Numerical Solutions to the Wave Equation
Numerical Solutions to the Wave Equation: This chapter introduces some popular numerical methods for approximating solutions to the acoustic and elastic wave equations. Such solutions include all events from primary and multiple scattering, and so are used for reverse time migration and waveform inversion. We will give an overview of the finite-difference method on regular grids and on staggered grids. Higher-order stencils are needed for better accuracy, so spectral collocation methods have been developed that are more accurate than the finite-difference method. Trefethen writes “If one wants to solve an ODE or PDE to high accuracy on a simple domain, and if the data defining the problem are smooth, then spectral methods are usually the best tool. They can often achieve ten digits of accuracy where a finite-difference or finite element method would get two or three. At lower accuracies, they demand less computer memory than the alternatives.” Therefore, this chapter also describes the pseudospectral and the spectral element methods. The pseudospectral method (has the highest accuracy, but can suffer from significant numerical errors at interfaces with large impedance contrasts. This problem can be mostly overcome with the spectral element method, and still retain high-order accuracy and reasonable eficiency in the solution.
Figures & Tables
This book describes the theory and practice of inverting seismic data for the subsurface rock properties of the earth. The primary application is for inverting reflection and/or transmission data from engineering or exploration surveys, but the methods described also can be used for earthquake studies. I have written this book with the hope that it will be largely comprehensible to scientists and advanced students in engineering, earth sciences, and physics. It is desirable that the reader has some familiarity with certain aspects of numerical computation, such as finite-difference solutions to partial differential equations, numerical linear algebra, and the basic physics of wave propagation (e.g., Snell’s law and ray tracing). For those not familiar with the terminology and methods of seismic exploration, a brief introduction is provided in the Appendix of Chapter 1. Computational labs are provided for most of the chapters, and some field data labs are given as well. Matlab and Fortran labs at the end of some chapters are used to deepen the reader’s understanding of the concepts and their implementation. Such exercises are introduced early and geophysical applications are presented in every chapter. For the non-geophysicist, geophysical concepts are introduced with intuitive arguments, and their description by rigorous theory is deferred to later chapters.