# Method of Suppressing Rayleigh Waves Based on the Technology of Time-Domain Differential Detection
## 2 Differential Detection Method for Suppressing Rayleigh Waves at Common Shot Points  
### Method Principle  
Figure 1 illustrates the principle of the differential detection method. Seismic waves are assumed to be generated at a surface shot point (Source) in a two-layer horizontal medium. The seismic wave records received by the main detector R\({}_{2}\) and the secondary detector R\({}_{1}\) contain numerous interfering waveforms, focusing on Rayleigh waves and reflected longitudinal waves from the target interface (Reflector).  
Figure 1: Principle schematic diagram of the common shot time-domain differential detection technology.  
The propagation paths of the Rayleigh waves received by detectors R1 and R2 are S1 and S2, respectively, whereas the reflected waves P1 and P2 are illustrated through a virtual source. The overburden thickness H has Rayleigh wave group velocity VR, with shear wave velocity Vs1 and longitudinal wave velocity VP1 in the upper medium, while the underlying medium demonstrates velocities Vs2 and VP2 for shear and longitudinal waves, respectively.  
The two-layer medium model is characterized by the following parameters: an upper layer thickness H = 8.0 m with P-wave velocity \(Vp_{1}=600\) m/s, S-wave velocity Vs1 = 250 m/s (approximately equal to the Rayleigh wave velocity \(V_{R}\)), and density \(\rho_{1}=1.9\) g/cm\({}^{3}\). The underlying half-space is defined by \(Vp_{2}=2000\) m/s, \(V_{S2}=800\) m/s, and \(\rho_{2}=2.3\) g/cm\({}^{3}\). The device parameters are set as S1 = 6.0 m and S2 = 6.5 m, indicating that the distance between the primary and secondary adjacent detectors is 0.5 m. The Rayleigh wave time differential between receivers is calculated as  
\[\Delta T=\Delta x/V_{R}\approx(S2-S1)/V_{S1}=2.0\,ms \tag{1}\]  
The travel time difference for the reflected longitudinal wave from the Reflector interface arriving at the two adjacent sensors is as follows:  
\[\Delta t=\left(\sqrt{S_{2}^{2}+4H^{2}}-\sqrt{S_{1}^{2}+4H^{2}}\right)/V_{\rm P 1}=0.5145\,\rm ms \tag{2}\]  
The Rayleigh wave recorded by the primary detector R2 (which is also the reference recording point for seismic reflection waves) is used as a reference in the time-domain differential approach. The record from the secondary detector R1 is adjusted backward in time by \(\Delta T\) (dynamic correction). After this adjustment, the initial arrival time of the reflected longitudinal wave is changed by \(\Delta T\) because \(\Delta T\) is approximately 1.5 ms greater than \(\Delta t\), resulting in the reflected longitudinal wave from the Reflector interface being \"overly time-shifted\".  
Despite the frequency dispersion and attenuation over a horizontal distance of 0.5 m, the amplitude-frequency and phase-frequency characteristics of Rayleigh waves change minimally; the group velocity and multi-frequency phase velocities remain relatively constant. Therefore, minimal amplitude compensation can ensure that the Rayleigh waves in both recordings are essentially similar. If the record from R1 is dynamically corrected and then inverted and combined with the record from R2, the Rayleigh waves can be effectively \"surgically stripped\", allowing the excessively time-shifted reflected longitudinal wave to maintain its modified version after \"damage\". Similarly, using the technical prototype depicted in Figure 1, various component recordings in the shot gather can also suppress Rayleigh waves through time-domain differential processing with analogous detection principles.