Seismic wave simulation using a trapezoid grid pseudo-spectral method
TAN Wenzhuo1, WU Bangyu1, LI Bo2, LEI Jun3
1. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China; 2. Sinopec INOPEC Geophysical Research Institute, Nanjing, Jiangsu 211103, China; 3. The Sixth Gas Production Plant, Changqing Oilfield Company, PetroChina, Xi'an, Shaanxi 710018, China
Abstract:The numerical solution to wave equation is the computational engine of many high-precision imaging and inversion methods in seismic exploration.In order to ensure the accuracy,regular methods generally set a fixed spatial sampling interval according to the minimum velocity of the model.This always causes over sampling in high velocity layers,and the calculation is redundant to some extent.Under the compaction of rocks cause by gravity,the wave propagation speed usually increases along with depth.The trapezoid coordinate transformation can incorporate the general increasing trend of wave velocity.So,we propose seismic wave simulation using a trapezoid grid pseudo-spectral method.We use the trapezoid grid based on trapezoid coordinate transformation to divide medium.The fine grid is used in the shallow low velocity region and the coarse grid is used in the deep high velocity region,which can effectively reduce the number of sampling points.Meanwhile,considering the accuracy of wave field simulation,we use pseudo-spectral method to solve the acoustic wave equation with variable coefficients after coordinate transformation and use the perfectly matched layer to eliminate the fictitious return wave caused by artificial boundary.The test on Marmousi model shows that comparing with the general method,the number of grids generated by trapezoidal grid can reduce 69%,and comparing with the regular grid pseudo-spectral method and the high order finite difference method,the calculation time of trapezoidal grid pseudo-spectral method can respectively reduce 58% and 60%.Therefore,this method is an efficient and high-precision seismic wave simulation method.
徐佼,张智,董超,等.几种典型地质模型的地震波场数值模拟[J].桂林理工大学学报,2014,34(3):416-422.XU Jiao,ZHANG Zhi,DONG Chao,et al.Seismic wave field simulation of several typical geological models[J].Journal of Guilin University of Techno-logy,2014,34(3):416-422.
[2]
王保利,高静怀,陈文超,等逆时偏移中用Poynting矢量高效地提取角道集[J].地球物理学报,2013,56(1):262-268.WANG Baoli,GAO Jinghuai,CHEN Wenchao,et al.Extracting efficiently angle gathers using Poynting vector during reserve time migration[J].Chinese Journal of Geophysics,2013,56(1):262-268.
[3]
韩如冰,郎超.频率域八阶NAD有限差分模拟及全波形反演[J].石油地球物理勘探,2019,54(6):1254-1266.HAN Rubing,LANG Chao.The eight-order frequency-domain NAD method and full-waveform inversion[J].Oil Geophysical Prospecting,2019,54(6):1254-1266.
[4]
Liu Y.Globally optimal finite-difference schemes based on least-squares[J].Geophysics,2013,78(4):T113-T132.
[5]
Wang E J,Liu Y,Sen M K.Effective finite-difference modeling methods with 2-D acoustic wave equation using a combination of cross and rhombus stencils[J].Geophysical Journal International,2016,206(3):1933-1958.
[6]
Liu J,Wei X C,Ji Y X,et al.Second-order seismic wave simulation in the presence of a free-surface by pseudo-spectral method[J].Journal of Applied Geophysics,2015,114:183-190.
[7]
孙献果,张东.地震波模拟中有限差分法与伪谱法的对比研究[J].中国科技论文,2018,13(17):2005-2008.SUN Xianguo,ZHANG Dong.Study on the diffe-rences between finite difference method and pseudo-spectral method algorithm in seismic numerical mo-deling[J].Chinese Science Paper,2018,13(17):2005-2008.
[8]
Moczo P,Kristek J,Gails M,et al.On accuracy of the finite-difference and finite-elements schemes with respect to P-wave and S-wave speed ratio[J].Geophysical Journal International,2010,182(1):493-510.
[9]
Moczo P,Kristek J,Gails M,et al.3-D finite-difference,finite-element,discontinues-Galerkin and spectral-element schemes analysed for their accuracy with respect to P-wave and S-wave speed ratio[J].Geophysical Journal International,2010,187(3):1645-1667.
[10]
Liu Y S,Teng J W,Lan H Q,et al.A comparative study of finite element and spectral element methods in seismic wavefield modeling[J].Geophysics,2014,79(2):T91-T104.
Chen F,Xu S.Pyramid-shaped grid for elastic wave propagation[C].SEG Technical Program Expanded Abstracts,2012,31:1-5.
[13]
高静怀,徐文豪,吴帮玉,等.深度均匀采样梯形网格有限差分地震波场模拟方法[J].地球物理学报,2018,61(8):3285-3296.GAO Jinghuai,XU Wenhao,WU Bangyu,et al.Tra-pezoid grid finite difference seismic wavefield simulation with uniform depth sampling interval[J].Chinese Journal of Geophysics,2018,61(8):3285-3289.
[14]
Wu B Y, Xu W H, Li B, et al.Trapezoid coordinate finite difference modeling of acoustic wave propagation using the CPML boundary condition[J].Journal of Applied Geophysics,2019,168:101-106.
[15]
Guan H,Dussaud E,Denel B,et al.Techniques for an efficient implementation of RTM in TTI media[C].SEG Technical Program Expanded Abstracts,2011,30:3393-3397.
[16]
Gao Y J,Song H J,Zhang J H,et al.Comparison of artificial absorbing boundaries for acoustic wave equation modeling[J].Exploration Geophysics, 2017,48(1):76-93.
[17]
Berenger J P.A perfectly matched layer for the ab-sorption of electromagnetic waves[J].Journal of Computational Physics,1994,114(2):185-200.
[18]
刑丽.地震声波数值模拟中的吸收边界条件[J].上海第二工业大学学报,2006,23(4):272-278.XING Li.Absorbing boundary conditions for the numerical simulation of acoustic waves[J].Journal of Shanghai Second Polytechnic University,2006,23(4):272-278.
[19]
胡建林,宋维琪,张建坤,等.交错网格有限差分正演模拟的联合吸收边界[J].石油地球物理勘探,2018,53(5):914-920.HU Jianlin,SONG Weiqi,ZHANG Jiankun,et al.Joint absorbing boundary in the staggered-grid finite difference forward modeling simulation[J].Oil Geophysical Prospecting,2018,53(5):914-920.
[20]
姜礼尚,陈亚浙,刘西垣,等.数学物理方程讲义[M].北京:高等教育出版社,2007,1-239.
[21]
Dablain M A.The application of high-order differencing to the scalar wave equation[J].Geophysics,1986,51(1):54-66.
[22]
Liu Y,Sen M K.A new time-space domain high-order finite-difference method for the acoustic wave equation[J].Journal of Computational Physics,2009,228(23):8779-8806.
[23]
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(1):71-80.TANG Huaigu,HE Bingshou.Pseudo spectrum method of first-order acoustic wave equation finite-difference schemes with fourth-order time difference accuracy[J].Oil Geophysical Prospecting,2017,52(1):71-80.
[24]
Koene E F M,Robertsson J O A,Broggini F,et al.Eliminating time dispersion from seismic wave modeling[J].Geophysical Journal International,2018,213(1):169-180.
[25]
Wu B Y,Zhang G W,Zhou Q B.Beamlet-domain premigration deghosting on variable-depth streamer seismic data[C].SEG Technical Program Expanded Abstracts,2016,35:4761-4765.