Frequency-domain Finite-difference Seismic Wave Modeling On Non-uniform Grids

Seismic wave numerical modeling is an important tool for understanding wave phenomenon in realistic underground media and is also an essential foundation of full waveform inversion(FWI)and migration.Because of easy implementation and high efficiency,finite-difference method has been more popular than other methods,for in-stance,finite-element method,pseudo-spectral method,and spectral-element method.Besides the time-domain finite-difference(TDFD)method,the frequency-domain finite-difference(FDFD)method has been more and more popular in recent years.Compared with TDFD method,FDFD method has several advantages:1.the stability is absent;2.seismic attenuation is easily implemented;3.the modeling at different frequencies is independent and can be performed simultaneously by parallelization.Therefore,it is reasonable to employ the FDFD method to perform numerical modeling when dealing with frequency-dependent physical problems with limited frequency bandwidth.Nevertheless,a crucial disadvantage of the FDFD method is that it uses prohibitive memory and computational time when handling large-scale models,especially 3D mod-els.In contrast to the TDFD method,every single-frequency wavefield is computed by solving equations independently in the FDFD method.A large sparse impedance ma-trix needs to be used when we solve the system of linear equations.The dimension of the impedance matrix,which is determined by the number of grid points,is a key factor controlling the computational efficiency.To reduce the memory and computa-tional time,fewer grid points should be used in the numerical computation.The use of a non-uniform grid is a common strategy for improving the computational efficiency.Quite a few studies regarding non-uniform grids have been published in recent years.Most studies of non-uniform grids focus on the TDFD operators,and the FDFD opera-tors are almost not considered.Therefore,in this study,we devote our effort to perform frequency-domain seismic modeling on non-uniform grids and improve the efficiency of the FDFD method.In this article,we investigate how to perform frequency-domain acoustic wave modeling on continuous non-uniform grids.We propose a generalized average-derivative optimal finite-difference scheme(GADOS)for frequency-domain acoustic wave mod-eling on continuous non-uniform grids.We regard the continuous non-uniform grids as an assembly of non-uniform nine-point stencils and select the proper coefficients from the huge "Dictionary" for every stencil to ensure that the numerical dispersion is mini-mal in the entire area.After optimization,all the phase-velocity errors of the GADOS for different grid spacing ratios are less than± 1%even if the number of grid points per wavelength is as small as four.Simulatiing seismic waves with the GADOS on non-uniform grids is more efficient in terms of computational time and memory than on uniform grids with the premise of ensuring sufficient accuracy.Since the grid spacings can't vary flexibly in the non-unioform grids,we introduce how to implement 2D frequency-domain acoustic wave modeling on discontinuous non-uniform grids.In the transition region between the finer grid and the coarser grid,sev-eral finite-difference schemes are used to perform acoustic wave modeling and the nu-merical examples demonstrate that using the second-order finite-difference scheme can obtain a better result than other schemes.Then,we generalize this strategy which is used to handle the transition area in the 2D case to the 3D case and implement 3D frequency-domain acoustic wave modeling on discontinuous non-uniform grids.Com-pared with the uniform grids,modeling the acoustic wave propagation on discontinuous non-uniform grids is more efficient.To improve computational effcicncy further,we perform the 21)frequency-domain acoustic wave modeling on the frequency-vclocity-adaptive(FVA)grids.We compare several different point source implementations to handle the situation when the location of a point source doesn't coincide with the grid node.Compared with the fixed grids,modeling acoustic wave propagation on the FVA grids reduces the accuracy slightly and improves the efficiency obviously.