Photometry of particulate mixtures: What controls the phase curve?

Icarus 250 (2015) 188–203

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Photometry of particulate mixtures: What controls the phase curve? C. Pilorget a,⇑, J. Fernando b,c, B.L. Ehlmann a,d, S. Douté e a

Division of Geological and Planetary Sciences, Caltech, Pasadena, CA 91125, USA Université Paris-Sud, GEOPS, UMR8148, Orsay 91405, France c CNRS, Orsay 91405, France d Jet Propulsion Laboratory, Caltech, Pasadena, CA 91109, USA e Institut de Planétologie et d’Astrophysique de Grenoble, Grenoble 38041, France b

a r t i c l e

i n f o

Article history: Received 4 August 2014 Revised 26 October 2014 Accepted 25 November 2014 Available online 11 December 2014 Keywords: Photometry Radiative transfer Regoliths Terrestrial planets

a b s t r a c t The amplitude and angular distribution of the light scattered by planetary surfaces give essential information about their physical and compositional properties. In particular, the angular variation of the bidirectional reflectance, characterized through the phase curve, is directly related to the grain size, shape and internal structure. We use a new radiative transfer model that allows specifying the photometric parameters of each grain individually to study the evolution of the phase curve for various kinds of mixtures (spatial, intimate and layered), mimicking different situations encountered for natural surfaces. Results show that the phase curve evolution is driven by the most abundant/brightest/highly anisotropic scattering grains within the mixture. Both spatial and intimate mixtures show similar trends in the phase curves when varying the photometric parameters of the grains. Simple laws have been produced to quantify the evolution of these variations. Layered mixtures have also been investigated and are generally very sensitive to the photometric properties of the top monolayer. Implications for the interpretation of photometric data and their link with the phases identified by spectroscopy are examined. The photometric properties of a few planetary bodies are also discussed over a couple of examples. These different results constitute a new support for the interpretation of orbital/in situ photometric datasets. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction As solar light penetrates into a surface, it is partially reflected back by interaction with its constituents and structures. The amplitude and angular distribution of this signal, as well as its evolution with the wavelength of light give essential information about the physical and compositional properties of this surface. Models have been developed and experimental work has been performed to better understand how the surface properties affect the light scattering (e.g. Hapke, 1981, 1984, 1986, 2002, 2008; Hapke and Wells, 1981; Douté and Schmitt, 1998; Shkuratov et al., 1999; Stankevich et al., 1999; Mishchenko et al., 1999; Stankevich and Shkuratov, 2004; Shkuratov and Grynko, 2005; Pilorget et al., 2013). In particular, it has been shown that analysis of the amplitude and angular distribution of the scattered light at a single wavelength (typically in the visible) constrains: (1) the absorptivity of the medium ⇑ Corresponding author. E-mail addresses: [email protected] (C. Pilorget), jennifer.fernando@u-psud. fr (J. Fernando), [email protected] (B.L. Ehlmann), [email protected] (S. Douté). http://dx.doi.org/10.1016/j.icarus.2014.11.036 0019-1035/Ó 2014 Elsevier Inc. All rights reserved.

(at this wavelength), (2) the size, shape and internal structure of the grains and (3) the spatial organization of the grains. Key parameters include the single scattering albedo, phase function, surface roughness, opposition surge and porosity. Photometry has therefore been used for decades to characterize the surface of the Moon (e.g. Helfenstein and Veverka, 1987; Shkuratov et al., 1999, 2011; Hapke et al., 2012), asteroids (e.g. Helfenstein and Veverka, 1989; Helfenstein et al., 1994, 1996; Domingue et al., 2002; Newburn et al., 2003; Hillier et al., 2011) and planets (e.g. Veverka et al., 1988; Johnson et al., 2006a,b; Jehl et al., 2008; Fernando et al., 2013, 2014). In particular, the angular variations of the scattered signal, characterized through the phase curve, are directly related to the grains’ size, shape and internal structure (McGuire and Hapke, 1995; Grundy et al., 2000; Grynko and Shkuratov, 2003; Shepard and Helfenstein, 2011; Souchon et al., 2011; Hapke, 2012). These grain properties are therefore critical for interpreting the geological and climatic processes that were or are currently occurring on the parent body. However, little is known about what controls the overall phase curve in a natural sample made of different grains with specific composition, phase function and grain size dis-

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tribution. Here, we use numerical modeling to simulate the radiative transfer within different kinds of mixture, and analyze the variations of the scattering behavior, in particular the angular variations of the signal, characterized by the phase curve. Our main objectives are: 1. To study the evolution of the bidirectional reflectance for different kind of mixtures (spatial, intimate and layered), and particulates (complex index of refraction, size and phase function). 2. To quantify the evolution of the observed phase curve (i.e. the angular and amplitude variations of the bidirectional reflectance) with these parameters and derive simple laws when possible. 3. To determine which particulates control the overall phase curve. 4. To discuss the implications for the interpretation of photometric data. 2. Model description The radiative transfer model from Pilorget et al. (2013) is used to perform the different simulations. The model simulates light scattering in a compact granular medium using a Monte–Carlo approach. The wavelength of the light is set to 750 nm. The approximation of geometric optics is assumed, which is a reasonable assumption for grain sizes greater than a few microns here. The physical and compositional properties of the sample are specified for each grain individually, thus allowing simulation of different kinds of heterogeneities/mixtures within the sample. Radiative transfer is then calculated using a ray tracing approach between the grains and probabilistic physical parameters such as a single scattering albedo and a phase function at the grain level. All scattering orders are taken into account. The incidence angle is set to an intermediate value of 45°. The bidirectional reflectance is then computed for different geometries covering the entire upper hemisphere. Because geometric optics is assumed, the diffraction peak of each particle taken individually can be neglected in the case of a compact granular medium (Hapke, 1993). The single scattering albedo is calculated for each grain using Hapke (1993) and thus is a function of the complex index of refraction of the material, the grain size and the potential inclusion of internal scatterers. A two-lobe Henyey–Greenstein phase function, as a function of phase angle, b, is attributed to each grain in the model. It can be expressed as follows:

Fig. 1. Cumulative distribution function for different two-lobe Henyey–Greenstein phase functions. Parameters b and c were chosen following McGuire and Hapke (1995).

