A photometric function for analysis of lunar images in the visualand infrared based on Moon Mineralogy Mapper observations

M. D. Hicks,1 B. J. Buratti,1 J. Nettles,2 M. Staid,3 J. Sunshine,4 C. M. Pieters,2 S. Besse,4

and J. Boardman5

Received 31 August 2010; revised 22 December 2010; accepted 9 February 2011; published 16 June 2011.

[1] Changes in observed photometric intensity on a planetary surface are caused byvariations in local viewing geometry defined by the radiance incidence, emission, andsolar phase angle coupled with a wavelength‐dependent surface phase function f (a, l)which is specific for a given terrain. In this paper we provide preliminary empiricalmodels, based on data acquired inflight, which enable the correction of Moon MineralogyMapper (M3) spectral images to a standard geometry with the effects of viewing geometryremoved. Over the solar phase angle range for which the M3 data were acquired ourmodels are accurate to a few percent, particularly where thermal emission is notsignificant. Our models are expected to improve as additional refinements to thecalibrations occur, including improvements to the flatfield calibration; improved scatteredand stray light corrections; improved thermal model corrections; and the computation ofmore accurate local incident and emission angles based on surface topography.

Citation: Hicks, M. D., B. J. Buratti, J. Nettles, M. Staid, J. Sunshine, C. M. Pieters, S. Besse, and J. Boardman (2011),A photometric function for analysis of lunar images in the visual and infrared based on Moon Mineralogy Mapper observations,J. Geophys. Res., 116, E00G15, doi:10.1029/2010JE003733.

1. Introduction

[2] The Moon Mineralogy Mapper (M3) imaging spec-trometer was one of two NASA‐provided instruments on theChandrayaan‐1 Indian Space Research Organization (ISRO)spacecraft. Launched in 2008, M3 is an imaging spectrom-eter covering the visible and infrared spectral range (∼0.4 to∼3 mm) at high spectral resolution. M3 operated in two modes:a “targeted” mode with 10nm spectral sampling for detailedobservations and a “global” mode with 20 and 40 nm spectralsampling for global coverage. The work presented here usesM3’s global mode data, as the mission unfortunately termi-nated before many targeted data could be acquired.[3] The scientific goals of M3 are to identify and map

minerals and volatiles on the surface of the Moon, and toplace these components within the context of the geophys-ical evolution of the Moon. Perhaps the greatest discoveryof the M3 mission so far is the discovery and mapping ofthe OH absorption features near 2.8 to 3.0 mm on the surface

of the Moon [Pieters et al., 2009]. Future science returnsfrom the M3 database promise to be greatly enhanced bya synergistic incorporation of other recent space‐based datasets, specifically NASA’s Lunar Reconnaissance Orbiter[Robinson and the LROC Team, 2010] and JAXA’s Kaguya[Kato et al., 2009] missions.[4] Most of the change in photometric intensity on a

planetary surface is not intrinsic but is caused by changes inlocal viewing geometry defined by the radiance incidence,emission, and solar phase angle. The goal of this paper isto provide a preliminary but accurate model to correct M3

images to a standard geometry with all the effects of viewinggeometry removed, based on data acquired inflight. Thismodel will provide a procedure for producing mosaics, indi-vidual spectra, and other products from M3 data that arefree of the effects of viewing geometry. The current modelis expected to improve as additional refinements to thecalibrations occur, including improvements to the flatfieldcalibrations; improved scattered light corrections; improvedthermal model corrections; and the computation of moreaccurate incident and emission angles based on derivedsurface topography. However, the fundamental model willnot change, and the changes in the numerical correctionfactors are expected to be small, a few percent at most forthe most extreme geometries at the poles and at the largestsolar phase angles.

2. The Model

[5] As a low‐albedo object, it has long been recognizedthat the Moon exhibits a surface reflectance that is domi-

1Jet Propulsion Laboratory, California Institute of Technology,Pasadena, California, USA.

2Department of Geological Sciences, Brown University, ProvidenceRhode Island, USA.

3Planetary Science Institute, Tucson, Arizona, USA.4Department of Astronomy, University of Maryland, College Park,

Maryland, USA.5Analytical Imaging and Geophysics LLC, Boulder, Colorado, USA.

