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

The analysis of elastic wave propagation and scattering is an important issue in fields such as earthquake engineering, nondestructive testing, and exploration for energy resources. Since the 1980s, the boundary integral equation method has played an important role in the analysis of both forward and inverse scattering problems. For example, Colton and Kress (1998) presented a survey of a vast number of articles on forward and inverse scattering analyses. 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). Several applications of the volume integral equation to scattering analysis for classical mechanics have also appeared. For example, Hudson and Heritage (1981) used the Born approximation of the solution of the volume integral equation obtained from the background structure of the wave field for the seismic scattering problem. Recently, Zaeytijd, Bogaert, and Franchois (2008) presented the MLFMA-FFT method for analyzing electro-magnetic waves, and Yang, Abubaker, van den Berg et al. (2008) used a CG-FFT approach to solve elastic scattering problems. These methods were used to establish a fast algorithm to solve the volume integral equation via a Fast Fourier transform, which is used for efficient calculation of the convolution integral. In this chapter, another method for the volume integral equation is presented for the direct forward and inverse elastic wave scattering problems for 3-D elastic full space. The starting point of the analysis is the volume integral equation in the wavenumber domain, which includes the operators of the Fourier integral and its inverse transforms. This starting point yields a different method from previous approaches (for example, Yang et al., 2008). By replacing these operators with discrete Fourier transforms, the volume integral equation in the wavenumber domain can be treated as a Fredholm equation of the 2nd kind with a nonHermitian operator on a finite dimensional vector space, which is to be solved by the Krylov subspace iterative scheme (Touhei et al, 2009). As a result, the derivation of the coefficient matrix for the volume integral equation is not necessary. Furthermore, by means of the Fast