atmospheric effects in satellite imaging of mountainous...

Reprinted from Applied Optics, Vol. 22, page 1702, June 1, 1983Copyright © 1983 by the Optical Society of America and reprinted by permission of the copyright owner.

Atmospheric effects in satellite imaging ofmountainous terrain

Robert W. Sjoberg and Berthold K. P. Horn

It is possible to obtain useful maps of surface albedo from remotely sensed images by eliminating effects dueto topography and the atmosphere, even when the atmospheric state is not known. A simple phenomenolog-ical model of earth radiance that depends on six empirically determined parameters is developed given cer-tain simplifying assumptions. The model incorporates path radiance and illumination from sun and skyand their dependencies on surface altitude and orientation. It takes explicit account of surface shape, repre-sented by a digital terrain model, and is therefore especially suited for use in mountainous terrain. A num-ber of ways of determining the model parameters are discussed, including the use of shadows to obtain pathradiance and to estimate local albedo and sky irradiance. The emphasis is on extracting as much informa-tion from the image as possible, given a digital terrain model of the imaged area and a minimum of site-spe-cific atmospheric data. The albedo image, introduced as a representation of surface reflectance, providesa useful tool to evaluate the simple imaging model. Criteria for the subjective evaluation of albedo imagesare established and illustrated for Landsat multispectral data of a mountainous region of Switzerland. Themethod exposes some of the limitations found in computing reflectance information using only the image-forming equation.

I. IntroductionEarth-sensing satellites are widely used to help map

natural resources, a very large fraction of which areconcentrated in mountainous terrain. Such areas areoften inaccessible to direct survey methods, and onemust rely on remote sensing to provide needed infor-mation. Since there are insufficient skilled photo-grammetrists to handle the vast quantity of satellitedata made possible through such programs as Landsat,automated preprocessing is essential. The develop-ment of computational schemes to help automate theinterpretation process requires an understanding of theimaging process.

The acquisition of information about mountainousterrain is important, but there have been relatively fewsuccessful applications of remotely sensed images forsuch areas. Explicit slope and elevation data have beenused several times as separate channels in automatedclassification.1"3 Hoffer and his co-workers were amongthe first to at least recognize the role of cast shadows inimages of rugged topography.2 Later studies have

The authors are with Massachusetts Institute of Technology, Ar-tificial Intelligence Laboratory, Cambridge, Massachusetts 02139.

Received 17 May 1982.0003-6935/83/111702-15$01.00/0.© 1983 Optical Society of America.

recognized the role of slope and elevation in the imagingprocess. Shadowsky and Malila4 used slope informa-tion to account for foreshortening and self-shadow. Arecent paper by Holben and Justice5 described fieldexperiments and satellite simulations to explicitly studythe topographic effect on remote imaging. The simu-lations were, however, specifically designed to minimizeand ignore atmospheric interactions by assuming clearsky conditions and concentrating on spectral channelsin which the atmospheric effects are not normally sig-nificant.

The goal of multispectral remote sensing is to recoverinformation about the imaged scene. This informationis usually surface reflectance, an intrinsic, property ofthe material comprising the surface that is independentof the particulars of illumination, topography, andsensor position. Reliable recovery of reflectance de-mands an accurate model of the imaging process, em-bodied in what is called the image-forming or imageirradiance equation. This equation relates image ir-radiance to local surface reflectance, incident illumi-nation, optical properties of the sensor, and other ra-diometric quantities. To compute the desired surfacedescription as a function of image irradiance and otherparameters, it is necessary to invert the image-formingequation.

Inverting the image-forming equation is a difficulttask in high-altitude satellite sensing of the earth dueto the presence of the atmosphere. The atmosphereaffects satellite imaging in at least three ways that must

1702 APPLIED OPTICS / Vol. 22, No. 11 / 1 June 1983

be accounted for in the imaging equation: it attenuatesenergy passing through, it confounds the desired signalwith irrelevant or spurious path radiance, and it imposesa distributed surface illuminant in the form of skylight.A substantial literature has developed on the relation-ship of atmospheric optics to remote sensing. A sampleof that literature can be found in the extensive bibli-ography of Howard and Garing.6 The reader is referredto McCartney's book on atmospheric optics7 andRozenberg's treatise on light scattering8 for introduc-tory material. LaRocca and Turner9 have publisheda comprehensive state-of-the-art review of methods forcomputing atmospheric quantities applicable to remotesensing.

The interaction of atmospheric effects with those oftopography makes remote sensing in mountainousterrain especially difficult. One effect results fromvariations in elevation across the scene. The air massbetween the sensor and the surface diminishes withaltitude, and atmospheric effects are consequently re-duced. There is less attenuation of solar illuminationand reflected radiance, less skylight, and less extraneouspath radiance at higher altitude. Another effect is dueto wide variation in surface orientation across the scene.The slope of a surface element determines its exposureto sunlight and skylight. An element that slopes awayfrom the sun sufficiently is self-shadowed and receivesno sunlight. Even surfaces that would otherwise besunlit may lie in shadows cast by surrounding terrainfeatures. Shadows are abundant in Landsat images ofmountainous regions due to the low solar elevation re-sulting from the early morning overflight time. Bymodeling the contribution of skylight to the reflectedsurface radiance, it is possible to recover informationabout the ground even in shadows.

The thrust of the research reported in this paper wasto render a more faithful representation of surface re-flectance than an original satellite image can deliver byexplicitly accounting for elevation and slope variationand atmospheric effects within the scene. The threegoals of the research were (1) to develop a computa-tionally simple form of the image-forming equationsuitable when viewing rough terrain through an atmo-sphere; (2) to explore ways of determining the modelparameters directly from a satellite image when aminimum of site-specific data is available; and (3) topresent the albedo map as a representation of surfacereflectance and to evaluate it as a tool in determiningmodel parameters.

An albedo image or albedo map is a synthetic imagethat characterizes the reflectance at each point on thesurface by a single scalar value. This is possible undercertain circumstances (such as assuming a Lambertiansurface) since the only difference in reflectance of anytwo points is the difference in the relative amounts ofincident energy reflected from each. The albedo rep-resents an intrinsic property of the terrain cover andshould be invariant with respect to terrain shape, sunand sensor position, and atmospheric state. Severalalbedo images from different spectral channels can becombined to produce a false-color map of terrain cover.

The composite albedo data may also be fed to a patternclassification system. Since the albedo map moreclosely represents surface properties than the rawimage, better classification results are expected, al-though this particular extension was not explored.

An albedo map can be generated only after theimage-forming equation has been inverted to expresslocal albedo in terms of topographic, atmospheric, andsensor optical properties. Effects due to the sensor it-self are presumably known. Those due to topographycan be determined from a suitable digital terrain modelof the area imaged. Atmospheric effects, on the otherhand, are not easily ascertained. A complete treatmentinvolves the solution of the 3-D nonlinear integro-dif-ferential radiative transfer equation appropriate to aspherical earth.9-12 This is computationally infeasible,even if the required information were available.

A number of assumptions simplify the sensor radi-ance equation to one depending on only six parameters.These parameters are empirically determined from theimage and auxiliary data. Without site-specific infor-mation, however, it is difficult to obtain trustworthyvalues.

A trial-and-error approach is explored here. An al-bedo image is generated using a trial set of model pa-rameters and evaluated according to subjective criteriaof acceptability. Adjustments are made in the valuesof the parameters, another albedo image is produced,and the evaluation is repeated. Although no formalismfor determining an optimal albedo image is given, ex-periments indicate that satisfactory results can be ob-tained.

