**3. Synthetic Data Application**

To validate the effectiveness of the proposed method (SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> ), we used least squares parabolic Radon transform (LSPRT) and sparse parabolic Radon transform based on *L*<sup>1</sup> regularization (SPRT*L*1) as the control group to perform a multiple attenuation test on the noisy synthetic data. The velocity model of the synthetic seismic data is shown in Figure 3a. Figure 3b is the CMP gather of a full wavefield. There are 750 sampling points in the time direction, the sampling interval is 4 ms and the offset range is 0 to 2000 m with an interval dx = 25 m.

**Figure 3.** (**a**) Velocity model; (**b**) CMP gather; (**c**) CMP gather after NMO.

In order to make the kinematic characteristics of seismic events closer to parabolas, the CMP gather should be normal movement corrected first. The primary reflections are flat and multiple reflections, which are parabolic because of inadequate correction after normal movement correction with velocities of the primary reflections in the CMP gather, as shown in Figure 3c.

After the Radon transform, the seismic events of primaries are mapped in the region with a negative q value and q = 0, and the seismic events of multiples are mapped in the region with a positive q value, as shown in Figure 4. In Figure 4a, due to the influence of the noise, the seismic events do not converge to a point well in the Radon domain, and finite seismic acquisition aperture causes severe smearing. In Figure 4b, although the smearing is alleviated, the result fell short. Figure 4c shows the result of the SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> method. Because the primary and multiple reflections are sparsely constrained, respectively, only the results of the primary seismic events are left in the Radon domain, and the mapping of multiples is removed directly. There is almost no smearing in Figure 4c, and it has higher resolution. The transformation accuracy and focusing ability are improved.

**Figure 4.** Radon domain results: (**a**) LSPRT method; (**b**) SPRT*L*<sup>1</sup> method; (**c**) SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> method.

Figure 5b is the multiple attenuation result obtained by LSPRT; Figure 5c is the result obtained by SPRT*L*1. Due to the overlapping energy of multiple and primary reflections in the Radon domain, there are many artifacts at near offset. They still have some residual multiple energy, especially at near offset. Figure 5d is the result obtained by SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> . There are almost no residual multiples at the arrow, and the reconstructed data have great consistency with the original data. Figure 6 shows the difference between the suppression results and those without multiple wavefield data. It is proved that the SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> method is effective in suppressing multiples, especially at the near-offset position. In order to quantitatively analyze the consistency between the reconstructed data and the original data, the following formula is applied to calculate the reconstruction error [42]:

$$s = \frac{\left\|{m'} - \hbar \right\|\_2^2}{\left\|{m'}\right\|\_2^2} \times 100\% \tag{26}$$

where *m* is no multiple wavefield data, *m*ˆ is data after multiple attenuation and *s* is reconstruction error.

Based on the examples of synthetic data, Table 1 lists the reconstruction error of three methods. It is obvious that compared with the LSPRT and SPRT*L*<sup>1</sup> methods, the SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> method is superior in reconstruction capability.

**Table 1.** Reconstruction error comparison.


**Figure 5.** Multiple attenuation result of the theoretical model data: (**a**) without multiple wavefield data; (**b**) LSPRT method; (**c**) SPRT*L*<sup>1</sup> method; (**d**) SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> method. The red arrows indicate the multiples suppression effect of the three algorithms, and there is no residual multiples in Figure 5d.

**Figure 6.** Difference image map: (**a**) by LSPRT; (**b**) by SPRT*L*1; (**c**) by SPRT*Lq*<sup>1</sup> − *Lq*2.

In order to verify whether the proposed method is suitable for seismic data with missing traces, we randomly selected 30%, 50% and 70% seismic trace from the synthetic seismic record containing noise and filled them with zero. We applied the proposed method to suppress multiples of missing seismic trace data. As can be seen from Figure 7a,b, the result of suppressing multiples is not affected, and there are also no residual multiples at the near offset. With the increase in the percentage of missing traces, the reconstruction effect becomes worse. When the percentage of missing traces reaches 70%, the phase distortion appears in the suppression result. Based on the reconstruction results, it can be proved that the SPRT*Lq*<sup>1</sup> − *Lq*<sup>2</sup> method is also suitable for missing trace data.

**Figure 7.** Multiple attenuation results of missing trace data by SPRT*Lq*<sup>1</sup> − *Lq*2: (**a**) 30% missing traces; (**b**) 50% missing traces; (**c**) 70% missing traces.
