Lunar opposition effect as inferred from Chandrayaan-1 M3 data

Authors


Corresponding author: V. Kaydash, Astronomical Institute of Kharkov V.N. Karazin National University, Sumskaya 35, Kharkov 61022, Ukraine. (vgkaydash@gmail.com)

Abstract

[1] The shadow-hiding and coherent backscattering enhancement mechanisms are considered to be the major contributors to the brightness opposition effect of the Moon. However, the actual proportions of the mechanisms at different phase angles still remain not well determined. In order to assess the lunar phase function across small phase angles, we utilize imaging spectrometer data acquired by the Moon Mineralogy Mapper (M3) onboard the Chandrayaan-1 spacecraft. We calculated phase functions of apparent reflectance and color ratios in the wavelength range 541–2976 nm for several mare and highland areas. As inferred from changes in the wavelength dependence of the phase curves, the shadow-hiding effect is a major component of the brightness opposition surge at phase angle > 2°. The coherent backscattering enhancement may contribute some to the opposition effect at phase angles < 2°. We found nonmonotonic behavior of color-ratio phase curves that reveal the minimum at ~ 2–4°. Using lunar observations, this is the first reliable evidence of the colorimetric opposition effect found earlier for lunar samples.

1 Introduction

[2] The brightness dependence on phase angle α (phase function f(α)) carries information about the surface composition and microstructure, and therefore photometry is used to study the Moon with telescopic and orbital spectrophotometric surveys [Muinonen et al., 2011; Shkuratov et al., 2011; Hapke et al., 2012]. Among the remarkable photometric properties of the Moon is the strong backscattering or brightness opposition effect (BOE). This fairly wide backscattering surge of the Moon at phase angles α ≤ 7° can be caused by three lightscattering phenomena: (1) single particle scattering, (2) shadow-hiding phenomenon with incoherent multiple scattering, and (3) the coherent backscattering enhancement [e.g., Shkuratov et al., 2012].

[3] The phase functions of single particles typically have a small backscatter lobe if the particle is not too small as compared to wavelength. Usually, the backscattering surges of aggregated or irregularly shaped particles have a width nearly 15° and amplitude ~ 1.5 [e.g., Zubko et al., 2008]. The second phenomenon relates to shadows cast by regolith particles on each other and incoherent multiple scattering that reduces the shadows. The third phenomenon results from constructive interference of rays propagating in a particulate media along direct and time reversal trajectories. When α approaches zero the constructive interference obtains a preference, which produces a backscattering spike if single scattering albedo is high enough to generate multiple scattering [e.g., Shkuratov et al., 2011, 2012]. Each of the three phenomena depends on physical properties of the regolith, such as its porosity and sizes of particles. Thus, it is important to interpret photometric measurements in terms of which mechanism dominates the lunar BOE [e.g., Shkuratov et al., 2011, 2012; Hapke et al., 2012].

[4] The Moon is not observed from the Earth at α < 1° out of eclipse, thus studies of the BOE were substantially intensified after retrieving data from orbital surveys of the last decades, e.g., Clementine, SMART-1, Chandrayaan-1, Kaguya, Change'E-2, and Lunar Reconnaissance Orbiter (LRO). Comparison of the reports on the space-derived photometry obtained by several groups of researchers reveals rather contradictory results. The conclusion by Buratti et al. [1996] on the main cause of the lunar opposition surge (0–5°) was thought to be due to the shadow-hiding phenomenon while the coherent backscattering effect plays only a minor contribution. Also, Hillier et al. [1999] deduced that only a small coherent backscatter component exists for the brightest regions over the 0.75–1.00 µm wavelength range. Hapke et al. [1998] and Helfenstein et al. [1997] found the contribution of coherent backscatter enhancement to be significant at phase angles < 5°. Kreslavsky et al. [2001] reported that the coherent component manifests itself only in the near-infrared (NIR) range where lunar albedo is high enough [see also Shkuratov et al., 2012].

[5] Recent analysis of the LRO data for highland regions by Hapke et al. [2012] showed that the coherent backscatter enhancement contributes nearly 40% in the UV, increasing to over 60% in the red. A significant contribution of this enhancement to the BOE at α < 10° is also assessed in Muinonen et al. [2011]. Thus, several authors concluded that the BOE contains a large component of coherent backscattering. On the other hand, a significant dependence of the angular width of the BOE on wavelength, which was theoretically predicted, has not been found [Hapke et al., 2012]. Previous studies of the wavelength dependence of the BOE angular width over a range of 415–1000 nm [Buratti et al., 1996; Hillier et al., 1999; Shkuratov et al., 1999; Kreslavsky et al., 2000] also did not detect wavelength dependence. Shkuratov et al. [2011, 2012] summarizing results of lunar BOE studies noted that to estimate the contributions of the shadowing and coherent enhancement phenomena to the backscattering peak at each phase angle is a difficult problem, even if theoretical models easily provide this. The problem is that models, using the coherent backscattering effect in an ad hoc manner, can force the effect to compensate for shortcomings of other model components to achieve the best fit.

