4162 - Korokhin, V. V., Y. I. Velikodsky, Y. G. Shkuratov, V. G. Kaydash
ARTICLE IN PRESS Planetary and Space Science 58 (2010) 1298–1306
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Removal of topographic effects from lunar images using Kaguya (LALT) and Earth-based observations Viktor V. Korokhin a,n, Yuriy I. Velikodsky a, Yuriy G. Shkuratov a, Vadym G. Kaydash a, Sergey Y. Gerasimenko a, Nikolai V. Opanasenko a, Gorden Videen b, Carle Pieters c a
Institute of Astronomy, Kharkiv V.N. Karazin National University, 35 Sumskaya Street, Kharkiv 61022, Ukraine Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, CO 80301, USA c Department of Geological Sciences, Brown University, Providence, RI 02912, USA b
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 February 2010 Accepted 20 May 2010 Available online 26 May 2010
Lunar images acquired at non-zero phase angles show brightness variations caused by both albedo heterogeneities and local topographic slopes of the surface. To distinguish between these two factors, altimetry measurements or photoclinometry data can be used. The distinction is especially important for imagery of phase-function parameters of the Moon. The imagery is a new tool that can be used to study structural anomalies of the lunar surface. To illustrate the removal of the topographic effects from photometric images, we used Earth-based telescopic observations, altimetry measurements carried out with the Kaguya (JAXA) LALT instrument, and a new photoclinometry technique that includes analysis of several images of the same scenes acquired at different phase angles. Using this technique we have mapped the longitudinal component of lunar topography slopes (the component measured along the lines of constant latitude). We have found good correlations when comparing our map with the corresponding data from Kaguya altimetry. The removal of the topographic surface properties allows for the study of the phase-function parameters of the lunar surface, not only for flat mare regions, but for highlands as well. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Lunar surface Photometry Albedo Phase function Topography Photoclinometry
1. Introduction An important goal of photometric studies of planetary images is to identify the physical properties of the material covering the surface of planetary bodies. One approach to this problem is to determine the photometric function, i.e., the function of how the intensity of light reflected from planetary areas varies with illumination and observation angles. The angle between vectors from the observed point to the light source and to the observer is the phase angle a. This is the most important angle variable in planetary photometry. The phase function is a component of the photometric function, which depends only on a (Hapke, 1993). The phase functions are determined by the complexity of the structure of the surface. Usually these phase functions have a single sharp maximum at a ¼0 being locally smooth at other a. Therefore, they can be described with a few parameters. The simplest parameters are ratios of phase-function values at different phase angles a; these characterize the slopes of phase curves. The slope also may be characterized with parameters of an analytical curve used for fitting to brightness measurements (see below). Imagery of the Moon can be made in terms of phase-angle ratios for different pairs of a
n
Corresponding author. Tel.: + 38 057 707 5064. E-mail address: [email protected] (V.V. Korokhin).
0032-0633/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2010.05.013
using original well calibrated brightness images. Such imagery is a relatively new and effective tool that can be used to study structural anomalies of the lunar surface (e.g., Shkuratov et al., 1994; Korokhin and Akimov, 1997; Kreslavsky and Shkuratov, 2003; Kreslavsky et al., 2003; Kaydash et al., 2009a, b). For instance, using photometric images acquired with Earth-based telescopes, weak swirls were found in the southern portion of Oceanus Procellarum (Shkuratov et al., 2010). The phase-angle-ratio images allow one to estimate spatial variations of the complexity of unresolved surface roughness and microtopography. However, the resolved topography spoils the images hampering their interpretation. Brightness variations on images of the lunar surface depend on the spatial distributions of local topographic slopes, albedo (that is, a characteristic of the surface material composition), and the global illumination/observation geometry. The lunar resolved topography influences brightness spatial variations, since the local surface slopes, or height gradients, change the local incident i and emergent e angles. This effect can be especially significant at large phase angles. Removing the influence of the resolved topography requires data on local slopes. Suitable global lunar height distributions have been obtained recently with the Laser Altimeter (LALT) aboard the spacecraft Kaguya (JAXA) (Araki et al., 2009), which we use in our analysis. Photogrammetry, photoclinometry (shape-from-shading) or their combination are techniques that allow one to retrieve
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information about the relief and albedo distributions directly from photometric images. Photogrammetry is based on the use of mutual parallaxes of surface details observed at two different angles. It can only be used if stereo images are available. For instance, photogrammetry cannot be practically applied to telescopic images of the Moon acquired from Earth, as the viewing geometry weakly varies for the Moon. However, this method has been used to retrieve the topographic information on different planets from images obtained during space missions (e.g., Connors, 1995; Oberst et al., 1999; Cook and Robinson, 2000; Herrick and Sharpton, 2000). Attempts have been made to recover the albedo distribution on a surface from images using the photometric stereo approach (e.g., Chen et al., 2002). Recently Clementine and Hubble Space Telescope images of the Apollo-17 landing area were used to map heights using the photogrammetric technique (Opanasenko et al., 2007). The Clementine and HST images have almost the same resolution, similar phase angles, and effective wavelength; however, the viewing angles are different. An automatic procedure to find the parallaxes, based on the determination of mutual shifts of the same lunar surface details by finding the maximum of the cross-correlation function of small portions of both images in a ‘‘running window,’’ was used. When photogrammetry is not applicable, photoclinometry has been exploited. This technique takes advantage of the fact that surface facets with orientations more nearly perpendicular to the illumination direction appear brighter than those facets facing away from the Sun. There are many different varieties of photoclinometry. In the simplest approach, a single image is used. By integrating slopes along a line parallel to the illumination direction, one can assemble a topographic profile. This approach ignores albedo and illumination/observation geometry variations over a surface. If the geometry can be taken into account, the albedo is unknown and may produce ambiguities. Thus, the albedo distribution should be estimated together with the determination of topography. When there are many images of the same scene at a fixed angle of view and different phase angles, one can find not only albedo and topography, but also the phase function for each point of the studied surface. The photoclinometric technique and its applications have a long history. Recovering lunar topography using photoclinometry was proposed by van Diggelen (1951). Later a series of studies devoted to lunar and planetary surface photoclinometry using Earth-based and space mission observations was carried out (e.g., Lambiotte and Taylor, 1967; Watson, 1968; Tyler et al., 1971; Bonner and Schmall, 1973; Parusimov and Kornienko, 1973; Howard et al., 1982; Muinonen et al., 1989; Wildey, 1973, 1975; McEwen, 1991; Jankowski and Squyres, 1991; Watters and Robinson, 1997; Kirk et al., 2003; Lohse et al., 2006). The shapefrom-shading method also has been developed in research unrelated directly to planetology (e.g., Ramachandran, 1988; Horn, 1989; Leclerc and Bobick, 1991; Vega and Yang, 1993; Tsai and Shah, 1994; Zhang et al., 1999; Wohler and Hafezi, 2005). Parusimov and Kornienko (1973) and later Davis and Soderblom (1984) pointed out that the topography determination should be carried out simultaneously with the determination of surface albedo using several different images. One of the main sources of photoclinometry errors is random noise of images. To provide an optimal filtration of the noise, Parusimov and Kornienko (1973), Hung et al. (1988), Kornienko et al. (1994), and Dulova et al. (2008) proposed the Bayesian approach to determine topography and optical characteristics of a planetary surface from photometric imagery. This technique uses a statistical approach and makes it possible to formulate a well posed mathematical problem in the presence of measurement noise. Note that the algorithm by Parusimov and Kornienko
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(1973) uses the fact that the height distribution must be a gradient field; whereas, the noise component may be a gradient or rotor field. Thus, canceling the rotor component of the resulting signal can noticeably suppress the noise. In this manuscript, we present results of a new photoclinometry technique that includes analysis of many calibrated images of the same scenes acquired at different phase angles. We demonstrate the capability of the method using high-quality absolute photometric imagery of the Moon obtained with the Kharkiv 15-cm refractor at Maidanak Observatory (Uzbekistan) (Velikodsky et al., 2010). An earlier implementation of the method (Korokhin and Akimov, 1997) used a linear approach that implies that the slopes are small. As we deal with many high-quality images that provide low noise, the methods of optimal filtration (e.g., Dulova et al., 2008) were not used. In addition, we present results of removing the topographic effects from lunar images of the photometric-function parameters. We use altimetry measurements carried out with the Kaguya (JAXA) LALT and results of our photoclinometry technique.
2. New technique in lunar photoclinometry The reflectance R of a planetary surface can be expressed through the photometric function F(i, e, a) Rði,e, aÞ ¼ A0 Fði,e, aÞ,
ð1Þ
where i and e are the incident and emergent angles, respectively. The value A0 is the reflectance (albedo) at a standard illumination/ observation geometry. In lunar photometry another suite of angles is often used: the photometric longitude g, and the photometric latitude b. They can be found from the system as (Hapke, 1993) cos g ¼ cos e=cos b,
cos b ¼ cos i=cosðgaÞ
ð2Þ
The values b and g are functions of the lunar surface coordinates. As has been noted, the photometric function F(a, b, g) can be presented as (Hapke, 1993) Fða, b, gÞ ¼ f ðaÞDða, b, gÞ,
ð3Þ
where f(a) is the phase function and D(a, b, g) is the disk function. The latter describes the global brightness distribution over the lunar disk, e.g., the brightness trend from the limb to terminator. We here use the lunar disk function suggested by Akimov (1979, 1988a, b) Dða, b, gÞ ¼ cosða=2Þðcos bÞna=ðpaÞ cosððga=2Þp=ðpaÞÞ=cos g,
ð4Þ
where a is measured in radians and n is the roughness coefficient (Akimov et al., 1999, 2000); n ¼0.34 for maria and n ¼0.52 for highlands, and we used below the average value n ¼0.43. This function ((4)) describes the scattering by the lunar surface more precisely than the Lommel–Zeeliger and Minnaert scattering laws (Akimov, 1988b; Kreslavsky et al., 2000). To find the phase function from reflectance, one needs to calculate f ðaÞ ¼ Fobs ða, b, gÞ=Dða, b, gÞ:
ð5Þ
Using formulas (3) and (4), photometric data for each lunar point can be transformed to the same photometric conditions. In particular, the data can be brought to the so-called mirror illumination/observation geometry at the photometric equator, i.e., when b ¼0, g ¼ a/2. Then, Eqs. (3) and (4) are significantly simplified Aeq ðaÞ Rða=2, a=2, aÞ ¼ A0 f ðaÞ:
ð6Þ
The value Aeq(a) is termed the equigonal albedo, which can be found for each point of the lunar disk, allowing a photometric comparison between all the points (Korokhin et al., 2007). Evidently,
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the branches of Aeq(a) corresponding to a with different signs should be symmetrical. To describe the observed phase dependence Aeq(a) of the lunar areas, different model functions may be used. The simplest phase function, which is very appropriate for the phaseangle range 10–401, is Aeq ðaÞ ¼ A0 expðmaÞ
f ðaÞ ¼ Fobs ða, bu, guÞ=Dða, bu, guÞ,
ð8Þ
where g0 ¼ g + Dg and b0 ¼ b + Db are the photometric coordinates accounting for the topography. The values Dg and Db can be expressed through the local slopes of the resolved lunar relief. !
