# Radon Transform Based on Waveform for AVO-Preserving Data Construction
## 2 Methods
### Waveform Radon Transform  
We propose a new method based on the waveform to realize the hyperbolic Radon transform. First, we provide an interpretation of low resolution in Radon transform in the time domain. We simply illustrate the forward transform process in Figure 1. In Figure 1, (a) presents the \(t-x\) domain, and (b) presents the Radon domain. There are three black curves representing the reflected waves in (a). The hyperbolic Radon transform stacks the \(t-x\) data gather along the hyperbolic trajectory represented by the red dotted lines in a, b. The correct stacking trajectory a matches the reflection waves in the time domain, which forms an energy value point A in the Radon model. The wrong stacking trajectory b forms a smaller energy point B, smearing the Radon model and affecting the resolution. To obtain a high resolution, it is necessary to reduce the influence of the wrong stacking trajectory. Therefore, in the forward transform, the ideal situation is for all stacking trajectory to match the reflected waves. To achieve this, we introduced the information of the energy point of each curvature in the Radon panel and the width of the wave cycle. The main difference between the correct trajectory and the wrong trajectory is their corresponding energy points in the Radon domain: the correct trajectory's energy point is larger, and the shape is closer to the waveform. Besides, the curvature of each cycle of the waveform is consistent, with every cycle corresponding a correct curvature. However, some factors that influence the accuracy of the energy point, the noise and the negative values of the wavelet, such as the trough. In addition, the waveform has a complex shape, so the wave cycle can be difficult to obtain. To overcome these problems, we used the waveform mode decomposition (WMD) to process the wavelet before the Radon transform [24]. We used the upper envelope in the WMD to present the wavelet. Taking the Rick wavelet as an example, the waveform and the upper envelope are shown in Figure 2. The upper envelope can reflect the position of the wave crest and the wave cycle, \(t_{w}\). Therefore, obtain high-resolution results in the Radon domain, two Radon transformations are needed. In the first Radon transform, the correct stacking trajectory are obtained  
Figure 1: The forward transform of the hyperbolic Radon transform: (**a**) \(t-x\) domain; (**b**) Radon domain. Figure 2: The waveform of Ricker wavelet and its upper envelope. using the energy points in the Radon domain. In the second transform, the \(t-x\) data are stacked along the correct stacking trajectory selected in the first transform. The specific steps are as follows:  
1. Make the first forward hyperbolic Radon transform in the time domain: stack the current \(t-x\) data gather along the given curvature \(p_{i}\) trajectory. After the transformation, the energy values of all samples in Radon panel are obtained. 2. Arrange the samples in the Radon panel in order of large energy to small energy and obtain an array \(S=\left(S_{1},S_{2},\ldots\ldots S_{nT\times np}\right)\). Each sample in array \(S\) has an intercept time \(\tau\) and curvature \(p\). The sample \(S_{1}\) is the largest energy sample in the Radon model and corresponds to \(\left(\tau_{S_{1}},p_{S_{1}}\right)\). \(S_{n}(n=1,2\ldots\ldots n\tau\times np)\) corresponds to \(\left(\tau_{S_{n}},p_{S_{n}}\right)\). 3. Make the second forward Radon transform in the time domain and the transform sequence accord to array \(S\). First, transform the \(\left(\tau_{S_{1}},\ \ p_{S_{1}}\right)\) in the \(t-x\) data gather and eliminate the amplitude energy of the stacking path. Use \(p_{S_{1}}\) to transform \(\left(\tau_{S_{1}}-\frac{t_{x}}{2},\ \ \tau_{S_{1}}+\frac{t_{x}}{2}\right)\) and eliminate these trajectories. Then, proceed in sequence by array \(S\).