tion on the overall phase curve for given grain size distributions, compositions and spatial organization is performed, using the b and c parameters. Several prior studies had shown that for natural particles, the parameters b and c describe a ’’L shape’’ in a c vs. b diagram: c tends to increase as the internal structure of the grains is filled with internal scatterers and b tends to decrease for rougher grains (e.g. McGuire and Hapke, 1995; Souchon et al., 2011; Hapke, 2012). We test in what follows realistic values of parameters b and c with regards to these constraints. The single scattering albedo is also affected by the roughness and the internal structure of the grains but is computed independently of the phase function in the framework of this study in order to highlight the effect of each parameter separately. Here, the single scattering albedo is calculated for each grain using Hapke (1993) analytical expressions. Simulation results with mixtures are then inverted by fitting the phase curves with the ones of pure homogeneous samples with given single scattering albedo and phase function parameters (parameters x; b and c). Uncertainties are estimated to be within the ±0.02 range for the parameter c, ±0.03 for the parameter b and ±0.02 for the single scattering albedo x. In what follows, the 0 derived photometric parameters are designated x0 ; b and c0 .

2

pðbÞ ¼

1þc 1b 1c þ 2 ð1  2b cos b þ b2 Þ3=2 2

3. Results

2



1b

2 3=2

ð1Þ

ð1 þ 2b cos b þ b Þ

with 0 6 b < 1 and 1 6 c 6 1. Here, b is equal to 0 if the photon is scattered backward and equal to 180 if the photon is scattered forward. The first term describes a back scattered lobe and the second term a forward scattered lobe. The parameter b describes the angular width of each lobe, whereas the parameter c describes the amplitude of the back scattered lobe relative to the forward. Thus, a positive value of c indicates that the particle is predominantly back scattering, and a negative value implies a forward scatterer (see Fig. 1). This phase function is commonly used for photometric studies and parameters b and c have been derived for various planetary bodies (e.g. Domingue and Hapke, 1992; Domingue and Verbiscer, 1997; Fernando et al., 2013; Sato et al., 2014). Two-compound mixtures are assumed in what follows. A systematic analysis of the influence of each compound’s phase func-

3.1. Spatial mixtures Spatial mixtures are mixtures where the areas occupied by the different compounds are segregated. The photons therefore only interact with one of the compounds (except when close to the boundary). The resultant number of scattered photons in a specific direction is equal to the sum of the photons scattered by each fraction of the sample in this direction. The overall phase curve is therefore a linear combination of the phase curve of each fraction of the sample. In a first series of tests, we simulate the radiative transfer in a sample made of two compounds, A and B, in a spatial mixture (or ’’checkboard’’). Both fractions are set to have the same grain size distribution (log-normal law centered on 70 lm with 72 volume% between 50 and 110 lm) and the same material (complex index of refraction n = 1.4 + 2.104i). With these parameters, the single scattering albedo of each fraction is estimated to be 0.68 (xA ¼ xB ¼ 0:68). Each grain belonging to fractions A and B

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are assigned a set of 2-lobe Henyey–Greenstein phase functions /A and /B with b = 0.2 and c varying between 0.8 and 0.8 depending on the simulation (see Section 2). The range of parameters b and c tested corresponds roughly to the vertical branch of the ’’L shape’’ described in Hapke (2012). A porosity of 0.5 is assumed. Fig. 2A shows that in the case of a 50–50% mixture, the derived parameter c0 of the overall phase curve is, at first order the mean value of the one of both fractions (small discrepancies can occur due to stochastic variations in the simulations). The derived 0 parameters b and x0 remain constant (within the inversion uncertainties) when changing c. The overall phase curve is therefore 0 such that x0 ¼ xA ¼ xB (=0.68), b ¼ bA ¼ bB (=0.2) and c0 ¼ ðcA þ cB Þ=2. A second series of tests was run at 75–25% spatial mixture (or 25–75% since the situation is symmetric) and shows 0 similar results: b ¼ bA ¼ bB ; x0 ¼ xA ¼ xB , and c0 ¼ f A cA þ f B cB , with f A and f B (f B ¼ 1  f A ) the volume fraction of materials A and B (Fig. 2B). Assuming a similar parameter b for each fraction, the forward/backward scattering properties of a spatial mixture can be directly derived from the one of each fraction as a linear combination of the different parameters c (Fig. 2C). The grain phase function represents the probability that the scattered photons are scattered into a specific direction (Fig. 3). The parameter c therefore describes the weight that this probability being forward or backward. By extension the coefficients q1 ¼ 1þc and q2 ¼ 1c from Eq. (1) have the same role. The probabil2 2 ity for a photon to be scattered in a given direction is therefore a linear function of the parameter c. Thus, the observed probabilistic behavior of the photons is given by the linear combination of the probabilistic behavior of the photons interacting with materials A or B, weighted by the fraction of each material. As a result, the derived parameter c0 is also a linear combination of cA and cB with c0 ¼ f A cA þ f B cB . The overall phase curve of a spatial mixture, when both compounds have similar complex index of refraction and grain size (and thus the same single scattering albedo), is therefore equivalent to (1) the one of a pure homogeneous sample with c0 ¼ f A cA þ f B cB , (2) the linear combination of the phase curves of the different compounds (Fig. 4). In a third series of tests, we keep parameter b fixed but assume that A and B are made of a different material, B being more absorbing than A (nA ¼ 1:4 þ 105 i and nB ¼ 1:4 þ 103 i). For the grain size distribution we use, the derived single scattering albedos are xA ¼ 0:98 and xB ¼ 0:27. Fig. 5 shows that the brighter fraction A of the mixture dominates the overall photometric behavior of the sample. Indeed, since fraction B of the mixture is made of a material that is more absorbing than the one of fraction A, the reflectance level of fraction A is higher than the one of fraction B, and therefore dominates the overall scattered signal. At the grain