[6] There is even a point of view presented recently by Hapke et al. [2012]: “our current understanding of the coherent backscatter opposition effect is incomplete or perhaps incorrect.”

[7] Thus, we may conclude that a detailed study of the lunar phase function in a wide spectral range at angles α < 5° is crucial for the testing of the BOE mechanisms and assessing the physical properties of the lunar surface. In this paper, we utilize spectral image cubes acquired by the Chandrayaan-1 Moon Mineralogy Mapper (M3) in a spectral range 541–2976 nm to estimate the wavelength dependence of the lunar phase function at small phase angles.

2 M3 Data Used

2.1 General Description

[8] The M3 is an imaging spectrometer, which was selected as a NASA Discovery Mission of Opportunity in 2005 and was launched on 22 October 2008 on board the Chandrayaan-1 spacecraft of the Indian Space Research Organization [Pieters et al., 2009]. The M3 instrument operates as a pushbroom imaging spectrometer with a field of view of 24°. Key spectral and radiometric characteristics of the instrument are the following: (1) spectra are measured in the 430–3000 nm range with 10 nm constant sampling and FWHM < 15 nm (low-resolution mode combines twice lower spatial and 20–40 nm spectral resolution), (2) 12 bits sampling dynamic range, (3) response linear to 1%, (4) absolute accuracy within 10%, and (5) SNR > 400 when imaging equatorial regions [Lundeen et al., 2011; Green et al., 2011]. Optimal surface lighting at image acquisition and numerous technical challenges during the flight limited the entire orbital survey to several optical periods. Each period is characterized by different spatial coverage, illumination/observation geometry, and spatial resolution [Lundeen et al., 2011; Boardman et al., 2011]. Mostly nadir observations were carried out in the low-resolution mode during the mission.

[9] Science data calibration for M3 includes a sequence of processing algorithms applied to calibrate downlinked data spectrally, radiometrically, and spatially [Green et al., 2011]. The M3 Level 1b calibration algorithm converts the measured raw digital numbers to spectral radiance units of spatially coregistered images of each wavelength, calculating the surface coordinates and illumination/observation geometry of all pixels. The M3 Level 2 processing converts the at-sensor radiance data (Level 1b) to radiance factor values at an incidence angle i of 30° and an emission angle e of 0°. This level includes a photometric correction, an algorithm for removing the thermal contribution to spectra (mainly after 2000 nm for warm regions), and an optional Ground Truth correction based on laboratory spectra of the mature highland soil 62231 from the Apollo 16 landing site [Lundeen et al., 2011]. The Ground Truth calibration allows one, for instance, to relate M3 and Clementine UV-Vis data. This paper does not exploit this calibration. We used M3 images radiometrically calibrated in spectral radiance (W/m2/µm/sr), then converted into radiance factor values, applying the thermal correction. The M3 images we used are publicly available in the Planetary Data System at http://pds-imaging.jpl.nasa.gov/data/m3/.

[10] The data presented here were acquired in the M3 Global Mode; thus, they are of a lower spectral resolution than in the M3 Target Mode. Taking into account the elimination of degraded spectral channels [Lundeen et al., 2011], there are in total 83 usable spectral bands from 540.84 to 2976.2 nm with a spectral resolution of 20 nm for the bands from 730 to 1548 nm and 40 nm for other spectral ranges.

2.2 Source M3 Data

[11] In the current study, we focus on M3 images with the so-called opposition geometry (i.e., at α ~ 0). Because of the near nadir-looking mode (e ~ 0), this illumination/observation geometry naturally occurs at the near-equatorial regions of the Moon, where i ~ 0. Among several optical periods of the mission only the period OP2C1 (20 May to 23 June 2009) satisfies the required opposition geometry along with large coverage of the Moon. All M3 images of the OP2C1 are taken in the low-resolution Global imaging mode at 280 m/pixel [Lundeen et al., 2011].

[12] Figure 1 presents a portion of the M3 mosaic for the reflectance (radiance factor) at a wavelength λ = 1489 nm. This is adopted from http://target.lroc.asu.edu/da/qmap.html. The data resulted from OP2C1 images calibrated with the Level 2 pipeline; the Ground Truth calibration is not applied. We note the diffuse latitudinally elongated spots pointed by arrows in Figure 1. The spots are associated with every image constituting the mosaic. The centers of these features that were observed in both mare and highland regions correspond to zero-phase-angle (opposition) points. A photometric correction has been applied to these data [Besse et al., 2013]; however, it does not perfectly account for the opposition surge, as one may see in Figure 1.

