等效交错网格高阶有限差分法标量波波场模拟
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摘要
密度是岩石物性参数的重要组成部分,对储层评价、岩性解释和油藏描述等具有重要作用,因此实际资料应用中含有密度信息的正演建模是必不可少的.作为波动理论应用最广泛的实际资料处理技术——逆时偏移和全波形反演,其正演建模一般采用经典的二阶标量波方程,且不考虑密度的空间变化.而一阶速度-应力控制方程通过交错网格有限差分法正演建模,其精度高于二阶方程但计算成本过高,很难在三维实际资料中应用.鉴于非均质正演在实际生产中所面临问题,本文借用交错网格的思想,充分考虑空间变密度对波传播的影响,提出基于等效交错网格的高阶有限差分建模算法并应用于非均质标量波正演.文中从数学上给出一阶方程和二阶方程在数值模拟中的等价性证明,并分析了震源、边界条件和稳定性.最后通过简单层状模型进行数值测试,对比不同建模方法以验证新方法的准确性;利用高阶精度方法对Sigsbee-2a模型做正演测试,验证本文提出的方法的稳定性.
Density is an important component of petrophysical parameters and plays an important role in reservoir evaluation, lithology interpretation and reservoir description. Therefore, forward modeling with density information in practical data is essential. As the most widely used processing technique of wave theory-reverse time migration and full waveform inversion, its forward modeling usually adopts the classical second-order variable-density scalar wave equation, and does not consider the spatial variation of density. The first-order velocity-stress control equation is modeled by staggered grid of finite difference method. Its accuracy is higher than that of the second-order equation, but the calculation cost is too high, so it is difficult to apply in three-dimensional actual data. In view of the problems encountered in actual production of heterogeneous forward modeling, using the idea of staggered grid, we fully considers the influence of spatial density on wave propagation, proposes a high-order finite difference modeling algorithm based on equivalent staggered grid and apply to non-homogeneous scalar wave forward. In the paper, the equivalence proofs of first-order and second-order equations in numerical simulation are given mathematically and the sources, boundary conditions and stability are analyzed. Finally, the numerical simulation is carried out by using the simple layered model. The accuracy of the new method is verified by comparison with different modeling methods. The Sigsbee-2 a model is tested by the high-order precision method to verify the stability of the proposed method.
Density is an important component of petrophysical parameters and plays an important role in reservoir evaluation, lithology interpretation and reservoir description. Therefore, forward modeling with density information in practical data is essential. As the most widely used processing technique of wave theory-reverse time migration and full waveform inversion, its forward modeling usually adopts the classical second-order variable-density scalar wave equation, and does not consider the spatial variation of density. The first-order velocity-stress control equation is modeled by staggered grid of finite difference method. Its accuracy is higher than that of the second-order equation, but the calculation cost is too high, so it is difficult to apply in three-dimensional actual data. In view of the problems encountered in actual production of heterogeneous forward modeling, using the idea of staggered grid, we fully considers the influence of spatial density on wave propagation, proposes a high-order finite difference modeling algorithm based on equivalent staggered grid and apply to non-homogeneous scalar wave forward. In the paper, the equivalence proofs of first-order and second-order equations in numerical simulation are given mathematically and the sources, boundary conditions and stability are analyzed. Finally, the numerical simulation is carried out by using the simple layered model. The accuracy of the new method is verified by comparison with different modeling methods. The Sigsbee-2 a model is tested by the high-order precision method to verify the stability of the proposed method.
引文
Alford R M,Kelly K R,Boore D M.1974.Accuracy of finite-difference modeling of the acoustic wave equation [J].Geophysics,39(6):758-918,doi:10.1190/1.1440470.
Alterman Z S,Kornfeld P.1968.Finite difference solution for pulse propagation in a sphere[J].Israel Journal of Technology,6:138-149.
Alterman Z S,Loewenthal D.1970.Seismic waves in a quarter and three-quarter plane [J].Geophysical Journal International,20(2):101-126,doi:10.1111/j.1365-246 X.1970.tb06058.x.
Alterman Z S,Rotenberg A.1969.Seismic waves in a quarter plane [J].Bulletin of the Seismological Society of America,59(1):347-368.
Di Bartolo L,Dors C,Mansur W J.2012.A new family of finite-difference schemes to solve the heterogeneous acoustic wave equation [J].Geophysics,77(5):1 SO-Z132.
