VTI介质交错网格FCT有限差分数值模拟
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摘要
波动方程有限差分数值模拟是研究地震波在地下介质中的波场特征和传播机理的重要手段。对于常规有限差分技术,当采用大网格对计算空间进行差分离散时会出现严重的数值频散问题,降低了计算精度。通量校正(Flux-corrected transport method,FCT)技术能够有效压制粗网格情况下有限差分的数值频散。本文研究了具有垂直对称轴横向各向同性(Vertical Transverse Isotropy,VTI)介质的交错网格FCT有限差分技术。首先从一阶速度—应力弹性波方程出发,在交错网格空间中给出了该方程的高阶有限差分法格式及稳定性条件,在此基础上研究了波动方程正演过程中的数值频散FCT压制技术,二者结合实现了该方程的高精度有限差分数值模拟。同常规算法相比,本文算法不额外增加内存需求,少量增加计算量,但可有效压制VTI介质中弹性波动方程正演的数值频散现象。当采用大网格进行数值模拟时,本文方法明显提高了波场模拟精度。
Wave equation finite difference numerical simulation is an important method of studying wave field characteristics during spreading in underground medium.In the traditional wave equation finite difference methods,it causes severe numerical frequency dispersion when using a long spatial step length,which reduced the calculation precision.FCT method can reduce the numerical dispersion producing in the coarse grid.This paper studies staggered grids FCT finite difference technology in transversely isotropic media with a vertical symmetry axis(VTI).Starting from the first order velocity-stress elastic wave equation,high order finite difference method format of the equation and stability conditions were given in staggered grids space,on that basis the numerical frequency dispersion FCT pressing technology was studied in wave equation modeling process,and high precision finite difference numerical simulation of the equation was realized by combining these two methods.Compared with the conventional algorithm,the algorithm in this paper does not need additional memory,a small increase calculation,but can effectively suppress numerical frequency dispersion phenomenon of elastic wave equation in VTI medium.While adopting a big mesh to carry on numerical simulation,the method proposed in this paper obviously improves the wave field simulation accuracy.
Wave equation finite difference numerical simulation is an important method of studying wave field characteristics during spreading in underground medium.In the traditional wave equation finite difference methods,it causes severe numerical frequency dispersion when using a long spatial step length,which reduced the calculation precision.FCT method can reduce the numerical dispersion producing in the coarse grid.This paper studies staggered grids FCT finite difference technology in transversely isotropic media with a vertical symmetry axis(VTI).Starting from the first order velocity-stress elastic wave equation,high order finite difference method format of the equation and stability conditions were given in staggered grids space,on that basis the numerical frequency dispersion FCT pressing technology was studied in wave equation modeling process,and high precision finite difference numerical simulation of the equation was realized by combining these two methods.Compared with the conventional algorithm,the algorithm in this paper does not need additional memory,a small increase calculation,but can effectively suppress numerical frequency dispersion phenomenon of elastic wave equation in VTI medium.While adopting a big mesh to carry on numerical simulation,the method proposed in this paper obviously improves the wave field simulation accuracy.
引文
[1]梁锴.TI介质弹性波传播特征及qP波深度偏移方法研究[D].中国东营,中国石油大学,2006.
[2]裴正林,牟永光.地震波传播数值模拟[J].地球物理学进展,2004,19(4):933～941.
[3]牟永光,裴正林.三维复杂介质地震数值模拟[M].北京:石油工业出版社,2005.
[4]吴国忱,王华忠.波场模拟中的数值频散分析与校正策略[J].地球物理学进展,2005,20(1):58～65.
[5]董良国,李培明.地震波传播数值模拟中的频散问题[J].天然气工业,2004,24(6):53～56.
[6]李庆春,邵广周,李斌.弹性波数值模拟的混合边界与频散抑制[J].煤田地质与勘探,2005,33(4):73～77.
[7]Book D L,Boris J P,Hain K.Flux-correctedtransport II:generalization of the method[J].Jour-nal Of Computational Physics,1975,18:248～283.
[8]Fei Tong,Larner K.Elimination of numerical dis-persion in finite-difference modeling and migrationby flux-corrected transport[J].Geophysics,1995,60(6):1830～1842.
[9]杨顶辉,滕吉文.各向异性介质中三分量地震记录的FCT有限差分模拟[J].石油地球物理勘探,1997,32(2):181～190.
[10]成景旺,顾汉明,刘琳,等.海上粘弹介质FCT有限差分正演模拟[J].石油天然气学报,2011,33(12):83～87.
[11]董良国,马在田,曹景忠,等.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):411～419.
[12]董良国,马在田,曹景忠.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,2000,43(6):856～864.
[13]Berenger J P.A perfectly matched layer for the ab-sorption of electromagnetics waves[J].J ComputPhys.,1994,114(2):185～200.
[14]韩令贺,何兵寿.VTI介质一阶准P波方程正演模拟及边界条件[J].石油地球物理勘探,2010,45(6):819～825.
[2]裴正林,牟永光.地震波传播数值模拟[J].地球物理学进展,2004,19(4):933～941.
[3]牟永光,裴正林.三维复杂介质地震数值模拟[M].北京:石油工业出版社,2005.
[4]吴国忱,王华忠.波场模拟中的数值频散分析与校正策略[J].地球物理学进展,2005,20(1):58～65.
[5]董良国,李培明.地震波传播数值模拟中的频散问题[J].天然气工业,2004,24(6):53～56.
[6]李庆春,邵广周,李斌.弹性波数值模拟的混合边界与频散抑制[J].煤田地质与勘探,2005,33(4):73～77.
[7]Book D L,Boris J P,Hain K.Flux-correctedtransport II:generalization of the method[J].Jour-nal Of Computational Physics,1975,18:248～283.
[8]Fei Tong,Larner K.Elimination of numerical dis-persion in finite-difference modeling and migrationby flux-corrected transport[J].Geophysics,1995,60(6):1830～1842.
[9]杨顶辉,滕吉文.各向异性介质中三分量地震记录的FCT有限差分模拟[J].石油地球物理勘探,1997,32(2):181～190.
[10]成景旺,顾汉明,刘琳,等.海上粘弹介质FCT有限差分正演模拟[J].石油天然气学报,2011,33(12):83～87.
[11]董良国,马在田,曹景忠,等.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):411～419.
[12]董良国,马在田,曹景忠.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,2000,43(6):856～864.
[13]Berenger J P.A perfectly matched layer for the ab-sorption of electromagnetics waves[J].J ComputPhys.,1994,114(2):185～200.
[14]韩令贺,何兵寿.VTI介质一阶准P波方程正演模拟及边界条件[J].石油地球物理勘探,2010,45(6):819～825.
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