Vision Through the Atmosphere 9781487586140

Table of contents :
PREFACE
CONTENTS
FIGURES
TABLES
TABLE OF NOTATION
CHAPTER ONE. INTRODUCTION
CHAPTER TWO. ELEMENTARY PHOTOMETRIC THEORY
CHAPTER THREE. THE EXTINCTION OF LIGHT IN THE ATMOSPHERE
CHAPTER FOUR. THE ALTERATION OF CONTRAST BY THE ATMOSPHERE
CHAPTER FIVE. THE RELEVANT PROPERTIES OF THE EYE
CHAPTER SIX. THE VISUAL RANGE OF OBJECTS IN NATURAL LIGHT
CHAPTER SEVEN. VISUAL RANGE OF LIGHT SOURCES AND OF OBJECTS ILLUMINATED BY ARTIFICIAL LIGHT
CHAPTER EIGHT. THE COLORS OF DISTANT OBJECTS, AND THE VISUAL RANGE OF COLORED OBJECTS
CHAPTER NINE. INSTRUMENTS FOR MEASURING VISUAL RANGE AND RELATED QUANTITIES
CHAPTER TEN. THE SPECIAL PROBLEMS OF THE METEOROLOGIST
CHAPTER ELEVEN. A NEW VISUAL SCIENCE-CONCLUSION
BIBLIOGRAPHY
INDEX

Citation preview

VISION THROUGH THE ATMOSPHERE

IN recent years, the problem of seeing through the atmosphere has been given intensive and costly consideration in several quarters, but particularly in the United States and Great Britain. A problem which once concerned mainly the meteorologists has become of great importance in military tactics as well as in peacetime transportation. The present volume is the only full account in English of the physical, physiological, and psychological factors which lie at the basis of the calculation of the range of vision through the atmosphere. There is an extended chapter on instruments and one on the author’s own theory of the colours of distant objects. The figures are from many sources although many of them have been drawn specially for this book. The bibliography contains 420 entries nearly all of which are directly referred to in the text. has recently retired after sixteen years with the Canadian Meteorological Service followed by seventeen years with the National Research Council of Canada. He is a Fellow of the Royal Society of Canada and of the Optical Society of America. In 1959 the latter Society awarded him its Ives Medal for his work in meteorological optics, photometry, and colorimetry. He is the author of the well-known Meteorological Instruments as well as of more than seventy papers in optics and meteorology.

W. E. KNOWLES MIDDLETION

... Jes grandes lunettes ne font pas voir plus loin que des petites a proportion de leur grandeur. . .. plus elles sont grandes, et plus elles grossissent les vapeurs, la poussiere et les autres petits corps dont !'air est to1ljours plein: et aussi on ne voit !'object que comme au travers d'un voile. -From a summary of a letter from Adrien Auzout to the Abbe Charles, in the Journal des s,avans for 1666, pp. 21-22

VISION THROUGH THE ATMOSPHERE

BY

W. E. KNOWLES MIDDLETON

UNIVERSITY

OF TORONTO PRE S S

Copyright, Canada, 1952 by University of Toronto Press Reprinted with corrections, 1958 Reprinted, 1963, 1968, 2017 in the United States of America ISBN

978-1-4875-8710 -9 (paper)

PREFACE IN THE decade which has elapsed since the appearance of the second edition of the author's Visibility in Meteorology, the problem of seeing through the atmosphere has been given intensive and very costly consideration in several quarters, particularly in the United States and in Great Britain. What originally concerned mainly the meteorologists has become of great im­ portance in military tactics as well as in peacetime transportation, and it is not too much to say that the subject has entirely changed its aspect in the last ten years. For these reasons this book has been written from an entirely new stand­ point, emphasizing the physical and psychological bases of the problem. There are no purely climatological or synoptic chapters, such as the last two in the former book; the author has not been associated with a meteoro­ logical service since 1946 and no longer feels competent to discuss such questions; but it is believed that meteorologists may still read the present work with profit, if not with much self-satisfaction. Apart from this omission the plan of the work is much the same, beginning with physics, continuing with physiology, and developing their inter­ relations to the point of predicting the visual range of objects and of lights. The discussion of photometric theory and nomenclature has been expanded into an entire preliminary chapter, which may well be passed over by any photometrists who may read the book. There is a fairly extended chapter on instruments. Every effort has been made to avoid speculative hypotheses and extrava­ gant claims for the utility of the new knowledge of the past ten years. Even at a fairly high rate of discount, however, it is difficult not to be impressed with the possibility of predicting a good many of the results of visual observation. A word about the bibliography. The approximately 420 entries are selected from a much larger number of references gathered by the writer, because they deal more or less directly with the subjects covered by this book. Purely climatological and synoptic papers have been purposely omitted. He has personally verified every entry so that any errors are likely to be mistakes in copying or typography. Inclusive page numbers have been given where applicable in the belief that readers are helped by knowing in advance how long a paper they will have to consider. The figures are from many sources and have, for the most part, been drawn or re-drawn for this book, often with changed units. Acknowledg­ ments should be made to the Council of the Royal Meteorological Society for permission to reproduce Figs. ·3.1, 6.1, and 6.2; to the Illuminating Engineering Society (London) for Figs. 3.13, 3.14, 3.15, 3.16, 3.17, 7.3, and 7.4; to the Journal of Aeronautical Sciences for Figs. 3.19 and 3.20; to the V

Vl

PREFACE

Journal of Mathematics and Physics for Fig. 2.1; to the Optical Society of America for Figs. 6.3 to 6.11, inclusive; to the United States Department of Agriculture for Figs. 6.16, 6.17, and 9.10; to the Journal of Scientific Instru­ ments for Fig. 9.29; and to the United States Office of Naval Research for permission to publish Fig. 7.1. Many people have been generous with information and figures, among whom the author especially wishes to thank Mr. J.M. Waldram, Professor H. R. Blackwell, Dr. H. W. Rose, and Dr. H. Dessens. He also would acknowledge the patient and expert co-operation of the staff of the Uni­ versity of Toronto Press, especially Miss Eleanor Harman. Finally, the author wishes to thank Professor S. Q. Duntley for many highly stimulating discussions and especially for infinite encouragement, friendship, and inspiration without which this book might never have been completed. W.E.K.M. Ottawa, May 27, 1952

CONTENTS

,. 3 4

2.3

18 3.3

60

75 81

83

104 128

138 vii

Vlll

CONTENTS

FIGURES

a vs n vs vs

V

opposite opposite opposite opposite C

X

FIGURES

5.5 Critical visual angle as a function of adaptation luminance. (After Blackwell, 1946) 90 5.6 Variation of liminal contrast with the width of a diffuse boundary (Middleton, 1937) 92 5.7 Threshold illuminance from a fixed, achromatic point source, as a function of background luminance 97 5.8 Apparent intensity of a flashing source which would have an intensity of 1 candle if allowed to appear without interruption (after Hampton, 1934) 101 6.1 Illustrating the calculation of the luminance of a vertical white object under an overcast sky 106 6.2 Luminance of a vertical white object in terms of that of the horizon sky, for various values of n and R' 107 6.3 Sighting range in yards of objects against the sky, background luminance 1000 foot-lamberts (full daylight), based on the Tiffany data for circular targets, at a probability of detection of 95 per cent 110 6.4 Sighting range of circular objects against the sky, background luminance 100 foot-lamberts (overcast day) 111 6.5 Sighting range of circular objects against the sky, background luminance 10 footlamberts (very dark day) 112 6.6 Sighting range of circular objects against the sky, background luminance 1 113 foot-lambert (twilight) 6.7 Sighting range of circular objects against the sky, background luminance 10-1 foot-lambert (deep twilight) 114 6.8 Sighting range of circular objects against the sky, background luminance 10-1 115 foot-lambert (full moon) 6.9 Sighting range of circular objects against the sky, background luminance 10-1 116 foot-lambert (quarter moon) 6.10 Sighting range of circular objects against the sky, background luminance 10-• 117 foot-lambert (starlight) 6.11 Sighting range of circular objects against the sky, background luminance 10-1 118 foot-lambert (overcast starlight) 6.12 Dependence of visual range on background luminance, for objects of area 119 10 1 to 106 sq. m. Vi - 10,000 m. Co = ± 1 6.13 Siedentopf's G-function plotted on a logarithmic scale against r / Vi as abscissa 124 for various values of the parameter Bh/Bo 6.14 Sighting range in yards of circular objects on the ground, seen from the air in full daylight, based on the Tiffany data for circular targets, at a probability of 129 detection of 95 per cent 6.15 Increase in the visual range of a small object, produced by a perfect telescope of magnification M, for three values of rand three values ofu. (After Hardy, 1946, 132 except that M has been plotted on a logarithmic scale) 6.16 Effect of direction of illumination on the luminance of a smoke column (after 133 Buck and Fons, 1935) 6.17 Effect of relative direction of sun and smoke on the time of discovery of experi134 mental forest fires in conditions of great visual range (after Buck, 1938) 7.1 Nomogram for the sighting range of lights, based on the Tiffany data (probability 139 of detection 95 per cent) 140 7.2 Illustrating the theory of searchlight illumination 7.3 Beam distribution from a 44-inch naval searchlight (after Chesterman and 141 Stiles, 1948) 7.4 Mean relative scattering function /(cj,), derived from night observations over 142 the sea. (After Chesterman and Stiles, 1948) 8.1 Chromaticity diagram and spectrum locus 147 151 8.2 Position of standard illuminants, and a diagram to illustrate color mixture 154 8.3 Transformation from trichromatic to monochromatic system 8.4 Perspective transformation of C.I.E. chromaticity diagram giving approximately uniform spacing to colors of luminance factor about 0.2 (after Farnsworth);

