Manifold regression for subsurface contaminant characterization based on sparse concentration data

In this paper we develop three manifold regression approaches for estimating quantitative metrics characterizing subsurface zones contaminated by Dense Non-Aqueous Phase Liquids (DNAPLs) based on sparse down-gradient concentration data. We are particularly interested in estimating source zone characteristics related to the distribution of contaminant mass in highly saturated pool regions as well as more diffuse ganglia areas regions. Source zone characterization, a necessary first step in the development of any remediation strategy, is challenging due to practical constraints associated with the data available for processing. We use manifold methods for jointly representing labeled training data comprised of known metrics as well as features derived from the corresponding data sets. We then employ an integrated approach to the problems of (a) robustly embedding the sparse test data into the manifold when the metrics are not available and (b) constructing a regression function operating directly in manifold space for metric estimation. The utility of the approach is enhanced by the explicit incorporation of a physical constraint associated with the metrics into the problem formulation. We apply our manifold regression approaches to a simulated data set whose the hydraulic conductivity fields were generated using a Transition Probability Markov Chain (TP/MC) model. Using densely sampled concentration data for training but sparsely sampled data for testing, the results demonstrate the potential of our approaches.