Let us define the local slope r as the declination of the local !
!
!
! !
surface normal n from the global one N, i.e., we consider r ¼ Nn !
where vectors n and N are normalized to unity. The local slopes of the surface affect the local illumination/observation geometry and, hence, the photometric coordinates g and b of a given point. The slope disturbance of a small surface site (planar element) is equivalent to the displacement of the site to the point with new photometric coordinates g0 and b0 and, therefore, new selenographic coordinates l0 ¼ l+ Dl and b0 ¼ b+ Db, where l and b are the selenographic longitude and latitude, respectively. This new point has a normal to the undisturbed surface parallel to the original local normal. We denote the longitudinal and latitudinal compo ! nents of the local slope r as rl and rb: ! rl ¼ r sin c,
! rb ¼ r cos c,
ð9Þ
where c is the azimuth angle of the projection of the local normal on the average plane, which is measured from the direction to the north pole. The quantities rl and rb are related to Dl and Db by the obvious equations rl ¼ Dl cos b,
Da ¼ WL cos b0 sin g0 ,
ð16Þ
Dg ¼ WL sin g0 ð1þ cot a0 cot g0 ðsin b0 Þ2 Þ=cos b0
at
g0 a0,
ð17Þ
ð7Þ
where m characterizes the slope of the function. Eq. (1) ignores the resolved lunar topography. To take this into consideration, one needs to use the following formula:
!
where g0 and b0 are the photometric coordinates of the lunar disk center,
rb ¼ Db:
ð10Þ
Formulas (9) and (10) are approximate (Korokhin and Akimov, 1997); they work well for the case of relatively small topography o slopes (up to 10 % ), when spherical triangles may be replaced by planar ones. Such conditions are well suited for lunar observations, since larger slopes on the 3 km bases provided by Earthbased telescope observations of the Moon are rare. In this work we develop a more sophisticated algorithm without such restrictions. The relations between the selenographic and photometric coordinates are defined by the expressions cos a0 ¼ sin bobs sin bS þcos bobs cos bS cosðlS lobs Þ,
ð11Þ
cos i ¼ sin b sin bS þ cos b cos bS cosðlS lÞ,
ð12Þ
cos e ¼ sin b sin bobs þ cos b cos bobs cosðlobs lÞ,
ð13Þ
cos j ¼ ðcos acos e cos iÞ=sin e sin i,
ð14Þ
where a0 is the phase angle at the lunar disk center, and j is the azimuthal difference between the incident and reflected rays at the point with selenographic coordinates (l, b). The values lS and bS are, respectively, selenographic longitude and latitude of the Sun, lObs and bObs are the same for the observer. The system (2) relates the angles i and e with g and b. To take into account the variations of the photometric conditions over the lunar disk, the following formulas are used
g ¼ g0 þ Dg, b ¼ b0 þ Db, a ¼ a0 þ Da,
ð15Þ
Dg ¼ WL cot a0 ðsin b0 Þ2 =cos b0
at
g0 ¼ 0,
Db ¼ WL sin b0 cos g0 ð1-cot a0 tan g0 Þ,
ð18Þ ð19Þ
where WL is the angular radius of the Moon seen from the Earth. For calculation of selenographic coordinates of points on images of the Moon given in the perspective projection, some equations from (Shalygin et al., 2003) were used. The formulas and code are available on the web-site http://www.astron. kharkov.ua/dslpp/cartography/index.html. This short introduction to terminology and definitions enables us to state the gist of our approach to retrieve information about the relief, albedo, and the parameters of phase-function slope from photometric images. We suppose that there is a rather large set of absolutely calibrated images obtained for the same scene at different phase angles. Hence, for each point of the lunar surface a phase dependence of the equigonal albedo can be plotted. We may numerically minimize the standard deviation of the observed Aeq(a) from the model function (7) varying simultaneously the parameters rl, rb, A0, and m. This automatically provides minimization of brightness variations caused by topographic slopes. For the minimization, we use the Nelder–Mead method (Nelder and Mead, 1965). The described procedure is possible for the slope component oriented along the illumination direction. For Earth-based observations at large phase angles, the incident solar rays are nearly co-linear with the lines of selenographic latitude. Hence, the reconstructed relief profiles pass along latitude lines, and the coordinates of points on each profile are values of longitude. Therefore, we term this slope component longitudinal, rl. Note, that in geography such a component is named ‘‘zonal’’. Thus, we fit the parameters rl, A0, and m, using a set of observations of the Moon carried out at various phase angles before and after full-moon. We map simultaneously the topographic slopes, albedo A0, and the parameters of phase function compensating for the influence of the resolved topography.
3. Mapping the topography slopes and parameters of phase function For the mapping we used data of absolute photometry of the Moon carried out in 2006 at the Maidanak Observatory (Uzbekistan) with a 15-cm refractor at red light 610 nm (Velikodsky et al., 2010). The data set of 12 maps of absolute albedo at a ¼ 12.811, 14.731, 16.211, 17.781, 18.741, and 21.021 before full-moon and at a ¼12.331, 13.421, 14.191, 15.071, 16.191, and 21.791 after fullmoon were used in our analysis. The observation data were selected to have maximally symmetric values of phase angles before and after full-moon. The range of phase angles approximately from 121 to 221 allows us to use a relatively simple (one exponent) model phase function (7). The maximal phase angle 221 allows us to map almost the full visible disk in contrast to our previous work (Korokhin and Akimov, 1997). The resolution element of our data is approximately 3.2 3.2 km2 in the lunar disk center. This is the minimal base of roughness. Before calculations all images have been additionally co-registered using an algorithm of soft co-registration (Kaydash et al., 2009a) to compensate for the influence of Earth’s atmospheric turbulence that deforms the lunar images. Fig. 1 presents a map of A0, i.e., the albedo distribution that is not influenced by topography. We note that this is
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Fig. 1. A map of A0 calculated in the range of phase angles 12–221.
Fig. 2. A map of longitudinal topographic slopes determined on a 3.2 km base; the map is calculated using 12 images in the range of phase angles 12–221.
a result of albedo extrapolation defined with formula (7) to a ¼0 using 12 images obtained at phase angles 121–221. The distribution of A0 is similar to that of the equigonal albedo Aeq(a) at any moderate phase angle and characterizes the reflection properties of the lunar surface materials. It should be emphasized that the albedo map in Fig. 1 does present real brightness distribution at a ¼0, as it ignores variations of opposition-effect amplitudes that are not described by formula (7). The opposition spike has components of shadow-hiding and coherent backscattering effects (e.g., Hapke, 1993; Shkuratov et al., 2004). To include these in the analysis, a more complicated approximation of the phase function is needed (Velikodsky et al., 2010). Fig. 2 shows a distribution of the longitudinal component rl of the topographic slope calculated on a 3.2 km base. The distribution in Fig. 3 shows realistic values of slope up to 81. This is in agreement with our assumption that topographic slopes on bases provided by Earth-based telescope observations are small. Some sites demonstrate slopes reaching 201. The maximal value of slope we obtained in this analysis is 221, observed at the selenographic coordinates l ¼0.21, b¼21.61 in Montes Apenninus. This new map of longitudinal slopes coincides well with our previous results (Korokhin and Akimov, 1997). Shown in Fig. 4 is a map of the parameter m of function (7) determined in the range of phase angles 12–221. The distribution describes the steepness of the phase dependence of the lunar surface brightness. The parameter m can be determined using many calibrated images; whereas, a phase-angle-ratio image includes only two components. In any case, the images of the phase ratio and of the parameter m calculated for the same phase angle range should be very similar. As one can see, the parameter m demonstrates an inverse correlation with albedo: the higher the steepness m, the lower the albedo. This is in agreement with our previous results (Shkuratov et al., 1994; Korokhin and Akimov, 1997; Kreslavsky et al., 2000; Kreslavsky and Shkuratov, 2003; Kaydash et al., 2009b). The albedo A0 and steepness m maps calculated using the described algorithm are almost free from the influence of the
lunar topography (see Figs. 1 and 4). Figs. 5 and 6 show the case, if we ignore the compensation for the topography, i.e., when the phase-function parameters (7) A0 and m are calculated without accounting for the lunar relief. The difference, especially between the steepness distributions (see Figs. 4 and 6), seems to be dramatic. In the case of Figs. 1 and 4 some traces of relief are observed near the limb and the polar regions. This probably is caused by neglecting the latitudinal component of local slopes and the presence of areas on the source images with too large emergence and incidence angles. Errors in using the approximation with the function (7) are shown in Fig. 7. Typical values of the error (the standard deviation divided by A0 in percents) are 0.35–0.5% for maria, 0.4–1.0% for highlands, and up to 4% for bright craters. Areas near the limb have maximal values of errors (up to 20%). Small errors are observed in maria, because their surface is relatively flat. Moreover, the function (7) works for maria better than for highlands and bright craters. Thus, higher errors for highlands and bright craters are caused by more complicated topography in addition to the simplified function with one exponent (7) being not representative of the phase curves of highlands. High errors near the limb appear due to the presence of real shadows in addition to poor image co-registration caused by neglecting real heights of the lunar sites in the compensation for libration. In spite of some shortcomings of the technique, the approximation of phase function without accounting for the influence of the lunar relief gives much larger errors.
