The wavelength dependence of the lunar phase curve as seen by the Lunar Reconnaissance Orbiter wide-angle camera

Authors


Abstract

[1] The Lunar Reconnaissance Orbiter wide-angle camera measured the bidirectional reflectances of two areas on the Moon at seven wavelengths between 321 and 689 nm and at phase angles between 0° and 120°. It is not possible to account for the phase curves unless both coherent backscatter and shadow hiding contribute to the opposition effect. For the analyzed highlands area, coherent backscatter contributes nearly 40% in the UV, increasing to over 60% in the red. This conclusion is supported by laboratory measurements of the circular polarization ratios of Apollo regolith samples, which also indicate that the Moon's opposition effect contains a large component of coherent backscatter. The angular width of the lunar opposition effect is almost independent of wavelength, contrary to theories of the coherent backscatter which, for the Moon, predict that the width should be proportional to the square of the wavelength. When added to the large body of other experimental evidence, this lack of wavelength dependence reinforces the argument that our current understanding of the coherent backscatter opposition effect is incomplete or perhaps incorrect. It is shown that phase reddening is caused by the increased contribution of interparticle multiple scattering as the wavelength and albedo increase. Hence, multiple scattering cannot be neglected in lunar photometric analyses. A simplified semiempirical bidirectional reflectance function is proposed for the Moon that contains four free parameters and that is mathematically simple and straightforward to invert. This function should be valid everywhere on the Moon for phase angles less than about 120°, except at large viewing and incidence angles close to the limb, terminator, and poles.

1. Introduction

[2] The opposition effect (OE) is the sharp peak in the brightness of reflected sunlight centered on 0° phase angle (the angle between the incidence and viewing vectors). The OE is a property of almost all airless bodies in the solar system. The first astronomical observation of it was in the rings of Saturn by Seeliger [1895]. It was subsequently detected in the phase curve of the Moon by Gehrels et al. [1964]. The first observation of the OE in powders in the laboratory was by Oetking [1966].

[3] Initially, it was assumed that the effect was caused by shadow hiding [Seeliger, 1887]. In the shadow hiding opposition effect (SHOE), shadows cast by the particles of the lunar regolith on each other are visible at all angles except zero phase angle, when each particle hides its own shadow. However, another effect that can cause a brightness surge near zero phase angle, coherent backscatter, was pointed out by Watson [1969] in connection with the scattering of radio waves by plasmas and later by Shkuratov [1988] for planetary regoliths. It was observed experimentally in colloidal suspensions of submicroscopic particles by Kuga and Ishimaru [1984], in planetary analog powders by Hapke and Blewett [1991], and in lunar soil samples by Hapke et al. [1993]. In the coherent backscatter opposition effect (CBOE), two portions of a wave that travel the same, multiply scattered path through a regolith, but in opposite directions, combine randomly if the phase angle is large but interfere coherently and positively at near zero phase (see Hapke [1990] for a detailed explanation).

[4] Many persons have argued that the lunar OE is dominated by shadow hiding because the CBOE requires multiple scattering and the low lunar albedo implies that this process is not important. The relative contributions of the two phenomena to the lunar phase curve, and specifically to the lunar OE, have been a matter of considerable debate [e.g., Helfenstein et al., 1997; Hapke et al., 1998; Kaydash et al., 2010; Shkuratov et al., 2011]. It is important for remote studies of the Moon and other low-albedo bodies to ascertain which phenomenon dominates the lunar OE because each depends on different microscopic properties of the regolith.

[5] Since its discovery, the Moon's OE has been studied from the Earth by many persons, including Wildey [1978] and more recently by Shkuratov and his associates, who have also carried out laboratory studies (see Shkuratov et al. [2011] for a detailed historical review). Only the portion of the brightness curve with phase angles larger than 1° can be observed from the Earth because at smaller phase angles the lunar surface is eclipsed by the Earth's shadow. Since the OE is highly nonlinear, observations at smaller phase angles, which can only be acquired using spacecraft, are essential to accurately characterize the phase curve.

