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Icarus 173 (2005) 3–15www.elsevier.com/locate/icaru

Interpreting photometry of regolith-like surfaces with differenttopographies: shadowing and multiple scattering✩

Yuriy G. Shkuratova,∗, Dmitriy G. Stankevicha, Dmitriy V. Petrova, Patrick C. Pinetb,Aurélien M. Cordb, Yves H. Daydoub, Serge D. Chevrelb

a Astronomical Institute of Kharkov National University, 35 Sumskaya St., Kharkov 61022, Ukraineb Observatoire Midi-Pyrénées. 14 Av. E. Belin, 31400 Toulouse, France

Received 28 June 2003; revised 29 November 2003

Available online 4 February 2004

Abstract

The effects of various types of topography on the shadow-hiding effect and multiple scattering inparticulate surfaces are studied. Twbounding cases were examined: (1) the characteristic scale of the topography is much larger than the surface particle size,characteristic scale of the topography is comparable to the surface particle size. A Monte Carlo ray-tracing method (i.e., geometricproximation) was used to simulate light scattering. The computer modeling shows that rocky topographies generated by randomlystones over a flat surface reveal much steeper phase curves than surface with random topography generated from Gaussian statistand slopes. This is because rocks may have surface slopes greater than 90◦. Consideration of rocky topography is important for interpretrover observations. We show the roughness parameter in the Hapke model to be slightly underestimated for bright planetary surfamodel neglects multiple scattering on large-scale topographies. The multiple scattering effectalso explains the weak spectral dependences othe roughness parameter in Hapke’s model found bysome authors. Multiple scattering between different parts of a rough surface suppresthe effect of shadowing, thus the effects produced by increases in albedo on the photometric behavior of a surface can be compwith the proper decreases in surface roughness. This defines an effective (photometric) roughness for a surface. The interchangeability ofalbedo and roughness is shown to be possible with fairly high accuracy for large-scale random topography. For planetary surfacea hierarchically arranged large-scale random topography, predictions made with the Hapke model can significantly differ from realroughness. Particulate media with surface borders complicated by Gaussian or clumpy random topographies with characteristicparable to the particle size reveal different photometric behaviors in comparison with particulate surfaces that are flat or the scatopographies is much larger than the particle size. 2004 Elsevier Inc. All rights reserved.

Keywords: Photometry; Regoliths; Radiative transfer; Computer techniques

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1. Introduction

Interpretations of spectroscopic, photometric, andlarimetric observations of planetary regoliths are currebased on theoretical modeling and computer and labtory simulations of light scattering by particulate medA widely accepted theoretical model describing such scatteing has been developed byBruce Hapke (1981, 1984, 1981993, 2002)and has been used to analyze the surfaces perties of the Moon, Mercury, asteroids, and satellites. H

✩ This paper was presented at the Symposium “Solar System ReSensing,” September 20–22, 2002, Pittsburgh, PA, USA.

* Corresponding author.E-mail address: [email protected] (Y.G. Shkuratov).

0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2003.12.017

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ever, Hapke’s model has simplifications and limitations tshould be studied and understood. Attempts to improvetheory by making the basic formulas more rigorous areimpractical for application to planetary observations. Forstance, the theory of multiple scattering byMishchenko etal. (1999), which is suggested instead of Hapke’s modparadoxically ignores the shadow-hiding effect and thefect of coherent backscattering enhancement, though teffects are very significant for both dark and bright platary regoliths. Computer and laboratory experiments aremost effective way to verify any photometric model andinvestigate how it can be improved. Our computer simutions, providing an exact consideration of light scatterin particulate surfaces in geometric optics approximationhave demonstrated(Stankevich and Shkuratov, 2002)fairly

http://www.elsevier.com/locate/icarus

4 Y.G. Shkuratov et al. / Icarus 173 (2005) 3–15

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good accordance of the Hapke model with benchmarkof computer experiments. This paper studies the roles oshadow-hiding effect and multiple scattering for regolilike surfaces with different boundary topographies in orto clear up how the Hapke model could be improved.

