1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. Dll, PAGES 11,917-11,923, JUNE 16, 2001 Deviation from reciprocity in bidirectional reflectance Marc ...

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RESEARCH, VOL. 106, NO. Dll,

PAGES 11,917-11,923, JUNE 16, 2001

Deviation from reciprocity in bidirectional reflectance Marc Leroy Centred'l•tudes Spatiales dela Biosphere, UMR CNES-CNRS-UPS, Toulouse, France

Abstract. The subjectof the paperis to discuss and quantifydeviationsfrom reciprocity of the bidirectionalreflectancedistributionfunction(BRDF), i.e., the differenceof BRDF obtainedwhen invertingilluminationand viewingdirections.Directionalreciprocityis not valid in general,becausewhen the illuminationbeam hasa spatialextensionlarger than the viewedarea (as is mostoftenthe casefor BRDF measurements), someof the scatterersbuildingup the observedradianceare locatedat differentplacesin reciprocal measurements. The physicalsystems under considerationin the two experimentsare different,hencethe breakdownof reciprocity.The paper developsa theory aimingat a quantitativeestimationof deviationsfrom directionalreciprocitydue to thisfactor.The theoryis basedon integralformsof the radiativetransferequationin a horizontalslabof heterogeneous absorbing and scatteringmedia.The observedsceneradianceis expanded in a seriesof scatteringorders.Integral expressions of the single-and multiple-scattering radianceare derivedand put in a form suitablefor the analysisof the reciprocityproblem. The first-orderexpression leadsto an estimateof the order of magnitudee of the relative

deviations fromreciprocity, e • h/D 8Q/Q x/tan2 0i + tan2 0v - 2 tan oi tan Ovcos whereD is the sizeof the viewedarea (pixel sizefor imagingsensors), h is the vertical photonmeanfree path, 15Q/Qis a measureof the sceneheterogeneity, and 0i, Ov,and are the illuminationand view zenith anglesand the relative azimuthbetweenillumination and view directions.It is arguedthat this order of magnitudeshouldremain approximately valid if all ordersof scatteringare taken into account.A discussion of practical applications in variousfields,laboratoryopticalreflectometry,Earth radiationbudget monitoring,and terrestrialsurfacesremote sensingis given. angular models of radiance to flux conversions.In remote sensingof terrestrial surface,this assumptionpermits to conThe principleof reciprocity,also referred to as Helmoltz strainthe designof BRDF modelsusedto normalizethe efreciprocity,is widelyacceptedin variousbranchesof physics. fectsof different Sun-sensorgeometriesin seriesof measureIts modernand mostgeneralform [Van derHulst, 1980]states ments, or to derive the albedo or other surfacecharacteristics. that "in any linear physicalsystem,the channelswhich lead The theoretical foundation of the directional reciprocity from a causeat one point to an effect at anotherpoint canbe problemwasestablished manyyearsago.Chandrasekhar [1947, equallywell traversedin the oppositedirection.Let the cause 1960] demonstratedwith integralforms of the radiativetransbe first placedat P and the effect measuredat Q; and in a fer equationthat directionalreciprocityholds rigorouslyfor secondexperiment,carriedout in the samephysicalsystem,let horizontallyhomogeneous absorbingand scatteringslabs.Case the causebe at Q and the effectat P. The reciprocityprinciple [1957] showed,also in the framework of radiative transfer is then expressedby the proportionality:effect at Q/causeat theory,that the emergentradiationin directionv at one point P = effect at P/causeat Q." The subjectof the paper is to rv of a closedsurface(boundinga heterogeneous medium), analyzethe deviationfrom reciprocityof the BidirectionalRe- due to an incidentbeam in direction-i at anotherpoint ri of flectanceDistributionFunction(BRDF), i.e., the variationof the surface,is equalto the emergentradiationat ri in direction BRDF occurringwhen illumination and view directionsare i dueto an incidentbeamat rv in direction-v. De Hoop [1960] inverted.We call this problem,after Davies[1994],the direc- establishedwith the more general framework of Maxwell's tional reciprocityproblem. equationsthat when a plane electromagnetic wave is incident BRDF is a key variablein remote sensing.It characterizes upon a scattering obstacle of finite dimensions, the far-zone surface bidirectional reflectance for all combinations of illumiscatteredfield satisfiesa reciprocityrelation. The result was nation and viewingdirections.It is an opticalpropertyof the shownto be valid under little restrictiveassumptions of elecsurfacewhichdoesnot dependon externalfactors,suchasthe tromagneticpropertiesof the obstacle(in particular,continuity distributionof downwellingradiances.Thermal emissionis a and symmetryof the dielectrictensorwithin the obstacle). function of BRDF properties through Kirchhoff's law. If The directionalreciprocityproblemwasinvestigatedexperBRDF can be assumedreciprocal,this reducesby half the imentallyfor a variety of scalesand target typeswith apparnumberof measurements necessaryfor its completedetermiently contradictoryresults.At the centimeterscale,in optical nation. This assumptionis often applied in Earth radiation reflectometrylaboratoryexperiments, Clarkeand Parry [1985] budgetstudiesto fill in missingvaluesfor the generationof and Kim [1988] showedan insignificantdeviationfrom reci1.

