Abstract
Based on the pure quasi-P wave equation in transverse isotropic media with a vertical symmetry axis (VTI media), a quasi-P wave equation is obtained in transverse isotropic media with a tilted symmetry axis (TTI media). This is achieved using projection transformation, which rotates the direction vector in the coordinate system of observation toward the direction vector for the coordinate system in which the z-component is parallel to the symmetry axis of the TTI media. The equation has a simple form, is easily calculated, is not influenced by the pseudo-shear wave, and can be calculated reliably when δ is greater than ε. The finite difference method is used to solve the equation. In addition, a perfectly matched layer (PML) absorbing boundary condition is obtained for the equation. Theoretical analysis and numerical simulation results with forward modeling prove that the equation can accurately simulate a quasi-P wave in TTI medium.
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References
Alkhalifah, T., 2000, An acoustic wave equation for anisotropic media: Geophysics, 65(4), 1239–1250.
Dellinger, J., and Etgen, J., 1990, Wave-field separation in two-dimensional anisotropic media: Geophysics, 55(7), 914–919.
Du, Q. Z., Guo, C. F., and Gong, X. F., 2015, Hybrid PS/ FD numerical simulation and stability analysis of pure P-wave propagation VTI media: Chinese J. Geophys, (in Chinese), 58(4), 1290–1309.
Duvenck, E., Milcik, P., and Bakker, P. M., 2008, Acoustic VTI wave equations and their application for anisotropic reverse-time migration: 78th Annual International Meeting, SEG, Expanded Abstracts, 2186–2190.
Fletcher, R. P., Du, X., and Fowler, P. J., 2009, Reverse time migration in tilted transversely isotropic (TTI) media: Geophysics, 74(6), WCA179–WCA187.
Fowler, P. J., Du, X., and Fletcher, R. P., 2010, Coupled equations for reverse time migration in transversely isotropic media: Geophysics, 75(1), S11–S22.
Han, L. H., and He, B. S., 2010, First-order Quasi-P wave equation forward modeling and boundary conditions in VTI media: Chinese J. Oil Geophysics Prospecting, (in Chinese), 45(6), 819–825.
Hestholm, S., 2007, Acoustic VTI modeling using highorder finite differences: 77th Annual International Meeting, SEG, Expanded Abstracts, 139–143.
Huang, Y. J., Zhu, G. M., and Liu, C. Y., 2011, An approximate acoustic wave equation for VTI media: Chinese J. Geophys, (in Chinese), 54(8), 2117–2123.
Kosloff, D. D., and Baysal, E., 1982, Forward modeling by a Fourier method: Geophysics, 47(10), 1402–1412.
Li, B., Li, M., and Liu, H. W., 2012, Stability of reverse time migration in TTI media: Chinese J. Geophys, (in Chinese), 55(4), 1366–1375.
Liu, Y., Li, C. C., and Mou, Y. G., 1998, Finite-difference numerical modeling of any even-order accuracy: Chinese J. Oil Geophysics Prospecting, (in Chinese), 33(1), 1–10.
Pestana, R. C., Ursin, B., and Stoffa, P. L., 2011, Separate P-and SV-wave equations for VTI media: Geopysics, 75(4), T163–T167.
Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51(10), 1954–1996.
Xu, S., and Zhou, H., 2014, Accurate simulation of pure quasi-P-waves in complex anisotropic media: Geophysics, 79(6), 341–348.
Xu, S., Tang, B., Mu, J., and Zhou, H., 2015, Quasi-P wave propagation with an elliptic differential operator: 85th New Orleans Annual Meeting, SEG, Expanded Abstracts, 4380–4384.
Yan, J., and Paul, S., 2009, Elastic wave-mode separation for VTI media: Geophysics, 74(5), WB19–WB32.
Yan, J., and Paul, S., 2012, Elastic wave-mode separation for tilted transverse isotropic media: Geophysical Prospecting, 60, 29–48.
Zhan, G., Pestana, R. C., and Stoffa, P. L., 2013, An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation: Journal of Geophysics and Engineering, 10(2), 25004–25011.
Zhou, H., Zhang, G., and Bloor, R., 2006, An anisotropic acoustic wave equation for modeling and migration in 2D TTI media: 76th Annual International Meeting, SEG, Expanded Abstracts, 194–198.
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We gratefully acknowledge all reviewers for their constructive comments and suggestions, and express our thanks also to AGED for their patience and tolerance.
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This work is supported by the National Natural Science Foundation of China (No. 41674118), the national science and technology major project (No. 2016ZX05027-002).
Zhang Jian-Min is a graduate student of geophysical prospecting at the Ocean University of China. Her research interests are seismic wave theory, forward modeling, and reverse-time migration imaging.
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Zhang, JM., He, BS. & Tang, HG. Pure quasi-P wave equation and numerical solution in 3D TTI media. Appl. Geophys. 14, 125–132 (2017). https://doi.org/10.1007/s11770-017-0613-1
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DOI: https://doi.org/10.1007/s11770-017-0613-1