Figure 1.

A portion of M3 reflectance mosaic at 1489 nm rendered at 125 m/pixel. Observations within the optical period OP2C1 are used. Mosaic was provided by Brown University / ACT Corporation. Opposition spots are marked with arrows.

[13] In total the M3 instrument has imaged the Moon at the opposition geometry about 40 times during the OP2C1 phase of the mission. These M3 frames being distributed in the ±7° latitude band along the equator could be used in order to study the opposition effect. However, not all of the images are suitable because of close location of the opposition spot center to the frame edges. In this case, the number of pixels corresponding to the opposition spot is greatly reduced, and hence the phase function determination becomes poor.

[14] We here are interested in a separate investigation of dark and especially bright sites in order to detect and quantify the effect of coherent backscatter enhancement. However, some of the M3 frames with the opposition spot are located in the mare-highland transition zone, or consist of large bright and dark patches thus being inhomogeneous in albedo. We restricted our analysis to several bright highland and dark mare (for comparison) sites with an appropriate location of the opposition spot inside the frame.

[15] We selected six M3 subframes presented in Figures 2a–2f. The top row of these figures shows images of three mare areas in a southeastern portion of Oceanus Procellarum: south to the Reiner-γ feature (Figure 2a), near the craters Encke B and C (Figure 2b), and south to the 30 km sized crater Reiner (Figure 2c). The bottom row represents images of two nearside highland areas: near the 18 km crater Riccioli H (Figure 2d), south to the 14 km crater Hevelius A (Figure 2e), and an area of farside highland with the opposition point at 110.6°W, 5.5°S (Figure 2f). Each image is a subframe of the corresponding M3 image, information about which can be found in Table 1. The second column of the table contains links to the appropriate images shown in Figures 2, 6, 7, 9-11 with curves or their subparts. The α = 0 points are marked with stars in the images. Table 1 presents the selenographic coordinates of these points. The cross-track (longitudinal) image width is 304 pixels, which corresponds to 85 km swath. The detector had fairly high temperature, but of close values (Table 1); this significantly reduces the difference in detector responses due to thermal nonlinearity [Green et al., 2011; Lundeen et al., 2011].

Figure 2.

Six areas under study: (a, b, c) upper row and (d, e, f) lower row present mare and highland areas, respectively. The left part of Figures 2a–2f presents reflectance at 1489 nm and the right part the corresponding distributions of masked pixels. Opposition spots are marked with stars. North is up, west to right elsewhere in the images. The M3 image numbers and coordinates of zero-phase angle spots are given in the Table 1.

Table 1. The M3 Images Selected; All of Them Were Acquired in the Suboptical Period OP2C1
M3 Image IDMare/Highland (Figures 2, 6, 7, 9-11)Detector T°KOpposition Point Longitude, (°)Opposition Point Latitude, (°)
M3G20090613T032520Mare (a)164.96−58.73
M3G20090611T090220Mare (b)164.12−35−2.5
M3G20090612T183813Mare (c)165.3−54.13.5
M3G20090614T090712Highland (d)165.9−75.31.0
M3G20090613T200943Highland (e)165.38−68.21.6
M3G20090617T005342Highland (f)168.71−110.6−5.5

2.3 M3 Data Corrections Used

[16] There is a problem related to residual band-to-band artifacts in the spectra caused by residual variability in the M3 detector response that was not compensated with the flat field [Nettles et al., 2011; Green et al., 2011]. In Figure 3a we present three spectra extracted from a fragment of the calibrated M3 image M3G20090417T155652. The imaged scene is a mare region in Oceanus Procellarum with several young craters with immature regolith around. Spectra SP1–SP3 are shown in Figure 3a with open circles and their sampling locations are pointed in Figures 3b–3e. The spectra SP1 and SP3 exhibit pronounced 1 and 2 µm features due to craters with low maturity, while the mare spectrum SP2 is almost featureless. Figures 3b and 3d present the 950 nm reflectance and 950/750 nm color ratio, respectively. The latter is known to be sensitive to the maturity degree of the lunar regolith [e.g., Lucey et al., 2000; Pieters et al., 2006]. Vertical striping over the M3 frames largely affects color-ratio images. This is also revealed in the spectral curves (Figure 3a) that exhibit some systematic and variable noise. To reduce this noise, we smoothed the spectra with a Gaussian filter using a kernel size of 40 nm. The filtered spectra are shown in Figure 3a with lines. Corrected images corresponding to 950 nm band and 950/750 nm color ratio are shown in Figures 3c and 3e. We note a substantial decrease in the noise. We applied the same filtering to all M3 data analyzed in this paper.

Figure 3.