Introduction

Copyright2001 by the AmericanGeophysicalUnion. Paper number 2000JD900667. 0148-0227/01/2000JD900667509.00

procitywith unpolarizedlight. Okayamaand Ogura [1985] found oppositeresults,but their experimentalprotocol was criticizedby Venable[1985].The measurements of Clarkeand 11,917

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LEROY: DEVIATION

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IN BRDF

Parry[1985]permittedalsoto showthat directionalreciprocity is stillvalid with polarizedlight, if one considerscorresponding statesof polarizationfor incidentand emergingfluxes.At the 10 m scale in land surface remote sensingstudies,Kriebel [1996] found some deviationto directionalreciprocityusing

the incomingirradianceE cos Oi, is denoted(L(-i, v)) (the anglebracketsrecall the radianceaveragingover S). (L(-i, v)) is dimensionless andis in essence the BRDF of the areaS. Directional reciprocityholds if and only if (L(-i, v)) = (L(-v, i)). One maydecompose (L(-i, v)) in multipleorders

airborne data. At the 50 km scale, clear scenesof Earth radi-

of scatteringwithin the slab,

ation budgetsatellitesensordata showedan insignificantdeviation from reciprocity[Davies,1994; Capderou,1998],while significantdeviationswere seenwith scenescontainingclouds [Davies,1994;LoebandDavies,1997].Lunarphotometrymeasurementsover large areas of the moon gave satisfactorydirectionalreciprocityresultsat the scaleof hundredsof kilometers[Minnaert,1941]. In the context of Earth radiation budget analysis,Di Girolamo et al. [1998]pointedout that a reciprocitybreakdowncan occurwhen the illuminated area has a spatialextensionlarger than that of the viewed area. This conditionis alwaysfulfilled in remote sensing,most often valid in laboratoryexperiments, but hasnot been consideredin previouswork (as for example in the recentdiscussion by Snyder[1998]).It impliesthat horizontal transportof radiation, enteringfrom outsideand exiting within the viewed area, producesa contributionto the measuredradiance, a priori different when illumination and view directionsare inverted.By contrast,photonsenteringand exitingthe viewed area are expectedto obey directionalreciprocity.The authorsperformed numericalsimulationexperiments on the basis of the Monte Carlo method, which indeed

supportedthese argumentsand producedviolationsof directional reciprocity,increasingas the sensorpixel sizedecreases or the sceneheterogeneityincreases. The presentpaper discusses this phenomenonfrom a theoretical point of view. It producesan explicitexpressionof the deviationfrom directionalreciprocityas a functionof the illumination/viewgeometryand of the three-dimensionaldistribution of the scattering and absorbingefficiencieswithin the media. It is based on integral forms of the radiative transfer equation,suitablefor the analysisof the reciprocityproblem, in a horizontal slabof heterogeneousabsorbingand scattering media.The approachdiffersfrom that of Chandrasekhar [1947, 1960] or Case[1957] in that here the sceneradianceobserved by the sensoris expandedin a seriesof scatteringorders,and integralforms of the radiativetransferequationare obtained for eachscatteringorder separately.An order of magnitudeof the deviationsfrom directionalreciprocityis derivedfrom the first-order integral form. It is then argued that this order of magnitudeshouldremain approximatelyvalid if all orders of scattering are taken into account. Practical applicationsof theseresultsat variouslengthscalesare discussed at the end of the paper.

