  By the use of Green's second integral identity we determine the field
scattered from a two-dimensional randomly rough isotropic or anisotropic
Dirichlet or Neumann surface when it is illuminated by a scalar Gaussian beam.
The integral equations for the scattering amplitudes are solved
nonperturbatively by a rigorous computer simulation approach. The results of
these calculations are used to calculate the full angular distribution of the
mean differential reflection coefficient. For isotropic surfaces, the results
of the present calculations for in-plane scattering are compared with those of
earlier studies of this problem. The reflectivities of Dirichlet and Neumann
surfaces are calculated as functions of the polar angle of incidence, and the
reflectiveties for the two kinds of surfaces of similar roughness parameters
are found to be different. For an increasing level of surface anisotropy, we
study how the angular intensity distributions of the scattered waves are
affected by this level. We find that even small to moderate levels of surface
anisotropy can significantly alter the symmetry, shape, and amplitude of the
scattered intensity distributions when Gaussian beams are incident on the
anisotropic surfaces from different azimuthal angles of incidence.
