Inverse modelling of in situ soil water dynamics: accounting for heteroscedastic, autocorrelated, and non-Gaussian distributed residuals

Inverse modelling of in situ soil water dynamics is a powerful tool to test process understanding and determine soil hydraulic properties at the scale of interest. The observations of soil water state variables are typically evaluated using the ordinary least squares approach. However, the underlying assumptions of this classical statistical approach of independent, homoscedastic, and Gaussian distributed residuals are rarely tested in practice. In this case study, we estimated the soil hydraulic properties of a homogeneous, bare soil profile from field observations of soil water contents. We used a formal Bayesian approach to estimate the posterior distribution of the parameters in the van Genuchten–Mualem (VGM) model of the soil hydraulic properties. Three likelihood models that differ with respect to assumptions about the statistical features of the time series of residuals were used. Our results show that the assumptions of the ordinary least squares did not hold, because the residuals were strongly autocorrelated, heteroscedastic and non-Gaussian distributed. From a statistical point of view, the parameter estimates obtained with this classical statistical approach are therefore invalid. Since a test of the classic first-order autoregressive (AR(1)) model led to strongly biased model predictions, we introduced an modified AR(1) model which eliminates this critical deficit of the classic AR(1) scheme. The resulting improved likelihood model, which additionally accounts for heteroscedasticity and nonnormality, lead to a correct statistical characterization of the residuals and thus outperformed the other two likelihood models. We consider the corresponding parameter estimates as statistically correct and showed that they differ systematically from those obtained under ordinary least squares assumptions. Moreover, the uncertainty in the parameter estimates was increased by accounting for autocorrelation in the observations. Our results suggest that formal Bayesian inference using a likelihood model that correctly formalizes the statistical properties of the residuals may also prove useful in other inverse modelling applications in soil hydrology.