The relationship between “direct, discrete” and “iterative, continuous” one‐dimensional inverse methods

Abstract

Two distinct approaches to solving the one‐dimensional seismic inverse problem are compared. These are (1) the “direct” method of Goupillaud (1961), applied to discretely varying media, and (2) the “iterative” methods of Gjevik et al (1976), or Gray and Hagin (1982), applied to discretely or continuously varying media. These two approaches are shown to be equivalent in two important respects. First, each method can be recovered from the other [e.g., the discretized version of the iterative methods yields the same set of equations as the direct method]. Second, because of the first equivalence, each method uses the same amount of information in reconstructing a profile to a certain depth z or traveltime τ into the medium. This information is the reflection data received for times less than 2τ. In particular, neither approach uses the “redundant data” received after time 2T in an inversion for a profile which is known to vary only for depths which correspond to traveltime T. In this sense the methods are as economical as possible, using the minimum amount of information required to solve the idealized problem. The key to relating the discrete, direct inversion to the continuous, iterative inversion is the Bremmer (1951) series for the reflected wave field. By using this series, it is possible to show that the equivalent inversion methods invert the same equation for the unknown acoustic impedance variations. The difference in the approaches used to solve this equation is analogous to the difference between solving a system of linear equations “directly” or “iteratively.”