## 3 Theory and Methods
### Modal Decomposition Methods
#### 3.1.2 Variational Mode Decomposition (VMD)  
The nonlinearity and volatility of water quality parametric data are key characteristics. In this study, the VMD method was utilized for decomposing the changes in water quality parametric data. VMD, which stands for variational mode decomposition, is based on solid theoretical principles. Unlike the EMD method, VMD employs a non-recursive strategy to extract signal components. By iteratively adjusting the center frequency and bandwidth, VMD effectively resolves challenges such as error transmission and modal overlap which are commonly encountered in traditional modal decomposition methods. The fundamental mathematical formulations of the VMD method are presented as the following:  
\[\min_{\{u_{k}\}\{u_{k}\}\{u_{k}\}}\left\{\sum_{k}\left\|\partial_{t}\left[ \left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j\omega_{k}t}\right\| _{2}^{2}\right\} \tag{5}\]  
\[\text{s.t.}\sum_{k}u_{k}=f \tag{6}\]  
where \(u_{k}(t)=\{u_{1},u_{2},\cdots u_{k}\}\) is the decomposed K modes, \(\omega_{k}\) denotes the central frequency of each modal component, \(\partial_{k}\) denotes the partial derivation of moment t, \(\delta_{k}\) denotes the impulse function, \(j\) represents an imaginary unit, and the sum of the signals of every modal component is denoted as \(f\).  
Figure 7 presents the results of the component modes obtained from the VMD decomposition of the DO data and the frequency spectra for each component. Key factors in using VMD include determining the penalty parameter \(\alpha\) and the number of modes K. Despite the VMD algorithm's strong performance in signal decomposition, it has several notable limitations. VMD is highly sensitive to parametric choices, making it difficult to determine optimal values for \(\alpha\) and K, which directly impacts the quality of the decomposition. Additionally, VMD's computational demands are high, especially when processing high-frequency complex signals, which can result in extended computation times. It is also susceptible to noise and initial conditions, potentially causing mode mixing or generating spurious components, thereby affecting the decomposition accuracy. This study calculated the central frequencies for each mode after decomposing the original water quality data using VMD. When the central frequencies of the decomposed modes are very close, this suggests that the number of modes has reached the desired level. Using the central frequency method, the number of decomposed modes was determined to be K = 8, with \(\alpha\) set to 2500.  
Mode mixing means that in EMD, the spectra of different IMFs overlap each other, resulting in the inability of each IMF to be clearly separated, resulting in a poor decomposition effect. This phenomenon can clearly be observed in Figure 6b,d,f in Section 3.1.1, where the spectrum overlap of each IMF component is serious. The VMD method improves the problem of mode mixing by introducing regularization terms and parametric adjustment mechanisms. VMD can separate each IMF more accurately, making the decomposition results more accurate. The examples shown as Figure 7b,d well proved this result. The overlap degree of the spectrum of the eight IMF components obtained by VMD decomposition significantly improved, which was beneficial to the data modeling prediction.