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

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 simu-lation 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 char-acterized 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 pseu-dospectral 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 homoge-neous 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.Nu-merical 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.

Hybrid absorbing boundary condition based on transmitting boundary and its application in 3D fractional viscoacoustic modeling

An accurate numerical simulation for wave equations is essential for understanding of wave propagation in the earth's interior as well as full waveform inversion and reverse time migration. However, due to computational cost and hardware capability limitations, numerical simulations are often performed within a finite domain. Thus, an adequate absorbing boundary condition (ABC) is indispensable for obtaining accurate numerical simulation results. In this study, we develop a hybrid ABC based on a transmitting boundary, which is referred to as THABC, to eliminate artificial boundary reflections in 3D second-order fractional viscoacoustic numerical simulations. Furthermore, we propose an adaptive weighted coefficient to reconcile the transmitting and viscoacoustic wavefields in THABC. Through several numerical examples, we determine that the proposed THABC approach is characterized by the following benefits. First, with the same number of absorbing layers, THABC exhibits a better ability in eliminating boundary reflection than traditional ABC schemes. Second, THABC is more effective in computation, since it only requires the wavefields at the current and last time steps to solve the transmitting formula within the absorbing layers. Benefiting from a simple but effective combination between the transmitting equation and the second-order wave equation, our scheme performs well in the 3D fractional Laplacian viscoacoustic numerical simulation.

Viscoacoustic prestack reverse time migration based onthe optimal time-space domain high-order finite-difference method

Prestack reverse time migration （RTM） is an accurate imaging method ofsubsurface media. The viscoacoustic prestack RTM is of practical significance because itconsiders the viscosity of the subsurface media. One of the steps of RTM is solving thewave equation and extrapolating the wave field forward and backward; therefore, solvingaccurately and efficiently the wave equation affects the imaging results and the efficiencyof RTM. In this study, we use the optimal time-space domain dispersion high-order finite-difference （FD） method to solve the viscoacoustic wave equation. Dispersion analysis andnumerical simulations show that the optimal time-space domain FD method is more accurateand suppresses the numerical dispersion. We use hybrid absorbing boundary conditions tohandle the boundary reflection. We also use source-normalized cross-correlation imagingconditions for migration and apply Laplace filtering to remove the low-frequency noise.Numerical modeling suggests that the viscoacoustic wave equation RTM has higher imagingresolution than the acoustic wave equation RTM when the viscosity of the subsurface isconsidered. In addition, for the wave field extrapolation, we use the adaptive variable-lengthFD operator to calculate the spatial derivatives and improve the computational efficiencywithout compromising the accuracy of the numerical solution.