Icarus141, 132–155 (1999)

Article ID icar.1999.6154, available online at http://www.idealibrary.com on

Opposition Effect from Clementine Data and Mechanisms of Backscatter

Yu. G. Shkuratov, M. A. Kreslavsky, A. A. Ovcharenko, D. G. Stankevich, and E. S. Zubko

Astronomical Observatory, Kharkov State University, Sumskaya Street, 35, Kharkov 310022, UkraineE-mail: shkuratov@ygs.kharkov.ua

C. Pieters

Department of Geological Sciences, Brown University, Providence, Rhode Island 02912

and

G. Arnold

Deutsche Forschungsanstalt fur Luft- und Raumfahrt., DLR Institut fur Planetenerkundung, Rudower Chaussee 5, D-12484 Berlin, Germany

Received April 9, 1998; revised February 18, 1999

An analysis of Clementine data obtained from a UVVIS cam-era and simulating laboratory photometric and polarimetric mea-surements is presented with the use of a new photometric three-parameter function combining the shadow-hiding and coherentbackscatter mechanisms. The fit of calculated curves to the averagebrightness phase function of the Moon derived from Clementinedata indicates that the coherent backscatter component is nonzero.The average amplitude of the opposition surge of the Moon in therange of phase angles 0◦–1◦ is approximately 10%. The Clementinedata also show a flattening of phase-dependent brightness at an-gles less than 0.25◦ that is caused by the angular size of the so-lar disk. The lunar brightness phase curves at small phase anglesare nearly the same in different wavelengths even though at largerphase angles (5◦–50◦) the lunar surface becomes distinctly redderwith increasing phase angle. According to the model, the lack ofwavelength-dependent brightness variations at small phase anglescan be due to quasifractal properties of the lunar surface. Resultsof related laboratory measurements suggest that: (1) besides thenarrow coherent backscatter opposition spike there is a broad com-ponent which can contribute to phase angles up to 10◦ and (2) acomponent of coherent backscatter can be important even for lowalbedo surfaces. The latter testifies the opposition effect of the lu-nar surface to be substantially formed by the coherent backscattermechanism. c© 1999 Academic Press

Key Words: Moon surface; photometry; regoliths.

1. INTRODUCTION

Photometry of the lunar surface is a classical branch of platt

Hapkeet al. 1993, 1998) were largely based on (1) obtainingClementine data and (2) taking new approaches to their inter-pretation. The latter includes an application of the interference

ment

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etary physics. Among the problems of lunar photometry,brightness opposition effect is of special interest. Recent sies of this effect (e.g., Nozetteet al. 1994, Shkuratovet al.1994a, Burattiet al.1996, Helfensteinet al.1997, Hillier 1997,

1

0019-1035/99 $30.00

Copyright c© 1999 by Academic PressAll rights of reproduction in any form reserved.

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mechanism called often the coherent backscatter enhanceor weak localization of light.

The opposition effect manifests itself as a rapid increasesurface brightness when phase angleα approaches zero. Belowabout 10◦ the brightness of the Moon grows more sharply that large phase angles. The lunar opposition effect is aboutat 1◦–10◦. Compared to other atmosphereless celestial bodthe Moon reveals a rather wide opposition surge. Indeed,asteroids of E-type (Harriset al. 1989) and bright satellites oJupiter (Thompson and Lockwood 1992), the point wherebrightness changes from linear growth to a steeply sloped sis in the range 2◦–3◦. It is assumed usually that a narrow spikis due to interference enhancement while a wide surge is asult of the shadow-hiding effect (e.g., Mishchenko and Dluga1992, 1993, Helfensteinet al.1997). To estimate contributionof both effects at each phase angleα is a difficult problem whichis studied recently rather intensively. For instance, Burattiet al.(1996) concluded that for the Moon at the phase angles 0◦–5◦:“... the principal cause of the lunar opposition surge is shadhiding, while coherent backscatter, if present, makes only anor contribution.” However, Hapkeet al.(1993, 1998) studyinglunar regolith samples have shown the contribution of cohebackscatter to be significant at small phase angles. Helfenset al. (1997) reached a similar conclusion analyzing telescolunar data.

Here we report our findings concerning the lunar oppositeffect derived from a detailed analysis of Clementine data.interpret the results, we use a semiempirical photometric mofor regolith-like surfaces (Shkuratov and Ovcharenko 1998) asimulating laboratory measurements in two ranges of ph

2

OPPOSITION EFFECT FROM CLEMENTINE 133

FIG. 1. The integral brightness phase functions of the Moon nearside at 0.45µm according to Rougier (1933), Shorthillet al. (1969), and Lane and Irvine(1973) (symbols) and to the average data by Akimov (1988a) (solid line).

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angles, 0.2◦–3.5◦ and 3◦–45◦. We focus on estimation of contrbutions of the shadow-hiding and coherent backscatter efin lunar phase function.

2. SOME REMARKS AND A SHORT REVIEW

Our starting point is a short review of some papers devoto the opposition effect. This review selects known workswell as rarely-cited ones to correct some stereotyped viewthe history of these studies. In addition, we present here aphotometric model used below for processing and interprelunar observations.

2.1. Observations

The strong backscatter of light from the Moon at a wide raof phase angles was discovered by Barabashev (1922) by mof the method of photographic photometry. This effect was tstudied in detail by his disciple Fedorets (1952). At the tithere was the widespread opinion that all surfaces shoulapproximately described by Lambert’s law, which Barabasand Fedorets showed was clearly incorrect for the Moon. Gehet al.(1964) were the first to emphasize the opposition effeca separate phenomenon with possibly a different explanafrom the general backscattering property of the lunar surfImportant references up to 1971 can be found in the excereview by Hapke (1971).

Akimov (1988a), another Barabashev disciple, used hisphotoelectric lunar observations carried out during 25 years

the data by Rougier (1933) to propose an integral phase func

cts

edasonew

ing

geans

enebe

evrelsasionce.ent

wnnd

of the Moon atλ= 0.45µm (λ is the wavelength). This functiois presented in Fig. 1 together with other data.

Progress in the study of the lunar opposition effect is relatethe mapping of its parameters. Wildey (1978) produced an imof the phase ratioI (2.0◦)/I (4.5◦), whereI (α) is lunar surfacebrightness at a phase angleα. He found interesting features witthis image. In particular, the overall mare/highland contrasthis parameter appeared to be very weak. High values oratio were observed in regions with moderate albedos. Simstudies of the phase ratioI (3.2◦)/I (14.5◦) images of the Moonfor two spectral ranges (λ= 0.55 and 0.38µm) were performedby Akimov and Shkuratov (1981). The ratioI (3.2◦)/I (14.5◦)at λ= 0.55µm is largely higher for regions with intermediaalbedo. As to the ultraviolet range, the ratio increases sysatically with albedo. Recent studies of lunar phase-ratio imahave been carried out by Shkuratovet al. (1994a).

The Moon cannot be observed from the Earth at phase anless than 1◦ out of eclipse. However, during the Apollo (Pohet al.1969, 1971, Whitaker 1969, Wildey 1972) and Clement(Nozetteet al.1994, Burattiet al.1996) missions, data on the oposition surge at these small phase angles were obtained. Ifound that the brightness ratioI (0)/I (8◦) is basically within thelimits 1.3–1.4 (Pohnet al.1969, Whitaker 1969). Surface varitions of the ratioI (0)/I (1.5◦) were also investigated. No corrlations with albedo or other surface characteristics were fo(Pohnet al.1971, Wildey 1972). Nozetteet al. (1994) reportedthat Clementine discovered a narrow lunar opposition spikeamplitude 20% in the last 0.25◦. However, as has been shown

tionShkuratov and Stankevich (1995) and Shkuratovet al.(1997b),

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134 SHKURAT

such a spike cannot be observed because the Sun at 1 AUan angular size of about 0.5◦. Each point of the Sun’s disk is aindependent light source which has its own phase angle.makes it impossible to observe any details of brightness pcurves the widths of which are less than the angular radiuthe Sun. Later Burattiet al.(1996) found from Clementine datthat: (1) the lunar brightness grows more than 40% betwphase angles of 4◦ and 0; (2) the opposition surge is somewhlarger at 0.41µm than at 1.00µm (difference is 3–4%); and (3the amplitude of the surge depends noticeably on terrain typeis about 10% greater in the lunar highlands.

2.2. Laboratory Measurements

Laboratory photometric studies of surfaces with complicastructure were started many years ago. In the context ofmeasurements, the well-known nameAngstrom (1885) shouldbe recalled. He was a pioneer in laboratory photometry of rosurfaces by visual methods. Barabashov and Chekirda (1carried out the first laboratory measurements with a photoment in a planetology context. From this, precise photometrsimulators of planetary surfaces began.

The first laboratory measurements of the narrow opposisurge were performed by Oetking (1966) who achieved a mmal phase angle about 1◦. Oetking measured samples of smokmagnesium oxide (MgO), which possesses a very high albHe found that a well-expressed opposition surge with amplitabout 25% started at approximately 10◦. It was surprising to findsuch a surge for a sample with suppressed shadow-hiding e(Hapke 1966).

