Abstract Few problems in elastodynamics have a closed-form analytical solution. The others can be investigated with semianalytical methods, but in general one is not sure whether these methods give reliable solutions. The same happens with numerical techniques: for instance, finite difference methods solve, in principle, any complex problem, including those with arbitrary inhomogeneities and boundary conditions. However, there is no way to verify the quantitative correctness of the solutions. The major problems are stability with respect to material properties, numerical dispersion, and the treatment of boundary conditions. In practice, these problems may produce inaccurate solutions. In this paper, the study of complex problems with two different numerical grid techniques in order to cross-check the solutions is proposed. Interface waves, in particular, are emphasized, since they pose the major difficulties due to the need to implement boundary conditions. The first method is based on global differential operators where the solution is expanded in terms of the Fourier basis and Chebyshev polynomials, while the second is the spectral element method, an extension of the finite element method that uses Chebyshev polynomials as interpolating functions. Both methods have spectral accuracy up to approximately the Nyquist wave number of the grid. Moreover, both methods implement the boundary conditions in a natural way, particularly the spectra element algorithm. We first solve Lamb’s problem and compare numerical and analytical solutions; then, the problem of dispersed Rayleigh waves, and finally, the two-quarter space problem. We show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist. The combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics.

Abstract Transverse isotropy with a vertical symmetry axis (VTI media) is the most common anisotropic model for sedimentary basins. Here, we apply P-wave processing algorithms developed for VTI media to a 2-D synthetic data set generated by a finite difference code. The model, typical for the Gulf of Mexico, has a moderate structural complexity and includes a salt body and a dipping fault plane. Using the Alkhalifah-Tsvankin dip-moveout (DMO) inversion method, we estimate the anisotropic coefficient η responsible for the dip dependence of P-wave NMO velocity in VTI media. In combination with the normal-moveout (NMO) velocity from a horizontal reflector [V nmo (0), the argument “0” refers to reflector dip], η is sufficient for performing all P-wave time-processing steps, including NMO and DMO corrections, prestack and poststack time migration. The NMO (stacking) velocities needed to determine V nmo (0) and η are picked from conventional semblance velocity panels for reflections from subhorizontal interfaces, the dipping fault plane and the flank of the salt body. To mitigate the instability in the interval parameter estimation, the dependence of V nmo (0) and η on the vertical reflection time is approximated by Chebyshev polynomials with the coefficients found by “global” fitting of all velocity picks. We perform prestack depth migration for the reconstructed anisotropic model and two isotropic models with different choices of the velocity field. The anisotropic migration result has a good overall quality, but reflectors are mispositioned in depth because the vertical velocity for this model cannot be obtained from surface -wave data alone. The isotropic migrated section with the NMO velocity V nmo (0) substituted for the isotropic velocity also has the wrong depth scale and is somewhat inferior to the anisotropic result in the focusing of dipping events. Still, the image distortions are not significant because the parameter η, which controls NMO velocity for dipping reflectors, is rather small (the average value of η is about 0.05). In contrast, the isotropic section migrated with the vertical velocity has a poor quality (although the depth of the subhorizontal reflectors is correct) due to the fact that in VTI media V o can be used to stack neither dipping nor horizontal events. The difference between v o and the zero-dip stacking velocity V nmo (0) is determined by the anisotropic coefficient δ, which is greater than η in our model (on average δ ≈ 0.1).