## 2 Problem Analysis of Linear Spectral Unmixing
### Imperfection in Approximating Adjacent Land Region
Unfortunately, employing a window size of 5 includes pixel C, which is challenging to decompose due to its adjacent region with pixel D. The abundances corresponding to these 3 \(\times\) 3 windows for pixels A, B, C, and D are listed in Table 1.  
Figure 1: Examples of selection coarse pixels in adjacent regions of pixel A, B, C, and D within a 3 \(\times\) 3 pixel window, where the dash box illustrated the boundary of the A window. Based on the land cover map, the value of abundance in each coarse pixel can be derived by calculating the proportion of area occupied by every land cover type to the total coarse pixel area. The abundance value of adjacent pixels located in windows of pixels A, B, C, and D has been shown in Table 1, each row corresponding to the distribution of land convert types and their abundance within a pixel. These arrangements of land cover types and their abundance can be seen as the coefficient matrix of the linear systems for pixels A, \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**1-1**} & & & & & & & & \\ \hline
**Num.** & **Pixel Code** & **Crop** & **Forest** & **Grass** & **Shrub** & **Wetland** & **Water** & **Built-Up** & **Bare Land** \\ \hline
1 & A & 42 & 17 & 3 & 9 & 0 & 0 & 7 & 22 \\
2 & b2 & 78 & 2 & 20 & 0 & 0 & 0 & 0 & 0 \\
3 & b3 & 82 & 0 & 0 & 0 & 0 & 0 & 18 & 0 \\
4 & b4 & 88 & 4 & 0 & 0 & 0 & 0 & 8 & 0 \\
5 & c2 & 83 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
6 & c4 & 1 & 61 & 5 & 0 & 28 & 3 & 1 & 0 \\
7 & c2 & 90 & 0 & 0 & 0 & 0 & 0 & 10 & 0 \\
8 & c3 & 85 & 15 & 0 & 0 & 0 & 0 & 0 & 0 \\
9 & c4 & 38 & 13 & 1 & 0 & 8 & 40 & 0 & 0 \\ \hline \multicolumn{1}{c}{**1-2**} & & & & & & & & & \\ \hline
**Num.** & **Pixel Code** & **Crop** & **Forest** & **Grass** & **Shrub** & **Wetland** & **Water** & **Built-Up** & **Bare Land** \\ \hline
1 & B & 5 & 12 & 69 & 2 & 4 & 8 & 0 & 0 \\
2 & b7 & 35 & 10 & 26 & 0 & 9 & 20 & 0 & 0 \\
3 & b8 & 32 & 25 & 31 & 0 & 0 & 12 & 0 & 0 \\
4 & b9 & 69 & 12 & 12 & 0 & 0 & 8 & 0 & 0 \\
5 & c7 & 3 & 14 & 71 & 0 & 5 & 7 & 0 & 0 \\
6 & c9 & 50 & 19 & 15 & 0 & 0 & 0 & 4 & 13 \\
7 & d7 & 18 & 4 & 78 & 0 & 0 & 0 & 0 & 0 \\
8 & d8 & 75 & 0 & 25 & 0 & 0 & 0 & 0 & 0 \\
9 & d9 & 87 & 5 & 4 & 0 & 2 & 2 & 0 & 0 \\ \hline \multicolumn{1}{c}{**1-3**} & & & & & & & & & \\ \hline
**Num.** & **Pixel Code** & **Crop** & **Forest** & **Grass** & **Shrub** & **Wetland** & **Water** & **Built-Up** & **Bare Land** \\ \hline
1 & C & 26 & 2 & 0 & 13 & 40 & 1 & 10 & 8 \\
2 & d5 & 0 & 0 & 0 & 0 & 36 & 64 & 0 & 0 \\
3 & d6 & 5 & 0 & 10 & 0 & 70 & 15 & 0 & 0 \\
4 & d7 & 19 & 4 & 77 & 0 & 0 & 0 & 0 & 0 \\
5 & e5 & 17 & 0 & 0 & 0 & 44 & 39 & 0 & 0 \\
6 & e7 & 43 & 2 & 56 & 0 & 0 & 0 & 0 & 0 \\
7 & f5 & 97 & 0 & 2 & 0 & 1 & 0 & 0 & 0 \\
8 & f6 & 95 & 0 & 5 & 0 & 0 & 0 & 0 & 0 \\
9 & f7 & 64 & 1 & 35 & 0 & 0 & 0 & 0 & 0 \\ \hline \multicolumn{1}{c}{**1-4**} & & & & & & & & & \\ \hline
**Num.** & **Pixel Code** & **Crop** & **Forest** & **Grass** & **Shrub** & **Wetland** & **Water** & **Built-Up** & **Bare Land** \\ \hline
1 & D & 81 & 0 & 19 & 0 & 0 & 0 & 0 & 0 \\
2 & e7 & 43 & 2 & 56 & 0 & 0 & 0 & 0 & 0 \\
3 & e8 & 94 & 1 & 5 & 0 & 0 & 0 & 0 & 0 \\
4 & e9 & 80 & 0 & 2 & 0 & 5 & 1 & 12 & 0 \\
5 & f7 & 64 & 1 & 35 & 0 & 0 & 0 & 0 & 0 \\
6 & f9 & 85 & 0 & 8 & 0 & 0 & 0 & 7 & 0 \\
7 & g7 & 93 & 0 & 7 & 0 & 0 & 0 & 0 & 0 \\
8 & g8 & 77 & 0 & 22 & 0 & 0 & 0 & 1 & 0 \\
9 & g9 & 93 & 0 & 3 & 0 & 0 & 0 & 5 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The inner part types and abundance of the coarse pixels (1-1, 1-2, 1-3, and 1-4 show A, B, C, and D in Figure 1, respectively, unit: %). B, C, and D. By combining the principle of the existence of solutions to linear systems, the solutions of pixels A (Table 1(1-1)), B (Table 1(1-2)), and C (Table 1(1-3)) are unavailable.