Minimum Relative Entropy Inversion: Theory and Application to Recovering the Release History of a Groundwater Contaminant

Abstract

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.

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

“…The entropy theory has been commonly used in hydrology for various applications, including probability inferences [37,38], parameter estimation [25,39], frequency analysis [1], geo-statistical modeling [6,[40][41][42], uncertainty or variability analysis [43,44], regionalization [45] and network design [46][47][48]. For a detailed review of applications of entropy in hydrology and water resources, the readers are referred to [6,[8][9][10].…”

Application of Simulated Annealing and Adaptive Simulated Annealing in Search for Efficient Optimal Solutions of a Groundwater Contamination related Problem

“…Other approach for source identification consists of solving the differential equations backwards in time (inverse problem). The random walk particle method [10,11], the quasi-reversibility technique [12], the minimum relative entropy method [13], the Bayesian theory and geostatistical techniques [14], and genetic algorithm [15,16] are some examples of this approach.…”

Linear Programming Method for Investigating the Disposal Histories and Locations of Pollutant Sources in an Aquifer

“…Neupauer et al (2000) compared the relative effectiveness of the methods from studies of Skaggs and Kabala (1994) and Woodbury and Ulrych (1996). Sidauruk et al (1998) presented an inverse method based on analytical solutions of contaminant transport problems.…”