The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
4 n
5 €X (*,r) = —\_d(t s ,x,r)-d{t Mt x,r)]
4n £4 , x
~ ~~r z-i v *,*+i (^*+1 — K)
^ k=M
(3)
For a given pixel, let V be a vector (of size (S'-l)xl ) of
successive velocities (i.e. L r =[v 0] v l2 L v s _ 2s _,]), Z a
parameter of DEM errors, and R a vector of interferogram
range changes in the line of sight (of size N x 1). Equation (2)
can be generalized into a matrix equation for the entire set of
interferograms:
T C
JVx(S-l) Nxt
B,
B,
/? =
rsin# rsin#
2i
(4)
4n
4 n
8$
4 n
where the N x (5 -1) matrix T references time intervals of
each interferogram. If all the acquisitions are well-connected
(i.e. they belong to a single sub-network), we should
have N>S , and A =
T
Vx(S-l)
is an S-rank matrix.
Therefore, Equation (4) is a well-determined (N = S ) or an
over-determined (N > S) system, and its solution can be easily
obtained in a least squares sense.
This InSAR time series with water vapour correction (InSAR
TS + PWV) technique allows us to map surface deformation as
it evolves in time together with a mean velocity field, with two
key features: (1) With water vapour correction, no a priori
deformation model is required in InSAR time series analysis; (2)
The capability to provide spatially dense deformation maps is
preserved by only using SAR pairs with small baselines. Note
that DEM errors are taken into account when using
interferograms with a relatively long perpendicular baseline (e.g.
>100 m) to increase the amount of data for the time series
analysis.
4.2 InSAR time series analyses for postseismic motions
after the 2003 bam earthquake
Figure 3 shows 25 ENVISAT images (indicated by red triangles)
collected under cloud free conditions in the three years since the
2003 M w 6.6 Bam (Iran) earthquake, from which 109
interferograms with a perpendicular baseline shorter than
400 m were produced using the JPL/Caltech ROI PAC
software (version 2.3) (Rosen et al., 2004). A subset around
Bam (over a 72 km by 72 km area) was unwrapped with the
SNAPHU program (Chen and Zebker, 2000).
Two different time series analyses were performed for the 109
unwrapped interferograms: (1) InSAR time series without
MERIS water vapour correction. To separate water vapour
effects from deformation signals in InSAR time series analysis,
an a priori temporal deformation model is usually required for
most existing InSAR time series approaches (i.e. permanent
scatterers and small baseline InSAR) (Raucoules et al., 2007);
otherwise atmospheric signals can be estimated by filtering only
(Hooper et al., 2007). In this study, no assumption was made for
this purpose. (2) InSAR time series with MERIS water vapour
correction, as demonstrated in Section 4.1.
Year
Figure 3. ENVISAT images from descending track 120
(denoted by red triangles with a label of date in format:
YYYYMMDD). Black dashed lines connecting triangles
represent radar interferograms with a perpendicular baseline
shorter than 400 m.
Figures 4(a) and 4(b) show InSAR time series results without
and with MERIS water vapour correction respectively on the
date of 20041222. It is clear that some phase variations in the
Northwest and Northeast comers (indicated by black ovals) in
Figure 4(a) were removed and the real geophysical signals
(indicated by red open rectangles) were enhanced after
correction (Figures 4(b)).
Figure 4. InSAR time series results: the LOS range changes on
date: 20041222 (relative to date: 20040211). (a) without
MERIS water vapour correction; (b) with MERIS water vapour
correction. Note: (1) Thick black lines indicate fault ruptures
mapped in the field by Talebian et al. (Talebian et al., 2004); (2)
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