Journal of Geophysical Research: Planets

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

Authors


Abstract

[1] Changes in observed photometric intensity on a planetary surface are caused by variations in local viewing geometry defined by the radiance incidence, emission, and solar phase angle coupled with a wavelength-dependent surface phase function f(α, λ) which is specific for a given terrain. In this paper we provide preliminary empirical models, based on data acquired inflight, which enable the correction of Moon Mineralogy Mapper (M3) spectral images to a standard geometry with the effects of viewing geometry removed. Over the solar phase angle range for which the M3 data were acquired our models are accurate to a few percent, particularly where thermal emission is not significant. Our models are expected to improve as additional refinements to the calibrations occur, including improvements to the flatfield calibration; improved scattered and stray light corrections; improved thermal model corrections; and the computation of more accurate local incident and emission angles based on surface topography.

1. Introduction

[2] The Moon Mineralogy Mapper (M3) imaging spectrometer was one of two NASA-provided instruments on the Chandrayaan-1 Indian Space Research Organization (ISRO) spacecraft. Launched in 2008, M3 is an imaging spectrometer covering the visible and infrared spectral range (∼0.4 to ∼3 μm) at high spectral resolution. M3 operated in two modes: a “targeted” mode with 10nm spectral sampling for detailed observations and a “global” mode with 20 and 40 nm spectral sampling for global coverage. The work presented here uses M3's global mode data, as the mission unfortunately terminated 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 to place these components within the context of the geophysical evolution of the Moon. Perhaps the greatest discovery of the M3 mission so far is the discovery and mapping of the OH absorption features near 2.8 to 3.0 μm on the surface of the Moon [Pieters et al., 2009]. Future science returns from the M3 database promise to be greatly enhanced by a synergistic incorporation of other recent space-based data sets, 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 in local viewing geometry defined by the radiance incidence, emission, and solar phase angle. The goal of this paper is to provide a preliminary but accurate model to correct M3 images to a standard geometry with all the effects of viewing geometry removed, based on data acquired inflight. This model will provide a procedure for producing mosaics, individual spectra, and other products from M3 data that are free of the effects of viewing geometry. The current model is expected to improve as additional refinements to the calibrations occur, including improvements to the flatfield calibrations; improved scattered light corrections; improved thermal model corrections; and the computation of more accurate incident and emission angles based on derived surface topography. However, the fundamental model will not change, and the changes in the numerical correction factors are expected to be small, a few percent at most for the most extreme geometries at the poles and at the largest solar phase angles.

2. The Model

[5] As a low-albedo object, it has long been recognized that the Moon exhibits a surface reflectance that is dominated by singly scattered light and can be described by the equation [Chandrasekhar, 1960]:

equation image

where I is the specific intensity, πF is the incidence solar flux, f(α, λ) is the wavelength-dependent surface phase function and μ and μ0 are the cosines of the emission and incident angles. This function is known as the “Lommel-Seeliger” or lunar scattering law. The surface phase function f(α) describes changes in the intensity on the surface due to phase angle alone and contains the physical attributes of the surface, including the roughness, single particle phase function, the single scattering albedo, and the compaction state of the optically active portion of the regolith. In our empirical treatment, we concatenate all these physical parameters into a single function; further work will derive the physical photometric parameters, which are described in the literature [Irvine, 1966; Hapke, 1981, 1984, 1986, 1990; Buratti, 1985; Buratti and Veverka, 1985]. The incident and emission angles were calculated from spacecraft navigation routines which assumed a spherical Moon. The values for the incident and emission angles can be improved with knowledge of surface topography for the determination of local slopes on a per pixel basis.

[6] The Moon's surface scatters closely according to equation (1), and the most detailed published analysis of the lunar surface shows that this equation describes the lunar surface well [Hillier et al., 1999]. Some preliminary work by McEwen [1996] on Clementine data showed that equation (1) does a “good job” of describing the lunar surface, although the addition of a Lambert scattering factor dependent on the cosine of the emission angle could provide some improvement to the model. This addition involves the use of eleven adjustable parameters [McEwen, 1996] and thus suffers some of the unwieldiness and nonuniqueness of a full Hapke model. The Lambert portion of the photometric function is conceptually the treatment of isotropic multiple scattering, which has been shown by both spacecraft data and laboratory measurements to be unimportant until albedos reach 0.3–0.6 [Veverka et al., 1978; Buratti, 1984]. McEwen's analysis states that the limb-darkening function varies with the incident, emission, and solar phase angles rather than with albedo, a result that suggests the contribution of multiple scattering is negligible for the Moon.

