## 4 Discussion
### Comparison of Different Inertia Weight Decreasing Methods  
From the principle of the C-AWPSO algorithm, its parameter updates mainly consist of two parts. One is the decreasing inertia weight, and the other is the update of learning factors based on the distance between particles and their historical best positions. There are generally several approaches to handling inertia weight: linear decrease, Gaussian decrease, exponential decrease, and logarithmic decrease [12; 21; 22; 23]. We used these four inertia weight decrease methods to invert simulated data with 5% noise, 10% noise, and 15% noise. The corresponding fitness convergence curves are shown in Figure 5 (all four  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **[PERSON] and** & **[PERSON] et al.** & **[PERSON]** & \multirow{2}{*}{**C-AWPSO Method**} \\  & **Essa’s Method (2015) [6]** & **[PERSON] (2019) [17]** & **[PERSON] (2021) [9]** & \\ \hline \(M\) (mGal) & 2.4 & 19.42 & 15.7 & 19.18 \\ \(z\) (m) & 202.26 & 162 & 178.4 & 153.6 \\ \(\theta\) (\({}^{\circ}\)) & 57.74 & 65 & 61.8 & 88.5 \\ \(w\) (m) & — & 315 & 331.5 & 560.53 \\ \(x_{0}\) (m) & — & — & \(-\)0.54 & 4.99 \\ RMS (mGal) & 4.57 & 5.39 & 1.91 & 0.68 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Result comparison.  
Figure 4: The Gazelle Fault anomalies predicted by the C-AWPSO algorithm are compared with the previous results [6; 9; 17].  
methods exhibit excellent convergence speed with zero RMS error in noise-free cases, so they are not displayed here).  
As shown in Figure 5, when using four different inertia weight decrease methods for inversion, there are noticeable differences in both RMS error and convergence efficiency. This phenomenon is likely related to the inherent characteristics of each decrease method. The Gaussian decrease method offers fast convergence but is sensitive to parameters and prone to getting stuck in local optima, resulting in a higher RMS error compared to the other three methods. The logarithmic decrease method is insensitive to parameters, has strong noise resistance, and can balance convergence efficiency with the ability to search for global optimal solutions [24]. The linear decrease method outperforms nonlinear decrease methods in terms of stability and is more universally applicable than other methods. When the characteristics of the inversion problem are not distinct, the linear decrease method is a safer choice [21]. Therefore, we can observe that, although the linear decrease method does not excel in convergence efficiency, its performance is relatively stable.  
To sum up, although the Gaussian decay method has a fast convergence speed, it is prone to getting trapped in local optima. The exponential decay method has similar issues and performs unstably. The logarithmic decay method, while having the advantage of maintaining search capability over the long term, has a slower convergence speed compared to the linear decay method and is more suitable for complex optimization problems. In fault inversion problems, the shape and type of faults are what need to be solved, and their characteristics are not obvious. The linear decay method is precisely suitable for such  
Figure 5: The fitness convergence curves of different methods with decreasing inertia weights. (**a**) 5% noise, (**b**) 10% noise, (**c**) 15% noise.  
situations. Therefore, we choose the stable linear decay method to control the change in inertia weight.