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

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...