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Properties of the far field operator in the inverse conductive scattering problem
Abstract
In this paper, we have studied the properties of the far field operator in the inverse scattering problem with the conductive boundary condition. The far field operator which maps the boundary data into the far field pattern of the solution to a boundary value problem plays a fundamental role in inverse scattering theory. In scattering theory the boundary data are given by the traces of plane waves. We have characterized the closure of the image in L2 space under the far field operator of the span of the traces of these incident plane waves.
In this paper, we derive a numerical solution of an inverse obstacle
scattering problem with conductive boundary condition. The aim of the
direct problem is the computation of the scattered field for a given
arbitrarily shaped cylinder with conductive boundary condition on its
surface.The inverse problem considered here is the reconstruction of the
conductivity function of the scatterer from meausurements of the far
field. A potential approach is used to obtain boundary layer integral
equations both for the solution of the direct and the inverse problem.
The numerical solutions of the integral equations which contain
logarithmically singular kernels are evaluated by a Nyström method
and Tikhonov regularization is used to solve the first kind of integral
equations occuring in the solution of the inverse problem. Finally,
numerical simulations are carried out to test the applicability and the
effectiveness of the method.
In this paper, we consider the inverse problem of the scattering of a plane acoustic wave by a multilayered scatterer; especially we study the properties of the corresponding far field operator. This problem models time-harmonic acoustic or electromagnetic scattering by a penetrable homogeneous medium with an impenetrable core. The discussion is essential for numerical approximation of the corresponding inverse problem, i.e., from the knowledge of the far field patterns to model the shape of the scatterer.
The inverse scattering problem under consideration is to determine the shape of an acoustically soft obstacle in from a knowledge of the time harmonic incident wave and the far field pattern of the scattered wave. It is assumed that the frequency is in the resonance region, i.e. high frequency asymptotic methods are not available. A method for solving this problem is presented which is based on the theory of Herglotz wave functions and avoids the use of integral equations. Numerical examples are given showing the practicality of this approach.
In a previous paper we presented a new method for determining the shape of an acoustically soft obstacle from a knowledge of the time-harmonic incident wave and the far field pattern of the scattered wave. The method given there was based on knowing the far field pattern for an interval of values of the square of the wave number k such that this interval contained the first eigenvalue lambda //1 of the interior Dirichlet problem. The purpose of this paper is to extend the methods of our earlier paper to treat the case when the far field pattern is only known for a single value of the wave number such that k**2 is not equal to lambda //1. In addition we show how to combine the results of this paper with the first to obtain a method that is valid for any fixed value of k independent of whether or not k**2 is an eigenvalue. We also extend our analysis to include the case of the impedance boundary value problem. Numerical examples are given showing the practicality of our method.
The inverse scattering problem considered in this paper is to determine the shape of an obstacle from a knowledge of the time-harmonic incident field and the phase and amplitude of the far field pattern of the scattered wave. A method is given which is based on determining the first eigenfunction of the unknown obstacle and which, in addition, avoids the use of integral equations. Numerical examples are given showing that the proposed method is both accurate and simple to use.
It is noted that one of the main difficulties in studying the inverse scattering problem for acoustic waves is obtaining information on the class of far field patterns corresponding to boundary conditions of a given type. The first step in such an investigation involves constructing the mapping taking given boundary data onto the corresponding far field pattern, the usual approach in this regard being through the use of integral equations. It is pointed out, however, that this approach is not always the optimal choice for obtaining the desired information. Attention is given here to the method of least squares for solving time harmonic scattering problems, the aim being to first present a contribution to the existing theoretical basis for this method and then to apply the results obtained to the problem of classifying those functions that can be far field patterns corresponding to a given physically realizable scattering problem. The acoustic scattering problem to be addressed here is formulated. Attention is restricted to two-dimensional scattering problems, although all of the results can be immediately generalized to the case of three dimensions.
Using Isakov's approach to uniqueness in inverse scattering, Hettlich has obtained a uniqueness result for the conductive boundary condition via weak solutions. Following ideas of Kirsch and Kress, using an integral equation approach in the present paper we give a simplified proof for Hettlich's uniqueness theorem which also includes a simpler version of the uniqueness proof of Kirsch and Kress for the transmission problem.
The far‐field operator for an exterior boundary value problem for the Helmholtz equation maps the boundary data onto the far‐field pattern of the solution. This paper computes the L ² ‐adjoint of this operator for various choices of boundary conditions. In scattering theory the boundary data are given by the traces of plane waves. We characterize the closure of the span of the images of these plane waves under the far field operator.
Finally, the results are extended to more general topologies including Sobolev and Hölder norms.