[6] Early studies of the lunar OE from spacecraft were done at visual wavelengths using Apollo photographs [Whitaker, 1969; Pohn et al., 1969]. The first spacecraft measurements of the wavelength dependence of the lunar OE were acquired with the Clementine ultraviolet-visible (UVVIS) imaging system [Buratti et al., 1996; Hillier et al., 1999; Shkuratov et al., 1999, 2001]. More recent observations have been made by the Chandrayaan spacecraft [e.g., Hicks et al., 2011]. Little evidence was found for wavelength dependence over a range of 415–1000 nm. Since present theories of the CBOE predict that it should vary strongly with wavelength, these workers concluded that the lunar OE was due mostly or completely to SHOE. However, it is important to verify these results by independent observations, particularly since questions remain concerning the calibration of the Clementine cameras [Shkuratov et al., 1999, 2001; Hillier et al., 1999].

[7] In this paper we report the first analyses of the lunar opposition effect as observed by the Lunar Reconnaissance Orbiter Camera (LROC) wide-angle camera (WAC). It will be shown that coherent backscatter makes a major contribution to the OE in the observed wavelengths (321 to 689 nm) and dominates at long wavelengths and that phase reddening is caused by multiple scattering.

2. Observations

[8] The LROC WAC [Robinson et al., 2010] is a push frame charge-coupled device array imager that in its normal operating mode views a repeating 60° wide swath (as seen from the spacecraft) centered on the nadir and measures reflected sunlight through seven narrowband filters. The central wavelengths of the filters are listed in Table 1. Note that the LROC WAC acquires observations into the UV (321 nm center, 32 nm full width at half maximum (FWHM)). By contrast, the UVVIS camera on the Clementine spacecraft could only observe down to 415 nm (40 nm FWHM). The images were radiometrically calibrated to give the radiance factor I/F(i, e, g), the radiance relative to a perfectly diffusing Lambert surface illuminated and viewed normally, where, i is the angle of incidence, e is the angle of emission or viewing, and g is the phase angle. The radiance factors and nominal photometric angles are given for each pixel in the online auxiliary material associated with this paper. The images used in the analyses are also listed. Two different types of observations of the brightness, as the phase angle and wavelength vary, are analyzed in this paper.

Table 1. Average Parameter Values
λ (nm)wbrR(0, 0, 0)gHaBC0BS0AN
  • a

    Given in degrees.

3210.159 ± 0.0160.310 ± 0.0310.293 ± 0.0294.330.456 ± 0.0470.783 ± 0.0780.146 ± 0.015
3600.193 ± 0.0190.308 ± 0.0310.316 ± 0.0314.490.456 ± 0.0460.617 ± 0.0620.158 ± 0.016
4150.226 ± 0.0230.284 ± 0.0280.348 ± 0.0355.440.456 ± 0.0460.60 ± 0.06170.174 ± 0.017
5660.333 ± 0.0330.294 ± 0.0290.480 ± 0.0484.730.456 ± 0.0460.399 ± 0.0400.240 ± 0.024
6040.357 ± 0.0360.298 ± 0.0300.507 ± 0.0514.680.456 ± 0.0460.358 ± 0.0360.254 ± 0.025
6430.387 ± 0.0390.304 ± 0.0300.553 ± 0.0554.020.456 ± 0.0460.325 ± 0.0330.276 ± 0.028
6890.420 ± 0.0420.307 ± 0.0310.602 ± 0.0603.690.456 ± 0.0460.293 ± 0.0290.301 ± 0.030

[9] In Data Set S1, light scattered from various points within a narrow east-west band in a single image passing through the zero phase point of each filter was measured as a function of the photometric angles. The image sequence covered a region in the highlands centered at 10.4°E, 3.0°S. Within this region the incidence angle changes by a relatively small amount around zero, so that the variation in phase angle is caused chiefly by variation in viewing angle across the field of view. The maximum phase angle in Data Set S1 is about 30°, the half width of the image.

[10] In Data Set S2, light scattered from various points within a rectangular region in the highlands north of M. Orientale at 2.5°S–2.5°N by 270°E–273°E was measured in a large number of images taken at various times and phase angles. Within these images the variation in phase angle is due chiefly to changes in angle of incidence while observing the same region from near vertical. The largest phase angle in Data Set S2 is about 120°, the maximum that can be observed without rolling the camera off-nadir.