2.

Why Directional Reciprocity is Not Valid

in General

Consider an infinite horizontal slab made of a heteroge-

neousscatteringand absorbingmedium(Figure 1 (top)). We assume,without lossof generality,that this medium is limited at the bottom by a black surface.The slabis uniformlyilluminated in direction -i, and the irradianceincidentnormallyto the slabisE cosOi,where Oiis the incidenceangleof the beam on the slab, and E is the irradiance of the beam. Let a sensor view an area S, locatedon top of the slab,in directionv. The radianceseen by the sensor,averagedover S, normalizedby

(/.(-i,

v))=

v)) + (œ('-)(-i, v)) +...

+

v)) +....

Weconsider firstthefirstorderof scattering term(L (•)( -i, v)). This term takesthe followingform (proofin AppendixA):

(L (1)(--i,¾))

cos Oicos Ov • Vvt(-i, r)P(r,-i,v)t(r,v)dr, (2) where 0v is the viewzenithangle;t(- i, r) is the dimensionless transmissionof light along a path in direction -i down to a point in the slablocatedat r; similarly,t(r, v) is the transmissionof light alonga path in directionv from a point locatedat r up to the top of the slab;P(r, -i, v) is the scatteringphase functionevaluatedat r, for an incidentlight in direction -i and a scattered radiation in direction v, and normalized so that

f4• P(r, 1, 1') d• l, = its(r), where trs(r) is the volume

scattering coefficient (m-•) at position r, and1is anarbitrary direction;Vv is the volumicspaceoccupiedby scattererswhich are at the origin of the light collectedby the sensorin direction v. It is a cylinderof baseS, of heighth, and whosegeneratrix

is alongv (seeFigure1 (top)). Assumenow that the viewingand illuminationdirectionsare inverted(Figure 1 (middle)). The radiancecollectedby the sensorin directioni becomes,from (2),

L(•)(-v, i) 1

1

cos 0 v COS0i S

fvit(-v, r)P(r,-v, i)t(r, i)dr, (3)

whereVi is now a cylinderof baseS and of heighth but whose generatrixis alongi (Figure 1 (middle)). Since

t(-v, r) = t(r, v),

(4)

t(-i, r) - t(r, i)

(5)

(the transmission of light betweentwo pointsdoesnot depend on light direction),and

P(r,-v,

i) = P(r,-i,

v)

(6)

(reciprocity oflightscattering byanelementary volume accordingto Hemholtzprinciple),the integrandsin (2) and (3) are identical. However, the two volumic spaceson which the integrationis performedare not equivalentin general,Vi 4: Vv. The relative deviationfrom reciprocityis

LEROY: DEVIATION

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IN BRDF

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V

V

A

Figure 1. Schematicrepresentation of the horizontalslab:(top) illuminationalong -i and viewingalongv; (middle)illumination along-v andviewingalongi; (bottom)identification of thevolumes•, V• andV, CI V•

f v'v Q(r) dr-fV$ Q(r) dr fVv Q(r) dr where Q(r) is definedas

Q(r) - t(-v, r)P(r,-v, i)t(r, i), (8) invariantwhen i and v are inverted,and V',: and V'i are scatteringvolumes,definedbelowandshownin light grayin Figure 1 (bottom), (7)

V'•= V•- V, 0 V•, V•'• Vi -- V, A Vv.

(9) (10)

In (7), J'vQ(r) dr mustbe interpretedasthe contributionto the radiancecollectedby the sensorof all photonsfor which the scatteringoccursin volume V.

