A finite-difference scheme in frequency-space domain to solve heterogeneous acoustic wave equation
WU You1,2, WU Guochen1,2, LI Qingyang1,2, YANG Lingyun1,2, JIA Zongfeng1,2
1. School of Geosciences, China University of Petroleum (East China), Qingdao, Shandong 266580, China; 2. Laboratory for Marine Mineral Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao, Shandong 266071, China
Abstract：To accurately and efficiently simulate the propagation of acoustic wave in heterogeneous media, this paper constructed a general framework of finite-difference simulation of heterogeneous acoustic wave equations in frequency-space domain using a staggered grid and a hybrid grid. The finite-difference schemes of the staggered grid and the hybrid grid were respectively derived and extended to higher-order forms. The weighted avera-ge method was employed to approximate the mass acceleration term, and the perfectly matched layer (PML) absorbing boundary condition was used to effectively suppress the artificial boundary reflection. The accuracy of the method was verified by a layered model, and its stability was examined by the Marmousi model. Numerical experiments show that under the same space subdivision precision, the simulation accuracy of the hybrid grid and the fourth-order staggered grid is much higher than that of the second-order staggered grid. Although the simulation accuracy of the hybrid grid is slightly lower than that of the fourth-order staggered grid, it has higher computational efficiency. Thus, the hybrid grid method can be the first choice for the forward modeling of acoustic wave in heterogeneous media in a frequency domain.
任浩然,王华忠,龚婷.标量地震波频率-空间域有限差分法数值模拟[J].石油物探,2009,48(1):20-26,15.REN Haoran,WANG Huazhong,GONG Ting.Seismic modeling of scalar seismic wave propagation with finite difference scheme in frequency-space domain[J].Geophysical Prospecting for Petroleum,2009,48(1):20-26,15.
[2]
李青阳,吴国忱,段沛然,等.基于互相关目标函数的反射波波形反演[J].石油地球物理勘探, 2020,55(4):754-765.LI Qingyang,WU Guochen,DUAN Peiran, et al.Reflection waveform inversion based on cross-correlation misfit function[J].Oil Geophysical Prospecting,2020,55(4):754-765.
[3]
HUSTEDT B,OPERTO S,VIRIEUX J.Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling[J].Geophysical Journal International, 2004,157(3):1269-1296.
[4]
LUO Y,SCHUSTER G.Parsimonious staggered grid finite-differencing of the wave equation[J].Geophysical Research Letters,1990,17(2):155-158.
[5]
PRATT G R, WORTHINGTON M H.Inverse theory applied to multi-source cross-hole tomography.part 1:acoustic wave-equation method[J].Geophysical Prospecting,1990,38(3):287-310.
[6]
PRATT R G.Inverse theory applied to multi-source cross-hole tomography.part 2:elastic wave-equation method[J].Geophysical Prospecting,1990,38(3):311-329.
[7]
JO C H,SHIN C,SUH J H.An optimal 9-point, finite-difference,frequency-space,2-D scalar wave extrapolator[J].Geophysics,1996,61(2):529-537.
[8]
ŠTEKL I,PRATT R G.Accurate viscoelastic mode-ling by frequency-domain finite differences using rotated operators[J].Geophysics,1998,63(5):1779-1794.
[9]
SHIN C, SOHN H.A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operator[J].Geophysics,1998,63(1):289-296.
[10]
吴国忱,梁锴.VTI介质频率-空间域准P波正演模拟[J].石油地球物理勘探,2005,40(5):535-545.WU Guochen,LIANG Kai.Quasi P-wave forward modeling in frequency-space domain in VTI media[J].Oil Geophysical Prospecting,2005,40(5):535-545.
[11]
殷文,印兴耀,吴国忱,等.高精度频率域弹性波方程有限差分方法及波场模拟[J].地球物理学报,2006,49(2):561-568.YIN Wen,YIN Xingyao,WU Guochen,et al.The method of finite difference of high precision elastic wave equations in the frequency domain and wave-field simulation[J].Chinese Journal of Geophysics,2006,49(2):561-568.
[12]
梁锴,吴国忱,印兴耀.TTI介质qP波方程频率-空间域加权平均有限差分算子[J].石油地球物理勘探,2007,42(5):516-525.LIANG Kai,WU Guochen,YIN Xingyao.Weighted mean finite-difference operator of qP wave equation in frequency-space domain for TTI medium[J].Oil Geophysical Prospecting,2007,42(5):516-525.
[13]
吴建鲁,吴国忱.频率域声-弹耦合地震波波动方程有限差分方法[J].地球物理学报,2018,61(6):2396-2408.WU Jianlu,WU Guochen.Finite difference method for acoustic-elastic coupled equations of seismic waves in the frequency domain[J].Chinese Journal of Geophysics,2018,61(6):2396-2408.
[14]
李青阳,吴国忱,段沛然.准规则网格高阶有限差分法非均质弹性波波场模拟[J].石油地球物理勘探, 2019, 54(3):539-550.LI Qingyang, WU Guochen, DUAN Peiran.Elastic wavefield forward modeling in heterogeneous media based on the quasi-regular grid high-order finite difference[J].Oil Geophysical Prospecting,2019,54(3):539-550.
[15]
张衡,刘洪,刘璐,等.基于平均导数方法的声波方程频率域高阶正演[J].地球物理学报, 2014, 57(5):1599-1611.ZHANG Heng,LIU Hong,LIU Lu,et al.Frequency domain acoustic equation high-order modeling based on an average-derivative method[J].Chinese Journal of Geophysics, 2014, 57(5):1599-1611.
[16]
刘璐,刘洪,刘红伟.优化15点频率-空间域有限差分正演模拟[J].地球物理学报, 2013, 56(2):644-652.LIU Lu,LIU Hong,LIU Hongwei.Optimal 15-point finite difference forward modeling in frequency-space domain[J].Chinese Journal of Geophysics,2013,56(2):644-652.
[17]
马晓娜,李志远,谷丙洛,等.2D声波频率域数值模拟中几种有限差分方法的对比分析[J].地球物理学进展, 2015, 30(2):878-888.MA Xiaona,LI Zhiyuan,GU Bingluo,et al.Comparisons and analysis of several optimization finite-differencing schemes in 2D acoustic frequency-domain numerical modeling[J].Progress in Geophysics,2015,30(2):878-888.
[18]
范娜,成景旺,秦雷,等.一种优化的频率域三维声波有限差分模拟方法[J].地球物理学报,2018,61(3):1095-1108.FAN Na,CHENG Jingwang,QIN Lei,et al.An optimal method for frequency-domain finite-difference solution of 3D scalar wave equation[J].Chinese Journal of Geophysics,2018, 61(3):1095-1108.
[19]
BERENGER J P.A perfectly matched layer for the absorption of electromagnetic waves[J].Journal of Computational Physics, 1994, 114(2):185-200.
[20]
VIRIEUX J.SH-wave propagation in heterogeneous media:velocity-stress finite-difference method[J].Exploration Geophysics,1984,15(4):265.
[21]
SAENGER E H, GOLD N, SHAPIRO S A.Mode-ling the propagation of elastic waves using a modified finite-difference grid[J].Wave Motion, 2000, 31(1):77-92.