Fig. 3. Illustration of the scattering behavior of a photon within a granular medium. For each grain encountered by the photon, the probability to be scattered in a given direction (/A ) is drawn in brown dashed lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

level, the difference between materials A and B remains in the fraction of photons that get scattered with regard to the ones that get absorbed, which is quantified by x. Thus, the overall parameter c of such mixture is given by:

c0 ¼

f A xA c A þ f B xB c B f A xA þ f B xB

ð2Þ 0

as illustrated in Fig. 5D. The overall derived parameter b remains 0 the same, such that b ¼ bA ¼ bB , whereas the single scattering albedo varies depending on the fraction of each material (x0 ¼ 0:58 for 25% A–75% B, x0 ¼ 0:80 for 50% A–50% B and x0 ¼ 0:90 for 75% A–25% B). This result can also be applied in case of two compounds with different grain sizes, since it will result in a difference in the single scattering albedos, similar to the case where both compounds have different composition (nA – nB ). It is interesting to note that some small differences appear in these results when the number of photons that interact with both compounds is not negligible in comparison to the total number of scattered photons, i.e. when we digress from the ideal spatial mixture case. This situation is only encountered when the two compounds have different absorption properties (xA – xB ): photons that first reach the bright side can be scattered to dark side and be absorbed, which results in a lack of scattered photons on the bright side. The forward and backward peaks are attenuated, 0 resulting in a slightly lower value of b for slightly higher values

Fig. 2. A and B: retrieved values of parameter c0 for a spatial mixture made of materials A and B. Parameter b is 0.2 and parameter c varies from 0.8 to 0.8 for each fraction. Materials A and B have the same complex index of refraction n = 1.4 + 2.104i. Their grain size distribution is also the same and follows a log-normal law centered on 70 lm. C: evolution of the derived parameter c0 with the fraction of material A.

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Fig. 4. Black lines: phase curves for a pure homogeneous sample with b = 0.2 and c = 0.4 (continuous line), c = 0.0 (dotted line), c = 0.4 (dashed line). Blue lines: phase curves for a spatial mixture made of materials A and B, with bA ¼ bB ¼ 0:2; cA ¼ 0:8 and cB ¼ 0:8. 25% A–75% B (continuous line), 50% A–50% B (dotted line), 75% A–25% B (dashed line). Black crystals: linear combination of the phase curves (/A and /B ) of pure homogeneous samples made of materials A and B, such that 25% /A –75% /B . Black squares: same except with 50% /A –50% /B . Black triangles: same except with 75% /A –25% /B . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

of c0 . When the grains have the same absorption properties (xA ¼ xB ), the situation is ’’symmetric’’ and such behavior thus does not occur. Similar to previous tests, we simulate spatial mixtures with various parameters b, assuming that both materials have the same complex index of refraction and grain size distribution. For these simulations, the parameter c is set at 0.6 (forward scattering) and the parameter b varies between 0 (isotropic case) and 0.9 (narrow scattering lobe). A linear combination of Henyey–Greenstein phase functions with different parameters b cannot be expressed as a Henyey–Greenstein phase function (contrary to a combination of phase functions with different parameters c) (Eq. (1)). The overall phase curve of the mixture thus cannot be perfectly fitted by the phase curve obtained with a pure homogeneous sample with given 0 parameters x0 ; b and c0 . The best fit is, however, generally quite close. Fig. 6A shows that the anisotropy of the lobes is driven non-linearly by the fraction with the highest anisotropy. For example, for a 50–50% mixture, if one of the materials has a high parameter b 0 (highly anisotropic lobes), the overall parameter b remains very high, whatever the parameter b of the other fraction is. This trend tends, however, to attenuate as the fraction of the material with the highest parameter b decreases (Fig. 6B). The retrieved overall parameter c0 is equal to the one specified for both materials (c = 0.6), within the uncertainties of the model, except when the parameter b is very low (b < 0.1). In the latter case, the scattering behavior is close to the isotropic case: the parameter c0 has almost no influence on the results and is therefore difficult to constrain. The overall single scattering albedo remains the one of both materials (x0 ¼ 0:68). The observed behavior comes from the non-linearity of the number of photons scattered in a given direction with the parameter b. Fig. 7 shows for instance the evolution of the probability for

191

a photon to be scattered within a 10° and 30° range around the forward direction as the parameter b increases (c is set at 0.6 here). The convexity of the curve implies that the combined probability is driven by the one with the higher parameter b, as illustrated by the results in Fig. 6. The use of different extinction coefficients for materials A and B has the same effect as changing the fractions of both compounds (Fig. 8), similar to results obtained previously: the parameter b of the brighter material dominates. As for previous results, the overall derived parameter c0 remains at 0.6, except for very low values of b, whereas the derived single scattering albedo only varies as a function of materials complex index of refraction and fractions, similar to previous simulations. Finally we test the case where both parameters b and c are different for the two compounds. Here both compounds have the same grain size and the same complex index of refraction (nA = nB = 1.4 + 2.104i). While the anisotropy of one of the compound remains low 0 (b 6 0.4), both overall parameters b and c0 tend to evolve quasi-linearly as the fraction of the compounds changes (Fig. 9), consistent with previous results. As the parameter b of one of the fraction 0 increases, both overall parameters b and c0 are affected and tend to be driven by the parameters of the fraction with the highest b. This confirms previous results that show that the compound with the highest parameter b tends to dominate (to a certain extent) the overall scattering behavior. 3.2. Intimate mixtures Intimate mixtures are mixtures where the grains with different compositions are homogeneously mixed. As a result, each photon can potentially interact with several different compounds (Fig. 10). In the first series of tests, we simulate the radiative transfer in a sample made of two compounds, A and B intimately mixed with parameters identical to Section 3.1. The different grains (phase A or B) are distributed randomly so that both fractions are homogeneously mixed. Fig. 11 shows that the derived parameter c0 of such intimate mixtures is given by:

c 0 ¼ f A cA þ f B c B

ð3Þ

(with f A and f B the fraction of materials A and B, with f B ¼ 1  f A ), 0 identical to the spatial mixture case. We also obtain b ¼ bA ¼ bB and x0 ¼ xA ¼ xB (=0.68). To understand this result, it is interesting to separate the scattered signal into the different orders of scattering: the nth order corresponds to photons that contribute to the measured signal after interacting with n grains. The first order corresponds to the contribution of photons that have only interacted with one grain, and is thus identical to the spatial mixture case because each photon has only interacted with one kind of grains. For n > 1, the photons may have interacted with i grains of material A (with i between 0 and n; n being the order) and n  i grains of material B in a given order. The probability for a photon to be scattered in a given direction after n interactions corresponds to the combination of the n probabilities to be scattered in a given direction, and therefore the combination of the different particle phase functions (Fig. 10). Fig. 12, indeed, shows that for the different mixtures, the contribution of each order corresponds to the linear combination of the corresponding orders for pure homogeneous samples. Interestingly, Fig. 12 shows that the curves for the orders of scattering greater than 1 are very similar to isotropically-scattering particles for the whole range of parameters c explored. Further tests with more anisotropic-scattering particles, however, show that these 2nd and greater scattering orders tend to differ from the isotropic case when the anisotropy of the lobes (parameter b)

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Fig. 5. A, B and C: retrieved values of parameter c0 for a spatial mixture made of materials A and B. Parameter b is 0.2 and parameter c varies from 0.8 to 0.8 for each fraction. Material A has a complex index of refraction nA = 1.4 + 105i and material B nB = 1.4 + 103i. Their grain size distribution is the same and follows a log-normal law centered on 70 lm. D: evolution of the derived parameter c0 with the fraction of material A.

