IEEE Geoscience and Remote Sensing Magazine - March 2013 - 15
The elements of [S] are the four complex scattering
amplitudes S IJ = | S IJ | exp (i{ IJ) where the subscripts horizontal (H) or vertical (V) indicate associated received and
transmitted polarization. The factor exp (- ikr)/r, where
k = 2r/m is the wave number, expresses the phase shift and
attenuation for a spherical wave of a radius that equals the
distance between the scatterer and the radar. In SAR interferometry this absolute phase term is used to estimate the
three-dimensional location of the scatterer. In SAR polarimetry the absolute phase is in most cases neglected and
only the relative phases exp (i ({ IJ - { MN)) between the
matrix elements are considered.
The scattering matrix can be measured by transmitting in two orthogonal polarizations on a pulse-to-pulse
basis and receiving the scattered waves in two orthogonal polarizations (commonly the same basis as used for
transmission). Most polarimetric systems operate in the
linear H-V basis: By transmitting a H polarized wave (i.e.,
Ev t /| Ev t| = [1, 0] T where the superscript T indicates the transpose operation) and receiving in H (i.e., Ev r /| Ev r| = [1, 0] T)
and V (i.e., Ev r /| Ev r| = [0, 1] T) polarization the S HH and S VH
elements are measured. The two remaining coefficients
S VH and S VV are measured in a second step by transmitting
a V polarized wave (i.e., Ev t /|Ev t| = [0, 1] T) and receiving in
H and V. However, [S] can be measured also by using any
other basis of orthogonal polarizations, as for example
left and right circular polarizations as well as by using different bases for transmit and receive (e.g., left-right circular polarization on transmit and linear H-V polarization
on receive).
In this sense it is important to note that the information content of the [S] matrix is independent of the
basis used for its measurement, but its representation
of course depends on the chosen reference frame (i.e.,
the bases used for the measurement). Accordingly, once
the full scattering matrix is measured, any arbitrary
complex scattering amplitude can be reconstructed as a linear combination of the elements of the measured scattering matrix. This is the great advantage of fully polarimetric
radar systems over conventional single- or dual-polarized
configurations.
In monostatic configurations, where receiver and transmitter are co-located, the [S] matrix becomes symmetric,
i.e., S HV = S VH = S XX, for all reciprocal scattering media.
In this case, and ignoring the absolute phase, the number
of independent parameters in [S] is reduced to five: Three
amplitudes and two relative phases. In the bistatic case,
where receiver and transmitter are spatially separated,
S HV ! S VH and [S] contains seven independent parameters:
Four amplitudes and three relative phases. While bistatic
polarimetry is widely used in optics, in SAR remote sensing the majority of the polarimetric systems is operated
in a monostatic mode. Very few experiments have been
successful in collecting fully
polarimetric bistatic data
providing some very first
insight in the physical interSAR polarimetry is
pretation of bistatic SAR
essential for extracting
polarimetry [37]. In the folgeo/bio-physical
lowing, all formulations will
parameters for land,
refer to the monostatic case.
snow and ice, ocean
The set of observables
and urban applications.
derived from the scattering
matrix and used in remote
sensing applications is summarized in Table 3.
The scattering matrix is able to completely describe
deterministic (point-like) scatterers that change the
polarization of the incident wave, but fails to describe
the depolarization of the incident wave as it happens in
the case of distributed scatterers. Distributed scatterers
are considered to be composed of a large number of randomly distributed deterministic scatterers (see(8)). The
measured scattering matrix consists then of the coherent
superposition of the individual scattering matrices of all
scattering centers within the resolution cell. In order to
fully describe the polarimetric scattering behavior of
distributed scatterers a second-order statistical formalism is required.
The most common formalism to fully characterize distributed scatterers is the 3 # 3 coherency [T] (or
covariance [C]) matrix defined by the outer product of a
three-dimensional scattering vector in the Pauli kv P (or lexicographic kv L) formulation:
1
6S HH + S VV , S HH - S VV , 2S XX@T , (11)
kv P =
2
and the coherency matrix is given by (12), shown at the
bottom of the page, where the superscript + indicates the
conjugate transpose operation. While the lexicographic formulation is more appropriate for system related considerations, the Pauli formulation is of advantage when it comes
to the interpretation of scattering processes [15].
Both matrices are by definition hermitian positive semi
definite, have the same real non-negative eigenvalues but
different orthonormal eigenvectors, and are in general of
full rank 3. With respect to the physical information content
the rank of [T] (or [C]) expresses the number of independent
scattering contributions in which [T] (or [C]) can be decomposed. In the most general case both matrices contain nine
R
V
1 S HH + S VV 2 2
1 ^S HH + S VV h^S HH - S VV h* 2 2 1 ^S HH + S VV h S*XX 2W
S
1S ^
2
+
*
*
v
v
[T] = 1 k P $ k P 2 = 2 1 S HH - S VV h ^S HH + S VV h 2
1 S HH - S VV 2
2 1 ^S HH - S VV h S XX 2W (12)
W
S
2 1 S XX ^S HH + S VV h* 2
2 1 S XX ^S HH - S VV h* 2
4 1 S XX 2 2
T
X
march 2013
ieee Geoscience and remote sensing magazine
15
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