. Mackin, J. H., “Concept of a graded river,” Geological Society of America Bulletin, 59: 463–512, 1948. Melton, M. A., “A derivation of Strahler’s channel-ordering system,” Journal of Geology, 67: 345–346, 1959. Morisawa, M. E., “Measurement of drainage-basin outline form,” Journal of Geology, 66: 587–591, 1958. Miller, V. C., “A quantitative geomorphologic study of drainage basin characteristics in the Clinch Mountain area, Virginia and Tennessee,” Project NR 389-042. Technical Report 3, Columbia University, Department of Geology, New York, 1953. Owens, P. N., R. J. Batalla, A. J. Collins, B. Gomez, D. M. Hicks, A. J. Horowitz, G. M. Kondolf, et al., “Fine-grained sediment in river systems: environmental significance and manage issues,” River Research Application, 21: 693–717, 2005. Radoane, M., N. Radoane, and D. Dumitriu, “Geomorphological evolution of longitudinal profiles in the Carpathians,” Geomorphology, 50: 293–306, 2003. Reddy, J. R., A Textbook of Hydrology, Laxmi Publications, New Delhi, India, 2005, p. 93. Renschler, C. S., M. W. Doyle, and M. Thoms, “Geomorphology and ecosystems: challenges and keys for success in bridging disciplines,” Geomorphology, 89: 1–8, 2007. Ries, K. G., III, “The national streamflow statistics program: a computer program for estimating streamflow statistics for ungaged sites,” U.S. Geological Techniques and Methods Report TM Book 5, Chap. A6, 2006, p. 45. Ritter, D. F., R. C. Kochel, and J. R. Miller, Process Geomorphology, 4th ed., McGraw Hill, Boston, MA, 2002. Robertson, D. M., G. E. Schwarz, D. A. Saad, and R. B. Alexander, “Incorporating uncertainty into the ranking of Sparrow model nutrient yields from Mississippi/Atchafalaya River Basin watersheds,” Journal of the American Water Resources Association, 45 (2): 534–549, 2009. Rosgen, D. L., “Watershed assessment of river stability and sediment supply (WARSSS),” Wildland Hydrology, Fort Collins, CO, 2006. Schumm, S. A., “Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey,” Geological Society of America Bulletin, 67: 597–646, 1956. Schumm, S. A., “Geomorphologic thresholds and complex response of drainage systems,” edited by M. Morisawa, Fluvial Geomorphology, S.U.N.Y. Binghamton Pubs., Binghamton, NY, 1973, pp. 299–310. Schumm, S. A., The Fluvial System, John Wiley and Sons, Chichester, 1977. Schumm, S. A. and R. W. Lichty, “Time, space, and causality in geomorphology,” American Journal of Science, 263: 110–119, 1965. Seaber, P. R., F. P. Kapinos, and G. L. Knapp, “Hydrologic unit maps,” U.S. Geological Survey Water Supply Paper 2294, 1987, p. 63. Sear, D. A., M. D. Newson, and C. R. Thorne, “Guidebook of applied fluvial geomorphology: Defra/Environment Agency,” Flood and Coastal Defense R&D Programme, R&D Technical Report FD1914, London, UK, 2003. Shreve, R. L., “Infinite topologically random channel networks,” Journal of Geology, 75: 178–186, 1967.

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Simon, A. and C. R. Hupp, “Channel evolution in modified Tennessee channels,” Proceedings of the 4th Federation of Interagency Sediment Conference, 2 (5): 71–82, 1986. Retrieved from https://www.sciencebase.gov/catalog/ item/50579e95e4b01ad7e02863d8. Smith, R. A., G. E. Schwarz, and R. B. Alexander, “Regional interpretation of water-quality monitoring data,” Water Resources Research, 33: 2781–2798, 1997. Strahler, A. N., “Dynamic basis of geomorphology,” Bulletin of the Geological Society of America, 63 (9): 923–938, 1952a. Strahler, A. N., “Hypsometric (area-altitude) analysis of erosional topography,” Geological Society of America Bulletin, 63 (11): 1117–1142, 1952b. Strahler, A. N., “Quantitative analyses of watershed geomorphology,” Transactions American Geophysical Union Journal, 38 (6): 913–920, 1957.

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Strahler, A. N., “Dimensional analysis applied to fluvially eroded landforms,” Geological Society of America Bulletin, 69: 279–300, 1958. Strahler, A. N., “Quantitative geomorphology of drainage basins and channel networks, Section 4-II,” edited by V. T. Chow, Handbook of Applied Hydrology, McGraw-Hill Book Company, New York, NY, 1964, pp.  4-39– 4-76. Thorne, C. R., R. D. Hey, and M. D. Newson, Applied Fluvial Geomorphology for River Engineering and Management, John Wiley & Sons, 1998. Turnipseed, D. P. and K. G. Ries, III, “The national streamflow statistics program: estimating high and low streamflow statistics for ungaged sites,” U.S. Geological Survey Fact Sheet 2007–3010, 2007, p. 4.

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Chapter

45

Infiltration Modeling BY

C. CORRADINI, R. MORBIDELLI, AND R. S. GOVINDARAJU

ABSTRACT

Many mathematical models of vertical infiltration of rainfall into a soil matrix with horizontal surface are described. Firstly, classical approaches developed for regular rainfalls which keep continuous saturation at the soil surface and a more general formulation suitable for any type of rainfall pattern are considered for applications at the local (point) scale in homogeneous soils. Then, a simple model for point infiltration into a two-layered soil with a more permeable upper layer and a more complex model for any two-layered soil type are presented. Two field-scale infiltration models are also analyzed. One is of semi-empirical type and represents the spatial variability of saturated hydraulic conductivity, Ks, the other relies upon a semi-analytical/conceptual framework and combines the spatial heterogeneity of Ks with that of rainfall rate. Open problems linked with rainfall infiltration into surfaces with significant slopes are also discussed. 45.1 INTRODUCTION

Rainfall infiltration plays an important role in the reduction of surface runoff and provides subsurface water that governs water supply for agriculture, transport of pollutants through the vadose zone, and recharge of aquifers. The spatiotemporal evolution of infiltration rate under natural conditions cannot be currently deduced by direct measurements at any scale of interest in applied hydrology, therefore, the use of infiltration modeling allowing to express it through measurable quantities is of fundamental importance. Even though the representation of the main natural processes in applied hydrology requires areal infiltration modeling for both flat and sloping surfaces, research activity has been limited for many years to the development of local, or point, infiltration models for vertically homogeneous soils (Green and Ampt, 1911; Philip, 1957a; 1957b; 1957c; Smith and Parlange, 1978; Corradini et al., 1997). Natural soils are rarely vertically homogeneous. In hydrological simulations, local infiltration for the estimate of effective rainfall can be reasonably schematized by a two-layered vertical profile (Mualem et al., 1993; Taha et al., 1997). Soils representable by a sealing layer over the parent soil or a vertical profile with a more permeable upper layer are frequently found. Some models for infiltration into crusted soils were developed under significant approximations by adapting the Green-Ampt model. A general semianalytical/conceptual model for crusted soils was later formulated by Smith et al. (1999), and was extended by Corradini et al. (2000) to represent infiltration and reinfiltration after a redistribution period under any rainfall pattern and for any twolayered soil, where either layer may be more or less permeable than the other. For a much more permeable upper layer, and under more restrictive rainfall patterns, a simpler semiempirical/conceptual model was presented by Corradini et al. (2011a). Under conditions of surface saturation, a simple Green-Ampt–based model was also proposed (Chow et al., 1988). In applied hydrology, upscaling of point infiltration modeling to the field scale is required to estimate the areal-average infiltration. This is a complex task because of the natural spatial heterogeneity of hydraulic soil properties (Nielsen et al., 1973; Warrick and Nielsen, 1980; Greminger et al., 1985; Sharma et al., 1987; Loague and Gander, 1990) and particularly of the soil

saturated hydraulic conductivity (Russo and Bresler, 1981; 1982) that may be assumed as a random field with a lognormal univariate probability distribution (Smith and Goodrich, 2000). In addition to the heterogeneity of ­saturated hydraulic conductivity, rainfall is also characterized by spatial variability (Goodrich et al., 1995; Krajewski et al., 2003) and their interaction makes in principle the estimate of infiltration at the field scale very complex (Morbidelli et al., 2006). A model of the expected areal-average infiltration into a much more permeable upper layer of a two-layered soil was also proposed by Corradini et al. (2011b) considering only a spatial horizontal random field of the saturated hydraulic conductivity. It involves the solution of a set of algebraic equations obtained by upscaling simple local infiltration equations to the field scale. The same quantity, but for a vertically nonuniform soil characterized by a saturated hydraulic conductivity decreasing with depth according to a power law, was derived by Govindaraju et al. (2012) and upscaled to the field-scale by the same semianalytical technique used in previous papers (Govindaraju et al., 2001; 2006). Most of the aforementioned models consider a surface placed horizontally or with a low slope that does not affect the infiltration process. However, in most real situations, infiltration occurs in surfaces characterized by different gradients (Beven, 2002; Fiori et al., 2007) and the role of surface slope on infiltration is not clear. In fact, the results obtained by some theoretical and experimental investigations (Sharma et al., 1983; Poesen, 1984; Philip, 1991; Chen and Young, 2006; Essig et al., 2009; Morbidelli et al., 2015) lead to rather contrasting conclusions, suggesting an improved understanding and modeling of infiltration on sloping surfaces are required. When macropore flow plays a significant role in determining infiltration, then other simplified approaches may be used (e.g., Arabi et al., 2006; Stillman et al., 2006). Finally, a method widely used for the representation of infiltration in ungaged basins, together with interception and depression storage, was developed by the United States Department of Agriculture (USDA)-soil-conversion service (SCS) to estimate rainfall excess (Soil Conservation Service, 1972). 45.2  BASIC EQUATIONS FOR VERTICAL INFILTRATION

Under conditions of horizontally homogeneous soils, the water movement in the vertical direction is governed by one-dimensional soil water flow and continuity equations. The flow rate, q, per unit cross-sectional area is described by Darcy’s law as:  ∂ψ  q = −K  − 1 (45.1)  ∂z  where K is the hydraulic conductivity, ψ the soil water matric capillary head, and z the vertical soil depth assumed positive downward. The infiltration rate, q0, is given by Eq. (45.1) applied at the soil surface. The mass conservation equation, in the absence of changes in the water density and porosity, ϕ, as well as of sinks and sources, is expressed by: ∂θ ∂q =− (45.2) ∂t ∂z 45-1

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45-2    Infiltration Modeling

where θ is the volumetric water content and t the time. Substitution of Eq. (45.1) in Eq. (45.2) leads to the well-known Richards equation: C1

∂ψ ∂  ∂ψ  ∂ψ = − C2 K ∂t ∂z  ∂z  ∂z

(45.3)

Equation (45.8) applies starting from the assumption of immediate ponding. Under more general conditions, with a constant rainfall rate r > Ks, that begins at the time t = 0, surface saturation is reached at a time tp > 0. For t ≤ tp the infiltration rate q0 is equal to r and later to the infiltration capacity. Mein and Larson (1973) formulated this process through Eq. (45.7) as:

where C1 = dθ/dψ and C2 = dK/dψ under the assumption that θ and K are unique functions of ψ with hysteresis in these functions being neglected (see also Dagan and Bresler, 1983; Smith, 2002). The initial condition at time t = 0 for z ≥ 0 is ψ = ψi, the upper boundary conditions at the soil surface are: z = 0 z = 0 z = 0

0 < t ≤ tp q0 = r(45.4a) tp < t ≤ tr θ0 = θs saturation stage (45.4b) tr < t q0 = 0 redistribution stage (45.4c)

where r is the rainfall rate, tp the time to ponding, tr the duration of rainfall; hereafter, the subscripts i and s denote initial and saturation quantities, respectively, and subscript 0 stands for quantities at the soil surface. The lower boundary conditions at a depth zb, which is not reached by the wetting front is ψ(zb) = ψi for t > 0. The soil water hydraulic properties can be represented by the following parameterized forms (Smith et al., 1993): − c /λ  1/c θ −θ  ψ = ψb  * r  − 1  + d  θ s − θ r   c −(bλ+a)/c   ψ −d   K = K s∗ 1 +     ψb  

(45.5a)

(45.5b)

where θ s∗ and K s∗ are used as scaling quantities, ψb is the air entry head, θr is the residual volumetric water content, c, λ, and d are empirical coefficients, b = 3 and a = 2 according to Burdine’s method (Brooks and Corey, 1964). For particular values of parameters, Eqs. (45.5a and 45.5b) reduce to the wellknown equations proposed by Brooks and Corey (1964) and van Genucthen (1980). For two-layered soils two additional conditions are required at the interface between the two layers:

ψ1 ( Zc ) = ψ2 ( Zc ) = ψc (45.6a)  ∂ψ  K1  1  −  ∂z Zc

  ∂ψ  1 = K 2  2  −   ∂z Zc

 1 

(45.6b)

where hereafter the subscripts 1, 2, and c denote variables in the upper layer, lower layer, and at the interface, respectively, and Zc is the interface depth. 45.3  CLASSICAL MODELS FOR POINT INFILTRATION INTO VERTICALLY HOMOGENEOUS SOILS

A variety of local infiltration models for vertically homogeneous soils with constant initial soil water content and over horizontal surfaces have been proposed (Green and Ampt, 1911; Kostiakov, 1932; Horton, 1940; Holtan, 1961; Swartzendruber, 1987; Philip, 1957a; 1957b; 1957c; Smith and Parlange, 1978; Broadbridge and White, 1988; Dagan and Bresler, 1983; Smith et al., 1993; Corradini et al., 1994; Corradini et al., 1997, Kacimov and Obnosov, 2013). Furthermore, for isolated regular storms and when ponding is not achieved instantly, extended forms of the Philip model (Chow et al., 1988), Green–Ampt model (Mein and Larson, 1973; Chu, 1978) and the Smith and Parlange model (Parlange et al., 1982) have been widely used, while for erratic rainfalls the Corradini et al. (1997) model can be properly selected. These four models have been extensively used in applied hydrology and as building blocks in the development of infiltration approaches at the field scale. 45.3.1  Extended Green–Ampt Model

The original Green–Ampt model represents infiltration into homogeneous soils under the conditions of continuously saturated soil surface and uniform initial soil moisture as:  ψ (θ − θ )  fc = K s 1 − av s i  (45.7) F   where ψav is the soil water matric capillary head at the wetting front, F the cumulative depth of infiltrated water, and fc the soil infiltration capacity. To express the infiltration as a function of time, this equation can be solved after the substitution fc = dF/dt. The resulting equation (Chow et al., 1988) is:   F F = K st − ψav (θ s − θi ) ln1 −   ψav (θ s − θi ) 

45_Singh_ch45_p45.1-45.10.indd 2

(45.8)

r − Ks = −

K sψav (θ s − θi ) tp

∫ r dt

  t = tp(45.9)

0

which leads to determine tp as: tp = −

K sψav (θ s − θi ) (45.10) r (r − K s )

Furthermore Eq. (45.8) becomes F = Fp − ψav (θ s − θi ) ln

F − ψav (θ s − θi ) + K s (t − t p ) Fp − ψav (θ s − θi )

  t > tp(45.11)

Equation (45.11) can be solved at each time, for example, by successive substitutions of F which is then substituted in Eq. (45.7) to obtain the corresponding value of fc. Alternatively, through a series expansion of the log-term and simplifying, the following explicit approximation to F is obtained (Govindaraju et al., 2001): 12 12 1  2 K s3  3 2 3 2 2 F = Fp +2 K sψav (θ s − θi ) (t 1 2 − t 1p 2 )+ K s (t − t p )+   (t − t p ) 3 18 ψ (θ s − θi ) 

t > tp(45.12)



Indicative values of |ψav | and Ks, together with porosity, φ, are shown in Table 45.1 (see also Rawls and Brakensiek, 1983). 45.3.2  Extended Philip Model

Philip (1957a; 1957b; 1957c; 1969) proposed an infiltration model obtained through an analytical series solution of the Richards equation under the conditions of vertically homogeneous soil, constant initial moisture content, and saturated soil surface with immediate ponding. For early to intermediate times, the series solution can be truncated after the first two terms and the infiltration capacity can be expressed as: 1 fc = St −1 2 + A (45.13) 2 where S is the sorptivity, depending on soil properties and initial moisture content, and A is a quantity ranging from 0.38 Ks to 0.66 Ks. For t→∞ Eq. (45.13) is replaced by fc = Ks. Integration of Eq. (45.13) yields the cumulative infiltration: F = St 1 2 + At (45.14) Philip’s model was extended for applications to less–restrictive c­ onditions. For constant rainfall rate r > Ks, surface saturation occurs at a time tp > 0 and Table 45.1  Green–Ampt Infiltration Parameters for Various USDA Soil Classes Saturated hydraulic conductivity, Ks (mmh-1)

Porosity, ϕ

Wetting front soil suction head, ψav (mm)

Sand

0.437

49.5

235.6

Loamy sand

0.437

61.3

59.8

Sandy loam

0.453

110.1

21.8

Loam

0.463

88.9

13.2

Silt loam

0.501

166.8

6.8

Sandy clay loam

0.398

218.5

3.0

Clay loam

0.464

208.8

2.0

Silty clay loam

0.471

273.0

2.0

Sandy clay

0.430

239.0

1.2

Silty clay

0.479

292.2

1.0

Clay

0.475

316.3

0.6

Soil texture class

[Source: Rawls and Brakensiek, 1983]

8/22/16 2:14 PM

Classical Models For Point Infiltration Into Vertically Homogeneous Soils     45-3 

following Chow et al. (1988) infiltration can be described through an equivalent time origin, t0, for potential infiltration after p ­ onding as:  A S 2 r −   2 tp = 2 (45.15) 2r (r − A) 1  2 S + 4 AFp 4 A2 

(

)

12

2

− S 

(45.16)

1 −1 2 fc = S (t − t0 ) + A   t > tp(45.17) 2 For unsteady rainfall under the condition of a continuously saturated surface for t > tp, the infiltration process can be represented adopting a similar procedure. Generally, S and A are derived from calibration of hydrological models; however, S can also be approximated through the parameters of the Green–Ampt equation (see Youngs, 1964) as: 12

S = 2 (φ − θi ) K s ψav 

(45.18)

Parlange et al. (1982) presented a three-parameter model obtained through an analytical integration of the Richards equation and expressed as:    α fc = K s 1 +  αF′   exp − 1   G (θ s − θi )  

      

where p is a parameter linked with the profile shape of θ, and G(θi,θ0) is expressed by Eq. (45.20) modified by the substitution of θs with θ0 and Ks with K0. Equation (45.22) can be applied for 0 < t < t2, the profile shape of θ(z) is approximated (Corradini et al., 1994) by  β z (θ − θ ) − F ′  θ ( z ) − θi 0 i = 1 − exp  (45.23) 2 ′ θ 0 − θi  (β − β ) F 

(45.19)

where F' = F – Kit is the cumulative dynamic infiltration rate, α a parameter linked with the behavior of hydraulic conductivity and soil water diffusivity, D, as functions of θ, and G is the integral capillary drive defined by 1 Ks

Functional forms for β and p were obtained by calibration using results provided by the Richards equation applied to a silty loam soil, specifically:

β (θ 0 ) = 0.6

θs

∫ D (θ ) dθ (45.20)

θi

Equation (45.19) includes as limiting forms (Smith and Goodrich, 2000) the Smith and Parlange (1978) equation (α = 1) and the Green–Ampt equation (α→0). It can be applied to determine tp and fc for any rainfall pattern and for t > tp can be rewritten under the condition of surface saturation in a time dependent form as:  ′   αKs  exp α F − 1 +  Kd  (θ s − θi ) G  ′ ′ (θ s − θi ) K sG ln  [(1 − α ) K s − Ki ](t − t p )= F − Fp −  α Fp′  Kd αKs   exp (θ − θ ) G  − 1 + K d  s i  

      

β p = 0.98 − 0.87exp (−r K s )  dθ0/dt ≥ 0

(45.24b)

β p =1.7 dθ0/dt < 0

(45.24c)

Rainfall rate

where Kd = Ks – Ki and Fp′ = F'(tp). The quantities Fp′ and tp are the values of F' and t, respectively, at which Eq. (45.19) with fc = r(tp) is first satisfied. The value of α usually ranges from 0.8 to 0.85 (Smith, 2002). 45.3.4  A Semianalytical/Conceptual Model

(a)

t1

tp’

t2

tp”

t3

Soil water content θi θs

t4

Time

Soil water content θi θs tp”

Depth

For complex rainfall patterns involving rainfall hiatus periods or a rainfall rate after time to ponding less than soil-infiltration capacity, the models presented in the previous subsections cannot be applied in principle because the assumption of uniform initial soil moisture cannot be met for successive storms. Alternatively, an approach for the application of the aforementioned classical models was developed (Mls, 1980; Péschke and Kutílek, 1982; and Verma, 1982) starting from the time compression approximation proposed by Reeves and Miller (1975) for post-hiatus rainfall producing immediate ponding (see also Brutsaert, 2005). However, Smith et al. (1993) by comparison with results of the Richards equation showed that the last approach was not sufficiently accurate because it neglects the soil water redistribution process, which is particularly important when long periods with a light rainfall or a rainfall hiatus occur. Models combining infiltration and redistribution are, therefore, the best solution when complex rainfall patterns, which are rather common under natural conditions, have to be considered (see also Basha, 2011a; 2011b). Dagan and Bresler (1983) developed an analytical model along this line starting from depth integrated forms of the Darcy and continuity equations and using simplifications in the initial and surface boundary conditions that make easier areal investigations, but reduce practical applications at the local scale. In any case, the model is not addressed toward local studies with erratic rainfalls producing successive infiltration-redistribution cycles.

θ s − θi + 0.4 (45.24a) θ s −θr

Equation (45.22) can be solved numerically. For q0 = r, with F' = (r - Ki)t, it gives θ0(t) until time to ponding, t ′p, corresponding to θ0 = θs and dθ0/dt = 0, then for t ′p < t ≤ t1, with θ0 = θs and dθ0/dt = 0, it provides the infiltration

(45.21)

45_Singh_ch45_p45.1-45.10.indd 3

 (θ0 − θi ) β (θ0 ) (θ0 − θi ) G (θi ,θ0 ) β (θ0 ) p K 0  q0−K 0 −    F′ d β θ   ( ) 0 F ′ (θ 0 − θi ) + β (θ 0 ) dθ 0  

(45.22)

45.3.3  A Three-Parameter Semianalytical Model

G=

dθ 0 = dt

tp’ t1

Depth

t0 = t p −

A more general model was formulated by Corradini et al. (1997) starting from the same integrated equations then combined with a conceptual representation of the wetting soil moisture profile. To highlight the structure of the last model, a specific rainfall pattern which allows to describe all the involved components is used here. The application to erratic rainfall is then a straightforward problem. Let us consider a stepwise rainfall pattern involving successive periods of rainfall with constant r > Ks, separated by periods with r = 0 (see Fig. 45.1). We denote by t1 the duration of the first pulse, t2 and t3, the beginning and end of the second pulse, respectively, and t4 the beginning of the third pulse. The model was derived considering a soil with a constant value of θi and combining the depth-integrated forms of the Darcy law and continuity equation. In addition, as the event progresses in time, a dynamic wetting profile, of lowest depth Z and represented by a distorted rectangle through a shape factor β(θ0) ≤ 1, was assumed. The resulting ordinary differential equation is

t1 t2

(b)

(c)

t3

Figure 45.1  (a) Rainfall pattern selected to describe the semianalytical/conceptual model for point infiltration. [(b) and (c)] Profiles of soil water content at various times indicated in (a) and associated with different infiltration-redistribution stages. For symbol, see text.

8/22/16 2:15 PM

45-4    Infiltration Modeling

capacity (q0 = fc) and for t1 < t < t2, with q0 = 0, it gives dθ0/dt < 0 thus ­describing the redistribution process. The second rainfall pulse leads to a new time to ponding, t ′′p, but reinfiltration occurs according to two alternative approaches determined by a comparison of r and the downward redistribution rate, DF(t = t2), expressed by:  1 dβ 1  ′ dθ 0 − Ki   t = t2(45.25) DF =  − − F  β dθ 0 θ 0 − θi  dt Specifically, for r ≤ DF the reinfiltrated water is distributed to the whole dynamic profile and θ0(t) can be still computed by Eq. (45.22), while for r > DF the profile of θ(z, t = t2) is assumed temporarily invariant and starts a superimposed secondary wetting profile which advances alongside the preexisting profile according to Eq. (45.22) modified substituting θi with θ0(t2) and Fʹ with Fʹ2t accumulated for t ≥ t2. If the secondary profile reaches the depth of the steady one, the compound profile reduces to a single profile and then Eq. (45.22) can be again applied [see Fig. 45.1 for a schematic representation of θ(z,t)]. On the other hand, if at t = t3 the secondary profiles has not caught up with the first one, in the successive rainfall hiatus redistribution is first applied to the secondary profile and then reestablished to the single profile. Finally, in the case at t = t4 the θ(z) profile is still compound and r is larger than DF(t4), a procedure of consolidation that merges the composite profile is applied early to avoid the formation of a further additional profile. The model incorporates all the components required for the application to any natural rainfall pattern. It was calibrated by Corradini et al. (1997) for a silty loam soil (denoted as Soil C), and then tested using different soils (denoted as A, B, J, D, E, and SL) from clay loam to sandy loam soil types. Table 45.2 shows the parameters required to define the hydraulic soil properties of these soils. Weighted implicit finite difference solutions of the Richards equation were used as a benchmark. For each soil, the model accuracy was found to be acceptable in terms of both infiltration rate and soil water content, even though better results were obtained for fine-textured soils. In addition the model was found to simulate fairly well the θ(z,t) profiles observed in laboratory and field experiments (Melone et al., 2006; 2008).

conditions, the model by Corradini et al. (2000) appears to be accurate and in any case requires a fairly limited computational effort. 45.4.1  Green–Ampt-Based Model for a More Permeable Upper Layer

Chow et al. (1988) presented a model for infiltration into a two-layered soil with a more permeable upper layer under the condition of continuously saturated soil surface. The classical Green–Ampt equations are applied until the wetting front is in the upper layer, then the following equations are used: L2

(θ 2s − θ 2i ) + K 2s

1  (θ 2 s − θ 2i ) Zc K 2 s − (θ 2 s − θ 2i ) K1s (ψav 2 + Zc ) K1 s K 2 s

 L2  ln1 + =t  ψav 2 + Zc 

A two-layer approximation, with each layer being schematized as homogeneous, is frequently used to set up models of infiltration for natural soils. The process of formation of a sealing layer was accurately examined by Mualem and Assouline (1989) and Mualem et al. (1993), and that of disruption was considered by Bullock et al. (1988), Emmerich (2003) and Morbidelli et al. (2011). Evidence of the role of crusted soils in semiarid regions has been recently provided by Chen et al. (2013). Green–Ampt-based models for infiltration into stable crusted soils were proposed by Hillel and Gardner (1970), Ahuja (1983), Vandervaere et al. (1998), and Moore (1981). An efficient approach which represents transient infiltration into crusted soils was proposed by Smith (1990), while Philip (1998) formulated a model, which describes upper layer dynamics, but in the limits of a ponded upper boundary. On the other hand, vertical profiles with a more permeable upper layer are observed in the hydrological practice and can be also used, for example, as a first approximation in the representation of infiltration into homogeneous soils with grassy vegetation (Morbidelli et al., 2014). In the last layering type, the simple model presented by Chow et al. (1988) can be usefully applied for infiltration into a saturated soil surface. Under more general

(45.26)

F = Zc (θ1s − θ1i ) + L2 (θ 2 s − θ 2i ) (45.27) fc =

K1 s K 2 s (ψav 2 + Zc + L2 ) (45.28) Zc K 2 s + L2 K1s

where L2 is the depth of the wetting front below the interface. Equation (45.26) can be solved at each time for L2 by successive substitutions; L2 is then used in Eqs. (45.27) and (45.28) to determine F and fc, respectively. 45.4.2  A Semianalytical/Conceptual Model for Any Two-Layered Soil

Corradini et al. (2000) formulated a semianalytical/conceptual model applicable to any horizontal two-layered soil, where either layer may be more permeable. It relies on the same elements previously used by Corradini et al. (1997), but adopted here in each layer and integrated at the interface between the two layers by the boundary conditions expressing continuity of flow rate, and capillary head [Eqs. (45.6a and 45.6b)]. In addition at the lower boundary for t > 0, we have ψ2 = ψi. The initial condition is ψ1 = ψ2 = ψi constant at t = 0, and at the interface q(Zc) is approximated through the downward flux in the upper layer as: q ( Z c ) = K1 s

45.4  MODELING OF POINT INFILTRATION INTO VERTICALLY NONUNIFORM SOILS



G1 (ψc ,ψ10 ) + K1c (45.29) Zc

where G(ψc,ψ10) is expressed by Eq. (45.20) modified by the substitutions of D(θ)dθ with K(ψ)dψ, θs with ψ10 and θi with ψc. Until water does not infiltrate in the lower layer, the Corradini et al. (1997) model is used, then starting from the time tc when the wetting front enters the lower layer, the following system of two ordinary differential equations may be applied:  K1s G1 (ψc ,ψ10 )  dψ10 1 = q10 − K1c −  dt γ Zc C1 (ψ10 )  Zc  −

(1 − γ )C1 (ψc ) dψc γ C1 (ψ10 )

  for t ≥ tc(45.30)

dt

dψ c 1 = dt PL (ψ c ,t )

 K G (ψ ,ψ ) β (θ ) p (θ − θ 2i ) K 2 s G2 (ψ i ,ψ c )  ×  K1c + 1s 1 c 10 − K 2c − 2 2c 2 2c  Z F2 c   for t ≥ tc(45.31)



Table 45.2  Hydraulic Properties and Parameters for a Variety of Representative Soils Soil A (silty)

Soil B (clay loam)

Soil C (silty loam)

Soil J (sandy)

Soil D (silty loam)

Soil E (clay loam)

Soil SL (sandy loam)

Ks (mmh ) Ks (mmh-1)

0.4000 0.3979

0.4000 0.4000

1.6000 1.6000

0.4000 0.2785

0.4000 0.3998

1.6000 1.6000

25.0000 25.0000

Φ

0.3500

0.3500

0.3500

0.3500

0.3500

0.3500

0.4530

θs* θs

0.3325 0.3324

0.3325 0.3325

0.3325 0.3325

0.3325 0.3219

0.3325 0.3325

0.3325 0.3325

0.4120 0.4120

θr

0.1225

0.1225

0.1225

0.1225

0.1225

0.1225

0.0410

Ψb (mm)

–25

–800

–400

–200

–400

–800

–300

Characteristic property *

-1

λ

0.2

0.2

0.4

0.5

0.4

0.2

0.5

c

5

5

5

5

5

5

5

d (mm)

10

100

0

185

100

0

0

G(θr,θs) (mm)

27.8

1103.9

526.6

110.7

426.9

1203.8

376.1

[Source: Corradini et al., 1997]

45_Singh_ch45_p45.1-45.10.indd 4

8/22/16 2:15 PM

Models for Rainfall Infiltration Over Heterogeneous Areas     45-5 

Two ­models characterized by significant differences in complexity and application area are presented here.

with PL (ψc , t ) and F2 defined as:   dβ F2 C2 (ψc ) PL (ψc , t ) = β 2 (θ 2c ) + 2 (θ 2c − θ 2i ) dθ 2c   (θ 2c − θ 2i ) β 2 (θ 2c )

(45.32)

F2 = F − Zc γ (θ1s − θ1i ) + (1 − γ ) (θ1c − θ1i ) − K 2it

(45.33)

and C1(ψ10) = dθ10/dψ10, C1(ψc) = dθ1c/dψ1c. The quantity γ represents a conceptual proportion of the upper layer, where θ is increasing due to rainfall and is assumed equal to 0.85 (Smith et al., 1999), β2 and p2 are determined by Eqs. (45.24a–45.24c), but applied using θ20, θ2s, θ2i, θ2r , and substituting r with q(Zc). On the basis of the same stepwise rainfall pattern earlier adopted to explain the Corradini et al. (1997) model, Eqs. (45.30) and (45.31) may be used for tc < t < t2. Then, the two-layer model has to be applied in each layer by analogy with the procedure described for homogeneous soils, in particular, compound and consolidated profiles develop in each layer. In the underlying soil, the generation of additional profiles occurs through q(Zc) in substitution of r. On the basis of the described steps, model application to erratic rainfall patterns is straightforward. The solution of the above system, Eqs. (45.30) and (45.31), may be obtained by a library routine for the Runge–Kutta–Verner fifth-order method with a variable time step. Calibration and testing of the model were performed through a comparison with numerical solutions of the Richards equation. Three soils (clay loam, silty loam, and sandy loam) with a variety of thicknesses were combined to realize two-layered soils, where either layer was more permeable, that were selected as test cases. In all instances, the simulations involved the cycle of infiltration-redistribution-reinfiltration. The infiltration rate as well as the water content at the surface and interface were found to be very accurately estimated by the semianalytical/conceptual model.                                                       

45.5  MODELS FOR RAINFALL INFILTRATION OVER HETEROGENEOUS AREAS

The mathematical problem of infiltration at the field scale is not analytically tractable, while using accurate Monte-Carlo (M-C) simulation techniques imposes an enormous computational burden for routine applications. Considering the spatial heterogeneity of saturated hydraulic conductivity, M-C simulations were used, for instance, by Sharma and Seely (1979), Maller and Sharma (1981), and Saghafian et al. (1995) in specific studies addressed to describe field-scale infiltration. M-C simulations were also used by Sivapalan and Wood (1986) to validate a relation they developed for areal average and variance of infiltration rate under a time-invariant rainfall rate; however, the averaging procedure was applied in space over a single realization and the behavior of the resulting errors was not specified. An alternative, but not much used, method to M-C sampling for the representation of the random variability of a soil property is the Latin Hypercube sampling (McKay et al., 1979) adopted by Smith and Goodrich (2000) to develop a simple parameterized approach for areal-average infiltration. Even though M-C simulations, performed by many realizations of the random variable, are too expensive in terms of computational effort for practical applications, they are useful as a tool to parameterize simple semiempirical approaches or to serve as a benchmark for validating semianalytical models. Along these lines, Govindaraju et al. (2001) developed three versions of a semianalytical/­conceptual models for estimating the expected areal-average infiltration into vertically homogeneous soils under a uniform rainfall spatial distribution, but with random horizontal values of the saturated hydraulic conductivity (see also Corradini et al., 2002). A shortcoming of this model is that the process of infiltration of overland flow running over pervious downstream areas (run-on process) is neglected. On the other hand, the importance of run-on was shown in a few investigations concerning the effects of horizontal variability of saturated hydraulic conductivity on Hortonian overland flow (Smith and Hebbert, 1979; Saghafian et al., 1995; Woolhiser et al., 1996; Corradini et al., 1998; Nahar et al., 2004), but the process has generally been disregarded in hydrological models. Some studies combining the random variability of saturated hydraulic conductivity and rainfall rate were performed by Wood et al. (1986), Castelli (1996), and Govindaraju et al. (2006). The last paper presented a semianalytical model developed under less-restrictive conditions, even though runon was not incorporated, and based upon the use of cumulative infiltration as the independent variable that was linked with an expected time. Subsequently, Morbidelli et al. (2006) formulated a more complete mathematical model for the expected areal-average infiltration, which considers both the saturated hydraulic conductivity and rainfall rate as random variables and combines the aforementioned semianalytical approach with a semiempirical/conceptual component to represent the run-on process. Some models for infiltration at the field-scale have recently been developed and represent a useful support for practical hydrological purposes.

45_Singh_ch45_p45.1-45.10.indd 5

45.5.1  A Semiempirical Approach

Smith and Goodrich (2000) proposed a semiempirical model to determine the areal-average infiltration rate into areas with random spatial variability of Ks. They assumed a lognormal probability density function, PDF, of Ks with a mean value and a coefficient of variation CV(Ks), and considered one realization of the random variable. Then adopting the Parlange et al. (1982) model (see also Smith et al., 1993) and the Latin Hypercube sampling method, through a large number of simulations performed for many values of CV(Ks) and rainfall rates, they developed the following effective areal relation for the scaled areal-average infiltration rate, I e∗, linked with the corresponding scaled cumulative depth, Fe∗: I e∗

= 1+

(

)

re∗ − 1

− cc

1   ∗ cc  re − 1 α Fe∗ e −1   1 +      α

(

with ca ≅ 1 +

)

  re∗ > 1

 0.8 (−0.85(rb∗ −1))  1 − e   CV ( K s )1.3 

(45.34)

(45.35a)

where I e∗ = I e K e , Fe∗ = Fe G (θ s − θi ) , re∗ = r K e and rb∗ = r < K s > . The quantity Ke denotes the areal effective value of Ks given by Ke =



r



r

∫ K f K ( K )d K + 1 − ∫ f K ( K ) d K r (45.35b) s

0



0

s



where f K s ( K ) is the PDF of Ks. Finally, Eq. (45.34) may be also applied for r variable with time. 45.5.2  A Semianalytical/Conceptual Model

Govindaraju et al. (2006) formulated a semi-analytical model to estimate the expected field-scale infiltration rate under the condition of negligible effects of the run-on process. The model incorporates heterogeneity of both Ks and r assumed as random variables with a log-normal and a uniform PDF, respectively. The quantity is estimated through the averaging procedure over the ensemble of two-dimensional realizations of Ks and r. For the sake of simplicity, we first examine the model under a steady-rainfall condition, and then provide the guidelines for applications to a time-varying rainfall rate. Starting from the extended Green–Ampt model and choosing F as the independent variable, at a given F can be written as: =

∞ Kc 

∞ ∞

∫ ∫ r fr (r ) f K ( K ) dr d K + ∫ ∫ 1 + s

0 Kc

0

0

ψ (θ s − θi )   K fr (r ) f K s ( K ) dr dK F 

(45.36) where fr(r) and f Ks ( K ) are the PDFs of r and Ks, respectively, with Kc which denotes the maximum value of Ks leading to surface saturation in the ith cell, determined by F ri Kc = = Fc ri (45.37) ψ (θ s − θi ) + F Equation (45.36) may be expressed as: < In ( F ) > =

1 M K s ( rmin + R ) Fc ,2  − M K s [ rmin Fc ,2 ] 2 R Fc2

{

−

}

2 rmin M K s ( rmin + R ) Fc ,0  − M K s [ rmin Fc ,0 ] 2R

{

}

R  +  rmin +  1 − M K s ( rmin + R ) Fc ,0   2

{

}

+

1  rmin + R    M K s ( rmin + R ) Fc ,1 − M K s ( rmin Fc ) ,1 Fc  R 

−

1 1  M K s ( rmin + R ) Fc ,2  − M K s ( rmin Fc ) ,2  Fc  R Fc 

{

{

}

}

1 M K s [ rmin Fc ,1] Fc  (45.38) +

8/22/16 2:15 PM

45-6    Infiltration Modeling

with rmin and rmin + R extreme values of the PDF of r and M K s given by M K s ( K a ,ω ) =

Ka

∫ K ω fK (K )d K

(45.39)

0

where Ka and ω stand for the first and the second argument, respectively, of the M K s function. To relate time to F an implicit relation between the e­ xpected value of t, , and F is provided as: < t (F ) > =

the effective rainfall. The depth of effective rainfall from a storm as a whole is expressed by (see also Chow et al., 1988): RE =

R > 0.2 S(45.47) R + 0.8 S with initial abstraction assumed to be 0.2 S, R storm rainfall depth, and S potential maximum retention which is computable, in mm, through the dimensionless CN as:

  ψ (θ − θ  F {1 − M Ks [ Fc ,0 ]} + F + ψ (θ s − θi ) ln  ψ (θ −s θ i+) F 

  s i)  ∞

× { M K s [ Fc , −1 ]} + ψ ∆θ ∑

1

j + 1) j+1 j=1 (

 1000  S = 25.4  − 10 (45.48)  CN 

{ M K [< r > Fc , j ]} s



(45.40)

To extend the model by incorporating the run-on effect, an additional empirical term is used in the form (Morbidelli et al., 2006):  t b  t  < I (t ) > ≅ < In (t ) > + < r > a   exp−c  t  pa   t pa 

0.36

(45.42)

b = 5.35 − 6.32 CV (r ) CV ( K s ) (45.43) 0.3

 < K >  s c = 2.7 + 0.3   CV (r ) CV ( K s ) 

(45.44)

where Eq. (45.42) holds for θi = 1 − M K s ( K c ,0) + F + ψ (θ s − θi ) ln  r   ψ (θ s − θi ) + F  ∞

+ M K s ( K c , −1) + ψ (θ s − θi ) ∑ i=1

M Ks ( K c ,i )

(i + 1) r i+1

(45.45)

 (45.46)

The last formulation was also extended for applications involving r variable in time. The same method can be used to adapt Eqs. (45.38) and (45.40) for unsteady rainfall patterns. Furthermore, the additional empirical term for run-on could be adapted for unsteady rainfalls following the guidelines indicated by Morbidelli et al. (2006). The solution of the model even in the conditions of coupled spatial variability of r and Ks is fairly simple and requires limited computational effort. The model for coupled heterogeneity of r and Ks [Eqs. (45.38), (45.40), and (45.41)] was validated by comparison with the results derived starting from M-C sampling and using a combination of the extended Green–Ampt formulation at the local scale with the kinematic wave approximation (Singh, 1996) that is required to represent run-on. Through a wide variety of simulations it was shown that: (1) the model produced very accurate estimates of over a clay loam soil and a sandy loam soil that are very similar to Soil B and Soil SL of Table 45.2, respectively; (2) the spatial heterogeneity of both r and Ks can be neglected only when >> or for storm durations much greater than tpa; (3) the effects on produced by significant values of CV(Ks) and CV(r) are similar; (4) run-on has considerable role for moderate storms and high values of CV(Ks) and CV(r); and (5) the model can be simplified using Eqs. (45.45) and (45.46) for CV(r) considerably less than CV(Ks) and steady rainfalls. 45.6  SOIL CONSERVATION SERVICE RUNOFF CURVE NUMBER MODEL

The SCS-curve number (CN) model involves an empirical formulation which provides an overall representation of the losses to be considered in estimating

45_Singh_ch45_p45.1-45.10.indd 6

where CN (in the range 0–100) can be associated with three antecedent moisture classes denoted by I, II, III, and determined using the 5-day antecedent rainfall depth from Table 45.3. Then, for class II, the value of CN (II) can be deduced from Table 45.4 on the basis of soil type and land use. For dry and wet conditions CN(I) and CN(III), respectively, can be estimated as: CN( I) =

(45.41)

where tpa is the time to ponding [see Eq. (45.10)] associated with and . The parameters a, b, and c are expressed by a = 2.8 CV ( r ) + CV ( K s ) 

( R − 0.2 S )2  

4.2 CN( II) (45.49a) 10 − 0.058CN( II)

CN (III) =

23 CN (II) (45.49b) 10 + 0.13CN (II)

The value of CN corresponding to the determined class is then used in Eq. (45.48). In applied hydrology, the temporal distribution of effective rainfall depth within a storm is generally required. The described approach can be extended for this purpose. Once the value of CN has been determined, Eq. (45.47) can be rewritten by substituting RE and R with the corresponding cumulative values RE(t) and R(t) at a time between the beginning, and the end of the storm. Then the effective rainfall depth within a time interval, Δt, is obtained by the difference RE(t + Δt) – RE(t). Finally, a physical support for the basic equations of the model was provided by Moore (1987) and Steenhuis et al. (1995), and a model adaption for the initial condition was proposed by Michel et al. (2005). 45.7  OPEN PROBLEMS

In spite of the continuous developments of infiltration modeling, the estimate of infiltration at different spatial scales, that is, from the local to watershed scales, is a complex problem because of the natural spatial variability of soil hydraulic characteristics and that of rainfall. An important issue to be addressed when areal estimates are involved is that concerning the determination of , CV(Ks), , and CV(r) together with the corresponding quantities for soil moisture content (Morbidelli et al., 2012). The models presented here apply to infiltration into a soil matrix, when macropore flow is significant the problem becomes much more complicated even though simplified approaches have been proposed. Two practical approximations to describe infiltration into soils with macropores rely upon the use of modified values of the saturated hydraulic conductivity of the soil matrix (Maidment, 1993) or on the representation of the two processes of infiltration controlled by the matrix potential and the macropore volume (Bronstert and Bardossy, 1999). Finally, all these models are formulated for horizontal land surfaces. Extensions of the classical infiltration theory to inclined surfaces were proposed by Philip (1991) and Chen and Young (2006); however, these theories do not explain the results of laboratory experiments, for example, those performed on bare soils in the absence of erosion and a sealing layer (Essig et al., 2009; Morbidelli et al., 2015). The modeling of the slope effects has to be, therefore, considered as an open problem, in particular when surfaces with vegetation are involved. Table 45.3  Classification of Antecedent Moisture Classes (AMC) for the SCS-CN Method Total 5-day antecedent rainfall (cm) AMC group

Dormant season

Growing season

I

5.33

[Source: Soil Conservation Service, 1972]

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references    45-7  Table 45.4  Runoff Curve Numbers for Selected Land Uses and Antecedent Moisture Class II Hydrologic soil group Land use description

A

B

C

D

without conservation treatment

72

81

88

91

with conservation treatment

62

71

78

81

poor condition

68

79

86

89

good condition

39

61

74

80

30

58

71

78

thin stand, poor cover, no mulch

45

66

77

83

good cover

25

55

70

77

good condition: grass cover on 75% or more of the area

39

61

74

80

fair condition: grass cover on 50–75% of the area

49

69

79

84

Commercial and business areas (85% impervious)

89

92

94

95

Industrial districts (72% impervious)

81

88

91

93

65%

77

85

90

92

38%

61

75

83

87

30%

57

72

81

86

25%

54

70

80

85

20%

51

68

79

84

98

98

98

98

paved with curbs and storm sewers

98

98

98

98

gravel

76

85

89

91

dirt

72

82

87

89

Cultivated land:

Pasture or range land:

Meadow: good condition Wood or forest land:

Open spaces, lawns, parks, golf courses, cemeteries, etc.

