Comparing hard and soft prior bounds in geophysical inverse problems

Details Comparing hard and soft prior bounds in geophysical inverse problems Comparing hard and soft prior bounds in geophysical inverse problems

Aug 1988

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1988geoj...94..249b&link_type=abstract

Geophysical Journal (ISSN 0952-4592), vol. 94, Aug. 1988, p. 249-261. Previously announced in STAR as N88-13820.

Mathematics

Probability

31

Bayes Theorem, Inversions, Prediction Analysis Techniques, Probability Distribution Functions, Stochastic Processes, Geophysics, Linear Programming, Vectors (Mathematics)

Scientific paper

In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about X sub 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. Two forms of prior information are compared: a soft bound on X sub E is a probability distribution p sub x on X which describes the observer's opinion about where X sub E is likely to be in X; a hard bound on X sub E is an inequality Q sub x(X sub E, X sub E) is equal to or less than 1, where Q sub x is a positive definite quadratic form on X. A hard bound Q sub x can be softened to many different probability distributions p sub x, but all these p sub x's carry much new information about X sub E which is absent from Q sub x, and some information which contradicts Q sub x. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound p sub x. If that probability distribution was obtained by softening a hard prior bound Q sub x, rather than by objective statistical inference independent of y, then p sub x contains so much unsupported new information absent from Q sub x that conclusions about z obtained with SI or BI would seen to be suspect.

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