There are at least two motivations to study light sctering in particulate surfaces with complicated boundariesOne of them is the necessity for deeper analysis of phcal mechanisms contributed tolight scattering in planetarsurfaces. The second one is of practical interest; it relto interpretation of photometric observations with martand lunar rovers, like Lunokhod-1, -2, the Mars Pathfinand MER-A, -B vehicles. In situ photometric observatiodeal with surface resolution of about 0.1–1 cm. One canexpect a priori that any photometric model applied satistorily to integral planetary observations or to large portioof a planetary surface with moderate resolutions (> 100 mper pixel) will give good results on roughness scales 01 cm (Cord et al., 2003c; Shkuratov, 2002; Stankevichal., 2002). The martian and lunar surfaces have micro-meso-topography that is complicated with stones and fasteep undulations (clumps) of the particulate surface boary. We consider the mentioned motivations in more det

1.1. Physical mechanisms

The important mechanisms for interpreting planetphotometric data are shadowing in particulate media (asas on random surface topographies) and multiple scaing (incoherent and coherent). This study concentrateshadowing and incoherent multiple scattering effects.coherent backscatter enhancement effect has been studdetail elsewhere (e.g.,Hapke, 1990, 1993, 2002; Shkuratet al., 1999, 2002).

Shadowing in particulate media is the important contritor to the photometric properties of the planetary regoliths aall phase angles even for highly reflective regolith surfa(e.g., Hapke, 1981, 1986, 1993). Although, the shadowhiding effect in regolith-like surfaces has been well studthere are still unresolved and insufficiently studied iss(e.g.,Shkuratov, 2002). For instance, the influence of shaowing on the different orders of multiple scattering hasbeen well investigated.

At large scales (> 100 m), planetary regolith surfacehave topographies that can be described by random sivalued functions (seeFig. 1, case 1). Such topographiinfluence brightness phase curves primarily at large pangles. First approximation models accounting for thefluence were presented bySmith (1967)andHapke (19841993)and used in many applications. Improvements ofanalytical models are very difficult, as the rigorous desction of shadowing even for random single-valued surfademands continual integrations (e.g.,Shkuratov et al., 200and references therein).

The shadowing effect for surface and particulate sttures that are hierarchically-arranged (Fig. 1, cases 2 and

in

-

Fig. 1. Types of topographies: (1) simple random Gaussian topograph(2) hierarchically-arranged randomGaussian topography, (3) rocky toporaphy, (4) particulate topography, (5) simple random Gaussian topograpof particulate surface, (6) rocky topography for particulate surface, a(7) clumpy random topography of particulate surface.

7, respectively) have also been examined(Shkuratov, 1995Shkuratov and Helfenstein, 2001; Shkuratov et al., 192003), since planetary surfaces have some attributes of pical fractals.

The domain when the characteristic scale of surftopography is comparable with the size of surface parti(cf. cases 4 and 5 inFig. 1) has been insufficiently studied both in terms of theoreticaland experimental approach(Helfenstein and Shepard, 1999; Pinet et al., 2001). For thissize domain, details of the surface topography can be stransparent for direct rays from the light source producadditional problems with calculating the shadow-hidingfect.

If a regolith-like surface is weakly absorbing, incohermultiple scattering can contribute considerably to photomric properties over the whole range of phase angles.scattering may take place inside and between particlesparticulate surface, as well as between elements of theface’s topography. Multiple scattering in a particle, whis implied as a sequence of light reflections inside theticle, may be considered when the particle size is sigcantly larger than the wavelength. Geometric optics for sparticles is applicable (e.g.,Grynko and Shkuratov, 2003Shkuratov and Grynko, 2004). We study here only interpaticle incoherent multiple scattering.

The radiative transfer equation is usually used to calcuphase curves of a particulate media that account for inparticle multiple scattering (e.g.,Hapke, 1981, 1993, 2002).

Interpreting photometry of regolith-like surfaces 5

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However, this equation does not take into considerationcorrelation in the propagation of rays in the medium, whis responsible for the shadow-hiding effect (e.g.,Shkuratovet al., 2000). The effect has been approximated in thediative transfer theory for single scattered component (eHapke, 1981, 1993, 2002), however, the correlation produing the shadow-hiding effect also affects higher orders oscattering(Stankevich and Shkuratov, 2002).

Another important aspect of multiple scattering isscattering between surface topography elements. The Hmodel does not consider this. The aspect is importanbright planetary surfaces. The multiple scattering betwsurface topography elements results in a decrease oeffective surface roughness (e.g.,Shkuratov et al., 1994Shepard and Campbell, 1998; Shepard et al., 2001).

The most complicated case is when the characterscale of the surface topography is close to the particle sThis case is rather typical as any real particulate medhas a transition upper layer that is formed due to verticafluctuations of particles near the average level (seeFig. 1,case 4). The transition layer can strongly influence calated phase curves of multiple and especially single scaing (Stankevich and Shkuratov, 2000).