II. Sensor Radiance EquationA. Image-Forming Equation

The signal generated by a remote imaging systemdepends on the irradiance striking the photosensitivesurface of the sensor. It is assumed that the charac-teristics of the optical system are known sufficiently wellthat one can recover this irradiance from the signal byinverting the sensor's transfer function. For a smallaperture, the sensor irradiance is the product of thedirectional irradiance13 and the solid angle subtendedby the aperture 0<^. The directional irradiance is thesum of the attenuated radiance of the surface and at-mospheric path radiance. The general image-formingequation is

Ei(Xi,Vi) = \Lt(lt,tm)Tu{tt,Vm) + Lp(Tt,Vm)}6w, (1)

the sensor irradiance at image coor-dinates (X(,y,)

where Ei(xi,yi)

Lt(Tt,Tm)

Ta(rt,rm)

Lp(rt,Vm)

: the radiance of the imaged surfaceelement (called target) in the direc-tion of the sensor;

: the atmospheric transmission fromthe target to the sensor;

•• the atmospheric radiance introducedin the path between the target andthe sensor;

1 June 1983 / Vol. 22, No. 11 / APPLIED OPTICS 1703

f t = the position of the target in a globalcoordinate system;

Vm = the position of the sensor in the glo-bal coordinate system; and

6w = the solid angle subtended by thesensor aperture.

(The above equation represents intensity only; polar-ization is ignored in this paper.) Although one neces-sarily works with the sensor signal from which imageirradiance can be computed, the subsequent discussionis more concerned with the radiance components thatconstitute the directional irradiance. The term sensorradiance will be used below to denote this directionalirradiance:

Lm (Tt ,r^) = ̂ "-^ = Lt (Tt ,!•„,) T^ (ft ,r,n) + Lp (r, ,r,n). (2)6w

1. Target RadianceThe target radiance Ltdct^m} depends on the pho-

tometric properties of the surface and the distributionof the illumination. Surface photometry is describedby the bidirectional reflectance-distribution function(BRDF) fr{Tt\9i,4>i;9r,<t>r), where (6,,^) and (0rA),respectively, specify the angles of energy incidence andemittance.13 The illumination is a complex combina-tion of attenuated solar irradiance, diffuse sky and cloudirradiance, and reflected ground radiance. Any of theseelements may be missing due to shadowing.2. Transmission

The reflected target radiance is attenuated by itspassage from the target at f t to the satellite at Tm by atransmission factor Tu(rt,r,n). The transmission iseasily expressed as a function of optical thickness r,which is a measure of the extinction properties (scat-tering and absorption) of the light path

Tu(Vt,Vm) = exp[—7-(rt,r,n)],

where r(Vt,rm) is the optical thickness between the twopoints F( and tm- Transmission appears explicitly hereas affecting only the target radiance, although it im-plicitly appears in the expressions for sky and sun ir-radiation of the target.3. Path Radiance

The path radiance Lp(rt,r,n) includes light fromoutside the target that is scattered into the sensor pathso as to appear to come from the target. Such atmo-spheric radiance is made of of light reflected from theground outside the target (the background), a portionof the solar beam passing through the target-sensorpath, and multiply scattered skylight. The path radi-ance does not include light from clouds that obscurepart of the scene, a problem which is better handled asa modification to the scene rather than a property of thelight path.

B. SimplificationsThe sensor radiance equation in the form of Eq. (2)

represents an extremely general formulation of theradiative transfer problem. Since the goal here is to

invert this equation to obtain a measure of surface re-flectance in terms of sensor radiance, a more simplifiedequation is required. Several basic assumptions makethe equation mathematically tractable. Other quiteliberal assumptions are made in lieu of more detailedinformation about the particular application domainand result in an equation that depends on only six pa-rameters.

First, the atmosphere is assumed to be a semi-infiniteplane-parallel horizontally homogeneous air mass.Variations occur only in the z direction (vertical).Specific contributions to the radiative behavior due tomultiple scattering and absorption are ignored. Byadopting a phenomenological model based on empiri-cally determined parameters, the major influences ofthese effects are incorporated as part of the aggregatebehavior. The radiance from clouds is ignored.

Second, the target surface is assumed to be illumi-nated only by (1) a distant point sun of extraterrestrialirradiance Eo whose incident rays make an angle 60 withthe zenith, and (2) a uniform hemispherical sky (see thefollowing paragraph). Illumination of a target by re-flected ground radiance (as one side of a valley lit by theother side) is ignored. Without loss of generality, it isalso assumed that the satellite sensor is a distant pointdirectly over the scene.

Third, it is assumed that the distribution of sky ra-diance can be replaced by an equivalent uniform dis-tribution, in the sense of producing the same irradianceon the target. This assumption permits an analyticexpression for sky irradiance on target surfaces that arenot horizontal and therefore see only a portion of thesky. Evaluation of the expression is straightfor-ward14'15:

£;sky(ri) = EAz)h(0n,<t>n) = £,(z)V2(l + C0s8n),

where Es(z) is the sky irradiance on an unobstructedhorizontal surface at elevation z, and h(9n,<f>n) = V2(l+ cosOn) is the factor by which this irradiance is di-minished for a surface whose normal makes an angle Onwith the zenith. The function h has no azimuthal de-pendence although the argument (j>n is retained forgenerality.

Fourth, the earth's surface is assumed to reflect ac-cording to Lambert's law, where the albedo is allowedto vary from point to point. The radiance of a Lam-bertian surface is independent of the observation angleand depends only on the total irradiance and the surfacealbedo (bihemispherical refectance p).13 The BRDFin this case is

p(rt)fr(tt',8i,<l>i',ffm,<f>m) =

Such a model of earth reflectance has been popular inmany investigations, even though its appropriatenesshas been seriously questioned.16 It is adopted here forthree reasons. First, since the reflectance of a Lam-bertian surface is completely characterized by its albedop, it is possible to compactly represent reflectance as animage, where intensity is proportional to albedo. Sec-ond, the sensor radiation equation is easily inverted for


(3)

Fig. 1. Comparison of the two scattering components of a referenceatmosphere (labeled Ref. in the figure) and their sum to exponentialfunctions (labeled Exp.) derived from least-squares fitting. The

parameters of the exponentials are given in the text.

a Lambertian reflector. Third, for applications of thetype considered in this paper, the ground cover is notknown a priori. In lieu of more specific information onthe geometric reflectance properties of the viewed scene,a Lambertian surface is assumed. The problem of de-termining the local albedo remains.

Under the conditions and assumptions stated above,Eq. (2) becomes

Lm(Xt,yt) = p(x^ T^(z)[EoTd(z)R(0n,<l>n)

+ E,(z)h(0n,<t>n)\ + Lp(z),

where Lm(xt,yt) = the sensor radiance associated withthe target at (x(,y();

p(Xt,Vt) = the albedo of the target at (xt,ytY,Tu{z) = exp[-T(z)], the vertical transmis-

sion from altitude z up to thesensor;

EQ = the extraterrestrial solar irradianceon a surface oriented normally tothe incident rays;

Td(z) = exp[-T(z)/cos(?o]> the slant-pathtransmission from sun to altitude2;

R(ftn,(S>n) = a function that captures the geo-metric dependence of the BRDFand shadow information;

= the sky irradiance on a horizontalsurface at altitude z;

E,(z)

h^n^n) = the geometric dependence of skyirradiance on surface orientation asdescribed above;

Lp(z) = the path radiance between thesensor and the surface at altitude z;and

T-(z) = The optical depth from the sensorto altitude z.

Altitude is obtained from the digital elevation model z= z(xt,y>t) and slope and aspect from the digital slopemodel 0n = 9n(Xt,yt) and 0n = (f>n(xt,yt). The functionR(ffn,<l>n) describes the foreshortening of the surface asseen by the sun and is zero if the target is shadowed:

|0, if target is self- or cast-shadowed,|cos0o, i f0^0o^"- /2 ,

R(8n,<t>n) =

where 0'o is the angle between the local surface normal(0n,<An) and the solar direction (0o»0o):

cos^o = cos0n cosBy + sm6n smOo cos(0n - 0o).

Equation (3) is easily inverted to obtain albedo, the onlyquantity depending explicitly on the target's (x,y) po-sition:

v[Lm(xt,yt) - Lp(z)] (4)p(Xt,Vt) =-,T^(z)[EoTd(z)R(0n,<t>n) + E,(z)h(Bn,<l>n)]

In applying this equation to a real image, the predictedradiance Lm is replaced by the radiance L computedfrom the recorded signal.

