A new look at photometry of the Moon

Icarus 208 (2010) 548–557

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A new look at photometry of the Moon Jay D. Goguen a,*, Thomas C. Stone b, Hugh H. Kieffer b,1, Bonnie J. Buratti a a b

Jet Propulsion Laboratory, MS 183-401, Pasadena, CA 91109, United States US Geological Survey, 2255 N. Gemini Dr., Flagstaff, AZ 86001, United States

a r t i c l e

i n f o

Article history: Received 23 January 2010 Revised 16 March 2010 Accepted 28 March 2010 Available online 1 April 2010 Keywords: Radiative transfer Moon, Surface Regoliths Photometry Spectrophotometry

a b s t r a c t We use ROLO photometry (Kieffer, H.H., Stone, T.C. [2005]. Astron. J. 129, 2887–2901) to characterize the before and after full Moon radiance variation for a typical highlands site and a typical mare site. Focusing on the phase angle range 45° < a < 50°, we test two different physical models, macroscopic roughness and multiple scattering between regolith particles, for their ability to quantitatively reproduce the measured radiance difference. Our method for estimating the rms slope angle is unique and model-independent in the sense that the measured radiance factor I/F at small incidence angles (high Sun) is used as an estimate of I/F for zero roughness regolith. The roughness is determined from the change in I/F at larger incidence angles. We determine the roughness for 23 wavelengths from 350 to 939 nm. There is no significant wavelength dependence. The average rms slope angle is 22.2° ± 1.3° for the mare site and 34.1° ± 2.6° for the highland site. These large slopes, which are similar to previous ‘‘photometric roughness” estimates, require that sub-mm scale ‘‘micro-topography” dominates roughness measurements based on photometry, consistent with the conclusions of Helfenstein and Shepard (Helfenstein, P., Shepard, M.K. [1999]. Icarus 141, 107–131). We then tested an alternative and very different model for the before and after full Moon I/F variation: multiple scattering within a flat layer of realistic regolith particles. This model consists of a log normal size distribution of spheres that match the measured distribution of particles in a typical mature lunar soil 72141,1 (McKay, D.S., Fruland, R.M., Heiken, G.H. [1974]. Proc. Lunar Sci. Conf. 5, Geochim. Cosmochim. Acta 1 (5), 887–906). The model particles have a complex index of refraction 1.65–0.003i, where 1.65 is typical of impact-generated lunar glasses. Of the four model parameters, three were fixed at values determined from Apollo lunar soils: the mean radius and width of the log normal size distribution and the real part of the refraction index. We used FORTRAN programs from Mishchenko et al. (Mishchenko, M.I., Dlugach, J.M., Yanovitskij, E.G., Zakharova, N.T. [1999]. J. Quant. Spectrosc. Radiat. Trans. 63, 409–432; Mishchenko, M.I., Travis, L.D., Lacis, A.A. [2002]. Scattering, Absorption and Emission of Light by Small Particles. Cambridge Univ. Press, New York. ) to calculate the scattering matrix and solve the radiative transfer equation for I/F. The mean single scattering albedo is x = 0.808, the asymmetry parameter is hcos Hi = 0.77 and the phase function is very strongly peaked in both the forward and backward scattering directions. The fit to the observations for the highland site is excellent and multiply scattered photons contribute P80% of I/F. We conclude that either model, roughness or multiple scattering, can match the observations, but that the strongly anisotropic phase functions of realistic particles require rigorous calculation of many orders of scattering or spurious photometric roughness estimates are guaranteed. Our multiple scattering calculation is the first to combine: (1) a regolith model matched to the measured particle size distribution and index of refraction of the lunar soil, (2) a rigorous calculation of the particle phase function and solution of the radiative transfer equation, and (3) application to lunar photometry with absolute radiance calibration. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction

* Corresponding author. Fax: +1 818 354 5148. E-mail address: [email protected] (J.D. Goguen). 1 Present address: Celestial Reasonings, 180 Snowshoe Lane, Genoa, NV 89411-1057, United States. 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.03.025

Analysis of the change in brightness of the lunar surface as the Moon moves through its phases is one of the classical problems in planetary astronomy. The Moon is one of the most studied of planetary surfaces and the only one for which we have well documented samples from known provenance returned to Earth. Indepth understanding of the process of light scattering in the lunar

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regolith is at the heart of reliable quantitative interpretation of remote sensing radiance measurements acquired from telescopes and spacecraft for all planetary surfaces. This is not a review paper and we cite only a small fraction of the many important published works on lunar photometry. We encourage interested readers to explore the references in the papers cited here. To place our research within the context of prior work, it is useful to consider two distinct classes of models based on different assumptions. One class of models which we will call ‘fractal’ models assumes a hierarchical structure of scattering elements. These include Shepard and Campbell (1998), Helfenstein and Shepard (1999), Shkuratov and Helfenstein (2001), and Shkuratov et al. (2003). As evidenced in these papers, fractal models can successfully reproduce aspects of lunar photometry. For example, Shkuratov et al. (2003) show that in the limit of an infinite number of self-similar hierarchical levels, the distribution of brightness on the disk of a spherical planet will obey Akimov’s formula (Akimov et al., 2000) which Kreslavsky and Shkuratov (2003) use to analyze lunar photometry from the Clementine mission. Shepard and Campbell (1998) contend that photometric estimates of the roughness of fractal surfaces will always be dominated by the ‘‘smallest faceted scale”. A second class of models treats the surface as an optically thick layer composed of a size distribution of particles and assumes that the scattered radiance is related to the solution of the radiative transfer equation (RTE) (Chandrasekhar, 1960). This model is an extension of the techniques used for the study of planetary atmospheres (e.g. Hansen and Travis, 1974) to surfaces where the particles are in physical contact. Modifications of the RTE solution to account for the effects of particle proximity, e.g. particle–particle shadowing to explain the opposition surge (Hapke, 1993) and related ‘‘structure factors” (Mishchenko, 1994), are often included. To account for the effects of macroscopic topography including mountains and craters, a roughness model is typically employed (Hapke, 1993; Buratti and Veverka, 1985; Shkuratov et al., 2000). Helfenstein and Veverka (1987) applied a version of Hapke’s model to photometry of the Moon to estimate particle albedo (a key parameter influencing multiple scattering), macroscopic roughness and several other parameters. One class of these approaches is not ‘‘right” and the other ‘‘wrong”. Both have their advantages and disadvantages and which class of model is appropriate depends on the nature of the investigation, somewhat analogous to particle and wave models of an electron. In this paper, we are focused on the class of RTE-based models for scattering from a particulate lunar regolith to better understand how the particle size distribution, composition and topography influence the radiance. From this point on, we focus on the RTE class of models, but we fully acknowledge that there exist other approaches, like the fractal models, that have much to contribute to the subject. This paper investigates in detail one intriguing aspect of lunar photometry: the difference between the brightness of a fixed location on the Moon at the same phase angle measured before full Moon and after full Moon. We model high fidelity measurements of this difference using two different plausible physical explanations, but restrict our analysis to the class of RTE-based particulate regolith models.

