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Xiaobo Liu, Yun Wang, Jianlei Zhang, Haifeng Chen, Hua Zhang. Propagation and Dispersion Characteristics of Scholte Wave in VTI Media with the Presence of Irregular Seabed. Journal of Earth Science, 2024, 35(2): 722-725. doi: 10.1007/s12583-024-1965-0
Citation: Xiaobo Liu, Yun Wang, Jianlei Zhang, Haifeng Chen, Hua Zhang. Propagation and Dispersion Characteristics of Scholte Wave in VTI Media with the Presence of Irregular Seabed. Journal of Earth Science, 2024, 35(2): 722-725. doi: 10.1007/s12583-024-1965-0

Propagation and Dispersion Characteristics of Scholte Wave in VTI Media with the Presence of Irregular Seabed

doi: 10.1007/s12583-024-1965-0
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  • Corresponding author: Yun Wang, wangyun@mail.gyig.ac.cn
  • Received Date: 19 Dec 2023
  • Accepted Date: 04 Jan 2024
  • Available Online: 11 Apr 2024
  • Issue Publish Date: 30 Apr 2024
  • Conflict of Interest
    The authors declare that they have no conflict of interest.
  • Surface waves propagating along the seafloor are generally called Scholte waves, and were first discovered and studied in the early 1950s (Kugler et al., 2005; Buchen and Ben-Hador, 1996). Scholte waves exhibit a dispersion phenomenon, which implies that their velocity varies with frequency. Low-frequency Scholte waves can propagate over long distances on the seabed with little attenuation (Bohlen et al., 2004). The particle motion of Scholte waves in a solid medium changes from a retrograde to a prograde ellipse (Klein et al., 2005; McMechan and Yedlin, 1981).

    Seismic forward modeling is the primary method for simulating underground seismic wave propagation (Wu et al., 2018; Yilmaz, 1987). Researchers have utilized various theoretical and numerical techniques for seismic forward modeling, such as the finite-element, spectral element, and finite-difference operators (Nilsson et al., 2007; Vossen et al., 2002; Komatitsch et al., 2000). Although the finite difference method is easy to perform, it cannot be directly applied to models with irregular surfaces. Appelôand Petersson (2009) proposed curvilinear coordinates to address the irregular free-surface problem, which has proven to be a good way to simulate the propagation of seismic waves in models with rough surfaces and layers (Sun et al., 2017; Wang et al., 2017; Randall, 1983). In the case of marine seismic exploration, the continuous fluid-solid boundary conditions must be dealt with separately (Qu et al., 2018; Choi et al., 2008; Zhang, 2004). The propagation and dispersion characteristics of Scholte waves have been studied previously. Carcione and Helle (2004) simulated Scholte wave propagation at the ocean bottom. Klein et al. (2005) acquired Scholte waves and analyzed their dispersion characteristics.

    The dispersion equations are analytical solutions of the Scholte wave. However, they are suitable only for elastic media with two flat layers (Huang et al., 2022; Sofronov et al., 2015). Transversely isotropic (VTI) media with a vertical axis of symmetry are widely used in seismic forward modeling, imaging, and inversion (Lu et al., 2023; Zhang et al., 2022). The anisotropy of the media leads to greater complexity in the wave propagation and dispersion characteristics of Scholte waves. In addition, the seabed is typically irregular, particularly in marginal seas (Zhang et al., 2023; Li et al., 2019). Understanding the behavior of Scholte waves in VTI media in the presence of an irregular seabed is crucial for their application in seismic inversion and imaging.

    In this study, we investigated numerical seismic forward modeling using a fluid-solid model in the presence of an irregular seabed. Two-dimensional second-order acoustic and elastic VTI wave equations were applied to fluid and solid media, respectively. A fluid-solid boundary condition was used at the irregular seabed. These equations were converted into curvilinear coordinates to handle an irregular seabed. The comprehensive propagation and dispersion characteristics of Scholte waves in VTI media with irregular seabeds were numerically studied.

    The ocean model consists of acoustic seawater and a solid elastic VTI seabed. The entire seismic wavefield is simulated in the time domain. In the Cartesian coordinate system, the 2D second-order acoustic wave equations in the fluid media are (Levander, 1988)

    ρf2uft2=λfx[ufx+vfz],
    (1a)
    ρf2vft2=λfz[ufx+vfz],
    (1b)

    where uf and vf denote the displacements in the x and z directions in the fluid medium, respectively, ρf(x,z) is the density of seawater, and λf is the Lamé constant of the fluid medium.

