Parasite inversion for determining the coefficients and time‐validity of Philip's two‐term infiltration equation

Many different equations have been proposed to describe quantitatively one‐dimensional soil water infiltration. The unknown coefficients of these equations characterize soil hydraulic properties and may be estimated from a n record, {t∼i,I∼i}i=1n$\{ {\tilde t_i},{\tilde I_i}\} _{i = 1}^n$, of cumulative infiltration measurements using curve fitting techniques. The two‐term infiltration equation, I(t)=St+cKst$I(t) = S\sqrt t + c{K_{\rm{s}}}t$, of Philip has been widely used to describe measured infiltration data. This function enjoys a solid mathematical–physical underpinning and admits a closed‐form solution for the soil sorptivity, S [L T−1/2], and multiple, c [−], of the saturated hydraulic conductivity, Ks [L T−1]. However, Philip's two‐term equation has a limited time validity, tvalid [T], and thus cumulative infiltration data, I∼(t∼)$\tilde I(\tilde t)$, beyond t=tvalid$t = {t_{{\rm{valid}}}$ will corrupt the estimates of S and Ks. This paper introduces a novel method for estimating S, c, Ks, and tvalid of Philip's two‐term infiltration equation. This method, coined parasite inversion, use as vehicle Parlange's three‐parameter infiltration equation. As prerequisite to our method, we present as secondary contribution an exact, robust and efficient numerical solution of Parlange's infiltration equation. This solution admits Bayesian parameter estimation with the DiffeRential Evolution Adaptive Metropolis (DREAM) algorithm and yields as byproduct the marginal distribution of Parlange's β parameter. We evaluate our method for 12 USDA soil types using synthetic infiltration data simulated with HYDRUS‐1D. An excellent match is observed between the inferred values of S and Ks and their “true” values known beforehand. Furthermore, our estimates of c and tvalid correlate well with soil texture, corroborate linearity of the c(β)$c({{\beta}})$ relationship for 0≤t≤tvalid$0 \le t \le {t_{{\rm{valid}}}$, and fall within reported ranges. A cumulative vertical infiltration of about 2.5 cm may serve as guideline for the time‐validity of Philip's two‐term infiltration equation. Core Ideas We describe a method that can determine the time validity of infiltration equations. This method infers as well the hydraulic properties of unsaturated soils. We present a robust and efficient numerical solution of Parlange's three‐parameter infiltration equation. Posterior distribution determines how well parameters are defined by calibration to data. Two and a half centimeters of c