JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

The Effect of Atmospheric Scattering Upon the Appearance of a Dark Object Againsta Sky Background*

G. A. FRY, C. S. BRIDGMAN, AND V. J. ELLERBROCKSchool of Optometry, The Ohio State University, Columbus, Ohio

(Received March 8, 1947)

A theoretical investigation has been made of the "ground-glass-plate effect" and the "edgeeffect" which are associated with the appearance of the dark object seen against a sky back-ground. The problem has been simplified by assuming that the dark object and the observerare placed in an unbounded space filled with an atmosphere having uniform properties at allpoints. Rectangular and disk-shaped objects have been considered. The dark object producesits effect by intercepting the rays of light which would be transmitted directly to the observerand also rays which would be scattered one or more times between the object and the observer.In computing the distribution of brightness across that part of the field of view occupied bythe object, consideration has been given to the rays which would have been transmitted directlyto the eye and rays which would have been scattered once between the object and the eye,but not to scattering of a higher order. Equations for computing the brightness in a givendirection with respect to the object have been developed.

INTRODUCTION

THEORETICAL and experimental investi-gations have been made to determine the

distribution of brightness at different parts of anobject, and its immediate surroundings, when itis viewed against a sky background through hazeor fog.

Some investigators have considered the possi-bility that the size of an object affects the ap-parent brightness at its center and have referredto this influence as the "edge effect." There issome doubt whether this effect is sufficientlypronounced to warrant consideration.

It has also been considered that the atmospherewill make the edge of an object appear blurred.This has been referred to as the "ground-glass-plate effect."

In 1941 Middleton' presented a summary ofthe investigations of these effects which had beenmade up to that time. Since then Middleton2 haspresented a new paper on the theory of the sub-

* The work described in this paper was carried out in partunder Contracts OEMsr-66 and OEMsr-687 between theOffice of Scientific Research and Development and TheOhio State University Research Foundation. The work hasalso received assistance through a grant for research inphysiological optics from the Bausch and Lomb OpticalCompany.

I W. E. K. Middleton, Visibility in Meteorology (Uni-versity of Toronto Press, Toronto, Canada, 1941), pp. 40-41, 56-57.

2 W. E. K. Middleton, "The 'diffusing effect' of fog,"J. Opt. Soc. Am. 32, 139-143 (1942).

ject, and more recently an experimental investi-gation has been reported by Duntley.3

The present paper represents a theoreticalanalysis of the "edge effect" and the "ground-glass-plate effect" for the case of a dark objectsurrounded on all sides by sky.

The dark object produces its effect by inter-cepting and absorbing rays of light that impingeupon it. Some of these rays would have beentransmitted to the eye without being scattered,and their interception reduces the apparentbrightness at all parts of the object but has noeffect upon the appearance of the sky background.

Some of the rays, which are intercepted by theobject, would have reached the eye after beingscattered at one or more points in the atmosphere,and this is true for rays directed both toward theeye and also in other directions. The interceptionof these rays affects the apparent brightness notonly at all parts of the object, but also at all partsof the sky background adjacent to the object.

In this investigation we have considered onlythe rays which would have been scattered oncebetween the object and the eye. A higher degreeof accuracy could be achieved by considering alsothe light which is scattered more than once, butcalculations based on light scattered only oncebring us close enough to the final answer tojustify ignoring the scatterings of a higher order.

3 S. Q. Duntley, "The visibility of objects seen throughthe atmosphere," J. Opt. Soc. Am. 36, 359 (1946).

635

VOLUME 37, NUMBER 8 AUGUST. 1947

FRY, BRIDGMAN, AND ELLERBROCK

The real problem is whether it is necessary toconsider scattering even of the first order.

FORMULATION OF THE PROBLEM

Let us consider the case of a rectangular darkobject (Fig. 1) which is perpendicular to the lineET, which connects the point of observation (E)with the center (T) of the rectangle. The distancefrom E to T is designated by s. The distance fromthe center to the top edge of the rectangle isdesignated by y', and the distance to the lateraledge by x'.

