# A Numerical Model to Estimate the Soil Thermal Conductivity Using Field Experimental Data
## 3 Results and Discussions  
Table 1 shows the soil thermal conductivity and the empirical parameter calibrated by the different methods with the experimental data of the Pedras Altas-RS site. The results of \(\lambda\) obtained using Equation (2) by the different methods (\(M1\), \(M2\), \(M3\), and \(M4\)) show variability of almost 25%, with a reduction of the RMSE of approximately 40% between \(M1\) and \(M4\) and an increase of \(R^{2}\) of approximately 12% for the daily period. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Calibration**} & \multicolumn{4}{c}{**Validation**} \\ \cline{2-9}  & \(\lambda\) & \(\epsilon\) & \multicolumn{4}{c}{_RMSE (\(Wm^{-2}\))_} & \multicolumn{4}{c}{\(R^{2}\)} \\  & **(\(Wm^{-1}K^{-1}\))** & **(\(Wm^{-2}\))** & **Daily** & **Day** & **Night** & **Daily** & **Day** & **Night** \\ \hline Method 1 (\(M1\)) & 1.331 & - & 5.56 & 7.31 & 2.92 & 0.82 & 0.71 & 0.77 \\ Method 2 (\(M2\)) & 1.148 & - & 3.86 & 5.08 & 1.97 & 0.92 & 0.86 & 0.89 \\ Method 3 (\(M3\)) & 1.014 & - & 3.24 & 3.97 & 2.28 & 0.94 & 0.91 & 0.86 \\ Method 4 (From all analyzed methods and periods, \(M1\) presented the worst statistical indices. This result is perhaps due to the fact all data were used in the simulation, which can generate numerical divergences when \(|\Delta T|\approx 0\), consequently, on average, can overestimate \(\lambda\). For \(M2\) were eliminated data when \(|\Delta T|\leq 0.1\), representing 5% of total data and resulting in a decrease of 30% in RMSE compared to \(M1\) (Table 1). However, in periods when \(G\) is around zero, this methodology can cause unrealistic \(\lambda\) values. The \(M3\), estimating \(\lambda\) value for a reference hour (13:00--local time), had a decrease of RMSE around 15%. However, also in this method, some unrealistic \(\lambda\) values can be obtained. An example is a day when precipitation occurs, or the sky is very cloudiness, increasing hardly the solar radiation at the surface. If in this hour \(G\) is close to zero or the soil temperature in both depths are similar, the same problems described for \(M1\) and \(M2\) can happen. The methods \(M3\) and \(M4\) presented results in which both the \(\lambda\) and the statistical indices were very close, suggesting that using the hour that represents the largest difference between the temperatures can be considered a good alternative for the estimation of \(\lambda\). However, using \(M3\), we do not have information about the diurnal cycle of \(\lambda\), as mentioned by [30]. Here we do not analyze the diurnal cycle of \(\lambda\) for anyone method. The \(M5\) presented the best statistical indices for all analyzed periods, proving to be the best method for \(\lambda\) estimation. In the daily period, a decrease of 3.13 Wm\({}^{-2}\) in RMSE between \(M1\) and \(M5\) is observed. The greatest errors were found in the day periods for all methods since it is during this period that the greatest temperature variations occur (Table 1). The results of \(\lambda\) obtained by Equation (2) and Equation (3) by the LSM (\(M4\) and \(M5\), respectively) show very close values. However, the RMSE of \(M4\) is higher than that of \(M5\), indicating that \(M5\) improved the estimate of \(G_{z}\). Additionally, the parameter \(\varepsilon\) represents almost 20% of the average annual value of \(G_{exp}\) that was 12.8 Wm\({}^{-2}\). Figure 4 shows the dispersion diagrams of \(G_{z}\) using the \(\lambda\) values in Table 1 with the validation data. In general, the \(M1\) and \(M2\) models underestimate the observations when they are negative and overestimate them when they are positive. The \(M3\) and \(M4\) models show an overestimation of the values for both situations. Finally, the \(M5\) model shows a behavior very similar to the observed data.