The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
102 
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