## 2 Problem Analysis of Linear Spectral Unmixing  
### Inversion of Inner Part Values  
Various land surface objects within the remote sensing pixel size are recorded in these low-spatial-resolution pixels. At the pixel scale, these various land surface objects are considered the internal components of the pixel, possessing a finer spatial resolution compared to the pixel itself. In different studies, these objects within the pixel have been referred to as endmembers, components, land cover types, class types, or sub-pixels. In this study, considering the spatial relation between the pixel and its internal components as well as the nature of these objects, we refer to them as the inner parts of the pixel. The main objective of unmixing approaches is to estimate the values of these inner parts based on the pixel value. The mathematical foundation of linear spectral unmixing approaches is the linear spectral mixing model. According to this model, the value of a coarse pixel can be expressed as a linear combination of several inner parts and their respective abundances, as shown in Formula (1), where \(R\) represents the pixel value, \(r\) denotes the inner part value, \(f\) represents the abundance of the inner part, \(n\) denotes the total number of inner parts in the pixel, and \(\varepsilon\) refers to the residual error of the model. Typically, the abundance of the inner parts is derived from a land cover map or land use database by calculating the area proportion of each class type relative to the pixel.  
\[R=\sum_{i=1}^{n}f_{i}\cdot r_{i}+\epsilon, \tag{1}\]  
The pixel value and the abundance of the inner parts are considered known parameters, while the values of several inner parts are treated as unknown parameters. Each coarse pixel is represented using a linear equation. If more pixels are available, a linear system can be constructed by combining these linear equations, as shown in Formula (2). Taking into account that the residuals remain constant within each equation, the coefficient matrix of this linear system can be expressed as Formula (3). According to the computational principles of linear systems, the number of equations \(m\) should be equal to or greater than the number of unknown parameters \(n\). The optimal solution of the linear system can be obtained using the least squares method under this condition [41]. Otherwise, the linear system would have no unique solution, which is referred to as an \"underdetermined problem\". The solvability of a linear system depends largely on its coefficient matrix. Therefore, the crucial aspect of inverting the values of inner parts for mixed pixels in linear spectral unmixing lies in creating linear systems with appropriate coefficient matrices.  
\[\begin{array}{l}f_{1,1}\cdot r_{1}+f_{1,2}\cdot r_{2}+\cdots+f_{1,\text{n}} \cdot r_{n}+\epsilon_{1}=R_{1}\\ f_{2,1}\cdot r_{1}+f_{2,2}\cdot r_{2}+\cdots+f_{2,\text{n}}\cdot r_{n}+ \epsilon_{2}=R_{2}\\ \vdots\\ f_{\text{m},1}\cdot r_{1}+f_{\text{m},2}\cdot r_{2}+\cdots+f_{\text{m},\text{ n}}\cdot r_{n}+\epsilon_{m}=R_{m}\end{array}, \tag{2}\]  
\[\mathbf{F}=\begin{bmatrix}f_{1,1}&f_{1,2}&\cdots&f_{1,\text{n}}\\ f_{2,1}&f_{2,2}&\cdots&f_{2,\text{n}}\\ \vdots&\vdots&\vdots&\vdots\\ f_{\text{m},1}&f_{\text{m},2}&\cdots&f_{\text{m},\text{n}}\end{bmatrix}, \tag{3}\]