Comparing hard and soft prior bounds in geophysical inverse problems

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Abstract
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 xE 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 xE is a probability distribution px on the model space X which describes the observer's opinion about where xE is likely to be in X; a ‘hard’ bound on xE is an inequality QX(xE, xE) 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 xE 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 xE besides Qx(xE, xE). And all the px's which preserve the rotational symmetry of Qx assign probability 1 to the event Qx(xE, xE) =∞. 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.

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