Reducing Numerical Dispersion with High-Order Finite Difference to Increase Seismic Wave Energy
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DOI:
https://doi.org/10.5614/j.eng.technol.sci.2023.55.4.5Keywords:
acoustic wave, forward modeling, high-order finite difference, numerical dispersion, proportional grid method, Taylor seriesAbstract
The numerical dispersion of 2D acoustic wave modeling has become an interesting subject in wave modeling in producing better subsurface images. Numerical dispersion is often caused by error accumulation with increased grid size in wave modeling. Wave modeling with high-order finite differences was carried out to reduce the numerical error. This study focused on variations in the numerical order to suppress the dispersion due to numerical errors. The wave equation used in modeling was discretized to higher orders for the spatial term, while the time term was discretized up to the second order, with every layer unabsorbed. The results showed that high-order FD was effective in reducing numerical dispersion. Thus, subsurface layers could be distinguished and observed clearly. However, from the modeling results, the wave energy decreased with increasing distance, so the layer interfaces were unclear. To increase the wave energy, we propose a new source in modeling. Furthermore, to reduce the computational time we propose a proportional grid after numerical dispersion has disappeared. This method can effectively increase the energy of reflected and transmitted waves at a certain depth. The results also showed that the computational time of high-order FD is relatively low, so this method can be used in solving dispersion problems.
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Chen, H., Zhou, H. & Sheng, S., General Rectangular Grid Based Time-Space Domain High-Order Finite-Difference Methods for Modeling Scalar Wave Propagation, Journal of Applied Geophysics, 133, pp. 141-156, 2016. doi: 10.1016/j.jappgeo.2016.07.021.
Kalyani, V.K., Pallavika & Chakraborty, S.K., Finite-Difference Time-Domain Method for Modelling of Seismic Wave Propagation in Viscoelastic Medium. Applied Mathematics and Computation, 237, pp. 133-145, 2014. doi: 10.1016/j.amc.2014.03.029.
Ren, Z., Li, Z., Liu, Y. & Sen, M.K., Modeling of the Acoustic Wave Equation by Staggered-Grid Finite-Difference Schemes with High-Order Temporal and Spatial Accuracy, Bulletin of the Seismological Society of America, 107(5), pp. 2160?2182, 2017. doi: 10.1785/0120170068.
Zheng, T., Wang, Y. & Chang, X., Wave Equation Based Microseismic Source Location and Velocity Inversion, Physics of the Earth and Planetary Interiors, 261, pp. 46-53, 2016. doi: 10.1016/ j.pepi.2016.07.003.
Cao, D. & Liao, W., A Computational Method for Full Waveform Inversion of Crosswell Seismic Data Using Automatic Differentiation, Computer Physics Communications, 188, pp. 47-58, 2015. doi: 10.1016/ j.cpc.2014.11.002.
Wang, Y., Zhou, H., Yuan, S. & Ye, Y., A Fourth Order Accuracy Summation-by-Parts Finite Difference Scheme for Acoustic Reverse Time Migration in Boundary-Conforming Grids, Journal of Applied Geophysics, 136, pp. 498-512, 2017. doi: 10.1016/ j.jappgeo.2016.12.002.
Artman, B., Podladtchikov, I. & Witten, B., Source Location Using Time-Reverse Imaging, Geophysical Prospecting, 58, pp. 861-873, 2010. doi: 10.1111/j.1365-2478.2010.00911.x
Haris, A., Silaban, S.P., Riyanto, A., Syahputra, R. & Mardiyati, S., Time Reverse Modeling of Hydrocarbon, International Journal of Geomate, 13(39), pp. 185-190. 2017. doi: 10.21660/ 2017.39.32648.
Lambert, M.A., Schmalholz, S.M., Saenger, E.H. & Steiner, B., Low-Frequency Microtremor Anomalies at an Oil and Gas Field in Voitsdorf Austria, Geophysical Prospecting, 57(3), pp. 393-411, 2008. doi: 10.1111/ j.1365-2478.2008.00734.x.
Steiner, B., Saenger, E.H. & Schmalholz, S.M., Time Reverse Modeling of Low-Frequency Microtremors: Application to Hydrocarbon Reservoir Localization, Geophysical Research Letters, 35, L03307, 2008. doi: 10.1029/ 2007GL032097.
