基于NAD算法的声波方程时间四阶差分解法
|
摘要
波动方程数值模拟是研究地震波传播机理的重要工具,有限差分求解波动方程是当前地震波数值模拟的主要方法之一。当地下介质中的地震波速度较低或地震波高频成分丰富时,常规有限差分技术常常产生严重的数值频散误差,这种误差会降低数值模拟的精度,影响对地震波传播机理的分析。为压制地震波数值模拟时产生的数值频散误差,提高波场模拟精度,提出了基于NAD算子的时间四阶精度波动方程差分格式。根据对应的差分格式,分析了该差分格式的数值频散关系。与常规四阶精度差分算法的频散曲线相比,基于NAD时间四阶精度差分方法不但能够实现时间频散的有效压制,同时其基于更多网格点的位移分量和位移梯度分量空间微分求解方法还能够实现空间频散的有效压制。另外在相同模型条件下,基于NAD算法的声波方程时间四阶差分解法可采用大网格对模拟空间进行差分离散,减少网格数,提高计算效率。
The wave equation numerical simulation is a major tool in seismic wave travel mechanism analysis. The finite difference solution for wave equation is one of principal methods in seismic wave numerical simulation at present. When seismic wave velocity is rather low or high frequency components abundant in underground medium,conventional finite difference techniques can often produce serious numerical frequency dispersion errors. This kind of errors will lower down numerical simulation precision and impact seismic wave travel mechanism analysis. To suppress dispersion errors in seismic wave numerical simulation,improve wave field simulation precision,a time fourth-order precision wave equation difference scheme based on NAD operator has been put forward. Based on corresponding difference scheme,analyzed the numerical dispersion relationship of the scheme. Compared with conventional fourth-order precision difference algorithm dispersion curves,to based on NAD algorithm not only can realize effective suppression of time dispersion,meanwhile,its displacement and displacement gradient components spatial differential solution method based on more mesh points can also realize spatial dispersion effective suppression. In addition,under the same model condition,based on NAD algorithm acoustic wave equation time forth-order difference solution can use bigger mesh to carry out difference dispersion on simulated space,reduce mesh number and improve computational efficiency.
The wave equation numerical simulation is a major tool in seismic wave travel mechanism analysis. The finite difference solution for wave equation is one of principal methods in seismic wave numerical simulation at present. When seismic wave velocity is rather low or high frequency components abundant in underground medium,conventional finite difference techniques can often produce serious numerical frequency dispersion errors. This kind of errors will lower down numerical simulation precision and impact seismic wave travel mechanism analysis. To suppress dispersion errors in seismic wave numerical simulation,improve wave field simulation precision,a time fourth-order precision wave equation difference scheme based on NAD operator has been put forward. Based on corresponding difference scheme,analyzed the numerical dispersion relationship of the scheme. Compared with conventional fourth-order precision difference algorithm dispersion curves,to based on NAD algorithm not only can realize effective suppression of time dispersion,meanwhile,its displacement and displacement gradient components spatial differential solution method based on more mesh points can also realize spatial dispersion effective suppression. In addition,under the same model condition,based on NAD algorithm acoustic wave equation time forth-order difference solution can use bigger mesh to carry out difference dispersion on simulated space,reduce mesh number and improve computational efficiency.
引文
[1]Bai B,He B S,Li K R,Tang H G,Yang J J. Numerical simulation of seismic wavefields in TTI media using the rotated staggered-grid compact finite-difference scheme[J]. Earthquake Science,2018,31(02):75-82.
[2]张晶,何兵寿,张会星.含直立裂隙介质的弹性波动方程正演模拟[J].中国煤炭地质,2008(01):50-54.
[3]何兵寿,陈婷,王胜.任意广角方程逆时偏移的脉冲响应及模型试算[J].中国煤炭地质,2014(2):49-54.
[4]胡楠,何兵寿.三维各向同性介质矢量波场保幅分离方法[J].煤炭学报,2017(9).
[5]史才旺,何兵寿.基于炮采样的多尺度全波形反演[J]. Applied Geophysics,2018,15(02):261-270.
[6]李凯瑞,何兵寿,胡楠.基于一阶速度-胀缩-旋转方程的多分量联合逆时偏移[J].煤炭学报,2018,43(04):1072-1082.
