  Rough surface scattering problems are always very challenging both
theoretically and numerically. In this paper, we adopt the Bloch transform and
the perturbation theory to investigate a special case, i.e., when the rough
surface is a slight perturbation of a periodic one. Based on known results from
the Bloch transform, the problem could be written into an equivalent bounded
variational problem in higher dimensional spaces. The first step is to consider
the non-perturbed problem. From the perturbation theory, the solution could be
written as a Neumann series. The second step is to consider the slightly
perturbed problems. The key point in the process is the relationship between
the Dirichlet-to-Neumann map. With the study between these two operators, it is
proved that when the right hand side belongs to a certain space, the problem is
equivalent to a modified one. Thus a high regularity result is obtained.
