## 2 Methodology
### MOGP Models Parametrization  
The MOGP processing was entirely conducted within the Multi-Output Gaussian Process Toolkit (MOGPTK) [43] version 0.3.2. The MOGPTK is a Python toolkit for performing multi-output GP regression with kernels utilizing the cross-correlation information between channels to better model time-series signals. The set of MOGP kernels implemented is particularly pertinent for signal reconstruction in the case of missing data. The toolkit provides the functionality for training and interpreting GP models with multiple data channels and includes plotting functions for the case of single input with multiple outputs. More information can be found at: [[https://github.com/GAMES-UChile/mogptk/](https://github.com/GAMES-UChile/mogptk/)]([https://github.com/GAMES-UChile/mogptk/](https://github.com/GAMES-UChile/mogptk/)) (accessed 11 January 2023).  
The preprocessed S1 and S2 time series were first loaded into MOGPTK creating two and three-channel models: (i) S2 GP VWC (CH-1), (ii) S1 orbit 68 RVI (CH-2), (iii) S1 orbit 141 RVI (CH-3). Then we removed a range of data from the CH-1 to simulate the S2 data loss corresponding to the winter wheat cycles (From the beginning of September to the end of December, we removed 9 S2 images for the 2020 campaign and 6 S2 scenes related to the 2021 winter wheat crop campaign) and additionally removed 10% of the data points for CH-2 and CH-3 randomly. Regarding 2020 and 2021 crop campaigns, 35 S2 (CH-1) + 92 S1-orbit 68 (CH-2) + 53 S1-orbit 141 (CH-2) images and 46 S2 (CH-1) + 103 S1-orbit 68 (CH-2) + and 59 S1-orbit 141 (CH-3) images were synergistically used to train the MOGP models, respectively. Aiming to improve training results each data channel was linearly detrended and normalized to have zero mean and unit variance utilizing the transformations provided by the MOGPTK. We analyzed four MOGP spectral kernels: (i) the Spectral Mixture Kernels for Multi-Output Gaussian Processes (MOSM) [70], (ii) the Cross-Spectral Mixture (CSM) [71], (iii) the Linear Model of Coregionalization (LMC) [39; 40], and (iv) the Convolutional Model (CONV) [72; 73] with four spectral components per channel (Q = 4). The MOGP kernels' parameters were randomly instantiated to establish an initial set of reasonable values for the S1 & S2 dataset using Independent Spectral Mixture (SM) as an estimation method. SM fits an independent GP model for each channel of the dataset with a spectral mixture kernel. Posteriorily it uses the tuned parameters as initial values of the MOGP kernels. For each channel, the noise was initialized with 1/30 of the variance. The MOGP models were then trained using the Adam optimizer [74]. During the training stage, the hyperparameters of the kernel are optimized to approach the training data. The learning rate was set up to 0.1 and the number of iterations for initialization in 500.  
In addition, we also considered the SOGP SM kernel using the Bayesian Nonparametric Spectral Estimation (BNSE) [75] as the initialization method to estimate the kernel parameters from the S1 & S2 dataset. BNSE aims to assess the power spectral density of a time series signal and then select the Q greater peaks in the estimated spectrum. The kernel's mean, magnitude, and variance are initialized based on the peak's position, magnitude, and with of the detected spectral components, respectively.