## 4 Atmospheric Correction Results and Analysis
### Comparison of Images Before and After Atmospheric Correction
\(w\) is the weighting function, \(w_{i}=0.5(\cos(\frac{\pi}{\sqrt{x^{2}+y^{2}}}\sqrt{\left(x_{i}-x_{c}\right)^{ 2}+\left(y-y_{c}\right)^{2}})+1)\), \(x\) and \(y\) represent the width and height of the local image region, (\(x_{c}\), \(y_{c}\)) denotes the center point of the local image region, and (\(x_{i}\), \(y_{i}\)) refers to the position of the \(i\)-th pixel in the local image region. \(DN_{i}\) is the grayscale value of the \(i\)-th pixel in the local image region, while \(DN\) is the weighted mean of the grayscale values in the local image region, \(DN=\sqrt{\frac{1}{\sum_{i=1}^{N}w_{i}}{\sum_{i=1}^{N}w_{i}DN_{i}}}\). The calculation of contrast primarily relies on texture and edge information. Therefore, when comparing two images with the same content, an image with less texture, overall blurriness, and weakened edges typically has a lower contrast, while a clearer image with well-defined features generally exhibits higher contrast. (b) Clarity  
Clarity, often referred to as image sharpness, is directly represented by the grayscale gradient, and different gradient operators can be used to calculate the clarity for image quality assessment [27; 28]. In this study, the Sobel operator is employed to compute clarity, as defined by the following equations:  
\[S_{x}= \begin{bmatrix}-1&0&1\\ -2&0&2\\ -1&0&1\end{bmatrix} S_{x}= \begin{bmatrix}1&2&1\\ 0&0&0\\ -1&-2&-1\end{bmatrix}  \tag{10}\]  
The Sobel operator is applied sequentially through convolution with the image grayscale values to obtain the Sobel clarity:  
\[S=\sqrt{\left|DN(i,j)*S_{x}\right|^{2}+\left|DN(i,j)*S_{y}\right|^{2}} \tag{11}\]Here, \(S\) denotes the Sobel clarity, where a higher value indicates better clarity. When comparing two images with identical content, the image with higher quality will have richer details and appear clearer, resulting in a higher clarity value. Conversely, the clarity value will be lower for an image with less detail and lower quality. (c) The calculation result  
Figures 7-9 display the calculated contrast and clarity values before and after atmospheric correction for three locations: Daxing Airport in China, the Indian Plains, and Xianning in Hubei, China. \"AC-B\" represents images before atmospheric correction, \"AC-A\" represents images after atmospheric correction, and \"Improvement\" quantifies the percentage enhancement. A 100% improvement means the corrected image has double the contrast or clarity of the uncorrected image. As shown, the contrast and clarity improvements vary by location, with the highest values observed at Xianning, followed by the Indian Plains and Daxing Airport. This variation corresponds to differences in aerosol optical depth across locations, as higher AOD leads to greater atmospheric effects and more pronounced improvements after correction. Additionally, the improvements in contrast and clarity decrease with an increasing wavelength, indicating that shorter wavelengths in the visible spectrum are more sensitive to atmospheric scattering. This finding aligns with the Rayleigh scattering theory, which predicts stronger scattering effects at shorter wavelengths. Overall, the atmospheric correction algorithm significantly enhances image quality, improving the visibility of surface features and supporting a more accurate analysis of remote sensing data.