All these issues are difficult to be solved analyticahowever, they can be well studied via computer simulatio

1.2. Resolution effects

Substantial increase in the spatial resolution of platary optical experiments (e.g., multispectral measuremfor the Mars Pathfinder, Mars Express, and MER-A,missions) requires the analysis of the effects of obsetion/illumination conditions on the bidirectional reflectancof surfaces composed of rocks and rock-and-soil mixtuTo model observations of different surfaces including rocksurfaces, a new spectral imaging facility has been desigat the Observatory Midi-Pyrenees(Pinet et al., 2001). Firstresults concerning the influence of the meso-scale textopography on the roughness parameter have been obt(Cord et al., 2002, 2003a, 2003b, 2003c). In particular, at-tempts to interpret laboratory measurements carried outthe facility have been made using the Hapke model.ting of the model parameters to the laboratory measuremproduces, in some cases, nonunique solution resulting inbiguous physical interpretation; the same was found eafor planetary data (e.g.,Helfenstein et al., 1988). Indeed,the Hapke model implicitly assumes the scale of surftopography to be much larger than the size of regolith pacles composing a particulate medium. Besides, this mdescribes topography with a single-valued random fution. However, in laboratory and in situ planetary measuments one may deal with other types of surface topograFor instance, topographies formed with rocks and/or pacle clumps, generally, have characteristic scales comparablewith the size of regolith particles (seeFig. 1, cases 3, 6, 7)Initial results from the facility(Cord et al., 2003c)show that

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for a flat powdered surface with grain size< 75 µm the sur-face roughnessθ (the angle of the surface RMS slope in tHapke model) ranges between 6◦ and 15◦, thus demonstrating that the roughness cannot be neglected.

Our study demonstrates the importance of surface toraphy in interpreting photometry of regolith-like surfaceWe use here new computer modeling results based onmetric optics. Single and incoherent multiple scatteringthe following surfaces is examined:

(1) random single-valued topography with the characterscale much larger than the particle size (Fig. 1, case 1);

(2) random hierarchically-arranged topography withminimal characteristic scale much larger than the pticle size (Fig. 1, case 2);

(3) random rocky topography with the characteristic scmuch larger than particle size (Fig. 1, case 3);

(4) statistically flat particulate surface (Fig. 1, case 4);(5) random single-valued topography with the characte

tic scale somewhat larger than the particle size (Fig. 1,case 5);

(6) random rocky topography with the characteristic scsomewhat larger than the particle size (Fig. 1, case 6);

(7) random clumpy topography with the characterisscales somewhat larger than the particle size (Fig. 1,case 7).

When the characteristic scale of the topography is mlarger than the particle size, the surface can be regaas continuous with a known indicatrix of surface elemeThis study uses a Lambertian indicatrix, though any indtrix could be incorporated. If the characteristic scale oftopography is comparable with the particle size, the enveing surface cannot be considered as continuous. This cafairly realistic, e.g., the hierarchical particulate topograp(seeFig. 1, case 7) is a model of media composed of agggated particles like the lunar regolith that contains muchglutinates. The rocky topography is simulated using sphrandomly embedded in a surface. This case can be attribto continuous surfaces (Fig. 1, case 3), when the spherand distances between them are much larger than theticle size, and to particulate surfaces (Fig. 1, case 6), when“rocks” and distances between them are somewhat lathan particle size. Another interesting example of compparticulate surfaces is media consisting of particles withferent albedo forming optically contrasting domains of dferent scales. However, this example, being rather diverequires a special consideration (e.g.,Stankevich and Shkuratov, 2004) and is not examined in this study.

2. Computer models

We use two ray-tracing models to study single and muple light scattering of light by both continuous and particlate surfaces with complicatedboundaries. Particles are co


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sidered to be opaque with the Lambertian indicatrix of thsurface elements (continuous surfaces are also describthe Lambertian indicatrix). It should be noted that the ingral scattering indicatrix of such a particle is elongatedthe backward direction. This elongation decreases theof multiple scattering in the media. Indeed, even at cservative (nonabsorbing) scattering, the contribution of horders of interparticle scattering to the total medium refltivity equals zero, if the particle indicatrix is theδ-functiondirected rigorously to/from the light source. Multiple sctering is computed using single scattering albedoω of theparticle surface element.

