THE MAXIMUM ENTROPY APPROACH TO THE INVERSION OF ONE‐DIMENSIONAL SEISMOGRAMS 1

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.