Towards Generalized Boundary Conditions in DISORT

We describe a considerably generalized treatment of boundary conditions in the Discrete Ordinate Radiative Transfer Code (DISORT). The non-Lambertian surface reflection option in previous versions has been replaced by a more general and realistic bidirectional reflectance distribution function (BRDF). This formulation of the lower boundary now allows for a BRDF that depends on the incident polar angle, reflected polar angle, and azimuth angle between the incident and reflected directions. This enables us to separate the Fourier components, needed in DISORT, by expanding the bidirectional reflectivity in a Fourier cosine series. To maintain high accuracy the coefficients of the Fourier expansion are calculated from the defining equation by Gaussian quadrature. Because the integrand of this equation contains the cosine function whose frequency increase as the number of streams used in the intensity calculations increases, the integrand becomes a highly oscillating function for large number of streams. This requires a high-order Gaussian quadrature in the integral, which combined with the need to evaluate the Fourier coefficients separately at a large number of incident and reflected directions, makes the process computationally expensive. Alternative ways of calculating the coefficients, that include discrete fast Fourier Transforms, have been explored, and the relationship between computational accuracy and speed has been established. In previous versions of DISORT, the top boundary condition is restricted to be a constant, with the idea that this may usefully approximate a thermally emitting upper boundary, or a highly scattering one such as a cloud. This restriction has now been relaxed using a treatment similar to that applied for the surface. This allows for an input of a radiation field at the top that is a function of the incident polar angle. This feature is useful, for example, in the long-wavelength part of the spectrum in a cloudy atmosphere, where the radiation incident at the top of the cloud can be computed outside of DISORT using a computationally less expensive model. Details of the implementation and computational considerations of the generalized boundary conditions are presented in the paper. The updated version of DISORT uses the Linear Algebra PACKage (LAPACK) and supporting routines from the Basic Linear Algebra Subprograms (BLAS) for solving systems of simultaneous linear equations.