Frequency‐domain acoustic‐wave modeling and inversion of crosshole data: Part I—2.5-D modeling method

Song, Z. M.; Williamson, Paul

doi:10.1190/1.1443817

Cited by 83 publications

(46 citation statements)

References 4 publications

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“…STOCKWELL (1995) tackled the acoustic problem in the time domain in an approximate way using asymptotic ray theory and restricted himself to in-plane propagation. RANDALL (1991) and OKAMOTO (1994) presented more elaborate and accurate out-of-plane time domain finite difference 2.5D methods for acoustic and elastic waves in isotropic media, whereas SONG and WILLIAMSON (1995) and GREEN-HALGH (1997, 1998) described frequency domain finite difference approaches to the problem in acoustic only media. GREENHALGH (1998a, b, 2006) presented frequency domain finite element acoustic algorithms, while YANG and HUNG (2001) gave a hybrid finite element, infinite element elastic formulation.…”

Section: Introductionmentioning

confidence: 99%

“…But for 2D and 3D models no analytic expressions (infinite integrals) exist and solutions can only be obtained numerically, so a different approach is required. SONG and WILLIAMSON (1995) describe 2.5D numerical modelling for acoustic waves in heterogeneous media but do not mention singularities. To form the inverse spatial Fourier transform, they sample along the wavenumber axis from 0 to slightly beyond the critical wavenumber associated with the P-wave (k c = x/c), where c is the speed of the Pwave near the surface, so they are actually integrating through the singularity.…”

Section: Introductionmentioning

confidence: 99%

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Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

Sinclair

Greenhalgh

Zhou

2011

Pure Appl. Geophys.

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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-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 spikelike behaviour still occurs and can be damped out without distorting the spectrum elsewhere, through the introduction of slight attenuation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

Sinclair

Greenhalgh

Zhou

2011

Pure Appl. Geophys.

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“…Nonlinear seismic diffraction tomography (GAN et al, 1994;GELIUS 1995) and frequency-domain full-waveform inversion (SONG and WILLIAMSON, 1995;PRATT, 1999;GREENHALGH, 1998, 2003;GREENHALGH and ZHOU, 2003;ALI et al, 2008) are two well established techniques for high resolution subsurface imaging. Both are based on a firstorder perturbation equation of the displacement vector due to small changes in the model.…”

Section: Introductionmentioning

confidence: 99%

“…This means that all the model parameters are functions of only the x-and z-coordinates. To process such data or interpret the dynamic characteristics of the data, one may apply a 2.5-D wave modeling technique (SONG and WILLIAMSON, 1995;FURUMURA and TAKENKA, 1996;ZHOU and GREENHALGH, 1998;NOVAIS and SANTOS, 2005;SINCLAIR et al, 2007;ZHOU and GREENHALGH, 2006) or a 2.5-D full-waveform inversion technique (ZHOU and GREENHALGH, 1998;GREENHALGH and ZHOU, 2003). This enables the computation of the 3-D seismic wavefield generated by a point-source in a 2-D geological model.…”

Section: Introductionmentioning

confidence: 99%

Computing the Sensitivity Kernels for 2.5-D Seismic Waveform Inversion in Heterogeneous, Anisotropic Media

Zhou

Greenhalgh

2010

Pure Appl. Geophys.

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2.5-D modeling and inversion techniques are much closer to reality than the simple and traditional 2-D seismic wave modeling and inversion. The sensitivity kernels required in full waveform seismic tomographic inversion are the Fréchet derivatives of the displacement vector with respect to the independent anisotropic model parameters of the subsurface. They give the sensitivity of the seismograms to changes in the model parameters. This paper applies two methods, called 'the perturbation method' and 'the matrix method', to derive the sensitivity kernels for 2.5-D seismic waveform inversion. We show that the two methods yield the same explicit expressions for the Fréchet derivatives using a constantblock model parameterization, and are available for both the linesource (2-D) and the point-source (2.5-D) cases. The method involves two Green's function vectors and their gradients, as well as the derivatives of the elastic modulus tensor with respect to the independent model parameters. The two Green's function vectors are the responses of the displacement vector to the two directed unit vectors located at the source and geophone positions, respectively; they can be generally obtained by numerical methods. The gradients of the Green's function vectors may be approximated in the same manner as the differential computations in the forward modeling. The derivatives of the elastic modulus tensor with respect to the independent model parameters can be obtained analytically, dependent on the class of medium anisotropy. Explicit expressions are given for two special cases-isotropic and tilted transversely isotropic (TTI) media. Numerical examples are given for the latter case, which involves five independent elastic moduli (or Thomsen parameters) plus one angle defining the symmetry axis.

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“…Song & Williamson 1995) and elastic wavefields (e.g. Okamoto 1993); the indirect boundary method (e.g.…”

Section: Introductionmentioning

confidence: 99%

2.5-Dmodelling of elastic waves using the pseudospectral method

Furumura

Takenaka

1996

Geophysical Journal International

103

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The pseudospectral code bamps is used to evolve axisymmetric gravitational waves. We consider a one-parameter family of Brill wave initial data, taking the seed function and strength parameter of Holz et. al. A careful comparison is made to earlier work. Our results are mostly in agreement with the literature, but we do find that some amplitudes reported elsewhere as subcritical evolve to form apparent horizons. Related to this point we find that by altering the slicing condition, the position of the peak of the Kretschmann scalar in these supercritical data can be controlled so that it appears away from the symmetry axis before the method fails, demonstrating that such behavior is at least partially a coordinate effect. We are able to tune the strength parameter to an interval of range 1 − A/A 10 −6 around the onset of apparent horizon formation. In this regime we find that large spikes appear in the Kretschmann scalar on the symmetry axis but away from the origin. From the supercritical side disjoint apparent horizons form around these spikes. On the subcritical side, down to this range, evidence of power-law scaling of the Kretschmann scalar is not conclusive, but the data can be fitted as a power-law with periodic wiggle.

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Frequency‐domain acoustic‐wave modeling and inversion of crosshole data: Part I—2.5-D modeling method

Cited by 83 publications

References 4 publications

Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

Computing the Sensitivity Kernels for 2.5-D Seismic Waveform Inversion in Heterogeneous, Anisotropic Media

2.5-Dmodelling of elastic waves using the pseudospectral method

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