The goal aimed at is to determine the distri-bution of apparent brightness (B) along the lineYY' across the center of the dark object. P repre-sents any point on the line YY' at an angulardistance () from the center of the rectangle.

In order to simplify the problem, it is assumedthat both the observer and dark object are placedin an atmosphere which is unbounded and uni-form in all of its optical characteristics at allpoints. It is also assumed that if it were not forthe dark object, the apparent brightness meas-ured at any point for any direction would beconstant and would be equal to the general levelof sky brightness (B') measured at E.

THE APPARENT BRIGHTNESS AT POINTS ABOVEAND BELOW THE OBJECT

If we pick out a point (P) which lies on the lineYY' above or below the object, the apparent

brightness (B) in the direction of P is determinedby the light scattered toward the eye at all thepoints along the line EP extended to infinity.

Although the line EP passes above or below thedark object, the amount of light scattered towardthe eye at any point II on the line EP is cut downby the dark object, because the object interceptspart of the light which is directed toward H andwhich is scattered at H toward E.

In order to determine the effect of the darkobject upon the apparent brightness (B) in thedirection of P, it is necessary to redefine B asequal to the general level of sky brightness (B')minus the loss in brightness produced by theobject. In order to determine this loss in bright-ness, it is necessary to consider first of all theeffect produced by a small area (AA) of theobject which we may assume to be centered atthe point (K) with coordinates (x, y).

The flux directed toward H which is interceptedby this small portion of the object is equivalentto the flux emitted by a surface having the sameconfiguration and a brightness equal to B'.

The illumination Al at H is given by the follow-ing formula in which (t) is the distance betweenK and H, and (em) is the transmission factor ofthe atmosphere between K and -I, and **represents the angle of incidence of the flux at thefront or the back surface of the object as the casemay be:

AI = (B' cosoe-"AA)/t2.

FIG. 1. Geometrical basis for calculating apparent bright-ness. A rectangular dark object centered at T is observedagainst a sky background from the point E. The figureillustrates specifically the loss in apparent brightness in thedirection of P caused by the interception of light incidenton the back surface of the object at K, which is directedtoward H and which would have been scattered at Htoward E if it had not been intercepted.

(1)

Let AL represent the total number of lumensattenuated by a small volume of atmosphere(A V) at H and scattered in all directions.

AL=AV-AI -a. (2)

In this equation 0- represents the attenuationcoefficient of the atmosphere. Although it ispossible for part of the attenuated light to beabsorbed, it is assumed in this investigation thatit is all scattered.

The candlepower of the small volume ofatmosphere in the direction of E, produced bylight received from K and scattered toward E, isgiven by the following equation:

AC=f(4) *AL. (3)

** For small values of X it is satisfactory to assume thatEK is perpendicular to the surface of the object.

636

ATMOSPHERIC SCATTERING 637

In this equation f(k) represents the scatteringfunction which is nothing more than the ratiobetween the candlepower, at an angle 4) from thedirection of incidence, and the total amount oflight attenuated and scattered in all directions.

For the purpose of defining the relation of theapparent brightness at P produced by the lightfrom K which is scattered toward E, it is satis-factory to assume that the small volume ofatmosphere at H has the form of a truncated coneor pyramid which has its apex at E, as shown inFig. 1, has bases perpendicular to EP with anarea (AN), and has a height (r). The volume(A V) is given by the following equation:

AV=Ar-AN. (4)

The apparent brightness (AB) in the directionof P is given by the following equation:

e-ra.ACAB = (5)

AN

It follows from Eqs. (1)-(5) that

AB e-(t+r),f(4)) cosoAAA= B' . (6)AA -Air t2

Since the quantity, (e-(t+r)1f(0) cosjI)/t2 , is afunction of , let us substitute the symbol (r) forit. Hence

(7)

If we define as B the loss in apparent bright-ness at P produced by the effect of the entireobject upon the light scattered toward the eye atall points between E and P,

oL0 60 120 180

FIG. 2. Wiener's scattering function f(O) for fog droplets.

