# Data Interpretation Technology of GPR Survey Based on Variational Mode Decomposition
## 3 Comparison of VMD, EMD, and EEMD  
A synthetic signal s is taken as an example in order to test the performance of VMD decomposition for signal. The synthetic signal \(s\) contains the two high frequency components under the background of the low frequency component. The high frequency components need to be separated from background for testing the effect of the VMD decomposition. The synthetic signal \(s\) is overlapped by \(s_{1}\) and \(s_{2}\), where \(s_{1}\) is the sinusoidal periodic function and \(s_{2}\) is the segmented sine function (Figure 3a). The function is as follows:  
\[s_{1}=\sin(20\pi t) \tag{17}\]  
\[s_{2}=\left\{ \begin{array}{cc}0.2\times\sin(300\pi t)&\quad\mbox{when }0.05\leq t\leq 0.1\\ 0.4\times\sin(100\pi t)&\quad\mbox{when }0.15\leq t\leq 0.25\\ 0&\quad\mbox{other}\end{array} \right. \tag{18}\]  
\[s=s_{1}+s_{2} \tag{19}\]  
The synthetic signal \(s\) was decomposed while using VMD (Figure 3a). Figure 4b-d shows the results of the decomposition. Figure 3: Synthetic signal and intrinsic mode function (IMF): (**a**) signal s; (**b**) the first IMF; (**c**) the second IMF; and, (**d**) the third IMF. According to the results of Figure 3b-d, the signal \(s\) was decomposed to obtain three IMFs. The three IMFs correspond to the three sine functions of signal \(s\). IMF1 corresponds to \(s_{1}\), IMF2 corresponds to \(s_{2}\) in 0.15-0.25 s, and IMF3 corresponds to \(s_{2}\) in 0.05-0.1 s. The results show that the VMD can properly separate IMFs from the signals. In order to study the performance advantages of VMD, the signal \(s\) was decomposed using VMD, EMD, and EEMD, respectively, and the instantaneous spectrum of the signal \(s\) was obtained by using these three methods (Figure 4). The results for the instantaneous frequency of \(s\) in Figure 4 show the performances of these three methods. In Figure 4a, the EMD method only obtained a component at a frequency of 10 Hz, but failed to get the components at the 50 Hz and 150 Hz frequencies, most especially the components at the 150 Hz frequency. In Figure 4b, EEMD can separate the different components into the three frequencies. However, the component at 10 Hz frequency shows some fluctuations in the 0.15-0.25 s range, and the other two components have some bending corresponding to the frequency section. We found that EEMD can separate the different components, although the accuracy is not high. According to Figure 4c, the VMD method clearly separates the three different components with accurate frequencies for each time period. Thus, VMD's effectiveness in the elimination of mode aliasing is significantly superior to that of the other two methods. During the mode decomposition process, the endpoint effect is usually a serious problem in EMD, since it causes the energy to be changed after decomposition. The signals \(u(i)\) decomposition generates and the IMF energy can be expressed, as follows:  
\[E=\sqrt{\frac{\sum\limits_{i=1}^{n}u^{2}(i)}{n}} \tag{20}\]  
where \(E\) is the IMF energy of n samples, \(u(i)\) is the signal, and \(n\) is the sample number of signals. Figure 4: Frequency division section: (**a**) frequency division section based on empirical mode decomposition (EMD); (**b**) frequency division section based on ensemble empirical mode decomposition (EEMD); (**c**) frequency division section based on variational mode decomposition (VMD). The energy leakage can be reflected by defining the deviation between the total energy of the IMF and the original signal energy. The evaluation index can be expressed, as follows:  
\[\xi=\frac{\sqrt{\sum\limits_{i=1}^{k}E_{i}^{2}}-E_{u}}{E_{u}} \tag{21}\]  
where \(E_{u}\) is the signal energy, \(E_{i}\) is the IMF energy of \(i\) number, and \(k\) is the total number of IMFs. According to Table 1, the energy leakage when using VMD is smaller than that when using EMD or EEMD, while the end effect of using VMD is not obvious. The VMD is composed of a plurality of adaptive Wiener filters.