Wavenumber sampling strategies for 2.5-D frequency-domain seismic wave modelling in general anisotropic media

The computational efficiency of 2.5-D seismic wave modelling in the frequency domain depends largely on the wavenumber sampling strategy used. This involves determining the wavenumber range and the number of the sampling points, and overcoming the singularities in the wavenumber spectrum when taking the inverse Fourier transform to yield the frequency-domain wave solution. In this paper, we employ our newly developed Gaussian quadrature grid numerical modelling method and extensively investigate the wavenumber sampling strategies for 2.5-D frequency-domain seismic wave modelling in heterogeneous, anisotropic media. We show analytically and numerically that the various components of the Green's function tensor wavenumber-domain solutions have symmetric or antisymmetric properties and other characteristics, all of which can be fully used to construct effective and efficient sampling strategies for the inverse Fourier transform. We demonstrate two sampling schemes-called irregular and regular sampling strategies for the 2.5-D frequency-domain seismic wave modelling technique. The numerical results, which involve calibrations against analytic solutions, comparison of the different wavenumber sampling strategies and validation by means of 3-D numerical solutions, show that the two sampling strategies are both suitable for efficiently computing the 3-D frequency-domain wavefield in 2-D heterogeneous, anisotropic media. These strategies depend on the given frequency, elastic model parameters and maximum wavelength and the offset distance from the sourc

Reply to comment by Jonás D. De Basabe on ‘3-D frequency-domain seismic wave modelling in heterogeneous, anisotropic media using a Gaussian quadrature grid approach'

In his comment, De Basabe criticises our paper ignoring the advantages of unstructured element mesh used in the finite element method, and argues that the Gaussian quadrature grid (GQG) approach is limited to a homogenous or constant layered geological model and does not have the spectral accuracy. In this reply, we give our response to his criticism and comment, and further clarify the accuracy and capability of the GQG approac

Non-linear traveltime inversion for 3-D seismic tomography in strongly anisotropic media

We have developed two, new non-linear traveltime inversion schemes for 3-D seismic tomography in anisotropic media. They differ from the traditional linearized inversion approach and offer five significant improvements: (1) they are based on an alternative form of the first-order traveltime perturbation equation, derived so as to simplify the inversion formulae and overcome the quasi shear wave singularity problem; (2) robust 3-D ray tracing is employed which enables the simultaneous computation of the first-arrival traveltimes and ray paths for the three body waves (qP, qS1 and qS2) in arbitrary anisotropic media; (3) the Jacobian matrix used in the update is based on an efficient computation for a 3-D anisotropic model, so that the inversion is applicable to both weakly and strongly anisotropic situations, unlike most previous approaches which assume weak anisotropy; (4) a local-search, constrained minimization is applied to the non-linear inversion which makes anisotropic tomographic imaging an iterative procedure; (5) there is an option to invert for the elastic moduli directly or the Thomsen parameters directly in heterogenous, tilted transversely isotropic media, using any source-receiver recording geometry. We have examined the imaging capability of the non-linear solver with individual body-wave modes using a 3-D synthetic anisotropic model incorporating two targets, a ‘high velocity' and a ‘low velocity' anomaly, embedded in a titled transversely isotropic medium. The model is illuminated by means of azimuthal VSP and crosshole measurements. The experimental results show that the two non-linear inversion schemes successfully image the ‘targets' and yield satisfactory 3-D tomograms of the elastic moduli and the Thomsen parameter

Reply to comment by Jonas D. De Basabe on '3-D frequency-domain seismic wave modelling in heterogeneous, anisotropic media using a Gaussian quadrature grid approach'

ISSN:0956-540XISSN:1365-246

Traveltime approximation for strongly anisotropic media using the homotopy analysis method

Traveltime approximation plays an important role in seismic data processing, for example, anisotropic parameter estimation and seismic imaging. By exploiting seismic traveltimes, it is possible to improve the accuracy of anisotropic parameter estimation and the resolution of seismic imaging. Conventionally, the traveltime approximations in anisotropic media are obtained by expanding the anisotropic eikonal equation in terms of the anisotropic parameters and the elliptically anisotropic eikonal equation based on perturbation theory. Such an expansion assumes a small perturbation and weak anisotropy. In a realistic medium, however, the assumption of small perturbation likely breaks down. We present a retrieved zero-order deformation equation that creates a map from the anisotropic eikonal equation to a linearized partial differential equation system based on the homotopy analysis method. By choosing the linear and nonlinear operators in the retrieved zero-order deformation equation, we develop new traveltime approximations that allow us to compute the traveltimes for a medium of arbitrarily strength anisotropy. A comparison of the traveltimes and their errors from the homotopy analysis method and from the perturbation method suggests that the traveltime approximations provide a more reliable result in strongly anisotropic media.publishedVersio

