Geology Reference
In-Depth Information
polrimetric data representations can be found in a number
of references including
Zebker and van Zyl
[1991],
Cloude
and Pottier
[1997],
Cloude
[1997],
Freeman and Durden
[1998],
Touzi et al
. [2004], and
Touzi
[2007].
The basic expression of the the scattering matrix [
S
]
describes the transformation of the electric field of the
incident wave to the scattered wave:
*
SS
r
hh
vv
(7.86)
hh vv
*
*
SS SS
hh
hh
vv
vv
where Re and Im are the real and imaginary components
of the complex number, ⟨ ⟩ denotes the average over a
number of neighboring pixels, * denotes complex conju-
gate, and | | is modulus of complex number. The co‐
polarized phase difference is an indicator of the dominant
scattering mechanism as described above.
With the reciprocity assumption of the cross‐polariza-
tion elements (
S
hv
=
S
vh
), two commonly used scattering
vector expressions have been derived from the scattering
matrix. The rationale behind their development is pre-
sented in
Hellman
[1999]. The first vector is known as the
scattering vector:
s
i
SS
SS
i
E
E
E
E
E
E
h
h
hh
hv
h
S
(7.79)
s
i
i
v
v
vh
vv
v
where
E
is the electric field intensity, superscripts
i
and
s
denote the incident signal (transmitted) and the scat-
tered signal (i.e., received), respectively, and subscripts
h
and
v
denote the vertical and horizontal polarizations,
respectively. The effective backscattering coefficients
0
rt
in the transmitted polarization
t
and received polariza-
tion
r
are related to the elements of the scattering matrix
by the following equations:
T
KSSS
C
,
2
,
(7.87)
hh
hv
vv
A well‐known matrix that can be derived from this vector
is called a covariance matrix C. It is constructed in the
power domain by multiplying the vector by its conjugate
transpose. This is an example of a second order matrix
from which parameters can be derived based on certain
decompositions to characterize the scattering mecha-
nisms of the imaged scene.
*
/
0
4
SS A
(7.80)
rt
tr
tr
where
A
is the area illuminated by the incident radar pulse,
⟨⋯⟩ denotes the ensemble averages of a number of pixels,
and * is the complex conjugate. In the above equation the
elements of the scattering matrix have to be calibrated.
Conversion of the scattering elements to backscatter coef-
ficient is available in commercial software packages.
A few useful parameters derived directly from the
elements of the scattering matrix are commonly used:
(these represent the second set referred to above) the
total power (SPAN), the co‐polarization and cross‐polar-
ization ratios (
R
hh
/
vv
and
R
hh
/
hv
, respectively), the depolari-
zation (cross‐polarization) correlation coefficient
R
depol
,
the co‐polarization phase difference
ϕ
hh
−
vv
, and co‐
polarization correlation coefficient
r
hh
−
vv
:
2
*
*
S
2
S SSS
hh
hh
hv
hh
vv
2
*
*
*
CKK
T
2
SS S
2
2
S S
(7.88)
CC
hv
hh
hv
hv
vv
2
*
*
SS
2
SS S
vv
vv
hh
hv
vv
where superscripts * and T denote the conjugate and
transpose, respectively.
The other vector expression is also derived from the
scattering matrix using combinations of its elements
(known as the Pauli spin elements). This vector is known
as the Pauli target vector;
*
*
*
SPAN
SS SS
2
SS
(7.81)
hh
hh
vv
vv
hv
hv
1
2
*
*
SS
T
K
(
S SSSS
),
,
2
(7.89)
R
hh
hh
P
hh
vv
hh
vv
hv
(7.82)
hh vv
/
SS
vv
vv
Roughly speaking, the square of the first, second, and
third elements represent contributions from the odd‐bounce
and double‐bounce scattering mechanisms, respectively.
In general, cross‐polarization return is associated with
random scattering, although under some special geomet-
rical arrangement it can be caused by double‐bouncing
scattering mechanisms.
A color composite image, known as Pauli image, can be
formed from the square of the three elements of Pauli vec-
tor. Assignment of the elements to the three colors (RGB)
is implemented such that the color distribution in the
*
*
SS
R
hh
hh
(7.83)
hh hv
/
SS
hv
hv
*
SS
R
vh
hv
(7.84)
depol
*
*
SS SS
vv
vv
hh
hh
*
*
Im
SS
1
tan
hh
vv
(7.85)
hh vv
Re
SS
hh
vv