Z is a factor whose value is determined entirelyby the scattering function f(o). Q is a factorwhose value depends upon the size and shape ofthe object and the distance (y") of P above orbelow T.

If we define B 1 as the loss in apparent bright-ness at P produced by the effect of the entireobject upon the light scattered toward the eye atall points on the extension of EP beyond P,

Bii = 2B'TfJJ f l(r)drdxdy. (12)-u 0 r'

The final equation for the apparent brightness(B) in the direction of a point P which lies on theline YY' above or below the object is:

B = B -Bn -BiII. (13)

The value of BnI is small with respect to thatof BI and hence may be dropped from Eq. (13),which reduces to

B =B'-BII. (14)

Bii = 2B'oJ jjf j l(r)drdxdy.-at 0

(8)

By making certain approximations which aredescribed in Appendix I, Eq. (8) reduces to,

Bii = 2B'-eo f(4)0) [ZQ - ]in which

fr27 f(4))Z= J sinO tanq5-do,

and 0 f(4)Y .1Z dxdy

Q=l Y I V )2X2]!

B

(9)

(10)

Loc

.9s

.98

97

.96

.95

I I I I - I I I I l I

I General Level of Sky Bri htnes

BE~~~~~

B o c

l Edge of ObjectI t J FJI I _I I I l I I I

U .04 .08 .12 .16Angular Distance () from the Center

Object. (Radians)

.20of the

FIG. 3. Distribution of apparent brightness along the line(11) YY' across the center of an object having an angular widthof 0.06 radians.

ABI (AA -Ar) == B'a -1(r).

FRY, BRIDGMTVIAN, AND ELLERBROCK

B

L..J

.99

.98

.97

.96

o .02 .04 .06 .08 .10 J2 .14 .16 .18.20Distance (A) from Center of Object(Rad)

Fi(;. 4. Distribution of apparent brightness (B) along theline YY' across the center of an object having an angularwidth of 0.01 radians.

APPARENT BRIGHTNESS IN THE DIRECTION OFA POINT BETWEEN THE TOP AND BOTTOM

EDGE OF THE DARK OBJECT

When P falls on the line YY' anywhere be-tween the upper and lower border of the darkobject, all of the light directed toward E origi-nating at points on the extension of EP beyond Pis intercepted by the dark object and hence inthis case,

B=B'-Bi-Bii. (15)

In this equation,

BI Be-Sa (16)

and represents the loss in brightness produced bythe interception of rays which are directed to-ward the eye and would have been transmittedthrough the atmosphere without scattering.

BI, represents the loss in brightness producedby the interception of light which would havebeen scattered toward the eye at the variouspoints along the line EP. BII in Eq. (15) is thesame as in Eq. (14) and is defined by Eq. (9).

.04 .08 .12 .16 .20VERTICAL DIAMETER

FIG. 5. Effect of varying the vertical diameter of the objecton B11 at the center of the object.

AN EXAMPLE

Let us consider a specific case in which the darkobject has a vertical angular width of 0.06radians and a horizontal angular length of 0.12radians. The distance (s) from the eye is 88meters, the attenuation coefficient (a) is 0.0391per meter, and the f(4) values are those given byWiener4 for fog droplets. (See Fig. 2.) For this setof f(4) values, Z=0.3733. The value for B' is1 c/M 2 .

In Fig. 3, the solid curve represents the dis-tribution of brightness values (B) at points on theline YY' at various distances from the center ofthe object. These values have been calculated inaccordance with Eqs. (14) and (15).

Figure 4, which is similar to Fig. 3, representsdata for an object which has a vertical angularwidth of 0.01 radians, but is similar in all otherrespects to the object described above.