11,920

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IN BRDF

Weseefrom(7)that(1,•,there isnocontribution toeofthe 3. Scaling of Deviations From Reciprocity

scattererslocated in Vi IIVv, the medium being homogeWe startfrom (7) anddefinethe averagecontributions Q to neousor heterogeneous; (2) a horizontallyhomogeneous me- the observedradianceoriginatingfrom V'v, V• and Vv as Q v, diumis suchthat thereis no deviationfrom reciprocity(e - 0), Q i, and Q, respectively;that is, becauseto anypoint rv belongingto V'v,thereis a corresponding point ri belongingto V• with the samescatteringproperQv •

ties,Q(rv) = Q(ri); (3) deviationsfrom directionalreciprocity occur,in general,in horizontallyheterogeneous media.The examinationof (7) showsthat the deviationfrom reciprocity existsbecausepart of the scattererswhich participateto the

vol(V'v) V'•Q(r)dr;

Q'= vol(V$)V$Q(r)dr;

constructionof the observedsignal are not the same when inverting illumination and view directions.A compensation mechanismexistsfor horizontallyhomogeneousmedia but is Q= vol(Vv)VvQ(r)dr (11) not operatingin general. The considerationof higherordersof scatteringn, although more complex,do not modifyqualitatively,and alsoquantita- wherevol(V) standsfor the volumeof the volumicspaceV. Then tivelyin an approximatesense,the conclusions obtainedabove. It is shownin Appendix B, followingthe samereasoningas vol(V'v)Qv- vol(V$)Q, vol(V'v) (Qv- Qi) above, that there is no contribution to e if the first and last

If

scatterings arelocated in Vi [•1 Vv,thescattering medium being homogeneousor heterogeneous.A horizontallyhomo-

e=

vol(Vv)Q

= vol(Vv) Q

vol(V'•) 8Q

geneous(planeparallel)mediumhasno deviationfrom direc= vol(Vv) Q' (12) tional reciprocityfor anyn. This latter resultis consistent with that of Chandrasekhar [1947, 1960]. AppendixB showsalso where/SQ is definedas Qv - Q i. We haveusedin the above that deviationsfrom reciprocityoccurin heterogeneous media equationthe fact that vol(V'v) = vol(V•). Let D be a typical dimensionof the surfaceS (for example,the diameterif S is a at any order n. The ratio between the nth and the first-order scattering circle),h be the height of the horizontalslab, and A be the distancebetweenthe projectionsalongv and i of the centerof S at the bottom surfaceof the horizontalslab(Figure 1 (botwhere to and r are typical values of the single-scattering tom)), albedo and optical thicknessof the medium (Appendix B). The sequenceof successive orders is therefore expectedto A = h x/tan 20i "3tan20v- 2 tanO•tanOvcosqb, (13) decrease very fast for either optically thin (r << 1) or stronglyabsorbing(to << 1) media. Equation (7) should where qbis the relative azimuthbetweenilluminationand view accuratelydescribethe deviationsfrom reciprocityin these directions. cases.For media that at the same time are optically thick If A > D, thenvol(V'v) • vol(Vv), e • $Q/Q, and large (r -> 1) and little absorbing(to • 1), the sequencedecreases deviationsfrom directionalreciprocitycanoccur(e • 1) in the media. slowly.We assumein this casethat the relative contribution caseof stronglyheterogeneous If A < D, then vol(V'v)/vol(Vv) • A/D (for example,if S of thenthorderto deviations fromreciprocity (•L(n)/L©) is roughly equivalent to thatof thefirstorder(15L (1)/L(1)).This is a circle of diameterD, vol(V'v)/vol(Vv) = 2/rr A/D + O(x) means"termsof theorder assumptionimplies that the ratio of absolute deviations O(A/D) 3, wherethenotation from reciprocity,up to the order of n, dividedby the total of x or higher").Equation(12) becomes,in this case, radiance, up to the order of n, is independentof n and is h •Q equalto the e of (7). Thus in all cases,we consider(7) as a basis for an estimation of the order of magnitude of the deviationsfrom directional reciprocity even if all scattering

radiancesscalesroughlyas L(n)/L(1) • [to(1 - e-*)]n-l,

e•D Q x/tan20•+tan 20v-2tan0•tan0vcosqb. (14)

orders

are included.

