# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 3 Numerical Examples
### Synthetic Data 2 Test  
In this section, a complex synthetic data with several multiples are generated to further verify the method's efficiency. Figure 6a,b show the designed synthetic CMP data before and after the NMO correction. It is apparent that multiples have a parabolic shape and are more easily to detect after the NMO correction. The size of the synthetic data is set as 1001 \(\times\) 100, with a sampling interval of 2 ms and a trace interval of 20 m.  
We perform the LSRT and EH-SRT to the NMO-corrected data and the estimated Radon model is shown in Figure 7a,b, respectively. The Radon model has the same time sampling and interval with seismic gather, with 101 slowness sampling points ranging from \(-\)0.2 to 0.8. It is apparent in Figure 7a that the Radon model of the LSRT is divergent in shallow layers, which is caused by the stack of limited traces after muting the NMO stretching [25], while the Radon model of EH-SRT is sparser than the LSRT result not only in the low layers but also in the deep region. The velocity for NMO correction is known to be absolutely accurate, resulting in the correction of primaries from parabolic to flat. Figure 5: The amplitude comparison before and after multiple suppression. (**a**) the near-offset trace at x = 420 m, (**b**) the fat-offset trace at x = 1220 m.  
Figure 6: The synthetic CMP gather data (**a**) before and (**b**) after NMO correction. Consequently, the Radon coefficients of the primary are aligned along a line where its horizontal ordinate is \(q=0\). However, the Radon coefficients of multiple vary with depth and slowness, and the deeper the layer is, the closer the Radon coefficients of multiples are to primaries. To verify the help of the EH-SRT to GMD-R, we apply GMD-R to the Radon model of LSRT and EH-SRT, and the result is shown in Figure 8a,c, respectively. The four columns in Figure 8 represent the separated primary modes, the separated multiple modes, the recovered primaries by performing the inverse RT to the decomposed primary modes, and the suppressed multiples. It is important to note that the fourth column does not represent the recovered multiples achieved by performing the inverse Radon transform on the separated multiple modes. Instead, it is the suppressed multiples obtained by subtracting the recovered primaries from the synthetic data. Comparing Figure 8a,c, we can observe that GMD-R achieves the separation of multiple and primary modes for both the LSRT and EH-SRT models. The decomposed primary modes entirely keep the linear features of primary coefficients in the Radon domain. However, due to the sparsity of the Radon model, the primary modes separated from the LSRT model contain many multiple Radon coefficients. As a result, some residual multiples are present in the recovered primaries. Meanwhile, in the shallow region, due to the lack of convergence of the LSRT model, there are many primary Radon coefficients decomposed in the multiple modes, leading to amplitude attenuation of the recovered primaries. The leakage of primaries in the difference profile indicates the issues. In contrast, these problems are greatly reduced in the EH-SRT processing result. It can be seen from the residual profile in Figure 8c that the recovered primary amplitudes in the shallow layers are less damaged. To further demonstrate the superiority of the GMD-based method over the filter method, we also applied the filter method to the EH-SRT model to suppress multiples. After several filtering parameter adjustments, the best multiple suppression results are obtained when the Radon coefficients with \(q>0.05\) are taken as multiples and muted, and the result is shown in Figure 8b. The four columns in Figure 8b represent the retained primary Radon coefficients, the muted multiple Radon coefficients, the recovered primaries, and the differences between the recovered primaries and the synthetic data. From the difference profile, it is evident that the filter method effectively suppresses multiples.