Varying the vertical angular width has noeffect on B1 but does affect BII. Figure 5 showsthe effect of varying the vertical diameter of theobject on the value of Bil at the center of theobject. This constitutes the "edge effect."

If we define contrast by comparing the bright-ness measured at the center of the object and at asecond point at a considerable distance from theobject, the difference between the quantitiesmeasured corresponds to BI+BII. On the otherhand we may define contrast in terms of theabrupt drop at the border. Since this drop at theborder is equal to BI, it is possible to use Eq. (16)to predict the drop from a and vice versa. Inorder to measure the drop it is necessary tomeasure the brightness both of the object and ofthe sky at points as close as possible to the edgeof the object.

The use of the drop at the edge of the object asan index of contrast is quite satisfactory in thecase of small objects, because the B, distributionis superposed on the relatively flat top of the BIIdistribution. When the object is large, however,the BII component is stronger in comparison withBI; the edge of the object falls closer to theinflection point of the BI, distribution; and thegradients on the two sides of the border are morepronounced. It is conceivable that these gradientsmight have some effect on the appearance of the

4 C. Wiener, Nova Acta d. k. Leop. Carol. Akad. d.Naturf. Halle 73, 1-240 (1900); 91, 1-292 (1909).

VD I Geea III! J General Level of k rghtnss

- r

Edge of 0 ect11. , I I I I II

InrI

638

ATMOSPHERIC SCATTERING 639

object and perhaps also upon the threshold ofvisibility. The effects of such gradients upon theappearance of the "textural quality" of colorhave been discussed in a previous paper.

The answer to the question of the existence ofthe "ground-glass-plate effect" is that it is repre-sented in the Bi, distribution, but this "blurred-edge effect" is masked more or less by the"sharp-edge effect" of the B distribution, de-pending on the size of the object.

EFFECTS OF VARYING THE DISTANCE (s) ANDTHE SIZE OF THE OBJECT AND THE ATMOS-

PHERIC ATTENUATION ((r) ON B ANDBii AT THE CENTER OF

THE OBJECT

In order to demonstrate these effects it is moreconvenient to use a disk-shaped object (Fig. 6)than a rectangular one. It is assumed that thedisk-shaped object is perpendicular to a line TEconnecting its center and the point of observa-tion. The disk can be divided into zones Av wideas shown in Fig. 6.

Area of any zone = 27rv v. (17)

The apparent brightness (AB) at the center ofthe disk which is produced by a single zone isobtained by substituting 27rvAv for AA in Eq. (7).

AB= 27rvBYS(r). (18)

Av*Ar

Since B is the sum of the AB values for all ofthe zones,

BII=27rB'af J I l(r) vdrdv. (19)

By making certain approximations describedin Appendix I, Eq. (19) reduces to,

BII =B o-e 8of(0 o)r 72Zv -- ) (20)

When the size of the object is specified in termsof radians, v'= 's, and hence

BI =B's e--8f(oo) r(2ZO' -O'2) (21)

B, is independent of the size and shape of the

6 G. A. Fry, "The stimulus correlate of bulky color,"Am. J. Psych. 43, 618-620 (1931).

object and is given by Eq. (16). Dividing Eq. (21)by Eq. (16) we obtain

BIIIBI = osf(q/o) (2ZO' 0'2).

The ratio (BII/BI) is directly proportional tothe distance (s) if the angular size of the object iskept constant. Figure 7 shows the relationshipbetween B 1 and B, at various distances.

Equation (22) also shows that the ratio(BiI/BI) is directly proportional to the attenua-tion coefficient (a).

The angular size of the object does not affectBI, but it does affect B. The effect on B isshown in Fig. 8.