C. Exponential FormsThe choice of functions for optical depth, sky irra-

diance, and path radiance is an open question. In theexperiments described below, T, Eg, and Lp were takento be independent exponential functions of altitude:

r(z) = roexp(-z/H},

Lp(z) = Lpo exp{-z/Hp),

E,(z}= Esoexpi-z/Hs),

where the sky irradiance Eg is on a horizontal target.The base constants TO, Eso, and Lpo and the scale heightsH, Hs, and Hp were determined from the evaluation ofsynthetic albedo images as described below.

The choice of exponential functions was inspired bytheir simplicity and their good match to several theo-retical forms advanced in the literature. Optical depthvalues computed by Valley17 for Rayleigh (pure mo-lecular) and aerosol components of the atmosphere areshown in Fig. 1 as functions of altitude for the spectralband 0.5 < \ < 0.6 ̂ m. Superimposed on these curvesare the respective best-fit (in the least-squares sense)exponential forms. The exponentials agree very wellwith the individual components. The third pair ofcurves is the sum of the Rayleigh and aerosol compo-nents and its best-fit exponential. The fit is not verygood here since the Rayleigh and aerosol componentshave distinct characteristic (scale) heights. The best-fitexponential parameters are

Rayleigh: TOR = 0.09917 HR = 8232 m,

HA = 1211 m,

H = 2529 m.

aerosol: TOA = 0.19

sum: TO = 0.2619

Some justification for the use of exponential formsfor sky irradiance is available in the literature. Whenoptical depth is a single exponential function of altitude,assuming that sky irradiance varies exponentially withaltitude is equivalent to assuming InE., varies linearlywith Inr. This hypothesis was examined using com-putation of sky irradiance values for a Rayleigh atmo-sphere from several published sources. Figure 2 showsthe variation of ln{Es/Eo) with Inr at sea level (z = 0)for X = 0.55 /xm and an average ground albedo of p =0.25. It is easily discerned that the points fall fairly


0 COULSON 196Bc COULSON, DAVE, t SEKERA 1960• DEIRMENDJIAN 1. SEKERA 1954• OTTEBMAN 1978-1« TURNER » SPENCER 1972

£-3-^

ft

-4 -3 -1-2

InT

Fig. 2. Variation of relative sky irradiance values with optical depthfor a Rayleigh atmosphere. The approximations made in the textamount to assuming a linear form for this variation. See notes forderivations: Coulson 1968,18 Coulson et al. I960,19 Deirmendjian and

Sekera 1954,20 Otterman 197821 Turner and Spencer 1972.22.23

range of Inr, however, each set of values is approxi-mately linear. Again, an exponential form for pathradiance will fit the Turner-Spencer theoretical formfor a combined Rayleigh and aerosol medium very well(Fig. 5).

III. Description of the Test AreaThe region selected for the research lies in south-

western Switzerland, between 7° I/ and 7° 15' E and4608/30// and 46021/5// N. The region, map titles Dentde Morcles and Les Diablerets, has been used in previ-ous studies.24'25 A digital elevation model (DEM) wasobtained as an array of 174 X 239 values on a 100-m griddigitized from contour maps. The vertical quantizationis 10 m. Altitudes in the scene range from 410 m in thevalley of the Rhone River (southeast corner of the area)to 2310 m on the Sommet des Diablerets (northeast

n COULSON, DAVE, & SEKERA 1960• OTTERMAN 1978« TURNER 1. SPENCER 1972

3.0A

K̂ -3-\

-4 -3 -2

InT

Fig. 3. Comparing sky irradiance on a horizontal surface as a func-tion of altitude for the Turner-Spencer model22 and the exponentialapproximation thereof at X = 0.55 jum. The calculation for theTurner-Spencer curve used a forward-scattering coefficient i] = 0.796,average background albedo Jo = 0.15, and a Rayleigh single-scatteringphase function. The exponential curve E,(z) = Eso exp(-z/Hs) wasfit by least-squares to yield £.,o = 3.04 mW cm"2 and Hs = 2944.9 m.

close to a straight line, particularly within the range ofoptical depths expected in mountainous terrain, say,from Inr = -2.3 to Inr = -0.49 for the case of a Rayleighatmosphere and an altitude range of from z = 0 m to z= 4000 m.

Figure 3 exhibits the closeness of an exponential fitmore directly. Here the Turner-Spencer formulationof sky irradiance for a mixed Rayleigh and aerosol at-mosphere22 is compared to a best-fit exponentialfunction of altitude. The agreement is excellent.

A similar if less satisfactory justification can be foundfor path radiance. Figure 4 shows data for ln(7rLp/£o)vs InT computed from three published sources, again for\ = 0.55 i^m and p = 0.25, in the zenith direction for asea-level target. The three sets do not agree as readilyas the five sky irradiance sets. Within the interesting

Fig. 4. Variation of relative path radiance values with optical depthin a Rayleigh atmosphere. The approximations made in the textamount to assuming a linear form for this variation. See notes forderivations: Coulson et al. I960,19 Otterman 1978,21 Turner and

Spencer 1972.22-23

Fig. 5. Comparing path radiance as a function of altitude for theTurner-Spencer model22 and the exponential approximation thereofat X = 0.55 i^m. The calculation for the Turner-Spencer curve useda forward-scattering coefficient T] = 0.796, average background albedoJo = 0.15, and a Rayleigh single-scattering phase function. The ex-ponential curve Lp(z) = Lpo exp(—z//fp) was fit by least-squares to

yield Lpo = 0.376 mW cm-2 sr-1 and Hp = 2732.56 m.


Fig. 6. Digital elevation model of the Dent de Morcles test regionis displayed here as an image. Elevation is encoded as brightness,the brighter the area, the higher it is. The figure to the right is ahistogram of elevation values, 0 m on the far left, 5110 m on the farright. The lowest point in the scene is 410 m, the highest is 3210 m.The peculiar disparity of adjacent altitudes in the histogram is anartifact of the process used to interpolate elevations during digitiza-tion of the contour maps (even altitudes were favored over odd ones).There are two histograms here: the smaller shaded one representsshadowed targets and the larger solid one sunlit targets (see text).

corner). A model of local slopes was generated from theDEM by modified first differences,26 providing the slopef f n and aspect 0ra for each target in the scene.

Figure 6 is an image of the area generated by trans-lating altitude into brightness, accompanied by a his-togram of altitudes present in the DEM. North is upin the photograph. The solid histogram representstargets that are sunlit, while the shaded one belowrepresents targets that are shadowed. This distinctionbetween sunlit and shadowed targets proved usefulduring the construction of albedo images describedbelow. (In this particular histogram, the solid upperportion has many spikes, giving it a shaded appearance.This is an artifact of the process used to interpolate el-evations during digitization of the contour maps, whereeven altitudes were favored over odd ones and thusappear more often.) The top of the solid histogramgives the total number of targets at a given altitude.The number of sunlit targets is the difference betweenthat value and the top of the shaded histogram.