[11] The minimum phase angle was different for each filter and data set, but the smallest minimum was 0.004°. The nominal pixel size on the surface was 100 m in the visible filters and 400 m in the UV. Local slopes were determined from the Kaguya digital terrain map [Araki et al., 2009], which has a resolution of 2 km/pixel using a 3 × 3 pixel box. To minimize effects of topography on the nominal photometric angles in Data Set S2, the images were overlaid onto the map, and all WAC pixels with slopes greater than 1° were discarded. To eliminate gores and minimize shadowing effects, all pixels with zero data values were also discarded. However, this slope and shadow filtering was not done for Data Set S1 because g, i, and e are sufficiently small that the photometric effects of roughness are minimal.

[12] Because theoretical photometric functions of particulate media based on the equation of radiative transfer contain the Lommel-Seeliger function,

display math

as a common factor, all values of I/F were divided by LS. This ratio is called the reduced reflectance and denoted by rR(i, e, g). The utility of the reduced reflectance is that for low-albedo bodies, such as the Moon, rR(i, e, g) is dominated to first order by the phase angle g [McEwen, 1996; Korokhin et al., 2007; Buratti et al., 2011], with i and e having only a second-order effect.

3. Results

[13] A typical result for Data Set S1 is shown in Figure 1, which plots rR(i, e, g) versus g for the 689 filter. The points correspond to individual areas within the scanned region and have varying values at the same angles because of differing albedos. To average the albedo variations, the data were smoothed with a 0.5° wide moving spatial filter, and the result is shown as the line in Figure 1. The filter width of 0.5° is the angular width of the Sun as seen from the Moon and is a limiting value for the angular resolution of the phase curve of a body at the distance of the Moon from the Sun. For g > ∼5°, the curve increases almost linearly as g decreases. However, for g < ∼5°, the curve rises nonlinearly to a value at g = 0 approximately 30% greater than its value at g = 5°. This is the OE.

Figure 1.

Reduced reflectances at 689 nm of the areas corresponding to Data Set S1 plotted against phase angle. The points are the individual reflectances. The line is a running average 0.5° wide.

[14] Figure 2 shows the smoothed curves of rR(i, e, g) for all the filters of Data Set S1. Figure 3 shows the smoothed curves normalized at 5°. Within the OE the normalized curves are very similar. Outside the OE the shorter-wavelength curves have steeper slopes than the curves taken through the longer-wavelength filters. This is the so-called phase reddening, first noted by Gehrels et al. [1964].

Figure 2.

Averaged reduced reflectances at all wavelengths in Data Set S1 versus phase angle. The residual standard error of each average is smaller than the line width.

Figure 3.

Averaged reduced reflectances of Figure 2 normalized at 5° phase angle. To avoid overcrowding the averaged reflectances are shown as thin lines without error bars.

[15] Similar curves are shown in Figures 46 for Data Set S2. These data extend out to much larger phase angles than Data Set S1, g ≈ 120°, yet they display similar behavior: quasi-linear at phase angles >∼5° with slopes that decrease as wavelength increases and an OE whose relative amplitude and angular width vary only weakly with wavelength. The latter observation confirms the lack of strong wavelength dependence of the OE found by Buratti et al. [1996] and Shkuratov et al. [2001] in their analyses of Clementine data. The scatter of data points at small phase angles is caused primarily by variations in albedos, but at large phase angles near the terminator in Data Set S2 the scatter is increased by the effects of subpixel roughness.

Figure 4.

Reduced reflectances at 689 nm of the areas corresponding to Data Set S2 plotted against phase angle. The points are the individual reflectances. The line is a running average 1° wide.

Figure 5.

Averaged reduced reflectances at all wavelengths in Data Set S2 versus phase angle. The residual standard error of each average is smaller than the line width.

Figure 6.

Averaged reduced reflectances of Figure 5 normalized at 5° phase angle. To avoid overcrowding, the averaged reflectances are shown as thin lines without error bars.

4. Quantitative Analysis

4.1. Reflectance Equations

[16] Data Set S2 was analyzed quantitatively by fitting a simplified version of the bidirectional reflectance model of Hapke [1993], as subsequently modified by Hapke [2002, 2008; see also Hapke, 2012]. Unfortunately, a similar analysis could not be done for Data Set S1 because the range of phase angles is insufficient to constrain the model parameters. The full theoretical function for the bidirectional reflectance is

display math

[17] Hapke [1993, 2002, 2008, 2012] should be consulted for the derivation and detailed discussions of the quantities in equation (2). In this equation, K is the porosity factor and is a monotonically increasing function of the filling factor ϕ; w is the volume average single scattering albedo; ie and ee are effective angles of incidence and emission, respectively, and are functions of the mean subpixel slope angle math formula; p(g) is the average single particle scattering function, which is assumed to be described by the three-parameter double Henyey-Greenstein function,

display math

M(ie, ee) is the contribution of interparticle multiple scattering,

display math

and S(i, e, g) is the roughness shadowing function.

