Inverse modeling of frequency domain-based one-dimensional soil water flow in layered soils

Soils are heterogeneous at multiple scales. Estimates of spatially varying saturated hydraulic conductivity (K_s) of soils dictate the resolution of modeling water flux based on the Richardson-Richards equation (RRE), but direct measurements of K_s everywhere in a field are practically impossible. As a result, this study develops an inverse approach to estimate spatially varying K_s utilizing soil moisture observations driven by periodic weather variability at different frequencies. Using a numerical solution to vertical, one-dimensional RRE based on Gardner-Kozeny (GK) model in a heterogeneous soil with periodic flux, we investigate the spatial cross-correlation between K_s and observations (the amplitude or phase shift) of soil moisture fluctuation at different frequencies without high computational cost in the conventional time-domain model. These correlation patterns vary with the spatial location and frequency, suggesting that soil moisture fluctuations at different frequencies at one monitoring location carry non-redundant information about the heterogeneous distribution of K_s. Guided by cross-correlation maps, several inverse experiments reveal the effectiveness of multi-frequency data of soil moisture fluctuation for mapping the heterogeneous distribution of K_s. Monte Carlo simulations suggest that multi-frequency data can provide non-redundant and useful information about the estimation of heterogeneous K_s. Synthetic data based on a commonly-applied nonlinear van Genuchten-Mualem (VG) model is then tested in our inversion algorithm. The results indicate that the spatial pattern of the inverted GK K_s is very similar to that of the VG K_s, although these two parameters are inherently not identical since they are used in two different models. In addition, the inverted GK K_s can successfully predict the soil moisture fluctuation of frequencies that were not used in the inversion. Despite having no field experiments to validate this model, we believe this is the first attempt to conduct frequency-domain inverse modeling in vadose zone hydrology using soil moisture fluctuation.