Issue 
A&A
Volume 531, July 2011



Article Number  A150  
Number of page(s)  8  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201016115  
Published online  05 July 2011 
Lunar mare singlescattering, porosity, and surfaceroughness properties with SMART1 AMIE
^{1} Department of Physics, PO Box 64, 00014 University of Helsinki, Finland
email: karri.Muinonen@Helsinki.Fi
^{2}
Finnish Geodetic Institute, PO Box 15, 02431 Masala, Finland
^{3} Instituto de Astrofísica de Canarias, Calle vía Láctea, 38205 La Laguna (Tenerife), Spain
^{4}
Space Exploration Institute, Case postale 774, 2002 Neuchâtel, Switzerland
^{5}
Microcameras & Space Exploration, PuitsGodet 10a, 2000 Neuchâtel, Switzerland
^{6}
IRAP/Observatoire MidiPyrénées/CNRS/Université Toulouse III, 14, Avenue Edouard Belin, 31400 Toulouse, France
^{7}
European Space Agency ESA, ESTEC/SRES, postbus 299, 2200 AG Noordwijk, The Netherlands
^{8}
European Space Agency ESA, ESAC/SREOS, Apdo. de correos 78, 28691 Villanueva de la Cañada, Madrid, Spain
Received: 10 November 2010
Accepted: 19 April 2011
A novel shadowing and coherentbackscattering model is utilized in the analysis of the singlescattering albedos and phase functions, local surface roughness, and regolith porosity of specific lunar mare regions imaged by the AMIE camera (Advanced Moon microImager Experiment) onboard ESA SMART1 mission. Shadowing due to the regolith particles is accounted via raytracing computations for denselypacked particulate media with a fractionalBrownianmotion interface with free space. The shadowing modeling allows us to derive the scattering phase function for a ~100μm volume element of the lunar mare regolith. The volumeelement phase function is explained by coherentbackscattering modeling, where the fundamental single scatterers are the wavelengthscale particle inhomogeneities or the smallest fraction of the particles on the lunar surface. The phase function of the fundamental scatterers is expressed as a sum of two HenyeyGreenstein terms, accounting for increased backward scattering as well as increased forward scattering. Based on the modeling of the AMIE lunar photometry, we conclude that most of the lunar mare opposition effect is caused by coherent backscattering within volume elements comparable in size to typical lunar particles, with only a small contribution from shadowing effects.
Key words: Moon / scattering / techniques: photometric
© ESO, 2011
1. Introduction
The Moon exhibits an opposition effect (Rougier 1933), that is, a nonlinear increase of diskintegrated brightness with decreasing solar phase angle, the angle between the Sun and the observer as seen from the object. Whereas the opposition effect is a ubiquitous phenomenon for atmosphereless solarsystem objects at large, the lunar opposition effect is of particular significance as we can witness the brightness of the full Moon with our own bare eyes. In the opposition night, the Moon is roughly twice as bright as in the nights just before and after the opposition.
The lunar opposition effect lacks a widelyaccepted physical explanation. It has been traditionally explained by mutual shadowing (Bowell et al. 1989; Hapke & van Horn 1963; Lumme & Bowell 1981; Lumme & Irvine 1982) among regolith particles (sizes in several tens of microns) large compared to the wavelength of incident light (SM; shadowing mechanism): the particles hide their own shadows at exact opposition. More recently, the coherentbackscattering mechanism (CBM) has been introduced as a possible explanation for the opposition effects of selected solarsystem objects (e.g., Shkuratov 1988; Muinonen 1990; Hapke 1990; Hapke et al. 1993; Mishchenko & Dlugach 1993). CBM is a multiplescattering interference mechanism, where electromagnetic waves propagating through the same scatterers in opposite directions interfere constructively in the backwardscattering direction but with varying interference characteristics in other directions.
For accruing knowledge on the properties of fundamental scatterers within the lunar regolith, SM needs to be accurately modeled. SM is affected by surface roughness, that is, the stochastic geometry of the interface between free space and regolith (Parviainen & Muinonen 2009, 2007; Peltoniemi 1993; Lumme et al. 1990). The dependence of SM on the angles of incidence and emergence (as measured from the outward surface normal) as well as the azimuth between the planes of incidence and emergence allows for the estimation of the surfaceroughness parameters.
In order to simulate the geometry of the lunar regolith, we make use of a droppingbased algorithm with spherical particles in modeling the bulk particulate medium, as well as fractionalBrownianmotion surfaces (fBm) and Gaussian random surfaces (Gs) in modeling the interface between free space and the particulate medium (Parviainen & Muinonen 2009, 2007; Muinonen et al. 2002, 2001; Peitgen & Saupe 1988). The single spherical particle and its immediate vicinity mimics what we call a volume element of the lunar regolith. The size of the volume element is thus of the order of 100 μm, that is, much larger than the wavelength of incident visible sunlight. We extract the effects of the stochastic geometry from the lunar photometry and, thereby, obtain the volumeelement scattering phase function of the lunar regolith locations studied. The volumeelement phase function allows us to constrain the physical properties of the typical regolith particles much larger than the wavelength.
We assume, hierarchically, that the volume element consists of a finite random medium of fundamental scatterers and that multiple scattering among these scatterers gives rise to coherent backscattering. In order to compute coherent backscattering by a finite medium large compared to the wavelength, such as the inhomogeneous lunar particles, we rely on the theoretical and computational methods put forward by Muinonen (2004) and Muinonen et al. (2010). To support the validity of the present approach, we refer the reader to the recent comparison to exact electromagnetic methods by Muinonen & Zubko (2010).
Due to the absence of polarimetric observations, we adopt a scalar model for coherent backscattering as summarized in Muinonen et al. (2010). Mishchenko & Dlugach (1992) point out the limited accuracy of the scalar model: the model can result in backscattering enhancement factors overestimated by 20% at maximum. Nevertheless, for the present preliminary and limited analysis, we consider the scalar model to be adequate in the evaluation of the leading scattering effects.
The Clementine space mission has been the first mission to obtain highresolution photometry over the entire lunar surface. Hillier et al. (1999) have carried out a Clementinebased diskresolved multispectral photometric study of the Moon. They have analyzed the photometric properties of the lunar surface with the UV/Vis camera in four different photometric channels between 400–1000 nm at a phaseangle range of 0–85° mainly in the nadirpointing geometry (zero angle of emergence). Hillier et al. have divided the lunar surface into two different types of terrain, mare and highlands, and derived synthesized phase curves for both terrain types over the entire lunar surface for, e.g., photometric corrections in mineralogical studies. According to them, the function fits suggest that the mare regions exhibit broader opposition effects, claimed to be indicative of surfaces more compacted than on the highlands regions. They have also used the Clementine data to study the physical properties of the lunar surface by fitting a photometric model that they have derived from that by Hapke (1984). For the mare regions, they have explained the entire opposition effect by SM with a halfwidth of ~8°.
Shkuratov et al. (1999) have carried out an analysis of the lunar opposition effect analyzing its causes using theoretical, observational, and experimental methods. They have reached the conclusion that the lunar opposition effect is at least partly caused by CBM. Hapke et al. (1993) have measured socalled polarization ratios for a number of lunar samples and suggested diagnostic tools for evaluating the importance of coherent backscattering and shadowing for the lunar opposition effect. Hapke et al. have concluded that SM and CBM contribute to the lunar opposition effect in roughly equal shares.
Kaydash et al. (2009) have analyzed SMART1 AMIE images of the lunar mare regions and swirls using socalled phaseratio images and empirical modeling of the photometric dependences in observation and illumination geometries away from opposition. They have indicated regions that show anomalous photometric dependences and have offered interpretations related to the geological evolution of the Moon. Kaydash et al. offer an uptodate assessment of the various phases of the SMART1 mission and of the AMIE imaging geometries as well as the status of the image processing.
In what follows, we carry out physical modeling of the shadowing and coherent backscattering phenomena and seek for a plausible interpretation in terms of fundamental physical parameters. In Sect. 2, we describe the SMART1 AMIE imaging data (Pinet et al. 2005) utilized in our assessment of lunar scattering, porosity, and surface roughness. Section 3 includes the theoretical modeling for the lunar mare scattering law and the inverse methods to derive the model parameters from the imaging data. In Sect. 4, we apply the methods to the chosen set of SMART1 AMIE data and discuss the results. We close the article with conclusions and future prospects in Sect. 5.
2. Observations
We have used selected data obtained by the AMIE instrument onboard the ESA SMART1 spacecraft specially operated to provide a wide range of observation geometries (Racca et al. 2002; Foing et al. 2006; Josset et al. 2006). We include four different lunar mare regions in our study (Fig. 1). Mare regions were selected for this study as they offer, in general, a relatively homogeneous surface with little topographic tilts (e.g., crater rims) that could pose difficulties in the determination of the observation geometry. Each of the four regions covers several hundreds of square kilometers of lunar surface. When selecting the regions, we have required that they have been imaged by AMIE across a wide range of phase angles (α), including the opposition geometry. The phaseangle range covered is 0–109°, with incidence and emergence angles as counted from the outward normal vector (ι and ϵ) ranging within 7–87° and 0–53°, respectively. The pixel scale varies from 288 m down to 29 m during the extended mission phase ended by the SMART1 spacecraft crashing into the lunar surface on September 3, 2006. Note that offnadirpointing observations have allowed for the extensive phaseangle coverage. In total, 220 images are used for the present study.
Fig. 1 The lunar regions observed by AMIE onboard ESA SMART1 (Advanced Moon microImager Experiment) and analyzed in the present study overlaid on the Clementine albedo map: 1) Oceanus Procellarum, around Reiner Gamma; 2) Oceanus Procellarum, between Mons Rümker and the Mairan crater; 3) Mare Imbrium, north of Copernicus crater; and 4) Mare Serenitatis. 