2. Lunar photometry and the single scattering solution of the radiative transfer equation A good starting point for this study is the exact solution of the radiative transfer equation for photons that been scattered only once (Chandrasekhar, 1960). Choosing his convention that the incident beam flux is pF through the plane perpendicular to the

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incident beam direction, the single scattering solution for the radiance (intensity) I is

I ¼ ðx=4Þðl0 =ðl þ l0 ÞÞPðaÞF

ð1Þ

where l0 = cos(i), l = cos(e), i and e are the angles between the normal to the surface and the direction to the incident beam and the observer, respectively. The single scattering albedo x is defined as the probability that a photon is scattered (not absorbed) and the single particle phase function P(a) describes the probability that a photon is scattered in the direction a, the phase angle, defined as the angle between the direction to the incident beam and the direction to the observer. For scattering from any surface, Hapke (1993) refers to the quantity I/F as the ‘‘radiance factor”; for a Lambert surface I/F = l0. As seen from Earth, all points on the lunar surface are at very nearly the same phase angle which defines the ‘phase of the Moon’: full Moon corresponds to a = 0, first and last quarters to a = 90°, and new Moon to a = 180°. For x  1 the contribution to I/F from photons scattered more than once (multiple scattering) is small and a low albedo surface will approach the eq. (1) solution. The quantity (I/F) (l + l0)/l0 will be a function of a only. Hillier et al. (1999) used this fact to summarize lunar photometric results from the Clementine mission. Kieffer and Stone (2005) report the results of an extensive program of precisely calibrated lunar photometric imaging obtained with the RObotic Lunar Observatory (ROLO) on site at USGS Flagstaff, Arizona. They made a small subset of this data, the ‘chips’ data set, available for this work. We used two of the chips sites representative of the lunar highlands (Site 9, 17.21° lat., 20.01° lon.) and maria (Site 0, Mare Serenitatis, 19.06° lat., 20.47° lon.) for this study which are at nearly the same longitude. Hereafter ‘‘highlands” will be used to refer to ROLO Site 9 and ‘‘mare” refers Site 0. Fig. 1 plots (I/F) (l + l0)/l0 as a function of a for the filter 23 (553 nm nominal wavelength, width 18.1 nm) ROLO data for these two sites. For both sites, the before full Moon (BFM, waxing phases) data follow a familiar trend commonly observed for many planetary surfaces: an approximately log-linear (i.e. constant magnitudes/° ‘phase coefficient’) decrease with increasing phase angle for a > 15°. The after full Moon (AFM, waning phases) data for these same sites in Fig. 1 do not retrace the BFM data as would be expected if Eq. (1) held, but are systematically lower at the same a with the difference increasing with increasing a. This means there is a l and l0 dependence that is not captured by Eq. (1). For these two sites, the Sun is closer to vertical illumination (smaller i) BFM than AFM while e remains nearly constant for all phases (Table 1). In the following sections, two quantitative physical models for this effect will be tested. The restricted phase angle range 45° < a < 50° will be used for these tests because: (1) this range is far removed from the a < 15° ‘opposition’ region where coherent backscatter and the shadow-hiding opposition effect can influence I/F; (2) a can be thought of as nearly constant over this narrow range so that the precise knowledge of the a-dependence of some quantities is not a consideration; and (3) the before and after full Moon difference effect is large and well-constrained by the ROLO photometry. Table 1 gives the 45° < a < 50° ROLO data subset and scattering geometries for the filter 23 observations at both sites. These data characterize the effect investigated here and are used in many of the figures. The ROLO data set is rich not only in the density of the time sampling over many years, but also in the range and number of wavelengths sampled. The full data set includes 23 filters with the VNIR camera and an additional 9 filters with the SWIR camera (Kieffer and Stone, 2005). For the purposes of this study, we use only the VNIR data set.

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Fig. 1. ROLO photometry (filter 23, 553 nm) (I/F) (l + l0)/l0 for representative lunar highlands (top) and mare (bottom) sites. Note the divergence with increasing phase angle between the before (+ symbols) and after (filled circles) full Moon observations. The origin of this divergence is the topic of this paper. The dotted lines indicate the region between 45° and 50° phase angle where the before and after full Moon difference is large and well characterized by the observations.

