Minimum relative entropy inversion: Theory and application to recovering the release history of a groundwater contaminant

In this paper we show that given prior information in terms of a lower and upper bound, a prior bias, and constraints in terms of measured data, minimum relative entropy (MRE) yields exact expressions for the posterior probability density function (pdf) and the expected value of the linear inverse problem. In addition, we are able to produce any desired confidence intervals. In numerical simulations, we use the MRE approach to recover the release and evolution histories of plume in a one-dimensional, constant known velocity and dispersivity system. For noise-free data, we find that the reconstructed plume evolution history is indistinguishable from the true history. An exact match to the observed data is evident. Two methods are chosen for dissociating signal from a noisy data set. The first uses a modification of MRE for uncertain data. The second method uses ''presmoothing'' by fast Fourier transforms and Butterworth filters to attempt to remove noise from the signal before the ''noise-free'' variant of MRE inversion is used. Both methods appear to work very well in recovering the true signal, and qualitatively appear superior to that of Skaggs and Kabala [1994]. We also solve for a degenerate case with a very high standard deviation in the noise. The recovered model indicates that the MRE inverse method did manage to recover the salient features of the source history. Once the plume source history has been developed, future behavior of a plume can then be cast in a probabilistic framework. For an example simulation, the MRE approach not only was able to resolve the source function from noisy data but also was able to correctly predict future behavior.