Residential (average impervious)

Paved parking lots, roofs, driveways, etc. Streets and roads

Legend: Group A—Deep sand, deep loess, aggregated silts Group B—Shallow loess, sandy loam Group C—Clay loams, shallow sandy loam, soils low in organic content, soils usually high in clay Group D—Soils that swell significantly when wet, heavy plastic clays, certain saline soils [Sources: Soil Conservation Service, 1972; Chow et al., 1988]

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45-8    Infiltration Modeling

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Chapter

46

Soil Moisture and Vadose Zone Modeling BY

BINAYAK P. MOHANTY

ABSTRACT

Vadose Zone, connecting surface, ground, and atmospheric water, serves as the gate keeper for terrestrial water cycle. Nonlinear partially-saturated soil water flow and retention based on heterogeneous soil texture, structure, organic matter, and other large scale land attributes contribute to the spatiotemporal dynamics of vadose zone hydrology and resultant soil moisture. In this chapter, we summarized the developments in vadose zone hydrology over the past three decades. New modeling concepts including single poro­ sity, dual porosity, dual permeability, and equivalent continuum approaches have emerged to address the challenges related to soil heterogeneity at continuum scale. Based on these concepts, several multi-dimensional vadose zone hydrology and land surface (connecting soil-vegetation-atmospheretransfer) numerical models are now available for different applications, including agriculture, hydrology, environment, weather, and climate assessment and forecast. As one of the primary variables of interest, near-surface soil moisture measurement and modeling at various spatio-temporal scales has drawn much attention in the Earth science community. Using inistu and remotely sensed soil moisture data, evolution of dominant physical controls on soil moisture at different spatial scales has been unraveled. In addition, various bottom-up aggregation (upscaling) or top-down disaggregation (downscaling) schemes in simple and complex vadose zone scenarios were developed that reflect the need and the challenges. In addition, a number of forward and inverse data assimilation schemes have been formulated to translate the space-time soil moisture (modeled and observed) data series for generating effective unsaturated soil water retention and hydraulic conductivity functions for vadose zone at the scale of interest, key inputs to any land surface models. 46.1  BACKGROUND

The “Critical Zone” extending from the top of the vegetative canopy through the depth of weathered bedrock plays a central role in Earth’s climate system and provides essential ecosystem services, thereby affecting virtually every aspect of society and the environment. This zone is where air, water, carbon, minerals, and organisms interact, forming and evolving the soil and plant ecosystem. The soil or “Vadose Zone” (Fig. 46.1) is the heart of the Critical Zone; it is the medium through which all relevant fluxes pass, and its pulse regulates the ecosystem as a whole. Humans are dependent on soil ecosystem services, particularly soil fertility and water holding capacity, for their roles in food and fiber production (Palm et al., 2007; Janzen et al., 2010), the hydrologic cycle (Loescher et al., 2007; Ines and Mohanty 2008a; 2009), the carbon cycle (Amundson, 2001), and the climate system (Lal, 2004; Brevik, 2012). Soil moisture in the partially saturated root zone (approximately top ~1–3 m and loosely defined as shallow vadose zone in this chapter) is the natural state variable of the land surface and subsurface critical to climate feedback, hydrology, and agriculture, and a key component of global water cycle.

Transpiration

Rain

Irrigation Soil evaporation

Run-off

Run-on

Near surface soil (0–5 cm) moisture data

Soil profile Z depth

Downward flux

q(h) K(h)

Upward flux

q(zi, t > 0) Shallow water table

Deep percolation Figure 46.1  Exchange of fluxes through the boundaries in a one-dimensional (1D) soil water balance model across the land-atmosphere interface and vadose zone-groundwater interface (Ines and Mohanty, 2008a). θ(Zi, t) reflects the near-surface soil moisture time series, θ(h) defines the soil water retention function, and K(h) defines the soil hydraulic conductivity function of the vadose zone (between land surface and groundwater).

Its  temporal and spatial variability over catchment areas affects surface and subsurface runoff, modulates evaporation and transpiration, determines the extent of groundwater recharge, and initiates or sustains feedback between the land surface and the atmosphere (NRC, 1991). At a particular point in time, root zone soil moisture content is influenced by: (1) the precipitation history, (2) the texture of the soil, which determines the water-holding capa­ city, (3) the slope of the land surface, which affects runoff and infiltration, and (4) the vegetation and land cover, which influences evapotranspiration (ET) and deep percolation. In other terms, the partitioning of vadose zone soil moisture to recharge to the groundwater, ET to the atmosphere, and surface/ subsurface runoff to the streams at different spatiotemporal scales and under different hydroclimatic conditions pose one of the greatest challenges to 46-1

46_Singh_ch46_p46.1-46.14.indd 1

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46-2     Soil Moisture and Vadose Zone Modeling

weather and climate prediction, water resources availability, sustainability, quality, and variability in agricultural, range and forested watersheds, and hydroclimatic conditions. Soil system or vadose zone reflects a three-phase system including: (1)  solid phase constituting soil matrix or minerals and organic matters, (2) liquid phase consists of soil water or soil solution with dissolved substances, and (3) gaseous phase or soil air. Soil water and soil air vary in composition in space and time. Soil water or moisture can be represented gravimetrically (as a ratio of mass of soil water to mass of solid) or volumetrically (as a ratio of volume of soil water to total or bulk volume of soil). Degree of saturation in the soil system reflects the percentage of porosity occupied by soil water. One way to represent the soil system is as a pore bundle, which forms the basis for using Poiseuille’s Law of capillarity to describe unsaturated flow. In most cases, soil pores are highly irregular, having an intricate geometry which prohibits microscopic description of flow pathways. For this reason, flow in soils is generally described using macroscopic or averaging terms. Although in the past three decades, advancements have been made to measure and model pore scale soil water phenomena, soil water retention and hydraulic fluxes through bulk soil system are mostly modeled at the continuum (Representative Elementary Volume) scale. 46.2  CONTINUUM-SCALE MODELS FOR PARTIALLY SATURATED FLOW IN THE VADOSE ZONE

In past decades, significant progress has been made in vadose zone hydrologic process understanding, conceptual modeling (Fig. 46.2), unsaturated hydraulic property characterization, and predictive vadose zone/soil vegetation atmosphere transfer (SVAT) modeling. The most popular model to describe flow in vadose zone has been pore-bundle capillary theory. While vadose zone hydraulics can be described with a number of closed-form expressions for soil water retention and hydraulic conductivity (Leij et al., 1997), few equations (such as Richards’ equation and Green and Ampt equation) for variably saturated flow remain as some of the most popular governing equations at the continuum scale. Among others, Arora et al. (2011) evaluated the characteristic features of different conceptual continuum scale models [single porosity model (SPM), mobile immobile model (MIM), and dual porosity model] for a hypothetical infiltration scenario of a soil system with matrix and macropore/fracture (Fig.  46.2). While the details can be found in Arora et al. (2011) and Simunek et al. (2012), a summarized description of the popular concepts used for continuum scale flow and related transport (for n ­onreactive ­solutes) models is given as follows. Note, however, solute transport ­modeling

Model concepts Are: (axisymmetrical-)3D

DPM: (2 domains-) 1D

ECM: (1 domain) 1D

in the vadose zone is discussed in more detail in a separate chapter. In the following sections, as different models may use same mathematical notations for different variables, in occasions we redefine the variables for better clarity. 46.2.1  Single Porosity Model

A unimodal pore size distribution is assumed sufficient in describing the closed form expressions for the hydraulic conductivity functions for the ­equilibrium SPM. In the 1D SPM, Richards’ Eq. (46.1) is used for describing variably saturated flow and Convection Dispersion Equation CDE (46.2) for modeling solute transport:



46.2.2  Mobile Immobile Model

Pore space heterogeneity in the subsurface also leads to nonequilibrium preferential flow movement at the continuum scale. Biological macropores and structural fractures provide preferential flow paths for water and solutes to reach groundwater (Mohanty et al., 1997; 1998; Jarvis et al., 2007). Soil- and crop-management practices (e.g., tillage operations, multiple cropping, etc.) have been found to modify agricultural soil structure and alter macropore densities. Variations in density, connectivity, and geometry of soil pores change preferential flow and transport characteristics of structured soils (Arora et al., 2011). The MIM approach represents the flow field through the macropore (mobile) domain and allows for water and solute transfer between mobile and immobile regions. As opposed to SPM, the mobile-immobile model describes soil hydraulic functions using the macropore (mobile) domain parameters, and utilizes information on matrix (immobile) domain for quantifying the interdomain mass transfer. Richards’ equation is used to simulate mobile water, and a source/sink term is used to account for water exchange with the soil matrix (immobile region) (Šimůnek et al., 2001; Köhne et al., 2006):

f



Z

Z

f

m

m

(f, m)

r f K

K

m

h



K

m

h

h

Figure 46.2  Some conceptual models [discrete fracture and axisymmetrical flow, dual permeability model (DPM), equivalent continuum model (ECM)] for describing soil hydraulics in biporous (contrasting textures, fracture/matrix) media (Sources: Kohne and Mohanty, 2005; Arora et al., 2011). Mohanty et al. (1997, 1998) showed the effectiveness of ECM concept for field-scale preferential flow.

46_Singh_ch46_p46.1-46.14.indd 2

 ∂h  ∂θm ∂ = K (hm ) m + 1 − G wMIM (46.3)  ∂z  ∂t ∂z 

∂θ im = G wMIM = w Sem − Seim  (46.4) ∂t

where ΓwMIM is the water-transfer rate from mobile to immobile region (T–1), w  is a first-order rate coefficient (T–1), and Sem and Seim are effective fluid saturations in the mobile and immobile regions, respectively. Convectivedispersive solute transport is assumed for the mobile region and analogous to  water flow, first-order solute exchange process is employed between the two regions (Šimůnek et al., 2003):

(f, m)

f

∂θ c ∂ρ s ∂  ∂c  ∂qc + = θ D  − − µ (θ c + ρ s ) + γθ + γρ (46.2) ∂t ∂t ∂z  ∂z  ∂z

where t is time (T), z is the vertical coordinate positive upward (L), θ is the water content (L3L–3), h is the pressure head (L), K is the unsaturated hydraulic conductivity (LT–1), S is a sink term (e.g., root water uptake rate by plants), c and s are solute concentrations in the liquid (ML–3) and solid phases (MM–1), respectively, ρ is the soil bulk density (ML–3), q is the volumetric flux density (LT–1), μ is a first-order rate constant (T–1), γ is a zero-order rate constant (ML–3T–1), and D is the dispersion coefficient (L2T–1). This formulation allows SPM to describe flow and transport that is uniform and at local equilibrium (Šimůnek et al., 2003; 2006; Köhne et al., 2009).

Z

 ∂h  ∂θ ∂ = K (h) + 1 − S (46.1)  ∂z  ∂t ∂z 



∂θmcm ∂ρm sm ∂ ∂c  ∂q c + = θm Dm m  − m m − µθmcm − G sMIM ∂t ∂t ∂z  ∂z  ∂z

(46.5)

∂θ imcim ∂ρim sim + = − µθ imcim + G sMIM (46.6) ∂t ∂t G sMIM = w s (cm − cim ) + G wMIMc* (46.7)

where Γ sMIM is the solute transfer rate between the two regions (ML–3T–1), ws is the constant first-order diffusive solute mass transfer coefficient (T–1), and c* is equal to cm for ΓwMIM > 0 and cim for ΓwMIM < 0.

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Numerical Vadose Zoneand Land Surface Models      46-3  46.2.3  Dual Permeability Model

Another concept to describe water flow in macroporus media is DPM. This approach uses two different hydraulic functions, one for each domain, for describing flow through the macroporous media. Exchange between the matrix and macropore domains is established through a first- or secondorder coupling term. In the dual permeability model, water flow in both macropore (subscript f ) and matrix (subscript m) domains is described by two coupled Richards’ equations (Gerke and van Genuchten, 1993): ∂θ f

=

∂t



 G ∂  ∂h f + K f  − w (46.8) K f ∂z  ∂z  wf



where wf is the dimensionless volume factor defined as the ratio of the ­macropore domain volume (Vf) relative to the total soil volume (Vt):

wf =



Vf Vt

in which aw is a first-order mass transfer coefficient for water (L–1T–1) given by

αW =



β K aγw (46.12) a2

where b is a dimensionless geometry-dependent shape factor, a is the characteristic length of the aggregate (L), Ka is the hydraulic conductivity of the fracture/matrix interface region (LT–1), and gw is a dimensionless scaling ­factor. Without fracture coatings for artificial macropores, Ka can be evaluated as follows: K a = 0.5[K f (h f ) + K m(hm )] (46.13)



DPM with a second-order term (DPM2) for inter-domain mass transfer of water can also be considered (Köhne et al., 2004): G wDPM2 =

β K a (h f − hm )[|hm − hi | − |h f − hi |] (46.14) 2a 2 |hm − hi |

where hi is the initial pressure head assumed to be equal for matrix and ­macropore (L). For DPM2, Ka is evaluated as: Ka =



pKm(hm ) + Km(h f ) p+1

(46.15)

where p is a weighting factor for which an average value of 17 was found to be  suitable for a range of hydraulic properties and initial conditions (Köhne et  al., 2004). For both DPM1 and DPM2, geometrical parameters can be derived according to Gerke and van Genuchten (1996) as:

β=

with ς =

1 ; 1 < ς < 100 (46.16) [0.19ln(16ς )]2

a+b (46.17) b

where, b is the radius of the cylindrical macropore (L). In addition to the MIM, DPM, and other multidomain model concepts, Durner et al. (1994) came up with multiple superimposed (combination) hydraulic functions to describe nonequilibrium flow in the vadose zone. Furthermore, by superimposing and joining different hydraulic functions across different ranges of pressure head (h) for capillary-dominated versus noncapillary dominated flow, Mohanty et al. (1997, 1998) formulated a novel piecewise-continuous hydraulic function efficient for simulating

46_Singh_ch46_p46.1-46.14.indd 3

θ = θs(c = 1)                 h >  hθ*   (46.19)

c

= ∑c w c



G wDPM1 = α w (h f − hm ) (46.11)



h ≤  hθ*   (46.18)

K (h) = ∑wc K c (h)

(46.10)

Γw is the rate of water exchange between the two domains (T–1) described with first-order mass transfer for DPM1 as:

 θ s ,c − θ r ,c  θ (h) =  Σc wcθ c (h) =  Σc wc θ r ,c +   n mc   1 +  (α c h) c  



 ∂θm ∂  ∂h Gw =  Km m + Km  + (46.9)  1 − wf ∂t ∂z  ∂z





preferential flow and solute transport in complex multidimensional field conditions.

(1 − (αch)n −1[ 1 + (αch)n ]−m ) c

c

mc /2

1 + (α c h)nc 

c

2

, where (mc = 1 − 1/ nc ) 

h ≤ hK*  (46.20)

 (|h−h*|)δ  K nc (h) =  K *+   K *  exp − 1                           hK* < h   ≤ 0 (46.21)   * K nc (h) =   K *+  K *   exp(−h )δ − 1                                h   > 0 (46.22)

where Kc  the hydraulic conductivity for capillary dominated flow domain c (LT–1); Knc the hydraulic conductivity for noncapillary dominated flow domain nc (LT–1); θs,c  the saturated water content for capillary dominated flow domain c (L3L–3); θr,c  the residual water content for capillary dominated flow domain c (L3L–3); h the equilibrium soil water pressure head for bulk soil (across all flow domains) (L); hK* ≈ hθ* = h* the critical or break-point soil water pressure head where flow changes from capillary dominated to noncapillary dominated flow or vice-versa (L); * K the hydraulic conductivity corresponding to h* (LT–1); δ a fitting parameter representing effective macroporosity or other structural features contributing to noncapillary dominated flow (L–1); αcnc the van Genuchten fitting parameters (van Genuchten, 1980) for the capillary dominated flow domain c (L–1, -); c the number of capillary dominated flow domains, where for c  = 1 the sum type multimodal van Genuchten–Mualem hydraulic functions Eqs. [(18) and (20)] reduce to the original unimodal van Genuchten–Mualem-type functions; nc the noncapillary dominated flow domain; and wc the weighting factor for capillary dominated flow domain c(-), subjected to ∑wc = 1 and 0 < wc < 1. Although aforementioned conceptual modeling techniques at the continuum scale for soil systems are more suitable for various hydrologic applications, pore network models, percolation theory, and other thermodynamics-based multiphase flow and transport approaches have been used in specific environmental applications. Besides these process-based models, other statistical and pattern recognition models for soil hydrology (e.g., pedo transfer functions, Sharma et al., 2006; Jana et al., 2008) are being developed and used for simpli­ city in various occasions.

46.3  NUMERICAL VADOSE ZONE AND LAND SURFACE MODELS

Several vadose zone and land surface models [e.g., HYDRUS, Soil-WaterAtmosphere-Plant (SWAP), Noah, and Community Land Model (CLM)] have been developed to simulate flow in shallow subsurface. Note, however, some of these models were developed keeping various applications in mind. For example, HYDRUS has been extensively applied for various complex flow and transport processes at local scale, while SWAP has been used for many field scale water-management applications, and Noah and CLM land surface models have been used in the context of large scale hydrology and weather and climate models. As readers are referred to the original sources for details, we provide brief summaries and salient features of some of these models.

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46-4     Soil Moisture and Vadose Zone Modeling 46.3.1  HYDRUS Suite of Models

A suite of software packages [HYDRUS _1D, 2D, and 3D (Simunek et al., 2006, 2012)] has been developed over the past two decades by accounting for the movement of water, heat, and multiple solutes in the vadose zone. These finite element numerical models solve multidimensional mixed-form Richards’ equation with sink or source term for water flow and Convection Dispersion equation with various decay chain reaction terms for solute transport processes, and can adopt various uniform or time-dependent boundary conditions, and hydraulic conductivity/soil water retention functions (Brooks and Corey, 1964; van Genuchten, 1980; Vogel and Cislerova, 1988; Durner, 1994; Kosugi, 1996). Conceptual models, such as SPM, MIM, and DPM (described in the previous section), are included in the HYDRUS package. Although uncoupled, water flow in HYDRUS is given as a sum of isothermal liquid flow, isothermal vapor flow, gravitational liquid flow, thermal liquid flow, and thermal vapor flow. System-independent and system-dependent boundary conditions for water, heat, and solute transport are featured in this model. Root water and nutrient uptake with various compensation terms, root growth, hysteresis in soil hydraulic properties, ET using radiation ­budget, subsurface drainage, seepage, and surface runoff are the salient features of this popular vadose zone model. Other important features of the HYDRUS model include carbon dioxide transport and production, major ion, and carbonate chemistry involving mass and charge balance equations, complexation reactions, cation exchange and selectivity, precipitation-­ dissolution, and various kinetic models. While water, solute, and heat transport modeling features are quite comprehensive in HYDRUS suite of models, some tools are available for colloid, virus, and bacteria transport in the vadose zone. In addition to conducting simulation in the forward mode, HYDRUS model is capable of inverse modeling for estimating optimum soil hydraulic and transport parameters, described in Sec. 46.2. Advanced numerical schemes to solve governing equations used in HYDRUS minimize mass balance errors and numerical instability. Space-time discretization based on a fully implicit discretization of the time derivative is solved with a Picard iterative solution facilitated by Gaussian elimination. Peclet number and Courant number are used as guiding principles for optimizing the space-time discretization minimizing the tradeoff between accuracy of the numerical solutions and computational burden. Atmospheric boundaries are simulated by applying prescribed head, flux, or gradient boundary conditions as well as other system-dependent conditions as surface ponding including run on/ runoff. Various hydrologic conditions including seepage face, deep drainage based on ground water table position, free drainage, and tile drainage can be simulated using different boundary conditions. Typical applications of HYDRUS model includes agricultural (irrigation design, drainage design, water management for crop production, root and crop growth simulation, salinization and reclamation processes, nitrogen dynamics and leaching to groundwater, pesticide transport and chain products, nonpoint source pollution, etc.), and nonagricultural (land-atmosphere interaction, surface waterground water interaction, environmental solute transport and contamination assessment, capillary barrier design and waste disposal, landfill cover design, tunnel and highway design, recharge basin analysis, etc.). 46.3.2  Soil-Water-Atmosphere-Plant Model

SWAP (Van Dam et al., 1997) has been used for simulating soil water flow between the soil, water, plant, and atmosphere system (Agnese et al., 2007; Ying et al., 2011). This model contains physical processes for soil water flow, potential and actual ET, crop growth, and irrigation. Daily potential ET is estimated using the Penman–Monteith method with daily meteorological data or crop factors (i.e., minimum resistance, leaf area index, and crop height), and the actual ET rate is calculated using the root water uptake reduction and maximum soil evaporation flux (Van Dam et al., 1997). As HYDRUS, this model simulates soil moisture dynamics in the soil profile using the Richards’ Eq. (46.1), and the soil hydraulic properties of Mualem and van Genuchten (Mualem, 1976; van Genuchten, 1980),



m θ (h) − θ res  1  (46.23)  Se = = θ sat − θ res 1 + α h n 

K (h) = K sat Seλ 1 − 1 − Se1/m 

(

46.3.3  Noah Land Surface Model

Noah land surface model [Noah LSM v2.7.1 (Ek et al., 2003)] has been widely used in both coupled (integrated with other models) and uncoupled (standalone) modes for simulating water and energy fluxes at various spatial scales. In uncoupled mode, it can be used as a 1D physically based model for estimating soil moisture dynamics at field-scale. Noah LSM calculates the total ET by summing the direct evaporation from top soil layer, canopy evaporation, and potential Penman–Monteith transpiration (Rosero et al., 2010). The  model has typically four soil layers with the thicknesses of 10, 30, 60, and  100  cm (total soil depth of 200 cm). It adapts a diffusion form of the Richards’ Eq. (46.25) for estimating soil moisture. Hydraulic conductivity and soil ­matric potential are calculated based on the Clapp and Hornberger (1978) [Eqs. (46.26) and (46.27)],





)

 (46.24)

∂θ ∂ ∂θ  ∂K (θ ) =  D(θ )  + + Q (46.25) ∂t ∂z  ∂z  ∂z −b  θ  ψ = ψsat   (46.26)  θ sat 

 θ  K (θ ) = K sat    θ sat 

2b + 3

(46.27)

where θ is the volumetric soil water content (L3L–3), z is the soil depth (L) 

∂ψ 

taken positive upward, D(θ) is the soil water diffusivity (L2T–1) K (θ ) ∂θ , K(θ)

is the unsaturated hydraulic conductivity (LT–1), Q is a soil moisture sink term, ψ and ψsat are the soil matric potential and saturated soil matric potential (L), b is the curve fitting parameter related to the pore size distribution (-), and θsat and Ksat are the saturated soil moisture content (L3L–3) and saturated hydraulic conductivity (LT–1), respectively. Noah LSM has been enhanced to achieve the better performance incorporating complex canopy resistance, bare soil evaporation, surface runoff, and higher-order time integration schemes. 46.3.4  Community Land Model

Community land model [CLM, Oleson et al. (2010)] is the LSM that provides the land surface forcing as the physical boundary for atmospheric model in the community climate system model. This model estimates bare soil evaporation based on the Philip and de Vries (1957) diffusion model and calculates transpiration using an aerodynamic approach using the biosphere atmosphere transfer scheme model (Dickinson et al., 1993). CLM has a 10-layered soil column with the fixed layer thickness of 1.75, 2.76, 4.55, 7.5, 12.36, 20.38, 33.60, 55.39, 91.33, and 113.7 cm (total depth of 343 cm). The vertical soil water flow can be solved by the modified Richards’ Eq. (46.28) (Zeng and Decker, 2009). This equation was derived by subtracting the hydrostatic equilibrium soil moisture distribution from the original Richards’ equation for improving the mass-conservative numerical scheme when the water table is within the soil column,



∂θ ∂   ∂(ψ − ψe )  =  − Q (46.28) K   ∂t ∂z   ∂z

where ψe is the equilibrium soil matric potential (L). The hydraulic conductivity is derived from Eq. (46.27), and equilibrium soil matric potential and equilibrium volumetric water content were shown in Eqs. (46.29) and (46.30) based on Clapp and Hornberger (1978),



−b θ (z )  ψe = ψsat  e  (46.29)  θ sat  1

m2

where h is soil matric potential (L), n(-), m(-), l(-), and α(cm–1) are the empirical shape factors of the soil water retention and hydraulic conductivity functions, m = 1-1/n, Se is the relative saturation (-), θres is the residual water content (L3L–3), Ksat is the saturated hydraulic conductivity (LT–1), K(h) is the

46_Singh_ch46_p46.1-46.14.indd 4

unsaturated hydraulic conductivity (LT–1) at matric potential h. SWAP simulates water flow, solute transport, heat flow, and crop growth simultaneously at field scales.



− ψ + z − z  b θ e ( z ) = θ sat  sat ∇  (46.30) ψsat  

where z is the soil depth of interface of two adjacent layers from the soil surface (L), and θe(z) is the equilibrium (e) volumetric water content (L3L–3) at depth z (z∇ is the water table depth). In CLM, 10 soil layers discretized

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Soil Moisture AcrossSpatial-Temporal Scales     46-5 

Oklahoma

1-44 SGP97 flight line area

LB-81

Little Washita watershed

Central facility

LW21 Scale km

EI Reno

Little Washita

Figure 46.3  Hierarchy of spatial scales (pore, field, landscape, watershed, region, and continent) important for understanding fundamental processes in soil hydrology and root zone soil moisture dynamics. (Source: Zhu et al., 2006.)

unevenly include a thin top soil layer (1.75 cm) needed to better simulate infiltration and evaporation fluxes (Sahoo et al., 2008). Furthermore, CLM considers the dynamics of water table as the lower boundary using the SIMple Groundwater Model [SIMGM, Niu et al. (2007)]. A groundwater component is defined as an unconfined aquifer below the soil column (343 cm). Other land surface models, such as variable infiltration capacity (Liang et al., 1994), Mosaic land surface model (Mosaic LSM) (Koster and Suarez, 1996), transport of unsaturated groundwater and heat (Pruess et al., 2012), parallel watershed flow model (Maxwell, 2013), and subsurface transport over multiple phases (White et al., 2012) are also capable of simulating multiphase vadose zone flow processes. The global land data assimilation systems uses these hydrological models for validating pixel-scale soil moisture from satellite platforms and evaluating water/energy cycle and fluxes near the land surface (Liu et al., 2009). The North American land data assimilation system has monitored and predicted hydrological drought conditions using state variables (e.g., soil moisture dynamics, runoff, evaporation, etc.) estimated from various hydrological models (Ek et al., 2011). However, these models incorporated with their own parameterization schemes and simplified processes that might not consider adequately the real-world conditions indicating that each model has its own strengths and drawbacks for certain processes (Hsu et al., 2009). Thus, as discussed in Kirchener (2006), inherent model structures might produce different model outputs and cause uncertainties due to different model structures and input parameters (i.e., atmospheric forcings, soil textures, vegetation covers, initial and bottom boundary conditions, etc.). To circumvent some of these model-specific uncertainties, multimodel averaging techniques with different weighting schemes (Kim et al., 2015) have been developed in the recent years. 46.4  SOIL MOISTURE ACROSS SPATIAL-TEMPORAL SCALES

While challenges persist to provide in-depth process understanding or to upscale the characteristic properties for describing soil moisture and vadose zone flow processes at various societal applications (e.g., agricultural management, hydrologic prediction, environmental protection, weather, and climate feedback) scales (e.g., field, catchment, watershed, river basin, or region), modern ground based and remotely sensed platforms encompassing large land areas on a time-continuous basis provide tools to fill in some of these gaps lately. To date, very few studies (e.g., Das and Mohanty, 2006, 2008; Das et al., 2008a, 2008b, 2010) have been made to quantitatively understand the

46_Singh_ch46_p46.1-46.14.indd 5

­ ultiscale dynamics of root zone soil moisture in land-surface hydrologic m systems (Fig. 46.3). Traditionally, soil moisture spatial variability studies using ground-based point-scale (i.e., sampling area of cm2) measurements are limited to small fields with uniform soil characteristics, topographic features, and vegetative conditions. Lately, the use of various active and passive microwave remote sensors has enhanced the capability to monitor near-surface soil moisture in large land areas (i.e., hundreds of square meters to thousands of square kilometers) encompassing various soil types (e.g., texture), topographic features (e.g., slope), vegetation/land cover, and climatic conditions. These remote-sensing signals give average values over an area usually known as a footprint. For larger footprints, only predominant soil and vegetation types are used for calibration purposes. An inherent difficulty of remotely sensed soil moisture measurement is relating soil moisture variability at the scale of the footprint to larger or smaller scale soil moisture variability (Stewart et al., 1996; Kumar, 1999). In the recent past, concern has been raised that footprint-scale measurements may be too crude for providing a good understanding of hydrologic systems at the subgrid (field, catchment, basin, and watershed) scales. Hence errors at the level of catchments, basins, and watersheds may add up to providing inaccurate regional-scale hydrologic fluxes important for water resources assessment, agriculture, and hydroclimatic predictions. The temporal and spatial distribution of soil moisture and the resultant fluxes to different hydrologic reservoirs are affected by interactions between soil, topography, vegetation, and climate (Reynolds, 1970; Sharma and Luxmoore, 1979; Moore et al., 1988; Loague, 1992; Charpentier and Groffman, 1992; Rodriguez-Iturbe et al., 1999; Mohanty and Skaggs, 2001; Joshi and Mohanty, 2010; Gaur and Mohanty, 2013). In-depth reviews of soil moisture spatiotemporal variability-related studies are given by Mohanty and Skaggs (2001), Western et al. (2002), Das and Mohanty (2006), Joshi and Mohanty (2010), Crow et al. (2012), Caldwell et al. (2012), and Gaur and Mohanty (2013; 2015, submitted to Water Resources Research), while a brief summary is given as follows. 46.4.1  Root Zone Soil

Heterogeneity in root zone soil affects the distribution of soil moisture through variations in texture, organic matter content, porosity, structure, and macroporosity (Fig. 46.4). Each of these factors affects the fluid transmission and retention properties in different ways (i.e., Entekhabi, 1994; Ek and Cuenca, 1994; Kim et al. 1997; Kim and Stricker 1996; Mohanty et al., 1997; Liu and Dickinson 2003; Joshi et al., 2011; Shin et al., 2012; Pollacco and Mohanty, 2012).

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46-6     Soil Moisture and Vadose Zone Modeling

Biological pores

Different soil textures

Structural cracks

Karst geology

Figure 46.4  Various biotic or abiotic preferential flow paths in vadose zone including: (1) biological pores, (2) structural cracks, (3) varying soil textures and their adjacency, and (4) karst geology.

Variability in soil hydraulic conductivity and soil water retention in the presence of contrasting textures/macropores/fractures greatly influences the vertical and lateral water transmission properties (Mohanty et al., 1997). Heterogeneities in soil particles and pore sizes can lead to significant soil moisture variations over very small distances. At the field plot scale, Das Gupta et al. (2006a) showed the significant effect of small scale soil texture and macroporosity variations on soil moisture distribution across depth as well as partitioning of precipitation between surface runoff, subsurface baseflow, ET, and groundwater recharge in a limestone Karst environment (see Fig. 46.4). Just as soil moisture variation depends on soil properties, spatial variation in the properties and thickness of the organic and mineral soil layers depends on long-term local soil moisture conditions. This may further be complicated by the dynamic nature of preferential flow paths caused by swelling and shrinking clay (e.g., Das Gupta et al., 2006b). The dual permeability and equivalent continuum modeling (discussed earlier) that have been applied to simulate preferential flow systems in contrasting soil textural or fractured media have demonstrated the importance of these processes in soils. 46.4.2 Topography

Topography (macro or micro) also plays an important role in the spatial organization of soil moisture at different scales. Variations in slope, aspect, curvature, upslope contributing area, and relative elevation all affect the distribution of soil moisture near the land surface. At the small catchment and hill-slope scales, soil moisture varies as a result of water-routing processes,

radiative (aspect) effects, and heterogeneity in vegetation and soil characteristics. Charpentier and Groffman (1992) studied the effects of topography and moisture content on the variability of soil moisture within remote sensing pixels during the First International Satellite Land Surface Climatology Project field experiment. They showed that within-pixel (66 m × 66 m) soil moisture variability increased with increased topographic heterogeneity. A  flat pixel had significantly lower standard deviations and fewer outlier points than a sloping or a valley pixel. Furthermore, they observed that remote ­sensing reflected soil moisture conditions less accurately on pixels with increased topographic variability and less precisely when soil is dry. Mohanty et al. (2000a) showed the dominance of a slope-effect on the diurnal soil moisture distribution in a gentle slope within the Little Washita agricultural watershed, Chickasha, Oklahoma. Several other studies also showed that the slope position is very important in determining soil moisture variation (Joshi and Mohanty, 2010; Joshi et al., 2011), suggesting that a simple averaging of soil moisture values over the slope may lead to errors at different time scales. While a number of studies concluded that in some landscapes topo­ graphy is a relatively poor indicator of soil moisture patterns and variability [Western et al. (1999), Kim and Barros (2002), Bindlish and Barros (2002), and Chang and Islam (2003), Jawson and Niemann (2007)], other studies, including Jacobs et al. (2004), Joshi et al. (2011), and Jana and Mohanty (2012a; 2012b) show that soil properties (i.e., percent silt, percent sand, and soil texture), and topography (elevation and slope) are significant physical controls jointly affecting the spatiotemporal evolution and time stability of soil moisture at both point- and remote sensing footprint-scale. Jana and Mohanty (2012a; 2012b; 2012c) conducted a comprehensive numerical study using a three-dimensional (3D) soil hydrologic model (HYDRUS_3D with surface flow routing) under same atmospheric forcing followed by field testing to investigate various topographic variations (e.g., microheterogeneity with mild slope, steep slope, concave slope, and convex slope) and the consequential soil moisture distributions across hill slope, and watershed. They suggested that topographic variations result in different behavior of soil moisture distribution at subgrid-scale, and boundary fluxes, such as seepage and recharge (see Fig. 46.5). This finding has significant implications for regional hydrology as boundary fluxes at the subgrid-scale will propagate to adjacent units and accentuate larger differences and scale behavior of root zone soil moisture and hydrologic fluxes across field, hill slope, catchment, to watershed. 46.4.3 Vegetation

Land cover is also critical for understanding the soil moisture regimes, as it affects infiltration, runoff, and ET through root water uptake. Vegetation type, density, and uniformity are some of the associated features that contribute to soil moisture variation at different space and time scales. Furthermore, the influence of vegetation on soil moisture and temperature is more dynamic as compared to soil and topographic factors. Previous studies show that the ­variability of soil moisture is lowest with full canopy cover and highest with partial coverage. Hawley et al. (1983) demonstrated that various vegetation-­ topography-soil combinations lead to temporal persistence (clustering) of soil moisture patterns in complex terrains with mixed vegetation. They also

Figure 46.5  Soil moisture distributions (modeled using 3D vadose zone model, HYDRUS_3D) under different topographical conditions, clockwise from top left (a) microtopography with mild slope, (b) steep slope, (c) concave slope, and (d) convex slope under same boundary conditions. (Source: Jana and Mohanty, 2012a.) Results signify soil moisture and local/boundary fluxes will vary for different landscape configurations for the same atmospheric forcing.

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Soil Moisture AcrossSpatial-Temporal Scales     46-7 

Iowa (Point scale-soil moisture)

0.6

SMEX02 SMEX05 Vegetation dominated

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∆ Entropy

0.4 0.2 0 –0.2 –0.4 165

170

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SGP 97 (ESTAR)

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SMEX02 (PSR)

0.5

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Figure 46.6  Time series of entropy difference of soil moisture with error bars for (a) point support scale, and (b) airborne footprint scale. For Iowa: ∆ Entropy = Soil-based entropy − Vegetation-based entropy, for Oklahoma: ∆ Entropy = Soil-based entropy − Topography-based entropy. Entropy analysis shows the dominance of different physical factors (soil, vegetation, and topography) under different land covers (agricultural vs. natural), wetness conditions, and measurement scales (point scale vs. airborne footprint). (Source: Gaur and Mohanty, 2013.)

s­ uggested that the presence of vegetation tends to diminish the soil moisture variations caused by topography. Mohanty et al. (2000b) showed that the vegetation dynamics (growth/decay), land management (tillage), and precipitation events controlled the intraseasonal soil moisture spatial structure for the pixel with flat topography and uniform soil texture. Past studies in temperate climate have suggested that soil, topography, and vegetation conjointly control the root zone soil moisture spatiotemporal variability and temporal stability. Furthermore, the influence of vegetation on soil moisture was found to be more dynamic across season and plant growth stages (Mohanty et al., 2000b; Pollacco and Mohanty, 2012; Gaur and Mohanty, 2013) as compared to soil and topographic factors. In an agricultural watershed in Iowa, using entropy based analysis (see Fig. 46.6), Gaur and Mohanty (2013) showed that at the point support scale during the relatively wet Soil Moisture Experiment (SMEX 2005), vegetation appeared to dominate the root zone soil moisture spatial distribution. However, during the drier year (SMEX 2002), the controls shifted between soil and vegetation during precipitation events. On the other hand, in a natural (grassland) watershed in Oklahoma, topography and soil played dominant roles for root zone soil moisture spatiotemporal distribution. 46.4.4  Climate

Precipitation, solar radiation, wind, and humidity are some of the important climatic factors that contribute to the space-time dynamics of root zone soil moisture (Das and Mohanty, 2008; Joshi and Mohanty, submitted; Shin et al., 2013b) and resultant fluxes. Precipitation is the single most important climatic forcing for soil moisture content and its distribution. As shown by Sivapalan et al. (1987), the dominant runoff producing mechanism may vary with storm characteristics and antecedent soil moisture conditions resulting in the spatiotemporal variability in soil moisture. During the Southern Great Plains 1997 (SGP97) hydrology campaign, Famiglietti et al. (1999) found a distinct trend in mean soil moisture for Little Washita (in southern Oklahoma), El Reno (in central Oklahoma), and DOE-ARM Central Facility (in north Oklahoma) locations with a south-to-north precipitation gradient. Within an analytical framework, Kim and Stricker (1996) showed the significance of rainfall pattern on partitioning of water over the budget terms for different climatic conditions. In a study, using statistical distribution and soil

46_Singh_ch46_p46.1-46.14.indd 7

vegetation atmosphere transfer (SVAT) modeling, Salvucci (2001) showed conditional dependence of soil moisture storage, drainage+runoff, and ET with amount of precipitation in Illinois. A multiscale analysis by Crow and Wood (1999) revealed a qualitatively different relationship between soil moisture means and soil moisture spatial variances when variability is sampled at fine (10 km) spatial scales. The threshold between these two scale regimes may represent a transition between organized coarse scale spatial heterogeneity imposed by land surface response to rainfall and disorganized fine scale variability produced by local variations in soil, topography, and vegetation. Peters-Lidard et al. (2008), Pollacco and Mohanty (2012), and Gaur and Mohanty (2015) further reflected the significance of quality and accuracy of precipitation data in soil hydrology. 46.4.5  Dominant Physical Controls on Soil Moisture and Scale

Based on the aforementioned discussion of various physical controls of soil moisture, a general conclusion could be drawn that soil moisture spatiotemporal patterns reflect a conjoint variability of soil, topography, vegetation, and precipitation attributes (Fig. 46.7). Jana (2010) suggested that soil moisture variability is “dominated” by soil properties at the field scale, topographic features at the catchment/watershed scale, vegetation characteristics, and precipitation patterns at the regional scale and beyond. Ensemble hydrologic fluxes (including ET, infiltration, surface runoff, subsurface baseflow, streamflow, and groundwater recharge) within and across the root zone reflect the evolution of soil moisture at a particular spatial scale (point, field, watershed, or region) and can be “effectively” represented by one or more linear/nonlinear hydrologic scale parameters reflecting dominant heterogeneity of the landscape. Mohanty and Skaggs (2001), Jacobs et al. (2004), Joshi and Mohanty (2010), Joshi et al. (2011), and Gaur and Mohanty (2013) conducted analyses of ground and remote sensing soil moisture data collected during the Southern Great Plains Hydrology Campaigns in 1997 and 1999 (SGP97 and GP99), Soil Moisture Experiments 2002, 2003, and 2005 (SMEX02, SMEX03, and SMEX05), and Cloud and Land-Surface Interaction Campaign (CLASIC 2007) and concluded that characteristic differences were observed in the space-time dynamics of soil moisture within selected remote sensing ­footprints with various combinations of soil texture, slope, vegetation, and precipitation.

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Regional scale

Watershed scale

Climate forcings

Field scale

46-8     Soil Moisture and Vadose Zone Modeling

Land cover patterns Topographic features Soil texture and structure Point scale

Coarsening scale

Continental scale

Figure 46.7  Multiple/dominant physical controls for soil moisture at different spatial scales ranging from local, field, watershed, regional, to continental scale by Jana (2010).

From a physical control perspective, soil moisture dynamics is related to transient upward (ET), downward (infiltration, groundwater recharge), and lateral (surface runoff, subsurface baseflow) fluxes. Brutsaert (1982) recognized two-stages of soil profile drying: (1) an atmosphere-controlled stage followed by (2) a soil-controlled stage. In the firststage, the moist soil profile can fully supply all the water demanded by the atmosphere. As the soil near the surface dries out, the moisture delivery rate is limited by the properties of the soil profile (Pollacco and Mohanty, 2012; Shin et al., 2012). Similarly, following any precipitation event, vertical and lateral subsurface flow and its connectivity is driven by soil hydraulic properties and their anisotropy resulting in root zone moisture distribution. With the recent launch of Soil Moisture Active Passive (SMAP) satellite, near-surface and root zone soil moisture products will be available globally. While near-surface (top 5 cm) soil moisture (L2 and L3 products) will be available at 3, 9, and 36 km spatial resolutions on 1–2 days intervals, root zone (top 1 m) soil moisture (L4) products will be available weekly at 9 km resolution. These SMAP soil moisture products will provide the basis for better understanding of land surface hydrologic stores and fluxes (ET, infiltration, surface runoff, subsurface baseflow, stream flow, and groundwater recharge) leading to improved weather, climate, hydrologic, and agricultural models. 46.5  INVERSE MODELING—SOIL HYDRAULIC PROPERTIES AT THE MODEL GRID SCALE

Vadose zone and land surface models described above have played a vital role in understanding the processes of the SVAT system. However, the success of our SVAT models is highly dependent on the quality (appropriateness for working scale) and availability of data required for the model applications in a specific hydrologic domain. Both the physical (e.g., soil hydraulic data) and nonphysical (management practices) properties of a hydrologic system are necessary to be defined when applying models for simulation studies. The root zone soil hydraulic properties, defined mainly by the water retention θ(h) and hydraulic conductivity K(h) functions, are necessary inputs to hydrologic models; where θ is the volumetric soil moisture, K the hydraulic conductivity and h the soil water pressure. They describe the potential of the soil to store and release, and to transmit water under different environmental and boundary conditions. Among various soil hydraulic and water retention functions, Brooks and Corey (1964), Clapp and Hornberger (1978), van Genuchten (1980), and Russo (1988) are some of the most popular in SVAT models. Parameter optimization, minimizing the difference between model predictions and observations, lies in the heart of inverse modeling. Genetic algorithms [GAs, Holland (1975)], Shuffled Complex Evolution-University of Arizona [SCE-UA, Duan et al. (1992)], and Particle Swarm Optimization [PSO, Kennedy and Eberhart (2001)] have been applied in estimating effective model parameters in hydrology. Bayesian Model Averaging [BMA, Hoetting et al. (1999)], Hydrological Uncertainty Processor (Krzysztofowicz, 1999; Krzysztofowicz and Kelly, 2000), Ensemble Model Output Statistics E-MOS (Gneiting et al., 2005), DiffeRential Evolution Adaptive Metropolis— Markov Chain Monte Carlo (DREAM-MCMC) algorithm (Vrugt et al., 2009), and Model Conditional Processor (Todini, 2008; Coccia and Todini, 2011) have been used to account for the model structural uncertainties. GAs have been used to minimize errors in searching optimized model parameters based on inversion model (Reed et al., 2000; Ines and Mohanty, 2008a, 2009;

46_Singh_ch46_p46.1-46.14.indd 8

Zhang et al., 2009; Shin et al., 2012; Shin et al., 2013a). Zhang et al. (2008) integrated several global optimization algorithms (i.e., GA, SCE-UA, PSO, etc.) with soil and water assessment tool and compared their performances in calibrating model input parameters. They showed that the GA found better optimized model parameters than others, although a large number of computational resources were required. Further the near-surface (Ines and Mohanty, 2008) and layer-specific data assimilation (Shin et al., 2012) approaches using the GA coupled with SWAP based on inversion model were developed for quantifying effective soil hydraulic properties in the homogeneous and heterogeneous soil profiles. Their findings indicated that the estimated effective soil parameters at the near-surface and subsurface layers can be adequately conditioned by the GA. However, although model parameter uncertainties for a single model can be minimized by simulation-optimization schemes (e.g.,  GA-SWAP, etc.), bias due to different model structures still remain (­considerably) in model outputs. 46.5.1  Top-Down Aggregated Approach

Remote sensing data have been used successfully in extracting biophysical data for regional hydrological modeling (Kite, 2001). Recently Ines and Mohanty (2008a) and Shin et al. (2012), respectively, presented near-surface soil moisture assimilation scheme for homogenous (no soil layers) and heterogeneous (with soil layers) soil texture, flat topography, and uniform land cover that can be used to quantify effective soil hydraulic parameters in the root zone soil profile. The (top-down) approach (review by Mohanty, 2013) included an indirect way of assimilating the near-surface soil moisture data into soil hydrologic models to derive saturated and unsaturated hydraulic conductivity K(h) and soil water retention θ(h) properties of the soil at the footprint/grid/pixel-scale. 46.5.2  Top-Down Stochastic Approach

Bresler and Dagan (1983) concluded that in spatially variable fields, stochastic modeling represents the actual flow phenomena in the porous media more realistically and provides the main statistical moments (mean and variances) by employing simplified flow models, which can be used with confidence in the applications. Along the line, using a Noisy Monte Carlo Genetic Algorithm (NMCGA) in conjunction with a SWAP model, Ines and Mohanty (2008b) suggested that the main assumption used in the (top-down) inverse modeling of RS near-surface soil moisture data is that the temporal RS data contain (enough) information that can possibly describe the effective (root zone) hydrologic conditions of a pixel, such that when inverted, it (i.e., the RS data) will provide a set of soil hydraulic parameters representative of that pixel. This resulted in a more general approach of a scale-dependent para­ meter estimation concept (Fig. 46.8) such that both the effective soil hydraulic parameters and their uncertainties could be determined simultaneously at a remote sensing footprint/model grid. Ines and Mohanty (2008b) determined the set of effective van Genuchten soil hydraulic parameters p*= {α*, n*, qres*, qsat*, Ksat*} to accommodate the uncertainty terms (e.g., analogous to horizontal heterogeneity in a pixel) of each parameter. However, soil hydraulic properties at the near-surface may not be always representative of the sub-surface (root zone) unless the soil profile is statistically homogenous. Ines and Mohanty (2008a, 2008b) showed that effective soil hydraulic parameters derived from the near-surface (0–6 cm) are applicable to some degree in representing the soil hydrologic processes in

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Inverse Modeling—Soil Hydraulic Properties at the Model Grid Scale     46-9 

Remote sensing

h

(km scale)

RS footprint scale

Scale 2

h

θ

Top-down approach

h

θ

Bottom-up approach

Scale 3

In situ

(cm scale) Point scale

Scale 1 θ

Figure 46.8  Top-down approach: panels show the concept of effective soil water retention properties (θ - h) and their uncertainties using soil moisture time series [θ(z, t)] at ­different scales using stochastic NMCGA. (Source: Ines and Mohanty, 2008b.)

the subsurface (root zone). But under highly heterogeneous soils, they are less successful. Recently, Shin et al. (2013b) recommended that additional information (e.g., deeper layer soil moisture data and ET from RS) should be used in addition to the near-surface soil moisture to characterize better the effective soil hydraulic properties of the soil profile. Pollacco et al. (2013) formulated a selector algorithm by devising multiobjective functions for improved para­ meter optimization using remotely sensed soil moisture and ET. 46.5.3  Bottom-Up Approach

In a series of study, using a combination of stochastic and aggregated approach, Zhu and Mohanty (2002a; 2002b; 2003; 2004; 2006), Zhu et al. (2004; 2006), and Mohanty and Zhu (2007) investigated the suitability of various soil hydraulic parameter upscaling schemes (bottom-up approach) by matching their prediction performances with ensemble vadose zone fluxes (evaporation and infiltration) under different hydroclimatic scenarios for horizontally and vertically heterogeneous soil systems (see Fig. 46.9). Their synthetic experiment results showed that soil texture, geological layering, groundwater depth, surface/profile soil moisture status, vertical flux direction (evaporation vs. infiltration), hydraulic parameter statistics, correlations, and spatial structures, root distribution in soil profile, and topographic features/ arrangements conjointly determines the “upscaled” pixel-scale soil hydraulic parameters for the equivalent homogeneous medium that delivers the same amount of flux (evaporation or infiltration) as the natural h ­eterogeneous medium. Thus, different homogenization algorithms (rules) for different hydrologic scenarios and land attribute complexities were suggested for para­ meter upscaling. The link between fine scale and coarse scale soil hydraulic conductivity for root zone soil water flow processes is nontrivial and one needs to take into account the effects of all the heterogeneities present at the fine scale

(Fig.  46.10). In more recent efforts, based on Efendiev and Hou (2009), ­multiscale finite element schemes with nonlinear basis function are being developed for linking soil water flow in finer grids within coarse grid in a computationally cost effective manner, while preserving the small as well as large scale landscape (heterogeneity) features in the flow domain. 46.5.4  Multiscale Modeling

Flexible multiscale modeling and uncertainty quantification tools play a critical role in performing predictive multiscale simulations. Traditionally, conceptual flow models at various scales, such as pore, continuum, field, catchment, watershed, and region, are considered in hydrology. Bridging scales is often accomplished via up/downscaling. Upscaling techniques form coarse-scale equations with a prescribed analytical form that may differ from underlying fine-scale equations. Coarse-scale parameters or functional relations are evaluated by averaging appropriate variables obtained from local solutions. Multiscale methods have emerged as powerful simulation techniques that can be used to link models at different scales. Multiscale methods share similarities with upscaling methods, though there are important differences. In multiscale methods, the fine-scale information is carried throughout the simu­ lation, while coarse-scale equations are generally not expressed ­analytically, but rather are formed and solved numerically. Multiscale methods allow downscaling of the processes at the regions of interest and extracting the finescale features of the processes. Moreover, multiscale methods are scalable and the simulations can be performed perfectly parallel. Predictive multiscale simulation techniques are needed, which can pass the information about subscale effects and associated uncertainties from one scale to another. Some of these approaches are developed for simpler applications and can be extended to more complex critical zone.

q– 1.20

4.00

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h=1.1 h=2

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h=0.5 h=0.7 h=0.99

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Figure 46.9  Bottom-up approach: soil hydraulic properties heterogeneity orientation (horizontal vs. vertical) and its impact on upscaled effective parameter coefficient (ECA) for Gardner’s α for different land surface pressure head (h) and coefficient of variation of α (CVA). (Source: Mohanty and Zhu, 2007.)

