IEEE Geoscience and Remote Sensing Magazine - March 2013 - 24

cm
−15

cm/month
0

-2.5

(a)

m
0.0

(b)

40
(c)

FIGURE 15. (a) Estimated subsidence over Mexico City obtained with two TerraSAR-X images acquired with a 6-month difference (overlay
of reflectivity and phase). Low coherence areas have been masked out. (b) Mean deformation velocity estimated over Mexico City using the
PS technique. (c) Zoom over the city of the refined DEM retrieved as an additional product to the deformation velocity, where the individual
buildings can be observed. The size of the PSs has been enlarged for visualization purposes. The scene size is approximately 8 km # 8 km.
Radar illumination from the right.

vapor content in the troposphere. The delay is in the order
of 2-4 meters, but its gradient at a local scale can be in the
order of up to 1cm/km or more [54], [86], hence limiting
the accuracy of the conventional differential SAR interferometry approach, and being only of interest for cases
where the displacements are
larger than this value. At
lower frequency bands like
Differential SAR
L- or P-band, the total elecinterferometry can
tron content (TEC) inside
the ionosphere results in a
detect displacements
further non-negligible path
of the Earth surface at a
delay, which can potensubwavelength scale.
tially vary within the synthetic aperture time, and
thus introduce undesired
defocusing and impulse response shifts in the azimuth
dimension [87], [88].
Figure 15(a) shows the subsidence over Mexico City
estimated with two TerraSAR-X images acquired with
a 6-month difference, where the SRTM DEM [89] was
used to remove the topographic phase. The maximum
displacement is about 15 cm in some city areas, corresponding roughly to a factor 5 of the wavelength. The
areas with a low coherence have been masked out for
visualization purposes. The subsidence is due to ground
water extraction and it is a well-known problem in Mexico City [90].
The exploitation of image time series is the solution
to the limitations of conventional differential SAR interferometry. By using a large stack of images acquired at
24

different time instants, the aforementioned contributions can be separated. The signal model of a single point
in the image stack for a given interferometric pair can be
written as

{ i,j = { topo + { disp + { atm + { n, (35)

where { topo represents the residual topographic component
after removing the external DEM, { atm the atmospheric
component, { n the phase noise, and i and j represent the
image indexes within the stack. But the main challenge is,
similar as with InSAR, the fact that the phases are wrapped
modulo 2r, which makes the problem non-linear.
Several approaches have been developed in the literature in order to extract the different contributions of
interest through different advanced filtering schemes.
The first step in these approaches is the selection of the
image pixels with enough quality to be used in the estimation process, being the most commonly approaches
the ones based on the amplitude dispersion, i.e., the socalled permanent scatterers (PS) technique [17], [69], or
on the interferometric coherence [91]-[93]. A PS appears
in the image as a pixel showing a stable amplitude behavior in time and usually corresponds to point-like targets
such as buildings or rocks, while distributed scatterers
are discarded. On the other hand, the selection using
the coherence further includes distributed scatterers that
remain coherent in time, e.g., surfaces, hence increasing the potential number of candidate pixels. Nevertheless, the coherence estimation implies a resolution loss
in opposition to the PS technique, which works at full
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - March 2013