Wang, Y., Zhou, H., Chen, H., Shengm, S. & Yuan, S., Acoustic Reverse Time Migration and Perfectly Matched Layer in Boundary-Conforming Grids by Elliptic Method, Journal of Applied Geophysics, 122, pp. 53-61, 2015. doi: 10.1016/j.jappgeo.2015.08.013.
Liu, Y. & Sen, M.K., A New Time-Space Domain High-Order Finite-Difference Method for the Acoustic Wave Equation, Journal of Computational Physics, 228, pp. 8779-8806, 2009a. doi: 10.1016/ j.jcp.2009.08.027.
Liu, Y. & Sen, M.K., Acoustic VTI Modeling with a Time?Space Domain Dispersion-Relation-Based Finite-Difference Scheme, Geophysics, 75, pp. A7-A13, 2010. doi: 10.1190/1.3374477.
Muelas, A., Salete, E., Benito, J.J., Ure, F., Gavete, L. & Ure, M., The Application of the Generalized Finite Difference Method (GFDM) for Modelling Geophysical Test, Journal of Geoscience and Environment Protection, 7, pp. 1-17, 2019. doi: 10.4236/gep.2019.74001.
Finkelstein, B. & Kastner, R., Finite Difference Time Domain Dispersion Reduction Schemes, Journal of Computational Physics, 221, pp. 422-438, 2007. doi: 10.1016/j.jcp.2006.06.016.
Wang, E. & Liu, Y., An Implicit Spatial and High-Order Temporal Finite Difference Scheme for 2D Acoustic Modelling, Exploration Geophysic, 49(2), pp. 187-201, 2017. doi: 10.1071/EG16094.
Yang, L., Yan, H. & Liu, H., An Optimal Implicit Staggered-Grid Finite-Difference Scheme Based on the Modified Taylor-Series Expansion with Minimax Approximation Method for Elastic Modeling, Journal of Applied Geophysics, 122, pp. 40-52, 2017. doi: 10.1016/j.jappgeo.2017.01.020.
Jing, H., Chen, Y., Wang, J. & Xue, W., A Highly Efficient Time-Space-Domain Optimized Method with Lax-Wendroff Type Time Discretization for The Scalar Wave Equation, Journal of Computational Physics, 393, pp. 1-28, 2019. doi: 10.1016/j.jcp.2019.04.066.
Liu, Y. & Sen, M.K., A Practical Implicit Finite-Difference Method: Examples from Seismic Modeling, Journal of Geophysics and Engineering, 6, pp. 231-249, 2009b. doi: 10.1088/1742-2132/6/3/003.
Liu, Y. & Sen, M.K., 3D Acoustic Wave Modeling with Time?Space Domain Dispersion-Relation-Based Finite-Difference Schemes and Hybrid Absorbing Boundary Conditions, Exploration Geophysics, 42, pp. 176-189, 2011a. doi: 10.1071/EG11007.
Tan, S. & Huang, L., An Efficient Finite-Difference Method with High-Order Accuracy in Both Time and Space Domains for Modelling Scalar- Wave Propagation, Geophysical Journal Intertational, 197, pp. 1250-1267, 2014a. doi: 10.1093/gji/ggu077.
Yan, H., Yang, L. & Li, X.Y., Optimal Staggered-Grid Finite-Difference Schemes by Combining Taylor-Series Expansion and Sampling Approximation for Wave Equation Modeling, Journal of Computational Physics, 326, pp. 913-930, 2016. doi: 10.1016/j.jcp.2016.09.019.
Quan, L.W., Chun, Y.C., Fei, W.Y. & Wei, L.H., Acoustic Wave Equation Modeling with New Time-Space Domain Finite Difference Operators, Chinese Journal of Geophysics, 56(6), pp. 840-850, 2013. doi: 10.1002/cjg2.20076.
Liu, Y. & Sen, M.K., An Implicit Staggered-Grid Finite-Difference Method for Seismic Modelling, Geophysical Journal International, 179, pp. 459-474, 2009. doi: 10.1111/j.1365-246X.2009.04305.x.
Liu, Y. & Sen, M.K., Scalar Wave Equation Modeling with Time?Space Domain Dispersion-Relation-Based Staggered-Grid Finite-Difference Schemes, Bulletin of the Seismological Society of America, 101, pp. 141-159, 2011. doi: 10.1785/0120100041.
