IEEE Geoscience and Remote Sensing Magazine - March 2013 - 18
dihedral component is described by two parameters: The
scattering amplitude
Table 5. POSSIBLE RADAR OBSERVABLES ESTIMATED
FROM THE SECOND-ORDER STATISTICAL PARAMETERS AND
THEIR RELATION TO THE ENVIRONMENTAL PARAMETERS/
APPLICATION EXAMPLES [15].
Second-order
Statistical Parameter
Application Examples
Entropy (H)
Segmentation and classification
Anisotropy (A)
Segmentation and classification
Surface roughness estimation
(bare surfaces)
Alpha angle (a)
Segmentation and classification
Soil moisture estimation
(bare surfaces)
Beta angle (b)
Surface slope estimation
(azimuth direction)
and the ratio
indicates dihedral (or helix type) scattering mechanisms.
The beta scattering angle, b i = arctan (|e 3i|/|e 2i|), ranges
between - r and r and describes the line-of-sight rotation
(given by b/2) of the scatterer represented by [Ti] . The existence of three such a i and b i scattering angles motivates a
statistical interpretation in terms of mean scattering angles,
given by
a = p1 a1 + p2 a2 + p3 a3
b = p1 b1 + p2 b2 + p3 b3 .
(21)
The entropy/(mean) alpha (H/a) space has been used
primarily for the initialization of unsupervised classifications [15]. In Table 5 second-order statistical parameters
and their corresponding applications are listed.
2) Model-Based Decomposition: Model-based decompositions have been proposed and applied, in order to
decompose the individual scattering contribution into
a set of predefined elementary scattering contributions
[45]-[48].
One of the first model-based decompositions used to fit
and to interpret SAR data is the Three-Component Decomposition proposed by A. Freeman and S.L. Durden in 1998
[45]. The model decomposes the measured coherency
matrix [T] into a Bragg scattering component (first matrix),
a (smooth) dihedral reflection (second matrix), and into a
volume of randomly oriented dipole scattering elements
(third matrix)
R 2
V
R
V
*
2 0 0
S a a 0W
S1 b 2 0W
fv
*
[T] = fs Sb b 0W + fd S a 1 0W + 4 >0 1 0H. (22)
S 0 0 0W
S0 0 0W
0 0 1
44 2 44 3
1 4T44 2 4 44 3X 1 4T44 2 4 44 3X 1Volume
scattering
Surface scattering
Dihedral scattering
The surface contribution is described by two parameters: The real ratio b = (R s + R p) / (R s - R p) and the backscattering contribution fs = R s - R p 2 where R s and R p are the
Bragg scattering coefficients perpendicular and parallel to
the incident plane and depend on the dielectric constant
of the surface and the incidence angle [15], [35]. Also the
18
1
fd = 2 R sh R th + R sv R tv exp (iz) 2 (23)
a=
R sh R th - R sv R tv exp (iz)
. (24)
R sh R th + R sv R tv exp (iz)
The horizontal and vertical Fresnel coefficients for the
soil R sh and R sv and for the trunk R th and R tv depend on the
soil (trunk) dielectric constants and the corresponding incidence angle [15], [35]. The volume component fv is often
approximated by a cloud of ellipsoids [35]. The phase component z accounts for any differential propagation effects
in the volume. Hence, the scattering properties of the layer
(neglecting multiple scattering interactions) are controlled
by three parameters: The electromagnetic density of the
volume that affects the scattered power, the shape and
the orientation distribution of the volume particles. The
exact volume formulation is described in [35], [49]. The
scattering power of the surface Ps, dihedral Pd and volume
Pv component are given by the trace of the corresponding
coherency matrices
Ps = fs $ (1 + b 2), Pd = fd $ (1 + a 2), Pv = fv . (25)
The total power is obtained as
Ptot = Ps + Pd + Pv
= fs $ (1 + b 2) + fd $ (1 + a 2) + fv .
(26)
Eq. (22) leads to an under-determined inversion problem of 5 (3 real and one complex) observables for six (real)
unknown parameters. For this, the non-dominant of the
two components, either surface or dihedral, is set to a con*
stant value according to the sign of Re {1 S HH S VV
2} after
removal of the volume contribution. Surface scattering is
dominant if this term is positive otherwise dihedral scattering is dominant.
The described canonical models have been widely
used as elements of different decomposition approaches.
Beyond this, several modifications have been proposed
in order to improve the interpretation of real scattering
scenarios by means of more realistic model components
[49], [50]. Besides the interpretation of polarimetric data
over vegetated terrain, model-based decompositions
have been used to initialize and support polarimetric
segmentation and classification approaches [15]. Fig. 9
shows on the top the Freeman and Durden decomposition applied on quad-polarimetric L-band data acquired
by DLR's E-SAR system over the Oberpfaffenhofen test
site. Red, green and blue correspond to fd, fv and fs . In
a second step, the dominant of the three contributions
has been used to classify the scene into three classes:
Dihedral, volume and surface like scatterers. These three
classes are then used to initialize a statistical Wishart
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