Determining groundwater-surface water exchange from temperature-time series: Combining a local polynomial method with a maximum likelihood estimator

The use of temperature-time series measured in streambed sediments as input to coupled water flow and heat transport models has become standard when quantifying vertical groundwater-surface water exchange fluxes. We develop a novel methodology, called LPML, to estimate the parameters for 1-D water flow and heat transport by combining a local polynomial (LP) signal processing technique with a maximum likelihood (ML) estimator. The LP method is used to estimate the frequency response functions (FRFs) and their uncertainties between the streambed top and several locations within the streambed from measured temperature-time series data. Additionally, we obtain the analytical expression of the FRFs assuming a pure sinusoidal input. The estimated and analytical FRFs are used in an ML estimator to deduce vertical groundwater-surface water exchange flux and its uncertainty as well as information regarding model quality. The LPML method is tested and verified with the heat transport models STRIVE and VFLUX. We demonstrate that the LPML method can correctly reproduce a priori known fluxes and thermal conductivities and also show that the LPML method can estimate averaged and time-variable fluxes from periodic and nonperiodic temperature records. The LPML method allows for a fast computation of exchange fluxes as well as model and parameter uncertainties from many temperature sensors. Moreover, it can utilize a broad frequency spectrum beyond the diel signal commonly used for flux calculations.