A multispectral image consisting of four spectralbands from Landsat 1 was obtained for the given area.The image, number 1078-09555, was acquired ~9:55-a.m. GMT on 9 Oct. 1972. For the experiments de-scribed below, only MSS band 4(0.5-0.6 ̂ m) was used,where the influence of the atmosphere is greater thanin the other bands. During the overflight, the sun wasat an elevation of 34.2° with an azimuth of 154.8°, cor-responding to an incident solar direction (0o,<f>o) givenby cos0o = cos(90°-34.2°) = 0.562, 0o = 295.2° coun-terclockwise from geographic east. The Landsat imagewas registered with the digital elevation model,24 andradiometric corrections were applied to eliminate

striping.27 Figure 7 shows the actual MSS band 4 asused in the experiments. Cast shadow information forthe given sun position was generated by the applicationof a hidden surface display algorithm.28 The resultingshadow masks, one for cast shadows and one for self-shadows, are shown in Fig. 8. Figure 9 is a syntheticimage of the area created from the digital elevationmodel. The image shows how the surface would appearat the time of the overflight if the surface were a perfectLambertian reflector of unit albedo and there were nointervening atmosphere.26

IV. Determining Equation Parameters

A. Calibration TargetsThe task of determining the parameters for the al-

bedo equation would be relatively easy if there weresuitable calibration points in a given satellite image.For example, Ahern et al.30 describe the use of clearlakes as calibration targets to determine path radiance.Since the lakes have very low albedo, the sensor radi-ance recorded over them is essentially all path radiance.This technique can be profitably applied in mountain-ous terrain, provided a number of clear lakes appearover a wide range of altitudes in the scene. A regressionanalysis applied to the recorded sensor signals wouldyield a set of best-fit parameters for path radiance.Indeed, in the ideal case, six independent sensor radi-ance values from targets of known elevation, orienta-tion, and albedo should be sufficient to determine all sixmodel parameters.

Fig. 7. Landsat 1 multispectral scanner image for the yellow-greenchannel 4, 500-600 nm. The original raw image was destriped andrectified to be commensurate with the digital terrain models as de-scribed in the text. Note the presence of clouds in the upper leftcorner of the image and the pronounced hazy appearance due to at-mospheric path radiance. The histogram records digitized imagebrightness. A value of 0 corresponds to a sensor radiance of 0.0 mWcm"2 sr-'1. A value of 511 corresponds to a sensor radiance of 2.48mW cm"2 sr~1. (As provided on computer compatible tape, LandsatMSS channels 4,5, and 6 are represented by 7-bit bytes, channel 7 by6-bit bytes. These data were scaled to 9-bit bytes during destripingand rectification, hence the maximum value of 511.) The peak at thehigh end of the histogram is due to saturation on clouds and snow.


CAST SHADOWS SELF-SHADOWS

Fig. 8. Figure on the left is a binary map of those targets in the Dentde Morcles region that were in cast shadow under the illuminationconditions of the Landsat overflight. The figure on the right is a bi-nary map of those targets determined to lie in self-shadow (orientedaway from the sun) under the given illumination. The shadowcomputation was performed using a method derived from hidden-

surface plotting.28'29

Regrettably, the Dent de Morcles region did notsatisfy this rather stringent requirement. There werefew lakes of sufficient clarity and they were not dis-tributed throughout the scene's elevation range, a sit-uation not uncommon in mountainous regions. Noother targets of known albedo (for example, large areasof blacktop or dark tilled soil) were present.

B. Role of ShadowsOn the other hand, the Landsat image contained a

large number of shadowed areas that could be exploitedas calibration targets. Shadows are especially attractivein that they exist at nearly every altitude in mountain-ous terrain and can provide elevation-dependent in-formation. They are prevalent in Landsat images dueto the early morning overflight and consequent low sunelevation. Piech and Schott31 used shadows advanta-geously in densitometric studies of specular and diffusereflection from lake surfaces. They found a linear re-lationship between the scene radiance from sunlit areasand that from areas in cast shadow. From the rela-tionship's slope and intercept values they were able torecover path radiance over the lakes as well as sky ir-radiance. There was no attempt to compare lake sur-faces at different altitudes or nonhorizontal surfaces.In more recent work, Woodham29 has examined thedifferences between sunlit and shadowed targets inLandsat images of a lake and a flat area of coniferousforest in British Columbia. He exhibited altitudeprofiles of sensor radiance that clearly showed a regularvariation of path radiance with altitude. This variationcan be exploited to help find path radiance.

C. Determining Optical DepthThe most satisfactory way of determining optical

depth is through direct measurements of transmissionat various altitudes. Such measurements were notavailable, however, for the experimental area on the

date the Landsat image was made. In lieu of site-spe-cific data, published tables of Rayleigh and aerosoloptical depth were used17 at least initially. In practice,an exponential form fitted to the altitude profiles of thesum of the two components was used, as describedabove.

D. Determining Path RadianceThe regular variation in sensor radiance pointed out

by Woodham can be used to obtain information aboutpath radiance. The scattered points in Fig. 10 comprisean altitude profile of the minimum sensor radiance overthe test region. These minima occur almost entirely inshadowed targets. If one assumes that the minimumradiances are from shadowed targets with very smallalbedo, this profile approximates the path radiancefunction Lp (z). One can determine a pair of values forthe parameters Lpo and Hp by fitting an exponentialcurve to the profile. Two similar exponential curves aresuperimposed on the minimum sensor radiance profilein Fig. 10.^

Normalized Sensor Radiance

Fig. 9. Synthetic image of the test region with the sun in the sameposition as during the Landsat overflight. The ground is assumedto be a Lambertian reflector of uniform albedo p = 1. No atmosphereis assumed. The peak in the center of the histogram is from radianceswithin the Rhone Valley (the nearly uniform gray area at bottomright), a relatively flat region of almost constant altitude. The peakat the extreme left of the histogram is zero radiance for all shadows.

Fig. 10. Two possible exponential models of path radiance as in-ferred from minimum sensor data Lp(z) = Lpo exp(-z/Hp), whereeach curve was fitted manually to the data. For the upper curve, Lpo= 0.33 mW cm-2 sr-1; for the lower curve, Lpo = 0.315 mW cm-2 sr-1.

Hp = 4720 m for both curves.


assumed and snow albedo is estimated at perhaps 0.90,a very coarse value for Es can be obtained for a targetat altitude z with sensor radiance L:

EAz}^^[L-Lp(z)}

pT^(z)h(6n,d)n)

Fig. 11. Altitude profile of average albedo from sunlit targets in theDent de Morcles test area. Albedo was calculated by assuming nointervening atmosphere and examining only sunlit targets at all al-titudes. In such targets, reflected solar irradiance generally domi-nates, and computed albedo should approximate the true value. Thedata have been smoothed but show a distinct increase of albedo with

altitude.

One must be careful when fitting model path radiancecurves to the minimum sensor data. The model func-tion is exponential, decreasing monotonically with al-titude. Although the data points in Fig. 10 clearly dropoff with elevation in the early part of the graph, thetrend reverses at higher 2. This trend is not too sur-prising, since the average albedo increases with altitudedue to the presence of snow. Even in shadow, theminimum radiance is substantial. This increase is ex-hibited in Fig. 11, where the average albedo for all sunlittargets was computed by subtracting path radiance andignoring sky irradiance. The high albedo of snowmeans that reflected skylight and the effects of mutualillumination cannot be neglected. When fitting themodel curve it is necessary to use sensor radiances fromtargets only up to some arbitrary maximum altitude,discarding those beyond. In doing this, however, onealso discards information about the behavior ofLp(z)at higher altitudes that may be needed to find H p .

Fitting path radiance to minimum sensor data re-quires satisfying physical constraints as well as choosingthe data carefully. The exponential curves must lieentirely below the sensor radiance values, except fornoise, since path radiance cannot be negative. Onecannot therefore just fit an exponential through the databy least-squares. The two curves of Fig. 10 were bothfitted by hand. The leftmost sensor data points in thefigure stand markedly above the curve and were ignoredin fitting the curve. There were few targets at this al-titude extreme, all of which, on closer examination, laynear a shadow boundary in a river. It is likely that thesetargets actually contributed some sun glint and werethus brighter than expected.

One could argue that a line will fit the altitude profileas well as the displayed exponentials. Certainly, anyof a host of functions would be suitable, but exponen-tials were chosen here for the reasons stated earlier.

E. Determining Sky IrradianceSky irradiance estimates and their variation with

altitude are very hard to make. The presence ofsnow-covered targets in shadow suggests one method.If suitable optical depth and sensor radiance models are

where h(0n,<f>n) = V2(l + cosffn), the geometric factor fora hemispherical sky, p = 0.90, and Tu(z) and-Lp(z) arepresumed known. If several such values are obtainedover a range of altitudes, Ego and Hs can be esti-mated.

