Theoretical derivation of the depth average of remotely sensed optical parameters

The dependence of the reflectance at the surface on the vertical structure of optical parameters is derived from first principles. It is shown that the depth dependence is a function of the derivative of the round trip attenuation of the downwelling and backscattered light. Previously the depth dependence was usually modeled as being dependent on the round trip attenuation. Using the new relationship one can calculate the contribution of the mixed layer to the overall reflectance at the surface. This allows one to determine whether or not to ignore the vertical structure at greater depth. It is shown that the important parameter to average is the ratio of the backscattering and absorption coefficients. The surface reflectance is related to the weighted average of this parameter, not the ratio of the weighted average of the backscattering and the weighted average of the absorption. Only in the special case of "optical homogeneity" where the ratio of the backscattering and absorption coefficients does not vary with depth, can the vertical structure be ignored. Other special cases including constant backscattering and variable absorption are also investigated.