1887
Volume 73, Issue 9
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

Accurately and effectively handling undulating interfaces, including both free surfaces and internal interfaces, remains a key challenge in seismic exploration. Traditional scalar wave equations typically neglect the influence of internal interfaces on wave propagation. To address this, the wave equation in layered media (WEILM) is established by introducing the Dirac delta function. For undulating interfaces, existing methods are mostly grid‐based, and often require additional grid processing to achieve accurate description, which increases computational cost. Therefore, by introducing the meshless method combined with free surfaces boundary conditions, this article proposes the method for dealing with undulating surfaces on the basis of the radial‐basis‐function‐generated finite difference (RBF‐FD) method. Theoretical analysis indicates that the stability of the proposed method is affected by the sum of the stencil node weights. Numerical experiments show that, compared with the scalar wave equation, the interfaces term introduced in WEILM effectively adjusts waveform amplitudes. Moreover, relative to the classical Lax–Wendroff correction (LWC) method, our approach can avoid spurious diffraction waves caused by grid discretization when handling undulating surfaces. By applying RBF‐FD to solve WEILM in conjunction with the method for dealing with undulating surfaces, complex seismic wavefield, including undulating surfaces, can be simulated with higher accuracy.

© 2025 European Association of Geoscientists & Engineers. © 2025 European Association of Geoscientists & Engineers.
Loading

Article metrics loading...

