The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems
Abstract
In this paper, we are concerned with interpreting some classical inverse-scattering theories so that they are relevant to the one-dimensional inverse impulse-response problem which arises in reflection seismology
First the Gelfand-Levitan integral equations (which arise in the inverse scattering theory for the Schrödinger equation) are derived strictly in the time domain. Originally these celebrated equations were derived as a means of solving an inverse spectral problem, which is naturally posed in the frequency domain. We show not only that these equations have a time-dependent formulation, but that their derivation is actually simpler here than in its original context. Next we give a similar derivation for the Marchenko integral equation. We then obtain, by an independent method, the integral equation of Gopinath and Sondhi, which is not unlike the Gelfand-Levitan linear equation. It was used by them as a means of solving a time-dependent inverse problem arising in speech synthesis. A new integral equation, similar to the Marchenko equation is also derived. Finally the integral equations are related to each other by direct transformation independent of the corresponding differential equations. The present paper opens with a section in which the equation of one-dimensional elastic waves and the corresponding seismic inverse impulse-response problem are transformed into a form to which the Gelfand-Levitan theory applies, and then into the equations which arose in Gopinath and Sondhi's work. We regard the integral equation of Gopinath and Sondhi as being directly applicable to the interpretation of seismic reflection data. The Gelfand-Levitan theory is also applicable but only after considerable transformation. Our calculations are entirely in the time domain. The resulting equations have the merit that in order to recover the unknown coefficient on a finite interval (0, L) we need to use the boundary data also only on a finite segment of the reflection time series. The length of the record is just the two-way travel time associated with length L. By contrast, frequency-domain approaches require that long records be Fourier transformed. We distinguish results which are true only when the unknown coefficient is continuous from those true when it is bounded but only piecewise continuous. In this work we have not addressed the important problem of deconvolution.- Publication:
-
Wave Motion
- Pub Date:
- 1980
- DOI:
- 10.1016/0165-2125(80)90011-6
- Bibcode:
- 1980WaMot...2..305B