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46-10     Soil Moisture and Vadose Zone Modeling

0.52 0.5

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Figure 46.10  Example of multiscale soil water flow modeling using upscaled soil hydraulic parameters, (left panel) soil moisture on the fine grid, (middle panel) fine-scale soil moisture averaged over a coarse grid, and (right panel) coarse-scale soil moisture using coarse-scale model with upscaled soil hydraulic parameters. Results signify the differences in soil moisture and hydrologic flux distribution based on grid size and model parameter treatment. (Source: Mohanty, unpublished work.)

46.5.5  Soil Moisture Upscaling Based on Multiscale Observation, Modeling, and Data Assimilation

46.5.6  Soil Moisture Downscaling

Data-assimilation schemes have been gaining ground over the past decade as a means for improving soil moisture predictability using process-based SVAT models in conjunction with spatiotemporal observations from different in situ/remote sensing platforms. Data from different sources are injected at different times into model outputs and, moreover, the boundary and initial conditions as well as forcing terms. Consequently, our computations become dynamic data-driven simulations. Data assimilation can be broadly divided into two categories: variational techniques with adjoint model and (ensemble or extended) Kalman filter techniques. Dunne and Entekhabi (2005) provides a review of applications of these techniques in land data assimilation including comparative advantages and limitations. Among others, Das and Mohanty (2006), Das et al. (2008a; 2008b; 2010), and Pollacco and Mohanty (2012) report on some of the significant data-assimilation studies related to land surface hydrologic modeling including soil moisture dynamics in general and several in the Great Plains region (with legacy data of many soil moisture field campaigns) in particular. The main idea of the procedure consists of updating model parameters, such as fine-scale hydraulic conductivity field at each time step when fresh data is injected (Das et al., 2008a, see Fig. 46.11). For the data representing large (coarse) scale information, adequate upscaled model is used and upscaled parameters are estimated. Further, the fine-scale model is updated using the coarse-scale information and hierarchical Bayesian ­ methods. Data-assimilation techniques with Markov Chain Monte Carlo (MCMC) schemes provide an efficient platform for deriving upscaled hydrologic parameters using process-based models and remote sensing data. Bayesian scheme further provide uncertainty bound for these parameters.

46.6  SUMMARY

In summary, modeling unsaturated flow and transport in vadose zone has made significant progress in the past three decades. However, as in any other subdisciplines of hydrology, challenges remain in terms of dealing with scale,

AMSR-E measurements

θ

Time series

θr

Posterior distribution of upscaled hydraulic

Likelihood

θs Prior Z

n Ksat β

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Upscaling parameter

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0.8 0.6 0

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Relaxed PDF chosen from dominant soils type within AMSR-E footprint

Multivariate distribution

Satellite-based soil moisture footprints provide available large-scale applications, because of their spatiotemporal extents. However, modeling for catchment/ watershed scale hydrology, agricultural water management, environmental fate  and transport, and ecosystems services particularly occur at finer-scale ranging from several hundred meters to several kilometers. Ines et al. (2013) developed a stochastic unmixing method for soil moisture using a simulationassimilation scheme. This approach extracts soil-type identification (representing soil hydraulic properties) and subarea fractions corresponding to soil-vegetation combinations within a remote sensing footprint scale soil moisture product. However, the stochastic disaggregation method estimates only the soil characteristics (soil texture–based ID values) and ­subarea fractions (% by the soil-vegetation combinations) within a pixel in a probabilistic sense without their specific locations practically recognized. In order to realize the full potential, a new deterministic downscaling algorithm was developed for estimating fine-scale soil moisture within remotely sensed soil moisture and ET products using genetic algorithm (Shin and Mohanty, 2013a, 2013b). This approach was evaluated under various synthetic and field (Little Washita, Oklahoma) conditions including homogeneous and heterogeneous land surface composed of different soil textures and vegetation. Analysis for various landscapes provides deeper insight to dominant hydrologic controls for root zone soil moisture at different scales and landscape complexities.

0.04

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2 3 Iterations Mean posterior plot

4

5 x 104

0.02 0 0.4

0.6

Mean

0.8

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Figure 46.11  MCMC-based scheme for deriving upscaled soil hydraulic parameters that provides a basis for multiscale soil moisture data-assimilation. (Source: Das et al., 2008a.)

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References    46-11 

heterogeneity, and uncertainty in the vadose zone. Although the formalized capillary-based governing flow equation and related numerical forward/ inverse models may be best suited for local flow processes, the physical basis of their adoption at the larger scale is debated. Alternative formulations based on thermodynamic principles, hydrologic functionality rules, or scale-­ appropriate water balance models across different irregular hydrologic units (e.g., catchments, watersheds, and river basins) may prove to be more effective for describing large scale soil hydrology. With the advent of new groundbased sensor networks and remote sensing (satellite) measurement techniques at multiple resolutions and frequency, development of novel scale-appropriate process modeling and data assimilation/fusion tools will be the key for improved understanding of water, carbon, nutrient, and energy cycle in the terrestrial system in the coming decades. REFERENCES

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and multiple solutes in variably-saturated media,” User Manual, Version 1.0, PC Progress, Prague, Czech Republic, 2012, p. 303. Sivapalan, M., K. Beven, and E. F. Wood, “On hydrologic similarity 2. A  scaled model of storm runoff production,” Water Resources Research, 23: 2266–2278, 1987. Stewart, J. B., E. T. Engman, R. A. Feddes, and Y. Kerr, “Scaling up in hydrology using remote sensing,” Wiley, Chischester, UK, 1996, p. 255. Todini, E., “A model conditional processor to assess predictive uncertainty in flood forecasting,” International Journal of River Basin Management, 6 (2): 123–137, 2008. Van Dam, J. C., J. Huygen, J. G. Wesseling, R. A. Feddes, P. Kabat, P. E. V. van Waslum, P. Groenendjik, et al., “Theory of SWAP version 2.0: simulation of water flow and plant growth in the soil-water-atmosphere-plant environment,” Technical Document, DLO Winand Staring Center, Wageningen Agricultural University, Wageningen, The Netherlands, 1997, p. 45. Van Genuchten, M. Th., “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils,” Soil Science Society of America Journal, 44: 892–898, 1980. Vinnikov, K .Y., A. Robock, N. Speranskaya, and C. A. Schlosser, “Scales of temporal and spatial variability of midlatitude soil moisture,” Journal of Geophysical Research, 101: 7163–7174, 1996. Vrugt, J. A., C. J. F. ter Braak, C. G. H. Diks, B. A. Robinson, J. M. Hyman, and D. Higdon, “Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling,” ­ International Journal of Nonlinear Sciences and Numerical Simulation, 10 (3): 271–288, 2009. Vogel, T. and M. Císlerová, “On the reliability of unsaturated hydraulic conductivity calculated from the moisture retention curve,” Transport in Porous Media, 3: 1–15, 1988. Western, A., R. B. Grayson, G. Bloschl, G. R. Willgoose, and T. A. McMahon, “Observed spatial organization of soil moisture and its relation to terrain indices,” Water Resources Research, 35: 797–810, 1999. Western, A., R. B. Grayson, and G. Bloschl, “Scaling of soil moisture: a  hydrologic perspective,” Annual Review of Earth and Planetary Sciences, 30: 149–180, 2002. Western, A., R. B. Grayson, G. Bloschl, and D. J. Wilson, “Spatial variability of soil moisture and its implications for scaling,” Scale Issues in Soil Physics, edited by Pachepsky, Y., Radcliffe, D. E., and Selim, H. M. CRC Press, Boca Raton, FL, 2003, pp. 119–142. White, M. D., D. H. Bacon, B. P. McGrail, D. J. Watson, S. K. White, and Z. F. Zhang, “STOMP: Subsurface transport over multiple phases: STOMP-CO2 and STOMP-Coe Guide,” Version 1.0 PNNL-21268, Pacific Northwest National Laboratory, Richland, WA, 2012. Ying, M., S. Feng, Z. Huo, and X. Song, “Application of the SWAP model to simulate the field water cycle under deficit irrigation in Beijing, China,” Mathematical and Computer Modelling, 54: 1044–1052, 2011. Zeng, X. B. and M. Decker, “Improving the numerical solution of soil moisture-based Richards equation for land models with a deep or shallow water table,” Journal of Hydrometeorology, 10: 308–319, 2009. Zhang, X., R. Srinivasan, K. Zhao, and M. V. Liew, “Evaluation of global optimization algorithms for parameter calibration of a computationally intensive hydrologic model,” Hydrological Processes, 2008, doi:10.1002/ hyp.7152. Zhang, X., R. Srinivasan, and D. Bosch, “Calibration and uncertainty analysis of the SWAT model using Genetic Algorithms and Bayesian Model Averaging,” Journal of Hydrology, 374: 307–317, 2009. Zhu, J., B. P. Mohanty, “Upscaling of soil hydraulic properties under steady state evaporation and infiltration,” Water Resources Research, 38, 2002a, doi:10.1029/2001WR000704. Zhu, J., B. P. Mohanty, “Spatial averaging of van Genuchten hydraulic parameters for steady state flow in heterogeneous soils,” Vadose Zone Journal, 1: 261–271, 2000b. Zhu, J., B. P. Mohanty, “Effective hydraulic parameters for steady state vertical flow in heterogeneous soils,” Water Resources Research, 39 (8), 2003, doi:10.1029/2002WR001831. Zhu, J., B. P. Mohanty, “Soil hydraulic parameter upscaling for steady state flow with root water uptake,” Vadose Zone Journal, 3: 1464–1470, 2004. Zhu, J., B. P. Mohanty, A. W. Warrick, and M. Th. van Genuchten, “Correspondence and upscaling of hydraulic functions for steady-state flow in heterogeneous soils,” Vadose Zone Journal, 3: 527–533, 2004. Zhu, J. and B. P. Mohanty, “Effective scaling factor for transient infiltration in heterogeneous soils,” Journal of Hydrology. 319: 96–108, 2006. Zhu, J., B. P. Mohanty, and N. N. Das, “On the effective averaging schemes of hydraulic properties at the landscape scale,” Vadose Zone Journal, 5: 308–318, 2006.

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Chapter

47

Hydrogeologic Characterization BY

KAVEH ZAMANI, TIMOTHY R. GINN, AND MOHAMED K. NASSAR

ABSTRACT

The subsurface is the unseen part of the hydrologic cycle that is critical for municipal, industrial, agricultural, and/or ecosystem water use especially in times of drought. The occurrence and mobility of subsurface water depends on a host of hydrogeological properties that are typically very difficult to measure and sometimes not even well-defined. Furthermore, these properties vary with location, often on multiple scales. In this chapter, we summarize techniques for characterization of the primary properties controlling groundwater flow in the saturated subsurface. Our focus is on only the most basic methods, with measurement scale defined by the sampling technique, in order to provide a simple background for field work approaches to quantifying the properties governing large-scale groundwater flow. 47.1  INTRODUCTION

Understanding groundwater for engineering, agricultural, or environmental activities require sufficient knowledge of subsurface geological and hydrological characteristics. Difficulties arise quantifying hydrogeological processes mainly due to incomplete characterization of subsurface properties at the required spatial scale. This scale ranges from the order of tens of meters for geothermal applications to kilometers for contamination problems to hundreds of kilometers for regional groundwater sustainability studies. Historically, many subsurface characterization techniques in hydrogeology were developed originally for petroleum and mining industries and subsequently adopted for groundwater engineering. Detailed references from these and related fields about hydrogeologic characterization exist, for example, Charbeneau (2006), EPA (2015), Fetter (1999), Mitchell and Soga (2005), Rubin and Hubbard (2005), and Todd and Mays (2005). The purpose of this chapter is to give a summary overview of some of the practicable techniques to characterize geological facies, structure, and permeability as well as groundwater occurrence hydraulic head, and flow velocity. Due to scope restrictions many related techniques, such as ground-penetrating radar, selfpotential, remote sensing and GRACE data, inclinometry, environmental tracers, and seismic techniques are not covered here, and the treatment is restricted to saturated subsurface materials. 47.2  BOREHOLE SAMPLES AND GROUNDWATER MONITORING WELLS

Drilled wells are main source of subsurface hydrogeological information. In general, location selection, design, well drilling, and sampling phases of wells have to be conducted based on the purpose of the well, geologic structure of the site, expected groundwater occurrence and contamination, and location of the site. Drilling techniques used for well construction include hollow-stem augers, solid-stem augers, water/mud rotary, pneumatic rotary, sonic (vibrator), direct push, and casing or cable. While hydraulic/pneumatic rotary and cable tool are the more common approaches for well construction, the investigation borehole drilling methods useful for obtaining core samples are ­hollow-stem auguring, rotary drilling, and direct push sampling. At contaminated sites, well casing and screens must be chemically compatible with the

substances being monitored. For more information on monitoring well design, drilling, installation, and maintenance, see ASTM D 5876 and 5978 standards (ASTM, 2011, 2012a; Aller et al., 1991). 47.3  INVESTIGATION OF BOREHOLE DRILLING

Site investigation with boreholes provides one-dimensional (1D) information on geologic structure and material properties via collection of core samples (e.g., ASTM, 2012a; Hunkeler, 2010). Rotary boring is a method of advancing a borehole in both rock and sedimentary formations up to depths of 350 m, and generates core samples of diameters 10–25 cm. The drill shaft and bit are advanced within a metal core barrel that is advanced in sections to provide a temporary or permanent borehold casing. In direct rotary drilling, drilling mud or fluid is pumped down the drilling core to the bits and brings up the cuttings to the surface. In reverse-circulation rotary drilling, used for larger boreholes, the drilling fluid is pumped down the annulus between the borehole and the casing and returns through the drill barrel. This provides the advantage of positive pressure to maintain the stability of the borehole during drilling. The method is called air-rotary drilling if instead of water, compressed air is being used. Subsurface materials returning to the surface with the drilling fluid provide qualitative if disturbed sample that can be used to characterize strata at depth. To acquire core samples, the drilling rod is raised and the drilling shaft and bit is replaced with a core sampler. Some advanced bits are also able to obtaining a core sample when they are used for drilling. Disadvantages of this method compared to auger drilling and direct push techniques include potential contamination by drilling mud/fluid that may impact groundwater chemistry, and the relatively higher cost and infrastructure required. Auger boring involves a spiral drill (augur) and is commonly used in relatively shallow and unconsolidated formations to depths up to 100 m (e.g., Hackett, 1987). A conventional auger would bring materials to the surface generally disturbed. The hollow-stem auger involves placement of the spiral drill inside a metal barrel, and core samples can be obtained as with rotary rigs by retrieving the drill and temporarily replacing it with a single or double sampling core. Advantages of this method include no involvement of drilling mud/fluid, lighter infrastructure, and faster sampling at lower costs than rotary methods. Direct-push sampling is an even lighter-footprint approach that involves driving a metal barrel into the subsurface often using high-­ frequency hammering or vibration. Core sample retrieval may be done conti­ nuously using this method, by driving core barrel segments that are retrieved, or at intervals wherein a driven penetration tip is replaced at intervals with a segment of sample core barrel that is retrieved after it is driven till it is filled. Advantages of this method are the light equipment and rapid advancement, while disadvantages include the small diameter core samples obtained and the requirement for penetrable subsurface materials. Percussion boring (or cable-tool drilling) is a method in which soil and rock formations are broken by repeated blows of heavy chisel or bit hanging from a cable or drill rod. A steel casing is driven into the borehole behind the bit utile the rock formation is reached. When the bottom of the hole is filled with broken rocks and sediments, water is added to the hole, if not already below the water table and the slurry of pulverized material is removed with 47-1

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47-2    Hydrogeologic Characterization

bottom-loading bailer. An advantage of percussion drilling is unlike rotary drilling, no drilling fluid/mud circulated through the well, hence, induced contamination issue does not exist. The other advantage is that the percussion drilling is suitable for remote location due to low fuel consumption and small needs for water. This method is generally slower than other alternatives drilling techniques; however, the drilling equipment are less expensive than rotary methods (ASTM, 2012a). 47.4  CONE PENETROMETRY, PERMEAMETRY, AND ELECTRICAL CONDUCTIVITY LOGGING

The static penetration or cone penetration test (CPT) is a method to determine geotechnical properties of relatively shallow subsurface materials and to delineate stratigraphy of fine sand, clays and organic soils in particular (e.g., Rogers, 2006). The standard test apparatus consists of a still cone with an apex angle of 55° to 60° and cross-section area of 10 cm2 (other variations exist) that is driven at a speed of 1–2 cm/s, while the tip resistance qc and friction fc are recorded as the instrument moves down at intervals, which can be as small as 5 mm. Generally, two types of friction-cone penetometers are used to measure qc and fc  : mechanical and electrical, the latter provides more continues profile compare to the former (cf. below, e.g., Harrington and Hendry, 2006). Several empirical curves that are useful in estimating the properties of subsurface have been suggested for the point resistance qc and the friction ratio Fr, defined as the ratio of frictional resistance over cone resistance (e.g., Lapham et al., 1997; ASTM, 2012b). Some electrical CPT has passive pressure transducer to record water pressure that is termed piezocone penetration test or “CPTU.” Advantages of this technology are low cost and rapid characterization of strata and resistive force of geomaterials; disadvantages include indirect relation of the measured quantity to hydrogeological properties and penetrometer blocking by strongly consolidated or hard-rock materials. For more information on conduction and interpretation of CPT for site investi­gation, see ASTM (1995). Butler and co-workers have extended the passive measurement of hydraulic pressure using direct-push equipment classically associated with cone penetrometry as done in the CPTU to active permeametry by inducing a hydraulic pressure at the porous cone tip and measuring the resulting pressure at different distances along the pushed shaft (Butler et al., 2002, 2007; Dietrich et al., 2008; Zshornack et al., 2013). This technique, termed DPP for direct push permeametry, measures the resulting pressure at can yield data on vertical variations in hydraulic conductivity with a resolution scale of 0.4 m vertical separation for conductivity values associated with aquifer materials. The advantages of this technique are the low cost and minimal disturbance of the subsurface with high-resolution data. Disadvantages include penetrometer/ permeametry refusal in tight materials, errors induced in the presence of very thin (less than 0.5 m) layering structure and errors associated with bypass flow-up along the penetrometer shaft; however, these latter errors can be controlled by careful push strategies that take into account the structural properties expected to be encountered in the materials (e.g., Zshornack et al., 2013). Chapuis and Chenai (2010) provide a list of improvements derived from a geotechnical context for the hydrogeologic extensions of direct-push technology including using a cone tip that is the same or smaller diameter than the shaft in order to minimize bypass flow, using shape factors in order to more accurately determine the local conductivity values, and allowing sufficient time for hydraulic equilibration to be attained prior to the collection of pressure data for identification of conductivity. Direct-push electrical conductivity logging is an alternative direct push technology that involves driving an electrical conductivity meter at sequential depths to obtain a vertical record of subsurface electrical conductivity. The device involves a metal rod with a pair of electrodes to incur an electrical current and two inner electrodes used to monitor the resulting voltage (e.g., Harrington and Hendry, 2006; Hunkeler, 2010). The advantages of this characterization approach are that electrical conductivity reflects water content and material type, with silts and clays and high water contents associated with a higher electrical conductivity than sands or gravels and lower water contents. When correlated with direct samples of the subsurface materials, this technique can be used to distinguish identifiable stratigraphic units and thus to infer hydrogeologic properties. Combination of direct-push Electrical Conductivity (EC) profiling with DPP or with small-scale slug tests provide combined data for the same bore that is useful for cross-validation of interpreted stratigraphy (e.g., Sellwood et al., 2005). 47.5  ELECTRICAL RESISTIVITY SURVEY

Surface electrical resistivity (reciprocal of conductivity) test is a geophysical method for hydrogeological exploration that applies the same electrostatics of the direct push EC probe to larger scales, with current induced between two

47_Singh_ch47_p47.1-47.4.indd 2

electrodes emplaced at the surface and monitoring electrodes measuring the difference in electrical potential (i.e., voltage), nearby and usually between the inducing electrodes. The current used is direct current, commutated direct current (i.e., a square-wave alternating current), or alternating current with low frequency (about 20 Hz). The resulting potential (volt) distribution can be related directly to electrical resistivity in several idealized cases, such as the case of perfectly layered strata or the case of homogeneous materials separated by a vertical dike (e.g., EPA, 2015). In the 1D case, the electrical resisti­ vity ρ of soil material is expressed as:

ρ=

A ∆V (47.1) L I

where, A is the cross-section area of current path, L is the length of flow path, ∆V is voltage, that is, change in electrical potential in volts, and I is the electrical current. Electrical resistivity is measured in the units of Ω. m or Ω. cm. In case measurement is conducted over a real heterogeneous earth, the subsurface profile may consist of various layers with different resistivity and the symbol ρ is replaced with apparent resistivity ρa. The electrical resistivity of various soils and rocks depends largely on the pore water content; however, ion concentration is an important factor too. The general electrical resistivity ranges from less than 100 for clays to 200–3000 for sands, to 1500–2500 for fractured rock, to 3000–30,000 for coarse gravels (e.g., Mitchell and Soga, 2005). There are several common procedures for measuring electrical resistivity of a soil profile with electrodes that are driven into the ground. Some of those include: Wenner array, Schlumberger array, dipole–dipole, Lee, halfSchlumberger, polar dipole, bipole–dipole, and gradient arrays. Here, we briefly describe the first three methods which are widely utilized, relying heavily on EPA (2015). The Wenner array method consists of four in-line, equally spaced electrodes. The outer two electrodes are typically the current source and the inner two electrodes are the receiver electrodes (Loke et al., 1996; ASTM, 2010). In the Schlumberger array, the potential electrodes are installed at the center of the electrode array with a small separation relative to the spacing between the current-inducing electrodes. The third common resistivity method involves the dipole–dipole array, that involves dipoles (two closely spaced electrodes), to measure the changes of electrical properties with depth. In depth discussion of implementations, limitations, and interpretation of electrical resistivity methods can be found in ASTM (2010) and in EPA (2015). 47.6  HYDRAULIC METHODS FOR IN SITU CONDUCTIVITY MEASUREMENT

In situ hydraulic conductivity is classically viewed as the primary control on groundwater flow in response to gradients in hydraulic head, and essentially all groundwater textbooks include chapters on its characterization in place by perturbation of the hydraulic head in one borehold and its observation in the same or adjacent boreholes screened in the same aquifer material where the perturbation takes place, when identifiable (e.g., Fetter, 2000; Todd and Mays, 2005; Charbeneau, 2006). These techniques can be categorized into slug tests and pumping tests. A slug test involves the instantaneous change in the water level in a borehole followed by monitoring its return to preperturbation conditions. Most commonly a cylinder (slug) is emplaced (or removed) from the borehole, causing a sudden increase (or decrease) in the water level in the borehole. The return of water level in the borehole to initial conditions is monitored (water level in the borehole as a function of time, best obtained through a pressure transducer deep in the borehole) and analyzed to determine the local hydraulic conductivity. The local hydraulic conductivity is inferred by inversion of an approximation of Darcy’s law that results in the conductivity being proportional to the slope of the natural log of hydraulic head versus time. The advantages of this approach are that only one borehole is needed, and no power source is required; the primary disadvantage is that the hydraulic property measured is particular to the local neighborhood of the borehole being used for the test. Pumping tests involve extension of the slug test concept to one or more observation wells and continuous pumping at a fixed rate at a control well. Observation wells for this experiment are best as narrow-diameter wells that are screened only in the aquifer being pumped, termed “piezometers,” in order to eliminate artifacts due to storage in the monitoring well and adjacent flows. Because the hydraulic head change in this approach is typically measured in separate monitoring wells, the hydraulic conductivity values obtained are representative of the well-to-well separation scale, significantly larger than that of the slug test. The analyses of the hydraulic head drop (drawdown) data observed as a function of time in the monitoring well involves different equations depending on the nature of the pumping

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references    47-3 

(­constant or pulsed in steps), the confined versus unconfined nature of the aquifer studies, and the impact of hydraulic boundaries (e.g., Charbeneau, 2000). For instance, in the confined case with constant pumping, assuming homogeneous and perfectly layered aquifer materials, the Theis solution is used, and this can be simplified to the Jacob solution for later time parts of the observed drawdown. The more complex unconfined case is fortunately solved in Neuman (1974), and these and a suite of related solution approaches for pumping test data interpretation is available in numerous available codes (e.g., www.aqtesolve.com). 47.7  CHARACTERIZATION OF THE HYDRAULIC GRADIENT AND FLOW RATES

As important as the stratigraphic understanding of the aquifer-aquitard structure of the subsurface and the corresponding hydraulic conductivities is the hydraulic gradient occurring within these formations, because the gradient and the hydraulic conductivity together dictate the natural groundwater flow direction and magnitude. The hydraulic gradient is the slope of the hydraulic head surface. In confined aquifers, the hydraulic head is the level to which water will rise in a piezometer screened in the aquifer, and in unconfined aquifers the hydraulic head is generally well approximated by the water table. In the two-dimensional (2D) case with perfectly layered aquifer materials, the hydraulic head can be visualized as a surface just like topographic elevation, and the gradient is the slope of this surface. A convenient means of evaluating the slope in a given region is to emplace three piezometers in a right triangle layout, ABC, with B being the piezometer located at the right angle corner. Then the gradient is given by the two slopes, the head difference between piezometers at A and B, divided by the distance between the two, and the head difference between piezometers at B and C, divided by the distance between those two, respectively. It must be remembered that the 2D assumption allows an approximation of what may be in reality be a 3D gradient. A means of evaluating the vertical component of the gradient (that is, the third dimension when the first two are horizontal), is via a “piezometer nest.” This is a series of piezometers installed in the same location, but each screened at different depths. Vertical components of hydraulic gradients are particularly important in determining vertical fluxes across low-conductivity formations and in recharge areas. Tracer testing provides an alternative approach to characterizing flow rates. The single-well point dilution tracer test involves emplacement of a tracer (typically dye or salt) in a known concentration in a well followed by monitoring of the change in concentration of the tracer in the well as a function of time. These data can be analyzed to estimate the ambient flow rate through its flow component through the well itself. This test typically takes a day and the analyses should take account of the well geometry and the artificially high hydraulic conductivity associated with the gravel pack associated with the well installation (e.g., Hunkeler, 2010). When more observation wells are available, this test can be generalized to a multiwell tracer test, in which the tracer is injected in one well and observed at other downstream wells. The disadvantages of this approach are the increased costs associated with installation of the additional monitoring wells and the challenge in capturing the tracer when the flow is relatively fast compared to the rate of dispersion of the salt around the injection well. This latter aspect can lead to the tracer plume following a narrow path that may not be intercepted by the downstream well. Also this method requires knowledge of the downstream direction a priori. Finally, vertical flows monitored within boreholes can provide useful ­information on hydraulic head and on conductivity profiles along the borehole (e.g., Molz et al., 1989; Paillet, 2000; USGS, 2015). In the absence of pumping, passive flow may occur in a borehole screened in multiple aquifer formations (or in one very thick one) due to vertical hydraulic gradients. In this case, the measurement of vertical flow rates, as a function of elevation within the b ­ orehole, reveals the difference in hydraulic head between the contributing and receiving formations or the hydraulic head difference between them, given the other. Such monitoring can be combined with pumping of a given well, in which case the flowmeter gives the relative flux rates entering (or leaving) the borehole as a function of distance. Given the knowledge of the hydraulic head within the formation, such data give the hydraulic conductivity variations along the borehole. Flowmetering in these cases can be done using a simple impeller, a heat tracer, or electromagnetic induction monitoring (e.g., USGS, 2015). 47.8  RECHARGE ESTIMATION

Characterization of the large-scale behavior of subsurface flow systems is incomplete without quantification of the boundary fluxes of water entering or  leaving the aquifer system. In fact, if flow rates at all inflow boundaries

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are  known for steady-state flow systems, then only relative conductivity ­information is needed to allocate the flow to different portions of the aquifer, and if that aquifer is homogeneous with a constant conductivity then only the aquifer geometry is needed to complete the understanding of the flow. Recharge-estimation methods are many, beyond the current scope, and include the broad field of environmental tracer interpretation. USGS (2014) provides a summary tabulation of methods with references, advantages, and limitations. Here, we briefly summarize two basic approaches to quantifying areal recharge resulting from a surface influx of water. The unsaturated zone water balance method uses unsaturated zone water saturation and tension (negative of pressure head) measurements before and after a recharge event (rainfall or irrigation) to quantify the increase in water content that occurs below the depth above which water is extracted by evapotranspiration. This depth is identified by changes in the water tension with depth, and the net increase in water content fellow this depth is areally averaged to determine net recharge. In the water table fluctuation method, the data collected is the increase in water table elevation and the recharge is estimated as the areally averaged change in saturated thickness multiplied by the specific yield. Advantages of these methods are the high-resolution identification of eventbased recharge that is a big improvement over annual water-balance estimates, and disadvantages are the requirement for multiple soil water data or water table data over the extent of the area in question. This latter requirement is the main limiter on the utility of the method for identification of event-based recharge on regional scales. 47.9  SUMMARY

This chapter provides only the briefest survey of selected methods associated with the range of quantities requiring characterization for understanding subsurface hydrological cycles. Continued innovation in the numerous fields providing such data, including geophysical, hydrological, electrical, is necessary to overcome our inability to characterize subsurface hydrogeological processes. The inaccessibility of the subsurface that makes hydrogeology the most challenging of the areas of environmental fluid mechanics. REFERENCES

Aller, L., R. J. Petty, J. H. Lehr, H. Sedoris, D. M. Nielsen, and J. E. Denne, Handbook of Suggested Practices for the Design and Installation of GroundWater Monitoring Wells, Environmental Monitoring Systems Laboratory, Office of Research and Development, US Environmental Protection Agency, Las Vegas, NV, 1991. ASTM, Designation D 5778-95(2000): Standard Test Method for Performing Electronic Friction Cone and Piezocone Penetration Testing of Soils, ASTM International, West Conshohocken, PA, 2000. ASTM, Designation D 6431: Standard Guide for Using the Direct Current Resistivity Method for Subsurface Investigation, ASTM International, West Conshohocken, PA, 2010. ASTM, Designation D 5978: Standard Guide for Maintenance and Rehabilitation of Groundwater Monitoring Wells, ASTM International, West Conshohocken, PA, 2011. ASTM, Designation D 6286: Standard Guide for Selection of Drilling Methods for Environmental Site Characterization, ASTM International, West Conshohocken, PA, 2012a. ASTM, Designation D 5876: Standard Guide for Use of Direct Rotary Wireline Casing Advancement Drilling Methods for Geoenvironmental Exploration and Installation of Subsurface Water-Quality Monitoring Devices, ASTM International, West Conshohocken, PA, 2012b. www.aqtesolve.com, accessed August 2015. Butler J. J., J. M. Healey, G. W. McCall, E. J. Garnett, and S. P. Loheide, “Hydraulic tests with direct-push equipment,” Ground Water, 40: 25–36, 2002. Butler, J. J., P. Dietrich, V. Wittig, and T. Christy, “Characterizing hydraulic conductivity with the direct-push permeameter,” Ground Water, 45 (4): 409–419, 2007. Chapuis, R. P. and D. Chenai, “Comment on ‘Characterizing hydraulic conductivity with the direct-push permeameter,’ by James J Butler Jr. Peter Dietrich, Volker Witting, and Tom Christy, July–August 2007, 45 (4): 409– 419,” Ground Water, 48 (6): 792–795, 2010. Charbeneau, R. J., Groundwater Hydraulics and Pollutant Transport, Waveland Press, Long Grove, IL, 2006, p. 593. Dietrich P, J. J. Butler, and K. Faiss, “A rapid method for hydraulic profiling in unconsolidated formations,” Ground Water, 46: 323–328, 2008. Environmental Protection Agency, Environmental Geophysics (8/8/2015), 2015, accessed online at: http://www.epa.gov/esd/cmb/GeophysicsWebsite/ pages/basicInformation.htm.

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47-4    Hydrogeologic Characterization

Fetter, C. W., Contaminant Hydrogeology (2nd ed.), Prentice Hall, Upper Saddle River, NJ, 1999, p. 500. Fetter, C. W., Applied Hydrogeology (4th ed.), Prentice Hall, Englewood Cliffs, NJ, 2000, p. 691. Hackett, G., “Drilling and constructing monitoring wells with hollow‐stem augers. Part 1: Drilling considerations,” Groundwater Monitoring & Remediation, 7 (4): 51–62, 1987. Harrington G. A. and M. J. Hendry, “Using direct-push EC logging to delineate heterogeneity in a clay-rich aquitard,” Ground Water Monitoring Review, 26: 92–100, 2006. Hunkeler, D., “Geological and hydrogeological characterization of the ­subsurface,” Handbook of Hydrocarbon & Lipid Microbiology, edited by K. N. Timmis, Springer, Berlin, 2010. Jones, S. B., M. W. Jon, and O. Dani, “Time domain reflectometry measurement principles and applications,” Hydrological Processes, 16: 141–153, 2002. Lapham, W. W., F. D. Wilde, and M. T. Koterba, “Guide-lines and standard procedures for studies of groundwater quality selection and installation of wells, and supporting documentation,” U.S.G.S. Water-Resources Investigations Report, 96–4233, 1997, p. 110. Mitchell, J. K., and K. Soga, Fundamentals of Soil Behavior, 3rd Edition John Wiley & Sons, New York, 2005, p. 592. Molz, F. J., R. H. Morin, A. E. Hess, J. G. Melville, and O. Guben, “The impeller meter for measuring aquifer permeability variations: Evaluations

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and comparisons with other tests,” Water Resources Research, 25: 1677– 1683, 1989. Neuman, S. P., “Effect of partial penetration on flow in unconfined aquifers considering delayed gravity response,” Water Resources Research, 10 (2): 303– 312, 1974. Paillet, F. L., “A field technique for estimating aquifer parameters using flow log data,” Ground Water, 38 (4): 510–521, 2000. Rogers, J. D., “Subsurface exploration using the standard penetration test and the cone penetrometer test,” Environmental & Engineering Geoscience, 12 (2): 161–179, 2006. Rubin, Y. and S. Hubbard, Hydrogeophysics, Springer, Delft, The Netherlands, 2005, p. 405. Sellwood, S. M., J. M. Healey, S. Birk, and J. J. Butler, “Direct-push hydrostratigraphic profiling: coupling electrical logging and slug tests,” Ground Water, 43 (1): 19–29, 2005. Todd, D. K. and L. W. Mays, Groundwater Hydrology (3rd ed.), JohnWiley & Sons, New York, 2005, p. 636. USGS, 2014, www.water.usgs.gov/ogw/gwrp/methods, accessed August 2015. USGS, 2015, www.water.usgs.gov/ogw/bgas/flowmeter, accessed August 2015. Zshornack, L., G. C. Bohling, J. J. Butler Jr., and P. Dietrich, “Hydraulic profiling with the direct-push permeameter: assessment of probe configuration and analysis methodology,” Journal of Hydrology, 496: 195–204, 2013.

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Chapter

48

Groundwater Modeling BY

HAI V. PHAM AND FRANK T.-C. TSAI

ABSTRACT

This chapter discusses a general procedure to conduct groundwater modeling. One of the most challenging steps in groundwater model development is generating valid computational grids to be consistent with geological information. The second challenge is model calibration, which can be very time consuming and become intractable for complex groundwater models. We introduce a grid generation technique to create MODFLOW grids from complex hydrostratigraphic architectures. The study first uses well logs to construct the hydrostratigraphic architecture with fine vertical discretization through indicator kriging. Then, an upscaling procedure is introduced to convert hydrostratigraphic architecture to MODFLOW grids. To improve model calibration, we introduce the covariance matrix adaptation-evolution strategy (CMA-ES) to estimate groundwater model parameters. The CMA-ES is a stochastic optimization method, which provides a covariance matrix of estimated parameters for uncertainty analysis. Moreover, the CMA-ES uses a global-local derivative-free algorithm, which is easy to be parallelized for calibrating computationally demanding groundwater models. The techniques are applied to groundwater modeling in the Baton Rouge area, southwestern Louisiana. Reconstruction of the Baton Rouge aquifer system using a vast amount of well logs is unprecedented. Specific groundwater models for the “1,200-foot” sand to the “2,000-foot” sand are developed and calibrated by the parallel CMA-ES. Groundwater head prediction uncertainty is derived through the Monte Carlo realizations of model parameters. 48.1 INTRODUCTION

Groundwater resources are vital for sustainable economic and demographic growth (Gleeson et al., 2012). Reliable groundwater modeling is necessary for successful management of groundwater resources. However, groundwater simulation models are subjected to various uncertainties in their predictions. The uncertainties may come from naturally occurring randomness (aleatory uncertainty), insufficient data and knowledge (epistemic uncertainty), and lack of technology to translate known data/knowledge into valid groundwater models (technological uncertainty). Aleatory uncertainty is generally small and not possible to eliminate. Epistemic uncertainty caused by a lack of data and knowledge (Hora, 1996; Senge et al., 2014) is significant and can be effectively reducing by collecting informative data from the field. Technological uncertainty can be significantly minimized by using better numerical methods and computing resources. This chapter focuses on groundwater modeling and introduces a general procedure for groundwater model development based on existing data. Aleatory and epistemic uncertainties will remain the same throughout the model-development process. However, technological uncertainty can be reduced by using a better grid-generation technique and high-performance computing for model calibration. A groundwater equation describing flows in saturated sedimentary rocks can be derived by introducing Darcy’s law to the mass conservation law (Bear, 1972). A widely accepted groundwater equation based on groundwater head (hydraulic head of groundwater) is (Freeze and Cherry, 1979)

Ss

∂h ∂  ∂h  ∂  ∂h  ∂  ∂h  = K y  +  K z  + Q (48.1)  Kx  + ∂t ∂ x  ∂ x  ∂ y  ∂ y  ∂z  ∂z 

where h is the groundwater head (L) in the porous medium; Q is the volumetric flux at sources or sinks per unit volume of the porous medium (T–1); Ss is the specific storage (L–1) for the porous medium; Kx, Ky , and Kz are anisotropic hydraulic conductivity (LT–1) for the porous medium in x, y, and z directions, respectively; and t is time (T). Equation (48.1) is derived under isothermal condition, compressible groundwater and soil matrix, and no spatial variation in water density. The coordinates are aligned with principal directions of anisotropy of the porous medium. Variations of Eq. (48.1) including formulation in terms of water pressure, full anisotropy of hydraulic conductivity, and densitydependent flow can be found in Bear and Cheng (2010). Groundwater modeling aims at solving the groundwater equation along with an initial condition, boundary conditions, sinks, and sources. Many numerical methods have been successfully implemented, including the finite difference method (Harbaugh, 2005), the finite element method (Voss, 1984; Lin et al., 1997; Diersch, 2002), the finite volume method (Panday et al., 2013), the analytic element method (Haitjema, 1995), and the lattice Boltzmann method (Servan-Camas and Tsai, 2009; 2010). The USGS MODFLOW (Harbaugh, 2005) is one of the most popular groundwater solvers to be used for modeling groundwater flows. Many graphical user interfaces, commercials [e.g., Groundwater Modeling System (GMS) (Aquaveo, 2014), Visual MODFLOW (Waterloo, 2005), Groundwater Vistas (Rumbaugh and Rumbaugh, 2001), and Argus One (Liskov et al., 1987)] and noncommercials [e.g., ModelMuse (Winston, 2009) and PMWIN (Chiang and Chiang, 2001)], have been developed to facilitate required MODLFOW packages generation for real-world case studies. 48.2  GROUNDWATER MODEL DEVELOPMENT

Developing a groundwater model requires geological and geophysical data as well as hydrologic and hydrogeologic data. Figure 48.1 shows a flowchart of groundwater model development. Some components may switch the order without loss of generality. A numerical model may be selected before a computational grid is generated. Errors and uncertainties exist in every component in the flowchart. Propagation of the errors and uncertainties through the components will eventually affect groundwater calibration and prediction results. One of the most challenging components is grid generation, which governs most of the technological uncertainty. Grid generation is the first step to construct a groundwater model structure after data are collected. A valid groundwater model needs to have a valid computational grid that is consistent with geological information whenever available. Otherwise, error from geologically unsupported computational grid can produce significant systematic error in groundwater prediction. Another challenging component is model calibration, which is the last step in groundwater model development before the developed model is applied to tasks. Model calibration can be very time consuming for complex groundwater models. Without advanced search algorithms and computing resources, searching for better parameter values can end prematurely. Moreover, grid error amplifies parameter estimation error. Model calibration may result in overparameterization and unrealistic 48-1

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48-2    Groundwater Modeling

Figure 48.1  Flowchart of groundwater model development.

model parameter values in order to reduce fitting error produced by model structure error due to an invalid grid. This chapter focuses on discussions of grid generation and model calibration for groundwater modeling in order to reduce model structure uncertainty and better model parameter estimation. 48.3  GRID GENERATION

The literature shows two approaches commonly used to construct a computational grid from borehole data: one is the solid approach and the other is the predefined grid approach. Before implementing either approach, one needs to obtain geological information (e.g., lithology, bed boundary elevation, formation dip, etc.) from well logs. Readers are referred to some classic books for well-log-interpretation techniques (Schlumberger, 1972; Hilchie, 1982; Bassiouni, 1994). Using the solid approach, one needs to manually correlate well logs and label distinct hydrofacies for each well log. Once the correlation between well logs is established, interpolation methods are used to generate the surfaces of hydrofacies. These surfaces represent the hydrofacies boundaries and construct a solid model. Jones et al. (2002) and Lemon and Jones (2003) developed a grid generator to generate computational grids for MODFLOW (Harbaugh et al., 2000; Harbaugh, 2005) from solid models. The beauty of this approach is the creation of nonuniform computational layers that match well with the generated surfaces, including pinch-outs. Due to its simplicity, the algorithm was adopted in several commercial software packages, for example, GMS (Aquaveo, 2014) and RockWorks (RockWare, 2010). The biggest challenge in using the solid model approach is to perform manual correlation between well logs, which is subjective and can become laborious and impractical when dealing with a vast amount of well logs in areas known to be highly complex (e.g., fluvial depositional environments). Manually correlating well logs often results in inconsistency with geological deposition, forces correlation of unrelated hydrofacies, and produces erroneous hydraulic connections of discontinuous hydrofacies. The predefined grid approach usually generates uniform, relatively coarse layers directly to be used for groundwater modeling. Examples of this approach include using T-PROGS (Carle, 1999) and geostatistical tools [e.g., GSLIB (Deutsch and Journel, 1997)]. This approach does not force generating surfaces of hydrofacies, and therefore, avoids the issues caused by manual correlation. Since geostatistical approaches have been well received in the literature, they are commonly adopted by commercial software, for example, T-PROGS with GMS (Jones et al., 2005; Faulkner et al., 2012; Aquaveo, 2014). However, the greatest concern of using predefined grids is of losing the vertical resolution of hydrofacies geometries if layers are not fine enough. Using very fine layers can improve vertical resolution to better capture vertical hydrofacies geometry, but will significantly increase computation time, since predefined grids are directly used for the groundwater modeling purpose. This chapter presents a grid-generation technique that maintains high vertical resolution of hydrofacies geometries with a reasonable number of nonuniform boundary-fitted layers. This is done by vertically upscaling a very-fine hydrostratigraphic architecture into a computational grid for groundwater modeling. The method avoids correlation error by using manual correlation and avoids gridded error by using a predefined grid. The grid-generation technique is able to facilitate simulation model development and reduce technological uncertainty by constructing geologically valid computational grids. While this chapter presents computational grid generation for MODFLOW, the technique is not restricted to any numerical models.

