The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
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change detection, all the SAR images have to be co-registered
into the space of the selected reference image by maximizing
correlation between SAR acquisitions. As the subsequent PS
detection is based on statistical calculation of SAR data, all the
SAR amplitude images are calibrated in a similar way as Lyons
& Sandwell (2003). The unique radiometric calibration factor of
each image is defined and calculated as a ratio of the amplitude
of each image (mean of all pixels) to the mean amplitude of the
entire dataset. Each SAR amplitude image is divided by this
ratio to make comparable brightness between images.
It should be pointed out that any regionalized variable follows a
fundamental geographic principal; that is the samples that are
spatially closer together tend to be more alike than those that
are farther apart. The concept of neighbourhood differencing is
hence often employed to compensate some spatially correlated
errors or offsets. Differential GPS is a good example. Likewise,
our FCN strategy benefits deformation analysis via differencing
operation along each connection (arc) of two PSs in the FCN as
some spatially-correlated errors such as atmospheric effects and
other biases may be cancelled out more or less.
Each initial interferogram is derived by a pixel-wise conjugate
multiplication (equivalent to phase differencing) between the
master SAR image and the co-registered slave SAR image. In
theory, such direct phase difference at each pixel is due to
several contributions, i.e., flat-earth trend, topography, ground
motion, atmospheric delay and decorrelation noise. To highlight
ground deformation, both the precise orbital data and the
external digital elevation model (DEM) are utilized to remove
flat-earth trend and topographic effects from each initial
interferogram, thus resulting in M differential interferograms. It
should be emphasized here that no any filtering is performed
during differential processing to maintain independency in
phase data.
Let us assume that the available DEM has errors and the ground
motion in radar line-of-sight (LOS) direction is of linear and
nonlinear accumulation in time. The differential interferometric
phase at a pixel from the z'th interferogram can be modelled as,
A jr An
a D -Bi-s+—T r v+4TV l ) (i)
A ■ R ■ sin 0 A
where 5, 1 = spatial (perpendicular) baseline
T. = temporal baseline (time interval)
A = radar wavelength (5.66 cm for ERS)
R = sensor-target distance
0 = radar incident angle
s = elevation error
v = linear LOS deformation velocity
(j)'f = residual phase
2.2 PS Networking
As PSs will be used to form an observation network similar to a
levelling or GPS network, they need to be picked out from the
decorrelated pixels or areas. Using time series of the calibrated
SAR amplitude data, we basically follow the strategy by
Ferretti et al. (2001) to identify PS candidates on a pixel-by
pixel basis. Any pixel with amplitude dispersion index (ADI)
less than 0.25 is determined as a PS candidate.
After selection of all the PSs, we connect the neighbouring PSs
to form a network. It will be seen that such network can provide
a good framework for data modelling and parameter estimating
by LS method. Unlike a triangular irregular network (TIN) as
applied by Mora et al. (2003), we freely link the adjacent PSs
using a given threshold of Euclidian distance. Any two PSs are
connected only if their distance is less than a give threshold
(e.g., 1 km). The PS network formed in this way is hereafter
referred to as freely-connected network (FCN).
3. MODELLING AND ESTIMATING
3.1 Modelling and estimating with PS Network
As discussed in section 2.2, our data modelling is based on the
idea of neighbourhood differencing along each arc of the FCN.
For the z'th interferometric pair, the differential interferometric
phase increment between two adjacent PSs of each arc can be
derived from equation (1) and expressed as,
An —_L Ayr
Ad> i (7’ < ) = —- - • B, ■ As + —— •T i • Av + A</>f ( 7] ) (2)
A ■ R ■ sin 6 A
—i
where B, = mean of perpendicular baselines at two PSs
R = mean of sensor-target distances
0 - mean of radar incident angles
Ae = increment of elevation errors
Av = increment of linear LOS deformation velocities
A </>f s = increment of residual phases
The increment of residual phases A</>f can be viewed as a sum
of several components, i.e., nonlinear-motion phase increment
A(/)f def , atmospheric phase increment A<j>f m and decorrelation-
noise phase increment A <f>f .
The investigation by Ferretti et al. (2000, 2001) indicated that if
A<j>f s is small enough, say |A$ res | < n , both As and Av can
be indeed derived directly from the M wrapped differential
interferograms. In fact, the solution of As and Av can be
obtained by maximizing the following objective function.
7 =
1 M
— '¿T (cos A4T + j • sin A<f>f )
maximum
(3)
where y = arc’s model coherence (MC)
r
A<j>f = difference between measurement and fitted value
Although the above objective function is highly nonlinear and
the phase dataset is measured in a wrapped version, the two
unknowns As and Av can be determined by searching a pre
defined solution space to maximize the MC value. It should be
noted that the phase unwrapping can be avoided through the