counted for the one-way light time to the surface to determine
the selenocentric surface coordinates of the laser pulse. The
timing of the arrival of the laser pulse on the lunar surface
was converted to selenocentric coordinates by interpolating
the spacecraft orbital trajectory to the time of the laser mea-
surement and accounting for the one-way light time to the
surface. We thus transformed the measured range from the
spacecraft to the surface to a lunar radius in a center-of-mass
reference frame. Our orbits were characterized by a formal
uncertainty in radial position of about 10 m and have an ac-
curacy with respect to the lunar center of mass of order 100
m, which is comparable to the single shot ranging precision of
the lidar. The radial orbit accuracy determines to first order
the accuracy of the global topographic model. We also in-
cluded a correction for spacecraft pointing errors, which were
at the milliradian level (Regeon et al. 1994), resulting in
changes to measured ranges of up to 4 m.
The lidar typically ranged once per 1.6 sec, expending nearly
a half million laser shots during the two month Clementine
lunar mapping mission. The instrument detector triggered
on 19% of the shots. To distinguish valid ranges from noise
hits for these ~100,000 triggers, we applied a Kalman filter
(Tarantola and Valette 1982) based on a stochastic model
of topographic errors (Goff and Jordan 1988). We developed
the filter (Zuber et al. 1996a) on the basis of the observation
that planetary surface topography displays a fractal distribu-
tion (Turcotte 1987), which makes it possible to predict the
likely dynamic range of topographic power over a specified
spatial distance such as the along-track or cross-track laser
shot spacing. Filtering the data yielded 72548 " valid" ranges.
2.2 Global Topographic Model
The filtered data were assembled into a 10.25? x 0.25?
grid, corresponding to the minimum spacing between orbital
passes. We note that most major lunar basins were sampled
by Clementine altimetry. The LIDAR did not return much
ranging information poleward of 78?. Consequently, before
performing a spherical harmonic expansion of the data set
it was necessary to fill in the polar regions (+ 2% of the
planets surface area) by interpolation. For this purpose we
used the method of splines with tension (Smith and Wessel
1990) to continue the data smoothly across the poles. We
then performed a 72"? degree and order spherical harmonic
expansion of the data to yield Goddard Lunar Topography
Model-2 (GLTM-2) (Zuber et al. 1996a). The elevations
were referenced to a spheroid with flattening of 1/3234.93,
which corresponds to the flattening term, Cao, we obtained
for the lunar gravity field (Lemoine et al. 1996). This is the
observed dynamical flattening of the planet.
2.3 Fundamental Shape
To first order the present shape of the Moon is a sphere
with maximum positive and negative deviations of ~ 8 km,
both occurring on the far side (240°E, 10°S; 160°E, 75°S)
in the areas of the Korolev and South Pole-Aitken basins.
These departures from sphericity are the combined result of
various processes in the Moons early history (Zuber et al.
1996b). The two largest global-scale features are the center-
of-mass/center-of-figure (COM/COF) offset and the polar
flattening, both of which are of order 2 km. In addition there
are smaller wavelength deviations, due primarily to impact
basins. The power spectrum of the topographic shows more
power at longer wavelengths as compared to previous mod-
els due to more complete sampling of the surface, particularly
the far side basins. The power of topography follows a simple
general relationship of 2 km /spherical harmonic degree.
Another fundamental characteristic of the lunar shape is that
the topographic signatures of the near side and far side are
very different. As shown in Figure 1, the near side has a
gentle topography with an rms deviation of only about 1.4
km with respect to the best-fit sphere compared to the far
side, which is twice as large. The shapes of the histograms
of the deviations from the sphere show a peaked distribution
slightly skewed toward lower values for the near side, while
the far side is broader but clearly shows the massive far side
South Pole-Aitken Basin as an anomaly compared to the rest
of the hemisphere. The sharpness of the near side histogram
is a result of the maria.
We compared elevations derived from the Clementine LIDAR
to control point elevations from the Apollo laser altimeters
(Davies et al. 1987). 'A summary of the attributes of the
Clementine and Apollo data sets is presented in Table 2.
Where Apollo and Clementine coverage overlap, measured
relative topographic heights generally agree to within ~ 200
m, with most of the difference due to our more accurate or-
bit corrections (Lemoine et al. 1995) and to variations in
large-scale surface roughness (Zuber et al. 1996a). In con-
trast, Clementine topography often differs from landmark el-
evations on the lunar limb (Head et al. 1981) by as much as
several km.
Differences in lunar shape parameters derived from Clemen-
tine vs. Apollo altimetry are mostly due to better coverage
associated with the former: the greatest variations in lunar
topographic height are on the far side over a broad latitude
band, much of which was not sampled by the Apollo laser
instruments.
Another notable charasteric of the Moon is the lack of any
significant ellipticity in the equatorial plane. Figure 2 shows
equatorial radii along with low degree and order spherical
harmonic terms evaluated at the equator. The (2,2) terms
in the spherical harmonic model indicate an amplitude in the
equatorial plane of about 800 m with a maximum ~ 40°E
longitude, smaller than the COM/COF offset, but aligned in
the same general direction. In contrast, the Moons equatorial
gravity field is aligned almost exactly with the Earth-Moon
line. Figure 2 also illustrates that the (1,1) terms are more
than a factor of two larger than the (2,2) terms, which in-
dicates that the largest topographic effect around the lunar
equator is the COM/COF offset.
3 REFERENCES
Arthur, D. W. G. and P. Bates 1968. The Tucson selen-
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Baldwin, R. B. 1963. The Measure of the Moon, Ed., Uni-
versity of Chicago Press, Chicago, 88.
Brown, W. E. J., G. F. Adams, R. E. Eggleton, P. Jackson,
R. Jordan, M. Kobrick, W. J. Peeples, R. J. Phillips, L.
J. Porcello, G. Schaber, W. R. Sill, T. W. Thompson,
S. H. Ward and J. S. Zelenka 1974. Elevation profiles
of the Moon. Proc. Lunar Sci. Conf. 5th, 3037-3048.
Davies, M. E., T. R. Colvin and D. L. Meyer 1987. A Unified
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