LIMITATIONS

Although the relationships demonstrated inthis analysis apply to a large variety situations,certain limitations must be recognized:

1. In order for Eqs. (9) and (16) to holdrigidly, the line of observation EP must be ap-proximately parallel to the surface of the earth.The reason for this is that the atmosphere is notlikely to be uniform at different levels above theearth's surface, and if EP is inclined, a- is notconstant for the various points along EP. Fur-thermore the amount of solar illumination reach-ing the different layers of atmosphere parallel tothe earth's surface varies considerably.

2. It cannot always be assumed that the

ET=r"=s

FIG. 6. Geometrical basis for calculating the apparentbrightness at the center of a disk-shaped object.

(22)

FRY, BRIDGMAN, AND ELLERBROCK

atmosphere between the object and the point ofobservation is uniform even though the line of

observation is parallel to the surface of the earth.3. The assumption that the illumination reach-

ing the front or back surface of the object isuniform from all directions does not necessarily

conform to reality. Particularly when the objectis-near the earth's surface, the light reflectedfrom the earth's surface toward the object will beconsiderably less under most circumstances thanthe sky light above the horizon. Furthermore,sky brightness will vary at different angles ofelevation, so that at least in the vertical directionthe illumination reaching the front or back sur-face of the object is normally not uniform. In thehorizontal direction there is greater chance ofhaving uniform illumination from all directions,particularly when the sun is directly overhead.

4. Wiener's scattering data for different direc-tions have been used in the above analysis.

Wiener's data were worked out for fog dropletsand are characterized by a preponderance offorward scattering. On the basis of this assump-

tion, component Bil becomes negligibly small. In

an atmospheric condition involving a larger

c/m 2 .50

Distance (s) from Point of Observation to theObject.

amount of backward scattering, it may becomenecessary to take BiII into account.

5. It has been assumed that the attenuation isaccomplished entirely by scattering. The theorycould be modified to take absorption into ac-

count, merely by breaking the attenuation factordown into two components, using the sum of the

two components in calculating attenuation, andusing only the scattering factor in computing thetotal amount of light scattered by an element ofvolume.

6. The theory also fails to take into accountthe complications dependent upon the polariza-tion and wave-length composition of sky light.

7. The object used in this investigation con-

sisted of a small segment of a plane surface

perpendicular to the line connecting its centerand the point of observation. This type of objectis most effective in intercepting flux directed

parallel to the line of observation. It has littleeffect on the light flux perpendicular to the line of

observation. It differs in this respect from a three-dimensional object such as a sphere which would

be equally effective in intercepting light from all

directions. The effect of using a surface instead of

a three-dimensional object has been, therefore, tokeep the BII distribution more concentrated.

APPENDIX I

Evaluation of f 1(r) dr

This integral can be evaluated by the followingmethod which is based on the proposition that

f (r)dr =f l(r)dr- f l(r)ir.0 - 00

(1)

As long as 0 is small and H falls between --

and E, it may be assumed that

fAO) =f ko), (2)

(3)

(4)

(5)

CosT = 1,

andt =s - r.

Hence,

FIG. 7. The effect of distance on BI and BIi at thecenter of a disk-shaped object when the angular size is keptconstant.

(s - r) 2 (6)

640

e-(r+t)o' = e-30,

ATMOSPHERIC SCATTERING

r°l(r)dr e--f(q o)

S (7II

E 11

-As long as 0 is small and H falls between E andP, it may be assumed that

e-(r+t)o, = e-sa'

1/t = sink/v,

cos# = cosq5,

and

Vdr =- d4.

COS2k

Consequently,

e-of(q) iI (r) = coso sin lo,V2

and

e-sof(qo) r2 f(O)J 1(r) = I sino tan - dq.f_ V JO ffo)

(8)

(9)FIG. 8. The effect of changing the size of a disk-shaped

(10) object upon the value of B1i at the center.

(11)

It follows from Eqs. (1), (7), and (13) that

l I(r)dr = e-sof(Oo) (Z/v- 1/s)0

(14)

in which(12)

Z= f sino tano dq.0 f()

(15)

(13) The value of Z is entirely determined by thescattering function f(k).

and641

(7)