Vis Camera

Icarus 141, 205–225 (1999) Article ID icar.1999.6184, available online at http://www.idealibrary.com on

Multispectral Photometry of the Moon and Absolute Calibration of the Clementine UV/Vis Camera John K. Hillier, Bonnie J. Buratti, and Kathryn Hill Jet Propulsion Laboratory, MS 183-501,4800 Oak Grove Drive, Pasadena, California 91109 E-mail: [email protected] Received October 5, 1998; revised June 11, 1999

INTRODUCTION We present a multispectral photometric study of the Moon between solar phase angles of 0 and 85◦ . Using Clementine images obtained between 0.4 and 1.0 µm, we produce a comprehensive study of the lunar surface containing the following results: (1) empirical photometric functions for the spectral range and viewing and illumination geometries mentioned, (2) photometric modeling that derives the physical properties of the upper regolith and includes a detailed study of the causes for the lunar opposition surge, (3) an absolute calibration of the Clementine UV/Vis camera. The calibration procedure given on the Clementine calibration web site produces reflectances relative to a halon standard and further appear significantly higher than those seen in groundbased observations. By comparing Clementine observations with prior groundbased observations of 15 sites on the Moon we have determined a good absolute calibration of the Clementine UV/Vis camera. A correction factor of 0.532 has been determined to convert the web site (www.planetary.brown.edu/clementine/calibration.html) reflectances to absolute values. From the calibrated data, we calculate empirical phase functions useful for performing photometric corrections to observations of the Moon between solar phase angles of 0 and 85◦ and in the spectral range 0.4 to 1.0 µm. Finally, the calibrated data is used to fit a version of Hapke’s photometric model modified to incorporate a new formulation, developed in this paper, of the lunar opposition surge which includes coherent backscatter. Recent studies of the lunar opposition effect have yielded contradictory results as to the mechanism responsible: shadow hiding, coherent backscatter, or both. We find that most of the surge can be explained by shadow hiding with a halfwidth of ∼8◦ . However, for the brightest regions (the highlands at 0.75–1.0 µm) a small additional narrow component (halfwidth of <2◦ ) of total amplitude ∼1/6 to 1/4 that of the shadow hiding surge is observed, which may be attributed to coherent backscatter. Interestingly, no evidence for the narrow component is seen in the maria or in the highlands at 0.415 µm. A natural explanation for this is that these regions are too dark to exhibit enough multiple scattering for the effects of coherent backscatter to be seen. Finally, because the Moon is the only celestial body for which we have “ground truth” measurements, our results provide an important test for the robustness of photometric models of remote sensing observations. °c 1999 Academic Press Key Words: Moon; Moon, surface; photometry.

In 1994 the Clementine spacecraft obtained multispectral observations covering almost the entire lunar surface. The primary focus of the mission at the Moon was mineralogic studies of the lunar surface and a wealth of new information about the Moon has been obtained. However, while the primary focus was mineralogy, the data provided by the Clementine spacecraft also has provided observations of the Moon useful in photometric studies. Clementine provided the first comprehensive multispectral data set of the entire lunar surface, and they provide us with an opportunity to study both the global and regional properties of the Moon. The Clementine data are typically at a resolution of decameters to kilometers. As such they provide an important regional context for the measurements made at the Apollo sites and provide an important intermediate link between these observations and Earth-based observations at the kilometer scale. They thus provide an opportunity to check the results of photometric modeling of remote sensing observations against “ground truth” obtained at the Apollo sites (e.g., Helfenstein and Shepard 1998). Our goals for this study are threefold. First we derive empirical photometric functions for the two major terrain types (maria and highlands) on the Moon. Such functions are needed to perform photometric corrections in a variety of studies including mineralogic studies and photoclinometry. Second, we fit the Clementine data for the different lunar terrains to a rigorous photometric model which expresses the light reflected by the surface as a function of radiance geometry in terms of the physical properties of the surface. Such modeling allows us to determine various properties of the surface such as its compaction state, particle albedo, particle size and size distribution, and surface macroscopic roughness. It is these results that can be compared to “ground truth” of the Apollo measurements as a general test of photometric modeling theories. Our third goal is a detailed study of the lunar opposition effect. The Clementine data set includes many images obtained at or near zero degrees phase angle providing by far the best data set currently available for studying this phenomenon on any celestial body. As an

205 0019-1035/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.

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ancillary part of our study, we have determined a new absolute calibration for the Clementine UV/Vis camera. The wealth of data provided by the Clementine spacecraft on the lunar opposition surge has sparked renewed interest in this phenomenon on the Moon (e.g., Buratti et al. 1996, Helfenstein et al. 1997, Hapke et al. 1998, Shkuratov et al. 1999) and, by extension, on other bodies in the Solar System. Until recently, the opposition surge was thought to be well understood in terms of a shadow-hiding mechanism in which particles hide their own shadows at opposition but the shadows quickly become visible as one moves away from opposition (Irvine 1996, Hapke 1986). However, recent observations of extremely narrow surges (at phase angles less than 1◦ ) on several Solar System bodies (Buratti et al. 1992, Thompson and Lockwood 1992, Domingue et al. 1991) require extremely porous surfaces to be explained by the shadow-hiding mechanism. This has led to the suggestion that a second mechanism, coherent backscatter, may be responsible for the observed surges (Kuga and Ishimaru 1984, Shkuratov 1989, Muinonen 1990, Hapke 1990, Mishchenko 1992, Mishchenko and Dlugach 1992). In this mechanism, photons following identical but reversed paths can interfere constructively in the backscattering direction, leading to up to a factor of two increase in brightness. Previous studies of the lunar opposition surge have led to conflicting results. Based on polarization measurements of lunar soil samples, Hapke et al. (1993) concluded that the lunar opposition surge was dominated by coherent backscatter. However, Buratti et al. (1996) examined Clementine observations of the opposition surge and found little wavelength dependence to the surge, leading them to conclude that the surge is predominantly due to shadow hiding. Helfenstein et al. (1997), examining a variety of lunar data sets, concluded that the surge could best be explained by a combination of a narrow (∼2◦ ) coherent backscatter surge superimposed on a broader shadow hiding surge. Hapke et al. (1998), performing a further analysis of their polarization data, came to a similar conclusion to Helfenstein et al. They also pointed out that the larger surge in the highlands compared to the maria observed by Buratti et al. (1996) could be explained by coherent backscatter (which would be more important in the brighter highlands terrain). Also, since the publication of our original paper (Buratti et al. 1996) it has been realized that the wavelength dependence of coherent backscatter may be more complicated than originally thought. While the angular width of the coherent backscatter peak should be proportional to wavelength, it is inversely proportional to the mean optical path length of a photon (which will be smaller in lower albedo surfaces). Thus it is possible for these two effects to substantially cancel out. In light of this fact and the conclusions of the more recent studies, we have performed a further examination of the Clementine data. In the prior study, no attempt at image calibration was performed. This limited the study to phase angles observable within a single image ◦ (phase angles < ∼4 ), and, obviously, only relative photometry could be performed. In the present study, our absolute calibra-

tion of the Clementine images allows us to perform absolute photometry, and to combine data from different images to examine a much larger range of phase angles than that examined by Buratti et al. (1996). The paper is divided into three major sections. Good relative calibrations of the Clementine UV/Vis camera have been obtained (for example as given on the Clementine calibration web site at www.planetary.brown.edu/clementine/calibration.html). While this is sufficient for mineralogic studies, an absolute calibration is required for our photometric study and the determination of photometric quantities such as the geometric and bond albedo. Therefore in our first task we use the lunar observations of Shorthill et al. (1969) to determine the absolute calibration of the Clementine UV/Vis camera. Our calibration procedures are presented in the Appendix. In the second section we derive empirical photometric functions from the Clementine data. These should be useful for performing photometric corrections to lunar data necessary in mineralogic studies as well as in other areas such as photoclinometry and producing image mosaics. Finally, in the third section of the paper we examine the surface scattering properties of the lunar surface using Hapke’s (1981, 1984, 1986) photometric model and in particular examine the lunar opposition effect to ascertain the mechanism responsible for its formation. IMAGE SELECTION

