N
C = 7 2
Sd s Ya; Q)
[=1
where d,, is the original image rotated clockwise by an angle
O0 and N isthe / th column of the rotated image.
3.2 Anisotropy of Atmospheric signal
To systematically examine the anisotropy of atmospheric
signature, we calculate the Radon transform of each differential
interferogram. In the analysis, we use a normalized version of
the Radon transform (Jónsson, 2002)
Rd, } = R, 1d, }/N (3)
It gives the average atmospheric effect values along lines
perpendicular to profile direction. The Radon transform of
atmospheric signatures in the three interferograms are shown in
Figures 4, 5 and 6.
0 50 100 150
Degree
Figure 4. Radon transform of atmospheric signatures over
Shanghai study region (unit: mm)
! - ë 2 fr
Degree
Figure 5. Radon transform of atmospheric signatures over Hong
Kong study region (unit: mm)
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
100 159
Degree
(5)
Figure 6. Radon transform of atmospheric signatures over New
South Wales study region (unit: mm)
The Radon transforms of the three study regions show variable
anisotropy. In the first transform, while the whole image shows
a nearly opposite symmetry along the center of the profile, the
extremely strong trends are visible in the two sides of the
profile when the angle varies from 0 to 90 degrees. This implies
that small areas of positive and negative atmospheric signals
locate in the southwest and northeast corner of the original
differential fields, respectively (Figure 1). The second
transform (Figure 5) also shows significant anisotropy with a
strong trend at about [35° The third transform (Figure 6) shows
very interesting patterns. From 0° to 90° the values change
from positive extremes to negative extremes slowly while from
91° to 180° thet change oppositely. This suggests that in all the
four corners there are positive or negative phase concentrations.
The original atmospheric fields (Figure 3) confirm that in the
northeast and southeast corners there is a small amount of
negative phases concentration while there are positive phase
concentrations in the northwest and southwest corners.
Of the three transforms, the second shows much complicated
variations. This may be due to the fact that the Shanghai and
New South Wales regions are relatively flat, while there are
several mountains in the Hong Kong region, with the highest
elevation being about 1 km. The complication in mountainous
region is due mainly to: (1) the vertical stratification effect or
“static” effect of the troposphere (Delacourt et al., 1998;
Willianms et al, 1998) related to significant topographic
variations, and (2) effect of the mountains on the local weather
conditions as the weather conditions can be quite different in
the two sides of a mountain.
4 Non-Gaussianity of Atmospheric Signals
4.1 Statistics for Hinich Non-Gaussianity Test
If a signal is purely Gaussian, its bispectrum will be zeros.
Therefore the Gaussianity of the atmospheric signals can be
checked by examining the deviation of its bispectrum from
zero. To this end, a consistent estimator of bispectrum from
finite samples is needed. Several such estimators have been
proposed in the past decades: (1) smoothing the sample
bispectrum in the bifrequency domain; (2) dividing the sample
into a number of segments and doing bifrequency smoothing in
each segment and then averaging the piecewise smoothed
bispectra; and (3) parametric method, etc. (Hinich, 1982; Lii
and Rosenblatt, 1982; Nikias and Raghuveer, 1987). We will
adopt the second method here.
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