Open with DEXTER 
Biases and dark currents were subtracted from the images in the usual way, followed by a flatfield correction. New darkcurrent reduction procedures have recently been derived from inflight measurements to replace the groundcalibration images that were rendered practically useless by the large radiation dose that AMIE experienced during the 13month journey of SMART1 to the Moon (Grieger 2008). The clear (or panchromatic) filter was chosen for the present study as it provides the largest field of view, consisting of 512 × 512 pixels, and usually also the best signaltonoise ratio.
Since the surface footprint of a single AMIE image could span several degrees in selenocentric longitude and latitude, the illumination geometry, that is, ι, ϵ, α, and the azimuth angle between the incident and emergent directions φ, was computed separately for each pixel in the images used in the photometric analysis by utilizing the NASA Navigation and Ancillary Information Facility SPICE software toolkit with the latest and corrected SMART1 AMIE SPICE kernels.
The photometric data points were extracted as follows. First, on average, 50 sample areas of 10 × 10 pixels were chosen by hand from each image, to exclude large craters and albedo anomalies from the analysis. Second, the surface normal, ι, ϵ, φ, and α were computed for each pixel in each sample area. Finally, the illumination angles and the observed intensity were averaged over each sample area. In total, the images used in the study resulted, for the four mare regions, in approximately 12 000 photometric sample points (Fig. 2). We note that the lunar opposition effect in Fig. 2 agrees in steepness over the phaseangle of 0.0–2.5° with the earlier lunar mare analysis using AMIE data by Kaydash et al. (2008).
Fig. 2 The multiangular photometry reduced from the lunar mare regions depicted in Fig. 1 as a function of the phase angle α. The opposition effect shows up as abrupt nonlinear brightening towards opposition. The viewing geometries of the AMIE lunar images are also depicted. Each geometry is fixed by three angles, for example, the angles of incidence and emergence (ι and ϵ) and the azimuth angle φ (that can be replaced by α). 