Examples of the spectra extracted from calibrated M3 image M3G20090417T155652 (acquired at 17 April 2009, detector temperature is 147.03 °K, α = 60°). (a) Spectra SP1–SP3: open circles present Level 2 calibrated data, lines are the same after the Gaussian filtering. (b and c) Radiance factor at λ = 950 nm before and after the Gaussian filtering. (d and e) Same for the color ratio 950/750 nm. The SP1–SP3 sampling locations are pointed with circles in images. The dynamic range of pairs Figures 3b, 3c and 3d, 3e is the same. The center of the area shown in Figures 3b–3e is located at 325.06°W, 5.2°S.

[17] The residual cross-track gradient in the M3 data and methods for reducing this likely instrumental effect were recently described and discussed in Besse et al. [2013]. The empirical correction contains scalar values that can be added to the Level 2 data to reduce the boundary differences between images constituting the large-scale mosaic. We assessed the cross-track effect on our results showing that it is negligibly small (see section 3.1).

[18] The photometric correction of M3 images consists of two parts: the Lommel-Seeliger function described limb-darkening and the wavelength-dependent phase function [Lundeen et al., 2011]. As we are interested in quantification of the wavelength dependence of BOE, we did not apply the Level 2 M3 phase function [Besse et al., 2013], but corrected the data with the Lommel-Seeliger function for consistency with many previous works [Nettles et al., 2011; Hicks et al., 2011; Buratti et al., 2011; Hapke et al., 2012; Besse et al., 2013]. This function is widely applied for planetary data, though it does not provide satisfactory accuracy at arbitrary illumination and observation [Shkuratov et al., 2011, 2012]. However, the M3 data have been acquired at the opposition geometry, when the variations related to the global brightness distribution are small.

2.4 Processing the M3 Photometric Data

[19] There are different methods to retrieve phase functions near opposition from data of orbit imaging. A simple averaging over different images to diminish local albedo variations was used by Buratti et al. [1996]. Moving of the opposition point between consecutive orbital images and calculation of the logarithmic derivative of the phase function from phase-ratio images was used by Shkuratov et al. [1999]. Mapping of the phase ratio f(α1)/f(α2) at α1 ~ 0 and α2 > 0 was also applied [Shkuratov et al., 1994, 1999, 2011; Kreslavsky et al., 2000; Kaydash et al., 2008; Velikodsky et al., 2011].

[20] Before choosing a method to retrieve phase functions from M3 data, we checked for the distributions of the incidence, emergence, and phase angles over the selected scenes. Examples of the distributions are shown along with scale bars in Figures 4b–4d for the area presented in Figure 4a (see also Figure 2b). Variations of the emission angle in the horizontal direction are rather large, from 3° to 15°; this is a consequence of 24° cross-track of M3 field of view [Lundeen et al., 2011]. This results in 0°–18° variations of α over the scene. Taking into account the substantial α variation in a single scene, which provides significant changes of f(α), we used the simple averaging method, when apparent reflectance values are averaged over an image in concentric elliptical bins of constant phase angles. When averaging the reflectance we have masked mare and highland craters, low reflectance areas (for highlands), and bright features in mare areas, e.g., the Reiner-γ formation. Maps of masked areas are presented in the right subimages of Figures 2a–2f. The maps are initially produced with a filter that exaggerates local albedo contrasts calculating the second derivative in the central pixel of a small window sliding over the images. This procedure effectively filters out tiny craters, steep-slope details, bright rays, etc. Then we have outlined manually large objects that disturb albedo homogeneity over the scene, e.g., Reiner-γ (Figure 2a), dark units, craters (Riccioli H in highland scene, see Figure 2d) etc. Some of the band-to-band artifacts in the M3 spectra caused by the residual variability in detector response are seen as thin along-track vertical strips. We applied to the M3 data a procedure, which largely reduces this variability, though some strips remain not entirely compensated. These objects also are manually outlined in frame masks (e.g., Figures 2a and 2c).

Figure 4.

Distributions of photometric angles over the selected part of the M3 image M3G20090611T090220 (Figure 2b). (a–d) Apparent reflectance at 1489 nm, incidence angle i, emission angle e, and phase angle α. Angle ranges in degrees are shown in scale bars. The opposition point is marked with stars.