4. Comparison with LALT data As has been noted, recent altimetric data have been obtained with the LALT (laser altimeter) aboard the spacecraft Kaguya (Araki et al., 2009). The LALT was able to obtain a range of data on a global scale along the satellite0 s trajectory including the high latitude region above 751 that has never been measured by an
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Fig. 3. A histogram of longitudinal topographic slopes on a 3.2 km base.
Fig. 4. A map of the parameter m that characterizes steepness of phase function in the range of phase angles 12–221.
altimeter. Although the data set is somewhat raw yet (Korokhin et al., 2010), we used the global LALT topography for comparison with our photoclinometric data in order to compensate for the topographic effect for the phase-angle-ratio maps. Before the comparison we processed the LALT data. We transformed the data array to put the point with zero selenographic coordinates in the center of the image. Then, we re-sampled the LALT map to the resolution of our ground-based observations. The next step was calculating the longitudinal component of the surface slopes from the height distribution. Then, we transformed the LALT map from a cylindrical onto an orthographic projection. Finally, the image was smoothed by a Gaussian filter with s ¼0.8 pix to make the resolution of the LALT and our maps similar. We computed the map of longitudinal topographic slopes determined on a 3.2 km base (see Fig. 8) from the LALT distribution of heights. Both the maps show a very
Fig. 5. A map of A0 calculated in the range of phase angles 12–221 ignoring the influence of the lunar topography.
similar distribution of the longitudinal slopes over the lunar disk (cf. Figs. 2 and 8), including the maximal value of the slope 221 at the point with the selenographic coordinates l¼ 0.21, b¼21.61. However, more detailed analysis reveals differences. For instance, our map demonstrates a weak residual influence from albedo (see the Copernicus ray system) and worsening resolution to the limb. The LALT map exhibits many small topographic artifacts. Fig. 9a and b show maps of longitudinal slopes for the central portion of the lunar disk ( 501 501) in a cylindrical projection: the upper panel correspond to LALT data and the lower one to our map. The most typical artifacts are displaying some craters as hummocks (see, e.g., marks 1–3, 7–9 in Fig. 9a, b) or disappearance of craters (marks 4–6). Sometimes, the LALT map reveals nonexistent peaks (see marks 10 and 11). The most probable reasons for these artifacts are errors of interpolation at mosaicing. We also note the presence of longitudinal modulation
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Fig. 6. A map of the steepness m of phase function calculated in the range of phase angles 12–221 ignoring the lunar relief.
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Fig. 8. A map of longitudinal topographic slopes determined on a 3.2 km base, retrieved from the LALT data.
5. Topography correction of photometric data Although we have mapped the longitudinal component only, this allows for correction of Earth-based photometric observations, because illumination of the lunar surface varies mainly due to changing selenographic longitude of the Sun. For the correction of maps of albedo, it is necessary to apply the following correcting coefficient for each point of the lunar disk k ¼ Dða, b, gÞ=Dða, bu, guÞ,
Fig. 7. A map of errors (standard deviation in albedo units) of the approximation (7).
caused by spacecraft orbital motion; this can be seen in highcontrast images of the altimetric map (Araki et al., 2009). Thus, the considered version of the LALT map should still be used with caution. In particular, our map could be proposed for an independent control of the LALT data.
ð20Þ
where D(a, b, g) and D(a, b0 , g0 ) are the disk functions, respectively, without and with taking into account topography slopes. In the numerator of Eq. (20), photometric coordinates g and b of the point are calculated using its current selenographic coordinates l and b. In the denominator of Eq. (20), l0 and b0 are calculated using the obvious formula l0 ¼l+ rl/cos b. In Fig. 10a, b we present, respectively, topographically corrected and uncorrected maps of the phase ratio 21/211. It can be seen that the topography in Fig. 10a is significantly suppressed. Despite the phase angle ranges 10–201 and 2–201 being somewhat different, the resemblance the phase-ratio map and the map of the parameter m is clearly seen (cf. Figs. 4 and 10a). As can be anticipated, the phase-ratio images reveal a correlation with albedo. Low albedo areas (dark mantle deposits) in Mare Tranquillitatis, Mare Vaporum, and Sinus Aestuum (see also Taurus Littrow and Sulpicius Gallus) reveal the most steep phase function, while bright craters and their rays (e.g., Tycho, Copernicus, and Proclus) have the smallest slope of the function. This is a somewhat unexpected result as the craters are rather bright and, hence, at small phase angles ( o51), the coherent backscattering effect should prominently manifest itself (e.g., Hapke, 1993). This result could be explained by the small width of the opposition spike from the effect of coherent backscattering. However, the phase ratio image 11/51 (it will be published) demonstrates the same behavior: bright craters and their rays have small phase-ratio slope.
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Fig. 9. Comparison of maps of longitudinal slopes retrieved from photometric (upper (a)) observations and the LALT data (lower (b)).
We also note that there is very noticeable difference between Mare Serenitatis and Mare Tranquillitatis. We note that differences were observed for color ratio in the visible spectral range (e.g., Pieters and McCord, 1976; Shkuratov et al., 1999). Detailed comparison of the phase ratio map 21/211 with albedo shows that there are many anomalous regions that do not exactly coincide with the albedo boundary. For instance, the Aristarchus Plateau and Marius Hills (pyroclastic formations) clearly show up on the phase-ratio images. We note that the topographic correction helps in visualizing these anomalies. Corrected photometric data allow us to study the phase dependence of the lunar surface brightness not only for flat mare regions, but for highlands and other areas with rough resolved topography. For instance, relatively small swirls in the region of the central nearside highlands (Blewett et al., 2007) are more apparent if the topographic effect is removed.
6. Conclusion 1. We have studied two approaches to remove the resolved topography effect from images of phase-function parameters
Fig. 10. Phase ratio 21/211 with (a) and without (b) topography corrections.
of the Moon using altimetry measurements carried out with the Kaguya (JAXA) LALT instrument and new photoclinometry data obtained with Earth-based telescopic observations. Using these techniques, we have mapped the longitude component of the lunar topography slopes with a resolution of 3.2 km/pix. Overall, we have found good correlation when comparing our map with the corresponding data obtained from Kaguya altimetry. 2. Our approach to retrieve information about the topography, albedo, and the parameters of phase function from photometric images requires using a robust set of absolutely calibrated images obtained for the same scene at different phase angles. We numerically minimized the standard deviation of the observed Aeq(a) from the model function (7) varying
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the parameters rl, A0, and m (see designations above). This procedure minimizes the effect of topographic slopes. In applying this technique, we used new data of absolute photometry of the Moon obtained at the Maidanak Observatory (Uzbekistan). 3. Although the LALT and photoclinometry maps show good correlation, more detailed analysis reveals differences. In particular, the photoclinometric map demonstrates a weak residual influence from albedo. The LALT map exhibits many small topographic artifacts, displaying some craters as hummocks. Thus, our map would be proposed for an independent control of the LALT data. 4. Corrected photometric data allow one to study phase dependence of the lunar surface brightness not only for flat mare regions, but for highlands and other areas with expressed topography. This is useful, for example, in studying the swirls in the region of the central nearside highlands (Blewett et al., 2007).
Acknowledgment This study was supported by CRDF Grant UKP2-2897-KK-07.
References Akimov, L.A., 1979. On the brightness distribution over the lunar and planetary disks. Sov. Astron. 23 (2), 231–235. Akimov, L.A., 1988a. Reflection of light by the Moon. 1. Kinemat. Fiz. Nebesykh Tel. 4 (1), 3–10 (in Russian). Akimov, L.A., 1988b. Reflection of light by the Moon. 2. Kinemat. Fiz. Nebesykh Tel. 4 (2), 10–16 (in Russian). Akimov, L.A., Velikodsky, Yu.I., Korokhin, V.V., 1999. Dependence of lunar highland brightness on photometric latitude. Kinemat. Phys. Celest. Bodies 15 (4), 232–236. Akimov, L.A., Velikodsky, Yu.I., Korokhin, V.V., 2000. Dependence of latitudinal brightness distribution over the lunar disk on albedo and surface roughness. Kinemat. Phys. Celest. Bodies 16 (2), 137–141. Araki, H., Tazawa, S., Noda, H., Ishihara, Y., Goossens, S., Sasaki, S., Kawano, N., Kamiya, I., Otake, H., Oberst, J., Shum, C., 2009. Lunar global shape and polar topography derived from Kaguya-LALT laser altimetry. Science 323, 897–900. Blewett, D.T., Hughes, C.G., Hawke, B. Ray, Richmond, N.C., 2007. Varieties of the lunar swirls. In: Proceedings of the 38th Lunar and Planetary Science Conference, Abstract no. 1232. Bonner, W.J., Schmall, R.A., 1973. A photometric technique for determining planetary slopes from orbital photographs. US Geological Survey Professional Paper 812-A, pp. 1–16. Chen, C.-Y., Klette, R., Kakarala, R., 2002. Albedo recovery using photometric stereo approach. In: Proceedings of the 16th International Conference on Pattern Recognition (ICPR ’02), Quebec, Canada, August 2002, vol. 3, pp. 700–703. Connors, C., 1995. Determinimg heights and slopes of fault scarps and other surfaces on Venus using Magellan stereo radar. J. Geophys. Res. E 107 (7), 14281–14361. Cook, A.C., Robinson, M.S., 2000. Mariner 10 stereo image coverage of mercury. J. Geophys. Res. E 105 (4), 9429–9444. Davis, P.A., Soderblom, L.A., 1984. Modeling crater topography and albedo from monoscopic viking orbiter images. I—Methodology. J. Geophys. Res. B 89 (11), 9449–9457. Dulova, I.A., Skuratovskii, S.I., Bondarenko, N.V., Kornienko, Yu.V., 2008. Retrieval of the surface topography from single images using the photometric method. Sol. Syst. Res. 42 (6), 522–535. Hapke, B., 1993. Theory of Reflectance and Emittance Spectroscopy. Cambridge University Press 450 pp. Herrick, R.R., Sharpton, V.L., 2000. Implications from stereo-derived topography of venusian impact craters. J. Geophys. Res. E 105 (8), 20245–20262. Horn, B.K.P., 1989. Height and Gradient from Shading. MIT Technical Report no. 1105A /http://people.csail.mit.edu/people/bkph/AIM/AIM-1105A-TEX.pdfS. Howard, A.D., Blasius, K.R., Cutts, J.A., 1982. Photoclinometric determination of the topography of the martian north polar cap. Icarus 50, 245–258. Hung, Y.-P., Cooper, D.B., Cernuschi-Frias, B., 1988. Bayesian estimation of 3-d surfaces from a sequence of images, International Conference on Robotics and Automation. IEEE, pp. 906–911. Jankowski, D.G., Squyres, S.W., 1991. Sources of error in planetary photoclinometry. J. Geophys. Res. 96 (E4), 20,907–20,922.