Open with DEXTER 
3. Theoretical and numerical methods
3.1. Scattering model for mare regions
Consider diffuse scattering of light from a semiinfinite particulate regolith with a rough interface towards free space. The reflection coefficient R relates the incident flux density πF_{0} and the emergent intensity I as (1)In order to model the scattering from the dark lunar mare regions, we utilize the LommelSeeliger reflection coefficient corrected for shadowing as (2)where is the singlescattering albedo of the volume element V, P_{V} is the normalized volumeelement scattering phase function, and S accounts for shadowing. For phase angle α = 0°, we have S ≡ 1.
Without the shadowing function S, the reflection coefficient in Eq. (2) coincides with the LommelSeeliger reflection coefficient, which is the firstorder multiplescattering solution to the radiative transfer equation for a semiinfinite planeparallel medium of scatterers. The reflection coefficient in Eq. (2) is applicable to dark media such as the lunar mare regolith: the contributions proportional to , k ≥ 2 are assumed negligible.
Consider next the definition of the volume element. It is well known that the radiative transfer theory succeeds in describing the angular characteristics of regolith reflection coefficients outside the opposition regime. This occurs in spite of the fact that the theory is known to be, strictly, inapplicable to closepacked regoliths of particles. This allows us to assume that the volume element V extends over a typical lunar particle and its immediate vicinity so that the size of the element is of the order of ~100 μm.
The geometric albedo p is the ratio of the diskintegrated brightness of an object and the diskintegrated brightness of a normally illuminated Lambertian disk in the exact backscattering direction α = 0°. Considering a spherical object with surface scattering characteristics described by the reflection coefficient in Eq. (2), we have (3)This coincides with the geometric albedo p_{V} of the volume element V: p_{V} = p.
3.2. Coherentbackscattering modeling for p, , and P_{V}
We model the lunar mare geometric albedo p (Eq. (3)) as well as the volumeelement albedo and phase function P_{V} (Eq. (2)) in terms of scalar coherent backscattering by a finite medium of fundamental scatterers (Muinonen et al. 2010). We denote the singlescattering albedo and phase function of the fundamental scatterers by and P_{0}. We further introduce the extinction mean free path ℓ = 1/k_{e} (k_{e} is the extinction coefficient) for the finite medium, which we typically express in a dimensionless form kℓ = 2πℓ/λ, where k and λ are the wave number and wavelength. We assume that the volume element V is spherical and that the radius of the volume element is 60 μm, mimicking a cubic element with an edgelength of 100 μm.
For describing the phase function P_{0}, we utilize double HenyeyGreenstein (HG) scattering phase functions P_{2HG}: (4)where θ = π − α is the scattering angle, g_{1} and g_{2} describe the forward and backward asymmetries, w is the normalized weight of the first HG term, and g is the asymmetry parameter of the phase function. The first HG term describes the common increase towards the forwardscattering direction, whereas the second term describes the increase from intermediate scattering angles towards the backwardscattering direction. Note that it remains as an open question what the increase towards the forwardscattering direction means in the case of fundamental scatterers constituting lunar particles much larger than the wavelength.
In the coherentbackscattering modeling, we require that the resulting lunar mare geometric albedos (Eq. (3)) be p < 0.2 and that the volumeelement singlescattering albedos be . Such singlescattering albedos are comparable to the corresponding albedos measured in the laboratory for single lunar analog particles large compared to the wavelength (Piironen et al. 1998). The geometricalbedo range for the mare regions has been estimated with the help of the study by Lumme & Irvine (1982).
3.3. Raytracing modeling for R
A numerical geometricoptics raytracing code has been developed to derive the volumeelement scattering phase function P_{V} and the combined mutual and roughsurface shadowing function S in Eq. (2) for a particulate medium mimicking the lunar regolith (Parviainen & Muinonen 2007, 2009). The code uses MonteCarlo ray tracing to compute the reflection coefficient R in Eq. (2) for a medium composed of opague spherical particles (Fig. 3).
For deriving P_{V} and S from the lunar photometry, we assume that the particle surfaces, that is, the surfaces of the volume elements, follow the LommelSeeliger scattering law (mathematical form equivalent to that in Eq. (2) without the shadowing function; e.g., Bowell et al. 1989). The scattering phase function for the fictituous particlesurface scatterers P_{∂V} is also modeled using a double HG phase function, P_{∂V} = P_{2HG}. To discriminate between the double HG parameters of the fundamental scatterers and those of the particlesurface scatterers, we utilize w_{∂V}, g_{1, ∂V}, and g_{2, ∂V} for P_{∂V}. The current shadowing modeling is an advanced version of earlier work in Muinonen et al. (2001) and Stankevich et al. (1999). According to our experience, the present parameterization of the volume element results in excellent fits to lunar mare photometry. We return to the physical interpretation of the modeling below.
Instead of limiting ourselves to the principal plane with varying ι and ϵ and φ = 0° or 180°, we have computed the shadowing function over the entire hemisphere, allowing us to account for azimuthal shadowing effects arising from varying φ. Recent studies (e.g., Shepard & Campbell 1998; Shkuratov et al. 2005; Parviainen & Muinonen 2007; Jehl et al. 2008) have demonstrated that azimuthal shadowing contributes significantly to the photometric response by a planetary regolith surface.
The simulation has consisted of two parts: the generation of the particulate media and the MonteCarlo ray tracing in the media. First, using a droppingbased randompacking method for five volume densities v = 0.2, 0.3, 0.4, 0.5, and 0.55, we have generated a medium consisting of 10^{6} spherical particles with mean radii of 1/500 of the width of the medium. Second, we have intersected the particulate media by twodimensional random fields to model the surface roughness with two parameters. We parameterize the fractionalBrownianmotion roughness with the Hurst exponent H and the standard deviation of heights of the whole field σ, while Gaussian roughness is parameterized by the correlation length l and σ. We thus describe the regolith analog media with three parameters v, H or l, and σ. Also, when considering media without external largescale surface roughness, we remove 20% of the medium in thickness from the surface in order to remove a density gradient arising from the dropping algorithm. For an example fBm surface, see Fig. 3.
Fig. 3 Porous random medium of spherical particles with fractionalBrownianmotion boundary surface: Hurst exponent H = 0.4, standard deviation of heights σ = 0.06, and volume density v = 0.35. 