3. Physical model 1: macroscopic roughness One obvious potential explanation for the radiance difference BFM and AFM is the macroscopic roughness of the surface, where macroscopic is understood to refer to any topography whose scale is greater than the larger of the scale of the particles comprising the surface or the wavelength of the light. Slopes that are tilted away from the incident beam contribute less to the radiance and may be shadowed from the illumination completely. Slopes tilted towards the observer may contribute more to the radiance. The net effect integrated over all slope orientations is a l, l0 and a dependence that can differ from that of Eq. (1). Numerous studies have tackled the problem of scattering of light by rough surfaces, including Cox and Munk (1954), Kivelson and Moszkowski (1965), Saunders (1967), Smith (1967), Buratti and Veverka (1985), Hapke (1993), Shkuratov et al. (2000) and references cited by these authors. For this study, we use the formulation detailed in van Ginneken et al. (1998) which builds on some of these earlier works. They describe a complete model for the radiance scattered from a randomly rough Gaussian surface described by a single parameter, the rms slope. Appendix A summarizes the relevant equations from van Ginneken et al. (1998) as implemented here and consistent with the notation and conventions adopted in this paper. An important advantage of the van Ginneken et al. model is the explicit integration over the radiance contributions from slopes at each orientation which avoids the additional approximation of ‘‘effective incidence and emission angles” in Hapke (1993). The first step in applying this model to the ROLO data shown in Fig. 1 is to select a phase angle bin, e.g. 45° < a < 50°, then fit a line to the BFM data within that narrow bin. Specifically, we fit

ðI=FÞðl þ l0 Þ=l0 ¼ 2A  100:4ba

ð2Þ

where A quantifies the intercept and b is the phase coefficient in magnitudes/degree. We then use Eq. (2) as the initial model I/F for each slope facet in the van Ginneken et al. roughness model.

We assume that the roughness has a negligible effect on the BFM data when the Sun is highest (i is smallest) and revisit this assumption later. Note that Eq. (2) is equivalent to a large number of possible Hapke models with a small contribution from multiple scattering and no roughness that can fit I/F over the limited 45° < a < 50° bin. The parameter A incorporates the effects of all combination of Hapke parameters that give the measured mean I/F at a = 47.5° (the center of the bin) and b includes all combinations of phase function choices and opposition surge parameters that reproduce the observed a dependence across the narrow bin. Here we use the van Ginneken et al. model for the effects of roughness, but we will show that the conclusions regarding the roughness are similar for either the van Ginneken et al. or the Hapke (1993) formulation of the roughness. Using Eq. (2) for the radiance contribution from each slope facet, we then use the van Ginneken roughness model to fit for the rms slope angle (and its error) that minimizes sum of the squares of the differences between the measured and modeled I/F for the l, l0 and a values of each observation (BFM and AFM) within the phase angle bin. If this roughness model is realistic, we should get the same rms slope angle independent of which phase angle bin is used for the analysis. Fig. 2 shows that this is indeed the case. For 5° wide phase angle bins spanning the range 10° < a < 65° the model fit rms slope angle is the same within the errors. The error in the rms slope angle is smallest for bins in the 40° < a < 50° range where the divergence of the BFM and AFM measurements is largest and the data sampling is dense. The remainder of this paper will adopt the 45° < a < 50° (dashed lines in Fig. 1) as the focus range for testing models of the BFM and AFM radiance difference. Fig. 3 shows the initial fit of the roughness model to the observations for the highland and mare sites for the focus range 45° < a < 50°. Parameter values for all roughness fits are detailed in Table 2. The roughness model can reproduce the observed radiance difference, but a close examination of Fig. 3 reveals that the

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Table 1 ROLO data subset, filter 23, 45° < a < 50° (all angles measured in degrees, a < 0° is before full Moon). Site 0

Site 9

i

e

a

I/F

i

e

a

I/F

30.04 29.81 29.48 29.28 27.94 65.28 65.57 66.68 66.92 72.89 73.11 73.34 73.49 73.73 73.20 73.44 73.68 73.91 28.30 28.08 27.89 27.72 27.13 26.94 26.75 26.58 65.56 65.78 66.04 66.25 66.50 66.75 67.00 67.25 38.41 38.16 37.96 68.81 69.04 69.30 27.64 27.45 27.25 26.88 26.69 26.49 26.31 26.13 25.94 25.75 25.57 66.44 66.66 68.34 68.55 68.79 69.04 69.26 69.48 69.72

35.16 35.23 35.35 35.43 33.68 18.66 18.73 31.01 31.04 32.57 32.64 32.72 32.77 32.86 28.39 28.46 28.54 28.62 30.52 30.60 30.67 30.74 30.99 31.07 31.15 31.22 21.43 21.48 21.54 21.60 21.67 21.75 21.83 21.91 28.66 28.68 28.70 29.01 29.09 29.18 27.80 27.88 27.97 28.16 28.26 28.37 28.46 28.56 28.65 28.74 28.82 26.17 26.23 26.76 26.83 26.90 26.97 27.03 27.08 27.13

48.77 48.60 48.34 48.17 49.52 49.60 49.85 46.64 46.80 45.38 45.51 45.65 45.74 45.87 46.90 47.08 47.24 47.41 49.77 49.58 49.41 49.26 48.76 48.59 48.42 48.25 47.83 48.05 48.30 48.48 48.69 48.90 49.10 49.29 47.95 47.76 47.60 45.04 45.16 45.31 47.68 47.51 47.33 47.02 46.86 46.69 46.53 46.37 46.20 46.01 45.84 45.75 45.96 47.33 47.48 47.66 47.85 48.02 48.19 48.39

0.0340 0.0332 0.0329 0.0333 0.0329 0.0178 0.0177 0.0187 0.0185 0.0147 0.0146 0.0145 0.0143 0.0142 0.0142 0.0139 0.0137 0.0135 0.0318 0.0323 0.0327 0.0329 0.0332 0.0334 0.0334 0.0337 0.0182 0.0181 0.0178 0.0176 0.0174 0.0175 0.0172 0.0170 0.0306 0.0307 0.0307 0.0179 0.0179 0.0177 0.0330 0.0333 0.0334 0.0336 0.0334 0.0336 0.0339 0.0340 0.0340 0.0344 0.0342 0.0189 0.0185 0.0171 0.0171 0.0168 0.0166 0.0164 0.0163 0.0160

