On the continuity dependence of elastic scattering amplitudes upon the shape of the scatterer

Abstract

The transmission problem of linear elasticity in R2 is considered. The authors assume a system of quasi-Fredholm singular integral equations which describes the scattering process and they use an asymptotic analysis to derive relations for the far-field patterns. They establish a continuity dependence of the far-field patterns on the scatterer's shape. This result holds for a set of admissible functions which are considered as parametrization of the boundary of the inclusion. Continuity properties of this nature secure the stability of the inverse scattering problem.

A linear sampling method for near-field inverse problems in elastodynamics

“…v S = P − Q), one can infer that v S is also a radiating solution of the homogeneous Navier equation in − . Now, with the aid of ( 1) and ( 2) in (22), it is seen that v = v F (ξ) + v S (ξ), ξ ∈ − where v F and v S are given, respectively, by (18) and (19).…”

“…Noninvasive identification of subterranean obstacles using elastic waves with frequencies in the resonance region is a long-standing problem in mechanics and engineering driven by its relevance to exploration seismology, nondestructive material testing, environmental remediation, medical diagnosis and defence applications. For this class of inverse scattering problems, employed imaging solutions are often based on nonlinear optimization which requires an initial approximation of the geometry and topology of the scattering obstacle [18,28,35,38]. 0266-5611/04/030713+24$30.00 © 2004 IOP Publishing Ltd Printed in the UK Over the past decade, developments in sonar and radar technologies have led to the introduction of an alternative technique for solving inverse scattering problems in the resonance region called the linear sampling method.…”

The Far-Field Equations in Linear Elasticity — an Inversion Scheme

“…So, the corresponding meaning of the radiation conditions for the dyadic displacement field is that each first vector from the left side of each dyad in (10) has to satisfy the Kupradze's radiation conditions at infinity. These radiation conditions hold uniformly over all directions ofx x.…”

“…Among the methods which are developed in this field are those presented in [1], [4], and [12] for the acoustical case and in [9] and [10] for the elastic case. In order to treat them Born or physical optics approximation cannot be used, so usually a nonlinear optimization scheme is proposed.…”

An Inversion Algorithm in Two-Dimensional Elasticity

“…Among the methods which are developed in this field are those presented in [5] and [14] for the acoustical case and in [10] and [11] for the elastic case. More recently in a series of papers, Colton et al have developed a very simple and numerically rapid method for solving inverse scattering problems in the resonance region for the acoustical case [4,6].…”