Stability considerations for one-dimensional inverse problems

Summary. We investigate the issues of stability and conditioning for the one-dimensional seismic inverse problem. We show that these issues are distinct; i.e. that numerically stable implementations of solutions to the inverse problem will not give accurate results if the problem is ill-conditioned. In addition, we identify the factors which determine the condition of the inverse problem. These are the total variation of the acoustic impedance profile being sought and the accuracy of the low-frequency content of the reflection data. We illustrate these results on implementations of two numerically stable algorithms for the inverse problem, one of which has a reputation for being unstable. A comparison shows nearly identical results for the two methods on noise-contaminated and frequency band limited reflection data. In fact, we conjecture that all of the well-known 'layer-stripping' inverse scattering methods share the same mathematical stability characteristics. On the other hand, we also show that ill-conditioning can lead to failure of such algorithms, through amplification of error due either to inaccurate data or to discretization or roundoff. Finally, we observe that appropriate smoothing of the seismic data for an ill-conditioned inverse problem (high-variation impedance profile) can cause the problem to become well-conditioned (lower-variation profile). As is typical with regularizations, the price paid for the newly-acquired ability to solve the problem is a loss of accuracy in the solution.