28.59 28.34 27.98 27.76 28.43 65.36 65.65 65.53 65.78 71.36 71.58 71.82 71.96 72.21 72.52 72.76 73.00 73.24 29.51 29.29 29.10 28.93 28.35 28.16 27.98 27.80 65.91 66.13 66.40 66.61 66.85 67.10 67.36 67.61 36.60 36.34 36.11 67.28 67.51 67.77 28.73 28.53 28.34 27.97 27.78 27.58 27.40 27.22 27.03 26.84 26.67 66.62 66.84 68.51 68.73 68.97 69.22 69.44 69.67 69.91

27.05 27.18 27.36 27.46 31.55 28.18 28.17 20.96 21.05 29.75 29.85 29.95 30.01 30.11 33.47 33.53 33.58 33.63 33.70 33.78 33.84 33.90 34.09 34.15 34.20 34.25 27.70 27.67 27.63 27.62 27.60 27.60 27.60 27.61 18.16 18.27 18.35 27.11 27.21 27.32 34.83 34.88 34.93 35.02 35.07 35.12 35.17 35.21 35.26 35.30 35.33 26.58 26.55 26.60 26.63 26.68 26.72 26.76 26.80 26.84

48.77 48.60 48.34 48.17 49.52 49.60 49.85 46.64 46.80 45.38 45.51 45.65 45.74 45.87 46.90 47.08 47.24 47.41 49.77 49.58 49.41 49.26 48.76 48.59 48.42 48.25 47.83 48.05 48.30 48.48 48.69 48.90 49.10 49.29 47.95 47.76 47.60 45.04 45.16 45.31 47.68 47.51 47.33 47.02 46.86 46.69 46.53 46.37 46.20 46.01 45.84 45.75 45.96 47.33 47.48 47.66 47.85 48.02 48.19 48.39

0.0676 0.0657 0.0653 0.0660 0.0660 0.0353 0.0347 0.0353 0.0348 0.0294 0.0292 0.0287 0.0285 0.0280 0.0281 0.0273 0.0271 0.0263 0.0654 0.0661 0.0669 0.0673 0.0683 0.0683 0.0681 0.0693 0.0355 0.0347 0.0344 0.0338 0.0334 0.0335 0.0328 0.0326 0.0618 0.0620 0.0623 0.0358 0.0357 0.0355 0.0688 0.0694 0.0695 0.0701 0.0700 0.0703 0.0703 0.0708 0.0714 0.0721 0.0716 0.0356 0.0349 0.0321 0.0320 0.0314 0.0310 0.0305 0.0301 0.0299

highlands BFM model points lie systematically below the observations while the AFM model points lie systematically above the observations. The reason for this systematic trend is the assumption that the A, b values obtained by fitting Eq. (2) to the BFM data in the 45° < a < 50° range can be used to calculate the model I/F for a surface with zero roughness, i.e. the I/F for the slope facets in the roughness model. If the model rms slope angle were small enough, this approximation would work better, as it does for the mare site

Fig. 2. Initial fits (highlands site, filter 23) of the rms slope angle and errors (vertical error bars) to the BFM and AFM difference (Fig. 1) for phase angle bins whose width is indicated by the horizontal ‘error’ bars. This test shows that the estimates of the rms slope do not depend on the phase angle bin used, but phase angles from 40° to 50° give the smallest errors.

Fig. 3. ROLO filter 23 (I/F) (l + l0)/l0 (filled circles) and the initial fit of the roughness model (open circles) for the highlands and mare sites. Short lines connect the observed points to the associated model points calculated for the same l, l0 and a values. Notice that the highlands site BFM model points lie systematically below the observations while the AFM model points lie systematically above the observations. This is discussed in the text in more detail and corrected in the final roughness fits.

with the smaller rms slope angle. But for rms slope angles 30° like those for the highland site, the I/F is reduced from that for a zero roughness surface even for near zenith illumination (small i) and the approximation is not as good. The roughness reduces the model I/F for the BFM points from its starting values which were from a fit to the observations with the result that the model points lie below the observed points in Fig. 3. As the fit tries to increase the roughness to accommodate the AFM points, it reaches the chi-squared minimum prior to achieving the full amplitude of the radiance difference causing the AFM model points to lie above the observed points in Fig. 3. To correct for this effect, we calculate (I/F)smooth/(I/F)rough, where (I/F)smooth is the (Eq. (2)) fit to the BFM data that was used for the initial fit and (I/F)rough is calculated from the roughness model with

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Table 2 Macroscopic roughness fits to ROLO photometry in VNIR filters. ID

k (nm)

Site 0: Mare Serenitatis (19.06° lat., 20.47° lon.) 

16 17 18 2 19 4 20 21 6 22 8 23 10 24 25 12 26 27 14 28 29 30 31 Mean

350 355 405 412 414 441 465 475 486 544 549 553 665 693 703 745 763 774 865 872 882 928 939



Site 9: Highlands (17.21° lat., 20.01° lon.) 



hrms

herror

A

b mag./°

hrms

herror

A

b mag./°

24.38 24.87 23.47 21.37 23.50 21.62 24.56 22.80 21.48 22.49 22.87 22.05 21.99 21.65 21.78 20.96 22.31 22.07 19.42 21.96 21.69 19.34 20.47

2.89 3.07 2.88 3.21 3.04 2.52 3.34 2.76 2.91 2.39 3.37 2.38 2.40 2.26 2.80 3.09 3.43 2.18 2.57 2.50 2.74 5.11 6.18

0.0344 0.0399 0.0618 0.0637 0.0605 0.0545 0.0554 0.0601 0.0581 0.0679 0.0744 0.0681 0.0889 0.0943 0.0813 0.0986 0.0783 0.0866 0.0946 0.1114 0.1059 0.0870 0.0825

0.0137 0.0180 0.0204 0.0207 0.0212 0.0168 0.0166 0.0170 0.0156 0.0167 0.0179 0.0165 0.0159 0.0179 0.0146 0.0172 0.0117 0.0130 0.0140 0.0185 0.0168 0.0129 0.0130