1305
Kaydash, V., Kreslavsky, M., Shkuratov, Yu., Gerasimenko, S., Pinet, P., Josset, J.-L., Beauvivre, S., Foing, B., AMIE SMART-1 Team, 2009a. Photometric anomalies of the lunar surface studied with SMART-1 AMIE data. Icarus 202 (2), 393–413. Kaydash, V.G., Gerasimenko, S.Y.u, Shkuratov, Yu.G., Opanasenko, N.V., Velikodskii, Yu.I., Korokhin, V.V., 2009b. Photometric function variations observed on the near side of the Moon: mapping. Sol. Syst. Res. 31 (2), 143–152. Kirk, R.L., Howington-Kraus, E., Redding, B., Galuszka, D., Hare, T., Archinal, B.A., 2003. High-resolution topomapping of candidate MER landing sites with MOC: new results and error analyses. In: Proceedings of the 34th Lunar Planetary Sciences, Abstract no. 1966, LPI Houston. Kornienko, Yu.V., Dulova, I.A., Nguyen Xuan, Anh, 1994. Wiener approach to the determination of optical characteristics of a planetary surface from photometric observations. Kinemat. Fiz. Nebesnykh Tel. 10 (5), 69–76 (Kinemat. Phys. Celest. Bodies (Engl. Transl.). 10, 69–76). Korokhin, V., Akimov, L., 1997. Mapping of phase parameters of the lunar-surface brightness. Sol. Syst. Res. 31 (2), 143–152. Korokhin, V.V., Velikodsky, Yu.I., Shkuratov, Yu.G., Mall, U., 2007. The phase dependence of brightness and color of the lunar surface: a study based on integral photometric data. Sol. Syst. Res. 41 (1), 19–27. Korokhin, V.V., Velikodsky, Yu.I., Shkuratov, Yu.G., Kaydash, V.G., Gerasimenko, S.Yu., Opanasenko, N.V., 2010. Comparison between Kaguya (Selene) altimetry data and lunar topography retrieved from photoclinometry. In: Proceedings of the 41st Lunar and Planetary Science Conference, 1–5 March, Abstract no. 1356, LPI Houston, USA. Kreslavsky, M., Shkuratov, Yu., 2003. Photometric anomalies of the lunar surface: results from Clementine data. J. Geophys. Res. 108 (E3) (1-1–1-13). Kreslavsky, M.A., Shkuratov, Yu.G., Velikodsky, Yu.I., Kaydash, V.G., Stankevich, D.G., Pieters, C.M., 2000. Photometric properties of the lunar surface derived from Clementine observations. J. Geophys. Res. Planets 105 (E8), 20,281–20,295. Lambiotte, J.J., Taylor, G.R., 1967. A photometric technique for deriving slopes from lunar orbiter photography. In: Proceedings of the Conference on the Use of Space Systems for Planetary Geology and Geophysics, Boston, MA, 25–27 May. Leclerc, Y.G., Bobick, A.F., 1991. The direct computation of height from shading, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 3–6 June. IEEE, pp. 552–558. Lohse, V., Heipke, C., Kirk, R.L., 2006. Derivation of planetary topography using multi-image shape-from-shading. Planet. Space Sci. 54, 661–674. McEwen, A.S., 1991. Photometric functions for photoclinometry and other applications. Icarus 92, 298–311. Muinonen, K., Lumme, K., Irvine, W.M., 1989. Statistical photoclinometry and surface topography of atmosphereless bodies. In: Proceedings of the 20th Lunar and Planetary Science Conference, LPI Houston, USA, pp. 729–730. Nelder, J., Mead, R., 1965. A simplex method for function minimization. Comput. J. 7, 308–313. Oberst, J., Jaumann, R., Zeitler, W., et al., 1999. Photogrammetric analysis of horizon panoramas: the pathfinder landing site in viking orbiter images. J. Geophys. Res. E 104 (4), 8927–8934. Opanasenko, A.N., Shkuratov, Y.G., Opanasenko, N.V., 2007. Topography of three localities on the Moon from combined Clementine and Hubble space telescope images. In: Proceedings of the 38th Lunar and Planetary Science Conference, Abstract no. 1564, LPI Houston, USA. Parusimov, V.G., Kornienko, Yu.V., 1973. About finding the most probable relief of a planet surface by its optical image. Astrometriya Astrofiz. 19, 20–24 (Astrometry Astrophys. (Engl. Transl.), 19, 20–24). Pieters, C., McCord, T., 1976. Characterization of lunar mare basalt types. In: Proceedings of the Seventh Lunar and Planetary Science Conference, LPI Houston, pp. 2677–2690. Ramachandran, V.S., 1988. Perceiving shape from shading. Sci. Am. 159, 76–83. Shalygin, E.V., Velikodsky, Yu.I., Korokhin, V.V., 2003. Formulas of the perspective cartographic projection for planets and asteroids of arbitrary shape. In: Proceedings of the Third Lunar and Planetary Science Conference, Abstract no. 1946, LPI Houston, USA. Shkuratov, Yu.G, Starukhina, L.V., Kreslavsky, M.A., Opanasenko, N.V., Stankevich, D.G., Shevchenko, V.G., 1994. Principle of undulatory invariance in photometry of atmosphereless celestial bodies. Icarus 109, 168–190. Shkuratov, Yu.G., Kaydash, V.G., Opanasenko, N.V., 1999. Iron and titanium abundance and maturity degree distribution on lunar nearside. Icarus 137, 222–234. Shkuratov, Yu., Videen, G., Kreslavsky, M.A., Belskaya, I.N., Ovcharenko, A., Kaydash, V.G., Opanasenko, N., Zubko, E., 2004. Scattering properties of planetary regoliths near opposition: photopolarimetry in remote sensing. Videen, G., Yatskiv, Ya., Mishchenko, M. (Eds.), NATO Science Series. Kluwer Academic Publishers, London. pp. 191–208. Shkuratov, Yu., Kaydash, V., Gerasimenko, S., Opanasenko, N., Velikodsky, Yu., Korokhin, V., Videen, G., Pieters, C., 2010. Probable swirls detected as photometric anomalies in Oceanus Procellarum. Icarus 208, 20–30. Tsai, P.S., Shah, M., 1994. Shape from shading using linear approximation. Image Vision Comput. J. 12 (8), 487–498. Tyler, G.L., Simpson, R.A., Moore, H.J., 1971. Lunar slope distributions: comparison of bi-static radar and photographic results. J. Geophys. Res. 76 (11), 2790–2795. van Diggelen, J., 1951. A photometric investigation of the slopes and heights of the ranges of hills in the maria of the Moon. Bull. Astron. Inst. Netherlands 11, 283–289.
ARTICLE IN PRESS 1306
V.V. Korokhin et al. / Planetary and Space Science 58 (2010) 1298–1306
Vega, O.E., Yang, Y.H., 1993. Shading logic: a heuristic approach to recover shape from shading. IEEE Trans. Pattern Anal. Mach. Intell. 15 (6), 592–597. Velikodsky, Yu.I., Opanasenko, N.V., Akimov, L.A., Korokhin, V.V., Shkuratov,Yu.G., Kaydash,V.G., 2010. Kharkiv absolute photometry of the Moon. In: Proceedings of the 41st Lunar and Planetary Science Conference, Abstract no. 1760, LPI Houston, USA. Watson, K., 1968. Photoclinometry from spacecraft images. US Geological Survey Professional Paper 599-B, pp. 1–10. Watters, T.R., Robinson, M.S., 1997. Radar and photoclinometric studies of wrinkle ridges on Mars. J. Geophys. Res. E 102 (5), 10889–10903.
Wildey, R.L., 1973. Theoretical autophotogrammetry: I. The method of photometric potential. Mod. Geol. 4, 209–215. Wildey, R.L., 1975. Generalized photoclinometry for mariner 9. Icarus 25, 613–626. Wohler, C., Hafezi, K., 2005. A general framework for three-dimensional surface reconstruction by self-consistent fusion of shading and shadow features. Pattern Recognition 38 (7), 965–983. Zhang, R., Tsai, P.-S., Cryer, J.E., Shah, M., 1999. Shape from shading: a survey. IEEE Trans. Pattern Anal. Mach. Intell. 21 (8), 690–707.