[18] The remaining quantities in equation (2) describe the two opposition effects: BS0 and BC0 are the amplitudes of the SHOE and CBOE, respectively; BS(g) and BC(g) are the angular shape functions of the SHOE and CBOE, respectively;

display math

where hS is the SHOE angular width parameter; and

display math

where hc is the CBOE angular width parameter. Equation (6) is based on the model derived by Akkermans et al. [1986] using diffusion theory (which is closely related to scalar radiative transfer theory). This model was adopted and slightly modified by Hapke [2002].

[19] The angular width parameter of the SHOE is related to the properties of the regolith by Hapke [1993]

display math

where 〈a〉 is the mean radius of a particle of the medium and ΛE is the extinction mean free path. All theoretical models of the CBOE predict that its width is proportional to the wavelength divided by a length that is essentially proportional to the horizontal projection of the mean distance between the first and last scatterers. The Akkermans et al. [1986] diffusion theory model gives

display math

where λ is the wavelength and ΛT is the transport mean free path. (The transport mean free path may be thought of as the average distance a photon travels before scattering changes its direction by a large angle.) However, although equation (8) and similar expressions are widely accepted, there is considerable experimental evidence that they may not be valid for media composed of complex particles larger than the wavelength in contact with each other. This is discussed further in section 5.

[20] The half width at half maximum of the SHOE is related to hS by

display math

and the half width of the CBOE is related to hC by

display math

[21] The amplitude of the SHOE is

display math

where R(0) is the amount of light scattered directly back into zero phase angle from the fraction of the particle cross-sectional area at or near the surface of a typical particle that is directly illuminated by the incident light. It consists of the Fresnel coefficient for specular reflection at normal incidence from the particle surface plus light scattered by near-surface imperfections, such as cracks and asperities within that area. There is no generally accepted expression for BS0. Physical considerations restrict the OE amplitudes to the range 0 ≤ BC0 ≤ 1, 0 ≤ BS0 ≤ 1.

[22] Note that the CBOE amplifies the single scattering term as well as the multiple scattering term because complex individual particles can exhibit a CBOE. This will be discussed in detail in section 6. There is some uncertainty as to whether the CBOE amplifies the SHOE in addition to the single scattering and, if so, whether the CBOE parameters are the same. Equation (2) assumes that it does and that the parameters are similar.

[23] Equation (2) has 10 free parameters: K, w, b1, b2, c, BC0, BS0, hS, hC, and math formula. The five parameters w, b1, b2, c, and BS0 (or R(0)) describe the light scattering properties of the regolith particles. The four parameters K, hs, hC, and math formula describe the structure of the regolith, including particle size (through hs and hC), porosity (through K, hs, and hC), and subpixel roughness (through math formula). What controls BC0 is uncertain, although there are indications that it is influenced by the albedo [Nelson et al., 2004].

[24] Under many circumstances this equation can be simplified considerably. The effects of macroscopic roughness are appreciable only near the limb or terminator, where e or i, respectively, is large. For the observations analyzed in this paper the camera was nadir-looking, so e is small. The range of phase angles extends out to 120°, which is near the terminator where roughness effects are important. However, the pixels in Data Set S2 were selected to minimize effects of roughness and shadows. Points with large slopes or reflectances of zero that are completely in shadows have been omitted from the plots so that only illuminated, horizontal areas on the surface are shown. Although these areas contain unresolved shadows, which decreases their average brightnesses, and subpixel sloping surfaces, the slopes that are illuminated tend to be tilted preferentially toward the Sun, which increases the mean brightness. The two effects cancel each other to first order, so that the brightnesses of the selected pixels are not too different from those of a smooth horizontal surface. Moreover, the number of data points at large phase angles is smaller than at small angles, so that points near the terminator have only a minor effect on the overall phase curve. Hence, in the first approximation the pixels used in the analyses describe a smooth, locally horizontal surface, so that math formula can be set equal to zero. Then iei, eee, and S(i, e, g) ≈ 1.

