Abstract
Numerical simulations of a seismic wavefield are important to analyze seismic wave propagation. Elastic-wave equations are used in data simulation for modeling migration and imaging. In elastic wavefield numerical modeling, the rotated staggered-grid method (RSM) is a modification of the standard staggered-grid method (SSM). The variable-order method is based on the method of variable-length spatial operators and wavefield propagation, and it calculates the real dispersion error by adapting different finite-difference orders to different velocities. In this study, the variable-order rotated staggered-grid method (VRSM) is developed after applying the variable-order method to RSM to solve the numerical dispersion problem of RSM in low-velocity regions and reduce the computation cost. Moreover, based on theoretical dispersion and the real dispersion error of wave propagation calculated with the wave separation method, the application of the original method is extended from acoustic to shear waves, and the calculation is modified from theoretical to time-varying values. A layered model and an overthrust model are used to demonstrate the applicability of VRSM. We also evaluate the order distribution, wave propagation, and computation time. The results suggest that the VRSM order distribution is reasonable and VRSM produces high-precision results with a minimal computation cost.
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References
Alterman, Z., and Karal, F. C., 1968, Propagation of elastic waves in layered media by finite difference methods: Bulletin of the Seismological Society of America, 58(1), 367–398.
Aminzadeh, F., Burkhard, N., Nicoletis, L., Rocca, F., and Wyatt, K., 1994, SEG/EAGE 3-D modeling project: 2nd update: The Leading Edge, 13(9), 949–952.
Chen, H. M., Zhou, H., Lin, H., and Wang, S. X., 2013, Application of perfectly matched layer for scalar arbitrarily wide-angle wave equations: Geophysics, 78(1), T29–T39.
Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications, California, 47–49.
Dankbaar, J. W. M., 1985, Separation of P-waves and S-waves: Geophysical Prospecting, 33(7), 970–986.
Dellinger, J., and Etgen, J., 1990, Wave-field separation in two-dimensional anisotropic media: Geophysics, 55(7), 914–919.
Duan, Y. T., Hu, T. Y., Yao, F. C., and Zhang, Y., 2013, 3D elastic wave equation forward modeling based on the precise integration method: Applied Geophysics, 10(1), 71–78.
Hendrick, N., and Brand, E., 2006, Practical evaluation of P- and S-wave separation via elastic wave field decomposition: Exploration Geophysics, 37(2), 139–149.
Hustedt, B., Operto, S., and Virieux, J., 2004, Mixedgrid and staggered-grid finite-difference methods for frequency-domain acoustic wave modeling: Geophysical Journal International, 157(3), 1269–1296.
Jackson, J. D., 1998, Classical electrodynamics, 3rd Edition: John Wiley & Sons, Inc, United States of America, 295–299.
Jastram, C., and Tessmer, E., 1994, Elastic modeling on a grid with vertically varying spacing: Geophysical Prospecting, 42(4), 357–370.
Jia, Y., and Paul, S., 2009, Elastic wave mode separation for VTI media: Geophysics, 74(5), 19–32.
Li, Z. C., Zhang, H., Liu, Q., and Han, W., 2007, Elastic wave high order staggered grid wave separation forward modeling: Oil Geophysical Prospecting, 42(5), 510–515.
Liu, Y., and Sen, M. K., 2011a, Finite-difference modeling with adaptive variable-length spatial operators: Geophysics, 76(4), T79–T89.
Liu, Y., and Sen, M. K., 2011b, Scalar wave equation modeling with time-space domain dispersion-relationbased staggered-grid finite-difference schemes: Bulletin of the Seismological Society of America, 101(1), 141–159.
Liu, Y., and Sen, M. K., 2009, An implicit staggeredgrid finite-difference method for seismic modeling: Geophysical Journal International, 179(1), 459–474.
Moczo, P., 1989, Finite-difference technique for SHwave in 2D media using irregular grids-application to the seismic response problem: Geophysical Journal International, 99(2), 321–329.
