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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