## 3 Synthetic Data Test
Based on Equation (5), the spectral values in Figure 8a are the superposition values of amplitudes along a straight line. However, based on Equation (7), they are the ratios of the superposition amplitudes to the variance of the amplitudes along a stack path. When the amplitudes along a certain stack path are similar, the variance value is close to zero. Thus, compared with the traditional \(\tau\)-\(p\) transform, we can get more discriminative spectral values in Figure 8b, which are then employed to constrain the median filtering. In the results of the modified Radon transform (Figure 8b), the yellow box shows the region of the upgoing P-wave energy spectrum. By the linear fitting of the extremum points of this region, we can get \(a=-53.3\), \(b=0.0088\). According to Equations (11) and (12), the upgoing P-wave can be obtained by aligning and median filtering, as shown in Figure 9a. Similarly, by the linear fitting of the extremum points in the region of the S-wave in the red box, we get \(a=89\), \(b=0.003\). The upgoing S-wave was obtained through Equation (10) and is shown in Figure 9d. Figure 7: Results for the downgoing wavefield and theoretical wavefield. (**a**) Downgoing P-wave obtained by our method; (**b**) Downgoing P-wave obtained by the high-resolution Radon transform; (**c**) Theoretical downgoing P-wave; (**d**) Downgoing S-wave obtained by our method; (**e**) Downgoing S-wave obtained by the high-resolution Radon transform; (**f**) Theoretical downgoing S-wave. Figure 8: The results for the (**a**) traditional and (**b**) modified linear Radon transform based on the VSP data after subtraction of the downgoing wavefields in the original synthetic VSP records. Figure 9: Results for the upgoing wavefield and theoretical wavefield. (**a**) Upgoing P-wave obtained by our method; (**b**) Upgoing P-wave obtained by the high-resolution Radon transform; (**c**) Theoretical upgoing P-wave; (**d**) Upgoing S-wave obtained by our method; (**e**) Upgoing S-wave obtained by the high-resolution Radon transform; (**f**) Theoretical upgoing S-wave. In order to further test the performance of the proposed method, we adopted a VSP record with more complicated wave interference. The parameters of model 2 are listed in Table 2. As shown in Figure 10, the distance between the source and well is 600 m. The geophones are located at the depth of 600-795 m with an interval of 5 m. In the modelling, we applied the staggered-grid finite difference simulation. The peak frequency of Ricker wavelet was set as 50 Hz. As shown in Figure 11, the synthesized record presents more reflections and multiples, which intersect and interfere with each other. Our separation results (Figure 12a-d) were compared with the results of the high-resolution Radon transform (Figure 12e-h). They are displayed with the same gain parameters. It can be seen in Figure 12f that in the vicinity of 0.8 s, there are a few mixed upgoing wavefield events marked by the green arrow. Figure 12f,h contain the weak P-wave first arrivals marked by the blue arrows. As shown in Figure 11, there are two parallel events over 1.6 s indicated by the red arrows, which are both upgoing P-waves. In our method, they are well separated and shown in Figure 12c and marked by the red arrows. However, the high-resolution Radon transform method cannot well separate these events as shown by the red arrows shown in Figure 12g,h.