[25] Since we do not know what the in situ value of ϕ is for the uppermost layer of undisturbed lunar soil, K is set equal to 1 for simplicity, although it is probably >1.

[26] Although it is highly probable that p(g) has a forward scattering lobe, laboratory experiments [Hapke, 1999] indicate that it does not become large enough to have a detectable effect on the reflectance until g > ∼120°, for which we have no data. Hence, we set the second term of equation (3) equal to zero by putting c = 1. In that case, b1 = 〈cos g〉, the mean cosine of the single particle scattering phase angle; for simplicity let b1 = b.

[27] Dividing equation (2) by 1/π, the normal reflectance of a perfect Lambert surface, converts r to I/F, and dividing the result by LS gives the reduced reflectance. With these approximations equation (2) becomes

display math

where

display math

and

display math

Equation (12) contains six free parameters.

4.2. Analysis

[28] Typically, BC(g) is only a few degrees wide, while media with narrow particle size distributions have BS(g) of the order of 20° wide. In this case the two OEs can often be untangled. However, for media with wide particle size distributions, like the lunar regolith, BS(g) can be quite narrow, comparable with BC(g). Further, the shapes of the two peaks are similar, and it becomes difficult to discriminate between the two. In order to study the OE further, we separately fitted the two end-members of equation (12), in which the OE is assumed to be either entirely SHOE or entirely CBOE, to Data Set S2. In that case the equation contains only four parameters to be fitted.

[29] Equation (12) with either BS0 or BC0 set equal to zero was fitted to Data Set S2 using the Levenberg-Marquardt regression analysis algorithm. This algorithm requires initial guesses for the four parameters. These guesses cannot simply be assigned random values because if they are too far from the best fit values, the algorithm may find solutions that are not optimum and may even be grossly nonphysical. In addition, different parameters may have similar effects on the solution. Hence, it is important to restrict the range of the trial values as much as possible using a priori knowledge of the nature of the solution. The initial values were found as follows. The OE is appreciable only at small phase angles. Thus, equation (12) with the OE set equal to zero was fitted to the portion of the smoothed curve for g > 20° to give preliminary values of w and b. Because w affects mainly the height of the curve while b controls the shape, these values are probably close to optimal. This solution was then extrapolated to smaller phase angles and removed from the smoothed curve, leaving only the OE. The OE equation was then fitted to this remainder, giving preliminary values for the amplitude and width parameters. Again, these parameters are likely to be close to optimal because they affect the phase curve differently. The values of the four parameters obtained in this way were then used as starting values to find the best fits using the points (not the smoothed curve) of Data Set S2.

[30] Typical results of the analysis are shown in Figure 7, which compares the fitted curves of the two end-member models and the smoothed curve of the data points for the 689 nm filter of Data Set S2. The best fit values of the parameters w and b, the reduced reflectance at zero phase rR(0, 0, 0), and the half width at half maximum gH, calculated from the angular width h parameters, are plotted in Figure 8 for all filters. Figure 8 shows that these parameters are robust and virtually independent of the particular OE model. Consequently, each quantity was averaged over the two end-member models at each wavelength; these average values are listed in Table 1.

Figure 7.

Reduced reflectances at 689 nm of Data Set S2 versus phase angle. The black line is the running average from Figure 4. The solid blue line is the all-CBOE model fitted to the data points. The dashed red line is the best fit of the all-SHOE model.

Figure 8.

Retrieved model parameters for Data Set S2 plotted against wavelength. The dots connected by solid lines are the values for the CBOE-only model. The triangles connected by dashed lines are the values for the SHOE-only model. The error bars are the formal fitting errors.

[31] The retrieved values of the OE amplitudes for the end-member models are plotted in Figure 9. Note that for the all-SHOE model BS0 > 1 at all wavelengths (in particular, BS0 = 1.99 at λ = 321 nm), and for the all-CBOE model BC0 > 1 at shorter wavelengths. Since physically neither parameter can exceed 1, this plot shows unequivocally that neither end-member model alone can account for the observations, so that both CBOE and SHOE must contribute to the OE, even in the UV where the albedo is extremely low. Moreover, the amplitudes decrease as w increases, as predicted by equation (11), showing that the SHOE cannot be negligible.

