VTI介质qP波方程高精度有限差分算子
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摘要
波动方程有限差分法是一种使用广泛的地震波数值模拟方法.但是有限差分法本身固有存在着数值频散问题,会降低地震波场模拟的精度与分辨率.为了克服常规有限差分算子的数值频散,本文针对VTI介质地震波数值模拟问题,构造了频率-空间域qP波波动方程高精度有限差分优化算子,根据最优化理论中高斯-牛顿法确定了高精度有限差分算子的优化系数.利用常规差分算子和高精度优化差分算子对归一化相速度的频散关系精度进行了对比分析,并对均匀各向同性介质和均匀VTI介质中的qP波地震波场进行了有限差分数值模拟,通过频散关系精度分析和波场数值模拟结果表明:有限差分优化算子具有较高的波场数值模拟精度,有效压制了传统有限差分算子数值模拟中的数值频散现象,提高了有限差分算子精度,为VTI介质频率-空间域qP波正演模拟奠定了基础.
Finite-difference method for wave equation is widely implemented in seismic wavefield numerical simulation.Finite-difference method has inherent numerical dispersion,which reduces the accuracy and resolution of seismic wavefield simulation.In order to decrease the numerical dispersion of conventional finite-difference operators,we present high precision finite-difference operators of qP wave equation in frequency–space domain for seismic wavefield numerical simulation in VTI media,and obtain the optimal coefficients of high precision finite-difference operators according to the Gauss-Newton method in optimization theory.We analyze dispersion relation accuracy of normalized phase velocity using conventional finite-difference operators and high precision optimal finite-difference operators separately,and do finite-difference numerical simulation for qP wave in homogenous isotropic media and VTI media.The results of dispersion relation accuracy analysis and numerical simulation indicate that optimal finite-difference operators have high precision in wavefield numerical simulation,and can efficiently decrease the numerical dispersion in conventional finite-difference numerical simulation,and improve accuracy of the finite-difference operators,which can provide the foundation of qP wave simulation in frequency–space domain in VTI media.
Finite-difference method for wave equation is widely implemented in seismic wavefield numerical simulation.Finite-difference method has inherent numerical dispersion,which reduces the accuracy and resolution of seismic wavefield simulation.In order to decrease the numerical dispersion of conventional finite-difference operators,we present high precision finite-difference operators of qP wave equation in frequency–space domain for seismic wavefield numerical simulation in VTI media,and obtain the optimal coefficients of high precision finite-difference operators according to the Gauss-Newton method in optimization theory.We analyze dispersion relation accuracy of normalized phase velocity using conventional finite-difference operators and high precision optimal finite-difference operators separately,and do finite-difference numerical simulation for qP wave in homogenous isotropic media and VTI media.The results of dispersion relation accuracy analysis and numerical simulation indicate that optimal finite-difference operators have high precision in wavefield numerical simulation,and can efficiently decrease the numerical dispersion in conventional finite-difference numerical simulation,and improve accuracy of the finite-difference operators,which can provide the foundation of qP wave simulation in frequency–space domain in VTI media.
引文
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[2]王德利,何樵登,韩立国.裂隙型单斜介质中多方位地面三分量记录模拟[J].地球物理学报,2005,48(2):386~393.Wang D L,He Q D,Han L G.Multi-azimuth three-componentsurface seismic modeling for cracked monoclinic media[J].Chinese J.Geophys.(in Chinese),2005,48(2):386~393.
[3]刘恩儒,岳建华,刘彦.具有离散裂缝空间分布的二维固体中地震波传播的有限差分模拟[J].地球物理学报,2006,49(1):180~188.Liu E R,Yue J H,Liu Y.Finite difference simulation of seismicwave propagation in 2-D solids with spatial distribution of discretefractures[J].Chinese J.Geophys.(in Chinese),2006,49(1):180~188.
[4]何峰江,陶果,王锡莉.贴井壁声波测井仪的有限差分模拟研究[J].地球物理学报,2006,49(3):923~928.He F J,Tao G,Wang X L.Finite difference modeling of the acous-tic field by sidewall logging devices[J].Chinese J.Geophys.(in Chi-nese),2006,49(3):923~928
[5]杜启振,王延光,付水华.方位各向异性粘弹性介质波场数值模拟[J].地球物理学进展,2006,21(2):502~504.Du Q Z,Wang Y G,Fu S H.Wavefield forward modeling with thepseudo-spectral method in viscoelastic and azimuthally anisotropicmedia[J].Progress in Geophysics(in Chinese),2006,21(2):502~504.
[6]冯德山,戴前伟,何继善等.探地雷达GPR正演模拟的时域有限差分实现[J].地球物理学进展,2006,21(2):630~636.Feng D S,Dai Q W,He J S,et al.Finite difference time domainmethod of GPR forward simulation[J].Progress in Geophysics(inChinese),2006,21(2):630~636.
[7]周竹生,刘喜亮,熊孝雨.弹性介质中瑞雷面波有限差分法正演模拟[J].地球物理学报,2007,50(2):567~573.Zhou Z S,Liu X L,Xiong X U.Finite-difference modelling of Ray-leigh surface wave in elastic media[J].Chinese J.Geophys.(in Chi-nese),2007,50(2):567~573.
[8]Jo Churl-Hyun,Shin Changsoo,Suh Jung Hee.An optimal 9-point,finite-difference,frequency-space,2-D scalar wave extrapolator[J].Geophysics,1996,61(2):529~537.
[9]Changsoo S,Heejeung S.A frequency-space 2-D scalar wave extrap-olator using extend 25-point finite-difference operator[J].Geophys-ics,1998,63(1):289~296.
[10]殷文,印兴耀,吴国忱,梁锴.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 W,Yin X Y,Wu G C,Liang K.The method of finite differ-ence of high precision elastic wave equations in the frequency domainand wave-field simulation[J].Chinese J.Geophys.(in Chinese),2006,49(2):561~568.
[11]tekl,Pratt.Accurate viscoelastic modeling by frequency-domain fi-nite differences using rotated operators[J].Geophysics,1998,63(5):1779~1794.
[12]Min D J,Shinz Changsoo,Kwon Byung-Doo,et al.Improved fre-quency-domain elastic wave modeling using weighted-averagingdifference operators[J].Geophysics,2000,65(3):884~895.
[13]Tariq Alkhalifah.Acoustic approximation for processing in trans-versely isotropic media[J].Geophysics,1998,63(2):623~631.
[14]Tariq Alkhalifah.An acoustic wave equation for anisotropic media[J].Geophysics,2000,65(4):1239~1250.
[15]Thomsen L.Weak elastic anisotropy[J].Geophysics,1986,51(10):1954~1966.
[16]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.Wu G C,Wang H Z.Analysis of numerical dispersion in wave-fieldsimulation[J].Progress in Geophysics(in Chinese),2005,20(1):58~65.
[17]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 G C,Liang K.Quasi P-wave forward modeling in frequency-space domain in VTI media[J].Oil Geophysical Prospecting(in Chi-nese),2005,40(5):535~545.
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