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The Scattering Matrix

The Stokes vector of a scattered wave in the far-field, ${\b I}$, can be expressed in terms of the incident Stokes vector, ${\b I}_o$, by means of a linear transformation,

\begin{displaymath}[I,Q,U,V]^{T} = \frac{\sigma_s}{4\pi r^2} {\b P}
[I_o,Q_o,U_o,V_o]^{T}
\end{displaymath} (3.22)

where ${\b P}$ is a 4$\times$4 matrix having, in general, 16 real independent elements, and $\sigma_s$ is the scattering cross-section. At optical wavelengths, it is not possible for an instrument to introduce non-linear effects and so there is no loss of generality by using a linear transformation (Hansen and Travis, 1974). When the scatterers are isotropic, homogeneous spheres, the scattering matrix reduces to four independent elements,

 \begin{displaymath}{\b P} =
\left[
\begin{array}{c c c c}
P_{11} & P_{12} & 0 &...
...} & -P_{34} \\
0 & 0 & P_{34}& P_{33} \\
\end{array}\right].
\end{displaymath} (3.23)

Two important types of scattering for which equation (2.23) is applicable are Rayleigh and Mie.

In order to conserve energy upon a scattering event, one component of the phase matrix must be normalized. This constraint is expressed as,

\begin{displaymath}\int_0^{4\pi} \frac{P_{11}}{4\pi} d\Omega = 1
\end{displaymath} (3.24)

where the P11 element, also known as the phase function, relates scattered radiance to incident radiance.


next up previous
Next: Types of Scattering Up: Basic Processes in Earth's Previous: Scattering
Chris McLinden
1999-07-22