This paper extends the parabolic integral equation method, which is very
effective for forward scattering from rough surfaces, to include backscatter.
This is done by applying left-right splitting to a modified two-way governing
integral operator, to express the solution as a series of Volterra operators;
this series describes successively higher-order surface interactions between
forward and backward going components, and allows highly efficient numerical
evaluation. This and equivalent methods such as ordered multiple interactions
have been developed for the full Helmholtz integral equations, but not
previously applied to the parabolic Green's function. In addition, the form of
this Green's function allows the mean field and autocorrelation to be found
analytically to second order in surface height. These may be regarded as
backscatter corrections to the standard parabolic integral equation method