Abstract
The theory of a number of inverse problems requires that the input source be ``impulsive'' in time, i.e., it has an initial singularity like a delta-function or a jump discontinuity. In real data, this rarely is the case. The purpose of this study is to investigate the effect which the type of initial singularity in the source has on the convergence of numerical methods for inverse problems. We take as our test problem the one-dimensional inverse problem of reflection seismology. Using a standard second-order difference scheme to approximate the forward problem, we show that second-order convergence can be obtained in the inverse problem if either the source or its first derivative in time has an initial jump discontinuity. First order convergence can be obtained if the second, third, or fourth derivative of the input source has an initial jump discontinuity. Surprisingly, first order convergence can be obtained if any derivative of the input source has an initial jump discontinuity, provided the response is sampled and processed appropriately. We conclude that there is theoretically enough information in the sampled response to smooth input sources to solve the inverse problem to first-order; numerically, the solution of the inverse problem may be very poorly conditioned. However, since the information is present in the response, regularized formulations for solving the inverse problem have a reasonable chance of success even with smooth input sources.