[7] Our initial model is based on images acquired during Observing Period 1 and Observing Period 2 (OP1 and OP2). Separate f(α, λ) fits for maria and highlands were generated. Figure 1 shows the regions from which data were selected for the initial maria fits. The yellow circles are centered in the lunar maria, and avoid regions above ±60 degrees latitude to eliminate extreme shadowing. The earliest data, which are the strips near the equator centered near 200° longitude, were also avoided in the fits due to initial high M3 detector temperatures. The pixels in each spectral band were first corrected for the effects of the incident and emission angle by multiplying by the factor (μ + μo)/μo (the “Lommel-Seeliger correction”). The data were then processed with the application of a running median filter of 0.1° phase angle resolution in order to minimize the effects of bad pixels, small-scale local topography, and inadvertent mixing of terrain types. The resulting f(α, λ) running median was fit with a sixth-order polynomial:

equation image

where the solar phase angle α is expressed in degrees. The formulation is similar to the model used by Hillier et al. [1999] to describe Clementine lunar photometry. This empirical fit was generated independently for each of the 84 M3 spectral channels. We note that this equation does not include an exponential term to treat the lunar opposition surge [Buratti et al., 1996; Hillier et al., 1999] because the solar phase angle range in OP1 and OP2 (24°–90°) does not cover the canonical range of the opposition surge (α < 12°). The resulting lunar maria model coefficients are listed in Table 1.

Figure 1.

A mosaic of M3 observations obtained during OP1. The yellow circles represent regions from which data were extracted for the derivation of the maria solar phase function. The red and green boxes represent regions extracted for comparison with the ROLO Chip 0 (Mare Serenitatis) and Chip 9 (highlands), respectively.