For best results, a photometric study requires data from the largest range of viewing and illumination geometries possible. However, several geometries are of particular importance. Observations at low phase angles (0–10◦ ) are crucial for characterizing the opposition surge. Observations at high phase angles as well as observations of limb-darkening at lower phase angles are important for examining the surface roughness. Observations at high phase angles (>130◦ ) are also important for examining any forward scattering exhibited by the surface particles while observations at a range of phase angles are necessary to constrain the phase function of surface particles. Unfortunately, Clementine did not obtain any observations at high phase angles (>90◦ ) and most of the Clementine observations are nadir pointing (with emission angle near 0◦ ) though a few nonnadir pointing observations were obtained. Thus, in selecting images to use in the study it was important to include images from as wide a range of viewing and illumination geometry as possible and particular attention was paid to include observations near 0◦ phase as well as observations away from 0◦ emission angle. Clementine obtained almost two million images of the lunar surface. Therefore a screening process was employed to obtain a manageable number of images to use for the study. We split the data into the two major terrain types (maria and highlands) as follows: USGS maps of the lunar surface were examined and rectangular boxes (in latitude/longitude space) chosen which contain only one or the other terrain. For the maria, 84 such boxes were chosen which cover all the maria regions on the Moon. For

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TABLE I Search Criteria Used to Screen Images for Use in the Present Study e

α

σDN /Mean DN

>5◦ <5◦ 0◦ –90◦

0◦ –180◦ 0◦ –180◦ 65◦ –75◦

Maria 0.08 0.07 0.10

<5◦ >5◦ ◦ 0 –90◦ 0◦ –90◦

0◦ –180◦ 0◦ –180◦ 40◦ –180◦ <5◦

Highlands 0.055 0.065 0.070 0.100

Mean DN

No. of images

>75 >75 >75

59 185 10

75–200 75–200 75–200 75–200

109 74 52 22

the highlands, 23 boxes were chosen which covered most of the highlands regions on the Moon. The Clementine Navigator website (available via the Planetary Data System at http://wwwpdsimage.jpl.nasa.gov/PDS/public/clementine/frontpage.html) was then utilized to find all of the images of nominal image quality taken through the UV/Vis B filter within each box. In order to reduce digitization errors, images with an average DN (data number) of less than 75 were excluded. The images were then screened for those for which the standard deviation/mean DN were under a given threshold. Such a procedure will screen for the most uniform images and thus will obtain those exhibiting the least albedo variegations. The search criteria were loosened somewhat for the few non-nadir-pointing sequences to ensure a significant sample of data from these important images. In addition, for the highlands, little data at mid to high phase angles and near zero degrees were obtained while for the maria no data at phase angles from 65◦ to 75◦ were obtained. Therefore, three additional screens were done with loosened constraints to obtain data in these regions. The criteria used for screening images in the maria and highlands are shown in Table I. With these criteria, 254 images in the maria and 257 images in the highlands were selected for the study. In addition to the B filter, images from the same sequence as the B-filter images were used for the A, C, D, and E filters as well to ensure that the same regions were surveyed in all the Clementine filters. The Clementine UV/Vis camera filters cover the visible to near infrared. The effective wavelengths of the Clementine UV/Vis camera filters are shown in Table II for reference. Our procedure for obtaining TABLE II Clementine UV/Vis Camera Effective Wavelengths Filter

λeff (µm)

A B C D E

0.415 0.75 0.90 0.95 1.00

an absolute calibration of the Clementine images and extracting the data from the images is given in the Appendix. EMPIRICAL PHOTOMETRIC FUNCTION

While sophisticated photometric models exist which can provide us with important information about a planetary surface, for many purposes a simple empirical function will suffice. Examples include photoclinometry and image mosaicking. Of particular importance to the Clementine mission is support for mineralogic studies (the primary focus of the mission). For such studies it is necessary to employ a photometric correction to the observations yielding reflectances at a standard viewing and illumination geometry. Without such a correction, observations at different viewing and illumination geometries cannot be meaningfully compared. Thus, to increase the scientific value of such studies, an accurate photometric function is needed for good calibration of images. We employed the following empirical photometric function: µ0 I f (α). = F µ + µ0

(1)

This functional form assumes that the scattering dependence on incidence and emission angle can be adequately accounted µ0 . In general, f (α) will for by the Lommel–Seeliger factor, µ+µ 0 have some dependence on incidence and emission angle due to multiply scattered light and macroscopic roughness. However, it has long been known that the lunar reflectance approximately follows Eq. (1) (e.g., Hapke 1963, 1971) and this is the dependence expected for singly scattered light. Thus it should be a good approximation for dark objects such as the Moon where singly scattered light dominates the reflectance. For the phase function f (α), we initially attempted to fit a fourth-order polynomial. However, for the highlands regions it was found that the polynomial did not provide a good fit to observations near zero degrees where the opposition surge results in a large nonlinear increase in brightness. Therefore, we added an exponential term to better account for the opposition surge, f (α) = b0 e−b1 α + a0 + a1 α + a2 α 2 + a3 α 3 + a4 α 4 ,

(2)

where α is the phase angle and b0 , b1 , and a0 –a4 are parameters to be fit. EMPIRICAL PHOTOMETRIC FUNCTION: RESULTS 0 Figures 1 and 2 show sample fits of f (α) to the data FI · µ+µ for µ0 the B filter in the maria and highlands, respectively. The results for the other filters are similar. The functions are only applicable over the range of phase angles for which data are available (0–75◦ and 0–85◦ for the highlands and maria, respectively) and should not be considered reliable beyond this range. As can be seen,

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FIG. 1. Plot showing the best fit empirical function f (α) against the data. The data have been corrected for variations in the incidence and emission angle by the Lommel–Seeliger factor, data = I /F × (µ0 + µ)/µ. Results shown are for the maria and the B filter. Results for the other filters are similar.

FIG. 2. Plot showing the best fit empirical function f (α) against the data. The data have been corrected for variations in the incidence and emission angle by the Lommel–Seeliger factor, data = I /F × (µ0 + µ)/µ. Results shown are for the highlands and the B filter. Results for the other filters are similar.

the empirical functions provide a good fit over the range of data available and we recommend its use for performing photometric corrections in this phase angle range. Figures 3 and 4 show the data binned in 1◦ intervals of phase angle plotted against the best fit curves. Most of the noise in the data is due to albedo variations between images used in the study. The effects of variations in albedo were reduced by ratioing the data and fits to the B filter (Figs. 5 and 6). Still, some significant variations in the wavelength dependence of the albedo is seen. This is not surprising: the Moon exhibits diverse mineralogic content and the filters were chosen specifically to be diagnostic of such variations. It does, however, suggest that

some caution needs to be exercised in interpreting our results. For example, the notable drop in the E-filter highlands phase function near 15◦ phase angle is likely, at least partly, to be due to a selection effect in which the data at lower phase angles comes predominantly from regions relatively dark in the E filter while that above 15◦ comes from relatively bright regions. This contention is supported by Fig. 7, which shows the ratio of E-filter to B-filter albedo for a typical image (lue3091i.087 and lub3101i.087) centered near 16◦ phase. Phase angle variations of several degrees are seen in the images so any significant variation as suggested by the fits should be readily observed. However, while craters, which are expected to consist of

FIG. 3. Best fit empirical phase function f (α) for the maria. The data have been binned in 1◦ increments in phase angle and corrected for variations in incidence and emission angle using the Lommel–Seeliger factor (data = I /F × (µ0 + µ)/µ).

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FIG. 4. Best fit empirical phase function f (α) for the highlands. The data have been binned in 1◦ increments in phase angle and corrected for variations in incidence and emission angle using the Lommel–Seeliger factor (data = I /F × (µ0 + µ)/µ).

relatively immature soils exhibiting strong absorption bands near 1 µm, do appear dark in the ratio image, no significant variations across the image are observed. Similarly, no variation was seen in either of the other two image sequences that we examined: (lue1814i.218, lub1824i.218) and (lue1844i.226, lub1854i.226), both centered near 12◦ phase. Therefore, in addition to the fits to the individual filters near 1 µm, we also fit the data for the C, D, and E filters combined in an effort to smooth out such effects and thus provide a more representative phase function for use with these filters. Our best fit parameters for the maria and highlands are shown in Table III. Figure 8 shows the best fits ratioed to the B-filter results normalized to one at 0◦ phase. Immediately apparent is the well known phase reddening effect (e.g., Lane and Irvine

1973) previously observed for the Moon. For both the maria and the highlands, the brightness in the violet (A filter, 0.415 µm) decreases by close to 30% relative to the B filter (0.75 µm) as the phase angle is increased from 0◦ to 70◦ . For the highlands, the phase reddening effect is seen to extend out to 1 µm, where the relative reflectance increases by about 15% as the phase angle is increased from 0◦ to 70◦ . However, this is not observed for the maria where the reflectance at 1 µm exhibits behavior very similar to that at 0.75 µm though perhaps with a slight increase in relative reflectance toward higher phase angles. As found by Buratti et al. (1996), no dramatic wavelength dependence in the opposition surge region (near 0◦ phase angle) is observed though, as suggested by the previous study, the phase reddening effect seen at higher phases does extend down to zero degrees,