Figure 9.

Retrieved model opposition effect amplitudes for Data Set S2 plotted against wavelength. The dots connected by solid lines are the values for the all-CBOE model. The triangles connected by dashed lines are the values for the all-SHOE model. The error bars are the formal fitting errors.

[32] Thus, the problem is to find the individual OE parameters. At g = 0 equation (12) becomes

display math

Using the values of w and b from Table 1, wp(0) and wM(0, 0) were calculated at each wavelength. Then all quantities in equation (15) are known except the two OE amplitudes.

[33] Let rC(0, 0, 0) be the continuum reduced reflectance at zero phase, i.e., the reflectance that would be observed in the absence of any OE,

display math

and Δr be the total height of the OE at zero phase,

display math

Making these substitutions and using equation (11) for BS0, equation (15) becomes

display math

[34] In Figure 10 Δr is plotted against rC(0, 0, 0) for the various filter wavelengths. Figure 10 shows that the points fall close to a straight line with slope 0.456 and vertical axis intercept 0.123; the statistical R value of the fit is 0.992. This line should not be literally interpreted as implying that Δr is finite while rC(0, 0, 0) = 0 anywhere, which is physically impossible, but rather that over the limited WAC range of wavelengths the relation between Δr and rC(0, 0, 0) is linear. If it is assumed that neither BC0 nor R(0) are strongly varying functions of w over the WAC range of wavelengths, then Bc0 ≈ 0.456 and R(0) ≈ 0.338; BS0 can be calculated from either R(0) or equation (16). The two OE amplitudes are listed in Table 1.

Figure 10.

Plot of Δr versus rC(0, 0, 0) for Data Set S2. The points are the values at the filter wavelengths. The solid line is the best fit straight line and has the equation Δr = 0.123 + 0.456rC(0, 0, 0).

[35] Unfortunately, because the shadow hiding and coherent backscatter peaks have similar shapes, the separate angular width parameters hC and hS could not be found with any degree of confidence.

4.3. Calibration and Errors

[36] The calibration of the LRO WAC is a challenging, continuing project. Prelaunch relative calibration errors are less than ±1%, and absolute errors are less than ±10% [Robinson et al., 2010]. The analysis of this paper requires the absolute reflectance, which is notoriously difficult to measure because of the possibility of systematic errors. In-flight calibrations in the visible are being done using Earth-based telescopic data, primarily the high-quality Robotic Lunar Observatory (ROLO) data set [Kieffer and Stone, 2005]. Unfortunately, there are few measurements of absolute lunar reflectance in the UV, and observations by the Hubble Space Telescope [Robinson et al., 2007] are being used for short-wavelength in-flight calibration. The residual standard errors in the smoothed phase curves in Figures 2 and 4 are less than the widths of the lines, while the formal RMS fitting errors of the model photometric parameters are shown in Figures 8 and 9. Thus, the absolute reflectance probably is the largest present source of error. On the basis of the best current estimates of the spacecraft team, we conservatively estimate the errors of the parameters to be ±10%, and these are the errors listed in Table 1.

5. Discussion

[37] The single scattering albedos retrieved from the model increase with wavelength (Figure 8), consistent with the reddish spectrum of the Moon. The retrieval method used (section 4.1) gives unambiguous values. However, it should be recalled that we have arbitrarily set the porosity factor K = 1, which tacitly assumes that the filling factor is small. If ϕ is not small, K will be increased and the retrieved w will be decreased accordingly at each wavelength. Also, if data at g > 120° become available, the forward scattering lobe of p(g) must be included; in that case c ≠ 1, which will also affect w.

[38] The value of the Lommel-Seeliger function at i = e = 0 is 1/2. Thus, the retrieved mean normal albedos of the highlands area of Data Set S2 are equal to one-half the reduced reflectances at zero phase and are given in Table 1 under AN. They vary from about 15% in the UV to 30% in the red. It should be noted these are the values of AN that would be measured if the Sun were a point source. The actual values are slightly smaller because of the slight rounding off of the theoretical phase curve by the finite angular width of the Sun. However, this difference is well within the ±10% errors.

[39] The retrieved values of b = 〈cos g〉 are virtually independent of wavelength with an average value of b = 0.30. This value of 〈cos g〉 means that p(g) has a large backscattering lobe, which is expected for particles that are larger than the wavelength, nearly opaque, and structurally complex.

