非均匀介质中地震波应力场的WNAD方法及其数值模拟
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摘要
通过对近似解析离散化(NAD)方法的分析,给出了一种求解声波和弹性波方程的带权重的近似解析离散化(WNAD)方法,并用WNAD方法、Lax-Wendroff修正格式(LWC)和二阶中心差分方法计算了二维波动方程初值问题的应力场数值误差.结果表明WNAD方法具有更高的数值精度.用WNAD方法、LWC和四阶交错网格法对二维非均匀介质中弹性波传播的应力场进行了数值模拟.应力场快照和地表地震记录表明,即使是在粗网格条件下WNAD方法的模拟结果仍无可见的数值频散和源噪声.另一方面,由于WNAD方法同时计算了地震位移和梯度场,使得应力的计算更为便捷和精确.而且WNAD方法中波位移梯度局部连接关系的使用使得应力在间断处能够自动近似地满足应力连续性.
In this article,we first present the weighted nearly analytic discrete (WNAD) method for acoustic and elastic wave equations.Then we compute numerical error of the stress-field for the 2-D acoustic initial value problem using the WNAD method,the conventional finite-difference method and the fourth-order Lax-Wendroff correction (LWC) scheme.Numerical calculations of the relative errors show that WNAD method has the highest accuracy among these methods. Finally,we present the stress-field snapshots generated and compare the numerical results computed by the WNAD method with those of the fourth-order LWC scheme and the fourth-order staggercd-grid FDM for the 2-D inhomogeneous medium case.Promising numerical results illustrate that the results of the WNAD method have no visible numerical dispersion or source-noises even though too coarse grids are used.What's more,the WNAD method computes not only the values of the displacement U,but also the gradients of U,which makes it more convenient and accurate for computing the stress fields.The continuity of the stress is satisfied automatically even when the models have large velocity contrast between adjacent layers because of using local connection relations of the displacement U and its gradients.
In this article,we first present the weighted nearly analytic discrete (WNAD) method for acoustic and elastic wave equations.Then we compute numerical error of the stress-field for the 2-D acoustic initial value problem using the WNAD method,the conventional finite-difference method and the fourth-order Lax-Wendroff correction (LWC) scheme.Numerical calculations of the relative errors show that WNAD method has the highest accuracy among these methods. Finally,we present the stress-field snapshots generated and compare the numerical results computed by the WNAD method with those of the fourth-order LWC scheme and the fourth-order staggercd-grid FDM for the 2-D inhomogeneous medium case.Promising numerical results illustrate that the results of the WNAD method have no visible numerical dispersion or source-noises even though too coarse grids are used.What's more,the WNAD method computes not only the values of the displacement U,but also the gradients of U,which makes it more convenient and accurate for computing the stress fields.The continuity of the stress is satisfied automatically even when the models have large velocity contrast between adjacent layers because of using local connection relations of the displacement U and its gradients.
