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

Abstract

The problem of reconstructing underground obstacles from near-field, surface seismic measurements is investigated within the framework of a linear sampling method. Although the latter approach has been the subject of mounting attention in inverse acoustics dealing with far-field wave patterns in infinite domains, there have apparently not been any attempts to apply this new method to the interpretation of near-field elastic wave forms such as those relevant to the detection of subterranean objects. Aimed at closing this gap, a threedimensional inverse analysis of elastic waves scattered by an obstacle (or a system thereof ), manifest in the surface ground motion patterns, is formulated as a linear integral equation of the first kind whose solution becomes unbounded in the exterior of the hidden scatterer. To provide a comprehensive theoretical foundation for this class of imaging solutions, generalization of the linear sampling method to near-field elastodynamics and semi-infinite domains is highlighted in terms of its key aspects. A set of numerical examples is included to illustrate the performance of the method. On replacing the featured elastodynamic half-space Green function by its free-space counterpart, the proposed study is directly applicable to infinite media as well.

Three-Dimensional Near-Field Microwave Imaging Using Hybrid Linear Sampling and Level Set Methods in a Medium With Compact Support

“…Moreover, to increase the grid points in level set method, interpolation of sampling points can be adopted. Because equation (25) updates the near boundary nodes, the re-initialization procedure is used for other nodes [7], [32]. Actually, this special feature of the sign distance function Φ is used and in each iteration step, other boundary points [after updating the near boundary nodes by using equation (25)] are re-initialized.…”

Inverse scattering analysis of elastic half space by means of the fast volume integral equation method

“…Colton et al [2] presented a linear sampling method for the inverse scattering analysis of electromagnetic wave field by means of the far field operator. Fata and Guzina [3] proposed the linear sampling method by defining a near field operator to identify the scattering object hidden in an elastic half space.…”

A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena

“…They also presented integral equation methods for acoustic and electromagnetic wave propagation, based on the theory of operators (1983 and 1998). Recently, Guzina, Fata and Bonnet (2003), Fata and Guzina (2004), and Guzina and Chikichev (2007) have dealt with inverse scattering problems in elastodynamics. The type of volume integral equation known as the Lippmann-Schwinger equation (Colton & Kress, 1998) has been an efficient tool for theoretical investigation in the field of quantum mechanics (see, for example, Ikebe, 1960).…”