[40] Figures 3 and 6 show that the relative slopes of the phase curves outside of the OE increase as the wavelength increases. This is the well-known phase reddening discovered by Gehrels et al. [1964], in which the slope of the lunar reflectance spectrum increases with phase angle. Its cause has been uncertain, but one hypothesis is that it is due to light transmitted through regolith particles, which were thought to become more transparent and less backscattering with increasing wavelength and albedo. However, Figure 8 shows that the parameter b, which controls the shape of the particle phase function, is virtually independent of wavelength, so this explanation must be incorrect. The only remaining part of the lunar reflectance equation that can affect the slope at large phase angles is M(i, e), which describes interparticle multiple scattering. Hence, the phase reddening must be due to the increased contribution of multiple scattering at large phase angles as the wavelength and albedo increase. This conclusion demonstrates that interparticle multiple scattering cannot be neglected when analyzing and modeling lunar photometric observations, especially at longer wavelengths.

[41] The unique capabilities of the LROC WAC, which measures the reflectance over a large range in wavelength, reflectance, and phase angle, allow the relative contributions of the CBOE and SHOE to be separated. The fraction of CBOE, BC0/(BS0 + BC0), and the SHOE fraction, BS0/(BS0 + BC0), contributing to the reflectance as a function of wavelength were calculated from Table 1 and plotted in Figure 11. Figure 11 shows that coherent backscatter is not negligible and accounts for nearly 40% of the OE in the UV increasing to over 60% in the red.

Figure 11.

Fractions of CBOE and SHOE for Data Set S2 versus wavelength. The error bars are the ±10% errors listed in Table 1 and discussed in the text.

[42] Although this detailed analysis was carried out for only one place on the Moon, the area was typical highlands terrain, so that it is highly likely that the conclusions of this paper, in particular, the important contribution of the CBOE and interparticle multiple scattering as the cause of phase reddening, apply to most lunar regions.

[43] Many workers have argued that the low lunar albedo does not allow the appreciable multiple scattering that is necessary for a CBOE and, thus, that the lunar OE is entirely or almost entirely SHOE [e.g., Buratti et al., 1996; Hillier et al., 1999; Shkuratov et al., 1999]. However, this argument is invalid and results from confusion about the meaning of multiple scattering [Hapke, 2002]. In the radiative transfer equation and its solutions, multiple scattering refers to scattering between particles of the medium. However, in the coherent backscatter phenomenon, multiple scattering is any event that changes the direction of a light wave. These events can be caused by scattering from a particle as a whole but also by refraction as a wave passes through the surface of a particle, by external or internal reflection from a particle surface, and by scattering from inhomogeneities, such as cracks, inclusions, or voids inside a particle, or from irregularities on its surface. Thus, multiple scattering of the type required for coherent backscatter can take place within a single particle, provided it is sufficiently complex. This is why the CBOE augments the single scattering term, as well as the multiple scattering term, in the reflectance model on which equation (2) is based.

[44] These arguments are supported by Zubko et al. [2008], who studied numerical models of scattering by complex single particles and found that they exhibited coherent backscatter. They are also in accord with Shkuratov and Helfenstein [2001]. The argument that the low lunar albedo implies that the OE must be almost exclusively a SHOE has no basis.

[45] The finding that the CBOE makes an important contribution to the lunar OE, especially in the red, is consistent with laboratory measurements of the polarization ratios of Apollo soil samples [Hapke et al., 1993]. The polarization ratios are defined as follows. Suppose a material is illuminated with linearly polarized light. Then the linear polarization ratio is defined to be the ratio of the reflected intensity with the polarization perpendicular to that of the incident light to the reflected intensity with the same direction of polarization as the incident light. If the sample is illuminated with circularly polarized light, the circular polarization ratio is defined as the ratio of the reflected intensity with the same helicity as the incident light to the reflected intensity with the opposite helicity as the incident light.