FIGURES

Xl

and showing effect of small subtense on twenty such colors (Middleton and Holmes, 1949) 162 8.5 The recognition of colored light signals at an illuminance of 1 lumen per 163 square mile (simplified from Hill, 1947a) 8.6 Key diagram showing regions of the chromaticity chart displayed in Figs. 8.7 165 to 8.11 inclusive 168 8.7 Apparent chromaticity of achromatic objects at a distance, in pure air 169 8.8 The same data as in Fig. 8.7, but for a light haze ( the seattering function {3' (cf>) will be proportional to the seattering coefficient b

THE ALTERATION OF CONTRAST

63

(d)

FIG. 4 .1. Illustrating Koschmieder's theory.

2

=

2

=

A.

(h); h·

•we have modernized Koschmieder's argument, and we hope clarified it, by the intro­ duction of some of the ideas of Chapter II. There is no change in its content.

64

VISION THROUGH THE ATMOSPHERE

Since the object is supposed intrinsically black, none of the light entering the eye actually originated at the object. We need not assume this con­ dition. Suppose that the object, observed close up, has a luminance Bo; then to the air-light given by equation (4.7) we have only to add the quantity B 0e-br, obtaining (4.8) Equation (4.7) is certainly the most important theoretical formulation in our entire subject. In this form, however, true absorption by the atmosphere is not taken into account, and as absorption is frequently of comparable magnitude to scattering, its effect is often included by writing (4.9) But it is not immediately obvious how this extension is justified, especially as there is no uniform relationship between the magnitudes of band k. An entirely distinct theory which makes use of more information and may be called the "two-constant theory" has been evolved by a number of authors, starting from a discussion by Schuster (1905) of the distribution of radiation in stellar atmospheres, and applied to the terrestrial atmosphere by Dietzius (1923) and others. 4.4 The "Two-Constant" Theory. We may introduce this section by a passing reference to the many investigations of the behavior of light in artificial turbid media, such as opal glass, paint films, and paper, of which we may cite those of Ryde (1931) as typical. The optical properties of such media may require as many as eight constants for their complete specification (Duntley, 1942); but such complication is not necessary in the discussion of the reduction of contrast by the atmosphere, where "two specially defined constants suffice under most of the commonly encountered seeing conditions" (Duntley, 1948a, p. 180). In the paper just quoted, Duntley has made several very illuminating applications of the two­ constant theory to the problems of vision through the atmosphere, appli­ cations which are not restricted to horizontal directions of sight, but boldly invade the almost unknown country of "non-horizontal visibility." The present writer wishes to acknowledge our debt to Duntley for making these advances, before stating his conviction that the derivation of the funda­ mental equations in the paper in question is incorrect, and that in fact the use of Schuster's theory in this particular problem is entirely inadmis­ sible. The equations themselves (nos. 10 and 11 on p. 182 of the paper) are quite correct, with one reservation which will be pointed out below. The paper as a whole is, in the opinion of the present author, the most useful single contribution yet made to the theory of the reduction of contrast by the atmosphere.

THE ALTERATION OF CONTRAST

65

Consider Fig. 4.2, in which an observer at R is looking down on an extended object at 0, the path of sight being inclined at an angle 0 with the horizontal. Imagine a plane lamina dr at a distance r from 0, and perpendicular to OR. Duntley considers two diffused bodies of flux moving in opposite directions through the lamina, each being augmented by space Light and by back-scattered flux from the other, and attenuated by absorption and by back-scattering. The upward-moving flux is of such a magnitude as to produce a diffused irradiance H1 on the lower. side of dr, and the other produces a diffused irradiance H. on its upper side. By making reasonable assumptions, Duntley arrives at a pair of differential equations of the

FIG. 4.2. Illustrating the two-constant theory.

Schuster type, very simple, and which need not (under one further assump­ tion) be solved simultaneously; one of these refers to an upward direction of view, the other to an observer looking downward. When rewritten in terms of luminance, the solutions of these lead to equations similar to (4.8). Now the writer believes that it is incorrect to introduce the absorption and scattering coefficients for diffuse flux into this problem. The only portion of the light from an object which goes to form an image of the object is that which comes to th� eye of the observer from the direction of the object. In the atmosphere, other light (the air-light) may come from the same direction, and of this a small portion may originally have come from the object; but as far as the air-light is concerned the object is only a part of the illuminated surroundings of any given parcel of air. We therefore present here a different and simpler-and, we hope, adequate-derivation of the equations in question.

66

VISION THROUGH THE ATMOSPHERE

Let us refer again to Fig. 4.2. Let the object have an intrinsic luminance Bo in the direction of the observer.* When seen from a distance r, the apparent luminance of the object will now be B(r). This is at the lower side of the lamina dr; at its upper surface the apparent luminance will be

B(r)

+ dB(r) dr. dr

The change in apparent luminance will be due to two phenomena acting together: (1) B(r) will be diminished by extinction in dr; (2) it will be increased by light coming from all directions and scattered in the direction of the eye by that part of dr which is between the object and the observer. Considering only the first of these processes, it will be evident from equations (2.1), (2.9), and (2.20) that the decrease in apparent luminance is pro­ portional to B(r). Considering the second, let Ba be the luminance of a lamina of air one meter thick as seen by an observer nearby looking in the direction of the object under the prevailing conditions of illumination and atmosphere. Then we may write immediately

(4.10)

dB(r) -dr

= - u(r)B(r) + Ba(r) .

Note that u and Ba have both been written down as functions of r, as well they may be. Let us first consider the extinction coefficient u. If 8 = 0, that is to say if the path of sight is horizontal, u may be constant, unJess indeed the air is inhomogeneous, in which event no simple theory can be devised. If 8 � 0, the extinction coefficient will in general be a function of the height, and consequently a function of r. We have already seen some examples of this in Waldram's results (e.g. Fig. 3.16), and it is evident that u may be no simple function of the height, or even analytic at all. As to Ba, this is plainly proportional to the total illumination on the lamina dr, and to some function of the scattering coefficient bin. the lamina. In fact

(4.11)

Ba

=

4r

J0

.B,f3'dw,

where .B, is the luminance of the surroundings as seen in any direction from a point in the lamina dr, {3' the volume scattering function for the angle between this direction and the direction OR, and dw an element of solid angle. However, we need not pay much attention to this, since it is not calculable in practice; we need only note that if the total (spatial) illuminance is the same at all heights, .Ba will be proportional to b, which is, like u, a function of r. Duntley speaks of the radiant density, and quotes experiments which showed that this quantity is constant up to at least 18,000 feet with solar altitudes of about 50 °, but speculates that it should *Our notation has to be almost entirely different from that of Duntley's paper, to conform to that used elsewhere in this book.

67

THE ALTERATION OF CONTRAST

a

small a

r,

0

a a

+

a a R

BR

a

I

BR

a

I

o J

R

a

R. a

R

a

a

= -

68

VISION THROUGH THE ATMOSPHERE

R."

=

0R

,j .... (") ,-i

>t'"'

r' .... C'l :::i:: ,-i

>-­ w co

FtG. 7 .1.

Nomogram for the sighting range of lights, based on the Tiffany data (probability of detection 95 per cent). The nine curves refer to different values of background luminance in foot-lamberts, from 103 to 10-•.

140

VISION THROUGH THE ATMOSPHERE

0

R

Fr G. 7. 2. Illustrating the theory of searchlight illumination.

1

=

2

2

0

0

4

2

1• 3

=

I., .

