Comparing hard and soft prior bounds in geophysical inverse problems

Abstract

Summary In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about the correct earth model x   E is essential if y is to supply useful limits for z. The one exception occurs when all the prediction functionals are linear combinations of the data functionals. We compare two forms of prior information: a ‘soft’ bound on x   E is a probability distribution px on the model space X which describes the observer's opinion about where x   E is likely to be in X; a ‘hard’ bound on x   E is an inequality QX (x   E , x   E ) 1, where Qx is a positive definite quadratic form on X. A hard bound Qx can be ‘softened’ to many different probability distributions px , but all these px 's carry much new information about x   E which is absent from Qx , and some information which contradicts Qx. For example, all the px 's give very accurate estimates of several other functions of x   E besides Qx (x   E , x   E ). And all the px 's which preserve the rotational symmetry of Qx assign probability 1 to the event Qx (x E , x   E ) =∞. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound px. If that probability distribution was obtained by softening a hard prior bound Qx , rather than by objective statistical inference independent of y, then px contains so much unsupported new ‘information’ absent from Qx that conclusions about z obtained with SI or BI would seem to be suspect.