Inverse Scattering for Goupillaud Horizontally Layered Earth Model
The Goupillaud horizontally layered earth model and its inverse scattering problem is considered in this paper: the determination of the reflection coefficient’s from an excitation-response pair. This problem is the prototype for the inverse scattering problems arising in various fields, obtaining the equations of the inverse scattering problem and mentioning that these equations are direct consequences of the causality property of the signal propagation model. The discrete analogues of the classical equations of Gelfand-Levitan, Marchenko and Krein are obtained with special choices of scattering data pairs for free-surface and non-free-surface reflections.
With the Goupillaud model, it can be seen that an efficient layer-recursive inverse scattering procedure arises more directly than do the linear equations; this result only invokes the causality of signal propagation and a simple rule for recursively computing the discrete waveforms at increasing depths in the scattering medium. This approach provides fast recursive inversion procedures, that at the beginning were called dynamic predictive deconvolutions and afterwards direct layer-peeling or Schur-type algorithms.
KeywordsReflection Coefficient Inverse Scattering Power Series Expansion Inversion Algorithm Reflection Function
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