# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 3 Numerical Examples
### Synthetic Data 2 Test
However, as a drawback, some of the primary coefficients are unintentionally suppressed as well. This excessive suppression of multiples leads to damage to primaries. There are fewer primary residuals in the difference panel in Figure 8c, which shows the advantage of GMD-R in preserving primaries. Instead of directly cutting off to separate primaries and multiples in the filter method, GMD-R uses a Wiener filter to cluster and separate modes. Figure 7: (**a**) The Radon model estimated by (**a**) LSRT and (**b**) EH-SRT. By gradually decaying the modes within the Radon domain at the boundary, GMD-R avoids the truncation effect, allowing for better recovery of the primaries while maintaining their amplitude. In this synthetic dataset, it would not be scientifically rigorous to compare amplitude variation before and after multiple suppression with only one trace. Therefore, a single trace, which is generated by superimposing 20 traces in the range from 0 to 500 m, is used to showcase the amplitude variation. Figure 9 displays the stacked trace, in which the black line represents the amplitude of synthetic data, and the red line and blue line indicate the recovered primaries with the Radon filter method and the GMD-R method, respectively. To provide a reference for comparison, the green line represents the stacked trace obtained by integrating synthetic data without multiples. This serves as a means to demonstrate the abilities of the two methods in preserving amplitude and suppressing multiples. It is evident that after multiple suppression, the amplitude at 1.12 s, 1.22 s, and 1.38 s decreased, but the amplitude of the GMD-R result is closer to the multiple-free data, validating the superior multiple suppression capability of GMD-R. Additionally, the demultiple result obtained with GMD-R showcases similar amplitudes as the multiple-free data at other times, further supporting the claim that GMD-R effectively preserves the amplitude of the primary signals. A conclusion can be drawn from the test that the proposed GMD-R can  
Figure 8: (**a**) The process results by performing GMD-R to the LSRT model, (**b**) the process results with the Radon filter method to the EH-SRT model, and (**c**) the process results by performing GMD-R to the EH-SRT model. The four columns represent the decomposed or retained primary modes, the decomposed or muted multiple modes, the recovered primaries using the inverse RT, and the differences profile between the recovered primaries and the synthetic data. not only achieve a better multiple removal than the filtering method but also preserve the amplitude of primaries. Multiples in seismic data exhibit low velocities compared to primaries, and they show different patterns in the velocity scanning spectrum. Therefore, we can evaluate the effectiveness of the two demultiple methods through velocity scan analysis. We perform velocity scanning to the synthetic data in Figure 6a and the velocity spectrum is shown in Figure 10a. One can find from the velocity spectrum that in addition to the primary energy clusters, which are linearly increasing with depths, that there are several low-velocity multiple energy clusters. Then, we perform velocity scanning to the demultiple result of the Radon filter method and GMD-R in Figure 8b,c, and the velocity spectrum is displayed in Figure 10b,c, respectively. Note that before velocity scanning, the multiple suppression result in Figure 8 is processed with the inverse NMO correction. It is evident that the multiple energy clusters in the velocity spectrum of raw synthetic data are both significantly reduced in Figure 10b,c. The reduced multiple energy clusters are primarily at 1.12 s, 1.22 s, and 1.38 s, which aligns with the findings presented in Figure 9. Moreover, the primary energy clusters of the demultiple result from GMD-R are more converged and focused, providing additional evidence for the efficacy of the proposed method in suppressing multiples and preserving primary amplitude.