/content/journals/10.1111/1365-2478.70098
2025-11-01
2026-01-24

  • From This Site

    /content/journals/10.1111/1365-2478.70098
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Appelo, D., and N. A.Petersson. 2009. “A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries With Free Surfaces.” Communications in Computational Physics5, no. 1: 84–107.
     [Google Scholar]
  2. Bayona, V., N.Flyer, and B.Fornberg. 2019. “On the Role of Polynomials in RBF‐FD Approximations: III. Behavior Near Domain Boundaries.” Journal of Computational Physics380: 378–399.
     [Google Scholar]
  3. Bayona, V., N.Flyer, B.Fornberg, and G. A.Barnett. 2017. “On the Role of Polynomials in RBF‐FD Approximations: II. Numerical Solution of Elliptic PDEs.” Journal of Computational Physics332: 257–273.
     [Google Scholar]
  4. Bohlen, T., and E. H.Saenger. 2006. “Accuracy of Heterogeneous Staggered‐Grid Finite‐Difference Modeling of Rayleigh Waves.” Geophysics71, no. 4: T109–T115.
     [Google Scholar]
  5. Briani, M., A.Sommariva, and M.Vianello. 2012. “Computing Fekete and Lebesgue Points: Simplex, Square, Disk.” Journal of Computational and Applied Mathematics236, no. 9: 2477–2486.
     [Google Scholar]
  6. Chaljub, E., D.Komatitsch, J. P.Vilotte, Y.Capdeville, B.Valette, and G.Festa. 2007. “Spectral‐Element Analysis in Seismology.” Advances in Geophysics48: 365–419.
     [Google Scholar]
  7. Chavent, G.1974. “Identification of Functional Parameters in Partial Differential Equations.” Joint Automatic Control Conference12: 155–156.
     [Google Scholar]
  8. Chen, H. M., K. J.Chen, L. Q.Wang, Y. H.Han, and H.Zhou. 2025. “Reverse‐Time Migration Method Based on Immersed Absorbing Boundary for Undulating Surfaces.” Chinese Journal of Geophysics68, no. 3: 1069–1086.
     [Google Scholar]
  9. Dablain, M. A.1986. “The Application of High‐Order Differencing to the Scalar Wave Equation.” Geophysics51, no. 1: 54–66.
     [Google Scholar]
  10. Dahlen, F. A., and J.Tromp. 1998. Theoretical Global Seismology. Princeton University Press.
     [Google Scholar]
  11. Dong, X. P., and D. H.Yang. 2017. “Numerical Modeling of the 3‐D Seismic Wavefield With the Spectral Element Method in Spherical Coordinates.” Chinese Journal of Geophysics60, no. 12: 4671–4680.
     [Google Scholar]
  12. Duan, P. R., B. L.Gu, Z. C.Li, and Q. Y.Li. 2024. “QR Radius‐Basis‐Function Finite Difference Method With Irregular Topography and Its Application to Elastic Reverse Time Migration.” Chinese Journal of Geophysics67, no. 3: 1181–1207.
     [Google Scholar]
  13. Flyer, N., G. A.Barnett, and L. J.Wicker. 2016. “Enhancing Finite Differences With Radial Basis Functions: Experiments on the Navier–Stokes Equations.” Journal of Computational Physics316: 39–62.
     [Google Scholar]
  14. Flyer, N., B.Fornberg, V.Bayona, and G. A.Barnett. 2016. “On the Role of Polynomials in RBF‐FD Approximations: I. Interpolation and Accuracy.” Journal of Computational Physics321: 21–38.
     [Google Scholar]
  15. Fornberg, B., and N.Flyer. 2015. “Fast Generation of 2‐D Node Distributions for Mesh‐Free PDE Discretizations.” Computers & Mathematics with Applications69, no. 7: 531–544.
     [Google Scholar]
  16. Fornberg, B., E.Lehto, and C.Powell. 2013. “Stable Calculation of Gaussian‐Based RBF‐FD Stencils.” Computers & Mathematics with Applications65, no. 4: 627–637.
     [Google Scholar]
  17. Gardner, G. H. F., L. W.Gardner, and A. R.Gregory. 1974. “Formation Velocity and Density—The Diagnostic Basics for Stratigraphic Traps.” Geophysics39, no. 6: 770–780.
     [Google Scholar]
  18. Hayashi, K., D. R.Burns, and M. N.Toksöz. 2001. “Discontinuous‐Grid Finite‐Difference Seismic Modeling Including Surfaces Topography.” Bulletin of the Seismological Society of America91, no. 6: 1750–1764.
     [Google Scholar]
  19. Jih, R. S., K. L.McLaughlin, and Z. A.Der. 1988. “Free‐Boundary Conditions of Arbitrary Polygonal Topography in a Two‐Dimensional Explicit Elastic Finite‐Difference Scheme.” Geophysics53, no. 8: 1045–1055.
     [Google Scholar]
  20. Kindelan, M., D.Álvarez, and P.Gonzalez‐Rodriguez. 2018. “Frequency Optimized RBF‐FD for Wave Equations.” Journal of Computational Physics371: 564–580.
     [Google Scholar]
  21. Komatitsch, D., and J. P.Vilotte. 1998. “The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures.” Bulletin of the Seismological Society of America88, no. 2: 368–392.
     [Google Scholar]
  22. Lan, H. Q., and Z. J.Zhang. 2011. “Three‐Dimensional Wave‐Field Simulation in Heterogeneous Transversely Isotropic Medium With Irregular Free Surfaces.” Bulletin of the Seismological Society of America101, no. 3: 1354–1370.
     [Google Scholar]
  23. Levander, A. R.1988. “Fourth‐Order Finite‐Difference P‐SV Seismograms.” Geophysics53, no. 11: 1425–1436.
     [Google Scholar]
  24. Li, B., Y.Liu, Y. M.Zhao, and X.Liu. 2017. “Hybrid Absorbing Boundary Condition for Piecewise Smooth Curved Boundary in 2D Acoustic Finite Difference Modelling.” Exploration Geophysics49, no. 4: 469–483.
     [Google Scholar]
  25. Lines, L. R., R.Slawinski, and P. R.Bording. 1999. “A Recipe for Stability of Finite‐Difference Wave‐Equation Computations.” Geophysics64, no. 3: 967–969.
     [Google Scholar]
  26. Liu, L. B., P. R.Duan, Y. Y.Zhang et al. 2020. “Overview of Mesh‐Free Method of Seismic Forward Numerical Simulation.” Progress in Geophysics35, no. 5: 1815–1825.
     [Google Scholar]