2.1. Continuous surfaces

We investigate random single-valued continuous surfawith Gaussian statistics of heights and slopes. In ation, we study hierarchically-arranged continuous surfacefor which each generation level is presented by a simGaussian random surface. As a prototype we incorporatcomputer model exploited for calculations of the shadhiding effect(Shkuratov and Stankevich, 1992; Stankevand Shkuratov, 1992). Portions of the surfaces (finite reaizations), which are large enough, are used for the followray-tracing procedure. An initial ray is traced from a randpoint of the surface in the direction of a light source. Thiscan either leave the surface without interruption (and hethe point is illuminated) or intersects the surface (and hethe point is shaded). Next, a new ray is traced from the pin the direction of an observer. If this ray does not intersthe surface, the point is visible. If the point is both visiband illuminated, it contributes to the intensity of the sinscattered light. To calculate the second scattering ordetrace a ray from a visible point (point 1) in a random dirtion. If the ray intersects the surface at an illuminated p(point 2), this contributes to the intensity of the double sctered light. To calculate higher scattering orders, we rethese steps taking into account that if a ray does not intethe surface or intersect it at a nonilluminated point, its ctribution equals zero. The procedure is repeated many timfrom the very beginning for different surface points to reacdesired accuracy of the ray-tracing process at calculationall needed scattering orders.

2.2. Particulate surfaces

We have used and described many times a ray-tramodel for particulate surfaces with macroscopicallyboundaries to calculate the shadow-hiding effect and mtiple scattering (e.g,Stankevich et al., 1999, 2000, 200Stankevich and Shkuratov, 2000, 2002; Muinonen et2001). To take roughness of a particulate surface intocount we produce an update computer model that is abdeal with larger number of particles.

We simulate with the models what happens in naturlaboratory studies. A large random system of opaque

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fusely scattering particles is generated in a computer mory and is “illuminated” by rays that are traced in the stem from a light source to an observer. Unlike the previmodel, the updated model is able to effectively simumultiple scattering in semiinfinite particulate media withbitrary surface topographies having the characteristic sclittle larger than the size of the medium’s particles. Forsake of simplicity the particles are assumed to be spheand uniform. The updated model has two separate stag

(1) generation of a random system of particles with a cervolume densityρ, and

(2) Monte Carlo ray tracing.

We use a system of particles randomly distributed in a cThe upper border of this cube can be complicated by a toraphy. The lateral and bottom sides are considered tcyclically closed, i.e., if a ray leaves the cubic volume, ethrough a lateral side, it comes into the same volume fthe opposite side. This is a way to simulate a semiinfimedium using a finite volume.

The quality of our previous model(Stankevich et al.1999, 2000, 2003; Stankevich and Shkuratov, 2000, 200) islimited by statistics. There are two main sources of statiserrors: the limited number of particles in the model volu(it was N = 8000) and the limited number of traced ra(normallyR = 10000). Increases of these numbers werestricted as computations become too long. The main probis that each ray is inspected for intersection with each mparticle, giving the total number of such inspectionsRN =8 × 107 for each phase angle. We improved the modeimplementing some internal arranging of the model volusubdividing the cube with a big numbern3 of small sub-cubes. Then a ray going through the model volume croonly a small number of the subcubes (∼ n). Hence we needto examine the ray intersection with only the particles insthe current subcube and some adjacent ones. This givetotal number of the inspections� RN . Using this we haveelaborated an improved algorithm allowing us atn = 256 todeal withR = 106 and the maximalN = 16×106. Typicallywe useN = 1.5×106, i.e., numbers of rays and particles aapproximately equal to balance the two sources of statiserrors. As the radius of the largest particle cannot exceeedge of small cube, the model does not incorporate partwith substantially different sizes.

3. Results and discussion

Although our model allows for calculations at arbitraillumination/observation geometries, this study focusestwo of them:

(1) a mirror geometry (i = ε = α/2, ϕ = π , wherei andε

are the angles of incidence and emergence, respective


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(2) an azimuthal geometry (i = ε = 85◦, ϕ ≈ α).

Below we present the results for single and multiple stering by continuous and particulate surfaces at thesediverse geometries. To characterize a simple random suwith Gaussian statistics we use the parameterθ that corre-sponds to the surface RMS slope.