Copyright 2011 by the American Geophysical Union.0148‐0227/11/2010JE003733

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nated by singly scattered light and can be described by theequation [Chandrasekhar, 1960]:

I=F ¼ f �; �ð Þ �o= �þ �oð Þ½ � ð1Þ

where I is the specific intensity, pF is the incidence solarflux, f (a, l) is the wavelength‐dependent surface phasefunction and m and m0 are the cosines of the emission andincident angles. This function is known as the “Lommel‐Seeliger” or lunar scattering law. The surface phase functionf (a) describes changes in the intensity on the surface dueto phase angle alone and contains the physical attributes ofthe surface, including the roughness, single particle phasefunction, the single scattering albedo, and the compactionstate of the optically active portion of the regolith. In ourempirical treatment, we concatenate all these physical param-eters into a single function; further work will derive thephysical photometric parameters, which are described in theliterature [Irvine, 1966; Hapke, 1981, 1984, 1986, 1990;Buratti, 1985; Buratti and Veverka, 1985]. The incident andemission angles were calculated from spacecraft navigationroutines which assumed a spherical Moon. The values for theincident and emission angles can be improved with knowl-edge of surface topography for the determination of localslopes on a per pixel basis.[6] The Moon’s surface scatters closely according to

equation (1), and the most detailed published analysis of thelunar surface shows that this equation describes the lunarsurface well [Hillier et al., 1999]. Some preliminary work byMcEwen [1996] on Clementine data showed that equation (1)does a “good job” of describing the lunar surface, althoughthe addition of a Lambert scattering factor dependent on thecosine of the emission angle could provide some improve-ment to the model. This addition involves the use of elevenadjustable parameters [McEwen, 1996] and thus suffers someof the unwieldiness and nonuniqueness of a full Hapke model.

The Lambert portion of the photometric function is concep-tually the treatment of isotropic multiple scattering, which hasbeen shown by both spacecraft data and laboratory measure-ments to be unimportant until albedos reach 0.3–0.6 [Veverkaet al., 1978; Buratti, 1984]. McEwen’s analysis states that thelimb‐darkening function varies with the incident, emission,and solar phase angles rather than with albedo, a result thatsuggests the contribution of multiple scattering is negligiblefor the Moon.[7] Our initial model is based on images acquired during

Observing Period 1 and Observing Period 2 (OP1 and OP2).Separate f (a, l) fits for maria and highlands were generated.Figure 1 shows the regions from which data were selectedfor the initial maria fits. The yellow circles are centered inthe lunar maria, and avoid regions above ±60 degrees lati-tude to eliminate extreme shadowing. The earliest data, whichare the strips near the equator centered near 200° longitude,were also avoided in the fits due to initial high M3 detectortemperatures. The pixels in each spectral band were firstcorrected for the effects of the incident and emission angleby multiplying by the factor (m + mo) /mo (the “Lommel‐Seeliger correction”). The data were then processed with theapplication of a running median filter of 0.1° phase angleresolution in order to minimize the effects of bad pixels,small‐scale local topography, and inadvertent mixing of ter-rain types. The resulting f (a, l) running median was fit witha sixth‐order polynomial:

f �; �ð Þ ¼ Ao þ A1�þ A2�2 þ A3�

3 þ A4�4 þ A5�

5 þ A6�6

ð2Þ

where the solar phase angle a is expressed in degrees. Theformulation is similar to the model used by Hillier et al.[1999] to describe Clementine lunar photometry. This empiri-cal fit was generated independently for each of the 84 M3

Figure 1. A mosaic of M3 observations obtained during OP1. The yellow circles represent regions fromwhich data were extracted for the derivation of the maria solar phase function. The red and green boxesrepresent regions extracted for comparison with the ROLO Chip 0 (Mare Serenitatis) and Chip 9 (high-lands), respectively.

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Table 1. “Mare” f (a) Model Coefficients

SpectralChannel

Wavelength(nm) A0

A1

(×10−2)A2

(×10−4)A3

(×10−6)A4

(×10−8)A5

(×10−10)A6

(×10−12)

