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,

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

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

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

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