Damped Wave Equation-based Traveltime Calculation using Embedded Boundary Method for Irregular Topography

Embedded Boundary Method를 이용한 불규칙한 지형에서의 감쇠 파동장 기반 초동주시 계산

  • Hwang, Seongcheol (Department of Energy Resources Engineering, Inha University) ;
  • Lee, Ganghoon (Department of Energy Resources Engineering, Inha University) ;
  • Pyun, Sukjoon (Department of Energy Resources Engineering, Inha University)
  • 황성철 (인하대학교 에너지자원공학과) ;
  • 이강훈 (인하대학교 에너지자원공학과) ;
  • 편석준 (인하대학교 에너지자원공학과)
  • Received : 2019.01.11
  • Accepted : 2019.02.14
  • Published : 2019.02.28


The first-arrival traveltime calculation method based on the damped wave equation overcomes the shortcomings of ray-tracing methods. Since this algorithm needs to solve the damped wave equation, numerical modeling is essential. However, it is not desirable to use the finite-difference method (FDM), which has good computational efficiency, for simulating the land seismic data because of irregular topography. Thus, the finite-element method (FEM) which requires higher computational cost than FDM has been used to correctly describe the irregular topography. In this study, we computed first-arrival traveltimes in an irregular topographic model using FDM incorporating embedded boundary method (EBM) to overcome this problem. To verify the accuracy and efficiency of the proposed algorithm, we compared our results with those of FEM. As a result, the proposed method using EBM not only provided the same accuracy as the FEM but also showed the improved computational efficiency.

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Fig. 1. Grid structures of FDM modeling scheme in irregular topography. (a) An example of Cartesian grid and a difference stencil for 9-point FDM. The black line indicates free surface with irregular topography. The green line indicates the assumed free surface in the standard FDM and the red points indicate the nodal points with free surface conditions. (b) An example of grid points used for EBM. The red line and points are related to extrapolation and the blue line and points are related to interpolation.

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Fig. 2. Modeling results using 9-point FDM and FEM in a homogeneous velocity model with irregular topography. (a)-(d): Snapshots of FDM at 0.04 s, 0.28 s, 0.52 s and 0.76 s, respectively. (e)-(f): Snapshots of FEM at 0.04 s, 0.28 s, 0.52 s and 0.76 s, respectively. (i)-(l): Wavefields extracted from the snapshots along the horizontal line at the depth of source location (0.04 s, 0.28 s, 0.52 s and 0.76 s, respectively). The blue solid lines indicate FDM results and the orange dashed lines indicate FEM results.

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Fig. 3. Homogeneous velocity model having a tilted flat surface (16.7o). The red box indicates enlarged grid structure for FDM.

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Fig. 4. Calculated traveltimes in the velocity model shown in Fig. 3: The dashed contour lines indicate the traveltimes calculated by EBM and the solid contour lines indicate the analytical traveltimes.

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Fig. 6. Comparison of the traveltimes calculated by EBM and FEMs with structured and unstructured meshes. (a) Homogeneous velocity model with a flat surface. Yellow star indicates the shot location. The p-wave velocity is 4000 m/s. (b) Traveltime error curves between analytical solution and respective numerical results. The blue, orange and green lines indicate the traveltime errors for EBM, FEM with structured mesh, and FEM with unstructured mesh, respectively.

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Fig. 7. Modified 2D Canadian Foothills velocity model. The actual velocity of the air layer is 330 m/s, but it is represented as blue color (complementary color for red or orange) to clearly distinguish the free surface boundary.

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Fig. 8. Results of traveltime calculation using the 2D Canadian Foothills velocity model shown in Fig. 7: (a) Contour map of traveltimes calculated by EBM. (b) Traveltime curves calculated with various source locations (1.5 km, 3.75 km, 6 km, 8.25 km and 10.5 km).

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Fig. 9. Comparison of computational time between EBM (blue) and FEM with unstructured mesh (orange). The graphs show the elapsed times when the grid spacing are (a) 15 m and (b) 5 m, respectively. In the case of FEM, the grid spacing means the reference grid spacing for mesh generation.

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Fig. 5. (a) Comparison of traveltime curves. The blue solid line indicates the traveltime curve computed by EBM, the red dotted line indicates staircase discretization and the orange dashed line is the analytical solution. (b) Traveltime residuals between analytical solutions and EBM results. (c) Traveltime residuals between analytical solutions and staircase discretization results.


