Analysis of the Scattering from Targets on Ocean like Rough Surfaces

INTRODUCTION

The electromagnetic scattering from rough surfaces such as ocean like surfaces has been extensively found in the literature. A recent review can be found in a special issue about this topic (Brown, 1998). Most recent advances have been focused on the direct numerical simulation of the scattering problem. Numerical techniques based on integral equation formulations such as the well-known method of moments (MoM) (Harrington, 1993) are apparently some of the have played an increasingly important role. Different methods have been developed in recent years in order to reduce the number of computer operations required to analyze the rough surface scattering problem via the method of moments. Recently, a new and powerful iterative numerical technique called the forward backward (FB) method has been proposed by Holiday, et.al., [3] [4] for solving the magnetic field integral equation (MFIE), which describes the current induced on a perfect electrically conducting (PEC) surface. A hybrid approach has been presented in (West, et. al., 1998) where the scattering from water waves of different degrees of breaking is numerically examined by combining the MoM and the geometrical theory of diffraction (GTD). The technique is implemented using impedance-surface boundary conditions to handle scattering media of conductivity such as sea water. In this paper, we present a generalization of the FB method which allows us to study the scattering from composite surfaces that can include one or more large arbitrarily shaped obstacle on the ocean surface. The new approach, called the generalized forward backward (GFB) method, is based on a combination of the conventional. FB method with the MoM where the MoM is only applied to the region close to the obstacle. The GFB could be combined with modern integral equation acceleration algorithms such as could reduce the fast multipole method (Burkholder and Kwon, 1996) or the novel spectral acceleration method, which could reduce the computational cost of O(N)

Let us consider now the composite problem depicted in the Fig. 1, where one or more PEC obstacles (like a ship or large rogue breaking wave) are included in the surface contour S. For this kind of problem the conventional FB method does not exhibit convergent behavior, because the

the obstacle and the nearby ocean like surface, and within the obstacle itself, all of which may not be taken into account with the conventional formulation involved in the standard FB method. The GFB method consists of a generalization of the conventional FB approach which enhances the scope of application of the previous method to composite scattering problems as the one depicted in Fig. 1. The GFB formulation will be presented for a simple problem consisting of a sea surface containing only one obstacle (Fig. 1) The current is expressed as the sum of forward and backward contributions. but now the impedance matrix is split in a different way Where the Zsg matrix is the diagonal part of Z with an additional block including the impendence submatrix corresponding to the ship and nearby sea region; while Zfg and Zbg are, respectively, the lower triangular part and the upper triangular part of Z but excluding the matrix Zsg, as illustrated. With this decomposition, matrix Zsg contains both the self (diagonal) terms and the interaction of the whole obstacle and nearby sea region together. Then, the original system is transferred in a similar way as in the conventional FB method, yielding the following matrix equation : Which can be iteratively solved for I, and I, as starting with The solution of differs from other solutions because neither Zsg, + Zsg nor Zsg, + Zsg are triangular matrices. Nevertheless, the equation can also easily solved by combining forward or backward substitution together with the direct factorization of the square block of Zsg whose dimension depends only on number of current elements in Regions 2, namely the slip and nearby sea.

A description of the procedure used to solve is presented. It must be pointed out that neither Z + Z are triangular matrices, nevertheless, these equations can be easily solved by combining forward or backward substitution together with the direct factorization of a square block submatrix whose dimension is just the extent of the MoM region. We explain only the resolution of which concerns the matrix Zsg + Zfg. Equation can be solved in a similar manner. Equation (5) has the general form where A = Zsg + Zfg . x = If,(i) and b = V – Zfg . Ib,(i-1) . A is a ―quasi‖ lower triangular matrix. In order to solve these equation is an efficient way, the matrices A, x, and b of (7) are subdivided into blocks. Thus (7) can expressed in terms of the blocks as follows :- The solution is then obtained as follows : (1) Eq. (8) is a lower triangular matrix, so x1 can be easily obtained by forward substitution. (2) Once x1 has been calculated, x2 can be obtained from (9) by the direct solution of LU decomposition may be used so that A22 needs to be factorized only once for a given Region 2

3) Finally, once x1 and x2 have been calculated, x3 can be obtained from (10) again by forward substitution from because is a lower triangular matrix.

In this paper, we have presented a new computational algorithm, the GFB method, that enhances the scope of application of the conventional FB method to composite 2-D scattering problems consisting of a randomly varying smooth ocean-like surface with abrupt target obstacles present like ships, rogue waves, etc. the new approach has been shown to provide very accurate results and maintains the same fast convergence and O(N2) computational cost associated with the conventional FB solution. The numerical results show that different random ocean surfaces greatly affect the backscatter patterns of a ship-like target although the patterns tend to have about the same average level. The purpose of the GFB method introduced here is to provide a numerical tool for studying this class of scattering problems.

1. Brown, E.G.S. (1998). Special issue on low grazing-angle backscattering from rough surfaces, IEEE Antennas Propagat., Vol. 46, pp. 2028-2031. 2. Harrington, R.F. (1993). Field Computation by Moment Method, New York, IEEE Press. 3. Holiday, D., DeRaad, L.L. and G. J. St-Cyr (1995). Volterra approximation for low grazing angle shadowing ocean-like surfaces, IEEE Trans. Antennas Propagat., Vol.43, pp. 119-1206. 4. Holiday, D., Deraad, L. L. and G. J. St-Cyr (1996). Forward-backward : A new method for computing low-grazing angle scattering, IEEE Trans. Antennas Propagat, Vol. 44, pp. 72-729. 5. West, J.C. Sturrm, J.M. and S.J. Ja (1998). Low-Grazing scattering from breaking water waves using an imepedance boundary 6. Burkholder, R. J. and D.H. Kwon (1996). High-frequency asymptotic acceleration of the fast multiple method Radio Sci., Vol. 51, pp. 1199-1206.

Research Scholar, Department of Computer Science, OPJS University, Rajasthan