0

Fig. 6. A and B: retrieved values of parameter b for a spatial mixture made of materials A and B. Parameter c is 0.6 and parameter b varies from 0 (isotropic) to 0.9 (highly anisotropic) for each fraction. Materials A and B have a complex index of refraction nA = nB = 1.4 + 2.104i. Their grain size distribution is also the same and follows a lognormal law centered on 70 lm.

increases. These results are consistent with Douté and Schmitt (1998) that noticed that when the anisotropy of the phase function remains low (b < 0.5), the Isotropic Multiple Scattering Approximation (IMSA) used in Hapke (1993) model could satisfactory reproduce the phase curve of low to moderate absorbing media. In the case where one of the material is more absorbing than the other, the probability for a photon to be scattered in a given direction after n interactions is weighted by the probability to be

scattered vs. the probability to be absorbed at each grain, thus by the single scattering albedo of the different grains. The overall parameter c0 of an intimate mixture made of two materials with different complex index of refraction but similar grain size distributions is therefore given by:

c0 ¼

f A xA c A þ f B xB c B f A xA þ f B xB

ð4Þ

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Fig. 7. Evolution of the probability for a photon to be scattered within a 10° (red crosses) and a 30° (blue squares) range around the forward scattering direction, for various parameter b. Parameter c is set at 0.6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

with f A and f B the fraction of the different compounds (Fig. 13). This result is identical to the spatial mixture case and implies that the overall parameter c0 is driven by the brighter grains in the mixture. In a third series of tests, we investigated the case of two materials with different grain sizes. We first simulate a mixture with material A made of 10 lm grains whereas material B follows a log-normal law centered on 70 lm. The mean size factor between these two materials is therefore 7. Both materials have the same complex index of refraction (n = 1.4 + 2.104i), which means that the single scattering albedo of A is greater than the one of B (x70 lm ¼ 0:70 and x10 lm ¼ 0:95). Fig. 14 shows that the fraction with the smaller grain size clearly drives the evolution of the parameter c0 . Interestingly, we can notice differences in the 0 retrieved parameters b and c0 , depending on what grain size distribution is assumed in the inversion. The differences between the results come from geometric reasons: photons that exit a large grain atop the sample surrounded by small grains have a lower probability to exit the sample with a high emergence angle than when surrounded by grains of similar size. This tends to limit the forward scattering behavior of small/large grains mixtures. Also, for the backscattering area, the phase curve obtained in Fig. 14A2 and B2 could also be mimicked by samples with a higher parame0 0 ter b (b  0:25—0:3Þ and a lower parameter c0 , within the uncertainties of the model. As the fraction of 10 lm grains decreases, this effect tends to disappear (Fig. 14E) and inversion results

0

0

Fig. 9. Evolution of the retrieved parameters b and c0 for three spatial mixtures. In all cases, the mixtures are made of materials A and B, each with a complex index of refraction nA = nB = 1.4 + 2.104i and a log-normal grain size distribution centered on 70 lm.

Fig. 10. Illustration of the scattering behavior of a photon within a heterogeneous granular medium (intimate mixture with two compounds here). For each grain encountered by the photon, the probability laws to be scattered in a given direction (/A and /B ) are drawn in dashed lines.

obtained with both grain size distributions tend to be similar. We also simulated the cases where fraction A and B have different

Fig. 8. Retrieved values of parameter b for a spatial mixture made of materials A and B. Parameter c is 0.6 and parameter b varies from 0 (isotropic) to 0.9 (highly anisotropic) for each fraction. Material A has a complex index of refraction nA = 1.4 + 105i and material B nB = 1.4 + 103i. Their grain size distribution is also the same and follows a log-normal law centered on 70 lm.

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Fig. 11. A, B, C, D and E: retrieved values of parameter c0 for an intimate mixture made of materials A and B. Parameter b = 0.2 and parameter c varies from 0.8 to 0.8 for each fraction. Materials A and B have the same complex index of refraction n = 1.4 + 2.104i. Their grain size distribution is also the same and follows a log-normal law centered on 70 lm. F: evolution of the derived parameter c0 with the fraction of material A.

Fig. 12. Phase curves of intimate mixtures samples made of materials A and B. Both materials have the same grain size distribution (A and B follow a log-normal law centered on 70 lm) and the same complex index of refraction (nA = nB = 1.4 + 2.104i). Continuous black lines: phase curves by taking into account all scattering orders; dotted lines: 1st order; dashed lines: 2nd order; mixed lines: 3rd order. In blue have also been added the phase curves of pure homogeneous samples with c given by Eq. (3) (crosses), the 1st order (diamonds), the 2nd order (triangles), 3rd order (squares). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

complex index of refraction: one case where the small grains are less absorbing than the large ones, and another case where the small grains are more absorbing than the large ones. In both cases, the results are consistent with previous results. The difference of single scattering albedo only cannot explain the photometric behaviors of the mixtures; the grain size difference also plays an important role. The behavior of parameter c0 can be estimated with:

c0 ¼

f A cxA cA þ f B xB cB f A cxA þ f B xB

ð5Þ

where c is a coefficient related to the grain size difference between both material.