However, estimates obtained in the way describedwere not reliable. The exponential optical depth(Rayleigh and aerosol) described above and both pathradiance curves shown in Fig. 10 were adopted to com-pute approximate sky irradiance. The values obtainedwere particularly sensitive to variations in Lp(z) sincethe term [L - Lp (z)] is usually small. They also dependon knowing h (On ,<f>n) accurately. It was not discovereduntil late into the research that the digital terrain modelhad errors in many parts, including the even-altitudeartifact shown in Fig. 6. These errors were magnifiedin the digital slope model. As a result, the method ofestimating sky irradiance suggested here was notespecially useful.

V. Subjective Evaluation of Albedo ImagesThe success of any method depends heavily on the

quality of the data used. Although the determinationof path radiance parameters from the minimum sceneradiance profiles generates reasonable values, no suchprocedure is possible for the sky irradiance parameters.Estimates of sky irradiance for shadowed snow-coveredtargets depend strongly on the accuracy of surface slope,as well as the accuracy of the path radiance model andthe assumption of the hemispherical nature of the sky.As mentioned above, the slope model was not reliablein many areas of the scene. Furthermore, because thesatellite image had been resampled during rectification,the sensor values for nearby sunlit and shadowed targetswere smeared together. Since noisy conditions are therule rather than the exception in remote sensing, abetter method was sought to determine the parametersof optical depth and sky irradiance.

The scarcity of reliable particulars about the imagedscene prompted an alternative procedure based on trialand error. One makes an educated guess at the atmo-spheric model parameter values and generates an albedoimage using these values. By applying a set of subjec-tive criteria, the acceptability of the image as an albedomap is determined. One can then refine the modelparameter values, generate another image, and reapplythe criteria. This process, while admittedly involvinga bit of art, has been found to produce acceptable albedomaps.A. Criteria for Judging Albedo Image

An albedo image ideally represents the spatial vari-ation of surface albedo over the scene and is invariantwith respect to the imaging situation and surface to-pography. To judge the quality of albedo maps gen-


Fig. 12. Albedo image, and its associated histogram, generated usingmodel parameters for exponential approximations to functional formsof path radiance and sky irradiance computed from the Turner-Spencer model:

Lpo = 0.376 mWTO = 0.26185, £,o = 3.04 mWcm~2,

Hz = 2945 m.

cm"2 sr~1,

Hp = 2734 m,H = 2529 m,

The average computed albedo for sunlit targets was p = 0.117 and forshadowed targets p = 0.276.

erated as described above, three subjective criteria wereestablished.

(1) There should be no visible evidence of surfaceshape (topography) that results from the imaging pro-cess. One expects the ground cover to change with el-evation as the nature of the surface material changes.On occasion, it may even change with surface aspect(azimuth of the surface normal); for example, vineyardstend to be planted on south-facing slopes more thannorth-facing. But there should be no shading differ-ence between targets of like albedo and different or-ientations.

(2) The computed albedo of sunlit and adjacentshadowed targets should be comparable, with no evi-dence of shadows. This is especially true if the shadowis cast by some nearby topographic feature. Theboundary of such a shadow arbitrarily cuts across a re-gion that is typically more or less homogeneous, that is,of similar elevation, surface slope, and ground cover onboth sides of the boundary. A substantial deviation inalbedo is probably due to an inaccurate atmosphericmodel, although bad image and topographic data alsocontribute.

(3) The dynamic range of the set of computed albedofor a scene ought to be reasonable. If the sensor radi-ance equation and the model underlying it were exactand the data were precise and error-free, every albedocalculated would lie between 0.0 and 1.0. Unfortu-nately, such ideal conditions are rarely realized inpractice. A certain percentage of targets will have ap-parent albedo outside the physically possible range.The remainder should be distributed mostly within therange expected for natural surfaces of the kind being

observed, a few percent for clear lakes and dark fieldsor dense forest and perhaps 85-95% for snow-coveredslopes.

B. Example Albedo ImagesThe above criteria were used to judge synthetic al-

bedo images generated for a number of parameter sets,of which only three are presented here. The differencesbetween acceptable albedo images were often quitesubtle, and it was thought they would not survive thereproduction process. However, the difference betweenthe albedo image and the original Landsat multispectralimage is substantial. In all experiments, an extrater-restrial solar irradiance of£o = 17.7 mW cm~2 was as-sumed for MSS channel 4.32'33

For comparison, the first albedo image is based on theTurner-Spencer calculations of path radiance and skyirradiance.22 Since no visibility measurements wereavailable for the Dent de Morcles test area, from whichaerosol optical depth is often inferred, the exponentialforms fitted to Valley's molecular and aerosol opticaldepths17 were used. From these, the Turner-Spencercoefficient for forward scattering is

,.O^L±0^. o.796,TR + TA

where TR = 0.09917 and TA = 0.19. (Using Valley'ssea-level numbers directly results in r) = 0.792, a dif-ference of only a half-percent.) Scattering was pre-sumed to be dominated by the molecular componentand a Rayleigh single-scattering phase function wasused. The average background albedo ~p is 0.15. Al-bedo was computed from Eq. (4) using the exponentialforms fitted to the Turner-Spencer model (shown in Fig.5 for path radiance and Fig. 3 for sky irradiance).

There are several noteworthy features of the albedoimage made using this model (Fig. 12). Certain areasthat appear quite bright in the original Landsat MSSband 4 image are muted in the albedo image. This isparticularly so for the snow-covered mountains in thenortheast (upper left) corner and of the clouds in thenorthwest corner. It is unfortunate but well-knownthat the MSS sensors aboard and Landsat satellitessaturate on snow, clouds, and other highly reflectivesurfaces in sunlight. In the conditions assumed for thepresent research, this saturation means that targets withan albedo greater than ~0.8 would be indistinguishablein the original MSS image. The recorded radiances forhigh-albedo targets such as snow and clouds are lessthan they should be, and the computed albedo is toolow.

A second fact is that it is quite easy to distinguishshadowed from sunlit areas. The shadowed areas arebrighter, that is, have a higher apparent albedo thanneighboring regions in sun. This is emphasized in thehistogram to the right of the image, where the peak forshadowed targets lies to the right of the peak for sunlittargets, although it is not as pronounced. This differ-ence is clear in the enclosed valley in the left center ofthe image. The cast shadow boundary running throughthe middle separates sun on the left from shadow on the


Eso = 3.0 mW

Fig. 13. Improved albedo image and its associated histogram. Al-bedo was computed using the following atmospheric model parame-ters:

TO = 0.26185,

H = 2529 m,

Lpo = 0.315 mWcm 'sr cm

H, = 4720m.Hp = 4720 m,


right. This visible distinction between shadows andsunlit areas violates the second of the criteria set forthabove for an acceptable albedo map. It is possible toimprove the image by adjusting the six model parame-ters appropriately.

The third feature of interest, and one that suggestshow to modify the model parameters, is the apparentincrease with elevation of albedo in shadows. Thenorthwest-facing slopes of the southeast ridge boundingthe central valley lie in shadow. Near the valley center,the computed albedo is ~0.25-0.30. It increases withelevation (and with slope) to 0.80-0.90 just below theridge crest. This contrasts with the opposite side of thevalley, where the sunlit southeast-facing slopes show nosuch differential: the albedo has a more or less uniformvalue of 0.20-0.30. A similar phenomenon can be seenin the vicinity of the summit of Dent de Morcles andother places in the image.

The albedo map also reveals an artifact due to thepresence of clouds in the image. Since the sensor ra-diance model ignores clouds and cloud shadows, thealbedo computed for targets obscured by clouds are toohigh and those computed for targets lying in cloudshadows are too low. This is evident in the upper leftcorner of the image, the cloud shadow lying slightlyabove and to the left of the bright area that is thecloud.