1 460.99 0.074 −0.044 −0.021 0.011 0.025 0.017 −0.0172 500.92 0.084 −0.057 −0.025 0.016 0.035 0.023 −0.0233 540.84 0.106 −0.098 −0.036 0.036 0.065 0.041 −0.0444 580.76 0.119 −0.115 −0.042 0.042 0.076 0.048 −0.0515 620.69 0.126 −0.126 −0.044 0.049 0.086 0.054 −0.0576 660.61 0.134 −0.135 −0.048 0.052 0.092 0.058 −0.0617 700.54 0.146 −0.152 −0.052 0.060 0.104 0.065 −0.0708 730.48 0.153 −0.159 −0.056 0.060 0.107 0.067 −0.0719 750.44 0.159 −0.170 −0.059 0.066 0.116 0.073 −0.07710 770.40 0.162 −0.178 −0.060 0.072 0.124 0.077 −0.08211 790.37 0.153 −0.157 −0.053 0.063 0.109 0.068 −0.07312 810.33 0.152 −0.153 −0.053 0.061 0.106 0.066 −0.07113 830.29 0.151 −0.148 −0.052 0.058 0.102 0.064 −0.06814 850.25 0.156 −0.157 −0.054 0.062 0.108 0.067 −0.07215 870.21 0.153 −0.146 −0.052 0.056 0.100 0.063 −0.06716 890.17 0.154 −0.149 −0.052 0.059 0.104 0.065 −0.06917 910.14 0.159 −0.158 −0.056 0.061 0.108 0.068 −0.07218 930.10 0.167 −0.172 −0.061 0.065 0.116 0.073 −0.07719 950.06 0.172 −0.185 −0.062 0.075 0.129 0.080 −0.08620 970.02 0.178 −0.194 −0.065 0.078 0.135 0.084 −0.09021 989.98 0.188 −0.214 −0.071 0.087 0.149 0.092 −0.09922 1009.95 0.195 −0.227 −0.074 0.094 0.159 0.099 −0.10623 1029.91 0.204 −0.238 −0.080 0.096 0.165 0.103 −0.11024 1049.87 0.223 −0.272 −0.088 0.113 0.191 0.118 −0.12725 1069.83 0.228 −0.277 −0.090 0.116 0.196 0.121 −0.13026 1089.79 0.236 −0.288 −0.093 0.121 0.204 0.126 −0.13627 1109.76 0.246 −0.302 −0.096 0.129 0.215 0.132 −0.14428 1129.72 0.251 −0.305 −0.098 0.128 0.216 0.133 −0.14429 1149.68 0.264 −0.326 −0.103 0.140 0.234 0.143 −0.15630 1169.64 0.266 −0.324 −0.104 0.136 0.229 0.141 −0.15331 1189.60 0.266 −0.320 −0.102 0.136 0.228 0.140 −0.15232 1209.57 0.277 −0.336 −0.108 0.142 0.239 0.147 −0.15933 1229.53 0.280 −0.342 −0.108 0.147 0.245 0.150 −0.16334 1249.49 0.280 −0.340 −0.107 0.147 0.244 0.150 −0.16335 1269.45 0.286 −0.346 −0.111 0.146 0.245 0.151 −0.16436 1289.41 0.282 −0.336 −0.108 0.141 0.238 0.147 −0.15937 1309.38 0.298 −0.365 −0.116 0.155 0.259 0.160 −0.17338 1329.34 0.290 −0.348 −0.111 0.148 0.248 0.152 −0.16539 1349.30 0.301 −0.368 −0.117 0.158 0.263 0.162 −0.17640 1369.26 0.295 −0.350 −0.114 0.145 0.245 0.152 −0.16341 1389.22 0.303 −0.362 −0.117 0.152 0.256 0.158 −0.17142 1409.19 0.296 −0.342 −0.113 0.141 0.240 0.149 −0.16043 1429.15 0.305 −0.362 −0.116 0.154 0.258 0.159 −0.17344 1449.11 0.309 −0.369 −0.118 0.157 0.263 0.162 −0.17545 1469.07 0.318 −0.384 −0.121 0.165 0.275 0.169 −0.18446 1489.03 0.318 −0.380 −0.121 0.163 0.272 0.167 −0.18147 1508.99 0.319 −0.378 −0.121 0.161 0.270 0.166 −0.18048 1528.96 0.328 −0.390 −0.126 0.165 0.277 0.171 −0.18549 1548.92 0.331 −0.397 −0.127 0.169 0.283 0.174 −0.18950 1578.86 0.344 −0.417 −0.133 0.178 0.298 0.183 −0.19951 1618.79 0.344 −0.413 −0.131 0.178 0.296 0.182 −0.19852 1658.71 0.356 −0.428 −0.136 0.183 0.306 0.188 −0.20453 1698.63 0.368 −0.442 −0.142 0.187 0.315 0.194 −0.21054 1738.56 0.374 −0.446 −0.143 0.190 0.318 0.196 −0.21355 1778.48 0.380 −0.457 −0.147 0.194 0.326 0.201 −0.21756 1818.40 0.375 −0.437 −0.142 0.184 0.310 0.191 −0.20757 1858.33 0.387 −0.461 −0.149 0.195 0.328 0.202 −0.21958 1898.25 0.399 −0.475 −0.154 0.200 0.338 0.208 −0.22559 1938.18 0.397 −0.461 −0.149 0.195 0.328 0.202 −0.21960 1978.10 0.400 −0.464 −0.149 0.196 0.330 0.203 −0.22161 2018.02 0.404 −0.463 −0.150 0.195 0.329 0.203 −0.22062 2057.95 0.416 −0.480 −0.156 0.201 0.340 0.210 −0.22763 2097.87 0.413 −0.470 −0.154 0.195 0.331 0.205 −0.22164 2137.80 0.429 −0.505 −0.162 0.215 0.360 0.222 −0.24065 2177.72 0.432 −0.497 −0.163 0.207 0.351 0.217 −0.23466 2217.64 0.442 −0.517 −0.165 0.221 0.370 0.227 −0.24767 2257.57 0.445 −0.512 −0.166 0.215 0.363 0.224 −0.24368 2297.49 0.463 −0.536 −0.175 0.223 0.378 0.234 −0.25269 2337.42 0.482 −0.567 −0.183 0.240 0.403 0.248 −0.26970 2377.34 0.492 −0.582 −0.186 0.248 0.416 0.256 −0.27871 2417.26 0.498 −0.582 −0.189 0.244 0.412 0.254 −0.275