Open with DEXTER 
The MonteCarlo simulation consists of ray tracing a large number of geometry samples from the medium for each photometric sample point. A single geometry sample consists of information on whether the sample point on the particle surface is illuminated and on the local angles of incidence and emergence as well as the local azimuth angle. Thus, the set of geometry samples represents the distribution of angles visible to the observer (the AMIE camera) for the given observation geometry. Now, the final model reflection coefficient R for a given photometric point can be obtained as a sum of the contributions computed from the illuminated geometry samples. This way of computing the observed angle distribution for a given observation geometry (instead of direct ray tracing for R) allows us to fit arbitrary reflectioncoefficient models to the photometric data without a need for computationally expensive additional ray tracing. For each photometric sample point, we have computed 50 geometry samples for 20 roughsurface realizations, yielding 1000 geometry samples in total.
Based on the work by Muinonen et al. (2001) and Stankevich et al. (1999), the shadowing function S can be assumed constant over the surface of an individual spherical particle. This allows us to discriminate between the two parts, P_{V} and S, in the reflection coefficient R derived by leastsquares fitting of the raytracing model to the lunar photometry. Since the numerical raytracing simulations utilize constituent spherical volume elements with LommelSeeliger surface scattering laws, the true volumeelement scattering phase function P_{V}(θ) is proportional to the product of P_{∂V}(θ) and an additional phaseangle dependent function that results from the integration of the LommelSeeliger cosine part over the surface of the spherical volume elements (this integration can be done analytically and independently of S as explained above): (5)The functional form of the shadowing function S now follows from Eq. (2) by dividing the model reflection coefficient by the phase function of Eq. (5) and the LommelSeeliger cosine part. Note that the particlesurface scattering phase function P_{∂V}(θ) emerges as an auxiliary computational tool to derive P_{V}(θ) and S.
3.4. Inversion
Inversion of the observations for the lunar surface properties consists of two steps. First, the shadowing function S and the volumeelement phase function P_{V} (in the relative sense) are derived with the help of the aforedescribed shadowing modeling for particulate media using leastsquares analysis of the observations. This step will give us P_{V} as well as the surface roughness parameters H and σ and the regolith porosity v. Second, the resulting P_{V} is explained using scalar coherentbackscattering modeling with realistic assumptions for the geometric albedo of the lunar mare regions or, in other words, . This step allows us to constrain the singlescattering albedo , the singlescattering phase function P_{0}, and the mean free path in the form kℓ for the finite medium of fundamental scatterers mimicking a lunar volume element.
Fig. 4 The multiangular AMIE photometry of the mare regions fitted using the fBmparticulatemedium model with H = 0.80, σ = 0.05, and v = 0.30, and a double HenyeyGreenstein particlesurface phase function with w_{∂V} = 0.001432, g_{1, ∂V} = −0.9228, and g_{2, ∂V} = −0.0546 (uppermost points). Also shown are the residuals shifted downward by 0.2 vertical units for clarity (lowermost points). See Table 1. 

