Journal of the

O PTICALof

SOCIETYAMERICA

VOLUME 54, NUMBER 3 MARCH 1964

Apparent Contrast of Objects on the Earth's Surface as Seen fromabove the Earth's Atmosphere

ROBERT S. FRASERSpace Tecirnology Laboratories, Inc., Redondo Beach, California

(Received 18 July 1962)

The apparent contrast of objects lying on the surface of the earth, when observed in the visible spectrumfrom above the earth's atmosphere, is calculated for three model atmospheres. The earth is illuminated bysunlight, and light is reflected from the earth's surface according to Lambert's law. The apparent contrastincreases with increasing wavelength. The apparent contrast is lower when aerosols are in the atmosphere,than when the atmosphere is free of aerosols. The apparent contrast can be enhanced significantly, if thealbedo of the object space is low, when an analyzer, such as a piece of Polaroid, is used in the optical systemof the receiver.

I. INTRODUCTION

RECONNAISSANCE satellites are now viewingobjects on the surface of the earth to obtain in-

formation about them. The objects are obscured by theearth's atmosphere, which acts as a veil over the ob-jects. This obscuration can be diminished by utilizingthe fact that the light scattered from the atmosphere ispolarized. To do this, one can place an analyzer in theoptical system of a receiver; then one-half or more ofthe obscuring skylight does not pass through the ana-lyzer. If the skylight is 100% plane-polarized, theanalyzer can be oriented so that no skylight passesthrough the analyzer, and the surface objects are notobscured at all by the atmosphere. On the other hand,if the skylight is unpolarized and the light reflected fromthe object space is neutral also, an analyzer eliminatesnot only one-half of the skylight but also eliminatesone-half of the light coming directly from the objectspace. In the latter case an analyzer is of no help inobtaining more information from the object space.Rather than use such qualitative descriptions of theobscuration, one can use the apparent contrast for aquantitative measure. The purpose of this paper is topresent calculations of the apparent contrast as a func-tion of the orientation of an analyzer in a reconnaissancereceiver- for three cases of model atmospheres.

One model atmosphere contains the principal atmos-pheric gases but omits the aerosol particles that are

always present in the atmosphere. This model atmos-phere is called a Rayleigh atmosphere. Because excellenttables of the Stokes parameters of light scattered fromthe earth and a Rayleigh atmosphere exist (Coulson,Dave, and Sekeral), the chief problem of calculating theapparent contrast for this model is getting these tabu-lated data into a computer. The Cornell AeronauticalLaboratory2 has already used these data to calculate theapparent contrast for visible light, in the direction ofthe nadir, and assuming no analyzer is in the receiver.The calculations for a Rayleigh atmosphere are ex-tended in this report to show the apparent contrast foran arbitrary direction of observation and for light thathas passed through an analyzer. As to be expected, theapparent contrast increases with increasing wavelength.Significant gains in the apparent contrast occur with theuse of an analyzer when looking at the earth on the sideof the nadir that contains the sun. The apparent con-trast is a maximum on the solar side of the nadir and notat the nadir. The apparent contrast is nearly independ-ent of the use of an analyzer for a large field of viewsurrounding the antisolar point, though.

1 K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related toRadiation Emerging from a Planetary Atmosphere with RayleighScattering (University of California Press, Los Angeles, 1960),pp. 275-467.

2 "A Scientific Investigation into Photographic Reconnaissancefrom Space Vehicles," Appendix B of Report No. VF-1260-P-8,Cornell Aeronautical Laboratory, Inc., Buffalo, New York (1961);ASTIA No. AD 260694.

289

ROBERT S. FRASER

SUN

0 OBSERVER

ANTISOLAR POINT

9 7 I TOP OF3 , \ 2 ATMOSPHERE

It r 1 ) SURFACE7 77/7/ 7 , ' = 0

\TARGET OF EARTH

FIG. 1. Coordinates in the vertical plane perpendicular to theearth and through the sun and observer. The target lies in thevertical plane in this particular case.

The other two atmospheric models contain aerosolparticles of sizes that are always present in the atmos-phere. The apparent contrast is less when an aerosol isin the atmosphere than when the atmosphere is free ofaerosols. Nevertheless, the apparent contrast can beincreased by the use of an analyzer when the atmos-phere contains aerosols. The advantages of using ananalyzer are greatest when the sun is near the horizon,and for low surface albedo. For example, when one lookstoward the nadir and the ground reflects 0.25 of theincident light, the apparent contrast achievable with ananalyzer is about 0.3 greater than the apparent contrastwhich would be observed without an analyzer. However,the highest apparent contrasts achieved with an analy-zer in an arbitrary direction of observation at wave-length X = 0.503 A are less than corresponding valueswithout an analyzer at X= 0.625 A.

The specific intensity of light that is reflected fromthe object space according to Lambert's law is pre-sented, for it is necessary to know to design a reconnais-sance system.

