  This paper is concerned with direct and inverse scattering by a locally
perturbed infinite plane (called a locally rough surface in this paper) on
which a Neumann boundary condition is imposed. A novel integral equation
formulation is proposed for the direct scattering problem which is defined on a
bounded curve (consisting of a bounded part of the infinite plane containing
the local perturbation and the lower part of a circle) with two corners and
some closed smooth artificial curve. It is a nontrivial extension of our
previous work on direct and inverse scattering by a locally rough surface from
the Dirichlet boundary condition to the Neumann boundary condition [{\em SIAM
J. Appl. Math.}, 73 (2013), pp. 1811-1829]. In this paper, we make us of the
recursively compressed inverse preconditioning (RCIP) method developed by
Helsing to solve the integral equation which is efficient and capable of
dealing with large wave numbers. For the inverse problem, it is proved that the
locally rough surface is uniquely determined from a knowledge of the far-field
pattern corresponding to incident plane waves. Further, based on the novel
integral equation formulation, a Newton iteration method is developed to
reconstruct the locally rough surface from a knowledge of multiple frequency
far-field data. Numerical examples are also provided to illustrate that the
reconstruction algorithm is stable and accurate even for the case of
multiple-scale profiles.
