S1. Parameter Optimization to Define ha
First, the SWRC described in equation (9) in the main text is fitted with observation to derive the parameters. The objective function to be minimized is defined as:
where Nθ is the number of water content observation, θi and are the measured and the fitted water content, respectively. p = (α , n ,m , θs ) is the parameter vector used for the optimization. θs is only optimized when there is no observation.
After the parameters of the SWRC are determined, we further fit the HCC, as described in equation (8) in the main text, with conductivity observations to determine the optimal value forha . The objective function is written as:
where NK is the number of hydraulic conductivity observation; Ki and are the measured and the fitted conductivity, respectively. For the saturated hydraulic conductivity Ks , the observed value is applied. Parameter l is set to 3.5 according to Wang et al. (2018).
Equations (S1) and (S2) are optimized by applying the shuffled complex evolution method developed at the University of Arizona (SCE-UA), as proposed by Duan et al. (1992). For a detail description of the setting of the SCE-UA method, we refer the reader to Wang et al. (2022a).