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Open Access Publications from the University of California

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

  • Author(s): Jaiswal, Parakh
  • Advisor(s): Vrugt, Jasper A
  • et al.

The topic of infiltration of water into variably saturated soils has received much

attention in the soil physics literature in the past decades. Many different equations

have been proposed to describe quantitatively the infiltration process. These equations

range from simple empirical equations to more advanced deterministic descriptions

of the infiltration process and semi-analytical solutions of Richards’ equation. The

unknown coefficients in these infiltration functions signify hydraulic properties and

must be estimated by curve fitting to measured cumulative infiltration data, Ie(t).


From all available infiltration functions, the two-term equation, I(t) = S

t + cKst of

Philip [1957] has found most widespread application and use. This popularity has not

only been cultivated by detailed physical and mathematical analysis, the two-term

infiltration equation is also easy to implement and admits a closed-form solution

for the soil sorptivity, S (L T1/2

), and multiple c (−) of the saturated hydraulic

conductivity, Ks (L T−1

). Yet, Philip’s two-term infiltration function has a limited

time 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). The

theoretical treatise in Philip [1957] provides a closed-form solution for the maximum

time validity, t


valid, of the two-term infiltration equation. It is not particularly easy to

experimentally corroborate these theoretical findings as this demands prior knowledge

of c, S and Ks

. What is more, the maximum time validity, t


valid may not characterize

properly the actual time validity, tvalid. In this paper, we introduce a new method to

determine simultaneously the values of the coefficient c, hydraulic parameters, S and


, and time validity, tvalid, of Philip’s two-term infiltration equation. Our method is

comprised of two main steps. First, we determine independently the soil sorptivity,

S, and saturated hydraulic conductivity, Ks by fitting the semi-implicit infiltration

equation of Haverkamp [Haverkamp et al., 1994] to measured cumulative infiltration

data. This step uses the DiffeRential Evolution Adaptive Metropolis (DREAM)

algorithm of Vrugt [2016] and returns as byproduct the marginal distribution of the


parameter β in Haverkamp’s infiltration equation. In the second step, the maximum

likelihood values of S and Ks are used in Philip’s two-term infiltration equation, and

used to determine the optimal values of c and tvalid via model selection using the

Bayesian information criterion. To benchmark, test and evaluate our approach we

use cumulative infiltration data simulated by HYDRUS-1D [Simunek et al., 2008]

for twelve different USDA soil types with contrasting textures. This allows us to

determine whether our procedure is unbiased as the inferred S and Ks of the synthetic

data are known before hand. Results demonstrate that the estimated values of S and

Ks are in excellent agreement with their ”true” values used to create the artificial

infiltration data. Furthermore, our estimates of c and tvalid are dependent on soil

texture and fall within the ranges stipulated in the literature.

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