    The 2D second-order elastic VTI seismic wave equations in solid media in Cartesian coordinates are (Lan et al., 2011)

    ρs2ust2=x[c11usx+c13vsz]+z[c44usz+c44vsx],
    (2a)
    ρs2vst2=z[c33vsz+c13usx]+x[c44vsx+c44usz],
    (2b)

    where us and vs represent the displacements along the x and z directions in the solid media, respectively; ρs(x, z) denotes the density in the solid media; c11, c13, c33, and c44 are the anisotropic parameters of the VTI media.

    At the interface between the fluid and solid, seismic waves are converted into elastic waves and travel through the solid material. To connect these two wavefields, a fluid-solid coupling boundary condition was used for the uneven seafloor. On the seabed, the normal components of the traction and particle velocity are consistent, whereas the tangential component is discontinuous (Qu et al., 2018; Zhang, 2004).

    Boundary-conforming grids have been utilized in numerical simulations to handle irregular seabeds (Thompson et al., 1985). The conversion of the Cartesian physical space (x and z) into the curvilinear computational space (q and r) was utilized for the wavefield simulation (Rao and Wang, 2018; Hvid, 1994). Spatial derivatives in Cartesian coordinates can be calculated using chain rules (Liu et al., 2023).

    x=qxq+rxr,
    (3a)
    z=qzq+rzr,
    (3b)

    where, qz, qx, rz, and rx denote the partial derivatives of q and r with respect to z and x, respectively.

    Using the coordinate conversion equations, the seismic wave equations in the elastic VTI medium in curvilinear coordinates can be written as

    ρs2ust2=q{qx[c11(qxqus+rxrus)+c13(qzqvs+rzrvs)]+qz[c44(qxqvs+rxrvs)+c44(qzqus+rzrus)]}+r{rx[c11(qxqus+rxrus)+c13(qzqvs+rzrvs)]+rz[c44(qxqvs+rxrvs)+c44(qzqus+rzrus)]},
    (4a)
    ρs2vst2=q{qz[c33(qzqvs+rzrvs)+c13(qxqus+rxrus)]+qx[c44(qxqvs+rxrvs)+c44(qzqus+rzrus)]}+r{rz[c33(qzqvs+rzrvs)+c13(qxqus+rxrus)]+rx[c44(qxqvs+rxrvs)+c44(qzqus+rzrus)]}.
    (4b)

    Figure 1 shows a sketch of the wavefield simulation of an ocean with an irregular seabed. To suppress artificial reflections caused by truncated boundaries, a convolutional perfectly matched layer (CPML) absorbing boundary was utilized at the left, right, and bottom edges of the model. Memory variables were incorporated into the seismic wave equations for the CPML implementation.

    Figure  1.  Sketch of wavefield simulation in ocean with irregular seabed.

    The characteristics of Scholte waves in VTI media with irregular seabeds were studied through seismic forward modeling using two models. The sizes of Models 1 and 2 were both 2.5 × 3.5 km and were discretized using 500 × 350 grid nodes with a grid interval of 5 m. The model parameters are listed in Table 1. In the numerical experiment, a Ricker wavelet was applied as the source signature in the vertical direction with a peak frequency of f0=15Hz. In the wavefield simulation, the sampling interval was 1 ms. The Thompson parameters of the VTI media were given as ε=0.2 and δ=0.5 for both models. High-resolution Radon transformation was applied to the shot records to observe the dispersion characteristics of the Scholte waves.

    Table  1.  Parameters for models
    Layer vp (m/s) vs (m/s) ρ (kg/m3)
    Water 1 500 0 1 025
    Model 1 3 000 1 800 2 000
    Model 2 2 000 800 1 900
     | Show Table
    DownLoad: CSV

    Figure 2 shows the snapshots of displacement in the vertical plane in Model 1 with VTI media at 0.6 and 0.8 s, respectively. Scholte waves propagate along irregular seabeds. Figure 3 shows the shot records of the horizontal and vertical components of the VTI media. Dispersed Scholte waves were mainly observed in vertical shot recordings. Figure 4 compares the dispersion spectra of the horizontal and vertical components in Model 1 with elastic and VTI media. The fundamental and higher modes of the Scholte waves were observed in the vertical and horizontal components, respectively. Anisotropy had little effect on the dispersion spectra of the models with high velocities.

    Figure  2.  Snapshots of displacement in the vertical plane in Model 1 with VTI media at 0.6 s (a) and 0.8 s (b).
    Figure  3.  Shot records of horizontal (a) and vertical (b) components in Model 1 with VTI media.
    Figure  4.  Dispersion spectra of horizontal and vertical components in Model 1 with elastic (a) and (b) and VTI (c) and (d) media.