33.54 35.05 34.56 26.17 34.66 26.93 34.91 34.64 24.23 35.30 36.64 34.50 32.30 34.97 35.05 30.64 35.42 35.15 24.17 35.36 35.21 30.45 30.59

3.03 2.54 2.49 5.39 2.29 5.37 2.61 2.58 4.53 3.95 3.02 2.18 4.25 2.51 2.34 3.93 3.37 2.35 5.51 2.51 2.49 4.85 5.68

0.0625 0.0732 0.1167 0.1165 0.1109 0.0973 0.1054 0.1170 0.1053 0.1580 0.1482 0.1371 0.1849 0.1755 0.1630 0.1673 0.1448 0.1652 0.1729 0.2090 0.2010 0.1598 0.1550

0.0113 0.0155 0.0181 0.0181 0.0183 0.0135 0.0144 0.0153 0.0127 0.0191 0.0165 0.0156 0.0166 0.0156 0.0141 0.0135 0.0096 0.0115 0.0113 0.0156 0.0141 0.0095 0.0102

22.18

1.26

34.09

2.56



hrms is the rms slope angle in degrees.  herror is the error in the rms slope angle in degrees. Filter 23 data (boldface) is featured in many of the figures.

the initial fit rms slope angle evaluated for the l, l0 and a values corresponding to the BFM geometry for the site at the center of the 45° < a < 50° range. The value of A is then increased (‘corrected’) by multiplying by (I/F)smooth/(I/F)rough and the fit iterated until convergence to the final corrected rms slope angle is reached. The final corrected fit of the roughness model to the ROLO filter 23 observations BFM and AFM is shown in Fig. 4 for the highland and mare sites. The correction process adjusted the (I/F)smooth model upward slightly from the BFM data points as indicated by the lines with only a very small correction (0.5%) for the less rough mare site and a significant but modest correction (3.3%) for the highland site. The systematic offsets between the modeled and observed points are gone.

We would like to emphasize that the van Ginneken et al. (1998) roughness model is the only model used here and that the zero roughness or ‘smooth’ I/F is determined directly from the BFM observations themselves which are selected for the minimum influence of roughness on I/F. The use of Eq. (2) requires only that the BFM data can be approximated by a straight line between the two vertical dashed lines in Fig. 1. The approach is to just find what rms slope angle will reproduce the BFM and AFM radiance difference. Specifically, no assumptions are needed or used regarding the particle albedo, phase function, porosity, etc. For macroscopic roughness on a scale that is large compared to the wavelength, as assumed by the roughness model used, the rms slope angle is expected to be independent of wavelength. Fits for

Fig. 4. Same as Fig. 3, except the open circles are the corrected fit of the roughness model to the ROLO filter 23 observations BFM and AFM. The lines are the corrected zero roughness model (Eq. (2)) after correcting the value of A as described in the text.

Fig. 5. Roughness model fits for the rms slope angle to the ROLO observations of the highlands and mare sites for each of the ROLO VNIR filters. The horizontal line and dashed lines show the weighted mean and error for all of the fits for each site. For macroscopic roughness on a scale large compared to the wavelength, the rms slope angle is expected to be independent of wavelength.

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the rms slope angle for each of the ROLO VNIR filters for both sites are plotted against the filter nominal wavelength in Fig. 5. The rms slope angle is indeed independent of wavelength across the whole VNIR range to the accuracy of this study. We find the weighted mean and error from the fits to all of the filters (see Table 2). The final values for the rms slope angles for the highland site (34.1° ± 2.6°) and mare site (22.2° ± 1.3°) are large and indicate very rough surfaces.

ferent model for the details of the roughness calculation, we use a new and different data set and a focused technique that does not depend on assumptions for other model parameters. Yet our analysis and the earlier work of others lead to the same conclusion. Interpretation of the BFM and AFM radiance difference (Fig. 1) as due to the effects of macroscopic roughness leads to the inescapable conclusion that it is the result of sub-mm scale micro-topography with large rms slope angles.

4. Context and implications of the roughness model results

5. Physical model 2: multiple scattering

Helfenstein and Shepard (1999) analyzed Apollo 11, 12 and 14 ALSCC stereo camera images of in situ lunar soils. They used direct parallax measurements from these image pairs to construct elevation maps of the 8 cm diameter image area with 85 lm spatial resolution. They analyze these stereo-derived elevation maps to determine the rms slope angle as a function of the size-scale (Eqs. (4)–(7) in Helfenstein and Shepard (1999)). Fig. 6 shows a version of their Fig. 9b modified to put the results of our roughness model fits in the context of their analysis. For both the highland and mare sites, the rms slope angles are consistent only with size-scales in the sub-mm range. This suggests that the roughness that photometry is sensitive to might be caused by the micro-topography of smaller particles clumping together, but that it has little to do with the visible craters and mountains that we normally associate with topography. Helfenstein and Veverka (1987) used an early version of Hapke’s photometric model to analyze photometry of the Moon and to estimate a slope parameter. Helfenstein and Shepard (1999) revisit that earlier work and revise the 8° estimate for the mare up to 24°. The highlands rms slope angle of 27° in Helfenstein and Shepard (1999) is smaller than our 34° estimate likely due in part to their inclusion of a modest contribution from multiple scattering for these brighter (A = 0.137) lunar terrains. They reach the same conclusion that the values of the slope parameter in Hapke’s (1993) model as derived from fits to lunar photometry refer to the sub-mm size-scale. Our analysis is completely independent from previous efforts to determine roughness from photometry of the Moon. We use a dif-

A different process that alters the l and l0 (and a) dependence of I/F from that in Eq. (1) is multiple scattering. To model whether multiple scattering between particles within the lunar regolith could reproduce the BFM and AFM radiance difference, we need to match the known particle size distribution of a typical lunar soil. McKay et al. (1974) give detailed particle size analyses of many lunar soil samples. For this analysis we selected 72141,1 a ‘‘typical mature soil” with graphic mean grain size Mz = 4.13u (57.1 lm) and inclusive graphic standard deviation rI = 2.08. The precise definitions and our use of these sedimentary petrology parameters are detailed in Appendix B. Following Hansen and Travis (1974, Eq. (2.60)), we define the log normal particle radius number distribution as

Fig. 6. A version of Fig. 9b from Helfenstein and Shepard (1999) modified to put the results of our roughness model fits in the context of their analysis of the Apollo ALSCC stereo camera images of in situ lunar soil. The two shaded areas mark the intersection of our weighted mean rms slope angle determinations (horizontal lines) for the highlands and mare sites with their characterization of the small scale lunar topography. Only sub-mm size-scales (for example, the vertical line) are compatible with the large rms slope angles derived from the BFM and AFM radiance difference.