TABLE III Best Fit Parameters for the Empirical Photometric Function to the Maria and the Highlands Filter

b0

b1

a0

a1 (×10−3 )

a2 (×10−5 )

a3 (×10−7 )

a4 (×10−9 )

A B C D E 1 µm

−0.0198 −0.0661 −0.0633 −0.0558 −0.0486 −0.0557

0.600 0.359 0.356 0.373 0.320 0.350

0.226 0.362 0.366 0.358 0.328 0.351

Maria −11.08 −20.01 −19.76 −18.73 −15.23 −17.89

30.82 61.78 60.27 55.84 44.10 53.34

−39.25 −81.46 −78.78 −71.52 −56.31 −68.79

17.89 37.16 35.63 31.61 24.76 30.63

A B C D E 1 µm

0.1053 0.1718 0.1598 0.1589 0.3545 0.1857

0.541 0.374 0.450 0.498 0.194 0.337

0.316 0.414 0.451 0.461 0.193 0.401

Highlands −9.65 −4.48 −6.72 −7.50 19.80 −1.18

23.57 −7.42 3.81 7.44 −89.65 −16.06

−37.46 18.75 −3.47 −9.37 136.01 26.76

24.18 −9.26 5.36 8.42 −69.15 −11.13

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FIG. 5. Color ratio plot for the maria. The phase functions f (α) and data have been ratioed to the B filter. The data have been binned in 1◦ increments in phase angle and corrected for variations in incidence and emission angle using the Lommel–Seeliger factor (data = I /F × (µ0 + µ)/µ).

leading to a somewhat larger surge in the A filter than the others. Interestingly, the E filter shows significantly different behavior. For the maria, the E filter shows a marked broad peak around 25◦ –30◦ phase where the brightness is 15% greater relative to the B filter than at 0◦ phase angle. A similar effect is seen for the highlands except that the peak is shifted toward higher phase angles (50◦ –60◦ ). Variations between the highlands and maria are shown in Fig. 9, where the best fit functions for the highlands are ratioed to the maria. The highlands exhibit a stronger more nar-

rowly peaked opposition surge at low phase angles (α < 4◦ ), 15–17% greater than the maria between 0◦ and 4◦ solar phase angle. These results are in full agreement with the earlier study of Buratti et al. (1996) who found the highlands to exhibit a roughly 10% greater surge between 0◦ and 1.25◦ phase angle. However, beyond 4◦ phase angle the trend reverses itself and then highlands become relatively brighter as the phase angle is increased to 20◦ –25◦ . This suggests that the maria exhibit a broader shadow-hiding surge indicative of a more compacted surface than the highlands and is in agreement with

FIG. 6. Color ratio plot for the highlands. The phase functions f (α) and data have been ratioed to the B filter. The data have binned in 1◦ increments in phase angle and corrected for variations in incidence and emission angle using the Lommel–Seeliger factor (data = I /F × (µ0 + µ)/µ).

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maria) and thus it is unclear if the notable dip in the curve in this phase angle range is a true effect or not. PHYSICALLY BASED PHOTOMETRIC MODELING

The empirical photometric functions derived in the previous section are useful for providing photometric corrections to data. However, photometric observations also provide important information about the physical nature and structure of a surface. Extraction of this information requires a more sophisticated model whose parameters are related to the physical properties of the surface. One such model which has been widely applied to Solar System objects is the model developed by Hapke (1981, 1984, 1986). Hapke’s photometric model is given by FIG. 7. Color ratio image of the E-filter image LUE3091i.087 to the Bfilter image LUB3101i.087. The image center is near 16◦ phase angle. No significant variation across the image is observed.

Helfenstein and Veverka (1987) who found the dark terrain of Shorthill et al. (1969) to exhibit a broader shadow-hiding surge (Hapke’s (1986) width parameter h = 0.12) than the average (h = 0.06) or bright terrains (h = 0.05), or the Moon as a whole (h = 0.07). It is also in agreement with Hapke et al. (1998), who found that the low albedo maria materials exhibit a relatively greater broad shadow hiding opposition effect than the higher albedo highlands materials they examined. The increase in the ratio at phase angles beyond 60◦ is consistent with a greater amount of forward scatter in the highlands particles, as would be expected from their higher albedo. Beyond 25◦ phase, the maria again become relatively brighter than the highlands as the phase angle is increased. It should be noted that there is relatively little data at phase angles from ∼50◦ to 70◦ (particularly for the

R(i, e, α) =

ω˜ 0 µ0 {[1 + B(α; h, B0 )]P(α; g1 , g2 , f ) 4 µ0 + µ + [H (µ0 ; b)H (µ; b) − PLG (α, b)]}S(i, e, α; θ¯ ),

(3)

where µ and µ0 are, respectively, the cosines of the emission (e) and incidence (i) angles (after adjustments for macroscopic roughness (Hapke 1984)), α is the phase angle, and R is the radiance factor. This function contains a number of constituent functions which describe various aspects of the scattering. The factor [1 + B(α; h, B0 )]P(α; g1 , g2 , f ) accounts for the light which has been singly scattered. P is the single particle phase function, which we model using a two-term Henyey–Greenstein phase function which consists of a linear combination of two one-term Henyey–Greenstein phase functions, P = (1 − f )PHG (α; g1 ) + f PHG (α; g2 ),

FIG. 8. Best fit empirical phase functions f (α) ratioed to the B filter and normalized to one at zero degrees phase angle.

(4)

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FIG. 9. Ratio of the best fit empirical phase functions f (α) for the highlands to the best fit functions for the maria.

where PHG (α; g) =

(1 − g 2 ) (1 + 2g cos(α) + g 2 )3/2

(5)

is the Henyey–Greenstein phase function and g1 and g2 are the asymmetry parameters for the two functions and f is the relative strength of the two components. The shadow hiding opposition effect is accounted for by the function B (Hapke 1986), which enhances the singly scattered light (the shadow hiding opposition effect is only expected to affect the singly scattered light significantly). This function incorporates two parameters, h and B0 , which describe the angular width and amplitude, respectively, of the opposition surge. The second factor in the braces, H (µ0 ; b)H (µ; b) − PLG (α, b), accounts for the multiply scattered light. H is Chandrasekhar’s H functions. In Hapke’s (1981, 1984, 1986) original model, the multiple scattering is approximated by that of isotropic particles. Here we relax this assumption somewhat and solve for the multiple scattering assuming that the particle phase function is given by a one-term Legendre polynomial: PLG (b) = 1 − b cos(α).

(6)

b is related to the phase function parameters by b = 3[(1− f )g1 + f g2 ] to yield the same effective asymmetry parameter as the two-term Henyey–Greenstein single particle phase function used above. Finally, S is a function to account for shadowing caused by macroscopic roughness (Hapke, 1984). It incorporates one parameter, θ¯ , which descibes the average slope angle of subresolution topographic relief. Since the inception of Hapke’s original model, a second possible mechanism for the opposition effect has been recognized: coherent backscatter (Kuga and Ishimaru 1984, Shkuratov 1989,

Muinonen 1990, Hapke 1990, Mishchenko 1992, Mishchenko and Dlugach 1992). Currently, there is some debate over how coherent backscatter should be incorporated into scattering models such as Hapke’s. Coherent backscatter affects only light that has been multiply scattered. Therefore, one might include it as an enhancement to the multiply scattered component alone. However, it may not be unreasonable for coherent backscatter to affect the singly scattered component if the light scattered by a single particle has undergone multiple scatterings within the particle, for example off of internal defects or surface asperities (technically, in this case the enhancement due to coherent backscatter could be construed as an enhancement to the particle phase function). Helfenstein et al. (1997) chose to include it as an enhancemnt to the total scattered light. Helfenstein et al. did this because the formulation for coherent backscatter they used (and adopted in this paper), based on the model of Ozrin (1992), calculates the enhancement to the total scattered light (including the singly scattered component). However, Ozrin’s (1992) calculations were for a surface consisting of nonabsorbing Rayleigh scatterers, dominated by multiply scattered light. For an absorbing medium, one would expect a relatively larger contribution from singly scattered light reducing the contribution of coherent backscatter. We have therefore made one modification to the model employed by Helfenstein et al. (1997). While we allow the multiply scattered light to be fully enhanced by the coherent backscatter effect we assume that only a fraction of the “singly” scattered light will be so enhanced. Thus our model, including the modifications for accounting for coherent backscatter, is given by R(i, e, α) =

ω˜ 0 µ0 {[1 + B(α; h, B0 )]P(α; g1 , g2 , f ) 4 µ0 + µ ¢ ¡ × 1 + B0c ξ (α, 1αCB ) + [H (µ0 ; b)H (µ; b) − PLG (α, b)](1 + ξ (α, 1αCB ))}S(i, e, α; θ¯ ). (7)

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FIG. 10. The coherent backscatter function, ξ , as a function of phase angle used in this study (based on the model of Ozrin (1992)).

