Extensions to minimum relative entropy inversion for noisy data

Minimum relative entropy (MRE) and Tikhonov regularization (TR) were compared by Neupauer et al. [Water Resour. Res. 36 (2000) 2469] on the basis of an example plume source reconstruction problem originally proposed by Skaggs and Kabala [Water Resour. Res. 30 (1994) 71] and a boxcar-like function. Although Neupauer et al. [Water Resour. Res. 36 (2000) 2469] were careful in their conclusions to note the basis of these comparisons, we show that TR does not perform well on problems in which delta-like sources are convolved with diffuse-groundwater contamination response functions, particularly in the presence of noise. We also show that it is relatively easy to estimate an appropriate value for epsilon, the hyperparameter needed in the minimum relative entropy solution for the inverse problem in the presence of noise. This can be estimated in a variety of ways, including estimation from the data themselves, analysis of data residuals, and a rigorous approach using the real cepstrum and the Akaike Information Criterion (AIC). Regardless of the approach chosen, for the sample problem reported herein, excellent resolution of multiple delta-like spikes is produced from MRE from noisy, diffuse data. The usefulness of MRE for noisy inverse problems has been demonstrated.