## 3 Wave Head Calibration Based on Parameter Optimization MVMD and Morphological Gradient
### Mpa  
MPA is an optimization meta-heuristic optimization algorithm that explores the optimal solution by simulating the predation behavior between predators and prey [18]. It is inspired by the efficient predation strategy of marine organisms. In the framework of MPA, the ocean is regarded as the global search space of the optimization problem, and the elite predator matrix represents the set of optimal solutions found in this space. At the same time, the prey matrix records the optimal candidate solutions obtained in the current iteration rounds. Based on the principle of complex interaction between predators and prey, the iterative process of MPA covers three core links: population initialization, iterative optimization, and marine memory update. Regarding the specific details of these processes, readers can refer to the literature [18], as this article will not be discussed here. Permutation entropy serves as a quantitative measure for evaluating time series signals' randomness and complexity, distinguished by its computational efficiency and robust resistance to interference [19]. The magnitude of permutation entropy correlates directly with the complexity of traveling wave intrinsic mode function components--higher entropy values indicate greater complexity and noise content. To optimize wave head calibration accuracy, the intrinsic mode function component exhibiting the lowest permutation entropy is selected as the traveling wave characteristic signal, effectively minimizing noise-induced influence. The permutation entropy calculation proceeds through the following steps:  
The phase space reconstruction of the \(m\)th traveling wave intrinsic mode function component \(\{\text{IMF}_{m}(n),n=1,2,\cdots,N\}\) is carried out, and Formula (10) is obtained. \[ \begin{pmatrix}\text{IMF}_{m}(1)&\text{IMF}_{m}(1+\tau)&\cdots&\text{IMF}_{m}(1 +(q-1)\tau)\\ \text{IMF}_{m}(2)&\text{IMF}_{m}(2+\tau)&\cdots&\text{IMF}_{m}(2+(q-1)\tau)\\ \text{IMF}_{m}(n)&\text{IMF}_{m}(n+\tau)&\cdots&\text{IMF}_{m}(j+(q-1)\tau)\\ \vdots&\vdots&&\vdots\\ \text{IMF}_{m}(K)&\text{IMF}_{m}(K+\tau)&\cdots&\text{IMF}_{m}(K+(q-1)\tau) \end{pmatrix}  \tag{10}\]  
In the formula, \(n=1,2,\cdots,K\), q is the embedding dimension, and \(\tau\) is the delay time. Each reconstructed component \(\{\text{IMF}_{m}(n),\text{IMF}_{m}(n+\tau),\cdots,\text{IMF}_{m}(n+(q-1)\tau)\}\) is arranged in ascending order according to its size, and Formula (11) is obtained. \[\text{IMF}_{m}(i+(n_{1}-1)\tau)\leqslant\cdots\leqslant\text{IMF}_{m}\big{(}i +\big{(}n_{q}-1\big{)}\tau\big{)} \tag{11}\]  
Here, \(n_{1},n_{2},\cdots,n_{q}\) is the column of the elements in the reconstructed component. For each row of the matrix obtained by reconstructing the intrinsic mode function component \(\text{IMF}_{m}(n)\), a set of symbol sequences can be obtained, as shown in Formula (12). \[S(l)=\big{(}n_{1},n_{2},\cdots,n_{q}\big{)} \tag{12}\]  
In the formula, \(l=1,2,\cdots,k\), and \(k\leqslant q!\). There are a total of \(q!\) different symbol sequences \((n_{1},n_{2},\cdots,n_{q})\) of q-dimensional phase space mapping, and the symbol sequence \(S(l)\) is one of the permutations. If the probabilities of \(k\) different symbol sequences are \(P_{1},P_{2},\cdots,P_{k}\), then the permutation entropy can be expressed as shown in Formula (13). \[H_{P}(q)=-\sum\limits_{n=1}^{k}P_{n}\ln P_{n} \tag{13}\]  
The permutation entropy \(H_{pm}\) of the \(m\)th traveling wave intrinsic mode function component can be obtained by normalizing it, as shown in Formula (14). \[H_{pm}=H_{P}/\ln(q!) \tag{14}\]  
The kurtosis value is a dimensionless parameter to measure the kurtosis of signal waveform, which can effectively describe the mutability of complex signals [20]. The kurtosis value of the signal obeying the normal distribution is about 3, while the singularity of the traveling wave fault characteristic signal is significant and obviously deviates from the normal distribution, and its kurtosis value should be greater than 3.