The function ξ accounts for the enhancement due to coherent backscatter. As noted above, we use the model developed by Ozrin (1992) and adopted by Mishchenko (1994) and Helfenstein et al. (1997). It incorporates one parameter, 1αCB which describes the width of the coherent backscatter opposition surge. It is shown plotted as a function of α/1αCB in Fig. 10. As can be seen, it predicts an enhancement factor of ∼0.54 at zero degrees phase angle. While calculated for Rayleigh scatterers, Mishchenko (1992b) shows that the enhancement factor does not depend greatly on particle size1 and thus this factor should be reasonable for larger particles as well. The parameter B0C accounts for the reduced coherent backscatter expected for the “singly” scattered component. Physically, it is the fraction of light (compared to Ozrin’s case of nonabsorbing Rayleigh scatterers) which undergoes multiple scattering within a particle and thus is subject to the coherent backscatter effect. Note that our model reduces to that employed by Helfenstein et al. (1997) when B0C = 1.0. The parameters of the model, and their physical significance, are summarized in Table IV. Unfortunately, the Clementine data cannot adequately constrain all of the model parameters. In particular, no observations at high phase angles were obtained (due to technical difficulties the planned observations during the lunar phasing orbits could not be made), making it impossible to constrain the properties of a forward scattering lobe to the particle phase function. Thus, g2 = 0.65 and f = 0.45 were assumed, based on the results of Helfenstein et al. (1997). Further, the macroscopic roughness is best determined from high phase angle data as well as limb darkening information. While there are a few observations away from e = 0◦ , most of the Clementine observations were nadir pointing 1 Unlike the total amplitude, the width of the coherent backscatter effect is dependent on the particle size.

making them ill-suited for constraining the surface macroscopic roughness, a result borne out in initial attempts to fit the data. Recently, Helfenstein and Shepard (1999) directly examined the lunar surface roughness by applying stereophotogrammetry to Apollo Lunar Surface Closeup Camera pictures to obtain topographic relief maps of undisturbed lunar soil. They found that a macroscopic roughness of 24◦ for the maria and 27◦ for the highlands found from fits to data from Shorthill et al. (1969) are consistent with the topographic relief revealed in the stereophotogrammetry. In addition to its normal mapping sequence, Clementine obtained observations of selected sites at a range of phase and emission angles. Figure 11 shows fits assuming various values of the surface macroscopic roughness to such Clementine data taken of the Apollo 16 landing site. Because the Apollo 16 landing site is in the highlands the photometric properties (other than the roughness) of the highlands (derived below) were assumed. The Apollo 16 landing site data suggest a slightly smaller value of 22◦ for the roughness. It is possible that the results of Helfenstein and Shepard, found for Fra Mauro, are not indicative of the highlands in general. However, the uncertainties in our measurements are large enough to be consistent with the roughness of 27◦ determined by Helfenstein and Shepard (1999) and we have adopted their values for the surface roughness here. This agreement between “ground truth” measurements and our remote sensing results, while not definitive, is reassuring. In initial attempts to fit the data it was found that the fitting routines often yielded an extremely broad (width >40◦ ) coherent or shadow hiding opposition surge: clearly these parameters were being used to fit aspects of the scattering beyond the opposition surge. This tendency was eliminated using the following procedure. The data at phase angles greater than 15◦ , where the opposition surge should be substantially reduced, was used to

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TABLE IV Summary of Photometric Model Parameters and Their Physical Significance Parameter

Definition

Description and geologic significance

ω˜ 0

Particle single-scattering albedo

Fraction of light scattered to light incident upon a single particle. It is related to the particle composition, size, and microstructure.

g1 , g2 , f

Particle phase function

Determines the directional scattering properties of individual particles. g1 < 0 describes the backscattering lobe. g2 > 0 describes the forward scattering lobe. f determines the relative strength of the two components. It is related to particle size, shape, structure, and composition.

h

Angular width of the shadow hiding opposition surge

Angular half width of the shadow hiding opposition surge is given by 1αSH = 2h. h is related to the porosity of and particle size distribution in a surface.

B0

Amplitude of the shadow hiding opposition surge

Fraction of light scattered near the surface of a single particle and thus subject to the shadow hiding opposition effect. It is related to particle composition and structure.

B0c

Amplitude of coherent backscatter opposition surge

Fraction of light multiply scattered within a single particle. Related to particle structure.

1αCB

Angular width of the coherent backscatter opposition surge

Related to porosity and particle size distribution.

θ¯

Surface macroscopic roughness

Average topographic slope angle of subresolution scale roughness.

constrain the particle single-scattering albedo and the phase function parameter g1 (for this calculation the opposition surge parameters were held at h = 0.06, B0 = 1.0, 1αCB = 2.0, and B0C = 1.0, typical of previous results for the lunar opposition

surge). With ω˜ 0 and g1 held constant, the opposition surge parameters were then fit to the data at phase angles less than 20◦ . The procedure was then repeated once (ω˜ 0 and g1 recalculated using the new best fit opposition surge parameters which were,

FIG. 11. The effect of roughness on model fits to Clementine data of the Apollo 16 landing site. The two data points near 10◦ phase angle were obtained during the regular mapping with e ∼ 0◦ . The remaining data points were obtained during a special sequence during orbit 30. For these points, the incidence angle is ∼26.5◦ , while the emission angle ranges from 40◦ to 80◦ . While the roughness is not well constrained, the Clementine data are consistent with the value of 27◦ determined by Helfenstein and Shepard (1998).

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TABLE Va Best Fit Hapke Model Parameters for the Highlands Filter

ω˜ 0

g1

B0

h

B0c

1αCB (◦ )

A

0.354 ± 0.010

−0.325 ± 0.016

1.00 ± 0.13

0.047 ± 0.012

0.41+0.29 −0.10

6.0+3.4 −2.6

B

0.512 ± 0.010

−0.338 ± 0.015

1.00 ± 0.10

0.055 ± 0.011

0.11

1.1+3.2 −0.7

C

0.552 ± 0.011

−0.320 ± 0.015

1.00+0.00 −0.08

0.060+0.010 −0.005

0.06

0.7+1.0 −0.6

D

0.565 ± 0.015

−0.311 ± 0.021

0.84+0.10 −0.08

0.063 ± 0.012

0.14+0.20 −0.13

3.1+3.1 −1.7

0.128+0.08 −0.04

1.00+0.00 −0.33

1.1+0.5 −0.1

0.041+0.013 −0.009

0.09

0.9+1.3 −0.8

E

0.577 ± 0.010

−0.312 ± 0.015

0.32+0.06 −0.04

1 µm

0.569 ± 0.010

−0.329 ± 0.015

0.82 ± 0.12

Note. B0c is not well constrained in the b, c, and 1 µm filters. However, relatively low values are preferred (see text).

in turn, recalculated). As expected, the second iteration did not substantially change the calculated parameters. Theoretically, the parameter B0 should not exceed one. However, in many previous works this requirement has been relaxed and optimum solutions found with B0 > 1 (see for example the discussion in Helfenstein et al. 1997). This relaxation was particularly important for finding optimal solutions to data for relatively dark objects such as the Moon (Helfenstein and Veverka 1987), Mercury (Veverka et al. 1988, Bowell et al. 1989), Phobos (Simonelli et al. 1996), and Deimos (Thomas et al. 1996). Helfenstein et al. (1997) found that the inclusion of coherent backscatter in their model allowed them to fit lunar data optimally without exceeding this limit. Following Helfenstein et al. we restrict B0 and B0C to their theoretically admissible values. PHOTOMETRIC MODELING: ERROR ANALYSIS

