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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
where, XYZ stands for the ground coordinates.
Based on standard deviations, the measurements’ weights are
assigned. In this study, à priori standard deviation of image
coordinates of tie point is taken as reference variance with unit
weight, 1.
5. RESULTS AND EVALUATION
This section presents and evaluates the results of the bundle
adjustment. First, the amount of correction of measurements is
examined in three sites according to the types of measurements.
Secondly, error propagation is theoretically computed, namely,
the variances of ground coordinates are calculated from the
normal equations in the least squares solution. Last, the MOLA
registration is carried out using adjusted exterior orientation
coefficients, and the improved registration results are visually
presented.
5.] Corrections and residuals
Table 2 presents the amount of correction of ground
coordinates. Overall, average correction of tie points is larger
than those of MOLA profiles. Because MOLA ground
coordinates are treated with relatively high weights in the
bundle adjustment, the corrections of MOLA ground
coordinates show from 4 mm to 0.1 mm in all three sites. That
means the MOLA ground coordinates are almost not changed.
For differences among sites, the ground coordinate corrections
of tie points show small changes in Gusev Crater and Isidis
Planitia sites, but in Eos Chasma site, the corrections of tie
ground coordinates show 91m, 107m and 55m in X, Y, and Z
respectively. Furthermore, RMSs of ground coordinates are so
small that the corrections of all points are quantitatively similar
except tie points in Eos Chasma. Therefore, the ground
coordinates of tie points in Eos Chasma are adjusted quite large
amount and the corrections show large differences among the
tie points as well.
Table 2. Statistics of ground coordinates correction (in meters)
X Y Z
Site Type (Mean (Mean (Mean
/RMS) /RMS) /RMS)
MOLA | 0.001/0.256 | -0.001/0.366 | 0.004/0.188
Eos
Chasma |. Te -91.232 106.913 54.799
/117.676 /151.426 /59.585
Gusev | MOLA | -0.001/0.299 | 0.001/0.098 | 0.001/0.135
Crater Tie -1.794/2.723 | 0321/0340 | 6.750/1.10
Isidis | MOLA | 0.0004/0.056 | 0.0001/0.222 | 0.023/0.746
Planitia | — ji. 0.222/0.189 | -1.786/1.011 | 3.144/3.148
The reason of the manifest differences in Eos Chasma is
presumed to be caused by the image acquisition date difference.
Navigation data for exterior orientation, kernel, are determined
according to each image acquisition time. In Eos Chasma site,
unlike other two study sites, the two stereo pair images are
acquired on quite different dates; one image from March 2001
and the other from May 2001. In this case, different navigation
data are used to estimate sensor orientation and position for
each image. Because exterior orientation from two different
MGS orbits is inconsistent with each other, Eos Chasma shows
incompatible results with other two study sites.
In Table 3, the amount of correction and residuals of image
coordinates are large in x image coordinates in all three sites,
which mean image coordinates are mainly changed along the
flight direction. Consequently, the reported MOLA mis-
registration along the flight direction in the previous research is
corrected after the bundle adjustment. This is later proven with
figures showing MOLA profiles overlaid with MOC stereo
images. The image coordinates of tie points showing relatively
quite small residuals are not changed at all on the images of all
sites. The results of Gusev Crater and Isidis Planitia show
similar pattern: large amount of correction in x direction, small
amount of correction in y direction and small RMS in both
directions. In the Eos Chasma site, however, RMSs of MOLA
image corrections have a large variance in both directions. The
different pattern in Eos Chasma site can be explained by the
same reason presented before.
Table 3. Statistics of image coordinates: correction for MOLA
profiles and residuals of tie points (in pixels)
Un
Image 1 Image 2
Sit Ty x J x y
ie ype Mean Mean Mean Mean
/RMS /RMS /RMS /RMS
12.189 4.040 -8.419 47.103
Eos MOLA /61.724 | /15.915 | /-18.116 | /40.515
f Tie -0.009 -0.002 0.009 -0.002
/0.615 /0.055 /0.418 /0.030
-24.176 3.286 24.895 -2.695
Glisev MODA /0.774 /0.254 /0.104 /0.163
Tie 0.086 -0.012 -0.089 0.009
/0.603 /0.084 /0.621 /0.069
35.044 -2.711 -24.789 2.891
Isidis MOEA /0.082 /0.198 /1.519 /0.357
| Tie -0.162 0.012 0.119 -0.018
/0.453 /0.035 /0.322 /0.038
MOLA ranges play a role of constraining the geometric
relationship between exterior orientation and ground
coordinates. Residuals of MOLA ranges vary from about 1
meter up to about 2 meters. Along with other measurements,
exterior orientation coefficients are also adjusted through the
bundle adjustment. MOLA and MOC registration results
presented later will show the effect of adjusted exterior
orientation coefficients.
5.2 Theoretic analysis
A posteriori reference standard deviations ( G, ) are computed
from the residuals of measurements. In all three study sites, a
posteriori reference standard deviations are close to a priori
(co71) reference standard deviations: 1.037, 1.059, and 1.014
pixels as shown in Table 4.
Subsequently, a posteriori standard deviation is used to
compute the variance-covariance of the ground coordinates of
MOLA and tie points. Table 4 shows the standard deviations of
the ground coordinates of MOLA and tie points. Generally, the
standard deviations of MOLA profiles show relatively smaller
than those of tie points. À posteriori standard deviations of
MOLA points are consistent with a priori estimation, 10 meters.
In the Eos Chasma site, the standard deviations of tie ground
coordinates are lager than the standard deviations of MOLA
points and the magnitudes are quite larger than those of other
two sites.
