A Digital Filtering Approach to Infer Vertical Fluxes from Streambed Temperature Profiles: Reconciling Process-Based and Time-Series Analyses

Heat is commonly used as a tracer to infer groundwater/surface-water exchange fluxes, where temperature is measured at multiple locations in a vertical streambed profile over time and analyzed to infer vertical fluid flux. Approaches to estimating time-varying flux from temperature time series can be broadly divided between (1) calibration of process-based models (i.e., numerical models) and (2) application of various steady-state analytical solutions to temperature signals' amplitude and/or phase and stitching results together over time. The first approach allows for rigorous treatment of heat-transport physics, non-ideal and non-periodic boundary conditions, and hydraulic/thermal heterogeneity; however, analysis of long time series and estimation of time-varying flux is computationally cumbersome. Although the second approach is efficient for long time series, it commonly focuses on an idealized signal of diurnal heating/cooling and cannot capitalize on non-periodic forcing (e.g., dam releases, storm events, active heating); further, it tends to smooth true flux transitions over several days.

Here, we present and demonstrate a digital filtering approach that uses a process-based model within a time-series framework. We apply the Extended and Unscented Kalman Filters to estimate time-varying flux based on temperature time series. The heat-transport equation is linearized around the current flux estimate, and the digital filter recursively updates system states corresponding to spatially distributed temperatures and a single, time-varying vertical flux. The approach can also estimate thermal or hydraulic parameters. Compared to existing approaches, the digital filtering approach (1) includes a rigorous process-based model yet is highly efficient for estimation of time-varying flux, (2) readily capitalizes on active-heating and episodic forcing, (3) allows for real-time estimation as new data become available, and (4) can be used for forecasting future system states.