SHDOM: Method in Brief

The SHDOM uses an iterative process to compute the source function (including the scattering integral) on a grid of points in space. The angular part of the source function is represented with a spherical harmonic expansion. Solving for the source function instead of the radiance field saves memory, because there are often parts of a medium where the source function is zero or angularly very smooth (hence few spherical harmonic terms). The other reason for using spherical harmonics is that the scattering integral is more efficiently computed than in discrete ordinates. A discrete ordinate representation is used in the solution process because the streaming of radiation is more physically (and correctly) computed in this way. An adaptive grid that chooses where to put grid points is useful in atmospheric radiative transfer because the source function is usually rapidly varying in some regions and slowly varying in others.

The iterative method is equivalent to a successive order approach. For each iteration

  1. the source function is transformed to discrete ordinates at every grid point,
  2. the integral form of the radiative transfer equation is used to compute the discrete ordinate radiance at every grid point,
  3. the radiance is transformed back to spherical harmonics, and
  4. the new source function is computed from the radiance in spherical harmonics.
As with all order of scattering methods the number of iterations increases with the single scattering albedo and optical depth. A sequence acceleration method is used to speed up convergence, which is typically achieved in under 50 iterations. During the solution process, the grid cells with the integral of the source function difference above a certain limit are split in half, generating new grid points. The number of spherical harmonic terms kept at each grid point also changes as the iterations proceed (i.e. adaptive spherical harmonic truncation).