The question of how to separate shadowing and the inteence effect arose immediately after the first application ofeffect in optics of solid planetary surfaces (Shkuratov 1985)teresting laboratory experiments with lunar samples usingearly and circularly polarized laser light, which were carriedby Hapkeet al. (1993, 1998), showed that the zero-phase pof the lunar regolith is caused by both shadow-hiding and coent backscatter effects in roughly equal amounts. To understhe nature of the narrow and wide opposition surges, laborameasurements at the phase angles 0.2◦–3.5◦ were performedby Shkuratovet al. (1997b). In these measurements, a samformed by MgO smoked deposits suggested two surperpopposition surges (α <1.5◦ andα <10◦), both with amplitudesof about 20%.

2.3. Theoretical Studies

The lunar opposition effect from its discovery almost upnow was explained as being caused by the shadow-hiding manism (Hapke 1981, 1984, 1986, Lumme and Bowell 1981981b, Bowellet al.1989, Hillier 1997). This mechanism is understood rather well, though its rigorous description faces la

difficulties, because the theoretical solution of the problem rquires estimations of Feynman integrals.

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The glory effect (Akimov 1980) and light focusing (O’Kee1957, Shkuratov 1983) produced by spherical regolith partiwere considered as additional mechanisms of the lunar option effect, but at present only interference is recognized asmain rival for the shadow-hiding effect. The history of the meanism began with the works of Gnedin and Dolginov (1963)Watson (1969). The first cited paper is devoted to the multiscattering problem in quantum physics while the second coners the problem of electromagnetic wave scattering from plafluctuations. Since then, this phenomenon has been studiedpendently by many physicists. There exists a review by Kravand Saichev (1982) on pre-1982 interference enhancementies. That the interference enhancement can explain the option effect of natural surfaces was casually mentioned by Kand Ishimaru (1984).

Despite this long and rich history, the interference mecnism was a big surprise for astronomers who studied the opption effect of celestial bodies. The first “astrophysical” worksthis field are Shkuratov (1985, 1988), Muinonen (1989, 199and Hapke (1990). Since then, the approach has been isively developed by Mishchenko (1992) and MishchenkoDlugach (1992, 1993). The “interference” explanation for bthe opposition effect and the negative polarization was giveShkuratov (1985, 1988, 1989) and independently by Muino(1989). There are reviews on the backscatter phenomenamosphereless celestial bodies (Muinonen 1993, Shkuratovet al.1994b).

3. NEW PHOTOMETRIC MODEL

The photometric models by Hapke (1981, 1984, 1986)Lumme and Bowell (1981a, 1981b) which have been widused in many works do not take into account coherent backter enhancement, although attempts to frame models worwith both the shadow-hiding effect and coherent enhancemexist (Shkuratovet al. 1991, 1996, 1998, Zhukovet al. 1994,Helfensteinet al.1997). These attempts, however, have notresulted in a convenient agreed-upon model.

We present here a new heuristic model combining the inference and shadow-hiding effects. This model operatesthree parameters. The model is semiempirical because wapproximate simple expressions at the derivation.

3.1. General Description

First, let us describe a model of scattering surface. Therethree classes of irregular surface structures. The first can bequately presented by a random single-connected single-vafunction. The second is described by a random single-connebut multiple-valued function. Finally, the third correspondsdisruption of the surface interface. This class is described bydom multiple-connected functions. Surfaces of the third care particulate. They may be considered as media. From

e-standpoint of the shadow-hiding effect, the second class is closeto the third. Actually, in both these cases, surface slopes can

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OPPOSITION EFFEC

be significant, more thanπ/2, and this allows very prominenshadow-hiding phase dependences to be obtained. The proity of shadow-hiding effects for the last two classes of surfais intuitively understandable. Indeed, on section of a randsingle-connected multiple-valued surface by any plane (e.gthe scattering plane), a two-dimensional “particulate” mediis formed in this plane. The same will be observed in the cof the third class of surfaces. Thus, in both cases, it is necesto calculate the shadow-hiding effect in particulate 2D medi

The lunar surface has a complicated structure. The strucis different for different scales. At bases more than 1 cm,surface can be mainly described by a random single-connesingle-valued function. For the scales less than 1 mm, the lusurface becomes a particulate medium. The lunar regolithface consists of opaque or semitransparent particles with son the order of 100µm; much more than the wavelengthlight. In turn, the regolith particles are mainly aggregatesmicrometer–submicrometer grains. To describe such a sursome simplifications are necessary.

Consider a discrete semi-infinite medium consisting of scterers with sizes of order and less than the wavelength. Supthat the boundary of the medium has a physical fractal struccreated by superposition of many surface undulations at diffescales: each large-scale undulation can be considered a refesurface for the next smaller-scale undulation (Shkuratov 19Even if each elementary undulation is small, their cumulateffect can be significant, when the number of such undulatis large. This procedure allows obtaining a steep-slope relieparticular, this relief can be non-single-valued; i.e., it may hslopes steeper than 90◦, like in the case of a particulate mediumWe suppose that the characteristic scale of the smallest-sundulation is greater than the size of medium discretenessthe wavelength. This permits using the scattering indicatrixthe semi-infinite medium with the plane boundary to characize the minimal surface element of the physical fractal structFor the scale of forming this intrinsic surface indicatrix, we ctake into consideration the interference effect. It is necesto emphasize that discreteness of the acting portion of sufractal surface can be formed by rather large fragments withtricate structure. It allows modeling of the shadow-hiding effas if we deal with a particulate medium. This approach makepossible to avoid traditional problems concerning calculatiof the shadow-hiding effect in particulate surfaces (e.g., Ha1981, 1986, Hillier 1997). Finally, this permits the simplificatioof the results. Besides, our model takes into account the acobserved hierarchy inherent in the lunar surface: the smthe scale of consideration, the larger the average slopes osurface.

Our model suggests the photometric function in the fof (α, b, l )= H (α)D(α, b, l ). The first factorH (α) accounts forstrong dependence of the photometric function onα. The sec-ond factorD(α, b, l ) presents dependences on the photomelatitudeb and longitudel . The photometric coordinates (α, l , b)

can easily be transformed into traditional coordinates (i, ε, ψ),

FROM CLEMENTINE 135

xim-esmbymseary.ureheted

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ric

wherei andε are the angles of incidence and emergence,spectively, andψ is the azimuth angle—the angle between tplains of incidence and emergence:

cosi = cosbcos(l − α)

cosε = cosbcosl (1)

cosα = cosi cosε + sini sinε cosψ.

3.2. Coherent Backscatter Enhancement

To estimate the interference effect, we develop an approsuggested by Shkuratov (1988), Shkuratovet al. (1991), andZhukovet al. (1994).

Let there be a discrete semi-infinite medium with a flat-oaverage boundary. Suppose that the medium consists of rsmall scatterers, which have a scattering amplitudea and theisotropic indicatrix. Let a ray from an infinitely remote sourcross the boundary at a point 1, the incident angle beingi . Aftermultiple scattering in the medium the ray crosses the boundat a point 2, moving from the medium with the emergent anε to an infinitely remote observer. The distance between thpoints 1 and 2 iss. Let us consider direct and time-reversal ligtrajectories connected the points 1 and 2 in the medium.contribution of such trajectories in the scattered electromagnfield looks like

E ∼ ζ12 exp(−i δ12)+ ζ21 exp(−i δ21), (2)

whereζ12 andζ21 are the amplitudes of multiple scattering fthe direct and time-reversal trajectories, andδ12 andδ21 are thecorresponding electromagnetic phases. Owing to sphericitthe indicatrix of scatterers, the conditionζ12= ζ21= ζ is valid,ζ =an, wheren is the multiplicity of scattering (n≥ 2). Theamplitudeζ does not depend on mutual position of the pointand 2. Thus, the intensity of interfering waves which correspto the direct and time-reversal trajectories is

E E∗ ∼ ζ ζ ∗(1+ cos1), (3)

where (∗) denotes the complex conjugation and1= δ12− δ21.The mutual shift of electromagnetic phases1 for the trajectoriesdepends upon mutual position of the points 1 and 2, whiccharacterized by vectorS linking the points.

Coherent backscattering enhancement occurs, since neadirectionα= 0 the trajectories (source)→ (point 1)→ (point 2)→ (observer) and (source)→ (point 2)→ (point 1)→ (ob-server) are almost equal—photons going along the trajectointerfere constructively. The electromagnetic phase shiftscribed is1= (k i − kε) ·S, wherek i andkε are the wave vectorsof the incident and emergent rays. In a coordinate presenta1= 2πsλ/u, whereλ is the wavelength andu= sini cosϕ−sinε cos(ψ −ϕ), ψ being the angle between the planes of indence and emergence andϕ being the polar coordinate ofS. If

ψ = 0 andi = ε (the case whenα= 0), the value1= 0.

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136 SHKURAT

One can see that the electromagnetic phase shift1 does notdepend on any properties of the light-scattering medium. Thfore, Eq. (3) holds for all possible trajectories of light crossthe points 1 and 2 and all multiplicities of scattering (of courexcluding the single scattering). Thus,

E E∗ ∼ 2m(1+ cos1), (4)

where E is the sum of fields corresponding to all orders (≥2)of scattering and2m is the multiple-scattered componentthe incoherent (“classical”) part of scattered flux. It shouldemphasized that this convenient factorized form for the cbination of the interference and “classical” effects presenteEq. (4) is due to sphericity of the scattering indicatrix of mediparticles, although at rather small phase angles this is alsofor an arbitrary indicatrix in some approximation.