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spectral channels. We note that this equation does notinclude an exponential term to treat the lunar oppositionsurge [Buratti et al., 1996; Hillier et al., 1999] because thesolar phase angle range in OP1 and OP2 (24°–90°) does notcover the canonical range of the opposition surge (a < 12°).The resulting lunar maria model coefficients are listed inTable 1.[8] A second function to represent the averaged lunar

highlands was fit by expanding the yellow circles by a factorof three, extracting I/F values from the all the regions notwithin these larger circles and at latitudes less than ±60°,applying the Lommel‐Seeliger correction, and finally fittingthe resulting values to equation (2). The values for the fit

coefficients for the lunar highlands (which we call “notmare”) are listed in Table 2 for 84 M3 spectral bands. The85th M3 band gave consistently nonphysical fits (i.e., I/Fless than zero) for both “mare” and “not mare” and wasexcluded from analysis. Figures 2a and 2b show the Lommel‐Seeliger corrected I/F values (the f (a) plotted as a func-tion of solar phase angle along with a running‐boxmedian and our empirical f (a) fits. Also plotted are curvesillustrating Clementine models for lunar mare and highlands[Hillier et al., 1999] and ground‐based observations of theMoon using the USGS Robotic Lunar Observatory (ROLO)[Kieffer and Stone, 2005]. Our M3‐based model agrees wellwith our preflight ROLO model [Buratti et al., 2011] at

Table 1. (continued)

SpectralChannel

Wavelength(nm) A0

A1

(×10−2)A2

(×10−4)A3

(×10−6)A4

(×10−8)A5

(×10−10)A6

(×10−12)

72 2457.19 0.515 −0.606 −0.196 0.255 0.430 0.265 −0.28673 2497.11 0.535 −0.634 −0.205 0.266 0.449 0.277 −0.29974 2537.03 0.547 −0.649 −0.211 0.271 0.458 0.283 −0.30575 2576.96 0.566 −0.661 −0.218 0.271 0.463 0.287 −0.30876 2616.88 0.587 −0.688 −0.228 0.281 0.481 0.298 −0.32077 2656.81 0.611 −0.718 −0.238 0.293 0.501 0.310 −0.33378 2696.73 0.627 −0.727 −0.245 0.291 0.502 0.312 −0.33379 2736.65 0.646 −0.741 −0.255 0.288 0.504 0.315 −0.33480 2776.58 0.665 −0.759 −0.266 0.288 0.510 0.320 −0.33781 2816.50 0.685 −0.777 −0.276 0.289 0.517 0.326 −0.34182 2856.43 0.729 −0.816 −0.297 0.294 0.534 0.339 −0.35183 2896.35 0.770 −0.855 −0.317 0.299 0.551 0.352 −0.36284 2936.27 0.816 −0.889 −0.339 0.297 0.561 0.360 −0.367

