In the preceding chapter we examined the problem of tracing rays and calculating travel time curves from a known velocity structure. We derived expressions for ray tracing in a one-dimensional (1-D) velocity model in which velocity varies only with depth; ray tracing in general three-dimensional (3-D) structures is more complex but follows similar principles. We now examine the case where we are given travel times obtained from observations and wish to invert for a velocity structure that can explain the data. As one might imagine, the inversion is much more complicated than the forward problem. The main strategy used by seismologists, both in global and crustal studies, has generally been to divide the problem into two parts:

1. A 1-D “average” velocity model is determined from all the available data. This is generally a non-linear problem but is tractable since we are seeking a single function of depth. Analysis often does not proceed beyond this point.

2. If sufficient 3-D ray coverage is present, the 1-D model is used as a reference model and a travel time residual is computed for each datum by subtracting the predicted time from the observed time. A 3-D model is obtained by inverting the travel time residuals for velocity perturbations relative to the reference model. If the velocity perturbations are fairly small, this problem can be linearized and is computationally feasible even for large data sets. This is the basis for tomographic inversion techniques.