A hybrid method for the solution of seismic wave propagation problems

Abstract

Continuing improvements in data acquisition techniques result in an increase in quantity and quality of data available for interpretation in terms of earth models. At the same time increasing performance of computers used in the interpretation process allows more detailed models to be used in computer simulation and inversion. As a result, our knowledge of the structure and evolution of the earth is continuously increasing. Apart from the theoretical benefits, our growing abilities in computer simulation (modeling) allow us to make more precise predictions and risk assessments. More detail and greater flexibility in earth models, require the implementation of new computational techniques, to simulate the models numerically. Many computational techniques are based on a separation of variables, which limits their applicability to relatively simple models. On the other hand, techniques that do not have these limitations, such as discretization techniques like the finite difference and finite element method, are limited in their applicability by practical restrictions, such as finite computer memory and speed of computation. In this thesis a method is presented to compute the seismic response of a class of models, consisting of a regular background medium, with an irregular inclusion of finite extent. In this method the wave field on the bounded inclusion is fonnulated using a finite element description. For the wave field in the - possibly infinite - regular background medium we use an integral representation. The integral representation involved contains integral kernels related to the Green's function of the background medium. We define a background medium to be regular when an algorithm for the Green's function is available. Because this hybrid method relies only partly on a finite element formulation, the restrictions imposed by finite computer resources are less severe then for a full finite element method. As an illustration of the specific advantages of the method, consider the following problem. Suppose we want to model the effects of a local anomaly on the ground motion at the earth's surface, in the presence of an incident wave, excited by an earthquake at a teleseismic distance. If the dimensions of the anomaly are small compared to the radius of the earth and the distance between the anomaly and the earthquake source, we can use as a model a plane wave, incident from a regular halfspace containing a localized anomaly near the free surface. The presence of an irregular anomaly excludes the use of analytical methods to solve this problem. If we want to use a discretization method, the fact that we are restricted to finite computer resources forces us to truncate the model of the medium, introducing artificial boundaries. This leaves us with the problem how to deal with the undesirable boundary effects and how to specify the model excitation in the finite model. Using the hybrid method for this type of problem we only have to discretize the localized anomaly and a truncation of the model will not be necessary (assuming an anomaly of limited volume). The effect of the background medium and the wave field excitation are both included in the integral representation, therefore we no longer have undesirable boundary effects and the wave field excitation is handled correctly. We derive the formalism for the hybrid finite element / integral equation technique for the general case of elastic wave propagation problems in a three dimensional (3-D) general background medium, with an inclusion of finite volume. The method has been implemented for the more restricted case of SH waves in a homogeneous 2-D medium with a finite irregular inclusion. A concise description of the formalism of the hybrid method has been published in (van den Berg, 1984) and more recently, including computational results, in (van den Berg, 1987)