Derived c is slightly less than the ratio of the grain sizes (c  5—6 in all three cases). Other simulations with a bimodal grain size distribution where the grain size ratio is 2 (70 lm and 35 lm diameter grains only) exhibit similar results, with c  2. These results directly derive from the probability for a photon to encounter a grain with given properties. The grains have a geometric cross-sections rA and rB . Since we consider the case of a compact particulate medium, the extinction efficiency is assumed to be 1 (we neglect the effect of diffraction). The extinction cross-section is therefore equal to the geometric cross-section. In a given volume dV, the volume extinction coefficients for A and B are given by EA ¼ N A rA and EB ¼ N B rB with NA and NB the number of particles in dV (Hapke, 1993). The ratio of the two volume extinction coefficients, and the coefficient c, is thus equal to the

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0

195

0

Fig. 13. Same as Fig. 11, with nA = 1.4 + 105i and nB = 1.4 + 103i. The derived parameter b is such that b ¼ bA ¼ bB ¼ 0:2, whereas the single scattering albedo x0 evolves as follows: x0 ¼ 0:62 for 50% A, x0 ¼ 0:44 for 25% A, x0 ¼ 0:33 for 10% A, x0 ¼ 0:28 for 5% A and x0 ¼ 0:26 for 1% A.

Fig. 14. Same as Fig. 11, with A made of 10 lm diameter grains and B with grains following a log-normal distribution centered on 70 lm. Fig. A and B show the inversion results when using a similar sample as the mixture (50% of 10 lm grains and 50% of 70 lm grains). Fig. A2 and B2 show the inversion results by assuming a sample with a grain size distribution centered on 70 lm, as for previous tests. The overall single scattering albedo x0 evolves as follows: x0 ¼ 0:86 for 50% A, x0 ¼ 0:81 for 25% A, x0 ¼ 0:77 for 10% A, x0 ¼ 0:74 for 5% A and x0 ¼ 0:68 for 1% A. Results from Fig. A, B, C, D, and E have been used for Fig. F to highlight the evolution of parameter c0 with the fraction of small grains.

ratio of the grain sizes. We interpret the slightly lower values found for the coefficient c in the first series of tests to result from the rather broad grain size distribution of material B (log-normal law centered on 70 lm with 72 volume% between 50 and 110 lm).

Note that the dependency of the overall photometric properties on the grain size distribution and the opacity of the different grains within an intimate mixture are consistent with analytical expressions from Hapke (1993) (Eq. 10.62).

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0

Fig. 15. Retrieved values of parameter b for an intimate mixture made of materials A and B, each with grain size distribution a log-normal law centered on 70 lm and a complex index of refraction nA = nB = 1.4 + 2.104i. Parameter c is 0.6 and parameter b varies from 0 (isotropic) to 0.9 (highly anisotropic) for each fraction.

Simulations of intimate mixtures with various parameters b show similar results as the spatial mixtures: the overall parameter 0 b is driven by the fraction with the highest parameter b (Fig. 15). This trend tends, however, to attenuate as the fraction of the material with the highest parameter b decreases. The retrieved overall parameter c0 is also equal to the one specified for both materials (c = 0.6), within the uncertainties of the model (except when the parameter b is very low as explained previously) and the single scattering albedo remains the one of both materials (x0 ¼ 0:68). This behavior comes from the non-linearity of the number of photons scattered in a given direction with the parameter b, as explained in Section 3.1. The combined probability to be scattered in a given direction is therefore highly influenced by the grains with the highest anisotropy. The use of different extinction coefficients for materials A and B has the same effect as changing the fractions of both compounds (Fig. 16). Increasing the brightness of one material can therefore be counterbalanced by decreasing its fraction. As in previous results, the overall parameter c0 remains at its original value, whereas the single scattering albedo only varies as a function of materials’ optical properties (complex index of refraction) and fractions. When mixing grains of different sizes (rA = 10 lm and rB = 0 70 lm, similar to previous simulations), the overall parameter b is clearly driven by the smaller grains (fraction A), as shown in Fig. 17. The influence of the large grains on the derived parameter 0 b remains small when the larger grains fraction is not at least several times greater than the smaller ones. Finally we test the case where both parameters b and c are different for the two compounds. Here both compounds have the same grain size and the same complex index of refraction (nA = nB = 1.4 + 2.104i). While the anisotropy of one of the compound 0 remains low (b 6 0.4), both overall parameters b and c0 tend to evolve quasi-linearly as the fraction of the compounds changes

(Fig. 9), consistent with previous results. For higher values of b, 0 both retrieved parameters b and c0 are affected and tend to be driven by the parameters of the fraction with the highest b. This confirms previous results that show that the compound with the highest parameter b tends to have a greater influence on the overall scattering behavior, similar to the case of a spatial mixture (see Fig. 18). In the case of a semi-infinite plane-parallel medium, Hapke (1993) has also derived analytical expressions of the average single scattering albedo and phase function for intimate mixtures. The results presented in this section are well suited to test their validity. The average single scattering albedo and phase function of various mixtures have been calculated following Hapke (1993). We then run our radiative transfer code on a pure homogeneous sam0 ple with these parameters and derived overall parameters x0 ; b and c0 , similar to previous tests. In the various simulations we performed, theoretical predictions by Hapke (1993) are consistent with our results. In the case of intimate mixtures with similar grain size distributions for the different phases but different forward vs. backward scattering behaviors, anisotropic properties and complex 0 index of refraction, all derived parameters b ; c0 and x0 are quite similar to our previous results, both qualitatively and quantitatively (Fig. 19A to be compared to Fig. 11A, Fig. 19B to be compared to Fig. 13A and Fig. 19D to be compared to Fig. 15A). To compute the average phase function in the case of a mixture made of grains with two different sizes (with material A made of 10 lm diameter grains and material B with grains following a log-normal distribution centered on 70 lm), we use a grain size ratio of 5 to be consistent with our previous results. Even in this case, small grains (material A) tend to dominate slightly more the overall phase curves than in our results (Fig. 19C to be compared to Fig. 14A). The computed average single scattering albedo x (=0.86) is also slightly lower than the derived parameter x0 (=0.90) obtained in our simulations. These small differences, however, do not occur

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Fig. 16. Same as Fig. 15, with nA = 1.4 + 105i and nB = 1.4 + 103i.

Fig. 17. Same as Fig. 15, with A made of 10 lm diameter grains and B with grains following a log-normal distribution centered on 70 lm.

when dealing with a mixture where the grain size ratio is lower. Tests with a grain size ratio of 2 show a very good agreement between the results obtained with Hapke (1993) computed average photometric parameters and the ones obtained when simulating the corresponding mixtures, both qualitatively and

quantitatively. We attribute the small differences that occur for larger grain size ratios to (1) geometric reasons already invoked in the previous paragraphs and that tend to slightly change the local photometric behavior around the larger grains and (2) small inhomogeneities in the samples we generate. Hapke (1993)

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analytical expressions, therefore, appear to be quite appropriate to get average single scattering albedo and phase functions in the case of homogeneous mixtures (semi-infinite medium case). In the case of small inhomogeneities or large grain size variations within the mixtures, differences can, however, occur. 3.3. Layered mixtures

0

Fig. 18. Evolution of the retrieved parameters b and c0 for three intimate mixtures. In all cases, the mixtures are made of materials A and B, each with a complex index of refraction nA = nB = 1.4 + 2.104i and a log-normal grain size distribution centered on 70 lm.

