## 3 Wave Head Calibration Based on Parameter Optimization MVMD and Morphological Gradient  
### Mvmd  
MVMD was first proposed by [PERSON] et al. (1998) in 2019. It is a multivariate extension of a novel VMD. Compared with traditional VMD, MVMD can process signals of multiple channels at the same time. In order to minimize the sum of the bandwidth of all intrinsic mode functions on all channels, and the sum of each mode can accurately recover the original signal, the correlation between each channel is fully utilized to improve the accuracy of signal decomposition and shorten the time of fault feature extraction, so as to quickly realize fault location ([PERSON] et al., 2019). In this paper, MVMD is used to decompose line-mode voltage and zero-mode voltage at the same time. The specific steps are as follows:  
Suppose that both the line-mode voltage and the zero-mode voltage are decomposed into \(m\) intrinsic mode function (IMF) components, as shown in (6).  
\[x_{c}(t)=\sum_{i=1}^{m}u_{m}(t),\;\;c=1,0 \tag{6}\]In the formula, \(x_{c}(t)=[x_{1}(t),x_{0}(t)]\), \(x_{1}(t)\) is the line-mode voltage signal, and \(x_{0}(t)\) is the zero-mode voltage signal. \(u_{m}(t)=[u_{1}(t),u_{2}(t),\ldots,u_{m}(t)]\) and \(u_{m}(t)\) are the \(m\)th IMF component.  
The Hilbert transform is performed on the component \(u_{m}(t)\) to obtain the analytical representation of the vector, which is denoted as \(u_{+}^{m}(t)\), and the bandwidth of each mode \(u_{+}^{m}(t)\) is estimated by the L\({}_{2}\) norm square of the gradient function after the harmonic conversion. Then, the related optimization problem under the constraint condition is changed into Formula (7).  
\[\begin{array}{l}\min\limits_{\{u_{m,c}\},\{\omega_{m}\}}\left\{\sum\limits_{ m}\sum\limits_{c}\left\|\partial_{t}\left[u_{+}^{m,c}(t)e^{-j\omega_{m}t} \right]\right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{m}u_{m,c}(t)=x_{c}(t),c=1,2,\ldots,C\end{array} \tag{7}\]  
In the formula, \(u_{+}^{m,c}(t)\) denotes the corresponding channel \(c\) and the analytic signal with mode \(m\). \(\omega_{m}\) denotes the central frequency of the \(m\)th mode, and \(\partial_{t}\) denotes the time-dependent partial derivative.  
By introducing the Lagrange multiplication operator, (7) is transformed from a constrained problem to an unconstrained problem, and the corresponding augmented Lagrange expression is shown in (8).  
\[\begin{array}{l}\mathcal{L}\big{(}\{u_{k,c}\},\{\omega_{k}\},\lambda_{c}\big{)} =\alpha\sum\limits_{k}\sum\limits_{c}\left\|\partial_{t}\left[u_{+}^{k,c}(t) e^{-j\omega_{k}t}\right]\right\|_{2}^{2}+\\ \sum\limits_{c}\left\|x_{c}(t)-\sum\limits_{k}u_{k,c}(t)\right\|_{2}^{2}+\sum \limits_{c}\left\langle\lambda_{c}(t),x_{c}(t)-\sum\limits_{k}u_{k,c}(t) \right\rangle\end{array} \tag{8}\]  
In Formula (8), \(\left\langle\cdot\right\rangle\) presents the inner product. Finally, the above unconstrained problem is solved by the alternating direction multiplier algorithm, and the center frequency update is expressed as (9), so as to complete the adaptive decomposition of multivariate signals through the updated relationship.  
\[\omega_{k}^{n+1}=\frac{\sum\limits_{c}\int_{0}^{\infty}\omega\left|\hat{u}_{k,c}(\omega)\right|^{2}d\omega}{\sum\limits_{c}\int_{0}^{\infty}\left|\hat{u}_{ k,c}(\omega)\right|^{2}d\omega} \tag{9}\]