4.

It is important to note that the deviationfrom reciprocity is tightly linked to the assumptionof illuminationof the slab over an area larger than S. The theoretical development assumesa uniform illumination extendingto infinity for the sakeof simplicity.The resultswould have been the samefor a uniform illumination over an area S' larger than S chosen such that photonsentering the slab outsideS' would have very low probability to exit within S. If illumination takes place only over S, the developmentaboveshowsthat directional reciprocityshouldbe strictlyvalid, the medium being homogeneousor heterogeneous,for all ordersof scattering. This result is consistent with the results of Di Girolamo

et al.

[1998] and of Case [1957] mentioned in section1, when one considersfor the latter a collection of points ri and rv satisfying the Case theorem and covering the area S.

Practical Applications Equation (14) providesa practicalframeworkto decide

whether BRDF modelsshouldbe consideredreciprocalor not in varioussituations,or whetherit is legitimateto complement a data set by its directionalreciprocalfor the retrieval of angularsignatures.Deviationsfrom reciprocitymaybe viewed as significantif they exceedthe experimentalerrors in the retrieval of BRDF, which are generallyof the order of a few 0.01 in relativevalue.To derivean order of magnitudeof e, we reinterpret h as the vertical photon mean free path in the absorbingand scatteringmedium,D as the sizeof the viewed

area(pixelsizefor animagingsensor),and8Q/Q asa measure of the sceneradianceheterogeneityon the sidesof the sensor pixel. Horizontallyhomogeneoussceneshave 8Q/Q = O. A fair order of magnitudefor highlyheterogeneous scenes(high differencesbetweenthe two sidesof the sensorpixel) may be

LEROY:

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IN BRDF

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8Q/Q • 1. One obtains, for example, 8Q/Q = 1 for a mediumthat is homogeneous within the volumeVv and filled with vacuumin the volumeV'i. It is then clear from (14) that reciprocitycan be assumedwhen h << D. For example: 1. In optical reflectometrylaboratoryexperiments,where D • 1-10 cm, BRDF measurements of flat surfaces(h • 1 /xm) shouldbe reciprocal;however,if the sampledtarget includesobjectsof the sizeof a few centimeters,a breakdownof directionalreciprocityshouldbe observedif the illumination beam is larger than the viewed area. 2. In remotesensingof land surfaces,if the area is flat, and if the atmospherecan be consideredas reasonablyhomogeneousoverthe pixel(no partialcloudiness, for example,on the pixel),one may statethat h • 10-30 m for tree covers,h • 10 cm-1 m for herbs or cultures,h • 1 •m-1 cm for soil surfaces(dependingon rugosity).Equation(14) predictsthat the observationshouldbe highly reciprocal if D > 500 m-1

that the area covered by the illumination beam exceedsthat viewed by the sensor.

km for tree covers, D > 20-100

teringradiance L (•)(r, v) atpointr insidetheslab,goingin an

D

>

1-10

m for herbs and cultures,

m for soil surfaces. If the area is not flat and

Appendix A Equation (2) is demonstratedin this appendix.Assumea horizontal scatteringand absorbingslab, horizontallyheterogeneous,boundedby a black surface,uniformly illuminated with incidenceangleOialongdirection-i (Figures1 (top) and A1). All radiancesconsideredhere are, as in the main text, normalizedby the illuminationirradianceon the top surfaceE cos Oi. The incomingradiancein direction1, evaluatedabove the slab, is 1

L(ø)(I)= COS0t 6(1 + i) ' where

6 denotes

the Dirac

distribution.

