3D Heterogeneous Staggered-Grid Finite-Difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities (original) (raw)
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Geophysical Journal International, 2016
We investigate the problem of finite-difference approximations of the velocity-stress formulation of the equation of motion and constitutive law on the staggered grid (SG) and collocated grid (CG). For approximating the first spatial and temporal derivatives, we use three approaches: Taylor expansion (TE), dispersion-relation preserving (DRP), and combined TE-DRP. The TE and DRP approaches represent two fundamental extremes. We derive useful formulae for DRP and TE-DRP approximations. We compare accuracy of the numerical wavenumbers and numerical frequencies of the basic TE, DRP and TE-DRP approximations. Based on the developed approximations, we construct and numerically investigate 14 basic TE, DRP and TE-DRP finite-difference schemes on SG and CG. We find that (1) the TE secondorder in time, TE fourth-order in space, 2-point in time, 4-point in space SG scheme (that is the standard (2,4) VS SG scheme, say TE-2-4-2-4-SG) is the best scheme (of the 14 investigated) for large fractions of the maximum possible time step, or, in other words, in a homogeneous medium; (2) the TE second-order in time, combined TE-DRP second-order in space, 2-point in time, 4-point in space SG scheme (say TE-DRP-2-2-2-4-SG) is the best scheme for small fractions of the maximum possible time step, or, in other words, in models with large velocity contrasts if uniform spatial grid spacing and time step are used. The practical conclusion is that in computer codes based on standard TE-2-4-2-4-SG, it is enough to redefine the values of the approximation coefficients by those of TE-DRP-2-2-2-4-SG for increasing accuracy of modelling in models with large velocity contrast between rock and sediments.
Discontinuous-Grid Finite-Difference Seismic Modeling Including Surface Topography
Bulletin of the Seismological Society of America, 2001
We have developed a two-dimensional P-SV viscoelastic finite-difference modeling technique for complex surface topography and subsurface structures. Realistic modeling of seismic wave propagation in the near surface region is complicated by many factors, such as strong heterogeneity, topographic relief, and large attenuation. In order to account for these complications, we use an O(2,4) accurate viscoelastic velocity-stress staggered-grid finite-difference scheme. The implementation includes an irregular free surface condition for topographic relief and a discontinuous grid technique in the shallow parts of the model. Several methods of free surface condition are bench marked, and an accurate and simple condition is proposed. In the proposed free surface condition, stresses are calculated so that the normal stresses perpendicular to the boundary and shear stresses on the free surface are zero. The calculation of particle velocities at the free surface does not involve any specific calculations, and the particle velocities are set to zero above the free surface. A discontinuous-grid method is introduced, where we use a 3 times finer grid in the near surface or low velocity region compared to the rest of the model. In order to reduce instability, we apply averaging or weighting to the replacement of the coarse-grid components within the fine grid field. The method allows us to avoid any limitation of the shape of the grid-spacing boundary. Numerical tests indicate that approximately 10 grid points per shortest wavelength, counted in coarse-grid spacing, with the discontinuous grid method results in accurate calculations as long as a small number of time steps is concerned.
Stable discontinuous staggered grid in the finite-difference modelling of seismic motion
2010
We present an algorithm of the spatial discontinuous grid for the 3-D fourth-order velocity-stress staggered-grid finite-difference modelling of seismic wave propagation and earthquake motion. The ratio between the grid spacing of the coarser and finer grids can be an arbitrary odd number. The algorithm allows for large numbers of time levels without inaccuracy and eventual instability due to numerical noise inevitably generated at the contact of two grids with different spatial grid spacings. The key feature of the algorithm is the application of the Lanczos downsampling filter.
Bulletin of the Seismological Society of America, 2003
We address the basic theoretical and algorithmic aspects of memoryefficient implementation of realistic attenuation in the staggered-grid finite-difference modeling of seismic-wave propagation in media with material discontinuities. We show that if averaging is applied to viscoelastic moduli in the frequency domain, it is possible to determine anelastic coefficients of the averaged medium representing a material discontinuity. We define (1) the anelastic functions in a new way, as being independent of anelastic coefficients, and (2) a new coarse spatial distribution of the anelastic functions in order to properly account for material discontinuities and, at the same time, to have it memory efficient. Numerical tests demonstrate that our approach enables more accurate viscoelastic modeling than other approaches.
Geophysical Journal International, 2016
The possibility of applying one explicit finite-difference (FD) scheme to all interior grid points (points not lying on a grid border) no matter what their positions are with respect to the material interface is one of the key factors of the computational efficiency of the FD modelling. Smooth or discontinuous heterogeneity of the medium is accounted for only by values of the effective grid moduli and densities. Accuracy of modelling thus very much depends on how these effective grid parameters are evaluated. We present an orthorhombic representation of a heterogeneous medium for the FD modelling. We numerically demonstrate its superior accuracy. Compared to the harmonic-averaging representation the orthorhombic representation is more accurate mainly in the case of strong surface waves that are especially important in local surface sedimentary basins. The orthorhombic representation is applicable to modelling seismic wave propagation and earthquake motion in isotropic models with material interfaces and smooth heterogeneities using velocity-stress, displacement-stress and displacement FD schemes on staggered, partly staggered, Lebedev and collocated grids.
