## 3 Results
### Gravity Data Denoising Results
To further analyze the differences among the denoising methods, Figure 10 presents comparative histograms of the estimated results of terrestrial gravity anomalies (in mGal) obtained by different methods. In each subfigure of Figure 10, green bars represent the true values, while bars in other colors represent the estimated values of each algorithm. From the U-Net result in the upper-left corner, it can be observed that this method has the highest overlap with the true values. The frequency distribution of the U-Net almost completely covers the true values and exhibits excellent fitting ability, especially for small gravity anomaly values. In contrast, the wavelet method (upper-right) shows a noticeable deviation from the true values, mainly concentrated in a small range of low-frequency gravity anomalies and fails to capture the medium- and high-amplitude anomaly variations. The moving average method (lower-left) shows poor consistency, particularly in the high-frequency region. The low-pass filter (lower-right) is accurate only in certain regions, indicating significant limitations in capturing medium- and high-frequency anomalies. In summary, the U-Net method shows outstanding performance in the detection and frequency matching of various types of gravity anomalies, highlighting its advantage in gravity data denoising applications. To further illustrate the effectiveness of different denoising methods, Figure 11 presents the spectral analysis results. The upper part of the figure shows the spectral curves of various denoising methods. It can be observed that the amplitude of the noisy data (orange curve) in the high-frequency region is significantly higher than that of the original signal (blue curve). For some traditional methods (e.g., purple and brown curves), the amplitude in the high-frequency region remains higher than that of the U-Net (green curve). A comparison of these curves reveals that the denoising performance of the U-Net network is superior to other methods. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Noise Level** & **PSNR (Noisy)** & **PSNR (Denoised)** & **PSNR (Wavelet)** & **PSNR (Moving Avg)** & **PSNR (Low-Pass)** \\ \hline
0.05 & 41.30 & 59.13 & 25.58 & 54.78 & 31.68 \\
0.1 & 33.77 & 52.03 & 20.78 & 47.59 & 30.76 \\
0.15 & 29.27 & 48.62 & 19.53 & 43.23 & 27.68 \\
0.3 & 25.49 & 48.81 & 25.73 & 39.48 & 30.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of PSNR values for different denoising methods. Figure 11: Comparative analysis of SSIM for different denoising methods. Figure 10: Histogram of error analysis of different noise reduction methods. Meanwhile, examining the spectral heatmaps (lower part of Figure 11), the high-frequency noise in the U-Net-denoised result (bottom-right subfigure of Figure 11) is significantly reduced. The \"cleanliness\" of its spectrum is closer to that of the original signal (bottom-left subfigure of Figure 11), which intuitively demonstrates the noise suppression effect of the U-Net. To validate the roles of the attention module and physical constraint term in network-based denoising, ablation experiments were designed. The experiments consisted of four groups: the baseline model (without physical constraints or attention mechanism), the model with only physical constraints added, the model with only the attention mechanism added, and the full model (incorporating both physical constraints and the attention mechanism). All experiments were conducted on data with 5% Gaussian noise added, and the results are presented in Figure 12. The left panel of the figure shows the PSNR comparison, while the right panel displays the SSIM comparison. It can be observed from the figure that the full model achieves significantly higher PSNR and SSIM values than the other models. These experimental results further demonstrate the critical role of adding physical constraint terms and attention modules in enhancing the network's denoising performance.