48_Singh_ch48_p48.1-48.8.indd 2

48.3.1  Well Log Interpretation

The primary sources of information used to establish hydrofacies geometries are wire-line spontaneous potential (SP) and electrical resistivity logs for boreholes. Spontaneous potential and resistivity log responses are controlled largely by the ratio of sand to clay minerals. They have long been used to interpret sedimentary depositional environments. Galloway (1977) used SP and resistivity curve morphologies to identify fluvial facies for channel fill, levee, crevasse splay and floodplain, and established a meandering stream facies. Kerr and Jirik (1990) adapted Galloway’s (1977) facies model and provided examples of SP and resistivity responses that match known fluvial facies for the middle Frio formation, South Texas. Sands deposited by braided streams produce jagged, wedge-shaped curve morphologies (Miall, 2010). Based on these established relationships between log responses and fluvial facies, Chamberlain et al. (2013) used SP and resistivity data to study depositional environments of siliciclastic sediments in the Baton Rouge area, Louisiana. Following Chamberlain et al. (2013), SP, resistivity, and gamma ray (when available) are used to identify the location of sand facies at depth. Figure 48.2a shows a typical SP-resistivity log from fluvial depositional environments. Based on deviations from a visually estimated shale baseline, boundaries of sands can be drawn on inflection points of SP curves. A cut-off value generally fell between 10 and 35 ohm-m for resistivity curves is assigned to determine boundaries of sands. Low long-normal resistivity generally indicates the occurrence of salty water. Low gamma ray response generally indicates a sand facies. Sand boundaries can be well identified by correlating SP, resistivity, and gamma ray curves (Schlumberger, 1972; Hilchie, 1982; Bassiouni, 1994). For example, four sand facies are picked and many thin sands are ignored as shown in Fig. 48.2a. Nonsand intervals are assumed clay (shale or mudstone) facies. Well log interpretation is inevitably subjected to an individual’s experience and the purpose of the work. This chapter does not intend to discuss the uncertainty of computational grids due to different log interpretations. Moreover, it is possible to use the established relationships between log responses and fluvial facies from Galloway (1977), Kerr and Jirik (1990), and Miall (2010) to infer different fluvial facies. For example, Fig. 48.2a shows some identified fluvial facies based on the established relationships. However, identifying specific fluvial facies is not the scope of this chapter. Instead, this chapter focuses on sand and clay facies identification in well logs. 48.3.2  Indicator Kriging

The study constructs hydrostratigraphic architecture of sand and clay facies using information from well logs. Well log data are first transformed into binary indicator values. The indicator value for sand facies is 1 and for clay facies is 0. To honor a regional geological dip as shown in Fig. 48.2b, indicator kriging needs to be performed on inclined surfaces where indicator data are obtained at the intersections with boreholes. To make it easier for operating indicator kriging, all boreholes are translated vertically to a nondipped domain. To do so, the vertical translation distance depends on the dip angle and the distance from well log location to a strike that serves as a pivot as illustrated in Fig. 48.2c. Then, indicator kriging is performed on horizontal surfaces given a two-dimensional grid. The two-dimensional grid can have irregular boundaries and its cells can be generated by any methods available in the literature. This chapter focuses on generating computational grids to be used by MODFLOW. The same discretization is applied to all horizontal surfaces at different depths. Then, a detailed three-dimensional hydrostratigraphic architecture can be

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Case study: Groundwater modeling in Baton Rouge, southeastern Louisiana     48-3  Depth (ft) SP (mv) Resistivity (ohm-m)

1100

–150–100–50

1300 B

1700

100

Imaginary horizontal plane Clay

Dip angle

Sand

Clay

Clay

1500 1600

10

Sand

1200

1400

1

Clay

A Shale baseline

1000

S

C

E

trike

Sand

D

Clay

Inclined surface

1800 Sand

Clay

Dip d

irectio

n

Sand Sand

Clay Sand

Clay Clay

Clay

Horizontal surface

1900

(a)

(b)

(c)

Figure 48.2  (a) Interpretation of fluvial facies for a well log: A—amalgamated braided channel-fill with brackish water, B—channel-fill point bar sand with brackish water, C— stacked/amalgamated channel-fill with very salty water, D­—floodplain, and E—natural levee (Kerr and Jirik, 1990; Miall, 2010); and translation of borehole positions from (b) dipped domain to (c) nondipped domain.

achieved by assembling a large number of horizontal surfaces with a fine interval. Indicator kriging is adopted to derive sand-clay distribution in horizontal surfaces of one-foot intervals. The resulting indicator data from horizontal surfaces are used to compute experimental variograms. Then, a variogram model can be derived by fitting to the experimental variograms. It is noted that the grid generation technique is not limited to indicator kriging. Any geostatistical method can be used to estimate hydrofacies for a surface. The expected value of an indicator function at an unobserved location is obtained by

S

v( x 0 ) = ∑λi I ( x i ), (48.2) i =1

where v( x 0 ) is the expected value at unobserved location x 0 , S is the number of boreholes for a horizontal surface, and λi are the indicator kriging weights. Indicator kriging has been well documented in the literature. Readers are referred to Olea (1999) for more information. The expected value of indicator function represents the probability that facies at a location x fall into sand facies or clay facies. By giving a cut-off a as follows, distributed sand and clay facies on a horizontal surface can be achieved:

 1  : sand if  v ( x ) ≥  α   I (x) =   0  : clay if   v ( x ) <  α

(48.3)

Determination of a defensible cut-off value is challenging. A value of 0.5 is commonly used for a neutral selection. However, a better cutoff can be determined in a calibration process where facies estimates are subject to additional information, for example, driller’s logs, total volume of sand or clay facies from electrical logs, etc. (Elshall et al., 2013). Once a hydrostratigraphic architecture with very fine vertical discretization is generated by indicator kriging, the hydrostratigraphic architecture is upscaled to a MODFLOW grid by merging the same hydrofacies in the vertical direction to reduce the number of layers. 48.4  MODEL CALIBRATION

The use of optimization algorithms for model calibration in groundwater model is a common practice. The optimization algorithms include local derivative algorithms, global heuristic algorithms, and hybrid global-local algorithms. Local derivative algorithms are computationally efficient, but at the cost of trapping at local solutions. Global heuristic algorithms are generally time consuming, but are able to escape local optima and improve solutions through a simple to complex forms of learning, such as genetic algorithm (Harrouni et al., 1996; Solomatine et al., 1999; Karpouzos et al., 2001; Bastani et al., 2010) and particle swarm optimization (Scheerlinck et al., 2009; Jiang et al., 2010). Hybrid global-heuristic local-derivative algorithms (Tsai et al., 2003; Blasone et al., 2007; Matott and Rabideau, 2008; Zhang et al.,

48_Singh_ch48_p48.1-48.8.indd 3

2009) run a global heuristic algorithm for exploring the search landscape followed by a local derivative algorithm for exploiting favorable search regions. We adopt the CMA-ES (Hansen and Ostermeier, 2001; Hansen et al., 2003) as a global-local stochastic derivative-free algorithm for groundwater model calibration. The important features in the CMA-ES are global-local search capability, parallelization, and uncertainty quantification for estimated parameters, which are especially great for calibrating computationally demanding groundwater flow models (Elshall et al., 2015). The enhanced search properties of the CMA-ES stems from its complex learning techniques with high level of abstract description. The CMA-ES adapts a covariance matrix representing the pair-wise dependency between decision variables, which approximates the inverse of the Hessian matrix up to a certain factor. The solution is updated with the covariance matrix and an adaptable step size, which are adapted by two conjugates that implement heuristic control terms. The covariance matrix adaptation uses information from the current population and from the previous search path. CMA-ES utilizes multiple solutions in iteration that do not exchange information and allows for embarrassingly parallel computation. This is the most efficient parallel technique since the solutions in iteration do not communicate. The parallel CMA-ES superiorly improves the calibration speed over the sequential CMA-ES. In addition, the speedup of parallel runs scales variably with increasing the number of processors up to a certain limit. In addition to the global-local search capabilities and parallelization, another favorable feature of CMA-ES is to quantify model parameter uncertainty due to estimation error. The solutions of the CMA-ES, which consist of a maximum likelihood estimate and a full covariance matrix, can be used for Monte Carlo sampling. Several algorithms have utilized the covariance matrix for Monte Carlo sampling (Haario et al., 1999; Haario et al., 2001; Kavetski et al., 2006; Smith and Marshall, 2008; Cui et al., 2011; Zhang and Sutton, 2011). As pointed out by Müller and Sbalzarini (2010) and Mueller (2010), the CMA-ES shares many common concepts and features with the derivative-free Markov chain Monte Carlo (MCMC) sampling algorithms (Haario et al., 1999; Haario et al., 2001; Haario et al., 2006; Andrieu and Thoms, 2008; Müller and Sbalzarini, 2010). Elshall et al. (2015) showed that the adapted covariance matrix of the maximum-likelihood estimates is precise and can be used for Monte Carlo sampling. Their work is one of a few studies that used CMA-ES to quantify model parameter uncertainty. 48.5  CASE STUDY: GROUNDWATER MODELING IN BATON ROUGE, SOUTHEASTERN LOUISIANA

This section illustrates the implementation of the flowchart of the groundwater model development by using a case study in the Baton Rouge area, southeastern Louisiana. Figure 48.3 shows the Baton Rouge area and the groundwater modeling domain. The Baton Rouge aquifer system consists of a succession of south-dipping siliciclastic sand units and mudstones of Upper Miocene through Pleistocene age, is extended to a depth of 3000 ft (914 m), and is highly complex as a result of fluvial deposition (Chamberlain et al., 2013). Sand deposition is nonuniform due to spatial and temporal variations

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48-4    Groundwater Modeling

Figure 48.3  Map of the Baton Rouge area and borehole locations. The box in the middle is the groundwater modeling area.

in fluvial processes as well as large amount of missing sand possibly due to erosional unconformity. From the region-scale study (Griffith, 2003), the Baton Rouge aquifers dip south towards the Gulf of Mexico. The east–west trending Baton Rouge fault and Denham Springs–Scotlandville fault crosscut the aquifer/aquitard sequence in the study area (McCulloh and Heinrich, 2013). Bense and Person (2006) suggested that the Baton Rouge fault is a lowpermeability fault and acts as a horizontal flow barrier (HFB) that separates the freshwater to the north from the saline water to the south. 48.5.1  Hydrostratigraphic Architecture

We analyzed wireline well logs from 583 boreholes shown in Fig. 48.3. The well log data were used to construct a hydrostratigraphic architecture of the Baton Rouge aquifer system, which includes the Mississippi River Alluvial Aquifer (MRAA), the “400-ft” sand, the “600-ft” sand, the “800-ft” sand, the “1000-ft” sand, the “1200-ft” sand, the “1500-ft” sand, the “1700-ft” sand, the “2000-ft” sand, the “2400-ft” sand, and the “2800-ft” sand. These sand units were classified and named by their approximate depth below ground level in the Baton Rouge Industrial District (Meyer and Turcan, 1955). The indicator kriging was adopted to construct a hydrostratigraphic architecture with a dip angle of 0.29° and a cutoff value of 0.40 (Elshall et al., 2013). The resulted hydrostratigraphic architecture is illustrated in Fig. 48.4. The model domain in the planar direction is discretized into 93 rows and 137 columns with a cell size 200 m by 200 m. The hydrostratigraphy shows the complexity of the fluvial depositional environment in the Baton Rouge area. There are 166 pumping wells in the model area. The majority of the pumping wells in the Industrial District were screened at the “1200-ft” sand and the “2000-ft” sand. Public supply wells area scattered, except for the cluster of Lula wells that were screened at the “1500-ft” sand. Saltwater intrusion has been a serious issue in the “1500-ft” sand and the “2000-ft” sand (Tomaszewski, 1996). The connector well (EB-1293) was built in 1999 to raise hydraulic head in the “1500-ft” sand by injecting groundwater of the “800-ft” sand through pressure difference. A newly scavenger well (EB-1424) was built in 2014 to extract salt water out of the “1500-ft” sand. Evaluation of the effectiveness of such groundwater remediation approaches requires a valid geological model and later a valid computational grid. The validity of the Baton Rouge geological model was verified by the up-to-date well log data and the depths of the screens of the pumping wells as shown in Fig. 48.4. 48.5.2  Groundwater Modeling

The constructed hydrostratigraphic architecture from the previous section is used to generate a MODFLOW grid. The interested sands for groundwater modeling are the “1200-ft” sand, the “1500-ft” sand, the “1700-ft” sand, and the “2000-ft” sand. The model domain and the horizontal discretization are

48_Singh_ch48_p48.1-48.8.indd 4

kept the same as in the hydrostratigraphic architecture. Using the upscaling approach, a MODFLOW grid of 968,316 cells given by 93 rows, 137 columns, and 76 layers accurately matches the complex hydrostratigraphic architecture and preserves layer continuity. Each cell is 200 m by 200 m with cell thickness ranges from 3.05 to 13.4 m. The average thickness of the layers is 5.2 m. The “1200-ft,” the “1500-ft,” and the “1700-ft” sands are from layer 6 to layer 46. The “2000-ft” sand is from layer 47 to layer 76. The hydrostratigraphic architecture shows the “1200-ft” sand, the “1500-ft” sand, and the “1700-ft” sand between the two faults are interconnected. Therefore, these sands were modeled together. The “2000-ft” sand is separated from the “1500-ft” sand and “1700-ft” sand by a thick clay layer and was simulated separately. Monthly pumping data are available from the Louisiana Capital Area Ground Water Conservation Commission (CAGWCC) from 1975 to 2010. The “1200-1500-1700-ft” sands model has 87 pumping wells. The average pumping rate of these wells from 1975 to 2010 was 112,556 m3/day (29.73 million gallons per day, MGD). The connector well (EB-1293) was injecting about 2600 m3/day (0.69 MGD) of groundwater from the “800-ft” sand to the “1500-ft” sand. The “2000-ft” sand model has 29 pumping wells extracting about 78,457 m3/day (20.73 MGD) in December 2010. The initial head on January 1, 1975, and the time-varied head boundary condition were estimated through extrapolation of the nearby head observation data. The “1500-ft” sand and the “1700-ft” sand are considered to have the same hydrogeological parameter values, which are different from the “1200-ft” sand. The two faults are considered as HFBs and their permeability values are characterized by the hydraulic characteristics (HCs) (Hsieh and Freckleton, 1993). The “1200-ft” sand and the combination of the “1500-ft” and the “1700-ft” sands have four homogeneous model parameters (hydraulic conductivity, specific storage, and two HCs for the two faults) to be estimated. Thus, the “1200-1500-1700-ft” sands model has eight unknown model parameters. In addition to the four homogeneous model parameters, the “2000-ft” sand model considers a vertical anisotropy ratio of hydraulic conductivity and a boundary head adjustment factor for the eastern boundary between the two faults to be estimated. The “2000-ft” sand model has six unknown model parameters. The MODFLOW Well (WEL) package is used to take into account the pumping rates of the pumping wells and the injection rate of the connector well. The Time-Variant Specified-Head (CHD) package is used to simulate the time-dependent boundary condition, which is assigned to all active cells at the model boundaries. Inactive cells represent the clay unit. The Baton Rouge fault and the Denham Springs–Scotlandville fault are represented as HFBs using the Horizontal Flow Barrier package. The model calibration period is from January 1, 1975, to January 1, 2010, 432 monthly stress periods. The time step is 1 month.

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Summary    48-5 

Lula wells

Industrial district wells

Scavenger well (EB-1424) Connector well (EB-1293)

Mississippi river 0.0

Elevation (x103) (ft)

–0.5 –1.0 –1.5 –2.0 –2.5

3.380

3.378

3.376

3.374 3

.372 3.3 70

UTM_Y

(x10 6)

3.368 3

(m)

.366

3.364 3

.362

665

670

680

675 UTM

3)

10 _Y (x

685

690

(m)

Figure 48.4  Constructed architecture of the Baton Rouge aquifer system. Cells underneath land surface are sand. Clay is blanked. Vertical white columns are pumping wells with screens at the bottom.

48.5.3  Model Calibration Using Parallel CMA-ES

To calibrate a groundwater model and estimate model parameters, the CMAES (Hansen and Ostermeier, 2001; Hansen et al., 2003) was adopted to minimize the root mean square error (RMSE) between the calculated and observed groundwater heads. The execution time for a single model simulation is around 1.89 ± 0.10 h for the “1200-1500-1700-ft” sands model and 1.28 ± 0.1 h for the “2000-ft” sand model by a personal computer equipped with a Pentium Dual Core CPU 3.0 GHz and 2GB RAM. It is impractical to conduct model calibration in a single processor. For example, using a population size of 80 and 40 iterations in CMA-ES, the computation time would be about 252 days for calibrating the “1200-1500-1700-ft” sands model. To accelerate the model calibration process, a parallel CMA-ES (Elshall et al., 2015) was implemented to SuperMike-II, a supercomputer at Louisiana State University. To maximize the efficiency of the parallel implementation, the number of core-based processors used is the same as the population size λ. For the best performance, Hansen and Ostermeier (2001) and Hansen et al. (2003) recommended the optimal population size 4 + [ 3ln(n)] ≤ λ ≤ 10n, where n is the number of unknown parameters. Thus, the number of processors for the “1200-1500-1700-ft” sands model is 80 ( λ = 80 = 10n) and for the “2000-ft” sand model is 64 ( λ = 64 ≈ 10.67n ). For both groundwater models, the optimal population size is at the upper limit λ = 10n. Given that the communication overhead is less than 1 s per iteration, the speedup of the parallel CMA-ES is roughly the population size λ.

station is the center of the cone of depression in the “1500-ft” sand. Heavy pumping has caused significant groundwater level decline north of the Baton Rouge fault and has caused saline groundwater flowing northward across the Baton Rouge fault toward the Lula pumping station and the Industrial District. 48.5.5  Uncertainty Quantification

Monte Carlo simulation was adopted to quantify groundwater head uncertainty due to model parameter estimation uncertainty. Parameter realizations were generated using the mean and the covariance matrix of the estimated model parameters obtained by the CMA-ES after model calibration was completed. For the illustration purpose, the study analyzed the simulated groundwater head at a USGS observation well EB-146 in December 14, 2010. EB-146 was screened at the “1200-ft” sand, close to the Government St. pumping station. Figure 48.6a shows the convergence of the RMSE using CMA-ES with the population size of 80. The RMSE was not improved after 60 iterations; yet the calibration process was not terminated to ensure convergence of the covariance matrix. The standard deviation of simulated groundwater head was calculated based on 290 realizations at 10, 30, 50, and so forth up to 170 iterations as shown in Fig. 48.6a. The result indicates that the covariance matrix was converged within 170 iterations. Figure 48.6b shows the mean simulated head and its one-standard deviation bounds at EB-146 for the last iteration. After reaching the optimal solution, the head variance quickly converged in less than 150 realizations.

48.5.4  Model Calibration Results

48.6 SUMMARY

The “1200-1500-1700-ft” sands model was calibrated using 2805 groundwater head data from 20 USGS observation wells from January 1975 to December 2010. The “2000-ft” sand model was calibrated using 1285 head data from 18 USGS observation wells for the same time period. The model calibration results show a good matching between the observed and the simulated groundwater heads. The RMSEs are 1.44 and 2.95 m for the “1200-1500-1700-ft” sands model and the “2000-ft” sand model, respectively. These RMSEs are acquired after 59 and 40 iterations for the “1200-1500-1700-ft” sands model and the “2000-ft” sand model, respectively. The good fitting results in groundwater head data although using homogeneous model parameters are because a valid geological model was built and correctly converted to a MODFLOW grid, without which complex heterogeneous model parameter fields may be needed. Table 48.1 presents the estimated parameter values for all sands. The Baton Rouge fault and the Denham Springs–Scotlandville fault are found to be lowpermeability faults that restrict horizontal flow. The “2000-ft” sand has hydraulic conductivity and specific storage higher than other three sands. Figure 48.5 presents the groundwater head distributions at the end of the model calibration period, December 31, 2010, using the estimated parameters in Table 48.1. The results show that the Industrial District is the center of the cone of depression in the “1200-ft” sand and the “2000-ft” sand and the Lula pumping

This chapter presents a general procedure for groundwater model development with a focus on grid generation and model calibration, which are two crucial modeling steps in order to reduce model structure uncertainty and to improve model parameter estimation. To do so, a better grid generation technique is needed in order to create a valid computational grid supported by geological data. Second, a better optimization algorithm is needed for efficient model calibration and parameter uncertainty estimation. In this chapter, we present a grid generation technique using indicator kriging to work on a vast amount of well log data. The technique can preserve facies geometries of a complex hydrostratigraphic architecture by using fine vertical discretization and regional geological dip. In addition, the technique avoids a possible overwhelming number of computational cells by using an upscaling approach. This chapter introduces a parallel CMA-ES as an efficient optimization method for model calibration. The important features in the CMA-ES include global-local search capability, parallelization, and uncertainty quantification for estimated parameters, which are especially great for calibrating computationally demanding groundwater flow models. The procedure for groundwater model development was illustrated by the groundwater modeling study in Baton Rouge, southeastern Louisiana. A

48_Singh_ch48_p48.1-48.8.indd 5

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Figure 48.5  Simulated groundwater head distributions on December 31, 2010 for (a) “1,200-ft” sand, (b) “1,500-ft” sand and “1,700-ft” sand, and (c) “2,000-ft” sand.

48_Singh_ch48_p48.1-48.8.indd 6

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References    48-7 

Figure 48.6  Convergence profiles for (a) RMSE (meters) and standard deviation (meters) of simulated groundwater heads at EB-146, and (b) simulated groundwater head and its one-standard deviation bounds at EB-146.

Table 48.1  Estimated Model Parameter Values for Different Sands Parameter

“1200-ft” sand

“1500-ft” and “1700-ft” sands

“2000-ft” sand

Hydraulic conductivity (m/d)

23.13

25.64

144.86

Specific storage (1/m)

5.27×10–6

2.82×10–6

1.86×10–5

Vertical anisotropy ratio

1

1

1

HC of the Baton Rouge fault  (1/d)

2.64×10–3

2.48×10–4

4.20×10–3

HC of the Denham Springs– Scotlandville fault (1/d)

6.08×10–3

5.00×10–2

1.34×10–6

Boundary condition adjustment factor (m)

–

–

1.36

significant amount of effort was spent on interpreting a great number of well logs in order to reconstruct a valid geological model before groundwater models were developed. The grid generation technique successfully reveals the complexity of the fluvial depositional environment in the Baton Rouge area including two geological faults. The validity of the Baton Rouge geological model was verified by the up-to-date well log data and the depths of the screens of the pumping wells. With a focus on the “1200-ft” sand, “1500-ft” sand, “1700-ft” sand, and “2000-ft” sand groundwater modeling, this chapter demonstrated the advantages of using parallel CMA-ES for not only model calibration, but also parameter uncertainty quantification. Using parallel computing, we were able to make model calibration time tractable for our computationally heavy groundwater models. The covariance matrix provided by the CMA-ES enables us to quantify groundwater head uncertainty through Monte Carlo realizations of model parameters. Finally, we iterate the importance of developing a valid computational grid to reduce systematical uncertainty in groundwater modeling caused by a lack of advanced technology to translate geophysical data into valid groundwater models. As demonstrated in the case study, highly parameterized groundwater models are not needed and model calibration can be simplified as long as the groundwater model structure is supported by geological data. ACKNOWLEDGMENTS

The study was supported in part by the LSU Economic Development Assistantship (EDA), the National Science Foundation under Grant No. EAR1045064, and the United States Geological Survey under Grant/Cooperative Agreement No. G10AP00136. LSU High Performance Computing is acknowledged for providing access to supercomputers. REFERENCES

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Chapter

49

Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes BY

FRED L. OGDEN AND GLENN S. WARNER

ABSTRACT

This chapter is intended to educate the reader regarding processes that produce flowing water on the surface of the earth. It emphasizes drivers that influence these processes, namely: climate, weather, groundwater, and includes some discussions on the mathematical methods used to predict runoff generation. The relative importance of physical system properties such as topography, land-cover, and soils is presented in the context of the different models of runoff generation, and evidence from the literature is used to draw conclusions on those important factors. Some classic examples from the literature are shown to illustrate longstanding unsolved problems in runoff generation. The chapter concludes with a discussion of those poorly understood aspects of runoff generation that are ripe for further study. 49.1  INTRODUCTION

This chapter deals with surface water, not groundwater. Surface water is special. Any water can be groundwater (GW) as the downward pull of gravity is omnipresent, and porous media are ubiquitous on the Earths’ surface. Something special must happen for water to become surface water. Either it must be returned to the surface from GW, or something must have prevented this water from becoming GW once it fell as rain, or melted from soil ice, hail, graupel, snow, or ice. The flow path can affect the geochemical makeup and ratios of light stable isotopes in surface water, which can serve as a diagnostic. Surface water is inherently more useful than GW because it is more easily accessed and quantified, and little to no pumping is required to use it. It is also inherently more dangerous because of the hazards posed by extreme hydrologic events, such as floods and droughts, which lead to dangerous excesses or shortages and associated negative socioeconomic consequences. Runoff is loosely defined as water from rain or snow melt that flows over the surface and into streams. In this chapter, we strictly define runoff as the volume of water (L3) that flows from a given area (L2). Therefore, runoff has the units of depth of water (L). The rate of runoff has units of (L T–1). Runoff is a convenient metric because it allows comparison of hydrological response per unit area. Engineers and hydrologists predict the amount or rate of runoff in response to real or hypothetical water input events. We use the generic term “water input” because water can be introduced from a variety of sources such as: rainfall, melting snow and ice, thaw of frozen soils, and irrigation. Hydrologic design requires the peak runoff rate for a given flood probability for sizing structures or establishing flood plain limits. Increasingly, predictions of runoff over a season or longer are required, including the changes in the timing and shape of the hydrographs for assessments of water availability, in-stream flows for aquatic habitat or for water rights considerations. In the larger context, the varied factors that affect runoff generation often depend on human activities as expressed by land use, water use, and by b ­ iological factors, the study of

which is known as ecohydrology. Although ­statistical estimation of peak runoff can be straightforward given data from stream gauging stations with a substantial period of record, prediction of the variation of runoff, and water quality over time as a function of land use changes, biological factors, and climate change is much more complex. Difficulties arise because these changes introduce nonstationarities that invalidate common statistical approaches, and predictions require an i­n-depth understanding of dynamic processes, ecological connections, and feedback involved in runoff generation. Hence, the emphasis in this chapter is on understanding fundamental processes that cause runoff to occur, not just being able to quantify peak runoff rate. Given the strong links between climate and floods and droughts, we can expect water resources to change with climate changes, particularly with increasing human population and associated demands for fresh water (Huntington, 2006; Vörösmarty et al., 2013). Furthermore, human-driven land use changes can produce significant impacts on runoff generation and the hydrological cycle. Often, land use, population, and climate change effects are confounded. It is difficult to separate the natural influence of climate variation from the effects of land use and water-use changes, particularly against the backdrop of anthropogenic climate change that is occurring (United States National Academy of Sciences, 2014). The accuracy of water-quality predictions with changing inputs of both water and chemical or biological agents depends on the physical processes and flow path that water follows to the point of interest. Accurate predictions of the amount and timing of flow, as well as flow path, are prerequisite to identification of sources, concentrations, and the ultimate fate of chemicals, sediment, and biological materials in watersheds. Predictions of runoff in response to precipitation and other forcings, such as evapotranspiration (ET) in the context of varying pre-event or antecedent conditions, requires understanding of mechanisms/processes within the system that are generating runoff and streamflow over a range of spatial and temporal scales. Our understanding of the important factors that affect runoff generation is limited (Beven and Germann, 2013). There are factors that we know are important, but we do not understand their effects on the physical mechanisms that control runoff generation to include them in a quantitative way in our models without introducing a large amount of uncertainty into the models as well. This is one reason why some claim that parsimony or stinginess in model parameters is desirable. However, the model complexity and hence the number of model parameters depends on the question being asked and parsimony should not interfere with predictive ability afforded by additional process descriptions or parameters when possible. Some seemingly parsimonious models have been demonstrated as inadequate in certain circumstances and there are important differences between conceptual and physics-based models (Schoups et al., 2008). 49.1.1  Runoff Ratio

Surface water results from the process of runoff generation. The primary objective of this chapter is to describe the factors that cause runoff, which 49-1

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49-2     Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes

include a number of theories that are complicated by poorly understood ­factors known to be important, but that do not have determinant m ­ athematical descriptions. This is one of the reasons why the study of runoff generation processes and mathematical process descriptions is an active and compelling area of research. Runoff generation is a nonlinear function of water input and soil properties, and is often discontinuous in space and time. This makes analytical predictions of runoff difficult except in simple situations, such as from small impervious areas over short time periods. Surface runoff sometimes follows a tortuous path, involving multiple exchanges between surface and GW. Surface flow paths include diffuse overland flow, and concentrated flow in rills, ­gullies, and small streams before reaching a river. One way to quantify runoff generation is the runoff ratio ρ, which represents the fraction of basin-average precipitation P that becomes surface runoff Rs:

ρ=

Rs (49.1) p

Figure 49.1 shows average annual ρ computed using data up to 2013 from the United States. Geological Survey (USGS) Historical Climatological Data Network (HCDN) stream gauges (Slack et al., 1993), for watersheds where the USGS computes annual mean areal precipitation (AMAP). HCDN stream gauge data come from watersheds with minimal human impacts on flows. The ρ values shown in Fig. 49.1 illustrate some very important spatial variations in runoff generation over the continental United States. at the annual time scale. The first observation is the region of low ρ (= 5

Arid

5 > φ >= 2

Semiarid

2 > φ >= 0.75

Subhumid

0.75 > φ >= 0.375

Humid

Source: Ponce et al., 2000.

intermediate ρ values because of high PET. The northwest aspect of the Appalachian mountains in the mid-Atlantic region has values of ρ > 0.6 that extend through Pennsylvania and upstate New York and into New England because of the increasing amount of annual precipitation that falls as snow, and reduced PET. Southeast of the Appalachian mountains, annual ρ values are generally between 0.25 and 0.35 because of high PET and a lower percentage of annual precipitation falling as snow. Watershed area is the dominant physiographic factor affecting runoff behavior (Gupta and Dawdy, 1995). However, the aridity index f (Budyko, 1974), which is defined as the ratio of annual PET to annual precipitation, is another important climate related factor:

φ=

PET (49.2) P

Ponce et al. (2000) provided hydroclimatological definitions (Table 49.1) as a function of the aridity index (f). Figure 49.2 shows a plot of the aridity index (f) vs. drainage area (km2) for the HCDN stations used in this analysis, assuming annual PET is 1400 mm for all watersheds. Also shown on Fig. 49.2 are the aridity index boundaries for different hydroclimatic regimes. Figure 49.2 shows that most of the stations used in the analysis are in the subhumid hydroclimatic regime, and have drainage areas from 10 to 10,000 km2. Drainage areas represented in the data shown in Fig. 49.3 range from 1.5 to 25,600 km2 (Fig. 49.2). The mean and median length of record for these stations are 62.5 and 61 years, respectively, with a standard deviation of 18.5  years. The longest period of record is 156 years, and the shortest is 24 years. Figure 49.3a shows the annual average ρ versus AMAP. Figure 49.3a shows that ρ has a large variance for AMAP < 1600 mm. For AMAP > 1600 mm, ρ values generally exceed 0.5 with few exceptions. Figure 49.3b shows annual precipitation minus annual runoff versus AMAP. The difference between annual precipitation and runoff in a water balance context represents losses to ET or changes in storage. In this analysis, each point in Fig. 49.3 represents the average of the period of record for each gauging station. Because of the length of record at most stations, changes in storage should be nearly zero; and changes in storage

Runoff ratio 0.0−0.1 0.1−0.2 0.2−0.3 0.3−0.4 0.4−0.5 0.5−0.6 0.6−0.7 0.7−0.8 0.8−0.9 0.9−1.0 Figure 49.1  Mean annual runoff ratio for approximately 600 stream gauging stations from the USGS HCDN for which AMAP estimates are available in the continental United States. [Source: USGS.]

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Dominant Runoff Generation Mechanisms    49-3  49.2   DOMINANT RUNOFF GENERATION MECHANISMS

6

Aridity index φ (−)

Arid 4

Semi arid

2

Sub humid Humid

0 100

101

102

103

Watershed area

104

105

(km2)

Figure 49.2  Aridity index versus catchment area for 633 HCDN stations. Data courtesy of USGS. The ρ values and data used to calculate them are plotted in Fig. 49.3 against annual mean areal precipitation (AMAP). Data source: USGS HCDN stations throughout the continental United States, Alaska, Hawaii, and Puerto Rico.

should be small compared to total amounts of rainfall, streamflow, and ET over the long period of record. Therefore, Fig. 49.3b represents actual ET plus losses to deep GW that underflows the site of the stream gauging station. Losses to deep GW have been documented, and Schaller and Fan (2009) called these GW exporting basins. Interbasin GW exchanges identified by Schaller and Fan (2009) were primarily a function of geology, and to a lesser extent, basin size, with smaller basins more likely to gain or lose GW than larger ones. Using data from nested stream gauges on the Contoocook River in New Hampshire, Dingman (2015, p. 440) showed that GW flow either as underflow through the alluvial aquifer at the gauge site or as general recharge that leaves the surface watershed as GW can be significant. Defining underflow as G/Q, where G and Q represent the underflow and measured surface discharge values, respectively, Dingman (2015) reported underflow values of 0.35, 0.18, 0.19, and 0.07 for nested watersheds having drainage areas of 176, 953, 1106, and 1984 km2, respectively. This finding supports the notion that underflow decreases with increasing catchment size. The quantity of actual ET in Fig. 49.3b is tightly constrained to the amount of precipitation for AMAP less than about 1400 mm. Beyond this, actual ET is limited to about 1400 mm, except in the case of a small number of outliers. Note that 1400 mm year−1 corresponds to about 3.8 mm day-1. 1.0

(a)

0.5 Average annual runoff ratio (−)

0.0

(b) Precipitation – runoff (mm)

4000 3000 2000 1000

4000

(c) Mean annual runoff (mm)

3000

0

2000 1000 0

0

1000 2000 3000 4000 Annual mean areal precipitation (mm)

5000

Figure 49.3  (a) Mean annual runoff ratio, (b) excess precipitation defined as mean annual precipitation minus mean annual runoff, and (c) mean annual runoff, all plotted against mean annual rainfall for 635 USGS HCDN stream gauges where mean annual precipitation estimates are available.

49_Singh_ch49_p49.1-49.12.indd 3

This section discusses the mechanisms that determine runoff at the time scale of individual precipitation or water input events. There is a continuum of runoff generation mechanisms depending on limiting values of the parameter i Ks−1, where i is the maximum rain rate (L T−1) that persists for a time and Ks is the saturated hydraulic conductivity (L T−1) of the soil. If this parameter is near 0, it is likely because of consistent low rain rates, such as those experienced in cloud forests, or because of very large values of Ks such as those found in the sand hills of Nebraska, or for many undisturbed, well-vegetated areas. When i Ks−1 is near 0 then subsurface stormflow will dominate the response of a catchment. Whether or not this subsurface stormflow causes saturation at the soil surface and direct runoff will depend upon the ratio of the subsurface capacity to convey water relative to the input flux at a point. One parameter that describes this likelihood is the Topographic Index (Beven and Kirkby, 1979). Subsurface stormflow is one end-member of the runoffgeneration continuum. Saturation excess runoff generation (Dunne and Black, 1970) is a special case of subsurface stormflow, where the capacity of the subsurface to convey water becomes less than the flux at a locus of points along a hillslope. When the cumulative amount of water input exceeds the available soil storage such that all pore space is saturated, the runoffgeneration mechanism is called saturation excess. In some cases when saturation excess runoff-generation occurs, GW can flow back onto the land surface in lower topographic regions, a process called exfiltration or return flow. This water becomes surface runoff at seepage faces and fills surface depressions. At the other end of the runoff-generation continuum are cases where the parameter i Ks–1 > 1 and the infiltration-excess mechanism (Horton, 1933) can occur. Whether or not it does indeed occur depends upon a host of factors including the time series and duration of precipitation, antecedent soil moisture conditions, and other factors such as microtopography, vegetation, etc. In between these two limiting cases of: i Ks–1 ≈ 0 and i Ks–1 > 1, are a variety of other behaviors that complicate evaluation of the runoff generation. Most often, these are a function of soil heterogeneity or layering. We call those factors “profile-related” runoff-generation mechanisms. The following sections discuss the infiltration-excess, saturation-excess, and profile-related runoff generation mechanisms. 49.2.1  Overland and Channel Phases of Runoff Generation

Runoff–generation processes can be divided into “land” and “channel” phases. Runoff in the land phase is conceptualized as diffuse or non-concentrated flow over a plane or hillslope. The channel phase consists of topographically confined or channelized flow. Channelized flow occurs over a large range of scales: from rills, small ephemeral gullies, and small tributaries/first-order streams up to large rivers. Channelized flow is most often measured because the shallow and transient nature of overland flow makes it hard to measure. Direct runoff is defined as runoff of water that never entered the soil or GW domain before reaching a channel. The amount of direct runoff is primarily controlled by the land phase of runoff generation where “losses”, such as interception, infiltration, and evaporation reduce rates of water input and consequently reduce the amount of water that becomes runoff. At larger scales, the channel phase of runoff generation dominates the runoff rate and shape of the hydrograph due to attenuation of the discharge through in-channel diffusive processes, bank storage, and attenuation of flood peaks by overbank flow. Changes in runoff volume within channel flow are usually small compared to the infiltration losses during the land phase, although seepage out of the channel (losing stream) or into the channel (gaining stream) do occur depending on the geology and climate as well as the permeability of nonalluvial aquifers and depth to water table. For the most part, the effects of channel phase processes are better understood compared to land phase processes. The attenuation of discharge with increasing area is shown in Fig. 49.4 (b)–(e) (modified from Dunne and Leopold, 1978). As the drainage area increases in the case of nested watersheds, the lag time to peak increases, and the peak runoff rate decreases. These changes can be attributed to diffusive attenuation as the flood wave moves downstream. The spatial extent of rainfall may have been a factor, but the absolute peak discharges shown on the right hand-side of each hydrograph figure increases with the drainage area, but with a scaling exponent of 0.59. Comparing watersheds of different sizes, scaling exponents less than 1.0 are the rule, and indicate that peak flows do not increase linearly with increasing area. Rather they increase at a rate less than linearly. 49.2.2  Subsurface Stormflow Runoff Generation

Knowledge of flow paths and residence times is essential for predicting the biogeochemical and geomorphological processes and feedbacks in a watershed. Many different pathways exist for water to travel from upland areas to

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Runoff rate (cm h−1)

49-4     Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes 49.2.3  Saturation-Excess Runoff-Generation Mechanism

15 (a)

10

Total rain: 7.13 cm

5 0 0.3

Qpeak 0.37 m3 s−1

(b) 0.52 km2

0.2

0.1

Runoff rate (cm h−1)

Runoff: 1.38 cm

0 0.1 0.05

Runoff: 0.60 cm

0 0.05

(d) 42.9

0 0.05 0

Qpeak 1.92 m3 s−1

(c) 8.3 km2

km2

Runoff: 0.35 cm

(e) 111 km2 0

Runoff: 0.23 cm

5

10

Qpeak 5.35 m3 s−1 Qpeak 8.50 m3 s−1 15

Time (h) Figure 49.4  Changes in runoff amount and timing of nested watersheds in saturation-excess Sleepers River watershed in Vermont. Panel (a) shows 15-minute rainfall. Panels (b) through (e) show hydrographs at different scales. [Source: Modified from Dunne and Leopold (1978).]

a  stream. Figure 49.5 depicts a hillslope with a shallow soil profile in a ­semihumid or humid hydroclimatic zone. During times of high water input and low ET a near-surface water table can exist. Surface water, such as vernal pools, seepage faces, and streamflow, are all in contact with the GW table. Infiltration in upland areas may percolate through the soil to deeper geologic strata where it becomes GW. These deeper strata typically offer more resistance to flow and provide a slower pathway to the stream, which is often the source of baseflow in streams, that is, the persistent flow during periods without water input events to the upland areas.

Rainfall amounts in excess of the available pore space in the presence of a nearsurface water table can cause the water table to rise to the land surface. When this happens, additional rainfall on those sites becomes direct runoff through the saturation-excess runoff-generation mechanism. Saturation excess runoffgeneration is a special case of the subsurface stormflow mechanism where the subsurface does not have the capacity to convey the flow at a point, which causes the water table to rise up to the land surface. For this reason, the saturation-excess mechanism is sometimes called saturation from below. Sklash and Farvolden (1979) and O’Brien (1982) wrote that precipitation on soils near streams can cause a rapid water table rise known as groundwater ridging, which can be considered a special case of saturation-excess runoff generation. Groundwater ridging has been observed in riparian zones where the capillary fringe is near the land surface at the beginning of a water input event. However, this phenomena is less likely to occur in the cases of high permeability soils (Zaltsberg, 1983), soils with preferential flow paths such as macropores (Beven and Germann, 1982; German, 1990; Youngs et al., 1996), forested hillslopes (Buttle and Sami, 1992), and channels incised substantially more than the height of the capillary fringe (Bonnell, 1993). The setting for saturation-excess runoff generation is shown in Fig 49.6. Preferential flow or percolation shown in Fig. 49.6 represents flow through contiguous large diameter pore spaces in the medium called macropores. Macropores in some situations represent the dominant flow path through soils (Nimmo, 2012). Vertical macropore flow occurs in soils when the Bond number Bo, which is defined as the ratio of gravity to capillary forces at the pore scale exceeds approximately 0.05, or when the pore diameter exceeds about 1.2 mm (Or, 2008). Biological actors, such as roots, microbes, micorrhyzea and fungi, and insects and abiotic processes such as freeze/thaw and shrink/swell cycles routinely produce contiguous, interconnected pores or apertures that exceed this dimension. Water can enter macropores through local ponding at the land surface in microtopography that funnels ponded water (Trojan and Linden, 1992) or as pore water that is forced into macropores by positive pressure. Because macropores are so common in soils, (Weiler and Naef, 2003; Bastardie et al. 2005) preferential non-Darcian flow is ubiquitous in undisturbed soils.

49.2.4  Infiltration Excess Mechanism

In soils with reduced infiltration capacities, such as disturbed soils, surface runoff is generated whenever the water input rate from rainfall, snowmelt, irrigation, or runon from upland areas exceeds the infiltration capacity of the soil. This runoff generation mechanism is called infiltration excess. Infiltration excess runoff generation at the hillslope scale is shown in Fig. 49.7. Runoff is produced when the water input rate exceeds the potential infiltration rate fp of the soil. The potential infiltration rate fp is transient, and

Figure 49.5  Hillslope-scale cross-section of potential pathways for water movement from upland areas to streams for a humid or subhumid hydroclimatic regime, with soil profile. Note groundwater contributions to stream flow.