Other apparent artifacts are the isolated very brightor very dark targets scattered about the image. It isunlikely that these are due to aliasing, since resamplingof the image during rectification would have blurredthem over several adjacent targets. It is most probablethat these isolated events result from either: (1) an

incorrect indication of shadow, for example, a sunlittarget may in fact be shadowed or vice versa; or (2) anincorrect surface orientation computed from an inac-curate digital slope model. The latter would be mostnoticeable in shadowed targets and in sunlit targets thatare near grazing solar incidence.

One clarification should be made in passing. Al-though saturation of the MSS sensor over high-albedoareas such as snow and clouds in principle means thatno target should have a computed albedo >0.8, highervalues will result due to errors in calculated surfaceorientation or shadow status. Such targets are revealedin the histogram of Fig. 12, where the rightmost bincontains a very large number of shadowed targets. (Acomputed albedo of greater than unity was clamped to1.0 before generating the albedo image and histo-gram.)

The behavior of the computed albedo in shadow inFig. 12 suggests that the elevation dependence of thepath radiance or sky irradiance components or both maybe wrong. The generally larger shadow albedo indicatesthat perhaps too little path radiance is being removed.A larger value would reduce shadow albedos while af-fecting sunlit target albedo less (the relative contribu-tion to sky radiance of reflected sunlight is about twicethat of path radiance for a horizontal surface under theconditions assumed here). By increasing the scaleheight Hp, the path radiance is made larger at higherelevations.

A similar argument applies to sky irradiance. Byincreasing the scale height Hs, the irradiance fromskylight becomes larger with altitude and results insmaller computed albedo. The ratios of reflectedskylight to path radiance to reflected sunlight for theconditions assumed here are ~2:5:9, respectively. In-creasing sky irradiance therefore makes only a smalldifference in most computed sunlit albedo values, whilesubstantially decreasing those for shadowed targets.

A second albedo image was generated from a differentset of parameters (Fig. 13). A single exponential opticaldepth function derived from Valley's data as describedearlier was used. Both the sea-level value Lpo and scaleheight Hp for path radiance were taken from the ex-ponential model fitted to minimum scene radiances asdiscussed above. Sea-level sky irradiance was arbi-trarily set at 3.0 mW cm"2, a value slightly under theTurner-Spencer value used in the first albedo image.Lacking other data, sky irradiance scale height wastaken to be the same as for path radiance.

The appearance of this image is better than that ofthe first one. Shadowed areas more closely match ad-jacent sunlit areas. The histogram of shadow albedomore closely parallels that of sunlit albedo. However,there is still marked disparity between the two. Fur-thermore, there is some evidence that the albedo gra-dient observed on shadowed northeast-facing slopesmay actually be a property of the scene. The divisionbetween the brighter, higher area and the darker valleyfloor is fairly distinct, and each area appears approxi-mately homogeneous. The larger albedo may indicatethe presence of moderately reflective surface material


such as unvegetated rock, perhaps interspersed withsnow. The south-facing slope is exposed to more sunand may support more vegetation and less snow. Thisis one example of an aspect-dependent effect thatshould be preserved in the albedo image, since it is ac-tually a property of the surface.

The parameter values used to obtain the albedoimage of Fig. 13 can be varied somewhat without sig-nificantly changing the appearance of the image. An-other very similar appearing albedo image was gener-ated from a moderately different set of parameter values(Fig. 14). Sea-level path radiance was increased to avalue between that of the two preceding models in aneffort to decrease shadow albedo over that computedin the second model. To partially compensate for re-ducing the sunlit albedo, sea-level optical depth wasdecreased, while scale height was increased. Thechange in optical depth in effect simulated a cleareratmosphere (less aerosol) than Valley's reference at-mosphere.

This last albedo image is subtly different from thesecond one in appearance, but the histograms of the twodiffer enough to favor this last one. If one compares thealbedo image in Fig. 14 with the synthetic image in Fig.9, it is apparent that virtually all shading due to varyingtopography has been obliterated. The match betweensunlit and shadowed target albedo is very good. Onenotes that the boundaries between sun and shadow areevident as strings of dark targets. These result frominaccuracies in the digital elevation model used to cal-culate slopes and the locations of cast shadows. Finally,

the range of computed albedo fits well within thephysically required range, for sunlit and shadowedtargets individually and together, and their respectivehistograms agree well. Tuning the parameters furthercould undoubtedly improve the image slightly, but thiswas not done.

VI. DiscussionThe simple model of sensor radiance advanced above

is based on an engineering approach designed to expe-dite the extraction of useful data from images ofmountainous terrain. In this regard, it enjoys severaladvantages. It is easily inverted to obtain albedo as alocally computable quantity. That is, the albedo of asurface target depends only on the target's own eleva-tion, orientation, and sensor radiance and not that ofother targets in the scene. It is computationally cheapto generate albedo images, providing fast feedback onthe appropriateness of the model parameters. It de-pends on a minimum of site-specific data and thereforecan be readily applied to existing images for which verylittle information on the atmospheric state is available.To be sure, a digital terrain model is required, but thisis generally much easier to produce post facto thanmeasurements of atmospheric state. The process ofsubjective evaluation allows one to develop an intuitionfor tuning if a small number of model parameters areused. Finally, by explicitly accounting for sky irra-diance and path radiance it is possible to recover re-flectance information from shadowed targets.

Fig. 14. Third example albedo image and its associated histogram.Albedo was computed using the following atmospheric model pa-rameters:

TO = 0.23,

H = 4000 m,

Lpo = 0.33 mW

Hp = 2734 m,

Eso = 3.04 mWcm"2,

H, = 2945 m.



On the other hand, by adopting this simple model oneabandons fidelity to actual environmental conditions.First, the atmosphere is not horizontally homogeneous,When combined with the spatial variation of surfacereflectance across the scene, this inhomogeneity meansthat path and sky radiance depend on horizontal posi-tion as well as altitude. The effect can be significanteven in flat terrain34'35 and is emphasized in moun-tainous areas, where large topographical features tendto decouple portions of the atmosphere.

Second, while it is conceivable that multiple scat-tering and absorption can be incorporated implicitlyinto the sensor radiance equation, neglect of the par-ticulars of particle scattering lead to an incorrecttreatment of, for example, sky irradiance. The hemi-spherical sky model used above ignores the increasedradiance of the horizon sky. Targets inclined towardthe horizon are therefore assumed to receive less skyirradiance than they really do, resulting in a largercomputed albedo. The computation of shadow albedois especially sensitive to this omission. The model alsoignores the presence of the solar aureole that resultsfrom the preferentially forward-scattering atmosphericaerosols. Surfaces inclined toward the sun receivesignificantly greater sky irradiance than computed inthe model here. However, this omission is not serious,since the aureole radiance ought to be included in thedirect solar irradiance term.

Third, radiative transfer theory clearly shows thatoptical depth, path radiance, and sky irradiance aretightly coupled, a fact ignored here by the assumptionof independent model parameters. A more carefulchoice of functional forms could exploit the constraintson the solution space afforded by this coupling. It wasnot considered appropriate for the current investigationas it would complicate the mathematics and would bedifficult to evaluate. However, the introduction ofoptical depth as the coupling element between the at-mospheric components is one refinement that shouldbe fairly straightforward.

Fourth, the exponential dependence on altitude ofthe functions r(z},Lp(z), and Es(z) are not faithful totheir respective behaviors in the real atmosphere, eventhough they agree with theoretical functional forms fora Rayleigh atmosphere. Recent measurements ofaerosol scattering coefficients over Europe show theydecrease slowly or are constant through an altitude of~2300 m and decrease rapidly thereafter.36'37 For theexperimental conditions here, this implies a split aerosoloptical depth, linear at low altitude and dropping offperhaps exponentially at the upper elevations in thescene. Of course, the variation of this profile across thescene due to changing topography may be larger thanthe discrepancy between this split form and an expo-nential form.

