Reflection Matrix

The surface reflected Stokes vector is a function of the downward Stokes vector incident upon it. They are related via,

$${\b I}(\tau_1;\mu,\phi) = \int_0^{2\pi} \int_0^1 {\b R}(\mu,\phi;-\mu',\phi') {\b I}(\tau_1;-\mu',\phi') \mu' d\mu' d\phi'$$ (3.82)

where $\b{R}$ is a 4$\times$4 reflection matrix relating each reflected Stokes parameter component to each incident component. This expression is analogous to the scattering source function vector. Also, ${\b Q} \, {\b R} \, {\b Q}$ (where ${\b Q}$ is a diagonal matrix of elements [1,1,-1,1]), must be used instead of simply $\b{R}$ in equation (2.82) to account for the change in symmetry when the atmosphere is illuminated from the bottom (Hovenier, 1969).

If the surface is a pure specular reflector, then the full reflection matrix can be written in terms of the Fresnel equations (e.g.: Haferman et al., 1997). Generally, however, it is assumed that the surface is depolarizing so that all matrix elements other than R₁₁ are zero. This reduces the matrix to a single element which is commonly referred to as the bidirectional reflectance distribution function (BRDF). Equation (2.82) becomes,

$$I(\tau_1;\mu,\phi) = \int_0^{2\pi} \int_0^1 R_{11}(\mu,\phi;-\mu',\phi') I(\tau_1;-\mu',\phi') \mu' d\mu' d\phi'$$ (3.83)

and $Q(\tau_1;\mu,\phi)=U(\tau_1;\mu,\phi)=V(\tau_1;\mu,\phi)=0$. In addition to its directional dependence, the BRDF function may also be a function of solar zenith angle and surface conditions such as windspeed. Two examples of surfaces in which a functional form of the BRDF exist are specular reflection off a wind-roughened sea surface and cloud surfaces. Both types of surfaces are discussed in some detail in Appendix B. In addition, BRDFs for several theoretical surfaces have been derived (Settle, 1996).