Application of Recursive Estimation to Heat Tracing for Groundwater/Surface‐Water Exchange

We present and demonstrate a recursive‐estimation framework to infer groundwater/surface‐water exchange based on temperature time series collected at different vertical depths below the sediment/water interface. We formulate the heat‐transport problem as a state‐space model (SSM), in which the spatial derivatives in the convection/conduction equation are approximated using finite differences. The SSM is calibrated to estimate time‐varying specific discharge using the Extended Kalman Filter (EKF) and Extended Rauch‐Tung‐Striebel Smoother (ERTSS). Whereas the EKF is suited to real‐time (“online”) applications and uses only the past and current measurements for estimation (filtering), the ERTSS is intended for near‐real time or batch‐processing (“offline”) applications and uses a window of data for batch estimation (smoothing). The two algorithms are demonstrated with synthetic and field‐experimental data and are shown to be efficient and rapid for the estimation of time‐varying flux over seasonal periods; further, the recursive approaches are effective in the presence of rapidly changing flux and (or) nonperiodic thermal boundary conditions, both of which are problematic for existing approaches to heat tracing of time‐varying groundwater/surface‐water exchange. Plain Language Summary We present and demonstrate a new approach to infer groundwater/surface‐water exchange based on temperature time series collected at different depths below the sediment/water interface. Algorithms for real‐time (“online”) and “offline” applications are presented. The new algorithms are effective in the presence of rapidly changing flow across the sediment/water interface, which has posed challenges to existing approaches. Key Points Recursive filtering applied to heat tracing enables real‐time estimation of groundwater/surface‐water exchange Recursive filtering and smoothing applied to heat tracing improve estimation of groundwater/surface‐water exchange Recursive filtering applied to heat tracing allows quantification of uncertainty in groundwater/surface water exchange estimates