    Figure 5 shows the snapshots of displacement in the vertical plane in Model 2 with VTI media at 0.6 and 0.8 s, respectively. Figure 6 shows the shot records of the horizontal and vertical components of the VTI media. Dispersed Scholte waves were observed in both shot records. Figure 7 compares the dispersion spectra of the horizontal and vertical components in Model 2 with those of the elastic and VTI media, respectively. One can observe that the anisotropy of the solid medium changed the energy distribution in the dispersion spectra of both components. In particular, the higher modes were difficult to identify, as shown in Figure 4c. Compared with the dispersion spectra of Model 1, the VTI medium caused more variations in the dispersion curves of the Scholte waves in Model 2.

    Figure  5.  Snapshots of displacement in the vertical plane in Model 2 with VTI media at 0.6 s (a) and 0.8 s (b).
    Figure  6.  Shot records of horizontal (a) and vertical (b) components in Model 2 with VTI media.
    Figure  7.  Dispersion spectra of horizontal and vertical components in Model 2 with elastic (a) and (b) and VTI (c) and (d) media.

    The ocean seismic wavefield was simulated numerically considering the effects of an irregular solid VTI seabed. The fluid-solid boundary condition, as well as the acoustic and elastic seismic wave equations, were successfully converted from Cartesian to curvilinear coordinates. In curvilinear coordinates, these equations are discretized using finite-difference operators. The propagation and dispersion characteristics of Scholte waves in VTI media with irregular seabeds were studied using numerical simulations. The dispersion phenomenon can be observed in the snapshots and shot gathers. A Scholte wave propagated along an irregular seabed. Compared to the dispersion spectra of the model with high velocities, the anisotropy of the media caused more variation in the dispersion curves in the model with low velocities.