Layered mixtures are referred here to as mixtures where one compound is set on top of the other, with a 100% cover. This means that the covering grains are in contact with each other, but there are still some space between the grains because of the porosity. In the ’’layered mixture case’’, the properties of the grains change with the depth (Fig. 20). As for every kind of sample, the scattered signal is composed of photons last scattered by the grains on top and photons scattered by grains underneath which are able to reach the surface through the spaces between the grains. The latter highly depends on the porosity and the spatial organization of the grains. As a result, the photometric curve of a layered mixture where both kinds of grains have similar phase functions but different complex index of refraction can be very different that the photometric curve of a pure homogeneous sample where the grains have a similar phase function (Fig. 21). In general, as long as the top layer is less absorbing than the material underneath, the angular variations of the bidirectional reflectance are similar

Fig. 19. A: retrieved values of parameter c0 for an intimate mixture made of materials A and B. The average single scattering albedo and phase function are calculated with Hapke (1993) and then used for all grains within the sample. Parameter b is 0.2 and parameter c varies from 0.8 to 0.8 for each fraction. Materials A and B have the same complex index of refraction n = 1.4 + 2.104i. Their grain size distribution is also the same and follows a log-normal law centered on 70 lm. B: same as A, but with nA = 1.4 + 105i and nB = 1.4 + 103i. C: same as A, but with A made of 10 lm diameter grains and B with grains following a log-normal distribution centered on 70 lm. D: 0 0 retrieved values of parameter b for an intimate mixture made of materials A and B, similar to A, but with parameter b that varies from 0. to 0.9 and parameter c0 ¼ 0:6.

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Fig. 20. Illustration of the layered mixtures used in the simulations. The ’’irregular cover’’ case corresponds to the case where all grains at the surface (from a sample with a grain size distribution following a log-normal law centered on 70 lm) are made of material A. The ’’regular cover’’ case corresponds to the case where a regular cover of 70 lm grains made of material A is set on top of the sample (whose grain size distribution follows a log-normal law centered on 70 lm).

Fig. 21. Phase curves for various layered mixtures where both materials have the same phase function (parameters b and c). nA = 1.4 + 105i and nB = 1.4 + 103i (A); nA = nB = 1.4 + 2.104i (B); nA = 1.4 + 103i and nB = 1.4 + 105i (C). See Fig. 20 for the different covers. Phase functions obtained for pure homogeneous samples with the same parameters b and c and with comparable reflectance level have been added for comparison (only their angular variations should be considered here).

to the one of a pure homogeneous sample with similar parameters b and c, with some minor differences, however, as shown in Fig. 21. In particular, when a bright material covers a dark one, the amplitude and angular variations of the reflectance are quite sensitive to the organization of the grains. In the case of an irregular monolayer (cf Fig. 20), more photons reach the grains underneath (which are absorbing), than in the case of the regular monolayer. This tends to lower the reflectance value and thus the single scattering albedo, but also leads to some differences in the angular variations of the reflectance (especially at high emergence angles). The case where absorbing material lays on top of bright material is more complex. The signal from the top absorbing layer is relatively weak, whereas the signal coming from the underneath material is quite strong in a particular distribution of directions that results from the spatial organization of the grains. The shape of the phase curve as well as its reflectance level are therefore strongly affected by the contribution from the underlying material, leading to some complex scattering behavior that cannot generally be described with a reasonable fit by a pure homogeneous sample 0 with a given single scattering albedo x0 and parameters b and c0 (e.g. Fig. 21C). In general, photons coming from the material underneath cannot exit the sample with large emergence angles, as these photons generally encounter the top layer absorbing grains, leading to a deficit of photons at these angles. In addition, the photometric behavior of these samples varies a lot with the spatial organization of the top layer (Fig. 21C).

The case where both materials have different photometric responses (parameters b and c) is rather similar to the previous case. As long as the top material is less or as absorbing as the one underneath, the majority of the scattered photons come from the grains atop the sample. The angular distribution of the scat0 tered photons, described by b and c0 , is therefore strongly driven by the parameters b and c of the grains atop the sample (the first monolayer). As above, small differences at high emergence angles may also occur for certain grain organizations when a bright material covers a dark one (Fig. 21A). In the latter case, the best fit is 0 obtained by using a more anisotropic phase function (b ¼ 0:3) than the one of the materials within the mixture (b = 0.2) (Fig. 22B2). Interestingly, having a more translucent material on top does not mean being more sensitive to the parameter b and c of the material underneath. It is actually quite the opposite since a brighter material on top contributes to a larger extent to the signal measured which leaves a smaller fraction of the signal coming from underneath. In case the material on top is less absorbing than the one underneath, complex scattering behaviors tend to occur, as explained previously, and no satisfying fit with a phase curve obtained with a pure homogeneous sample can generally be obtained. Increasing the porosity of the sample leads to the situation where more photons coming from the underlying material can exit the sample. However, as long as the grains above keep creating

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Fig. 22. A, B, B2, C, and D: retrieved values of parameter c0 for a layered mixture made of materials A and B. Material A (on top) is made of 70 lm large grains that have a complex index of refraction nA = 1.4 + 2.104i (A and C) and nA = 1.4 + 105i (B, B2, D). Material B (covered material) has a complex index of refraction nB = 1.4 + 2.104i (A and C) and nB = 1.4 + 103i (B, B2, D). Its grain size distribution follows a log-normal law centered on 70 lm. Parameter b is 0.2 and parameter c varies from 0.8 to 0.8 for each 0 0 fraction. A good fit can generally be obtained with b ¼ 0:2. However, in certain cases (B), a more anisotropic phase function (b ¼ 0:3) provides a better fit, even though 0 forcing b ¼ 0:2 (B2) gives pretty similar phase curves (Fig. 21A) whose evolution can be more easily compared to the evolution of other layered mixtures (A, C, D). The overall single scattering albedo x0 is as follows: x0 ¼ 0:68 (A), x0 ¼ 0:74 (B and B2), x0 ¼ 0:68 (C) and x0 ¼ 0:84 (D).