[21] The reflectance averaging in concentric elliptical bins of constant α requires obviously the true location of opposition point. We checked whether the actual location of the point as inferred from the reflectance images is consistent with the zero-phase-angle point obtained from the illumination/observation geometry supplied with the Level 1b calibration. We averaged reflectance in the bins of constant α in four quadrangles that are produced by crossing the image by vertical and horizontal lines intersecting in the probable opposition point (Figure 5a). If the point coincides with the actual opposition point, then the four phase curves f(α) should maximally coincide with each other. We searched for the most probable center of the opposition spot in the neighborhood of the visible opposition spot each time calculating four phase curves comparing them. An example of calculation of the actual zero-phase-angle point from the M3 reflectance data is given in Figure 5b. Phase curves in the upper plot of Figure 5b are derived with location of zero-phase point from supplied M3 geometry information; phase curves in the lower plot are derived with location of the point as inferred from the reflectance images. We note that for all six sites the coincidence of Q1..Q4 phase curves is better for calculated values of zero-phase point than for one supplied with M3 calibrated data. In such a way for the six scenes under study we found shifts between the calculated and supplied opposition points. We introduced the shifts to the spatial distributions of i, e, and α before calculations of the phase dependences of radiance factor. The shifts vary from 3 to 10 km with major components oriented along-track. We consider that these shifts are owing to a problem with the Chandrayaan-1 pointing after both star trackers were lost and the orbit altitude was raised [Boardman et al., 2011; Lundeen et al., 2011]. The finally determined opposition points are marked by stars in Figures 2a–2f and listed in Table 1.

Figure 5.

(a) A sketch of the division of a subframe (see Figure 2a) into four quadrangles in the probable point α = 0. Filled circle marks the opposition point as inferred from the geometry of illumination and observation supplied with the Level 1b calibrated data. Star marks the point calculated in the current study. The distance between these points is ~10 km. (b) Phase curves for Figure 5a calculated in four quadrangles. Q1, Q2, Q3, and Q4 curves are derived for corresponding quadrangles. Curves marked with “supplied” are derived for location of zero-phase point from supplied M3 geometry information, curves marked “calculated” are derived for calculated location of zero-phase point.

[22] Phase curves were sampled in every 0.3° with a bin width of 0.3°. The first sampling bin including the opposition point integrates all pixels inside the 0–0.3° range. This bin consists of the minimal number of pixels; in our case it was ~400–600 depending on the number of masked pixels for each scene. The width of bins is a trade-off between the resolution of phase curve and the number of pixels falling in each bin, which influences the photometric accuracy.

3 Spectral Behavior of BOE: Results and Discussion

3.1 Normalized Phase Function

[23] In Figure 6 we present spectra of radiance factor for the six analyzed areas. The spectra are produced by averaging the radiance factor values for all 83 wavelengths inside the phase angle bin centered at α = 10°. Bars for each spectrum show the standard deviation in this bin. Thus, the bars do not characterize errors of spectral measurements; they only describe albedo variations in the bin. As can be anticipated the mare and highland areas form two albedo groups.

Figure 6.

Spectra of radiance factor for the six analyzed sites. Spectral intensities are derived by averaging pixel values in the phase angle bin centered at α = 10°. The bars for each spectrum are equal to the standard deviation of albedo variations in a selected bin.

[24] We calculated the radiance factor as a function of α for the six areas shown in Figures 2a–2f at all 83 wavelengths and show the phase curves in Figures 7a–7f. The plots in upper and lower rows of Figures 7a–7f correspond to the mare and highland regions, respectively. Each curve is normalized on its own value at α = 10°. The legend of the rainbow-style color coding for all 83 curves is presented in Figure 7g, so that bluish colors refer to shorter wavelengths and the reddish ones represent longer wavelengths. Numbers shown in Figure 7g are wavelengths in nanometers. Figures 7a–7f show the phase curves with small peaks and wavy features. Their reasons are the residual vertical streakiness remained after the Gaussian spectral smoothing (see Figures 3b–3e) and real albedo variations over the scenes. Although averaging over α bins decreases these effects, some noise still presents on the final phase curves and can be seen as false details of phase functions simultaneously in all spectral bands. Owing to real differences in illumination for the six studied areas, the maximal values of α for each scene vary from 13° to 18°. We also have taken into account the decreasing number of pixels (and hence quality of phase function) when α bins approach the frame edges. This typically occurs for α > 11° when the number of pixels in proper α bins decreases twice as compared to most populated bins. Thus we stopped all phase functions at 11° for their reliable comparison.

Figure 7.

Phase functions for areas shown in Figures 2a–2f. (a–c) Mare and (d–f) highland sites. Each curve is normalized on its own value at α = 10°. (g) For the rainbow-style color coding all 83 curves from the wavelength range 541–2976 nm are used.

[25] In Figure 8 we present an example of f(α) with and without compensation of residual cross-track gradient in the M3 data discussed in Besse et al. [2013]. The correction intends to improve mosaicked images by reducing boundaries between them. This is dependent on λ, see http://pds-imaging.jpl.nasa.gov/data/m3/CH1M3_0004/EXTRAS/CROSSTRACK_CORR_TABLES/. As can be seen in Figure 8, the curves reveal very little differences between two sets of data. The result is obtained for the region shown in Figure 2a, but the same is observed for all areas under study. Very small influence of the correction is also found for color ratios C(α) = f(λ1,α)/f(λ2,α).