Open with DEXTER 
Fig. 5 The lunar mare volumeelement phase function P_{V} (see Eqs. (2) and (5)). The phase function follows from dividing more than 12 000 SMART1 AMIE original data points by bestfit model S(μ,μ_{0},φ)/(μ + μ_{0}) in Eq. (2). 

Open with DEXTER 
4. Results and discussion
In order to derive the lunar mare volumeelement phase function, a bestfit solution to the photometric observations was searched from the computed fBm scattering models using the differentialevolution globaloptimization method (Storn & Price 1997). The effects due to different values of v, H, and σ were small but noticeable (Table 1). After the fit, the firstorder approximation for the volumeelement phase function was obtained from Eq. (5) based on the fit shown in Fig. 4. The volumeelement phase function is depicted in Figs. 5 and 6.
In detail, the bestfit fBm parameters are as follows: v = 0.30, H = 0.80, and σ = 0.05 with the rms value of 2.001 in the units of Fig. 2. For comparison, the bestfit Gaussian model gives v = 0.30, l = 0.50, and σ = 0.03 with the rms value of 2.083, whereas the socalled smooth model gives v = 0.50 with the rms value of 2.151. This offers a preliminary indication that the fBm model is indeed the leading model for the lunar regolith. Table 1 shows how the particlesurface phase function P_{∂V}(θ) and the volumeelement phase function P_{V}(θ) (Eq. (5)), as well as the rms values of the leastsquares fits vary as a function of the fBm parameters.
Bestfit particlesurface phase functions P_{∂V}(θ) modeled as double HenyeyGreenstein phase functions (HG) for media with fractionalBrownianmotion surface roughness.
The lunar volumeelement phase function exhibits a narrow backscattering intensity surge. The most notable result of the comparison of the photometric observations and numerical SM modeling is that SM does not explain the intensity surge, even for the most porous media considered in the study, v = 0.2. Note that the LommelSeeliger scattering law corrected for shadowing in Eq. (2) is largely validated by the fact that, after the division of the observational data with the model data for S(μ,μ_{0},φ)/(μ + μ_{0}), the remaining variation can be explained, to a precision compatible with the observational errors, by a function that depends on the phase angle α only. This function is then proportional to P_{V}(θ).
The resulting volumeelement phase functions differ from those experimentally measured for, e.g., large individual Saharan sand particles (Munoz et al. 2007). This underscores the difference of lunar volumeelement scattering characteristics as compared to those of compact, optically rather homogeneous particles with surface roughness.
Fig. 6 The lunar mare volumeelement phase function of Fig. 5 averaged using bin size of 0.5°. We show the bestfit volumeelement scattering phase function P_{V} (solid line). For the largest phase angles, there is a deviation that is probably due to the incapability of the present model volumeelement phase function to fit the observations and to increasing real brightness variegation or correlated observational errors. 

Open with DEXTER 
Fig. 7 The binned lunar mare volumeelement phase function with the corresponding coherentbackscattering modeling including a variation envelope. Double HG singlescattering phase functions for the fundamental scatterers give rise to coherentbackscattering peaks capable of matching the observations. For the largest phase angles, there is a deviation between the model and the observations potentially due to simplified modeling. 

Open with DEXTER 
Fig. 8 As in Fig. 7 for the phaseangle range of 0° ≤ α ≤ 30°. 

Open with DEXTER 
Fig. 9 Lunar mare singlescattering phase function P_{0} of the fundamental scatterers (see Eq. (4)) expressed as a double HG function with a variation envelope. 