Saenger, E.H., Wave Propagation in Fractured Media: Theory and Applications of the Rotated Staggered Finite-Difference Grid, Dissertation, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2000.
Yan, H., Yang, L. & Li, X.Y., Optimal Staggered-Grid Finite-Difference Schemes by Combining Taylor-Series Expansion and Sampling Approximation for Wave Equation Modeling, Journal of Computational Physics, 326, pp. 913-930, 2016. doi: 10.1016/j.jcp.2016.09.019.
Yang, L., Yan, H. & Liu, H., Optimal Rotated Staggered-Grid Finite-Difference Schemes for Elastic Wave Modeling in TTI Media, Journal of Applied Geophysics, 122, pp. 40-52, 2015. doi: 10.1016/ j.jappgeo.2015.08.007.
Liu, Y. & Sen, M.K., Time?Space Domain Dispersion-Relationbased Finite-Difference Method with Arbitrary Even-Order Accuracy for the 2D Acoustic Wave Equation, Journal of Computational Physics, 232, pp. 327-345, 2013. doi: 10.1016/j.jcp.2012.08.025.
Xu, S., Baoa, Q., Rena, Z. & Liu, Y., Applying an Advanced Temporal and Spatial High-Order Finite-Difference Stencil to 3D Seismic Wave Modeling, Journal of Computational Physics, 436, 110133, 2021. doi: 10.1016/ j.jcp.2021.110133.
Jastram, C. & Behle, A., Acoustic modeling on a grid of vertically varying spacing. Geophysical Prospecting, 40, pp. 157-170, 1992. doi: 10.1111/j.1365-2478.1992.tb00369.x
Jastram, C. & Tessmer, E., Elastic Modelling on a Grid with Vertically Varying Spacing: Geophysical Prospecting, 42, pp. 357-370, 1994. doi: 10.1111/j.1365-2478.1994.tb00215.x
Robertsson, J.O.A. & Holliger, K., Modeling of Seismic Wave Propagation Near the Earth?s Surface. Physics of the Earth and Planetary Interiors, 104, pp. 193-211, 2017. doi: 10.1016/S0031-9201(97)00045-9.
De Lilla, A., Finite Difference Seismic Wave Propagation Using Discontinuous Grid Sizes, M. Sc. Thesis, Massachusetts Institute of Technology, 1997, Boston, USA
Pitarka, A., 3D Elastic Finite-Difference Modeling of Seismic Motion Using Staggered Grids with Nonuniform Spacing, Bulletin of the Seismological Society of America, 89, pp. 54-68, 1999. doi: 10.1785/ BSSA0890010054.
Aoi, S. & Fujiwara, H., 3D Finite-Difference Method Using Discontinuous Grids, Bulletin of The Seismological Society of America, 89, pp. 918-930, 1999. doi: 10.1785/BSSA0890040918.
Kristek, J., Moczo, P. & Galis, M., Stable Discontinuous Staggered Grid in the Finite-Difference Modelling of Seismic Motion, Geophysical Journal International, 183(3), pp. 1401-1407, 2010. doi: 10.1111/ j.1365-246X.2010.04775.x.
Fan, N., Zhao, L.F., Xie, X.B. & Yao, Z.X., A Discontinuous-Grid Finite-Difference Scheme for Frequency-Domain 2D Scalar Wave Modelling. Geophysics, 83(4), T235-T244. 2018. doi: 10.1190/ geo2017-0535.1.
Chen, Y., Yang, G., Mad, X., He, C. & Song, G., A Time-Space Domain Stereo Finite Difference Method for 3D Scalar Wave Propagation, Computers & Geosciences, 96, pp. 218-235, 2016. doi: 10.1016/ j.cageo.2016.08.009.
Dai, W., Fowler, P. & Schuster, G.T., Multi-Source Least-Squares Reverse Time Migration, Geophysical Prospecting, 60, pp. 681-695, 2012. doi: 10.1111/j.1365-2478.2012.01092.x
Murthy, N., Guddati, M.N. & Heidari, A.H., Migration with Arbitrarily Wide-Angle Wave Equations, Geophysics, 70(3), pp. S61-S70, 2005. doi: 10.1190/1.1925747.
