Shape deformations in rough-surface scattering: cancellations, conditioning, and convergence.
We analyze the conditioning properties of classical shape-perturbation methods for the prediction of scattering returns from rough surfaces. A central observation relates to the identification of significant cancellations that are present in the recurrence relations satisfied by successive terms in a perturbation series. We show that these cancellations are precisely responsible for the observed performance of shape-deformation methods, which typically deteriorates with decreasing regularity of the scattering surfaces. We further demonstrate that the cancellations preclude a straightforward recursive estimation of the size of the terms in the perturbation series, which, in turn, has historically prevented the derivation of a direct proof of its convergence. On the other hand, we also show that such a direct proof can be attained if a simple change of independent variables is effected in advance of the derivation of the perturbation series. Finally, we show that the relevance of these observations goes beyond the theoretical, as we explain how they provide definite guiding principles for the design of new, stabilized implementations of methods based on shape deformations.