Conjugate-Gradient and Quasi-Newton Methods
Conjugate-Gradient and Quasi-Newton Methods: We now will discuss two gradient-optimization methods commonly used in geophysical inversion: the conjugate-gradient (CG) method and the quasi-Newton (QN) method. Unlike the Newton method, these two methods do not explicitly compute the inverse to the Hessian; instead, they iteratively move along descent directions that reduce the data residual. Each iteration costs only O(N2) operations of a matrix-vector multiply. Another strength is that, in the case of CG and low-memory QN methods, no Hessian matrix needs to be stored or inverted explicitly. Their weakness is that fast convergence is not guaranteed. However, they are generally faster than the nonpreconditioned steepest-descent method.
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.