Generalized functions approach to the derivation of the high-frequency radar cross-section of ocean surfaces with electromagnetically-large waves

Abstract

The radar cross-section (RCS) of the ocean surface mathematically describes the interaction between electromagnetic fields and ocean waves, allowing for a better understanding of observations made by coastal high-frequency (HF) radars. However, theoretical limitations have restricted the calculation of the RCS of the ocean surface to electromagnetically-small waves. This thesis proposes an approach to study and evaluate the scattering of electric fields over ocean surfaces with electromagnetically-large waves, as well as an extension of the generalized functions approach to the one-body scattering problem to curvilinear coordinates with variable basis vectors. The present work enables the analysis of HF radar signals scattered by the ocean surface during storms and electromagnetically-high sea states, expanding the capabilities of HF radars in radio oceanography. First, a system of equations for the electric field in curvilinear coordinates in the presence of a single scatter is proposed. It is shown that the derived equations can be applied to coordinate systems with variable basis vectors, being reduced to the form presented in the literature [1] when Cartesian coordinates are considered. As a proof of concept, the system of equations for curvilinear coordinates is applied to a perfectly-electrically conductive (PEC) sphere, yielding the Stratton-Chu integral equation for the electric field, a general solution in classic electromagnetics. Then, the electric field scattering over a time-varying conductive random surface is obtained for an ocean surface with electromagnetically-large waves. First, the electric field expressions are obtained by applying the proposed system of equations to a time-varying conductive random surface, removing the restrictions to the growth of the roughness scales presented in the literature. This allows for the study of ocean surfaces with electromagnetically-large waves. The derivation results in additional electric field expressions that correct the height-restricted expression accounting for the energy of electromagnetically-large waves. The resulting expression is reduced to the height-restricted formulation if the roughness scale is sufficiently small. From the expression of the electric field, the RCSs of the correction terms are derived. These derivations yield expressions that can be reduced to a general form, allowing the generation of correction terms to the radar cross-section at a given order by a product with the desired correction factor. In a morphological analysis of the correction factors, it is shown that at small roughness scales, the correction terms do not affect the total cross-section for the ocean surface; however, by increasing the roughness scale beyond the limits presented in the literature, the contribution of the correction terms to the total RCS become significant. Evidence of the presence of the proposed correction terms was approved in field measurements; however, definitive proof of the effects observed here can be the object of future research.