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3.2 Orbital Parameters and Atmospheric Effects
The last parameter of the InSAR model is the
interferometric constant Die that appears in the
interferometric equation. This term usually represents the
unwrapping integration constant, to be determined using
at least one GCP. In our procedure, the parameterization
of Die is of the form:
D| C = d 0 + d 1 • col + d 2 • lin + d 3 • col • lin + d 4 • col 2 + d 5 • lin 2 (6)
where the coefficients do, di, 62, d 3 , d 4 and ds have to be
estimated in the adjustment.
It includes the following components:
- the phase unwrapping integration constant;
- the effect of orbit errors on the interferometric distance;
- the low frequency atmospheric effects on the
interferometric phase.
There is a single unwrapping constant for the entire scene
only if one integration zone is created during the
unwrapping. Otherwise, for each separate zone a different
constant has to be estimated.
In our calibration, the orbit errors are not explicitly
modelled because the set of described parameters (Ro,
AR, To, AT, fD and Die) is sufficient to compensate for the
maximal errors we can expect in the orbits. The
effectiveness of the compensation was tested by
simulation assuming very conservative orbit biases. The
orbital errors of the master satellite are compensated for
by three parameters: the along-track component by To;
the radial component by Ro; and the out-of-plane
component, which affects both MP and the angle between
the vectors Vm and MP, by Ro and fo. The radial and out-
of-plane components of the slave orbital errors are
compensated for by setting the coefficients of Die as
parameters in the adjustment; the along-track component
has no effect on the positioning.
The interferometric constant Die can partially compensate
for the low frequency atmospheric effects on the
interferometric phase. Between the two image
acquisitions, changes in the refractive index of
atmosphere may occur. These changes are mainly due to
tropospheric disturbances (Hanssen and Feijt, 1996),
(Hanssen, 1998) and can result in very big phase shifts
(shifts up to 3 cycles are reported). Their consequences in
the generated DEMs can be very impressive: artefacts
(e.g. depressions) interpreted as relief can appear. Their
magnitude depends on the baseline length and can even
reach 100+200 m. Expressing the interferometric constant
as a second order polynomial and with an adequate
distribution of GCPs, the atmospheric effects on the
generated DEMs characterised by low spatial frequency
can be compensated for.
4. ESTIMATION OF THE InSAR PARAMETERS
The calibration of the InSAR geometry requires the
refinement of the model parameters through least squares
adjustment using GCPs. The calibration improves the
image to object geometric transformation of a specific
interferometric pair and has to be carried out for every
processed InSAR pair.
The input data for the LS adjustment are the precise
master and slave orbits and the GCPs, i.e. points whose
position in the image space (col, lin,
space (xp,y P ,zp) is known. For each GCP it is possible to
write one range equation, one Doppler equation and one
interferometric equation. These are the same equations
(2), (3) and (4) used for the grid generation, but in this
case the position of the pixel footprint P (xp,y P ,zp) is
known and the model parameters (Ro, AR, To, AT, f D o, ídi,
fü2, ÍD3, do, d-i, d2, d 3 , d 4 and ds) that have to be refined
are the unknowns.
Some model parameters are geometrically correlated.
Setting them as free parameters causes an ill-conditioned
normal matrix in the adjustment. In order to avoid a
singular or ill-conditioned normal matrix, the parameters,
which can not be determined with the given configuration,
have to be excluded from the adjustment. An effective
way to proceed in such a case is to add a so-called
pseudo-observation to the observation equations (in this
case the 3 equations written for each GCP) for each
unknown parameter:
Pk = Pk ! wk
where:
Pk is the K th unknown parameter (to be adjusted),
p K is the approximate value of the K th parameter p K
(treated as pseudo-observation),
w K is the weight associated to the K^ pseudo
observation.
The weight associated to each pseudo-observation is
employed to this purpose: those parameters whose
values can not be determined in the adjustment receive a
large weight in the relative pseudo-observation.
The InSAR geometry calibration requires the estimate of
not homogeneous (e.g. times, distances, frequencies) and
highly correlated parameters. We face two opposite
requirements: we need a stable estimate of the
parameters (this implies to avoid building an ill-
conditioned matrix and hence to reduce the number of
adjusted parameters); but we aim at accurate calibration
of the InSAR geometry, i.e. we have to compensate for all
distortions (this demands to increase the number of
parameters).
We choose to exclude from the adjustment only the
parameters that make very ill-conditioned the normal
matrix N. The selection of the set of parameters is based
on the following points:
- analysis of the normal matrix eigenvalues and of the
condition number;
- check of the coefficient of total correlation of each
parameter;
- evaluation of the correlation coefficient between pairs of
parameters.
The complete description of the least squares adjustment
adopted for the InSAR geometry calibration (e.g.
linearization of the equations, construction of the
stochastic model, solution estimation, etc.) is reported in
appendix A.
4.1 GCP Identification
The GCPs are usually measured on the amplitude SAR
images (see Figure 2). Depending on the land cover type
and the topography, the measure can be also performed
on coherence images (see Figure 3).
It is quite difficult to establish a general rule for the
number of GCPs required for the calibration. For areas up
to 50 by 50 km (comparable with those we analysed), a
set of 8+10 GCPs, evenly distributed in the whole scene,
should assure a sufficient redundancy for the LS
estimation.