Figure 2. (a) The solar phase function for the lunar maria extracted from the yellow regions depictedin Figure 1. The data are scattered because of albedo variations on the Moon and because of changesin the incident and emission angles due to local topography. (b) The solar phase function of the highlands(“not mare”) regions. The blue curves in each panel represent running boxcar medians of 0.1° width, usedto generate the f (a) fits. The bottom panel of both Figures 2a and 2b plots the number of data pointsincluded in each solar phase angle bin.

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Table 2. “Not Mare” f (a) Model Coefficients

SpectralChannel

Wavelength(nm) A0

A1

(×10−2)A2

(×10−4)A3

(×10−6)A4

(×10−8)A5

(×10−10)A6

(×10−12)

1 460.99 0.331 −0.760 0.424 0.421 −0.144 −0.427 0.1992 500.92 0.394 −0.951 0.594 0.538 −0.244 −0.607 0.3023 540.84 0.416 −0.937 0.542 0.513 −0.215 −0.561 0.2764 580.76 0.433 −0.951 0.548 0.515 −0.225 −0.575 0.2865 620.69 0.471 −1.060 0.626 0.581 −0.258 −0.652 0.3256 660.61 0.497 −1.133 0.686 0.625 −0.292 −0.715 0.3607 700.54 0.507 −1.127 0.671 0.617 −0.285 −0.703 0.3548 730.48 0.524 −1.163 0.700 0.636 −0.304 −0.736 0.3739 750.44 0.531 −1.167 0.691 0.635 −0.296 −0.727 0.36710 770.40 0.538 −1.191 0.705 0.650 −0.298 −0.739 0.37111 790.37 0.552 −1.228 0.723 0.672 −0.299 −0.754 0.37612 810.33 0.564 −1.266 0.751 0.694 −0.313 −0.783 0.39113 830.29 0.578 −1.306 0.781 0.719 −0.327 −0.814 0.40814 850.25 0.588 −1.325 0.794 0.730 −0.333 −0.827 0.41415 870.21 0.596 −1.333 0.790 0.731 −0.329 −0.824 0.41216 890.17 0.606 −1.361 0.808 0.749 −0.334 −0.841 0.41917 910.14 0.618 −1.399 0.847 0.771 −0.362 −0.885 0.44618 930.10 0.622 −1.398 0.840 0.767 −0.360 −0.880 0.44419 950.06 0.623 −1.371 0.791 0.746 −0.321 −0.826 0.40920 970.02 0.631 −1.394 0.820 0.762 −0.341 −0.857 0.42821 989.98 0.638 −1.405 0.836 0.767 −0.357 −0.878 0.44222 1009.95 0.637 −1.370 0.796 0.741 −0.335 −0.838 0.42023 1029.91 0.638 −1.360 0.793 0.734 −0.340 −0.839 0.42324 1049.87 0.644 −1.364 0.797 0.737 −0.343 −0.843 0.42525 1069.83 0.652 −1.369 0.795 0.737 −0.341 −0.842 0.42526 1089.79 0.664 −1.389 0.809 0.748 −0.348 −0.857 0.43227 1109.76 0.675 −1.398 0.803 0.750 −0.342 −0.852 0.42828 1129.72 0.683 −1.412 0.820 0.759 −0.355 −0.871 0.44029 1149.68 0.689 −1.407 0.802 0.753 −0.341 −0.853 0.42830 1169.64 0.700 −1.435 0.829 0.769 −0.359 −0.883 0.44631 1189.60 0.716 −1.481 0.866 0.798 −0.377 −0.920 0.46632 1209.57 0.719 −1.477 0.858 0.793 −0.373 −0.913 0.46233 1229.53 0.727 −1.491 0.862 0.800 −0.372 −0.916 0.46234 1249.49 0.743 −1.543 0.908 0.833 −0.397 −0.964 0.48935 1269.45 0.749 −1.552 0.913 0.837 −0.402 −0.971 0.49336 1289.41 0.762 −1.596 0.951 0.864 −0.422 −1.010 0.51537 1309.38 0.761 −1.570 0.918 0.844 −0.403 −0.977 0.49638 1329.34 0.769 −1.595 0.937 0.860 −0.412 −0.997 0.50639 1349.30 0.767 −1.567 0.897 0.837 −0.386 −0.956 0.48140 1369.26 0.785 −1.637 0.969 0.882 −0.432 −1.032 0.52741 1389.22 0.802 −1.688 1.012 0.914 −0.454 −1.076 0.55042 1409.19 0.815 −1.719 1.032 0.931 −0.463 −1.097 0.56243 1429.15 0.830 −1.755 1.054 0.953 −0.470 −1.118 0.57144 1449.11 0.834 −1.771 1.070 0.962 −0.480 −1.135 0.58145 1469.07 0.850 −1.802 1.085 0.980 −0.483 −1.150 0.58746 1489.03 0.856 −1.827 1.111 0.996 −0.500 −1.178 0.60447 1508.99 0.865 −1.842 1.112 1.003 −0.496 −1.178 0.60248 1528.96 0.858 −1.789 1.065 0.966 −0.477 −1.134 0.58049 1548.92 0.844 −1.754 1.043 0.946 −0.467 −1.112 0.56850 1578.86 0.853 −1.765 1.050 0.952 −0.471 −1.120 0.57251 1618.79 0.868 −1.792 1.061 0.966 −0.472 −1.130 0.57652 1658.71 0.878 −1.794 1.052 0.963 −0.467 −1.123 0.57253 1698.63 0.883 −1.770 1.018 0.943 −0.447 −1.090 0.55354 1738.56 0.905 −1.820 1.056 0.974 −0.465 −1.129 0.57355 1778.48 0.905 −1.813 1.049 0.969 −0.460 −1.120 0.56856 1818.40 0.915 −1.829 1.057 0.977 −0.464 −1.130 0.57357 1858.33 0.931 −1.859 1.071 0.993 −0.467 −1.144 0.57858 1898.25 0.949 −1.884 1.075 1.004 −0.465 −1.149 0.57959 1938.18 0.963 −1.927 1.118 1.033 −0.489 −1.193 0.60460 1978.10 0.976 −1.951 1.131 1.046 −0.493 −1.206 0.61061 2018.02 0.987 −1.958 1.128 1.046 −0.492 −1.205 0.60962 2057.95 0.995 −1.957 1.119 1.042 −0.489 −1.199 0.60663 2097.87 1.000 −1.963 1.122 1.044 −0.491 −1.203 0.60964 2137.80 1.017 −2.000 1.138 1.064 −0.492 −1.217 0.61365 2177.72 1.031 −2.014 1.147 1.069 −0.502 −1.231 0.62366 2217.64 1.052 −2.076 1.190 1.106 −0.520 −1.274 0.64467 2257.57 1.059 −2.088 1.211 1.114 v0.538 v1.298 0.66068 2297.49 1.068 −2.080 1.191 1.103 −0.530 −1.283 0.65269 2337.42 1.103 −2.184 1.273 1.167 −0.569 −1.366 0.69670 2377.34 1.114 −2.217 1.300 1.189 −0.581 −1.392 0.70971 2417.26 1.120 −2.211 1.293 1.180 −0.583 −1.389 0.71072 2457.19 1.148 −2.296 1.356 1.231 −0.614 −1.454 0.745