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Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
Removal of topographic effects from lunar images using Kaguya (LALT) and Earth-based observations Viktor V. Korokhin a,n, Yuriy I. Velikodsky a, Yuriy G. Shkuratov a, Vadym G. Kaydash a, Sergey Y. Gerasimenko a, Nikolai V. Opanasenko a, Gorden Videen b, Carle Pieters c a
Institute of Astronomy, Kharkiv V.N. Karazin National University, 35 Sumskaya Street, Kharkiv 61022, Ukraine Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, CO 80301, USA c Department of Geological Sciences, Brown University, Providence, RI 02912, USA b
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 February 2010 Accepted 20 May 2010 Available online 26 May 2010
Lunar images acquired at non-zero phase angles show brightness variations caused by both albedo heterogeneities and local topographic slopes of the surface. To distinguish between these two factors, altimetry measurements or photoclinometry data can be used. The distinction is especially important for imagery of phase-function parameters of the Moon. The imagery is a new tool that can be used to study structural anomalies of the lunar surface. To illustrate the removal of the topographic effects from photometric images, we used Earth-based telescopic observations, altimetry measurements carried out with the Kaguya (JAXA) LALT instrument, and a new photoclinometry technique that includes analysis of several images of the same scenes acquired at different phase angles. Using this technique we have mapped the longitudinal component of lunar topography slopes (the component measured along the lines of constant latitude). We have found good correlations when comparing our map with the corresponding data from Kaguya altimetry. The removal of the topographic surface properties allows for the study of the phase-function parameters of the lunar surface, not only for flat mare regions, but for highlands as well. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Lunar surface Photometry Albedo Phase function Topography Photoclinometry
1. Introduction An important goal of photometric studies of planetary images is to identify the physical properties of the material covering the surface of planetary bodies. One approach to this problem is to determine the photometric function, i.e., the function of how the intensity of light reflected from planetary areas varies with illumination and observation angles. The angle between vectors from the observed point to the light source and to the observer is the phase angle a. This is the most important angle variable in planetary photometry. The phase function is a component of the photometric function, which depends only on a (Hapke, 1993). The phase functions are determined by the complexity of the structure of the surface. Usually these phase functions have a single sharp maximum at a ¼0 being locally smooth at other a. Therefore, they can be described with a few parameters. The simplest parameters are ratios of phase-function values at different phase angles a; these characterize the slopes of phase curves. The slope also may be characterized with parameters of an analytical curve used for fitting to brightness measurements (see below). Imagery of the Moon can be made in terms of phase-angle ratios for different pairs of a
n
Corresponding author. Tel.: + 38 057 707 5064. E-mail address: [email protected] (V.V. Korokhin).
0032-0633/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2010.05.013
using original well calibrated brightness images. Such imagery is a relatively new and effective tool that can be used to study structural anomalies of the lunar surface (e.g., Shkuratov et al., 1994; Korokhin and Akimov, 1997; Kreslavsky and Shkuratov, 2003; Kreslavsky et al., 2003; Kaydash et al., 2009a, b). For instance, using photometric images acquired with Earth-based telescopes, weak swirls were found in the southern portion of Oceanus Procellarum (Shkuratov et al., 2010). The phase-angle-ratio images allow one to estimate spatial variations of the complexity of unresolved surface roughness and microtopography. However, the resolved topography spoils the images hampering their interpretation. Brightness variations on images of the lunar surface depend on the spatial distributions of local topographic slopes, albedo (that is, a characteristic of the surface material composition), and the global illumination/observation geometry. The lunar resolved topography influences brightness spatial variations, since the local surface slopes, or height gradients, change the local incident i and emergent e angles. This effect can be especially significant at large phase angles. Removing the influence of the resolved topography requires data on local slopes. Suitable global lunar height distributions have been obtained recently with the Laser Altimeter (LALT) aboard the spacecraft Kaguya (JAXA) (Araki et al., 2009), which we use in our analysis. Photogrammetry, photoclinometry (shape-from-shading) or their combination are techniques that allow one to retrieve
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information about the relief and albedo distributions directly from photometric images. Photogrammetry is based on the use of mutual parallaxes of surface details observed at two different angles. It can only be used if stereo images are available. For instance, photogrammetry cannot be practically applied to telescopic images of the Moon acquired from Earth, as the viewing geometry weakly varies for the Moon. However, this method has been used to retrieve the topographic information on different planets from images obtained during space missions (e.g., Connors, 1995; Oberst et al., 1999; Cook and Robinson, 2000; Herrick and Sharpton, 2000). Attempts have been made to recover the albedo distribution on a surface from images using the photometric stereo approach (e.g., Chen et al., 2002). Recently Clementine and Hubble Space Telescope images of the Apollo-17 landing area were used to map heights using the photogrammetric technique (Opanasenko et al., 2007). The Clementine and HST images have almost the same resolution, similar phase angles, and effective wavelength; however, the viewing angles are different. An automatic procedure to find the parallaxes, based on the determination of mutual shifts of the same lunar surface details by finding the maximum of the cross-correlation function of small portions of both images in a ‘‘running window,’’ was used. When photogrammetry is not applicable, photoclinometry has been exploited. This technique takes advantage of the fact that surface facets with orientations more nearly perpendicular to the illumination direction appear brighter than those facets facing away from the Sun. There are many different varieties of photoclinometry. In the simplest approach, a single image is used. By integrating slopes along a line parallel to the illumination direction, one can assemble a topographic profile. This approach ignores albedo and illumination/observation geometry variations over a surface. If the geometry can be taken into account, the albedo is unknown and may produce ambiguities. Thus, the albedo distribution should be estimated together with the determination of topography. When there are many images of the same scene at a fixed angle of view and different phase angles, one can find not only albedo and topography, but also the phase function for each point of the studied surface. The photoclinometric technique and its applications have a long history. Recovering lunar topography using photoclinometry was proposed by van Diggelen (1951). Later a series of studies devoted to lunar and planetary surface photoclinometry using Earth-based and space mission observations was carried out (e.g., Lambiotte and Taylor, 1967; Watson, 1968; Tyler et al., 1971; Bonner and Schmall, 1973; Parusimov and Kornienko, 1973; Howard et al., 1982; Muinonen et al., 1989; Wildey, 1973, 1975; McEwen, 1991; Jankowski and Squyres, 1991; Watters and Robinson, 1997; Kirk et al., 2003; Lohse et al., 2006). The shapefrom-shading method also has been developed in research unrelated directly to planetology (e.g., Ramachandran, 1988; Horn, 1989; Leclerc and Bobick, 1991; Vega and Yang, 1993; Tsai and Shah, 1994; Zhang et al., 1999; Wohler and Hafezi, 2005). Parusimov and Kornienko (1973) and later Davis and Soderblom (1984) pointed out that the topography determination should be carried out simultaneously with the determination of surface albedo using several different images. One of the main sources of photoclinometry errors is random noise of images. To provide an optimal filtration of the noise, Parusimov and Kornienko (1973), Hung et al. (1988), Kornienko et al. (1994), and Dulova et al. (2008) proposed the Bayesian approach to determine topography and optical characteristics of a planetary surface from photometric imagery. This technique uses a statistical approach and makes it possible to formulate a well posed mathematical problem in the presence of measurement noise. Note that the algorithm by Parusimov and Kornienko
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(1973) uses the fact that the height distribution must be a gradient field; whereas, the noise component may be a gradient or rotor field. Thus, canceling the rotor component of the resulting signal can noticeably suppress the noise. In this manuscript, we present results of a new photoclinometry technique that includes analysis of many calibrated images of the same scenes acquired at different phase angles. We demonstrate the capability of the method using high-quality absolute photometric imagery of the Moon obtained with the Kharkiv 15-cm refractor at Maidanak Observatory (Uzbekistan) (Velikodsky et al., 2010). An earlier implementation of the method (Korokhin and Akimov, 1997) used a linear approach that implies that the slopes are small. As we deal with many high-quality images that provide low noise, the methods of optimal filtration (e.g., Dulova et al., 2008) were not used. In addition, we present results of removing the topographic effects from lunar images of the photometric-function parameters. We use altimetry measurements carried out with the Kaguya (JAXA) LALT and results of our photoclinometry technique.
2. New technique in lunar photoclinometry The reflectance R of a planetary surface can be expressed through the photometric function F(i, e, a) Rði,e, aÞ ¼ A0 Fði,e, aÞ,
ð1Þ
where i and e are the incident and emergent angles, respectively. The value A0 is the reflectance (albedo) at a standard illumination/ observation geometry. In lunar photometry another suite of angles is often used: the photometric longitude g, and the photometric latitude b. They can be found from the system as (Hapke, 1993) cos g ¼ cos e=cos b,
cos b ¼ cos i=cosðgaÞ
ð2Þ
The values b and g are functions of the lunar surface coordinates. As has been noted, the photometric function F(a, b, g) can be presented as (Hapke, 1993) Fða, b, gÞ ¼ f ðaÞDða, b, gÞ,
ð3Þ
where f(a) is the phase function and D(a, b, g) is the disk function. The latter describes the global brightness distribution over the lunar disk, e.g., the brightness trend from the limb to terminator. We here use the lunar disk function suggested by Akimov (1979, 1988a, b) Dða, b, gÞ ¼ cosða=2Þðcos bÞna=ðpaÞ cosððga=2Þp=ðpaÞÞ=cos g,
ð4Þ
where a is measured in radians and n is the roughness coefficient (Akimov et al., 1999, 2000); n ¼0.34 for maria and n ¼0.52 for highlands, and we used below the average value n ¼0.43. This function ((4)) describes the scattering by the lunar surface more precisely than the Lommel–Zeeliger and Minnaert scattering laws (Akimov, 1988b; Kreslavsky et al., 2000). To find the phase function from reflectance, one needs to calculate f ðaÞ ¼ Fobs ða, b, gÞ=Dða, b, gÞ:
ð5Þ
Using formulas (3) and (4), photometric data for each lunar point can be transformed to the same photometric conditions. In particular, the data can be brought to the so-called mirror illumination/observation geometry at the photometric equator, i.e., when b ¼0, g ¼ a/2. Then, Eqs. (3) and (4) are significantly simplified Aeq ðaÞ Rða=2, a=2, aÞ ¼ A0 f ðaÞ:
ð6Þ
The value Aeq(a) is termed the equigonal albedo, which can be found for each point of the lunar disk, allowing a photometric comparison between all the points (Korokhin et al., 2007). Evidently,
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the branches of Aeq(a) corresponding to a with different signs should be symmetrical. To describe the observed phase dependence Aeq(a) of the lunar areas, different model functions may be used. The simplest phase function, which is very appropriate for the phaseangle range 10–401, is Aeq ðaÞ ¼ A0 expðmaÞ
f ðaÞ ¼ Fobs ða, bu, guÞ=Dða, bu, guÞ,
ð8Þ
where g0 ¼ g + Dg and b0 ¼ b + Db are the photometric coordinates accounting for the topography. The values Dg and Db can be expressed through the local slopes of the resolved lunar relief. !