3.1. Scattering by continuous random surfaces

First of all, we consider single scattering by facets ocontinuous Gaussian topography withθ = 45◦ (the RMSslope equals 1) and for rocky topography modeled wspherical “stones” randomly embedded in a substratea flat surface boundary (cf. cases 1 and 3 inFig. 1). Thevolume density of the stone layer is taken 0.3. This dsity provides almost the samenumber of details of structurin the model frame as in case of the Gaussian topograShown inFig. 2are phase curves of the probability that agoing from an illuminated point of the surface to an obseris not interrupted. Gaussian (G) and rocky (R) topograpat both the mirror (M) and azimuthal (A) geometries are psented inFig. 2. As one can see at the mirror geometryrocky topography demonstrates much steeper phase cat small phase angles than in case of the Gaussian topphy. This is quite reasonable as rocky topographies maysurface slope angles greater thanπ/2, producing shadows asmall phase angles for any illumination/observation geetry. In contrast, the Gaussian surface produces shadothe mirror geometry only when the values ofα/2 are greatethanπ/2− θ . At the azimuthal geometry the shadow-hidiprobability for these different types of topography reveaa similar phase angle behavior. This demonstrates thatype of meso-scale topography exhibited by a surface iimportant factor for calculating the shadow-hiding effectplanetary surfaces, especially in the context of coming roexplorations of Mars.

Fig. 2. Phase curves of the probability for simple random Gaussianand rocky (R) topographies at the mirror (M) geometry and azimuthalgeometry.

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Contributions from the different scattering orders tototal light flux from a randomly rough surface with Gaussstatistics decrease rapidly with increasing scattering oreven for rough surfaces atω = 1. Figures 3 and 4illustratesthis, showing phase curves for the mirror and azimugeometries, respectively, at surface slopes ofθ = 20◦ and40◦. As can be predicted the contribution from the higorders of scattering increases with increasing phase aThus, for the azimuthal geometry atα > 145◦ (θ = 40◦) thetotal scattered flux is fully controlled by the second ordescattering. Rougher surfaces are characterized by largertribution from the high scattering orders, as should be.

Multiple scattering of light between different parts ofrough surface suppresses the effect of shadowing. Hthe phase curve of a rough surface with high albedo hlower slope than that of a dark surface with the same rou

(a)

(b)

Fig. 3. Phase curves for separate orders and total sum atθ = 20◦ (a) and40◦ (b) for the mirror geometry (ω = 1).


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Fig. 4. Phase curves for separate orders and total sum atθ = 20◦ (a) and40◦ (b) for the azimuthal geometry (ω = 1).

ness. Therefore, we can expectthat the albedo increase mat least approximately be compensated for with an appriate decrease in surface roughness. This defines antive photometric roughnessθeff that is dependent on albed(e.g.,Shkuratov et al., 1994; Shepard and Campbell, 19Shepard et al., 2001). Introducing the parameterθeff allowsone to take the multiple scattering into account usingsingle scattering consideration. We examine this possibbelow for continuous random surfaces with Gaussian sttics including the case of fractal-like topography.

Figure 5presents phase curves (solid lines) for contin-uous random Gaussian surfaces withω = 1 and differentRMS slopes (θ = 20◦ and 40◦) in the mirror geometryThe first five scattering orders were taken into accountFig. 3). The single scattering order (points) appears to pvide good fits by small decreases ofθ . The least-square

-

(a)

(b)

Fig. 5. Examples of phase curves for random Gaussian surfacesθ = 20◦ (a) and 40◦ (b) at the mirror geometry (solid lines). The surfacare absolutely white with the Lambertian indicatrix. Points correspond tthe “equivalent” surfaces withθeff = 19.3◦ and 33.4◦ , respectively.

method is used for the fitting. The differences betweeθandθeff are fairly small: instead of 20◦ and 40◦ we obtain19.3◦ and 33.4◦, respectively. The quality of the fitting becomes worse with growth of roughness, however, eveθ = 40◦ the curves can still be very close to each othOne can observe the same for the azimuthal geometryFig. 6), however, the differences betweenθ andθeff in thiscase distinct from those for the mirror geometry: instea20◦ and 40◦ we obtain 18.7◦ and 36.7◦, respectively. Thuswe may conclude that for rough surfaces with rather smaθ ,the effects of albedo and roughness on phase angle care approximately interchangeable. We have calculatedpendences ofθeff on θ for random Gaussian surfaces wdifferent albedoω at the mirror and azimuthal geometri(seeFigs. 7 and 8). We note that for the mirror geometry th


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Fig. 6. Examples of phase curves for random Gaussian surfacesθ = 20◦ (a) and 40◦ (b) at the azimuthal geometry (solid line). The suface is absolutely white with the Lambertian indicatrix. Points correspondto the “equivalent” surface withθeff = 18.7◦ and 36.7◦, respectively.

difference betweenθ andθeff is more than for the azimuthageometry. The reason is that atthe azimuthal geometry thcontributions of high scattering orders are less than formirror geometry.

