Mathematical aspects of BRDF modeling : adjoint problem and Green ' s function

− Adjoint formulation of three−dimensional radiative transfer and the Green's function concept have been developed in neutron transport several decades ago. This is not merely yet another method of simulating the radiative transfer process, but a method of reformulating the problem to better incorporate existing radiation models into a particular research. In the case of photon transport in vegetation canopies, the Green's function is a canopy radiative response to a point monodirectional source located outside the canopy. The Green's function, therefore, has intrinsic canopy information. It can be evaluated by using existing canopy radiation models. The problem−dependent adjoint formulation of radiative transfer allows us to express a particular canopy radiation effect in terms of the Green's function and, as a consequence, to better adjust the existing models to the solution of a specific radiation problem. Application of this technique to the retrieval of biophysical parameters from remotely sensed data (the table look−up method) was discussed in (Kimes et al., this issue). In this paper, we will illustrate how this concept can be applied to the estimation of cloud optical properties from ground−based measurements of spectral zenith radiance above the vegetation canopy under broken cloud conditions. In spite of different physical formulations of these problems, both of them use the Green's function to describe radiation fields due to the interaction between the canopy ground and the canopy and the canopy−clouds interaction. This technique allows us not only to extend an applicability range of existing canopy−radiation models, but also to incorporate of various approaches developed in other fields of physics into BRDF modeling and its applications.