Equation (4) ignores the vector nature of light. To take iaccount polarimetric effects, we make use of the vector statefor the reciprocity principle (e.g., Cho 1990). Atα= 0 we have

(E0 · E12

) = (E0 · E21), (5)

whereE0 is the electric vector of incident field, andE12 andE21

are the electric vectors of the scattered field, correspondindirect and time-reversal trajectories. Equation (5) shows thacopolarized components for these trajectories are the samethey always interfere constructively. As to the cross-polaricomponents for the trajectories, they are arbitrary and mayterfere both constructively and destructively. This meansin backscattering, the contribution of the cross-polarized cponents is mainly incoherent. Only for a rather narrow se“symmetrical” trajectories for which the condition|E12| = |E21|is valid as well as Eq. (5) are both co- and cross-polarized cponents enhanced by interference.

Thus, one may rewrite Eq. (4) as

E E∗ ∼ 2m(1+ η cos1). (6)

The factorη∼= 1/2 is introduced to account for the interferenenhancement only of the copolarized component. For simpliwe use below the exact equalityη= 1/2.

Let us put the flux photometric functionχ (α, i, ε) of themedium as

χ (α, i, ε) = 21+2m

(1+ Ä

2

), (7)

where21 is the single-scattering component and the interferetermÄ can be determined as follows:

Ä =∫ 2π

0 dϕ∫∞

d cos1 ·W(s)sds∫ . (8)

2π ∞

0 W(s)sds

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A screening factorW(s) characterizes density of the probbility that a photon goes from the medium in the point 2, if thphoton came to the medium in the point 1. The valued is thesize (radius) of the medium volume, in which single scatteris formed. In this volume the coherent backscatter effect canoccur. In particular, it can be treated as an effective size (radof particles, if they are structureless and much larger thanλ. Ifthe characteristic size of particles is of the order or less thaλ,one can assume thatd ∼ λ. The valued is a semiempirical butconvenient characteristic.

We describe the dependence of the probability density osby the functionW(s) which presents an energy distributionscattering spot for an isotropic point source placed in a scattemedium consisting of weakly absorbing particles in the diffusapproximation (Ishimaru 1978),

W(s) = 1

sexp

(− s

L

), (9)

where L is the characteristic scale of light diffusing in thmedium. The valueL characterizes the attenuation of light frothe point source occurred due to absorption and scatter in meIf a medium consists of slightly absorbing scatterers,L is muchlonger than the mean free path of light transport in the mediIntegration in Eq. (8) overs gives

Ä = 1

2π

2π∫0

dϕexp

(− dL

) [cos(

2πdλ

u)− 2πL

λu · sin

(2πdλ

u)]

1+ ( 2πLλ

u)2 .

(10)

If phase angles are small enough, the term

cos

(2πd

λu

)− 2πL

λu · sin

(2πd

λu

)≈ 1. (11)

We consider this equality as the exact one to estimate the intein Eq. (10). In this case, integration overϕ yields

Ä = exp(− d

L

)√1+ ( 2πL

λ

)2[sin2 i + sin2 ε + 2(cosi cosε − cosα)]

.

(12)

Calculations show that at small anglesi and ε, the valueÄdepends on the scattering geometry only slightly; thereffor approximate estimates we can use any geometry, e.g.“mirror” one: i = ε=α/2. Then sin2 i + sin2 ε+ 2(cosi cosε−cosα)= 4 sin2 α/2, and hence

Ä = exp(− d

L

)√1+ ( 4πL

λsin α

2

)2 . (13)

We will use this simple approximate expression below in or

to describe the component of coherent backscatter enhancement.

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OPPOSITION EFFEC

The classical solution of the radiative transfer equation forisotropic indicatrix of elementary scattering volume with singscattering albedoω gives the expression for the indicatrix of lighflux scattered by the medium (e.g., Ishimaru 1978)

8(α, i, ε) = ω

4· ϕ(ω, i )ϕ(ω, ε)

cosi + cosεcosi cosε, (14)

whereϕ(ω, γ ) is the Ambartsumian–Chandrasekhar functioConsequently, one can write that

2m(α, i, ε) ∼[ϕ(ω, i )ϕ(ω, ε)− 1

cosi + cosε

]cosi cosε (15)

and

21(α, i, ε) ∼ cosi cosε

cosi + cosε. (16)

Thus, finally,

χ (α, i, ε) =1+ [ϕ(ω, i )ϕ(ω, ε)− 1]

×1+ exp(−d/L)

2 ·√

1+ ( 4πLλ

sin α2

)2 · cosi cosε

cosi + cosε. (17)

If the multiple scattering component dominates total scatteflux (we have suggested this when we considered the cohebackscatter enhancement), then

χ (α, i, ε) ≈1+ exp

(− dL

)2 ·√

1+ ( 4πLλ

sin α2

)2

· [ϕ(ω, i )ϕ(ω, ε)− 1] cosi cosε

cosi + cosε. (18)

We will use this intrinsic indicatrix below.

3.3. Shadow-Hiding Effect

Let us consider a random surface arranged as a physical fr(Feder 1988). This means that the surface is hierarchic andbe expanded on statistically similar structure generations wdiffering scales. Let the photometric function of such a muscale surface beF(ρn, α,b, l ) whereρn is the cumulative slopeof the surface,ρn= n

√θ2,√θ2 being the RMS slope for a singl

structure generation andn being the number of the generationThis function can be represented as

F(ρn, α,b0, l0) = 1∫χ [α, bn(u, v), ln(u, v)] du dv, (19)

S0S(iv)

n

FROM CLEMENTINE 137

he-

n.

edent

ctalcanithi-

.

whereS0 is a sufficiently large part of the main reference planl0 andb0 are the photometric coordinates for the main refereplanes;S(iv)

n is the part of the surface both illuminated and visib(iv), simultaneously;u andv are spatial coordinates on this suface; andχ [α, bn(u, v), ln(u, v)] is an elementary flux indicatrixof scattering (the intrinsic photometric function of a particulasurface),ln andbn being the local photometric coordinates. Thindicatrix can be described, for example, by Eq. (18).

Equation (19) represents an averaging the elementary fludicatrixχ (α, b, l ) over the part of the surface both illuminateand visible. As a rule, it is very difficult to obtain an explicexpression for the integral (19), and this is the main difficuof any shadow-hiding theory. However, if one assumes thatsurface is formed as the physical fractal, the averaging can bplaced by an equivalent procedure that can be reduced to soa relatively simple differential equation. Actually, in the casethe physical fractal formed byn generations, one can write tha

F(ρn, α,b0, l0)

= 1

S0

∫S0

F [ρn−1, α,b(u0, v0), l (u0, v0)]du0 dv0

cosθ (u0, v0), (20)

whereθ are the local slopes of the structure generation oflargest scale, andu0 andv0 are coordinates introduced on thmain reference plane.

If n tends to infinity keeping the fractal and topological dimesions equal, we come to mathematical objects named “fractowhich have been introduced by Shkuratov (1995). The transifrom a physical fractal to the corresponding fractoid (Shkura1995) makes it possible to obtain from Eq. (20) the differenequation

∂F(ρ, α,b, l )

∂ρ= 1

4

(∇2+ 2)F(ρ, α,b, l ), (21)

where∇2 is the Laplacian (in the variablesb andl ) andρ is thelimit of ρn whenn goes to infinity andθ2 goes to zero simulta-neously (the fractoid definition implies that this limit exists anis not equal to zero). Supplementing this differential equatwith proper initial and boundary conditions, one comes toclassical Sturm–Liouville problem on a spherical sector withfollowing boundaries:α−π/2≤ l ≤π/2 and−π/2≤ b≤π/2.Its solution has been presented in detail by Shkuratov (1995)Shkuratovet al.(1994a). The functionF(ρ, α,b, l ) is expandedin the Laplacian eigenfunctions for the spherical sector,

F(ρ, α,b, l )

=∞∑

m=0

∞∑n=0

Emn(α) · exp

(−ρ

4(Cmn(α)− 2)

)· cos

[km

(l − α

2

)]k

(1 1 2

)

· (cosb) · 2F1 n+ km +

2,−n;

2; sin b , (22)

O

t

e

atov

se

isuedthe

t wethein-gles.it a

g

ul-

-edolso

b of

ear

isk

s).

he88a,

ion

138 SHKURAT

where Cmn(α)= [km(α)+ 1/2+ 2n]2− 1/4 and km(α)=π (2m+ 1)/(π −α),m= 0, 1, 2, . . .. The function2F1(a, b; c;x) is the hypergeometric polynomials. The coefficientsEmm(α)can be found if the initial indicatrixχ (α, b, l ) is known,

Emn(α)

= 1

Mmn(α)

π/2∫α−π/2

dl

π/2∫0

db · (cosb)km+1 · cos

[km

(l − α

2

)]

·χ (α, b, l ) · 2F1

(n+ km + 1

2,−n;

1

2; sin2 b

), (23)

where

Mmn(α) =π/2∫

α−π/2dl

π/2∫0

db · (cosb)2km+1 · cos

[km

(l − α

2

)]

· 2F1

(n+ km + 1

2,−n;

1

2; sin2 b

)= π2n!(2m+ 1) · 0(km + n+ 1)

(2km + 4π + 1) · 0( 12 + n

) · 0 (km + 12 + n

) · km.

