## 2 Materials and Methods
### Acoustic Stability Method
Compared with other clustering algorithms such as hierarchical clustering, K-means is suitable for processing structured data with dense distribution and moderate dimensions, which is consistent with the distribution characteristics of Argo float data in the South China Sea [22]. Therefore, this paper uses the K-means clustering method to carry out cluster analysis on the EOF projection coefficients to analyze the acoustic stability factors and discuss their regional consistency, which allows us to produce more accurate SSP inversion results. In the process of obtaining the EOF projection coefficients, the high-dimensional original data are compressed into a small number of modes through EOF decomposition, where each SSP sample is represented by low-dimensional projection coefficients. Each EOF mode has a clear physical meaning. For example, the first mode reflects thermal changes in the sea surface, while the second mode reflects salinity effects in the middle layer. Cluster analysis of the EOF projection coefficients can be used to analyze the key factors of acoustic stability. To explore the consistency of the EOF we employed profile data of the South China Sea. After processing, a total of 3881 SSPs were obtained. Our method is described as follows. First, the background profiles of all data are created, the empirical orthogonal matrix is calculated, the EOF projection coefficients are calculated using SLA and SSTA data, and the SSPs are inversed using the background profiles and projection coefficients, which are also used to analyze the inversion accuracy. Second, a clustering analysis step is added to the above method. After obtaining the background profiles, empirical orthogonal matrix, and EOF projection coefficients, cluster analysis is performed on the EOF projection coefficients. In the case of completed classifications, combined with the EOF projection coefficients of different categories, the corresponding background profiles are used for SSP inversion. Finally, the inversion errors before and after classification are compared to determine the effectiveness of the clustering method for acoustic stability analysis. When using K-means to carry out cluster analysis of the EOF projection coefficients, the EOF projection coefficients calculated statistically are generally used as the training set, and the \(\mathbf{i}\)-th order projection coefficient, \(\mathbf{a}_{i}\), of the \(\mathbf{n}\)-th sample is used to calculate the weight of all samples in the \(\mathbf{i}\)-th order mode of each cluster [23]:  
\[A_{i}^{(k)}=\frac{1}{N_{k}}{\sum_{m=1}^{N}\omega_{kn}a_{i}^{(n)}}, \tag{5}\]  
where \(N_{k}\) is the total number of samples in the \(\mathbf{k}\)-th cluster; \(\mathbf{A}_{i}(\mathbf{k})\) represents the mean value of the \(\mathbf{i}\)-th order EOF projection coefficient in the \(\mathbf{k}\)-th cluster, that is, the value of the cluster center in the \(\mathbf{i}\)-th dimension; and \(\mathbf{\omega}_{\mathbf{kn}}\) is the cluster indicator function, which is used to determine whether sample \(n\) belongs to the \(\mathbf{k}\)-th cluster: \(\mathbf{\omega}_{\mathbf{kn}}\) = 1 when \(\mathbf{n}\) belongs to the \(\mathbf{k}\)-th cluster and it equals zero when it does not. After obtaining the value of \(\mathbf{A}\mathbf{i}(\mathbf{k})\), the Euclidean distance of the cluster is calculated as [23]  
\[\epsilon=\sum n=1^{N}\sum k=1^{K}\omega_{kn}\sum i=1^{G}\Big{(}a_{i}^{(n)}-A_{ i}^{(k)}\Big{)}^{2} \tag{6}\]  
where \(\epsilon\) is the sum of the squares of the Euclidean distances from all samples to their cluster centers, \(\mathbf{G}\) is the order of the EOF modes used for inversion, and \(\mathbf{K}\) is the total number of categories after clustering. \(\epsilon\) provides a measure of the compactness of the cluster, where smaller values indicate a better clustering effect. Figure 1 shows the results of the acoustic stability pre-clustering analysis. Red, blue, and yellow represent three different clustering results.