Premium
BAYESIAN ESTIMATION IN SEISMIC INVERSION. PART II: UNCERTAINTY ANALYSIS 1
Author(s) -
DUIJNDAM A. J. W.
Publication year - 1988
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1111/j.1365-2478.1988.tb02199.x
Subject(s) - a priori and a posteriori , covariance , inverse problem , probability density function , gaussian , mathematics , bayesian probability , maximum a posteriori estimation , estimation theory , inversion (geology) , mathematical optimization , algorithm , statistics , geology , mathematical analysis , maximum likelihood , paleontology , philosophy , physics , epistemology , quantum mechanics , structural basin
A parameter estimation or inversion procedure is incomplete without an analysis of uncertainties in the results. In the fundamental approach of Bayesian parameter estimation, discussed in Part I of this paper, the a posteriori probability density function (pdf) is the solution to the inverse problem. It is the product of the a priori pdf, containing a priori information on the parameters, and the likelihood function, which represents the information from the data. The maximum of the a posteriori pdf is usually taken as a point estimate of the parameters. The shape of this pdf, however, gives the full picture of uncertainty in the parameters. Uncertainty analysis is strictly a problem of information reduction. This can be achieved in several stages. Standard deviations can be computed as overall uncertainty measures of the parameters, when the shape of the a posteriori pdf is not too far from Gaussian. Covariance and related matrices give more detailed information. An eigenvalue or principle component analysis allows the inspection of essential linear combinations of the parameters. The relative contributions of a priori information and data to the solution can be elegantly studied. Results in this paper are especially worked out for the non‐linear Gaussian case. Comparisons with other approaches are given. The procedures are illustrated with a simple two‐parameter inverse problem.