Picture-based physics: Using POD derived process constraints to enhance imaging of groundwater systems

Geophysical imaging techniques are increasingly important tools for developing conceptual models of subsurface processes and characterizing hydrogeologic properties. In particular, recent studies have shown that coupled inversion, i.e., using coupled hydrologic process and geophysical instrument simulators as a single forward model, can lead to vast improvements in our ability to accurately constrain estimates of aquifer hydraulic properties with a limited amount of geophysical data. In this case, the hydrologic model acts as an implicit process constraint that regularizes the inverse problem. A major challenge with the coupled approach, however, is that the hydrologic process model is a hard constraint on the inverse problem; it is not possible to accurately reproduce subsurface behaviors when the conceptualization of the hydrologic model is incorrect. To overcome this issue, we take an image-processing perspective to develop physics-based constraints for inverse problems. Philosophically our approach is analogous to the use of images as training data for inferring multiple point geostatistics. In our case, however, we instead use training images to extract a set of basis vectors using proper orthogonal decomposition (POD) that can be used to constrain the inverse problem. Careful selection of a basis for an inverse problem is important as an appropriate basis can significantly improve estimation efficiency and accuracy. A key element of our approach is that the training images are generated using Monte Carlo simulations of the physical groundwater process of interest (e.g., flow and transport). As a result the training images and POD basis provide a site specific, non-parametric representation of the processes driving the groundwater system. A consequence is that a limited number of POD basis vectors can be used to compactly capture complex heterogeneities within the subsurface, thereby reducing the dimensionality of parameter space and effectively regularizing the inverse problem. In this talk we provide a conceptual understanding of POD and its application to inverse problems. We then provide examples where the POD inversion strategy is used to enhance geophysical imaging of unsaturated flow in the vadose zone and solute transport in an aquifer. Finally, we discuss how the POD approach can be extended for applications such as hydraulic tomography and general geostatistical estimation problems.