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Dominant Runoff Generation Mechanisms    49-5 

from above is sometimes used to describe this mechanism. A wetting front (Zf in Fig. 49.8) develops and moves downward in the soil as the total amount of infiltration increases as the water input continues. The wetting front in this case can be quite sharp in that above the wetting front the soil is saturated, with a very rapid decrease in water content across the front. The sharpest wetting fronts are observed in coarse-textured soils (Hillel, 1971). The sharp wetting front is the conceptual basis for the development of infiltration equations, such as the Green–Ampt (1911) method. However, in coarser soils, or heterogeneous soils with regions of coarseness, the wetting front can become unstable resulting in fingered flow—a jagged, nonuniform type of wetting front (Parlange and Hill, 1976). 49.2.5  Profile-Related Infiltration Mechanisms

Figure 49.6  Conceptual drawing showing details of the saturation excess runoff generation mechanism. Note that when water table rises to the land surface, infiltration will stop.

as discussed later depends upon the rainfall rate and the slope of the land ­surface. If the water input rate is less than the fp, then all water enters the soil; otherwise, the runoff rate at that point equals the difference between the water input rate and fp. This mechanism is sometimes called Hortonian Runoff after Horton (1933) who was one of the first to document infiltration and runoff from water inputs on agricultural lands. Two forces drive the vertical infiltration process: gravity and capillarity. Capillarity is a very nonlinear function of the water content, going from nearly zero near saturation, to very high values in the case of dry soils. Near saturation, the hydraulic conductivity approaches its maximum value of saturated hydraulic conductivity Ks, and decreases rapidly as the soil becomes less than saturated. Hence, at the beginning of rainfall, capillarity provides most of the drive, while as more and more rainfall infiltrates, hydraulic conductivity limits the infiltration rate. In the battle between capillarity and hydraulic conductivity, the hydraulic conductivity is the most important parameter, because its effect is not only on the capillary driven flow, but also on the gravity driven flow component, whereas capillarity does not affect the gravity driven flow. Rainfall on the nearly impervious rock outcrops shown in Fig. 49.7 is a limiting case of the infiltration excess mechanism where the ratio of the parameter i Ks–1 → • because the saturated hydraulic conductivity of impervious surfaces is 0 or very nearly so. Conceptually, under the infiltration-excess mechanism, water completely saturates the soil as it moves vertically downward hence, the phrase saturation

Most soils are heterogeneous and anisotropic. Therefore, critical soil parameters are seldom uniform either vertically or laterally, even within well-defined layered soils. Two critical parameters are: porosity, that is, the amount of pore space compared to solid particles, and hydraulic conductivity. Both of these parameters are likely to decrease with depth from the surface of the soil under natural conditions because of gravitational forces that cause soil consolidation. This leads to decreasing permeability with depth in the soil. There are several possible factors that can overcome this consolidation effect. Included in these are freeze/thaw, shrinkage cracking upon drying, biological activity or bioturbation, soil creep and downslope motion, etc. Freeze/thaw processes as well as soil shrink/swell cause soil cracking to various depths. While these cracks are open, sand, vegetation, and other detritus fall into them, and create anisotropy that is often conducive to flow in the subsurface. Bioturbation is the process by which biological actors overturn the soil over a long time span, and with decreasing intensity with increasing depth (Kirkby, 1988). Tree fall is a dominant form of bioturbation. Burrowing animals are a dominant factor in temperate and arid regions (Yair, 1995), while in the tropics, ants are a significant actor (Wilkinson et al., 2009). In soils that do not freeze earthworms and ants are responsible for up to 10–15 tonnes per hectare per year of soil movement (Madge, 1965; 1969). Niedzialek and Ogden (2012) identified rapid subsurface flow generating streamflow when the rainfall rate on a 17 ha tropical catchment was at least 40 mm h–1 for 20 min, with no observed surface runoff except for some minor return flow. This despite the fact that the soils in the study catchment were primarily weathered clays. Because of either natural layering, consolidation effects, or bioturbation, the infiltration rate may be limited at some depth in the soil. This can lead to saturated conditions and positive pore pressures developing at that depth, creating a transient perched water table. There are two possible outcomes from this condition. The first is that the perched water table rises to the land surface and results in saturation-excess overland flow. The second condition is that the positive pore pressures activate downslope macropores and soil pipes (Nieber and Warner, 1991; Jones 2010), leading to rapid subsurface storm flow through preferential flow paths. We refer to this condition as a profile-related runoff mechanism. Unlike saturation excess runoff where the water table is created by bedrock or some

Figure 49.7  Infiltration excess runoff-generation mechanism at the hillslope scale in a semiarid region. Note streamflow losses to groundwater. Infiltration excess runoff generation mechanism at the hillslope scale in a semiarid region.

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49-6     Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes

Figure 49.8  Schematic drawing of infiltration excess runoff generation mechanism. Note: Zf is the depth to the wetting front.

other impeding layer, this profile-related mechanism is temporary and can be rain rate dependent. Because significant infiltration capacity remains in the impeding layer, the transient perched water table will dissipate soon after the end of precipitation or water input. This mechanism was suggested by Ogden et al. (2013) as the primary cause of differences in runoff behaviors between two adjacent tropical watersheds with identical geomorphology and soils, but with different land uses. 49.3  INFILTRATION EQUATIONS

Given the importance of infiltration as it affects the runoff generation mechanism, it is perhaps a bit surprising that there is not a widely accepted solution method to calculate infiltration that is generally applicable. Dozen of infiltration calculation methods and several laws have been proposed since 1911. Some of them are empirical equations that allow fitting of a mathematical equation to observed infiltration data. Those methods are not discussed here. The most fundamental developments, in order of their development were: Green and Ampt (1911), Richards (1931), Philip (1957a; 1957b), Parlange et al. (1982), Haverkamp et al. (1990), Celia et al. (1990), and Ogden et al. (2015). The Green–Ampt (1911) method assumed a delta-function soil water diffusivity, homogeneous soil, and uniform initial water content profile. Green and Ampt (1911) derived their method from saturated flow conservation of mass using a Lagrangian description of wetting front dynamics with gravity and capillary-driven flow. However, the Green and Ampt method was put into an applicable formulation by Mein and Larson (1973). The Richards (1931) equation is the standard by which all other methods are compared. It is an infiltration law that is valid assuming that the flow of water through the soil is dominated by capillarity and gravity, in that order. A general numerical solution of the Richards (1931) equation was not published until Celia et al. (1990). However, this numerical solution, while general, is not guaranteed to converge or to conserve mass. The Richards (1931) equation has been criticized as overemphasizing capillarity (Germann, 2010), and for its general computational expense, particularly when simulating sharp fronts (Tocci et al., 1997). Another criticism of the numerical solution of Richards (1931) equation is that its performance in general soil-water dynamics modeling is unpredictable and computationally expensive with widely varied soil constituative relations (Short et al., 1995). Philip (1957a; 1957b) made mathematical advances under specific initial and boundary conditions towards analytical solution of Richards (1931) equation. The true utility of those solutions is limited by those required assumptions. Parlange et al. (1982) derived a three parameter infiltration equation that encompasses both the Green and Ampt (1911) solution, and the Talsma and Parlange (1972) limit of a rapidly increasing soil water diffusivity and unsaturated hydraulic conductivity function. Ogden et al. (2015) derived a novel “finite water-content” infiltration solution method. Compared to the numerical solution of Richards (1931) equation in a numerical test involving 2600 mm of rainfall on a loam soil with a fixed water table at 1 m below the land surface, the difference in cumulative infiltration over the 8 month

49_Singh_ch49_p49.1-49.12.indd 6

simulated period was only 7 mm. The finite water-content method of Ogden et al. (2015) is equivalent to Richards (1931) equation minus a term which is equal to the diffusive flux of capillary head divided by the slope of the wetting front: D(θ)(∂2θ/∂θ 2) (∂θ/∂ θ)–1 (Seo et al., in review). Note that the diffusive flux has a zero mean and therefore does not affect the net flux, and that the slope of the wetting front is infinite for a sharp front, which is common in hydrologically important coarser soil textures, which renders this term unimportant. Beven and Germann (2013) raised important questions on the adequacy of Darcy’s law and Richards (1931) equation in describing flow phenomena in macroporous soils. They suggest that a viscosity based Stokes law approach has merit to describe film flow in pores. The finite water-content solution method (Ogden et al., 2015) is guaranteed to converge and to conserve mass, and it has the potential to simulate non-Darcian flow in a portion of the porous medium. The need to develop methods to characterize the number, size distribution, and location distribution of macropores in soils is apparent. Developing methods to accurately simulate flows through macroporous soils is similarly important. 49.4  FACTORS AFFECTING RUNOFF

We have posited that the dominant type of runoff generation mechanism at a particular point, hillslope, or catchment is a function of a combination of climatological, physiographical, and biological factors. Because of our poor understanding of biological factors and how they affect runoff, we focus here on climatic and physiographic factors. 49.4.1  Climatic, Pedologic, and Vegatative Factors

Climate, pedologic, and vegetative effects on runoff include: (1)  the storm total precipitation Ps, peak rainfall intensity i, and duration tr of rainfall of a given frequency, (2) soil development over many millenia, (3) the variety and density of vegetation, and (4) the magnitude of soil water losses by ET. Furthermore, the largest effect of climate may be on potential infiltration rates. Dunne (1978) showed that a saturation-excess runoff is considerably more likely to occur in a humid climate, while infiltration-excess runoff is more likely in arid regions. This factor can be at least partially explained by the aridity index f. Canopy cover on well-vegetated soils prevents raindrop impact on bare soils with associated surface sealing and loss of infiltration capacity. Arid environments typically have lower plant densities with land surfaces exposed to the rain drop impact and surface sealing by wind-­ deposited clay particles. Some of the highest runoff rates and therefore erosion rates result from Hortonian runoff due to bare or sparse vegetated cover. 49.4.2  Physiographic and Hydraulic Factors

At the hillslope scale, we assume that the dominant factors are a function of geomorphology and soil hydraulic characteristics. In terms of geomorphology, the hillslope length L and plan-view radius of curvature of the elevation contours R, as well as the land-surface slope So are important. In terms of soil

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Poorly Understood Factors Affecting Runoff Generation     49-7 

The factor f3 is the aridity index previously defined (Ponce et al., 2000):

104 Rainfall PDFs Northern Mississippi Central Panama

10−2

Clay

10−1

Silt loam/loess

100

Fine sand

101 Peat

Ks , i (cm h−1)

102

Clay with macropores

Clean sand

Gravel

103

0.0 0.5 P(i)

Figure 49.9  Ranges of saturated hydraulic conductivity values for common soil textures together with rainfall probability density functions from northern Mississippi and central Panama. [Source: adapted from Freeze (1972).]

properties, the thickness D, porosity n, average initial water content θi, and the mean saturated hydraulic conductivity of the soil, Ks, are important. Laboratory analyses performed using packed disturbed columns to measure Ks can demonstrate a wide range of values. When infiltration capacity is measured in situ or using intact columns, Ks and the infiltration rate are much less dependent on soil texture than on recent disturbance and vegetation conditions. An undisturbed, vegetated soil can exhibit much greater infiltrability than a tilled sandy textured soil. Although changes in soil structure play a role, the major differences in Ks are due to the presence or lack of macropores. Figure 49.9 shows ranges of Ks for some typical soil types along with probablity distribution functions for rainfall calculated using data from northern Mississippi (temperate zone) and central Panama (seasonal tropics). Figure 49.9 provides an indication of the likelihood of infiltration excess runoff generation for several soils and for clay with macropores. When coupled with disturbances such as tillage, compaction from human activities, etc., wide ranges of Ks can be expected, even over 2 or 3 orders of magnitude. For example, data from Starr (1990) indicate ranges of Ks from 40 to about 110 mm hr–1, and 10 to about 220 mm hr–1 during a summer season for a field under conservation tillage and plow tillage, respectively. Gupta et al. (2006) found a range in Ks from about 0.1 to over 5 × 10−5 m s−1 in a vertisol soil over a 2-year period. Burwell et al. (2011) showed that as newly seeded Bermuda grass emerged and grew over a 3-month period, runoff rates decreased by a factor of 5 or more during 2 different years.

f3 = f.(49.6) The factor f4 is the ratio of the storm total precipitation ps to the average pore space available to store water, which is calculated from the soil porosity n, average initial water content θi and thickness of the vadose zone D: P f4 = (49.7) (n − θi )D The factor f5 is the slope plan view concavity, where R is the plan-view radius of curvature of the elevation contours, and L is the slope length.: f5 =

R .(49.8) L

The sixth and final factor f6 is the ratio of the average rainfall flux during a precipitation event LP/tr, with tr equal to the rainfall duration, divided by the ability of the soil to convey water downslope KsSoD (Ogden and Watts, 2000): f6 =

LP . (49.9) tr K s So D

Figure 49.10 shows a Bayesian synthesis of runoff generating factors expressed as dimensionless quantities. The distribution of points for f4 and f6 were informed from results in the papers by Ogden and Watts (2000) and Niedzialek and Ogden (2004), respectively. The other distribution functions were estimated based on physical understanding of the system. The abscissa on each plot ranges from 0 to 1 with 0 indicating subsurface stormflow dominated, and 1 indicating infiltration excess dominant. The spread of the data points are indicative of the uncertainty in the effect of a particular dimensionless parameter value on a particular runoff generation mechanism. Note that χ = 0 does not necessarily indicate the occurrence of saturation excess runoff, as it is a special case of subsurface stormflow where the topography and/or soil variability promotes exfiltration and/or the water table intersecting the land surface. Figure 49.11 shows results of a principal component analysis (PCA) of the distributional means of the points shown in Fig. 49.10. This revealed that concavity R L–1 and slope So are colinear but acting in opposite directions, while both are orthogonal to the runoff vector, giving great significance to these two parameters. The parameter i Ks–1 and the transport parameter: L P tr–1 Ks–1 So–1D–1 are equivalent, but are conceptually different. This difference arises from the fact that these two terms operate at time scales. The term iKs–1 operates at the time scale of individual rainfall events while the term LPtr–1Ks–1So–1D–1 is more meaningful at longer time scales. The affects of the aridity index PET P–1 and the ratio of storm total rainfall Ps to soil available storage capacity (n-θi)–1 D–1 are similar, but they also work at very different time scales. These two dimensions of the PCA shown in Fig. 49.11, describe 85% of the variance in the runoff dynamics expectation. This exercise demonstrates the equivalence of similar terms that operate on different time scales, and illustrates the complexity of the multidimensional hydrologic response surface. Given this complexity across scales, we should not expect a lumped one- or two-parameter model to correctly predict runoff, especially when the parameters are static in nature.

49.4.3  Dimensionless Parameters

49.5  POORLY UNDERSTOOD FACTORS AFFECTING RUNOFF GENERATION

From the previously described factors that were identified as important, we can hypothesize some functions that might help describe whether or not a particular hillslope will predominantly produce subsurface stormflow, or infiltration excess (surface) runoff as two end-members of the previously discussed continuum of runoff generation mechanisms, which we call χ. In this analysis, P varies from 0 (subsurface stormflow) to 1 (infiltration excess) mechanisms. We assume the following:

In this section, we discuss factors that are known to be important in runoff generation, including tillage, land use/land cover, soil wetness, slope, and rainfall intensity. Some of these factors have been recognized as important, at least in a qualitative way. However, the actual mechanisms/processes of how they affect runoff generation are poorly understood or have so far escaped quantitative description.

X = f1 f 2 f3 f 4 f5 f6 (49.3) .

Plants and plant residue affect fp in two major ways: (1) they provide protection to the soil surface from raindrop impact and (2) they provide mechanisms for macropore development. The macropore development may be direct, for example, in the form of root channels, or indirect due to the formation of organic matter which in turn promotes soil fauna activity (e.g. earthworms, ants, beetles, etc.) which promotes macropore development. The plant canopy of a well-vegetated soil surface intercepts rainfall and provides protection of the soil from surface sealing. The potential for soil sealing due to raindrop impact is highly associated with the amount and type of clay in the soil. When the surface is exposed, large raindrops having very high kinetic energy will dislodge and disintegrate larger aggregates of

Where the factor f1 is the ratio of peak rainfall rate to the soil saturated hydraulic conductivity (Saghafian et al., 1995): f1 =

i . (49.4) Ks

The factor f2 is the slope So: f 2 = So (49.5) .

49_Singh_ch49_p49.1-49.12.indd 7

49.5.1  Agricultural Effects of Tillage Practices

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49-8     Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes

0

2

4 6 i KS−1

8

10

0

30

Storm total rain/soil storage capy.

Subsurface

0

5

10–2

10

PET P−1

100

102

P (n−

q i)–1

104

D –1

Rain flux/subsurface transp. capy.

Subsurface

Surface

Slope concavity Dominant flow path

10 20 So (percent)

Surface

Aridity index Dominant flow path

Slope

Surface Subsurface

Dominant flow path

Peak rain rate/soil conductivity

10–1

100

101 R

102

L−1

103

0

2

4 −1

L P tr

K−1

6

So

−1

D−1

Figure 49.10  Bayesian posterior distributions of runoff-generation mechanism as affected by dimensionless parameters.

1.0 Ps (n − θi )D

Dim 2 (38.42%)

PET P

So

0.5

−1

Runoff

0.0

i Ks−1 L Ps t r K s So D

−0.5

R L−1 −1.0 −1.0

−0.5

0.0 Dim 1 (46.60%)

0.5

1.0

Figure 49.11  Variables factor map from principal component analysis on posterior distributions shown in Fig. 49.10.

49_Singh_ch49_p49.1-49.12.indd 8

clay ­particles. These small particles then form barriers to water entry to the soil, greatly reducing the fp of the soil. The fp of a surface sealed soil has been shown (e.g., Baumhardt et al., 1990; Assouline and Mualem, 1997; Cerdan et al., 2001) to be an order of magnitude or more lower than for the same soil under vegetative canopy. Agricultural operations may greatly impact fp through tillage, compaction, harvesting of crops and removal/additions of mulches. The impacts of tillage practice may include important seasonal changes, e.g. amount of wintertime infiltration (Dao, 1993). Changes due to tillage may dramatically change the soil surface from a vegetated to bare condition if the operation includes moldboard plowing or similar tillage that create enormous retention storage. Some types of tillage, such as conservation or no-till operations, may leave much of the residue of a previous crop on the surface as a mulch and thereby continue to provide a relatively high fp. Repeated agricultural tillage often results in formation of what is called a “plowpan” due to compaction by repeatedly driving farm implements over the soil, and the fact that farming implements have a limited tillage depth of 12–15 in (30.5–38 cm). The plowpan has a low Ks that creates an artificial barrier to an infiltration wetting front and tends to elevate nongrowing-­season soil moistures (Downer and Ogden, 2003). Figure 49.12 is a photograph taken down a fence line that clearly shows the effect of tillage practice on hydrologic response of farm land. The field on the left side of the fence is farmed using standard tillage practices. The field on the right-hand side of the fence line was farmed using no-till practices for three consecutive years (David Kusel, personal communication, May 20, 2015). This is evident by the presence of vegetation in a portion of the no-till

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Poorly Understood Factors Affecting Runoff Generation    49-9  49.5.4  Effect of Slope on Runoff

field. Notice that there is considerably more runoff from the field that is farmed using standard tillage practices, and runon infiltration as runoff from the field on the left flows onto the no-till field on the right. 49.5.2  Land Use and Land Cover

Land use and land cover impact the potential infiltration rate fp and runoff in two fundamental ways: (1) how important is ET, which has a large affect on the evolution of soil water content between water input events, and (2) what are the timing and magnitude of human activities associated with a particular land use and land cover. Of course, intense development often results in extensive impervious surfaces associated with buildings and roads. In agricultural operations, each crop will have a different time when planting, harvesting, etc. occurs and may result in temporary loss of vegetative cover or soil compaction from machinery. Due to growth stages (e.g., deciduous trees) and seasonal variations in ET by plants owing to growing season, some land covers may result in drier or wetter soil conditions than others at different times of the year. For example, Zhang et al. (2001) have documented that mature forests have higher ET than grasses in semiarid Australia, while Kelliher et al. (1993) found similar ET rates for coniferous forests and grasslands worldwide, despite the significant differences in canopy morphology and leaf area indices. The seasonality of the ET demand coupled with the seasonal patterns of the intensity and amount of precipitation will determine the overall influence of a particular land use. Urban soils are extensively modified by construction and compaction, then by placement of imported topsoils that have nothing to do with the base soils (Smith et al., 2005). This has two confounding effects. First, the hydraulic behavior of the topsoil is very unlike the underlying base soils. Second, the predevelopment soil texture map has very little in common with the actual postdevelopment soils. For this reason, the use of standard soils datasets may lead to large simulation errors, while the incorporation of dual-layer infiltration and specific urban soil hydraulic parameters can lead to significant improvements in process-based modeling (Ogden et al., 2013). 49.5.3  Antecedent Soil Wetness

The initial soil water content at the start of a precipitation event (θi) is one of the most important factors affecting runoff generation (Mishra et al., 2003). High θi can reduce infiltration and enhance infiltration excess runoff generation in finer soil textures. In coarser soil textures, high initial soil water is usually associated with a near surface water table, corresponding with less available pore space, and increased runoff. Under both mechanisms higher initial soil moisture leads to more runoff in less time. Calculation of θi at the start of a water input event using a water budget approach will improve runoff predictions, although verification, especially by position in the landscape, is difficult (Downer and Ogden, 2004). In areas subject to freezing soils, water budget predictions of soil moisture over an entire year are complicated by soil water movement upward into a freezing front.

49_Singh_ch49_p49.1-49.12.indd 9

49.5.5  Effect of Rainfall Intensity on Runoff

The common concept of infiltration excess runoff generation is that once runoff begins fp will decrease and therefore runoff will increase with time as long as the water input rate is higher than fp. In other words, the fp of the soil is not a function of the water input rate after the initiation of runoff. Yet several observers have shown increasing fp with increased rainfall intensities. Nassif and Wilson (1975; Fig. 49.14) show that fp increases as rainfall rates increase, which is contrary to the generally accepted theory that fp can only decrease as time proceeds within a storm after runoff begins. Observations by Bare soil 300

Infiltration rate (mm h−1)

Figure 49.12  Two fallow fields after an Iowa rain storm on May 11, 2002, with 50 mm (2 in) of rainfall in 2 h. Field on the left is cultivated using standard tillage practices, while field on the right is no-till. [Source: David Kusel, Carroll County, Iowa.]

Most textbooks have some statement to the effect of “the steeper the slope, the more runoff.” This axiom has been readily accepted by most lay people as well as professional hydrologists. Yet is it true? The answer depends on the runoff mechanism. When considering the saturation-excess runoff mechanism, Dunne (1978) and Dunne and Leopold (1978) showed that this mechanism favors areas of low slope. Conversely, they showed that saturation-excess runoff seldom occurs on steep slopes, but that subsurface stormflow occurs quite rapidly on steep slopes. In the case of saturation-excess runoff production, the axiom is NOT true. In the case of infiltration-excess runoff, Nassif and Wilson (1975), Fox et al. (1997), and Huang et al. (2013) performed rainfall simulator experiments on different slopes. Figure 49.13 shows up to a 24% decrease in infiltration capacity with increasing slope. This result supports the previously stated maxim. The results are based on a laboratory study for bare and grass covered “standard” soil that is classified as a loam, based on soil properties provided. Nassif and Wilson (1975) reported little effect of slope on infiltration into a peat or clayey sand soils that had much lower permeabilities than the loam soil. A thorough explanation was not included for the increase in runoff with increasing slope. Fox et al. (1997) found a declining rate of infiltration with slope, up to a slope of about 12%, after which there was no effect. They also measured surface storage volume (depression storage) and found that it had a very similar shaped response curve with slope as infiltration. Huang et al. (2013) found a slight initial decrease in runoff with slope, but then a significant increase in runoff at larger slopes. They attributed the initial decrease to possible faster and larger surface crust formation on the bare soil for the smallest slope. Both Nassif and Wilson (1975) and Huang et al. (2013) found the biggest factor was vegetative cover in predicting runoff rates. Laboratory studies do not necessarily represent the processes in the real world. First, they use packed trays of sifted soil in which soil structure and macropores have been destroyed; and second, they ignore the field/hillslope scale processes that involve runon, subsurface flow, and natural surface microtopography. Some field studies have assessed runoff with slope, for example, Ribolzi et al. (2011) who found in a study on very steep slopes (30 vs. 75%) in Laos, that the steeper slope produced less surface runoff. They attributed this to the formation of microterraces on the steeper slope that were more pervious than the tilled soil on the 1 m2 plots. The possible role of subsurface flow and differences with slope on the small plots was not ­mentioned.

Soil with grass cover 300

(a)

0% 8% 16% 24% 36%

200

200

100

0

(b)

Slope So

100

0

5

10

Time (min)

15

0

0

5

10

15

Time (min)

Figure 49.13  Effect of land surface slope on infiltration rate for rainfall simulator tests on soil pans (a) without grass cover; and (b) with grass cover. Legend is the same for both figures. [Source: Nassif and Wilson,1975.]

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49-10    Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes

300 Water input rate (mm h−1) 312 234 156 78

200

200

100

0

(b)

100

0

5

10

15

0

0

Time (min)

5

10

15

Time (min)

Figure 49.14  Infiltration rates for different rainfall rates for (a) bare and, (b) grass covered loam soils in laboratory study. Slope = 16%. Legend is the same for both figures. [Source: Nassif and wilson, 1975.]

Yu et al. (1997; 1998) based on field studies in Australia also documented increasing fp with rainfall intensities; they postulated that it is due to the spatial variation in Ks at the plot scale, and developed a one-parameter exponential distribution to characterize the spatial variation of maximum fp. Several researchers have provided hypotheses on the mechanisms of how infiltration capacity might increase with rainfall intensity. One hypothesis is that longer detention times and higher depths of flow on flatter slopes could lead to the phenomenon, but the difference in depths of flow are very small and would add little to the overall pressure gradient driving fp. Nachabe et al. (1997) provided a hypothetical rationale of the phenomenon by showing how different parts (e.g., upper, middle, and lower) portions of a watershed will produce ponding and runoff at different parts of a storm depending on whether Ks increases or decreases with elevation, that is, for the different zones. At the other end of the scale, Fox et al. (1997) advance the theory that variations in fp at the microscale can explain the phenomenon. They state that, even for their “smooth” packed soil, the surface was never entirely covered with water, and that small mounds and aggregates protruding above the flow may have higher Ks than where raindrop impact has created a more dense, less permeable surface seal. As indicated above, Yu et al. (1997; 1998) attribute it to spatial variations in Ks at a smaller, undefined scale. Extent and density of vegetation may be a key in explaining the phenomenon. In Fig. 49.14, the vegetated loam soil has much higher and greater range of fp than the bare soil for different rainfall intensities. As indicated earlier, many others have documented the connection between vegetation and fp, for example, Huang et al. (2013) and Dunne et al. (1991) who showed that a deeper wetting front formed directly beneath plants in an arid region of Kenya. At an intermediate scale, there may be a strong connection between extent of ponding in depressions and the role of macropores. Rainfall simulator experiments conducted in the semiarid Wyoming steppe indicate increasing infiltration with increasing rainfall rates that visually corresponds to rapid ponding of soil pans between vegetation clumps, followed by observed high infiltration down the shafts of wheatgrass into the soil (Ogden, unpublished data). In the real world, no surface is perfectly smooth. Rather, natural topographic depressions fill and grow in area as the rainfall intensity exceeds the fp at a point. As the area of ponding grows, the number of macropores encountered associated with vegetation and animals increases. On steeper slopes, the extent of ponding decreases for the same rainfall intensity and runoff will begin earlier compared to flat slopes. Increases in ponded area expose more surface to ponded water and allows more macropore activation, increasing the fp. Because ponded areas expand more on flatter topography than on steeper topography, fp will not increase as much on steep versus flat slopes and result in more runoff. So, we think there is a logical explanation for why “the steeper the slope, the more runoff ” is true, but only if the Hortonian runoff mechanism governs the runoff process. Further research is needed to prove this hypothesis. If true, there may be a need to describe the extent of ponding based on the microtopography of a slope in order to predict the ponded area as a function of the resultant rainfall minus fp. 49.5.6  Relationship of Runoff to Watershed Area

Dunne (1978) graphed the relationship of peak runoff rate per area versus drainage area using data from USDA research watersheds. Ogden and Dawdy

49_Singh_ch49_p49.1-49.12.indd 10

(2003) performed a flood quantile scaling study using annual peak flow data (1981–1996) from the 21 km2 USDA Goodwin Creek Experimental (GCEW) Watershed and found simple scaling up to 21 km2. The data from Dunne (1978) are shown in Fig. 49.15 together with the results shown by Ogden and Dawdy (2003), and a subsequent reanalysis of peak runoff quantiles up to 50-year recurrence interval using GCEW data from 1981 through 2006 (Ogden, unpublished data). The envelope drawn on the data by Dunne (1978) declines slightly until a drainage area of about 0.1 km2 (25 acres) is reached, at which point it declines rather sharply with an exponent of –0.38. This indicates a flood quantile scaling exponent of 0.62. Ogden and Dawdy (2003) reported a flood peak scaling exponent for the 20-year recurrence interval of 0.77 using 16-year data from the GCEW between 1981 and 1996. The ICOLD envelope curve of maximum observed flood peaks in watersheds less than 300 km2 has an exponent of 0.78 (ICOLD, 2002). A reanalysis value for the 50-year recurrence interval using 26 years data from 1981 to 2006 revealed a simple scaling exponent of 0.71. Scaling breaks have been identified by other researchers (Smith, 1992; Gupta and Dawdy, 1995). One potential reason for the break in the relationship at this point is that for at smaller scales, highintensity convective rainfall is probably the critical precipitation type; and this type of storm may have a typical area of coverage corresponding to the break point on the x-axis. Another possible reason is the attenuating effects of storage (e.g., channel and overbank) may start to become dominant around the observed break point, reflecting the different dominant hydraulic processes in the land versus channel phases. As noted in the title of Fig. 49.15, the data are from USDA watersheds where agriculture is the dominant land use, and where Hortonian-type runoff is the expected runoff mechanism. Goodrich et al. (1997) showed that in the semiarid Walnut Gulch watershed in southern Arizona, channel infiltration caused a scale break between 37 and 60 ha. 49.6 Conclusions

This chapter started by stating that hydrologists and engineers calculate peak flows for hypothetical storms, as well as flows over extended time periods for varied forcing, complex terrain and changing land use and climate conditions. Doing this requires broad predictive understanding of complex interrelated processes, particularly if the predictive model is to get the right answer for the right reasons (Klemes, 1986; Grayson et al., 1992; Kirchner, 2006; Paniconi and Putti, 2015; Fatichi et al., 2016). Model complexity aside, it is clear from the observations presented in this chapter that runoff generation is complex with numerous poorly understood and difficult to quantify processes. Many of the process-level unknowns are biological actors and their effects, including human land use change effects. Demonstrated lack of widely applicable predictive ability, in general, and a lack of progress in increasing the predictive ability of physics-based models even in “well-constrained” situations hints at a need for a fundamentally different approach (Semenova and Beven, 2015). Sivakumar et al. (2014) suggest that our lack of understanding of systemlevel complexity is manifested by our lack of ability to classify differences among catchments. We agree with this assertion and believe that improved process-level understanding is paramount. However, progress will only come when we acknowledge that the tools that are currently being used in standard practice, and that are loved by regulators because of their simplicity, are inadequate. These tools include the Curve Number method (Mockus, 1949), Unit Hydrograph (e.g., Sherman, 1932; McCarthy, 1940), and Lag time equations (e.g., Kirpich, 1940; Kerby, 1959). The problem is not the existence of these

−1

(a)

Peak runoff rate (cm h )

Infiltration rate (mm h−1)

300

10

1

10

0

10

USDA watershed data (from Dunne, 1978) Envelope (from Dunne, 1978) 20−year GCEW recurrence (Ogden and Dawdy, 2003) 50−year GCEW recurrence (based on 1981−2006 data)

−1

10

−4

10

−3

10

−2

−1

0

10 10 2 Drainage area (km )

10

1

10

2

Figure 49.15  Observed peak runoff rates for the Hortonian runoff-generation mechanism versus. catchment area. Data points from USDA study watersheds assembled by Dunne (1978). Curves from USDA-ARS GCEW at scales form 0.1 to 21 km2 after Ogden and Dawdy (2003).

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References    49-11 

methods. The problem is that their entrenched status as “standard practice” and regulatory requirements that they be used has stopped both researchers and practitioners from asking the question “How can we do this better?” Perhaps the more relevant question is: “When we evaluate their performance, why do our current tools perform so poorly?” Finally, we must acknowledge that advances in prediction will come from advances in predictive understanding made from well-designed process-level studies. Normally, exhaustive data-collection efforts tend to follow technological developments. Large systematic data collection campaigns such as county-level soil surveys undertaken in the 1950s through the 1970s in support of Curve Number hydrology. We need contemporary studies using technologies such as geophysical methods, remote sensing, light stable isotopes, geochemistry, and in situ testing of soil hydraulic behavior at meaningful scales. The need for improved prediction capabilities warrants a network of existing and newly established research catchments aimed at collecting high-quality full featured hydrologic data for formulating and testing processbased hydrologic prediction tools. ACKNOWLEDGMENTS

The USGS HDCN data were downloaded by Nels Frazier using his superior web services skills. Hernan Moreno assisted with statistical analysis of the Bayesian runoff factors using the R software package. Dave Kusel, Rick Cruse and Witold Krajewski all contributed to our acquisition of the photograph shown in Fig. 49.12. We also acknowledge the input of four anonymous reviewers. REFERENCES

Assouline, S. and Y. Mualem, “Modeling the dynamics of seal formation and its effect on infiltration as related to soil and rainfall characteristics,” Water Resources Research, 33 (7): 1527–1536, 1997. Baumhardt, R. L., M. J. M. Römkens, F. D. Whisler, and J-Y. Parlange, “Modeling infiltration into a sealing soil,” Water Resources Research, 26 (10): 2497–2505, 1990. Beven, K. and P. Germann, “Macropores and water flow in soils,” Water Resources Research, 18: 1311–1325, 1982. Beven, K. and P. Germann, “Macropores and water flow in soils revisited,” Water Resources Research, 49: 3071–3092, 2013, doi:10.1002/wrcr.20156. Beven, K. and M. J. Kirkby, “A physically-based variable contributing-area model of catchment hydrology,” Hydrological Science Bulletin, 24 (1): 43–69, 1979. Bonell, M., “Progress in the understanding of runoff generation in forests,” Journal of Hydrology, 150: 217–275, 1993 Budyko, M. I., Climate and Life, Academic Press, Orlando, FL, 1974, p. 508. Burwell, R. W. et al., “Losses of surface runoff, total solids, and nitrogen during Bermudagrass establishment on levee embankments,” 2011, doi:10.2134/jeq2010.0458. Buttle, J. M. and K. Sami, “Testing the groundwater ridging hypothesis of streamflow generation during snowmelt in a forested catchment,” Journal of Hydrology, 135: 53–72, 1992. Celia, M. A., E. T. Bouloutas, and R. L. Zarbo, “A general mass-conservative numerical solution for the unsaturated flow equation,” Water Resources Research, 26 (7): 1483–1496, 1990. Cerdan, O., V. Souchere, V. Lecomte, A. Couturier, and Y. Le Bissounnais, “Incorporating soil surface crusting processes in an expert-based runoff model: sealing and transfer by runoff and erosion related to agricultural management,” Catena, 46: 189–205, 2001. Dao, T. H., “Tillage and winter wheat residue management effects on water infiltration and storage,” Soil Science Society of America Journal, 57: 1586–1595, 1993. Dingman, S. L., Physical Hydrology, 3rd. ed., Waveland Press, 2015. Downer, C. W. and F. L. Ogden, “Prediction of runoff and soil moistures at the watershed scale: Effects of model complexity and parameter assignment,” Water Resources Research, 39 (3), 2003, doi:10.1029/2002WR001439. Downer, C. W. and F. L. Ogden, “GSSHA: model to simulate diverse streamflow producing processes,” Journal of Hydrologic Engineering, 9 (3): 161–174, 2004, doi:10.1061/(ASCE)1084-0699(2004)9:3(161) Dunne T. and L. B. Leopold, Water in Environmental Planning, Freeman Publishers, 1978. Dunne, T., Field studies of hillslope flow processes, Hillslope Hydrology, edited by M. J. Kirkby, John Wiley & Sons, 1978. Dunne, T. and R. D. Black, “Partial area contributions to storm runoff in a small New England watershed,” Water Resources Research, 6: 1296–1311, 1970.

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Dunne, T., W. Zhang, and B. F. Aubry, “Effects of rainfall, vegetation and microtopograhy on infiltration and runoff,” Water Resources Research, 27: 2271–2285, 1991. Fox, D. M., R. B. Bryan, and A. G. Price, “The influence of slope angle on final infiltration rate for interrill conditions,” Geoderma, 80: 181–194, 1997. Freeze, R. A., “Role of subsurface flow in generating surface runoff: 2. Upstream source areas,” Water Resources Research, 8: 1272–1283, 1972. Germann, P. F., Macropores and hydrologic hillslope processes, Process Studies in Hillslope Hydrology, edited by M. G. Anderson and T. P. Burt, 1990. Germann, P. F., “Comment on ‘Theory for source-responsive and freesurface film modeling of unsaturated flow,’” Vadose Zone Journal, 9 (4), 1000–1101, 2010. Goodrich, D. C., L. J. Lane, R. M. Shillito, S. N. Miller, K. H. Syed, and D. A. Woolhiser, Linearity of basin response as a function of scale in a semiarid watershed,” Water Resources Research, 33 (12): 2951–2965, 1997. Grayson, R. B., I. D. Moore, and T. A. McMahon, “Physically based hydrologic modeling: 2. Is the concept realistic?,” Water Resources Research, 28: 2659–2666, 1992. Green, W. H. and G. A. Ampt, “Studies on soil physics: 1. The flow of air and water through soils,” Journal of Agricultural Science, 4 (1): 1–24, 1911. Gupta, D. S., B. P. Mohanty, and J. M. Kohne,. “Soil hydraulic conductivities and their spatial and temporal variations in a vertisol,” Soil Science Society of America Journal, 70: 1872–1881, 2006. Gupta, V. K. and D. R. Dawdy, “Physical interpretations of regional variations in the scaling exponents of flood quantiles,” Hydrological Processes, 9: 347–361, 1995. Haverkamp, R., J-Y. Parlange, J. L. Starr, G. Schmitz, and C. Fuentes, “Infiltration under ponded conditions: 3. A predictive equation based on physical parameters,” Soil Science, 149 (5), 292–300, 1990. Hillel, D., Soil and Water: Physical Principles and Processes, Academic Press, 1971. Horton, R. E., “The role of infiltration in the hydrologic cycle,” Eos, Transactions of American Geophysical Union, 14: 446–460, 1933, doi:10.1029/ TR014i001p00446 Huang, J., P. Wu and X. Zhao, “Effects of rainfall intensity, underlying surface and slope gradient on soil infiltration under simulated rainfall experiments,” Catena, 104: 93–102, 2013. Huntington, T. G., “Evidence for intensification of the global water cycle: review and synthesis,” Journal of Hydrology, 380: 83–95, 2006. Jones, J. A. A.,“Soil piping and catchment response,” Hydrological Processes, 24: 1548–1566, 2010. Kelliher, F. M., R. Leuning, and E-D. Schulze, “Evaporation and canopy characteristics from coniferous forests and grasslands,” Oecologia, 95: 153–163, 1993. Kerby, W. S., Time of concentration for overland flow, Civil Engineering, ASCE, Reston, VA, Vol. 26, Suppl. 3, 1959, p. 60. Kirchner J. W., “Getting the right answers for the right reasons: linking measurements, analyses, and models to advance the science of hydrology,” Water Resources Research, 42: W03S04, 2006, doi:10.1029/2005WR004362. Kirkby, M. J., “Hillslope runoff processes and models,” Journal of Hydrology, 100: 315–339, 1988. Kirpich, Z. P., Time of concentration of small agricultural watersheds, Civil Engineering, Vol. 10, Suppl. 6, 1940, p. 362. Klemes, V., “Dilettantism in hydrology: transition or destiny?,” Water Resources Research, 22: S177–S188, 1986. Madge, D. S., “Leaf fall and litter disappearance in a tropical forest,” Pedobiologia, 5: 273–288, 1965. Madge, D. S., “Field and laboratory studies on the activities of two species of tropical earthworms,” Pedobiologia, 9: 188–214, 1969. McCarthy, G. T., Chap. V of “Flood Control,” The Engineer School, Fort Belvoir, VA, 1940, pp. 127–177. Mein, R. G. and C. L. Larson, “Modeling infiltration during a steady rain,” Water Resources Research, 9: 384–392, 1973. Mishra, S. K., J. V. Tyagi, and V. P. Singh, “Comparison of infiltration models,” Hydrological Processes, 17 (13): 2629–2652, 2003. Mockus, V., “Estimation of total (and peak rates of) surface runoff for individual storms. Exhibit A in Appendix B,” Interim Survey Report (Neosho) River Watershed USDA, 1949. Nachabe, M. H., T. H. Illangasekare, H. J. Morel-Seytoux, and L. R. Ahuja, “Infiltration over heterogenous watershed: influence of rain excess,” Journal of Hydrologic Engineering, 2: 140–143, 1997. Nassif, S. H. and E. M. Wilson, “The influence of slope and rain intensity on runoff and infiltration,” Hydrological Sciences Bulletin, 20: 539–553, 1975. Nieber, J. L. and G. S. Warner, “Soil pipe contributions to steady subsurface stormflow,” Hydrological Processes, 5: 329–344, 1991.

