Three-Dimensional Wavefield in a Two-Dimensional Basin Structure due to Point Source.

Fujiwara, Hiroyuki

doi:10.4294/jpe1952.44.1

Cited by 12 publications

(11 citation statements)

References 23 publications

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“…This is a Fredholm integral equation of the second kind for u i ( j ) but the first kind for t 2 (1) . The definitions of the matrices are given in Fujiwara (1996).…”

Section: Convergence Of Iterative Methodsmentioning

confidence: 99%

The fast multipole method for solving integral equations ofthree-dimensional topography and basin problems

Fujiwara¹

2000

Geophys. J. Int.

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Summary The fast multipole method is developed for the solution of the boundary integral equations arising in wave scattering problems involving 3‐D topography and 3‐D basin problems. When coupled with an iterative solver for linear equations, the fast multipole method can significantly reduce memory requirements. The order of operations for the product of the matrix obtained from the discretization of the integral kernel and a vector is reduced from N2 to the order of pαN, where p is the order of the multipole expansion and α depends on the details of the implementation. In order to achieve efficient implementation the translation operators for the multipole expansion need to be used in the diagonal form based on a spherical wave decomposition. For problems with topography, the number of iterations required of the linear equation solver to achieve convergence is small and no preconditioning is necessary. However, for a basin problem, block‐diagonal preconditioning is essential in the application of the iterative solver. Both the memory requirements and the CPU time are considerably reduced for topography problems. Although the memory requirement is reduced for a basin problem used in this numerical experiment, the CPU time would be still longer than that for the ordinary boundary element method if sufficient memory were available. These results indicate that the fast multipole method might be much more efficient than the ordinary method for 3‐D elastic wave scattering problems with more than several tens of thousands of unknown variables.

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“…This is a Fredholm integral equation of the second kind for u i ( j ) but the first kind for t 2 (1) . The definitions of the matrices are given in Fujiwara (1996).…”

Section: Convergence Of Iterative Methodsmentioning

confidence: 99%

The fast multipole method for solving integral equations ofthree-dimensional topography and basin problems

Fujiwara¹

2000

Geophys. J. Int.

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“…We do not treat the displacement and the traction on S (1) 2 b and S (2) 1 b as unknown values and replace the infinite integrals by known integral values approximately. This approximation technique was proposed to eliminate non‐physical waves caused by truncation of lateral boundaries in the BEM with the full‐space Green’s functions (Fujiwara & Takenaka 1993; Yokoi & Takenaka 1995; Fujiwara 1996a) and it is known as the ‘reference solution’ approximation. In this study, we first apply the reference solution approximation to the truncation of vertical boundaries.…”

Section: Calculation Of the Wavefield For A Whole Basinmentioning

confidence: 99%

“…For the comparison, we use the solutions obtained for this model by the BEM with the full‐space Green’s functions (Fujiwara, personal communication, 1997). In his method, the boundary is divided into triangular elements and the reference solution approximation is taken into account for the truncation of the free surface (Fujiwara 1996a). To compare the full wavefield, we use eq.…”

Section: Calculation Of the Wavefield For A Whole Basinmentioning

confidence: 99%

Excitation of secondary Love and Rayleigh waves in a three-dimensional sedimentary basin evaluated by the direct boundary element method with normal modes

Hatayama¹,

Fujiwara

1998

Geophys. J. Int.

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This paper aims to present a new method to calculate surface waves in 3-D sedimentary basin models, based on the direct boundary element method (BEM) with vertical boundaries and normal modes, and to evaluate the excitation of secondary surface waves observed remarkably in basins. Many authors have so far developed numerical techniques to calculate the total 3-D wavefield. However, the calculation of the total wavefield does not match our purpose, because the secondary surface waves excited on the basin boundaries will be contaminated by other undesirable waves. In this paper, we prove that, in principle, it is possible to extract surface waves excited on part of the basin boundaries from the total 3-0 wavefield with a formulation that uses the reflection and transmission operators defined in the space domain. In realizing this extraction in the BEM algorithm, we encounter the problem arising from the lateral and vertical truncations of boundary surfaces extending infinitely in the half-space. To compensate the truncations, we first introduce an approximate algorithm using 2.5-0 and 1-D wavefields for reference media, where a 2.5-D wavefield means a 3-D wavefield with a 2-D subsurface structure, and we then demonstrate the extraction. Finally, we calculate the secondary s urface waves excited on the arc shape (horizontal section) of a vertical basin boundary subject to incident SH and SV plane waves propagating perpendicularly to the chord of the arc. As a result, we find that in the SH-incident case the Love waves are predominantly excited, rather than the Rayleigh waves and that in the SV-wave incident case the Love waves as well as the Rayleigh waves are excited. This suggests that the Love waves are more detectable than the Rayleigh waves in the horizontal components of observed recordings.

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“…The Hankel functions contain singularities when the wavenumber k y = k p or k y = k s . In an earlier paper, FUJIWARA (1996) considers the more general problem of the frequency-wavenumber domain Green's functions for computing the 3D wavefield in a 2D basin structure of arbitrary cross section involving a free surface. The solution was obtained by a boundary element method but no specific mention is made of the critical wavenumbers.…”

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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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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Three-Dimensional Wavefield in a Two-Dimensional Basin Structure due to Point Source.

Cited by 12 publications

References 23 publications

The fast multipole method for solving integral equations ofthree-dimensional topography and basin problems

The fast multipole method for solving integral equations ofthree-dimensional topography and basin problems

Excitation of secondary Love and Rayleigh waves in a three-dimensional sedimentary basin evaluated by the direct boundary element method with normal modes

Wavenumber Sampling Issues in 2.5D Frequency Domain Seismic Modelling

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