THE MAXIMUM ENTROPY APPROACH TO THE INVERSION OF ONE‐DIMENSIONAL SEISMOGRAMS¹

Abstract

Determination of impedance or velocity from a stacked seismic trace generally suffers from noise and the fact that seismic data are bandlimited. These deficiencies can frequently be alleviated by ancillary information which is often expressed more naturally in terms of probabilities than in the form of equations or inequalities. In such a situation information theory can be used to include ‘soft’information in the inversion process. The vehicle used for this purpose is the Maximum Entropy (ME) principle. The basic idea is that a prior probability distribution (pd) of the unknown parameter(s) or function(s) is converted into a posterior pd which has a larger entropy than any other pd which also accounts for the information. Since providing new information generally lowers the entropy, this means that the ME pd is as non‐committal as possible with regard to information which is not (yet) available. If the information used is correct, then the ME pd cannot be contradicted by new, also correct, data and thus represents a conservative solution to the inverse problem. In the actual implementation, the final result is, generally, not the pd itself (which may be quite broad) but rather the expectation values of the desired parameter(s) or function(s). A general problem of the ME approach is the need for a prior pd for the parameter(s) to be estimated. The approach used here for the velocity is based on an invariance criterion, which ensures that the result is the same whether velocity or slowness is estimated. Unfortunately, this criterion does not provide a unique prior pd but rather a class of functions from which a suitable one must be selected with the help of other considerations.

Minimum relative entropy and probabilistic inversion in groundwater hydrology

“…In the ®eld of geophysics the familiar technique of high resolution spectral analysis (Haykin, 1983), autoregessive modeling (Ulrych and Bishop, 1975) and predictive deconvolution (Robinson and Treitel, 1980) are all closely related to the PME. This principle has also been applied by Dong et al (1984) and Shen and Mansinha (1983) to the problem of magnitude-frequency relationship of earthquakes and by Rietsch (1988) in the inversion of one dimensional seismograms. Applications in the ®eld of image reconstruction have met with great success (Gull and Skilling, 1984).…”

“…The model estimation problem can now be approached from the point of view of probability theory. Given an estimate of the pdf of the model, subject to constraints, we can obtain an estimate of the model by computing expected values (Rietsch, 1977). However, it is most common in applications of probabilistic inversion to assume Gaussian prior pdf 's and likelihoods.…”

“…The second example to be discussed in this paper was ®rst detailed by Rietsch (1977) as an example application of MaxEnt. The problem is also cited as a classic example of a grossly undetermined linear inverse problem (Parker, 1994).…”

“…Figure 4 shows the`extended' MaxEnt solution (bold), Rietsch's (1977) analytic solution (his Eq. 62, dots) and Bullen's (1975) earth model II (dashed).…”

On the Use of Information Theory to Quantify Parameter Uncertainty in Groundwater Modeling

“…The basic concept of Jaynes' maximum entropy approach is that when making inferences based on incomplete information for given expectations (prior information) of a univariate or multivariate random function, the probability distribution must have the maximum entropy permitted by the available information expressed in the form of constraints [10]. The maximum entropy approach was applied to various geophysical problems such as seismic spectral analysis [11], seismic deconvolution [12], and earthquakes [13]. Singh [10] reviewed the maximum entropy applications in the studies of hydrology and water resources.…”