Let us define the local slope r as the declination of the local !
!
!
! !
surface normal n from the global one N, i.e., we consider r ¼ Nn !
where vectors n and N are normalized to unity. The local slopes of the surface affect the local illumination/observation geometry and, hence, the photometric coordinates g and b of a given point. The slope disturbance of a small surface site (planar element) is equivalent to the displacement of the site to the point with new photometric coordinates g0 and b0 and, therefore, new selenographic coordinates l0 ¼ l+ Dl and b0 ¼ b+ Db, where l and b are the selenographic longitude and latitude, respectively. This new point has a normal to the undisturbed surface parallel to the original local normal. We denote the longitudinal and latitudinal compo ! nents of the local slope r as rl and rb: ! rl ¼ r sin c,
! rb ¼ r cos c,
ð9Þ
where c is the azimuth angle of the projection of the local normal on the average plane, which is measured from the direction to the north pole. The quantities rl and rb are related to Dl and Db by the obvious equations rl ¼ Dl cos b,
Da ¼ WL cos b0 sin g0 ,
ð16Þ
Dg ¼ WL sin g0 ð1þ cot a0 cot g0 ðsin b0 Þ2 Þ=cos b0
at
g0 a0,
ð17Þ
ð7Þ
where m characterizes the slope of the function. Eq. (1) ignores the resolved lunar topography. To take this into consideration, one needs to use the following formula:
!
where g0 and b0 are the photometric coordinates of the lunar disk center,
rb ¼ Db:
ð10Þ
Formulas (9) and (10) are approximate (Korokhin and Akimov, 1997); they work well for the case of relatively small topography o slopes (up to 10 % ), when spherical triangles may be replaced by planar ones. Such conditions are well suited for lunar observations, since larger slopes on the 3 km bases provided by Earthbased telescope observations of the Moon are rare. In this work we develop a more sophisticated algorithm without such restrictions. The relations between the selenographic and photometric coordinates are defined by the expressions cos a0 ¼ sin bobs sin bS þcos bobs cos bS cosðlS lobs Þ,
ð11Þ
cos i ¼ sin b sin bS þ cos b cos bS cosðlS lÞ,
ð12Þ
cos e ¼ sin b sin bobs þ cos b cos bobs cosðlobs lÞ,
ð13Þ
cos j ¼ ðcos acos e cos iÞ=sin e sin i,
ð14Þ
where a0 is the phase angle at the lunar disk center, and j is the azimuthal difference between the incident and reflected rays at the point with selenographic coordinates (l, b). The values lS and bS are, respectively, selenographic longitude and latitude of the Sun, lObs and bObs are the same for the observer. The system (2) relates the angles i and e with g and b. To take into account the variations of the photometric conditions over the lunar disk, the following formulas are used
g ¼ g0 þ Dg, b ¼ b0 þ Db, a ¼ a0 þ Da,
ð15Þ
Dg ¼ WL cot a0 ðsin b0 Þ2 =cos b0
at
g0 ¼ 0,
Db ¼ WL sin b0 cos g0 ð1-cot a0 tan g0 Þ,
ð18Þ ð19Þ
where WL is the angular radius of the Moon seen from the Earth. For calculation of selenographic coordinates of points on images of the Moon given in the perspective projection, some equations from (Shalygin et al., 2003) were used. The formulas and code are available on the web-site http://www.astron. kharkov.ua/dslpp/cartography/index.html. This short introduction to terminology and definitions enables us to state the gist of our approach to retrieve information about the relief, albedo, and the parameters of phase-function slope from photometric images. We suppose that there is a rather large set of absolutely calibrated images obtained for the same scene at different phase angles. Hence, for each point of the lunar surface a phase dependence of the equigonal albedo can be plotted. We may numerically minimize the standard deviation of the observed Aeq(a) from the model function (7) varying simultaneously the parameters rl, rb, A0, and m. This automatically provides minimization of brightness variations caused by topographic slopes. For the minimization, we use the Nelder–Mead method (Nelder and Mead, 1965). The described procedure is possible for the slope component oriented along the illumination direction. For Earth-based observations at large phase angles, the incident solar rays are nearly co-linear with the lines of selenographic latitude. Hence, the reconstructed relief profiles pass along latitude lines, and the coordinates of points on each profile are values of longitude. Therefore, we term this slope component longitudinal, rl. Note, that in geography such a component is named ‘‘zonal’’. Thus, we fit the parameters rl, A0, and m, using a set of observations of the Moon carried out at various phase angles before and after full-moon. We map simultaneously the topographic slopes, albedo A0, and the parameters of phase function compensating for the influence of the resolved topography.
3. Mapping the topography slopes and parameters of phase function For the mapping we used data of absolute photometry of the Moon carried out in 2006 at the Maidanak Observatory (Uzbekistan) with a 15-cm refractor at red light 610 nm (Velikodsky et al., 2010). The data set of 12 maps of absolute albedo at a ¼ 12.811, 14.731, 16.211, 17.781, 18.741, and 21.021 before full-moon and at a ¼12.331, 13.421, 14.191, 15.071, 16.191, and 21.791 after fullmoon were used in our analysis. The observation data were selected to have maximally symmetric values of phase angles before and after full-moon. The range of phase angles approximately from 121 to 221 allows us to use a relatively simple (one exponent) model phase function (7). The maximal phase angle 221 allows us to map almost the full visible disk in contrast to our previous work (Korokhin and Akimov, 1997). The resolution element of our data is approximately 3.2 3.2 km2 in the lunar disk center. This is the minimal base of roughness. Before calculations all images have been additionally co-registered using an algorithm of soft co-registration (Kaydash et al., 2009a) to compensate for the influence of Earth’s atmospheric turbulence that deforms the lunar images. Fig. 1 presents a map of A0, i.e., the albedo distribution that is not influenced by topography. We note that this is
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Fig. 1. A map of A0 calculated in the range of phase angles 12–221.
Fig. 2. A map of longitudinal topographic slopes determined on a 3.2 km base; the map is calculated using 12 images in the range of phase angles 12–221.
a result of albedo extrapolation defined with formula (7) to a ¼0 using 12 images obtained at phase angles 121–221. The distribution of A0 is similar to that of the equigonal albedo Aeq(a) at any moderate phase angle and characterizes the reflection properties of the lunar surface materials. It should be emphasized that the albedo map in Fig. 1 does present real brightness distribution at a ¼0, as it ignores variations of opposition-effect amplitudes that are not described by formula (7). The opposition spike has components of shadow-hiding and coherent backscattering effects (e.g., Hapke, 1993; Shkuratov et al., 2004). To include these in the analysis, a more complicated approximation of the phase function is needed (Velikodsky et al., 2010). Fig. 2 shows a distribution of the longitudinal component rl of the topographic slope calculated on a 3.2 km base. The distribution in Fig. 3 shows realistic values of slope up to 81. This is in agreement with our assumption that topographic slopes on bases provided by Earth-based telescope observations are small. Some sites demonstrate slopes reaching 201. The maximal value of slope we obtained in this analysis is 221, observed at the selenographic coordinates l ¼0.21, b¼21.61 in Montes Apenninus. This new map of longitudinal slopes coincides well with our previous results (Korokhin and Akimov, 1997). Shown in Fig. 4 is a map of the parameter m of function (7) determined in the range of phase angles 12–221. The distribution describes the steepness of the phase dependence of the lunar surface brightness. The parameter m can be determined using many calibrated images; whereas, a phase-angle-ratio image includes only two components. In any case, the images of the phase ratio and of the parameter m calculated for the same phase angle range should be very similar. As one can see, the parameter m demonstrates an inverse correlation with albedo: the higher the steepness m, the lower the albedo. This is in agreement with our previous results (Shkuratov et al., 1994; Korokhin and Akimov, 1997; Kreslavsky et al., 2000; Kreslavsky and Shkuratov, 2003; Kaydash et al., 2009b). The albedo A0 and steepness m maps calculated using the described algorithm are almost free from the influence of the
lunar topography (see Figs. 1 and 4). Figs. 5 and 6 show the case, if we ignore the compensation for the topography, i.e., when the phase-function parameters (7) A0 and m are calculated without accounting for the lunar relief. The difference, especially between the steepness distributions (see Figs. 4 and 6), seems to be dramatic. In the case of Figs. 1 and 4 some traces of relief are observed near the limb and the polar regions. This probably is caused by neglecting the latitudinal component of local slopes and the presence of areas on the source images with too large emergence and incidence angles. Errors in using the approximation with the function (7) are shown in Fig. 7. Typical values of the error (the standard deviation divided by A0 in percents) are 0.35–0.5% for maria, 0.4–1.0% for highlands, and up to 4% for bright craters. Areas near the limb have maximal values of errors (up to 20%). Small errors are observed in maria, because their surface is relatively flat. Moreover, the function (7) works for maria better than for highlands and bright craters. Thus, higher errors for highlands and bright craters are caused by more complicated topography in addition to the simplified function with one exponent (7) being not representative of the phase curves of highlands. High errors near the limb appear due to the presence of real shadows in addition to poor image co-registration caused by neglecting real heights of the lunar sites in the compensation for libration. In spite of some shortcomings of the technique, the approximation of phase function without accounting for the influence of the lunar relief gives much larger errors.
