In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0.
For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion.
An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light.
The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.