As shown previously, a given [S]-matrix can be represented by a scattering vector , for example in the Pauli basis
This observation leads to the following important Scattering Vector Reduction Theorem
[Cloude95]: Any polarimetric
back-scattering mechanism, represented by a complex unitary vector , obeying
reciprocity can be reduced to the identity ,
by a series of 3 matrix transformations
An important observation following from the algebraic properties of the Pauli spin matrices is that the angle in this decomposition represents the physical orientation of the scatterer about the line of sight. Hence by calculating , 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 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 and can be used to represent a wide variety of different scatterers as shown in Fig. 4.7.
At = 0, we obtains an isotropic surface ( the first Pauli matrix). However as increases, the surface becomes anisotropic (i.e. ). At = we have a dipole and the orientation of the dipole is determined by . If then one obtain an anisotropic dihedral i.e. and the phase difference is . In the limit = 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 between h and v. Note that is rotation invariant, i.e., it is decoupled from 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 vector, while the - and - 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.
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 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
Together with the entropy from (4.19) we can now form an H- -plane for classification purposes. The averaging inherent in this model implies that as the entropy increases the range of and is reduced. We can quantify the bounds for such a variation for by invoking symmetry arguments. Within this plane the values are constrained by two curves, I and II (see Fig. 4.8).
The canonical form for these bounding curves are, according to [Cloude95]
The resulting area of the H--plane in Fig. 4.8 is the basic feature space for the most classification approaches.