Bayesian Monitoring Design for Streambed Heat Tracing: Numerical Simulation and Sandbox Experiments

Heat tracing methods have been widely employed for subsurface characterization. Nevertheless, there were very few studies regarding the optimal monitoring design for heat tracing in heterogeneous streambeds. In this study, we addressed this issue by proposing an efficient optimal design framework to collect the most informative diurnal temperature signal for Bayesian estimation of streambed hydraulic conductivities. The data worth (DW) was measured by the expected relative entropy between the prior and posterior distributions of the conductivity field. An adaptively refined Gaussian process surrogate was employed to alleviate the computational burden, resulting in at least three orders of magnitude of speed-up. The applicability of the optimal experimental design framework was evaluated by both numerical and sandbox experimental cases. Results showed that the most informative locations centered in the transition zones among the main patterns of the hydraulic conductivity field, while the most informative times centered in a short period after the minimum/maximum temperature appeared. With the fixed number of measurements, extending the calibration period was more beneficial than increasing the monitoring frequency in improving the estimation results. To our best knowledge, this work is the first study on Bayesian monitoring design for streambed characterization with the heat tracing method. The method and results can provide guidance on selecting monitoring strategies under budget-limited conditions.