In the last two decades, boundary methods have gained increasing popularity. In particular, the boundary integral equation approaches and their discretizations into boundary-element methods (BEM) have produced successful solutions to various problems in elastodynamics. Recognized advantages over domain approaches are the dimensionality reduction, the relatively easy fulfillment of radiation conditions at infinity, and the high accuracy of results. There are two types of BEM formulations: direct and indirect. In the former, the unknowns are the physical values of displacements and tractions. This formulation arises from the discretization of reciprocal integral representation theorems. In contrast, in the indirect BEM one formulates the problems in terms of auxiliary layer densities, which can then be used to obtain physical quantities of interest. Both formulations are related; it can be shown that they are mathematically equivalent [40, 51].
Perhaps the first application of an integral formulation (a direct one) to study topography-related seismic amplification is due to Wong and Jennings [48]. They studied the seismic response of arbitrary canyon geometries under incident SH waves. A similar approach has been used by Zhang and Chopra [52], to consider the 3-D response of a canyon in an elastic half-space.
On the other hand, the combination of discrete wavenumber expansions for Green's functions [6] with boundary integral representations has been successful in various studies of elastic wave propagation in sedimentary valleys. Bouchon [7], Campillo and Bouchon [10], Campillo [9], Gaffet and Bouchon [18], Bouchon et al [8], and Campillo et al. [11] used source distributions on the boundaries, whereas Kawase [24], Kawase and Aki [25], and Kim and Papageorgiou [26] used the Somigliana representation theorem. These are discrete wavenumber versions of BEM, indirect and direct, respectively. However, such procedures require a considerable amount of computer resources and cannot deal readily with highly heterogeneous deposits.
In 1995, Sanchez-Sesma et al [42] presented a simplified indirect boundary-element method and applied it to simulate the seismic response of arbitrarily shaped 3-D models of alluvial valleys, based on the integral representation of the scattered (diffracted, reflected, and refracted) elastic waves in terms of single-layer boundary sources. Scattered waves are thus constructed at the boundaries from which they radiate. Hence, it can be regarded as a numerical realization of Huygens' principle. This approach is in fact a variation of a boundary method that has been used to deal with various problems of scattering and diffraction of elastic waves [29]. In its many variants, such a technique is based upon the superposition of solutions for sources with their singularities placed outside the region of interest. Then the single-layer boundary representation of elastic wave fields is described and applied to compute the response of various models of alluvial deposites for incident elastic waves in a half-space. Comparisons were done with other numerical solutions for a soft hemispherical inclusion under incident SH waves. The responses of 3-D valleys were computed in both frequency and time domains for incident P, S, and Rayleigh waves.
While boundary-element methods have been popular for moderately-sized linear models, the inhomogeneity, nonlinearity, and large scale basins such as the LA Basin preclude their use here.