A popular way to account for uncertainty in inverse problems is to use a
Bayesian approach and define a posterior distribution of the quantities of
interest. A conventional Bayesian treatment, however, requires assuming
specific values for parameters of the prior distribution and of the
distribution of the measurement errors (e.g., the standard deviation of the
measurement errors). In practice, these parameters are often poorly known
a priori and it is difficult for users to set their value. Moreover, the
posterior uncertainty is predicated on fixing these parameters; if they are
not well known a priori, the posterior uncertainties have dubious value.
This talk presents extensions to the conventional Bayesian treatment that
allow for uncertainty in the parameters defining the prior distribution and
the distribution of the measurement errors. These extensions are known in
the statistical literature as "empirical Bayes" and "hierarchical
Bayes." I will demonstrate the practical application of these approaches
to a simple linear inverse problem: using seismic travel times measured by
a receiver in a well to infer compressional wave slowness in a 1D
earth. These procedures do not require choosing fixed values for poorly
known parameters, but only a range of realistic values (e.g., a minimum and
maximum value for the standard deviation of the measurement
errors). Inversion is thus made easier for general users, who are not
required to set parameters they know little about.
Copyright. All Rights Reserved.