[7]郭鹏,何兵寿,沈骥千. VTI介质弹性波波场分解的空间域算法[J].石油地球物理勘探,2013,48(4):567-575.
[8]李佳珂,张会星,白冰,张建敏. TTI介质纯准P波一阶压力-速度方程及求解方法[J].中国煤炭地质,2018,30(07):72-78.
[9]陈可洋.高阶弹性波波动方程正演模拟及逆时偏移成像研究[D].黑龙江大庆:大庆石油学院,2009.
[10]常海明,何兵寿,杨佳佳,等.逆时偏移中的归一化加权互相关成像条件及应用[J].中国煤炭地质,2014,26(01):51-55.
[11]刘学义,何兵寿. VTI介质中弹性波方程正演的一阶混合吸收边界[J].中国煤炭地质,2015,27(03):59-63.
[12]Liang W Q,Wang Y F,Yang C C. Comparison of numerical dispersion in acoustic finite-difference algorithms[J]. Exploration Geophysics,2014,46(2):206-212.
[13]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(6).
[14]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.
[15]何兵寿,张会星.多分量波场的矢量法叠前深度偏移技术[J].石油地球物理勘探,2006,41(4):369-374.
[16]Zhang J H,Yao Z X. Optimized explicit finite-difference schemes for spatial derivatives using maximum norm[J]. Journal of Computational Physics,2013,250(10):511-526.
[17]Zhang J H,Yao Z X. Optimized finite-difference operator for broadband seismic wave modeling[J]. Geophysics,2013,78(1),A13-A18.
[18]薛东川,王尚旭.利用组合质量矩阵压制数值频散[J].石油地球物理勘探,2008,43(3):318-320.
[19]Fei T,Larner K. Elimination of numerical dispersion in finitedifference modeling and migration by flux-corrected transport[J]. Geophysics,1995,60(6):1830-1842.
[20]张省,何兵寿,王玉凤. VTI介质交错网格FCT有限差分数值模拟[J].工程地球物理学报,2012,9(05):565-571.
[21]李立平,何兵寿. TTI介质弹性波FCT有限差分数值模拟[J].地球物理学进展,2017(04):168-174.
[22]唐怀谷,何兵寿.一阶声波方程时间四阶精度差分格式的伪谱法求解[J].石油地球物理勘探,2017,52(01):71-80.
[23]Yang D H. Optimal Nearly Analytic Discrete Approximation to the Scalar Wave Equation[J]. Bulletin of the Seismological Society of America,2006,96(3):1114-1130.
[24]Yang D H,Song G,Lu M. Optimally Accurate Nearly Analytic Discrete Scheme for Wave-Field Simulation in 3D Anisotropic Media[J]. Bulletin of the Seismological Society of America,2007,97(5):1557-1569.
[25]Yang D H,Liu E,Song G,Wang N. Elastic wave modelling method based on the displacement–velocity fields:an improving nearly analytic discrete approximation[J]. Journal of Seismology,2009,13(2):209-217.
[26]Kondoh Y,Hosaka Y,Ishii K. Kernel optimum nearly-analytical discretization(KOND)algorithm applied to parabolic and hyperbolic equations[J] Computers&Mathematics with Applications,2010,27(27):59-90.
[27] Yang D H,Lu M,Wu R. An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations[J]. Bulletin of the Seismological Society of America. 2004,94(5):1982-1992.
[28]Yang D,Song G,Chen S,Hou B. An improved nearly analytical discrete method:an efficient tool to simulate the seismic response of 2-D porous structures[J]. Journal of Geophysics&Engineering. 2007,4(1):40-52.
[29]Berenger J P. A Perfect Matched Layer for the Absorption of Electromagnetic Waves[J]. Journal of Computational Physics. 1994,114(2):185-200.
[30]王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34.
[31]陈可洋.完全匹配层吸收边界条件研究[J].石油物探,2010,49(5):472-477.
[2]张晶,何兵寿,张会星.含直立裂隙介质的弹性波动方程正演模拟[J].中国煤炭地质,2008(01):50-54.
[3]何兵寿,陈婷,王胜.任意广角方程逆时偏移的脉冲响应及模型试算[J].中国煤炭地质,2014(2):49-54.
[4]胡楠,何兵寿.三维各向同性介质矢量波场保幅分离方法[J].煤炭学报,2017(9).