Table 1. “Mare” f(α) Model Coefficients
Spectral ChannelWavelength (nm)A0A1 (×10−2)A2 (×10−4)A3 (×10−6)A4 (×10−8)A5 (×10−10)A6 (×10−12)
1460.990.074−0.044−0.0210.0110.0250.017−0.017
2500.920.084−0.057−0.0250.0160.0350.023−0.023
3540.840.106−0.098−0.0360.0360.0650.041−0.044
4580.760.119−0.115−0.0420.0420.0760.048−0.051
5620.690.126−0.126−0.0440.0490.0860.054−0.057
6660.610.134−0.135−0.0480.0520.0920.058−0.061
7700.540.146−0.152−0.0520.0600.1040.065−0.070
8730.480.153−0.159−0.0560.0600.1070.067−0.071
9750.440.159−0.170−0.0590.0660.1160.073−0.077
10770.400.162−0.178−0.0600.0720.1240.077−0.082
11790.370.153−0.157−0.0530.0630.1090.068−0.073
12810.330.152−0.153−0.0530.0610.1060.066−0.071
13830.290.151−0.148−0.0520.0580.1020.064−0.068
14850.250.156−0.157−0.0540.0620.1080.067−0.072
15870.210.153−0.146−0.0520.0560.1000.063−0.067
16890.170.154−0.149−0.0520.0590.1040.065−0.069
17910.140.159−0.158−0.0560.0610.1080.068−0.072
18930.100.167−0.172−0.0610.0650.1160.073−0.077
19950.060.172−0.185−0.0620.0750.1290.080−0.086
20970.020.178−0.194−0.0650.0780.1350.084−0.090
21989.980.188−0.214−0.0710.0870.1490.092−0.099
221009.950.195−0.227−0.0740.0940.1590.099−0.106
231029.910.204−0.238−0.0800.0960.1650.103−0.110
241049.870.223−0.272−0.0880.1130.1910.118−0.127
251069.830.228−0.277−0.0900.1160.1960.121−0.130
261089.790.236−0.288−0.0930.1210.2040.126−0.136
271109.760.246−0.302−0.0960.1290.2150.132−0.144
281129.720.251−0.305−0.0980.1280.2160.133−0.144
291149.680.264−0.326−0.1030.1400.2340.143−0.156
301169.640.266−0.324−0.1040.1360.2290.141−0.153
311189.600.266−0.320−0.1020.1360.2280.140−0.152
321209.570.277−0.336−0.1080.1420.2390.147−0.159
331229.530.280−0.342−0.1080.1470.2450.150−0.163
341249.490.280−0.340−0.1070.1470.2440.150−0.163
351269.450.286−0.346−0.1110.1460.2450.151−0.164
361289.410.282−0.336−0.1080.1410.2380.147−0.159
371309.380.298−0.365−0.1160.1550.2590.160−0.173
381329.340.290−0.348−0.1110.1480.2480.152−0.165
391349.300.301−0.368−0.1170.1580.2630.162−0.176
401369.260.295−0.350−0.1140.1450.2450.152−0.163
411389.220.303−0.362−0.1170.1520.2560.158−0.171
421409.190.296−0.342−0.1130.1410.2400.149−0.160
431429.150.305−0.362−0.1160.1540.2580.159−0.173
441449.110.309−0.369−0.1180.1570.2630.162−0.175
451469.070.318−0.384−0.1210.1650.2750.169−0.184
461489.030.318−0.380−0.1210.1630.2720.167−0.181
471508.990.319−0.378−0.1210.1610.2700.166−0.180
481528.960.328−0.390−0.1260.1650.2770.171−0.185
491548.920.331−0.397−0.1270.1690.2830.174−0.189
501578.860.344−0.417−0.1330.1780.2980.183−0.199
511618.790.344−0.413−0.1310.1780.2960.182−0.198
521658.710.356−0.428−0.1360.1830.3060.188−0.204
531698.630.368−0.442−0.1420.1870.3150.194−0.210
541738.560.374−0.446−0.1430.1900.3180.196−0.213
551778.480.380−0.457−0.1470.1940.3260.201−0.217
561818.400.375−0.437−0.1420.1840.3100.191−0.207
571858.330.387−0.461−0.1490.1950.3280.202−0.219
581898.250.399−0.475−0.1540.2000.3380.208−0.225
591938.180.397−0.461−0.1490.1950.3280.202−0.219
601978.100.400−0.464−0.1490.1960.3300.203−0.221
612018.020.404−0.463−0.1500.1950.3290.203−0.220
622057.950.416−0.480−0.1560.2010.3400.210−0.227
632097.870.413−0.470−0.1540.1950.3310.205−0.221
642137.800.429−0.505−0.1620.2150.3600.222−0.240
652177.720.432−0.497−0.1630.2070.3510.217−0.234
662217.640.442−0.517−0.1650.2210.3700.227−0.247
672257.570.445−0.512−0.1660.2150.3630.224−0.243
682297.490.463−0.536−0.1750.2230.3780.234−0.252
692337.420.482−0.567−0.1830.2400.4030.248−0.269
702377.340.492−0.582−0.1860.2480.4160.256−0.278
712417.260.498−0.582−0.1890.2440.4120.254−0.275
722457.190.515−0.606−0.1960.2550.4300.265−0.286
732497.110.535−0.634−0.2050.2660.4490.277−0.299
742537.030.547−0.649−0.2110.2710.4580.283−0.305
752576.960.566−0.661−0.2180.2710.4630.287−0.308
762616.880.587−0.688−0.2280.2810.4810.298−0.320
772656.810.611−0.718−0.2380.2930.5010.310−0.333
782696.730.627−0.727−0.2450.2910.5020.312−0.333
792736.650.646−0.741−0.2550.2880.5040.315−0.334
802776.580.665−0.759−0.2660.2880.5100.320−0.337
812816.500.685−0.777−0.2760.2890.5170.326−0.341
822856.430.729−0.816−0.2970.2940.5340.339−0.351
832896.350.770−0.855−0.3170.2990.5510.352−0.362
842936.270.816−0.889−0.3390.2970.5610.360−0.367

[8] A second function to represent the averaged lunar highlands was fit by expanding the yellow circles by a factor of three, extracting I/F values from the all the regions not within these larger circles and at latitudes less than ±60°, applying the Lommel-Seeliger correction, and finally fitting the resulting values to equation (2). The values for the fit coefficients for the lunar highlands (which we call “not mare”) are listed in Table 2 for 84 M3 spectral bands. The 85th M3 band gave consistently nonphysical fits (i.e., I/F less than zero) for both “mare” and “not mare” and was excluded from analysis. Figures 2a and 2b show the Lommel-Seeliger corrected I/F values (the f(α) plotted as a function of solar phase angle along with a running-box median and our empirical f(α) fits. Also plotted are curves illustrating Clementine models for lunar mare and highlands [Hillier et al., 1999] and ground-based observations of the Moon using the USGS Robotic Lunar Observatory (ROLO) [Kieffer and Stone, 2005]. Our M3-based model agrees well with our preflight ROLO model [Buratti et al., 2011] at solar phase angles constrained by M3 data but diverges at low solar phase angles. The data in Figures 2a and 2b are scattered because of albedo variations on the Moon and because of changes in the incident and emission angles due to local topography, both of which are exhibited more intensely in highland regions.

