Numerical Methods for Analysis of Em Scattering from an Electrically Large Ocean Surface

Electromagnetic scattering is considered from finite and electrically large perfectly conducting rough surfaces. Three solution approaches are considered. The first two are based on a multi-level fast-multipole algorithm (MLFMA) formulation. In one MLFMA approach the rough surface resides in free space and is terminated by a resistive taper. In the second MLFMA approach the rough surface is terminated in an infinite perfectly conducting halfspace, with a smooth taper employed between the rough surface and the infinite halfspace, thereby avoiding the need for a resistive taper. The first MLFMA formulation is based on a free-space electric-field integral equation analysis, and the second uses the halfspace Green’s function and a combined-field integral equation. These two approaches, within the context of the MLFMA, allow consideration of electrically large surfaces, permitting comparison with high-frequency asymptotic solutions. The third method considered is a ray-based solution, with which even larger surfaces may be considered. Several numerical results are presented, with which the relative accuracy and computational efficiency of the approaches are compared. NUMERICAL METHOD A. Resistive Taper Oh and Sarabandi [1] developed a Taylor taper loading to suppress unwanted diffraction from surface truncation in the analysis of scattering from a rough surface. A power-law resistive loading was presented by Haupt and Liepa [2]. In many examples the Taylor taper has proven superior [3], and therefore it is used in our implementation. The rough surface is artificially extended in all directions, and a resistive taper is utilized in the extended area. By using a resistive taper R(r), the original electric-field integral equation (EFIE) is modified as [ ] ) ( ) ( ) ( ) ( r J r J r r E L R i + = (1) where Ei represents the incident electric field, J(r) the induced electric current on the target and L is an EFIE operator. Because we are considering a resistive taper an EFIE formulation is employed, which has well-known problems with conditioning [4]. In addition, this method requires a relatively large region for the resistive taper, increasing the number of unknowns (for the induced current) that must be evaluated. B. Halfspace Formulation Rather than considering a finite rough surface terminated in a resistive taper, an alternative approach is to taper the rough surface into an infinite halfspace background medium. This problem is considered by utilizing the halfspace Green’s function for analysis of the rough surface. By smoothly transitioning from the perfectly conducting rough surface to a perfectly conducting planar halfspace, edge effects at the termination of the rough surface are minimized, without the need for a resistive taper. In addition to avoiding the need for extra basis functions to handle the resistive taper, the halfspace formulation avails a combined field integral equation (CFIE) formulation, since a closed target is considered (see Fig. 1). C. Ray Tracing The formulations in the previous two sections are solved by expanding the currents induced on the rough surface in terms of a basis, where here we employ the RWG basis functions described in [5]. The associated integral equations are solved via the multilevel fast multipole algorithm (MLFMA) [6-8], with the use of image theory to address the halfspace case. The MLFMA formulation allows consideration of electrically large surfaces, the results from which may be compared to high-frequency asymptotic results. The purpose of such an analysis is to validate the accuracy Rough surface Image Infinite flat plate Figure 1. Rough surface and its image, placed in the presence of an infinite halfspace background.