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

INTRODUCTION

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.

WAVE EQUATIONS OF SOLID VTI MEDIA WITH ROUGH SEABED

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)

where ${u}^{f}$ and ${v}^{f}$ denote the displacements in the $x$ and $z$ directions in the fluid medium, respectively, ${\rho }_{f}\left(x, z\right)$ is the density of seawater, and ${\lambda }_{f}$ 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)

where ${u}^{s}$ and ${v}^{s}$ represent the displacements along the $x$ and $z$ directions in the solid media, respectively; ρ_s(x, z) denotes the density in the solid media; ${c}_{11}$, ${c}_{13}$, ${c}_{33}$, and ${c}_{44}$ 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).

where, ${q}_{z}$, ${q}_{x}$, ${r}_{z}$, and ${r}_{x}$ 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

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.

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CHARACTERISTICS OF SCHOLTE WAVE

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 $\times$ 3.5 km and were discretized using 500 $\times$ 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 ${f}_{0}=15\mathrm{ }\mathrm{H}\mathrm{z}$. In the wavefield simulation, the sampling interval was 1 ms. The Thompson parameters of the VTI media were given as $\varepsilon=0.2$ and $\delta =-0.5$ 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	${v}_{p}$ (m/s)	${v}_{s}$ (m/s)	$\rho$ (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

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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).

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Figure  3.  Shot records of horizontal (a) and vertical (b) components in Model 1 with VTI media.

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Figure  4.  Dispersion spectra of horizontal and vertical components in Model 1 with elastic (a) and (b) and VTI (c) and (d) media.

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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).

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Figure  6.  Shot records of horizontal (a) and vertical (b) components in Model 2 with VTI media.

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Figure  7.  Dispersion spectra of horizontal and vertical components in Model 2 with elastic (a) and (b) and VTI (c) and (d) media.

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CONCLUSIONS

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.