# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 3 Numerical Examples  
### Synthetic Data 1 Test  
In this section, we design a Radon model consisting of five primaries and several multiples as shown in Figure 3a, and its synthetic data are displayed in Figure 3b. The size of the synthetic data is set as 1001 \(\times\) 60, with a sampling interval of 2 ms and a trace interval of 20 m. The Radon coefficients of primaries in the true Radon model are located in the line where its horizontal ordinate is \(q=0\), and the multiple coefficients are distributed in the area where \(0.04\leq q\leq 0.1\). Figure 3b displays the fact that multiples and primaries exhibit superposition at near offset and cross-over at far offset in the synthetic data. To demonstrate the effectiveness of our proposed EH-SRT, we perform LSRT and EH-SRT to the synthetic data, and the estimated Radon models are displayed in Figure 3c,d. It is apparent that the Radon model of LSRT is not sparse, causing primaries and multiples to overlap with each other. While the Radon model of EH-SRT is very sparse and is very similar to the true Radon model.  
Subsequently, GMD-R is applied to the two Radon models to decompose modes, which are displayed in Figure 4a,b. The four columns in Figure 4 represent the decomposed primary modes, the decomposed multiple modes, the recovered primaries, and the recov  
Figure 3: (**a**)Ture Radon model, (**b**) the synthetic CMP data, (**c**) the Radon model of the LSRT, (**d**) the Radon model of the EH-SRT.  
Figure 2: Flowchart of the proposed GMD-based demultiple method.  
ered multiples, respectively. One can find from Figure 4a that the decomposition results of the LSRT model are unsatisfactory due to limited sparsity of the LSRT model. There are multiple coefficients in the separated primary modes, resulting in multiple residuals in the recovered primaries. While these issues are fixed in the decomposition results of the EH-SRT model, multiples and primaries are separated not only in the Radon domain but also in the time domain. And there are no residual multiples in the recovered primaries.  
To more clearly verify the effectiveness of the proposed approach in multiple removal, we select a near-offset trace at \(x=420\) m and a far-offset trace at \(x=1020\) m to conduct the amplitude comparison before and after multiple elimination. It is crucial to note that only the demultiple result based on the EH-SRT model is employed in this test. In Figure 5a, the amplitude comparison of the near-offset trace is presented, where the blue line represents the amplitude of the synthetic data, and the red line represents the amplitude of the recovered primaries. It is apparent that there are obvious multiples at 0.6 s, 1.05 s, and 1.25 s, while the multiples at 0.8 s and 1.5 s are not easily discovered. This is because multiples overlap with primaries, leading to waveform variations in both types of reflections. Through comparison, it can be observed that GMD-R can not only suppress the obvious multiples but also recover primaries affected by multiples at 0.8 s and 1.5 s. This achievement can be attributed to the precise separation of primary and multiple coefficients achieved in the Radon domain. The far-offset trace comparison is displayed in Figure 5b, where multiples exhibit a certain time shift compared with Figure 5a. The waveform variations in Figure 5b reaffirm the conclusion drawn from Figure 5a, highlighting the ability of GMD-R to effectively suppress multiples and recover primary signals.  
Figure 4: The decomposed results by performing GMD-R on (**a**) the LSRT model and (**b**) the EH-SRT model. The four columns represent the decomposed primary modes, the decomposed multiple modes, the recovered primaries, and the recovered multiples, respectively.