## 3 Results
### Validation of FDR Measurements
#### 3.1.3 Validation of Standard Functions  
For all plots, profiles and layers, standard calibration functions (Equations (1) and (2)) were applied and the coefficients of determination (R\({}^{2}\)) and predictive quality indices are listed in Tables S1-S5. The standard linear (LM) and quadratic models (QM) generally explained the measured topsoil water content rather poorly, indicated by R\({}^{2}\) as low as 0.005 in OFH layers of the WIJ site and explaining just 10 to 50% of the variance in mineral soil layers. With increasing depth, the standard calibration functions approximate the measured water contents slightly better. Figure 3 displays the positive or negative bias (MPE) and total prediction error (RMSPE) of the standard linear and quadratic functions when predicting the measured \(\theta_{\rm v}\). In general, the quadratic standard model generates a greater bias and total prediction error compared to the linear model. A paired \(t\)-test of all profiles and layers reveals that when the standard LM is used the bias is 0.015-0.019 m\({}^{3}\) m\({}^{-3}\) smaller compared to the QM for all plots but GON (Table 5). On the latter plot a positive bias (i.e., overestimation of \(\theta_{\rm v}\)) was observed attributed to the clayey subsoil layers (Figure 3). As expected, the random error (SDPE) does not differ significantly between standard LM and QM. The resulting overall prediction error is significantly smaller when using the standard LM, and its index of agreement is significantly greater. On the WIJ, RAV, BRA, and HOE plots a predominantly negative bias was found, especially in the topsoil: up to 0.16 m\({}^{3}\) m\({}^{-3}\) in mineral (M01) and up to 0.32 m\({}^{3}\) m\({}^{-3}\) of \(\theta_{\rm v}\) in OFH layers was underestimated (Tables S1-S5). For almost all layers the allowable measurement error threshold for \(\theta_{\rm v}\) of 0.03 m\({}^{3}\) m\({}^{-3}\) is exceeded except for two M48 layers in RAV, and two M48 layers in HOE. So, for only 4 out of 60 layers the (linear) standard function predicts \(\theta_{\rm v}\) according to the required level of accuracy. The mean total prediction error is greatest for the GON plot (RMSPE = 0.17 m\({}^{3}\) m\({}^{-3}\)) and smallest for the RAV plot (0.07 m\({}^{3}\) m\({}^{-3}\)). The RMSPE is similar for BRA and HOE. The greater overall prediction error on the sites WIJ and GON is mainly due to the large prediction errors of the OFH layers (Figure 3). If we omit the values for OFH, the mean RMSPE for WIJ becomes 0.09 m\({}^{3}\) m\({}^{-3}\) whereas it remains high (0.17 m\({}^{3}\) m\({}^{-3}\)) for GON here due to the high estimation errors in the clayey substrates. Figure 3: Predictive quality of standard linear and quadratic models for all plots, profiles and depth layers. Negative bias indicates underestimation of real soil water content. Red dashed lines show the allowable error threshold, black dotted line provides mean total prediction error (RMSPE). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SITE** & **MPE** & **SDPE** & **RMSPE** & **dr** \\ \hline WIJ & 0.019 *** & 0.0008 \({}^{\text{NS}}\) & \(-\)0.015 *** & 0.14 *** \\ RAV & 0.016 *** & 0.0006 \({}^{\text{NS}}\) & \(-\)0.007 *** & 0.09 *** \\ BRA & 0.016 *** & \(-\)0.0013 * & \(-\)0.009 * & 0.08 * \\ GON & \(-\)0.030 \({}^{\text{NS}}\) & \(-\)0.0050 \({}^{\text{NS}}\) & \(-\)0.040 * & 0.16 ** \\ HOE & 0.015 *** & \(-\)0.0002 \({}^{\text{NS}}\) & \(-\)0.009 ** & 0.07 * \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Paired difference of validation indices computed for standard linear compared to quadratic model (LM-QM). Significance level according to a paired \(t\)-test: ***\(p<0.001\), ***\(p<0.01\), *\(p<0.05\), \({}^{\text{NS}}\) denotes not significant. Prior to the use of the standard LM and QM, a temperature correction (TC) of the _PA_ signal was performed using Equation (3), and compared to the same model without TC (Table 6).