Figure 2.

(a) The solar phase function for the lunar maria extracted from the yellow regions depicted in Figure 1. The data are scattered because of albedo variations on the Moon and because of changes in 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, used to generate the f(α) fits. The bottom panel of both Figures 2a and 2b plots the number of data points included in each solar phase angle bin.

Table 2. “Not Mare” f(α) Model Coefficients
Spectral ChannelWavelength (nm)A0A1 (×10−2)A2 (×10−4)A3 (×10−6)A4 (×10−8)A5 (×10−10)A6 (×10−12)
1460.990.331−0.7600.4240.421−0.144−0.4270.199
2500.920.394−0.9510.5940.538−0.244−0.6070.302
3540.840.416−0.9370.5420.513−0.215−0.5610.276
4580.760.433−0.9510.5480.515−0.225−0.5750.286
5620.690.471−1.0600.6260.581−0.258−0.6520.325
6660.610.497−1.1330.6860.625−0.292−0.7150.360
7700.540.507−1.1270.6710.617−0.285−0.7030.354
8730.480.524−1.1630.7000.636−0.304−0.7360.373
9750.440.531−1.1670.6910.635−0.296−0.7270.367
10770.400.538−1.1910.7050.650−0.298−0.7390.371
11790.370.552−1.2280.7230.672−0.299−0.7540.376
12810.330.564−1.2660.7510.694−0.313−0.7830.391
13830.290.578−1.3060.7810.719−0.327−0.8140.408
14850.250.588−1.3250.7940.730−0.333−0.8270.414
15870.210.596−1.3330.7900.731−0.329−0.8240.412
16890.170.606−1.3610.8080.749−0.334−0.8410.419
17910.140.618−1.3990.8470.771−0.362−0.8850.446
18930.100.622−1.3980.8400.767−0.360−0.8800.444
19950.060.623−1.3710.7910.746−0.321−0.8260.409
20970.020.631−1.3940.8200.762−0.341−0.8570.428
21989.980.638−1.4050.8360.767−0.357−0.8780.442
221009.950.637−1.3700.7960.741−0.335−0.8380.420
231029.910.638−1.3600.7930.734−0.340−0.8390.423
241049.870.644−1.3640.7970.737−0.343−0.8430.425
251069.830.652−1.3690.7950.737−0.341−0.8420.425
261089.790.664−1.3890.8090.748−0.348−0.8570.432
271109.760.675−1.3980.8030.750−0.342−0.8520.428
281129.720.683−1.4120.8200.759−0.355−0.8710.440
291149.680.689−1.4070.8020.753−0.341−0.8530.428
301169.640.700−1.4350.8290.769−0.359−0.8830.446
311189.600.716−1.4810.8660.798−0.377−0.9200.466
321209.570.719−1.4770.8580.793−0.373−0.9130.462
331229.530.727−1.4910.8620.800−0.372−0.9160.462
341249.490.743−1.5430.9080.833−0.397−0.9640.489
351269.450.749−1.5520.9130.837−0.402−0.9710.493
361289.410.762−1.5960.9510.864−0.422−1.0100.515
371309.380.761−1.5700.9180.844−0.403−0.9770.496
381329.340.769−1.5950.9370.860−0.412−0.9970.506
391349.300.767−1.5670.8970.837−0.386−0.9560.481
401369.260.785−1.6370.9690.882−0.432−1.0320.527
411389.220.802−1.6881.0120.914−0.454−1.0760.550
421409.190.815−1.7191.0320.931−0.463−1.0970.562
431429.150.830−1.7551.0540.953−0.470−1.1180.571
441449.110.834−1.7711.0700.962−0.480−1.1350.581
451469.070.850−1.8021.0850.980−0.483−1.1500.587
461489.030.856−1.8271.1110.996−0.500−1.1780.604
471508.990.865−1.8421.1121.003−0.496−1.1780.602
481528.960.858−1.7891.0650.966−0.477−1.1340.580
491548.920.844−1.7541.0430.946−0.467−1.1120.568
501578.860.853−1.7651.0500.952−0.471−1.1200.572
511618.790.868−1.7921.0610.966−0.472−1.1300.576
521658.710.878−1.7941.0520.963−0.467−1.1230.572
531698.630.883−1.7701.0180.943−0.447−1.0900.553
541738.560.905−1.8201.0560.974−0.465−1.1290.573
551778.480.905−1.8131.0490.969−0.460−1.1200.568
561818.400.915−1.8291.0570.977−0.464−1.1300.573
571858.330.931−1.8591.0710.993−0.467−1.1440.578
581898.250.949−1.8841.0751.004−0.465−1.1490.579
591938.180.963−1.9271.1181.033−0.489−1.1930.604
601978.100.976−1.9511.1311.046−0.493−1.2060.610
612018.020.987−1.9581.1281.046−0.492−1.2050.609
622057.950.995−1.9571.1191.042−0.489−1.1990.606
632097.871.000−1.9631.1221.044−0.491−1.2030.609
642137.801.017−2.0001.1381.064−0.492−1.2170.613
652177.721.031−2.0141.1471.069−0.502−1.2310.623
662217.641.052−2.0761.1901.106−0.520−1.2740.644
672257.571.059−2.0881.2111.114v0.538v1.2980.660
682297.491.068−2.0801.1911.103−0.530−1.2830.652
692337.421.103−2.1841.2731.167−0.569−1.3660.696
702377.341.114−2.2171.3001.189−0.581−1.3920.709
712417.261.120−2.2111.2931.180−0.583−1.3890.710
722457.191.148−2.2961.3561.231−0.614−1.4540.745
732497.111.161−2.3201.3681.243−0.619−1.4670.751
742537.031.170−2.3551.3941.264−0.631−1.4930.765
752576.961.197−2.3961.4081.282−0.637−1.5110.774
762616.881.225−2.4921.4911.342−0.681−1.5960.821
772656.811.253−2.5511.5171.372−0.687−1.6230.832
782696.731.288−2.6721.6221.447−0.745−1.7310.893
792736.651.305−2.7051.6271.457−0.750−1.7430.900
802776.581.321−2.7901.7031.510−0.792−1.8210.944
812816.501.327−2.7901.6841.504−0.777−1.8020.932
822856.431.398−2.9811.8211.611−0.850−1.9491.013
832896.351.448−3.0831.8651.660−0.867−1.9991.037
842936.271.493−3.1361.8541.672−0.856−1.9971.033

