## 3 Results and Validation
### Validation
#### 3.2.1 Validation of SRGAN Model and Super-Resolved NDVI  
In this study, the evaluation metrics for image reconstruction quality serve as indicators reflecting the reconstruction accuracy of an SR model. The quality of image reconstruction is assessed by comparing the peak signal-to-noise ratio (\(PSNR\)) and structural similarity (\(SSIM\)) between the super-resolved image and the ground truth high-resolution image. PSNR is sensitive to the noise and distortion of an image and reflects the overall quality of the image. A higher PSNR value indicates that the reconstructed image is less different from the reference image at the pixel level, which can be defined using the mean square error (\(MSE\)) between a ground truth image \(I_{y}\) with N pixels and a reconstructed image \(I_{SR}\) as follows:  
\[PSNR=10\cdot log_{10}\left(\frac{L^{2}}{MSE}\right), \tag{10}\]  
\[MSE=\frac{1}{N}{\sum_{i=1}^{N}}{\left(I_{y}-I_{SR}\right)^{2}}, \tag{11}\]  
where \(L\) represents the peak signal value, such as \(L\) = 255 for an 8-bit grayscale image. On the other hand, \(SSIM\) serves as a perceptual model that evaluates the similarity between images by considering lightness, contrast, and structure. In the context of remotely sensed images, these factors are closely linked to the texture and shape features of the land surface; a higher SSIM value indicates that the reconstructed image closely resembles the reference image in terms of structure, with better preservation of details and boundaries of the land cover. It is defined as follows:  
\[SSIM=\left(l\left(I_{SR},I_{y}\right)^{\alpha}\cdot c\left(I_{SR},I_{y}\right)^ {\beta}\cdot s\left(I_{SR},I_{y}\right)^{\gamma}\right), \tag{12}\]  
where \(l\left(I_{SR},I_{y}\right),c\left(I_{SR},I_{y}\right),s\left(I_{SR},I_{y}\right)\) represent the similarity of lightness, contrast, and structure between the ground truth and reconstructed images, respectively, while \(\alpha\), \(\beta\), and \(\gamma\) are weight parameters. Table 2 presents the quantities of training and testing patches, PSNR, and SSIM for SRmodel_a1 and SRmodel_a2 as detailed in Section 2.3.1. The test images of SRmodel_a1 and SRmodel_a2 are the real 1 km and 250 m MODIS images, respectively. The average PSNR for both SR models is around 27, while the average SSIM exceeds 0.79. This aligns closely with the typical performance seen in most remote sensing image super-resolution studies, showing a slight improvement [49]. These results indicate that the reconstructed HR images closely match the original images not only in pixel quality and consistency but also in perceptual quality, aligning well with visual features and maintaining consistency in structure and information content. To assess the rationality and validity of the 30 m resampled data mentioned in Section 2.3.4, we calculated the PSNR and SSIM for the 30 m resampled data, 30 m super-resolution data with the real 30 m Landsat data (Table 3). The reconstruction accuracy decreased compared to previous results, with the quality of the 30 m resampled data being superior to the 30 m super-resolution data. Additionally, it was evident that after two stages of super-resolution and one resampling of the original data, the values of PSNR and SSIM gradually decreased, aligning with the logic of image reconstruction. Notably, in the Qilian Mountains, characterized by diverse vegetation and complex terrain, the unique surface characteristics and reflectance properties of snow caps and glaciers \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**SRGAN Model** & \multicolumn{2}{c}{**Image Number**} & **Patches Number** & **PSNR** & **SSIM** \\ \hline \multirow{2}{*}{SRmodel\_a1} & Training & 24 & 4320 & \multirow{2}{*}{27.55} & \multirow{2}{*}{0.7979} \\  & Test & 2 & - & & \\ \hline \multirow{2}{*}{SRmodel\_a2} & Training & 24 & 8556 & \multirow{2}{*}{26.51} & \multirow{2}{*}{0.7956} \\  & Test & 2 & - & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SRGAN model training and testing numbers, PSNR, and SSIM.