The first-order

(A1) scat-

upward direction v, satisfiesthe classicalradiative transfer includes,for example, mountains,a proper estimate of h equation shouldbe the height of the mountains.It is therefore to be 0 foreseenthat 1 km spatialresolutionsensorsshouldobservea V•rr L(1)(r, v)+ o-e(r)L(•)(r, v) breakdownof reciprocityon mountainousareas. 3. For Earth RadiationBudgetsatellitedata where broken cloud field scenesare involved,one may considerthat h • altitude of clouds • 1-10 km. Therefore reciprocityshould holdwith high accuracy,accordingto (14), ifD > 50-200 km, dependingon the cloudheight. where O-e(r)is the volume extinction(scatteringplus absorption) coefficient,andP(r, 1,1') is the scatteringphasefunction, normalizedsuchthat f4•P(r, 1, 1') d12•: O-s(r),where O-s(r) 5. Concluding Remarks is the volume-scatteringcoefficientat r. We have In conclusion,this paper presentsa theoreticalframework L(ø)(r,1) = t(-i, r)LIø)(I), (A3) for understanding and quantifyingdeviationsfrom directional reciprocity.The predictionsof (14) agreea posterioriwith the wheret(- i, r) is the transmission of lightbetweenan outside resultsof Monte Carlo simulations[Di Girolamo et al., 1998], point R and M(r) located at r, such that MR is along -i whichshowedthat the deviationsfrom reciprocity(1) increase (Figure A1), with sceneheterogeneity,(2) increaseas the view direction getsfartherfrom the illuminationdirection,(3) decreaseasthe t(-i, r) = e - f• o-e(r)ds. (A4) sensorpixelsizeincreases. The scalingderivedfrom the theory permitsto determinewhetherangularmodelsshouldbe con- The radiative transfer equation (A2)writes (with (A1) and sideredas reciprocalor not, dependingon the characteristics (A3)) of photonmean free path, sensorresolution,and sceneheterogeneityof the physicalproblemto be studied. This studyhasbeenperformedin the frameworkof radiative transfertheory.This frameworkis widelyusedto model radi-

=f4P(r, 1,v)L(ø)(r, 1) d12,,

ation-matter

interaction

(A2)

within natural surfaces and the atmo-

sphereand has therefore somedegreeof generality.On the other hand, radiative transfer theory does not account for phenomenasuchas diffractionor surfacediscontinuities (diopttic surfaces).It wasdemonstrated by Liet al. [1999]that it is indeedpossibleto conceiveexamplesof structuredsurfaces containingdioptric elementswhich do not satisfyreciprocity even when the illuminated

and viewed areas are the same.

It is emphasizedthat the presenttheory is not in contradiction with the Helmholtz reciprocityprinciple mentioned in section1. The Helmholtz principle holds if and only if the physicalsystems, whichcreatean effect at Q when a causeis appliedat P, are the samewheneffectand causeare inverted. As illustratedby Figure 1, the physicalsystemsunder consideration are clearly not the samewhen invertingillumination and view directions,sincethe scatterersconstructingthe signal Figure A1. Representationof the heterogeneous horizontal are locatedat differentplacesin the two experiments, provided slab boundedby a black surface.

11,922

LEROY: DEVIATION

d

1

-- L")(r v) + O'e(r)L (•)(r, v) = --P(r,-i,

dv

FROM RECIPROCITY

'

cos Oi

v)t(-i,

r),

(AS)

wherev countsdistancealongthe v direction.Let t(r, v) =

IN BRDF

and directionas l; rn is the lastscattering center;the integral overrn is 1-D (hencethe notationrn) andalongthe segment QM; the integralsoverr•... rn-• are 3-D integralsoverthe

volume of theentireslab;Q(n)isgivenbytheexpression

exp-f• O'e(r) ds bethetransmission of lightbetween M(r) andP locatedat the top of the slab,suchthat MP is alongv (FigureA1). (A5) may then be rewritten d

1

ß t(--i, r0t(r•, r2) ... t(rn, r)P(r•,-i, t(r, v)P(r,-i,

dv[t(r,v)Lø)(r, v)]= cos 0i

v)t(-i,

r),

(A6) which,by integrationalongthe path QP, gives

L(1)(P, v)=cos 10i f•t(r,v)P(r,-i,v)t(-i,r) dv.

1

Q(n)(-i, r•,r2,...,rn,r)= ir•r212 Ir2r3 2''' Irn_•rn 2

1

r•r2)

ßP(r2,-r•r2, r2r3)''' P(rn, -rn-•rn, v).