Resistivity inversion in 2-D anisotropic media: numerical experiments

Many rocks and layered/fractured sequences have a clearly expressed electrical anisotropy although it is rare in practice to incorporate anisotropy into resistivity inversion. In this contribution, we present a series of 2.5-D synthetic inversion experiments for various electrode configurations and 2-D anisotropic models. We examine and compare the image reconstructions obtained using the correct anisotropic inversion code with those obtained using the false but widely used isotropic assumption. Superior reconstruction in terms of reduced data misfit, true anomaly shape and position, and anisotropic background parameters were obtained when the correct anisotropic assumption was employed for medium to high coefficients of anisotropy. However, for low coefficient values the isotropic assumption produced better-quality results. When an erroneous isotropic inversion is performed on medium to high level anisotropic data, the images are dominated by patterns of banded artefacts and high data misfits. Various pole-pole, pole-dipole and dipole-dipole data sets were investigated and evaluated for the accuracy of the inversion result. The eigenvalue spectra of the pseudo-Hessian matrix and the formal resolution matrix were also computed to determine the information content and goodness of the results. We also present a data selection strategy based on high sensitivity measurements which drastically reduces the number of data to be inverted but still produces comparable results to that of the comprehensive data set. Inversion was carried out using transversely isotropic model parameters described in two different co-ordinate frames for the conductivity tensor, namely Cartesian versus natural or eigenframe. The Cartesian frame provided a more stable inversion product. This can be simply explained from inspection of the eigenspectra of the pseudo-Hessian matrix for the two model description

Scattering of plane transverse waves by spherical inclusions in a poroelastic medium

The scattering of plane transverse waves by a spherical inclusion embedded in an infinite poroelastic medium is treated for the first time in this paper. The vector displacement wave equations of Biot's theory are solved as an infinite series of vector spherical harmonics for the case of a plane S-wave impinging from a porous medium onto a spherical inclusion which itself is assumed to be another porous medium. Based on the single spherical scattering theory and dynamic composite elastic medium theory, the non-self-consistent shear wavenumber is derived for a porous rock having numerous spherical inclusions of another medium. The frequency dependences of the shear wave velocity and the shear wave attenuation have been calculated for both the patchy saturation model (inclusions having the same solid frame as the host but with a different pore fluid from the host medium) and the double porosity model (inclusions having a different solid frame than the host but the same pore fluid as the host medium) with dilute concentrations of identical inclusions. Unlike the case of incident P-wave scattering, we show that although the fluid and the heterogeneity of the rock determine the shear wave velocity of the composite, the attenuation of the shear wave caused by scattering is actually contributed by the heterogeneity of the rock for spherical inclusions. The scattering of incident shear waves in the patchy saturation model is quite different from that of the double porosity model. For the patchy saturation model, the gas inclusions do not significantly affect the shear wave dispersion characteristic of the water-filled host medium. However, the softer inclusion with higher porosity in the double porosity model can cause significant shear wave scattering attenuation which occurs at a frequency at which the wavelength of the shear wave is approximately equal to the characteristic size of the inclusion and depends on the volume fraction. Compared with analytic formulae for the low frequency limit of the shear velocity, our scattering model yields discrepancies within 4.0 per cent. All calculated shear velocities of the composite medium with dilute inclusion concentrations approach the high frequency limit of the host materia

Transient solution for poro-viscoacoustic wave propagation in double porosity media and its limitations

The analytical transient acoustic solution and dispersion characteristics for the double-porosity model are obtained over the whole frequency range for a homogeneous medium. The solution is also obtained by approximating the double porosity model with a uniform poro-viscoacoustic model and then constructing the transient response. The comparison between the results of the two models shows the likely validity and limitations of numerical solutions using a poro-viscoacoustic model to represent a double porosity medium in the heterogeneous case. Our calculations show that the dissipation by local mesoscopic flow of the double porosity model is very hard to fit over the entire frequency range by a single Zener element. However, since seismic exploration is normally restricted to a fairly narrow frequency band, this means that for frequency-dependent material properties, such as attenuation, the values around the centre frequency of the source will primarily determine the wave propagation characteristics. We choose the relaxation function that just approximates the dispersion behaviour of the double porosity model around the source centre frequency. It is shown that if the frequency is much lower than the peak attenuation frequency of the double porosity model, then wave propagation can be well described by the poro-viscoaoustic model with a single Zener element. For most water-filled sandstones having a double porosity structure, this holds true across the seismic frequency range. The transient solution for the heterogeneous double porosity medium is numerically obtained by a time splitting and Fourier pseudo-spectral staggered-grid method. As illustrative examples, the 2-D wavefield in a two-layer, water-saturated double porosity model are approximated by poro-viscoacoustic and poro-viscoelastic methods, respectivel

Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

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 ω-k y 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 spike-like behaviour still occurs and can be damped out without distorting the spectrum elsewhere, through the introduction of slight attenuatio