## 2 Methods
### Constraint under the Linear Radon Transform  
When the seismic wave propagates to the geophone in the well, the slowness of the wavefield in the z direction is calculated by [27]:  
\[p_{n}=\frac{\cos\varphi_{n}}{v_{n}}\,, \tag{3}\]  
where \(\varphi_{n}\) is the angle between the direction of wave propagation and the z-axis (downward is positive), \(v_{n}\) is the velocity of wave propagation, and \(n\) is the type of wavefield. \(p_{n}\) can also be considered the  
Figure 1: The process of separating a single wavefield using the conventional median filter in the case of non-parallel events in the target wavefield. \(\Delta t\) represents the maximum time shift of the first arrival; black arrows denote downgoing P-wave events, red arrows denote downgoing S-wave events, green arrows denote upgoing S-wave events, and blue arrows denote upgoing S-wave events. (**a**) Original seismic data; (**b**) Statically aligned data; (**c**) Target wavefield after median filtering. ray parameter of the wavefield events. When the length of the geophone array is much smaller than the depth of the geophones, the seismic wave can be regarded as a plane wave rather than a spherical wave. That means a direct wave or reflected wave from the same position reaches the geophones in a parallel direction. Therefore, the slowness of the wavefield is invariant with the geophones, and the events in the wavefield are approximately a straight line. The travel time of geophone level \(i\) is:  
\[t_{i}=t_{1}+\ i\Delta hp_{n}, \tag{4}\]  
In the case of a short array, the reflection time-distance curves are approximately straight lines, and, therefore, the seismic wavefield in the space-time domain can be transformed into the \(\tau\)-\(p\) domain by the linear Radon transform. The forward and inverse formulas of the sample angle stacking are [11; 28; 29]:  
\[v(p,\tau)=\sum\limits_{i=1}^{N_{x}}u(h_{i},t=\tau+h_{i}p), \tag{5}\]  
and  
\[u(h_{i},t)=\sum\limits_{i=1}^{N_{x}}v(p,\tau=t-h_{i}p), \tag{6}\]  
where \(v(p,\tau)\) is the \(\tau\)-\(p\) domain signal, \(t\) is the time, \(h\) is the depth, \(\tau\) is the intercept time, \(p\) is the slowness, \(u(h_{i},t)\) is the inverse result, \(N_{x}\) is the number of geophones, and \(h_{i}\) is the depth of the number \(i\) geophone. Different types of Radon forward transform can be used for wavefield separation. First, the seismic data in the space-time domain are transformed into the \(\tau\)-\(p\) domain by Equation (5); after that, the wavefield is distinguished by the characteristics of different types of wavefields in the \(\tau\)-\(p\) domain, and then the target wavefield is obtained by the Radon inverse transform shown in Equation (6). In addition to the linear Radon transform, there are a series of Radon transformations with different domains and different stack paths. In order to obtain the slowness values of the target wavefield, we only perform the Radon forward transformation without the Radon inverse transformation. To increase the focus of the calculated energy spectrum in the \(\tau\)-\(p\) domain, we add a time window to Equation (5) and improve it as:  
\[V(p,\tau)=\sum\limits_{m=-\frac{M}{2}}^{M}\frac{\sum\limits_{i=1}^{N_{x}}u^{2 }(h_{i},t=\tau+h_{i}p+m)}{\sigma_{m}{}^{2}[u(h_{i},t)]}, \tag{7}\]  
where \(M\) is the length of the time window and \(\sigma_{m}{}^{2}\) is the variance of the amplitude sequence in the stack path of the sampling point \(m\). Therefore, the modified \(\tau\)-\(p\) domain seismic data (slowness spectrum) obtained by Equation (7) can show the amplitude similarity and the magnitude of the seismic wave energy spectrum in the stack path corresponding to different intercept times and slowness. The extreme point on the slowness spectrum is an important basis for calculating the reasonable slowness of the target wavefield.