Figure 8.

Phase functions for selected spectral bands. Each curve is normalized on its own value at α = 10°. Dots show phase curves derived from M3 data subjected to the cross-track correction [Besse et al., 2013], solid curves present phase curves derived from noncorrected M3 data.

[26] Inspection of the phase functions in Figures 7a–7f and Figure 8 reveals an inverse wavelength dependence of their steepness: the slope of phase function is greater for shorter wavelengths both for highlands and mare areas. The slope and shape of phase curves slightly differ from area to area, but one may conclude that the phase ratio f(0.15°)/f(10°) is about 1.4–1.6 for all these areas, with an exclusion (~1.7–1.9) for the area displayed in Figure 2d.

3.2 Wavelength Dependence of Phase Ratios

[27] For a detailed investigation of the phase slope, we calculated several phase ratios f(0.15°)/f(10°), f(2°)/f(10°), f(5°)/f(10°), f(6°)/f(10°), f(7°)/f(10°) and plotted them as a function of wavelength λ for the studied areas (Figures 9a–9f). All the ratios decrease with λ, only f(0.15°)/f(10°) reveals almost neutral behavior for some areas. Since the lunar reflectance generally increases with λ, we may conclude that the slope of phase curves diminish with surface albedo increasing. Such a behavior of the phase curves is characteristic when incoherent multiple scattering weakens the shadow-hiding effect. If the coherent backscattering enhancement is significant, the opposite dependence of the slope on albedo should be observed [Shkuratov et al., 2012]. Indeed, increasing albedo implies the rising multiple scattering component that increases the amplitude of the coherent backscattering spike.

Figure 9.

Spectral dependences of ratios f(0.15°)/f(10°), f(2°)/f(10°), f(5°)/f(10°), f(6°)/f(10°), and f(7°)/f(10°) for six areas from Figures 2a–f. (a–c) Mare and (d–f) highland sites. Each spectrum obtained at a given phase angle is divided to spectrum at α = 10°. The curves are not normalized.

[28] To clarify the role of the coherent backscatter enhancement in the formation of phase function, a more subtle analysis is needed. Therefore, we study another set of phase ratios as functions of wavelength: f(0.15°)/f(1°), f(2°)/f(3°), f(5°)/f(7°), and f(6°)/f(10°). The result is presented in Figures 10a–10f. All curves are normalized to their own values at 541 nm, so all phase ratios equal unity at the shortest wavelength 541 nm. The reason for choosing these ratios is that the nonlinear increase of the reflectance related to the coherent backscattering enhancement begins with α ~ 5°. Thus, one may formally consider the phase ratios f(0.15°)/f(1°) and f(2°)/f(3°) as an assessment of the enhancement.

Figure 10.

Phase-angle ratios f(0.15°)/f(1°), f(2°)/f(3°), f(5°)/f(7°), and f(6°)/f(10°) plotted as a function of λ for areas from Figures 2a–2f. (a–c) Mare and (d–f) highland areas. Phase ratios are normalized to their own values at 540 nm. The color strip coding of the wavelengths is the same as in Figures 7a–7f.

[29] As before the plots for mare and highland areas are shown, respectively, in the upper and lower rows of Figures 10a–10f. The color strip near the wavelength axis is the same color coding as in Figures 9a–9f. The thin lines in the color legends correspond to the 20 nm spectral resolution for the bands from 730 to 1548 nm.

[30] The f(0.15°)/f(1°) plots (solid black lines) reveal an increase of the phase slope with λ for all areas from Figures 2a–2f. One exclusion is almost neutral dependence for the area in Figure 2d, for which the spectrum of reflectance has a minimal phase slope. The greatest increase—up to 6% for the longest wavelengths—is observed in Figures 10a and 10f. Such a behavior means that the BOE amplitude in the range 0–1° tends to increase with albedo in spite of the reverse influence of incoherent multiple scattering. Thus, the wavelength dependence of the ratio f(0.15°)/f(1°) is clear evidence that the coherent backscattering enhancement becomes important for the Moon at α < 2° for λ from 541 to 2976 nm. Overall, this result does not contradict the earlier measurements by Clementine [Buratti et al., 1996; Hillier et al., 1999; Shkuratov et al., 1999; Kreslavsky et al., 2000], where the authors did not find BOE spectral dependence, since they investigated a narrower spectral range 415–1000 nm.

