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JOURNAL OF GEOPHYSICAL
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  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  inverted.We call this problem,after Davies,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  budgetstudiesto fill in missingvaluesfor the generationof and Kim  showedan insignificantdeviationfrom reci1.
Copyright2001 by the AmericanGeophysicalUnion. Paper number 2000JD900667. 0148-0227/01/2000JD900667509.00
procitywith unpolarizedlight. Okayamaand Ogura  found oppositeresults,but their experimentalprotocol was criticizedby Venable.The measurements of Clarkeand 11,917
Parrypermittedalsoto showthat directionalreciprocity is stillvalid with polarizedlight, if one considerscorresponding statesof polarizationfor incidentand emergingfluxes.At the 10 m scale in land surface remote sensingstudies,Kriebel  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. 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).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 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.
Why Directional Reciprocity is Not Valid
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
v)) + (œ('-)(-i, v)) +...
Weconsider firstthefirstorderof scattering term(L (•)( -i, v)). This term takesthe followingform (proofin AppendixA):
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
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),
t(-i, r) - t(r, i)
(the transmission of light betweentwo pointsdoesnot depend on light direction),and
i) = P(r,-i,
(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
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.
In (7), J'vQ(r) dr mustbe interpretedasthe contributionto the radiancecollectedby the sensorof all photonsfor which the scatteringoccursin volume V.
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
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
scatterings arelocated in Vi [•1 Vv,thescattering medium being homogeneousor heterogeneous.A horizontallyhomo-
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
 and of Case  mentioned in section1, when one considersfor the latter a collection of points ri and rv satisfying the Case theorem and covering the area S.
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
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
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
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,,
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. 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.
-- L")(r v) + O'e(r)L (•)(r, v) = --P(r,-i,
wherev countsdistancealongthe v direction.Let t(r, v) =
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
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
Theupward first-order scattering radiance (L (•)( - i, v)) averaged over the exit surfaceS is
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' --
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
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
The result reads
t(r', r)rr•(r') dr'.
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