DIRECT AND INVERSE PROBLEMS RELATING REFLECTION COEFFICIENTS AND REFLECTION RESPONSE FOR HORIZONTALLY LAYERED MEDIA

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Abstract
If a plane wave is normally incident on the boundary of a horizontally layered medium, the reflection and transmission responses can be found in terms of the reflection coefficients at the interfaces of the medium. Recursion formulas for finding these responses have been given by various authors: Goupillaud (1961), Sherwood (1962), Kunetz and d’Erceville (1962), Sherwood and Trorey (1965), and Claerbout (1968). This paper gives explicit solutions of the recursion formulas for the numerator and denominator of the reflection and transmission operators. The solutions are expressed as polynomials in the variable z of the z-transform, with coefficients given as explicit sums of products of reflection coefficients. These formulas are equivalent to those given by Goupillaud. We also give the operator which removes all effects of any given set of layers at the top, the operator being equivalent to one given by Goupillaud. The inverse problem of finding the reflection coefficients from a reflection record or transmission record has been solved in two different, equivalent forms by Kunetz and d’Erceville and Claerbout. This paper gives further results derived along the lines of Claerbout’s paper. Since the solutions are developed for the noise‐free case, the effects of timing errors, scaling errors, and random noise on the reflection coefficient series computation are investigated. Examples are given which show the zones of stabilities. The Appendix gives the general form of the energy conservation equation in elastic wave propagation in strict analogy with the Poynting formulation for electromagnetic wave propagation. The main body of the paper includes a proof that Claerbout’s polynomial identity is an energy conservation equation for the case when no free surface is present.

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