2 2 1 nðrÞ ¼ pffiffiffiffiffiffiffi eðln rln rg Þ =ð2rg Þ 2prg r

ð3Þ

The two parameters for n(r) that result in a weight % distribution that matches Mz and rI for lunar soil 72141,1 are rg = 5.295  105 mm and rg = 1.448 (Appendix B). The resulting model weight % histogram is compared to the published histogram for 72141,1 in Fig. 7. The weight % distribution is dominated by the large particle tail of this broad size distribution due to the weighting by r3. Similarly, the light scattering cross-section is dominated by not quite as large particles due to the weighting by r2 and, as we shall see soon, reff  8 lm. The complex index of refraction, n  ik, at the wavelength of interest must be provided before I/F can be calculated for the model size distribution of particles. The finer particle sizes of lunar soils

Fig. 7. The measured weight % vs. particle diameter histogram for lunar soil 72141,1 (dashed lines, reproduced from McKay et al. (1974), Fig. 3c). The weight % histogram for the log normal particle number distribution that reproduces the measured Mz and rI for 72141,1 (solid lines) is shown for comparison. For reference, u = 10 corresponds to a 1 lm diameter particle.

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contain significant quantities of agglutinates and glass that are formed when impacts melt the soil and weld rock fragments together (McKay et al., 1974). Chao et al. (1970) report the Na D line (589 nm) index of refraction values for a large range of lunar glasses of impact origin. Although these n values span a large range (at least 1.57 < n < 1.75), a large portion of their samples concentrate around n = 1.65 and we adopt that value for our lunar soil model. It is more difficult to find a firm value for k, but a typical value for terrestrial basalts and basaltic glass at 550 nm wavelength is k  0.001 (Pollack et al., 1973). It is expected that k might be somewhat larger for lunar soil particles due to the inclusion of nm-scale pure Fe grains (Housley et al., 1973). In general, k can vary over many orders of magnitude for wavelengths near absorption bands and we choose to vary the precise value of k as the only free parameter in our lunar soil model. To calculate I/F for our lunar soil model, we use the set of three FORTRAN programs described in Mishchenko et al. (1999, 2002) that are available for downloading from http://www.giss.nasa.gov/staff/mmishchenko/brf/. For a size distribution of spherical particles, spher.f calculates the elements of the scattering matrix using Lorentz–Mie theory and saves the Legendre expansion coefficients. The second program, refl.f, calculates and saves the Fourier components of the reflection function. Finally, the third program, interp.f, calculates I/F for a specified set of scattering geometries. Table 3 gives the values of the parameters used in these codes and summarizes some of the relevant calculated results for our best fit model. For this study, we modeled the ROLO filter 23 photometry of the highlands site. Fig. 8 compares our best model fit to the observations in a similar format to Figs. 3 and 4. To achieve this excellent fit, we held the lunar soil model parameters rg, rg, and n fixed (Table 3), but increased k to 0.003 from its initial guess (0.001). Note that this model has only four parameters, three of which are set a priori to match relevant measured physical properties of Apollo soil samples, and employs a rigorous solution of the radiative transfer equation that respects the

Table 3 Parameters used and selected output from Mishchenko’s codes. Program

Parameter

Type

Value

spher.f

NMIE

Input

10,000

NDISTR AA BB

Input Input Input

2 5.295d-5 2.0967

R1 R2 LAM MRR MRI NK N DDELT NPNA REFF VEFF hCOSi ALBEDO

Input Input Input Input Input Input Input Input Input Output Output Output Output

1.d-5 0.1 0.55d-3 1.65 3.d-3 100 100 1.d-5 1801 8.16d-3 2.468 0.7682 0.80815

Output

1192

LMAX1

Input

2500

NG

Input Output

100

LMAX1

Input

2500

NG

Input

100

refl.f

interp.f

Units

Comments

mm

Max allowed Mie coefficients Log normal distribution rg  0.053 lm

Fig. 8. ROLO filter 23 photometry for the highlands site (filled circles: BFM, top; AFM, bottom). Model points (open circles) are the rigorous solution of the radiative transfer equation for the log normal number distribution that matches the measured weight % distribution for lunar soil 72141,1 (Fig. 7) and complex index of refraction 1.65–0.003i (see text). The contribution due to photons scattered once is indicated by the dashed line at bottom.

important physics. The fourth parameter k value that achieves the best fit is similar to that for terrestrial basalts and basaltic glasses and is also a physically reasonable value for the lunar soil mineralogy. The mean particle phase function, P(a), which is the integral over all particle sizes for this lunar soil model, is shown in Fig. 9. For once scattered photons, the scattering angle is H = 180°  a, but this relationship does not hold for subsequent scattering events. For continuity with the previous discussion, we plot P(a) – instead of the conventional P(H) – with the understanding that the distinction between a and H is clear. The mean albedo for single scattering by a particle is x = 0.808 and multiple scattering between particles contributes P80% of I/F in the focus 45° < a < 50° range of this study. The contribution due to single scattering (Eq. (1)) for the soil model x and P(a) is shown

r2g mm mm mm

mm

Smallest r = 0.01 lm Largest r = 100 lm Wavelength = 0.55 lm Real index n Imaginary index k # Gaussian pts/division # Divisions Convergence criterion P(a) at 0.1° intervals reff  8 lm Effective variance Asymmetry parameter x, single scattering albedo Max order Mie coefficients used All parameter statements # Gaussian pts refl.write 150 MB All parameter statements # Gaussian pts

Fig. 9. The mean particle phase function P(a) for the lunar soil model described in Table 2. Note that the ordinate is a logarithmic scale and the scattering is very anisotropic. The rigorous calculation for this anisotropic phase function shows that multiple scattering contributes P80% of I/F in the focus 45° < a < 50° range of this study (see Fig. 8).