Open with DEXTER 
The lunar backscattering intensity surge can be assigned to multiple interactions between the fundamental scatterers within the volume element V. We stress that this multiple scattering is presently included in what we call the lunar volumeelement phase function representing a scattering volume large enough to give rise to coherentbackscattering effects.
After carrying out thousands of scalar coherentbackscattering simulations for varying singlescattering albedos and phase functions of the fundamental scatterers, we have converged on computing volumeelement scattering characteristics using 31 singlescattering albedos , 0.61, 0.62, ..., 0.90 and 8 extinction mean free paths ℓ with kℓ = 240, 270, 300, ..., 450. We have utilized a total asymmetry parameter of g = 0.60, 0.65, 0.70, 0.75 or 0.8 in the double HG phase function. In full detail, for the forward and backwardscattering HG terms, we have assumed g_{1} = 0.95 and g_{2} = −0.33, –0.32, –0.31, ..., –0.17, respectively, with the weight factor for the forwardscattering term being determined by the total asymmetry parameter. Thus, we have used altogether 85 different singlescattering phase functions. For the final phase of coherentbackscattering modeling, we have thus computed altogether 21 080 different models for the lunar mare volume element.
The singlescattering albedo and total asymmetry parameter of the bestfit solution are and g = 0.60. In the HG phase function, we have g_{1} = 0.95 and g_{2} = −0.31, giving w = 0.722. The unitless mean free path is kℓ = 450. The resulting geometric albedo for the mare regions is p = 0.176. Figures 7 and 8 show the bestfit coherentbackscattering model with a variation envelope among the sequence of models for spherical media of fundamental scatterers mimicking the volume element in the lunar surface. For 180 data points of the binned volumeelement phase function with assumed observational error standard deviation of 0.03 for points with α < 60° and 0.1 for points with α ≥ 60°, the χ^{2} value of the best fit is 0.928. The variation envelope corresponds to models with χ^{2}values within a factor of 1.3 from the bestfit value. The threshold χ^{2}value has been chosen on the basis of the fluctuations of the HG models to offer a view of how well, overall, the models fit the observations. There are particularly good fits to the observational data all through the parameter phase space of , g, g_{1}, g_{2}, and kℓ and, given the heuristic characteristics of the present modeling, the bestfit solutions have been allowed to occur at the bounds of the parameters g and kℓ. Clearly, acceptable fits could be obtained with less forward scattering double HG phase functions (i.e., smaller gvalues) and longer mean free path lengths. Figure 9 shows the bestfit singlescattering phase function among the models utilized and its variation envelope corresponding to the aforedescribed χ^{2} analysis.
There is qualitative agreement between the chosen double HG function and the numerical lightscattering computations for wavelengthscale Gaussian random particles with the discretedipole approximation (Muinonen et al. 2007) as well as agglomerated debris particles (Zubko et al. 2006). In these cases, the phase functions show smooth increase of intensity towards the backwardscattering direction.
Note that the present HG solution (Fig. 9) resembles the phase function for solarsystem particles in Bowell et al. (1989, their Fig. 1). The phase functions from different sources should, however, be compared with caution as their interpretation can significantly differ. We stress that more detailed modeling including polarization is to be carried out in the future using the methods described in Muinonen et al. (2010), Muinonen (2004), and Boehnhardt et al. (2004).
5. Conclusions
Based on the present theoretical modeling of the lunar photometry from SMART1 AMIE, we conclude that most of the lunar mare opposition effect is caused by coherent backscattering within ~100μm volume elements comparable to lunar particle sizes, with only a small contribution from shadowing effects. We suggest that the lunar fundamental scatterers exhibit increase in scattered intensity towards the forward direction as well as the backward direction. These characteristics resemble those experimentally measured and theoretically computed for realistic single small particles in free space. Further interpretation for the phase functions of lunar mare volume elements and fundamental scatterers is beyond the scope of the present study.
We find that it is possible to derive information about submicrontomicronscale surface properties based on multiangular imaging of the target areas. We put forward a novel method where the stochastic surface geometry is derived from the imaging data, whereafter the reduced data allow the derivation of information on the smallscale physical properties. The present modeling paves the way to quantitative interpretation of the polarization ratios experimentally measured for lunar samples by Hapke et al. (1993). More generally, the modeling can be considered as a beginning of a synoptic approach to explaining all observational and experimental data on scattering by the lunar regolith.
The Moon has been imaged by AMIE onboard SMART1 with three color filters in addition to the panchromatic channel. A global diskresolved multifilter study similar to the one reported in Hillier et al. (1999) for Clementine images is possible and will remain as a topic for future studies. Note that Mare Serenitatis (area 4 in Fig. 1) is one of the Lunar International Calibration Targets (LISCT) proposed by Pieters et al. (2008).
Further theoretical and numerical development of the present methods can result in application to other lunar regions observed by various missions. In the recent years, the Moon has been studied by four space missions in addition to SMART1, namely Kaguya (Selene), Chandrayaan 1, Chang’e 1, and the currently operating Lunar Reconnaissance Orbiter and Chang’e 2. They have extensively mapped the lunar surface with various resolutions in UV/Vis wavelengths. As the calibrated photometric data from these missions is becoming available, applying the present theoretical modelling to the combined data from all possible lunar missions to obtain global diskresolved analyses of singlescattering phase functions and stochastic surface geometry is emerging as an attractive opportunity.
