IEEE Geoscience and Remote Sensing Magazine - March 2013 - 17
mechanisms used are surface, dihedral and volume (or
multiple) scattering. Scattering decompositions are widely
applied for interpretation, classification, segmentation
and/or as a pre-processing step for scattering parameter
inversion. In general, the decompositions of second-order
scattering matrices (i.e., [T] or [C]) are rendered into two
classes: Eigenvector and eigenvalue based decompositions and
model-based decompositions.
An extended review of scattering decompositions can
be found in [15]. The basic concepts of the eigenvector
and model-based decompositions are described in the
next section.
1) Eigenvector and Eigenvalue based Decomposition: Since
the coherency matrix [T] is hermitian positive semi-definite,
it can always be diagonalized by a unitary similarity transformation of the form [15], [35]
[T] = [U] [K][U] -1, (15)
where
Rm 0 0 V
e 11 e 12 e 13
S 1
W
K = S 0 m 2 0 W, [U] = >e 21 e 22 e 23H . (16)
S 0 0 m 3W
e 31 e 32 e 33
X
T
[K] is the diagonal eigenvalue matrix with elements
the real nonnegative eigenvalues, 0 # m 3 # m 2 # m 1 and
[U] = 6ve 1 ve 2 ev3@ is the unitary eigenvector matrix with columns the corresponding orthonormal eigenvectors ev1, ve 2
and ve 3 . Each of them can be parametrized in terms of five
angles [15], [35]
R
V
iW 1i
S cos a i e
W
ve i = Ssin a i cos b i e iW 2iW . (17)
S
W
sin a i sin b i e iW 3i
T
X
that-as it will be discussed later-characterize the associated scattering mechanism.
The idea of the eigenvector approach is to use the diagonalization of the coherency matrix [T] of a distributed scatterer, which is in general of rank 3, as a decomposition into
the non-coherent sum of three independent (i.e., orthogonal) coherency matrices [Ti]
[T] = [U] [K][U] -1 = [T1] + [T2] + [T3]
= m 1 $ (ve 1 $ ve 1+) + m 2 $ (ve 2 $ve 2+) + m 3 $ (ve 3 $ ve 3+).
(18)
The [Ti] matrices are of rank 1 implying a deterministic scattering contribution, characterized by a single
scattering matrix. There are two important statistical
parameters arising directly from the eigenvalues of the
coherency matrix. The first one is the polarimetric scattering entropy H defined by the logarithmic sum of the
eigenvalues of [T]
3
H = - / p i log 3 p i, p i =
i=1
mi
3
/ mj
, (19)
j=1
march 2013
ieee Geoscience and remote sensing magazine
Table 4. POLARIMETRIC RADAR OBSERVABLES
AND PARAMETERS DERIVED FROM THE ELEMENTS
OF THE COHERENCY MATRIX [T ].
3 # 3 Backscattering Matrix
Application Examples
Correlation coefficient
Crop phenology classification
HV basis (c HHVV )
Dielectric constant estimation
of bare soils/surfaces
Correlation coefficient
LR basis (c LLRR)
Surface roughness estimation
(bare surfaces)
where p i expresses the appearance probability for each
contribution. The entropy ranges from 0 to 1 and can be
interpreted as a measure of the randomness of the scattering process, or in other words, it expresses the number of effective scattering processes in which [T] can be
decomposed by means of (18). An entropy of 0 indicates
a rank 1 [T] matrix with only one nonzero eigenvalue,
i.e., m 2 = m 3 = 0, implying a non-depolarizing scattering
process described by a single scattering matrix. At the
other extreme an entropy of 1 indicates the presence of
three equal nonzero eigenvalues, i.e., m 1 = m 2 = m 3 and
characterizes a random noise scattering process, which
depolarizes completely the incidence wave regardless of
its polarization. However, most distributed natural scatterers lie in between these two extreme cases, having
intermediate entropy values.
The second parameter is the polarimetric scattering
anisotropy defined as the normalized difference of the second and third eigenvalues
A=
m2 - m3
. (20)
m2 + m3
A ranges also from 0 to 1 and expresses the relation
between the secondary scattering processes. For a deterministic scatterer with entropy H = 0 the anisotropy is
defined as zero, A = 0. The same is the case for a completely
depolarizing scatterer with H = 1. For scatterers characterized by intermediate entropy values, a high anisotropy
indicates the presence of only one strong secondary scattering process. In this sense, the anisotropy provides complementary information to the entropy and facilitates the
interpretation of the scatterer. The great advantage of these
two parameters arises from the invariance of the eigenvalue
problem under unitary transformations: The same scatterer
leads to the same eigenvalues and consequently to the same
entropy and anisotropy values independent of the basis
used to measure the corresponding scattering matrix.
The physical and geometrical interpretation of the scattering mechanisms represented by [T1], [T2] and [T3] is given
by the corresponding eigenvectors. For each eigenvector the
scattering alpha angle a i = arccos (|e 1i|) ranges between 0
and 90 degrees and is associated to the type of corresponding scattering mechanism: 0c # a i # 30c corresponds in
general to surface scattering processes, 40c # a i # 50c to
dipol-like scattering behavior and finally 60c # a i # 90c
17
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