Several sources contribute to uncertainties in our derived parameters. One source is the random error in the data and the consequent uncertainty in how well the parent distribution of the data are represented by the finite number of data points taken. This is the error calculated using standard error analysis (variances in the parameters given by the diagonal elements of the error matrix (Bevington 1969)). This error is quite small and we

believe it is dominated by additional sources of uncertainty. One such source is systematic errors in the data due to calibration errors. Systematic errors between images (for example between the long and short exposure images in an image sequence) are estimated to be around 1% on the Clementine calibration web site. An additional and perhaps more significant source of uncertainty is the fact that data have been combined from a number of regions which undoubtedly exhibit some photometric variations (and, in particular, albedo variations). While attempts have been made to correct for albedo variations between regions such effects cannot be completely eliminated. While it is difficult to quantify these sources of error, we believe the error bars for the parameters shown in Table V, calculated as the change in the parameters needed to increase the residuals by 5%, provide a reasonable estimate. For comparison, these are on the order of 20× those associated with the formal uncertainty calculated from the error matrix. In addition to the uncertainties associated with the data, the approximations and idealizations made in the modeling yield additional uncertainties. Hapke’s photometric model assumes that classical radiative transfer provides an adequate approximation (with corrections for surface roughness and the opposition effect) even for a compacted surface as long as the particles are large compared with a wavelength. Hillier (1997) has shown that this assumption may be significantly in error and thus the

TABLE Vb Best Fit Hapke Model Parameters for the Maria Filter

ω˜ 0

g1

B0

h

B0c

1αCB (◦ )

A

0.203 ± 0.003

−0.307 ± 0.010

1.00+0.00 −0.10

0.042+0.015 −0.005

1.00+0.00 −0.14

5.7+2.8 −1.2

B

0.333 ± 0.007

−0.226 ± 0.015

1.00+0.00 −0.05

0.068+0.015 −0.010

1.00+0.00 −0.08

8.3+2.2 −1.8

C

0.343 ± 0.008

−0.233 ± 0.017

1.00+0.00 −0.06

0.061+0.012 −0.010

1.00+0.00 −0.10

7.3+2.2 −1.5

D

0.339 ± 0.008

−0.231 ± 0.017

1.00+0.00 −0.05

0.067+0.013 −0.010

1.00+0.00 −0.08

8.0+2.2 −1.5

0.060+0.023 −0.009

0.57+0.18 −0.15

7.5+5.5 −3.0

0.096+0.020 −0.016

1.00+0.00 −0.05

11.7+3.3 −2.4

E

0.334 ± 0.007

−0.266 ± 0.017

0.99+0.00 −0.12

1 µm

0.326 ± 0.009

−0.215 ± 0.017

1.00+0.00 −0.03

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physical significance of the parameters derived should be viewed with caution. Fortunately, the discrepancy is smallest in the backscattering direction and thus the approximation is less serious for examining the opposition effect. Also, it should be noted that the inversion of reflectance measurements is an ill-posed problem where different sets of parameters can yield nearly identical reflectances. Because of this, the photometrist must be careful to ensure that a unique solution has been found and care must be taken in choosing the form of the particle phase function employed. For example, any light which is forward scattered by a particle will only be seen in the backscattered light if it is scattered by an additional particle. This means that for a dark surface such as the Moon, which exhibits little multiple scattering, essentially no signature of such light will be seen in the backscattered direction. Thus, a model which assumes that all such forward scattered light is absorbed rather than scattered (with an appropriately lowered particle single-scattering albedo) will yield nearly the same reflectance in the backscattered direction as a model which includes the forward scattered light. This is why (without any data at α > 90◦ ) we cannot constrain the forward scattered component of the phase function. Further, it shows that our derived particle single-scattering albedo may need to be revised if the phase function is more or less forward scattering than we have assumed. PHOTOMETRIC MODELING: RESULTS

Our best fit parameters using a nonlinear least squares approach are shown in Table V. The goodness of fit of the model is shown for the B filter in Figs. 12 and 13. The results for the other filters are similar. Because most of the data are nadir pointing, in these plots the modeled results are for the geometry e = 0, i = α. For the highlands, the model provides a good

fit to the data though there is some discrepancy at the highest phase data observed (α ∼ 70◦ ). These data were collected from three images near 4◦ S 268◦ E and 9◦ S 259◦ E. The discrepancy can be plausibly explained if this region is of higher albedo than the typical highlands. The fit to the maria data is not as good. The model predicts a somewhat steeper phase function than suggested by the data. This may be due to a selection effect. Because most of the data are nadir pointing, the low phase data is derived from regions near the equator while the higher phase data comes from regions at increasingly higher latitudes. The maria tend to be concentrated at lower latitudes. Thus it is likely that the relatively small maria regions at high latitudes may be contaminated by materials from nearby highlands regions while such contamination occurs to a lesser extent nearer the equator. Early results based on Clementine data suggesting a strong very narrow peak (width <0.25◦ ; Nozette et al. 1994) have been criticized because the finite size of the Sun (diameter of 0.5◦ at the Moon’s distance from the Sun) should smooth out any features less than 0.25◦ wide (Shkuratov et al. 1999). Indeed, the finite size of the Sun should cause the phase curve to round over and become flat at exactly zero degrees phase. It is difficult to see how such narrow features could be seen and we believe that this criticism is valid. The reported sharp surges are probably due to the location of a bright crater at exactly the zero phase point and thus are spurious. Fortunately, this does not affect the conclusions of our previous paper (Buratti et al. 1996). Indeed, their data do show a roundoff at low phase angles due to the finite size of the Sun, as expected (see their Fig. 7). Our data do not show such a roundoff. However, this may be due to the sampling technique we employed. As noted in the Appendix, each data point was averaged over a 24 × 24 box of pixels in an image. The photometric angles vary by about 0.5◦ across each

FIG. 12. Model fit to the data for the highlands. The results shown are for the B filter. The results in the other filters are similar. Because most of the data is near e = 0◦ , the model is calculated for the geometry e = 0◦ , i = α. Both the model and the data have been multiplied by the factor (µ0 + µ)/µ0 to correct for variations in emission and incidence angle.

CLEMENTINE PHOTOMETRY

217

FIG. 13. Model fit to the data for the maria. The results shown are for the B filter. The results in the other filters are similar. Because most of the data is near e = 0◦ , the model is calculated for the geometry e = 0◦ , i = α. Both the model and data have been multiplied by the factor (µ0 + µ)/µ0 to correct for variations in emission and incidence angle.

such box of pixels. Thus, our resolution is not good enough to see the expected roundoff at zero degrees phase. Our results for the particle single-scattering albedo are generally higher than earlier results based on models using a oneterm Henyey–Greenstein phase function or a Legendre polynomial (Helfenstein et al. 1997, Helfenstein and Veverka 1987). However, fits using a two-term Henyey–Greenstein phase function which allows a forward scattering component to the phase function tend to yield higher particle single-scattering albedos (Helfenstein et al. 1997, Hillier 1993) and our results are consistent with the value of 0.279 ± 0.002 found by Helfenstein et al. (1997) for the Moon as a whole in the V filter using this form of the phase function. Consistent with previous results we find the Moon to exhibit a moderate backscattering lobe to the phase function. The phase function shows some dependence on the terrain, exhibiting a somewhat narrower backscattering component in the highlands than in the maria, though this may be due to the selection effect affecting the maria data. The phase function is substantially independent of wavelength except for a somewhat more pronounced backscattering lobe at 0.415 µm in the maria. The Clementine spacecraft data provide the best data set currently available for studying the opposition effect on the Moon or any other Solar System object. Buratti et al. (1996) found that the opposition surge of the Moon exhibits little wavelength dependence and from this concluded that shadow hiding is the dominant mechanism on the Moon. However, more recent studies have called this conclusion into question. Helfenstein et al. (1997) found that their data could be best fit by a narrow surge (halfwidth = 2.0 ± 0.5◦ ), attributed to coherent backscatter, superimposed on a broader shadow hiding surge (h = 0.158 ± 0.001 or halfwidth = 18.1 ± 0.1◦ ). Hapke et al. (1998) examined the polarization of the scattered light from lunar samples in the laboratory and came to a similar conclusion: both mecha-

nisms contribute roughly equal amounts to the overall surge with coherent backscatter forming a narrow surge (less than a few degrees) and is important at small phase angles while shadow hiding forms a broader surge and dominates the observed effect at larger phase angles. The present study gives us an opportunity to reexamine this question. Because coherent backscatter is a multiple scattering phenomenon, we expect it to be strongest for the brightest regions. For these regions (the highlands in filters B–E) our best fit parameters do indicate a small narrow coherent backscatter opposition peak in agreement with the recent studies. The data and fit at low phase angles are shown in Figs. 14 and 15 for the B and D filters, respectively. In addition to the nominal model, the results of a fit assuming no coherent backscatter (labelled SHOE only) and a fit assuming B0C = 0.0 are shown. As can be seen, though the effect is not large, the data do indicate a small narrow surge which may be attributable to coherent backscatter. Our data suggest a somewhat narrower surge (typically around 1◦ ) but it is in agreement within the error bars of the result of Helfenstein et al. (1997). Our results for the shadow hiding opposition serge width (h ∼ 0.06) are in line with earlier results found with models incorporating shadow hiding alone (e.g., Helfenstein and Veverka 1987) and agree with the average halfwidth of ∼8◦ found by Hapke et al. (1998). The discrepancy with the results of Helfenstein et al. may be explained by the fact that our results are for the highlands while theirs incorporated data from a variety of sources. The broader lunar surge for the darker terrains found by Helfenstein and Veverka (1987) supports this suggestion. For the B, C, and 1-µm filter fits it was found that any value of B0C can fit the data. However, for higher values this is accomplished by narrowing the coherent backscatter peak to be nearly out of the range of the data examined. Therefore, relatively low values are preferred. Thus, except for the E-filter which, as noted above, may be anomalous,

