Wavenumber sampling issues in 2.5D frequency domain seismic modelling

Abstract

There are several important wavenumber sampling issues associated with 2.5D seismic modelling in the frequency domain, which need careful attention if accurate results are to be obtained. At certain critical wavenumbers there exist rapid disruptions in the mainly smooth oscillatory spectra. The amplitudes of these disruptions can be very large, and this affects the accuracy of the inverse Fourier transformed frequency-space domain solution. In anisotropic elastic media there are critical wavenumbers associated with each wave mode—the quasi-P (qP) wave, and the two quasi-shear (qS1 and qS2) waves. A small wavenumber sampling interval is desirable in order to capture the highly oscillatory nature of the wavenumber spectrum, especially at increasing distance from the source. Obviously a small wavenumber sampling interval adds greatly to the computational effort because a 2D problem must be solved for every wavenumber and every frequency. The discretisation should be carried out up to some maximum wavenumber, beyond which the field becomes evanescent (exponentially decaying or diffusive). For receivers close to the source, activity persists beyond the critical wavenumber associated with the minimum shear wave velocity in the model. Fortunately, for receivers well removed from the source, the contribution from the evanescent energy is negligible and so there is no need to sample beyond this critical wavenumber. Sampling at Gauss–Legendre spacings is a satisfactory approach for acoustic media, but it is not practical in elastic media due to the difficulty of partitioning the integration around the different critical wavenumbers. We found to our surprise that in transversely isotropic media, the critical wavenumbers are independent of wave direction, but always occur at those wavenumbers corresponding to the maximum phase velocities of the three wave modes (qP, qS1 and qS2), which depend only on the elastic constants and the density. Additionally, we have observed that intermediate layers between source and receiver can filter out to a large degree, the sharp irregularities around the critical wavenumbers in the x–ky spectra. We have found that, using the spectral element method, the singularities (poles) at the critical wavenumbers which exist with analytic solutions, do not arise. However, the troublesome spikelike behaviour still occurs and can be damped out without distorting the spectrum elsewhere, through the introduction of slight attenuation.