## 3 Method
### Model Parameter Setting  
How to determine the parameter \(M\)? To further assess the possible impact of function model setting, the following sensitivity experiment is carried out. Combinations of two Gaussian functions and four Gaussian functions are applied to the same eddy TA profiles, respectively. That is, a double-Gaussian model and quadruple-Gaussian model are designed for comparison.  
The fitting method employed in the procedure is based on non-linear least squares to identify the optimal parameters. However, during the comparison process, it was observed that a significant number of TA profiles were unable to identify the optimal fitting parameters after an upper limit on the number of iterations. This resulted in the fitting failing when it was attempted with quadratic Gaussian functions. This may be attributed to the fact that the quadratic Gaussian function necessitates the estimation of more uncertain parameters, which in turn requires more observed data points. As a result, only 35,849 (88.2%) CE TA profiles were successfully fitted with quadruple-Gaussian functions, while the number for AE was 37,920 (85.5%).  
The TA profiles that can be fitted by quadruple-Gaussian functions, are then fitted using both the triple-Gaussian model and the double-Gaussian model. Two measurements of \(R^{2}\) and the root mean square error (RMSE) are calculated for each profile, and Table 3 displays the average of the two indicators. \(R^{2}\) has been illustrated in the previous subsection, and RMSE, which indicates the magnitude of the error produced by the model in the fit or prediction, is shown in (4).  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\hat{T}_{i}^{\prime}-T_{i}^{\prime} \right)^{2}} \tag{4}\]  
where \(\hat{T}_{i}^{\prime}\) is the model-fitted temperature anomaly value at the \(i\)th depth level of a profile, \(T_{i}^{\prime}\) is the observed value or the interpolated one based on observation for the corresponding depth level, and \(n\) is the total number of depth levels.  
The results indicate that the double-Gaussian model is not appropriate for fitting all eddy TA profiles with the average \(R^{2}\) less than 0.8 for CEs, and less effective fitting results for AEs. The \(R^{2}\) of most AE TA profiles is less than zero, which may be due to the fact that the automatic curve extreme value extraction procedure fails to capture the primary fluctuation extremes of the curve. Instead, it may have replaced them with other jitter points that had larger anomaly amplitudes but were not the primary fluctuation components.  
The triple- and quadruple-Gaussian models had better fits with \(R^{2}\) values greater than 0.80 and RMSE values less than or equal to 0.22 \({}^{\circ}\)C, respectively. The quadruple-Gaussian model provides a better fit than the triple model, which is easy to understand, as the combination of more Gaussian functions results in a more accurate fit. It shows an average increase in \(R^{2}\) of 0.05/0.04 (CE/AE) and a corresponding decrease in RMSE of 0.03/0.04 \({}^{\circ}\)C. However, this improvement is limited.  
Considering that the computational complexity increases with the number of functions, and the previously mentioned problem that four functions would lead to convergence  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Double-Gaussian** & **Triple-Gaussian** & **Quadruple-Gaussian** \\ \hline Average \(R^{2}\) & CE & 0.72 & 0.81 & 0.86 \\  & AE & \(-\)21.82 & 0.83 & 0.87 \\ \hline Average \(RMSE\) (\({}^{\circ}\)C) & CE & 0.26 & 0.21 & 0.18 \\  & AE & 1.11 & 0.22 & 0.18 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Indicator results for different numbers of Gaussian function.  
failures for a large number of TA profiles, a maximum of three functions was chosen for this study. The aim of the study is to identify major fluctuations in the TA profiles rather than to capture all anomalies in detail. Therefore, the parameter M = 3 was chosen for the MG model based on the study objectives, the number of model parameters and the computational complexity.