=

2

+ •

r

h,

J

2 ½

½.. 0

0

= =

dy;

1r -

/rh]

I., [j�'

uh(r

J J

0 ½"

½,r

l,i, [j�'

+ h)

(r

+ yr2 +

2urh/(ry;

+ h)]d,J,,.

2urh/(ry;

+ h)]d,J,,.

2

]d,J,,.

VISUAL RANGE IN ARTIFICIAL LIGHT

C

=

(Bi

141

+ B2) - Ba Ba

I,;

lo

-{,

FIG. 7 .3. Beam distribution from a 44-inch naval searchlight. The points are experi­ mental, the line is from equation (7. 9). (After Chesterman and Stiles, 1948.)

= °

B2

=

/:i J:..

2

-

+

*Hampton, Hulburt, and Chesterman and Stiles all proceeded along essentially these lines, the last-mentioned authors probably with most success. Rocard's treatment is rather different, and (with many approximations) attempts to allow for secondary scattering. tReversed to suit the convention adopted here, that back scatter corresponds to °

142

VISION THROUGH THE ATMOSPHERE

These approximations are probably entirely justifiable. It is rather interesting that Hulburt (1946), working from entirely different data, found f3o' = a-/26.9, which differs from rr/8-rr by only 7 per cent. There is reason to believe that a calculation of this sort may be adequately precise in any weather in which a searchlight is likely to be of any use other than as a signal. In thicker weather the computations of Rocard (1932) suggest that the additional light diffused by secondary scattering will be im­ portant; certainly the illuminance on the target may be substantially increased by it, as shown in the calculations made by Middleton (1949) for another purpose, which will receive attention in Chapter IX.

FIG. 7 .4. Mean relative scattering function f(tp), derived from night observations over the sea. (After Chesterman and Stiles, 1948.)

7.2.3 THE SEARCHLIGHT DIRECTED UPWARDS. The photometric theory of the anti-aircraft searchlight is further complicated by several circumstances. First and foremost, we can no longer consider the atmosphere uniform over the length of the beam. Secondly, we cannot assume the axial symmetry of a beam·which is in fact scarcely even symmetrical about one plane. Thirdly, the phenomena seen near the termination of the beam are complicated by perspective. The problem has been dealt with in a remarkable paper by Beggs and Waldram (1948), and so concisely that it would be difficult to summarize it here. We shall therefore leave the reader with the reference as far as the theory is concerned, setting out some of the main results below. 7.2.4 VISUAL RANGE OF OBJECTS IN A SEARCHLIGHT BEAM. In the naval problem (horizontal beam) we have shown in outline how the contrast be­ tween the target and its surroundings may be calculated, as a function of r, h, J0, R, and rr. The remainder of the problem concerns the minimum con­ trast actually observable under the rather peculiar conditions of observation. It would seem unlikely that the Tiffany data could fit the problem; and at any rate they were not available at the time of Chesterman and Stiles' researches. These authors made a very careful investigation on a model scale,

VISUAL RANGE IN ARTIFICIAL LIGHT

using small ship outlines and simulated searchlight beams projected on a screen in the correct values of luminance.* The method of constant stimuli (see p. 84) was used, and the results referred to a probability of detection of 0.95.t Finally, numerous calculations of r0 (the visual range) were made, and certain approximate laws deduced therefrom, which we shall endeavor to summanze: (1) log ro varies very approximately inversely as ½ log u, but the rate of increase decreases a little with decreasing u. (2) ro naturally increases with increasing reflectance R of the target, but not by any means in proportion. A white target has only about six times the visual range of one with reflectance 0.01, three times the range of one with R�= 0.04. (3) log ro is approximately linear in log h, the slope of the relation being greater, the greater the value of (u/R). (4) log r0 is approximately linear in the logarithm of the beam-spread, decreasing as the spread increases at a rate which is greater, the greater the value of (u/R). (5) r0 depends very little on the peak intensity lo. The change of r0 with lo is greatest when the air is very clear, as _indeed is obvious without calculation. (6) No general statement was possible about th� effect of target size on r0, the relationships being extremely complex. In the anti-aircraft problem the photometric calculations depend on the total transmittance of the atmosphere between the target and the ground, typical values of which were reported by Waldram· (1945), as described in Chapter III. In calculating the range it was assumed that the aircraft acted as a point source of light, and existing threshold data were employed. The intensity of the source was estimated from experiments ori a model aircraft in various attitudes (Stevens and Waldram'1946, p. 62). It may be stated that the general conclusions regarding the range agree with those of Chesterman and Stiles, but are naturally less quantitative. Beggs and Waldram also in­ vestigated the visual range of an aircraft illuminated by a searchlight from the ground, seen from another aircraft flying at the same height. *For an excellent general account of such model investigations, their virtues and limita­ tions, see Stevens and Waldram (1946). Their defence of laboratory experimentation should convince the most sceptical "brass hat"; they conclude (p. 53): "For a full-scale trial to be valuable, therefore, the physical principles involved must be thoroughly understood and the probable results forecast; the trial is most valuable as a confirmation of predicted results. It is generally useless for discovering the principles." tDr. Stiles has kindly furnished the author with copies of two of his original drawings, and also another graph showing thresholds at 50 per cent probability of detection. These were compared with the Tiffany data, and show a general agreement to within about 0.2 log unit, in spite of the very special conditions, after the transmittance of the binoculars used by Chesterman and Stiles is allowed for.

144

VISION THROUGH THE ATMOSPHERE

One general criticism which may be made of these researches is that they ignore the effect of the natural luminance of the sky. A comparatively mode­ rate aurora, for instance, or the last remnants of twilight, would certainly modify the results considerably.* *Beggs and Waldram (1948) refer to this in discussing the visual range of a searchlight beam, seen from the air.

CHAPTER EIGHT

THE COLORS OF DISTANT OBJECTS, AND THE VISUAL RANGE OF COLORED OBJECTS 8.1 Extension of Theory to Colored Objects. In the first six chapters of this book we made the assumption that the non-self-luminous objects at which we had to look were invariably neutral in color; that is to say black, white, or grey, reflecting all wavelengths in equal measure. In practice this is not so; green forests and red barns have sometimes to be used as "visibility marks," and the conspicuity of a bright-colored aircraft crash-landed in a green wilderness may be a matter of life and death. This chapter will therefore deal with two questions, first treated of by the author in 1935(d) in a paper with a similar title: (1) what is the effect of the atmosphere on the apparent color of a colored object, and (2) to what extent does the visual range of a colored object differ from that of a neutral grey object of the same luminance factor? Obviously if we are to theorize about the colors of distant objects we must presuppose a method for the numerical description of colors. The existence of such a method is well known, but many physicists and meteorologists will be unfamiliar with it, so that a brief outline is desirable. For further details the reader may consult Le Grand (1949), Bouma (1948), or W. D. Wright

In 1931 the International Commission on Illumination* established a method of specifying colors which has the great advantage that by its use all computers will arrive at the same numerical description of a color if they start from the same set of spectrophotometric or spectroradiometric data, and since the adoption of this scheme it has been possible to give an unambiguous and internationally recognized notation for the color of any object or light source for which the necessary data has been determined, and under specified conditions of observation. The starting point of the C.I.E. color metric was the establishment of a hypothetical "standard observer," whose response to radiation of various wavelengths is near the average of a number of actual observers with normal color perception. All calculations are then made in terms of this "C.I.E. observer." 8.2 The Methods of Specifying Color. One of the most valuable facts about color is that all colors can be matched under appropriate experi­ mental conditions by a mixture of two or more lights of different color in the proper proportions. In particular, most colors can be matched by a combina*Now referred to in most countries as the C.I.E. (Commission Internationale de l' �clairage) and in Central Europe as the I.B.K. (Internationale Beleucluungs Kommission).

