Bayesian Determination of the Soil Hydraulic Parameters and the Time Validity of Philip’s Two-Term Infiltration Equation From Measured Infiltration Data

The topic of infiltration of water into variably saturated soils has received muchattention in the soil physics literature in the past decades. Many different equationshave been proposed to describe quantitatively the infiltration process. These equationsrange from simple empirical equations to more advanced deterministic descriptionsof the infiltration process and semi-analytical solutions of Richards’ equation. Theunknown coefficients in these infiltration functions signify hydraulic properties andmust be estimated by curve fitting to measured cumulative infiltration data, Ie(t).2From all available infiltration functions, the two-term equation, I(t) = S√t + cKst ofPhilip [1957] has found most widespread application and use. This popularity has notonly been cultivated by detailed physical and mathematical analysis, the two-terminfiltration equation is also easy to implement and admits a closed-form solutionfor the soil sorptivity, S (L T1/2), and multiple c (−) of the saturated hydraulicconductivity, Ks (L T−1). Yet, Philip’s two-term infiltration function has a limitedtime validity, tvalid (T), and consequently, measured cumulative infiltration data, Ie(t),beyond t = tvalid (T) should not be used to estimate S and Ks (among others). Thetheoretical treatise in Philip [1957] provides a closed-form solution for the maximumtime validity, t+valid, of the two-term infiltration equation. It is not particularly easy toexperimentally corroborate these theoretical findings as this demands prior knowledgeof c, S and Ks. What is more, the maximum time validity, t+valid may not characterizeproperly the actual time validity, tvalid. In this paper, we introduce a new method todetermine simultaneously the values of the coefficient c, hydraulic parameters, S andKs, and time validity, tvalid, of Philip’s two-term infiltration equation. Our method iscomprised of two main steps. First, we determine independently the soil sorptivity,S, and saturated hydraulic conductivity, Ks by fitting the semi-implicit infiltrationequation of Haverkamp [Haverkamp et al., 1994] to measured cumulative infiltrationdata. This step uses the DiffeRential Evolution Adaptive Metropolis (DREAM)algorithm of Vrugt [2016] and returns as byproduct the marginal distribution of the3parameter β in Haverkamp’s infiltration equation. In the second step, the maximumlikelihood values of S and Ks are used in Philip’s two-term infiltration equation, andused to determine the optimal values of c and tvalid via model selection using theBayesian information criterion. To benchmark, test and evaluate our approach weuse cumulative infiltration data simulated by HYDRUS-1D [Simunek et al., 2008]for twelve different USDA soil types with contrasting textures. This allows us todetermine whether our procedure is unbiased as the inferred S and Ks of the syntheticdata are known before hand. Results demonstrate that the estimated values of S andKs are in excellent agreement with their ”true” values used to create the artificialinfiltration data. Furthermore, our estimates of c and tvalid are dependent on soiltexture and fall within the ranges stipulated in the literature