[31] There are factors that may significantly suppress the very narrow coherent spike 0–1° of the Moon. First of all, this is the masking effect related to the Sun that has the angle size of 0.5°. Smoothing with such a “window” destroys a narrow spike decreasing its amplitude. Even a large spike that is very narrow can be almost eliminated if integrated (i.e., averaged) in a rather wide range of phase angles [Shkuratov et al., 1999]. Another reason that can destroy a narrow spike is the bin resolution that is 0.3° in our analysis. It should be emphasized that both these factors affect the coherent backscattering as well as shadow-hiding components of BOE at phase angles small enough. Figures 5b, 7a, 7f, and 8 demonstrate rounding of f(α) at α < 0.5°, as should be. The evenning can be partially compensated when a strong brightness spike takes place at α < 0.5°; perhaps this is observed in Figures 7b, 7d, 7e. However, there is always a phase angle at which the rounding must be formed; pure cusps in electromagnetic wave scattering are impossible. Thus, we may conclude that the Moon probably has a notable effect of the coherent backscattering enhancement at α < 1° in the NIR spectral range, but this effect is smoothed by the Sun whose angular size is 0.5°.

[32] The area presented in Figure 2d has the opposition point located in the closest proximity of the bright crater Riccioli H rim. Although we excluded the crater itself from the analysis when we calculated the phase curves (see Figure 2d), this area has the highest albedo observed in our study. It does not reveal the f(0.15°)/f(1°) increasing with λ and has large f(0.15°)/f(10°) values. These facts do not contradict each other. Indeed, the regolith of young crater rims consists of coarser particles. This provides larger interference bases in multiple lightscattering [Shkuratov et al., 1999], and owing to that the coherent backscattering spike may become narrower than 1°. The high ratio f(0.15°)/f(10°) can be caused by anomalous roughness of crater rims. Large blocks, boulders, and rocks covered with regolith produce a complicated hierarchical topography that can exhibit steep phase curves [Shkuratov and Helfenstein, 2001].

[33] The phase ratio f(2°)/f(3°) (dashed black lines) is almost constant over the wavelength range for mare areas (Figures 10a–10c) and even slightly decreases for highlands (Figures 10d–10f). Thus, one can conclude that the contributions of the coherent backscatter enhancement and incoherent multiple scattering equalize each other in the range 2–3°. This points out that the coherent backscattering spike can be somewhat wider than 2°.

[34] In Figures 10a–10f we also show the phase ratios f(5°)/f(7°) and f(6°)/f(10°). All areas presented in Figures 2a–2f show decreasing ratios with λ. For f(6°)/f(10°) the amplitude decrease is up to 6%. Hence, we confirm an inverse wavelength (albedo) dependence of the steepness of phase curves at the 5–10° phase range [e.g., Velikodsky et al., 2011; Shkuratov et al., 2011]. We note nonmonotonous behavior of phase ratios f(5°)/f(7°) and f(6°)/f(10°) at the wavelength near 1 µm and partially near 2 µm. These features of phase ratios resemble inversed absorption bands in typical lunar spectra. Yokota et al. [2011] also found this effect for ratios f(5°)/f(30°) and f(10°)/f(30°) calculated with Kaguya data for the 500–1600 nm range. The result is quite obvious, as in the absorption bands the incoherent multiple scattering decreases with decreasing albedo.

[35] Thus, taking into account the observed spectral behavior of the phase ratios, we may conclude that shadow-hiding and incoherent multiple scattering are the major components of light scattering at α > 2° even in NIR, where lunar albedo is substantially higher than in the visible spectral range. This is in sharp contrast to the conclusion by Hapke et al. [2012] who suggested that the coherent backscatter enhancement contributes nearly 40% in the UV, increasing to over 60% in the red light.

3.3 Phase Curve of Color Ratio

[36] The described spectral variations of phase functions urge us to investigate phase curves of color ratios C(α) = f(λ1,α)/f(λ2,α). The function C(α) is poorly studied. The color-ratio increasing when λ1 > λ2 (reddening) with α was found many years ago using telescope observations of the Moon [McCord, 1969; Mikhail, 1970; Lane and Irvine, 1973]. Recent telescope data confirmed this reddening [Korokhin et al., 2007; Kaydash et al., 2010]. The same effect has been found with space-derived observations [Buratti et al., 1996; Shkuratov et al., 1999; Kreslavsky et al., 2000; Denevi et al., 2010]. Hicks et al. [2011] provided an empirical photometric function in the 541–2976 nm and confirmed reddening for the highlands at large phase angles up to 80°, though their model was underconstrained at angles less than 35°. Yokota et al. [2011] developed another model for the lunar phase function in the 500–1600 nm range and examined the wavelength dependence of the function; they found phase reddening for the range of α = 5–75°.