some priviledged directions for the photons to exit, the contribution of the photons coming from the grains underneath remains complex and difficult to interpret. 4. Discussion 4.1. What grains tend to lead the photometric behavior? Previous studies have already shown that the reflectance level (and thus the single scattering albedo x) was controlled by the smallest and darkest grains in a intimate mixture (e.g. Clark, 1983; Clark and Roush, 1984; Clark and Lucey, 1984). For example, as little as a 2 percents of submicrometer carbon grains can lower the reflectance value of a clay (grain size 6 63 lm) from 0.6 to 0.15 (Clark, 1983). Here we have shown that the angular variations of the bidirectional reflectance (through the parameters b and c) are also controlled by the smallest grains but also the brightest grains, in contrast with the single scattering albedo. As a result, the angular variations of the bidirectional reflectance and its level may not be sensitive to the same fraction in an intimate mixture. Thus, the 0 retrieved parameters b and c0 and what they imply in terms of grain shape and structure may not be directly applied to the major phases derived from spectroscopy (x0 vs. wavelength). In order to directly link the physical properties to the phases derived from 0 spectroscopy, the photometric parameters b ; c0 and x0 must be driven by the same endmember(s). This typically requires that one compound is major within the mixture and that the relative albedo and grain size are close among all the endmembers. In the case of a spatial mixture, the overall single scattering albedo x0 is less sensitive to the presence of endmembers with different absorptivities than in an intimate mixture, since the various phases

have independant scattering behaviors (e.g. Fig. 12 of Pilorget et al. (2013)). Both spectral and photometric curves result from a linear combination of the one of the different endmembers. As a result, 0 the photometric parameters b ; c0 and x0 will be driven by the same endmember(s) if they are major within the mixture and at least as bright as the other ones or close. The layered mixture case is more complex and needs to be studied on a case by case basis. It has been previously shown that a good knowledge of the photometric properties (in particular the phase function) of the different materials within an intimate mixture allows better quantification of the different phases with spectroscopy (e.g. Mustard and Pieters (1989)). The coupling of photometry and spectroscopic data from orbit, therefore, appears to be an efficient way to improve the inversion of the spectroscopic data and refine endmember quantification. However, the link between the parameters 0 b and c0 and the phases derived from spectroscopy can only be made in specific conditions, as discussed above. Applying the overall parameters directly to the different endmembers may lead to some errors in the quantitative estimates. An estimate of the fractions of the different endmembers could, however, be first obtained using isotropic phase functions. Depending on these results, assumptions could be made on which endmember actually 0 drives the photometric parameters b and c0 and thus should be attributed with these photometric properties. 4.2. Covered materials: what can be inferred in terms of photometric properties? For the layered mixture case, the variability of grain properties with depth can lead to complex photometric behaviors. We have shown here that when the material on top was less or equally

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absorbing that the one underneath, the angular distribution of the photons exiting the sample was strongly driven by the photons coming from the top monolayer. The influence of the underlying material is small compared to the latter and strongly affected by the spatial organization of the grains on top. When the material on top is more absorbing than the material underneath, the relative proportion of the photons coming from the underlying grains increases and may dominate. Their angular distribution is, however, strongly affected by the structure of the top monolayer. In both cases, a good knowledge of the photometric parameters of the material on top as well as its structure will be a requirement to infer the photometric parameters b and c of the underlying material. Interestingly, compositional information from spectroscopy of the underlying material may be easier since the reflectance level, described by the single scattering albedo x0 , is more sensitive to the material underneath than the other parameters. In particular, in the case of dark material covered by bright material, a large fraction of the photons scattered by the grains atop the sample will reach the underlying grains, with the possibility to be absorbed. In the scope of this study, we focused on the case of 100% covered material (grains are in contact). Natural cases may also imply partially covered material, like for example hematite covering sulfates/basalt in Meridiani Planum on Mars (Fernando et al., 2014). These cases mix spatial and layered mixtures. However, as little as a couple of tens of percents of cover can be sufficient to impact significantly the phase curve, in particular at high emergence angle (e.g. Fig. 16 of Fernando et al. (2014)). These cases should therefore be interpreted cautiously. 5. Potential implications for the interpretation of the photometric behavior of various planetary bodies in the Solar System Results obtained in this study are critical for the interpretation of the photometric behavior from planetary bodies. In this section, we discuss their interpretation for a few examples taken from studies focused on the Moon, Mars and the icy satellites. 5.1. Moon The Moon is the planetary body whose photometric behavior has been the most studied (e.g. Helfenstein and Veverka, 1987; Shkuratov et al., 1999, 2011; Hapke et al., 2012; Souchon et al., 2013; Sato et al., 2014). It exhibits a general backscattering behavior, with an albedo that varies between the mare (x = 0.512, Hillier et al., 1999) and the highlands (x = 0.333, Hillier et al., 1999). It also exhibits a strong opposition surge. Several studies have also looked into more details at the spatial variations of the photometric parameters. In a recent study, Sato et al. (2014) mapped the various photometric parameters derived from the Hapke model using the Lunar Reconnaissance Orbiter Camera (LROC) Wide Angle Camera (WAC) data. In particular they observed a higher parameter b and lower parameter c for the mare relative to the highlands indicating decreased backscattering. They suggested that the difference in the photometric behaviors could be explained by the higher content of absorbing ilmenite and SMFe (submicroscopic metallic Fe) in the mare soil (McKay et al., 1991). Higher content of SMFe within the agglutinates results in a reduced backscattering behavior with regards to the other particulates and thus of the overall backscattering behavior and also leads to a lower albedo. Such interpretation is supported by our results that show that the weight of each fraction within an intimate mixture is directly linked to its single scattering albedo, and thus to its opacity. They also observed that mare with high and low ilmenite concentrations showed similar parameters b and c, but a lower albedo when high

