# Hydraulic Characterization of Green Roof Substrates by Evaporation Experiments
## 3 Results
### Estimation of Soil Hydraulic Functions by Inverse Method  
The retention and hydraulic conductivity curve parameters of three substrates estimated by the inverse method applied to the evaporation experiments are shown in Table 3. According to the results of the previous section, the \(\theta_{r}\) parameter was set at \(\theta_{150}\) and the \(\lambda\) parameter was estimated. Comparison between the mean soil hydraulic parameters obtained by the direct and inverse methods is reported in Table 4.  
The parameters of the water retention curves (i.e., \(\alpha\), \(n\), and \(\theta_{s}\)) obtained as the mean of the parameters of Equation (5) fitted to the individual samples were practically identical to those obtained by fitting Equation (5) to the mean water retention data. Minor differences in the two averaging approaches were found for hydraulic conductivity function parameters (\(K_{s}\) and \(\lambda\)). However, the differences in estimated \(K_{s}\) were reported within a factor of 0.96-1.04 that is negligible in practice. It was concluded that the two averaging approaches yielded coincident estimations of the substrate hydraulic properties as far as the unimodal \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Substrate** & **Sample** & \(K_{s}\) & \(\lambda\) & **RMSE** & **NSE** & **AIC** \\ \hline MIX & 1 & 1.08 & 1.51 & 0.00 & 0.96 & \(-\)12.82 \\  & 1 & 0.58 & 0.50 * & 0.00 & 0.94 & \(-\)12.57 \\  & 2 & 4.31 & 1.77 & 0.00 & 0.97 & \(-\)0.21 \\  & 2 & 1.74 & 0.50 * & 0.00 & 0.96 & \(-\)0.10 \\  & 3 & 1.25 & 1.32 & 0.00 & 0.90 & \(-\)0.12 \\  & 3 & 0.97 & 0.50 * & 0.00 & 0.88 & \(-\)0.06 \\  & 4 & 1.85 & 0.26 & 0.00 & 0.95 & \(-\)0.05 \\  & 4 & 1.99 & 0.50 * & 0.00 & 0.95 & \(-\)0.03 \\ \hline \hline \end{tabular}  
* \(\lambda\) fixed at 0.5.  
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Substrates** & **Sample** & \(\theta_{r}\) & \(\theta_{s}\) & \(\alpha\) & \(n\) & \(K_{s}\) & \(\lambda\) \\ \hline  & & [cm\({}^{3}\) cm\({}^{-3}\)] & [cm\({}^{3}\) cm\({}^{-3}\)] & [cm\({}^{-1}\)] & & [cm h\({}^{-1}\)] & \\ \hline TMT & 1 & 0.04 & 0.50 & 0.06 & 1.42 & 2.26 & 3.66 \\  & 2 & 0.04 & 0.46 & 0.04 & 1.36 & 0.73 & 3.10 \\  & 3 & 0.04 & 0.46 & 0.03 & 1.41 & 0.41 & 2.83 \\  & 4 & 0.04 & 0.49 & 0.03 & 1.58 & 10.10 & 1.54 \\ \hline ATV & 1 & 0.11 & 0.45 & 0.09 & 1.51 & 1.61 & 0.09 \\  & 2 & 0.11 & 0.62 & 0.09 & 1.66 & 13.26 & 0.00 \\  & 3 & 0.11 & 0.69 & 0.12 & 1.49 & 21.13 & 0.00 \\  & 4 & 0.11 & 0.70 & 0.08 & 1.44 & 100.00 & 4.23 \\ \hline MIX & 1 & 0.07 & 0.55 & 0.04 & 1.52 & 4.63 & 2.53 \\  & 2 & 0.07 & 0.61 & 0.05 & 1.42 & 4.15 & 0.97 \\  & 3 & 0.07 & 0.48 & 0.02 & 1.42 & 0.33 & 2.46 \\  & 4 & 0.07 & 0.58 & 0.05 & 1.43 & 17.78 & 2.11 \\ \hline \hline \end{tabular}  
* \(\lambda\) fixed at 0.5.  
\end{table} 
Table 3: Parameters of retention and hydraulic conductivity curves obtained by applying the inverse method approach. The retention curves and hydraulic conductivity functions obtained for the single samples of the three substrates with the direct and inverse method are compared in Figure 4. The plots confirm the good agreement of water retention curve estimates obtained by the direct and inverse methods with coefficients of determination, R\({}^{2}\), that ranged between 0.9768 and 1.00 (Table 5). On average, the best agreement between the water retention curve estimated by the two methods (i.e., direct and inverse) was observed for the MIX substrate (mean R\({}^{2}\) = 0.9997) and the worst for the TMT substrate (mean R\({}^{2}\) = 0.9973). The correlation for the hydraulic conductivity function was generally lower than the water retention curve, confirming that the two methods could yield different results for \(K(h)\) (Table 5). The largest discrepancies were observed for substrate TMT with a mean R\({}^{2}\) of 0.9741. The remaining substrates were characterized by similar mean R\({}^{2}\) values (R\({}^{2}\) = 0.9915 for ATV and R\({}^{2}\) = 0.9916 for MIX).