An analogous consideration can be carried out for hiechical (fractal-like) topographies. We study here a topogphy with two hierarchical levels (each is a random Gausrelief), when the large-scale topography is regarded as aerence surface for the small-scale topography (seeFig. 1,case 2).Figure 9shows phase curves (solid lines) for fractlike surfaces of 2-levels, each being a random Gaussianface with θ = 20◦ and 40◦, respectively. The calculationwere made for the mirror geometry atω = 1. Points corre-spond to the “equivalent” surfaces that are a simple (1-le

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Fig. 7. Dependences ofθeff on θ for random Gaussian surfaces with diffeent albedoω at the mirror geometry. The scales are given in degrees.

Fig. 8. Dependences ofθeff on θ for random Gaussian surfaces with diffeent albedoω at the azimuthal geometry. The scales are given in degree

random Gaussian topography withθeff = 25.5◦ and 46◦, re-spectively. The effective characteristic angle is larger thaθ ,as we use the 1-level surface to simulate the 2-levelsPresented inFig. 10 is the same for the azimuthal geomtry. It is amazing that for both these geometries simulacurves (points) provide fairly good fits to “actual” dependences (solid line). Like the simple topography, the 2-levsurface reveals difference in multiple scattering contritions for the mirror and azimuthal geometries.

As has been made above, we calculatedθeff as a functionof θ for 2-levels surfaces with different albedoω at the mir-ror and azimuthal geometries (seeFigs. 11 and 12). In thiscase we note again that for mirror geometry the differe


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Fig. 9. Phase curves (solid lines) for hierarchical surfaces of 2-levels,of which is a random Gaussian surface withθ = 20◦ (a) and 40◦ (b). Thesurfaces are absolutely white with the Lambertian indicatrix. Calculatwere made for the mirror geometry. Points correspond to the “equivasurfaces that are simple random Gaussian topography withθeff = 25.5◦ and46◦, respectively.

betweenθ andθeff is more prominent, than that for the aimuthal geometry.

Thus we may summarize that accounting for multiscattering on large-scale topographies with the charaistic angle> 30◦ and albedo> 20% is important. As theHapke model does not take multiple scattering into coneration, determinations of the roughness parameter withmodel will be slightly underestimated. For example, ifing a fit with the Hapke model one finds thatθ of a roughsurface with albedo 30% is 40◦, this actually means that thparameterθ equals approximately 42◦ (seeFigs. 7 and 8).Besides, in many fits with the Hapke model, workers hoften found that the roughness parameterθ reveals an albed

-

(a)

(b)

Fig. 10. Phase curves (solid lines) for hierarchical surfaces of 2-levels,is a random Gaussian surface withθ = 20◦ (a) and 40◦ (b). The surfaces arabsolutely white with the Lambertian indicatrix. Calculations were mfor the azimuthal geometry. Points correspond to the “equivalent” surfacethat are simple random Gaussian topography withθeff = 22◦ and 45◦ , re-spectively.

dependence, recently reinvestigated in terms of spectral dependence (e.g.,Cord et al., 2003a, 2003c). From the presenresults, we can offer an explanation for this puzzling fture with the contribution of multiple scattering. If a surfaunder study has a large-scale topography with the twomore-levels structure, predictions with the Hapke modelnoticeably differ from real values. Besides, it should be ephasized that the influence of albedo on estimates of roness increases when surface elements scatter light atphase angles more effectively, than in case of the Lambeindicatrix.

3.2. Scattering by particulate surfaces with roughboundary

We have also investigated the more complicated cwhere particulate surfaces have surface topographies w


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Fig. 11. Dependences ofθeff on θ for hierarchically-arranged surfaceformed with 2-levels which are random Gaussian surfaces with diffealbedoω at the mirror geometry. The scales are given in degrees.