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Table 2. (continued)

SpectralChannel

Wavelength(nm) A0

A1

(×10−2)A2

(×10−4)A3

(×10−6)A4

(×10−8)A5

(×10−10)A6

(×10−12)

73 2497.11 1.161 −2.320 1.368 1.243 −0.619 −1.467 0.75174 2537.03 1.170 −2.355 1.394 1.264 −0.631 −1.493 0.76575 2576.96 1.197 −2.396 1.408 1.282 −0.637 −1.511 0.77476 2616.88 1.225 −2.492 1.491 1.342 −0.681 −1.596 0.82177 2656.81 1.253 −2.551 1.517 1.372 −0.687 −1.623 0.83278 2696.73 1.288 −2.672 1.622 1.447 −0.745 −1.731 0.89379 2736.65 1.305 −2.705 1.627 1.457 −0.750 −1.743 0.90080 2776.58 1.321 −2.790 1.703 1.510 −0.792 −1.821 0.94481 2816.50 1.327 −2.790 1.684 1.504 −0.777 −1.802 0.93282 2856.43 1.398 −2.981 1.821 1.611 −0.850 −1.949 1.01383 2896.35 1.448 −3.083 1.865 1.660 −0.867 −1.999 1.03784 2936.27 1.493 −3.136 1.854 1.672 −0.856 −1.997 1.033

