Incorporating Hydrologic Insight into Geophysical Inversion: Resolution Limitations and Direct Estimation of Solute Plume Moments

Time-lapse geophysical tomography (e.g., electrical resistivity and radar) can provide valuable insights into hydrologic phenomena, including tracer transport, aquifer dynamics, and engineered remediation. Tomograms have been used to infer the spatial and temporal moments of solute plumes for model development and calibration. The reliability of inferred moment values is limited by tomographic resolution, which is a function of survey geometry, measurement physics, measurement error, and inverse problem parameterization and regularization. Here, we (1) assess the resolution-dependent reliability of moment inference based on results from conventional pixel-based inversion with Tikhanov-style regularization; and (2) investigate alternative parameterization/regularization techniques that capitalize on hydrologic insight to produce more reliable moment estimates. Conventional pixel-based parameterization and regularization criteria yield the simplest solution that satisfies the data, where solution simplicity is measured by deviations from a prior mean and/or the norm of the first or second spatial derivative (flatness and smoothness, respectively) between adjacent pixels. While effective for static imaging of large-scale geologic or aquifer structure, these measures of simplicity may be less appropriate for imaging transient hydrologic processes and non-stationary targets such as solute plumes. For underdetermined problems, tomograms may overpredict the extent and underpredict the magnitude of target plumes. We contend that, at best, conventional regularization criteria do not capitalize on valuable hydrologic information, such as the total mass of injected fluid or solute; at worst they are inconsistent with the physics underlying the transport process of interest and may lead to misleading estimates of plume moments. We explore strategies to incorporate hydrologic insight into tomographic inversion for time-lapse hydrologic monitoring: moment-based tomographic inversion (MBTI) and object-based tomographic inversion (OBTI). With these approaches, we seek to estimate directly the geometric parameters describing the plume distribution in space and/or time. MBTI is appealing in that the inversion parameters, i.e., the orthogonal moments of the image, are related to the geometric moments commonly used to characterize plume structure and identify controlling transport processes, such as dispersion and rate-limited mass transfer. Simple plumes can be described adequately by moments up to order 3 or 4, whereas complex plumes that are strongly affected by aquifer heterogeneity may require higher-order moments. Under OBTI, the target is parameterized by one or more shapes based on a conceptual model of flow and aquifer structure. Compared to conventional pixel-based parameterization, MBTI and OBTI may reduce the number of inversion parameters by a factor of 100 or more, producing more reliable estimates of plume moments while reducing or precluding common artifacts such as streaking.