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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
  
Suppose that the sample {x(0),x(1),--, x(N — bj is divided 
into L segments of M elements each (thus, N = LM ) and 
that the side for bifrequency smoothing is of length M,, the 
estimated bispectrum B(a, o) is (Hinich, 1982) 
I 
B(w,®,) = 3b" mn) 4) 
= 
1 mM, —1 nM, —1 
b? (m,n) 2 5 s DPF hc) 3 
M, k,=(m=l)M, k,=(n=1)M, 
  
Fk, ky) eur n, )X "(a ) X (a), ) and 
M-1 
X (0, )= NR y? (exp(- j0.1)- 
1=0 
The statistics for non-Gaussianity test is (Hinich and Wilson, 
1990) 
TCH -2N «N Y |B(o,. a.) - B(o,. e) "[Scop$ca, )$(o, * &) 
CELA) 
  
(5) 
where A, — LM,/N = M,/M , and S(q) is the estimate of 
power spectra. If A, = VAN , TCH is an approximate Chi- 
square distribution with a degree of freedom of 2P , where 
PUMA y^. 
4.0 Non-Gaussianity of Atmospheric Signals 
The null hypothesis for the test is: the atmospheric signatures in 
a SAR interferogram are Gaussian. Under the null hypothesis, 
B(w,,w,) =0 for all bifrequency pairs, and thus TCH is 
approximately 72.00). The a -level test is to reject null 
hypothesis if TCH > T,, , where a = Pr 2T. ; 
To satisfy the relationship of A, = TEES and to conduct 
more detailed tests, we divide each interferogram into a number 
of pieces along the azimuth direction and in each piece a test is 
carried out. The number of tests, i.e., the number of pieces, and 
the test results for each of the study regions are listed in Table 
2. Also shown are parameters used in bispectrum estimations 
for each study region. 
  
  
  
  
  
  
  
  
  
  
  
M L M, |Gauss. Nan: 
Gauss. 
Shanghai 50 1250 | 25 7 0 50 
Hong Kong 25 300 | 12 5 0 25 
New South = 
2 2 7 
Wales 25 500 | 20 5 0 25 
  
  
Interfero. | Number Parameters used in 
N : ; : Test Results 
of tests | Bispectrum estimation 
  
  
  
  
  
  
Table 2. Details of Hinich Non-Gaussianity tests. 
The Hinich non-Gaussianity tests show that for all the study 
regions the differential atmospheric signals show significant 
non-Gaussianity. This contradicts to assumptions made by 
some authors that the atmospheric signals are Gaussian 
(Ferretti, et al., 1999 and Yue, et al., 2002). The atmospheric 
effect on InSAR measurements and parameter estimation are, 
however, subject to further study. 
5 Spectral Analysis of Atmospheric Signals 
The mean differential atmospheric delays in each of the study 
areas are calculated and removed from the unwrapped 
interferograms. A 2D Fast Fourier Transform (FFT) is 
performed next for each of the areas and the results are squared 
to obtain the power spectra. The ID rotationally averaged 
power spectra thus obtained are given in Figures 7, 8 and 9. 
  
  
1 0° =. x 
\ 
4 A. 
5 10 lA, 
8 CAI 
De > > 
510 N. 
3 ™S 
n > 
10° S 
\ 
x 
= 
10° dad io xam 
10? 10° 10° 10! 10° 
Wavenumber (cycles/km) 
Figure 7. Power spectrum of differential atmospheric signals in 
the Shanghai study region. The dashed lines follow a slope of 
vis. 
  
  
  
10° s ' c | 
No | 
A, 
4 s 
10 uum 
5 N 
5 
Q 
$ S 
510° | 
T 
o 
oO 
x A 
10° N 
10? 107 10° 10’ 10° 
Wavenumber (cycles/km)