Comparing hard and soft prior bounds in geophysical inverse problems

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 p x 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 Q X ( x E , x E ) 1, where Q x is a positive definite quadratic form on X. A hard bound Q x can be ‘softened’ to many different probability distributions p x , but all these p x 's carry much new information about x E which is absent from Q x , and some information which contradicts Q x . For example, all the p x 's give very accurate estimates of several other functions of x E besides Q x ( x E , x E ). And all the p x 's which preserve the rotational symmetry of Q x assign probability 1 to the event Q x (x E , x E ) =∞. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound p x . If that probability distribution was obtained by softening a hard prior bound Q x , rather than by objective statistical inference independent of y , then p x contains so much unsupported new ‘information’ absent from Q x that conclusions about z obtained with SI or BI would seem to be suspect.