引文
[1] Chen K H.Propagating numerical model of elastic wave in anisotropic in homogeneous media finite element method.Symposium of 54th SEG,1984,54:631--632
[2] Yang D H,Liu E,Zhang Z J.Evaluation of the U-W finite element method in anisotropic porous media.J.Seism.Explor.,2008,17:273--299
[3] Cerveny V,Firbas P.Numerical modelling and inversion of travel times of seismic body waves in inhomogeneous anisotropicmedia.Geophys.J.R.Astr.Soc.,1984,76:41-51
[4] Booth D C,Crampin S. The anisotropic reflectivity technique:anomalous arrives from an anisotropic upper mantle.Geophys.J.R.Astr.Soc.,1983,72:767-782
[5] Kelly K R,Ward R W,Treitel S.Synthetic seismograms:a finite-difference approach.Geophysics,1976,41(1) :2-27
[6] Dablain M A.The application of high-order differencing to the scalar wave equation.Geophysics,1986,51(1) :54-66
[7] Virieux J.SH-wave propagation in heterogeneous media:velocity-stress finite difference method.Geophysics,1984,49 (11) :1933-1957
[8] Virieux J.P SV wave propagation in heterogeneous media:velocity stress finite-difference method.Geophysics,1986,51(4) :889-901
[9] 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.地球物理学报,2000,43(3) :411～419 Dong L G,Ma Z T,Coo J Z,et al.A staggered-grid high-order difference method of one order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(8) :411-419
[10] 王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法.清华大学学报(自然科学版),2002,42(8) :1128～1131 Wang S Q,Yang D H,Yang K D.Compact finite difference scheme for elastic equations.J.Tsinghua Univ.(Sci.& Tech.) (in Chinese),2002,42(8) :1128-1131
[11] Komatitsch D,Vilotte J P.The spectral element method:an efficient tool to simulate the seismic response of 2D and 3D geological structures.Bull.Seisrn.Soc.Am.,i998,88(2) :368-392
[12] Komatitsch D,Tromp J.Introduction to the spectral element method for three-dimensional seismic wave propagation.Geophys.J.Int.,1999,139(3) :806-822
[13] Yang D H,Teng J W,Zhang Z J,et al.A nearly-analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bull.Seism.Soc.Am.,2003,93(2) :882--890
[14] Yang D H,Lu M,Wu R S,et al.An optimal nearly-analytic discrete method for 2D acoustic and elastic wave equations.Bull.Seism.Soc.Am.,2004,94(5) :1982-1991
[15] Yang D H,Peng J M,Lu M,et al.A nearly-analytic discrete method for wave-field simulations in 2D porous media.Commun.Comput.Phys.,2006,1(3) :528-547
[16] Lu M,Yang D H.A modified nearly analytic discrete method and wave-field simulation in transversely isotropic media.Sci.inChina (Ser.D-Earth Sci.),2005,48(8) :1321-1328
[17] Yang D H,Peng J M,Lu M,et al.Optimal nearly analytic discrete approximation to the scalar wave equation.Bull.Seism.Soc.Am.,2006,96(3) :1114-1130
[18] Fei T,Larner K.Elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport.Geophysics,1995,60(6) :1830-1842
[19] Zheng H S,Zhang Z J,Liu E R.Non linear seismic wave propagation in anisotropic media using the flux corrected transport technique.Geophys.J.Int.,2006,165:943-956
[20] Zhang Z J,Wang G J,Harris J M.Muhi-component wave-field simulation in viscous extensively dilatancy anisotropic media.Phys.Earth Planet Inter,1999,114:25-38
[21] Yoshiomi Kondoh.On thoughts analysis of numerical scheme for simulation using a kernel optimum nearly-analytical discretization (KOND) method.J.Phys.Soc.Jpn.,1991,60(9) :2851-2861
[22] Kondoh Y,Hosaka Y,Ishii K.Kernel optimum nearly-analytical discretization algorithm applied to parabolic and hyperbolic equations.Computers Math.Appl.,1994,27(3) :59-90
[23] Yang D H,Liu E,Zhang Z J,et al.Finite-difference modeling in two-dimensional anisotropic media using a fluxcorrected transport technique.Geophys.J.Int.,2002,148:320-328
[24] Yang D H,Song G J,Chen S,et al.An improved nearly analytical discrete method:an efficient tool to simulate the seismic response of 2-D porous structures.J.Geophy.Eng.,2007,4:40-52
