## 4 Experiment
### Results and Discussion
#### 4.2.2 Statistics for Underdetermined Pixels in Globe Scale  
An unmixing test was conducted on all sixty land cover maps at ten different mixed scales, as described in Section 4.1. The test aimed to count the number of mixed pixels that encountered the underdetermined problem and calculate the proportion of these pixels among all mixed pixels. Figure 6 illustrates the proportions of underdetermined pixels for the ten mixed scales. The results indicate that the underdetermined problem is present in all land surface areas of the study when performing mixed-pixel unmixing based on the linear spectral mixing model. Additionally, Figure 6 shows that the underdetermined problem exists to varying degrees across the ten mixed scales.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Class Types**} & **Combinations of Class Types** & **Layouts of Combinations** & **Underdetermined Layouts** & **Underdetermined Proportion (\%)** \\ \hline
2 & 3 & 45 & 0 & 0 \\
3 & 7 & 3003 & 3 & 9.99 \\
4 & 15 & 319,770 & 342 & 10.70 \\
5 & 31 & 48,903,492 & 44,528 & 9.11 \\
6 & 63 & 9,440,350,920 & 8,098,340 & 8.58 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulated results within an adjacent region of 3 \(\times\) 3 pixels.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Num.**} & **WRS Path and Row** & **Day of Year** & **Num.** & **WRS Path and Row** & **Day of Year** \\ \hline
1 & 119034 & 2010256 & 8 & 121036 & 2010014 \\
2 & 119035 & 2010256 & 9 & 122034 & 2010117 \\
3 & 120034 & 2010312 & 10 & 122035 & 2010069 \\
4 & 120035 & 2010123 & 11 & 122036 & 2010117 \\
5 & 120036 & 2010231 & 12 & 123034 & 2010172 \\
6 & 121034 & 2010254 & 13 & 123035 & 2010172 \\
7 & 121035 & 2010014 & 14 & 123036 & 2010172 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The Landsat-8 OLI multi-spectral images covering the Shandong Province.  
For the mixed scale of 5, half of the land cover tiles have more than 21 underdetermined pixels in every 10,000 mixed pixels. As for the mixed scales ranging from 15 to 50, these numbers are 51, 92, 99, 119, 139, 170, 176, 193, and 194 underdetermined pixels in every ten thousand mixed pixels. Considering the spatial resolution of the land cover map is 30 m, this indicates that approximately 4.67 acres per 100-by-100 mixed pixels area cannot be inverted based on the linear spectral unmixing theory when the image pixel spatial resolution is 150 m. As the pixel spatial scale increases, more and more pixels become mixed pixels, and larger portions of the land surface become unsolvable. These unsolvable areas amount to 11.34, 20.45, 22.01, 26.46, 30.91, 37.80, 39.14, 42.92, and 43.14 acres when the image spatial resolutions are 300, 450, 600, 750, 900, 1050, 1200, 1350, and 1500 m, respectively, for half of the land cover maps over the study area. These results can serve  
Figure 5: The quantity of underdetermined layouts (**a**) and the proportions (**b**) among several class types.  
Figure 6: The proportion of underdetermined mixed pixels among land cover maps, where mixed scales range from 5 to 50.  
as a reference for estimating unsolvable areas when unmixing land surfaces with mixed pixels using remote sensors with low spatial resolutions.  
Figure 7 displays the average proportions of underdetermined pixels and mixed pixels for the ten mixed scales across all sixty land cover maps. The graph shows that the number of mixed pixels increases with the mixed scales, and a larger portion of these mixed pixels gradually encounters the underdetermined problem. Generally, the proportions of underdetermined pixels remain at a low level, accounting for 0.1% to 2% of mixed pixels. However, when considering the enormous number of mixed pixels from various remote sensing systems that image the global land surface day and night, this issue becomes significant.