Stable Q-compensated viscoelastic reverse-time migration based on the modified fractional Laplacian wave equations

The attenuation property of earth media can lead to amplitude loss and phase dispersion effects on multicomponent elastic data. Ignoring their impacts during imaging process will result in blurred and dislocated imaging profiles. To accurately characterize the attenuation effect in viscoelastic media, we first derive a new viscoelastic wave equation with decoupled fractional Laplacians. Numerical tests show that the proposed wave equation can accurately capture the propagation characteristics of seismic waves in viscoelastic media. Furthermore, our new wave equation can be modified to yield a decomposition equation, which enables the separated propagation of vector P- and S-wavefields. Building on the derived viscoelastic forward propagator, we develop a stable Q-compensated viscoelastic reverse-time migration approach. Usually, the inner product imaging condition is used to obtain imaging results. However, the result of inner product is affected by the angle between vectors, making the resulting images contaminated with the angle information. In this article, we introduce the magnitude- and sign-based imaging condition for PS imaging and conduct a cross-correlation imaging condition based on the scalar P-wavefield for PP imaging. In contrast to the inner product imaging condition, our imaging scheme is capable of overcoming the contamination by the angle information. In addition, high-frequency noise is amplified exponentially during the attenuation compensation process, affecting imaging precision. To address this problem, we derive the stabilized Q-compensation wave equations explicitly for vector- and scalar wavefields. Numerical examples demonstrate that the proposed Q-compensated viscoelastic reverse-time migration method can effectively correct the viscoelastic effects, yielding high-quality PP- and PS-imaging profiles.