[46] The circular polarization ratio in a CBOE increases as the phase angle is decreased, but in a SHOE it decreases. The reasons for this behavior are well understood [Hapke, 1990]. No other mechanism for making the circular ratio increase has been advanced in the more than 25 years since the CBOE was discovered. Hence, there are strong reasons for believing that this increase is a hallmark of the CBOE. When the polarization ratios of eight Apollo samples from different lunar regions were measured, it was found that the circular polarization ratio increased and the linear polarization ratio decreased as the phase angle decreased in the OE [Hapke et al., 1993]. This behavior is illustrated in Figure 12 for an Apollo soil sample measured at 633 nm. The upturn of the circular ratio is a strong independent argument that the lunar OE is dominated by coherent backscatter but was a major puzzle as long as the OE was thought to be caused by shadow hiding.

Figure 12.

Reflectance I/F and polarization ratio phase curves at 633 nm of a sample of lunar soil.

[47] One of the main arguments that the lunar OE is caused by shadow hiding is that its angular width is virtually independent of wavelength (Figures 3, 6, and 8 and Table 1). However, we suggest that this argument is not valid, even though all theoretical CBOE models that have been published to date predict that gSH should be proportional to λ (equation (10)), and laboratory studies [Van der Mark et al., 1988] have confirmed equation (10) for the case of colloidal suspensions of well-separated, nonabsorbing particles that are smaller than the wavelength.

[48] Theoretically,

display math

where n is the average number of particles per unit volume, σ is the mean particle cross-sectional area, QS is the scattering efficiency, and QE is the extinction efficiency. For regoliths of particles larger than the wavelength, all quantities in equation (17) are approximately independent of λ, except w, which for lunar regolith is linearly proportional to λ (Figure 8). Thus, the angular width of the Moon's CBOE should increase as λ2, contrary to observations.

[49] Moreover, a large number of laboratory studies of planetary regolith analog powders composed of particles larger than the wavelength in contact with each other found no wavelength dependence [Shkuratov et al., 1999; Nelson et al., 2002; Kaasalainen et al., 2005]. Also, equation (17) shows that ΛΤ is a strong function of particle size and spacing, yet none of these laboratory analog samples displayed the dependence predicted by equations (10) and (17) [Nelson et al., 1998, 2000; Shkuratov et al., 1997; Kaasalainen, 2003; Piatek et al., 2004]. Most of these samples had high albedos, so there is little doubt that their OEs had a large component of coherent backscatter.

[50] These major disagreements of laboratory measurements with predictions of OE theory had already cast serious doubt on the validity of equation (10) and similar expressions. Rather than implying that the Moon's photometric function is not a CBOE, the new observations by the LROC WAC add to the growing number of indications that our present understanding of the CBOE is incomplete or incorrect for media of particles larger than the wavelength in contact.

6. A Simplified Lunar Bidirectional Reflectance Function

[51] An approximate lunar bidirectional reflectance, which is useful and sufficiently accurate for many purposes, can be derived as follows. Equation (12) can be put into the form

display math

[52] Now, outside of the OE, [1 + BS0BS(g)]−1 = 1. At short wavelengths M(i, e) is small compared with p(g), so [1 + BS0BS(g)]−1 has little effect on rR. At long wavelengths BS0 is relatively small because of the inverse dependence of BS0 on w, which increases with λ, so that [1 + BS0BS(g)]−1 ≈ 1 there. Hence, to a first approximation the reflectance can be written

display math

where p(g) is given by equation (13), M(i, e) is given by (14), BE(g) is given by equation (6) for BC(g), and BE0BE(g) is to be interpreted as an effective CBOE that describes the combined effects of both the SHOE and CBOE. Equation (19) is essentially an all-CBOE model, except for the interpretation of BE0BE(g). It has four free parameters that affect the reflectance in different ways so that it is straightforward to invert and retrieves unambiguous parameter values (see section 4.2). It should be valid everywhere on the Moon except close to the limb and terminator, where roughness effects are important, or at phase angles larger than 120°, where the forward scattering lobe of p(g) becomes significant. If shadowed pixels are discarded, as described in section 2, it should then be fairly accurate near the terminator also. Equation (19) has been used by Sato et al. [2011] to produce a global color mosaic of the Moon from the LROC WAC data. The model should also be applicable to other bodies.

Acknowledgments

[53] We thank the hard work of the Goddard Space Flight Center LRO Mission Operations Center and the LROC Science Operations Center. We also thank Bonnie Buratti, Paul Helfenstein, and Yuri Shkuratov for constructive reviews that materially improved this paper. This study was funded by the National Aeronautics and Space Administration Lunar Reconnaissance Orbiter Project.

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