## 2 Methods and Process Procedures  
### Overview of the Reconstruction Method  
The reconstruction approach began by gathering all time-series of Sentinel-2 images for a specified period and a study area. Then, using a spatiotemporal image fusion model to predict cloud-free synthetic Sentinel-2 images for all dates when cloud-free observation images were not available with clear-sky Sentinel-2 and MODIS images. After that, we replaced those cloud and cloud shadow contaminated pixels of the Sentinel-2 observations with the corresponding pixels of the synthetic images predicted at the same acquisition dates. The reconstruction procedure consisted of four steps: (1) an image fusion model was used to predict Sentinel-2 images at acquisition dates when cloud-free images (with either partial or full cloud coverage) were not available; (2) the cloudy and cloud shadow pixels of the cloud contaminated images were identified based on analysis of the differences of the synthetic and observation image pairs; (3) the cloudy and cloud shadow pixels of the observations were replaced by the corresponding pixels of the synthetic image; (4) the pixel values from the synthetic image were calibrated to the pixel values of the observation image with a radiometric normalization process. After these processes, an entire cloud-free Sentinel-2 images time-series can be reconstructed.  
By using PSRFM [24] as the spatiotemporal image fusion model to generate the cloud-free synthetic Sentinel-2 image time-series, Figure 1 shows the main work flow of the reconstruction procedures. The procedural details are given as follows:  
1. [label=(0)]
2. Gather the Sentinel-2 image time-series of a study area's sub-period (\(t_{0}\), \(t_{1}\), \(t_{2}\),..., \(t_{n}\)). Without losing generality, assume this period starts at \(t_{0}\) and ends at \(t_{n}\), and the images at \(t_{0}\) and \(t_{n}\) are clear-sky ones. Between the two acquisition dates, all other images at dates \(t_{k}\) (\(k=1,\ 2,\ldots,\,n\)-1) are either partially cloud-covered or fully cloud-covered (with more than 50% cloud coverage). Therefore, all images at \(t_{k}\) need to be processed for cloud-free time-series reconstruction. An entire study period can consist of multiple such sub-periods.
3. Retrieve the time-series MODIS images at the Sentinel-2 image acquisition dates. They are then processed (resampled, re-projected, and geometrically registered with the Sentinel-2 images) for image fusion modelling.
4. Generate synthetic Sentinel-2 images for all acquisition dates \(t_{k}\) (\(k=1,\,2,\,\ldots,\,n\)-1) from the clear-sky MODIS and Sentinel-2 image pairs at \(t_{0}\) and \(t_{n}\), and MODIS images at \(t_{k}\) using PSRFM.
5. Make a cloud and cloud shadow mask for every cloud contaminated Sentinel-2 image at \(t_{k}\) by analyzing the reflectance differences between the observational and synthetic Sentinel-2 images.
6. Replace the cloudy and cloud shadow pixels of the Sentinel-2 observation images with the corresponding pixels of the synthetic Sentinel-2 images according to the cloud and cloud shadow mask.
7. Normalize the pixel values of the replacement by calibrating them to the reflectance values of the cloud-free pixels of the same Sentinel-2 observation.
8. Repeat the steps 3 to 6 until all the images are processed. Then, a cloud-free Sentinel-2 image time-series is reconstructed.