Dong L G,Ma Z T,Cao J Z.2000.A study on stability of the staggered-grid high-order diffe rence method of first-order elastic wave equation[J].Chinese Journal of Geophysics(in Chinese),43(06):856-864,doi:10.3321/j.issn:0001-5733.2000.06.015.
DU Qi-Zhen,GUO Cheng-Feng,GONG Xu-Fei.2015.Hybrid PS/FD numerical simulation and stability analysis of pure P-wave propagation in VTI media [J].Chinese Journal Of Geophysics(in Chinese),58(4):1290-1304,doi:10.6038/cjg20150417.
Kelly K R,Ward R W,Treitel W.1976.Synthetic seismograms:A finite-difference approach [J].Geophysics,41:2-27,doi:10.1190/1.1440605.
Liu Y,Sen M.2009a.Advanced finite-difference methods for seismic modeling [J] Geohorizons,14(2):5-16.
Liu Y,Sen M.2009b.An implicit staggered-grid finite-difference method for seismic modelling [J].Geophysical Journal International,179(1):459-474,doi:10.1111/j.1365-246 X.2009.04305.x1.
Loewenthal D,Mufti I.1983.Reversed time migration in spatial frequency domain [J].Geophysics,48(5):505- 650,doi:10.1190/1.1441493.
Luo Y,Schuster G.1990.Parsimonious staggered grid finite-differencing of the wave equation [J].Geophysical Research Letters,17(2):155-158,doi:10.1029/GL017 i002 p00155.
Magnier S,Mora P,Tarantola A.1994.Finite differences on minimal grids [J].Geophysics,59(9):1230-1465,doi:10.1190/1.1443700.
Ottaviani M.1971.Elastic-wave propagation in two evenly-welded quarter-spaces [J].Bulletin of the Seismological Society of America,61(5):1119-1152.
Saenger E H,Gold N,Shapiro S A.2000.Modeling the propagation of elastic waves using a modified finite-difference grid [J].Wave Motion,31:77-92,doi:10.1016/S0165-2125(99)00023-2.
Tong P,Yang D,Hua B.2011.High accuracy wave simulation-revised derivation,numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation [J].International Journal of Solids and Structures,48(1);,56-70,doi:10.1016/j.ijsolstr .2010.09.003.
Virieux J.1986.P-sv wave propagation in heterogeneous media:Velocitystress finite-difference method [J].Geophysics,51:889-901,doi:10.1190/1.1442147.
Yin Wen,Yin Xing-Yao,Wu Guo-Chen,et al.2006.The method of finite difference of high precision elastic wave equations in the frequency domain and wave-field simulation [J].Chinese Journal of Geophysics(in Chinese),49(2):561-568,doi:10.3321/j.issn:0001-5733.2006.02.032.
Yong Peng,Huang Jian-Ping,Li Zhen-Chun,et al.2016.Optimized staggered-grid finite-difference method in time-space domain based on exact time evolution schemes [J].Chinese Journal Of Geophysics(in Chinese),59(11):4223- 4233,doi:10.6038/cjg20161124.
董良国,马在田,曹景忠.2000.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,43(06):856-864,doi:10.3321/j.issn:0001-5733.2000.06.015.
杜启振,郭成锋,公绪飞.2015.VTI 介质纯 P 波混合法正演模拟及稳定性分析[J].地球物理学报,58(4):1290-1304,doi:10.6038/cjg20150417.
杨庆节,刘财,耿美霞,等.2014.交错网格任意阶导数有限差分格式及差分系数推导.吉林大学学报:地球科学版,44(1):375-385.
殷文,印兴耀,吴国忱,等.2006.高精度频率域弹性波方程有限差分方法及波场模拟.地球物理学报,49(2):561-568,doi:10.3321/j.issn:0001-5733.2006.02.032.
雍鹏,黄建平,李振春,等.2016.一种时空域优化的高精度交错网格差分算子与正演模拟[J].地球物理学报,59(11):4223- 4233,doi:10.6038/cjg20161124.
Alterman Z S,Kornfeld P.1968.Finite difference solution for pulse propagation in a sphere[J].Israel Journal of Technology,6:138-149.
Alterman Z S,Loewenthal D.1970.Seismic waves in a quarter and three-quarter plane [J].Geophysical Journal International,20(2):101-126,doi:10.1111/j.1365-246 X.1970.tb06058.x.