Fifth, a digital terrain model gives information on theshape of the ground, not always on the shape of theoverlying ground cover. In the case of a forest, for ex-ample, the tops of the trees are not constrained to followthe ground. Although this is a complication of remoteimaging in general, it is particularly important in ruggedterrain. The sensor radiance and albedo equationsaccount for the shape of the imaged surface, which inthis case is not necessarily that provided by the topo-graphic model. On the other hand, the difference intarget orientation this causes may be masked by evenlarger variations between the assumed Lambertiancharacter of the reflection of light from trees and its truebehavior.

In some respects, the departure of the model from realconditions is forced by the nature of the remote sensingproblem addressed here. In particular, lack of a prioriknowledge about the photometry of the surface was onereason cited for assuming a Lambertian surface. Thereflectance of most of the earth's surface that has beenmeasured shows strong anisotropy,38 especially at lowsun elevations.39 The computation of albedo for sunlittargets ought to account more carefully for the incidentsolar direction in the function R(6n,<i>n) in Eqs. (2) and(4). However, unless one already has some idea of thenature of the surface, a more complicated expression isnot justified. Similarly, the expression for target irra-diance does not include reflected ground radiance.Mutual illumination among targets can have a largeeffect as seen in Fig. 10 (if indeed that is a partial ex-planation for the rise in minimum sensor radiance withaltitude), but to include it in the target irradiance in-tegral presupposes knowledge of the albedo of the sur-rounding terrain.

The presence of sloping targets is a feature of ruggedterrain that demonstrates the importance of modelingboth sun and sky irradiance. For sunlit targets, thedirect solar component generally contributes a sub-stantially larger portion of total target irradiance thanskylight. This is certainly true for low optical thickness,high sun elevation, and moderate surface slopes.However, even in clear atmospheres, when the targetsurface slopes away from the sun so as to be near grazingincidence, the solar and sky irradiance components arecomparable. For medium to low sun elevations, themagnitude of the slope as measured by the angle On isnot that large. Under the sensor radiance modeladopted here and using the model parameters for Fig.14, the target irradiance from sun and sky are equalwhen the slope has the values in Table I. For surfacesfacing directly away from the sun ((pn ~ <Po w 180°), asmall slope will so increase the relative importance ofskylight that it becomes the dominant irradiant.

A. Sensitivity AnalysisIt is useful to briefly analyze the dependence of

computed albedo on the model parameters to see howsensitive it is to the values of the model parameters.This gives some indication of the importance of eachparameter in determining albedo.


Table I. Surface Slope Angle for Which the Direct Solar Contribution toTarget Irradlance Equals the Diffuse Sky Contribution; Values are Given atVarious Altitudes and Two Azimuthal Angles for the Atmospheric Model of

Fig. 14

On when 4>n - 00 = 180°Alt. (m) (degrees)

On when 0n - 0o = 90°(degrees)

0410

100020003210

712172327

4759697883

Table II. Relative Sensitivities {9p/QX)/(p/X) of Computed Albedo to the Path Radiance Model Parameters X = Lpo and X = Hp at Various Altitudesfor MSS Band 4; Other Parameter Values are Those of Fig. 14; Model Sensor Radiance L Is in mW cm"2 sr~1

Alt.(m)

0500

1000150020002500

Lpo-33.000-4.218-2.061-1.277-0.876-0.637

(L = 0.34)Hp0.000

-0.771-0.754-0.700-0.641-0.582

Lpo-0.306-0.242-0.194-0.156-0.127-0.104

(L

0.000-0.044-0.071-0.086-0.093-0.095

X == 1.41)

Hp(L

Lpo-0.153-0.125-0.102-0.083-0.068-0.056

= 2.48)Hp0.000

-0.023-0.037-0.046-0.050-0.052

Table III. Relative Sensitivities (9plQX)t(p/X} of Computed Albedo tothe Sky Irradiance X = E,n and X = H , and Optical Depth Model

Parameters X = To and X = H for Horizontal Sunlit Targets at VariousAltitudes for MSS Band 4; Other Parameter Values are Those of Fig. 14

Alt.(m)

0500

1000150020002500

Esd-0.278-0.240-0.207-0.177-0.151-0.129

X =H..0.000

-0.041-0.070-0.090-0.103-0.109

TO0.5260.4770.4320.3900.3500.314

H

0.0000.0600.1080.1460.1750.196

Table IV. Relative Sensitivities (9plQX)/{ptX) of Computed Albedo tothe Sky Irradiance X = E,o and X = H, and Optical Depth Model

Parameters X = To and X = H for Horizontal Shadowed Targets atVarious Altitudes for MSS Band 4; Other Parameter Values are Those of

Fig. 14

Alt.(m)

0500

1000150020002500

Eso-1.000-1.000-1.000-1.000-1.000-1.000

x=H,0.000

-0.170-0.340-0.509-0.679-0.849

TO0.2300.2030.1790.1580.1400.123

H

0.0000.0250.0450.0590.0700.077


Table II lists the relative sensitivities of computedalbedo to changes in the two path radiance parametersfor a low, a moderate, and a high sensor radiance value.The numbers in each column are the values of (Qp/9 X ) / ( p / X ) , where X is the model parameter heading thecolumn. The sensitivities of the sky irradiance andoptical depth parameters are listed in Tables III and IV,respectively, for horizontal sunlit and shadowed targets.The sensitivities were computed for the set of parametervalues used to generate the albedo image in Fig. 14.

Not surprisingly, the relative effects of each of opticaldepth, path radiance, and sky irradiance drop off withaltitude as the atmosphere diminishes. This is reflectedby the decreasing values of TO, Lpo, and Eso. The de-pendence on the scale heights H, Hp, and Hs, however,increases with altitude (it is proportional to the ratio ofz to the scale height). This emphasizes the importanceof using the radiance from high-altitude targets to helpdetermine the scale heights. As mentioned above, theuse of minimum sensor radiance to find a form for Lp (z)suffers, in this experimental situation at least, from theinability to use the higher elevation information.

It appears that the computation of albedo of a sunlittarget is overall most sensitive to changes in the opticaldepth parameters, except for targets of low sensor ra-diance at low altitude. For shadows, on the other hand,computed albedo is most sensitive to changes in skyirradiance and to path radiance at very low sensor valuesand low altitudes. But the sensitivity to changes inoptical depth is still very important for shadow al-bedos.

VII. ConclusionsIt is evident that the albedo images generated above

are more meaningful representations of intrinsic surfaceproperties than the original satellite image. An honestevaluation of their worth requires at least comparing thecomputed albedo maps to ground truth for the experi-mental area or examining the output of an automatedclassification program given an albedo map as input.Unfortunately, the former evaluation was not possiblein the present investigation since no ground truth wasavailable, and the latter was not attempted due to timeconstraints. Nonetheless, several facts emerge from theexperiments performed in the course of the research.

(1) It is especially important to model terrain andatmospheric effects in mountainous areas for threereasons: (a) to recover the reflectance of shadowedtargets they must be identified and sky irradiance mustbe adequately modeled; (b) the variation of ground slopemust be accounted for in both the solar and sky irra-diance components, and these two effects are compa-rable for targets near grazing incidence; (c) the pathradiance varies with altitude across the scene and mustbe modeled carefully to obtain reflectance in targets oflow sensor radiance (such as those in shadow and onsteep slopes).

(2) A simple model of sensor radiance can approxi-mate atmospheric effects well enough to generate rea-sonable, simple representations of surface reflectancefor mountainous terrain. Values of the model param-eters can be obtained by trial generation of albedo im-ages and subsequent subjective evaluation.

(3) The albedo image represents surface reflectance(under the Lambertian assumption) in a simple formimmediately comprehensible to a human evaluator. Itis therefore a useful tool in determining how the modelparameters should be set to obtain an acceptable albedomap.

(4) Although the presence of specific calibrationtargets in the scene would simplify the determinationof parameter values, shadows may be used almost aseffectively. Path radiance as a function of altitude canbe computed this way. But subjective evaluation is stillnecessary to assist in finding suitable values for theother components, especially sky irradiance.