4. Comparison with LALT data As has been noted, recent altimetric data have been obtained with the LALT (laser altimeter) aboard the spacecraft Kaguya (Araki et al., 2009). The LALT was able to obtain a range of data on a global scale along the satellite0 s trajectory including the high latitude region above 751 that has never been measured by an
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Fig. 3. A histogram of longitudinal topographic slopes on a 3.2 km base.
Fig. 4. A map of the parameter m that characterizes steepness of phase function in the range of phase angles 12–221.
altimeter. Although the data set is somewhat raw yet (Korokhin et al., 2010), we used the global LALT topography for comparison with our photoclinometric data in order to compensate for the topographic effect for the phase-angle-ratio maps. Before the comparison we processed the LALT data. We transformed the data array to put the point with zero selenographic coordinates in the center of the image. Then, we re-sampled the LALT map to the resolution of our ground-based observations. The next step was calculating the longitudinal component of the surface slopes from the height distribution. Then, we transformed the LALT map from a cylindrical onto an orthographic projection. Finally, the image was smoothed by a Gaussian filter with s ¼0.8 pix to make the resolution of the LALT and our maps similar. We computed the map of longitudinal topographic slopes determined on a 3.2 km base (see Fig. 8) from the LALT distribution of heights. Both the maps show a very
Fig. 5. A map of A0 calculated in the range of phase angles 12–221 ignoring the influence of the lunar topography.
similar distribution of the longitudinal slopes over the lunar disk (cf. Figs. 2 and 8), including the maximal value of the slope 221 at the point with the selenographic coordinates l¼ 0.21, b¼21.61. However, more detailed analysis reveals differences. For instance, our map demonstrates a weak residual influence from albedo (see the Copernicus ray system) and worsening resolution to the limb. The LALT map exhibits many small topographic artifacts. Fig. 9a and b show maps of longitudinal slopes for the central portion of the lunar disk ( 501 501) in a cylindrical projection: the upper panel correspond to LALT data and the lower one to our map. The most typical artifacts are displaying some craters as hummocks (see, e.g., marks 1–3, 7–9 in Fig. 9a, b) or disappearance of craters (marks 4–6). Sometimes, the LALT map reveals nonexistent peaks (see marks 10 and 11). The most probable reasons for these artifacts are errors of interpolation at mosaicing. We also note the presence of longitudinal modulation
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Fig. 6. A map of the steepness m of phase function calculated in the range of phase angles 12–221 ignoring the lunar relief.
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Fig. 8. A map of longitudinal topographic slopes determined on a 3.2 km base, retrieved from the LALT data.
5. Topography correction of photometric data Although we have mapped the longitudinal component only, this allows for correction of Earth-based photometric observations, because illumination of the lunar surface varies mainly due to changing selenographic longitude of the Sun. For the correction of maps of albedo, it is necessary to apply the following correcting coefficient for each point of the lunar disk k ¼ Dða, b, gÞ=Dða, bu, guÞ,
Fig. 7. A map of errors (standard deviation in albedo units) of the approximation (7).
caused by spacecraft orbital motion; this can be seen in highcontrast images of the altimetric map (Araki et al., 2009). Thus, the considered version of the LALT map should still be used with caution. In particular, our map could be proposed for an independent control of the LALT data.
ð20Þ
where D(a, b, g) and D(a, b0 , g0 ) are the disk functions, respectively, without and with taking into account topography slopes. In the numerator of Eq. (20), photometric coordinates g and b of the point are calculated using its current selenographic coordinates l and b. In the denominator of Eq. (20), l0 and b0 are calculated using the obvious formula l0 ¼l+ rl/cos b. In Fig. 10a, b we present, respectively, topographically corrected and uncorrected maps of the phase ratio 21/211. It can be seen that the topography in Fig. 10a is significantly suppressed. Despite the phase angle ranges 10–201 and 2–201 being somewhat different, the resemblance the phase-ratio map and the map of the parameter m is clearly seen (cf. Figs. 4 and 10a). As can be anticipated, the phase-ratio images reveal a correlation with albedo. Low albedo areas (dark mantle deposits) in Mare Tranquillitatis, Mare Vaporum, and Sinus Aestuum (see also Taurus Littrow and Sulpicius Gallus) reveal the most steep phase function, while bright craters and their rays (e.g., Tycho, Copernicus, and Proclus) have the smallest slope of the function. This is a somewhat unexpected result as the craters are rather bright and, hence, at small phase angles ( o51), the coherent backscattering effect should prominently manifest itself (e.g., Hapke, 1993). This result could be explained by the small width of the opposition spike from the effect of coherent backscattering. However, the phase ratio image 11/51 (it will be published) demonstrates the same behavior: bright craters and their rays have small phase-ratio slope.
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Fig. 9. Comparison of maps of longitudinal slopes retrieved from photometric (upper (a)) observations and the LALT data (lower (b)).
We also note that there is very noticeable difference between Mare Serenitatis and Mare Tranquillitatis. We note that differences were observed for color ratio in the visible spectral range (e.g., Pieters and McCord, 1976; Shkuratov et al., 1999). Detailed comparison of the phase ratio map 21/211 with albedo shows that there are many anomalous regions that do not exactly coincide with the albedo boundary. For instance, the Aristarchus Plateau and Marius Hills (pyroclastic formations) clearly show up on the phase-ratio images. We note that the topographic correction helps in visualizing these anomalies. Corrected photometric data allow us to study the phase dependence of the lunar surface brightness not only for flat mare regions, but for highlands and other areas with rough resolved topography. For instance, relatively small swirls in the region of the central nearside highlands (Blewett et al., 2007) are more apparent if the topographic effect is removed.
6. Conclusion 1. We have studied two approaches to remove the resolved topography effect from images of phase-function parameters
Fig. 10. Phase ratio 21/211 with (a) and without (b) topography corrections.
of the Moon using altimetry measurements carried out with the Kaguya (JAXA) LALT instrument and new photoclinometry data obtained with Earth-based telescopic observations. Using these techniques, we have mapped the longitude component of the lunar topography slopes with a resolution of 3.2 km/pix. Overall, we have found good correlation when comparing our map with the corresponding data obtained from Kaguya altimetry. 2. Our approach to retrieve information about the topography, albedo, and the parameters of phase function from photometric images requires using a robust set of absolutely calibrated images obtained for the same scene at different phase angles. We numerically minimized the standard deviation of the observed Aeq(a) from the model function (7) varying
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the parameters rl, A0, and m (see designations above). This procedure minimizes the effect of topographic slopes. In applying this technique, we used new data of absolute photometry of the Moon obtained at the Maidanak Observatory (Uzbekistan). 3. Although the LALT and photoclinometry maps show good correlation, more detailed analysis reveals differences. In particular, the photoclinometric map demonstrates a weak residual influence from albedo. The LALT map exhibits many small topographic artifacts, displaying some craters as hummocks. Thus, our map would be proposed for an independent control of the LALT data. 4. Corrected photometric data allow one to study phase dependence of the lunar surface brightness not only for flat mare regions, but for highlands and other areas with expressed topography. This is useful, for example, in studying the swirls in the region of the central nearside highlands (Blewett et al., 2007).
Acknowledgment This study was supported by CRDF Grant UKP2-2897-KK-07.
References Akimov, L.A., 1979. On the brightness distribution over the lunar and planetary disks. Sov. Astron. 23 (2), 231–235. Akimov, L.A., 1988a. Reflection of light by the Moon. 1. Kinemat. Fiz. Nebesykh Tel. 4 (1), 3–10 (in Russian). Akimov, L.A., 1988b. Reflection of light by the Moon. 2. Kinemat. Fiz. Nebesykh Tel. 4 (2), 10–16 (in Russian). Akimov, L.A., Velikodsky, Yu.I., Korokhin, V.V., 1999. Dependence of lunar highland brightness on photometric latitude. Kinemat. Phys. Celest. Bodies 15 (4), 232–236. Akimov, L.A., Velikodsky, Yu.I., Korokhin, V.V., 2000. Dependence of latitudinal brightness distribution over the lunar disk on albedo and surface roughness. Kinemat. Phys. Celest. Bodies 16 (2), 137–141. Araki, H., Tazawa, S., Noda, H., Ishihara, Y., Goossens, S., Sasaki, S., Kawano, N., Kamiya, I., Otake, H., Oberst, J., Shum, C., 2009. Lunar global shape and polar topography derived from Kaguya-LALT laser altimetry. Science 323, 897–900. Blewett, D.T., Hughes, C.G., Hawke, B. Ray, Richmond, N.C., 2007. Varieties of the lunar swirls. In: Proceedings of the 38th Lunar and Planetary Science Conference, Abstract no. 1232. Bonner, W.J., Schmall, R.A., 1973. A photometric technique for determining planetary slopes from orbital photographs. US Geological Survey Professional Paper 812-A, pp. 1–16. Chen, C.-Y., Klette, R., Kakarala, R., 2002. Albedo recovery using photometric stereo approach. In: Proceedings of the 16th International Conference on Pattern Recognition (ICPR ’02), Quebec, Canada, August 2002, vol. 3, pp. 700–703. Connors, C., 1995. Determinimg heights and slopes of fault scarps and other surfaces on Venus using Magellan stereo radar. J. Geophys. Res. E 107 (7), 14281–14361. Cook, A.C., Robinson, M.S., 2000. Mariner 10 stereo image coverage of mercury. J. Geophys. Res. E 105 (4), 9429–9444. Davis, P.A., Soderblom, L.A., 1984. Modeling crater topography and albedo from monoscopic viking orbiter images. I—Methodology. J. Geophys. Res. B 89 (11), 9449–9457. Dulova, I.A., Skuratovskii, S.I., Bondarenko, N.V., Kornienko, Yu.V., 2008. Retrieval of the surface topography from single images using the photometric method. Sol. Syst. Res. 42 (6), 522–535. Hapke, B., 1993. Theory of Reflectance and Emittance Spectroscopy. Cambridge University Press 450 pp. Herrick, R.R., Sharpton, V.L., 2000. Implications from stereo-derived topography of venusian impact craters. J. Geophys. Res. E 105 (8), 20245–20262. Horn, B.K.P., 1989. Height and Gradient from Shading. MIT Technical Report no. 1105A /http://people.csail.mit.edu/people/bkph/AIM/AIM-1105A-TEX.pdfS. Howard, A.D., Blasius, K.R., Cutts, J.A., 1982. Photoclinometric determination of the topography of the martian north polar cap. Icarus 50, 245–258. Hung, Y.-P., Cooper, D.B., Cernuschi-Frias, B., 1988. Bayesian estimation of 3-d surfaces from a sequence of images, International Conference on Robotics and Automation. IEEE, pp. 906–911. Jankowski, D.G., Squyres, S.W., 1991. Sources of error in planetary photoclinometry. J. Geophys. Res. 96 (E4), 20,907–20,922.
