## 3 Results
### Inversion Effect of Sound Speed Profiles After Clustering  
The root mean square error (RMSE), quantified as the difference between direct inversion without clustering and inversion after three-category clustering, is shown in Figure 2. It can be seen from the figure that the accuracy after acoustic stability clustering is significantly higher than before clustering at most depths. The average RMSE obtained after acoustic stability clustering is 1.67 m/s, while the average RMSE obtained by direct inversion is 2.03 m/s. Compared with the method of directly using the s-EOF-r for inversion, the acoustic stability pre-clustering method extracts the main types of disturbances, significantly reducing the RMSE at sea depths of 0-150 m; however, the improvement effect is relatively limited at water depths below 150 m, among which the RMSE after clustering in the region of 150-250 m shows a significant rebound. This phenomenon is because the sound speed characteristics in this layer are relatively stable, and the performance of the pre-clustering method will decrease when facing such sudden changes in acoustic stability. The maximum RMSE of the s-EOF-r inversion method at sea depths of 0-150 m reaches 4.79 m/s, and after pre-clustering, it is reduced to 3.30 m/s. Two peaks in the inversion error appear at 60 m and 200 m, which come from random, large gradient disturbances at these depths. Generally, the randomness of this kind of disturbance is difficult to explain using traditional physical analysis methods, resulting in certain challenges for the physical mechanism method of sEOF-r in solving related problems. Figure 3 shows the RMSE as a function of sample's serial number, the RMSE value of each sample point represents the average RMSE of all inversion results within a sea depth of 1000 m at that point. It can be seen that the RMSE after acoustic stability clustering is generally better than before clustering. The RMSE before clustering fluctuates around 3 m/s, including two prominent peaks at 5.22 m/s and 5.00 m/s. The RMSE of the SSP after acoustic stability clustering is generally below 2.5 m/s, and the maximum RMSE does not exceed 4 m/s. The extraction and classification of sample features by acoustic stability pre-clustering enable the sEOF-r to separate the main modes of sea surface disturbance more efficiently, further reducing the error in subsequent inversion. From the perspective of the mean value of the SSP inversion error, the accuracy of the acoustic stability pre-clustering method is slightly higher than that of the non-clustering method. From Figure 3, it can be seen that the RMSE peaks of the two methods basically match, indicating that the main components interfering with the inversion process are consistent. In terms of the maximum inversion error, the RMSE of the acoustic stability clustering method is significantly lower than that of the non-clustering method, while at the place where the errors of both are low, this performance is not obvious. The main source of the error peak is the change of the water body structure caused by seasonal circulation in the South China Sea, which has a significant impact on the SST and sea surface height. To study the impact of the seasonal characteristics of the sea surface parameters in the South China Sea on the acoustic stability clustering method further, Figure 4 shows the typical inversion effects of the SSP in each of the four seasons. Compared with the non-clustering method, the acoustic stability clustering method has significant advantages for the spring and winter data. Although the acoustic stability clustering method is based on statistical principles, making it generally difficult to explain the occurrence of random water masses and fronts from a physical perspective, it can still effectively reflect disturbances  
Figure 3: RMSE as a function of sample size for the one-category and three-category clustering results. Figure 2: RMSE with depth for the one-category (i.e., unclustered) and three-category clustering results. near the sea surface.