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49-12    Watershed Runoff, Streamflow Generation, and Hydrologic Flow Regimes

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Chapter

50

Snowmelt Runoff Generation and Modeling BY LEV S. KUCHMENT

ABSTRACT

The chapter presents information on present day understanding of processes of snowmelt runoff generation and physically based models of these processes. Special attention is paid to describing the processes in the snow pack (snow accumulation, the energy budget and snowmelt, water balance, and the heat and water transfer in melting snow cover). Opportunities of taking into account of stochastic spatial structure of snow characteristics and application of the fractal theory have been showed. The experience of constructing distributed physically based model of snowmelt runoff generation which includes except snow accumulation and snowmelt models of infiltration of melt water into soil and overland and channel flow has been discussed. Three case studies illustrate the possibilities of constructing distributed physically based models on the basis of analysis of leading physical processes, taking into account available experimental information and user requirements. 50.1 INTRODUCTION

A permanent snow cover is formed on about 20% of the Northern hemisphere and about 15% of the Southern hemisphere. A significant part of land is covered by snow several times during cold period. The presence of a snow cover on a drainage basin greatly influences the runoff generation. In many parts of the world, river runoff consists mainly of water yielded by the melting of snow. The snowmelt spring runoff of most large plain rivers of Russia and Canada exceeds half of annual runoff; at the same time, the portion of snowmelt runoff from mountain areas in arid regions can be significantly larger. Modeling of snow pack processes and snowmelt runoff generation plays an important role in understanding of these processes and developing a means for their prediction. The models of snowmelt runoff generation of different sophistication levels are now main tools in snowmelt runoff forecasting, water resources planning, and estimating the risk of extreme snowmelt floods. The choice of optimal structure of a snowmelt runoff generation model for a particular river basin depends not only on the problem at hand and the required specifications of processes but also to a significant extent on the availability of data needed for the determination of parameters of these models. As a result, according to proportion of using a priori and experimental information for model construction, the snowmelt runoff models like the other models of runoff generation can be conditionally divided into two types: (1) the conceptual models wherein the choice of the structure of models of conceptual understanding of processes is applied but measurements of the input and the output of the hydrological system is the main information for determining parameters of the models; (2) physically based models in which the choice of the structure is based on the fundamental laws of physics and hydrodynamics and a priori information on the processes of the hydrological cycle. The parameters of physically based models can be determined, in principle, on the basis of direct measurements of characteristics of the hydrological systems. The snowmelt runoff generation models can be separated also into lumped and distributed types. Lumped models are based on assumption that the river

basin is a single unit with averaged over basin input data and model variables and parameters. Distributed models use input data, model variables, and model parameters distributed in space. To decrease the number of fitting parameters, the conceptual models are usually assumed to be lumped. Lumped conceptual models contain aggregated empirical parameters with a complicated physical interpretation and a large range of variation. Conceptual lumped models with various degrees of sophistication are often used for operational snowmelt runoff forecasting and these models have been described in many papers and handbooks (e.g., WMO, 1975). In this chapter, we consider only the physically based models. 50.2  SNOW ACCUMULATION PROCESSES

Snowfall over an area is determined by meteorological conditions; however, snow accumulation is largely a function of area topography and elevation. A major role in the accumulation and distribution of snow cover on the ground is played by interception of snow by the vegetative cover. The ratio of accumulated snow in the forests and fields differs considerably, depending on forest types and the previous meteorological conditions (Fedorov, 1977; Kuusisto, 1984; Pomeroy and Gray, 1995). The interception by forest canopies may reach 45% in spruce and 30% in pine forests. Snow spatial redistribution is strongly affected by the interaction of wind and topography as well as by interaction of wind and vegetation. Much of the intercepted snow is blown off by wind and accumulates in open areas. During blowing and transport of snow, significant evaporation may occur [the evaporation losses may reach 40–50% of annual snowfall (Pomeroy et al., 1993)]. Gullies and surface depressions are filled up by snow first of all and can accumulate a considerable portion of the total river basin snow resources. The average snow-melting rate in a forest may be lower by a factor of 2–6 as compared with open sites. As a result, the snow water equivalent (the depth of water which would result from the melting of snow) in forest areas is usually 10–40% more than in the open areas (in some cases, a general increase of precipitation in the forest is possible). The Small-scale variations of snow cover, caused by the spatial change of relief, vegetation, and local meteorological conditions, are superimposed on large-scale variations associated with physiographic and climatic zonality. It lead to large spatial variability of snow cover characteristics, and they are often considered as random values. The coefficient of spatial variation of snow water equivalent ranges from 0.15–0.20 in the forest zone to 0.30–0.60 in the steppe zone. After snowfall, the snow pack undergoes significant transformation (metamorphosis) caused by compaction, action of thermal gradients, and change of the crystal structure resulting from the interactions of ice, liquid water, and water vapor (Anderson, 1976). The air humidity in the snow pack is usually close to saturation; however at rapid changes of air temperature there may be a significant transfer of water vapor from the lower layers of snow (thermodiffusion of water vapor) and its considerable sublimation on the ice crystals. In addition to the changes brought about by vapor, the freezing together of two or ice crystals takes place, especially in the presence of liquid water in a snow pack. This transformation of snow structure is called constructive metamorphosis. 50-1

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50-2     Snowmelt Runoff Generation and Modeling

The term “destructive metamorphosis” applies to the transformation of the form of snowflakes, which takes place during the first 1–2 day period after deposition as a result of mechanical interaction of snowflakes. Constructive and destructive metamorphosis results in producing a uniform and coarse structure of the snow (this process is called snow ripening). A snow cover is considered to be ripe when any additional melt or rain water cannot be held within the snow but will move through the pack and become outflow. This occurs when the snow cover is isothermal at 0°C and the liquid water storage capacity is full. The metamorphosis of snow significantly changes the density and other physical properties of snow. Snow at the time of fall may have a density as low as 10 to as high as 150 kg m–3; snowfall in the form of dry snow may vary in density between 70 and 150 kg m–3; average wind-toughened snow has a density of about 280–300 kg m–3. Ripe snow has a uniform density of 400–500 kg m–3. The influence of constructive metamorphosis on the snow density is usually small; however this process can considerably change the mechanical properties of snowpack and plays an important role in the heat transfer in snow. During melting period the snow density changes as a result of formation of liquid water and transformation of structure of ice crystals (especially when there is night refreezing of melt water). The greatest density which can be attained by shifting the snow grains around is about 550 kg m–3. Further densification, which can occur under the action of deformation, refreezing, recrystallization, produces a compact, dense material called firn. At a density of between 820 and 840 kg m–3, the air spaces disappear and the material can be defined as ice. Accumulation on land of ice results from recrystallization of snow and other forms of precipitation lead to formation of glaciers. Being a porous medium, the snow pack has much in common with the soil. In the dry snow, liquid water is retained mostly by film tension and capillary forces. The porosity of snow varies from 0.80–0.87 (for new snow) to 0.50–0.70 (for old coarse-grained snow). The liquid-water holding capability of snow (the maximum value of liquid-water content beyond which water will drain by gravity action) is about 0.13–0.15. The movement of water through snow pack begins when snow pack is saturated by liquid water beyond these values. In the period of snowmelt, part of the liquid water may refreeze. The proportion of solar radiation falling on snow cover and then reflected (albedo) is depending on the types of soil and vegetation and varies over the winter. The new snow has an albedo of 0.75–0.90 and after ripening the albedo can reach 0.35–0.40. The convective exchange of snow cover with the atmosphere consists of sensible and latent heat fluxes. Sensible heat flux is the heat rate at which heat is carried to or from the snow surface by atmospheric turbulence. Latent heat flux is the rate at which heat is released by evaporation of liquid water from snow and snow sublimation (the phase transition of water from solid to vapor) at the snow surface and carried to or from the snow surface by the turbulent movement of atmospheric masses. Evaporation of liquid water from snow occurs only when the vapor pressure at the dew point is lower than the snow surface temperature and when the air is unsaturated with water vapor. When the dew point is above 0 °C, condensation takes place with the release of latent head of vaporization. The portion of liquid water snow lost by evaporation may be very small during the active snowmelt period when dew points are above 0°C. When the dew points of snow are low and the air is unsaturated with water vapor, the evaporation may be significant even at the low air temperature. The latent heat of vaporization is extremely large, approximately 2.83 MJ kg–1. The energy required to sublimate 1 kg of snow is equivalent to that required to raise the temperature of 10 kg of liquid water by 67 °C. Energy of vaporization is released to the environment upon recrystallization of vapor to ice. The latent heat of snow fusion is approximately 333 kJ kg–1 at 0 °C. The energy required to melt 1 kg of snow (already a 0 °C) is equivalent to that required to raise the temperature of l kg of water to 79 °C. The latent heat of melting only represents 12% of the latent heat of vaporization. A very little evaporation of snow may occur without simultaneous melting, and usually much more snow will be melted than evaporated. The precipitation heat can be a considerable contribution to positive snow pack energy balance. However, in most cases the effects of rainfall on the ripening snow and a decrease of albedo are more important. 50.3  ENERGY BUDGET OF SNOW PACK AND SNOWMELT

The melting rate of snow pack S can be found as: S = (Qsw + Qlw + Qls + QH + QE + QP + QG)/ρwL(50.1) where Qsw is the net short-wave radiation; Qlw is the incoming long wave radiation; Qls is the outgoing long wave radiation; QH is the sensible heat exchange; QE is the latent heat exchange; QP is the heat content of liquid

50_Singh_ch50_p50.1-50.10.indd 2

precipitation; QG is the heat exchange at ground surface; L is the latent heat of ice fusion; and is rw the density of water. The net shortwave radiation is the most dominant energy component during snowmelt. In the process of the metamorphosis and ripening, the snow pack decreases its reflected capability and absorbs during snowmelt the most part of short wave radiation. The shortwave radiation that is not reflected by snow cover is absorbed largely in the top 30 cm of the snow pack. The degree to which this radiation penetrates varies with wavelength in general, the shorter wavelengths penetrate further than longer wavelengths. The long-wave radiation incident on a snow cover is absorbed and reradiated as thermal radiation. Short-wave radiation is calculated as: Qsw = Qsi (1 − r ) (50.2) where Qsi is the average incoming solar radiation which is a function of angle between sun and surface normal (according to latitude, season, cloudiness, and atmospheric absorption), and r is the snow albedo. Kuzmin (1961) has obtained for determination of the net shortwave radiation the following empirical dependency: Qsw = Q0 (1.0 − r )(1.0 − 0.20 N − 0.47 N 0 )

(50.3)

where Q0 = 17.46β is the short-wave radiation flux under clear sky conditions for the day and the hour in question; β is the angle of short-wave radiation above the horizontal in degrees, calculated as a function of local latitude (j), the declination (δ ), and the sun’s hour angle (ω ) by formulas sin β = sinϕ sinδ + cosϕ cosδ cosω , 2π δ = 23.5sin (td − 81), (50.4) Π π ω = (th − 12) 12 where π is the number of days in 1 year; td is the number of days from the 1st of January to the day under consideration; td is the local time (in hours from the midnight to the hour under consideration); and N and N0 are the total and the lower level cloudiness (ratiometric), respectively. To calculate snow albedo, the empirical relation derived from experimental data in Kuchment et al. (1983) is used: r = 1.03 − ρ s (50.5) where ρ s is the density of snow in gcm–3 . Incoming long-wave radiation from the atmosphere can be estimated by using the Stefan–Boltzmann equation. The estimation is usually carried out for a clear sky and then some corrections are added to account for the effect of cloud cover. There are many empirical equations to estimate the incoming long wave radiation for different meteorological conditions. Kuz’min (1963), for example, suggested the following empirical formula:

(

)

Qlw = εσ (Ta + 273) 0.61 + 0.05 ea (1.0 + 0.12 N + 0.12 N 0 ) (50.6) 4

where ea is the vapor prvessure in mb, ε is the effective emissivity of the atmosphere, and σ is the Stefan–Boltzmann constant. Outgoing long-wave radiation, Qls , can be calculated by the Stefan– Boltzmann equation, presuming insignificant reflectance of incoming long wave radiation: Qls = ε sσ (Ts + 273) (50.7) 4

where ε s is the effective emissivity of snow, Ts the snow temperature in degrees Centigrade. The fluxes of sensible and latent heat depend on the respective vertical gradients of air temperature and humidity. Calculation of these fluxes is normally based on the theory of atmospheric boundary layer and the balk transfer formulas: QH = ρaC ρ (Ta − Ts ) ra QE = ρa Ls( qs –q)/ ra 

(50.8)

where ρa is the air density, C ρ is the air heat capacity, q and qs are the air specific humidity and the snow surface specific humidity, ra is the aerodynamic resistance, and is Ls the latent heat of vaporization. Assuming a logarithmic boundary layer profile gives

K 2U 1 (50.9) = ra [ln( Z / Z0 )]2

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Heat and Water Transfer in Melting Snow Cover     50-3 

where U is the wind speed at height Z , Z0 is the roughness height (the height at which the logarithmic profile predicts zero velocity), and K is von Karman’s constant (0.4). Kuz’min (1961) obtained for the sensible heat and for the latent heat fluxes the following empirical formulas: QH =18.85 (Ta – Ts)(0.18 + 0.098u) QE =32.82 (es – ea)(0.18 + 0.098u)

(50.10)

where u is the wind velocity at 10 m height in ms–1 ; es is the saturated vapor pressure over the ice in mb; and ea is the air vapor pressure at 2 m height. The heat flux from ground surface to snow is commonly considered to be a small component of snow pack energy balance. However, it can have an important cumulative effect at calculating the time of snow melt start and the soil freezing. This flux can be estimated as: H g = −λg

∂Tg ∂Z

(50.11)

where λ g is the soil thermal conductivity, and Tg is the soil temperature. The soil thermal conductivity is a function of soil moisture content. Because the thermal conductivity of ice is about four times that of water, the thermal conductivity of saturated soil abruptly increases after soil freezing. The use of Eq. (50.1) requires measurements of radiation and of heat exchange components, which are often not available. In the absence of direct meteorological model estimates or observations, energy budget components can be estimated from empirical relationships with commonly observed driving meteorological parameters, such as air temperature and humidity, wind velocity, and cloudiness. Simple method of calculation of melting rate The most informative index of the snowmelt rate is the air temperature. The relation between these values can be presented as: S = a(Ta − Tb ) 

(50.12)

where Tb is the air temperature below which no melt occurs (it is commonly 0–2 °C), and a is an empirical coefficient (degree-day factor) which can be interpreted as the snowmelt per day at the change of air temperature per degree. The degree-day factors vary depending on climatic and physiographic conditions but in many cases it is possible to classify their variation according to the latitude, topography, and vegetation. Because forest cover has a significant effect on many of the variables affecting snow cover energy exchange, there is a considerable difference between degree-day factors for forest and open areas. The typical degree-day factors are usually 4–5 mm/day °C for mid-latitude open areas, are 3–4 mm/day °C for deciduous forest, and are 1.5–2.0 mm/day °C for dense coniferous forest. Differences in relief aspects are also important. In open mountain areas the degree-day factors reach 5–6 mm/day °C. Melt factors in arctic areas tend to be smaller than those at lower latitudes with similar physiographic conditions mainly due to lower radiation intensities and relatively little wind during the melt season. Windy areas typically have higher melt factors than areas where calm conditions prevail. In many cases, the degree-day factors increase during the progress of snowmelt as a result of the decrease of snow albedo, soil warming, and increasing solar radiation. For example, the degreeday factor, averaged for Finland, is 1.45 mm/day °C at the beginning of the snowmelt period and 4.75 mm/day  °C at the end of the snowmelt period. The maximum values of the degree-day factor reach 80–90 mm/day °C. The main difference in melting of snow and ice results from the low albedo of ice. Typical mid-latitude degree-day factors for ice melting are 5–10 mm/ day oC. To take into account the influence of the snow density on snow melting, the following formula is often used: S = rs β (T a –Tb )(50.13) where b is the empirical coefficient. 50.4  SIMULATION OF SNOW ACCUMULATION PROCESSES

The snow water balance equation can be written as follows: d (W ) = Rs + Rl − S − Es − Rw dt

(50.14)

where W is the snow water equivalent; Rs and Rl are the snowfall and rainfall rate, respectively; S is the snowmelt rate, Es is the rate of snow sublimation,

50_Singh_ch50_p50.1-50.10.indd 3

Rw is the water outflow from snowpack, and W= (rs/rw)Hs, where Hs, is the snow depth. In order to compute the depth of snow cover, it is necessary to calculate for each time step the changes in snow density caused by snow compaction and destructive metamorphism. Constructive metamorphism only changes the density profile of a snow cover. The density of new snow varies, based on meteorological conditions. The colder and drier the air mass, the lower is the density of new snow. In many publications, the new snow density ρ0 (in g sm–3) is estimated as:   1.8 Ta + 32   0.05 +   at Ta > –15°C  ρ0 =  100 (50.15)  0.05 at Ta ≤ –15°C  A model of the densification of snow cover suggested is in Anderson (1976). The rate of snow compaction (in sm s–1) is calculated by the formula V=

v1 ρs H s2  exp(v 2Ts + v3 ρs ) 2

(50.16)

where Hs is in sm, ρs is in g sm–3, ν1, and ν2 and ν3 are empirical coefficients which are equal 2.8 × 10–6 sm2 s–1 g–1, –0.08 °C–1, 21 cm3 g–1, respectively (Anderson, 1976). It is possible to assume that snow capacity, in which melt water can be retained, is defined from the empirical formula: w max = 0.11 − 0.11

ρs  ρw

(50.17)

For estimating snow accumulation and melt in forested areas to take into account the influence of canopy, two different approaches are commonly used. The first approach is based on simulating the mechanisms by which canopies transform radiation, turbulent exchange, and mass fluxes. The second approach is based on using empirical relationships between meteorological variables obtained from simultaneous measurements in the open and in the forest. For example, for a single snowfall event onto a snow-free canopy, Hedstrom and Pomeroy (1998) suggested the following simplified dependence: –C p X c   ∗ I = I ∗  1– e I    

(50.18)

where I ∗ = S p LAI(0.27 + 46 ρ0−1 ) is the interception capacity, Sp is the effective winter leaf area index, LAI (total horizontal area of stems and needles per unit area of ground times the clumping factor), and Cp is the maximum plan area of the snow-leaf contact per unit area of ground.

50.5  HEAT AND WATER TRANSFER IN MELTING SNOW COVER

When melt or rainfall water enters into a snow pack with a temperature of less than 0 °C, the snow pack experiences rapid metamorphic changes resulting in the growth of the mean ice grain size and the snow density and thereby in the decrease of snow permeability. Liquid water freezes on the ice grains and releases latent heat of fusion. The snow pack is divided by a wetting front into two zones: an upper layer of wet snow at 0 °C and a lower layer of dry snow with a temperature below 0 °C. Simultaneously, capillary storages above the wetting front are filled and capillary forces are not sufficiently large to transport the liquid water above the wetting front. Additional water moves the wetting front downward; however the rate of this movement is slower than that of the melt water above the wetting front. The development of wetting fronts, the re-freezing of melt water within the snow cover, the liquid water retained by snow capillaries influence the availability of melt water for runoff. To describe the heat and mass transfer processes in the melting of snow pack, the following system of equations can be applied (Colbeck, 1972): ∂θ 1 ∂qw ρ I ∂ I =– – ∂t ρ w ∂ z ρ w ∂t ∂T  ∂T  ∂I Cef s λef  + ρI L ∂z  ∂z  ∂t

(50.19)

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50-4     Snowmelt Runoff Generation and Modeling

where qw is the liquid water flow, Cef = CI ρ I I + ρ w Cwθ, θ is the volumetric liquid water content in snow, I is the volumetric content of ice, Cw is the water heat capacity, ρI is the ice density. The increase of ice content can be caused by liquid water flow from the upper snow layers at Ts < 0 or by freezing of liquid water as a result of cold transfer from the lower layers. The liquid water flow in the snow pack can be presented as

γ (h) = α h 2 H (50.25) where a and H are constants, this random variable is statistically self-­similar. By averaging S( x ) and variances for two embedded circles or squares with the center at a point X o and with areas Fk and F, we obtain mk − S0 = r H (mF − S0 ) (50.26)

 ∂Ψ S  qW = − K SS  − 1 (50.20)  ∂z  where Ψ S (WS ) is the snow capillary potential, K SS (WS ) is the snow hydraulic conductivity, WS = θ + I. Substituting Eqs. (50.20) in (50.19), we get ∂θ ∂  ∂Ψ S  ρ ∂I =  K ss − K ss  − s (50.21)  ρw ∂t ∂t ∂z  ∂z The snow hydraulic conductivity can be expressed as

σ k2 = r 2 H σ F2 (50.27) where S0 = S ( x 0 ); mk and σ k2 are the means and the variance of S( x ) over the area Fk, respectively; and mF and σ F2 are the means and the variance of S( x ) Fk . F As a measure of irregularity of a random surface and correlation of the large-scale and small-scale variations, Mandelbrot (1982) introduced the fractal dimension as over the area F respectively; r =

K SS = K oS sSn (50.22) where ss is the effective saturation of snow calculated as s −s ss = sw si (50.23) 1 − ssi

θ θ , s = si , θ smax = 1 − I , θ si is the irreducible snow water θ smax si θ smax saturation (the maximum liquid water content which is held by capillary forced and is not available for flow). Constant n in Eq. (50.22) is about 3.5, and K oS is the snow hydraulic conduction at Ss =1. The value of irreducible water saturation is estimated as 0.07–0.10. The effective amount of liquid water retained in the snow by capillary and tension forces (the liquid water holding capacity) relates to snow density. According to the Russian practice, in many cases it is possible to take θ si = 0.13 for ρ s ranged from 0.3 to 0.5 g cm–3. The snow water pressures gradients may be large at very low flow rates of water flux in the zone immediately above wetting fronts, and structural interfaces within the snow. However, the snow water pressure gradient is commonly very small and can be neglected in Eq. (50.21). Thus, Eq. (50.21) now becomes where ssw =

∂θ ∂ ρ ∂I (50.24) = ( K ss )– 1 ρw ∂t ∂t ∂ z

50.6  SPATIAL VARIABILITY OF SNOW COVER

The spatial variations of snow cover characteristics are usually considered as random fields. The spatial variabi1ity of snow density is significantly less than the variabi1ity of snow depth and hence the spatial variations of snow depth and snow water equivalent for a given area have similar probabilistic distributions, which commonly follow lognormal or two-parametric gamma probability laws. However, these random fields may be strongly heterogeneous, and statistical parameters needed for the construction of probabilistic distributions may vary in space and as a function of the area size. The number of measurement points needed for reasonable estimation of spatial variance or higher statistical moments is commonly sufficient only for areas which are significantly larger than the grid cells of numerical runoff models or only for a small part of these grid cells. Thus, it is necessary to assign the statistical parameters for domains without measurements or transfer these parameters from the larger or smaller domains. An opportunity of solution of this problem is associated with knowledge of regularities in the stochastic spatial structure of snow characteristics and searching relationships between variations of these characteristics for different spatial scales. The problem of transferring the parameters of statistical distributions of variations of snow characteristics is simplified if to assume that the snow field is statistically self-similar. The statistical self-similarity is such a property of a given random variable S( x ), when statistical distribution of S( x ) within any cell Fk of a random field F will be the same as the distribution over the whole area F, if a scaling transformation of this variable within Fk is made. Such a scaling transformation is done when the variable S( x ) is multiplied by a factor rH, where r is a constant depending on the ratio of Fk to F and H is a constant depending on a measure of spatial correlation of S( x ) (Mandelbrot, 1982). If the variogram of the value of S( x ) has power structure

50_Singh_ch50_p50.1-50.10.indd 4

D = E +1+ H

(50.28)

where E is the topological dimension of used data. Therefore, estimation of the fractal dimension of a fractal field is possible to carry out by studying the fractal dimension of its section (E = 2) or points along a straight line (E = 1). The available information about the spatial snow distribution includes two forms of snow cover data: the point measurements along straight-line snow courses (E = 1) and snow cover data received on the basis of averaging the courses measurements for different area scales (E = 2). Kuchment and Gelfan (2001) carried out the investigation of stochastic spatial structure of snow characteristics to test an assumption on statistical self-similarity of fields of snow characteristics using the snow depth measurements at the snow courses from tens to hundreds of meters situated in six different regions of the world and the maximum snow water equivalent data for area scales from tens to hundreds of kilometers from five regions of East Europe and the Kolyma River basin. Results of investigation are presented in Figs. 50.1 and 50.2 and in Tables 50.1 and 50.2. As can be seen from Figs. 50.1 and 50.2, the spatial variograms of snow depth and of maximum snow water equivalent in logarithmic coordinates are approximated quite well by linear functions and we can consider the variations of snow cover characteristics at the chosen snow courses and areas as fractals (the fractional Brownian processes). The corresponding fractal dimensions of these spatial variations are presented in Tables 50.1 and 50.2. 50.7  CONSTRUCTING GENERAL MODEL OF SNOWMELT RUNOFF GENERATION

To construct a snowmelt runoff generation model, it is necessary to add to the description of snow accumulation and snowmelt processes models of infiltration of melt water into soil as well overland and channel flow. Vertical water and heat transfer in the soil associated with soil freezing, thawing, and

g (cm2)

10,000

1000 b

a c

100

d e f

10

h(m) 1

10

100

1000

Figure 50.1  Variograms of snow depth for scales from tens to hundreds meters: a, Tien-Shan region; b, Alaska; c, Valday region; d, Oka River basin; e, Don River basin; f, Lower Volga region.

8/22/16 2:21 PM

CONSTRUCTING GENERAL MODEL OF SNOWMELT RUNOFF GENERATION     5 

Northern European Russia

10,000

g (mm )

1000

100

h(km) 10

100

Kama River Basin g

h(km)

100 10

1000

100

1000

Sosna River Basin

10,000

(mm2)

g

(mm2)

1000

1000

h(km) 10

100

1000

Carpatian Region

10,000

g

h(km)

100

1

10

100

1000

Kolyma River Basin

100,000

(mm2)

g

(mm2)

10,000

1000

100

g (mm2)

1000

10,000

100

Central European Russia

10,000

2

h(km) 10

100

1000

1000

h(km) 10

100

1000

Figure 50.2  Variograms of the snow water equivalent for scales from tens to hundreds of kilometers. Table 50.1 Fractal Dimension of the Snow Depth Field Region

Table 50.2 Fractal Dimension of the Maximum Snow Water Equipment Fields

Number of data points

Lower limit of distances (m)

Upper limit of distances (m)

Exponent of variogram ± standard deviation

Fractal dimension surface

Tien-Shan region (mountains)

100

10

2000

0.30±0.03

1.85

Northern European Russia

65

142

34

0.30±0.05

2.85

Valday region (forest)

50

10

1000

0.15±0.01

1.93

41

116

37

0.22±0.03

2.89

Don River basin (steppe)

100

10

2000

0.18±0.02

1.91

Central European Russia

114

33

0.16±0.03

2.92

100

100

1420

0.15±0.04

1.92

Kama River basin

50

North of Alaska (tundra)

82

32

0.28±0.02

2.86

280

25

7000

0.22±0.01

1.89

Sosna River basin

30

Oka River basin (open area)

40

50

32

0.31±0.05

2.84

Lower Volga region (steppe)

100

10

2000

0.36±0.03

1.82

Carpation Region Kolyma River basin

20

94

49

0.67±0.07

2.66

50_Singh_ch50_p50.1-50.10.indd 5

Region

Number Mean areally Standard Exponent of Fractal data averaged snow deviation of snow variogram dimension points water equivalent water equivalent ± standard (mm) (mm) deviation

8/22/16 2:21 PM

50-6     Snowmelt Runoff Generation and Modeling

infiltration of water can be described with the following equations (Gelfan, 2006; Kuchment et al., 1983):

ceff

c­ onductivity of unfrozen soil, the heat capacities and the thermal conductivities of frozen and unfrozen soil can be calculated by empirical formulas, for example, from Kuchment et al. (1983), Koren et al. (1999), and Koren (2006). ∂W ∂  ∂θ ∂I  To calculate overland and channel flow, the kinematic wave equations are =  D + DI − K  (50.29)  ∂t ∂ z  ∂ z ∂z commonly used. The general structure of the distributed physically based models for a par∂I ∂W ∂T ∂  ∂T   ∂θ  ∂T − K + ρw L cT =  λ  + ρ w cw  D + D I ticular river basin is chosen on the basis of analysis of leading physical pro ∂z  ∂z ∂z ∂t ∂t ∂ z  ∂ z  cesses, taking into account available experimental information and user requirements. A part of parameters of these models are the measured values, where W, q, and I are the total waterρcontent, liquid water content, and ice some parameters are determined on the basis of empirical dependencies content of soil, respectively (W = θ + i I ); K = K(q, I) is the hydraulic conobtained from field and laboratory investigations of soil physics and hydroρw logic relations in different physiographic zones. To improve the accuracy of ductivity of the frozen soil; T is the soil temperature; λ represents the thermal the model, some parameters have to be fitted (calibrated). For fitting the parameters, except measurements of runoff, the available measurements of  ∂ψ   ∂ψ  conductivity of soil; D = K  and DI = K  ; ψ = ψ (θ ,I ) is the capilevapotranspiration, soil moisture, and snow characteristics are often used. In  ∂ I  θ  ∂θ  I many cases, a priori information even on the possible range of parameter ∂θ + ρi ci Ican considerably decrease the uncertainty in their estimation. The lary potential of the frozen soil; cT = ceff + ρ w L ; ceff = ρ g c g (1 − P ) + ρ w cwθvalues ∂T physically based models of snowmelt runoff generation need usually calibra= ρ g c g (1 − P ) + ρ w cwθ + ρi ci I ; r and c are the density and the specific heat capacity, respection of 4–6 parameters. tively (indexes w, i, and g refer to water, ice, and soil matrix, respectively); and Numerical simulation of distributed models of runoff generation are based P is the soil porosity. on splitting up the river basin area into grid cells for which the model input To solve these equations, it is necessary to have also the empirical relations and coefficients can be considered as spatially uniform. This splitting can be ψ = ψ (θ ,I ) and K = K(q, I) as well as a relation between overcooled liquid realized by using, for example, a regular rectangular grid schematization (e.g., water content and negative soil temperature soil. However, for application of Abbott, 1996 or Xia et al., 2007) or finite element schematization which takes these equations, experimental data are usually not available and in a number into account the river basin topography or different basin characteristics of papers (e.g., Koren et al., 1999; Koren, 2006) it is assumed that melting (Kuchment et al. 1986; Kuchment and Gelfan, 2011). Depending on the probwater saturates the upper layer of soil just after the beginning of snow melting lem under consideration, the size of grid cells may vary over a wide range and and infiltration into frozen soil is constant and is equal to the saturated at strong spatial variability of meteorological inputs and land surface charachydraulic conductivity of the frozen soil. In this case, the following empirical teristics, ignoring the subgrid changes of these values, may considerably affect formula obtained by Kulik (1978) can be used: the hydrological cycle variables, even if we use very small grid domains. It may be justifiable to assume that the subgrid variations of the meteorological K uf inputs and model coefficients can be described by statistical distributions.  (50.30) Kf = 2 (1 + aI ) One usually assumes that snow water equivalent and saturated hydraulic conductivity inside each finite element are gamma or lognormal distributed where Kf is the rate of infiltration in frozen soil, Kuf is the saturated hydraulic variables and their variances are a function of the size of subgrid areas. conductivity of unfrozen soil, soil, a is an empirical coefficient, and I is the However, the available experimental information is often only sufficient to volumetric ice content of the frozen layer of soil. calculate the parameters of these distributions for the area that is significantly To calculate the depth of the soil freezing and the volumetric ice content, larger than the grid cells or, vice versa, for a small part of considered area. it is possible to use the equations which describe the movement of front of Thus, it is necessary to assign the statistical parameters for domains without freezing H(t) and the transfer soil moisture from the unfrozen layer of soil to measurements, or to transfer these parameters from the larger or smaller this front (Kuchment and Gelfan, 2011): domains (to make a spatial scaling of statistical parameters). In some cases, the statistical parameters in subgrid areas can be found from the relationships ∂T ∂  ∂T  Cf = λ between these parameters and the values that can be measured or determined  , 0 < z < H (t ) (50.31) ∂t ∂ z  f ∂ z  on the basis of available measurements. For example, in Kuchment et al. (1986) for taking into account subgrid effects in the generation of snowmelt ∂T ∂  ∂T  Cuf , H (t ) < z < Lg (50.32) =  λuf runoff, the empirical relationships of the coefficient of variation of snow  ∂t ∂ z  ∂z  water equivalent and the depth of soil freezing with the means of these values were used; Kuchment et al. (1996) used an empirical relationship between the T (0, t ) = T0 (t ); T ( H , t ) = 0; T ( Lg , t ) = TL ; T ( z ,0) = T ( z ) coefficient of variation of the saturated hydraulic conductivity and its mean ∂θ ∂  ∂θ  value to describe the subgrid effects in a rainfall–runoff generation model. =  Df − K  , H (t ) < z < Lg (50.33)  More general approaches for assigning the subgrid statistical parameters of ∂t ∂ z  ∂z input values and model coefficients can be developed on the basis of analysis θ ( L , t ) = θ L ; θ ( H ) = θ 0 ; θ (z,0) = θ (z) of stochastic structure fields of these values for different spatial scales. In Kuchment and Gelfan (2002) it has been shown that for describing snow dH ∂T ∂T (50.34) λf distribution in subgrid areas of snow water equivalent it is possible to use the z = H −0 = λuf z = H +0 + Lρ w (θ − − θ 0 ) dt ∂z ∂z relationships (50.27). where H (t ) is the depth of frozen soil; T ( z , t ) is the soil temperature at the depth z, λ f and λuf are the thermal conductivities of frozen and unfrozen layer of soil, respectively; C f and Cuf are the heat capacities of frozen and 50.8  CASE STUDIES unfrozen layers of soil, respectively; θ − is the liquid water content just above (The physically based distributed model of snowmelt runoff generation in the the freezing front; θ 0 is the liquid water content at a temperature near 0 °C Vyatka River basin (the catchment area is 124,000 km2). The model is based on (assumed to be equal to the wilting point); D f is the diffusivity of soil moisthe schematization of river basin by 477 finite elements (Fig. 50.3) and includes ture; K is the hydraulic conductivity of unfrozen layer of soil; Lg is the depth the description of the following hydrological processes: snow cover formation of the ground where the ground temperature and the volumetric moisture and snowmelt [Eq. (50.14)], freezing and thawing of soil [Eqs. (50.31)–(50.34)], content can be considered as constants equated TL and θ L , respectively. vertical soil moisture transfer [Eq. (50.33)] and evapotranspiration, water retenThe temperature T0 of the soil surface is calculated from tion in basin storage, overland, and channel flow (Kuchment et al., 1983; T −T ∂T Kuchment and Gelfan, 2011). λs a 0 = − λ f (50.43) Stochastic subgrid variations of snow cover and saturated hydraulic conducH ∂ z z =0 tivity are taken into account. The meteorological inputs of the model are where Hs and λs are the snow depth and thermal conductivity, respectively. measurements of snow cover, liquid precipitation, air temperature, and air To describe soil moisture transfer from the unfrozen layer to the freezing humidity. The hydrometeorological records obtained from 21 meteorological front, the Richards equation is used. This process plays an important role in stations and 66 snow courses were used. Most parameters were taken from the vertical redistribution of soil moisture during the cold period for soils, field measurements and empirical relationships. To determine the parameters which are typical for forested-steppe zone. The diffusivity, the hydraulic of subgrid variations for different subgrid areas, a scaling procedure based on

50_Singh_ch50_p50.1-50.10.indd 6

8/22/16 2:21 PM

Case studies    50-7 

Figure 50.3  Finite-element schematization of the Vyatka River basin (bold lines represent the channel network; thin lines represent the ­boundaries of finite elements; tree symbols mark the forested elements).

the fractal theory is applied. Six parameters (saturated hydraulic conductivities for open and forested terrains, two parameters of the evaporation model, maximum water detention by basin storage, and the celerity of the kinematic wave) are calibrated. Examples of comparison calculated and observed hydrographs are shown in Fig. 50.4. The Seim River basin (the catchment area to Kursk is 7460 km2)(Kuchment and Gelfan, 2002). It is a part of the Dnieper River basin. The relief of the basin is a rugged plain with many river valleys, ravines, gullies. The finite element schematization of the drainage area and the structure of the river network was chosen, taking into account the river basin topography, soils,

50_Singh_ch50_p50.1-50.10.indd 7

land use, and vegetation, and include 298 finite elements. To describe the snowcover formation and snowmelt, Eq. (50.14) was applied. The movement of the soil thawing front was described by the equations similar to ones used for the soil freezing description. It was assumed that melting water saturates the upper layers of soil just after the beginning of snow melting; so the infiltration rate into the frozen soil was assigned equal to the saturated hydraulic conductivity of the frozen soil [Eq. (50.30)]. The changes of unfrozen soil moisture content and infiltration into the soil during the warm period were calculated by the Richards equation. To model overland and channel flow, the one-dimensional kinematic wave equations were applied. The subgrid effects

8/22/16 2:21 PM

50-8     Snowmelt Runoff Generation and Modeling

10,000

Q,(m3/s)

1971

8000

8000

6000

6000

4000

4000

2000

2000

0 28.02

19.03

08.04

28.04

18.05

07.06

0 28.02

27.06

1973

10,000

8000

6000

6000

4000

4000

2000

2000 19.03

08.04

28.04

18.05

07.06

0 28.02

27.06

1975

10,000

8000

6000

6000

4000

4000

2000

2000 19.03

08.04

28.04

18.05

07.06

0 28.02

27.06

1977

10,000

8000

6000

6000

4000

4000

2000

2000 19.03

08.04

28.04

18.05

07.06

1979

10,000

0 28.02

27.06

8000

6000

6000

4000

4000

2000

2000

0 28.02

19.03

08.04

28.04

19.03

08.04

18.05

07.06

27.06

0 28.02

18.05

07.06

27.06

28.04

18.05

07.06

27.06

18.05

07.06

27.06

18.05

07.06

27.06

18.05

07.06

27.06

1976

19.03

08.04

28.04

1978

19.03

08.04

28.04

1980

10,000

8000

28.04

1974

10,000

8000

0 28.02

08.04

10,000

8000

0 28.02

19.03

10,000

8000

0 28.02

1972

10,000

19.03

08.04

28.04

Figure 50.4  Observed (solid line) and calculated (dashed line) hydrographs of the Vyatka River.

50_Singh_ch50_p50.1-50.10.indd 8

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REFERENCES    50-9 

caused by small-scale variations of snow water equivalent and saturated hydraulic conductivity were taken into account. Five parameters were calibrated against measured runoff hydrographs. The Kolyma River basin (the catchment area 99,400 km2) (Kuchment et al., 2000). It is a permafrost region. The model describes snow-cover formation and snowmelt, thawing of the ground, evaporation, basin-water storage dynamics, overland, subsurface, and channel flow. The main difference between this model and models of runoff generation for regions with moderate climate is the small role given to infiltration of water into soil and a larger dependence of runoff losses on the depth of thawed ground. The thaw of the frozen ground increases significantly the water input and the water storage capacity, changing the ratio between surface and subsurface flow. The choice of the structure of the model is based on the analysis of the long-term observations of runoff generation processes at the Kolyma water balance station. The following general scheme of runoff generation was accepted. The snowmelt water, first of all, fills up the free storage capacity in topographic depressions, the peat mats and the ground where this water freezes. It is assumed that the basin storage capacity is statistically distributed over the basin and the mathematical expectation of this capacity before snowmelt depends only on the water balance of the ground in the antecedent summer–autumn period (before the snowmelt ground is deeply frozen). Excess snowmelt and rainfall water over the free storage capacity forms overland flow. Ice melting and evaporation of soil moisture begin at the snow free areas of the river basin. The melt of ice in the ground and in the depressions produces the subsurface flow and increases the basin storage capacity. The subsurface flow occurs above the frozen layer of the ground. The water retained by the capillary forces does not take part in subsurface flow. The infiltration of rainfall water into the ground is quick and does not depend on the ground moisture conditions . The kinematic wave equations are applied to describe the overland and channel flow. The basin was represented by 316 finite-elements. Five parameters were calibrated against measured runoff hydrographs. REFERENCES

Abbott, M. B. and J. C. Refsgaad (Ed.), Distributed Hydrological Modeling, Kluwer Academic, Dordrecht, The Netherlands, 1996. Anderson, E. A., “A point energy and mass balance model of snow cover,” NOAA Technical Report, NWS19, Washington, D.C., 1976, p. 150. Colbeck, S. C., “A theory of water percolation in snow,” Journal of Glaciology, 11 (3): 369–385, 1972. Fedorov, S. F., “Study of the water balance components in the forest zone of the European part of the USSR,” Hydrometeoizdat, 1977, p. 264 (in Russian). Gelfan A. N., J. W. Pomeroy, and L. S. Kuchment, “Modelling forest cover influences on snow accumulation, sublimation, and melt,” Journal of Hydrometeorology, 5: 785–803, 2004. Gelfan, A. N., “Physically based model of heat and water transfer in frozen soil and its parametrization by basic soil data,” Predictions in Ungauged Basins: Promises and Progress, edited by M. Sivapalan et al., IAHS, Vol. 303, 2006, pp. 293–304. Hedstrom, N. R. and J. W. Pomeroy, “Measurement and modeling of snow interception in the boreal forest,” Hydrological Processes, 12: 1611–1625, 1998.

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Koren, V. J., K. Schaake, Q-Y. Mitchell, F. C. Duan, and J. M. Baker, “A parameterization of snowpack and frozen ground intended for NCEP weather and climate models,” Journal of Geophysical Research, 104 (D16): 19569– 19585, August 27, 1999. Koren, V., “Parametrization of frozen ground effects: sensitivity to soil properties,” Predictions in Ungauged Basins: Promises and Progress, IAYS, Vol. 303, 2006. Kuchment, L. S., V. N. Demidov, and Y. G. Motovilov, River Runoff Generation (Physically-Based Models), Moscow: Nauka (in Russian), 1983. Kuchment L. S, V. N. Demidov, and G. Y. Motovilov, “A physically-based model of the formation of snowmelt and rainfall-runoff,” Symposium on the Modeling Snowmelt-Induced Processes, International Association of Hydrological Sciences: IAHS, Budapest, 155: 27–36, 1986. Kuchment L. S., A. N. Gelfan, and V. N. Demidov, “A distributed model of runoff generation in the permafrost regions,” Journal of Hydrology, 240: 1–22, 2000. Kuchment, L .S. and A. N. Gelfan, “Statistical self-similarity of spatial variations of snow-cover: verification of the hypothesis and application for runoff modelling,” Hydrological Processes, 15: 3343–3355, 2001. Kuchment, L. S. and A. N. Gelfan, “Estimation of extreme flood characteristics using physically based models of runoff generation and stochastic meteorological inputs,” Water International, 27 (1): 77–86, 2002. Kuchment, L. S. and A. N. Gelfan, “Assessment of extreme flood characteristics based on a dynamic-stochastic model of runoff generation and the probable maximum discharge,” Journal of Flood Risk Management, 4 (2): 115–127, 2011. Kuusisto, E., Snow Accumulation and Snowmelt in Finland, Publication of Water Resources Institute, Helsinki, 1984, p. 151. Kuzmin, P. P., Melting of Snow Cover [English Translation by Israel Program for Scientific Translation, Jerusalem, 1972], 1961, p. 179. Kuzmin, P. P., Snow Cover and Snow Reserves [English Translation by Israel Program for Scientific Translation, Jerusalem], 1963, p. 139. Male, D. H. and D. M. Gray, Handbook of Snow: Principles, Processes, Management and Use, Pergamon Press, 1981, p. 745. Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman, New York, 1982, p. 461. Pomeroy, J. W, D. M. Gray, and P. G. Landine, “The Prairie Blow Snow Model: characteristics, validation, operation,” Journal of Hydrology, 144: 165–192, 1993. Pomeroy, J. W. and D. M. Gray, “Snowcover: accumulation, relocation and management,” NHRI Science Report No. 7, National Hydrology Research Institute, Saskatoon, Saskatchewan, Canada, 1995, p. 135. Kulik, V. Y., Water infiltration into soil, Moscow: Gidrometeoizdat, 1978, pp. 3–9 (in Russian). Xia Zhang, Shufen Sun, and Yongkang Xue, “Development and testing of a frozen soil parameterization for cold region,” Studies of Hydrometeorology, Special section volume 8: 691–671, August 2007. World Meteorological Organization (WMO), “Intercomparison of conceptual models used in operational hydrological forecasting,” Operational Hydrology Report No. 7, Geneva, Switzerland, 1975, p. 172.

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Chapter

51

Glacial Melting and Runoff Modeling BY

GLENN TOOTLE, ANGELA PELLE, AND GREG KERR

ABSTRACT

Alpine glaciers are important indicators of climate. These glaciers are incredibly sensitive to changes in temperature and snowpack, the primary drivers of glacial mass balance. Alpine glaciers uniquely influence watershed hydrology in many ways including contributions to streamflow in the form of glacial meltwater. Remote sensed products offer datasets which provide temporal changes in alpine glacier area and mass (volume). These datasets vary from aerial photographs to satellite imagery. When combined with empirical areavolume scaling techniques, remote sensed data of temporal glacier area change can estimate mass (volume) loss of glaciers, thus, providing an estimate of glacial meltwater contribution to late-summer / early-fall streamflow with uncertainty. Paired watershed analysis of similar (climatically and geologically) glaciated and non-glaciated watersheds provide a mechanism of assessing temporal changes in runoff hydrographs. The paired watershed analysis displays how alpine glaciers typically delay runoff peaks when compared to non-glaciated watersheds. This delay can have a critical impact on regions which have a limited growing season for agricultural production. Through the use of remote sensed data, empirical area-volume scaling techniques and paired watershed analysis, hydrologists can assess mass contributions and temporal variability of glacial meltwater. 51.1 INTRODUCTION

Glaciers and ice caps cover approximately 10–11% of the earth’s surface (Gleick, 1996) and store approximately 75% of the world’s freshwater (Bates, 2008). With the recent studies in the Northern and Central Rocky mountains of North America, and the findings that glaciers are retreating as a response to the regional climate warming (Marston et al., 1991; Key et al., 2002), the western United States has seen significant impacts on summer streamflows. These critical summer streamflows replenish reservoirs and provide irrigation water for agriculture purposes. Alpine glaciers make up about 4% of the world’s land ice area (Dyurgerov and Meier, 1997, 2000) and are often located in remote locations. However, research shows these alpine glaciers are important regional climate change indicators due to their high sensitivity to temperature and precipitation changes (Meier, 1984; Oerlemans et al., 1998; Granshaw and Fountain, 2006). Over the past century, annual air annual temperatures have risen by 0.74 ± 0.18°C globally (Trenberth et al., 2007). During the twentieth century, this increase in temperature has been associated with a continuous retreat of smaller alpine glaciers (Dyurgerov and Meier, 2000). The shrinkage of these alpine glaciers has a direct impact on the hydrologic cycle including fresh water supply, agriculture outputs, shortage of hydroelectric power, excessive flooding, and habitat loss. Glaciers serve as valuable frozen, fresh water reservoirs, storing water in the winter and releasing it during the warmer summer months (Marston, 1989). The melt water from these glaciers is thought to be important during the warmer summer and fall months to supplement flows needed for irrigation, fisheries, and

the fulfillment of interstate water compacts. This storage and releasing effect of water is important for various practical, scientific and educational fields including hydroelectric power, flood forecasting, sea level fluctuations, glacier dynamics, sediment transport, and formation of landforms (Jansson, 2003). As a result, management and prediction of future flows from glaciers has become important to water planners in many regions (Hutson, 2003). 51.2  REMOTE SENSING

Remote sensing is the science of obtaining information about objects or areas from a distance, usually from aircraft or satellites. Remote sensors are able to collect data by detecting the energy that is reflected from the Earth. There are two methods by which remote sensing products collect data, active and passive. Active sensors use internal stimuli to collect data. For example, a laserbeam remote sensing product projects a laser onto the surface of the Earth and measures the time that it takes for the laser to reflect back to its. sensor. Examples of active sensors are RADAR and LiDAR. Passive sensors, in contrast, collect data by responding to external stimuli. Passive sensors record radiation that is reflected from Earth’s surface, typically from the sun, and therefore, can only be used during daylight hours. Examples of passive sensors include film photography, infrared, and radiometers. Remote sensing has many applications in various fields of study, but this chapter will focus on the application of remote sensing in area and volume quantification of glaciers. The effort to map glaciers began as early as the late nineteenth century, with explorers mapping them through manual observations. Glacier mapping then evolved with the use of aerial imagery in the early twentieth century. This revolutionized glacial mapping because it allowed scientists to view those areas unreachable on foot. This method however, did not allow for any extensive physical quantification of a glacier. Therefore, the evolution of three-dimensional (3D) imagery has significantly improved the quality and accuracy with which glaciers can be studied today. In particular, it allows for the quantification of glaciers in terms of area and volume, pieces of critical information to those areas heavily dependent on glacial melt off for water supply. 51.3  APPLICATION OF REMOTE SENSING IN GLACIER QUANTIFICATION 51.3.1  Data Sources

While there are hundreds of sources of remote sense data, some sources are not publicly available. However, there are plenty of reliable publicly available data sources with various types of remote sensing products. For aerial imagery within the United States, 1–3 m resolution aerial photography is available from the U.S. Geological Survey’s National Map Site (http://nationalmap. gov/). Satellite imagery is a bit more difficult to acquire, especially if looking for fine resolution imagery. Typically open source satellite imagery will have resolution cell sizes greater than 100 m/pixel. NASA’s hosted Landsat GeoCover provides a collection of high resolution provided in a standardized, 51-1

51_Singh_ch51_p51.1-51.6.indd 1

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51-2     Glacial Melting and Runoff Modeling

orthorectified format, covering the entire land surface of the world except Antarctica (http://glcfapp.glcf.umd.edu/research/portal/geocover/). Purchased remote sensing products are readily available from sources such as the U.S. Geological Survey and can provide larger tracts of area, higher resolutions, and larger selections of remote sensing products. 51.3.2 Application

Geo-referencing The first method of glacier quantification will focus on the use of remote sensing products for direct volume calculations. Geo-referenced (spatial coordinates known) images are common and readily available, however, not all images have this feature built in. If this is the case, the user must generate spatial coordinates using ground control points (GCP) such as buildings, rocks, and other landmarks. In order to produce volume calculations, images must be geo-referenced to known spatial coordinates and elevations. This can be done by overlaying imagery over a digital elevation model (DEM). Analyzing the Geo-referenced Images Once the image is geo-referenced, it can be analyzed through either manual or digital interpretation with the use of various software packages. Both techniques offer advantages and disadvantages and must be considered. Manual image interpretation requires increased time and effort, but is generally considered more accurate because the human eye can depict differences whereas digital interpretation may not. Many times glaciers are either located in a “bowl like feature” or on the north side of a peak where they are shaded. For analysis of aerial imagery, shadows are a challenge with digital interpretation. Therefore, manual methods allow for better interpretation of the shadow influence. Satellite imagery is more conducive to the use of digital interpretation because the image is a collection of pixels capturing various land covers, allowing for easier interpretation between shadows, snow, rock, etc. (Edmunds et al., 2012). Area Error Estimation Possible sources of error in imagery that can cause discrepancies in area calculations include having poor spatial information for images (especially older images), having images that are not taken directly above the glacier (slightly oblique images), and user error while geo-referencing the image (Cheesbrough, 2007). Two methods to calculate error from area calculations using remote sensing data come from Hall et al. (2003) and DeBeer and Sharp (2007). Hall et al. (2003) determined that error could be determined by an equation using the aerial photo being analyzed and the base map used for georectifying. The total digitizing error (ed ) was calculated using Eq. (51.1) (Hall et al., 2003).

ed = rp2 + rb2 + er (51.1)

where rp is the pixel resolution of the georeferenced paper maps, rb is the pixel resolution of the base map, and er is the registration error of the summation of the georeferenced paper map root mean square error (RMSE) and the base map RMSE. Once the digitizing error was determined, the area uncertainty (ea) was measured using the following formula (Hall et al., 2003):



 2e  ea = ri2 ∗  d  (51.2)  ri 

where ri is the image’s pixel resolution and ed is the total digitizing error calculated in Eq. (51.2). The second method, total error (dQ), is a method by DeBeer and Sharp (2007) that combines the Hall method and DeBeer and Sharp methods.