O’Brien, G. S., 2010, 3D rotated and standard staggered finite-difference solutions to Biot’s poroelastic wave equations: Stability condition and dispersion analysis: Geophysics, 75(4), T111–T119.
Opršal, I., and Zahradník, J., 1999, Elastic finite-difference method for irregular grids: Geophysics, 64(1), 240–250.
Saenger, E. H., and Bohlen, T., 2004, Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid: Geophysics, 69(2), 583–591.
Saenger, E. H., Gold, N., and Shapiro, S. A., 2000, Modeling the propagation of elastic waves using a modified finite-difference method: Wave Motion, 31, 77–92.
Saenger, E. H., Ciz, R., Krüger, O. S., Schmalholz, S. M., Gurevich, B., and Shapiro, S. A., 2007, Finite difference modeling of wave propagation on microscale: A snapshot of the work in progress: Geophysics, 72(5), SM293–SM300.
Sun, R., McMechan, G. A., Hsiao, H. H., and Chow, J., 2004. Separating P- and S-waves in prestack 3D elastic seismograms using divergence and curl: Geophysics, 69(1), 286–297.
Tang, J., Li, J. J., Yao, Z. A., Shao, J., and Sun, C. Y., 2015, Reservoir time-lapse variations and coda wave interferometry: Applied Geophysics, 12(2), 244–254.
Tessmer, E., 2000, Seismic finite-difference modeling with spatially varying time steps: Geophysics, 65(4), 1290–1293.
Virieux, J., 1984, SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method: Geophysics, 49(11), 1933–1957.
Virieux, J., 1986, P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method: Geophysics, 51(4), 889–901.
Wang, S. X., Li, X. Y., Qian, Z. P., Di, B. R., and Wei, J. X., 2007, Physical modeling studies of 3-D P-wave seismic for fracture detection: Geophysical Journal International, 168(2), 745–756.
Wang, W. Z., Qin, Z., Hu, T. Y., Li, Y. D., and Zhang, Y., 2014, Elastic wave equation rotated-grid forward modeling and imaging based on wave separation method: 76th EAGE Conference and Exhibition, Extended abstract, Th G105 10.
Wang, X., Dodds, K., and Zhao, H., 2006, An improved high-order rotated staggered finite-difference algorithm for simulating elastic waves in heterogeneous viscoelastic/anisotropic media: Exploration Geophysics, 37(2), 160–174.
Yang, H. X., and Wang, H. X., 2013, A study of damping factors in perfectly matched layers for the numerical simulation of seismic waves: Applied Geophysics, 10(1), 63–70.
Yang, J. H., Liu, T., Tang, G., and Hu, T. Y., 2009, Modeling seismic wave propagation within complex structures: Applied Geophysics, 6(1), 30–41.
Zhang, Q. S., and George, A. M., 2010, 2D and 3D elastic wave field vector decomposition in the wave number domain for VTI media: Geophysics, 75(3), 13–26.
Zhang, Y., and Liu, Y., 2012, 3D poroviscoelastic rotated staggered finite-difference modeling with PML absorbing boundary conditions: 82nd SEG, Annual International Meeting, Expanded Abstracts, 1–5.
Zhou, H., Wang, S. X., Li, G. F., and Shen, J. S., 2010, Analysis of complicated structure seismic wave fields: Applied Geophysics, 7(2), 185–192.
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Wang Wei-Zhong received his B.Sc. in Geophysics from China University of Geosciences (Wuhan) in 2013. He is currently a PhD. candidate in the School of Earth and Space Sciences at Peking University. His interests include seismic wave forward modeling and imaging.
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Wang, WZ., Hu, TY., Lu, XM. et al. Variable-order rotated staggered-grid method for elastic-wave forward modeling. Appl. Geophys. 12, 389–400 (2015). https://doi.org/10.1007/s11770-015-0507-z
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DOI: https://doi.org/10.1007/s11770-015-0507-z