(24)

For values ofρ greater than 0.3, the series (22) convergvery quickly and, therefore, one may save the only term inexpansion. The brightness photometric function,f (α, b, l )=F(α, b, l )/(cosl · cosb), looks like

f (α, b, l ) = H (α) cos

[π

π − α(

l − α2

)](cosb)α/(π−α)

cosl, (25)

where

H (α) = 1

Mexp

(−ρα(3π − 2α)

(π − α)2

) π/2∫α−π/2

dl

π/2∫0

χ (α, b, l )

× (cosb)(2π−α)/(π−α) cos

[π

π − α(

l − α2

)]db, (26)

M(α) = π3/2

(π − α3π − α

)0(

ππ−α

)0(

3π−α2(π−α)

) . (27)

Note that in the paper by Shkuratovet al. (1994a) a misprintwas committed. A few formulas describe the latitude depdence as cos(πb/(π −α)), whereas it should be (cosb)π/(π −α).

Equation (26) is rather complicated. Even for simplest intial indicatrices, e.g., for the Lambert scattering law, the fin

V ET AL.

eshis

n-

expression is rather cumbersome (Shkuratov 1995, Shkuret al.1994a).

3.4. Further Simplification of the Model and Illustrations

A comparison of the functionH (α) calculated for a few typesof intrinsic indicatrix given by Eq. (18) with experimental phafunctions shows that the calculatedH (α) falls too sharply atα >40◦ for any ρ. This occurs for two reasons. The firstthat the phase functions of the hierarchy-arranged multivalfractal-like surfaces are actually somewhat steeper than incase of typical particulate media. The second reason is thado not incorporate multiple scattering between elements offractal-like surface in the model. This scattering should dimish the shadow-hiding effect essentially at large phase anThe simplest way to avoid these shortcomings is to explosemiempirical expression forH (α). We suggest the use of

H (α) = exp(−kα)

2+ exp(− d

L

)2+ exp

(− dL

)√1+ ( 4πL

λsin α

2

)2 , (28)

wherek is a formal parameter close in some sense toρ.The idea of using exp(−kα) to describe the shadow-hidin

effect was suggested by Akimov (1980). The coefficientk takesinto account diminution of the shadow-hiding effect due to mtiple scattering. We assume that the coefficientk depends onalbedoA ask= k0(1− A), wherek0 is a function of surface geometry characteristics and does not depended on alb(Akimov 1980). Other simple empirical expressions can abe used to approximate the functionH (α). For example, if wedeal with particulate media covered by a semitransparent slasmall particles, the function exp(−kα) is not optimal to describethe shadow-hiding effect (Hillier 1997, Stankevichet al.1999).In this case it is necessary to use a function which is nonlinon a logarithmic scale.

The part describing brightness distribution over the lunar d(see Eq. (25)),

D(α, b, l ) = cos[

ππ−α

(l − α

2

)]cosl

· (cosb)α/(π−α), (29)

has no parameters (like the Lambert or Lommel–Zeeliger lawIt presents dependences on the photometric latitudeb and lon-gitudel for fractal-like surfaces (Shkuratov 1995). Note that texpression (29) was suggested earlier by Akimov (1975, 191988b) and we often name it Akimov’s formula.

Thus, finally, we use the semiempirical photometric funct

f (α, b, l ) = exp(−kα)

2+ exp(− d

L

)2+ exp

(− dL

)√1+ [ 4πL

λsin(α2

)]2

cos[

π(l − α

)]

i-al

× π−α 2

cosl· (cosb)α/(π−α). (30)

T

aR

:

t

ob

cth

on

-otpa

e

os

delused:3

OPPOSITION EFFEC

It has been published without derivation by ShkuratovKreslavsky (1998) and Shkuratov and Ovcharenko (1998).member that the anglesα, b, andl are in radians.

From Eq. (30) the integral phase function can be derived

F(α) = 2√π· exp(−kα)

2+ exp(− d

L

)2+ exp

(− dL

)√1+ ( 4πL

λsin α

2

)2

×(

1− α

π

)0(

3π−α2(π−α)

)0(

4π−3α2(π−α)

) . (31)

Equation (31) can in particular be used to interpret asoid observations. To illustrate this, we present results ofting model curves to observations of the E-type asteroidAngelina (Harriset al. 1989) in Fig. 2 and Phobos (Avaneset al.1991) in Fig. 3. Brighter 64 Angelina is characterizedsuppressed shadow-hiding (k= 0.62) and larger light-diffusionvolume (L = 9.06µm) than darker Phobos (k= 1.36, L =3.97µm). The different values ofd for Phobos (d= 0.5µm)and 64 Angelina (d= 4.14µm) can be treated as a differenbetween the average size and albedo of regolith particles ofbodies.

We also use Eq. (31) to understand the influence of the mparameters on phase dependences (Fig. 4). As one cagrowth of the parameterL leads to increasing the oppositioeffect amplitude and its narrowing. Whend increases, the amplitude decreases. Variations ofk result in total steepnessphase function: the larger thek, the steeper the curves. Nothat Eq. (30) predicts a decrease of the phase function slovery small phase angles. This decrease is absolutely naturany diffraction phenomena. However, the factor exp(−kα) canpartially compensate this effect.

FIG. 2. An illustration of fitting by Eq. (31). Observational data for astoids 64 Angelina (Harriset al.1989) and the best fit with Eq. (31):L = 9.06µm,

d= 4.14µm, andk= 0.62.

FROM CLEMENTINE 139

nde-

er-fit-64vy

eese

delsee,

n

fee atl for

r-

FIG. 3. An illustration of fitting by Eq. (31). Observational data for Phob(Avanesovet al.1991) and the best fit with Eq. (31):L = 3.97µm,d= 0.5µm,andk= 1.36.

FIG. 4. Results of calculations by Eq. (31) for different values of the moparameters. As a basis set the following values of the parameters areL = 7 µm, d= 3 µm, andk= 1 (curve 2 in all plots). The curves 1 andcorrespond to the casesL = 3 and 11µm (plot a),d= 0 and 6µm (plot b), and

k= 0.5 and 1.5 (plot c).

O

oeir

to

o

en

w

o

fau

et

tee

uls;

scfe

rov

d

ee be-

thethe

ngu-ated.ersse-

–20hingt con-look.bso-een.y ofthanper-hens, fores,

cali-amby

sult,We

nec-ingight-cor-

sedard

atiothe

hugees fored afromea of

tedacye

-givewith(see-ee

-; this

140 SHKURAT

One must pay attention to an important case. If the conditiL ∝ λ andd ∝ λ are met, the opposition spike has no sptral dependence. The conditions are satisfied, if the medpossesses quasifractal properties, when the surface structumains the same in some sense independently on variationsλ(i.e., on variations of the “consideration” scale).

The function (30) is used below for processing and interpretion of Clementine data and to analyze results of our laborameasurements.

4. LUNAR PHOTOMETRY FROM CLEMENTINE IMAGES

In this section we present some results of our study of phmetric properties of the lunar surface with images acquiredClementine spacecraft with the UVVIS camera. We focus hon conclusions about nature of the opposition effect of the luregolith. Details of the data processing and extended reviethe results will be published elsewhere (M. A. Kreslavskyet al.,manuscript in preparation).

As has been noted above, the illumination/observation geetry is described by three angular parameters:α, l , andb. Forobservations from the Earth, for each point of the lunar neside, parametersl and b are close to selenographic longitudand latitude. For the regular Clementine mapping, the surwas observed approximately from the local zenith, that is,derb ≈ l ≈ 0. However, a few sites were imaged under othillumination/observation geometries.

UVVIS camera images were taken in the visible and ninfrared range. In this work we analyze images taken withfilters A (0.41µm), B (0.75µm), and D (0.95µm). We expectthe photometric characteristics of the lunar surface in the filC (0.90µm) and E (1.00µm) to be very close to those in thfilter D (0.95µm) (Nozetteet al.1994).

In the following subsections we present average phase ftions for the Moon at the phase angles 3◦–50◦ in the three spectrabands 0.41, 0.75, and 0.95µm; an example of surface brightnedependence on all three geometrical parameters for one siteexamples of phase functions at the phase angles 0◦–2◦.

4.1. Average Phase Function at 3◦–50◦

The regular lunar mapping lasted about 2 months. Pairorbits separated by approximately 1 month followed close traon the surface. As a result, a set of stripes on the lunar surwas observed twice under different phase angles. This makpossible to derive the phase function of the lunar surface fa large set of image pairs at different phase angles (Kreslaet al.1997). This approach presented below could be namedmethod of linked phase slopes.

We restricted the analysis to the latitude zone of the Mobetween±50◦ and to the phase angle range of 3◦–50◦. At higherlatitudes and large phase angles the Sun is always low an

cal topography influences the apparent brightness too much.

V ET AL.

nsc-ume re-of

ta-ry

to-byrearof

m-

ar-ecen-er

arhe

rs

nc-

sand

ofes

aces itm

skythe

on

lo-

smaller phase angles a special processing is necessary (slow).

We listed all pairs of overlapping images obtained duringregular mapping under standard viewing geometry, whencamera was directed downward. Then, coordinates of rectalar areas, where images from each pair overlap, were calculWe found that accuracy of known coordinates of image cornis not enough to match the images. For several randomlylected pairs, error in matching by ballistic data is of order 15image elements. Automated computer procedures for matcgave bad results, because at different phase angles the mostrasted features on the images (e.g., craters) have a differentOn the other hand, the number of pairs is huge and it is alutely impossible to match images just on a computer scrThus we did not compensate errors resulting from inaccuraccamera pointing data. Pairs for which overlap area was less10% of image (that is, the pairs most sensitive to errors in suposition) were excluded. We believe that the errors vanish wdata are averaged over a huge number of frame pairs. Thueach filter we listed about 10,000 pairs of overlapping framwhere phase angle difference was greater than 3◦.

