## 3 Methodology
### Filling Data Gaps of Monthly Land-Water Mask  
Based on the long-term water probability mask and the area of interest (AOI) mask of Lake Ray Roberts, the remaining data gaps of all monthly mask are filled. The method for filling data gaps is exemplary demonstrated for Lake Ray Roberts in October 2012 (Figure 10(a)). First, the binary monthly mask (1: water, 0: land) with data gaps is multiplied with the long-term water probability mask (Figure 10(b)) of Lake Ray Roberts. Then, Figure 10(c) shows the water probability mask for all water pixels with existing data gaps. The Lake Ray Roberts has a maximum extent of 132.16 km\({}^{2}\). The monthly water mask of October 2012 contains data gaps of voids (0.72 km\({}^{2}\)), clouds (57.26 km\({}^{2}\)) and cloud shadows (14.99 km\({}^{2}\)) which are about 55% of the maximum extent. The area of all water pixels in the AOI which are not covered by data gaps is called initial area \(A_{initial}\) (46.85 km\({}^{2}\)). The dependency between surface area and water probability shown in Figure 10(f) is used in the gap-filling process. It provides information about the surface area for each percentage of the long-term water probability. This function reflects the surrounding topography and is always monotone decreasing. Based on all water pixels larger than a selected percentage of water, a new mask of the land-water distribution is created. Then, this is used for filling the data gaps. Each long-term water probability \(p\) is assigned to a fixed final surface area in the curve. For example, all water pixels with a water probability larger than 60% always cover a surface area larger than 104.59 km\({}^{2}\). \[|(A_{initial}+A_{fill}(p))-A(p)|\stackrel{{!}}{{=}}Min \tag{6}\]  
This information is used in an iterative approach where Equation (6) is minimized. Here, the initial area \(A_{initial}\) is introduced as constant value for the current month. Then, the fill area \(A_{fill}\) is computed iteratively using different water probabilities \(p\). Therefore, the water probability \(p\) is applied pixel-wise on the long-term water probability mask where all water pixels with a higher probability than the estimated water probability \(p\) are removed. Then, the remaining water pixels are intersected with the mask of data gaps in order to get the fill area \(A_{fill}\) which is computed again in each iteration step. Then, \(A_{fill}\) is added to the initial area \(A_{initial}\). This results in a monthly area land-water mask and a surface area. This is subtracted by the final area \(A(p)\) derived from Figure 10(f) in which the same water probability \(p\) is used as for the calculation of \(A_{fill}\). The remaining area residuals are then iteratively minimized with respect to the water probability \(p\). Finally, the areas \(A_{initial}\) (red) and \(A_{fill}\) (green) are merged which is shown in Figure 10(d). The resulting surface area \(A_{final}\) is 107.86 km\({}^{2}\) based on the water probability \(p\) of 49.0%. One can clearly see that using the optimized water probability \(p\) leads to a very good agreement at the lake shore without inconsistencies. Figure 10: (**a**) Number of water pixels of all accumulated monthly files (**b**) Number of water and land pixels of all accumulated monthly files (**c**) Water probability mask (**d**) Labeling of connected surfaces (**e**) Mask of interested water body (**f**) Dependency between water probability and area extent. For filling the data gaps, the percentage or size of the data gaps in the AOI is of minor relevance. The main requirement is that parts of the lake shore are visible as this information is used in the iterative gap-filling step. One can conclude, the less data gaps near the shore are available, the better is our approach to derive correctly filled masks and accurate surface areas. Finally, an error assessment is performed for each gap-filled monthly land-water mask, respectively surface area. This is essential in order to quantify the quality of each resulting surface area.