Travel time(like) variables and the solution of velocity inverse problems

It was demonstrated earlier that, in the solution of 1-D inverse problems, there is great advantage to changing the independent variable from, say, z (depth) to travel time [Formula: see text]. Essentially the advantage comes from the fact that the acoustic wave equation in travel time has the unknown c(z) appearing in a less critical position. The current paper takes a step toward applying these ideas to the much harder, but more interesting, inverse problem in three dimensions. There is no simple 3-D analog of the above definition of τ. However, it is shown that a surprisingly effective way of decomposing travel time into x, y, z components is straightforward. These are defined via line integrals from, say, (0, 0, 0) to an arbitrary point (x, y, z) along the straight line connecting the points, thus approximating the more natural integrals along the unknown raypath. These line integrals define the new coordinates, and the associated wave equation is derived and then simplified by dropping less important terms. The inverse problem is then attacked in this setting using the 3-D inversion techniques of Cohen and Bleistein (1979). The resulting algorithm is demonstrated to be very similar to those earlier results; however, it is shown that for a single reflecting plane the new results are of “second‐order” accuracy as oppose to first order (when the change in c is small relative to c itself). The algorithm also has some similarity to that of Raz (1982), and some comparisons between these two results are made.