145

146

VISION THROUGH THE ATMOSPHERE

tion of some spectral hue and white, and this is the basis of the so-called "monochromatic-plus-white" notation, in which_a color is defined by its HUE and PURITY.* The purity is the proportion of the pure spectral hue in the mix­ ture of hue and white. The purples cannot be matched in this way, but can be mixed with an appropriate spectral hue to match white, the purity being then considered negative. Most colors can also be matched by a mixture of lights of three colors, such as red, green, and blue. This is the basis of the trichromatic system of color specification. These three colors, or PRIMARIES, need not be real colors at all, but may be arbitrarily defined, as long as they have the property that no mixture of any two will match the third. The primaries of the C.I.E. systems are of this arbitrary sort, and none represents an actual physical color. They are denoted by the letters X, Y, and Z, and defined by three equations giving the amounts of each that will match three monochromatic stimuli of specified wavelengths. They were also chosen so that only the_ primary Y has any luminosity, a property .which simplifies certain calculations, but is not otherwise important to us here. In this system a color is completely-specified by the color-equationt (8.1)

Q +-+XX

+

yY

+ ZZ,

its luminance being equal to Y. The units of X, Y, and Z are so chosen that if X = Y = Z, Q matches a source which radiates energy equally throughout the spectrum. X, Y, and z are called the TRISTIMULUS VALUES of Q. In the special event that the sum of the coefficients of X, Y, and Z is unity, we have the unit equation (8.2)

Q+-+xX+ yY + zZ

of the color, in which Q represents one trichrcmatic unit (T.U.)t of Q, and (8.3) The quantities x, y, and z are called the CHROMATICITY CO-ORDINATES of the color, and their sum is, of course, equal to 1. It should further be noted that the luminance of one T.U. of a color is equal to its co-ordinate y. The qualities of a color that differentiate it from another color are its CHROMATICITY and its LUMINANCE. Since x y z = 1, only two of these

+ +

*The terminology throughout this chapter is that of the Optical Society of America, as set out in the Comparative list of Colar Terms published by the Inter-Society Color Council (1949), except for a few terms recently (1948) adopted by the C.I.E. tThe writer is well aware of the danger of confusing these so-called equations with ordinary algebraic equations, although he does not share the extreme views of some of his friends in the United States, who would abolish color equations altogether. It is felt that the use of the sign+-+, meaning "is a mixture of," will distinguish them from ordinary equations, while retaining their value for purposes of explanation. tThis term has no official sanction, or at least is not referred to in reference I.S.C.C. (1949); it was used by the early writers on the C.I.E. System (Smith and Guild, 1932, etc.) and is a very useful concept for teaching purposes.

CoLORS OF DISTANT OBJECTS

147

co-ordinates need be given to specify the chromaticity, and x and y are usually chosen, especially in North America. Thus if we draw axes OX and OY, the point x, y may represent the chromaticity of the color Q, as shown in Fig. 8.1. The luminance of Q is not shown in such a figure, and where it is important

X

FIG. 8.1. Chromaticity diagram and spectrum locus. The general location of the various colors is noted.

it is usually given numerically in any of the usual units; or it may be thought of as the third dimension, perpendicular to the plane of the paper. The colors of monochromatic spectral radiations can, of course, be matched by appropriate quantities of X, Y, and Z. The curved line in Fig. 8.1 is the locus of the spectral colors, generally called the SPECTRUM Locus, and all physically realizable colors lie within the closed figure formed by the

148

VISION THROUGH THE ATMOSPHERE

spectrum locus and the straight line joining its ends. A chart such as Fig. 8.1 is called a CHROMATICITY DIAGRAM. In the C.I.E. system the position of the spectrum locus is defined in terms of the standard observer referred to above; this observer also follows the international (C.I.E.) LUMINOSITY CURVE, the curve of luminosity of spectrally homogeneous lights, plotted as a function of wavelength, relative to the maximum luminosity. The ordinate of this curve at wavelength A is denoted by V;,,, and in terms of the LUMINOSITY K (see p. 7) V;,,

= Kx/Kmax·

This unambiguous set of definitions makes it possible to calculate the chro­ maticity co-ordinates of any color from a knowledge of the energy distribution throughout the spectrum of the light received from it.* We shall state the procedure without proof. Consider first a primary source (self-luminous object), and let WAdA be the radiant emittance from it within the wavelength interval (A- ½d>.., A + ½d>..). Let xA, yA, zA be the chromaticity co-ordinates of spectral light of wavelength >... We form quantities ix, fix, zx, known as the color mixture data for wavelength A, and given by ZA XA (8.4) xx= -A Vx, YA= V>., zA= - VA• Y YA

These are to be multiplied separately by WAd>.. and the products integrated over the whole of the visible spectrum, giving the tristimulus values of the source:

I I

X=Kmax Wxixd>..,

(8.5)

Y= Kmax WAyAd>..,

z = Kmax f

WAzAd>...

The chromaticity co-ordinates are then given by

(8.6) Since there is nothing to be gained by retaining the factor K max as far as the calculation of x, y, and z is concerned, we shall omit it in subsequent equations. This merely involves a change in the units of X, Y, and Z. Table 8.1 gives the adopted values of the color mixture data. Since these functions are not analytic, they must be integrated mechanically; summation at intervals of 10 mµ is entirely adequate for our purpose, especially as all the functions go to zero when Vx goes to zero, that is to say at the ends of the visible spectrum.

•There are certain restrictions, since the standard observer was established for a field of 2° diameter and, of course, for photopic vision. The colors of very small sources form a special study.

COLORS OF DISTANT OBJECTS TABLE 8.1 THE C.I.E. COLOR MIXTURE DATA FOR THE EQUI-ENERGY SPECTRUM, AFTER HARDING AND SISSON (1947)

149

150

VISION THROUGH THE ATMOSPHERE

The color of a secondary source (not self-luminous) depends on the spec­ tral constitution of the light which falls on it, and also on the spectral reflec­ tance of the object. Suppose a source producing a normal irradiance Hxd>.. in the wavelength interval (>.. - ½d>.., >.. + ½dX) at the surface of an object of spectral reflectance* f>,. The spectral radiant emittance in this interval wiil be (8.7)

and by an argument similar to the above, we may immediately write (8.8)

X1 =

J

Hxfxi\d>..,

J

Hxfxzxd>..,

Y1 = J Hx!xfjxd>.., Z, =

next finding the values of x, y, and z by means of equation (8.6). If the light passes once through a filter having spectral transmittance t, as for instance if such a filter is held in front of the eye, we shall have (8.9)

X = I Hxfxtxixd>.. etc.

For technical purposes, three standard light sources known as "A", "B", and "C" have been introduced by the C.I.E. Source "A" is a tungsten lamp at a color-temperature of 2854°K., "B" and "C" the same lamp used with specified filters which produce light corresponding to one of the yellower and one of the bluer phases of daylight respectively. The chromaticity of these sources, and also of the equi-energy source, is shown in Fig. 8.2 by the points A, B, C, and W respectively. Source "C" is useful in the present problem be­ cause it represents a color (and a distribution of energy) not very different from that of overcast daylight near sea level, and we have therefore repro­ duced a table (8.2) of values of Wxix, Wxih, and Wxzx throughout the visible spectrum. This table has been arranged so that the summation for Y gives the LUMINANCE FACTOR directly in per cent; that is to say, it is made so that Wxyx = 100.00,

L

and this may serve to emphasize that the complete numerical description of those attributes of a color which are susceptible to calculation requ;res three numbers, namely, x, y, and Y:t To complete this introduction to our subject we must describe very

*The usual symbol is r, as noted on p. 6; to use this in later equations would be very confusing unless other notations were changed, and we shall therefore usefx in this chapter only. tThe colorimetrist ought to be the first to admit that there are several other attributes of a color which do not at present lend themselves to such arithmetic (see R. M. Evans, 1948). It is to be regretted that he often forgets this.

CoLOR� OF DISTANT OBJECTS

151

briefly the result of mixing two colors; and by this we mean the mixture of two lights (additive mixture), not of two pigments. The new color produced by mixing two colors may readily be determined by algebraic methods or by the use of the chromaticity diagram. Let the two colors (see Fig. 8.2) be 0 1 and 02, and let their unit equations be (8.10)

X

FIG. 8. 2. Position of standard illuminants, and a diagram to illustrate color mixture.