II. EQUATION FOR THE APPARENT CONTRAST

The apparent contrast at some height above theearth's surface, which corresponds to a normal opticaldepth r from the top of the atmosphere, is defined as

It (T; AU, ma- , (T; A,U,50)

Is(T ,u,'p) (1)IS~ (I; Ja, P)

where It and Is denote the apparent specific intensitiesin the directions of the target and its surroundings,respectively; jA = cos6, where 0 is the zenith angle and yis the azimuth (see Fig. 1). The target is assumed to besmall, both horizontally and vertically, in which casethe A, so direction differs only slightly from the g', ,p'

direction. The specific intensities in Eq. (1) can refer to

where

C(T y(; 0,p) EI t (ri; 6,y); 0,eP)se[a se(

= Y (r I; 0, 0)C (Tl ;0 (A),

y (r1; O. so) = I.S (Tl; 0, (p)eTl~ sec0/ [Is (Tl;

the specific intensity of the light which is passed by ananalyzer, with an arbitrary orientation of its transmis-sion plane, or to the total specific intensity, which isequal to the sum of two mutually perpendicular com-ponents of the specific intensity. As one approaches thesurface of the earth, the normal optical thickness rincreases to the total normal optical thickness Ti, andthe apparent contrast becomes equal to the intrinsiccontrast in Eq. (1).

The specific intensity of the light flowing from thedirection of the target at Ti to the level T can be ob-tained by integrating the equation of radiative transferto obtain

I (T; 6, A) = I(ri; 0, p)e-(r1-r)secO

+J J(t; 6,,p)e7(tT-)s1csecldt, (2)

where J is a source function. T= 0 at the reconnaissancevehicle, if it is above the earth's atmosphere. The firstterm on the right of the equal sign of Eq. (2) representsthe specific intensity of the upward radiation at theearth's surface [I(Ti;6, p)], which is attenuated as itpasses directly through the atmosphere. If the opticalthickness of the atmosphere (Ti) is sufficiently large,very little radiation from the ground passes directlythrough the atmosphere. The second term on the rightof Eq. (2) represents the specific intensity of the sky-light, or of the light scattered toward the observer bythe atmosphere. (Absorption and emission processes canbe neglected in this study, which considers only visibleradiation.) Each level at T =t of the field of view ofoptical length sec6 dt scatters radiant energy of specificintensity secddt J(t; 0, (p) in the direction of the emergentintensity; but this differential of energy is attenuated asit passes through the atmosphere above it according tothe factor e-(t-r)scO.

In order to simplify the calculations of the contrastconsiderably, the intensity of the skylight at the top ofthe atmosphere is assumed to be independent of thereflection properties of the object being viewed, ortarget, but does depend on the reflectivity of the terrainsurrounding the object. As the size of the target ap-proaches zero, the dependence of the specific intensityof the skylight on the presence of the target becomesvanishingly small. If one assumes that the specificintensity of the skylight from above the target is thesame as that from above the surroundings to the target,and substitutes Eq. (2) into Eq. (1), then the apparentcontrast at the top of the atmosphere is

/ I.(T1; 6, p)e-?1 sece+ Tf J(t; 0 ,p)eL- secsecodt]

6 ,p)e T1 seck+ J(t; 0,p)et seC0sec0dt].

(3)

(4)

(5)

290 Vol. 54

March1964 OBJECTS ON EARTH'S SURFACE AS SEEN FROM ATMOSPHERE 291

C(Tr; O,2p) represents the intrinsic contrast at the objectspace. As radiation from the object space passes throughthe earth's atmosphere, the intrinsic contrast is attenu-ated at the rate y, which is called the attenuation co-efficient. As one approaches the object space, or as theatmosphere becomes more transparent to the radiationpassing through it, the specific intensity of the skylight,which is represented by the integral in Eq. (5), de-creases to zero. ESome light is always reflected from theterrain surrounding the object; I,(T1)>0.] Hence, theattenuation coefficient y approaches one, and theapparent contrast approaches the intrinsic contrast. Onthe other hand, if the specific intensity of the skylightbecomes large in comparison to the specific intensity ofthe light transmitted directly from the object spacethrough the atmosphere, for example when the atmos-phere contains considerable haze, the attenuationcoefficient becomes small; and the apparent contrastalso becomes small. The attenuation coefficient willalways lie in the range of zero to one, 0 < y < 1. Theattenuation coefficient, rather than the apparent con-trast, is emphasized in the following sections, since theattenuation coefficient is valid for an entire object spaceand is independent of the light reflected from the target.

Lambert's law of reflection is assumed for the objectspace, except for the target. Lambert's law states thatthe specific intensity of the reflected light is isotropic,and the light is unpolarized. The light that is reflectedfrom the small target can be of arbitrary polarizationand specific intensity. The reflectivity of the target isspecified by the value assigned to the intrinsic contrastand the albedo of the remaining object space.

If no light is reflected from the object space, exceptfor the target, Eq. (4) is indeterminate for the apparentcontrast. The intrinsic contrast goes to infinity, and theattenuation coefficient goes to zero. The apparentcontrast is calculated from Eq. (3) when the objectspace is black.

Since the specific intensity of the light reflected fromthe object space is not tabulated, the method of calcu-lating it is described. Chandrasekhar3 has expressed thespecific intensity of the light reflected from the lowerboundary at T, according to Lambert's law as follows:

PoA15(,,4) 1 A E[Y(TS,1)+Oy)r(Tr1,Mo)] (6)

which is valid when wr units of incident solar flux passthrough a unit area normal to the direction of propaga-tion.. The albedo A equals the ratio of the reflectedmonochromatic flux to the incident monochromatic fluxof skylight and attenuated direct sunlight. The Fyz, yr,and s functions are tabulated in Sekera and Ashburn4

S. Chandrasekhar, Radiative Transfer (Oxford UniversityPress, New York, 1950).

4 Z. Sekera and E. V. Ashburn, "Tables Relating to RayleighScattering of Light in the Atmosphere," NAVORD Report 2061,U. S. Naval Ordnance Test Station, Inyokern, California (1953).

and Sekera and Blanch.5 The specific intensity of thereflected light (Ia) is isotropic and depends on thealbedo, the total optical thickness of the atmosphere,and the solar zenith angle.