[9] The spectral dependence of our f(α) models is shown in Figures 3a and 3b for “mare” and “not mare. ” These curves show a model spectrum of the lunar maria and highlands in 10° increments of solar phase angle, with both raw fits and smoothed values shown. Figure 3c shows the f(α) model spectra normalized to 1.489 μm to illustrate substantial reddening for the lunar highlands for observations at large solar phase angles. Our mare model was underconstrained at solar phase angle less than 35° and cannot be used to estimate solar phase reddening, although recent telescopic studies suggest that the color index C(600/470 nm) grows more quickly with solar phase angle for highlands than mare at solar phase angles α less than 40–50° [Kaydash et al., 2010].

Figure 3.

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

[10] The values listed in Tables 1 and 2 can be used to correct M3 spectra and images to an arbitrary viewing geometry, with the caveat that our models were not well constrained at low solar phase angles below 35° and 25° for the “mare” and “not mare” models, respectively. For the initial analysis of M3 data the science team normalized the measurements to a solar phase angle of 30° to correspond to the geometry of the RELAB experiments [Pieters, 1983]. Figure 4 is a graphical rendition of these corrections, f(30°, λ)/f(α, λ).

Figure 4.

A graphical rendition of the “not mare” solar phase angle correction. These are the factors as a function of wavelength and solar phase that correct the “reflectances” (I/F corrected by the factor (μo + μ)/μo) to 30° solar phase.

3. The Performance of the Photometric Model

[11] As a test of the photometric model, a histogram of normal reflectances in the visible (0.54 μm) channel is shown in Figure 5 after the photometric corrections for two regions on the Moon: the highlands and the lunar maria. These two regions were extracted from focus areas selected for intensive calibration in the USGS's Robotic Lunar Observatory (ROLO) dedicated ground-based lunar calibration project [Kieffer and Wildey, 1996; Kieffer and Stone, 2005; Buratti et al., 2011]. The reflectance values for the highlands were extracted from the area of chip 9 in the ROLO database at a latitude of −17.21° and longitude of 20.10° and the reflectance values for the maria were extracted from chip 0 in Mare Serenitatis at a latitude of 19.06° and a longitude of 20.47°. (the precise ROLO chip 0 and chip 9 locations were not available in the M3 data set.) Figure 5 shows histograms of albedo (normal reflectance) for both regions. The mean reflectance of 0.06 for the maria and 0.11 for the highlands is low but reasonable, although we note that these numbers are based on a model extrapolation to a solar phase angle of 0° rather than on actual measurements at opposition. Subsequent modeling of the opposition surge will increase these numbers by a factor of 30–40%., as suggested by the behavior of the ROLO and Clementine models presented in Figures 2a and 2b.

