声波方程变网格有限差分正演模拟的虚假反射分析
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摘要
变网格正演模拟方法因其低存储、高效率等特点,在含低速带或小尺度异常体的地下介质的数值模拟中发挥了重大作用.变网格模拟算法的精度与模拟时采用的频率、细网格长度、网格比、介质速度等参数密切相关.当模拟参数选取不合理时,在粗、细网格的分界处会产生强能量虚假反射,严重降低波场模拟的精度及分辨率.因此,选择合理的参数以压制虚假反射的产生在变网格正演模拟中显得尤为重要.本文基于一维假设和平面波理论,结合变系数变网格算法,推导出在变网格声波方程正演模拟中由于网格变化所引起的虚假反射的反射率函数公式,该公式为反射率函数与频率、细网格步长、网格比、介质速度的函数.基于该函数表达式,本文通过理论分析和模型数值算例,系统分析了频率、细网格步长、网格比、介质速度等对于变网格数值模拟中虚假反射的影响.理论分析和数值模拟实验均表明:在声波方程变网格正演模拟中,到达临界位置(即虚假反射率函数为1)之前,频率越高、细网格步长越大、网格比越大、介质速度越低,虚假反射率越大,即虚假反射现象越明显;反之,虚假反射率越小,虚假反射越弱.本文推导出的反射率函数可为实际变网格数值模拟中的各参数选取提供理论指导.
Owing to its low memory and high efficiency, variable grid forward modeling has played important role in the numerical simulation of underground media which contained low velocity zones or small scale abnormal bodies. The accuracy of variable grid forward modeling is closely related to the parameters of the frequency, the fine mesh step, the grid ratio and the medium velocity. Therefore, it is very important to select reasonable parameters to suppress spurious reflections in variable grid forward modeling. Based on one-dimensional assumption and plane wave theory, and combined with variable coefficient variable grid algorithm, this paper derives the reflectivity function of spurious reflections caused by grid change in variable grid forward modeling of acoustic wave equation. The formula is a function of the frequency, the fine mesh step, the grid ratio and the medium velocity. And based on the reflectivity function, the paper analyses the influence of the frequency, the fine mesh step, the grid ratio and the medium velocity on the spurious reflections of variable grid simulation systematically by theoretical analysis and numerical examples. Both theoretical analysis and numerical examples show that: in the variable grid forward modeling of acoustic wave equation, before reaching the critical position(the value of reflectivity function is 1), the higher the frequency, the larger the fine mesh step, the larger the grid ratio and the lower the medium velocity, the larger the spurious reflectivity will be, which means the spurious reflections will be more obvious. And on the other hand, the lower the frequency, the smaller the fine mesh step, the smaller the grid ratio and the higher the medium velocity, the smaller the spurious reflectivity will be, which means the spurious reflections will be even weaker. The reflectivity function of spurious reflections derived in this paper can provide theoretical guidance for the selection of parameters in the actual variable grid numerical simulation.
Owing to its low memory and high efficiency, variable grid forward modeling has played important role in the numerical simulation of underground media which contained low velocity zones or small scale abnormal bodies. The accuracy of variable grid forward modeling is closely related to the parameters of the frequency, the fine mesh step, the grid ratio and the medium velocity. Therefore, it is very important to select reasonable parameters to suppress spurious reflections in variable grid forward modeling. Based on one-dimensional assumption and plane wave theory, and combined with variable coefficient variable grid algorithm, this paper derives the reflectivity function of spurious reflections caused by grid change in variable grid forward modeling of acoustic wave equation. The formula is a function of the frequency, the fine mesh step, the grid ratio and the medium velocity. And based on the reflectivity function, the paper analyses the influence of the frequency, the fine mesh step, the grid ratio and the medium velocity on the spurious reflections of variable grid simulation systematically by theoretical analysis and numerical examples. Both theoretical analysis and numerical examples show that: in the variable grid forward modeling of acoustic wave equation, before reaching the critical position(the value of reflectivity function is 1), the higher the frequency, the larger the fine mesh step, the larger the grid ratio and the lower the medium velocity, the larger the spurious reflectivity will be, which means the spurious reflections will be more obvious. And on the other hand, the lower the frequency, the smaller the fine mesh step, the smaller the grid ratio and the higher the medium velocity, the smaller the spurious reflectivity will be, which means the spurious reflections will be even weaker. The reflectivity function of spurious reflections derived in this paper can provide theoretical guidance for the selection of parameters in the actual variable grid numerical simulation.
引文
Frank J, Reich S. 2004. On spurious reflections, nonuniform grids and finite difference discretizations of wave equations [R]. MAS-E0406,Amsterdam:CWI.
