# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 3 Numerical Examples
### Field Data Test  
The practicality of the proposed approach is showcased in this section through the utilization of field data obtained from the open-source Madagascar. A CMP gather from the dataset is selected and shown in Figure 11a, and there are 751 time samplings and 60 traces, with a sampling interval of 4 ms and a trace interval of 50 m. We perform the LSRT and EH-SRT to the data and the estimated Radon model is shown in Figure 11b,c, respectively. The Radon model has the same time sampling and interval as the field data and with 161 slowness sampling points ranging from \(-0.1\) to \(0.2\). And there are numerous artifacts in the LSRT model in Figure 11b, which are caused by the stack of limited traces after muting the NMO stretching. It is apparent that the Radon model obtained by EH-SRT is sparser and has fewer artifacts than the LSRT result.  
Next, we perform GMD-R on the Radon model of EH-SRT. Figure 12a,b shows the decomposed primary modes and multiple modes, respectively. It is evident that the separated primary modes are mainly focused around the vicinity of slowness \(q=0\), and the multiple modes are out of the area, thus achieving the separation of primaries and multiples. And then we perform the inverse RT to the primaries modes and the recovered primaries are shown in Figure 13b. One can find that there are fewer multiples in the multiple elimination result of the GMD-R method. And then, the filter method is applied to the EH-SRT model to suppress multiples. After several tests, the optimal demultiple result is obtained when the Radon coefficients in the region of horizontal coordinates \(q>0.015\) and vertical coordinates \(t>0.7\)\(s\) are taken as multiple and muted. After that, by performing the inverse RT to the filtered Radon model, the demultiple result is obtained and shown in Figure 13a.  
We then perform inverse NMO correction to the demultiple result of the two methods in Figure 13. Subsequently, we conducted velocity scanning analysis to generate the velocity spectrum, as shown in Figure 14b,c. In the velocity spectrum of the raw data, as shown in Figure 14a, the velocity energy clusters mainly locate in the low-velocity region, which is due to the presence of the relatively low-velocity multiples in the field data. After multiple removal, these energy clusters in the low-velocity region significantly decrease, while the energy clusters of primaries become clear and exhibit a characteristic increase with depth. In comparison to the filter method, the primary energy clusters in Figure 14c are more focused, and exhibit fewer low-velocity residuals, indicating the advantage of the GMD-R method in multiple removal.  
Figure 11: **(a)**The CMP gather, **(b)** the Radon model of LSRT, and **(c)** the Radon model of EH-SRT.  
Figure 14: Velocity spectrum of (**a**) raw data, (**b**) the demultiple result obtained from the Radon filter method, (**c**) the demultiple result obtained from the GMD-R method.  
Figure 12: (**a**) The decomposed primary modes and (**b**) the decomposed multiple modes obtained by performing GMD-R on the Radon model of EH-SRT.  
Figure 13: (**a**) The demultiple result of performing the filter method on the EH-SRT model, and (**b**) the demultiple result of performing GMD-R on the Radon model of EH-SRT.  
Figure 15a-c shows the stack profile of the raw data, the demultiple result of the filter method and GMD-R method, respectively. Compared with the stack profile before multiple elimination, the filter method and GMD-R method both suppress several multiples. However, there are still some multiple residuals in Figure 15b while these multiples are absent in Figure 15c, especially in the areas pointed by the red arrows. From this point, we can recognize that the proposed method obtains a better result in multiple suppression.