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FIG. 14. Model fit to the low phase data. In addition to the nominal model, two additional fits are shown: one assuming no multiple scattering within a particle (B0C = 0.0) and one assuming no coherent backscatter. The results shown are for the highlands in the B filter. The data have been binned in 0.25◦ increments in phase angle. While not large, a narrow surge at small phase angles is observed. The error bars indicate one standard deviation of the data in each bin and shows the range of albedos seen. The error in the mean is a better measure of the uncertainties in the observations and is on the order of the size of the symbols used or less.

our results suggest a small value for B0C . This is not surprising, for relatively dark particles such as those which make up the lunar regolith, one would not expect much multiple scattering to occur within the particle. Because the model of Helfenstein et al. (1997) essentially assumes B0C = 1.0, our results suggest a significantly reduced coherent backscatter effect exhibiting a total enhancement of ∼10–15% about 1/6 to 1/4 that of the shadow hiding component (Table VI). Our results for the darker regions (the maria and the highlands in the A filter) suggest a broader and stronger coherent backscat-

ter surge. It is hard to understand why coherent backscatter would be stronger at lower albedos. Being a multiple scattering phenomenon it should exhibit the opposite effect. Figure 16 shows the fit to the B-filter maria data at low phase angles. The results for the other filters are similar. As can be seen, the fit is not very good. The data at low phase angles indicate a shallower opposition effect. It is clear that it is difficult to reconcile the high phase data (α > 20◦ ) with the data in the opposition surge region. The additional fit shown in Fig. 16 is to the low phase data only and assumes a shadow hiding opposition surge with h = 0.0725

FIG. 15. Model fit to the low phase data. In addition to the nominal model, two additional fits are shown: one assuming no multiple scattering within a particle (B0C = 0.0) and one assuming no coherent backscatter. The results shown are for the highlands in the D filter. The data have been binned in 0.25◦ increments in phase angle. While not large, a narrow surge at small phase angles is observed. The error bars indicate one standard deviation of the data in each bin and shows the range of albedos seen. The error in the mean is a better measure of the uncertainties in the observations and is on the order of the size of the symbols used or less.

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TABLE VI Contributions of Shadow Hiding and Coherent Backscatter Opposition Surge Mechanisms

Filter

AT

ASHOE

ACBOE

Shadow hiding enhancement A − ACBOE ( T ACBOE )

B C D E

0.286 0.293 0.285 0.301

0.255 0.265 0.247 0.196

0.168 0.177 0.185 0.244

0.701 0.655 0.544 0.232

0.121 0.109 0.152 0.536

0.143

0.093

0.077

0.854

0.536

Helfenstein et al. (1997)

Coherent backscatter enhancement A − ASHOE ( T ASHOE )

Note. AT is the model normal albedo. ASHOE and ACBOE are the normal albedos assuming only shadow hiding or coherent backscatter, respectively.

and ω˜ 0 = 0.452. Unfortunately, Clementine obtained essentially no data for the maria in the crucial phase angle range of ∼10◦ to 20◦ which would have helped to resolve this issue. Thus caution needs to be exercised in interpreting the results for the maria. Unlike the highlands, the data for the maria suggest little, if any, evidence for a narrow opposition surge. Interestingly, unlike the other filters, the data for the highlands in the A filter (which is of lower albedo than the other filters) also show little evidence for a narrow coherent backscatter peak (Fig. 17), a result not marred by the lack of data available for the maria. A plausible explanation for the lack of a narrow surge in the maria and A-filter highlands data is that these regions are too dark to exhibit an observable coherent backscatter surge. An alternative explanation is that the mean optical pathlength is shorter in the darker regions leading to a broader coherent backscatter peak which is difficult to distinguish from the shadow hiding peak. However, we be-

lieve that it is unlikely that the mean optical pathlength would vary by the necessary factor of several over the given range of albedos and thus favor the former explanation. In either case this provides a natural explanation for the observation by Buratti et al. (1996), confirmed in this paper, that the highlands exhibit a roughly 10% stronger surge between 0 and 1.25◦ phase angle. In contrast to our result, Hapke et al. (1998) did see evidence for coherent backscatter (albeit at a reduced level compared to the highlands) in the maria. The technique employed by Hapke et al. (1998) (examining the circular polarization of lunar samples) is probably more sensitive to detecting the presence of coherent backscatter (though perhaps not as good for determining its total strength; Paul Helfenstein, pers. commun.). Thus it is likely that there is some coherent backscatter in the maria but it is below the level of detectability using the techniques employed in this paper.

FIG. 16. Model fit to the low phase data. In addition to the nominal model, two additional fits are shown: one assuming no multiple scattering within a particle (B0C = 0.0) and one assuming no coherent backscatter. As can be seen, the fits suggest a narrower surge than the low phase data. A fourth fit with h = 0.073 and no coherent backscatter only to the low phase data is shown as well. The results shown are for the maria in the B filter. The data have been binned in 0.25◦ increments in phase angle. Unlike the highlands, little evidence for a narrow surge is seen. The error bars indicate one standard deviation of the data in each bin and shows the range of albedos seen. The error in the mean is a better measure of the uncertainties in the observations and is on the order of the size of the symbols used or less.

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FIG. 17. Model fit to the low phase data. In addition to the nominal model, two additional fits are shown: one assuming no multiple scattering within a particle (B0C = 0.0) and one assuming no coherent backscatter. The results shown are for the highlands in the A filter. The data have been binned in 0.25◦ increments in phase angle. Unlike in the other filters, little evidence for a narrow surge is seen. The error bars indicate one standard deviation of the data in each bin and shows the range of albedos seen. The error in the mean is a better measure of the uncertainties in the observations and is on the order of the size of the symbols used or less.

The width of the shadow hiding opposition effect is predicted to depend only on the porosity and particle size distribution in the surface. Thus, it should be independent of wavelength. In contrast, because coherent backscatter opposition effect is an interference effect, its width should be proportional to wavelength. However, it is also expected to be inversely proportional to the mean photon optical path length. The mean photon optical path length will increase (due to increased multiple scattering) as the albedo is raised. Thus, higher albedo surfaces should exhibit

sharper opposition peaks. The wavelength dependence of the width of the opposition surge is shown in Fig. 18. As expected, the shadow hiding surge shows little wavelength dependence. The coherent backscatter peak also exhibits little wavelength dependence, though the error bars are large enough that some dependence cannot be ruled out. The opposition surge amplitude as a function of albedo is shown in Fig. 19. As expected for a single scattering phenomenon, the shadow hiding amplitude decreases with albedo.

FIG. 18. Opposition surge width as a function of wavelength. For clarity, only the error bars for the highlands observations are shown. The error bars for the maria are similar.

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FIG. 19. Opposition surge amplitude as a function of albedo. Results shown are for the shadow hiding opposition effect (SHOE) and the coherent backscatter opposition effect (CBOE).

