The Scattering Vector

As shown in the previous section, the complex [S]-matrix describes the scattering process and contains therefore the information about the target. Instead of the matrix notation we may use a four-element-complex vector $\vec{k}$ which contains the complete information of the [S]-matrix.

$$\left[ S \right] =\left[\begin{array}{cc} S_{HH} & S_{HV} S_{VH... ...rrow \vec{k} = \frac{1}{2} Trace([S]\Psi) = \left[ k_0, k_1, k_2, k_3 \right]^T$$ (3.11)

where $Trace([S])$ is the sum of the diagonal elements of [S] and $\Psi$ is a complete set of $2 \times 2$ complex basis matrices under a hermitian inner product [Cloude86b]. In principal, any complete orthonormal basis set of four $2 \times 2$ matrices can be used, as long as the Euclidian norm of the vector $\vec{k}$ is invariant. Two bases are widely referred in the literature. The first one, following straight forward from the lexicographic expansion of [S] is commonly referred as Borgeaud basis [Borgeaud89]

$$\Psi_B = \left\{ 2 \left[\begin{array}{cc} 1 & 0 0 & 0\end{arra... ...y} \right], 2 \left[\begin{array}{cc} 0 & 0 0 & 1\end{array} \right] \right\}$$ (3.12)

with the corresponding vector $\vec{k}_B$, containing the complex elements of $[S]_{hv}$

$$\vec{k}_B = \left[S_{hh},S_{hv},S_{vh},S_{vv}\right]^T$$ (3.13)

This vector is directly related to the system measurable. A basis which is more related to the physics of wave scattering, commonly known as the Pauli basis, is formed by the Pauli spin matrices [Cloude86b], [Cloude96].

$$\Psi_P = \left\{ \sqrt{2} \left[\begin{array}{cc} 1 & 0 0 & 1\e... ...t], \sqrt{2} \left[\begin{array}{cc} 0 & -i i & 0\end{array} \right] \right\}$$ (3.14)

As we shall see later, for the { $\vec{e}_h,\vec{e}_v$} basis the matrices represents isotropic scattering mechanisms, (surface, dihedral, 45 $^\circ\;$tilted dihedral, and crosspolariser). The corresponding vector $\vec{k}_P$ is then

$$\vec{k}_P = \frac{1}{\sqrt{2}}\left[S_{hh}+S_{vv},S_{hh}-S_{vv},S_{hv}+S_{vh},i(S_{vh}-S_{hv}) \right]^{T}$$ (3.15)

The factor 2 in (3.12) as well as the factor $\sqrt{2}$ in (3.14) arises from the restriction, that the norm of the scattering vector $\vec{k}$ , which is equal to the total scattered power

$$\vert\vec{k}\vert^{2} = \vec{k}_{P}^{*T} \cdot \vec{k}_{P} = \vec... ...{hh}\vert^2+\vert S_{hv}\vert^2+\vert S_{vh}\vert^2+\vert S_{vv}\vert^2 \right)$$ (3.16)

should be independent of the choice of the basis matrices of $\Psi$.

Both vectors contain the same information only in different representations. It is, therefore, possible to transform the scattering vector $\vec{k}$ from the linear { $\vec{e}_h,\vec{e}_v$} basis into any other elliptical polarisation basis { $\vec{e}_i,\vec{e}_j$} [Boerner81].

$$\vec{k}_{Bij}=[U]_{4B}\vec{k}_{Bhv}$$ (3.17)

The transformation basis $[U]_{4B}$ is defined as

$$[U]_{4B}= [U]_{2 \times 2} \otimes [U]_{2 \times 2}^T = \frac {1}... ... \rho & -\rho\rho^* & 1 & -\rho^* \rho^2 & \rho & \rho &1 \end{array} \right]$$ (3.18)

where $\otimes$ denotes the Kronecker product of matrices.

Since $\vec{k}_P$ is related to $\vec{k}_B$ by

$$\vec{k}_{P4} = \left[ D_4 \right]\vec{k}_{B4} = \frac{1}{\sqrt{2}} \left[ \begin{array}{cccc} 1 & 0 & 0 & 1 \\ 1... ...\\ 0 & 1 & 1 & 0 \\ 0 & i & -i & 0 \end{array} \right] \vec{k}_{B4} %%\quad$$

$$and$$

$$\vec{k}_{B4} = \left[ D_4 \right]^{-1} \vec{k}_{P4} = \frac{1}{\sqrt{2}} \left[\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -i \\ 0 & 0 & 1 & i \\ 1 & -1 & 0 & 0 \end{array} \right] \vec{k}_{P4}$$ (3.19)

we find $[U]_{4P}$ as

$$\left[ U_{4P}\right] = \left[ D_4 \right] \left[ U_{4B}\right]\left[ D_4 \right]^{-1}$$ (3.20)