[5]史才旺,何兵寿.基于炮采样的多尺度全波形反演[J]. Applied Geophysics,2018,15(02):261-270.
[6]李凯瑞,何兵寿,胡楠.基于一阶速度-胀缩-旋转方程的多分量联合逆时偏移[J].煤炭学报,2018,43(04):1072-1082.
[7]郭鹏,何兵寿,沈骥千. VTI介质弹性波波场分解的空间域算法[J].石油地球物理勘探,2013,48(4):567-575.
[8]李佳珂,张会星,白冰,张建敏. TTI介质纯准P波一阶压力-速度方程及求解方法[J].中国煤炭地质,2018,30(07):72-78.
[9]陈可洋.高阶弹性波波动方程正演模拟及逆时偏移成像研究[D].黑龙江大庆:大庆石油学院,2009.
[10]常海明,何兵寿,杨佳佳,等.逆时偏移中的归一化加权互相关成像条件及应用[J].中国煤炭地质,2014,26(01):51-55.
[11]刘学义,何兵寿. VTI介质中弹性波方程正演的一阶混合吸收边界[J].中国煤炭地质,2015,27(03):59-63.
[12]Liang W Q,Wang Y F,Yang C C. Comparison of numerical dispersion in acoustic finite-difference algorithms[J]. Exploration Geophysics,2014,46(2):206-212.
[13]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(6).
[14]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.
[15]何兵寿,张会星.多分量波场的矢量法叠前深度偏移技术[J].石油地球物理勘探,2006,41(4):369-374.
[16]Zhang J H,Yao Z X. Optimized explicit finite-difference schemes for spatial derivatives using maximum norm[J]. Journal of Computational Physics,2013,250(10):511-526.
[17]Zhang J H,Yao Z X. Optimized finite-difference operator for broadband seismic wave modeling[J]. Geophysics,2013,78(1),A13-A18.
[18]薛东川,王尚旭.利用组合质量矩阵压制数值频散[J].石油地球物理勘探,2008,43(3):318-320.
[19]Fei T,Larner K. Elimination of numerical dispersion in finitedifference modeling and migration by flux-corrected transport[J]. Geophysics,1995,60(6):1830-1842.
[20]张省,何兵寿,王玉凤. VTI介质交错网格FCT有限差分数值模拟[J].工程地球物理学报,2012,9(05):565-571.
[21]李立平,何兵寿. TTI介质弹性波FCT有限差分数值模拟[J].地球物理学进展,2017(04):168-174.
[22]唐怀谷,何兵寿.一阶声波方程时间四阶精度差分格式的伪谱法求解[J].石油地球物理勘探,2017,52(01):71-80.
[23]Yang D H. Optimal Nearly Analytic Discrete Approximation to the Scalar Wave Equation[J]. Bulletin of the Seismological Society of America,2006,96(3):1114-1130.
[24]Yang D H,Song G,Lu M. Optimally Accurate Nearly Analytic Discrete Scheme for Wave-Field Simulation in 3D Anisotropic Media[J]. Bulletin of the Seismological Society of America,2007,97(5):1557-1569.
[25]Yang D H,Liu E,Song G,Wang N. Elastic wave modelling method based on the displacement–velocity fields:an improving nearly analytic discrete approximation[J]. Journal of Seismology,2009,13(2):209-217.
[26]Kondoh Y,Hosaka Y,Ishii K. Kernel optimum nearly-analytical discretization(KOND)algorithm applied to parabolic and hyperbolic equations[J] Computers&Mathematics with Applications,2010,27(27):59-90.
[27] Yang D H,Lu M,Wu R. An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations[J]. Bulletin of the Seismological Society of America. 2004,94(5):1982-1992.
[28]Yang D,Song G,Chen S,Hou B. An improved nearly analytical discrete method:an efficient tool to simulate the seismic response of 2-D porous structures[J]. Journal of Geophysics&Engineering. 2007,4(1):40-52.
[29]Berenger J P. A Perfect Matched Layer for the Absorption of Electromagnetic Waves[J]. Journal of Computational Physics. 1994,114(2):185-200.
[30]王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34.
[31]陈可洋.完全匹配层吸收边界条件研究[J].石油物探,2010,49(5):472-477.