A new elastic wave equation for decoupling P-wave and S-waves and its application

Abstract

The imaging of P-wave and S-wave in reverse time migration (RTM) of elastic waves is often achieved by cross-correlating P-waves or S-waves with different propagation directions. This requires us to obtain the Poynting vector or optical flow vector of each imaging point at different times during the wavefield extrapolation process and use it to indicate the direction of wave propagation. But we can only obtain the Poynting vector of the mixed wavefield of P-wave and S- waves, and we cannot obtain the Poynting vector of pure P-wave or pure S-wave when using the existing velocity-stress elastic wave equations for the wavefield extrapolation process. Therefore, the propagation direction obtained is also a mixed wavefield rather than pure P-wave or pure S-wave, and this does not meet the requirements for elastic wave RTM and will cause errors. The existing first-order velocity-dilation-rotation elastic wave equation, although it overcomes the aforementioned issues, cannot accurately describe the law of wave propagation at the wave impedance interface due to the assumption of a homogeneous medium. Especially when the interface of P-wave and S-wave velocities is not consistent, it will lead to errors in the reflection, transmission, and conversion wavefields when using this equation for elastic wavefield extrapolation. In addition, severe energy leakage effects will occur at the interface of S-wave velocity when using this equation, which will lead to inaccurate S-wave imaging. In this paper, we propose a new elastic wave equation for decoupling P-wave and S-waves based on the assumption of an inhomogeneous medium, which not only gives the propagation direction of pure P-wave and pure S-wave, but also completely overcomes the above problems. Using the new equation of the Poynting vector in the elastic wave field to perform cross-correlation imaging, the model calculations show that the imaging results eliminate the noise generated by RTM, demonstrating the accuracy and applicability of the equation.