Huang C, Dong L G. 2009a. High-order finite-difference method in seismic wave simulation with variable grids and local time-steps [J]. Chinese Journal of Geophysics (in Chinese), 52(1): 176-186.
Huang C, Dong L G. 2009b. Staggered-grid high-order finite-difference method in elastic wave simulation with variable grids and local time-steps [J]. Chinese Journal of Geophysics (in Chinese), 52(11): 2870-2878, doi: 10.3969/j. issn.0001-5733.2009.11.022.
Jastram C, Behle A. 1992. Acoustic modelling on a grid of vertically varying spacing [J]. Geophysical Prospecting, 40(2): 157-169, doi: 10.1111/j.1365-2478.1992.tb00369.x.
Li S Z, Sun C Y, Peng P P. 2018. Seismic wave field forward modeling of variable staggered grid optimized difference coefficient method [J]. Geophysical Prospecting for Petroleum (in Chinese), 57(3): 378-388, doi: 10.3969/j.issn.1000-1441.2018.03.007.
Li Z C, Li Q Y, Huang J P, et al. 2014. A stable and high-precision dual-variable grid forward modeling and reverse time migration method [J]. Geophysical Prospecting for Petroleum (in Chinese), 53(2): 127-136, doi: 10.3969/j.issn.1000-1441.2014.02.001.
Li Z C, Zhang H, Zhang H. 2008. Variable-grid high-order finite-difference numeric simulation of first-order elastic wave equation [J]. Oil Geophysical Prospecting (in Chinese), 43(6): 711-716, doi: 10.13810/j. cnki. issn.1000-7210.2008.06.007.
Meng F S, Li Y S, Wang L, et al. 2013. Based on variable and staggered grids seismic numerical simulation [J]. Progress in Geophysics (in Chinese), 28(6): 2936-2943, doi: 10.6038/pg20130614.
Moczo P. 1989. Finite-difference technique for SH-waves in 2-D media using irregular grids-application to the seismic response problem [J]. Geophysical Journal International, 99(2): 321-329, doi: 10.1111/j.1365-246x.1989.tb01691.x.
Oliveira S A M. 2003. A fourth-order finite-difference method for the acoustic wave equation on irregular grids [J]. Geophysics, 68(2): 672- 676, doi: 10.1190/1.1567237.
Qu Y M, Li Z C, Huang J P, et al. 2016. Full waveform inversion based on multi-scale dual-variable grid in time domain [J]. Geophysical Prospecting for Petroleum (in Chinese), 55(2): 241-250, doi: 10.3969/j. issn.1000-1441.2016.02.010.
Sun C Y, Ding Y C. 2012. Accuracy analysis of wave equation finite difference with dual-variable grid algorithm [J]. Oil Geophysical Prospecting (in Chinese), 47(4): 545-551, doi: 10.13810/j.cnki.issn.1000-7210.2012.04.002.
Sun L J, Liu C C, Zhang S X. 2015. A variable grid finite-difference scheme with PML boundary condition in elastic wave of porous media [J]. Geophysical Prospecting for Petroleum (in Chinese), 54(6): 652- 664, doi: 10.3969/j.issn.1000-1441.2015.06.003.
Sun X D, Li Z C, Jia Y R. 2017. Variable-grid reverse-time migration of different seismic survey data [J]. Applied Geophysics, 14(4): 517-522, doi: 10.1007/s11770- 017- 0652-7.
Vichnevetsky R. 1981. Propagation through numerical mesh refinement for hyperbolic equations [J]. Mathematics & Computers in Simulation, 23(4): 344-353, doi: 10.1016/0378- 4754(81)90021-5.
Wang Y, Schuster G T. 1996. Finite-difference variable grid scheme for acoustic and elastic wave equation modeling [J]. SEG Technical Program Expanded Abstracts, 15: 674-677, doi: 10.1190/1.1826737.
Zhang H, Li Z C. 2011. Seismic wave simulation method based on dual-variable grid [J]. Chinese Journal of Geophysics (in Chinese), 54(1): 77-86, doi: 10.3969/j.issn.0001-5733.2011.01.009.
Zhu S W, Qu S L, Wei X C, et al. 2007. Numeric simulation by grid-various finite-difference elastic wave equation [J]. Oil Geophysical Prospecting (in Chinese), 42(6): 634- 639, doi: 10.13810/j.cnki.issn.1000-7210.2007.06.013.
黄超, 董良国. 2009a. 可变网格与局部时间步长的高阶差分地震波数值模拟[J]. 地球物理学报, 52(1): 176-186.