The above expression for the specific intensity Eq. (6)is valid only when the albedo A is constant over theentire horizontal plane at ri. Since the target is assumedto be small, Eq. (6) will give a good approximation tothe value of the specific intensity of the light reflectedfrom the object space surrounding the target; that is,

I s(T1s,po) = 7w(lj0

The specific intensity of the light reflected from thetarget (It) can be calculated from Eq. (1), if the in-trinsic contrast at Ti is known.

III. APPARENT CONTRAST AND INTENSITYFOR A RAYLEIGH ATMOSPHERE

An arbitrary small volume of the earth's atmospheredoes not scatter light according to Rayleigh's law.Rayleigh's law is valid only for particles which have anindex of refraction close to one and a size much smallerthan the wavelength of the light. Particles with a size ofthe same order as the wavelength are always present inthe atmosphere. However, measurements of skylightpolarization (Sekera6 ) indicate that the Rayleigh ap-proximation improves as the atmospheric aerosol con-tent decreases. Also, Cornell Aeronautical Laboratory2

made measurements of apparent contrast from highaltitudes of objects on the earth's surface. They foundthat their calculations of contrast for a Rayleighatmosphere agreed well with values measured on ex-ceptionally clear days. It was shown analytically bySekera6 that the Rayleigh approximation improves asthe aerosol content decreases, and the approximationimproves as one proceeds from the red to the nearultraviolet part of the spectrum, where, usually, theamount of radiant energy scattered by the atmosphericgas exceeds that scattered by the aerosol. It appearsthat one can apply rather accurately the calculations ofcontrast and specific intensity for a Rayleigh atmos-phere to a very clear earth's atmosphere, especially forthe bluer light.

The tables (Coulson et al.1) which are used to cal-culate the apparent contrast are given as function of thetotal normal optical thickness Ti. A unique relationshipexists between T, and the wavelength, if the atmosphereis free of aerosol particles and is a gas. The relationshipis given on p. vi of these tables and will be used in thissection whenever a wavelength is said to correspond toa total normal optical thickness.

5 Z. Sekera and G. Blanch, "Tables Relating to Rayleigh Scat-tering of Light in the Atmosphere," Contract No. AF 19 (122)-239,University of California, Department of Meteorology, Los Angeles(1952).

6 Z. Sekera, Advan. Geophys. 3, 43-104 (1956).

ROBERT S. FRASER

A. Contrast Attenuation Coefficients inthe Sun's Vertical

The principal modifications of the apparent contrastby an analyzer in an optical receiver can be seen bylooking at the contrast attenuation coefficients for thesun's vertical. The contrast attenuation coefficients,which are calculated from Eq. (5), are given in Fig. 2for the case that the albedo of the object space isA= 0.25, except for the target. The target can have anarbitrary albedo. The line designated by y representsthe contrast attenuation coefficient which would bedetected above a Rayleigh atmosphere without ananalyzer. The maximum value on that curve falls on thesolar side of the nadir, about 20° from the nadir. Thecoefficient decreases from the maximum to a value of0.2 in a direction of 10° above either horizon. Thehighest values of the attenuation coefficients in anarbitrary direction, except near the antisolar point,occur when the analyzer is oriented parallel to the sun'svertical. These values lie on the yz curve. The maximumvalue on the yz curve occurs slightly closer to the nadirthan the direction of the maximum degree of polariza-tion of the light scattered by the atmosphere alone(dashed curve). This maximum is 0.2 greater than theno-analyzer coefficient for the same direction. The

0.9

0.8

0.7

Z 0.60

z!R 0.5

!Q0U 0.4

0.3

0.2

0.1 L100 80 60 40 20 0 20 40 60

DIRECTION OF OBSERVATION IN DEGREES80 100

FIG. 2. Contrast attenuation coefficients in the sun's vertical fora Rayleigh atmosphere. The wavelength' X=0.495,u (r1=0.15);the albedo A = 0.25; the solar zenith angle Go= 53.10.

lowest possible values of the attenuation coefficients inany direction, except near the antisolar point, occurwhen the analyzer is oriented perpendicular to the sun'svertical. The yr data have the following useful applica-tion: if an analyzer in a surveillance system developeda malfunction so that the transmission plane of theanalyzer were not perpendicular to the plane of polari-zation of the skylight, the attenuation coefficients wouldbe greater than, or equal to, the yr values, except thatthe yz values are slightly smaller near the antisolar point.The apparent contrast is almost independent of theorientation of the analyzer in directions near the anti-solar point, because the degree of polarization of the

FIG. 3. The contrast attenuation coefficients that are valid whenno analyzer is used (left) and when the transmission plane of theanalyzer is perpendicular to the plane of polarization of the sky-light (right). The total normal optical thickness r1=0.25, whichcorresponds to X= 0.436 ,u. The albedo of the object space A = 0.25,except that the albedo of the target is arbitrary. The solar zenithangle Oo = 53. 1°.

skylight is small in those directions. The data on Fig. 2show that when the earth's atmosphere is quite trans-parent, the apparent contrast can be increased by usingan analyzer, in which case the maximum occurs on thesolar side of the nadir.