[2] Yang D H,Liu E,Zhang Z J.Evaluation of the U-W finite element method in anisotropic porous media.J.Seism.Explor.,2008,17:273--299
[3] Cerveny V,Firbas P.Numerical modelling and inversion of travel times of seismic body waves in inhomogeneous anisotropicmedia.Geophys.J.R.Astr.Soc.,1984,76:41-51
[4] Booth D C,Crampin S. The anisotropic reflectivity technique:anomalous arrives from an anisotropic upper mantle.Geophys.J.R.Astr.Soc.,1983,72:767-782
[5] Kelly K R,Ward R W,Treitel S.Synthetic seismograms:a finite-difference approach.Geophysics,1976,41(1) :2-27
[6] Dablain M A.The application of high-order differencing to the scalar wave equation.Geophysics,1986,51(1) :54-66
[7] Virieux J.SH-wave propagation in heterogeneous media:velocity-stress finite difference method.Geophysics,1984,49 (11) :1933-1957
[8] Virieux J.P SV wave propagation in heterogeneous media:velocity stress finite-difference method.Geophysics,1986,51(4) :889-901
[9] 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.地球物理学报,2000,43(3) :411～419 Dong L G,Ma Z T,Coo J Z,et al.A staggered-grid high-order difference method of one order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(8) :411-419
[10] 王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法.清华大学学报(自然科学版),2002,42(8) :1128～1131 Wang S Q,Yang D H,Yang K D.Compact finite difference scheme for elastic equations.J.Tsinghua Univ.(Sci.& Tech.) (in Chinese),2002,42(8) :1128-1131
[11] Komatitsch D,Vilotte J P.The spectral element method:an efficient tool to simulate the seismic response of 2D and 3D geological structures.Bull.Seisrn.Soc.Am.,i998,88(2) :368-392
[12] Komatitsch D,Tromp J.Introduction to the spectral element method for three-dimensional seismic wave propagation.Geophys.J.Int.,1999,139(3) :806-822
[13] Yang D H,Teng J W,Zhang Z J,et al.A nearly-analytic discrete method for acoustic and elastic wave equations in anisotropic media.Bull.Seism.Soc.Am.,2003,93(2) :882--890
[14] Yang D H,Lu M,Wu R S,et al.An optimal nearly-analytic discrete method for 2D acoustic and elastic wave equations.Bull.Seism.Soc.Am.,2004,94(5) :1982-1991
[15] Yang D H,Peng J M,Lu M,et al.A nearly-analytic discrete method for wave-field simulations in 2D porous media.Commun.Comput.Phys.,2006,1(3) :528-547
[16] Lu M,Yang D H.A modified nearly analytic discrete method and wave-field simulation in transversely isotropic media.Sci.inChina (Ser.D-Earth Sci.),2005,48(8) :1321-1328
[17] Yang D H,Peng J M,Lu M,et al.Optimal nearly analytic discrete approximation to the scalar wave equation.Bull.Seism.Soc.Am.,2006,96(3) :1114-1130
[18] Fei T,Larner K.Elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport.Geophysics,1995,60(6) :1830-1842
[19] Zheng H S,Zhang Z J,Liu E R.Non linear seismic wave propagation in anisotropic media using the flux corrected transport technique.Geophys.J.Int.,2006,165:943-956
[20] Zhang Z J,Wang G J,Harris J M.Muhi-component wave-field simulation in viscous extensively dilatancy anisotropic media.Phys.Earth Planet Inter,1999,114:25-38
[21] Yoshiomi Kondoh.On thoughts analysis of numerical scheme for simulation using a kernel optimum nearly-analytical discretization (KOND) method.J.Phys.Soc.Jpn.,1991,60(9) :2851-2861
[22] Kondoh Y,Hosaka Y,Ishii K.Kernel optimum nearly-analytical discretization algorithm applied to parabolic and hyperbolic equations.Computers Math.Appl.,1994,27(3) :59-90
[23] Yang D H,Liu E,Zhang Z J,et al.Finite-difference modeling in two-dimensional anisotropic media using a fluxcorrected transport technique.Geophys.J.Int.,2002,148:320-328
[24] Yang D H,Song G J,Chen S,et al.An improved nearly analytical discrete method:an efficient tool to simulate the seismic response of 2-D porous structures.J.Geophy.Eng.,2007,4:40-52
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