黄超, 董良国. 2009b. 可变网格与局部时间步长的交错网格高阶差分弹性波模拟[J]. 地球物理学报, 52(11): 2870-2878, doi: 10.3969/j.issn.0001-5733.2009.11.022.
李世中, 孙成禹, 彭鹏鹏. 2018. 可变交错网格优化差分系数法地震波正演模拟[J]. 石油物探, 57(3): 378-388, doi: 10.3969/j.issn.1000-1441.2018.03.007.
李振春, 李庆洋, 黄建平, 等. 2014. 一种稳定的高精度双变网格正演模拟与逆时偏移方法[J]. 石油物探, 53(2): 127-136, doi: 10.3969/j.issn.1000-1441.2014.02.001.
李振春, 张慧, 张华. 2008. 一阶弹性波方程的变网格高阶有限差分数值模拟[J]. 石油地球物理勘探, 43(6): 711-716, doi: 10.13810/j.cnki.issn.1000-7210.2008.06.007.
孟凡顺, 李洋森, 王璐, 等. 2013. 基于可变交错网格的地震波数值模拟[J]. 地球物理学进展, 28(6): 2936-2943, doi: 10.6038/pg20130614.
曲英铭, 李振春, 黄建平, 等. 2016. 基于多尺度双变网格的时间域全波形反演[J]. 石油物探, 55(2): 241-250, doi: 10.3969/j.issn.1000-1441.2016.02.010.
孙成禹, 丁玉才. 2012. 波动方程有限差分双变网格算法的精度分析[J]. 石油地球物理勘探, 47(4): 545-551, doi: 10.13810/j.cnki.issn.1000-7210.2012.04.002.
孙林洁, 刘春成, 张世鑫. 2015. PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J]. 石油物探, 54(6): 652- 664, doi: 10.3969/j. issn.1000-1441.2015. 06.003.
张慧, 李振春. 2011. 基于双变网格算法的地震波正演模拟[J]. 地球物理学报, 54(1): 77-86, doi: 10.3969/j.issn.0001-5733.2011.01.009.
朱生旺, 曲寿利, 魏修成, 等. 2007. 变网格有限差分弹性波方程数值模拟方法[J]. 石油地球物理勘探, 42(6): 634- 639, doi: 10.13810/j.cnki.issn.1000-7210.2007.06.013.
Huang C, Dong L G. 2009a. High-order finite-difference method in seismic wave simulation with variable grids and local time-steps [J]. Chinese Journal of Geophysics (in Chinese), 52(1): 176-186.
Huang C, Dong L G. 2009b. Staggered-grid high-order finite-difference method in elastic wave simulation with variable grids and local time-steps [J]. Chinese Journal of Geophysics (in Chinese), 52(11): 2870-2878, doi: 10.3969/j. issn.0001-5733.2009.11.022.
Jastram C, Behle A. 1992. Acoustic modelling on a grid of vertically varying spacing [J]. Geophysical Prospecting, 40(2): 157-169, doi: 10.1111/j.1365-2478.1992.tb00369.x.
Li S Z, Sun C Y, Peng P P. 2018. Seismic wave field forward modeling of variable staggered grid optimized difference coefficient method [J]. Geophysical Prospecting for Petroleum (in Chinese), 57(3): 378-388, doi: 10.3969/j.issn.1000-1441.2018.03.007.
Li Z C, Li Q Y, Huang J P, et al. 2014. A stable and high-precision dual-variable grid forward modeling and reverse time migration method [J]. Geophysical Prospecting for Petroleum (in Chinese), 53(2): 127-136, doi: 10.3969/j.issn.1000-1441.2014.02.001.
Li Z C, Zhang H, Zhang H. 2008. Variable-grid high-order finite-difference numeric simulation of first-order elastic wave equation [J]. Oil Geophysical Prospecting (in Chinese), 43(6): 711-716, doi: 10.13810/j. cnki. issn.1000-7210.2008.06.007.
Meng F S, Li Y S, Wang L, et al. 2013. Based on variable and staggered grids seismic numerical simulation [J]. Progress in Geophysics (in Chinese), 28(6): 2936-2943, doi: 10.6038/pg20130614.
Moczo P. 1989. Finite-difference technique for SH-waves in 2-D media using irregular grids-application to the seismic response problem [J]. Geophysical Journal International, 99(2): 321-329, doi: 10.1111/j.1365-246x.1989.tb01691.x.
Oliveira S A M. 2003. A fourth-order finite-difference method for the acoustic wave equation on irregular grids [J]. Geophysics, 68(2): 672- 676, doi: 10.1190/1.1567237.
