Traveltime Calculation by Solution of the Eikonal Equation: This chapter presents the numerical solution of the eikonal equation by a finite-difference method, in which the output is the first-arrival travel-time field for a smoothly varying inhomogeneous velocity model. These traveltimes can be used for either traveltime tomography or reflection migration.
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.