All images from the pairs were decompressed and thenbrated. The calibration developed by the Brown University tewas used. We applied all three additive corrections, dividingthe exposure duration and the flat field correction. In the reimages calibrated in counts per millisecond were obtained.did not perform the absolute calibration, because it was notessary for derivation of the phase function. For the overlapprectangle of the calibrated images we calculated the mean brness and its standard deviation. The mean brightness wasrected for deviation of the photometric longitudel from zero us-ing the longitudinal multiplier from Eq. (30). We excluded thopairs where the difference between the variances (the standeviation divided by the mean) was abnormally high. The rof the mean brightness for two images in each pair gave usphase ratio for the corresponding phase angles. From thisset of the phase ratios we reconstructed average phase curveach spectral band. Our reconstruction procedure minimizsum of squares of deviations of approximated phase ratiosmeasured ones. Each phase ratio was weighted with the arthe corresponding overlapping rectangle.

The average lunar phase function for the B filter is presenin Fig. 5 (symbols). It can be approximated with great accurby Eq. (30) (solid line in Fig. 5). Unfortunately, a large volumin the space of parametersk, d, andL provides a good fit. Behavior of the phase function at the smallest phase angles canadditional constraints on the parameters. Good agreementsmall-phase-angle data, which we obtained for a few sitesbelow), can be achieved whend/λ= 1.5. Values of the parametersk andL providing the best fit under this constraint in the thrspectral bands are listed in Table I. One can see thatk increasesandL decreases with decreasingλ. This result is physically obvious: with decreasing wavelength the albedo decreases

Forleads to growth ofk (due to diminishing the multiple scattering)

T

os

e

eu

n

e

a

dex

asedineure-

.cess

ne

early

hernt isther by

the

s a

fornar

loserea

OPPOSITION EFFEC

FIG. 5. The average brightness phase function of the Moon derived frClementine data at 0.75µm by the method of linked phase slopes (symbolThe line is the best fit by Eq. (30). The parameters values areL = 4.51 µm,k= 0.871, andd= 1.12 µm. The dashed line corresponds the pure shadohiding component.

and to shortening of the light diffusion lengthL. Our estimationshows that the variations ofk (remember thatk= k0(1− A))are in quantitative accordance with spectral variations of albA. To estimate relative contributions of the shadow-hiding acoherent backscatter enhancement effects, we omitted thterference multiplier in Eq. (30) and then calculated the pshadowing dependence. The result is presented in Fig. 5 bydashed line. As one can see, for the wavelength 0.75µm therelative contribution of the coherent backscatter enhancemeabout 15% at the phase angle 3◦. Moreover, it is clearly seenthat the interference component of the lunar phase dependoccupies a rather wide range of phase angles. Existence of bcoherent backscatter surges is confirmed also by our laborameasurements (see below).

The average Clementine phase dependences of spectral ror the normalized color indices C(0.75/0.42µm) and C(0.95/0.75µm), are given in Fig. 6. A monotonic rise of the coloindices is observed: the lunar surface becomes redder with

TABLE ISpectral Dependence of the Model Parameters

k and L at d/λ= 1.5

λ, µm K L, µm

0.415 1.042 1.380.750 0.871 4.51

0.950 0.813 5.79

FROM CLEMENTINE 141

m).

w-

dond

in-rethe

t is

nceroadtory

tios,

rin-

FIG. 6. The average normalized phase dependences of color inC(0.750/0.415µm) and C(0.95/0.75µm) of the Moon derived from Clementinedata.

creasing phase angle. This is in agreement with the Earth-bobservations of the Moon by Mikhail (1970) and Lane and Irv(1973), but was not reproduced in some laboratory measments of lunar samples at phase angles less than 10◦ (Akimovet al.1979, Shkuratovet al.1996; Hapkeet al.1998; see below)

The monotonic behavior of the average lunar color indiC(0.75/0.42 µm) and C(0.95/0.75 µm) at phase angles lesthan 10◦ can be interpreted in terms of Eq. (30). Actually, if osuggests that the ratioL(λ)/λ andd(λ)/λ, which are the mainparameters of the coherent backscatter mechanism, are nconstant, the behavior is a function ofk, which is a monotonicfunction of albedo and, consequently, of wavelength. In otwords, for the Moon the coherent backscatter enhancemenot vanishingly small; however, in this case the growth ofcharacteristic interference base is almost compensated foincreasing the wavelength.

4.2. Dependence of Brightness on PhotometricLatitude and Longitude

In addition to the regular mapping of the lunar surface withobservation geometry corresponding tob= l = 0, Clementinestudied a few regions with different viewing angles. This givegood opportunity to check the dependence onl andb predictedby Eq. (30). Also, it allows us to find the phase dependencesparticular regions and to compare them with the average luphase function.

As an example, we present here the results for a region cto the Apollo 15 landing site. A small topographically even a

shown in Fig. 7 was selected for the analysis. Clementine took

142 SHKURATOV ET AL.

FIG. 7. The Clementine image LUE3909L.299 (filter E,λ= 1.00 µm) of the vicinity of Apollo 15 landing site. The scene is centered at 26.2◦N, 3.4◦E.d

dlynif

ee

np

agigst

m

dio

entdata

er E(29)

Marked is the area for which the photometric measurements are presente

a large series of images of this region at different anglesα, b,andl . Brightness of the area in this series of images is plotteFig. 8 as a function ofα. In the series the phase angle initialdecreases and then increases, producing two branches obrightness dependence. Brightness of the two branches is dent for the same phase angles because the other two anglesbandlare different. In order to correct for the dependence on photomric latitude and longitude, we divided the measured brightnby Akimov’s functionD(α, b, l ) (see Eq. (29)). This brings thdata to the uniform observation geometry (α, b= 0, l = 0). Theresult is presented in Fig. 8. One can see the two branchescoincide. This demonstrates, on the one hand, that Eq. (29)vides a good description of the surface reflectance and, onother hand, the good photometric quality of the Clementine dThe resulting phase dependence is almost linear and is in ament with the average lunar data. The data presented in Fcorrespond to the filter E (1µm). For the shorter wavelengththe branches after the correction do not coincide so well, budifference never exceeds 3%.

4.3. Opposition Spike

Among images of the lunar surface taken by the UVVIS caera during the Clementine mission there are images includingzero phase angle point or, in other words, the spacecraft shapoint. It gives a unique opportunity to study the lunar opposit

spike. This spike is clearly seen in the images as a diffuse bri

in Fig. 8.

in

thefer-

et-ss

owro-theta.ree-. 8

,the

-theown

FIG. 8. Normalized reflectance of the area marked in Fig. 7 at differobservation geometry plotted against the phase angle (triangles). Theseare extracted from 88 frames obtained by Clementine UVVIS camera in filt(λ= 1.00µm). The same data corrected for observation geometry using Eq.

ghtare presented by pluses—these present the phase function of the site shown inFig. 7.

T

o9

o

cnait

vuectwmarc

tgnoriuenat

hsova-)

i

r

idc

tldey

ge-Thisme

on168

of.300

darkragenglesthers iness

d the

ich, are

arac-h. Ins ofesize

ce of

theretude

ences

ach

es isritythat

arly

om-ere

er forcar-

and

OPPOSITION EFFEC

spot around the zero phase angle point (Nozetteet al. 1994).Images of such a kind were also taken during the Apollo 8 (Pet al.1969) and other Apollo missions. As an example, Fig.shows the frame LUC2275J.167 (0.90µm filter). The imageis centered at 1.3◦N, 3.9◦E, and includes the northeast partSinus Medii. This region is brightened by the ray systemCrater Triesnecker.

To extract the phase function using images with the spaceshadow point, different methods have been used. For instaNozetteet al.(1994) and Burattiet al.(1996) suggested usingsimple averaging over a number of different images to diminregional albedo variations. We use other approaches to exthe phase function from Clementine data (Shkuratovet al.1997a,Kreslavskyet al.1998).

Due to spacecraft motion the zero-phase-angle point moon the surface between consecutive images. Thus, after msubtraction (or division) of registered images taken in filtclose to each other, the albedo pattern of features is practieliminated, but the opposition spot is not quenched compledue to the shift of the spacecraft shadow point. Figure 9b shocontrasted composite image—the result of dividing of the fraLUD2271J.167 (0.95µm) by the frame presented in Fig. 9(0.90µm). One can clearly see in Fig. 9b a specific patternlated to the opposition spot, though the difference in the lophase angle for any point in this composite image does noceed 0.2◦. Knowing the phase angle in each point of both imawe calculated the logarithmic derivative of the phase functioeach point of the frame. Then we averaged the derivativeall points with the same phase angle. Integration of the detive provides a phase curve. As a rule, integrating procedsuppress experimental noise and, therefore, our resulted ddences (see Fig. 10) are smooth enough. The curves 1 acorrespond, respectively, to the mentioned pair of imagesthe pair of frames LUD3742J.150 and LUC3746J.150. The lapair is centered at 1.4◦N, 48.7◦E, in Mare Fecunditatis.