The advantage of using an analyzer may be pointedout with the following example. Assume that the thresh-old contrast for detection is reached when the attenu-ation coefficient is 0.73. The target can not be detectedwithout an analyzer for the conditions given on Fig. 2.The yz attenuation coefficient shows that the sametarget could be detected throughout a 600 arc of thesun's vertical, with a properly oriented analyzer.

292 Vol. 54

March1964 OBJECTS ON EARTH'S SURFACE AS SEEN FROM ATMOSPHERE 293

B. Azimuthal Dependence of the ApparentContrast

The azimuthal contrast data are plotted on polarcoordinate paper. The nadir datum is plotted at thecenter, and the zenith angle increases radially. The 00azimuthal direction of view occurs in the sun's verticalbelow the sun; and the 1800 azimuth coincides with thesolar vertical on the antisolar side of the nadir. Thedata are plotted for azimuths 0O-180°, because the dataare symmetrical about the sun's vertical.

First, the maximum attenuation coefficients are com-pared with the attenuation coefficients which applywhen no analyzer is used (Fig. 3). The maximum valuesof the attenuation coefficients, which are on the right of

l80o

FIG. 4. The maximum contrast attenuation coefficients, whichare valid when the transmission plane of the analyzer is perpen-dicular to the plane of polarization of the skylight, for low sun(left) and for sun at the zenith (right). The albedo of the objectspace is A =0.25, except that the albedo of the target is arbitrary.r1=0.25; X=0.4 3 6 ,u.

Fig. 3, are obtained by letting the transmission plane ofthe analyzer be perpendicular to the plane of polariza-tion of the skylight. The highest value of 0.79 occurs onthe solar side of the sun's vertical (00 azimuth). Thevalues decrease from the maximum as one proceeds at aconstant zenith angle of 0= 30° from 00 to 1800 azimuth.The contrast attenuation coefficients that apply whenno analyzer is used are shown on the left of Fig. 3. Theno-analyzer values are significantly less than the max-imum values at 00 azimuth, but are only slightly less at1800 azimuth and zenith angle 0> 30°. The attenuationcoefficients achievable with an analyzer exceed 0.6 for alarge field of view, whereas the coefficients do not attainthis value when no analyzer is used.

1800

FIG. 5. The max-imum contrast at-tenuation coefficients(transmission planeof analyzer is per-pendicular to theplane of polarizationof the skylight) whenthe albedo is high(A =0.8),except thatthe albedo of thetarget is arbitrary.Tr1=0. 2 5 ; X=0.4 36ju;Oo= 53.1°.

12003

.4

.5

900

The maximum attenuation coefficients, which applywhen the transmission plane of the analyzer is perpen-dicular to the plane of polarization of the skylight areillustrated for low and high suns in Fig. 4. When the sunis at the zenith (0o= 00), the attenuation coefficients areindependent of azimuth. The broad band of maximaextends almost from the nadir to a zenith angle of0= 60°. The maximum contrast attenuation coefficientson the left of Fig. 4 are valid for a solar zenith angle ofo= 78.5'. The attenuation coefficients show consider-

able asymmetry for low sun. The large values on thesolar side of the sun's vertical (00 azimuth) increase asone proceeds away from the vertical to an azimuth of600. One exception occurs, since the maximum value of0.741 still occurs on the sun's vertical at 0= 120. Theattenuation coefficients reach higher values for low sunthan high sun, but the contrast attenuation coefficientsare above 0.5 for a much larger solid angle of observa-tion for the high sun.

The azimuthal dependence of the maximum contrastattenuation coefficient for high albedo is shown in Fig.5. If one compares these data with that for lower albedoA=0.25 (Fig. 3), one sees that the coefficients for thehigher albedo exceed those of the lower albedo at cor-responding directions. The 0.5 line includes a larger

ROBERT S. FRASER

FIG. 6. The apparent contrast when the albedo of the target isA t = 0.25 and the albedo of the object space surrounding the targetis A =0. rl=0.25; X=0.436,u; Oo=53.10.

solid angle for the high albedo, especially on the anti-solar side of the nadir.

The actual contrast, instead of the contrast attenu-ation coefficients, are given in Fig. 6 for the case thatthe object space is black, except that the target reflects0.25 of the incident light according to Lambert's law. Inthis case the contrast is the ratio of the specific intensityof the light direct from the target to the receiver to thespecific intensity of the skylight [Formula (3)]. If noanalyzer is used in the optical system, the contrasts onthe left of Fig. 6 apply. If the transmission plane of theanalyzer is perpendicular to the plane of polarization ofthe skylight, the maximum contrasts on the right arevalid. The maximum values on the right for directionsnear 0° azimuth are several times greater than corre-sponding no-analyzer values. However, there is negli-gible advantage to using an analyzer to enhance the con-trast for a large solid angle about the antisolar point.