Fig. 12. Dependences ofθeff on θ for hierarchically-arranged surfaceformed with 2-levels that are random Gaussian surfaces with diffealbedoω at the azimuthal geometry. The scales are given in degrees.

characteristic scale larger than the particle size. We compafour surface topographies:

(a) a statistically flat particulate surface (Fig. 1, case 4);(b) a particulate surface undulated by random topogra

with a Gaussian distribution of heights and Gausscorrelation function (Fig. 1, case 5);

(c) a surface complicated with a layer of “rocks” radomly distributed (with random embedding) over a fingrained particulate medium (Fig. 1, case 6); and

(d) a topography formed with clumps of particles (Fig. 1,case 7).

It should be noted that our computer model takes automcally into account all mutual correlations of all trajectories

Fig. 13. Phase curves of reflectance for rocky (R) and flat-particulate (Ftopographies at the mirror geometry.

Fig. 14. Phase curves of reflectance for rocky (R) and flat particulatetopographies at the azimuthal geometry.

rays in particulate surfaces providing for the shadow-hidingeffect for all scattering orders.

Phase curves for semiinfinite particulate media (withvolume densityρm = 0.1), whose surfaces are statisticaflat or complicated by rocks distributed with the volume dsity ρr = 0.3, are presented inFigs. 13 and 14for the mirrorand azimuthal geometries, respectively. Multiple scatteis computed using five scattering orders. As a characttic of the rocky relief, we use the radius of opaque “stonR = 10r, wherer is the radius of medium particles. This isrepresentative morphometric distribution observed in soexperimental simulations (e.g.,Cord et al., 2003c). We con-sider also that the thickness of the rocky layer equals 2R (seeFig. 1, case 6). Multiple scattering between rocks, rocksmedium particles, and within the particulate medium is tainto account. The brightness surges seen inFig. 13are dueto the shadow-hiding effect. The surge is more prominenthe flat particulate medium (F). Opaque rocks superimpoon a flat particulate topography decrease the shadow-hsurge as their surface is characterized by the Lambertiadicatrix that has not the opposition surge. For the azimugeometry both curves are veryclose to each other, since


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Fig. 15. Normalized phase curves of different scattering orders and suma random Gaussian particulate surface (right panel) in comparison wflat particulate surface (left panel) at the mirror geometry.

this geometry a rocky topography can contribute to an opsition surge.

We also study the effect caused by surface topograpwhose particulate details with the characteristic sizeL aresmall enough to be semitransparent. Random topograwith the Gaussian statistics of heights and slopes (Fig. 1,case 5) and the so-called clumpy topographies formeaggregates of particles (Fig. 1, case 7) are investigated. Wexamine phase curves from different scattering ordersω = 1. Figure 15presents data for 10 orders of scatterand their sum calculated for the mirror geometry. The ripanel shows data for the random Gaussian particulate toraphy withL = 10r andθ = 45◦ at ρ = 0.3. The conditionsL = 10r andθ = 45◦ imply that details of the topographare semitransparent for direct rays from the light souFor comparison, the left panel shows the equivalent cfor a flat particulate surface. Predictably, curve (1) descing the single scattering is steeper for the surface withGaussian topography due to additional shadowing. In botthese compared cases the double scattering shows maxat moderate phase angles. For the flat surface this mmum is higher and placed at larger phase angles than icase of the particulate surface with the Gaussian topophy. At phase angles near 110◦–120◦ intensities of the firsand second scattering orders are equivalent for both parlate surfaces. This relationship between the first and secoscattering orders is also seen for continuous surfaces aproximately the same phase angles (cf.Figs. 3b and 15).Figure 15also shows that the role of the higher ordersscattering is more significant for rough surfaces. Moreoat phase angles smaller than 30◦ the curve corresponding tthe third scattering order lies higher than that of the secscattering order and has a steep slope.

s

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Fig. 16. Normalized phase curves of different scattering orders and suma random Gaussian particulate surface (right panel) in comparison wflat particulate surface (left panel) at the azimuthal geometry.

A similar comparison for these two types of particulsurfaces was carried out in the azimuthal geometryFig. 16). The data shown in the right panel are noisy becaof poor ray statistics in the azimuthal geometry (angles olumination and observation are 85◦). As was above, in thmirror geometry case, the shadow-hiding effect is stronfor the rough surface. In the azimuthal geometry the difence between the values of the high scattering orders foversus rough particulate surfaces is much more promithan in the previous mirror geometry. All orders are steefor rough particulate surfacesbecause of the shadow-hidineffect on large-scale topography elements.