Alterman Z S,Rotenberg A.1969.Seismic waves in a quarter plane [J].Bulletin of the Seismological Society of America,59(1):347-368.
Di Bartolo L,Dors C,Mansur W J.2012.A new family of finite-difference schemes to solve the heterogeneous acoustic wave equation [J].Geophysics,77(5):1 SO-Z132.
Dong L G,Ma Z T,Cao J Z.2000.A study on stability of the staggered-grid high-order diffe rence method of first-order elastic wave equation[J].Chinese Journal of Geophysics(in Chinese),43(06):856-864,doi:10.3321/j.issn:0001-5733.2000.06.015.
DU Qi-Zhen,GUO Cheng-Feng,GONG Xu-Fei.2015.Hybrid PS/FD numerical simulation and stability analysis of pure P-wave propagation in VTI media [J].Chinese Journal Of Geophysics(in Chinese),58(4):1290-1304,doi:10.6038/cjg20150417.
Kelly K R,Ward R W,Treitel W.1976.Synthetic seismograms:A finite-difference approach [J].Geophysics,41:2-27,doi:10.1190/1.1440605.
Liu Y,Sen M.2009a.Advanced finite-difference methods for seismic modeling [J] Geohorizons,14(2):5-16.
Liu Y,Sen M.2009b.An implicit staggered-grid finite-difference method for seismic modelling [J].Geophysical Journal International,179(1):459-474,doi:10.1111/j.1365-246 X.2009.04305.x1.
Loewenthal D,Mufti I.1983.Reversed time migration in spatial frequency domain [J].Geophysics,48(5):505- 650,doi:10.1190/1.1441493.
Luo Y,Schuster G.1990.Parsimonious staggered grid finite-differencing of the wave equation [J].Geophysical Research Letters,17(2):155-158,doi:10.1029/GL017 i002 p00155.
Magnier S,Mora P,Tarantola A.1994.Finite differences on minimal grids [J].Geophysics,59(9):1230-1465,doi:10.1190/1.1443700.
Ottaviani M.1971.Elastic-wave propagation in two evenly-welded quarter-spaces [J].Bulletin of the Seismological Society of America,61(5):1119-1152.
Saenger E H,Gold N,Shapiro S A.2000.Modeling the propagation of elastic waves using a modified finite-difference grid [J].Wave Motion,31:77-92,doi:10.1016/S0165-2125(99)00023-2.
Tong P,Yang D,Hua B.2011.High accuracy wave simulation-revised derivation,numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation [J].International Journal of Solids and Structures,48(1);,56-70,doi:10.1016/j.ijsolstr .2010.09.003.
Virieux J.1986.P-sv wave propagation in heterogeneous media:Velocitystress finite-difference method [J].Geophysics,51:889-901,doi:10.1190/1.1442147.
Yin Wen,Yin Xing-Yao,Wu Guo-Chen,et al.2006.The method of finite difference of high precision elastic wave equations in the frequency domain and wave-field simulation [J].Chinese Journal of Geophysics(in Chinese),49(2):561-568,doi:10.3321/j.issn:0001-5733.2006.02.032.
Yong Peng,Huang Jian-Ping,Li Zhen-Chun,et al.2016.Optimized staggered-grid finite-difference method in time-space domain based on exact time evolution schemes [J].Chinese Journal Of Geophysics(in Chinese),59(11):4223- 4233,doi:10.6038/cjg20161124.
董良国,马在田,曹景忠.2000.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,43(06):856-864,doi:10.3321/j.issn:0001-5733.2000.06.015.
杜启振,郭成锋,公绪飞.2015.VTI 介质纯 P 波混合法正演模拟及稳定性分析[J].地球物理学报,58(4):1290-1304,doi:10.6038/cjg20150417.
杨庆节,刘财,耿美霞,等.2014.交错网格任意阶导数有限差分格式及差分系数推导.吉林大学学报:地球科学版,44(1):375-385.
殷文,印兴耀,吴国忱,等.2006.高精度频率域弹性波方程有限差分方法及波场模拟.地球物理学报,49(2):561-568,doi:10.3321/j.issn:0001-5733.2006.02.032.
雍鹏,黄建平,李振春,等.2016.一种时空域优化的高精度交错网格差分算子与正演模拟[J].地球物理学报,59(11):4223- 4233,doi:10.6038/cjg20161124.