**1. Introduction**

A seismogram synthesis is the theoretically calculated ground motion that would be recorded for a given crust structure and seismic source, which is an important tool that can be applied to the study of earthquake source from strong motion records. At the early stage of seismology, the reflectivity method [1] and the generalized ray method [2] were widely used for simulating seismic wave excitation and propagation, where a laterally homogeneous model is considered. The real Earth's crust, however, is a laterally inhomogeneous structure indeed. To study the site effect of strong ground motion, a stratified structure with irregular interfaces is an eligible model for simulating seismic wave propagation. For the study of seismic waves propagating through a sediment filled basin in the case of rigid grains, the work [3-4] is noteworthy, where a model was proposed to reproduct a nonlinear effect experimentally observed for real seismic waves: site amplification decreases as the amplitude of the incident wave increases.

In the last three decades, the study of seismic wave excitation and propagation in laterally inhomogeneous media has become extensively important in seismology and geophysics, including both analytical solutions and non-analytical results. The analytical solutions, however, only exist for a few special cases [5], such as the semi-cylindrical canyon and alluvial valleys, semi-elliptical canyon, and hemispherical canyon. The methods based on high-frequency approximation [6](such as asymptotic ray theory, Gaussian beam method, and so on) can give a proper solution by including the contribution from the neighboring rays but are only appropriate for handling body wave propagation under relatively high frequency. The methods based on plane wave decomposition (such as discrete wave number method [7-10]) can provide a complete wave field for any finite frequency but usually involve the truncation of matrices and vectors having an infinite dimension. Nowadays, the

© 2012 Qian, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

major approaches to study a laterally inhomogeneous model mainly rely on numerical methods which include domain methods and boundary methods. The domain numerical methods (such as finite difference [11], finite element [12], pseudo-spectral method [13], and spectral element method [14]) have become the standard tools for seismic wave modeling, however, these methods do not explicitly consider the boundary continuity conditions between different formations. That disables the methods to sufficient accuracy for modeling the reflection/transmission across irregular interfaces. Furthermore, they have difficulties in dealing with a large-scaled model because they require the subdivision of the domain into elements, especially the case when the domain needs to be extended to infinity. Boundary numerical methods have emerged as good alternatives to the domain numerical methods in dealing with an infinite continuum model because the radiation condition at infinity is easily fulfilled. They also have an obvious advantage over the domain numerical methods: the dimensionality of the problem under consideration is reduced by one order, because only a boundary instead of a domain discretization is required. Among this type of methods, the boundary element method (BEM) has been extensively used in simulating seismic wave excitation and propagation [15]. However, for a stratified model with irregular interfaces, the dimension of the final system of simultaneous equations in the traditional boundary element metod is proportional to the production of the element number and the interface number, which leads to an exponential increase in computing time and memory requirement. In order to overcome that problem, different approaches have been tried, for example, Fu [16] proposed an improved block Gaussian elimination scheme and Bouchon et al. [17] introduced a sparse method. Recently, global matrix propagators were introduced to improve the efficiency of the traditional BEM for modeling seismic wave excitation and propagation in multilayered solids [18-19], which expects great savings in computing time and memory requirement.

An Efficient Approach for Seismoacoustic Scattering Simulation Based on Boundary Element Method 127

However, there are still difficuilties in using these methods to simulate wave propagation in models with highly irregular topography since these methods do not explicitly consider the boundary and continuity conditions between different formations. That disables the methods to sufficient accuracy for modeling the wave reflection/transmission across irregular interfaces. Furthermore, the two methods usually require very large computational

To overcome the problem, the reflection/transmission matrices for the fluid-solid irregular interface [20] were developed based on the discrete wavenumber representation of the wave field, i.e. the Aki-Larner method [25]. But their method is only suitable for a longwavelength irregularity and becomes unstable at very steep interfaces (e.g. a vertical interface), which is due to the intrinsic feature of the method itself. On the other hand, BEM is more suitable than the other methods to model the seismoacoustic scattering at very steep irregular interfaces[26], especially in the cases when better accuracy is required near boundaries and sources. However, the traditional BEM requires a tremendous increase in memory capacity and computational time when it's used to model the wave propagation in multilayered media. Motivated by the work in [19-20], we develop a method based on the combination of the traditional BEM and a global matrix propagator to simulate the seismoacoustic scattering due to an irregular fluid-solid interface. This method takes the advantage of the global matrix propagator to suppress the tremendous increase in computational resources required for simulating wave propagation in multilayered media, but also retains the merits of the boundary element method. This implies that higherfrequency seismoacoustic scattering in complex structures can be calculated with reduced

