Haiqiang Lan

Three-Dimensional Wave-Field Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface

Modeling of seismic-wave propagation in anisotropic medium with irregular topography is beneficial to interpret seismic data acquired by active and pas- sive source seismology conducted in areas of interest such as mountain ranges and basins. The major challenge in this context is the difficulty in tackling the irregular free-surface boundary condition in a Cartesian coordinate system. To implement sur- facetopography,weusetheboundary-conforminggridandmaparectangulargridonto a curved grid. We use a stable and explicit second-order accurate finite-difference scheme to discretize the elastic wave equations (in a curvilinear coordinate system) in a 3D heterogeneous transversely isotropic medium. The free-surface boundary con- ditions are accurately applied by introducing a discretization that uses boundary- modified difference operators for the mixed derivatives in the governing equations. The accuracy of the proposed method is checked by comparing the numerical results obtained by the trial algorithm with the analytical solutions of the Lambs problem, for anisotropicmediumandatransverselyisotropicmediumwithaverticalsymmetryaxis, respectively. Efficiency tests performed by different numerical experiments illustrate clearlytheinfluenceofanirregular(nonflat)freesurfaceonseismic-wavepropagation.

Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation

The finite-difference (FD) method is a powerful tool in seismic wave field modelling for understanding seismic wave propagation in the Earth’s interior and interpreting the real seismic data. The accuracy of FD modelling partly depends on the implementation of the free-surface (i.e. traction-free) condition. In the past 40 years, at least six kinds of free-surface boundary condition approximate schemes (such as one-sided, centred finite-difference, composed, new composed, implicit and boundary-modified approximations) have been developed in FD secondorder elastodynamic simulation. Herein we simulate seismic wave fields in homogeneous and lateral heterogeneous models using these free-surface boundary condition approximate schemes and evaluate their stability and applicability by comparing with corresponding analytical solutions, and then quantitatively evaluate the accuracies of different approximate schemes from the misfit of the amplitude and phase between the numerical and analytical results. Our results confirm that the composed scheme becomes unstable for the Vs/Vp ratio less than 0.57, and suggest that (1) the one-sided scheme is only accurate to first order and therefore introduces serious errors for the shorter wavelengths, other schemes are all of second-order precision; (2) the new composed, implicit and boundary-modified schemes are stable even when the Vs/Vp ratio is less than 0.2; (3) the implicit and boundary-modified schemes are able to deal with laterally varying (heterogeneous) free surface; (4) in the corresponding stability range, the one-sided scheme shows remarkable errors in both phase and amplitude compared to analytical solution (which means larger errors in travel-time and reflection strength), the other five approximate schemes show better performance in travel-time (phase) than strength (amplitude).

A High-Order Fast-Sweeping Scheme for Calculating First-Arrival Travel Times with an Irregular Surface

Abstract Topography‐dependent eikonal equation (TDEE) formulated in a curvilinear coordinate system has been recently established and is effective for calculating first‐arrival travel times in an Earth model with an irregular surface. In previous work, the Lax–Friedrichs sweeping scheme used to approximate the TDEE viscosity solutions was only first‐order accurate. We present a high‐order fast‐sweeping scheme to solve the TDEE with the aim of achieving high‐order accuracy in the travel‐time calculation. The scheme takes advantage of high‐order weighted essentially nonoscillatory (WENO) derivative approximations, monotone numerical Hamiltonians, and Gauss Seidel iterations with alternating‐direction sweepings. It incorporates high‐order approximations of the derivatives into the numerical representation of the Hamiltonian such that the resulting numerical scheme is formally high‐order accurate and inherits fast convergence from the alternating sweeping strategy. Extensive numerical examples are presented to verify its efficiency, convergence, and high‐order accuracy.

Seismic wavefield modeling in media with fluid-filled fractures and surface topography

We present a finite difference (FD) method for the simulation of seismic wave fields in fractured medium with an irregular (non-flat) free surface which is beneficial for interpreting exploration data acquired in mountainous regions. Fractures are introduced through the Coates-Schoenberg approach into the FD scheme which leads to local anisotropic properties of the media where fractures are embedded. To implement surface topography, we take advantage of the boundary-conforming grid and map a rectangular grid onto a curved one. We use a stable and explicit second-order accurate finite difference scheme to discretize the elastic wave equations (in a curvilinear coordinate system) in a 2D heterogeneous transversely isotropic medium with a horizontal axis of symmetry (HTI). Efficiency tests performed by different numerical experiments clearly illustrate the influence of an irregular free surface on seismic wave propagation in fractured media which may be significant to mountain seismic exploration. The tests also illustrate that the scattered waves induced by the tips of the fracture are re-scattered by the features of the free surface topography. The scattered waves provoked by the topography are re-scattered by the fractures, especially Rayleigh wave scattering whose amplitudes are much larger than others and making it very difficult to identify effective information from the fractures.