(B3)

The notationr•r2, for example,refersto a vectorgoingfromr• to r2. (B2) is obtainedby recurrence. Assumeit istrue at order (n - 1), insertthe corresponding expression in the transfer equation(B1), integrate(B1) alonga segmentQM. The result

(A7) reads

Theupward first-order scattering radiance (L (•)( - i, v)) averaged over the exit surfaceS is

(L")(-i,v))= •

1

L(•)(P, v)dS

t(r',1)dr' t(r, l)L(n)(r, 1) =COS 10•••(r) •4 •M' (r') d•l, P(r', 1', l)

1

COS 0i S dS t(r,v)P(r,-i,v)t(-i,r) dv

drn-•

ff• drl,...,drn-2Qn-l(-i, rl,...,rn-1, r'), slab

COS 0i COS 0vS Vvt(r,v)P(r-i, v)t(-i r) dr,

(B4)

wheret(r, 1) is the transmission of light alongI betweenr and

whichis the expression of (2).

thetop(bottom)of theslabif I isdirectedupward(downward). Q' is locatedat the bottom(top) of the slabif 1' is upward (downward)andsuchthatQ'M' isalong1'.M' islocatedat r'. The variablern-• is integratedover Q'M'. (B4) is equivalentto (B2). To showthis,set r' -= rn, l' --

Appendix B

rn_lrn,d•, drn_1 = drn_l/rn_lrnl2, andobserve thatt(r',

(A8)

The nth-order scatteringradiancecan be calculatedsimilarly to the first-orderradiance,usinga recurringreasoning. Assumeagain a horizontalslab uniformlyilluminatedwith incidenceangle 0i alongdirection-i, boundedat the bottom by a blacksurface.The radiativetransferequationto be solved at the n th order is 0

l)/t(r, 1) = t(r', r). The demonstration that (B2) is true for n = 2 is madewith a similarreasoning(makefirstan estimate

ofL (1)withintheslabandthensolve(B1)forn = 2). From(B2) it remainsto evaluatethe radianceon topof the slab,to choosethe directionI -= v, andto averagethe radiance over S as is done in AppendixA. The result is

(L(n)(-i, ¾)) --

I • L(n)(r, 1)+ O'e(r )L(n)(r, 1)

=f4P(r, 1', l)L(n-•)(r, 1') d•D_, l.

(B1)

COS 0i COS 0v•

drn

••• dr• dr2 .....drn_• Q(n)(-i, r•, r2 ....,rn, V) slab

The solutionof (B1) canbe put in the followingform,for n -> 2'

(B5) from whichit followsimmediatelythat

L(n)(r, 1) =

co•Oi••(r) drn•••

dr1 dr2, . . . , drn-1

(L(n)(-v, i))= cos 0icos 0v•

slab

ß Q(n)(-i,rl, r2,..., rn,r),

drn

(B2)

whereL(n)(r, 1)stands for theradiance evaluated withinthe slabat positionr, goingin direction1,for an illuminationabove

ff• dr•dr2,..., drn_• Q(n)(-v, r•, r2 .....rn, i). slab

the slabin direction-i; Q is a pointlocatedeitheron the top or at the bottom of the slab, suchthat QM has the same sense

(B6)

LEROY: DEVIATION

FROM RECIPROCITY

In (B5)and(B6),Q(") hasa formsimilarbutnotidentical to (B3): 1

11,923

ton University)and an unknownreferee for helpful commentson the manuscript.

1

Q(n)(-i, r•,r2,..., rn,V)= Ir•r2 2'" Irn_•rnl 2 X t(-i,

IN BRDF

References Capderou, M., Confirmation of Helmoltz reciprocityusing ScaRaB

r•) ... t(rn, v)

satellite data, Remote Sens.Environ., 64, 266-285, 1998.

