# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 4 Discussion  
### The Effect of the Number of Decomposed Modes  
Similar to EMD and VMD, the number of decomposed modes is a crucial parameter in GMD and must be accurately determined before processing. Setting the number of modes too high can lead to unnecessary computational overhead or the destruction of effective modes, while setting it too low can result in an incomplete decomposition of the modes. In this section, we present a simple example to demonstrate how to select the appropriate number of decomposed modes in GMD-R.  
Figure 16a depicts a synthetic seismic dataset containing a horizontal event and two parabolic events with similar curvature. We perform the LSRT and the EH-SRT to the synthetic data and the two estimated Radon models are displayed in Figure 16b,c, respectively. One can find that the Radon model of our proposed method exhibits a higher sparsity and contains fewer Radon coefficients compared to the LSRT result.  
Then, GMD-R with different decomposition numbers is applied to the EH-SRT Radon model to show the effect of the decomposition number selection. First, we perform GMD-R to the Radon model of EH-SRT with a decomposition number of three. Figure 17 showcases these decomposed modes, with the top row presenting the modes in the Radon domain, and the bottom row showing the three corresponding recovered events in the time domain. It is evident from Figure 17 that the three events in Figure 16a are taken as three modes and separated. Next, we repeat the same processing for Figure 16a, but this time with a decomposition number of four. Figure 18 depicts the result, with the top row indicating the four decomposed modes in the Radon domain and the bottom row displaying the four recovered events corresponding to the top row data. It can be observed that the first mode in Figure 17 is split into two modes in Figure 18 because the decomposition number exceeds the actual number of modes. The effectively separated modes are corrupted by the excessive decomposition number. In addition, we perform the same processing with a decomposition number of two, and the result is shown in Figure 19. The two decomposed Radon modes are displayed in Figure 19a,b, while Figure 19c,d present the recovered events  
Figure 15: Stack profile of (**a**) raw data, (**b**) primaries obtained from the Radon filter method, (**c**) primaries obtained from the GMD method.  
corresponding to Figure 19a,b. Figure 19 illustrates that the flat event is treated as one mode, whereas the other two parabolic events are considered as one mode due to their similar curvature. Conclusions can be drawn from this test that what GMD-R focuses more on is curvature, not intercept time.  
Figure 16: (**a**) The synthetic data, (**b**) the estimated Radon model of LSRT, (**c**) the estimated Radon model of EH-SRT.  
Figure 17: (**a**–**c**) denotes the three decomposed modes in the Radon space; (**d**–**f**) denotes the three recovered events corresponding to (**a**–**c**).  
In seismic processing, if the velocity is corrected, primaries are horizontal while multiples are not horizontal after NMO correction. The two types of reflection waves can be represented as \(q=0\) and \(q>0\) in the Radon domain. Therefore, we can apply GMD-R to the NMO-corrected gather to separate primary and multiple modes by selecting the decomposition number of 2.