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  1. Alkhalifah, T., and Fomel, S., 2001, Implementing the fast marching eikonal solver: spherical versus Cartesian coordinates, Geophys. Prospect., 49(2), 165-178.
  2. AlSalem, H., Petrov, P., Newman, G., and Rector, J., 2018, Embedded boundary methods for modeling 3D finitedifference Laplace-Fourier domain acoustic wave equation with free-surface topography, Geophysics, 83(5), T291-T300.
  3. Cerveny, V., Molotkov, I. A., and Psencik, I., 1977, Ray method in seismology, Univ. of karlova press.
  4. Gray, S. H., and Marfurt, K. J., 1995, Migration from topography: Improving the near-surface image, Can. J. Explor. Geophys., 31(1-2), 18-24.
  5. Julian, B. R., and Gubbins, D., 1977, Three-dimensional seismic ray tracing, J. Geophys. Res., 43(1), 95-114.
  6. Jo, C. H., Shin, C., and Suh, J. H., 1996, An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator, Geophysics, 61(2), 529-537.
  7. Kim, S., 2002, 3-D eikonal solvers: First-arrival traveltimes, Geophysics, 67(4), 1225-1231.
  8. Komatitsch, D., Martin, R., Tromp, J., Taylor, M. A., and Wingate, B. A., 2001, Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles, J. Comput. Acoust., 9(2), 703-718.
  9. Kreiss, H. O., and Petersson, N. A., 2006, A second order accurate embedded boundary method for the wave equation with Dirichlet data, SIAM J. Sci. Comput., 27(4), 1141-1167.
  10. Li, J., Zhang, Y., and Toksoz, M. N., 2010, Frequency-domain finite-difference acoustic modeling with free surface topography using embedded boundary method, 80th Ann. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts, 2966-2971.
  11. Lee, D., Park, Y., and Pyun, S., 2017, Analysis on the Reliability and Influence Factors of Refraction Traveltime Tomography Depending on Source-receiver Configuration. Geophys. and Geophys. Explor., 20(3), 163-175 (In Korean with English abstract).
  12. Miller, G. H., and Trebotich, D., 2012, An embedded boundary method for the Navier-Stokes equations on a time-dependent domain, Comm. App. Math. Comp. Sci., 7(1), 1-31.
  13. Park, Y., and Pyun, S., 2018, Refraction traveltime tomography based on damped wave equation for irregular topographic model, J. Appl. Geophy., 150, 160-171.
  14. Pereyra, V., Lee, W. H. K., and Keller, H. B., 1980, Solving two-point seismic-ray tracing problems in a heterogeneous medium: Part 1. A general adaptive finite difference method, Bull. Seismol. Soc. Amer., 70(1), 79-99.
  15. Pyun, S., Shin, C., Min, D. J., and Ha, T., 2005, Refraction traveltime tomography using damped monochromatic wavefield, Geophysics, 70(2), U1-U7.
  16. Pyun, S., Son, W., and Shin, C., 2011, 3D acoustic waveform inversion in the Laplace domain using an iterative solver, Geophys. Prospect., 59(3), 386-399.
  17. Pyun, S., and Park, Y., 2016, A Study on Consistency of Numerical Solutions for Wave Equation, Geophys. and Geophs. Explor., 19(3), 136-144 (In Korean with English abstract).
  18. Schwartz, P., Barad, M., Colella, P., and Ligocki, T., 2006, A Cartesian grid embedded boundary method for the heat equation and Poisson's equation in three dimensions, J. Comput. Phys., 211(2), 531-550.
  19. Sethian, J. A., and Popovici, A. M., 1999, 3-D traveltime computation using the fast marching method, Geophysics, 64(2), 516-523.
  20. Shin, C., Min, D. J., Marfurt, K. J., Lim, H. Y., Yang, D., Cha, Y., Ko, S., Yoon, K., Ha, T., and Hong, S., 2002, Traveltime and amplitude calculations using the damped wave solution, Geophysics, 67(5), 1637-1647.
  21. Shin, C., Ko, S., Kim, W., Min, D. J., Yang, D., Marfurt, K. J., Shin, S., Yoon, K., and Yoon, C. H., 2003, Traveltime calculations from frequency-domain downward-continuation algorithms, Geophysics, 68(4), 1380-1388.
  22. Sun, Y., and Fomel, S., 1998, Fast-marching eikonal solver in the tetragonal coordinates 68th Ann. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts, 1949-1952.
  23. Vidale, J. E., 1990, Finite-difference calculation of traveltimes in three dimensions, Geophysics, 55(5), 521-526.
  24. Vinje, V., Iversen, E., and Gjoystdal, H., 1993, Traveltime and amplitude estimation using wavefront construction, Geophysics, 58(8), 1157-1166.
  25. Zhang, X., and Bording, R., 2011, Fast marching method seismic traveltimes with reconfigurable field programmable gate arrays, Can. J. Explor. Geophys., 36(1), 60-68.
  26. Zhang, X., and Tan, S., 2015, A simple and accurate discontinuous Galerkin scheme for modeling scalar-wave propagation in media with curved interfaces, Geophysics, 80(2), T83-T89.