  27. Liu, S. L., D. H.Yang, X. W.Xu, X. F.Li, W. H.Shen, and Y. S.Liu. 2021. “Three‐Dimensional Element‐by‐Element Parallel Spectral‐Element Method for Seismic Wave Modeling.” Chinese Journal of Geophysics64, no. 3: 993–1005.
     [Google Scholar]
  28. Long, J., and C. G.Farquharson. 2019. “Three‐Dimensional Forward Modelling of Gravity Data Using Mesh‐Free Methods With Radial Basis Functions and Unstructured Nodes.” Geophysical Journal International217, no. 3: 1577–1601.
     [Google Scholar]
  29. Lucy, L. B.1977. “A Numerical Approach to the Testing of the Fission Hypothesis.” Astronomical Journal82, no. 12: 1013–1024.
     [Google Scholar]
  30. Mao, Q., Y.Zhong, Y.Liu, et al. 2024. “Simulate the Elastic Wavefields in Media With an Irregular Surfaces Topography Based on Staggered Grid Finite Difference.” Journal of Geophysics and Engineering21, no. 4: 1286–1301.
     [Google Scholar]
  31. Martin, B., and B.Fornberg. 2017. “Seismic Modeling With Radial Basis Function‐Generated Finite Differences (RBF‐FD)—A Simplified Treatment of Interfaces.” Journal of Computational Physics335: 828–845.
     [Google Scholar]
  32. Martin, B., B.Fornberg, and A.St‐Cyr. 2015. “Seismic Modeling With Radial‐Basis‐Function‐Generated Finite Differences.” Geophysics80, no. 4: T137–T146.
     [Google Scholar]
  33. Masson, Y.2023. “Distributional Finite‐Difference Modelling of Seismic Waves.” Geophysical Journal International233, no. 1: 264–296.
     [Google Scholar]
  34. Pasquetti, R., and F.Rapetti. 2004. “Spectral Element Methods on Triangles and Quadrilaterals: Comparisons and Applications.” Journal of Computational Physics198, no. 1: 349–362.
     [Google Scholar]
  35. Patera, A. T.1984. “A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion.” Journal of Computational Physics54, no. 3: 468–488.
     [Google Scholar]
  36. Pérez‐Ruiz, J. A., F.Luzón, and A.Garcı́a‐Jerez. 2005. “Simulation of an Irregular Free Surfaces With a Displacement Finite‐Difference Scheme.” Bulletin of the Seismological Society of America95, no. 6: 2216–2231.
     [Google Scholar]
  37. Priolo, E., J. M.Carcione, and G.Seriani. 1994. “Numerical Simulation of Interfaces Waves by High‐Order Spectral Modeling Techniques.” Journal of the Acoustical Society of America95, no. 2: 681–693.
     [Google Scholar]
  38. Robertsson, J. O. A.1996. “A Numerical Free Surfaces Condition for Elastic/Viscoelastic Finite‐Difference Modeling in the Presence of Topography.” Geophysics61, no. 6: 1921–1934.
     [Google Scholar]
  39. Robertsson, J. O. A., and K.Holliger. 1997. “Modeling of Seismic Wave Propagation Near the Earth's Surfaces.” Physics of the Earth and Planetary Interiors104, no. 1–3: 193–211.
     [Google Scholar]
  40. Sande, V. D. K., and B.Fornberg. 2021. “Fast Variable Density 3‐D Node Generation.” SIAM Journal on Scientific Computing43, no. 1: A242–A257.
     [Google Scholar]
  41. Song, G. J., Z. L.Wang, X. M.Zhang, Y. L.Chen, and J. D.Tian. 2019. “SG‐ONADM Hybrid Method for Simulation of the Viscous Acoustic Seismic Wave Field.” Chinese Journal of Geophysics62, no. 9: 3524–3533.
     [Google Scholar]
  42. Taylor, M. A., and B. A.Wingate. 2000. “A Generalized Diagonal Mass Matrix Spectral Element Method for Non‐Quadrilateral Elements.” Applied Numerical Mathematics33, no. 1–4: 259–265.
     [Google Scholar]
  43. Tessmer, E.2000. “Seismic Finite‐Difference Modeling With Spatially Varying Time Steps.” Geophysics65, no. 4: 1290–1293.
     [Google Scholar]
  44. Vaaland, U., H. N.Gharti, and J.Tromp. 2019. “Simulations of Seismic Wave Propagation Using a Spectral‐Element Method in a Lagrangian Framework With Logarithmic Strain.” Geophysical Journal International216, no. 3: 2148–2157.
     [Google Scholar]
  45. Virieux, J.1986. “P‐SV Wave Propagation in Heterogeneous Media: Velocity‐Stress Finite‐Difference Method.” Geophysics51, no. 4: 889–901.
     [Google Scholar]
  46. Virieux, J., H.Calandra, and R. E.Plessix. 2011. “A Review of the Spectral, Pseudo‐Spectral, Finite‐Difference and Finite‐Element Modelling Techniques for Geophysical Imaging.” Geophysical Prospecting59, no. 5: 794–813.
     [Google Scholar]
  47. Virieux, J., and S.Operto. 2009. “An Overview of Full‐Waveform Inversion in Exploration Geophysics.” Geophysics74, no. 6: WCC1–WCC26.
     [Google Scholar]
  48. Wang, J., Y.Liu, and H.Zhou. 2022. “High Temporal Accuracy Elastic Wave Simulation With New Time–Space Domain Implicit Staggered‐Grid Finite‐Difference Schemes.” Geophysical Prospecting70, no. 8: 1346–1366.
     [Google Scholar]
  49. Wu, H., C. Y.Sun, S. Z.LiJ.Tang, and N.Xu. 2021. “Mesh‐Free Radial‐Basis‐Function‐Generated Finite Differences and Their Application in Reverse Time Migration.” Geophysics86, no. 1: S91–S101.
     [Google Scholar]
  50. Yang, D. H.2002. “Finite Element Method of the Elastic Wave Equation and Wave‐Fields Simulation in Two‐Phase Anisotropic Media.” Chinese Journal of Geophysics45, no. 4: 600–610.
     [Google Scholar]
/content/journals/10.1111/1365-2478.70098
Loading
Meshfree Modelling Technique for Variable Parameters Wave Equation in Layered Media With Undulating Topography
Geophysical Prospecting 73, e70098 (2025); https://doi.org/10.1111/1365-2478.70098
/content/journals/10.1111/1365-2478.70098
/content/journals/10.1111/1365-2478.70098
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): interface interaction; numerical simulation; radial‐basis‐function‐generated finite difference (RBF‐FD); undulating surfaces; wave equation in layered media (WEILM)

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error