Modeling seismic wave propagation using staggered-grid mimetic finite differences
2017
Mimetic finite difference (MFD) approximations of continuous gradient and divergente operators satisfy a discrete version of the Gauss-Divergente theorem on staggered grids. On the mimetic approximation of this integral conservation principle, an unique boundary flux operator is introduced that also intervenes on the discretization of a given boundary value problem (BVP). In this work, we present a second-order MFD scheme for seismic wave propagation on staggered grids that discretized free surface and absorbing boundary conditions (ABC) with same accuracy order. This scheme is time explicit after coupling a central three-level finite difference (FD) stencil for numerical integration. Here, we briefly discuss the convergence properties of this scheme and show its higher accuracy on a challenging test when compared to a traditional FD method. Preliminary applications to 2-D seismic scenarios are also presented and show the potential of the mimetic finite differene method.
Bulletin of the Seismological Society of America, 2004
Kristek et al. (2002) developed a technique for simulating the planar free surface in the 3D fourth-order staggered-grid finite-difference (FD) modeling of seismic motion. The technique is based on (1) explicit application of zero values of the stress-tensor components at the free surface and (2) adjusted FD approximations (AFDAs) to vertical derivatives at and near the free surface. The technique was shown to be more accurate and efficient than the standard stress-imaging technique in 1D models. In this study, we tested accuracy of the AFDA technique in media with lateral material discontinuities reaching the free surface. We compared the FD synthetics with synthetics calculated by the standard finite-element (FE) method because the FE method naturally and sufficiently accurately satisfies the boundary conditions at the free surface and the traction interface continuity conditions at internal material discontinuities. The comparison showed a very good level of accuracy of the AFDA technique. We also demonstrated the very good sensitivity of our FD modeling to different positions of the same physical model in the spatial FD grid.
Finite-Difference Seismic Simulation Combining Discontinuous Grids with Locally Variable Timesteps
Bulletin of the Seismological Society of America, 2004
A locally variable timestep scheme that matches with discontinuous grids in the finite-difference method is developed for the efficient simulation of seismic-wave propagation. The first-order velocity-stress formulations are used to obtain the spatial derivatives using finite-difference operators on a staggered grid. In the case of a media interface with high velocity contrast, the computational domain consists of two regions with different grid spacings. Each region roughly covers the medium of the lower or higher wave propagation velocity. There is a small overlap of the two regions, called the transitional zone, within the higher velocity medium. A grid three times coarser in the high-velocity region compared with the grid in the low-velocity region is used to avoid spatial oversampling. Temporal steps corresponding to the spatial sampling ratio between both regions are determined based on local stability criteria. The wave field in the margin of the region with the smaller timestep is linearly interpolated in time using the values calculated in the region with the larger one within the transitional zone. Since the temporal interpolation is a 1D operation performed separately from the spatial interpolation strategy employed to connect two regions with different grid spacings, the proposed scheme is not restricted to 2D or 3D problems with a specific order of accuracy of the spatial finite-difference approximation. The use of the locally variable timestep scheme with discontinuous grids results in remarkable savings of computation time and reductions in memory requirements, with the efficiency depending on the simulation model.
A 3-D hybrid finite-difference-finite-element viscoelastic modelling of seismic wave motion
Geophysical Journal International, 2008
We have developed a new hybrid numerical method for 3-D viscoelastic modelling of seismic wave propagation and earthquake motion in heterogeneous media. The method is based on a combination of the fourth-order velocity-stress staggered-grid finite-difference (FD) scheme, that covers a major part of a computational domain, with the second-order finite-element (FE) method which can be applied to one or several relatively small subdomains. The FD and FE parts causally communicate at each time level in the FD-FE transition zone consisting of the FE Dirichlet boundary, FD-FE averaging zone and FD Dirichlet zone.
Solving the elastic wave equation using the Finite-Difference (FD) method on an equidistant spatial grid is inefficient if the geological model exhibits strong spatial variations in the seismic velocities or small scale structures that require a dense spatial sampling. The application of spatial variable grid spacings leads to a significant reduction of the computational requirements. In this work we show that previously published variable grid approaches, in which the grid size changes in all spatial directions, produce unstable solutions. The instabilities occur at larger simulation times. We show that the instability is caused by interpolation errors on the transition between the fine and the coarsely sampled domains and the application of symmetric FD operators. The FD scheme can be stabilized by using a modified asymmetric FD operator. The stability and accuracy of the new asymmetric FD scheme is demonstrated with two examples.