Suppose the luminance of 01 is B1 and that of 02 is B 2 ; then the amount of the two colors in T.U. will be B1/y1 and Bdy2 respectively, which we shall denote by B1' and B2'. If now we multiply each unit equation by the number

152

VISION THROUGH THE ATMOSPHERE TABLE 8.2 VALUES OF

W;,.X;,., W;,.y;,., AND W;,.Z). FOR C.1.E. SOURCE "C", AFTER HARDING AND SISSON (1947)

COLORS OF DISTANT OBJECTS

of T.U. of the corresponding color and add, we have the equation of the mixed color Q: (8.11)

(B1

+ B; )Q +--t (B� X1 + B; X2)X + (B� Y1 + B; Y2)Y + (B� z1 + B; z;)Z

and if we divide the coefficients on the right-hand side of this by (B 1' we obtain the chromaticity co-ordinates of Q: (8.12)

B� X1 x Bi

+ B; X +B 2

2

'y

= B� {'1 B1

+

+ B ') 2

B� Z1 + B; Z2 + B; Y + B; ' z - B + B . 2

i

2

Now equation (8.11) refers to (B 1' B2') trichromatic units of Q. One T.U. of it has luminance y, and thus the luminance of the new color is (8.13) We may, of course, use any convenient unit of luminance in all these calcula­ tions. In the present problem we shall find it convenient to choose as a unit the luminance of the horizon sky. Equations (8.12) are the expressions for the co-ordinates (x, y, z) of the center of gravity of two masses B1' at (xi, Yi, zi) and B2' at (x2, Y2, z 2). This suggests a graphical procedure. If we plot Qi and Q2 on the diagram (Fig. 8.2), the position of Q is given at once by dividing the straight line Q i Q2 in the ratio The process may be extended to any number of colors, and indeed the justi­ fication for summations such as those of equation (8.5) is to be found in this possibility. The transformation from the trichromatic to the monochromatic system is thus effected graphically (Hardy, 1936). In order to obtain the HUE-WAVE­ LENGTH a large-scale plot of the spectrum locus (or of the required part of it) is prepared. On this diagram (see Fig. 8.3) are plotted the color Q under consideration and also a point W representing the color which is being con­ sidered as white, which is not necessarily or even usually the equal energy point, but rather the color of the general illumination prevailing. The hue­ wavelength is then immediately found by joining Wand Q by a straight line and producing it to cut the spectrum locus at D. If WQ produced does not cut the spectrum locus, as in the case of WQ' in Fig. 8.3, it is produced back­ wards to D'. The colors for which this occurs are the purples. The EXCITATION PURITY (often simply called PURITY) is the ratio (8.14)

P. = WQ/WD

which may be measured on the diagram; but it is just as easy, and more accurate, to make use of the one of the formulas

154 (8.15)

VISION THROUGH THE ATMOSPHERE

P.

= (y - Yw)f(Yn - Yw),

P. = (x - xw)/(xn - Xw),

for which the numerator has the greater absolute value. The COLORIMETRIC PURITY P, is expressed as the ratio of the luminance

FIG. 8.3 Transformation from trichromatic to monochromatic system.

of the spectral component to the luminance of the spectral and white com­ ponents taken together. It may easily be calculated by the formula (8.16)

P, = P. · YnlY-

For the purple colors, P, and P, are considered negative. The excitation purity is more generally used in North America than the colorimetric purity. These properties of the chromaticity diagram extend to diagrams made

COLORS OF DISTANT OBJECTS

by simple projective transformations of the right triangle generally used. Several diagrams of this sort have been devised, which make distances on the diagram correspond more nearly to perceived differences in chromaticity. An example will appear later (Fig. 8.f). 8.3 The Colors of Distant Objects.* 8.3.1 THE MEAN EFFECTIVE EXTINCTION COEFFICIENT. Let us now go

back to equation (4.25)

and make the observation that the values of the contrast referred to therein were tacitly supposed to be measured in "white light," to which rather in­ definite illumination the extinction coefficient was supposed to apply. Actually, as was set out at length in Chapter III, the extinction coefficient varies with wavelength, and usually this variation is more pronounced when the air is very clear. A simple approximation to this variation was given in equation (3.28), in which the exponent n may vary from - 4.08 for pure air to about zero for dense fog. In a very light hazet once measured by the writer (1935c), permitting a visual range of some 150 km., n had the value - 2.09, and it probably never greatly exceeds this negative quantity. It is customary to place the "wavelength of white light" at about 0.57µ. This is probably justified in view of the limited accuracy obtainable in atmos­ pheric-optical measurements, but it is not strictly true. In reality the mean effective extinction coefficient is a function of the distance rand of the energy distribution of the illumination, as may be seen from the following brief analysis: For a black object we had, supposing horizontal vision, Bb

= Bh(l - e-")

and if this is true for "white light," it is certainly valid for a small range of wavelengths (X - ½dX, X + ½dX). Expressing this restriction by means of the subscript X, we may write (8.17)

Bb,>.

=

Bh ,>.(1 - e-">-').

Now Bh, the luminance of the horizon, is the limit approached by Bb when r - ro on the fundamental assumption (h), p. 62; the color of the sky at the horizon should then be that of the total illumination on the intervening atmosphere,t and its luminance will be (8.18)

Bh

=

C I W>.17>.dA,

*This treatment is substantially that of a recent paper by the author (1950), and is more general than that of the earlier paper (1935d). tReally an extremely clear atmosphere. The term "haze" is used only to distinguish it from theoretically pure air. tThe earth not being flat, this condition is not quite reached when the air is fairly clear.

156

VISION THROUGH THE ATMOSPHERE

and consequently

(8.19)

Dividing, we obtain

(8.20)

_ B b /Bh -

f Wxyx(1 - e-ax ' )d"X .' f Wxyxd"X

but this is equal to 1 - e-amr, where which we are searching, whence

.i>.(1 - e-">-')dX and the chromaticity co-ordinates may be found by means of equation (8.6). The quantity Y is the total luminance of the object in terms of that of the horizon sky.

8.3.3 THE APPARENT CoLoR OF AN ACHROMATIC OBJECT OF INHERENT CONTRAST Co. In Chapter VI it was pointed out that the visual range of a white or grey object could only be calculated if we could measure or assume its inherent luminance, and hence its inherent contrast Co with the horizon sky. This limitation is of necessity carried over into our discussion of the apparent colors of such objects. The color of a surface that does not reflect selectively is, of course, that of the illuminant, and therefore the chromaticity of such a surface at r = 0 will be that of the daylight. This will not be strictly true of a surface on which the sun is shining, because the color of direct sun­ light is a little yellower than that of the total daylight; but the exception is quantitatively unimportant, and we shall see later how to correct for this error if necessary. According to Dorno (1919, Table 50b) the value of Bo/Bh for a sunlit snowfield may rise to as much as 5; on an overcast day, as we saw from Fig. 6.2, something between 1 and 2 may be expected. For dark grey objects this quantity will of course be near zero, and since Co = (Bo/Bh) - 1, we have to reckon with values of Co from near - 1 to about+ 4. Now let us rewrite equation (4.16) in terms of monochromatic light and horizontal vision, adopting the symbol Bu for the apparent luminance of the grey object:

(8.25)

This may be written (8.26)

Bu .>.

=

Bu [1

or, in terms of inherent contrast, (8.27)

+

(!o. >. h.>.

1) e-">-,],

Remembering that Bh ,>. = W>.Y>., we may again use the reasoning that led to equations (8.24), and obtain for the tristimulus values of the achromatic object at distance r,

158

VISION THROUGH THE ATMOSPHERE

X = Y=

(8.28)

Z =

I I I

W1,i1,(l

+ Coe-u1,r)dX,

W1,y1,(l

+ Coe-�')dX,

W1,z1,(l

+ Coe-u:>..')dX,

the chromaticity co-ordinates being obtained by unification as before.*

8.3.4 THE APPARENT CoLoR OF A CoLORED OBJECT AT A DISTANCE.

Suppose an opaque surface of any color whatsoever, and let f1, be its spectral reflectance. If such a surface were placed beside an ideal white surface and the two were in the same plane, the ratio of the monochromatic luminances of the two surfaces at wavelength ).. would be J1,. If now the surfaces were observed horizontally against the horizon sky, we might measure or calculate the inherent luminance of the white object, Bo,:>.., as in the last section, and if f1, is known, it follows from equation (8.25) that the apparent (mono­ chromatic) luminance of the colored surface is

(8.29) The chromaticity co-ordinates might be derived from this, but it is probably more convenient to express them in terms of the inherent contrast of the ideal white object, which is more likely to be amenable to estimation or even measurement. Call it We may write (8.29) in the form

c...

(8.30)

B1,:>..

= Bh,:>..[l

+

�:.\:>.. - l)e-�'l

Now consider the quantity (!1, Bo,1,/Bh .:>..) - 1, which we shall write f1,Bo,:>.. - Bh ,:>.. Bh,:>.. Let us add and subtract f1,Bh .:>.. in the numerator:

so that, finally,

(8.31) The tristimulus values may now be written down at once: •These more general equations replace equations (26) of the original paper (Middleton, 1935d), which are of limited validity because of the considerations brought out on page 105 above. But we have to know C0, as always.