Figure 5.

A histogram of normal reflectances for both the highlands and maria, produced by the model 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 angles were not attained during OP1 and OP2.

[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 at 7.5° N and 59.0° W in Oceanus Procellarum. This archetypical lunar swirl exhibits large albedo variations, and is thus an ideal feature for applying and testing a photometric model. Figure 6a is a mosaic of Reiner Gamma produced without any photometric corrections, but with calibrated radiance data. Figure 6b shows the same mosaic corrected for photometric effects following the method outlined in Section 2, and using the correction for the maria regions.

Figure 6.

An example of the use of the photometric function described in this paper to correct an M3 mosaic of Reiner Gamma, a lunar swirl. (a) The 1.489 μm mosaic uncorrected for any photometric effects. (b) The mosaic correct with the maria photometric function. The Reiner Gamma swirl is about 70 km wide.

[13] A second example making use of the highlands photometric function (“not mare” model) is shown in Figure 7. This region covers the Orientale Basin, which is about 900 km in diameter and is located at 19.4°S and 92.8° W. Figure 7a is the uncorrected mosaic, while Figure 7b shows the correction for 0.54 μm. Figure 7c shows observations obtained at 1.489 μm corrected with the “not mare” model for that wavelength.

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 photometric corrections. (b) An image in the visible (0.54 μm) corrected for all photometric effects with our model and (c) the correction at 1.489 μm. (d) A 2.2/1.2 μm ratio map of the Orientale Basin showing a low-level residual striping pattern. The bright areas at the center and rim of Orientale are likely due to thermal radiation that has not been subtracted; these artifacts are in the lowest-albedo regions where the thermal flux should be highest.

4. Discussion and Future Work

[14] The photometric model presented in this paper is a purely empirical one that is useful for correcting spectra and multispectral mosaics in the 0.404 to 2.983 μm spectral range for the effects of viewing geometry. The functions presented apply only to the phase angle range of the observational set (24°–90°). Further work will present a more detailed function for the opposition surge. Another improvement would be to partition the correction between the mare and “not mare” functions on the basis of the albedo of the surface or geographical location. For example, Figure 7, the correction for the Orientale Basin, was made with the “not mare” photometric function. The correction is good except for the area in the center of the basin, the most mare-like region in the mosaic.

[15] Figure 7d shows the 2.1/1.2 μm ratio of the photometrically corrected mosaics for the Orientale Basin. The rippling effect of about <5% represents the total photometric integrity of the M3 data after all calibrations and photometric corrections. The pattern is due to a combination of all likely effects not corrected for in the M3 calibration procedures in addition to inaccuracies in our photometric model: residual scattered light from the M3 instrument and stray light from the Chandrayaan-1 spacecraft; imperfect flatfielding (a cross-track average to the flatfield combining a range of emission angles was used); errors in the calculation of the incident and emission angles due to surface topography and other factors, 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. Thermal emission becomes significant (∼a few % relative to reflected sunlight) at 2.4 μm for a typical lunar surface with an albedo of 0.1 [Hapke, 1993]. For lunar maria the thermal emission is even more substantial. Thermal radiation exhibits an isotropic scattering law proportional to the cosine of the emission 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 becomes less and less applicable if the thermal radiation has not been subtracted. With the emission angle changing in the cross-track direction of our scans, this factor can become significant, particularly if one is ratioing two photometrically corrected mosaics at different wavelengths (see Figure 7d). Future work will thus need to incorporate a thermal model, particularly beyond 2 μm; our model is most accurate for wavelengths where thermal emission is not significant. Figure 7d shows bright areas at the center and rim of the Orientale Basin which are consistent with a higher temperature and thermal emission at 2.2 μm in the low-albedo regions. Current limitations on the accuracy of wavelength-dependent calibrations (flatfielding and scattered light for example) may also cause wavelength-dependent artifacts in the data. Our model works particularly well (∼1–2% relative error) when producing mosaics of M3 images at individual wavelengths below 2 μm.

Acknowledgments

[17] The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

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