In this appendix the authors have set up the integral equation formulation o fthe scalar scattering problem. Complete solutions, such as those describing the reflection and transmission of radiation by a planar interface, can be extracted from this only in the most idealised of circumstances. Nonetheless, the integral equation formulation provides us with a framework within which many aspects of scattering by a rough surface can be developed systematically. Approximate descriptions of scattering by a perfectly conducting surface can be derived, both through iterative solution of the integral equation (physical optics and small height perturbation theory) and progressive refinement of the Green's function (half space and Lorentz reciprocity based calculations). The latter Lorentz reciprocity based approach also allows us to derive the small height perturbation theory result for an imperfectly conducting rough surface with relatively little trouble; this in turn can be modified to take account of the large scale swell structure of the sea surface through its incorporation into the composite model. Methods for the numerical solution of the scattering integral equations have been developed. Sufficient detail has been given for explicit expressions to be identified and coded up for both perfectly and imperfectly conducting surfaces. The modelling of low grazing angle scattering invariably encounters difficulties that arise from the finite size of scattering surface that can be accommodated within the computer; these has been ameliorated by introducing semi-infinite adjunct planes. A different and improved method for the calculation of the terms in the integral equations that arise from these planes has been described, which also allows us to treat the imperfectly conducting surface through the use of the impedance boundary condition that was derived.
