## 4 Experiment  
### Study Area and Data  
To investigate the theoretical existence of the underdetermined problem in linear spectral unmixing, a simulation experiment was conducted within a local neighborhood of coarse pixels. The local region had a size of \(3\times 3\) coarse pixels, and it was assumed that various land cover types appeared randomly around the center coarse pixel. The land cover layout of the entire region was ensured to be non-repetitive throughout the experiment. The underdetermined results were recorded for local regions with land cover types ranging from 2 to 6. Several issues of land cover types were taken into account, including the huge calculation workload among the global simulation, the relationship between the spatial resolution of coarse pixels and the spatial scale of land cover types within the corresponding coarse pixel extent, and the practical application of linear spectral unmixing. Following the simulation experiment, a statistical experiment was conducted on a global scale using the land cover data from GlobeLand30 (www.ngcc.cn (accessed on 18 June 2023)). The land cover types considered in this dataset included water, wetland, artificial surface, cultivated land, forest, shrubland, grassland, bare land, tundra, permanent snow, and ice. The world terrestrial ecoregion map (www.worldlifte.org (accessed on 18 June 2023)) was used as a reference to manually select map sheets from GlobeLand30. The sampling process aimed to achieve qualitative balance in terms of continent and ecological region factors, ensuring representative map sheets with both heterogeneous and homogeneous landscapes from around the globe. Most map sheets had a size of approximately 6 degrees in longitude and 5 degrees in latitude, while a few map sheets in high latitudes spanned 12 degrees in longitude. A total of sixty map sheets were used in the experiment, as shown in Figure 4 and Table 2. For each map sheet, ten different mixed pixel scales were chosen to simulate varying mixed situations within the same landscape region. The inner part values were then inverted based on these mixed pixels and their abundances, following the methodology described in the simulation test of Section 2.2. The spatial resolution ratio between the mixed pixel and the land cover pixel ranged from 5 to 50, with an increment of 5 (referred to as the mixed scale). A land cover map tile from the N11_60 region, consisting of 4000 \(\times\) 4000 pixels, was selected to analyze the performance of the proposed elastic-window-based algorithm for mixed scales ranging from 5 to 50. As a point of comparison, regular windows with fixed sizes of 3, 7, 11, 15, 21, 27, 33, and 45 were also selected. Additionally, an experiment was conducted to downscale MODIS-NDVI (Normalized Difference Vegetation Index) data in the Shandong Province of China using the proposed algorithm. Fourteen Landsat images from the year 2010 (shown in Table 3), covering the local region, were selected to generate a fine-resolution land cover map with a spatial resolution of 30 m. The ISO-DATA classification method in ENVI was employed for this purpose. The 16-day composite of MODIS-NDVI data (MOD13Q1 h27v05) for the Shandong Province in 2010 was collected to obtain the coarse pixel values. A total of 23 NDVI products were resampled to a resolution of 240 m to align with the resolution of the land cover map.