Acknowledgments
It is a pleasure to thank Kari Lumme, Jukka Piironen, Jani Tyynelä, Tatjana Tchumatchenko, Yurij Shkuratov, Dmitrij Stankevich, Evgenij Zubko, and Olli Wilkman for helping us to interpret the SMART1 AMIE observations of the Moon. We would like to thank Bruce Hapke for providing critical reviews on the present work. The research has been supported, in part, by the Academy of Finland, and in France by the French Space Agency CNES and PNP (Programme National de Planétologie). Patria Advanced Solutions contributed significantly to the Science Validation of the Flight Model AMIE Camera. The ESA SMART1 project and operations teams, and the STOC team are also acknowledged.
References
 Boehnhardt, H., Bagnulo, S., Muinonen, K., et al. 2004, A&A, 415, L21 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bowell, E. L. G., Hapke, B., Domingue, D., et al. 1989, in Asteroids II, ed. R. P. Binzel, T. Gehrels, & M. Matthews (Tucson: University of Arizona Press), 524 [Google Scholar]
 Foing, B. H., Racca, G. D., & Marini, A., et al. 2006, Adv. Space Res., 1, 6 [NASA ADS] [CrossRef] [Google Scholar]
 Grieger, B. 2008, ESA Technical Note, S1AMIESGSTN013 [Google Scholar]
 Hapke, B. 1984, Icarus, 59, 41 [NASA ADS] [CrossRef] [Google Scholar]
 Hapke, B. 1990, Icarus, 88, 407 [NASA ADS] [CrossRef] [Google Scholar]
 Hapke, B., & van Horn, H. 1963, J. Geophys. Res., 68, 4545 [NASA ADS] [CrossRef] [Google Scholar]
 Hapke, B. W., Nelson, R. M., & Smythe, W. D. 1993, Science, 260, 509 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Hillier, J. K., Buratti, B. J., & Hill, K. 1999, Icarus, 141, 205 [NASA ADS] [CrossRef] [Google Scholar]
 Jehl, A., Pinet, P., Baratoux, D., et al. 2008, Icarus, 197, 403 [NASA ADS] [CrossRef] [Google Scholar]
 Josset, J.L., Beauvivre, S., Cerroni, P., et al. 2006, Adv. Space Res., 1, 14 [NASA ADS] [CrossRef] [Google Scholar]
 Kaydash, V., Kreslavsky, M., Shkuratov, Y. G., et al. 2008, Lunar and Planetary Science XXXIX [Google Scholar]
 Kaydash, V., Kreslavsky, M., Shkuratov, Y., et al. 2009, Icarus, 202, 393 [NASA ADS] [CrossRef] [Google Scholar]
 Lumme, K., & Bowell, E. 1981, AJ, 86, 1694 [NASA ADS] [CrossRef] [Google Scholar]
 Lumme, K., & Irvine, W. M. 1982, AJ, 87, 1076 [NASA ADS] [CrossRef] [Google Scholar]
 Lumme, K., Peltoniemi, J. I., & Irvine, W. M. 1990, Transp. Theory Stat. Phys., 19, 317 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Mishchenko, M. I., & Dlugach, J. M. 1992, Astrophys. Space Sci., 189, 151 [NASA ADS] [CrossRef] [Google Scholar]
 Mishchenko, M. I., & Dlugach, J. M. 1993, Planet. Space Sci., 41, 173 [NASA ADS] [CrossRef] [Google Scholar]
 Muinonen, K. 1990, Ph.D. Thesis, University of Helsinki [Google Scholar]
 Muinonen, K. 2004, Waves in Random Media, 14, 365 [NASA ADS] [CrossRef] [Google Scholar]
 Muinonen, K., & Zubko, E. 2010, in Electromagnetic and Light Scattering XII, ed. K. Muinonen, A. Penttilä, H. Lindqvist, T. Nousiainen, & G. Videen (Helsinki: University of Helsinki), Conf. Proc., 194 [Google Scholar]
 Muinonen, K., Stankevich, D., Shkuratov, Y., Kaasalainen, S., & Piironen, J. 2001, J. Quant. Spec. Radiat. Transf., 70, 787 [NASA ADS] [CrossRef] [Google Scholar]
 Muinonen, K., Shkuratov, Y., Ovcharenko, A., et al. 2002, Planet. Space Sci., 50, 1339 [NASA ADS] [CrossRef] [Google Scholar]
 Muinonen, K., Zubko, E., Tyynelä, J., Shkuratov, Y., & Videen, G. 2007, J. Quant. Spec. Radiat. Transf., 106, 360 [NASA ADS] [CrossRef] [Google Scholar]
 Muinonen, K., Tyynelä, J., Zubko, E., & Videen, G. 2010, Light Scatt. Rev., 5, 477 [CrossRef] [Google Scholar]
 Munoz, O., Volten, H., Hovenier, J., et al. 2007, J. Geophys. Res., 112, [Google Scholar]
 Parviainen, H., & Muinonen, K. 2007, J. Quant. Spec. Radiat. Transf., 106, 398 [NASA ADS] [CrossRef] [Google Scholar]
 Parviainen, H., & Muinonen, K. 2009, J. Quant. Spec. Radiat. Transf., 110, 1418 [NASA ADS] [CrossRef] [Google Scholar]
 Peitgen, H., & Saupe, D. (eds.) 1988, The science of fractal images (New York: SpringerVerlag) [Google Scholar]
 Peltoniemi, J. I. 1993, J. Quant. Spec. Radiat. Transf., 50, 655 [NASA ADS] [CrossRef] [Google Scholar]
 Pieters, C. M., Head, J. W., Isaacson, P., et al. 2008, Adv. Space Res., 2, 248 [NASA ADS] [CrossRef] [Google Scholar]
 Piironen, J., Muinonen, K., Nousiainen, T., et al. 1998, Planet. Space Sci., 46, 937 [NASA ADS] [CrossRef] [Google Scholar]
 Pinet, P., Cerroni, P., Josset, J., et al. 2005, Planet. Space Sci., 53, 1309 [NASA ADS] [CrossRef] [Google Scholar]
 Racca, G. D., Marini, A., & Stagnaro, L., et al. 2002, Planet. Space Sci., 1323 [Google Scholar]
 Rougier, A. 1933, Ann. Obs. Strasbourg, 205 [Google Scholar]
 Shepard, M. K., & Campbell, B. A. 1998, Icarus, 134, 279 [NASA ADS] [CrossRef] [Google Scholar]
 Shkuratov, I. G. 1988, Kinematika i Fizika Nebesnykh Tel, 4, 60 [NASA ADS] [Google Scholar]
 Shkuratov, Y. G., Kreslavsky, M. A., Ovcharenko, A. A., et al. 1999, Icarus, 141, 132 [NASA ADS] [CrossRef] [Google Scholar]
 Shkuratov, Y. G., Stankevich, D. G., Petrov, D. V., et al. 2005, Icarus, 173, 3 [NASA ADS] [CrossRef] [Google Scholar]
 Stankevich, D., Shkuratov, Y. G., & Muinonen, K. 1999, J. Quant. Spec. Radiat. Transf., 63, 445 [NASA ADS] [CrossRef] [Google Scholar]
 Storn, R., & Price, K. 1997, J. Glob. Optim., 11, 341 [CrossRef] [MathSciNet] [Google Scholar]
 Zubko, E., Shkuratov, Y., Muinonen, K., & Videen, G. 2006, J. Quant. Spec. Radiat. Transf., 100, 489 [NASA ADS] [CrossRef] [Google Scholar]
All Tables
Bestfit particlesurface phase functions P_{∂V}(θ) modeled as double HenyeyGreenstein phase functions (HG) for media with fractionalBrownianmotion surface roughness.
All Figures
Fig. 1 The lunar regions observed by AMIE onboard ESA SMART1 (Advanced Moon microImager Experiment) and analyzed in the present study overlaid on the Clementine albedo map: 1) Oceanus Procellarum, around Reiner Gamma; 2) Oceanus Procellarum, between Mons Rümker and the Mairan crater; 3) Mare Imbrium, north of Copernicus crater; and 4) Mare Serenitatis. 