As one can see, the opposition effect of the Moon at the pangle 0.2◦–1.6◦ is rather inert with flattening at very small phaangles<0.25◦. This flattening is due to the angular dimensiof the Sun’s disk (Shkuratov 1991, Shkuratov and Stanke1995). For comparison we also present in Fig. 10 the phdependences adopted from Burattiet al. (1996). These dependences present the mare (curve 3) and highland (curve 4gions in average. The pronounced surges at phase angles<0.5◦

observed in the Burattiet al. (1996) data may be due to thedifferent approach to analysis.

4.4. Spectral Dependence of Opposition Spike

The approach to study the opposition effect presented inpreceding section is rather accurate, but it does not allow detion of information about the spectral dependence of the phfunction from the Clementine data set. In this section we consanother method for studying the opposition spots, the methophase-ratio image. It enables the determination of the spe

behavior of the lunar phase function near opposition from t

FROM CLEMENTINE 143

hna

fof

raftce,

shract

edtualrsallyelys ae

e-al

ex-esinverva-respen-d 2nd

ter

aseenichse

re-

r

theiva-aseder

oftral

Clementine data (Kreslavskyet al.1998). The method is like thaapplied to Earth-based telescope images of the Moon (Wi1978, Akimov and Shkuratov 1981, Shkuratovet al. 1994a).A phase-ratio image is prepared by dividing small- and larphase-angle images of the same region of the lunar surface.allows compensation of major albedo variations, although soartifacts originating from surface relief are introduced.

There are a few UVVIS images where this direct comparisis possible. The image shown in Fig. 11a (frame LUA1983J.(0.415µm)) is one of them. Coordinates of its center are 1.4◦Nand 1.3◦E. One can see the crater Blagg at the lower leftthe scene. We used the complementary image LUA0598J(0.415µm) taken at a phase angle of about 26◦ to calculate thephase-ratio image. The results are presented in Fig. 11b (shades correspond to low values of the ratio). Then, we avethe phase ratios over all points which have the same phase aat the small-phase-angle image excluding crater walls and otilted areas. In this way we obtained average phase curveblue, red, and IR filters (see Fig. 12). In this case, the brightnis referred to a phase angle of about 26◦. At the smallest phaseangles the averaged area is small, statistics are poor, anresults are not reliable.

In Fig. 11b some weak variations of the phase ratio, whare not related to changes of phase angle and relief slopesseen. These are actual variations of the opposition spike chteristics. They depend on albedo and structure of the regolitparticular, small bright craters have comparatively low valuethe ratio. In terms of Eq. (30), this implies rather high valuof L and/ord. In turn, these can result from large average sof regolith particles in the craters.

As one can see (Fig. 12), there is no wavelength dependenthe opposition spike for the phase angle range 0◦–1.4◦: all thesecurves are almost parallel. However, at larger phase anglesis a distinct wavelength dependence which affects the magniof these normalized reflectances: the phase ratioI (1◦)/I (26◦) isgreater at shorter wavelengths. The average phase dependderived from Clementine data for the phase angle range 3◦–50◦

also show this behavior (see Fig. 6). This independent approconfirms similar results of Burratiet al. (1996). The fact thatno conspicuous wavelength dependence of the phase curvobserved can be explained in terms of Eq. (30). The similaof the phase curves at different wavelengths means simplythe conditions of quasifractality,L ∝ λ andd ∝ λ, are approxi-mately satisfied; i.e., the structure of the lunar regolith is nethe same in some range of scales.

5. LABORATORY PHOTOMETRY

A series of laboratory measurements was carried out for cparison with the lunar data. Two instrument arrangements wused, one for intermediate to large phase angles and anothsmall phase angles. Measurements presented below wereried out to understand the relative roles of shadow-hiding

heinterference mechanisms on the lunar phase function.

144 SHKURATOV ET AL.

e images

FIG. 9. (a) The Clementine frame LUC2275J.167 showing the opposition spot in northeastern part of Sinus Medii. (b) Ratio of the Clementin LUD2271J.167 and LUC2275J.167.

T

a

.

t

d

iss

as

cOrure

d

hasehasloncesases ising. Asthe

ffer-that

pike.avetypi-fect

itionsur-es of-dow

ace,engthncectal-by

forms and

phase-

and-thannce-gen-ltipleualtheinundovuldrentder

veralatovlu-

OPPOSITION EFFEC

FIG. 10. Near-opposition phase functions derived by the “differentimethod for the sites 1.3◦N, 3.9◦E, Sinus Medii (curve 1), and 1.4◦N, 48.7◦E,Mare Fecunditatis (curve 2). The phase functions for “average” highlandmare regions (Burattiet al.1996) presented by curves 3 and 4, respectively

5.1. Measurements in the Phase-Angle Range 3◦–45◦

The measurements of samples with complicated surface sture were performed at the phase angle range 3◦–45◦ usingour laboratory photometer for large phase angles describeShkuratov and Akimov (1987). Some results have been plished by Shkuratovet al. (1987, 1989, 1991, 1992). In thseries of measurements we used two broad spectral rangeeffective wavelengths of 0.62µm (±0.06µm) and 0.42µm(±0.04µm). The apertures of the light source and receiverequal to 0.5◦ and 1◦, respectively. Thus, smoothing of phacurves by the apertures was small.

Light flux reflected from a bright surface is mainly formeby multiple scattering, which strongly diminishes the shadohiding effect. Thus, the interference mechanism dominatesphase function of a bright surface. Figure 13 (open boxes) shthe phase-angle dependence of normalized (atα= 10◦) reflec-tance for an even optically thick slab of smoked MgO with albenearly 95% (relatively the Halon standard atα= 5◦). As one cansee, a surge in brightness with the amplitude of about 20% ocin the range of 3◦<α<10◦. The same surge for smoked Mghad been earlier observed by Oetking (1966). A similar subut narrower and with smaller amplitude, has also been fofor a Halon sample (Shkuratovet al.1989). Since shadows fothe bright sample of MgO are nearly absent, the strong pobserved has a largely coherent backscatter origin.

We compared the pure coherent backscatter phase depen

with that of pure shadow hiding. For very dark surfaces multipscattering is negligible, and shadowing is the only cause of

FROM CLEMENTINE 145

l”

and

ruc-

byub-

with

ree

dw-the

ows

do

urs

ge,nd

ak

ence

opposition surge. Figure 13 (black boxes) presents the pfunction for soot deposited on an even surface. This samplean extremely low albedo (about 2.5% with respect to the Hastandard atα= 5◦). It is remarkable that the phase dependenfor soot and MgO almost coincide with each other in this phangle range, even though the origin of the phase functionquite different, the former being almost entirely shadow-hidand the latter being almost entirely coherent backscatterwill be shown in a later section, at very small phase anglesopposition effects of these two materials is quite different.

Figure 14 presents data for clear glass powders with dient sizes of particles and provides another demonstrationcoherent backscattering can produce a strong opposition sThe curves show that the samples with smaller particles hstronger peaks under the same albedos. This behavior iscal of the coherent backscatter origin of the opposition ef(Shkuratov 1985).

Let us consider the wavelength dependence of the opposspike parameters. The experimental data for MgO and sootfaces presented in Fig. 13 show that the phase dependencthe color index C(0.6/0.4 µm) is almost absent for both samples. For the soot surface it is not surprising, because shahiding does not depend on wavelength. For the MgO surfwe could anticipate some dependence, because the wavelλ is a parameter in Eq. (30). To explain why the dependeis absent, we can again use the assumption of the quasifraity. This assumption is quite natural for surfaces originatedsmoked deposits (Feder 1988), where very small particlesaggregates that in turn are elements of larger aggregateso on.

There are, nevertheless, surfaces that show conspicuousdependences of the color index C(α). As an example, let us consider results of measurements of a powder Fe2O3 in two spectralbands (Fig. 15). The powder has very different albedo in redblue light: 22 and 4%, respectively, atα= 5◦. At small phase angles the phase curve of this sample in the red light is steeperin the blue light because of the coherent backscatter enhament. At larger phase angles the curve in the red light has atle slope, because shadow hiding is suppressed by the muscattering component and forward scattering of the individparticles is increased with increasing albedo. That can giveminimum of C(α) seen in Fig. 15. The observed increasethe color index (red/blue) with decreasing phase angle arothe opposition we call a “color opposition spike” (Shkuratet al.1996). The phase angle of the color-index minimum cobe considered a characteristic of the strength of the cohebackscatter component of the opposition spike. For the powFe2O3 in red light this angle is very large, about 20◦.

Phase-angle dependences of color index (red/blue) for selunar samples have also been studied (Akimov 1980, Shkuret al.1996, Hapkeet al.1998). It has been shown that somenar samples exhibit minimum on the curves at the range 4◦–10◦.

letheFigure 16 demonstrates data corresponding to our measure-ments of lunar samples from landing site Luna 24 (coarse 94- to

146 SHKURATOV ET AL.

◦ ◦

FIG. 11. (a) The Clementine UVVIS frame LUA1983J.168 (0.415µm). Coordinates of the frame center are 1.4N and 1.3E. (b) The ratio of Clementineframes LUA1983J.168 and LUA0598J.300 for the same region.

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OPPOSITION EFFEC

FIG. 12. Near-opposition phase functions derived by the method of phratio for the wavelengths 1.000, 0.750, and 0.415µm. The dependences anormalized at the phase angle of about 26◦.

