International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
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with ¢¢ being the expectation value of the interferometric phase.
From the standard deviations the weight matrix Pp = diag -
is set up. The stochastic model derived in this way implicitly ac-
counts for noise introduced by temporal, thermal and geometric
decorrelation as well as errors originating from imperfect inter-
polation and co-registration procedures.
Additional variance and covariance values are usually introduced
by orbit errors and atmospheric effects. These additional error
sources are neglected in this study for the following two reasons:
(1) Orbit errors have been significantly reduced using control
point information. (2) Atmospheric effects showed to be small in
polar regions as their cold atmosphere appears to be very stable
and can not hold much water vapor. A closer look on atmospheric
effects is included in section 3.
3 ANALYSIS OF THE APPROACH
3.1 Accuracy analysis
The diagonal of Q;;, which contains information about the vari-
ances of the adjusted parameters, is used to derive theoretical
standard deviations of topography oop. and surface motion esp.
For analyzing the accuracy of the approach, simulated data sets
have been generated on the basis of existing DEM and velocity
maps. The coherency estimates ol each interferogram account
for the local surface slope. The dependency of Otopo ANd Odisp
on the number of independent data sets is shown in Figure 1. The
2 4 6 8 10 12 14
number of data sets
Figure I: Theoretical standard deviations oop. (black) and og),
(gray) dependent on the number of data sets.
improvement, which can be attained if more than two interfero-
grams are combined for estimating the unknown parameters, is
clearly visible.
Figure 2 shows how the standard deviations of the estimated to-
pography and displacement parameters depend on the observa-
tion geometry, which is mainly a function of the interferomet-
ric baseline B. The presented results are calculated on the ba-
sis of 3 simulated interferograms. In 256 simulation runs, the
baselines of the interferograms 1 and 2 are varied from 0 m to
400 m each. The baseline of the third interferogram is fixed at
200 m. In general, the standard deviations of both, topography
and motion, show distinct dependency on the baseline ratio of
effective baseline B, [m]
NT
effective baseline B.
N
Tho — 100 150 200 250 300 350 Ts — 100 150 200 2% 300 350
effective baseline B, [m] effective baseline B, [m]
a) b)
Figure 2: a) Mean standard deviation of topography (m) and 5)
mean standard deviation of displacement parameters (mm/day) as
a function of baseline constellation. The effective baseline length
of By ist set to 200 m.
|
1006
the involved interferograms. Baselines of similar length result in
a weak configuration of the adjustment model and finally, in in-
creased values for Cropos ANd Caisp. In case of identical baselines
the adjustment get's singular (this case is indicated by the cross
in Figure 2). If the baselines of the 3 interferograms are well
distributed, the topographic height may be estimated with an ac-
curacy of oopo = 3m and the surface motion with a standard
deviation of Odisp = 1 — 2mm/day. More detailed inspection
of Figure 2 shows however, that topography- and displacement-
uncertainties are not minimized by the same measurement setup.
Otopo 18 decreasing for well distributed long baselines, optimal
accuracies of surface motion arise if all baselines are short.
3.2 Sensitivity regarding model errors
The interferometric model presented in Equation (5) has been
simplified by neglecting the influences of atmospheric effects and
penetration depth. If these non-modeled influences are signifi-
cant, model errors are introduced resulting in a systematic falsifi-
cation of the estimated unknowns.
3.20.1 Atmospheric effects Because of the relative character
of an interferogram, the theoretical expectation value of atmo-
spheric effects will be zero for an arbitrary pixel. However, the
variance of the signal might be significant depending on the re-
spective weather conditions. If enough observations are com-
bined, the empirical expectation value, which is estimated from
the data, converges the theoretical value. Thus, for large amount
of observations, atmospheric effects will cancel out. The effect of
non-modeled atmospheric influences on the estimated unknowns
has been calculated for varying observation configurations. At-
mospheric phase screens have been simulated based on a method
presented in (Hanssen, 2001) considering the effect of the polar
atmosphere on the interferometric phase as described in (Gray
et al., 1997). Figure 3 shows the effect of the polar atmosphere
on the unknown topography /, and motion v as a function of the
number of multi-temporal data sets. The solid lines in Figure 3
| |
Me eO "5
number of multi-temporal data sets N
2 4 AL 10 nw nu
number of multi-temporal data sets N
a) b)
Figure 3: Effect of non-modeled atmospheric effects on the es-
timated unknowns topography (a)) and motion (5)) for polar re-
gions.
shows the systematic error of the estimated topography (Figure
3a)) and motion (Figure 35)). For a low number of data sets,
topography may be falsified up to Ah = 10m, the estimated mo-
tion up to Av = 0.4 cm/day. With increasing number of data sets
Ah and Av converge zero as expected. For investigating whether
Ah and/or Av differ significantly from zero, a significance test
is performed. The dashed lines in Figure 3 represent the upper
acceptance limit for the null hypothesis. Values lying above the
dashed line are significant, values below insignificant. Figure 3
shows that in arctic regions systematic errors of the adjusted un-
knowns due to atmospheric effects are insignificant for all tested
configurations.
3.2.2 Penetration depth — The penetration depth d into the gla-
cier surface depends mainly on its physical properties. As pre-
sented in (Hoen, 2001), C-band signals penetrate up to 275m
into cold firn. The impact ¢,q on the interferometric phase in-
creases with the interferometric baseline. Figure 4 shows the de-
pendence of ó,4 on d and B. A time independent d results in à
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