۱- Introduction

۲- Stiffness matrices for fluid and anisotropic soil layers

۳- Impedances of foundations on anisotropic soil

۴- Rayleigh and Love waves in anisotropic half space

۵- Vertical point load applied to layered half-spaces

۶- Discontinuity seismic sources

۷- Derivation of PML parameters

۸- Summary and conclusion

References

Abstract

In this study, we introduce and discuss features and improvements of the well-established stiffness matrix method that is used in simulation of wave propagation in layered media. More specifically, we present stiffness matrices for an acoustic layer and a vertically transverse isotropic (VTI) viscoelastic soil layer. Combining these stiffness matrices enables a straightforward technique for modeling of acousto-elastic wave propagation in layered infinite media. In addition, we propose a technique to simulate discontinuity seismic sources, which was not used earlier in the context of the stiffness matrix method. Finally, we propose a framework to derive a key parameter of the absorbing boundary domain technique Perfectly Matched Layer (PML). Numerical examples are presented in order to help understanding the features and improvements discussed in the study from the fields of geophysics and soil dynamics. It is believed that the features and improvements discussed herein will make the application of the stiffness matrix method even wider and more flexible.

Introduction

The stiffness matrix method is a well-developed approach for simulating wave propagation in layered media, and has been successfully applied to various problems during the last decades (e.g. [1–۳] and [4]). The method describes the wave motion in a layered medium in terms of symmetric and banded matrices and with straightforward and efficient solution procedure, producing the dynamic responses simultaneously at all layer interfaces and in all directions. The method has later been extended to acoustic layers ([5,6]). The discrete version of the stiffness matrix solution, called Thin-Layer Method (TLM), has also been developed and applied to various problems ([7–۹]). Recently, TLM has been combined with the so-called Perfectly Matched Layer method (PML) that enables calculation of wave motion in infinite domains [10]. Despite these extensions, there are still features and improvements of the stiffness matrix method that could advance the use of the method in theoretical and applied problems. The present study introduces and discusses some of those features, including • Vertically transverse isotropic (VTI) soil layer stiffness • Discontinuity seismic sources • Derivation of PML parameters For completeness, first we present the acoustic layer stiffness matrices in forms that can be used in offshore or fluid-soil-coupled applications (e.g. seismic wave in the ocean environment). We introduce three different formulations in terms of vertical displacement, velocity potential and pressure. Each formulation has its own advantages and disadvantages. For example, the second and third formulations make it straightforward to implement the so-called air-gun source that is used as explosive acoustic source in offshore seismic surveys. This is because the air volume injected by the air-gun is explicitly defined in the two formulations, which are shown later. On the other hand, the first formulation is more suitable for applying vertical disk load on seabed or within water column, because the disk load can be represented by a term that can be set directly in the matrix equations. It is also shown that the three solutions are interrelated such that one can be derived from the other two through relevant constitutive laws. Next, the soil stiffness matrices for the vertically transverse isotropic (VTI) layers are derived for both P-SV (in-plane) and SH (antiplane) wave modes.