On the Validity of Flux Estimates using Semi-Infinite Domains: Comparing Flux Estimates from Semi-Infinite and Bounded Domains using the LPMLEn - a Multi Frequency, Multi Sensor Method to Estimate Vertical Streambed Fluxes and Sediment Thermal Properties

When using heat as a tracer, one of the most commonly used models to estimate streambed Darcy fluxes is the 1D heat equation governed on a semi-infinite domain. For this solution the lower boundary condition is set at infinity. However, recent research has been questioning the validity of this approach. Here we explore whether the 1D heat equation governed on a semi-infinite domain is a good approximation of measured temperature time-series. Using the LPMLEn, our newly developed frequency domain method, we compare flux and thermal diffusivity estimates originating from the semi-infinite domain model with those originating from a bounded domain model, where the lower boundary condition is set locally at a temperature sensor.

Using simulations and conducting inverse modeling on field data, we show that for some practical field conditions, such as a thermal diffusivity changing over depth outside the probed domain, the flux estimates originating from the semi-infinite domain model can be in the wrong direction, i.e., downwelling instead of upwelling or vice versa. This is caused by the fact that all the assumptions for the semi-infinite domain should hold up to infinity. In contrast, the bounded model isolates the problem to the probed domain such that the assumptions of homogeneous units are more likely to hold in the probed domain, resulting in better estimates. Our research further demonstrates that the semi-infinite domain model is not for all circumstances the appropriate model choice and flux estimates can be misleading, a fact modelers should keep in mind in future studies.