The maximum contrast at each azimuth on the rightof Fig. 6 does not occur in the same direction where thedegree of polarization is a maximum. The maximumdegree of polarization for a Rayleigh atmosphere ofsmall optical thickness occurs along the dashed linemarked by E) = 900, where E) is the angle between theincident solar rays and the direction of propagation ofthe scattered light from the earth. The dashed linemarked E) = 840 extends from the maximum on the sun'svertical. The actual maximum contrast at each azimuthis indicated by the dotted line. The maximum values donot occur along the 840 line for two reasons. First, thespecific intensity of light from the target decreases with

increasing zenith angle according to

It(0)=It(0=00 )exp[-Tr(sec0- 1)].

Second, the minimum specific intensity of skylightpassed by an analyzer lies between the 900 line and thedotted line. Consequently, the direction of maximumcontrast at an arbitrary azimuth on the solar half of thesky lies between the nadir and the direction of themaximum on the sun's vertical (0° azimuth); also thedirection of maximum contrast occurs between thedirection of the maximum degree of polarization andthe nadir.

The azimuthal dependence of the maximum contrastattenuation coefficients are shown for low and hightotal normal optical thicknesses in Fig. 7. The data forthe lower total normal optical thickness, T1=0.02, or awavelength of X= 0.809/1 for an earth's atmospherewithout aerosols, are shown on the left of the polardiagram, and for the higher total normal optical thick-ness of rl= 1.0 (X= 0.312 A) are shown on the right. Themaxima are 0.99 for the lower optical thickness and0.23 for the higher optical thickness; both occur at aboutthe same zenith angle on the solar vertical. The attenu-ation coefficients for the large optical thickness ap-proaches zero at a zenith angle of 0=810. As to beexpected, the attenuation coefficients are the largest forthe smallest optical thickness in an arbitrary directionof observation.

C. Specific Intensity

The specific intensity of the light that is reflectedfrom the object space at the earth's surface and passes

FIG. 7. The contrast attenuation coefficients for small and largetotal normal optical thickness. The albedo of the object space isA = 0.25, except that the albedo of the target is arbitrary. Oo= 53.1°.

294 Vol. 54

March1964 OBJECTS ON EARTH'S SURFACE AS SEEN FROM ATMOSPHERE 295

0.

z

z

U.

0.

0 .0(0 1 2 3 4 5

(SEC 60)

FIG. 8. The specific intensity of light from the nadir. The light isreflected from the earth's surface, then passes through a Rayleighatmosphere and then an analyzer. The albedo of the object spaceis A = 0.25.

through the atmosphere is given by the numerator ofEq. (5):

0.51s(TlIAoA)eTrIP,

.2

0.020.10

10 \ 0.15 &0.25I

1.0\

\

0.809 /I

02 05461A,

0.495 )u

01 \1 0.1436 kL

0.312 ,uSUN'S ZENITH ANGLE I

036.9 53.1 78.5)2 I I I I I I -

0.4

(7)

where the factor of one-half has been introduced toaccount for the attenuation by an analyzer in the opticalreceiver. Since the assumption is made that the light isreflected from the object space according to Lambert'slaw, I. is identical to I in Eq. (6). Hence, the specificintensity of the reflected light reaching the receiver isindependent of the azimuthal direction of observation,but does depend on the zenith angle of observation (0),since the optical path length varies as ri sec6. In addi-tion, the specific intensity of the light reflected from theobject space at the object space depends on the sun'szenith angle, the total normal optical thickness of theatmosphere, and the albedo of the object space.

The specific intensity of the light that would bereceived by a reconnaissance satellite scanning towardthe nadir has been calculated from (7) and plotted inFig. 8. In order to convert the values given on Fig. 8 toabsolute values of specific intensity, the values on Fig. 8should be multiplied by Fx/7r, where F, is the flux ofsolar energy through a unit area normal to the propaga-tion direction, per unit bandwidth, per unit time. If noanalyzer were used in the optical system, the specificintensity of the received radiation would be twice aslarge as the values shown on Fig. 8. The wavelengthswhich are shown apply to the earth's atmosphere if no

0.1

z

Z 0.04U

0.01

0.004

(SEC 6.)

FIG. 9. Specific intensities of light reflected from object spacesof different albedos and passing through the atmosphere and ananalyzer 2 (I.e7-). At and A. indicate the albedos of the targetand of the object space surrounding the target, respectively.Observation direction is toward the nadir.

aerosols are present. The specific intensity of the radi-ation reaching the reconnaissance satellite increaseswith increasing wavelength, but decreases with increas-ing solar zenith angle. The latter fact depends largely onthe incident solar energy being reduced by the factorcos0o, where Oo is the zenith angle of the sun.

The specific intensity of radiation reflected from asmall target on the surface of the earth to a reconnais-sance vehicle above the atmosphere is shown for variousalbedos in Fig. 9. The lower two solid curves refer to analbedo of A= 0.25. The upper curve of the two applieswhen the albedo of the entire object space is A= 0.25.The lower curve applies when the albedo of the objectspace is zero, except for the small target with an albedoof A t= 0.25. When the albedo of the entire object spaceis 0.25, the atmosphere is diffusely illuminated at thelower boundary. The atmosphere scatters part of thisdiffuse light back down to the lower surface, therebyincreasing the amount of energy which is incident on it.The ratio of the upper to the lower continuous curve(ri=0.25) increases with increasing albedo and reaches1.16 when At=0.8. Since the ratio is not large, thespecific intensity of the light reflected from the smalltarget is not sensitive to the albedo of the remainingobject space, at least for moderate, or smaller, opticalthickness (rl<0.25). The dashed curves show how thespecific intensity of light reflected from the targetdepends on the albedo of the object space when the

ROBERT S. FRASER

optical thickness is large (Tl= 1.0). In this case whereA t= 0.8, the ratio of the upper to the lower dashed curveis 1.56. The specific intensity of the light reflected fromthe target shows greater dependence on the reflectivityof the object space at larger optical thickness.