[37] On the other hand, a nonmonotonic behavior of C(α) in visible spectral interval was noted by several authors using laboratory measurements of lunar regolith samples. These measurements revealed in some cases a minimum of the function C(α) in the range ~5–15° [O'Leary and Briggs, 1973; Akimov et al., 1979; Shkuratov et al., 1996, 2011, 2012]. Eight lunar samples measured by Hapke et al. [1998] exhibited such a minimum at 3–4° for red/blue color ratios. Korokhin et al. [2007] reported that the minimum would exist at 10° using reanalyzed integral observations of the Moon by Lane and Irvine [1973]. The increase of C(α) at small decreasing α was named the color opposition effect (COE) [Shkuratov et al., 1996].

[38] Using the M3 multispectral data, we are able to obtain new independent information on the phase dependence of color ratios. Figures 11a–11f show the functions C(λ,α) = f(λ,α)/f(λ0,α) calculated for the areas shown in Figures 2a–2f using the 83 color ratios. The rainbow-style color coding is presented in the legend (Figure 11g). As can be seen, the phase curves reveal a clear minimum of C(α) depending on area. At wavelengths below 1000 nm a shallow minimum at 4–6° (or no minimum at all) can be seen. For λ > 1000 nm this minimum is more pronounced and located at 2–4°. We again note an exclusion related to the area shown in Figure 2d that exhibits a monotonic increase of C(α) for all λ. Such a behavior has been observed for a highland sample of immature regolith from the landing site of Soviet Luna-20 [Shkuratov et al., 1996, 2012].

Figure 11.

Phase dependences of color ratio C(α) = A(λ,α)/A(λ0,α), where λ0 = 540.84 nm, for areas shown in Figures 2a–2f. (a–c) Mare and (d–f) highland sites. Each curve is normalized on its own value at α = 2°. (g) For the rainbow-style color coding 83 curves from the wavelength range 541–2976 nm are used.

[39] The minimum can be in principle due to the coherent backscattering enhancement and the spectral difference of the backscattering lobe of phase functions of single-particle scattering [Shkuratov et al., 2012]. The latter mechanism may be responsible rather for wide minima located for regolith samples at α = 10°–15° [Shkuratov et al., 1996]. As for the minimum at α = 2–4°, it is caused rather by the coherent backscattering enhancement. The area shown in Figure 2d does not have the minimum, which is consistent with the assumption that coherent backscattering enhancement peak for immature lunar soils is located at α < 1°.

[40] Our results reveal the colorimetric opposition effect (COE) of the Moon in NIR. While the observed αmin ~ 2–4° values are in accordance with the measurements of C(633/433 nm, α) for eight lunar samples by Hapke et al. [1998], this cannot be treated as evidence of a 40–60% contribution of the coherent backscattering to the total flux, which was deduced in Hapke et al. [2012]. We interpret our results as the coherent backscattering noticeably affects lunar phase curves only at α < 2°, and its values do not exceed 5–7% even in NIR spectrum range.

4 Conclusion

  1. [41] Using data of the Chandrayaan-1 imaging spectrometer M3, we suggest a new detailed description of the lunar opposition effect and its dependence on wavelength in the range 541–2976 nm. Our analysis showed that the data are suitable for this task if we reduce the noise using a Gaussian smoothing filter decreasing both spectral and spatial resolution. We found that errors in M3 pointing may change the actual position of the opposition point in images acquired during the suboptical period OP2C1 of the mission. Real locations were reconstructed from the images themselves and used for further photometric analysis.

  2. [42] For M3 data, phase dependences of the color ratios reveal a nonmonotonic behavior or COE. We found a minimum of phase functions of color ratio at αmin ~ 2–4°. At larger phase angles reddening of spectra are observed. At smaller phase angles our result clearly shows in general spectral blueing. This minimum would be considered as the first reliable indicator of a small coherent backscattering enhancement component at α < 1–2°.

  3. [43] We calculated phase curves of reflectance and color ratios for six mare and highland areas in the phase angle range 0–10°. The slope of color-ratio curves increases with wavelength at α from 0–1°. In the 2–3° range this slope is spectrally insensitive, and for the 5–10° range it decreases with wavelength. We interpret these results as evidence of dominating shadow-hiding and incoherent multiple scattering in the lunar opposition effect at α > 2°. At α < 2° the coherent backscattering enhancement may also contribute to scattering; however, its values are rather small, 5–7% even at λ = 3 µm, where lunar surface albedo is rather high.

  4. [44] Adding more sites imaged with M3 under opposition geometry and thus deriving the “mean” (in terms of albedo) phase dependence of color ratio is the goal for future work.

Acknowledgments

[45] This work was partially supported by Brown University of Providence, RI USA, with the attribution of visiting position to V. G. Kaydash. The M3 instrument was funded as a mission of opportunity through the NASA Discovery program. These analyses were supported through the NASA Lunar Science Institute (NNA09DB34A). Authors thank Yasuhiro Yokota and two anonymous reviewers for remarks that improved the paper.

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