Geology Reference
In-Depth Information
image is close to the surface color in nature. For example,
if the blue color is assigned to the first element (
S
hh
+
S
vv
),
the sea surface in the image will appear blue because it
triggers surface scattering (odd‐bounce) mechanism.
Another second order polarimetric description matrix
can be constructed from Pauli target vector. It is known
as coherency matrix (
T
):
use of this decomposition to decompose the coherency
matrix into three unit orthogonal eigenvectors related
to the three scattering mechanisms contributions to the
total received signal [
Cloude and Pottier
, 1997]. Both the
covariance and the coherence matrices should be aver-
aged over a window of neighboring pixels before deriving
the polarimetric parameters as described below. This
should further reduce the speckle in the data (make the
data multilook instead of single look).
TK
PP
*
(7.90)
1
(7.93)
[] [[[]
TVV
This is a 3 × 3 Hermitian positive semidefinite matrix,
which can be written as
where [
V
] is the matrix of the eigenvectors, and [
Λ
] is
the 3 × 3 diagonal matrix of the eigenvalues
λ
1
,
λ
2
and
λ
3
.
This equation can be rewritten in the form
2
*
2
2
*
2
S
2
Re
S SSS
2
Im
S SS
hh
hh
vv
vv
hh
hh
vv
vv
1
2
2
*
2
2
*
2
T
S
2
Im
S
SSS
2
Re
S SS
T
T
T
T
ee ee ee
(7.94)
hh
h
hh v
vv
hh
hh v
vv
111 222
3 33
*
*
*
2
SS SS
2
2
SS
*
2
SS
hv
hh
hv
vv
hv
h
h
hv v
where e
i
are the three orthogonal eigenvectors. The rela-
tive values of
λ
i
are indicators of the dominant scattering
mechanisms. For example, if the primary eigenvalue
λ
1
is
much larger than the secondary ones (
λ
2
and
λ
3
), it means
that one scattering mechanism is dominant. An eigenvec-
tor
e
i
can be written as
*
*
2
SS SS
2
hh
hv
vv
hv
*
*
2
SS SS
S
2
hh
hv
vv
hv
2
4
hv
(7.91)
T
i
j
j
(7.95)
e
i
e
cos
, sin os
e
,sin
sin
e
i
i
i
i
j i
i
i
i
i
The three diagonal elements of the coherence matrix are
the squares of the elements in the Pauli target vector
[equation (7.89)]. Therefore, they represent the three
scattering mechanisms as mentioned above.
The covariance and coherence matrices are linearly
related. The covariance matrix can be converted into coher-
ence matrix using the following linear transform. Note that
an average (i.e., multilook) of [
T
] and [
C
]should be used:
where
α
i
is an angle related to the scattering processes,
β
i
is the orientation angle of the target about the radar
line of sight, and
δ
i
and
γ
i
are phase angles related to the
target material. A few parameters derived from this
decomposition are commonly used in applications of
polarimetric data [
Cloude and Pottier
, 1997]. They include
entropy
H
, which takes values between 0 and 1; anisot-
ropy
A
, which takes values also between 0 and 1; and the
weighted‐average alpha‐angle
α
with values between 0
and 90°. The first two parameters are derived from the
eigenvalues and the third from the eigenvectors. Denoting
the probability obtained from the eigenvalues
λ
i
as
P
i
101
10 1
020
1
2
T
TNCN N
[]
where
(7.92)
3
(7.96)
P
i
/
Both covariance and coherency matrices contain the
same information about the scattering measurements
(amplitude, phase, and correlation) as well as mechanisms.
The trace of each matrix represents the total power of the
scattered wave. The coherency matrix is closely related to
the physical and geometrical properties of the imaged sur-
face and therefore allows better physical interpretation.
The study of the polarimetric scattering mechanism
from a given surface requires decomposition of the
covariance or the coherency matrix into a combination
of second order descriptors of the three mechanisms men-
tioned above; odd-bounce, double-bounce and random
scattering. A commonly-used approach is the well-known
Eigen decomposition. The following text illustrates the
i
j
j
1
The definitions of
H
,
A
, and
α
can be written as
HPPP PP P
1
log
log
log
(7.97)
3
1
2 32 3
3
3
(7.98)
A
(
)/(
)
2
3
2
3
PP P
11 22 33
(7.99)
where
α
i
is the arccos of the first element of the eigenvec-
tor
e
i
as indicated in equation (7.95). Equation (7.99) rep-
resents a weighted average of the
α
i
values in the three
eigenvectors. The above three parameters are related to
the number or the type of the scattering mechanisms.
