## 3 Methods
### Vmd  
VMD is an adaptive, fully non-recursive method used to address modal variation and signal processing problems [25]. VMD achieves signal decomposition by solving an optimization problem, which enhances its robustness and adaptability. It can automatically determine the central frequency and bandwidth of each mode, avoiding the issue of mode mixing. Additionally, it employs a constrained variational problem to minimize the bandwidth between modes, thereby improving the accuracy of the decomposition. Its core advantage lies in its ability to extract intrinsic modes from the signal, thereby revealing the underlying physical significance of the data. At a monthly scale, earthquake magnitudes exhibit non-stationarity, and VMD can decompose these complex seismic signals into more analyzable components. By eliminating noise and capturing key signal features, VMD provides a solid foundation for subsequent magnitude prediction and other seismic analyses. Assume that the original signal can be composed of \(K\) finite-bandwidth modal components \(\;v_{k}(t)\), with each IMF having a center frequency \(\omega_{(t)}\). The constraint is that  
Figure 4: Flow chart of the experiment.  
the sum of these \(K\) modal components equals the input signal. The specific decomposition process concept is as follows:  
(1) Obtain the analytic signal of \(v_{k}(t)\) through the Hilbert transform and compute its one-sided spectrum. By multiplying it with the operator \(e^{-j\omega_{k}t}\), shift the center band of \(v_{k}(t)\) to the corresponding baseband:  
\[\left[\left(\delta(t)+\frac{j}{\pi t}\right)\times v_{k}(t)\right]e^{-j\omega_ {k}t} \tag{11}\]  
(2) Calculate the squared norm \(L^{2}\) of the demodulation gradient and estimate the bandwidth of each modal component. The process is presented by the following formula:  
\[\underset{\{\mu_{k}\},\{\omega_{k}\}}{min}\left\{\sum_{K}\|d_{t} \bigg{[}\bigg{(}\delta(t)+\frac{j}{\pi t}\bigg{)}\times\mu_{K}(t)\bigg{]}e^{-j \omega_{K}t}\big{\|}_{2}^{2}\right\} \tag{12}\] \[s.t.\ \sum_{k}v_{k}=s \tag{13}\]  
In the above formula, \(\{v_{k}\}=\{v_{1},\ldots,v_{k}\}\) represents the decomposed IMF components, and \(\{\omega_{k}\}=\{\omega_{1},\ldots,\omega_{k}\}\) represents the center frequencies of the respective components.  
To find the optimal solution to the constrained variational problem, we first introduce the Lagrange multiplier \(\tau(t)\) and the second-order penalty factor \(\alpha\), which transform the constrained variational problem into an unconstrained variational problem. The second-order penalty factor \(\alpha\) ensures the accuracy of signal reconstruction in the presence of Gaussian noise. The Lagrange multiplier \(\tau(t)\) ensures the strictness of the constraint conditions. The extended Lagrangian expression is as follows:  
\[\begin{split} L(\{v_{k}\},\{\omega_{k}\},\tau)&:= \alpha\underset{k}{\sum}\Big{\|}\partial_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{ \pi t}\Big{)}\times v_{k}(t)\Big{]}e^{-j\omega_{k}t}\Big{\|}_{2}^{2}\\ &\quad+\Big{\|}s(t)-\underset{k}{\sum}v_{k}(t)\Big{\|}_{2}^{2}+ \bigg{\langle}\tau(t),s(t)-\underset{k}{\sum}v_{k}(t)\bigg{\rangle}\end{split} \tag{14}\]  
Using the alternating direction method of multipliers (ADMM) to iteratively update each component and its center frequency, the saddle point of the unconstrained model is ultimately obtained, which represents the optimal solution for the original data.