# Hydraulic Characterization of Green Roof Substrates by Evaporation Experiments
## 2 Materials and Methods
### Data Analysis  
Coupled \(h\) vs. \(\theta\) and \(K\) vs. \(h\) data obtained for each soil sample from the UHG and evaporation experiments were fitted by the unimodal van Genuchten-Mualem model [18] (VGM):  
\[S_{e}=\frac{\theta(h)-\theta_{r}}{\theta_{s}-\theta_{r}}=\frac{1}{\left(1+\left| \alpha h\right|^{n}\right)^{m}} \tag{5}\]  
\[K(h)=K_{s}S_{e}^{\lambda}\left[1-\left(1-S_{e}^{\frac{1}{n}}\right)^{m}\right] ^{2} \tag{6}\]  
where \(S_{e}\) is the effective water content, \(K_{s}\) (LT\({}^{-1}\)), is the saturated hydraulic conductivity, \(\theta_{r}\) and \(\theta_{s}\) (L\({}^{3}\)L\({}^{-3}\)) denote the residual and the saturated water contents, respectively, \(\lambda\) is a pore connectivity parameter, and \(\alpha\) (L\({}^{-1}\)), \(n\), and \(m\) (=\(1-1/n\)) are empirical parameters. Given that \(\theta_{r}\) and \(\lambda\) influence the dry end of the soil hydraulic functions [41; 42], it is likely that these parameters, when left unconstrained, could be poorly estimated. Therefore, it appeared worthwhile to evaluate if fixing \(\theta_{r}\) and \(\lambda\) to an a priori fixed value could influence the quality of estimated \(\theta(h)\) and \(K(h)\) functions. In order to verify the role of fixing/estimating \(\theta_{r}\) and \(\lambda\) parameters, the following preliminary analyses were conducted:  
* The retention data obtained by the evaporation method were fitted using Equation (5) considering \(\alpha\), \(n\), and \(\theta_{s}\) as fitting parameters, whereas \(\theta_{r}\) was either fitted or fixed at \(\theta_{150}\);
* The hydraulic conductivity data obtained by the evaporation method were fitted using Equation (6) considering \(\alpha\), \(n\), \(\theta_{s}\) and \(\theta_{r}\) values obtained from the previous step, \(K_{s}\) as a fitting parameter and \(\lambda\) either fitted or fixed at \(\lambda\) = 0.5. The quality of fitting was evaluated for each substrate sample in terms of the Root Mean Square Error (RMSE) and Nash-Sutcliffe Efficiency (NSE) [43; 44]. When the estimates obtained considering a different number of fitted parameters were compared, the Akaike's information criterion (AIC) was also considered. The best fitting approach was selected as the one with the smallest value of AIC in order to penalize overfitting [5; 45]. The possibility of estimating parameters of Equations (5) and (6) by the inverse method was also evaluated. To this aim, the sample weights and the pressure heads in the three compartments of the sample collected at 1 h intervals during the transient evaporation process were used as the input of the inverse method package of Hydrus-1D software [9]. Parameters \(\alpha\), \(n\), and \(\theta_{s}\) and \(K_{s}\) were estimated, while \(\theta_{r}\) and \(\lambda\) were fixed at the optimal values obtained in the preceding step. Soil hydraulic properties obtained by the inverse method were compared with those obtained by the direct Wind method. Comparison was conducted either considering each replicated sample separately or the average \(\theta(h)\) or \(K(h)\) curves obtained by the four replicates. Specifically, average \(\theta(h)\) or \(K(h)\) curves were obtained by two approaches: (i) \(\theta\) and \(K\) values were sampled at fixed \(h\) values (46 values for both \(\theta\) and \(K\)) and averaged, the parameters of averaged \(\theta(h)\) or \(K(h)\) curves were then obtained by fitting the VGM model; (ii) the VGM parameters of the four samples were averaged and a curve corresponding to the average parameters considered. Finally, the mean \(K(h)\) relationships obtained from direct and inverse methods were compared with the mean near-saturated hydraulic conductivity values measured at \(-6\), \(-12\) and \(-18\) cm by the UHG method. This last analysis aimed at evaluating the ability of the evaporation experiment when analyzed using the direct or inverse method to estimate accurate \(K\) values close to saturation.