δQ = (δq1 )2 + (δq2 )2 +  (δqn )2 (51.3)

where dq1,…, dqn represent each individual uncertainty in surface area occurring from the area uncertainty with the respect to georeferencing as well as the delineation process of individual glacier boundaries. Volume Analysis A common method of directly calculating volume from imagery is creating a stereo pair (aka stereoscopy). A stereo pair is a set of two or more photos with overlapping portions, which are positioned such that the parallax between the common objects allows the user to view the objects in 3D using either red/ blue anaglyph stereo glasses or LCD stereo glasses. The stereo paired, georeferenced imagery with elevation information (DEMs) is then transformed into a digital terrain model (DTM). An example of this product can be seen

51_Singh_ch51_p51.1-51.6.indd 2

in Fig. 51.1 (Cheesbrough, 2007). This allows for a digital volume calculation through the use of software packages. Also of interest is calculating the change in volume of glaciers over time, because it captures changing climate variables (temperature, radiation, etc.). Volume Error Estimation Errors in volume estimation is determined by evaluating overall elevation differences and errors associated with the DEM used in preparing the stereo pairs. This method involves first determining a base line elevation by finding the average elevation difference between two selected years. Volume change is then divided by the average elevation to determine an area used in error calculation. For example, if the elevation difference is 4.84 m and the corresponding volume difference is 1.29 million cubic meters, the area would be 1.29 ∗ 106 m3 = 266,529 m 4.84 m The second step is to multiply the DEM error (specific to your DEM) by a random number with a mean of zero and a standard deviation of one. This results in an error between the values of the negative of the error to the positive of the error. This value is then multiplied by the aforementioned area to create an error estimation. After repeating this process a user-determined number of times, the data is normally distributed and the confidence interval provides the associated error estimation. Ve = CI of (DEM error ∗ Area ∗ Random Number) (51.4) Volume estimation (Ve) is for one iteration and the confidence interval (CI) is found after all iterations determined by the user are calculated with a desired confidence level (Edmunds et al., 2012). 51.3.3  Advantages and Disadvantages

Remote sensing’s advantages lie in its ability to directly calculate volume with a higher levels of accuracy due to high resolution imagery. In addition, remote sensing allows for the production of valuable visual tools such as 3D mapping. However, remote sensing’s disadvantage is that in order to achieve high accuracy volume calculations, time intensive data preprocessing (geo-referencing) and manual/digitization methods are required. Another disadvantage is that imagery of a desired location or resolution might not be readily available. To overcome the issue of data availability, empirical methods have been developed that use more commonly available aerial imagery. 51.3.4  Empirical Methods: Volume–Area Scaling

Several methods are available to estimate glacier volume based on glacier area (volume–area scaling relationships). It should be noted that volume–area scaling relationships are most accurate when applied to glaciers that are in equilibrium with climate. These volume–area scaling methods determined that glacier volumes (V expressed in m3) and areas (A expressed in m2) could be related by a power law expressed as: V = α Aβ

(51.5) Chen and Ohmura (1990) derived a and b by analyzing 63 glaciers having known areas and volumes using topographic surveys and radio-echo soundings in North America, Europe and Asia. They derived a (0.120) and b (1.396) by using a regression analysis on an area versus volume plot of the glaciers. Driedger and Kennard (1986) used similar methods as Chen and Ohmura (1990) but focused more on examining the relationship between glacier flow and geometry of 25 glaciers in Washington and the Oregon Cascades. They found their relationship was most appropriate for small alpine glaciers less than 2.6 km long when using a (3.93) and b (1.124). Bahr et al. (1997) methodology was generally selected because of its ability to reasonably estimate smaller glacier volumes (Granshaw and Fountain, 2006). The Bahr et al. (1997) method is based on the width, slope, side drag, and mass balance of 144 glaciers located in Europe, North America, central Asia, and the Arctic (Bahr et al., 1997). Bahr et al. (1997) then tested the parameters against the known volumes (calculated with radio echo soundings) and the areas of each individual glacier. Bahr et al. (1997) empirically derived a and b from a regression analysis plot. Based on units, there are several variations of this equation: a = 0.226, when Area (A) is input as m2 and the resulting Volume (V) is in m3 a = 32.7, when Area (A) is input as km2 and the resulting Volume (V) is in million m3 a = 0.033, when Area (A) is input as km2 and the resulting Volume (V) is in km3 b = 1.36 for all variations based on the Bahr regression analysis plot

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Application in Streamflow Measurement     51-3 

50

50 0

)

)

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Dinwoody Glacier (2001)

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Dinwoody Glacier (1983)

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Figure 51.1  3D image of Dinwoody Glacier in 1983 (left image), 3-dimensional image of Dinwoody Glacier in 2001 (right image).

51.3.5  Errors in Area–Volume Estimations

It has been documented that when using volume–area scaling techniques, the error in estimated volume changes for individual glaciers can be large (Granshaw and Fountain, 2006). However, according to Demuth et al. (2008), when the volume–area scaling technique is applied to a large number of glaciers, the result should be a much smaller error since the error associated with Bahr et al.’s (1997) regression equation is randomly distributed. Area calculations with error estimates can be used in either of the three volume equations to produce minimum maximum and average volume. The area error estimates can be calculated using methods presented earlier (Hall et al., 2003 and Debeer and Sharp, 2007). Another method of estimating the errors associated with calculating volume is to take the average difference between volume estimates using Chen and Ohmura (1990) and Driedger and Kennard (1986) methods were taken as a measure of uncertainty for the volume calculated using Bahr et al.’s (1997) method. 51.3.6  Advantages and Disadvantages

The most significant advantage of volume–area empirical methods is the ease of its implementation. A simple delineation of glacier area can be seen in Fig. 51.2 (Maloof et al., 2014). However, this simplicity also comes with increased error over the more time and data intensive direct volume methods explained earlier. Errors culminate in the area–volume method beginning with delineation and area calculation of the glacier. While error can be minimized by choosing the correct coefficients based on the glacier’s physical characteristics, empirical methods still exhibit a ~12% underestimation when compared to measured values. Therefore, use of empirical methods must be used cautiously and well documented with the errors associated with it. 51.4  GLACIATED VERSUS NONGLACIATED WATERSHEDS

Hydrologists view glacier bodies as “frozen water reservoirs,” which store water in the form of ice and release it at future times (Pochop et al., 1990; Fountain and Tangborn, 1985; Jansson et al., 2003). Thus, watersheds with glaciers are shown to provide a more stable source of water than nonglaciated watersheds (Ferguson, 1973; Fountain and Tangborn, 1985; Braithwaite and Olsen, 1988). Glaciers also affect the annual hydrograph by delaying seasonal runoff through internal storage of liquid water, resulting from snow melt, for release later in the year. Since it has been documented that most North American glaciers have been in recession in recent years (Nylen, 2002; Granshaw and Fountain, 2006), water planners have become more interested in their contribution to the local hydrologic system. 51.5  APPLICATION IN STREAMFLOW MEASUREMENT 51.5.1 Data

When comparing glaciated and nonglaciated watersheds, it is critical that the precipitation characteristics of the watersheds are similar. Thus, snowpack accumulation must be similar, and, given the assumption that glacier meltwater will occur during July, August and September (JAS) season, the watersheds must display similar precipitation patterns during that season. Weather stations for comparison of climate conditions between glaciated and nonglaci-

51_Singh_ch51_p51.1-51.6.indd 3

ated watersheds are limited. Open source snowpack data can be found with Natural Resources Conservation Service (NRCS) SNOTEL (i.e., SNOwpack TELemetry) data source (http://www.wcc.nrcs.usda.gov/snow/). Observed precipitation data is preferred, but due to the limit in locations of such monitoring stations, modeled precipitation data is commonly used. Reliable streamflow data is critical in quantifying glacier impacts on streamflow. Therefore, it is advised that only unimpaired stream gage stations be used for analysis. An unimpaired stream gage station is defined as a station with minimal effects of anthropogenic uses including storage, diversion, and consumptive use. Unimpaired stations are identified on the Hydro-Climatic Data Network (HCDN) (Slack et al., 1993; Wallis et al., 1991). Once a station is identified, the United States Geological Survey (USGS) can be used to obtain stream gage information from the National Water Information System— NWIS (2011). 51.5.2 Methods

First, the mean specific runoff (“runoff year” annual runoff divided by the watershed area) (Fountain and Tangborn, 1985), should be implemented to remove differences resulting from scaling differences between the watersheds. This inspection verifies that the watersheds follow the same temporal trends in streamflow on a year-to-year basis. Next, monthly volumes for each watershed gage station are converted to a percentage of annual streamflow. This procedure allowed for two separate but important analyses: the qualitative observation of the hydrograph to determine how glaciated watersheds compare to the nonglaciated watersheds [i.e., higher late summer (JAS) percentages in the glaciated watershed] and also the quantitative comparison between the different watersheds to determine the glacier streamflow contribution. It can be assumed that a majority of glacial melt occurs during the late summer months (JAS) and that groundwater recharge and evaporation are negligible. Finally, to determine the percentage of late summer (JAS) streamflow attributed to the presence of a glacier, the non-glaciated JAS% is subtracted from the glaciated JAS% and this value is then divided by the glaciated JAS%. To calculate the variability of different watersheds (glaciated and nonglaciated) for the late summer months of JAS, a dimensionless coefficient of variation (CV) can be calculated.



CV =

σ X bar

(51.6)

where s is the standard deviation of the summed monthly average discharge volumes of the 3 months (JAS) for the period of record and Xbar is the average of the summed monthly average discharge volumes of the three months (JAS). The lower the variability of a watershed, the greater glacierization is and, theoretically, the more reliable it is as a source of runoff. To reinforce this theory, yearly deviation can be calculated dev =

( yi − yi ) yi

(51.7)

where yi = year average of JAS flow and yi = JAS flow of year i. The decomposition of the variability to a yearly basis allows for the determination of which

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51-4     Glacial Melting and Runoff Modeling

Figure 51.2  Delineation of Knife Point Glacier for 2006.

specific years account for the difference between glaciated and nonglaciated watersheds. Large ranges of yearly deviations (dev) are more often observed in nonglaciated watersheds, reinforcing that glaciated watersheds exhibit more stable sources of runoff. In addition, the deviation from the mean values could be compared to standardized snow water equivalent values from historic April 1st snow water equivalent data (Aziz et al., 2010; Hunter et al., 2006), obtained from the NRCS for SNOTEL stations. The presence of glaciers modifies streamflow in powerful, even unique ways important to both hydraulic and hydrologic engineering (Fernández et al., 1991; Fleming and Clarke 2005). The principal influences on streamflow are often contributions in volume during periods of recession, a delay of the maximum seasonal flow, storage of spring snowmelt in the form of liquid water for release later in the year, and a decrease in annual and monthly variation of runoff (Fountain and Tangborn, 1985). 51.6 CONCLUSION

Glaciers have the ability to dramatically impact the hydrologic characteristics of a watershed by storing ice and snowmelt for release at a future time. The impacts on streamflow are especially important during the late summer months of JAS. Thus, as long as the glaciers exist, the contributions of glaciers to late season watershed streamflows are significant whether the glaciers are retreating or assimilating. REFERENCES

Aziz, O. A., G. A. Tootle, S. T. Gray, and T. C. Piechota, “Identification of pacific ocean sea surface temperature influences of upper Colorado river basin snowpack,” Water Resources Research, 46: W07536, 2010.

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Bahr, D., M. Meier, and S. Peckman, “The physical basis of glacier volume area scaling,” Journal of Geophysical Research: Solid Earth, 102 (B9): 20355– 20362, 1997. Bates, B. C., Z. W. Kundzewicz, S. Wu, and J. P. Palutikof (Eds.), Climate change and water, Technical Paper of the Intergovernmental Panel on Climate Change, IPCC Secretariat, Geneva, 210, 2008. Braithwaite, R. J. and O. B. Olesen, “Effect of glaciers on annual run-off,” Journal of Glaciology, 34 (117): 200–207, 1988. Cheesbrough, K. S., “Glacial recession in Wyoming’s wind river range,” Master’s Thesis, University of Wyoming, Laramie, WY, 2007. Chen, J. and A. Ohmura, “Estimation of alpine glacier water resources and their change since the 1870s,” Hydrology in Mountainous Regions I: Hydrological Measurements; the Water Cycle, edited by H. Lang and A. Musy, International Association of Hydrological Sciences (IAHS), Wallingford, 1990, pp. 127–135. Debeer, C. M. and M. J. Sharp, “Recent changes in glacier area and volume within the southern Canadian Cordillera,” Annals of Glaciology, 46 (1): 215– 221, 2007. Demuth, M. N., V. Pinard, A. Pietroniro, B. H. Luckman, C. Hopkinson, P. Dornes, and L. Comeau, “Recent and past-century variations in the glacier resources of the Canadian rocky mountains: Nelson river system,” Terra Glacialis, 11 (248): 27–52, 2008. Driedger, C. L. and P. M. Kennard, “Ice volumes on Cascade volcanoes: Mount Rainier, Mount Hood, Three Sisters, and Mount Shasta,” U.S. Geological Survey Professional Paper 1365, USGS, Reston, VA, 1986. Dyurgerov, M. B. and M. F. Meier, “Year-to-year fluctuation of global mass balance of small glaciers and their contribution to sea level changes,” Arctic Antarctic and Alpine Research, 29 (4): 392–401, 1997.

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References    51-5 

Dyurgerov, M. B. and M. F. Meier, “Twentieth century climate change: evidence from small glaciers,” Proceedings of the National Academy of Sciences of the United States of America, 97 (4): 1406–1411, 2000. Edmunds, J., G. Tootle, G. Kerr, R. Sivanpillai, and L. Pochop, “Glacier variability (1967–2006) in the Teton Range,” JAWRA, Journal of the American Water Resources Association, 48 (1): 187–196, 2012. Ferguson, R. I., “Sinuosity of supraglacial streams,” Geological Society of America Bulletin, 84: 251–255, 1973. Fernández, P. C., L. Fornero, J. Maza, and H. Yañez, “Simulation of flood waves from outburst of glacier-dammed lake,” Journal of Hydraulic Engineering, 117 (1): 42–53, 1991. Fleming, S. W. and K. K. Clarke, “Attenuation of high frequency interannual streamflow variability by watershed glacial cover,” Journal of Hydraulic Engineering, 131 (7): 615–618, 2005. Fountain, A. G. and W. V. Tangborn, “The effect of glaciers on stream flow variations,” Water Resources Research, 21 (4): 579–586, 1985. Gleick, P. H., “Water resources,” Encyclopedia of Climate and Weather, edited by S. H. Schneider, Oxford, New York, , Vol. 2, 1996, pp. 817–823. Granshaw, F. D. and A. G. Fountain, “Glacier change (1958–1998) in the North Cascades National Park Complex,” Journal of Glaciology, 52 (177): 251–256, 2006. Hall, D. K., K. Bayr, R. A. Bindschadler, and Y. L. Chien, “Consideration of the errors inherent in mapping historical glacier positions in Austria from ground and space (1893–2001),” Remote Sensing of Environment, 86: 566–577, 2003. Hunter, T., G. A. Tootle, and T. C. Piechota, “Oceanic-atmospheric variability and western U.S. snowfall,” Geophysical Research Letters, 33: L13706, 2006. Hutson, H. J., “Technical memorandum,” Wyoming State Water Plan, 2003, http://waterplan.state.wy.us/plan/bighorn/techmemos/glaciers.html (accessed on March 23, 2009). Jansson, P., R. Hock, and T. Schneider, “The concept of glacier storage: a review,” Journal of Hydrology, 282 (1–4): 116–129, 2003. Key, C. H., D. B. Fagre, and R. K. Menicke, Glacier retreat in Glacier National Park, Satellite Image Atlas of Glaciers of the World, U.S. Geological Survey Professional Paper, MT, 2002, pp. J365–J375.

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Maloof, A., J. Piburn, G. Tootle, and G. Kerr, “Recent alpine glacier variability: Wind River Range, Wyoming, USA,” Geosciences, 4 (3): 191–201, 2014. Marston, R. A., L. O. Pochop, G. L. Kerr, and M. L. Varuska, “Recent trends in glaciers and glacier runoff, Wind River Range, Wyoming,” Proceedings of the Symposium on Headwaters Hydrology, edited by W. W. Woessner, and D. F. Potts, American Water Resources Association, Middleburg, VA, 1989. Marston, R. A., L. O. Pochop, G. L. Kerr, M. L. Varuska, and D. J. Veryzer, “Recent glacier changes in the Wind River Range, Wyoming,” Physical Geography, 12 (2): 115–123, 1991. Meier, M. F., “Contribution of small glaciers to global sea level,” Science, 51: 49–62, 1984. Nylen, T., “Spatial and temporal variations of glaciers (1913 and 1994) on Mt. Rainier and the relation with climate,” Master’s Thesis, Portland State University, Portland, OR, 2002, p. 128. Oerlemans, J., et al., “Modelling the response of glaciers to climate warming,” Climate Dynamics, 14 (4): 267–274, 1998. Pochop, L. O., R. A. Marston, G. L. Kerr, D. J. Veryzer, and R. Jacobel, Glacial icemelt in the Wind River Range, Wyoming, Watershed Planning and Analysis in Action, American Society of Civil Engineers, Durango, CO, 1990, pp. 118–124. Slack, J. R., A. Lumb, and J. M. Landwehr, Hydro-Climatic Data Network (HCDN) Streamflow Data Set, 1874–1998 (CD-ROM), U.S. Geological Survey, Reston, VA, 1993. Trenberth, K. E., P. D. Jones, P. Ambenje, R. Bojariu, D. Easterling, A. Klein Tank, D. Parker, F. Rahimzadeh, J. A. Renwick, M. Rusticucci, B. Soden, and P.  Zhai, 2007: Observations: Surface and Atmospheric Climate Change. In: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor and H. L. Miller (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. Wallis, J. R., D. P. Lettenmaier, and E. F. Wood, “A daily hydroclimatical data set for the continental United States,” Water Resources Research, 27 (7): 1657–1663, 1991. “What Is Remote Sensing?” What Is Remote Sensing? NOAA, n.d. Web. February 26, 2015, http://oceanservice.noaa.gov/facts/remotesensing.html.

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Chapter

52

Reservoir and Channel Routing BY

MUTHIAH PERUMAL AND ROLAND K. PRICE

ABSTRACT

Methods for reservoir and channel routing are important for reservoir design, flood control planning, flood forecasting and rainfall-runoff studies of basins. A wide range of methods has been developed, many based on hydrological principles which have been shown to be consistent with hydrodynamic assumptions. Various categories of routing methods are reviewed in this chapter. Whereas the majority of channel routing methods are primarily relevant to steep river reaches there are a few methods appropriate for flat or nearly flat reaches which can be important in the downstream reaches of larger rivers. 52.1 INTRODUCTION

A flood is a hydrological event characterized by high discharges and/or water levels that can lead to the inundation of land adjacent to streams, rivers, lakes, wetlands, and other water bodies. Floods are a natural part of the hydrological cycle. As extreme events they contribute to the morphology of streams and river channels which is formed over thousands of years such that the river channels are shaped to contain the more frequent flows within their banks. A flood occurs when the stream or river cannot convey the amount of water generated by runoff from a heavier than usual rainfall event, and consequently there is an overspill of water onto land adjacent to the channel. For larger less frequent flows, the channel has insufficient capacity to contain the flow and water inundates the flood plains adjacent to the channel. Given the lower frequency of the higher flood events, the flood plains tend to form a less well-defined river valley. A river flood therefore involves the inundation of normally dry land (or wetlands) adjacent to a channel due to the inadequate capacity of the channel to convey the current flow. The flood can be caused by the collection in the upstream channel network of excess rainfall-runoff over the catchment, or the failure of a dam or local embankment or the presence of obstructions downstream. The local capacity of the channel may be limited due to vegetation growth or sedimentation, or the morphological raising of the riverbed. Once a flood has been generated in a long river, it can be said to propagate along the river toward its mouth, though the nature of the propagation may be distorted by additional runoff entering the river along its length. Generally, a flood builds up rapidly in the headwaters of the river, but may take several days or even weeks to reach the sea or lake to which it discharges. It is important to distinguish between the travel time of the water and the travel time of the flood. Generally, the latter is faster than the former. The speed of propagation of the flood peak is dependent on the gradient of the riverbed and the extent of flooding. The flatter the river and the wider the extent of flooding on adjacent flood plains, the slower the speed of the flood peak. These factors affecting the speed of travel are a function of two important concepts: storage and conveyance. Water in the channel and on the flood plains can be said to be “stored” dynamically. Storage is significant in affecting the rate at which a flood peak decreases as it propagates downstream. Conveyance refers to the ease with which water (rather than the flood disturbance) moves downstream. Some flood plains convey floodwater in a downstream direction and therefore add to the conveyance of the river channel.

The degree of flood plain conveyance depends on the topography of the flood plain and obstructions such as hedges and boundary walls, embankments, etc. The propagation of the flood is intimately connected with the conveyance. It is in this context the subject of flood routing plays a key role for studying flood wave movement in natural rivers, manmade channels and reservoirs, and natural lakes. Flood routing is an important analysis in hydrological studies required for the purposes of reservoir design, flood control planning, flood forecasting and rainfall-runoff studies of basins. It is a mathematical procedure of determining the flood hydrograph at a location in a river or immediately downstream of a reservoir using the input hydrograph known at an upstream location of the river or reservoir. The routing process is characterized by attenuation and translation of the flood peak of the inflow hydrograph. Translation is the dominant characteristic exhibited by the flood wave when routed along a reach or through a reservoir. Meanwhile, attenuation is a characteristic of the flood hydrograph routed along a river reach: the phenomenon is even more obvious when the flood hydrograph is routed through a reservoir. Depending on the physical system through which the hydrograph is routed, flood routing can be classified as reservoir routing and river routing. In most practical situations the dynamic characteristics of an incoming flood wave are significantly damped while passing through a reservoir, and the outflow can be determined by assuming that the water surface in the reservoir is horizontal. Due to the nature of reservoir flood wave, the outflow hydrograph from the reservoir has to conserve mass while the shape of the hydrograph is dominantly influenced by the outlet and topographical characteristics of the reservoir zone. For these reasons the reservoir routing uses a lumped form of the continuity equation and the storage-discharge relationship of the reservoir outlets. However, the routed hydrograph estimated at a location of a river is not only influenced by the characteristics of the river reach but also of the inflow hydrograph such as rate of rise, magnitude of peak and time to peak, and the magnitude of lateral flow in the river reach. The attenuation and translation characteristic of a flood wave depend on the inflow hydrograph and river reach characteristics. Unlike the reservoir flood wave, the dynamic nature of a river flood wave is difficult to damp. Due to the persistence of the dynamic characteristics of river floods, a number of methods and governing equations are available in literature for routing floods in rivers. Similar to the reservoir routing methods, all the river routing methods also need to conserve mass and, therefore, the equation of continuity is not approximated, though this equation can be used either in a lumped or distributed form depending on the method. This chapter describes both the reservoir routing and the channel routing. 52.2  RESERVOIR ROUTING

Reservoir routing is a mathematical procedure by which the hydrograph immediately below the reservoir is determined for the given inflow hydrograph(s) of the river(s) contributing to the storage of the reservoir. The routed hydrograph is characterized by the attenuation of flood peak of the inflow hydrograph. In reservoir routing, the lumped continuity equation is invariably used along with the storage equation which expresses the storage 52-1

52_Singh_ch52_p52.1-52.16.indd 1

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52-2     Reservoir and Channel Routing

versus outflow relationship of the discharge outlet(s) of the storage structure. In practice, these relationships are nonlinear in form, and they are considered to be available, or to be developed using topographical information immediately upstream of the storage structure with information on flow through the outlets for different operating conditions, prior to the reservoir routing process. As flow depths at different flow sections of the reservoir just upstream of the storage structure are generally large, it may be safely assumed that the flow velocity is insignificant in the storage zone and the water surface is horizontal. Therefore, the momentum equation is not needed to describe the flow routing in the reservoir, except when there is a dam failure or a sudden massive landslide into the storage zone, resulting in an abrupt surge formation within the reservoir. Reservoir routing is required for solving various problems such as design flood estimation for spillways and other outlets of the storage structure, formulation of criteria for flood control, studying soil saturation conditions due to varying water levels along the periphery of the reservoir storage area for assessing landslide possibility, effects of reservoir operation on downstream flooding conditions, and for dam failure analysis. 52.2.1  The Modified Puls Routing Method

This method is based on the use of a lumped continuity equation expressed by I (t ) − O (t ) =



dS (t ) (52.1) dt

Equation (52.1) is approximated in finite difference form for a time interval ∆t by I 2 + I1 Q2 + Q1 S2 − S1 − = 2 2 ∆t



(52.2)

where, subscripts 1 and 2 denote the values at the beginning and end of the time interval ∆t. Equation (52.2) is now rearranged so that all known terms are on the left hand side:

(I 2 + I1 )

∆t  ∆t  ∆t   +  S1 − Q1  =  S2 + Q2   2 2  2

(52.3)

The routing process consists first of evaluating the left hand side of Eq. (52.3) for the known value of flow and storage at the beginning of routing time interval and then estimating the corresponding values of Q2 from the rela­ tionship between the storage and discharge. The modified Puls method consists of the following steps: 1. Develop the relationships for storage, S versus discharge, Q plot from the given reservoir elevation versus storage, and the reservoir elevation versus discharge. 2. Select the routing interval Δt in such a manner that the discharge varies linearly within the time interval. 3. Develop (S + QΔt/2) versus Q relationship with the help of the developed S versus Q relationship as developed in step (1). 4. Compute the total inflow volume into the reservoir by multiplying (I2 + I1)/2 by Δt. 5. Next, estimate the value of (S1 – Q1Δt/2) corresponding to the discharge Q1. Q1 at t = 0 is considered to be zero corresponding to the water level at the level of the spillway crest for the uncontrolled flow condition or a non-zero value corresponding to a controlled flow situation. 6. Now compute the value of (I2 + I1) Δt/2 + (S1 – Q1 Δt/2) to arrive at the estimate of (S2 + Q2Δt/2). 7. Using (S2 + Q2Δt/2), estimate Q2 from the relationship (S + QΔt/2) versus Q developed in step (3). 8. With the above computation steps, the value of discharge Q2 corresponding to the known I2 is computed. By subtracting Q2Δt from the (S2 + Q2Δt/2) the estimate of (S2 – Q2Δt/2) can be determined. 9. In the next routing time step Q2 becomes Q1 and the estimate (S2 – Q2Δt/2) becomes (S1 – Q1Δt/2). By repeating the computation steps 4 to 8, the entire routed outflow hydrograph from the reservoir can be derived. 52.3  RIVER ROUTING

Although the hydrodynamic equations governing one-dimensional flood wave movement in rivers and channels were developed by Saint–Venant in

52_Singh_ch52_p52.1-52.16.indd 2

1871 (Saint–Venant, 1871), it was not until 1954 that these equations were being solved using computers. However, as water resources development and flood control activities required the construction of various hydraulic structures across and along the rivers, engineers needed estimates of peak flood discharges and maximum water levels for the design and operation of these structures, and for flood estimation and flood forecasting. The need for these estimates brought about the need for the development of various empirical and hydrologic flood routing methods. While the development of empirical methods required the use of at-site specific observations, the hydrologic routing methods have a more rational basis for their development such as the use of a lumped storage, derived by spatially accumulating the distributed storage along the routing reach, and using the storage equation describing the water stored in the channel reach at any instant of time during the flood movement through the reach. The parameters involved with the storage equation can be evaluated using the observed inflow and the corresponding outflow hydrographs available for flood events (Singh, 1988). The fact that these hydrologic routing methods have been used extensively and have successfully served the hydrological design and operational requirements of water resources projects long before the arrival of computationally intensive hydraulic routing methods suggests that these hydrological methods are rationally and scientifically based. Hydrologic flood routing methods, like the hydraulic flood routing methods, have been developed based on the principle of conservation of mass. The lumped continuity equation used in all the hydrologic routing methods is expressed as:

I (t ) − O (t ) =

dS (t ) dt

(52.1) where I (t ), O(t ), and S(t ) denote the inflow, outflow and the storage in the channel reach, respectively, at any time t during unsteady flow. Since there are two unknowns, O (t ) and S (t ) , in Eq. (52.1), a second equation is needed to link the reach storage at time t with the outflow and/or inflow at time t and possibly their temporal derivatives. Due to the use of the storage equation in the hydrologic routing methods, these methods are also known as the storage routing methods. Depending on the form of the storage equation employed, hydrological routing methods are categorized as linear and nonlinear methods. Some of the storage equations commonly used in the hydrological routing methods are given as follows (Singh, 1988):

S(t ) = KO (t )

S(t ) = K [θ I (t ) + (1 − θ )O (t )] S(t ) = a0O (t ) + a1



(52.5)

d 2O (t ) dO (t ) + a2 (52.6) dt dt 2

S(t ) = a0O (t ) + a1 S(t ) = a0O (t ) + a1

(52.4)

dO (t ) + b0 I (t ) (52.7) dt

dO (t ) dI (t ) + b0 I (t ) + b1 (52.8) dt dt



S(t ) = KO (t + τ ) (52.9)



S(t ) = K [O (t )] (52.10)



S(t ) = K [θ I (t ) + (1 − θ )O (t )] (52.11)

m

m

where K is the storage constant or coefficient in time units, θ is the weighting factor, and a0, a1, a2 , b0, b1 are constants, τ is the translation time, and m is a nonlinear exponent. For the sake of brevity the time t is dropped henceforth, except in Eq. (52.9). Equations (52.4)–(52.9) are characteristic of the linear hydrological routing methods. Equations (52.10) and (52.11), respectively, form a nonlinear version of the linear storage Eqs. (52.4) and (52.5). Unlike the empirical flood routing methods developed exclusively using reach specific inflow and outflow data, the storage routing methods may be considered as being universally applicable. For this reason, such methods may be considered as being semiempirical in which the form of the storage equation selected is universal, but the parameters of the storage equation need to be calibrated using site specific data.

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THE CLASSICAL MUSKINGUM FLOOD ROUTING METHOD     52-3 

The hydrological routing methods have been based on the above storage equations because they are able to mimic the attenuation and translation characteristics inherent in flood wave movement along a river reach. The use of the above storage equations [with the exception of Eq. (52.9)] assumes that the output is realized for a given input instantaneously. This will not necessarily be true when these equations are applied to a long river reach in that the reach may need to be subdivided into a number of subreaches with the equations applied sequentially to successive subreaches to arrive at the routed hydrograph for the outlet of the whole reach. This perhaps prompted Meyer (1941) to advocate the lag and route method which uses the storage Eq. (52.9) for routing floods in long river reaches by explicitly incorporating the translation or lag-time between the instances of the discharge realized on the rise of the flood at the inlet (when dI dt > 0) and the corresponding discharges at the outlet of the reach (when dQ dt > 0). From the discussion earlier, the lumped continuity Eq. (52.1) and any of the above storage equations together may be considered as the lumped representation of the distributed continuity equation and the momentum equation forming the one-dimensional Saint–Venant equations. The verification of this analogy is established in the Section on Simplified Hydraulic Routing Methods below, specifically in terms of the storage Eqs. (52.4) and (52.5). The lag and route method attempts to model the translation and attenuation characteristics explicitly using a single linear channel and a single linear reservoir (SLR) (Dooge, 1973). Equations (52.4) and (52.5) are the storage equations used in the wellknown Nash cascade model proposed by Nash (1957) and the classical Muskingum method proposed by McCarthy (1938). Although the Nash Cascade model is widely used as a conceptual rainfall-runoff model, it can also be employed for flood routing in rivers and channels (Dooge, 1973). Based on the consideration that the channel reach may be subdivided into a cascade of linear reservoirs, the hydrodynamic basis of the hydrological routing method using the Nash cascade model was independently established in the Kalinin–Milyukov River routing method (Kalinin and Milyukov, 1957) as discussed later in the Section on Simplified Hydraulic Routing Methods. Among the hydrological channel routing methods, the Muskingum method is the most popular. Therefore, it is pertinent to discuss this classical method in detail here. 52.4  THE CLASSICAL MUSKINGUM FLOOD ROUTING METHOD

The classical Muskingum method (McCarthy, 1938), which is named after its application to the Muskingum River, a tributary of the Ohio River in USA, is a linear storage routing method. This method employs the lumped continuity equation and the linear storage equation expressed earlier, respectively, as I (t ) − O (t ) =



dS (t ) (52.1) dt

S(t ) = K [θ I (t ) + (1 − θ )O (t )] (52.5)



Applying the continuity equation at the center point “M” of the finite difference grid as shown in Fig. 52.1 results in the expression for the storage using a centered finite difference scheme, as

(I

j +1

) (

− O j +1 + I j − O j 2

)=S

j +1

− Sj

2

(52.12)

which, together with Eq. (52.5) and some algebraic simplification, leads to the Muskingum routing equation for the outflow discharge at the ( j + 1) time step

O j +1 = C1I j +1 + C2 I j + C3O j (52.13)

where the coefficients C1, C2 , and C3 are given by

C1 =

− Kθ + 0.5 ∆t K (1 − θ ) + 0.5 ∆t



C2 =

Kθ + 0.5 ∆t (52.14b) K (1 − θ ) + 0.5 ∆t



C3 =

K (1 − θ ) − 0.5 ∆t (52.14c) K (1 − θ ) + 0.5 ∆t

(52.14a)

Here S is the Muskingum reach storage volume, I is the inflow discharge, O is the outflow discharge, K is the travel time, θ is the weighting parameter, the suffix j denotes the ordinate at time j∆t , and ∆t is the routing time step. Note  that C1 + C2 + C3 = 1.0 , which identifies the mass conserving ability of the classical Muskingum method. Also, the lumped continuity Eq. (52.1) assumes that there is no lateral inflow or outflow in the Muskingum reach. The storage equation given by Eq. (52.5) was proposed by McCarthy (1938) with the notion that the reach storage at any instant of time during unsteady flow consists of two substorages, viz., a prism storage and a wedge storage as shown in Fig. 52.2. The prism storage S p is expressed as SP = KO (52.15)



and the wedge storage SW is expressed as SW = Kθ ( I − O )



(52.16) Combining these two storages leads to the Muskingum storage equation given by Eq. (52.5). A careful study of Eq. (52.16) reveals that the wedge storage is positive during the rising part of flood at a section when I > O in the considered routing reach, and it is negative during the falling part of the flood at a section when I  0. The discharge [θ I ((t )) + (1 − θ )O(t )] is known as the weighted discharge. Perumal and Price (2013) give credence to McCarthy’s heuristic notion of prism and wedge storages based on a simplification of the momentum equation of the Saint–Venant equations while proposing a fully mass conservative variable parameter McCarthy– Muskingum method (VPMM). The routing parameters K and θ of the Muskingum method remain constant during a given routing event and are computed by trial and error using a recorded flood event. In the traditional approach of estimating the value of θ , the reach storage S is plotted against the corresponding weighted discharge value [θ I (t ) + (1 − θ )O (t )] given in Eq. (52.5) for different trial values of θ resulting in various sizes of looped graphs; the value of θ which gives the narrowest loop is considered as the most appropriate for use in the method. A number of alternative ways of estimating the parameters θ and K using a set of observed inflow and outflow hydrographs are described by Singh and McCann (1980), including the method of moments derived by Dooge (1973). The latter method is more convenient for estimating the parameters of the classical Muskingum method as discussed earlier, and also for estimating the three parameters of the multi-reach Muskingum method (Strupczewski et al., 1989) required for routing in long channel reaches. Many textbooks (e.g., Chow et al., 1988) state that the value of θ varies between 0 and 0.5. But Perumal (2003) showed that further insight into the range of values that θ can assume, can be deduced by analyzing the storageweighted discharge relationship given by Eq. (52.5) as follows: Inflow section 1

Wedge storage Prism storage

j+1 ∆t

M

2

j ∆x 1

2

Figure 52.1  Computational grid of the Muskingum method.

52_Singh_ch52_p52.1-52.16.indd 3

Outflow section Figure 52.2  Definition sketch of the Muskingum reach explaining the prism and wedge storage concept.

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52-4     Reservoir and Channel Routing

From Eq. (52.5), the outflow can be written as O=−



θI S + (52.17) 1 − θ K (1 − θ )

Differentiating Eq. (52.17) with reference to t gives

θ dI dO 1 dS =− + (52.18) dt 1 − θ dt K (1 − θ ) dt



When the reach storage S is maximum, dS dt = 0, and Eq. (52.18) reduces to

θ dI dO =− dt 1 − θ dt



(52.19) Consider a typical flood routing case as shown in Fig. 52.3. Applying Eq.  (52.19) at the intersection of the inflow and outflow hydrographs as in Fig.  52.3a, where the storage is maximum, it follows that θ > 0. Similarly, Fig. 52.3b corresponds to the case when θ = 0, that is, when the storage as well as the outflow discharge is a maximum. In this case the storage is a function of the outflow only (and corresponds to the linear reservoir case). The last case, as shown in Fig. 52.3c, is one in which the peak outflow is less than the synchronous inflow. Here the channel reach length is less than the length corresponding to the linear reservoir case. For such a situation both dO dt and

Inflow

Discharge

Outflow

tp

Inflow Discharge

52.5  NASH CASCADE MODEL FOR RIVER ROUTING

By conceptualizing the routing river reach as a series of equal length subreaches with each subreach having the same storage characteristics governed by Eq. (52.4), the Nash model can be applied to river routing as well as rainfall-runoff modeling. Accordingly, the routed outflow from the first subreach becomes inflow to the second subreach, and so on. The outflow of the last subreach is the desired outflow of the whole reach. Though the use of a SLR to simulate the observed flood hydrograph at the outlet of a long reach is not appropriate in that the peak outflow discharge occurs on the recession of the inflow hydrograph, a cascade of such reservoirs in series can serve the intended purpose of simulating the outflow hydrograph such that the peak outflow does not occur on the recession of the inflow hydrograph, but after it. In this way, the Nash Cascade can appropriately account for both the attenuation and translation characteristics of a flood wave. The way in which the Nash model accounts for the attenuation and translation characteristics of a flood wave is depicted in the instantaneous unit hydrograph (IUH) of the Nash cascade model, shown in Fig. 52.4, derived for the Dirac delta input function applied at the inlet of the reach. The Nash model parameters, namely N, the number of linear reservoirs, and K, the linear reservoir storage coefficient, can be obtained using the method of moments as advocated by Nash (1957) for rainfall-runoff modeling. Alternatively, these parameters can also be estimated using a plethora of optimization techniques available in the literature (Singh, 1988). It may be noted that the method is not restricted to an integer number of linear reservoirs, as adopted by the Kalinin–Milyukov method (Kalinin–Milyukov, 1957), but a non-integer value of N may be used to achieve a close reproduction of the observed hydrograph of N, using either the method of moments or a suitable optimization technique. Because the IUH is developed using the zero initial flow condition, the simulation of the Nash model corresponds to an outflow hydrograph over and above the initial steady flow present in the reach which can be added to the simulated outflow using the Nash model to compare with the observed hydrograph at the reach outlet. 52.6  OTHER LINEAR STORAGE MODELS

T

(a)

dI dt are negative, and thus require θ < 0. Szilagyi (1992) confirmed this conclusion using some typical flood observations of flooding in the Danube River in Eastern Europe.

Outflow

tp

The storage equations of the SLR and the Muskingum method given by Eqs. (52.4) and (52.5), respectively, and the other lumped storage equations given by Eqs. (52.6) to (52.8) may be considered as particular cases of the generalized storage equation proposed by Kulandaiswami (1964) for rainfallrunoff modeling. Kulandaiswami et al. (1967) investigated the storage Eqs. (52.4) and (52.6) to (52.8) for flood routing studies using limited field data. Based on their study, these authors concluded that the storage Eq. (52.8) could simulate the routing process very well compared with the other three storage equations; however, this result should be acknowledged with the caution that the conclusion was based on limited data. Because the storage equations selected by Kulandaiswami et al. (1967), with the exception of Eq. (52.4), could not be linked to the governing equations of flood routing as in the case of Nash model and the Muskingum method, further investigations of these storage equations have not been reported in the literature. The same

T

(b)

0.14

n=1 n=2 n=3 n=4

0.12

Inflow

Outflow

0.06

0.02

T

Figure 52.3  Scenarios of the classical Muskingum method solutions for varying ranges of θ: (a) outflow hydrograph when θ > 0; (b) outflow hydrograph when θ = 0; (c) outflow hydrograph when θ < 0.

52_Singh_ch52_p52.1-52.16.indd 4

0.08

0.04

tp (c)

IUH u (0,t)

Discharge

0.1

0

0

20

40

60 Time

80

100

120

Figure 52.4  Modeling attenuation and translation characteristics of a flood wave using the Nash model.