    ACKNOWLEDGMENTS: This research was supported by the research project of the China National Petroleum Corporation (No. 2021ZG02); National Natural Science Foundation of China (Nos. 42004091, 62127815, 42150201); Beijing Natural Science Foundation (No. 8222030). This work was sponsored by the Chinese "111" project (No. B20011). The final publication is available at Springer via https://doi.org/10.1007/s12583-024-1965-0.
  • Appelö, D., Petersson, N. A., 2009. A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces. Communications in Computational Physics, 5: 84–107
    Bohlen, T., Kugler, S., Klein, G., et al., 2004. 1.5D Inversion of Lateral Variation of Scholte-Wave Dispersion. Geophysics, 69(2): 330–344. https://doi.org/10.1190/1.1707052
    Buchen, P. W., Ben-Hador, R., 1996. Free-Mode Surface-Wave Computations. Geophysical Journal International, 124(3): 869–887. https://doi.org/10.1111/j.1365-246x.1996.tb05642.x
    Carcione, J. M., Helle, H. B., 2004. The Physics and Simulation of Wave Propagation at the Ocean Bottom. Geophysics, 69(3): 825–839. https://doi.org/10.1190/1.1759469
    Choi, Y., Min, D. J., Shin, C., 2008. Two-Dimensional Waveform Inversion of Multi-Component Data in Acoustic-Elastic Coupled Media. Geophysical Prospecting, 56(6): 863–881. https://doi.org/10.1111/j.1365-2478.2008.00735.x
    Hvid, S. L., 1994. Three Dimensional Algebraic Grid Generation: [Dissertation]. Technical University of Denmark, Copenhagen
    Huang, P. D., Lu, J., Wang, Y., 2022. Second-Order Approximate Reflection Coefficients of Vertical Transversely Isotropic Thin Beds. Acta Geophysica, 70(3): 1155–1169. https://doi.org/10.1007/s11600-022-00758-y
    Komatitsch, D., Barnes, C., Tromp, J., 2000. Wave Propagation near a Fluid-Solid Interface: A Spectral-Element Approach. Geophysics, 65(2): 623–631. https://doi.org/10.1190/1.1444758
    Klein, G., Bohlen, T., Theilen, F., et al., 2005. Acquisition and Inversion of Dispersive Seismic Waves in Shallow Marine Environments. Marine Geophysical Researches, 26(2): 287–315. https://doi.org/10.1007/s11001-005-3725-6
    Kugler, S., Bohlen, T., Bussat, S., et al., 2005. Variability of Scholte-Wave Dispersion in Shallow-Water Marine Sediments. Journal of Environmental and Engineering Geophysics, 10(2): 203–218. https://doi.org/10.2113/jeeg10.2.203
    Lan, H., Zhang, Z., 2011. Three-Dimensional Wave-Field Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface. Bulletin of the Seismological Society of America, 101(3): 1354–1370. https://doi.org/10.1785/0120100194
    Li, Q. Y., Wu, G. C., Wu, J. L., et al., 2019. Finite Difference Seismic Forward Modeling Method for Fluid-Solid Coupled Media with Irregular Seabed Interface. Journal of Geophysics and Engineering, 16(1): 198–214. https://doi.org/10.1093/jge/gxy017
    Liu, X. B., 2023. Modeling Seismic Waves in Ocean with the Presence of Irregular Seabed and Rough Sea Surface. Journal of Geophysics and Engineering, 20(1): 49–66. https://doi.org/10.1093/jge/gxac093
    Lu, J., Ma, Z. J., Xiong, S., et al., 2023. Imaging of 3-D Three-Component Vertical Seismic Profile Data Based on Horizontally Layered Azimuthally Anisotropic Media. IEEE Transactions on Geoscience and Remote Sensing, 61: 3317140. https://doi.org/10.1109/tgrs.2023.3317140
    McMechan, G. A., Yedlin, M. J., 1981. Analysis of Dispersive Waves by Wave Field Transformation. Geophysics, 46(6): 869–874. https://doi.org/10.1190/1.1441225
    Nilsson, S., Petersson, N. A., Sjögreen, B., et al., 2007. Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation. SIAM Journal on Numerical Analysis, 45(5): 1902–1936. https://doi.org/10.1137/060663520
    Qu, Y. M., Sun, J. Z., Li, Z. C., et al., 2018. Forward Modeling of Ocean-Bottom Cable Data and Wave-Mode Separation in Fluid-Solid Elastic Media with Irregular Seabed. Applied Geophysics, 15(3): 432–447. https://doi.org/10.1007/s11770-018-0699-0
    Randall, C., 1983. Numerical Simulation of Acoustic Propagation at a Fluid-Solid Interface. The Journal of the Acoustical Society of America, 74(S1): S87–S88. https://doi.org/10.1121/1.2021196
    Rao, Y., Wang, Y. H., 2018. Seismic Waveform Simulation for Models with Fluctuating Interfaces. Scientific Reports, 8(1): 3098. https://doi.org/10.1038/s41598-018-20992-z
    Sofronov, I., Zaitsev, N., Dovgilovich, L., 2015. Multi-Block Finite-Difference Method for 3D Elastodynamic Simulations in Anisotropic Subhorizontally Layered Media. Geophysical Prospecting, 63(5): 1142–1160. https://doi.org/10.1111/1365-2478.12231
    Sun, Y. C., Zhang, W., Xu, J. K., et al., 2017. Numerical Simulation of 2-D Seismic Wave Propagation in the Presence of a Topographic Fluid-Solid Interface at the Sea Bottom by the Curvilinear Grid Finite-Difference Method. Geophysical Journal International, 210(3): 1721–1738. https://doi.org/10.1093/gji/ggx257
    Thompson, J. F., Warsi, Z. U. A., Mastin, C. W., 1985. Numerical Grid Generation Foundations and Applications: North Hollad Publishing Company, New York
    van Vossen, R., Robertsson, J. O. A., Chapman, C. H., 2002. Finite-Difference Modeling of Wave Propagation in a Fluid-Solid Configuration. Geophysics, 67(2): 618–624. https://doi.org/10.1190/1.1468623
    Wang, Y., Zhou, H., Yuan, S. Y., et al., 2017. A Fourth Order Accuracy Summation-by-Parts Finite Difference Scheme for Acoustic Reverse Time Migration in Boundary-Conforming Grids. Journal of Applied Geophysics, 136: 498–512. https://doi.org/10.1016/j.jappgeo.2016.12.002
    Wu, H., Shao, G. Z., Li, Q. C., 2018. Study of Scholte Wave Dispersion Curves and Modal Energy Distribution Using a Wavefield Numerical Simulation Method. Exploration Geophysics, 49(3): 372–385. https://doi.org/10.1071/eg16048
    Yilmaz, O., 1987. Seismic Data Processing, Investigations in Geophysics. Society of Exploration Geophysicists, Houston
    Zhang, J. F., 2004. Wave Propagation across Fluid-Solid Interfaces: A Grid Method Approach. Geophysical Journal International, 159(1): 240–252. https://doi.org/10.1111/j.1365-246x.2004.02372.x
    Zhang, Z., Lu, J., Zhang, X. Y., et al., 2022. Approximation of P-, S1-, and S2-Wave Reflection Coefficients for Orthorhombic Media. Geophysics, 87(4): C63–C76. https://doi.org/10.1190/geo2021-0400.1
    Zhang, H. M., Lu, J., Wang, Y., et al., 2023. PP- and PS-Wave Migration of OBN Seismic Data from Rugged Seabeds. Acta Geophysica, online first. https://doi.org/10.1007/s11600-023-01203-4
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