COLORS OF DISTANT OBJECTS

= = Z

j

=

+

w

+

+

w

+

+

w

+ w

w

x

Xq, yq,

b

b

=

w

=

a

a

B,.,

b

b

=

a

w

+

0,

q

o

1

+

2)

r ro - r

r .,, J

>

MEASURING VISUAL RANGE

Fig. 9.2 shows some results of such a calculation. The numbers I, II, III, and IV on the curves refer to the four sets of conditions set out in table 9.1, which also indicates that conditions II and IV differ in the size and number of the drops, but are otherwise similar, The direct light Ea will be the same in each of these two cases. ooo ,------,---.---,-----,

9.2. Calculations from equation 9.6 for two hypothetical atmospheres (see text) and two values of 0. The ordinate is the percentage error of a measurement with a telephoto­ meter having the half-angular opening shown by the abscissa.

FIG.

It will be seen that the percentage error is enormous. The value of u given in table 9.1 corresponds to a meteorological range Vm = 1000 m. (very nearly), TABLE 9.1 CONDITIONS DEALT WITH IN FIG.

9.2

so that case I represents a lamp at about the daylight visual range of ordinary objects. The error naturally decreases with the distance, and almost propor­ tionally, as shown in Fig. 9.3. But the most striking feature of the results is the much smaller error with the larger droplets (compare cases II and IV).

180

VISION THROUGH THE ATMOSPHERE

An actual fog will contain droplets of all sizes, and it is obvious that there will be no universal relation between this error and the value of u. For the practice of telephotometry, it is obvious then that the aperture ,i,-

0.5 Km. 1.0 FIG. 9.3. Interpolations from Fig. 9.2 (cases I to III). Error plotted against distance. of the instrument should be kept small. Further figures in the original paper show that the aperture 8 of the lamp is of less importance, and makes little change after it gets as large as 0.1 radian; but still a projector is better than an open lamp, if difficulties of alignment can be overcome. These results are confirmed by experiments (Middleton, 1949). Fig. 9.4

FIG. 9.4. Experimental values of error plotted against ,JI'. Dense fog; ro= 114 m., u = 3.1 X 10-i m- 1• 8 = 0.30 radians. Note that the scale of abscissae is five times as large as in Fig. 9. 2. is constructed from readings made with a telephotometer of variable aperture at two periods in a dense fog, at a distance comparable to the meteorological

MEASURING VISUAL RANGE

181

range. The error is about five times that calculated in case I, probably be­ cause of a preponderance of smaller droplets.* We may conclude that a large and variable error is inherent in such measurements as usually made. In a permanent installation it ought to be possible to provide a series of screens, as shown in Fig. 9.5, and eliminate nearly all this error; but the expense would be heavy.

FIG. 9. 5. Possible distribution of screens to minimize errors in telephotometry due to scattered light.

An interesting by-product of the above calculations is the luminance of the field of light around the lamp, obtained by differentiating E2 with respect toy, and multiplying by ½,r sin y,. Even in the monodisperse fogs for which the calculations were made, there is no distribution of luminance that would appear as definite rings, even in monochromatic light. The absence of a corona, then, is adequately accounted for by the fact that a large range of values of et, is associated with each value of y,. The best coronas are seen in plane distribu­ tions of droplets, as on a window; or in clouds of small extent in the direction of the light source. The theory dealt with in this section has nothing to do with the actual instrumental errors of the telephotometer, such as uncorrected non-linearity of scale or instability of calibration. It is the designer's role to minimize these.

9.3.2 VISUAL TELEPHOTOMETERS. 9.3.2.1 General Remarks. In most branches of photometry physical methods appear to be supplanting the earlier visual procedures, and this is true in telephotometry to some extent. Nevertheless visual telephotometers have their advantages. For measuring the illuminance from a distant source, for example, certain types of visual instruments can avoid the errors referred to in the preceding section; while for the sort of relative photometry dealt with below (section 9.3.2.2) they give very direct results. There are three kinds of visual telephotometers: those for measuring extended sources, those employing the so-called "Maxwellian view" in the photometry of a distant lamp, and finally those using an "artificial star." The last two classes have been dealt with at length in a very interesting paper by L. J. Collier (1938), to which the writer has frequently referred. 9.3.2.2 Visual Telephotometers for use with Extended Sources. There are two classes of telephotometer for use with extended sources. One of these, also dealt with by Collier (1938, p. 144), compares the luminance of such a *See also Smith and Hayes (1940).

182

VISION THRO.UGH THE ATMOSPHERE

distant source with that of a comparison source contained in the telephoto­ meter. The other measures only the rel,ative luminance of two distant sources, as, for example, a hill and the horizon sky above it, with the intention that the results should be put into some such equation as (4.25) for the calculation of cr. The "Sichtmesser" of Foitzik (1933, 1934, 1938, 1947) is a special type which may for convenience be treated in this section. The telephotometer of Koschmieder and Ruhle (Koschmieder, 1930b) had an integral comparison source, and is shown diagrammatically in Fig. 9.6. This was a telescope about two meters long with a special eyepiece assembly. The photometric cube* was made to reflect only in a small central patch, the

FIG. 9.6. Telephotometer of Koschmieder and Ruhle.

landscape showing in the remainder of the field as in an ordinary telescope. The control of the comparison field was by three Nicol prisms. Two very large instruments of this type have been described by Benford (1919), one of which had an objective with a diameter of 15 inches and a focal length of 30 feet; and in these the distant surface was seen through a central aperture, the remainder of the cube being reflecting. Obviously this type of field would be suitable for an artificially illuminated source at night, while that of Koschmieder is more suitable for daytime use. An exceedingly compact telephotometer of this sort, using a prismatic telescope, was designed by Schuil about 1943. This was intended specifically for the measurement from an aircraft of the luminance of objects on the ground, and has a large erect field on which appears a small U-shaped com­ parison field. There appears to be no published description other than a manufacturer's bulletin. In these instruments the primary image of the distant source is formed by the objective at the center of the photometer cube. The comparison sur­ face can be imaged at the same place by a lens, or it may be a relatively large diffusing surface near the cube. In either case it is necessary to ensure that the exit pupils of the two optical systems coincide, as otherwise the relative luminance of the two parts of the field may change if the eye is moved *Often called a Lummer cube, after its inventor. It will be assumed that the reader i s acquainted with the ordinary devices used in photometry, for which any standard treatise on optics may be consulted.

MEASURING VISUAL RANGE

183

slightly. Collier recommends that an artificial pupil should be provided in front of the eye, small enough to act as the aperture stop of the instrument even when it is focussed on a nearby object for calibration.* A slightly different arrangement has been adopted by Evans and Chivers (1933) in which the photometric cube does not act as the field stop, but merely serves to reflect the light from the comparison field into the eyepiece. Typically the distant object and the comparison field are illuminated semicircles of similar angular dimensions. Such an instrument needs to be designed with great care, and also needs corrections for distance unless an aperture stop, smaller than the objective, is provided. It is less generally useful than the first type.

FIG. 9. 7. Middleton's telephotometer.

When we come to "relative" telephotometers, we find a great number of different designs. An early one was that used in the author's researches (Middleton, 1935c) already referred to in Chapter III. It consists of a tele­ scope of about S cm. diameter and 75 cm. focal length, with the assembly shown in Fig. 9. 7 in front of the eyepiece. The element W consists of a neutral glass wedge and a piece of plain glass of the same index of refraction and thickness, side by side. The. bi-plate B juxtaposes two fields about 10' of arc apart. The instrument is directed at the horizon in such a manner that *There is an excellent treatment of this important matter in Koschmieder (1930b, p. 18).

184

VISION THROUGH THE ATMOSPHERE

the image of a portion of the sky falls on the wedge, that of a suitable object just below the horizon on the clear part of W. The focal point is the interior angle of the bi-plate, which appears as a fine line in the eyepiece. The scales S and F indicate the position of the wedge for balance. The value of B/Bh can be immediately obtained from the calibration of the wedge, and u calcu­ lated as described in Chapter IV. The corrections detailed in section 4. 7 must be kept in mind. In such an instrument the wedge should be calibrated in situ, by some arrangement like that shown in Fig. 9.8, which was used by the author.

FIG. 9. 8. Calibration equipment for relative telephotometry. There should be baffles, which are omitted for clarity.

