9-11 Nov. 1999
]V Adt + e. =
19-Fe.)4 (9)
wcosp+e,)+qs
o and q. Rearranging eq.
os =
» t qidRsin 9 (9.1)
];dRsin 0 cos 9 * q4
n the time error correction
c correction, both have the
r' which means they are
e q;. This should not be
change in elevation due to
rtant observation is that a
orbed by the translation
ction since they represent
. (nvolving dR and the
that the correction for the
are equivalent, thus dz is
1 leads to the the following
ly + dR + qRsing +
'S( COS Q t que, * qoe, t €;
volves the Taylor series
cos (sin 9
? following form:
9) +
SQ)tqi- (9.3)
p T He, + 92° y + €,
s follows. The terms (in
equation is the difference
Z;-R) and the computed
2Y+43). The difference is
ers on the right hand side
the distance between the
26, + €, ). This model is
- (G,caP !) (10)
International Archives of Photogrammetry and Hemote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
solved by
AEP YY An A BP BI qoi
To analyze the system we take the observation equation for the
first iteration. Here the approximations for @ andg will be a,
=@,=0. Thus the form will be:
(Zo - R) - (1X o + 42ŸY0 + 93) = qydx + qady + dR +
00 00 (11)
I + Ge, s qd2€ y + €;
We ignore the multiple error components and augment them all
into a single component, rewriting equation (11) in matrix
notation of the following form:
dz 90.023 1 qf. gk
= oes "Tamil (12)
dz, qd1 492 1 quR, q2R,, © €n
Equation (12) allows some important observations concerning
the recovery of the calibration parameters. Firstly, it can be
seen that the parameters are independent of the satellite
position at the ranging time (i.e. Xo or Yo are not part of any
coefficient). Secondly, flat surfaces or a surface tilted only in
one direction (i.e. either q; or q» equal to 0) cannot recover
some of the parameters (since the relevant columns turns to 0).
More important, analyzing the linear dependency between
parameters shows that the first three columns are linearly
dependent (different by a constant), columns 4 and 5 are
linearly dependent as well. The consequence is that a single
surface is insufficient to recover all of the parameters
regardless of the number of observations, and at most two
parameters can be resolved’. In order to recover all parameters
or even part of them (such as the rotation parameters and the
ranging bias) at least two surfaces will be required.
Analyzing the relations between columns 1, 2 and 4, 5 shows
that the difference between the coefficients is the additional
range component in column 4 and 5. Small relief variation (for
example due to small slope angle) will therefore result in high
correlation between the two pairs of columns. High correlation
indicates that the parameters in question have a very similar
effect on the system and therefore it is difficult to resolve the
actual effect of each of them separately when some of them are
involved. For calibrating spaceborne laser altimeters, where the
flying altitude is high (for example 600,000 m for the GLAS
satellite), reasonable relief variations will be too small to have
any significant effect, therefore the effect of dx, dy, will be
very similar to the one of q, ®. Experiments with different
! Here we assume that calibration using data from a single orbit. It has
been proposed to use ascending and descending orbits over the same
location, so that a single surface will be sufficient. This can be modeled
also as combination of two surfaces, but the more important question is
whether the biased do not change through time.
surface configurations have indeed shown that the correlation
approaches 1 for the satellites flying: altitude and remains high
even for lower altitude. The correlation matrix presented in
table 1 is an example for the correlation for a flying altitude of
600,000 [m]
1.00000 0.85234 0.75224 -0.999784 0.85667
-0.85234 1.00000 0.97063 0.85940 -0.999869
-0.75224 0.97063 1.00000 0.75974 -0.970570
-0.99978 0.85940 0.75974 1.00000 -0.86374
0.85667 -0.99986 -0.97057 . -0.863747 1.00000
Table 1. The correlation matrix between the 5 parameters for
600,000[m] orbital altitude
For three parameters only (two rotations and range bias) the
correlation for the same surface configuration was reduced
significantlly. Table 2 presents the correlation matrix between
these parameters for an orbital altitude of 600,000 [m]
1.00000 -0.519158 -0.390343
-0.519158 1.00000 -0.426124
-0.390343 -0.426124 1.00000
Table 2. The correlation matrix between the first 3 parameters
for 600,000] m] orbital altitude
4. OPTIMAL SURFACE CHARACTERISTICS FOR
RECOVERING THE GLAS CALIBRATION
PARAMETERS
We now examine surface characteristics that provide a robust
solution for the calibration parameters. It is clear that different
configurations provide solutions with different robustness
measures, for example set of surfaces with small deviation
from an horizontal plane are expected to produce weak solution
than other configurations. We analyze the robustness of the
solution by three criteria, the correlation matrix, the variances
of the estimated parameters and the condition number. The
properties of the correlation matrix were discussed above, high
correlation implies strong relation between parameters and less
confidence in the obtained values. The variance is a general
measure for the goodness of the estimation. The condition
number, defined as the ratio between the largest and the
smallest eigenvalue is an indication for the significance of the
parameters. As the ratio approaches 1 all parameters have
similar significance; as it approaches oo, the system of
equations is rank deficient (singular). In searching for a good
set of surfaces we are looking into the number of surfaces
required and their trend. An optimal solution will have a
minimal set of surfaces and reasonable slopes, the overall size
of the configuration site is also an issue to be addressed. To test
configurations we simulated a flight path over a set of surfaces
while introducing biases to the modeled parameters. In
addition, to assess the robustness of the solution and its
convergence to the correct values we also introduced random
“ noise to some of the parameters.
