Two exact first-order k-space formulations for low-rank viscoacoustic wave propagation on staggered grids

Abstract

Wave propagation in the viscoacoustic media is physically dispersive and dissipated. Completely excluding the numerical dispersion error from the physical dispersion in the viscoacoustic wave simulation is indispensable to understanding the intrinsic property of the wave propagation in attenuated media for the petroleum exploration geophysics. In recent years, a viscoacoustic wave equation characterized by fractional Laplacian gains wide attention in geophysical community. However, the first-order form of the viscoacoustic wave equation, often solved by the conventional staggered-grid pseudospectral method, suffers from the numerical dispersion error in time due to the low-order finite-difference approximation. It is challenging to completely eliminate the error because the viscoacoustic wave equation contains two temporal derivatives, which stem from the time stepping and the amplitude attenuation terms, respectively. To tackle the issue, we derive two exact first-order k-space viscoacoustic formulations that can fully exclude the numerical error from the physical dispersion. For the homogeneous case, two formulations agree with the viscoacoustic analytical solution very well and have the same efficiency. For the heterogeneous case, our second k-space formulation is more efficient than the first one because the second formulation significantly reduces the number of the wavenumber-space mixed-domain operators, which are the expensive part of the viscoacoustic k-space simulation. Numerical cases validate that the two first-order k-space formulations are effective and efficient alternatives to the current staggered-grid pseudospectral formulation for the viscoacoustic wave simulation.