Qu Y M, Li Z C, Huang J P, et al. 2016. Full waveform inversion based on multi-scale dual-variable grid in time domain [J]. Geophysical Prospecting for Petroleum (in Chinese), 55(2): 241-250, doi: 10.3969/j. issn.1000-1441.2016.02.010.
Sun C Y, Ding Y C. 2012. Accuracy analysis of wave equation finite difference with dual-variable grid algorithm [J]. Oil Geophysical Prospecting (in Chinese), 47(4): 545-551, doi: 10.13810/j.cnki.issn.1000-7210.2012.04.002.
Sun L J, Liu C C, Zhang S X. 2015. A variable grid finite-difference scheme with PML boundary condition in elastic wave of porous media [J]. Geophysical Prospecting for Petroleum (in Chinese), 54(6): 652- 664, doi: 10.3969/j.issn.1000-1441.2015.06.003.
Sun X D, Li Z C, Jia Y R. 2017. Variable-grid reverse-time migration of different seismic survey data [J]. Applied Geophysics, 14(4): 517-522, doi: 10.1007/s11770- 017- 0652-7.
Vichnevetsky R. 1981. Propagation through numerical mesh refinement for hyperbolic equations [J]. Mathematics & Computers in Simulation, 23(4): 344-353, doi: 10.1016/0378- 4754(81)90021-5.
Wang Y, Schuster G T. 1996. Finite-difference variable grid scheme for acoustic and elastic wave equation modeling [J]. SEG Technical Program Expanded Abstracts, 15: 674-677, doi: 10.1190/1.1826737.
Zhang H, Li Z C. 2011. Seismic wave simulation method based on dual-variable grid [J]. Chinese Journal of Geophysics (in Chinese), 54(1): 77-86, doi: 10.3969/j.issn.0001-5733.2011.01.009.
Zhu S W, Qu S L, Wei X C, et al. 2007. Numeric simulation by grid-various finite-difference elastic wave equation [J]. Oil Geophysical Prospecting (in Chinese), 42(6): 634- 639, doi: 10.13810/j.cnki.issn.1000-7210.2007.06.013.
黄超, 董良国. 2009a. 可变网格与局部时间步长的高阶差分地震波数值模拟[J]. 地球物理学报, 52(1): 176-186.
黄超, 董良国. 2009b. 可变网格与局部时间步长的交错网格高阶差分弹性波模拟[J]. 地球物理学报, 52(11): 2870-2878, doi: 10.3969/j.issn.0001-5733.2009.11.022.
李世中, 孙成禹, 彭鹏鹏. 2018. 可变交错网格优化差分系数法地震波正演模拟[J]. 石油物探, 57(3): 378-388, doi: 10.3969/j.issn.1000-1441.2018.03.007.
李振春, 李庆洋, 黄建平, 等. 2014. 一种稳定的高精度双变网格正演模拟与逆时偏移方法[J]. 石油物探, 53(2): 127-136, doi: 10.3969/j.issn.1000-1441.2014.02.001.
李振春, 张慧, 张华. 2008. 一阶弹性波方程的变网格高阶有限差分数值模拟[J]. 石油地球物理勘探, 43(6): 711-716, doi: 10.13810/j.cnki.issn.1000-7210.2008.06.007.
孟凡顺, 李洋森, 王璐, 等. 2013. 基于可变交错网格的地震波数值模拟[J]. 地球物理学进展, 28(6): 2936-2943, doi: 10.6038/pg20130614.
曲英铭, 李振春, 黄建平, 等. 2016. 基于多尺度双变网格的时间域全波形反演[J]. 石油物探, 55(2): 241-250, doi: 10.3969/j.issn.1000-1441.2016.02.010.
孙成禹, 丁玉才. 2012. 波动方程有限差分双变网格算法的精度分析[J]. 石油地球物理勘探, 47(4): 545-551, doi: 10.13810/j.cnki.issn.1000-7210.2012.04.002.
孙林洁, 刘春成, 张世鑫. 2015. PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J]. 石油物探, 54(6): 652- 664, doi: 10.3969/j. issn.1000-1441.2015. 06.003.
张慧, 李振春. 2011. 基于双变网格算法的地震波正演模拟[J]. 地球物理学报, 54(1): 77-86, doi: 10.3969/j.issn.0001-5733.2011.01.009.
朱生旺, 曲寿利, 魏修成, 等. 2007. 变网格有限差分弹性波方程数值模拟方法[J]. 石油地球物理勘探, 42(6): 634- 639, doi: 10.13810/j.cnki.issn.1000-7210.2007.06.013.
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