Hybrid modeling of spatial continuity for application to numerical inverse problems

A novel two-step modeling approach is presented to obtain optimal starting values and geostatistical constraints for numerical inverse problems otherwise characterized by spatially-limited field data. First, a type of unsupervised neural network, called the self-organizing map (SOM), is trained to recognize nonlinear relations among environmental variables (covariates) occurring at various scales. The values of these variables are then estimated at random locations across the model domain by iterative minimization of SOM topographic error vectors. Cross-validation is used to ensure unbiasedness and compute prediction uncertainty for select subsets of the data. Second, analytical functions are fit to experimental variograms derived from original plus resampled SOM estimates producing model variograms. Sequential Gaussian simulation is used to evaluate spatial uncertainty associated with the analytical functions and probable range for constraining variables. The hybrid modeling of spatial continuity is demonstrated using spatially-limited hydrologic measurements at different scales in Brazil: (1) physical soil properties (sand, silt, clay, hydraulic conductivity) in the 42 km² Vargem de Caldas basin; (2) well yield and electrical conductivity of groundwater in the 132 km² fractured crystalline aquifer; and (3) specific capacity, hydraulic head, and major ions in a 100,000 km² transboundary fractured-basalt aquifer. These results illustrate the benefits of exploiting nonlinear relations among sparse and disparate data sets for modeling spatial continuity, but the actual application of these spatial data to improve numerical inverse modeling requires testing.