Truncated pseudo-differential operator $\sqrt{-\nabla ^2}$ and its applications in viscoacoustic reverse-time migration

Summary The pseudo-differential operator with symbol |k|α has been widely used in seismic modeling and imaging when involving attenuation, anisotropy and one-way wave equation, which is usually calculated using the pseudo-spectral method. For large-scale problems, applying high-dimensional Fourier transforms to solve the wave equation that includes pseudo-differential operators is much more expensive than finite-difference approaches, and it is not suitable for parallel computing with domain decomposition. To mitigate this difficulty, we present a truncated space-domain convolution method to efficiently compute the pseudo-differential operator $\sqrt{-\nabla ^2}$, and then apply it to viscoacoustic reverse-time migration. Although $\sqrt{-\nabla ^2}$ is theoretically nonlocal in the space domain, we take the limited frequency band of seismic data into account, and constrain the approximated convolution stencil to a finite length. The convolution coefficients are computed by solving a least-squares inverse problem in the wavenumber domain. In addition, we exploit the symmetry of the resulting convolution stencil and develop a fast spatial convolution algorithm. The applications of the proposed method in Q-compensated reverse-time migration demonstrate that it is a good alternative to the pseudo-spectral method for computing the pseudo-differential operator $\sqrt{-\nabla ^2}$, with almost the same accuracy but much higher efficiency.