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D. M. Boore

Bio: D. M. Boore is an academic researcher. The author has contributed to research in topics: Dispersion (water waves) & Acoustic wave equation. The author has an hindex of 1, co-authored 1 publications receiving 604 citations.

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TL;DR: In this paper, the authors considered the effect of grid dispersion on the accuracy of finite-difference seismograms and showed that the grid can be twice as coarse (five or more grid points per upper half-power wavelength) and good results can still be obtained.
Abstract: two methods rapidly deteriorates. This effect, known as “grid dispersion,” must be taken into account in order to avoid erroneous interpretation of seismograms obtained by finite-difference techniques. Both seconti-order accuracy and fourth-order accuracy finite-difference algorithms are considered. For the second-order scheme, a good rule of thumb is that the ratio of the upper half-power wavelength of the source to the grid interval should be of the order of ten or more. For the fourth-order scheme, it is found that the grid can be twice as coarse (five or more grid points per upper half-power wavelength) and good results are still obtained. Analytical predictions of the effect of grid dispersion are presented; these seem to be in agreement with the experimental results. ence methods to that obtained by classical analytical methods for the simple case of an infinite twodimensional 90.degree wedge (quarter space) in an otherwise infinite homogeneous medium. The source field was that due to a line source distribution located parallel to the corner of the wedge (see Figure I). For analytical simplicity, the acoustic velocity of the wedge medium is taken to be zero. This is equivalent to having a perfectlj “soft” wedge medium such as a vacuum.

655 citations


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TL;DR: In this paper, a plane circular model of a frictional fault using numerical methods was studied and it was shown that the average corner frequency of S waves v s is related to the final source radius, a, by v s = 0.21 β/α.
Abstract: We study a plane circular model of a frictional fault using numerical methods. The model is dynamic since we specify the effective stress at the fault. In one model we assume that the fault appears instantaneously in the medium; in another, that the rupture nucleates at the center and that rupture proceeds at constant subsonic velocity until it suddenly stops. The total source slip is larger at the center and the rise time is also longer at the center of the fault. The dynamic slip overshoots the static slip by 15 to 35 per cent. As a consequence, the stress drop is larger than the effective stress and the apparent stress is less than one half the effective stress. The far-field radiation is discussed in detail. We distinguish three spectral regions. First, the usual constant low-frequency level. Second, an intermediate region controlled by the fault size and, finally, the high-frequency asymptote. The central region includes the corner frequency and is quite complicated. The corner frequency is shown to be inversely proportional to the width of the far-field displacement pulse which, in turn, is related to the time lag between the stopping phases. The average corner frequency of S waves v s is related to the final source radius, a , by v s = 0.21 β/α . The corner frequency of P waves is larger than v s by an average factor of 1.5.

1,628 citations

Journal ArticleDOI
Alan Levander1
TL;DR: The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson's ratio materials, with minimal numerical dispersion and numerical anisotropy.
Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.

1,429 citations

Journal ArticleDOI
TL;DR: The spectral element method as discussed by the authors is a high-order variational method for the spatial approximation of elastic-wave equations, which can be used to simulate elastic wave propagation in realistic geological structures involving complieated free surface topography and material interfaces for two- and three-dimensional geometries.
Abstract: We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as Δ tC < O ( nel−1/nd N −2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.

1,183 citations

Journal ArticleDOI
TL;DR: In this article, the authors proposed a convolution of a singular solution having abrupt stress drop with a "rupture distribution function" to spread out the rupture front in space-time.
Abstract: Propagation of plane strain shear cracks is calculated numerically by using finite difference equations with second-order accuracy. The rupture model, in which stress drops gradually as slip increases, combines two different rupture criteria: (1) slip begins at a finite stress level; (2) finite energy is absorbed per unit area as the crack advances. Solutions for this model are nonsingular. In some cases there may be a transition from rupture velocity less than Rayleigh velocity to rupture velocity greater than shear wave velocity. The locus of this transition is surveyed in the parameter space of fracture energy, upper yield stress, and crack length. A solution for this model can be represented as a convolution of a singular solution having abrupt stress drop with a ‘rupture distribution function.’ The convolution eliminates the singularity and spreads out the rupture front in space-time. If the solution for abrupt stress drop has an inverse square root singularity at the crack tip, as it does for sub-Rayleigh rupture velocity, then the rupture velocity of the convolved solution is independent of the rupture distribution function and depends only on the fracture energy and crack length. On the other hand, a crack with abrupt stress drop propagating faster than the shear wave velocity has a lower-order singularity. A supershear rupture front must necessarily be spread out in space-time if a finite fracture energy is absorbed as stress drops.

930 citations

Journal ArticleDOI
TL;DR: In this paper, the authors quantitatively compare finite-difference and finite-element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time-domain and frequency-domain techniques.
Abstract: Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite-difference and finite-element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time-domain and frequency-domain techniques. It is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors in reflection and transmission coefficients. This paper shows that in homogeneous media the explicit finite-element and finite-difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson's ratio less than 0.3. Finite-elements are superior to finite-differences when modeling elastic media with Poisson's ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central-differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency-domain finite-element solutions employing a weighted average of consistent and lumped masses yield the most accurate resuls, and they promise to be the most cost-effective method for CDP, well log, and interactive modeling.--Modified journal abstract.

861 citations