Interpretation of Scattering Mechanisms

A very essential advantage of full polarimetric data is the possibility to extract information about scattering mechanisms as discussed above. However, for classification purposes it is not only necessary to extract and separate different scattering mechanisms, but also to identify these scattering mechanisms.

As shown previously, a given [S]-matrix can be represented by a scattering vector $\vec{k}$, for example in the Pauli basis $\vec{k}_{P}$

$$[S] = \left[ \begin{array}{cc} S_{11} & S_{x} S_{x} & S_{22} \... ...array}{c} S_{11} + S_{22} S_{11} - S_{22} 2S_{x} \end{array} \right]$$ (4.21)

where $\vec{k}$ is a unitary complex vector. Since a three dimensional unitary complex vector has five degrees of freedom, it can be parameterized in terms of a set of five angles [Cloude95]

$$\vec{k} = \left[ \begin{array}{ccc} \cos(\alpha)e^{i\phi} \sin... ...(\beta)e^{i\delta} \sin(\alpha)\sin(\beta)e^{i\gamma} \end{array} \right]$$ (4.22)

A change of the angles $\alpha$ and $\beta$ by small amounts of $\Delta\alpha$ and $\Delta\beta$ corresponds to a differential change from one scattering mechanism $\vec{k}$ into another $\vec{k}'$.

$$\vec{k}'= [R_1]\vec{k}=\left[\begin{array}{ccc} 1 & 0 & 0 0 & ... ...ta} 0 & \sin{\Delta \beta} & \cos{\Delta \beta} \end{array}\right] \vec{k}$$

$$\vec{k}' = [R_2]\vec{k}=\left[\begin{array}{ccc} \cos{\Delta \alp... ...lta \alpha} & \cos{\Delta \alpha} & 0 0 & 0 & 1\\ \end{array}\right]\vec{k}$$ (4.23)

The corresponding transformation matrices $[R_1]$ and $[R_2]$ for changes in $\Delta\alpha$ and $\Delta\beta$ are simple plane rotations.

This observation leads to the following important Scattering Vector Reduction Theorem [Cloude95]: Any polarimetric back-scattering mechanism, represented by a complex unitary vector $\vec{k}$, obeying reciprocity can be reduced to the identity $[1,0,0]^T$, by a series of 3 matrix transformations

$$\left[\begin{array}{c} 1 0 0 \end{array}\right]= \left[\... ...0 0 & e^{i\delta} & 0 0 & 0 & e^{i\gamma} \\ \end{array}\right]{\vec{k}}$$ (4.24)

where the angles are interpreted as follows:

• $\alpha$-target scattering type $0 \leq \alpha \leq 90^o$
• $\beta$-orientation of target $-180 \leq \beta < 180^o$
• $\phi ,\delta ,\gamma$-target phase angles
  

An important observation following from the algebraic properties of the Pauli spin matrices is that the angle $\beta$ in this decomposition represents the physical orientation of the scatterer about the line of sight. Hence by calculating $\beta$, one may obtain a direct estimate of the target orientation angle. This is much simpler than using the polarimetric signature or Stokes reflection matrix to determine orientation.

The $\alpha$ angle is however not related to the target orientation, although it appears in the mathematical form of a plane rotation in (4.19). It represents an internal degree of freedom of the target and can be used to describe the type of scattering mechanism.

The alpha parameter is a continuous angle with a range of $0^{o} \le \alpha \le 90^{o}$ and can be used to represent a wide variety of different scatterers as shown in Fig. 4.7.

