Abstract 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.

Abstract Introduction to Seismic Inversion: Seismic inversion, or tomography, is the procedure for reconstructing earth properties from seismic data. The tomogram is presented graphically as a two- or three-dimensional (2D or 3D) grid of pixels, in which each pixel contains the model parameter of interest, such as the velocity value or its reciprocal value known as slowness. An example is shown in Figure 1.1, which is a vertical slice of a 3D tomogram inverted from first-arrival traveltimes picked from seismic data. These arrivals are assumed to propagate mostly along the dashed raypaths (see Figure 1.2a) from the source at s to the geophone g, and they are recorded as a seismic trace. An actual set of traces recorded for one shot is shown in Figure 1.2b, and Figure 1.3 depicts several rays associated with different types of events. Appendix 1A provides an overview of exploration seismology.

Abstract Introduction to Gradient Optimization: The unconstrained optimization problem for the real-valued functional f(x) is defined as finding the optimal model vector x*that satisfies the following condition.