1305
Kaydash, V., Kreslavsky, M., Shkuratov, Yu., Gerasimenko, S., Pinet, P., Josset, J.-L., Beauvivre, S., Foing, B., AMIE SMART-1 Team, 2009a. Photometric anomalies of the lunar surface studied with SMART-1 AMIE data. Icarus 202 (2), 393–413. Kaydash, V.G., Gerasimenko, S.Y.u, Shkuratov, Yu.G., Opanasenko, N.V., Velikodskii, Yu.I., Korokhin, V.V., 2009b. Photometric function variations observed on the near side of the Moon: mapping. Sol. Syst. Res. 31 (2), 143–152. Kirk, R.L., Howington-Kraus, E., Redding, B., Galuszka, D., Hare, T., Archinal, B.A., 2003. High-resolution topomapping of candidate MER landing sites with MOC: new results and error analyses. In: Proceedings of the 34th Lunar Planetary Sciences, Abstract no. 1966, LPI Houston. Kornienko, Yu.V., Dulova, I.A., Nguyen Xuan, Anh, 1994. Wiener approach to the determination of optical characteristics of a planetary surface from photometric observations. Kinemat. Fiz. Nebesnykh Tel. 10 (5), 69–76 (Kinemat. Phys. Celest. Bodies (Engl. Transl.). 10, 69–76). Korokhin, V., Akimov, L., 1997. Mapping of phase parameters of the lunar-surface brightness. Sol. Syst. Res. 31 (2), 143–152. Korokhin, V.V., Velikodsky, Yu.I., Shkuratov, Yu.G., Mall, U., 2007. The phase dependence of brightness and color of the lunar surface: a study based on integral photometric data. Sol. Syst. Res. 41 (1), 19–27. Korokhin, V.V., Velikodsky, Yu.I., Shkuratov, Yu.G., Kaydash, V.G., Gerasimenko, S.Yu., Opanasenko, N.V., 2010. Comparison between Kaguya (Selene) altimetry data and lunar topography retrieved from photoclinometry. In: Proceedings of the 41st Lunar and Planetary Science Conference, 1–5 March, Abstract no. 1356, LPI Houston, USA. Kreslavsky, M., Shkuratov, Yu., 2003. Photometric anomalies of the lunar surface: results from Clementine data. J. Geophys. Res. 108 (E3) (1-1–1-13). Kreslavsky, M.A., Shkuratov, Yu.G., Velikodsky, Yu.I., Kaydash, V.G., Stankevich, D.G., Pieters, C.M., 2000. Photometric properties of the lunar surface derived from Clementine observations. J. Geophys. Res. Planets 105 (E8), 20,281–20,295. Lambiotte, J.J., Taylor, G.R., 1967. A photometric technique for deriving slopes from lunar orbiter photography. In: Proceedings of the Conference on the Use of Space Systems for Planetary Geology and Geophysics, Boston, MA, 25–27 May. Leclerc, Y.G., Bobick, A.F., 1991. The direct computation of height from shading, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 3–6 June. IEEE, pp. 552–558. Lohse, V., Heipke, C., Kirk, R.L., 2006. Derivation of planetary topography using multi-image shape-from-shading. Planet. Space Sci. 54, 661–674. McEwen, A.S., 1991. Photometric functions for photoclinometry and other applications. Icarus 92, 298–311. Muinonen, K., Lumme, K., Irvine, W.M., 1989. Statistical photoclinometry and surface topography of atmosphereless bodies. In: Proceedings of the 20th Lunar and Planetary Science Conference, LPI Houston, USA, pp. 729–730. Nelder, J., Mead, R., 1965. A simplex method for function minimization. Comput. J. 7, 308–313. Oberst, J., Jaumann, R., Zeitler, W., et al., 1999. Photogrammetric analysis of horizon panoramas: the pathfinder landing site in viking orbiter images. J. Geophys. Res. E 104 (4), 8927–8934. Opanasenko, A.N., Shkuratov, Y.G., Opanasenko, N.V., 2007. Topography of three localities on the Moon from combined Clementine and Hubble space telescope images. In: Proceedings of the 38th Lunar and Planetary Science Conference, Abstract no. 1564, LPI Houston, USA. Parusimov, V.G., Kornienko, Yu.V., 1973. About finding the most probable relief of a planet surface by its optical image. Astrometriya Astrofiz. 19, 20–24 (Astrometry Astrophys. (Engl. Transl.), 19, 20–24). Pieters, C., McCord, T., 1976. Characterization of lunar mare basalt types. In: Proceedings of the Seventh Lunar and Planetary Science Conference, LPI Houston, pp. 2677–2690. Ramachandran, V.S., 1988. Perceiving shape from shading. Sci. Am. 159, 76–83. Shalygin, E.V., Velikodsky, Yu.I., Korokhin, V.V., 2003. Formulas of the perspective cartographic projection for planets and asteroids of arbitrary shape. In: Proceedings of the Third Lunar and Planetary Science Conference, Abstract no. 1946, LPI Houston, USA. Shkuratov, Yu.G, Starukhina, L.V., Kreslavsky, M.A., Opanasenko, N.V., Stankevich, D.G., Shevchenko, V.G., 1994. Principle of undulatory invariance in photometry of atmosphereless celestial bodies. Icarus 109, 168–190. Shkuratov, Yu.G., Kaydash, V.G., Opanasenko, N.V., 1999. Iron and titanium abundance and maturity degree distribution on lunar nearside. Icarus 137, 222–234. Shkuratov, Yu., Videen, G., Kreslavsky, M.A., Belskaya, I.N., Ovcharenko, A., Kaydash, V.G., Opanasenko, N., Zubko, E., 2004. Scattering properties of planetary regoliths near opposition: photopolarimetry in remote sensing. Videen, G., Yatskiv, Ya., Mishchenko, M. (Eds.), NATO Science Series. Kluwer Academic Publishers, London. pp. 191–208. Shkuratov, Yu., Kaydash, V., Gerasimenko, S., Opanasenko, N., Velikodsky, Yu., Korokhin, V., Videen, G., Pieters, C., 2010. Probable swirls detected as photometric anomalies in Oceanus Procellarum. Icarus 208, 20–30. Tsai, P.S., Shah, M., 1994. Shape from shading using linear approximation. Image Vision Comput. J. 12 (8), 487–498. Tyler, G.L., Simpson, R.A., Moore, H.J., 1971. Lunar slope distributions: comparison of bi-static radar and photographic results. J. Geophys. Res. 76 (11), 2790–2795. van Diggelen, J., 1951. A photometric investigation of the slopes and heights of the ranges of hills in the maria of the Moon. Bull. Astron. Inst. Netherlands 11, 283–289.
ARTICLE IN PRESS 1306
V.V. Korokhin et al. / Planetary and Space Science 58 (2010) 1298–1306
Vega, O.E., Yang, Y.H., 1993. Shading logic: a heuristic approach to recover shape from shading. IEEE Trans. Pattern Anal. Mach. Intell. 15 (6), 592–597. Velikodsky, Yu.I., Opanasenko, N.V., Akimov, L.A., Korokhin, V.V., Shkuratov,Yu.G., Kaydash,V.G., 2010. Kharkiv absolute photometry of the Moon. In: Proceedings of the 41st Lunar and Planetary Science Conference, Abstract no. 1760, LPI Houston, USA. Watson, K., 1968. Photoclinometry from spacecraft images. US Geological Survey Professional Paper 599-B, pp. 1–10. Watters, T.R., Robinson, M.S., 1997. Radar and photoclinometric studies of wrinkle ridges on Mars. J. Geophys. Res. E 102 (5), 10889–10903.
Wildey, R.L., 1973. Theoretical autophotogrammetry: I. The method of photometric potential. Mod. Geol. 4, 209–215. Wildey, R.L., 1975. Generalized photoclinometry for mariner 9. Icarus 25, 613–626. Wohler, C., Hafezi, K., 2005. A general framework for three-dimensional surface reconstruction by self-consistent fusion of shading and shadow features. Pattern Recognition 38 (7), 965–983. Zhang, R., Tsai, P.-S., Cryer, J.E., Shah, M., 1999. Shape from shading: a survey. IEEE Trans. Pattern Anal. Mach. Intell. 21 (8), 690–707.