Figure 3. (a) The f (a) as a function of wavelength for 10° intervals in solar phase angles for the maremodel. The red lines are wavelength‐dependent model fits, while the green line shows smoothing in thespectral domain done by a third‐order polynomial fit. (b) The f (a) for the “not mare” model. (c) The “notmare” data normalized to 1.489 mm to illustrate solar phase reddening.

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Figure 4. A graphical rendition of the “not mare” solar phase angle correction. These are the factorsas a function of wavelength and solar phase that correct the “reflectances” (I/F corrected by the factor(mo + m)/mo) to 30° solar phase.

Figure 5. A histogram of normal reflectances for both the highlands and maria, produced by themodel described in this paper (for normal reflectance, the incident and emission angle are both zero).The numbers were derived without including a lunar opposition surge, as these small solar phase angleswere not attained during OP1 and OP2.

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solar phase angles constrained by M3 data but divergesat low solar phase angles. The data in Figures 2a and 2bare scattered because of albedo variations on the Moon andbecause of changes in the incident and emission anglesdue to local topography, both of which are exhibited moreintensely in highland regions.[9] The spectral dependence of our f (a) models is shown

in Figures 3a and 3b for “mare” and “not mare. ” Thesecurves show a model spectrum of the lunar maria and high-lands in 10° increments of solar phase angle, with both rawfits and smoothed values shown. Figure 3c shows the f (a)model spectra normalized to 1.489 mm to illustrate sub-

stantial reddening for the lunar highlands for observationsat large solar phase angles. Our mare model was under-constrained at solar phase angle less than 35° and cannot beused to estimate solar phase reddening, although recent tele-scopic studies suggest that the color index C(600/470 nm)grows more quickly with solar phase angle for highlandsthan mare at solar phase angles a less than 40–50° [Kaydashet al., 2010].[10] The values listed in Tables 1 and 2 can be used to

correct M3 spectra and images to an arbitrary viewing geome-try, with the caveat that our models were not well constrainedat low solar phase angles below 35° and 25° for the “mare”and “not mare” models, respectively. For the initial analysisof M3 data the science team normalized the measurements toa solar phase angle of 30° to correspond to the geometry ofthe RELAB experiments [Pieters, 1983]. Figure 4 is a graph-ical rendition of these corrections, f (30°, l)/f (a, l).

3. The Performance of the Photometric Model

[11] As a test of the photometric model, a histogram ofnormal reflectances in the visible (0.54 mm) channel is shownin Figure 5 after the photometric corrections for two regionson the Moon: the highlands and the lunar maria. These tworegions were extracted from focus areas selected for inten-sive calibration in the USGS’s Robotic Lunar Observatory(ROLO) dedicated ground‐based lunar calibration project[Kieffer and Wildey, 1996; Kieffer and Stone, 2005; Burattiet al., 2011]. The reflectance values for the highlands wereextracted from the area of chip 9 in the ROLO database at alatitude of −17.21° and longitude of 20.10° and the reflec-tance values for the maria were extracted from chip 0 inMare Serenitatis at a latitude of 19.06° and a longitude of20.47°. (the precise ROLO chip 0 and chip 9 locations werenot available in the M3 data set.) Figure 5 shows histogramsof albedo (normal reflectance) for both regions. The meanreflectance of 0.06 for the maria and 0.11 for the highlandsis low but reasonable, although we note that these numbersare based on a model extrapolation to a solar phase angle of0° rather than on actual measurements at opposition. Sub-sequent modeling of the opposition surge will increasethese numbers by a factor of 30–40%., as suggested by thebehavior of the ROLO and Clementine models presented inFigures 2a and 2b.[12] Figure 6 shows the use of the photometric model for

a region on the Moon known as the Reiner Gamma Swirl.This albedo feature is about 70 km wide and is located at7.5° N and 59.0° W in Oceanus Procellarum. This arche-typical lunar swirl exhibits large albedo variations, and is thusan ideal feature for applying and testing a photometric model.Figure 6a is a mosaic of Reiner Gamma produced withoutany photometric corrections, but with calibrated radiancedata. Figure 6b shows the same mosaic corrected for pho-tometric effects following the method outlined in Section 2,and using the correction for the maria regions.[13] A second example making use of the highlands pho-

tometric function (“not mare” model) is shown in Figure 7.This region covers the Orientale Basin, which is about900 km in diameter and is located at 19.4°S and 92.8° W.Figure 7a is the uncorrected mosaic, while Figure 7b showsthe correction for 0.54 mm. Figure 7c shows observations

