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.

Abstract Steepest-Descent Method: This chapter introduces the optimization method known as steepest descent (SD), in which the solution is found by searching iteratively along the negative gradient-g direction, the path of steepest descent. Relative to the Newton method for large problems, SD is inexpensive computationally because the Hessian inverse is not needed, but it can suffer from slow convergence with ill-conditioned Hessians because it does not take into account information about curvature. Nevertheless, experience shows that iterative SD with preconditioning and regularization can be quite useful when combined with multiscale methods for solving large seismic inverse problems.