Minimum relative entropy and probabilistic inversion in groundwater hydrology |
| |
| Authors: | Allan D Woodbury Tadeusz J Ulrych |
| |
| Institution: | (1) Department of Civil and Geological Engineering University of Manitoba, Winnipeg, Manitoba, Canada, CA;(2) Department of Earth and Ocean Sciences University of British Columbia, Vancouver, B.C. Canada, CA |
| |
| Abstract: | The similarity between maximum entropy (MaxEnt) and minimum relative entropy (MRE) allows recent advances in probabilistic
inversion to obviate some of the shortcomings in the former method. The purpose of this paper is to review and extend the
theory and practice of minimum relative entropy. In this regard, we illustrate important philosophies on inversion and the
similarly and differences between maximum entropy, minimum relative entropy, classical smallest model (SVD) and Bayesian solutions
for inverse problems. MaxEnt is applicable when we are determining a function that can be regarded as a probability distribution.
The approach can be extended to the case of the general linear problem and is interpreted as the model which fits all the
constraints and is the one model which has the greatest multiplicity or “spreadout” that can be realized in the greatest number
of ways. The MRE solution to the inverse problem differs from the maximum entropy viewpoint as noted above. The relative entropy
formulation provides the advantage of allowing for non-positive models, a prior bias in the estimated pdf and `hard' bounds
if desired. We outline how MRE can be used as a measure of resolution in linear inversion and show that MRE provides us with
a method to explore the limits of model space. The Bayesian methodology readily lends itself to the problem of updating prior
probabilities based on uncertain field measurements, and whose truth follows from the theorems of total and compound probabilities.
In the Bayesian approach information is complete and Bayes' theorem gives a unique posterior pdf. In comparing the results
of the classical, MaxEnt, MRE and Bayesian approaches we notice that the approaches produce different results. In␣comparing
MaxEnt with MRE for Jayne's die problem we see excellent comparisons between the results. We compare MaxEnt, smallest model
and MRE approaches for the density distribution of an equivalent spherically-symmetric earth and for the contaminant plume-source
problem. Theoretical comparisons between MRE and Bayesian solutions for the case of the linear model and Gaussian priors may
show different results. The Bayesian expected-value solution approaches that of MRE and that of the smallest model as the
prior distribution becomes uniform, but the Bayesian maximum aposteriori (MAP) solution may not exist for an underdetermined
case with a uniform prior. |
| |
| Keywords: | |
| 本文献已被 SpringerLink 等数据库收录！ |
|