Case, K. M., Transfer problemsand the reciprocityprinciple, Rev. X P(rb -i, r•r2) ... P(rn, -rn-•rn, v). (B7) Mod. Phys.,29, 651-663, 1957. Chandrasekhar,S., On the radiative equilibrium of a stellar atmoThe integralsare volumeintegralsfor each of the n-scattering sphere(Part XVII), Astrophys. J., 93, 441-454, 1947. pointsrk. The lastscatteringtakesplacein V•, (equation(B5)) Chandrasekhar,S., Radiative Transfer,Dover, Mineola, N.Y., 1960. or in Vi (equation(B6)), but all other(n - 1) scatterings take Clarke, F. J. J., and D. J. Parry,Helmholtz reciprocity:Its validity and applicationto reflectometry,Ltg. Res. Technol.,17, 1-11, 1985. place anywherewithin the slab.It can be seenwith (B5) and Davies,R., Spatialautocorrelationof radiationmeasuredby the Earth (B6) that the n th order-scattering radianceis reciprocalif the RadiationBudgetexperiment:Sceneinhomogeneities and reciprocslabis horizontallyhomogeneous, while it is not reciprocalfor ity violation, J. Geophys.Res., 99, 20,879-20,887, 1994. a horizontallyheterogeneousslab.These equationsshowalso DeHoop, A. T., Reciprocity theorem for the electromagneticfield scatteredby an obstacle,Appl. Sci. Res., Sec.B, 8, 135-140, 1960. that there is no contributionto • if the first and last scattering Girolamo,L. D., T. Varnai, and R. Davies,Apparentbreakdownof centers r• andr, arelocated withinVi • V•. Finally, if the Direciprocity in reflectedsolar radiances,J. Geophys.Res.,103, 8795-

illuminationbeam is strictlyappliedon the area S, without any illumination outsideS, then the first scatteringtakes place in V• (illuminationalong -i, viewingalongv) and in V• (illumination along -v, viewingalongi); the expressions aboveshow that in this case,reciprocityholds at any order of scattering even for a heterogeneousmedium. Note that a rough dimensioningof the ratio between successivescatteringordersmay be obtainedby integrating(B1)

alonga segment QM and assuming L ("•) location

and direction.

L(•)(r, l)•

independent of

The result reads

L (•-•)'

(r)

t(r', r)rr•(r') dr'.

(B8)

2645-2646, 1988.

Kriebel, K.-T., On the limited validity of reciprocity in measured BRDFs, Remote Sens.Environ., 58, 52-62, 1996.

Li, X., J. Wang, and A. Strahler,Apparent reciprocityfailure in directional reflectanceof structuredsurface,Prog.Nat. Sci., 9, 747-752, 1999.

Loeb, N. G., and R. Davies,Angular dependenceof observedreflectances:A comparisonwith plane parallel theory,J. Geophys.Res., 102, 6865-6881, 1997.

Minnaert, M., The reciprocityprinciple in lunar photometry,Astrophys.J., 93, 403-410, 1941. Okayama,H., and Ogura,I., Experimentalverificationof nonreciprocal responsein light scatteringfrom rough surfaces,Appl. Opt., 23, 3349-3352, 1984.

Snyder,W., Reciprocityof the BRDF in measurementsand modelsof

Evaluate L(n)(r, l) ontopof theslab(M(r) -= P), poser =

j'• (re(r)dr andassume thatthesingle-scattering albedo to(rx(r)/rre(r) is independentof r. One finds

structured surface, IEEE TGARSS, 36, 685-691, 1998.

Van der Hulst, H. C., Multiple light scattering,Tables,formulasand applications,Academic, San Diego, Calif., 1980. Venable, W. H., Commentson reciprocityfailure,Appl. Opt., 24, 3943, 1985.

L© L (•-•) - to(1 - e-*).

8803, 1998.

Kim, M.-J., Verification of the reciprocitytheorem,Appl. Opt., 27,

(B9)

Acknowledgments.Thisworkwassupportedby CNES, CNRS, and the UniversityPaul Sabatier.The author thanksY. Knyazikhin(Bos-

M. Leroy, CESBIO, UMR CNES-CNRS-UPS, 18, avenue E. Belin, 31401 ToulouseCedex4, France. ([email protected])

(ReceivedFebruary24, 2000; revisedSeptember6, 2000; acceptedOctober 3, 2000.)

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