## 2 Problem Analysis of Linear Spectral Unmixing
### Imperfection in Approximating Adjacent Land Region  
The values of coarse pixels in different bands are distinct from one another. Therefore, the values of inner parts in different bands cannot be assumed to be the same unknown parameters in a linear system. The inversion of inner part values for a pixel needs to be performed within the current band. Considering the variations caused by natural or artificial influences, the same land cover type may exhibit inconsistent characteristics in different regions. For instance, crop growth can be affected by factors such as agrotype, surface water, and cultivation, resulting in spatially variable behavior. This phenomenon, known as spectral-spatial variability [42], can cause the inner part values of pixels in different locations to differ significantly from each other across the entire image band. Based on the assumption that \"Everything is related to everything else, but near things are more related than distant things\", known as Tobler's First Law of Geography [33], coarse pixels in adjacent regions are logically selected to form the linear system for the central pixel. Typically, the adjacent pixels are chosen using a regular window of a specific size. The window is defined using Formula (4), where \(S\) represents the odd length or width of the rectangular area and \(N\) represents the number of inner parts on the land surface. \[S^{2}\geq N, \tag{4}\]  
In most cases, this rectangular area is square-shaped, with a coarse pixel size as the unit--for example, a window region of \(3\times 3\) or \(5\times 5\) pixels centered on a coarse pixel. By using this formula, the window size \(S\) can be calculated based on the number of inner parts \(N\) in the mixed pixel. Then, the adjacent pixels within this window are selected to form the linear system, with pixel values and inner part abundances as known parameters and the inner part values as unknown parameters. An example is provided to illustrate the problem by combining a land cover map and a low-spatial-resolution image band. In Figure 1, the grid represents the pixels in the low-spatial-resolution image band, and specific pixels are labeled as A, B, C, and D. Pixel A encompasses six land cover types: crop, built-up area, bare land, grass, forest, and shrub. Pixel B includes five land cover types: crop, forest, grass, water, and wetland. Pixel C consists of seven land cover types: crop, built-up area, bare land, grass, forest, shrub, and wetland. Finally, pixel D contains two land cover types: crop and grass. According to Formula (4), the window size is 3 pixels by 3 pixels. A total of nine pixels inside the window are selected to form the linear system for the inversion of inner part values for each of these four pixels separately. However, the linear systems of pixels A, B, and C encounter the underdetermined problem. While bare land and shrubs are present in pixel A, they do not exist in the pixels within the window. The same situation occurs for pixel C, where built-up areas and shrubs cannot be separated in the unmixing process. Both cases, A and C, involve situations where undividable inner parts are present within the window. The pixels around pixel B can handle this unmixing scenario, but due to a limited number of adjacent pixels (e.g., pixel c9) included in the linear system of pixel B, the underdetermined problem arises (although a weighted method can help improve this case). The situation with pixel B involves undividable inner parts in the adjacent region within the specific window. When it comes to selecting enough pixels to decompose a given pixel, the appropriate window sizes for pixels A, B, and C are 5, 3, and 7, respectively. The inner part numbers of these pixels satisfy Formula (4), but the window sizes adopted are ill-suited. Thus, it appears that opting for a larger window size might be a suitable choice. However, the case of pixel D indicates that this assumption is incorrect. Pixel D encompasses grass and crop, which can be easily decomposed using a 3 \(\times\) 3 window.