A Separated Formulation of the Elastic Wave Equation in P- and S-potential Fields

Elastic wave imaging has been a significant challenge in the exploration industry due to the complexities in wave physics and in numerical implementation. In this paper, we derive the elastic wave equations without the assumptions of homogeneous Lamé parameters to capture the mode conversion between the P- and S-waves in an isotropic, constant-density medium. The resulting set of two coupled second-order equations for P- and S-potentials clearly demonstrates that mode conversion only occurs at the discontinuities of the shear modulus. Applying Born approximation to the new equations, we derive PP and PS imaging conditions as the first gradients of waveform matching objective functions. The resulting images are consistent with the physical perturbations of the elastic parameters, and hence are automatically free of the polarity reversal artifacts in the converted images. When implementing elastic reverse time migration (RTM), we show that scalar wave equations can be used to back propagate the recorded P-potential, as well as individual components in the vector field of the S-potential.