## 3 Solution  
### Iterative Approximation Procedure  
The diversity of land surface types and their various distributions result in significant complexity on the land surface. This complexity extends from large to local areas and is reflected in remote sensing images with a discrete pattern. As a result, many pixels cover multiple land cover types, and it is difficult to ensure that the same land cover types are present in other pixels within the adjacent region. Expanding the extent around the adjacent region is important and effective for selecting pixels to form the linear system with the same land cover types, but it is not essential. The presence of the required pixels to such an extent is uncertain. Additionally, the size of the extent calculated based on the number of land cover types in the unmixing pixel is only an estimated value and is not always reliable. A larger extent can introduce more pixels compared to a smaller extent, which can help reduce the possibility of the underdetermined problem to some extent. However, as shown in the case of pixel D in Figure 1, it is not a fundamental solution. Moreover, since each linear system determines a least squares solution by involving the same land cover type but different values to a large extent, the outputs can reduce the spectral variability. Therefore, more attention should be paid to the complexity of the adjacent region.  
Theoretically, each pixel has other pixels that can assist in the successful inversion of inner part values. These pixels are scattered in the adjacent region around the mixed pixel at various distances, implying a spatial distribution pattern of scattered pixels. Unmixing a pixel is a process that approximates this spatial distribution pattern using certain methods. However, due to the unknowns of these spatial distribution patterns, it is almost impossible to select all the pixels at once within a certain extent or specific shapes. Nevertheless, continuously approximating the spatial distribution pattern in the adjacent region is a preferable solution until knowledge of these patterns becomes available. Because of this idea, a strategy, as shown in Figure 2, is introduced to guide the approximation procedure. It involves utilizing the mixed pixel to estimate the initial spatial approximating mode and selecting pixels from the adjacent pixels within the estimation region. Based on the adjacent pixels, an optimization option can be chosen to adjust the initial approximating mode. The adjustment may involve assigning higher weights to similar or nearby pixels or directly excluding opposite pixels. Through iterative selection and optimization, the spatial approximating mode approaches the spatial distribution pattern of theoretical pixels. At the same time, it is ensured that the number of selected pixels is not less than the inner part numbers \(N\) of the mixed pixel, and the rank of the coefficient matrix \(\mathbf{F}\), which is composed of the abundances of these pixels, should be equal to \(N\) (as shown in Formula (5)). The strategy estimates the spatial distribution pattern of pixels based on the mixed pixel according to Tobler's First Law of Geography. Furthermore, the composition of land cover types and their abundance among adjacent pixels is utilized to optimize the previous spatial approximation mode to prevent failure when the assumption is not valid. The iterative procedure modifies the spatial approximation mode continuously and ensures the approximation of the spatial distribution pattern properly in the adjacent region. In extreme cases, an excessively large range may be traversed for the approximation process.  
\[rank\mathbf{F}=N\]