IV. TURBID ATMOSPHERES

A. Atmospheric Models

The atmosphere is never free of aerosol particles, aswas assumed to do the calculations which are given inthe previous sections, but always contains aerosolparticles. The apparent contrasts for two models ofturbid atmospheres are given in this section. The modelsused for these calculations, as well as the method andassumptions introduced to solve the equation of radi-

1 .0

0.9

0.8

0.7

z 0.6

z0!RD

U 0.4

0.3

0.2

0.1

0 I - I I I1000 800 600 400 20° 00 200 400 600

> = 00 U = 1 80°

DIRECTION OF OBSERVATION

80° 100°

FIG. 10. The contrast attenuation coefficients for three modelatmospheres. The data apply to the sun's vertical. The albedo ofthe object space is A =0.25, except that the albedo of the target isarbitrary. The sun's zenith angle oo=53.1°; the wavelengthX=0.625 A.

ative transfer, are presented in another paper by theauthor.' However, the principal characteristics of themodels and the type of solution to the equation ofradiative transfer are repeated here.

One model atmosphere is of the type frequentlyobserved over continental regions. The turbidity forthis model is low; that is, the total normal opticalthickness of the aerosol alone is rT=0.09 2 when thewavelength X=0.625 ,. The other model represents aturbid atmosphere of a type that is sometimes presentover Los Angeles. The Los Angeles model contains muchfewer particles less than 0.1 u radius but more particleswith radius larger than 0.3 u than does the continentalaerosol. The Los Angeles aerosol is moderately turbid,since the total normal optical thickness of the aerosolalone is ri= 0.19 2 (X=0.625 y), which is about twice aslarge as that for the continental model. The phasefunction, which gives the angular distribution of radi-ation scattered out of a neutral beam incident on a smallsample of the aerosol particles (excluding the gas), issimilar for both models. However, the Rubenson degreeof polarization of the scattered light is significantlydifferent for the two models. The Rubenson degree ofpolarization attains a maximum of 0.38, for the con-tinental aerosol, but is generally slightly negative forthe Los Angeles aerosol (X=0.625,4).

B. Calculation of Apparent Contrast

The approximate solution of the equation of radiativetransfer includes the primary scattering of the directsunlight by the aerosol particles plus all orders ofRayleigh scattering. The atmospheric gas scatters lightaccording to Rayleigh's law, and the Rayleigh phasematrix can be applied to part of the light scattered byeach aerosol particle. This solution improves in accuracyas the aerosol content decreases. Hence, this method isapplied to model atmospheres of moderate, or less,turbidity (Tr1<0.25).

The contrast attenuation coefficients that wouldapply to a reconnaissance satellite which scanned theearth through three different atmospheres separately isshown for the sun's vertical in Fig. 10. The azimuthaldependence of the apparent contrasts will be similar tothat which was just shown for a Rayleigh atmosphere.The coefficients have been calculated for red light(X=0.625 u), where the apparent contrast is high incomparison to that for the bluer part of the spectrum.The solid lines (yz) are calculated for the case where thetransmission plane of the analyzer in the optical systemis parallel to the sun's vertical. The solid lines show themaximum values of the attenuation coefficients, exceptnear the antisolar point where the plane of polarizationof the light from the earth is parallel to the sun's verti-cal. The dashed lines (y) show the attenuation coeffi-cients which are valid when no analyzer is used. Therelative advantages of using an analyzer for the three

7 R. S. Fraser, J. Opt. Soc. Am. 54, 157 (1964).

296 Vol. 54

Marchl964 OBJECTS ON EARTH'S SURFACE AS SEEN FROM ATMOSPHERE 297

TABLE I. The contrast attenuation coefficients and optical parameters for three aerosol models. The contrast attenuation coefficients anddegree of polarization of skylight apply to the direction where the contrast attenuation coefficients are a maxima. X=0.625 p; 0o=53.10.

Atmospheric Attenuation coefficient Rubenson degree ofmodel (A = 0.25) Total normal optical thickness polarization

AtmosphereNo Aerosol alone including

analyzer Analyzer (phase angle the aerosol(y) (yI) TlG T1A Te E=900 ) (A =0)

Rayleigh(no aerosol) 0.89 0.97 0.058 0 0.058 .*. 0.94

Continental 0.79 0.87 0.058 0.092 0.150 0.35 0.74Los Angeles 0.68 0.78 0.058 0.192 0.250 -0.06 0.61

atmospheres are similar: from about 200 above theantisolar point to the horizon below it the apparentcontrast is not changed significantly with use of ananalyzer; the apparent contrast can be increasedappreciably toward the nadir and on the solar side withthe use of an analyzer.

