  In this paper, we will study the Bloch transformed rough surface scattering
problems, and propose a numerical method based on the Bloch transformed
problems. Based on the mathematical theory of the scattering problems from
locally perturbed periodic surfaces, the same techniques will be applied to the
rough surface scattering problems, and an equivalent coupled family of
quasi-periodic scattering problems in one periodic cell will be established.
The most important result obtained in this paper is on the finite Fourier
series approximation of the Bloch transformed field with respect to the
quasi-periodicity parameter. It will be proved that the finite series is
exactly the Bloch transformed solution corresponds to truncated rough surfaces.
Thus the truncation provides a reasonable approximation, and could be applied
to the numerical solutions. Based on the approximation, a numerical method is
proposed for the rough surface scattering problems. The convergence of the
numerical method is proved and illustrated by the numerical experiments. The
method provides a completely new perspective for the rough surface scattering
problems. There is possibility that some high order method will be developed
based on this new method.
