# Discussions on the Processing of the Multi-Component Seismic Vector Field
## 2 Multi-Component De-Noising Methods with Respect to the Vector Seismic Field
### The Vector Separating Method of the P-wave and S-wave  
The separation of the P-wave and S-wave cannot only be used as a special de-noising method to eliminate the mutual interference between the P-wave and S-wave, but also as an important tool for the extraction of the pure wave and the reconstruction of the wavefield. For the conventional and scalar processing systems, separation of the P-wave and S-wave is the foundational step, while for the vector processing methods, separation of the P-wave and S-wave can reduce the complexity of the vector field data, which is favorable for the analysis of fast and slow S-waves and can increase the imaging quality. It should be emphasized that the subsequent multi-wave joint application requires the separated P-wave and S-wave, though the data must retain the original vector characteristics. The wavefield separation method based on the Radon transform independently processes the single-component seismic data, and cannot retain the vector characteristics of multi-component seismic data [75]. [PERSON] et al. [76] separated P-waves and S-waves according to the polarization direction of multi-component seismic data, but the separated P-waves and S-waves retrograded to the scalar field. [PERSON] et al. [77], [PERSON] et al. [78], and [PERSON] and [PERSON] [79] proposed a series of separation methods for P-waves and S-waves based on the divergence and curl of different media with homogeneous,  
Figure 6: (**a**) Original seismic data, (**b**) seismic data contaminated with random noise, (**c**) de-noised data using the group sparsity constrained time-frequency transform (blue line), and (**d**) de-noised data using the sparsity constrained time-frequency transform (blue line). inhomogeneous, and isotropic properties, but the separated P-waves became scalars. Moreover, the divergence and curl operation causes distortions in the phase and amplitude for the wavefield after separation; therefore, the results need to be corrected. To ensure, after separation, that the P-waves and S-waves still retain the relative vector characteristics as in the original input data, [PERSON] [35] proposed a wave equation method for the separation of P-waves and S-waves in the frequency-wavenumber domain, which first separated the P-wavefield and S-wavefield from the total wavefield using the divergence and curl operation according to the Helmholtz decomposition principle. Thereafter, the method projected two wavefields in the vertical and horizontal directions by multiplying them with the normalized wave numbers in the vertical and horizontal directions. This method barely changes the phase and amplitude information of the P-waves and S-waves. The derived P-waves and S-waves are still two components of x and z and retain the vector information of the original wavefield. However, this method is only applicable to the homogeneous isotropic surface medium with known P-velocity and S-velocity. For an inhomogeneous isotropic medium, [PERSON] et al. [20] proposed a vector separation method of P-waves and S-waves based on the wavefield extrapolation. In this method, the P- and S-wave-separated elastic equation was first used to downward extrapolate the elastic wavefield from the surface to a reference depth, and the P-waves and S-waves were separated in the process of downward extrapolation. Only the P-waves were saved at the reference depth; then, the elastic wave equation was used to upward extrapolate the saved P-wave to the surface to obtain the separated P-wave recording. The P-waves were subtracted from the original recording of the total wavefield to obtain the separated S-wave recording. [PERSON] et al. [80] proposed a similar vector separation method for P-waves and S-waves, but they first used the elastic wave equation to downward extrapolate the elastic wavefield from the surface to a reference depth. Thereafter, they used the P- and S-wave-separated elastic equation to extrapolate the wavefield upward to the ground surface to obtain the separated P-waves and S-waves.