Figure 4.7: The interpretation of the $\alpha$-angle

At $\alpha$ = 0, we obtains an isotropic surface ( the first Pauli matrix). However as $\alpha$ increases, the surface becomes anisotropic (i.e. $S_{hh} \neq S_{vv}$). At $\alpha$ = $45^{o}$ we have a dipole and the orientation of the dipole is determined by $\beta$. If $\alpha > 45^{o}$ then one obtain an anisotropic dihedral i.e. $S_{hh} \neq S_{vv}$ and the phase difference is $180^{o}$. In the limit $\alpha$ = $90^{o}$ we obtain the second Pauli matrix (and the third as well, which differs from the second only by a rotation). This point can also be used to represent targets which cause a phase shift about $2 \pi$ between h and v. Note that $\alpha$ is rotation invariant, i.e., it is decoupled from $\beta$ and so we can identify the scattering mechanism independently of its physical orientation in space. Note that the eigenvalues and therefore, the entropy H, are independent of the chosen $\vec{k}$ vector, while the $\alpha$- and $\beta$ - angles, which are derived from the eigenvectors, are not. Hence the interpretation above is only valid for the coherency matrix [T].In Table 4.3 we show a list of the parameter values for a range of canonical scattering mechanisms of interest in polarimetry.

Table 4.3: Examples of parameter values for canonical scatterers ($\infty$ means no fixed values)

Canonical	$\alpha$	$\beta$	$\gamma$	$\delta$	$\phi$
Scatterer
Sphere	$0^o$	$\infty$	$\infty$	$\infty$	$\psi^o$
Dihedral at $\theta^o$	$90^o$	$2\theta^o$	$- \delta$	$-\gamma$	$\psi^o$
Dipole at $\psi^o$	$45^o$	$2\theta^o$	$\psi^o$	$\psi^o$	$\infty$
Helix	$90^o$	$\pm45^o$	$\delta + 90^o$	$\gamma - 90^o$
Surface at $\theta^o$	$0^o$	$2\theta^o$	$\approx 0^o$	$\approx 0^o$	$\approx 0^o$

We can clearly see that the alpha parameter provides a powerful extension of the Pauli decomposition, as it frees us from having to consider only isotropic scattering mechanisms. Since isotropic scattering in this sense is normally associated with man-made calibration reflectors rather than with natural media. The importance of $\alpha$ is that it permits us to extend the decomposition into more practical remote sensing applications.

In the general case when azimuthal symmetry is not present (which will be the case when we have for example scatterers with preferred orientation), then we must resort to (4.17) and, by using a 3 - symbol Bernoulli model, we can define an average scattering mechanism $\bar{\alpha}$

$$\overline{\alpha} = P_1\alpha_1 + P_2\alpha_2 + P_3\alpha_3$$ (4.25)

and an average orientation angle $\bar{\beta}$

$$\overline{\beta} = P_1\beta_1 + P_2\beta_2 + P_3\beta_3$$ (4.26)

$P_1 , P_2 , P_3$ are the relative intensities in (4.19) and $\alpha_1 , \alpha_2 , \alpha_3$ and $\beta_1 , \beta_2 , \beta_3$ are the values calculated from the 3 eigenvectors $\vec{e}_{1} ,\vec{e}_{2} ,\vec{e}_{3}$ we obtain from (4.18).

Together with the entropy from (4.19) we can now form an H- $\bar{\alpha}$-plane for classification purposes. The averaging inherent in this model implies that as the entropy increases the range of $\bar{\alpha}$ and $\bar{\beta}$ is reduced. We can quantify the bounds for such a variation for $\bar{\alpha}$ by invoking symmetry arguments. Within this plane the values are constrained by two curves, I and II (see Fig. 4.8).

Figure 4.8: The H-$\alpha$-plane with the constraining curves

The canonical form for these bounding curves are, according to [Cloude95]

$$\mathbf{[T]_I}=\left[\begin{array}{ccc} 1 & 0 & 0 0 & m & 0 0 & 0 & m \end{array}\right] \mbox{\quad 0 $\leq$ m $\leq$ 1} \quad \begin{array}{ll} \mathbf{[T]_{II}}=\left[\begin{array... ...y}\right] \mbox{0.5 $\leq$ m $\leq$ 1} \end{array}%%\right.$$ (4.27)

their corresponding H and $\bar{\alpha}$ values are then given by

$$H(m)=\frac{1}{1+2m} log_3 \left( \frac{m^{2m}}{(1+2m)^{2m+1}} \ri... ...= \left\{\begin{array}{l} \frac{\pi}{2} \frac{\pi}{1+2m} \end{array} \right.$$ (4.28)

The resulting area of the H-$\alpha$-plane in Fig. 4.8 is the basic feature space for the most classification approaches.