200-µm and fine<74-µm regolith). The fine sample exhibitsminimum of C(α) at about 25◦. This phase angle is larger thain the case of coarse regolith. There are lunar samples thnot reveal such a minimum in the range of phase angles 1◦–60◦

(Akimov 1980, Shkuratovet al.1996). No minimum is exhibitedalso by our analysis of the Clementine data. One explanatio

FIG. 13. Normalized reflectance (0.62µm) (boxes) and color index(0.62/0.42 µm) (triangles) vs phase angle for a “shadow-free” bright s

face (sample of MgO) (open symbols) and pure “shadow-hiding” surface (ssmoked deposits on an even background) (black symbols).

FROM CLEMENTINE 147

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t do

of

r-

FIG. 14. Normalized reflectance (0.62µm) phase functions of colorlesglass powders with the different average sizes of the particles.

this could be that the conditions of quasifractality are apprimately valid for the pristine regolith on the lunar surface,not for the sorted samples. This may be also related to thethat the Luna 24 soils are immature.

Thus, data presented in Figs. 13 and 14 show that the cohbackscatter enhancement can contribute the opposition effea wide range of phase angles. Moreover, Figs. 15 and 16 evidthat the component can be observed even for comparativelysurfaces, e.g., for the lunar surface.

ootFIG. 15. Normalized reflectance (0.62µm) and color index (0.62/0.42µm) vs phase angle for a powder Fe2O3.

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FIG. 16. Color index (0.62/0.42µm) phase dependences of the lunar saples 24092,4-3,3 (coarse) and 24092,4-1,10 (fine) delivered from the Lunlanding site.

5.2. Measurements in the Phase-Angle Range 0.2◦–3.5◦

(Illumination by Nonpolarized Light)

Our laboratory photometer–polarimeter for small phasegles allows measurements in nonpolarized light in the rangphase angles 0.2◦–3.5◦ (Ovcharenkoet al.1996, Shkuratovet al.1997b). A halogen lamp is normally used as a source of lighthis instrument. We can also use linearly polarized light (see nsection). The source and detector angular apertures are 0◦;therefore, the aperture smoothing of opposition spikes is vsmall. The instrument permits measurements of samples in aspectral bands. In this series of measurements we use effewavelengthλ= 0.7µm (width±0.03µm) and 0.5µm (width±0.05µm).

With this instrument we have carried out measurementthe same samples which were studied at larger phase angleabove). Figure 17 demonstrates the phase-angle dependenbrightness and color index of the sample MgO. As one canthe sample exhibits not only the wide opposition surge shoin Fig. 13, but also a narrow strong spike. To understand thisus turn to the scheme in Fig. 18. It presents the backscattetwo scales when particles of a medium are aggregates coning of small grains. In this case, the interference enhancemof rays scattered within and between the aggregates givesto the wide and narrow opposition peaks, respectively. Ifdistribution function of the interference bases is multimodphase functions of brightness and/or color index can have sirregularities (Shkuratovet al.1996, 1997b). Note also that thexplanation is not the only one: for the broad peak one canexclude some influence of the shadow-hiding effect despite vhigh albedo of the MgO sample.

At small phase angles,α= 0.2◦–3.5◦, a significant differenceis seen in the brightness and color index phase dependencthe MgO and soot surfaces compared to the larger phase a

(compare Figs. 17 and 19 with Fig. 13). There is a very stro

V ET AL.

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brightness spike near opposition for MgO that is characterof coherent backscatter, but only a modest and roughly linincrease for soot that is characteristic of shadow hiding. Unthe case of large phase angles, the MgO sample shows abut noticeable phase dependence of color index (about 2%the phase angles 0.2◦–3.5◦. The soot surface does not give acolor index dependence.

Our measurements also document a strong particle-sizependence of opposition spike parameters in the phase arange 0.2◦–3.5◦ for well-sorted quartz samples. Figure 20 dmonstrates this effect. The same results have been obtainelier by Shkuratovet al. (1997b) and Nelsonet al. (1998). Notethat Nelsonet al. measured aluminum oxide powders. Theimportant findings are not consistent with the hypothesis (eMishchenko 1992, Mishchenko and Dlugach 1992, 1993)the coherent backscatter enhancement is only observable,particle size is of the order of the wavelength. Equation (gives a rather good fit to experimental data (see Fig. 20)dicting larger values ofL and d for the coarse particle powder. The fitting curves have the following values of the paraters:k= 0.2 (both curves),L = 220µm andd= 90µm (for the

FIG. 17. Normalized reflectance (0.7µm) and color index (0.7/0.5 µm)

ngvs phase angle for “shadow-free” surface (sample of MgO-smoked deposits onan even background).

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OPPOSITION EFFEC

FIG. 18. A scheme of two scale interference opposition spikes.

FIG. 19. Normalized reflectance (0.7µm) and color index (0.7/0.5 µm)

vs phase angle for pure shadow-hiding surface (sample of soot smoked oeven background).

FROM CLEMENTINE 149

FIG. 20. Normalized reflectance (0.7µm) of quartz samples with the average size of the particles of 150 and 15µm. The measurements were performby the small-phase-angle photometer. The fitting curves calculated by Eqare also given at the following values of the parameters:k= 0.2, L = 220µm,andd= 90µm (for the 150-µm curve) andk= 0.2, L = 29µm, andd= 9.5µm(for the 15-µm curve).

150-µm curve), andL = 29µm andd= 9.5µm (for the 15-µmcurve). In both cases the values ofd are on the order of theparticle size. As we have noted above, for structureless pcles larger thanλ the parameterd should be on the order of thparticle size.

Some samples give a minimum on the color index phase cin the range 0.2◦–3.5◦ similar to that observed for the iron oxidin Fig. 15. Shown in Fig. 21 are examples for palagonites, gspectrophotometric and photometric analogs of Martian soThe albedos (0.7µm) of the samples are 18% (left plots) an33% (right plots) atα= 3◦. One can see that location of the miimum depends on albedo: the minimum is broader and shitoward large phase angles for the palagonite sample withalbedo.

It is interesting to compare contributions of single-particand interparticle scattering in phase function. We have stua powder of spherical particles of barite colorless glass.average diameter of spherical particles is of about 50µm andthe refractive index equals 1.45. The albedo of the sampapproximately 110% relative to the Halon photometric standatα= 2◦; that means that the opposition spike of the spherparticle powder is stronger than that of Halon. The resultsmeasurements for a monolayer and a thick slab of the spheparticles are shown in Fig. 22. One can clearly see a promipeak on the phase curve near 0.5◦. According to calculationsmade by Mie’s theory (Ovcharenkoet al. 1996), this peak is aglory ring. The peak is observed both for the thick slab (curveand for the monolayer (curve 2). In the first case the contof the glory ring is smaller, which is evidently caused by t

n anmasking effect of interparticle scattering. However, the fact that

150 SHKURATOV ET AL.

FIG. 21. Normalized reflectance (0.7µm) and color index (0.7/0.5 µm) phase functions of a palagonite powders measured by the small-phase-anglea ◦

a

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photometer. The albedos (0.7µm) of the samples are about 18% (left plots)

we detect this feature for the thick slab shows that evensuch a bright surface single-particle scattering can noticecontribute to the phase dependence of brightness.

Of course, this contribution depends on the size and shathe particles. If powder particles are significantly smaller t

FIG. 22. Normalized reflectance phase dependences (0.7µm) for a thickslab (curve 1) and a monolayer (curve 2) of spherical particles. The depen

(3) corresponds to quartz powder with submicrometer particles (albedo is a80% atα= 3◦).

nd 33% (right plots) atα= 3 .

forbly

e ofan

ence

wavelength, the opposition spike of the powder caused by inparticle and even interaggregate scattering should be weak.can be seen in Fig. 22 (curve 3), which presents data for a qupowder with particle size about 10 nm. In terms of Eq. (3for this fine powder the parameterd is larger thanL; i.e., thecharacteristic scale of light diffusion in the surface is less ththe dimension of volume for which the cooperative effectsparticle interaction are important.

5.3. Measurements in the Range 0.2◦–3.5◦ (Illuminationby Linearly Polarized Light)

To investigate the mechanisms contributing to brightnephase functions, linear polarized light can be used. This approwas applied by Hapkeet al. (1993, 1998) to estimate shadowhiding and interference contributions for lunar samples.

There are four formally independent schemes to measphase dependences of brightness using linearly polarized lWe denote them (sp), (ss), (ps), and (pp). The first and secletters mark orientations of the polarizer and analyzer axesspectively. The marks “s” and “p” correspond to the cases wthe axes are perpendicular or parallel to the scattering plrespectively. In the cases (ss) and (pp), the copolarized comnents of scattered flux are measured. The measurements ocross-polarized components correspond to the (sp) and (ps) cbinations. It is shown (Hapkeet al.1998) that combinations (ss

boutand (pp) exhibit close phase dependences at small enough phase

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FIG. 23. Normalized reflectance (0.7µm) phase dependences of crospolarized (1) and copolarized (2) components for red water color samplean albedo of 60%. Also shown is the dependence of the normalized crco- ratio (3).

angles. It is true also for the pair (sp) and (ps). Our measuremconfirmed these results. We present below phase dependof co- and cross-polarized components only for the casesand (ps).

