International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
hardware coordinates disagreed with the NAC average values
by more than 50 m. The cause of these outlier estimates is under
investigation.
2.2 Elevation
The elevation for each object was derived using 2 m per pixel
digital terrain models (DTMs) produced from NAC stereo
images (Tran, 2010) tied to LOLA altitude tracks, which are
accurate in elevation to +2-5 m for the Apollo sites. At the
LRRR sites, we modified the DTM elevation values by taking
the difference in elevation between the NAC DTM’s LRRR
elevation and the elevation from laser ranging (Williams 2010),
and applying that difference to the whole NAC DTM. The
vertical offsets at all three LRRR sites were <2 m.
To calculate each object's coordinates, the corresponding image
is projected on a sphere with a radius equal to the object's
elevation. This strategy mitigates parallax errors from inexact
surface intersections due to small errors in spacecraft position or
pointing knowledge. To determine the coordinates of a pixel,
the instrument line of sight for that pixel is intersected with a
shape model. If that model is a DTM rather than a sphere, then
the differences in elevation around the correct point can
magnify ephemeris errors. This problem primarily affects the
longitude of coordinates in off-nadir images, but there are many
off-nadir images of the Apollo sites, as they were high-priority
targets that were imaged on many orbits that did not pass
directly over the sites.
2.3 Ground Coordinates
To compute the ground coordinates of an object from a line and
sample pair in an non-map-projected NAC image, we used the
CAMPT function in the Integrated System for Imagers and
Spectrometers (ISIS) (Anderson, 2004), which in turn uses
routines in the NASA Navigation and Ancillary Information
Facility's (NAIF) SPICE Toolkit. CAMPT calculates ground
coordinates from pixel coordinates using the relevant kernel
files, which define the absolute locations and orientations of the
spacecraft, instrument, and target. The two key time dependent
kernels in this case are the spacecraft kernel (SPK), which
contains the spacecraft position relative to the Moon for a given
time period, and the camera kernel (CK), which contains
orientation information for the spacecraft and instruments over
time. CAMPT uses the appropriate kernels to determine the
instrument location and pointing when each pixel was imaged,
and intersects the line of sight for that pixel with a shape model
of the Moon to determine the ground coordinates of that pixel.
After preparing an appropriate shape model as described earlier,
the largest remaining position uncertainty, on the order of 50 m,
is due to temperature dependent NAC pointing errors. These
errors were corrected using temperature- and slew-dependent
CK files that our team developed, which reduce the average
absolute positioning error to £15 m in both latitude and
longitude (Speyerer, 2012). This correction modifies the camera
pointing relative to the spacecraft body based on the
temperature of the structure to which the NACs are mounted,
and the angle the spacecraft is slewed towards or away from the
Sun. The latter may be a proxy for differential heating across
the mounting points, although the exact physical mechanism is
unknown. See Speyerer et al. (this volume) for further
discussion.
2.3.1 LRRR Sites: At the Apollo 11, 14, and 15 sites, the
LRRRs enable accurate control of the spacecraft pointing (Table
1). For each image, we updated the camera pointing using the
DeltaCK function of ISIS, which updates the cross-track and
down-track components of the camera pointing to align the
image with a known point, assuming the estimated spacecraft
position is correct. We then collected and averaged the locations
of each object from these corrected images. This method
restricted the usable images to those where the LRRR was
visible, but dramatically reduced the variation (to +2 m at most)
in calculated hardware positions as compared to the sites
without LRRRs (Table 2).
Object Latitude Longitude Radius (m)
All LRRR 0.673440 23.473073 1735472.7
Al4 LRRR -3.644170 | 342.521352 1736336.1
A1S LRRR 26.133396 3.628507 1735477.3
Lunokhod 1 38.315158 | 324.992036 1734928.7
Lunokhod 2 25.832307 30.922149 1734639.0
Table 1: Laser ranging derived coordinates for the five lunar
retroreflectors, from Williams (2008) (Apollos and Lunokhod 2)
and Murphy (2011) (Lunokhod 1). These values, and all others
reported in this paper, are in mean Earth/polar axis coordinates.
2.3.2 Sites with no LRRR: At the Apollo 12, 16, and 17
sites, there is no reference point with coordinates known to
similar or better accuracy than the NAC pixel size (the ALSEP
central station positions are only known to +30 m (Davies and
Colvin, 2000)). Thus, at these locations, hardware coordinates
were calculated by using the average latitude and longitude
from all NAC images in which each object is visible (Table 2).
3. RESULTS
The calculated positions for each object, along with the
uncertainties in each measurement, are shown in Table 2.
4. DISCUSSION
41.1 Accuracy of Davies and Colvin estimates: The
difference between coordinates derived by Davies and Colvin
(2000) and the those reported here were generally close to the
reported uncertainties from Davies and Colvin: 5 m for the
Apollo 14 and 15 ALSEPs, and 30 m for the Apollo 12, 16, and
17 ALSEPs. No error estimates were given for the LMs, but
their coordinates were also within 30 m of the NAC
coordinates. The worst lateral offset was the Apollo 16 LM, at
29 m southeast of the Davies and Colvin coordinates. Their
reported elevations, however, were high by 40 m at every site
without an LRRR, which is larger than the maximum vertical
error reported by Davies and Colvin for the VLBI data. A
possible source for some of this error is inaccuracy in Davies
and Colvin vertical and horizontal positions of the reference
ALSEPs at Apollos 14 and 15 relative to the LRRRs. At Apollo
14, the ALSEP position was off by 6 m in latitude, longitude,
and altitude, although the Apollo 15 ALSEP was only off by 3
m in longitude. We note that the published USGS maps (All,
A12, A14) and sketch maps in the mission preliminary science
reports (A15, A16, A17), which were used by Davies and
Colvin to locate the LMs relative to the ALSEPS, are good to
within 10-20 m over the 60-200 m distance between the ALSEP
and LM at each site.
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