MINIMUM RELATIVE ENTROPY INVERSION OF 1D DATA WITH APPLICATIONS1

Abstract

AbstractThe pioneering work of E. T. Jaynes in the field of Bayesian/Maximum Entropy methods has been successfully explored in a number of disciplines. The principle of maximum entropy (PME) is remarkably powerful and versatile and leads to results which are devoid of spurious structure. Minimum relative entropy (MRE) is a method which has all the important attributes of the maximum‐entropy (ME) approach with the advantage that prior information may be easily included. These ‘soft’ prior constraints play a fundamental role in the solution of underdetermined problems. The MRE approach, like ME, has achieved considerable success in the field of spectral analysis where the spectrum is estimated from incomplete autocorrelations. In this paper we apply the MRE philosophy to 1D inverse problems where the model is not necessarily positive, and thus we show that MRE is a general method of tackling linear, underdetermined, inverse problems. We illustrate our discussion with examples which deal with the famous die problem introduced by Jaynes, the question of aliasing, determination of interval velocities from stacking velocities and, finally, the universal problem of band‐limited extrapolation. It is found that the MRE solution for the interval velocities, when a uniform prior velocity is assumed, is exactly the Dix formulation which is generally used in the seismic industry.