The two surfaces of sandblasted glass S 1 and S 2, cut from the same sheet, can be placed at various distances from the projection lamp L, and their relative luminance calculated. Pairs of readings should be taken, inter­ changing the glasses. In view of the properties of ground glass* it is desirable to set up the equipment symmetrically. Telephotometers of this kind lend themselves to filter photometry; but if the above method of calibration is employed in such a procedure, the color­ temperature of the calibration lamp had better be raised (by means of filters) to that of average daylight. Another relative telephotometer, which has the advantage that the angular separation of the fields can be varied, is due to Lohle (1935a, 1940c, 1941b), and seems to have taken several outward forms, the final one having been made by Zeiss. The principle is shown in Fig. 9.9, in which will be seen two telescopes Fh and F., directed at the horizon sky and the distant object respectively. A hinge at D permits the adjustment of the separation. There is an adjustable diaphragm in front of the objective of Fh which makes it possible to equate the luminance of the two fields and so measure the ratio B/Bh. Our only reservation about this instrument is that at low settings the measuring diaphragm will act as the aperture stop, and errors due to the Stiles-Crawford effect (see p. 189) may result. Another interesting relative photometer of this type is that of Gurewitsch and Kastrow (1934), which employs a bi-prism to divide the field, much as in the well-known Martens polarizing photometer. In this connection it may be well to utter a warning; it will not do to use polarizing elements in a *See Middleton and Smith (1949).

MEASURING VISUAL RANGE

185

telephotometer of this sort, because of the partial polarization of the light from the sky and the landscape. It is, of course, quite legitimate to use them in an artificial comparison beam, as in the telephotometer of Koschmieder mentioned on page 182. The Re/ativphotometer of Weber (1913) was relative in the sense that it used the existing daylight to illuminate the comparison field, and by using

FrG. 9. 9. Principle of Liihle's telephotometer.

such an instrument twice, the ratio B/Bh can be measured. A very simple modern instrument of this type was described by Byram (1935a), and is shown diagrammatically in Fig. 9.10. The opal glass G 2 is held normal to the rays of the sun, and the illumination which it provides on the inner opal G 2 is controlled by a graduated slide H. A silvered spot on the clear glass plate S juxtaposes the images of part of G 1 and of a distant object, the latter being reflected by the mirror M. F 1 and F 2 are filters. A small artificial pupil is provided, to make further optical parts unnecessary in bright daylight. This

FIG. 9.10. Byram's relative telephotometer.

instrument was devised for forestry work, and its inventor has described two even simpler "haze meters," which are really relative telephotometers (Byram, 1935a, 1940; Byram and Jemison, 1948). The second of these is shown in cross-section in Fig. 9.11. In this instrument the landscape is seen through a slit Sand by reflection in the mirrors M1 and M2, which are fixed. Mirror M2 has the silvering re-

186

VISION THROUGH THE ATMOSPHERE

moved from 13 narrow separate horizontal strips, through which small por­ tions of the sky are seen. The light from the sky also passes through the neutral glass wedge W, so that 13 rectangles of graduated luminance are seen superimposed on the distant landscape. F is a blue filter. The observer records the number of the rectangle which matches the selected target and

FIG. 9. 11. Byram's haze meter (from Byram, 1940).

consults a table prepared from the calibration of the wedge and some such equation as 6.2 to derive the visual range. Byram estimates that "in the hands of a competent lookout man" the error in estimating the visual range with instruments of this sort should not exceed ± 12 per cent. If this accuracy can in fact be attained, the small size and low cost of the instrument should render it very interesting. In contrast to the telephotometer just described, the elaborate instru­ ment described by Foitzik (1933, 1934, 1938, 19476) is much more expensive and almost certainly much more precise. In his 1934 paper Foitzik gives credit for the idea of this Sichtmesser to Koschmieder, and in his book Lohle (1941, p. 99) refers to it as the Koschmieder-Zeiss meter. It is indeed built around the famous Zeiss "Pulfrich photometer," as shown in Fig. 9.12.* It

FIG. 9 .12. The Koschmieder-Zeiss Sichtmesser (after Foitzik, 1938).

has the great advantage of using the same light source for both the distant light and the comparison field, thereby eliminating errors due to fluctuations of the supply voltage. A beam of light from the source S, collimated by a lens system L1L2, is projected towards a distant mirror, whence it is reflected *There appear to have been at least three forms, but all on the same principle.

MEASURING VISUAL RANGE

187

back to the matt surface M1'- A constant fraction of the beam is taken out by the inclined glass plate G, and used to illuminate a second matt surface M2'- The remainder of the optical system is a Pulfrich photometer.* The mirror, it should be mentioned, is a. "triple mirror," consisting of three re­ flecting surfaces in the position of three adjacent faces of a cube, with the corner away from the light. If the angles are all 90°, the beam of light is reflected back on itself whatever the orientation of the mirrors. In practice a glass block is used, consisting of the corner of a cube limited by a plane cutting the three edges at equal distances from the corner. The reflections are internal, and cleaning is simplified. From an instrumental standpoint this telephotometer is very elegant. Small errors due to daylight may be corrected for by interrupting the beam of light going to the mirror, making a reading, and subtracting the result from the reading with light. To give the instrument an adequate range, mirrors are provided at 25 meters and at 250 meters, giving an effective length of path of SO or SOO meters. This should providet reasonably precise measure­ ments of meteorological ranges between SO m. and 20 km. The advantages of having only one light source are considerably reduced nowadays by the availability of alternating-current regulators in small sizes; and it is questionable whether such an elaborate instrument could or should compete with photoelectric telephotometers such as those referred to in section 9.3.3.3. 9.3.2.3 Visual Telephotometers using the "Maxwellian View." If a real image of a source of light is formed at the pupil of the eye by a lens, the whole surface of the lens appears uniformly bright. This principle was used by J. C. Maxwell (1860) in his famous color mixer, and the arrangement has come to be known as the Maxwellian view. It is of obvious utility in the photometry of small or distant sources of light, and we shall restrict our con­ sideration to this use of the principle, although it is also adaptable to the measurement of the luminance of an extended surface. If by means of a photometric cube we illuminate adjacent parts of a lens, or parts of two lenses, from the source under measurement and a comparison source respectively, forming images of the two sources in exactly the same position at the pupil of the eye, we have a photometer. The two most obvious ways of doing this are illustrated by the photometers of Gehlhoff and Schering (1920, 1922) and of Buisson and Fabry (1920, 1922). The editors of the Zeitschrift fiir Instrumentenkunde did their readers the service of printing long abstracts of these two papers on facing pages, and it is interesting to see how different may be the approaches to the same problem. In the photometer of Gehlhoff and Schering, shown diagrammatically in Fig. 9.13, a distant source L1 is imaged at P by an objective lens 01, the light passing through the transmitting portion of the photometric cube Won its *This photometer has exit pupils which vary in size with the setting of the measuring diaphragms, and may therefore run foul of the Stiles-Crawford effect (see p. 189). tSee the discussion on page 198.

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way to P. The small comparison source L 2 is also imaged at P, its light being reflected by the totally reflecting portion of W, after being varied for the purposes of measurement by the Nicol prisms N 1 and N 2- The observer is obliged to accommodate his vision to the distance WP. Assuming that the images of the sources L 1 and L 2 are smaller than the pupil of the eye, it can be shown that the retinal illuminance at balance is approximately (9.7) E1 = T1Td1JU(r1+f1)2h2, where I1 is the intensity of L 1 in the direction of the photometer, r1 its dis­ tance, f1 the focal length of 0 1, r 1 the distance L101, h the posterior nodal

FIG. 9.13. The photometer of Gehlhoff and Schering (diagrammatic).

distance of the eye, T1 the transmittance of the optical system (including the eye), and T2 that of the atmosphere between L 1 and the photometer. Other things being equal, E1 is proportional to the square of the focal length of 01. The immense gain in luminance over a photometer using a diffusing screen may easily be seen. If we imagine a perfectly diffusing screen of trans­ mittance (or reflectance) unity replacing the lens 01, and if p is the radius of the pupil, we can show that the retinal illuminance E2 is

/n

= E1 p2 if we assume the same state of accommodation and the same transmittances T 1 and T2. The luminance of a field large enough to appear as an extended surface is proportional to the retinal illuminance, and it will be found that if the pupil is about 5 mm. in diameter, a lens of only about 25 cm. focal length will produce a gain of 104, while ifJi were a meter the gain would rise to 160,000. The limit in this direction (apart from portability) is, of course, the necessity of having the image of the distant light very small. It is perhaps better to have the eye of the observer accommodated for infinity. The telephotometer of Fabry and Buisson (Fig. 9.14) takes care of this. The figure is almost self-explanatory if we note that J is the comparison source, H a neutral wedge, and P a totally reflecting prism, reflecting the light from the distant source X. Collier and Taylor (1938) at the National Physical Laboratory have de­ veloped this type of telephotometer somewhat further, with the measurement E2

MEASURING VISUAL RANGE

189

of atmospheric transmittance directly in mind. In their instrument (Fig. 9.15), La is the main objective that forms the image at the eye-ring R, L2 a colli­ mating lens, and L1 a focussing lens. The lenses L2 and Ls are fixed. This makes it possible to increase the luminance of the field without a corresponding increase in the over-all dimensions of the instrument, as the luminance is proportional to fi2fa2/h2, where f1 is the focal length of the lens L1, etc. In the instrument as constructed,Js was SO cm.,f2 10 cm., andf1 could be either 10 cm. or 25 cm. In the latter case the luminance could be as great as that which would be produced by an objective of 125 cm. focal length.