Open with DEXTER  
In the text 
Fig. 2 The multiangular photometry reduced from the lunar mare regions depicted in Fig. 1 as a function of the phase angle α. The opposition effect shows up as abrupt nonlinear brightening towards opposition. The viewing geometries of the AMIE lunar images are also depicted. Each geometry is fixed by three angles, for example, the angles of incidence and emergence (ι and ϵ) and the azimuth angle φ (that can be replaced by α). 

Open with DEXTER  
In the text 
Fig. 3 Porous random medium of spherical particles with fractionalBrownianmotion boundary surface: Hurst exponent H = 0.4, standard deviation of heights σ = 0.06, and volume density v = 0.35. 

Open with DEXTER  
In the text 
Fig. 4 The multiangular AMIE photometry of the mare regions fitted using the fBmparticulatemedium model with H = 0.80, σ = 0.05, and v = 0.30, and a double HenyeyGreenstein particlesurface phase function with w_{∂V} = 0.001432, g_{1, ∂V} = −0.9228, and g_{2, ∂V} = −0.0546 (uppermost points). Also shown are the residuals shifted downward by 0.2 vertical units for clarity (lowermost points). See Table 1. 

Open with DEXTER  
In the text 
Fig. 5 The lunar mare volumeelement phase function P_{V} (see Eqs. (2) and (5)). The phase function follows from dividing more than 12 000 SMART1 AMIE original data points by bestfit model S(μ,μ_{0},φ)/(μ + μ_{0}) in Eq. (2). 

Open with DEXTER  
In the text 
Fig. 6 The lunar mare volumeelement phase function of Fig. 5 averaged using bin size of 0.5°. We show the bestfit volumeelement scattering phase function P_{V} (solid line). For the largest phase angles, there is a deviation that is probably due to the incapability of the present model volumeelement phase function to fit the observations and to increasing real brightness variegation or correlated observational errors. 

Open with DEXTER  
In the text 
Fig. 7 The binned lunar mare volumeelement phase function with the corresponding coherentbackscattering modeling including a variation envelope. Double HG singlescattering phase functions for the fundamental scatterers give rise to coherentbackscattering peaks capable of matching the observations. For the largest phase angles, there is a deviation between the model and the observations potentially due to simplified modeling. 

Open with DEXTER  
In the text 
Fig. 8 As in Fig. 7 for the phaseangle range of 0° ≤ α ≤ 30°. 

Open with DEXTER  
In the text 
Fig. 9 Lunar mare singlescattering phase function P_{0} of the fundamental scatterers (see Eq. (4)) expressed as a double HG function with a variation envelope. 

Open with DEXTER  
In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.