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ilmenite concentrations are present. Our results support the fact that ilmenite grains should have a limited impact on the overall photometric behavior since they are quite opaque relative to the other particles and exhibit a rather symmetric scattering behavior (Sato et al., 2014). Using combined AMIE (Smart-1) and M3 (Chandrayaan-1) data, Souchon et al. (2013) have also studied the spectrophotometric properties of the pyroclastic deposits in Lavoisier crater. They observed several distinct photometric behaviors for different pyroclastic deposits, though similar spectral signatures, that they interpreted to result from distinct grain sizes, roughness, particle scattering properties or compaction state. Our results support the fact that small variations of the size of the various particles with given composition and scattering properties could have a significant impact on the overall photometric behavior, without having to invoke different scattering properties for the same group of particles for the different deposits. 5.2. Mars Both in situ and orbital measurements of the photometric properties of Mars have been obtained, in particular over the MERs (Mars Exploration Rovers) landing sites (Meridiani Planum for MER-Opportunity and Gusev Crater for MER-Spirit) (e.g. Johnson et al., 2006a,b; Jehl et al., 2008; Fernando et al., 2013, 2014). Various photometric behaviors have been observed over these regions. For example, measurements performed in situ by Pancam showed an important photometric variability depending on the nature of the soil (outcrop rocks, ripples, spherule soil, dusty soil, etc.) (Johnson et al., 2006a,b). In particular, they observed the important effect of dust deposition/removal on the photometric properties of the surface at Gusev Crater. Jehl et al. (2008) as well as Fernando et al. (2014) observed similar effects from orbit, with surface dust controlling the photometric behavior where it was present. Our results show that even a thin layer of material at the surface tends to control the overall photometric properties and are consistent with the observations. More realistic simulations in the case of Gusev Crater have been performed with this model and have confirmed the strong control of dust in mixtures made of small grain dust (10 lm in size that simulates dust agglutinates formed by an assemblage of 3 lm dust particles (Lemmon et al., 2004)) intimately mixed with coarse basaltic grains (500 lm in size (Herkenhoff et al., 2004)) as well as basaltic grains partially covered by a monolayer of dust (Fernando et al., 2014). Similar behavior was observed in Meridiani Planum where hematite-rich spherules that covered sulfate-rich deposits tend to control the photometric properties (Johnson et al., 2006a; Fernando et al., 2014). 5.3. Icy satellites The surface of most satellites in the outer Solar System are covered, at least in part, by water frost (Clark et al., 1986). The latter is highly forward scattering, as shown by numerous studies (e.g. Verbiscer and Veverka, 1990; Dumont et al., 2010). However, extensive analysis of the photometric behavior of various icy satellites have revealed that their surface exhibits strong backward scattering behaviors (Verbiscer et al., 1990). Verbiscer et al. (1990) suggested that the formation of microstructures in the frost could potentially lead to a backscattering behavior. So far, no experimental work that we know about has supported this hypothesis. The potential impact of a dark component, with backscattering properties, spatially or intimately mixed with the frost has also been investigated (Verbiscer et al., 1990). However, as highlighted in this paper, the impact of each component within the mixture on the overall photometric behavior is highly dependent on its albedo. Huge amounts of dark material would be required

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Fig. 23. Evolution of the photometric curves when adding backscattering (c = 0.8, b = 0.3) dark (x = 0.5) particles (monolayer of spherical particles) on top of a soil made of bright forward scattering particles representative of water frost (x = 0.99, c = 0.6, b = 0.3). Dark particles are set randomly on top of the water frost. The aerial fraction covered by the dark particles represents the surface occupied by the particles when looking at nadir. As the dark particles density increases, the reflectance drops at high emergence (the photons that come from the underlying water frost encounter the dark particles and are absorbed when escaping with high emergence angles).

grains. Simple laws have been produced to quantify the evolution of these variations, which can be used for future laboratory work to simulate a material with given scattering properties and for the interpretation of spaceborn datasets. We have also shown that in the case of layered mixtures, most of the signal that is measured is caused by photons that come from the grains atop the sample. The angular distribution of the bidirectional reflectance is therefore highly driven by these grains, which makes it difficult to assess for the underlying grains. When the top layer is made of a more absorbing material, complex scattering behaviors tend to occur where the organization of the grains plays an important role. These different results have important consequences in the interpretation of spectro-photometric data from planetary surfaces. In particular, we have shown that the angular variations of the bidirectional reflectance (through the parameters b and c) and its amplitude (through the single scattering albedo x) are not sensitive to the same kind of grains. As a result, caution should be taken when linking the physical properties derived from the angular variations of the bidirectional reflectance to the phases identified by spectroscopy. Several examples of photometric behaviors observed for various objects like the Moon, Mars and the icy satellites have been discussed in the framework of these results. In particular, it has been shown that when covering bright forward scattering medium with absorbing backscattering grains, as little as a 15–20% coverage is sufficient to make it look like a bright backscattering medium. This result opens a new field of investigation to explain the backscattering behavior of the icy satellites.

Acknowledgments to fit the angular variations of the phase curve, in contradiction with the compositional data and the overall albedo. Our results suggest a new possibility to explain this photometric behavior. Results obtained in the case of layered mixtures show that backscattering behavior could be produced by forward scattering water frost partially covered by a monolayer of dark backscattering grains. As shown in Fig. 23, the presence of this monolayer of backscattering grains on top of some highly bright and forward scattering material could create an overall backscattering behavior, while maintaining a high albedo. For example, as little as a 15–20% coverage is sufficient to make it look like a bright backscattering medium. The presence of these absorbing grains on top tend to strongly limit the contribution of the photons that exit the water frost with a high emergence angle, thus limiting the forward scattering properties of the surface. The backscattering properties of the dark component, potentially resulting from an aggregate structure, would then become dominant. Similar behavior has already been noticed in Fernando et al. (2014) when simulating the photometric behavior of hematite (absorbing and strongly backscattering) covering sulfate deposits (bright and forward scattering) in Meridiani Planum on Mars.

6. Conclusion Photometry and spectroscopic techniques give important information about the surface physical and compositional properties. Using a radiative transfer model, we have studied the evolution of the photometric behavior of various mixtures (spatial, intimate and layered) that are representative of what could be found in natural environments. We find that in the case of both spatial and intimate mixtures, the phase curve forward/backward scattering properties are driven by the most abundant, brightest (through the complex index of refraction or the grain size) and highly anisotropic scattering

We would like to thank our colleagues at Caltech, GEOPS and IPAG for inspiration and advice. We are grateful to B. Hapke and an anonymous reviewer for their detailed and useful comments, and to the editorial team of Icarus for manuscript preparation and publication.

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