# Radon Transform Based on Waveform for AVO-Preserving Data Construction
## 3 Examples  
### Synthetic Examples  
To evaluate the performance of the proposed method, we designed synthetic data with a Ricker wavelet which showed strong amplitude variation with offset. Figure 3a shows the synthetic CMP gather containing 70 offsets from 0 to 2760 m. There were three reflecting waves located at 500 ms, 900 ms and 1200 ms, respectively. Two the reflecting waves were variable amplitude events at 500 ms and 900 ms, respectively. Figure 3b shows the Ricker wavelet and its upper envelope of the first trace and the last trace in Figure 1a. To accurately describe the AVO characteristics of the reflections located at 500 ms and 900 ms are provided in Figure 4.  
To test the data reconstruction performance, we created some empty channels in the synthetic data, as shown in Figure 5a. We used the waveform Radon transform (WRT) and CGRT to reconstruct the data gather. The Radon panel results in Figure 5b,c show that the resolution of the WRT in the Radon panel was higher than the CCRT, and the WRT waveform was closer to the Ricker wavelet. Figure 6 shows the inverse transform result and its residual.  
Figure 3: (a) Synthetic CMP gather; (b) the waveform and upper envelope.  
Compared with the CGRT, the WRT inverse transform results recovered the original \(t-x\) data better, as shown in Figure 6. Figure 7 shows the AVO characteristic between the true and predicted data using the CGRT and proposed WRT. It shows that the WRT was able to preserve the AVO on the full range of offsets.  
Figure 8 shows the waveform of the original data at the first trace and one of the reconstruction traces. The black line represents the original data, the red line and the blue line represent the reconstruction data of the CGRT and WRT, respectively. Compared with the CGRT, the WRT kept an accurate waveform with little error. The waveform error of the WRT increased at the reconstruction trace. However, the WRT still kept an accurate waveform compared with the CGRT.  
Figure 4: (**a**) AVO of the reflection at 500 ms; (**b**) AVO of the reflection at 900 ms.  
Figure 5: (**a**) Seismic data with voids; (**b**) Radon panel result of the CGRT; (**c**) Radon panel result of the WRT.  
Figure 6: (**a**) Reconstruction result of the CCRT; (**b**) reconstruction result of the WRT; (**c**) reconstruction residual of the CCRT; (**d**) reconstruction residual of the WRT.  
Figure 7: (**a**) AVO of reflection at 500 ms; (**b**) AVO of reflection at 900 ms.  
In order to quantitatively describe the results of the data reconstruction, we use signal-to-noise ratio, defined as  
\[SNR=10\log_{10}\left(\frac{||x||_{2}^{2}}{||x-\hat{x}||_{2}^{2}}\right) \tag{19}\]  
where \(x\) represents original full data and \(\hat{x}\) represents reconstructed data. The SNR of the reconstructed results obtained by the CGRT and the WRT are 10.49 dB and 22.69 dB, respectively.  
We recorded the running time of the CGRT and WRT. The hardware of the computer used in the experiment included an intel(R) Core (TM) i7-4790 [EMAIL_ADDRESS] GHz. The computational time of the CGRT was 135 s, and that of the WRT was 15 s. These results prove that the WRT was faster than the CGRT.