The Effect of Discontinuities on the Order of Convergence of Numerical Inversion Methods

Abstract

We study numerical methods for a one-dimensional inverse problem in reflection seismology. An impulsive wave in pressure is applied to the surface of a piecewise-smooth stratified elastic half-space and the resulting particle velocity is measured at the surface. The characteristic impedance of the medium is to be determined as a function of travel time. It has been shown recently that if the impulse response is measured appropriately, then several numerical methods to solve this inverse problem approximately are uniformly second-order convergent for piecewise-smooth media with discontinuities occurring only on the grid, i.e., at integer multiples of the mesh width. Piecewise-constant examples show that if the discontinuities do not occur on the grid, uniform convergence is lost near travel times, either where discontinuities occur or which correspond to multiple reflections. In intervals bounded away from these points, convergence appears to be uniform and at least second order. We present an example here of a piecewise-polynomial medium with a discontinuity off the grid for which at best first-order convergence is obtained at any point past the discontinuity. We conclude that unless the location of the discontinuities is either known a priori or determined as part of the solution to the inverse problem, no numerical method for this inverse problem can be expected to converge beyond the first discontinuity any faster than first order. This conclusion carries over to many related reflection inverse problems, including inverse problems for the wave equation with unknown variable velocity.