However, coherent backscatter also appears to decrease with albedo, opposite of what is expected for a multiple scattering phenomenon. However, as noted above the low albedo data comes from the maria data which may be suspect. The above results suggest that the model parameters for the coherent backscatter surge for the maria and A-filter highlands may be an artifact of the model and may not be associated with a coherent backscatter surge at all. It is interesting to note that the model halfwidths of the shadow hiding and coherent backscatter surges are very similar for the maria and A-filter highlands (Fig. 18). This suggests that the surge seen in the maria and A-filter highlands may be due to a single mechanism with a single characteristic width of the surge. However, because the surge is stronger than can theoretically be explained by shadow hiding alone the model requires both mechanisms to fit it. As noted above, such strong surges beyond the theoretical limits of shadow hiding have been observed on many bodies and particularly on dark bodies in the Solar System. The cause of such strong surges is unclear. However, one possibility is that the individual particles themselves may exhibit a sharply peaked phase function in the backward direction. Another possibility is that the surface consists of both some strongly forward and some strongly backscattering particles. Such a surface would exhibit a relatively strong doubly scattered component to which coherent backscatter could contribute even for low albedo surfaces. The observations of the opposition surge provided by Clementine provide us with the opportunity to obtain accurate determinations of the Moon’s geometric albedo ( p), phase integral (q), and Bond albedo (A). By definition, the geometric albedo requires that the reflectance be known at opposition and thus requires an accurate determination of the opposition surge. The geometric albedo, phase integral, and Bond albedo for the maria and highlands (these assume a uniform planetary surface consisting of maria or highland materials) as well as the normal reflectance are shown in Table VII. The error bars shown

do not include possible errors due to the lack of data beyond ∼80◦ phase angle. For example, a 10% uncertainty in the model reflectance beyond 90◦ phase angle increases the error bar for the Bond albedo by roughly 50%. The normal reflectance is very near the calculated geometric albedo, which is not surprising: the Moon exhibits little limb darkening at opposition. The albedo and phase integral of the lunar nearside, calculated assuming a 32% areal coverage of maria and 68% highlands is shown in Table VIII. Also shown are the results of Buratti et al. (1996), who calculated the photometric quantities using the data of Lane and Irvine (1973) augmented by opposition surge data from Clementine. Our results show a slightly higher geometric albedo and slightly lower phase integral yielding Bond albedos similar to those found by Buratti et al. (1996), which indicates a somewhat stronger opposition surge than found in the previous study. The results are generally consistent, though the Clementine data TABLE VII Geometric Albedo, Phase Integral, Bond Albedo, and Normal Reflectance Derived from Clementine Data Filter

p

q

A

rn

A B C D E 1 µm

0.13 ± 0.01 0.17 ± 0.02 0.18 ± 0.02 0.17 ± 0.02 0.16 ± 0.01 0.16 ± 0.01

Maria 0.33 ± 0.03 0.44 ± 0.04 0.43 ± 0.04 0.44 ± 0.04 0.45 ± 0.04 0.48 ± 0.03

0.042 ± 0.001 0.074 ± 0.002 0.076 ± 0.002 0.076 ± 0.002 0.074 ± 0.002 0.075 ± 0.004

0.127 0.168 0.178 0.175 0.164 0.158

A B C D E 1 µm

0.20 ± 0.01 0.28 ± 0.01 0.29 ± 0.01 0.28 ± 0.02 0.29 ± 0.01 0.29 ± 0.02

Highlands 0.40 ± 0.03 0.45 ± 0.03 0.49 ± 0.03 0.52 ± 0.04 0.50 ± 0.03 0.50 ± 0.04

0.078 ± 0.002 0.127 ± 0.002 0.140 ± 0.003 0.144 ± 0.004 0.146 ± 0.003 0.143 ± 0.003

0.198 0.286 0.293 0.285 0.301 0.295

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TABLE VIII Photometric Parameters for the Lunar Near Side Filter

p

q

A

pa

qa

Aa

A B C D E 1 µm

0.176 ± 0.009 0.246 ± 0.010 0.252 ± 0.010 0.246 ± 0.014 0.231 ± 0.010 0.248 ± 0.012

0.38 ± 0.02 0.46 ± 0.02 0.47 ± 0.02 0.50 ± 0.03 0.53 ± 0.02 0.49 ± 0.03

0.066 ± 0.001 0.109 ± 0.002 0.119 ± 0.002 0.122 ± 0.003 0.122 ± 0.002 0.121 ± 0.002

0.116 0.233 0.232 0.230 0.230

0.45 0.50 0.53 0.54 0.55

0.052 0.117 0.123 0.124 0.143

Note. Results were calculated using an average areal coverage of 32% Maria and 68% highlands for the lunar nearside and the geometric and bond albedos from Table VII. a Results from Buratti et. al (1996).

support a higher albedo at 0.415 µm than found by Lane and Irvine (1973). CONCLUSIONS

In the Appendix of this paper, the absolute calibration of the Clementine UV/Vis camera is examined. The calibration procedure provided on the Clementine calibration web site (http:// www.planetary.brown.edu/clementine/calibration.html) provides good results for relative photometry. For many applications a relative calibration is sufficient. However, the procedure yields reflectances significantly higher than those predicted by groundbased observations. For absolute photometry, a further correction factor is required. Using the lunar observations of Shorthill et al. (1969) a correction factor of 0.532 ± 0.005 has been found, which brings the Clementine calibration using the procedure outlined on the web site in line with groundbased observations. Using this calibration the lunar Bond albedo calculated from the Clementine data is in good agreement with the disk-integrated groundbased observations of Lane and Irvine (1973), indicating that a good absolute calibration of the Clementine images has been obtained. In the second part of the paper empirical phase functions for the highlands and maria in each of the UV/Vis filters have been derived. The derived parameters are shown in Table III. These functions should prove useful for performing photometric corrections to the Clementine data necessary in many studies including mineralogic studies, photoclinometry, and image mosaicking. Finally, in the third section of the paper, we have used the absolutely calibrated Clementine data to perform a photometric study of the Moon. Currently, there is some debate over the cause of the opposition surge on the Moon: is it shadow hiding, coherent backscatter, or a combination of the two? The Clementine data are particularly useful for studying this effect. All the data show a relatively broad surge (typical halfwidth ∼8◦ ) which we attribute to shadow hiding. In addition, the highest albedo observations (highlands in the B–E filter) show a small narrow surge (halfwidth ∼1◦ ) superimposed on the broader surge which may be attributed to coherent backscatter. However, no compelling

evidence for a narrow surge is seen in the lower albedo observations (maria and A filter highlands). This is not surprising: coherent backscatter, as a multiple scattering phenomenon, is expected to be stronger for brighter surfaces. Thus, it is plausible that the maria and highlands in the A filter are too dark to exhibit an observable coherent backscatter peak. In conclusion, we find that the lunar opposition surge is due mainly to shadow hiding but coherent backscatter makes a significant contribution (total amplitude approximately 1/6 to 1/4 that of the shadow hiding component) at very low phase angles for the brighter terrains. APPENDIX 1: ABSOLUTE CALIBRATION OF THE CLEMENTINE UV/VIS CAMERA

A. Image Calibration The selected Clementine images were calibrated using the calibration procedure (excluding the photometric correction) outlined on the Clementine calibration web site (www.planetary.brown.edu/clementine/calibration.html) maintained at Brown University. As recommended on the web site, we use the 6-96 in-flight composite flat-fields derived by USGS (M. Robinson and A. McEwen). These were produced by averaging a very large number of images together (presumably, any albedo features are washed out by the averaging procedure). The calibration procedure ties Clementine observations of the Apollo 16 landing site to laboratory reflectance measurements of Apollo 16 soil samples to set the absolute calibration scale. Because the laboratory data are given as reflectance relative to a halon standard, the results of the calibration procedure will be the reflectance relative to the halon standard at the laboratory standard geometry (incidence angle, i = phase angle, α = 30◦ ; emission angle, e = 0◦ ). Thus, to scale the results to absolute reflectance requires multiplying the results by the bidirectional reflectance of halon at i, α = 30◦ , e = 0◦ : 0.91582 . However, this procedure yields a reflectance considerably higher than that predicted by Earthbased observations. This discrepancy between laboratory reflectance results (to which the Clementine calibration is tied) and remote sensing observations of the Moon have been noted previously (Blewett et al. 1997). The cause of the discrepancy is unclear. One possible explanation for this discrepancy is that the roughness and compaction states of the laboratory samples undoubtedly differ from those of the pristine lunar surface. In any case, it was decided to determine and apply a correction factor to bring the Clementine data in line with the diskresolved remote sensing observations of the Moon of Shorthill et al. (1969). This factor was determined as follows. Shorthill et al. observed a number of regions over a range of phase angles throughout a full lunation. We selected 15 of these regions (4 “dark,” 5 “average,” and 6 “bright” regions) chosen to minimize the difference in viewing/illumination geometry (and thus potential errors due to imperfect photometric corrections) between the Shorthill et al. data and the Clementine images of the Shorthill et al. sites (Table IX). We then performed the standard calibration procedures discussed above and collected reflectance data from the Clementine images as described in the next section and averaged the data together to obtain a resolution of ∼18 × 18 km, similar to the resolution obtained by Shorthill et al. We then compared the Shorthill et al. observations nearest in phase angle to the Clementine data (typically we chose two points,

2 The bidirectional reflectance of HALON was calculated using the procedure outlined by Paul Helfenstein (1994, pers. commun.). First, the bidirectional reflectance was evaluated using Hapke’s (1981, 1984, 1986) photometric model with the following parameters: ω˜ 0 = 0.99999714, h = 0.793, B0 = 1.51, g = 0.415, θ¯ = 0.0 obtained from least-squares fits to the data of Weidner and Hsia (1981). The predicted directional-hemispherical reflectance from these parameters slightly underestimates National Bureau of Standard values (Weidner and Hsia 1981) so a scale factor of 1.0091 was used to eliminate the small discrepancy.