Application of a perfectly matched layer in seismic wavefield simulation with an irregular free surface

Recently, an effective and powerful approach for simulating seismic wave propagation in elastic media with an irregular free surface was proposed. However, in previous studies, researchers used the periodic condition and/or sponge boundary condition to attenuate artificial reflections at boundaries of a computational domain. As demonstrated in many literatures, either the periodic condition or sponge boundary condition is simple but much less effective than the well-known perfectly matched layer boundary condition. In view of this, we intend to introduce a perfectly matched layer to simulate seismic wavefields in unbounded models with an irregular free surface. We first incorporate a perfectly matched layer into wave equations formulated in a frequency domain in Cartesian coordinates. We then transform them back into a time domain through inverse Fourier transformation. Afterwards, we use a boundary-conforming grid and map a rectangular grid onto a curved one, which allows us to transform the equations and free surface boundary conditions from Cartesian coordinates to curvilinear coordinates. As numerical examples show, if free surface boundary conditions are imposed at the top border of a model, then it should also be incorporated into the perfectly matched layer imposed at the top-left and top- right corners of a 2D model where the free surface boundary conditions and perfectly matched layer encounter; otherwise, reflections will occur at the intersections of the free surface and the perfectly matched layer, which is confirmed in this paper. So, by replacing normal second derivatives in wave equations in curvilinear coordinates with free surface boundary conditions, we successfully implement the free surface boundary conditions into the perfectly matched layer at the top-left and top-right corners of a 2D model at the surface. A number of numerical examples show that the perfectly matched layer constructed in this study is effective in simulating wave propagation in unbounded media and the algorithm for implementation of the perfectly matched layer and free surface boundary conditions is stable for long-time wavefield simulation on models with an irregular free surface.

An upwind fast sweeping scheme for calculating seismic wave first-arrival travel times for models with an irregular free surface

The topography-dependent eikonal equation formulated in a curvilinear coordinate system has recently been established and revealed as being effective in calculating first-arrival travel times of seismic waves in an Earth model with an irregular free surface. The Lax–Friedrichs sweeping scheme, widely used in previous studies as for approximating the topography-dependent eikonal equation viscosity solutions, is more dissipative and needs a much higher number of iterations to converge. Furthermore, the required number of iterations grows with the grid refinement and results in heavy computation in dense grids, which hampers the application of the Lax–Friedrichs sweeping scheme to seismic wave travel-time calculation and highresolution imaging. In this paper, we introduce a new upwind fast sweeping solver by discretising the Legendre transform of the numerical Hamiltonian of the topographydependent eikonal equation using an explicit formula. The minimisation related to the Legendre transform in the sweeping scheme is solved analytically, which proved to be much more efficient than the Lax–Friedrichs algorithm in solving the topographydependent eikonal equation. Several numerical experiments demonstrate that the new upwind fast sweeping method converges and achieves much better accuracy after a finite number of iterations, independently of the mesh size, which makes it an efficient and robust tool for calculating travel times in the presence of a non-flat free surface.

Simulating Seismic Wave Propagation in Viscoelastic Media with an Irregular Free Surface

In seismic numerical simulations of wave propagation, it is very important for us to consider surface topography and attenuation, which both have large effects (e.g., wave diffractions, conversion, amplitude/phase change) on seismic imaging and inversion. An irregular free surface provides significant information for interpreting the characteristics of seismic wave propagation in areas with rugged or rapidly varying topography, and viscoelastic media are a better representation of the earth’s properties than acoustic/elastic media. In this study, we develop an approach for seismic wavefield simulation in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming grid method, the 2D time-domain second-order viscoelastic isotropic equations and irregular free surface boundary conditions are transferred from a Cartesian coordinate system to a curvilinear coordinate system. Finite difference operators with second-order accuracy are applied to discretize the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, we select the convolutional perfectly matched layer boundary condition in order to effectively suppress artificial reflections from the edges of the model. The snapshot and seismogram results from numerical tests show that our algorithm successfully simulates seismic wavefields (e.g., P-wave, Rayleigh wave and converted waves) in viscoelastic isotropic media with an irregular free surface.