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NONLINEAR ROUTING METHODS    52-5 

reasoning may also be attributed to the lag and route method. Similarly, estimation of the lag time τ in a physically realizable sense is not possible as τ depends on the rate of rise of the inflow and the initial flow present in the routing reach. For these reasons, only the linear storage Eqs. (52.4) and (52.5), which can be related to the governing equations of the flood routing process, have been investigated extensively.

function input at the upstream end. The linearization of the Saint–Venant Equations about a reference discharge q0, applied to a unit width of channel carrying a unit discharge, leads to the following linear form:

52.7  LINEAR DIFFUSION ANALOGY ROUTING METHOD

in which, q = the discharge/unit width of the channel u0 = the velocity corresponding to the reference discharge, q0 y0 = the depth corresponding to the reference discharge, q0 g = acceleration due to gravity x = the distance from upstream end of the reach. Harley (1967) obtained the following unit response function due to the Diracdelta function as input at the upstream end of the reach:

The hydrologic routing methods are generally perceived to employ the storage concept due to the presence of the storage term in the lumped continuity equation. However, this category of methods can also be developed without directly adopting the storage concept as in the case of the diffusion analogy method advocated by Hayami (1951). He proposed a flow-depth hydrograph routing method based on a linear convection-diffusion (CD) equation:

∂y ∂y ∂2 y +c = D 2 (52.20) ∂t ∂x ∂x

where y is the depth, and c and D denote the wave celerity and diffusion coefficient respectively for the flood wave. In a similar manner, a linear CD equation using the discharge Q as the operating variable can be defined by replacing y by Q in Eq. (52.20). When the parameters c and D are considered time-invariant and are evaluated using the observed inflow and outflow hydrographs of a flood event, the above equation represents a linear hydrologic system (Dooge, 1973). Hayami (1951) reasoned that the routing method based on the diffusion analogy equation would take into account the storage effect of the routing reach, including the storage induced by the irregularities in the geometry of the channel reach. The linear diffusion wave is characterized by a IUH of the form:

u( 0,t ) =

L 4π Dt 3

e

−

( L−ct )2 4 Dt

(52.21)

where, L is the length of the routing reach. Using this IUH, the T-hour unit flood hydrograph of Hayami (1951) can be derived at the outlet of a river reach corresponding to a unit flood applied at the inlet. Here the unit flood is a fictitious wave having a constant depth and with a duration of the unit time interval T. Given that the depth hydrograph to be routed consists of a series of unit floods each with a duration of T-hours and with different magnitudes, Hayami (1951) convoluted the magnitudes with the estimated unit flood hydrographs using the principle of superposition to arrive at the required routed depth hydrograph at the outlet of the routing reach. While most of the hydrologic routing methods developed in the precomputer era employed the discharge as the operating variable due to the direct use of the lumped continuity equation by these methods, Hayami’s approach based on linear systems theory employs the flow depth as the operating variable. However, the discharge can also be used as the operating variable if the volume of flow is of interest, and the governing equation is characterized by the same IUH as in Eq. (52.21). Flow depth is the preferred variable over discharge for operational purposes such as flood forecasting at different locations along a river. Whereas the two parameters c and D can be estimated for a given set of observed inflow and outflow hydrographs using a plethora of optimization techniques, the easiest technique may be the method of moments approach advocated by Dooge (1973). Using his approach, the routing parameters are given by:



c=

D=

L

( M0′ − M1′ )

(52.22)

c 3 ( M0,2 − M1,2 ) (52.23) 2L

where M1′ and M 0′ are the first moments about the origin of the inflow and outflow hydrographs, respectively, and M1,2 and M 0,2 are the second moments about the centroid of the inflow and outflow hydrographs, respectively. 52.8  COMPLETE LINEARIZED MODEL

Linearized Saint–Venant Equation Harley (1967) obtained a general linear solution to the flood routing problem by solving the linearized equation of motion described by the Saint–Venant Equations, for a semi-infinite uniform open channel subject to a Dirac-delta

52_Singh_ch52_p52.1-52.16.indd 5





( gy

0

− u02

2

) ∂∂xq − 2u

0

2

∂2 q ∂2 q ∂q 2qS0 ∂q − = 3 gS0 + (52.24) ∂ x ∂t ∂t 2 ∂x u0 ∂t

x x  x u( x ,t ) = e − pxδ  t −  + h  −   c1 c2   c1 

sx −rt

I ( 2hm ) / m (52.25)

where

c1 = u0 + gyo (52.26)



c2 = u0 − gyo (52.27) F=



u0 (52.28) gyo



p=

S0 (2 − F ) (52.29) (2 y0 ( F 2 + F ))



r=

S0u0 (2 + F 2 ) (52.30) (2 y0 F 2 ) s=







h=

S0 (52.31) 2 y0

S0u0 (4 − F 2 )(1 − F 2 ) (52.32) (4 y0 F 2 )

 x  x m =  t −   t −  (52.33)  c1   c2 

and I (⋅) is a first order Bessel function of the first kind and δ is the Diracdelta function. The accuracy of this model is dependent on the magnitude of the reference discharge. The moments of the linear channel response are related with the channel and flow characteristics of the complete linearized solution model (Harley, 1967) as

U1′ =

x (52.34) c



2 2 F2   y   x  U 2 =  1 −   0    (52.35) 3 4   S0 x   c 



4  F2   F2   y   x  U3 = 1 −  1 +   0   (52.36) 3 4  2   S0 x   1.5u0 

2

3

in which, U1′ is the first moment about the origin, and U2 and U3 are the second and third moments, respectively, about the mean; c, the wave celerity obtained using Chezy’s equation. As this is a three parameter model, the first three moments are sufficient to estimate the parameters. 52.9  NONLINEAR ROUTING METHODS

It is generally assumed that flood wave propagation is strictly a nonlinear process, and therefore linear models are unlikely to be realistic in certain

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52-6     Reservoir and Channel Routing

Wave speed

circumstances and incapable of accounting for the nonlinear characteristics of the process. For this reason nonlinear forms of the storage Eqs. (52.4) and (52.5) were proposed, resulting in the storage Eqs. (52.10) and (52.11), respectively. Equation (52.10) represents the storage equation of a nonlinear reservoir. This equation has been advocated by Rockwood (1958), Laurenson (1962), and Mein et al. (1974) for river routing based on the consideration that flood waves are inherently nonlinear in nature because each segment of a flood wave hydrograph with different discharges travel at different celerities and the linear flood routing methods cannot account for the way in which the flood wave celerity varies with discharge. The US Army Corps of Engineers HEC-RAS model (USACE, 2010) recommends the use of this storage equation for routing floods in very steep river reaches using the modified Puls routing method. Virtually all forms of nonlinear models, whether based on the storage equation or the Saint–Venant Equations or one of their approximations, such as the kinematic wave (KW), involve a power function-type relationship with the average velocity increasing as the discharge increases (Pilgrim, 1986). However, there is now much field evidence available to suggest that, although small floods are grossly nonlinear as in the case of mountainous rivers, the power function-type of accounting for nonlinearity is not valid for large floods, and therefore linearity is often acceptable for the higher flows of interest (Kulandaiswamy and Subramanian, 1967; Bates and Pilgrim, 1983). Price (1973) with a small amount of data from streams in the UK and Wong and Laurenson (1983, 1984) with more data from Australian rivers found that the relationship between flood wave speed (wave celerity) and discharge takes the form as shown in Fig. 52.5, indicating that the use of a monotonic power function-type of relationship between storage and discharge as given by Eq. (52.10) is not appropriate for application in all ranges of flood routing in river reaches with large flood plains. Recognizing the inability of the classical Muskingum method with constant parameters to simulate the nonlinear behavior of flood wave movement in channels, Gill (1978) sought to modify the classical Muskingum method by replacing the linear storage equation by the nonlinear storage equation given by Eq. (52.11). In addition to the two parameters employed in the classical Muskingum method, Gill’s storage equation employs a nonlinear exponent m. When m = 1, the storage equation of Gill’s nonlinear Muskingum (NLM) method reduces to the storage equation of the classical Muskingum method. It may be noted that an increase of one more parameter in the form of the exponent “m” associated with the modified form of the storage equation employed in the NLM method increases the flexibility of the method in simulating the observed outflow hydrograph of the calibration flood event. However, the close reproduction of the outflow hydrograph achieved in calibration mode does not guarantee the set of estimated calibration parameters K, θ, and m to be universal or globally optimal for the considered channel reach in simulating all the possible flood events that may be routed in that channel reach. If a flood event with a higher magnitude and/or differing inflow hydrograph characteristics or a different initial flow along the reach are introduced compared with those of the original calibration event to be routed in the same channel reach, the parameters calibrated from the past event may not successfully reproduce the observed hydrograph corresponding to this new inflow hydrograph and initial flow. Apart from this issue of operating outside the range of the original calibration, the power function-type of nonlinear models are more difficult to apply consistently than the linear models. The analysis of Napiorkowski and Strupczewski (1981) shows that a mathematically involved nonlinear estimation of diffusion offers little improvements over the results obtained using linear theory. While comparing the performance of linear versus nonlinear Muskingum methods, Singh and Scarlatos (1987) concluded that the power function-type NLM method is less accurate than the linear Muskingum method. Also, where input errors are sufficiently large, linear models may perform better than the nonlinear models because they do not amplify the input errors (Singh and Woolhiser, 1976). Moreover, as long as the parameters

Discharge Figure 52.5  General form of wave speed-discharge relationship in rivers.

52_Singh_ch52_p52.1-52.16.indd 6

of both linear and nonlinear models are calibrated on the inflow and corresponding outflow hydrographs for recorded events, a nonlinear model is equally deficient as a linear model in that its calibrated parameters are not universal, implying that the calibrated parameters of the nonlinear model may not be able to closely reproduce the outflow hydrograph of an independent event not used in the calibration range. This general deficiency of models whose parameters are calibrated on recorded events, may be attributed to the fact that in the calibration the parameters are not only linked to the channel characteristics, but also to the dynamic input characteristics like the shape, rate of rise, peak and time-to-peak discharge of inflow hydrograph, and the magnitude of the initial flow present in the routing reach. 52.10  FLOW ROUTING USING HYDRAULIC METHODS

Strictly, unsteady flow in natural rivers is three dimensional with flow variables like flow rate, velocity, and depth varying across and over the flow section in the horizontal and vertical directions normal to the general direction(s) of the flow. Whereas this may be the micro-level scenario at a point in a section, the higher rate of variation of the flow variables in space and time in the longitudinal flow direction compared with that in the other two directions leads to the safe assumption that the flow is by and large one-dimensional along the longitudinal flow direction, except under circumstances when the flood flow exceeds the bank-full level and inundates extensive flood plains or there are highly nonlinear conditions on the flow such as those immediately downstream of a failed dam. These latter situations may become important for flood inundation studies consisting of flood forecasting, flood insurance studies, and planning evacuation measures during dam failures. A large body of literature is available on one-dimensional flow studies applied to the phenomenon of unsteady flow in rivers and channels. These studies primarily use the one dimensional flow equations governing the conservation of mass and conservation of momentum. Together these equations are known as the Saint–Venant Equations, and were proposed by Saint–Venant in 1871. The equations, with a lateral flow q1 per unit length of the channel reach that does not contribute any additional momentum to the flow, can be written as:

∂Q ∂ A + = ql   (Continuity equation) ∂ x ∂t

(52.37)

and S f = S0 −

∂ y v ∂v 1 ∂v   (Momentum equation) (52.38) − − ∂ x g ∂ x g ∂t

1 2 3

4

where, Q is the discharge, A is the area of cross section, v is the velocity, S f is the friction slope, S0 is the bed slope, y is the flow depth with reference to the channel bottom, and x and t are the independent variables denoting longitudinal distance along the river and time, respectively. If the depth variable is expressed in terms of water surface elevation h above an arbitrary datum, then the terms (1) and (2) together may be denoted as −∂h ∂ x signifying the body force. Equations (52.37) and (52.38) are also known as the dynamic wave equations or the Saint–Venant Equations. Unsteady flow in rivers and channels can be studied more accurately using the hydrodynamic principles embedded in the assumptions made when modeling shallow water waves, which is the regime that includes river floods. Many commercial as well as free computer packages based on numerical solutions of the full or approximate Saint-Venant equations for gradually varying unsteady flow in open channels are available for field applications. Detailed descriptions of the theory behind these hydrodynamic based methods have been dealt by many authors like Cunge et al. (1980), and Chaudhry and Mays (1993) to quote a few. One may opt for the use of such numerical based methods when accurate unsteady flow solutions are warranted. This chapter essentially focuses on the well-known hydrological routing methods which have been widely and successfully used to solve many practical problems and attempts to give a hydrodynamic basis for their success with a view of applying them without and with a limited calibration of parameters of these methods. In many cases of open channel flow all the terms (1)–(4) in the momentum equation are of similar significance referred to the bed slope S0. However, when using the equations for flood routing, some of the terms in the momentum equation are of less significance than others according to the nature of the routing reach being studied. Consequently, the momentum equation can be approximated by either eliminating or approximating some of the terms of the equation. Various simplified routing methods may be developed by combining

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SIMPLIFIED HYDRAULIC FLOOD ROUTING METHODS      52-7 

the modified momentum equation with the continuity equation. It may be noted that the simplification is made only to the momentum equation, and not to the distributed continuity equation: there are too few terms in the continuity equation to produce any meaningful approximation. It is desirable therefore to explore the practical relevance of simplified hydraulic methods and the types of simplified routing methods available in the literature. 52.11  BASIS FOR THE DEVELOPMENT OF SIMPLIFIED MOMENTUM EQUATIONS

The momentum equation in the Saint–Venant equations describes the dynamic effects of one-dimensional flood wave propagation according to the balance between the body, surface, and inertial forces of the control volume of the flow at any instant of time in the flow direction. These forces are represented by the terms (1), (2), (3), and (4) and Sf of the momentum equation shown in Eq. (52.38). When the effects of certain physical processes are small relative to other processes in a flood wave, the numerically labeled terms in Eq. (52.38), which identify different momentum forces on the control volume of the flow, are characterized by widely varying magnitudes (Ferrick, 1985) with reference to the gravity force characterized by the bed slope, S0. Relatively very small magnitudes of the component forces with reference to S0 can warrant their truncation or elimination, resulting in various simplified momentum equations. Henderson (1966) showed by an order of magnitude analysis that the terms (2), (3), and (4) are typically very small when compared with the term (1), the bed slope S0 for a very fast rising flood. Similarly, Kuchment (1972), as quoted by Dooge (1980), showed that terms (1) and (2) are of the same order of magnitude, but terms (3) and (4) are generally several orders smaller than the term (1). The inferences of Henderson (1966) and Kuchment (1972) were confirmed by Weinmann and Laurenson (1979), and Zoppou and O’Neill (1982) using observed flood waves in the rivers of Australia. Several researchers (e.g., Stoker, 1957; Cunge et al., 1980; Fread, 1981; Lai, 1986) argued in favor of using a numerical solution of the full Saint–Venant Equa­ tions, also known as the dynamic wave equations, for flood routing studies based on the accuracy of the results and technical advances in the computational methods and their increasing availability. Due to these arguments flood routing studies based on simplifications of the dynamic wave equations are perceived as being inherently less accurate (NERC, 1975). But the dynamic wave routing methods are important for studying river flood waves primarily when the inertia force is significant, and their importance diminishes as this force decreases with respect to gravity. When modeling flood wave propagation in rivers, the dynamic waves propagate away from their sources with speeds u ± gy and are insignificant in affecting the flood wave, whereas the bulk of the wave propagates with the KW speed. It was pointed out by Ferrick (1985) that when the magnitudes of different terms in the momentum equation are widely varying, the use of simplified momentum equations could be necessary because the use of all the terms in the momentum equation can lead to the problem of “stiffness” in the numerical solution. A case in point is the modeling of the movement of flood waves in steep reaches where the flow can be super-critical, using an implicit non-linear solution of the full Saint–Venant equations whether based on a four-point or six-point numerical solution of the nonlinear finite difference equations. Recognizing this problem, the HECHMS model (USACE, 2010) recommends the use of a variable parameter Muskingum–Cunge (VPMC) method proposed by Ponce and Yevjevich (1978) for studying flood waves in steep rivers. Similarly, the HEC-RAS model (USACE, 2010) advocates the use of the modified Puls routing method, which is generally used for storage routing in reservoirs. To avoid the problem of stiffness in the numerical solution of the Saint–Venant equations, the Flood Studies Report (NERC, 1975) and Ferrick (1985) recommend the use of appropriate wave type equations to obtain accurate solutions. Ferrick (1985) identified three major classifications of river waves, namely: (1) Bulk waves, (2) Dynamic waves, and (3) Gravity waves. Bulk waves are categorized as either diffusive or KWs. Flood waves in free flowing rivers are more commonly bulk waves, and they propagate with the KW celerity (Lighthill and Whitham, 1955; Ferrick et al., 1984). Lighthill and Whitham (1955) pointed out that, in flood wave propagation in rivers, the dynamic wave behavior of a flood wave becomes unimportant in relation to the behavior of the flood wave as a KW. The findings of Ponce and Simons (1977) on river behavior using linear stability theory substantiate the observation of Lighthill and Whitham (1955). Therefore, it is no wonder that storage routing methods which try to mimic the behavior of diffusion wave and KW have become successful in practice, except under situations where downstream control has a significant impact on the flow in the routing reach. A significant body of literature is available on the development and application of diffusive and KWs for flood routing studies (NERC, 1975; Ferrick et al., 1984; Ferrick, 1985).

52_Singh_ch52_p52.1-52.16.indd 7

52.12  SIMPLIFIED HYDRAULIC FLOOD ROUTING METHODS

The simplified hydraulic flood routing methods may be classified into two major groups as 1. Directly derived simplified methods 2. Indirectly derived simplified methods 52.12.1  Directly Derived Simplified Flood Routing Methods

This class of simplified methods is derived directly from the full Saint–Venant Equations after truncating or linearizing or approximating some of the terms in the momentum equation. The following simplified equations governing the propagation of a flood wave are derived directly from the Saint–Venant equations: 1. Convection-diffusion (CD) equations 2. Kinematic wave (KW) equations 3. Approximate convection-diffusion (ACD) equations 4. Linearized Saint–Venant equation Convection-Diffusion Equations CD equations are also referred in the literature as diffusive wave equations or zero-energy equations as they describe waves with perceptible attenuation during flood wave propagation along the river by ignoring the inertial terms of the momentum equation. CD equations can be formulated using either the flow depth or the discharge as the operating (or dependent) variable, and therefore depending on the operating variable used the solution would either be a discharge or a stage hydrograph; given one form of hydrograph, the other can be derived from it. The application of the CD equation for the flow depth described by Eq. (52.20) was first employed by Hayami (1951), though he did not derive this equation directly from the Saint–Venant Equations. In rivers with small bed slope, the magnitudes of convection and local accelerations can be assumed to be negligible; therefore, flood wave movement in these rivers can be described by the momentum equation with only the gravity and pressure forces described by the first two terms of Eq.  (52.38). By differentiating the expression of unsteady discharge using this curtailed momentum equation, the following modified forms of the momentum equations can be obtained:

∂Q ∂A ∂2 A =c − D 2 (52.39) ∂x ∂x ∂x



∂Q ∂A ∂2 A (52.40) =c −D ∂t ∂t ∂ x ∂t

The diffusion coefficient D is expressed as

D=

Q (52.41) 2S f B

and the generalized expression for the wave celerity is given by Singh (1996) as

  P dR   c = 1 + m  v  B dy   

(52.42)

where, B is the top width of the section corresponding to the flow depth y; P, and R, respectively, denote the wetted perimeter and the hydraulic radius of the flow section. Use of Eq. (52.39) with the continuity Eq. (52.37) leads to the CD equation in terms of the flow depth as:

∂y ∂y ∂2 y q +c =D 2 + l ∂t ∂x ∂x B



∂Q ∂Q ∂2 Q +c = D 2 + cql (52.44) ∂t ∂x ∂x

(52.43) Similarly, the use of Eq. (52.40) with the continuity Eq. (52.37) leads to the CD equation in terms of the discharge as

The details of the development of these equations can be found in Miller and Cunge (1975), Ponce (1989), and Singh (1996). The CD equations in their linear and nonlinear formulations may be solved using numerical methods (Thomas and Wormleaton, 1970, 1971; Price, 1973; NERC, 1975; Akan and Yen, 1977; Katopodes, 1982). The linear form of the equations may also be solved using analytical techniques (Hayami, 1951;

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52-8     Reservoir and Channel Routing

Dooge, 1973). The application of Eqs. (52.39) and (52.40) assumes that the routing reach is prismatic and the entire flow section of the routing reach can be considered as an active flow section in conveying the flow. However, this condition may not be fully met in practice and, therefore, this prompted Hayami (1951) to evaluate the diffusion coefficient and the wave celerity using a set of recorded hydrographs for the inflow and the corresponding outflow. This enabled him to estimate the wave celerity c and the diffusion coefficient D taking into account the non-prismatic section reach with and without an active flow area of cross section for the flood event analyzed. However, Hayami (1951) only used the linear form of the CD equation [Eq. (52.20)] in which he assumed that all the recorded flow depths of the flood wave propagate with the same celerity and diffuse with the same diffusion coefficient, which is well known not to be the case for flood wave propagation in natural rivers. To enable the study of flood wave movement in rivers and channels considering their nonlinear characteristics, Eqs. (52.43) and (52.44) may be studied using numerical methods or using the quasilinear or multilinear solution approach (Todini and Bossi, 1986). The latter technique uses Hayami’s approach with the parameters c and D remaining constant during a given routing time interval but varying from one routing time level to the next. Kinematic Wave Equations The seminal paper on KWs describing non-attenuating flood waves is by Lighthill and Whitham (1955). The mathematical description of KW assumes the absence of the pressure and inertial terms; that is, terms (2), (3), and (4) of the momentum equation given by Eq. (52.38). Accordingly, the governing momentum equation for this wave is

S f = S0 (52.45)

By equating the friction slope Sf to the bed slope S0, there is the danger that the KW may be erroneously interpreted as having a uniform flow. However, because Eq. (52.45) is used with the unsteady flow continuity equation given by Eq. (52.37), KW will not in general describe uniform flow. Using Eq. (52.45) with the unsteady discharge equation, one can arrive at the following alternative forms of the momentum equation:

∂Q ∂A (52.46) =c ∂x ∂x



∂Q ∂A (52.47) =c ∂t ∂t

The use of Eq. (52.46) with Eq. (52.37) leads to the KW equation in its depth formulation as

∂y ∂ y ql +c = (52.48) ∂t ∂x B

The use of Eq. (52.47) with Eq. (52.37) leads to the KW equation in its discharge formulation as

∂Q ∂Q +c = cql (52.49) ∂t ∂x

Application of KW theory for studying unsteady flow is more suitable for overland flow modeling rather than for flood routing in channels as river flood waves have a tendency to distort their profiles even if they do not attenuate their peak flows, resulting in the change of their hydrograph shape while propagating down the channel (Miller, 1984). Detailed theoretical descriptions of kinematic flood waves can be found in the works of Singh (1996) and Miller (1984). Approximate Convection-Diffusion Equations A new wave type known as the approximate diffusive wave was advocated by Perumal and Ranga Raju (1999). The reason for introducing this wave type stems from the necessity of understanding the hydrodynamic basis behind the success of applying storage-based routing methods, such as the Muskingum method and the Kalinin–Milyukov method, in the field over many years. The development of these storage routing methods employs the heuristic assumption that the water surface varies linearly along the routing reach, implying that the water surface slope (1/S0) ∂ y ∂ x is small, but not negligible, such that one may consider this ratio is approximately constant at any instant of time during unsteady flow in the considered short routing reach, but may vary from one routing time interval to the next. This further ∂2 y ∂2 Q implies the insignificant role of the terms ∂ x 2 and ∂ x 2 in describing these flood waves. Accordingly, the momentum equations governing this flood

52_Singh_ch52_p52.1-52.16.indd 8

wave in terms of flow depth and discharge can be deduced from the respective formulation of the momentum equations for the diffusive wave given by Eqs. (52.39) and (52.40) as:

∂Q ∂A (52.50) =c ∂x ∂x



∂Q ∂A (52.51) =c ∂t ∂t

Combining these simplified momentum equations with the continuity equation given by Eq. (52.37) results in the governing equations of the approximate diffusive wave in terms of flow depth and discharge, respectively, as

∂y ∂ y ql +c = (52.52) ∂t ∂x B



∂Q ∂Q +c = cql (52.53) ∂t ∂x

Comparing Eqs. (52.48) and (52.52), and (52.49) and (52.53) reveals that they are identical in the dependent variables of flow depth and discharge. However, Eqs. (52.52) and (52.53) are more generalized than the corresponding Eqs. (52.48) and (52.49) as the former set of equations has wider applicability than the latter, which are more restricted in modeling flood waves characterized by the magnitude of (1 S0 )(∂ y ∂ x ) ≈ 0. Further, as these equations have been developed based on the assumption of flow depth varies approximately linearly along the reach, they are capable of accounting for flood wave diffusion, which lies within the transition range between diffusive and KWs. Based on this consideration, Perumal and Ranga Raju (1999) and Perumal et al. (2004) justified the logic behind the well-known Jones formula used for estimating the flood discharge at a site, knowing the flood depth there. Equation (52.52) is used rather than Eq. (52.48) with ql = 0 as has been proposed by Henderson (1966). It may also be stated that Eq. (52.53) is more generalized than Eq. (52.49) as the range of applicability of the former equation for modeling flood waves is wider than the latter as is demonstrated in the following text. 52.12.2  Indirectly Derived Simplified Flood Routing Methods

Storage Routing Methods Indirectly derived simplified methods are not derived directly from the Saint– Venant equations, but instead use a lumped continuity equation in place of Eq. (52.37) and a linear or nonlinear storage equation. The storage equation expresses the storage as a function of inflow and/or outflow in a channel reach, but without describing how the storage is distributed within the reach. The concept of using a storage equation was first proposed by Zoch (1934) in the context of rainfall-runoff modeling using a SLR. This expresses the catchment storage due to the rainfall as a linear function of outflow. However, the use of a SLR to represent the channel storage is not suitable for modeling flood wave propagation in channels except in the case of short channel reaches. This is because the linear reservoir predicts that the peak outflow occurs on the recession of the inflow hydrograph, and therefore the model cannot accurately predict both the attenuation and translation of the flood wave. However, the channel storage can be represented realistically by a series of SLRs which is capable of accounting separately for the attenuation and the translation effects of a channel reach. In particular, the linear cascade model, proposed by Nash (1957) for basin rainfall-runoff modeling, is suitable for studying flood wave movement in rivers. This aspect was discussed in the Section on hydrologic routing methods. Dooge (1973) pointed out that the conceptualization of the Nash model appears in the Kalinin–Milyukov method (Kalinin and Milyukov, 1957) for modeling flood wave movement in a river reach, but they develop the concept from hydrodynamics principles. The linear storage equation proposed by McCarthy (1938), which expresses the storage as a linear weighted function of the inflow and outflow, forms the basis for the development of the classical Muskingum method. This method is a hydrological routing method and employs the routing parameters evaluated from a set of recorded inflow and outflow hydrographs for a river reach. These parameters remain constant over the duration of routing process. It was pointed out when describing the hydrological routing methods that these methods have limited practical applicability for modeling flood waves in that they are restricted to the range of past floods employed in the estimation (or calibration) of the parameters for these methods. Because the classical Muskingum method has been successfully used in the field for many years, various researchers have proposed different approaches to establishing the hydrodynamics link of this method to the equations governing unsteady

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The Muskingum Routing Method    52-9 

flow in rivers. Their objective is to relate the parameters of the method to the flow and channel characteristics so as to enable the method to be more applicable for routing floods which were not observed in the past and also for routing floods in ungauged river reaches. Well-known approaches seeking to address this issue are due to Apollov et al. (1964), Cunge (1969), Dooge (1973), Koussis (1976), and Dooge et al. (1982). The method proposed by Cunge  (1969) is widely referred in the literature as the Muskingum–Cunge (MC) method (NERC, 1975; Ponce, 1989). With the objective of accounting for the nonlinear behavior of the flood movement process, Ponce and Yevjevich (1978) introduced the VPMC method, which was later improved by Ponce and Chaganti (1994) to enable the variation of the parameters of the method at every time interval in a predefined manner. Recently, a number of improved physical interpretations of the Muskingum method have been proposed with the objective of varying the parameters of the method in a way consistent with the Saint–Venant equations, and in particular, for overcoming the mass conservation problem with the VPMC method. These methods include studies by Todini (2007), Price (2009), and Perumal and Price (2013). This Section describes the hydrodynamics basis of the Kalinin–Milyukov method (Kalinin and Milyukov, 1958) and the classical Muskingum method, as conceptualized by McCarthy in which the reach storage is a summation of prism and wedge storages. The hydrodynamics approach helps to establish the relationships between the parameters of the method and the channel and flow characteristics. As the flood waves are inherently nonlinear in nature with different discharges associated with the flood wave travelling at different celerities, the linear storage routing methods, even if they are developed from the equations describing one-dimensional flood wave movement in rivers, are of limited use for flood routing studies. The hydrodynamics based Muskingum method presented herein overcomes this deficiency. 52.13  KALININ–MILYUKOV METHOD

The Kalinin–Milyukov method subdivides a channel reach under consideration into a number of so-called “characteristic reaches” relating the storage within each characteristic reach linearly to the respective outflow discharge for the reach. The method is developed based on the following assumptions (Apollov et al., 1964; Perumal, 1992): 1. The discharge at any given section for any instant is a function of depth of flow and the slope of the water surface at that instant. 2. The looped rating curve for unsteady flow can be reduced to a single valued rating curve between the discharge at a given section for a given time and the depth of flow at that section after an interval ∆t, and consequently to a single-valued relationship between the discharge at the given section and the depth of flow some distance upstream from the discharge section for a given time. 3. The steady flow regime in the channel corresponding to a discharge, Q, is replaced by an unsteady flow regime in such a way that the discharge at the outflow section of a characteristic reach does not change, that is, dQ = 0. 4. The new slope of the water surface under unsteady conditions is constant throughout a reach during a time step. Using assumptions (1), (2), (3), and (4), the distance, d, between the outflow section and the mid-section of a reach, whose depth of flow is uniquely related with the discharge at the outflow section, can be shown to be (Miller and Cunge, 1975):

d=

∂Q ∂S f ∂Q ∂ y

(52.54)

where Sf is the friction slope and y is the depth of flow. If the unsteady flow is considered as a small perturbation from a steady condition, then the expression given by Eq. (52.54) can be written as (Miller and Cunge, 1975):

d=

Q0 dQ (52.55) 2S0 0 dy0

where S0 is the bed slope, Q0 is the reference discharge and y0 is the depth of flow corresponding to Q0. The characteristic length of the Kalinin–Milyukov method is given as 2d. Since the term dQ dy is derived from a single valued rating curve, it can be expressed as:

dQ dQ =B (52.56) dy dA

where B is the water surface width corresponding to the flow depth, y, at the given section and dQ dA represents the wave celerity, c.

52_Singh_ch52_p52.1-52.16.indd 9

Equation (52.55) may be rewritten using Eq. (52.56) as

d=

Q0 (52.57) 2S0 B0c0

where B0 and c0 are respectively the water surface width and wave celerity corresponding to Q0. The characteristic reach length of the Kalinin–Milyukov method is 2d. Therefore, the number of characteristic reaches required for routing a flood hydrograph in a given length of reach ∆x is

N=

S0 B0c0 ∆x (52.58) Q0

and lag time of each characteristic reach is given by

K=

Q0 (52.59) S0 B0c02

Using the moment matching technique, Dooge (1973) related the first and second moments of the IUH of the Nash model with the corresponding moments of the linearized Saint–Venant model, when F = 0, and arrived at the same relationships for the number of linear reservoirs in series and the storage coefficient as given by Eqs. (52.58) and (52.59), respectively. The Nash model is a conceptual representation of the Kalinin–Milyukov method when its parameter representing the number of linear reservoirs in a reach is an integer. Channel routing using the Nash model with the parameter evaluated using Eqs. (52.58) and (52.59) is more flexible than the Kalinin–Milyukov method because it can also operate with non-integer N values. This avoids the interpolation of the outflow hydrograph when the last characteristic reach of the Kalinin–Milyukov method does not coincide with the outflow section of the given routing reach (Koussis, 1980). 52.14  THE MUSKINGUM ROUTING METHOD

The hydrological literature is replete with the world-wide use of flood routing methods having a range of interpretations of the Muskingum routing method. Consequently, during the last several decades, the classical Muskingum method originally proposed by McCarthy in 1938 has been transformed from its semiempirical characteristics into a physically based model accounting for the nonlinear dynamics of flood wave movement. To gradually transform the classical Muskingum method into a physically based model by linking its routing parameters with the measurable flow and channel characteristics, several researchers have attempted a number of interpretations of this method. All such interpretations were grouped by Kundzwicz (1986) under the following approaches: (1) direct interpretation; (2) matching difference schemes; and (3) matching the impulse response of the Muskingum method with that of the linearized Saint–Venant equation using the method of moments approach (Dooge, 1973). However, with the recent developments in the Muskingum method, these interpretations can be further categorized as 1. Classical interpretation of Muskingum storage equation (Chow, 1959) 2. Direct interpretation (Brakensiek, 1963; Strupczewicz and Kundzewicz, 1980) 3. Matching the impulse response of the Muskingum method with that of the linearized Saint–Venant equation using the method of moments approach (Dooge, 1973) 4. Matching difference schemes (Harley, 1967; Cunge, 1969; Koussis, 1976; Dooge et al., 1982; Ponce and Yevjevich, 1978; Ponce and Chaganti, 1994; Todini, 2007) 5. Using the Kalinin–Milyukov routing method (Apollov et al., 1964; Perumal, 1992) 6. First-order approximations of the Saint–Venant Equations (approximations of CD equations) (Price, 2009) 7. Using the heuristic interpretation of McCarthy (Perumal and Price, 2013) A brief description of all these interpretations is discussed as follows. Classical Interpretation of Muskingum Storage Equation Chow (1959) interpreted the classical Muskingum storage equation using the following concepts: 1. Both inflow (I) and outflow (O) discharges at the inlet and outlet sections of the prismatic river reach are uniquely related to the respective stages (y). 2. The stages at the inlet and outlet sections of the Muskingum reach are uniquely related to the reach storages (S) established at the upstream reach of these respective sections.

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52-10     Reservoir and Channel Routing

Based on these concepts, and using the power function relationship, the dependent variables can be written as:

I = ay n , O = ay n , Si = by m , So = by m (52.60a–d)

where, Si and So are the storages at the inlet and outlet sections, respectively; and a, b, m, and n are the empirical values. The implicit assumption behind the set of Eqs. (52.60) is that the channel or river reach is prismatic. From Eqs. (52.60a–d), it follows that

 I Si = b    a

m/n

 O , So = b    a

m/n



(52.61a, b)

By weighting the reach storages at the inlet and outlet sections using the weighting factor θ:

S = θ Si + (1 − θ ) So (52.62)

Next, using Eqs. (52.61a, b) in (52.62), the expression for the storage becomes

(

)

S = K θ I m/n + (1 − θ )O m/n (52.63)

where K = ba −m n is the storage coefficient. If m n = 1, Eq. (52.63) yields the classical Muskingum storage equation. There is, however, a logical flaw in this interpretation, because the storage Eq. (52.63) is derived using the one-to-one steady state relationship between stage and discharge, which is not strictly valid for unsteady flow conditions. Further, there is no physical basis for m n = 1. In view of these shortcomings, the appropriateness of Chow’s (1959) interpretation of the classical Muskingum method should be reconsidered. Direct Interpretation from the Saint–Venant Equations Brakensiek (1963) was probably the first researcher publishing in English who attempted to interpret the Muskingum method using the hydrodynamic principle based on the following concepts: 1. Under subcritical flow conditions, unsteady flow in a river is closer to the KW movement. 2. The Muskingum method is more suitable for modeling KWs in a river reach which has a one-to-one relationship between stage and discharge. 3. The reach storage can be expressed as: Reach storage = Average of the flow areas of upstream and downstream boundaries × Length of channel reach, L. If the outflow discharge with mean velocity V is given by the power function: Q = aAb, then the Muskingum routing parameters can be derived using the hydrologic mass conservation equation assuming a uniform channel section as (Brakensiek, 1963):

Q = aAb , K =

L 1 , θ= V 2b

(52.64) where A = flow area corresponding to flow Q; and L = reach length. However, the interpretation by Brakensiek (1963) for the Muskingum parameters is not consistent with the parameter relationships established by the current theories. Subsequently, Strupczewski and Kundzewicz (1980) attempted to interpret the Muskingum method directly from the Saint– Venant equations, but they could not express the flood storage within the reach as a linear function of the weighted discharge due to their interpretation of a one-to-one relationship between the stage and the discharge for depicting the storage equation of the Muskingum method. Interpretation Based on the Moment Matching Approach By matching the first and second moments of the IUH for the linearized Saint–Venant equation with the corresponding moments of the Muskingum IUH, and using the Chezy or Manning’s friction law, Dooge (1973) arrived at the following expressions for the routing parameters K and θ of the Muskingum method applicable to a wide rectangular channel reach, respectively, as



52_Singh_ch52_p52.1-52.16.indd 10

K=

∆x co

(52.65)

Qo yr 1 1 − (m − 1)2 Fo2  (52.66) θ= − 2 2So Aoco ∆x 

where yr is the average hydraulic mean depth; Ao is the area of cross section corresponding to Qo; and m = 3/2 or 5/3 for the Chezy’s or Manning’s friction law used, respectively. Interpretation Based on the Matching Difference Schemes Dooge and Harley (1967) presented the Muskingum parameter relationships for a wide rectangular channel and flow characteristics based on the moment matching technique (Nash, 1957), and Dooge et al. (1982) later arrived at the same for any shape of prismatic channel and for any type of friction law. However, it is Cunge’s (1969) matched diffusivity approach which has become more popular as the MC method, the name first coined by Price (NERC, 1975). By matching the numerical diffusivity of the approximate linear KW equation, derived from the classical Muskingum routing equation with the physical diffusivity of the linear CD equation, Cunge (1969) arrived at the relationship for K as given in Eq. (52.65) and θ for wide rectangular channels as

1 Qo θ= − 2 2So Boco ∆x

(52.67)

On the basis of the state trajectory variation method applied to the Saint– Venant Equations, Dooge et al. (1982) established the generalized relationships for K and θ for prismatic channels having any shape of cross sections, in which the relationship for K is the same as Eq. (52.65) and the relationship for θ is expressed as given in Eq. (52.66). Todini (2007) proposed a mass conserving variable parameter Muskingum method, known as the Muskingum–Cunge–Todini (MCT) method based on Cunge’s matched diffusivity approach (Cunge, 1969), but estimated the parameter, K, of the Muskingum method based on the normal velocity (vo) corresponding to the unsteady flow depth at the middle of the Muskingum reach, rather than using the celerity as envisaged by Cunge (1969). The routing parameters of the MCT method are given as (Todini, 2007)

K =

∆x (52.68) vo

1  Qovo  θ = − (52.69) 2  2So Boco2 ∆x 

where Q0 is the average of the inflow and outflow discharges and is considered as the normal discharge corresponding to the flow depth y0 , at the midsection of the reach, B0 is the water surface width corresponding to y0 , and c0 is the wave celerity corresponding to Q0. The MCT method also enables the stage hydrograph of the routed hydrograph to be estimated; however, at the midsection of the last subreach adjacent to the routing reach outlet. Interpretation Based on the extension of the Kalinin–Milyukov method In describing an extension of the Kalinin–Milyukov method as an interpretation of the Muskingum method, Apollov et al. (1964) estimated the parameters K and θ of the classical Muskingum method in the same way as given by Cunge (1969). Interpretation Based on the Approximations of Convection-Diffusion Equations Price (2009) proposed a volume conservative routing method based on an approximation of the Saint–Venant Equations such that the parameters are determined using the channel and flow characteristics. This resulted in the same parameter relationships as developed by Todini (2007). Although the approaches followed by Todini (2007) and Price (2009) in arriving at the routing equations are different, the parameter relationships of these methods are identical. Therefore, both methods should yield the same routing results. Price (2009) established in particular a CD equation for flood wave propagation in steep rivers that is solved numerically using a four-point finite difference scheme which results in the same numerical scheme used to apply the Muskingum method. The routing equation is:

∂Q ∂Q ∂  aQ ∂Q  ∂ +c + εc  2 aq q + O ε 2 (52.70)  = cδ q + ε c ∂t ∂x ∂t  c ∂ x  ∂t

{ } ( )

where c is the celerity as a function of discharge, aQ is the attenuation parameter and aq is a lateral inflow parameter. Equation (52.70) is accurate to order ε 2 , where ε is the parameter defined as the ratio of the flow depth slope to the bed slope. The equation only requires an initial condition and an upstream boundary condition, namely the inflow discharge as a function of time. There is no need for a downstream boundary condition. Just as most of

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The Muskingum Routing Method    52-11 

the flood routing methods discussed in this chapter are independent of any downstream boundary, they are only valid strictly for flows in steep rivers. Greater effort should be given to approximate flood routing methods that are applicable to almost flat or nearly flat river reaches. Such reaches occur in many rivers at their downstream limits. Solutions of the non-inertial equations can provide a sound basis for flood routing methods applicable to flat or nearly flat river reaches. Such methods depend radically on the downstream boundary condition; Price (2015) (Private communication). Using the Heuristic Interpretation of McCarthy Although the VPMC method proposed by Ponce and Yevjevich (1978) has been widely used in practice (DAMBRK by Fread, 1990; HEC-HMS by USACE, 2010, 2013), this method is saddled with a perceptible loss of mass (Ponce and Chaganti, 1994; Perumal and Sahoo, 2008). Subsequently, the well-known mass conservation deficiency of the VPMC method has been tackled by Todini (2007), Price (2009), and Perumal and Price (2013). However, to date, only the VPMM method advocated by Perumal and Price (2013) has explained the validity of the heuristic assumption proposed by McCarthy (1938) when developing the classical Muskingum method that the reach storage consists of two component storages, namely, the prism and wedge storages as depicted in Fig. (52.2). The successful application of the classical Muskingum method in the past as well as in current practice implies the validity of McCarthy’s (1938) storage concept. The VPMM routing method is developed from an approximation of the momentum equation of the Saint–Venant equations. Subsequently, this approximation is applied directly to the one-dimensional continuity equation of the Saint–Venant equations leading to a fully mass conservative routing method (Perumal and Price, 2013). It turns out that the form of the routing equation for this method is the same as the classical Muskingum method proposed by McCarthy in 1938. The development of the method based on hydraulic principles enables the characterization of the storage in the considered channel reach as the prism and wedge storages envisaged by McCarthy (1938) in the classical Muskingum method. The form of the storage equation arrived at from the development of the VPMM method is identical to that of the classical Muskingum method with the travel time, K, and weighting parameter, θ, respectively, linked to the flow and channel characteristics (Perumal and Price, 2013). However, the applicability of the method is restricted by the assumptions of no lateral flow and downstream effects in the reach. A brief theoretical background of the method is appropriate herein for appreciating its capability. Theoretical Background of the VPMM Method The VPMM method assumes that at any instant of time during unsteady flow, a steady flow relationship is applicable between the stage at the middle of the routing reach and the discharge passing somewhere downstream of it. This assumption is also employed in the Kalinin-Milyukov method (Apollov et al., 1964; Miller and Cunge, 1975). Therefore, the normal discharge Q3 passing just downstream of the midsection of the computational reach (section 3 in Fig. 52.6) is uniquely related to the stage y M at the midsection (section M in Fig. 52.6) of the computational reach having the length ∆x.

The governing finite difference equation of the VPMM method derived as a simplification of the full Saint–Venant equations and developed on the basis of application of the hydraulic continuity equation at the midsection of the numerical grid as shown in Fig. (52.1) assuming no lateral flow in the reach can be given as (Perumal and Price, 2013):

 Q j +1 Q j   Q j +1 + Qij Qij++11 + Qij+1  ∆x  j3+1 − j 3  = ∆t  i −  (52.71)   2 2  v0,M v0,M 

where Q3j and Q3j +1, Qij and Qij +1 , Qij+1 and Qij++11, and v0,j M and v0,j+M1 are the normal discharge, inflow, outflow, and normal velocity at the midsection, where j denotes the temporal grid level; “M” denotes the midsection of the spatial grid size ∆x; i denotes the spatial grid location; and the suffix (0,M) associated with any variable denotes its value corresponding to normal depth at the midsection of the computational subreach. The explicit expression of the unknown variable of Eq. (52.71) leads to the classical Muskingum (McCarthy, 1938) routing equation as

Qij++11 = C1Qij +1 + C2Qij + C3Qij+1 (52.72)

where

∆t − 2 K j +1θ j +1 ∆t + 2 K jθ j −∆t + 2 K j (1 − θ j ) C1 = j +1 j +1 ; C2 = j +1 j +1 ; C3 = ∆t + 2 K j +1 (1 − θ j +1 ) ∆t + 2 K (1 − θ ) ∆t + 2 K (1 − θ ) (52.73a–c) and the travel time K at the time level (j+1) is expressed as

K j +1 =

∆x v0,j +M1

(52.74)

The weighting parameter, θ at the time level (j+1) can be given by j +1



 4  PdR dy  2  Q3j +1 1 − F 2     9  dA dy   M 1 j +1 θ = − (52.75) 2 2.So .BMj+1 .coj +,M1 .∆x

Assuming that the magnitudes of the inertial terms are negligible in natural flood waves (Henderson, 1966; Price, 1985), Eq. (52.75) can be modified assuming FM ≈ 0 , as

1 Q3j +1 θ j+1 = − (52.76) 2 2.So .BMj+1 .coj +,M1 .∆x

When a constant discharge is used as the reference discharge, the expressions for K and θ reduces to

K=

∆x (52.77) vo , M r

and 1

M

where, the suffix r of K and θ refers to the reference level discharge Qr. The flow depth yij++11 corresponding to the outflow Qij++11 is estimated using equation (Perumal and Price, 2013)

3 2

yu



I yM

QM y3

Q3 yd Q

∆x/2 ∆x

L

Figure 52.6  Definition sketch of the VPMM routing reach.

52_Singh_ch52_p52.1-52.16.indd 11

Qr 1 θ= − (52.78) 2 2.Sro . BoM r .co ,M .∆x r

yij++11 = y Mj+1 +

(Qij++11 − QMj+1 ) (52.79) BMj+1c Mj+1

where y Mj+1 and QMj+1 are the stage and discharge variables at the midsection of the routing reach, respectively. As a consequence of the ability of the VPMM method to explain the classical Muskingum method as conceptualized by McCarthy (1938), it may be inferred that the attenuation introduced in the classical Muskingum method can be attributed to the reach storage effect, without the need to explain it based on matched diffusivity theory as advocated by Cunge (1969). Based on the mathematical approach employed in the development of the VPMM method, Perumal and Price (2013) identified the applicability criterion for the method to be Abs.[(1/S0)(∂ y / ∂ x )] ≤ 0.5. This was based on the application of the Binomial series expansion in the development of the method to retain only the first-order term (1/S0) ∂ y / ∂ x .

8/22/16 2:23 PM

52-12     Reservoir and Channel Routing

River Reach Details and Establishment of Reach-Averaged Rating Curve and Stage-mean Flow Area Relationship The practical applicability of the VPMM method to route flood hydrographs in natural rivers was tested by routing the recorded flood hydrographs in the 15 km river reach between the Pierantonio and Ponte Felcino gauging stations of the Tiber River in Central Italy. The reach-averaged bed slope S0 for this reach was estimated as 0.0016. The existing actual rating curves at Pierantonio (upstream) and Ponte Felcino (downstream) stations were averaged to arrive at the reach-averaged rating curve for the Pierantonio-Ponte Felcino reach as shown in Fig. (52.7). The water surface widths (dA dy ) were extracted from the established stage-flow area relationship using the first order backward difference scheme. Similarly, the wave celerity, (dQ dA), and the flow velocity, v( y ), were extracted from the discharge-flow area relationships. Therefore, the look-up tables required for the reach-averaged channel and flow characteristics (dQ dA, dA dy , and v) versus the normal depth were generated using the reach-averaged rating curve and the stage-mean flow area information. The characteristic relationships between the discharge, wave speed and velocity versus the normal depth generated look-up tables for the Pierantonio-Ponte Felcino reach are shown in Fig. (52.8). The stage hydrograph at the outlet of the reach corresponding to the routed discharge hydrograph arrived at that section can be estimated using the VPMM method. It may be noted that the assumption of prismatic channel reach in the development of the VPMM method does not affect the routed discharge hydrograph

8

Pierantonio station Ponte Felcino station Reach-averaged

7

Stage (m)

6 5 4 3 2

Pierantonio station Ponte Felcino station Reach-averaged

300 250 200 150 100 50 0

0

2

4 Stage (m)

6

8

Figure 52.8  Flow depth versus flow area relationship of Pierantonio and Ponte Felcino gaging stations and the reach-averaged relationship.

at the outlet, because the discharge is a volume conserving variable. However, the stage variable is sensitive to the local geometrical variations and, therefore, a conversion equation is needed to convert the estimated stage hydrograph at a reach-averaged section to the corresponding actual stage hydrograph at that section. This required conversion equation is estimated as yactual = 0.927 * yequivalent + 0.062 and it is adopted to enable the conversion of equivalent stage hydrograph value estimated by Eq. (52.79) at the outlet of the reach to that of the actual stage value at that section. Details of Routing Simulations Ten recorded flood events consisting of floods of December 1996; April 1997; November 1997; February 1999; December 2000; April 2001; November 2005; December 3, 2005; December 5, 2005; and December 30, 2005 were routed in this reach. These flood events were routed from Pierantonio to Ponte Felcino by the VPMM method using a space step of ∆x = 1 km (i.e., 15 subreaches) and a routing time interval of ∆t = 1800 s. The pertinent characteristics of these simulated events are given in Table 52.1. It can be seen from this table that all the recorded discharge hydrographs at the reach outlet could be reproduced with ηq > 95.30% , except those of December 2000 and November 2005 flood events for which the Nash–Sutcliffe (Nash and Sutcliffe, 1970) model efficiency is ηq 90%. As noted earlier, these two flood events recorded significant lateral flows as given in Table 52.1. It can be inferred from this table that the errors in reproducing the peak discharge |qper | 3.46% (but