J.D. Goguen et al. / Icarus 208 (2010) 548–557

as the dashed line in Fig. 8. Rigorous calculation of the multiple scattering by such anisotropic P(H) clearly plays an important role in the physical interpretation of I/F. This model for the lunar soil has limitations. It assumes, as many studies do, that the particles are spheres of uniform composition, but the lunar soil particles include angular rock fragments and breccias. Non-spheres tend to have P(a) with more light scattered towards a  90° (less extreme anisotropy) than equivalent spheres (Hapke, 1993; Mishchenko et al., 2000). This may be partly responsible for the different slopes of the data and model points in Fig. 8. No modification of the RTE solution has been made to accommodate near-field effects such as particle–particle shadowing. In the best interest of focusing this research to a manageable topic, we have only examined the restricted 45° < a < 50° phase angle range and it is unlikely that this model with these parameters will work as well over the full range of a. In summary, our physical model of the lunar soil that includes a rigorous calculation of multiple scattering can also reproduce the measurements of the before and after full Moon radiance difference. 6. Discussion and conclusions We have shown that two very different models, one with little multiple scattering and a very rough surface and another with no roughness but physically realistic multiple scattering, can both model the BFM and AFM radiance difference equally well. We conclude that macroscopic roughness and multiple scattering can both influence I/F in a similar way. They are strongly correlated ‘‘parameters” in the sense that any assumptions regarding one of them will influence the value deduced for the other. For example, if I/F varies as l0/(l + l0) as in Eqs. (1) and (2), as will be nearly true for any model where the particle albedo is small, additional l, l0 variation of I/F like the BFM–AFM difference can be modeled as increased surface roughness. Similarly, in the lunar soil model, by adjusting the parameter k – which affects both x and P(H) – to match the observed I/F, we are implicitly finding a solution for a flat regolith layer with no roughness. We know that the lunar surface is rough and that the ‘‘truth” lies somewhere between these two extreme cases. But even if both models were combined to calculate I/F for tilted facets where each facet consisted of the flat lunar soil multiple scattering model, the rms slope and k parameters would still be strongly correlated and any fitting procedure would get caught in a ‘‘chi-squared trough” where neither parameter is well-determined and suitable convergence elusive. To resolve this dilemma requires either additional knowledge of one of the two parameters so it can be held fixed, or sufficient suitable additional observations, e.g. more observation geometries, higher signal-tonoise ratio, etc., that can help to resolve the correlation. A brief discussion of the large difference between the particle albedo found from our multiple scattering model, x = 0.808, and the values reported for Hapke model fits to lunar photometry at a similar wavelength is in order. Helfenstein and Veverka (1987) report w = 0.33 for the highlands. Helfenstein and Shepard (1999) reanalyze the same data set with a ‘‘more sophisticated” version of Hapke’s model and report w = 0.40 for the highlands. The probability of a photon being scattered n times is wn, so for such low w, the contribution to I/F from each successive scattering decreases rapidly and their models can be considered consistent with the suite of Hapke models that are encompassed by Eq. (2). This is corroborated by the similarity of their roughness parameter values to ours. But the primary reason for the difference in particle albedo parameters between their study and ours is due to different definitions. Hapke (1993) treats diffracted light as if it were not scattered and their w has ‘‘diffraction subtracted” while Mishchenko et al. (1999) include diffracted light as scattered in the conventional

555

RTE definition of x (Chandrasekhar, 1960). We emphasize this distinction by using two different symbols. For particles large compared to a wavelength and in the far-field, the cross-section due to diffraction is equal to the particle area and contributes 0.5 to x. For particles at the wavelength scale and smaller, ‘‘diffraction subtracted” is poorly defined. We conclude that: (1) The unique ROLO data set of lunar images (Kieffer and Stone, 2005) contains a wealth of new and precise absolute photometric measurements of the Moon that can be used to advance our knowledge of how light interacts with the lunar regolith. (2) Two independent and very different physical models, macroscopic roughness and multiple scattering, can reproduce the observations of the BFM and AFM radiance difference at a fixed location on the lunar surface. (3) Using a new technique, our estimates of the photometric roughness for highland and mare sites (Table 2) improve on, and are consistent with, previous studies. All photometrically determined rms slope angles to date are so large that they must refer to micro-topography on a sub-mm spatial scale. (4) A physical model of scattering from a flat regolith layer with the particle size distribution of a typical mature lunar soil and realistic complex index of refraction (1.65–0.003i) can also match the observations of the BFM and AFM radiance difference. A key component of this model is a solution of the radiative transfer equation including the rigorous treatment of all orders of scattering. (5) P80% of I/F can be contributed by multiple scattering between anisotropically scattering particles with x = 0.808 and asymmetry parameter hcos Hi = 0.77, yet still be consistent with ‘dark’ surfaces like the lunar regolith. (6) The surface of the Moon is rough and this roughness has a photometric signature, but unless multiple scattering by strongly anisotropic P(H) is carefully modeled, the photometric signature of the roughness can be masked – and vice versa to some extent. Clearly both phenomena contribute. It seems likely that the relatively small rms slopes of the crater and mountain large scale topography will have a smaller effect on I/F than multiple scattering, except perhaps near the limb and/or terminator. (7) Physical models of the lunar regolith, like the one used for our multiple scattering model, show potential for determining regolith physical properties from photometry and merit vigorous pursuit.