Figure 6. An example of the use of the photometric func-tion described in this paper to correct an M3 mosaic of ReinerGamma, a lunar swirl. (a) The 1.489 mm mosaic uncorrectedfor any photometric effects. (b) The mosaic correct with themaria photometric function. The Reiner Gamma swirl isabout 70 km wide.

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obtained at 1.489 mm corrected with the “not mare” modelfor that wavelength.

4. Discussion and Future Work

[14] The photometric model presented in this paper is apurely empirical one that is useful for correcting spectra andmultispectral mosaics in the 0.404 to 2.983 mm spectral

range for the effects of viewing geometry. The functionspresented apply only to the phase angle range of the obser-vational set (24°–90°). Further work will present a moredetailed function for the opposition surge. Another improve-ment would be to partition the correction between the mareand “not mare” functions on the basis of the albedo of thesurface or geographical location. For example, Figure 7, thecorrection for the Orientale Basin, was made with the “not

Figure 7. An example of the application of our photometric model using the highlands (“not mare”)function for the solar phase angle correction. (a) An M3 mosaic of the Orientale Basin with no photomet-ric corrections. (b) An image in the visible (0.54 mm) corrected for all photometric effects with our modeland (c) the correction at 1.489 mm. (d) A 2.2/1.2 mm ratio map of the Orientale Basin showing a low‐levelresidual striping pattern. The bright areas at the center and rim of Orientale are likely due to thermal radi-ation that has not been subtracted; these artifacts are in the lowest‐albedo regions where the thermal fluxshould be highest.

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mare” photometric function. The correction is good exceptfor the area in the center of the basin, the most mare‐likeregion in the mosaic.[15] Figure 7d shows the 2.1/1.2 mm ratio of the photo-

metrically corrected mosaics for the Orientale Basin. Therippling effect of about <5% represents the total photometricintegrity of the M3 data after all calibrations and photometriccorrections. The pattern is due to a combination of all likelyeffects not corrected for in the M3 calibration procedures inaddition to inaccuracies in our photometric model: residualscattered light from the M3 instrument and stray light from theChandrayaan‐1 spacecraft; imperfect flatfielding (a cross‐track average to the flatfield combining a range of emissionangles was used); errors in the calculation of the incidentand emission angles due to surface topography and otherfactors, and finally the subtraction of the thermal component.[16] Thermal radiation has not been subtracted from the

M3 measurements used to obtain our model fits. Thermalemission becomes significant (∼a few % relative to reflectedsunlight) at 2.4 mm for a typical lunar surface with an albedoof 0.1 [Hapke, 1993]. For lunar maria the thermal emissionis even more substantial. Thermal radiation exhibits an iso-tropic scattering law proportional to the cosine of the emis-sion angle. Our model is fit separately to each wavelength,but as the wavelength and thus thermal radiation increases,the Lommel‐Seeliger correction for reflected light becomesless and less applicable if the thermal radiation has not beensubtracted. With the emission angle changing in the cross‐track direction of our scans, this factor can become signif-icant, particularly if one is ratioing two photometricallycorrected mosaics at different wavelengths (see Figure 7d).Future work will thus need to incorporate a thermal model,particularly beyond 2 mm; our model is most accurate forwavelengths where thermal emission is not significant.Figure 7d shows bright areas at the center and rim of theOrientale Basin which are consistent with a higher temper-ature and thermal emission at 2.2 mm in the low‐albedoregions. Current limitations on the accuracy of wavelength‐dependent calibrations (flatfielding and scattered light forexample) may also cause wavelength‐dependent artifacts inthe data. Our model works particularly well (∼1–2% relativeerror) when producing mosaics of M3 images at individualwavelengths below 2 mm.

[17] Acknowledgments. The research described in this paper wascarried out at the Jet Propulsion Laboratory, California Institute of Technol-ogy, under contract with the National Aeronautics and Space Administration.

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