# Data Interpretation Technology of GPR Survey Based on Variational Mode Decomposition
## 2 Theory and Methodology  
### The Instantaneous Frequency Based on VMD Theory  
#### 2.1.1 VMD Theory  
VMD theory's overall framework, which includes the construction and solution of both variational models and problems, is a variational problem. The variational model is introduced to solve the signal decomposition when the VMD method is used to obtain the IMF components. The constraints are used to search for the optimal solution of the variational model. The center frequency and bandwidthof every modal function component are iteratively updated to achieve signal decomposition. Finally, the frequency band of the signal is adaptively decomposed to obtain the narrow-band IMF of the predetermined scale [23].  
In the construction of variational models, the mode function \(u_{k}(t)\) was taken as the Hilbert transform and every analytic signal was obtained. Subsequently, the central frequency of prediction was \(e^{-j\omega_{k}t}\) and the frequency spectrum of every mode was modulated to the corresponding frequency baseband. Finally, the \(L^{2}\) norm of the modulation signal was solved and the signal bandwidth of every mode was evaluated. The variational problem is the solution of the K mode function \(u_{k}(t)\), and the sum of the estimated bandwidth of every mode is minimal when the constraining condition is the sum of every mode equal to the input signal. The signal decomposed to N IMF classifications. The corresponding variational problem is defined, as follows:  
1. Every IMF component was taken as the Hilbert transform, so that the analytic signal could be obtained. \[\left(\delta(t)+\frac{j}{\pi t}\right)\ast u_{k}(t)\] (1) where \(\delta(t)\) is the Dirichlet function and \(u_{k}(t)\) is K IMF.
2. The analytic signal of the estimated central frequency can be changed to the baseband by using frequency shift. \[\left[\left(\delta(t)+\frac{j}{\pi t}\right)\ast u_{k}(t)\right]e^{-j\omega_{ k}t}\] (2) where \(\omega_{k}\) is the central frequency.
3. The \(L^{2}\) norm of the module signal was solved, and the bandwidth of every mode was estimated. The variational problem was constructed, as follows: \[\left\{\begin{array}{c}\min_{\{u_{k}\},\|u_{k}\}\left\{\sum_{k}\|\partial_{t }\left[\left(\delta(t)+\frac{j}{\pi t}\right)\ast u_{k}(t)\right]e^{-j\omega_{ k}t}\|^{2}\right\}\\ s.t.\sum_{k}u_{k}=f(t)\end{array}\right.\] (3)  
For solving the variational problem, the original signal was decomposed into K narrow mode components to obtain the optimal solution to the constrained variational problem. The augmented Lagrangian can be expressed, as follows:  
\[\begin{array}{c}L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum_{k}\|\partial _{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)\ast u_{k}(t)\right]e^{-j\omega _{k}t}\|^{2}\\ +\|f(t)-\sum_{k}u_{k}(t)\|^{2}_{2}+\left\langle\lambda(t),f(t)-\sum_{k}u_{k}( t)\right\rangle^{2}\end{array} \tag{4}\]  
where \(\alpha\) is the penalty factor, \(\lambda(t)\) is the Lagrange multiplier, \(\|\cdot\|^{2}_{2}\) is the norm, and \(\langle\cdot\rangle\) is the inner-product operator.  
The alternative direction multiplier operator was used. \(\omega_{k}\), \(u_{k}\), and \(\lambda\) were updated, so the variational problem could be solved.