First, we show the effect of terrain with gentle slopes. This may
serve as a reference configuration. We use three surfaces with
the following slopes: {q,=0.01, gq, = 0.0}, {q,=0.0, q,=0.01}
and {q,=0.0, q,=0.0}, the first two surfaces are tilted by 1% in
each direction and the third one is a horizontal planar surface.
The result in table 3 present the correlation between
parameters.
1.000000 0.316074 0.673140
0.316074 1.000000 0.847108
0.673140 0.847108 1.000000
Table 3. Correlation Matrix for gentle sloping terrain
The condition number for this configuration approaches a value
of 45849. These results confirm our intuition that similar trend
for all surfaces results in high correlation, leading to a weak
solution. Introducing ranging noise of +5m resulted in a
Standard Deviation of - ¢ = £2.95, however the parameters
variance were £40” for the mounting biases and +2.2m for the
range. The variance of the rotation angles is a function of the
slopes. The ranging noise is, in general, much smaller than the
one introduced, however, here we also modeled the effect due
to coarse surface segmentation.
Experimenting with large differences in slopes shows that the
robustness of the solution improves dramatically. The terrain
slopes are 10.1, 0.2), 10.3, 0,1}, 10.0, 0.1}, 10.1, 0.0}, 10.4,
0.3}, {0.2, 0.3}. For such configuration the condition number
was reduced to 225 only and the correlation matrix (presented
in table 4) was reduced as well.
1.000000 -0.519159 -0.390311
-0.519159 1.000000 -0.426157
-0.390311 -0.426157 1.000000
Table 4. Correlation Matrix for high sloping surfaces
With ranging noise the SD was still on the order of o= 42.96,
however, the variances are reduced dramatically with de, dœ =
+2” (for 600,000 m this is equivalent to +5m on the ground)
and +1.26m for the range.
Steep sloping terrain has, however, limitations. From a
practical point of view it is difficult to find areas with such
steep slopes in a large area. In addition, the returned waveform
shape is highly dependent on the slope of the terrain. The
higher the slope the weaker the waveform; background and
electronic noise become more influential and the ranging
reliability is reduced. We therefore seek a configuration with
less distinct slopes that will still yield a good solution.
Returning to the observation equations in (12) and analyzing
the normal equations we realize that having trends in all
directions, i.e. rising surfaces and descending surfaces might
compensate one another. In terms of the normal equations, such
configurations reduce some of the off-diagonal parameters,
consequently reducing the correlation between the parameters.
The following experiment involves a combination of 7 surfaces
with the following characteristics: {0.0, 0.3}, {0.3, 0.0}, {0.1 -
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
0.3}, {-0.3, 0.1}, {-0.2, 0.0}, (0.0, -0.2). The correlation
matrix for this configuration is presented in table 5.
1.000000 0.271940 0.190221
0.271940 1.000000 0.229318
0.190221 0.229318 1.000000
Table 5. Correlation Matrix for opposing slopes
The condition number of 39.7 indicates a well-balanced
solution. Note that without exceeding 3096 slope we obtain a
maximum correlation that is less than 20%, indicating that the
calibration parameters are independent of one another. In
conclusion, the obtained values are reliable. Introducing a
random ranging error of ~5m, the variances for the parameters
are as follows: dg, dw = +1” (equivalent to £5m on the ground
for 600,000 m this is) and variance of dr = +0.69m for the
ranging bias. The standard deviation (SD) is on the order of
€2.4m. Considering a random range error of -5m we obtain a
rapid convergence rate.
For the experiment with studying the effect of the position bias
(both in terms of time error), we introduced a bias of 500m in
the satellite position. The SD changed only few millimeters.
However, when introducing the five parameter model, the
adjustment diverged. That the solution diverged for a good set
of surfaces and for +5m noise level is just another indication
for the effect of highly correlated parameters. We also checked
the effect of a large random ranging error (50m). Using the
three parameters model, the solution still converged to the true
parameters.
We started minimizing the configuration both by reducing the
number of surfaces and the magnitude of the slopes. For a
configuration of three surfaces and following slopes {-0.2,
0.2}, {0.2, 0.2}, {0., -0.1}, the correlation was reduced to the
following figures (Table 6):
1.000000 0.003183 0.070234
0.003183 1.000000 -0.192688
0.070234 -0.192688 1.000000
Table 6. Correlation Matrix for 3 surfaces with opposing, steep
slopes
The condition number rose to 53.0 but this is still a reasonable
value. The SD was in the order of 6 = 42.8, and the variances
were the following: d@, d& = +2” and dr = £0.83. Remaining
with three surface (two are the minimal configuration) and
reducing the slopes to 15% at most - {-0.15, 0.15}, {0.15,
0.15}, {0., -0.1}, generated the following figures.
1.000000 0.002220 0.095204
0.002220 1.000000 -0.036995
0.095204 -0.036995 1.000000
Table 7. Correlation Matrix for 3 surfaces with opposing,
gentle slopes
Intern:
The correlation, p
condition number
of the decrease i
solution implies
independently. T
+2.6”, dr = 10.36
Reducing the slog
0.07} still yields :
but the correlat
significance of th
still maintained. '
do, dw = X4" wh
The rise of the r
reducing the sur
surfaces do not pi
in the condition
slope decrease, at
number will appre
The decrease in |
the opposing surf
is indeed the dor
solution. The inc
slope decreases
accuracy estimate
raise practical
requirements sin
descend. Even g
should not be too
for the size of th
obtained with 12-
not exceed the oi
spacing between
This would imph
the size of the «
robustness of the
reliability of the e
The two prevai
airborne and spac
correspondence b
the non-redundar
circumstance cau
become highly cc
the surface shap
proposed an algor
calibration parar
correspondence |
adaptive coarse-t
sequential refiner
all parameters ca
correlation), we d
parameter depen
general, we have
profiler.
