## 3 Method
### Multiple Gaussian Model
A value of 1 indicates that there is no difference between the predicted or fitted values and the true values, while a value of 0 indicates that the model is fitted as well as it would be if the fitted value is set directly to the mean of the observed data, and a negative value indicates a less effective model. \[R^{2}=1-\frac{\sum_{i=1}^{n}\left(\hat{T}_{i}^{\prime}-T_{i}^{\prime}\right)^ {2}}{\sum_{i=1}^{n}\left(\hat{T}_{i}^{\prime}-T_{i}^{\prime}\right)^{2}} \tag{3}\]  
where \(\hat{T}_{i}^{\prime}\) is the model-fitted value for depth level \(i\), \(T_{i}^{\prime}\) is the observed value for the corresponding level, and \(\hat{T}_{i}^{\prime}\) is the mean of the observed values for the profile. To illustrate the fitting procedure, we take the TA profiles of a CE (Figure 6) and an AE (Figure 7) as examples. In these examples, the MG fitting yields M = 3, which represents a combination of three Gaussian functions. The TA profile in Figure 6a identifies three extreme points: \(P_{1}\), \(P_{2}\), and \(P_{3}\), all of which have negative values. The corresponding depths are \(u_{1}\) = 30 m, \(u_{2}\) = 155 m, and \(u_{3}\) = 375 m, respectively, with \(T_{E1}^{\prime}\) = \(-\)4.39 \({}^{\circ}\)C, \(T_{E2}^{\prime}\) = \(-\)4.21 \({}^{\circ}\)C, and \(T_{E3}^{\prime}\) = \(-\)2.95 \({}^{\circ}\)C, whereas the three extreme points identified in Figure 7 exhibit different polarities. The \(P_{1}\) at a depth of 25 m has a negative value of \(-\)5.25 \({}^{\circ}\)C, while \(P_{2}\) and \(P_{3}\) at depths of 555 m and 820 m are both positive, with values of 5.107 \({}^{\circ}\)C and 3.13 \({}^{\circ}\)C, respectively. It can be seen that the TA profiles of actual mesoscale eddies do not ideally consist of a single peak, but are generally a composite of two or more peaks in a profile curve, where the polarity (positive or negative) of the peaks may be the same (Figure 6a) or opposite (Figure 7a). The three extreme points were found, combined with the low-pass filtered curves firstly, then the parameters of each individual Gaussian function were obtained, and at the end, the parameters were automatically adjusted when fitting via the MG model, which helped to fit the TA profile more accurately. The extreme points identified in Figures 6 and 7, which represent the centres of fluctuations at different depths in the TA profile, reflect the vertical variations in temperature anomalies caused by the mesoscale eddies. However, it can also be observed that not all fluctuations are significant. For example, the extreme point in Figure 6d is deeper than the other two extreme points. The Gaussian function fitted from this point has a smaller amplitude and the fluctuations are more muted. Consequently, this point does not contribute significantly to the overall profile. A similar situation is observed in Figure 7d, where the identified extreme point does not represent the main components of the TA profile and only contributes to improving the fitting accuracy. Then how to judge the primary fluctuation of eddy temperature anomalies? The fluctuation with the maximum extreme temperature anomaly does not necessarily indicate the primary components of the profiles. As shown in Figure 6, the first Gaussian function with an extreme point of \(-4.39\)\({}^{\circ}\)C (maximum temperature anomaly) is located at a depth of 30 m; however, it does not have the largest contribution. Instead, the second Gaussian function with an extreme of \(-4.21\)\({}^{\circ}\)C has the largest \(R^{2}\) value with 0.96, indicating the primary fluctuation curve. The same phenomenon occurs in Figure 7. The second Gaussian function with the extreme of 5.07 \({}^{\circ}\)C is the primary component instead of the first one with the largest anomaly of \(-5.25\)\({}^{\circ}\)C. The maximum extreme values may be caused by drastic changes in the seasonal characteristics of the sea water near the surface. In addition, the burr phenomenon may cause single or multiple anomalies, which can be affected by the external environment during buoy measurements.