## 3 Results
### Training MOGP Kernels for VWC Time Series Modelling  
Two S1 & S2 three-channel datasets were created for training the MOGP models. We first assessed quantitatively the performance of four spectral-mixture kernels for selecting  
Figure 4: Temporal profiles (mean value and standard deviation) of averaged selected ROI-2 over a winter wheat parcel belonging to the BVCR 2021 crop campaign. S2 GP VWC time series from October 2019 to the end of January 2022 (**a**). S1 RVI orbit 141 smoothed (dashed blue line) and original (dashed black line) time series (**b**). S1 RVI orbit 68 smoothed (dashed red line) and original (dashed black line) time series (**c**). S1 RVI Orbit 68 (red line) and S1 orbit 141 (blue line) RVI time series (**d**). the best MOGP model and also the outcomes of SOGP to establish a ranking between them. We then explored the contribution of the S1 RVI data at two distinct descendent orbits and its effect on the MOGP models' metrics. To do so, several trials were conducted considering 2 or 3 channels from the S1 & S2 datasets. The space of possible combinations is as follows: (a) S2 GP VWC and S1 RVI orbit 68; (b) S2 GP VWC and S1 RVI orbit 141; and (c) S2 GP VWC, S1 RVI orbit 68 and S1 RVI orbit 141. Table 2 presents the prediction statistics for the MOGP modeling when the S1 & S2 data streams are used to reconstruct time series data of VWC over the averaged BVCR 2020 winter wheat ROI-1. For each MOGP model and S1 & S2 data stream configuration, we evaluated the following error metrics: MAE, MAPE, RMSE, NRMSE and the training time. The MOGP modeling outcomes for the 2021 winter wheat averaged ROI-2 can be visualized in Table 3. A linear increase in runtime can be noted when the three channels are used to train the MOGP regression models. By eliminating S2 GP VWC samples from the S1 & S2 datasets, we artificially produced data gaps. Subsequently, we used the extracted values as a baseline for the evaluation of the MOGP models' predictions. For that reason, from the S2 GP VWC time series in Figures 3a and 4a, we deleted 9 samples from September 8 to December 12 of 2020 and 6 samples from August 19 to November 27 of 2021, which corresponds to the 2020 and 2021 complete phenological winter wheat cycles, respectively, at the study site. It is noteworthy that the number of S2 acquisitions is different for the 2020 and 2021 crop campaigns due to the cloud coverage dynamic over the study region (see Section 2.3). These latent S2 GP VWC observations exemplify realistically the cloudy image acquisition conditions of S2 at high latitudes. The MOGP regressions for the evaluated models are shown in Figure 5. The MOGP results are also compared to the SM SOGP predictions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & \\ \hline
**MOGP Kernel** & **MAE [g m\({}^{-2}\)]** & **MAPE [\%]** & **RMSE [g m\({}^{-2}\)]** & **NRMSE [\%]** & **Time [s]** \\ \hline MOSM & 828.85 & 56.42 & 927.56 & 44.34 & 10.58 \\ CSM & 242.7 & 15.43 & 360.55 & 17.24 & 17.85 \\ SM-LMC & 346.16 & 22.56 & 495.49 & 23.69 & 12.68 \\ CONV & 250.17 & 19.48 & 313.11 & 14.97 & 21.42 \\ SM & 881.4 & 58.91 & 1005.71 & 48.07 & 6.03 \\ \hline \multicolumn{6}{c}{**S2 GP VWC and S1 RVI orbit 141**} \\ \hline MOSM & 1025.79 & 69.92 & 1116.62 & 53.38 & 9.37 \\ CSM & 283.95 & 19.76 & 378.01 & 18.07 & 16.06 \\ SM-LMC & 482.25 & 31.99 & 580.76 & 27.76 & 11.49 \\ CONV & 255.42 & 25.25 & 419.36 & 20.05 & 19.25 \\ SM & 883.69 & 59.05 & 1009.05 & 48.23 & 4.98 \\ \hline \multicolumn{6}{c}{**S2 GP VWC, S1 RVI orbit 68 and S1 RVI orbit 141**} \\ \hline MOSM & 907.21 & 62.61 & 992.18 & 47.43 & 18.56 \\ CSM & 472.31 & 32.75 & 512.23 & 24.49 & 35.18 \\ SM-LMC & 463.04 & 30.75 & 546.85 & 26.14 & 22.67 \\ CONV & 249.3 & 21.83 & 336.74 & 16.1 & 40.27 \\ SM & 881.77 & 58.93 & 1006.25 & 48.1 & 10.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error metrics and training time of the MOGP and SOGP evaluated kernels for the 2020 winter wheat averaged ROI-1.