(5) The sensor radiance model is generally mostsensitive to changes in the optical depth parameters,except at low sensor radiances. It is expected that amore realistic form of optical depth as a function of al-titude, including a proper account of the couplingamong transmission, path radiance, and sky irradiance,would benefit the albedo computation even if othersimplifying assumptions were unchanged.

The engineering method presented in this paper il-lustrates the limitations in recovering surface reflec-tance strictly from the image-forming equation. Acoarse model of surface and atmospheric effects wasassumed here in order to find which aspects are mostimportant in mountainous terrain. As seen, a coarsemodel leads to coarse results. A more refined approachwould furnish a more satisfying analytic procedure thanthe subjective method discussed above; however, itmight require considerably more detailed input than isgenerally available. At one extreme, an exact model ofthe radiative transfer process would paradoxically re-quire an exact description of surface reflectance beforeit could be solved to find surface reflectance. It is clearthat human image analysts do not dynamically solve theradiative transfer equation when interpreting imagesbut apply more intelligent techniques. Until suchtechniques are better understood, one must be contentwith improving the satellite image by filtering it to re-move influences—such as topographic and atmosphericeffects—that are not part of the desired reflectanceinformation.

This paper is a revised version of a dissertation sub-mitted by R.W.S. to the Department of Electrical En-gineering and Computer Science, MIT, in May, 1981,in partial fulfillment of the requirements for the degreeof Master of Science. The research described hereinwas done at the Artificial Intelligence Laboratory of theMassachusetts Institute of Technology. Support forthe Laboratory's artificial intelligence research is pro-vided in part by the Defense Advanced ResearchProjects Agency under Office of Naval Research con-tract N00014-80-C-0505.


The authors would like to thank Kurt Brassel forproviding the original digital terrain model of the Dentde Morcles and Les Diablerets regions, Robert J.Woodham for timely comments and criticism, and J. M.Brady for useful comments on the paper. An unknownreferee is appreciatively acknowledged for his studiedreview of the original manuscript and the many excel-lent criticisms and comments that resulted. R.W.S.especially thanks B.K.P.H. for supervising the presentresearch and for many useful discussions and encour-agement.

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8. G. V. Rozenberg, Sov. Phys. Usp. 3, 346 (1960).9. A. J. LaRocca and R. E. Turner, "Atmospheric Transmittance

and Radiance: Methods of Calculation," IRIA State-of-the-ArtReport, ERIM 107600-10-T (Environmental Research Instituteof Michigan, Ann Arbor, 1975); also available from NTIS asAD-A017 459.

10. S. Chandrasekhar, Radiative Transfer (Dover, New York,1960).

11. S. Ueno, H. Kagiwada, and R. Kalaba, J. Math. Phys. 12, 1279(1971).

12. G. I. Marchuk et al.. The Monte Carlo Methods in AtmosphericOptics (Springer, Berlin, 1980).

13. F, E. Nicodemus et al., "Geometrical Considerations and No-menclature for Reflectance," Nat. Bur. Stand. U.S. Monogr. 160(1977).

14. P. Moon, Ilium. Eng. 37, 707 (1942).15. B. K. P. Horn and R. W. Sjoberg, Appl. Opt. 18,1770 (1979).16. J. A. Smith, T. L. Lin, and K. J. Ranson, Photogramm. Eng. Re-

mote Sensing 46,1183 (1980).17. S. L. Valley, Handbook of Geophysics and Space Environments

(McGraw-Hill, New York, 1965).18. K. Coulson, J. Atmos. Sci. 25, 759 (1968). Sky irradiance data

for jUo = cos0o = 0.562 were interpolated from the published tablesfor a surface albedo of A (or p) = 0.25. £, (r) = Fs + Fd of thepublished data, where the value for F^ for A =0.25 was interpo-lated from the values for A = 0.20 and A = 0.30.

19. K. L. Coulson, J. V. Dave, and Z. Sekera, Tables Related to Ra-diation Emerging from a Planetary Atmosphere with RayleighScattering (U. California Press, Berkeley, 1960). Sky irradiancedata were obtained by numerically integrating the tabulatedvalues of downward radiance for (UQ = cosOo = 0.6 and A (or p ) =0.25. Path radiance data were obtained from the tabulated valuesof upward radiance at p.o = ws6o = 0.6 and A = 0.25 by firstsubtracting the reflected target radiance. The reflected targetradiance is L( = (Td.EoMo + Es}A/Tv.

20. D. Deirmendjian and Z. Sekera, Tellus 6,382 (1954). Sky irra-diance data for p.o = cosOo = 0.562 at A (or p) = 0.25 were inter-polated from the published values of (KO = 1.0,0.6,0.1 at A =0.25.

21. J. Otterman, Appl. Opt. 17,3431 (1978). Sky irradiance valuesfor ô = 0.562 and p = 0.25 were computed directly from thepublished formula for a Rayleigh atmosphere. Otterman failsto give an explicit formula for path radiance (which would beinappropriate for his approach) but does give one for radiantexitance. The plotted path radiance value is actually the radiantexitance at ô = 0.562 and p = 0.25 divided by 27r.

22. R. E. Turner and M. M. Spencer, in Proceedings, Eighth Inter-national Symposium on Remote Sensing of the Environment(Environmental Research Institute of Michigan, Ann Arbor,1972), pp.895-934.

23. Sky irradiance and path radiance data were obtained directlyfrom the published formula for a Rayleigh atmosphere with p =0.25 and ô = 0.562.

24. B. K. P. Horn and B. L. Bachman, Commun. Assoc. Comput.Mach. 21, 914 (1978).

25. R. J. Woodham, "Looking in the Shadows," Working Paper 169(Artificial Intelligence Laboratory, MIT, 1978).

26. B. K. P. Horn, Proc. IEEE 69,14 (1981).27. B. K. P. Horn and R. J. Woodham, Comput. Graphics Image

Process. 10, 69 (1979).28. R. J. Woodham, "Two Simple Algorithms for Displaying Ortho-

graphic Projections of Surfaces," Working Paper 126 (ArtificialIntelligence Laboratory, MIT, 1976).

29. R. J. Woodham, Proc. Soc. Photo-Opt. Instrum. Eng. 238, 361(1980).

30. F. J. Ahern, D. G. Goodenough, S. C. Jain, and V. R. Rao, inProceedings, Eleventh International Symposium on RemoteSensing of the Environment (Environmental Research Instituteof Michigan, Ann Arbor, 1977), pp. 731-755.

31. K. R. Piech and J. R. Schott, Proc. Soc. Photo-Opt. Instrum. Eng.51,84(1974).

32. M. P. Thekaekara, J. Environ. Sci. 13, 6 (1970).33. J. Otterman and R. S. Fraser, Remote Sensing Environ. 5, 247

(1976).34. R. E. Turner, "Radiative Transfer in Real Atmospheres," Tech-

nical Report ERIM 190100-24-T (Environmental Research In-stitute of Michigan, Ann Arbor, 1974); also available from NTISas N74-31873.

35. R. E. Turner, "Atmospheric Effects in Multispectral RemoteSensor Data," Technical Report ERIM 109600-15-F (Environ-mental Research Institute of Michigan, Ann Arbor, 1975); alsoavailable from NTIS as N75-26474.

36. W. S. Hering, "An Operational Technique for Estimating VisibleSpectrum Contrast Transmittance," Scientific Report 16 (Visi-bility Laboratory, Scripps Institution of Oceanography, La Jolla,Calif., 1981).

37. R. W. Johnson and J. I. Gordon, "Airborne Measurements ofAtmospheric Volume Scattering Coefficients in Northern Europe,Summer 1978," Scientific Report 13 (Visibility Laboratory,Scripps Institution of Oceanography, La Jolla, Calif., 1980).

38. B. W. Fitch, J. Atmos. Sci. 38, 2717 (1981).39. B. Brennan and W. R. Bandeen, Appl. Opt. 9, 405 (1970).


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