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TABLE IX Shorthill et al. (1969) Regions Used as Calibration Sites

Region

Lat

Lon

Clementine image

Shorthill reflectance

Clementine reflectance

Shorthill/ Clementine

D13

24.4◦

9.8◦ W

lua29451.172

0.0818

0.567

D14

12.0◦

5.7◦ W

lua4081k.038

0.0711

0.612

D16

18.0◦

9.4◦ E

lua2968k.297

0.0490 0.0465 0.0421 0.0442 0.0441 0.0442 0.0614 0.0542 0.0399 0.0396

0.0691

0.614

lua2377k.165 D17

16.9◦

19.9◦ E

lua2937k.293

0.1017 0.0649

0.604 0.599 ± 0.011

Dark terrain average A2 A10

−16.13◦

−19.82◦

62.10◦ W 6.4◦ W

lua0229h.328 lua1104h.171

A12

−17.45◦

4.63◦ W

lua1301h.170

A18

5.73◦

A20

−7.82◦

14.5◦ E

lua2411j.295

23.57◦ E

lua3190i.159

0.0489 0.0809 0.0820 0.0750 0.0800 0.0688 0.0677 0.0910

0.0781 0.1386

0.616 0.583

0.1296

0.580

0.1202

0.584

0.1634

0.546 0.582 ± 0.011

Average terrain average B36 B37

−23.5◦ −29.0◦

60.0◦ W 2.0◦ W

lua1651g.327 lua2100g.037

B52

−8.0◦

19.0◦ E

lua3231i.161 lua3062i.029

B54

−12.0◦

21.0◦ E

lua3242h.160

B58

−28.0◦

11.0◦ E

lua2376g.032

B62

−27.5◦

10.5◦ E

lua2407g.032

0.0587 0.0695 0.0688 0.1379 0.1313 0.0846 0.0842 0.1281 0.1153 0.0639 0.0621 0.0694 0.0738

0.1044 0.1125

0.562 0.615

0.2195

0.587

0.1413 0.1936

0.629

0.1087

0.580

0.1203

0.595

Bright terrain average

0.595 ± 0.010

Total average

0.592 ± 0.006

Note. The Shorthill et al. reflectances shown have been corrected to the viewing and illumination geometry of the corresponding clementine image.

one each before and after full Moon) to the Clementine results, after performing a photometric correction using a nonabsolutely calibrated version of the photometric functions we calculated above. The results are shown in Table IX. A correction factor of 0.592 ± 0.006 was obtained as an average for the 15 regions. Finally, a correction factor of 0.8994 (calculated from the convolution of the lunar spectra of Pieters et al. (1991) available on the PDS Geosciences Spectroscopy subnode with the response of the Shorthill et al. instrument and the Clementine A filter) was applied to account for the slight difference in wavelength between the Clementine A filter (0.415 µm) and the Shorthill et al. data (0.445 µm). This yields a final multiplicative correction factor of 0.532 ± 0.005 which was applied to the Clementine data.

available by anonymous ftp at naif.jpl.nasa.gov under the path pub/naif/toolkit). Reflectance data from 48 × 48 box bins of pixels were averaged and the viewing and illumination geometry at the center of the box calculated. In order to remove any problems which might occur at the edge of the images, a 24-pixel band around the edge of the images was excluded. This produced a resolution somewhat less than 1◦ in incidence, emission, and phase angle, more than sufficient at most phase angles. However, at low phase angles (in the opposition surge region) a finer resolution is desired. Hence, for images whose center lies at a phase angle of 5◦ or less, the data were binned in 24 × 24 boxes (with a 12-pixel boundary). Therefore, 35 data points were obtained from each image at greater than 5◦ phase while 165 data points were obtained from each image at less than 5◦ phase.

B. Data Collection and Analysis After we selected and calibrated the images as described above, data was collected from each image using the following procedure. The pointing information was obtained using procedures adapted from NAIF’s SPICE library (toolkit

C. Normalization of Albedo Differences Unfortunately, for the highlands, the data from regions near opposition appear darker than elsewhere. All of the Clementine low phase data are obtained near

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HILLIER, BURATTI, AND HILL

TABLE X Normalization Factors for Region 105E Filter

Normalization factor

A B C D E Average

1.078 1.051 1.049 1.048 1.033 1.052 ± 0.005

an empirical phase function f (α), was calculated by binning the data (excluding the data for the image in question) in 1◦ increments of phase angle and using a linear interpolation between the binned data points. Only bins containing at least 15 data points were included. This was done to ensure that at least two images contribute data to the bin and thus prevent the computed fit from being thrown off by a single anomalous image. The data from the image being examined is then compared to the computed phase function and an average normalization factor to bring the image in line with the computed curve calculated, Correction Factor =

the equator and the highland regions near zero degrees phase are all near maria. Such small patches of highlands terrain interspersed among the maria are known to be darker than the albedo observed in extended highlands regions (Minnaert 1961). It is likely that these regions are contaminated with maria materials, darkening the surface. Thus, the following procedure was used to normalize the low phase data. The lowest phase data is from three regions: 0–3◦ N, 103–109◦ E; 1◦ S–3◦ N, ∼6◦ E; 0–5◦ N, 73–79◦ E (hereafter 105E, 6E, 75E). The data in region 105E covers phase angles from 2.7◦ to 8.3◦ while the second two extend from ∼7◦ down to 0◦ . First, a best fit to the Lommel–Seeliger photometric function, I µ0 f (α), = F µ + µ0

(8)

where µ0 = cos(i), µ = cos(e), and f (α) is a fourth-order polynomial, f (α) = a0 + a1 α + a2 α 2 + a3 α 3 + a4 α 4

(9)

was obtained to the data excluding these three regions. Then, the data in region 105E at phase angles greater than 7.5◦ (where there is some overlap in phase angle coverage with regions imaged at higher phase) was compared to the empirical fit and an average normalization factor needed to bring the data in line with the fit calculated (Table X). In order to not obscure any possible wavelength variations at low phase angles, the same normalization factor (an average of the normalization factor for all five filters) was applied for each filter to the data in the region 105E. A normalization factor for the still lower phase angle data found in regions 6E and 75E was found using a similar procedure with the new normalized 105E data added to extend the observations to lower phase. A new empirical fit (including the 105E data) was found. Then, an average normalization factor was found (Table XI) using data at phase angles greater than 5◦ (at a phase angle range overlapping data from region 105E) and an average of the normalization factors applied to data for each filter. Even within the maria or highlands, there is substantial variation between regions. It is likely that the observed variations are mainly due to albedo and not to the other properties of the surface. Therefore, for the fits to the rigorous photometric function, a normalization for albedo variations between regions within the same terrain classification was performed as follows. For each image,

TABLE XI Normalization Factors for Regions 6E and 75E Filter

C6E

C75E

A B C D E Average

1.312 1.298 1.326 1.329 1.310 1.315 ± 0.005

1.228 1.219 1.230 1.233 1.227 1.227 ± 0.002

N f emp (αi ) 1 X , N i=1 f (αi )

(10)

where N is the number of data points from the image (either 35 or 165), f emp is the computed phase function, αi is the phase angle of the ith data point from the image in question, and f (αi ) =

I cos(i i ) + cos(ei ) · , F cos(i i )

(11)

where i i and ei are the incidence and emission angle of the ith data point, respectively. Only data points within 0.5◦ in phase angle of a bin (i.e., only data which overlaps in phase angle data from at least two other images) were included in the average. The normalization factor is then applied to all of the data from the image in question. If no data points meet the criteria, i.e., there is no overlapping data between the image in question and the other images, no normalization is performed. The above procedure is performed for each image.

ACKNOWLEDGMENTS This work was performed while John Hillier held a National Research Council–JPL Research Associateship. Work performed at the Jet Propulsion Laboratory/California Institute of Technology under contract to NASA. Work supported by the NASA Planetary Geology, Astronomy, and Lunar and Asteroid Data Analysis Programs.

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