According to Hapkeet al. (1993, 1998), measurementsco- and cross-polarized components can give information arelative contributions of the shadow-hiding and coherent bascatter effects in the brightness phase functions. The followscheme was suggested. The phase dependence of the coized component is contributed by single- and multiple-scattefluxes, i.e., by both shadow-hiding and coherent backsceffects, while the cross-polarized component is formed onlymultiple scattering, i.e., by the coherent backscatter. Hapkeet al.(1993, 1998) consider that interference enhancement ariseboth components of light polarization. According to this schethe steeper phase behavior of the copolarized componentpared to the cross-polarized component implies that the shahiding effect dominates (Hapkeet al.1993, 1998). Our measurements and a simple physical consideration presented belowthat this schema should be somewhat corrected.

Actually, Eq. (5) shows that the copolarized componentsdirect and time-reversal trajectories are the same; i.e., azero phase angle they always interfere constructively, whethe cross-polarized components for the trajectories can inteboth constructively and destructively even at pure backscaing. This means that in coherent backscattering, the contribuof the cross-polarized components is smaller than that ofcopolarized ones.

Let us consider results of our polarimetric measuremeFigure 23 presents the phase dependences of co- and cpolarized components of light scattered by a red water c

sample (albedo is 60%, atλ= 0.7µm). The water color sample

FROM CLEMENTINE 151

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was obtained by drying of a water color suspension. Usingoptical microscope, we find a surface relief for this sample oon the scales less than 10µm. Such irregularities are rather tranparent in red light. Thus, the surface of the sample was eveto microscales and, hence, the shadow-hiding effect was mmal. As one can see in Fig. 23, the copolarized component hvery prominent peak, whereas the cross-polarized one is alconstant. This is in accordance with the occurrence of the inference enhancement mainly in the copolarized componenrequired by the reciprocity principle (see Eq. (5)).

According to Hapkeet al. (1993, 1998), the shadow-hidineffect must be basically revealed in the copolarized componWe agree with this only partially. Indeed, shadowings are eftive for large enough fragments of a given medium—whencan neglect their transparency. However, this imports thatshadow-hiding effect contributes both co- and cross-polarcomponents, since both these components are largely formthe limits of each fragment. This is confirmed by the followiexperiment. We used a sample of a hardened cement suwith random “macrorelief” with characteristic slope about 4◦.The phase dependences of co- and cross-polarized compowere measured for different illumination/observation geomtries. If the incidence was along the global normal of the sam(l = b= 0), there were no macroscopic shadows on the surfIn this case co- and cross-polarized components (see Figcurves 3) behave as in the case of the even water color su(see Fig. 23). When the sample tilts either in the scattering p(l = 60◦,b= 0) or in the perpendicular direction (l = 0,b= 60◦),

FIG. 24. Normalized reflectance (0.7µm) phase dependences of copolized and cross-polarized components for a hardened cement surface with ra“macrorelief.” Curves 1–3 correspond to the following illumination/observat

geometry: (b= 60◦, l = 0), (b= 0, l = 60◦), and (l = b= 0), respectively.

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152 SHKURAT

shadows appear on the surface of the sample and the shadfect manifests itself, making brightness phase functions of bco- and cross-polarized components steeper (see Fig. 24)interesting to note that if the copolarized curves 1 and 2normalized by curve 3, the result practically coincides withcurves corresponding to the cross-polarized components.

For the surface with extremely low albedo the multiple sctering is utterly small and the cross-polarized componentarise only due to asymmetry (anisometry or anisotropy) ofdividual scatterers. If the surface is formed by densely pacparticles with size on the order of the wavelength or less,single-particle scattering loses the physical sense, since, oto near-field electromagnetic interactions, scattering by aparticles can be considered as an effective single scattereing, generally speaking, cross-polarized components. Figurshows the phase dependences of the normalized cross- andlarized components measured for a sample of soot smokedeven surface (albedo is about 2.5% at 3◦). Particles of soot havesubmicrometer sizes and form fuzzy and, in general, asymric aggregates. Despite the extremely low albedo of the sple, there is a cross-polarized scattered component, whicabout 10 times less intensive than the copolarized one. Ascan see in Fig. 25, unlike the dielectric surfaces (Figs. 2324), the smoked soot shows the brightness dependence of cpolarized component steeper than that of the copolarized o

Thus, we could consider soot surfaces as a coherent-bscatter-free standard. This standard exhibits the cross-polacomponent steeper than the copolarized one (Fig. 25). Any cwhen this does not occur we can treat as a presence of thherent backscatter component.

The different behavior of the cross- and copolarized comnents for utterly dark and bright materials would be interes

FIG. 25. Normalized reflectance (0.7µm) corresponding to cross- ancopolarized components vs phase angle for soot smoked on an even su

The cross-polarized component is about 10 times less intensive than the clarized one.

V ET AL.

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FIG. 26. Ratio of cross- and copolarized components vsα for the lunarsample (61221) (Hapkeet al.1998) and mixtures of soot and chalk. The lunsample is presented by solid curve. Points, crosses, rhombs, and stars corrto the weight ratios (soot/chalk): 3/1, 1/5, 1/10, and 1/30 with albedos of 5,20, 47, and 65%, respectively. Our samples and the lunar sample were meatλ= 0.7 and 0.633µm, respectively.

in applications. Figure 26 shows the normalized ratio of croand copolarized components (cross-/co- ratio) vs phase angmixtures of soot and chalk. The mixture were prepared in wwith the following soot/chalk weight ratios : 3/1, 1/5, 1/10, and1/30. The presented data show that the slope of the phase-dependences of the cross-/co- ratio increases as the albethe mixture increases. The comparison of Figs. 24 and 26idences that even surfaces with albedo of about 5% (poinFig. 26) have some interference component. This agrees witconclusion made by Shkuratovet al. (1991) that a contributionof the coherent backscatter enhancement can be found evePhobos, a body darker than the Moon. Figure 26 (solid cushows also the phase dependence of cross-/co- ratio for thesample 61221 (Apollo16 landing site) measured by Hapkeet al.(1998) atλ= 0.633µm. The lunar sample has the dependeclose to that of the mixture 1/10 with albedo 47% (rhombesThis evidences that the opposition effect of the lunar surfacsubstantially formed by the coherent backscatter mechanis

6. CONCLUSION

At the present time, due largely to results of the Clemenmission, interest in photometry of the Moon is experienca second birth. A new approach to interpretation of the dis required. We brought together the experimental data (luobservations and simulating laboratory measurements) andoretical model deliberately, as such a complex approch senow to be most fruitful. This approach allows the followin

opo-comparatively reliable conclusions.

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1. We presented a photometric function (30), which combithe shadow-hiding and coherent backscatter mechanismsuse three nontraditional parameters, which have not been pously exploited. The parameters have rather clear physical sand the model is simple and gives a good fit to experimentalobservational data.

2. Using the average phase dependence of brightness defrom the Clementine data and the photometric model we shothat the best fit is observed when the coherent backscattertribution is not equal zero. Our calculations indicate that forMoon the relative contribution of the enhancement is about 1at the wavelength 0.75µm and phase angle 3◦. Besides, it wasshown that the width of the lunar interference surge is up to 1◦.

3. The average amplitude of the lunar opposition spikethe range of phase angles 0◦–1◦ is approximately 10%. TheClementine data indicate clearly flattening of phase dependeat phase angles less 0.25◦ that is a result of the size of the Sundisk at 1 AU (Shkuratov and Stankevich 1995).

4. Our analysis confirms that no noticeable dependenclunar brightness phase curves on wavelength is observed atphase angles. This can be explained by the coherent backsmechanism, if one suggests that the ratiosL(λ)/λ andd(λ)/λ,which are the main parameters of the model, are nearly conover the wavelengths. In other words, we propose that forMoon the coherent backscatter enhancement is not smallthe changes ofλ are practically compensated for by properchanging the characteristic interference base and the size ovolume where the interference mechanism is ineffective. Tsuggestion is natural, when we assume the lunar regolith tapproximately a quasifractal structure.

5. Results of our laboratory measurements document a bcoherent backscatter surge that can contribute to brightnesphase angles up to 10◦. Sometimes, e.g., for the MgO sampltwo interference features can be observed: the first (a wide spis caused by scattering in particles which are aggregatesubmicron dust and the second (a narrow spike) is produby interparticle scattering. It is interesting to note that phaangle behavior of the negative polarization, the phenomewhich as well as the opposition spike has the interference or(Shkuratov 1985), is also unusual for smoke-deposited Msamples. As has been shown (Lyot 1929, Shkuratovet al.1994b)such samples exhibit characteristics as if two superposedative branches of polarization were present. These phenomcan be also a result of a superposition of the interference eat two characteristic scales of surface structure.

6. Often phase dependences of color index reveal a minimat small phase angles. It can be treated as an indicator of cohbackscatter contribution. For example, such a minimum is cacteristic of some lunar samples and optical analogs of Margrounds (palagonites). Clementine observations have not yevealed this feature, but this difference may have a theoreexplanation.

7. Our laboratory polarimetric experiments, following th

Hapkeet al. (1998) approach, show that a coherent backsc

FROM CLEMENTINE 153

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ter component can be important even for surfaces with albas low as about 5%. This strengthens evidence that the option effect of the lunar surface is partially formed by the coherbackscatter mechanism.

ACKNOWLEDGMENTS

We are grateful to Bruce Hapke and John Hillier for helpful comments onmanuscript. This study was partially supported by CRDF (Grant UG2-295)

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