We studied the interchangeability of albedo and roughness for particulate surfaceslike it has been examined abofor continuous surfaces. It turns out that the interchangeity is not so well defined in this case. We cannot achieveisfactory fits varying albedo and roughness. A good fit wfound for the range of phase angles 10◦–30◦ only.Figure 17shows the phase ratio (10◦/30◦) as a function of the roughness parameterθ for the mirror and azimuthal geometriestwo values ofω. It is interesting to note the interchangeabity of albedo and surface roughness effects for the phaseto be possible in the azimuthal geometry only. This indicathat the phase ratio does not contain enough informatioretrieving surface characteristics.

We also investigated clumpy topographies that aretually hierarchically arranged media (seeFig. 1, case 7).A medium composed of clumps with the volume densρcl = 0.3 was generated and each clump is a piece


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Fig. 17. The phase ratioI (10◦)/I (30◦) as a function ofθ for a randomGaussian particulate surfaces with different albedoω at the mirror geome-try.

Fig. 18. Normalized phase curves of different scattering orders and sum fora clumpy particulate surface (right panel) in comparison with an equivalentflat particulate surface (left panel) at the mirror geometry.

Fig. 19. Normalized phase curves of different scattering orders and sum fa clumpy particulate surface (right panel) in comparison with an equivalenflat particulate surface (left panel) at the mirror geometry.

medium withρ = 0.3. Thus, the total volume density of thmedium isρ = 0.09. Figure 18represents phase curves fdifferent scattering orders and their sum for the hierarchmedium (right side) and for the equivalent medium withthe clumpy structure (uniform medium) withρ = 0.1 in themirror geometry. As one can see the hierarchical medexhibits more prominent shadow-hiding effect and weamultiple scattering in comparison to uniform media. Tsame comparison has been carried out for the azimugeometry (seeFig. 19). The behavior of all the studied dependences of the clumpy topography is similar to thosthe Gaussian topography of particulate surfaces.

4. Conclusion

Results from including surface topography in the moding the border effects at calculations of the shadowing efand multiple scattering in particulate media improvesinterpretation of photometricobservations of regolith-likesurfaces. This investigation demonstrates the following:

(1) The shadow-hiding effect originating from both the pticulate structure and border topography is significeven for the conservative case where the single scaing albedo is close to 1. Hence, any theories ignorthis fundamental effect (e.g.,Mishchenko et al., 1999)should be used with great caution.


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(2) The rocky surface topographies produce much stephase curves than those with a Gaussian surface toraphy. This is because rocks may create surface sangles greater than 90◦. The effect is important in ththeoretical interpretation of photometric data obtainin situ for the martian and lunar surfaces.

(3) The quantitative interchangeability between albedosurface roughness with large-scale topography is foThis allows one to introduce an effective (photometroughness. Unfortunately, for surface topographies wcharacteristic scale comparable to the particle sizerelationship between albedo and surface roughnesstheir effects on the phase curve is more complex.

(4) Multiple scattering on large-scale topographies wlarge characteristic angles is important even for sfaces with moderate albedo. Numerous determinatof the roughness parameter of planetary surfacesthe Hapke model appear to be slightly underestimaThis is also important in the analysis of laboratory ptometric measurements.

(5) Weak spectral dependences of the roughness parain the Hapke model can be explained by a wavelendependence in the multiple scattering component oftotal reflected flux. Comparison of theoretical versusservational data can be re-interpreted with this newderstanding.

(6) Prediction of surface roughness from the Hapke mocan differ significantly from actual surface roughnefor planetary surfaces with large-scale topographieshave several levels of topographic structure.

(7) Particulate media with surface borders describedGaussian or clumpy topographies with characteriscale slightly larger than the particle size reveal diffent photometric behaviors compared to flat particusurfaces.

(8) The considered surface topography effects do noquire radical modifications of Hapke’s theoretical modThis model provides a good first approximation forterpretation of planetary surface characteristics. Netheless, we hope that our numerical results will stimuattempts to improve the analytical approach providbenchmarks for verifications of theoretical results.

Acknowledgments

We are grateful to B. Hapke and D. Dominguemany useful remarks. This study was partially suppoby INTAS Grant #2000-0792 and by the French NatioProgram of Planetology (PNP/INSU/CNRS). It has abenefited from the support of Paul Sabatier UniversityToulouse and Midi-Pyrénées Observatory, with the attriion of visiting positions for Y. Shkuratov and D. Stankeviand the use of computational facilities of the laboratUMR 5562 “Laboratoire Dynamique Terrestre et Plataire.”

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