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Non-Deterministic Scatterers

In the previous section we discussed deterministic scatterers which are described completely by a single constant scattering matrix [S] or scattering vector $ \vec{k}$. For remote sensing applications the assumption of pure deterministic scatterers is not valid. Since, for SAR applications, the resolution cell is bigger than the wavelength, e.g. natural terrain surfaces contains many spatially distributed deterministic scattering centers, each of these centers is completely represented by an individual $ [S]_i$-matrix. Therefore the measured [S]-matrix for one resolution cell, consists of the coherent superposition of the individual $ [S]_i$-matrices of all scattering centers located within the resolution cell. To deal with statistical scattering effects and the analysis of partial scatterers the concept of a scatter covariance matrix or coherence matrix are introduced [Boerner81], [Cloude92], [Cloude99a], [Tragl90].

By forming the outer product $ \left< \vec{k} \vec{k}^\dagger \right>$ of the scattering vectors $ \vec{k}$ with its conjugate transposed vector $ \vec{k}^\dagger$ we get a $ 4 \times 4$ matrix. If we use the conventional scattering vector $ \vec{k}_B$ we get the so called polarimetric covariance matrix [Boerner81],

$\displaystyle [C]_{4 \times 4} = \left< \vec{k}_B \vec{k}_B^\dagger \right> = \...
...S_{vv} S_{vh}^*\right> & \left< \vert S_{vv}\vert^2 \right> \end{array} \right]$ (3.21)

where $ \left< ... \right> $ denotes a spatial ensemble averaging assuming homogeneity of the random scattering medium. Analogously, the so called polarimetric coherency matrix $ [T]_{4 \times 4}$ [Cloude92] is formed as

$\displaystyle [T]_{4 \times 4} = \left< \vec{k}_P \vec{k}_P^\dagger \right>$ (3.22)

The polarimetric covariance/coherence matrices are of full rank (rank 4 in the four-dimensional representation). Without the ensemble averaging both matrices have rank 1 and characterize a deterministic scattering process [Papathanassiou99]. Both matrices are by definition hermitian semi-definite matrices and have the same real non-negative eigenvalues. but different eigenvectors.

Since the interpretation of the physical scattering mechanisms is easier using the polarimetric coherency matrix, we will not use the polarimetric covariance matrix further in this treatment.

However, as discussed in section 4.10, for the backscattering case the reciprocity theorem yields $ S_{hv} = S_{vh} = S_{x}$ and hence, the $ 4^{th}$ Pauli matrix (iso-cross-polarizers) can not occur. Therefore the scattering vector from (3.15) reduces to the 3-dimensional case, without loss of information.

$\displaystyle \vec{k}_{P3} = \frac{1}{\sqrt{2}}\left[S_{hh}+S_{vv},S_{hh}-S_{vv}, 2 S_{x}\right]^{T}$ (3.23)

Constrained to the 3-dimensional subspace we can now use the new scattering vector $ \vec{k}_{P3}$ to form a $ 3 \times 3$ coherency matrix [T] as follows
$\displaystyle [T]_{3 \times 3} = \left< \vec{k}_{P3} \vec{k}_{P3}^{\dagger} \ri...
...array}{c} A=S_{hh }+S_{vv} B= S_{hh } - S_{vv} C=2S_{x}
\end{array} \right.$     (3.24)

where $ \left<{XY}\right>$ denotes a spatial ensemble averaging, assuming homogeneity of the random medium. $ {[T]}_{3 \times 3}$ contains the complete information on variance and correlation between all complex elements of [S].
next up previous contents
Next: Target Decomposition Up: Polarimetric Description of Scatterers Previous: The Scattering Vector   Contents