Both the y and yi attenuation coefficients depend sig-

0.9

0.8

0.7

0.6

z0

> 0.5

BY 0.4

0.3

0.2

0.1

80° 60° 400 200 0° 20° 400 6C

DC=I O 0=FEA

DIRECTION OF OBSERVATION

nificantly on the aerosol model. If no aerosol is presentin the atmosphere, the coefficients are the largest. Thecoefficients are larger for the continental model than forthe Los Angeles model for two reasons. The total opticalthickness of the continental aerosol is about one-halfthat of the Los Angeles aerosol and the Rubenson degreeof polarization of light scattered by the aerosol in thedirection of maximum contrast is much greater for thecontinental aerosol (Table I). The maximum apparent

z0

z

0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1 I I I I I I I I I 1 . 11000 800 600 400 200 0o 20° 400 600 800 100°

q5= 0 = 11000

FIG. 11. Contrast attenuation coefficients as a function of wave-length for a continental atmosphere and for the sun's vertical. Thealbedo of the object space is A = 0.25, except that the albedo of thetarget is arbitrary. The sun's zenith angle Oo= 53.1g.

DIRECTION OF OBSERVATION

FIG. 12. The contrast attenuation coefficients for low and highsun. The coefficients are calculated for the continental atmospheremodel and apply to the sun's vertical. The albedo of the objectspace is A = 0.25, except that the albedo of the target is arbitrary.The wavelength X=0. 6 2 5 g.

ROBERT S. FRASER

z0

z

I

U0

200 40° 60°

SUN'S ZENITH ANGLE, 0S800

FIG. 13. The contrast attenuation coefficients in the directionswhere the maximum occurs and towards the nadir. The coefficientsapply to the continental atmosphere model. The albedo of theobject space is A =0.25, except that the albedo of the target isarbitrary. The wavelength X =0.625 /i.

contrast achievable with an analyzer does not neces-sarily decrease as the total normal optical thickness ofthe aerosol increases, but also depends on the degree ofpolarization of the light scattered by the aerosol.

The wavelength dependence of the contrast attenu-ation coefficients, which are calculated for a continentalatmosphere, are shown in Fig. 11. The contrast attenu-ation coefficients are higher for the longer wave-length (X=0.625 g) than for the shorter wavelength(X=0.503 u), although the relative advantage (theratio yl/y) of using an analyzer is greater at X= 0.503 ,u.The attenuation coefficients in the longer wavelength,when no analyzer is used, exceed the maximum co-efficients for the shorter wavelengths, except for a 100arc of the sky.

The contrast attenuation coefficients are shown forlow and high sun in Fig. 12. When the sun is at thezenith (Oo= 0), the apparent contrasts without an analy-zer towards an arbitrary direction of view, except nearthe nadir, are much greater than with an analyzer whenthe sun is low. The apparent contrast is not enhancedby the use of an analyzer, when surveillance is towardsthe nadir and the sun is at the zenith. On the other hand,if the surveillance is towards the nadir when the sun islow, the apparent contrast is much greater with ananalyzer that has its transmission plane parallel to thesun's vertical than without an analyzer.

0.5-- X

z z0o 0.4 _ _

0.3-

0.2

0.1

0.0 - l l l I-1000 800 600 400 20° 0° 200 40° 600 800 1000

0= vDIRECTION OF OBSERVATION

FiG. 14. The contrast attenuation coefficients when the albedoof the object space is A = 0.8 for three curves and A = 0.25 for onecurve, except that the albedo of the target is arbitrary. The co-efficients apply to the continental atmosphere and the sun'svertical. The sun's zenith angle is Oo=53.1a. The wavelength isX=0.625 ,.

The contrast attenuation coefficients in the directionswhere the single maximum occurs (for a particularposition of the sun) and towards the nadir are shown as

TABLE II. The ratio of the apparent contrast which is observed when the transmission plane of the analyzer is parallel to the sun'svertical to the apparent contrast which is observed without an analyzer. The observer is above the earth's atmosphere. The object spaceis at the surface of the earth. The sun's zenith angle is 0o=53.10. The albedo of the object space, except for the target, is given by thevalue of A.

CL(r =0)/C(r=0)Model Wavelength Direction of observation toward Direction of observation

atmosphere the maximum contrast toward the nadirA=0 A=0.25 A =0.8 A=0 A =0.25 A =0.8

Rayleigh 0.809 A 28.5 1.03 1.01 1.84 1.02 1.00Rayleigh 0.546 11.3 1.16 1.03 1.78 1.08 1.02Continental 0.625 3.86 1.11 1.04 1.62 1.06 1.02Continental 0.503 4.09 1.24 1.07 1.62 1.13 1.04Los Angeles 0.625 2.42 1.14 1.04 1.47 1.08 1.02

298 Vol. 54

March1964 OBJECTS ON EARTH'S SURFACE AS SEEN FROM ATMOSPHERE 299

a function of the sun's zenith angle (0o) in Fig. 13. The 20

maximum coefficients, which apply when the trans- x 0.625 A

mission plane of the analyzer is parallel to the sun'svertical, do not change greatly with changes in the 10 -

sun's position, whereas the apparent contrast observed CONT NENTAL

without an analyzer decreases when the sun's zenith / \

angle increases beyond 500. Also, the maximum coeffi- ANTI-

cients are not significantly greater towards the direction SUN

where the maximum occurs than towards the nadir. Theadvantage of using an analyzer for high sun is small, x 0.503

but the advantage is great for low sun.The contrast attenuation coefficients for high albedo E 2.0 /\ Xz

of the object space, A = 0.80, except that the albedo for IU ~LOS ANGELESA-the target is arbitrary, are given on Fig. 14. The con- Z X 0.625\

trast attenuation coefficients are higher than when the < 1.0 -

albedo is less (A=0.25). There is little advantage for l

using an analyzer when the albedo is high, such as itwould be for snow fields.