For simplicity, the Chapter is organized as follows: the mathematical formulation for an SHwave (Shear wave polarized in the horizontal plane) propagation in a multilayered solid is first presented in an easily-understood way in Section 2, where some simple examples are calculated to test the formulation in solid layers; then we gratually go deep into the full formulation for the fluid-solid scattering simulation in Section 3, where two irregular fluidsolid models are used to test the validity of the fluid-solid formulation; based on the formulation in Section3, one of the two fluid-solid models is calculated to show the effects of water layer and the water reverberation in water layer in Section 4; finally summary is

The problem to be studied is illustrated in Figure 1. In this model, there are *L* homogeneous

irregular interfaces (*<sup>i</sup>*-1) and (*<sup>i</sup>*) (Throughout the section, the superscript with round brackets indicates layer index and the subscript with round brackets interface index). The uppermost interface is a free surface, and an arbitrary source is embedded in the *s*th layer. Assume each individual layer to be isotropic, linearly elastic material, so the tensor of elastic constants is

(*i*=1, 2, …, *L*) over a half-space, among which the *i*th layer is bounded by two

**2. SH-wave propagation modeling in mutilayered solids** 

resources (i.e. a large amount of memory and long calculation time).

computer resources.

drawn in Section 5.

layers (*i*)

**2.1. Methodology statement** 

On the other hand, the seismoacoustic scattering due to an irregular fluid-solid interface must be considered when we model the seismic wave propagation in oceanic regions or gulf areas. This kind of problem is related to a wide range of seismic research conducted at or close to oceanic regions or gulf areas [20], such as deep ocean acoustic experiments, ocean bottom seismic observations, or the interpretation of the effects of water layer on the observed surface wave traveling through gulf areas. Many metropolitan cities all over the world are located at or close to seaside, so it is important to take into account the effect of the sea water when conducting the earthquake resistant design for high-rise buildings in such big gulf cities. Also, the underground structures beneath sea bottom are known to be strongly inhomogeneous. Therefore, it is necessary to simulate the seismoacoustic scattering in irregularly multilayered elastic media overlain by a fluid layer due to some scenario earthquake event. Furthermore, ocean bottom observations are a key feature in the modern world-wide standardized seismograph network [21]. Correct seismic wave modeling can help to obtain valid interpretation of the observed waveforms. In the past three decades, many techniques, classified into different categories [7, 18], have been developed for calculating seismic wave excitation and propagation in laterally inhomogeneous media. Among those methods, the finite difference method (FDM) [22-23] and the finite element method (FEM) [24] become very popular in modelling seismoacoustic scattering due to irregular fluid-solid interface because of their flexibility in modeling complex media. However, there are still difficuilties in using these methods to simulate wave propagation in models with highly irregular topography since these methods do not explicitly consider the boundary and continuity conditions between different formations. That disables the methods to sufficient accuracy for modeling the wave reflection/transmission across irregular interfaces. Furthermore, the two methods usually require very large computational resources (i.e. a large amount of memory and long calculation time).

To overcome the problem, the reflection/transmission matrices for the fluid-solid irregular interface [20] were developed based on the discrete wavenumber representation of the wave field, i.e. the Aki-Larner method [25]. But their method is only suitable for a longwavelength irregularity and becomes unstable at very steep interfaces (e.g. a vertical interface), which is due to the intrinsic feature of the method itself. On the other hand, BEM is more suitable than the other methods to model the seismoacoustic scattering at very steep irregular interfaces[26], especially in the cases when better accuracy is required near boundaries and sources. However, the traditional BEM requires a tremendous increase in memory capacity and computational time when it's used to model the wave propagation in multilayered media. Motivated by the work in [19-20], we develop a method based on the combination of the traditional BEM and a global matrix propagator to simulate the seismoacoustic scattering due to an irregular fluid-solid interface. This method takes the advantage of the global matrix propagator to suppress the tremendous increase in computational resources required for simulating wave propagation in multilayered media, but also retains the merits of the boundary element method. This implies that higherfrequency seismoacoustic scattering in complex structures can be calculated with reduced computer resources.

For simplicity, the Chapter is organized as follows: the mathematical formulation for an SHwave (Shear wave polarized in the horizontal plane) propagation in a multilayered solid is first presented in an easily-understood way in Section 2, where some simple examples are calculated to test the formulation in solid layers; then we gratually go deep into the full formulation for the fluid-solid scattering simulation in Section 3, where two irregular fluidsolid models are used to test the validity of the fluid-solid formulation; based on the formulation in Section3, one of the two fluid-solid models is calculated to show the effects of water layer and the water reverberation in water layer in Section 4; finally summary is drawn in Section 5.