FIG. 9.14. The photometer of Fabry and Buisson (optical system).

The comparison field is produced by a source S' and a collimating lens L4, and its luminance is varied by means of an optical wedge W. For the practical details of construction and use, the reader is referred to Collier (1938), or to Collier and Taylor (1938). One important source of error affecting many telephotometers, but es­ pecially those employing the Maxwellian view, was discovered by Stiles and Crawford in 1933, and is now generally referred to as the Stiles-Crawford

FIG. 9.15. Diagram of the telephotometer of Collier and Taylor. ejfect. These investigators showed that the luminosity produced by a narrow

beam entering the eye and striking the retina depends on the region of the pupil through which the beam passes, being greatest when it enters at some point near the center of the pupil. One of their figures is reproduced as Fig. 9.16. Now in a Maxwellian-view instrument it cannot be assumed either that the images of the distant source and the comparison source coincide or that they are the same size. Even if by careful pointing their centers are made to coincide, a difficult matter in practice, their relative diameter will depend on

190

VISION THROUGH THE ATMOSPHERE

the size and distance of the distant source. An integration of the curve of Fig. 9.16 will determine the relative efficiency of an image of any given size, and obviously this will be less, the larger the image. Fig. 9.17, from Collier (1938, p. 150) shows the practical results. Collier concludes that the diameter of the image should not be greater than 0.5 mm. It is obviously very impor-

FIG. 9.16. The Stiles-Crawford effect. Relative luminous efficiency of a narrow beam enter­ ing the eye at various points along a horizontal traverse across the center of the pupil. One observer, right eye. (After Stiles and Crawford, 1933.)

tant that such instruments should be focussed properly, and an eyepiece ought to be provided, which could be slipped on, correctly positioned behind the eye-ring, for this purpose. The Maxwellian-view telephotometer, in the hands of a skilled observer, is obviously very suitable for measuring the transmittance of the atmosphere

IMAGE DIAMETER

FIG. 9.17. Results of the Stiles-Crawford effect. Dependence of measured intensity on image diameter in a telephotometer using the Maxwellian view.

over long paths at night, without the necessity of providing a source of very high intensity. Only an elaborate photoelectric telephotometer can be made as sensitive. The practical advantages of a bare lamp over a searchlight are enormous; it can be standardized accurately in the laboratory and does not have to be oriented as precisely.

191

MEASURING VISUAL RANGE

9.3.2.4 Visual Telephotometers using an "Artificial Star."

°

FIG. 9.18. Middleton's "artificial star" telephotometer.

y

I,

e-f3u 6

2

E,)-{3y].

r,

192

VISION THROUGH THE ATMOSPHERE

Routine measurements on a fixed light can be facilitated by a graph or table prepared from this equation. In the calibration of such an instrument, the quantity E. is, of course, not measured directly, but is deduced from measurements on a small calibrated source at a distance of a few meters in clear air. Such a simple instrument is naturally limited in its accuracy. In the first place it is necessary to be sure that the sources being compared are actually point sources, and the criteria set down in Chapter V may be used, perhaps with a "factor of safety" of two or three. LeGrand and Geblewicz (1939) have shown that serious errors can arise from neglect of this precaution. The same authors have also pointed out that two sources in different parts of the field of view will appear of different brightness even if the illuminance they produce at the eye is the same; this was also referred to by Collier (1938), who states (p. 142), "Generally, to ensure that two point sources are being compared under the same conditions, foveal vision only can safely be used in the comparison.'' Collier goes further, recommending that the two sources should always be viewed alternately, either by being presented alternately in the same place, or together but at such an angular separation that the eye cannot look at them simultaneously. The writer does not agree with this recommendation, finding that a separation of a few minutes of arc is adequate to produce a well-defined "double star" and that the precision of setting is not very largely dependent on the separation. It is highly desirable that any color difference should be small. This method of observation was used by the writer in his early work on the subject (1931, 1932). The precision obtainable is not in any case very satisfactory. Collier (1938) and the writer agree that ± 12 to 14 per cent is about what may be expected of a routine transmission measurement with an instrument of this type, on the pessimistic but necessary assumption that errors in calibration and mea­ surement may add up. But in spite of its low precision, the "point source" telephotometer has its advantages. Simplicity is one; but a more important advantage is that it is very little affected by scattered light, and the errors discussed in section 9.3.1.1 are therefore minimized. It is effectively a tele­ photometer with a very small angle of view-the subtense of the distant lamp itself. 9.3.3 PHOTOELECTRIC TELEPHOTOMETERS. 9.3.3.1 General Remarks. The modern tendency in photometry is to sub­ stitute physical methods for visual ones; in particular to replace the eye by the photocell. Such substitutions are indeed frequently made by people whose confidence in photoelectric devices is excessive, and much knowledge and experience is desirable if reliable results are to be obtained. This is not the place for a treatise on photoelectricity and its applications, which are dealt with in many special works and in every modern general textbook of optics. We shall merely remind the reader that light-sensitive devices are of three types: photo-conductive (which we shall not consider further), photo-emissive,

MEASURING VISUAL RANGE

193

and photo-voltaic. For brevity, photo-emissive cells will be referred to as phototubes, photo-voltaic cells as barrier-layer cells or merely photocells. In using all these devices, we should never forget that their spectral sensitivity is often very different to that of the eye. Some types can be corrected rather closely to the normal luminosity curve with simple filters; others only with very complex combinations of glass and liquid; but at any rate, one must be entirely clear about what sort of radiation he is actually measuring. The problem is complicated by the notable variations in spectral sensitivity between light-sensitive devices otherwise apparently identical, as for example several phototubes of the same type and manufacture. There is also an important change in spectral characteristics with temperature. Telephotometers using light-sensitive cells may be divided broadly into two classes: those which measure the luminance of extended surfaces, and those which measure the illuminance from a distant lamp or projector. 9.3.3.2 Photoelectric Telephotometers measuring the Luminance of Extended Surfaces. One of the earliest applications of physical photo­ metry to our problem was made by Lohle (1929) whose telephotometer con­ sisted of a phototube at the focus of an objective, a battery, and an electro­ meter to measure the photo-current. Stray light was allowed for by a method essentially the same as that discussed in section 4. 7 above. This instrument was used to test Koschmieder's conclusions.

FrG. 9.19. Schematic diagram of the photoelectric telephotometer of Coleman et al.

The technique of such measurements has come a long way since that time, thanks to the astonishing development of vacuum tubes and their use. The telephotometer used by Coleman and his associates (1949) is the most elabo­ rate modern example. In principle this is a telescopic system which forms an image of a distant object in the plane of a diaphragm in front of a phototube (Fig. 9.19). The optical quality of the system is good enough to permit a field of view of only about half a minute of arc. (The black targets used by Coleman* subtended 1.86 minutes.) Baffles inside the telescope reduce stray light, further reduced *One of these is shown in Fig. 4.6, p. 78.

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VISION THROUGH THE ATMOSPHERE

by large baffles in front of the instrument, as shown in Fig. 9.20, which is a plan of the large van which was adapted to house the telephotometer and its accessories. There is little doubt that the success of this instrument is largely due to the precautions taken to eliminate stray light. The phototube is actually a photo-multiplier tube, the anode current being measured by an exceedingly elaborate potentiometer circuit due to Nottingham, into the details of which we shall not go. The entire equipment is, of course, extremely expensive.

FIG. 9.20. The large van used to house Coleman's telephotometer; plan view.

A somewhat less elaborate photo-multiplier tube circuit is being used by Duntley at the time of wri ting. This follows a circuit developed by Sweet (1947) for a well-known commercial densitometer, which operates on the principle of keeping the anode current constant automatically, and measuring the voltage on the dynodes. This procedure gives a logarithmic scale which can extend to four log units, a very convenient feature. By the use of a gain control it is possible to obtain a direct reading of the contrast between two surfaces, measured one after the other. Pollak and Gerlich (1932, 1933) also used a photoelectric telephotometer to measure the apparent luminance of a black box. 9.3.3.3 Photoelectric Telephometers measuring the Illuminance from a Distant Light. The instruments described in the last section are for use

chiefly in the daytime, and while th