Tocheport, A., Rivera, L. & Chevrot, S., A Systematic Study of Source Time Functions and Moment Tensors of Intermediate and Deep Earthquakes, Journal of Geophysical Research, 112, B07311, 2007. doi: 10.1029/2006JB004534.
Vavry?uk, V., Moment Tensor Decompositions Revisited, Journal of Seismology, 19(1), pp. 231-252, 2014. doi: 10.1007/s10950-014-9463-y.
Kang, T.S. & Baag, C.E., Finite-Difference Seismic Simulation Combining Discontinuous Grids with Locally Variable Timesteps, Bulletin of the Seismological Society of America, 94, pp. 207-219, 2004a. doi: 10.1785/0120030080.
Society of Exploration Geophysicists(SEG) Wiki, AGL Elastic Marmousi, from : https://wiki.seg.org/wiki/ AGL_Elastic_Marmousi (15 April 2022).
Wu, J., Liu, Q., Zhang, X., Zhou, C., Yin, X., Xie, W., Liang, X. & Huang, J., Attenuation Characteristics of Impact-Induced Seismic Wave in Deep Tunnels: An In Situ Investigation Based on Pendulum Impact Test, Journal of Rock Mechanics and Geotechnical Engineering, 14, pp. 494-504, 2022. doi: 10.1016/ j.jrmge.2021.12.005.
Dutta, A., Biswas, R., Singh, C., Kumar, M.R., Jana, N. & Singh, A., Depth-Wise Attenuation Mechanism of Seismic Waves in the Andaman Region, Soil Dynamics and Earthquake Engineering, 151, 107000, 2021. doi: 10.1016/j.soildyn.2021.107000.
Finkelstein, B. & Kastner, R., A Comprehensive New Methodology for Formulating FDTD Schemes with Controlled Order of Accuracy and Dispersion, IEEE Transactions on Antennas and Propagation, 56, pp. 3516?3525, 2008. doi: 10.1109/CEM.2007.4387655.
Fan, N., Zhao, L.F., Gao, Y.J. & Yao, Z.X., Discontinuous Collocated-Grid Implementation for High-Order Finite-Difference Modelling, Geophysics, 80(4), P. T175-T181, 2015. doi: 10.1190/ geo2015-0001.1.
Kang, T.S. & Baag, C.E., An Efficient Finite-Difference Method for Simulating 3D Seismic Response of Localized Basin Structures, Bulletin of the Seismological Society of America, 94, pp. 1690-1705, 2004. doi: 10.1785/012004016.
Moczo, P., Lab, P., Kristek, J. & Hron, F., Amplification and Differential Motion Due to an Antiplane 2D Resonance in The Sediment Valleys Embedded in a Layer Over the Half-Space, Bulletin of the Seismological Society of America, 86, pp. 1434-1446, 1996. doi: 10.1785/BSSA0860051434.
Wang, Y.B. & Takenaka, H., A Multidomain Approach of the Fourier Pseudospectral Method Using Discontinuous Grid for Elastic Wave Modeling, Earth, Planets and Space, 53, pp. 149-158, 2001. doi: 10.1186/BF03352372.
Wu, C., Harris, J.M., Nihei, K.T. & Nakagawa, S., Two-Dimensional Finite-Difference Seismic Modeling of an Open Fluid-Filled Fracture: Comparison of Thin-Layer and Linear-Slip Models, Geophysics, 70(4), pp. T57?T62, 2005. doi: 10.1190/1.1988187.
Zhang, W., Shen, Y. & Chen, X.F., Numerical Simulation of Strong Ground Motion for the M s8. 0 Wenchuan Earthquake of 12 May 2008: Science in China, Series D: Earth Sciences, 51, pp. 1673?1682, 2008. doi: 10.1007/s11430-008-0130-4.
Hayashi, K., Burns, D.R. & Toksoz, M.N., Discontinuous-Grid Finite-Difference Seismic Modeling Including Surface Topography, Bulletin of the Seismological Society of America, 91(6), pp. 1750-1764, 2021. doi: 10.1785/0120000024.
Tan, S. & Huang, L., A Staggered-Grid Finite-Difference Scheme Optimized in the Time?Space Domain for Modeling Scalar-Wave Propagation in Geophysical Problems, Journal of Computational Physics, 276, pp. 613-634, 2014b. doi: 10.1016/j.jcp.2014.07.044.