The apparent contrasts are given for the otherextreme of albedo in Fig. 15. The object space is black,except that the albedo of the target is At=0.25. The Zapparent contrast can be enhanced considerably byusing an analyzer when the albedo is low. The ratio of 0.2

the maximum apparent contrast which can be observed

20.0 1000 800 600 400 200 0 200 400 600 800 1000

DIRECTION OF OBSERVATION

10 FIG. 16. The maximum apparent contrast for two model atmos-pheres when the object space is black, except that the albedo of the

Cs target is At=0.25. The target is in the sun's vertical and is ob-served through an analyzer with its transmission plane parallel to

/ c,- -OLAR the sun's vertical. The sun's zenith angle is Oo = 53.10.

/ \POINT

/ .C / , with an analyzer in the optical receiver to the apparentcontrast which is observed without an analyzer is given

2.0 I / / \_ _- as a function of albedo in Table II. (The assumption isZ / Cr made that the intrinsic contrast is independent of the

,/ borientation of an analyzer, which is true if the target andIi lthe remaining object space reflect light according to

< 1.0 -_ _ _ _

Lambert's law.) The apparent contrast increases only afew percent by using an analyzer when the albedo ishigh, but it is advantageous to use an analyzer when thealbedo is low.

F zThe change in the maximum apparent contrast withI Iwavelength and aerosol model for a black object space,

except for the target, is shown in Fig. 16. The apparent0.2 contrast is about twice as large for X =0.625 p as it is for

X = 0.503 A. Also, the apparent contrast is about twiceas large for the continental atmosphere as it is for the

0.1 Los Angeles atmosphere (X=0.625 A).I00° 80° 600 400 200 00 200 400 60° 80° 100°

=o4 C. Intensity of Light ReflectedDIRECTION OF OBSERVATION from Object Space

FIG. 15. The maximum apparent contrast for a continental The specific intensity of light that has been reflectedatmosphere when the object space is black, except that the albedo from the object space according to Lambert's lawof the target is At=0.25. The target is in the sun's vertical. Thewavelength is X=0.625 u. The sun's zenith angle is 00= 53.10. [Eq. (7)] and passes through the atmosphere is shown

RO BERT S. FRASER

0.2

ZzU.

W

(SEC e0).

FIG. 17. The specific intensity of light that is reflected from theobject space with an albedo of A= 0.25 and passes through theearth's atmosphere and an analyzer. 7r units of incident solar fluxpass through a unit area normal to the direction of propagation.The wavelength is X=0.625 p.

on Fig. 17. The assumption was made that the aerosolparticles scatter light according to the law given by theRayleigh phase matrix. If no analyzer were used, thespecific intensities would be twice those which are shownon Fig. 17. The specific intensity of the reflected lightthat would be observed by a reconnaissance satellitedecreases with increasing solar zenith angle, and alsodecreases as the total normal optical thickness of theaerosol alone (ri'1) increases.

V. CONCLUSION

It is well to review the principal restrictions to apply-ing these apparent contrast and contrast attenuationcoefficient data. The data cannot be applied to atmos-pheres which are too turbid or where the aerosol con-centration does not decrease exponentially with in-creasing height above the surface. Furthermore, cloudscannot be near the cone extending from the sensor andincluding the object space, since the atmosphere is

assumed to be horizontally homogeneous. The objectspace, except for the target, is assumed to reflect lightaccording to Lambert's law. However, the character ofthe skylight is not changed greatly if the light is re-flected from the ground according to a much differentlaw, namely Fresnel's law.8 The target is small, but atleast a few feet across. The edges of the target appear tomove during the viewing time, because of atmosphericscintillation. As a result, the recorded intensity of thelight from near the edges is intermediate between theintensities of the lights from the target and from itsadjacent surroundings.

Within these limitations it was shown that the ap-parent contrast of objects on the surface of the earth asviewed above the atmosphere in visible light is greatestfor the longest wavelengths of the visible spectrum.However, a reconnaissance system may operate moreefficiently at shorter wavelengths where more solarenergy may be reflected from the object space. Theapparent contrast can be enhanced by the use of ananalyzer in the optical receiver, but an attitude controlof the reconnaissance vehicle is required for optimumperformance. The transmission plane of the analyzershould be oriented perpendicular to the plane ofpolarization of the skylight. Also, the direction ofobservation should be near the nadir, or even better, ina direction between the nadir and the perpendicular tothe solar rays. However, the maximum apparent con-trast is not much greater than the apparent contrastseen through an analyzer towards the nadir, exceptwhen the albedo is small. When looking near the nadir,the apparent contrast is not enhanced by the use of ananalyzer when the sun is near the zenith, but an analyzerimproves the apparent contrast considerably when thesun is near the horizon. Finally, an analyzer does notincrease the apparent contrast significantly when thealbedo of the object space is high, such as for snowsurfaces, but a properly oriented analyzer increases theapparent contrast considerably when the albedo is low.

8 Z. Sekera, Union Geod6sique G6ophys. Intern. Monogram 10,66 (1961).

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