Acknowledgments We would like to acknowledge M. Mishchenko for making his radiative transfer software publicly available and constructive reviews from Yu. Shkuratov and an anonymous referee. This research was supported in part by grants from the NASA Planetary Science Research Program. Appendix A. Summary of the roughness model equations This appendix summarizes the equations used from the Gaussian surface roughness model in van Ginneken et al. (1998). Refer to that reference for a more thorough derivation. For clarity, we adopt their symbols where possible. Light is incident at an angle i and the surface is viewed at an angle e, both measured relative to the vertical z axis. The azimuth of

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the viewer ur is measured counter-clockwise from the x axis (as viewed from above) which is normal to z and in the plane defined by z and i. These three angles define the photometric geometry and are related to the phase angle a by

cos a ¼ cos i cos e þ sin i sin e cos ur

ðA1Þ

(We note that this azimuth convention differs by 180° from that used in Mishchenko et al. (1999) and many radiative transfer texts which results in a change of sign between the two terms on the right hand side.) Each point on the rough surface is characterized by the tilt angle ha and tilt azimuth ua that describe the direction of local normal to the rough surface at that point. The probability distribution of tilt (slope) angles is 2

Pa ðha ; rÞdha ¼

sin ha e r2 cos3 ha

 tan ha 2r2 dha

ðA2Þ

Piv ði; e; /r ; rÞ  1=f1 þ Kðr; max½i; eÞ þ nKðr; min½i; eÞg

4:41ur 4:41ur þ 1

ðA4Þ

and 2    cot i r 1 cot i Kðr; iÞ ¼ pffiffiffiffiffiffiffi e 2r2  erfc pffiffiffi 2 2p cot i 2r

ðA5Þ

Shkuratov et al. (2000) have published a solution for this problem that avoids the approximation in Eq. (A4). I/F for the rough surface is obtained by integrating the contributions of each illuminated and visible surface element over all slope angles and azimuths. For i > e 4 Z X m¼1

tm

sm

i>e 1 2 3 4 e>i 5 6 7 8

sm

tm

am

bm

0

p/2  i p/2  e

p/2  i p/2  e p/2

s3

hk

p u2  p max[u2  p, ur + u3  p] u2

p p  u2 min[p  u2, p + ur  u3] u3  2 p

s1 s3 s2 s3

t2 t1 t3 t4

a1 ur + u3  p a3 a4

b1 ur  u3 + p b3 b4

where

cos u2 ¼ cot i cot ha cos u3 ¼ cot e cot ha and hk is the solution of

Z bm 

2A100:4ba

am

 cos h0i cos h0r dua  Pa ðha ; rÞdha 0 0 cos hr þ cos hi cos ha cos e 2p

Appendix B. The sedimentary petrology description of lunar soil grain size distributions

ðA3Þ

where an approximation is introduced through the expression for the parameter n

I=F ¼ Piv ði; e; /r ; rÞ

m

arccosðcot i cot hk Þ þ arccosðcot e cot hk Þ ¼ ur :

where the local slope is tan ha and r is the rms slope that characterizes the rough surface. For r < 1, the probability that a point is both illuminated and visible is

n¼

Table A1 Integration limits in Eq. (A6) for the roughness model.

ðA6Þ

Here h0i and h0r are the incidence and emission angles relative to the local surface element normal and are given by

cos h0i ¼ cos ua sin i sin ha þ cos i cos ha

ðA7Þ

cos h0r ¼ cosðua  ur Þ sin e sin ha þ cos ha

ðA8Þ

Note that the quantity in square brackets in Eq. (A6) is the contribution to I/F from each facet of the surface according to Eq. (2) for the local scattering geometry relevant to that tilted facet. Other scattering laws can be investigated by substituting for the expression in square brackets in Eq. (A6). The integration limits sm, tm, am and bm are given in Table A1. For the case e > i, Eq. (A6) is evaluated for the sum over m = 5–8 and the corresponding integration limits for m = 5–8 in Table A1. These limits were determined by combining the conditions for self-shadowing and self-masking of slopes as described in van Ginneken et al. (1998), but the explicit expressions in Table A1 are our contribution. This formulation as a sum of integrals takes into account the two disconnected ranges of ua that occur when m = 4 or 8. The double integrals in Eq. (A6) were evaluated numerically using the int_2d function in IDL.

The standard sedimentary petrology technique used to analyze the grain size distribution of the Apollo lunar soil samples was to sieve them through a stack of graded sieves, weigh the contents of each sieve, and plot the cumulative weight % vs. u, where u = log2 Dmm, and Dmm is the grain diameter in mm. As defined in Folk and Ward (1957), the first two moments of the size distribution are the graphic mean grain size Mz and the inclusive graphic standard deviation rI

Mz ¼ ðu16 þ u50 þ u84 Þ=3

ðB1Þ

and

rI ¼ ðu84  u16 Þ=4 þ ðu95  u5 Þ=6:6

ðB2Þ

where u50 is the diameter for which 50% of the weight is in larger particles, etc. Because light scattering by grains depends on the particle cross-section (area) and not the mass (volume), we need to find the log normal particle radius number distribution (n(r) in Eq. (3)) that will give the mass distribution corresponding to a measured Mz and rI. Using exactly the same procedure used for lunar soils, i.e. constructing the cumulative weight % distribution from r3 n(r) and using the definitions in Eqs. (B1) and (B2) (and assuming uniform density and spherical particles), we find the two parameters for the log normal n(r) in Eq. (3) that result in a weight % distribution that matches Mz and rI for lunar soil 72141,1. These are rg = 5.295  105 mm and rg = 1.448. The small size of rg = 0.05 lm should not concern the reader because rg is only used as a log normal number distribution parameter; actual particles of this scale and smaller do not contribute significantly to either the weight % or the scattering cross-section.

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