Abstract

Seismograms are routinely used to extract information about the internal structure of the subsurface. The basic philosophy is to idealize the earth with a simple model characterized by resolvable parameters. Further, we assume that the seismograms obey simple mathematical relations, such as the wave equation, and that these seismograms are uniquely determined by the given parameters.The computation of seismograms for a given model whose parameters are specified gives rise to the 'forward problem'. The determination of the parameters of the model from the seismogram generally constitutes the 'inverse problem'. While the forward problem is often relatively simple, the inverse problem tends to be much more involved.One of the few cases for which the inverse problem has a simple direct solution is given for a model consisting of a homogeneous one-dimensional layered medium. The layers overlie a homogeneous half-space, and are excited by a normally incident plane wave. Such direct solutions to the inverse problem are in contrast to the iterative inversion techniques of the Gilbert-Backus type. We have chosen to treat the inverse problem for the seismogram escaping into the homogeneous substratum, i.e., the transmission seismogram. We do so because this time series is a simple autoregressive, or all-pole process. The problem is studied both with and without white noise.The Wiener-Levinson algorithm has been found to be well suited for the direct inversion of the noise-free case, while the 'maximum entropy' algorithm is generally more appropriate for noisy data. Numerical examples serve to clarify and illustrate these points.

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