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The air/earth interface is accurately represented in a 3D finite-difference elastic wave propagation algorithm merely by assigning material properties of air to the spatial grid nodes above the earth's surface. Computational stability is maintained by making the boundary gradational. Synthetic seismic traces calculated by this approach compare favorably with those computed by imposing an explicit stress-free condition on the surface.

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... In this paper, I use the simplest example of a heterogeneous model — that of two half-spaces. I embark on this investigation bearing in mind the documented occurrences of numerical instabilities at interfaces with strong material contrasts Crase, 1990; Seron et al., 1996; Bartel et al., 2000; Saenger et al., 2000 . These instabilities arise even though the stability criterion derived using conventional von Neumann analysis may be satisfied for each gridpoint in the numerical model. ...

... For a pure velocity contrast, as the velocity in medium 2 becomes smaller, the stability value never exceeds unity and actually decreases. In contrast, the pure density contrast goes unstable for small values of density in medium 2. This agrees with the observation that instabilities arise at strong density contrasts when using the standard staggered grid Bartel et al., 2000; Saenger et al., 2000. ...

Evaluating the performance of finite-difference algo-rithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In prac-tice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numeri-cal model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at inter-faces with strong material contrasts. These interface instabil-ities occur even though the conventional von Neumann sta-bility criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann anal-ysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard stag-gered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coef-ficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near inter-faces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into ac-count the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.

... Although not strictly speaking a boundary condition, a common method of implementing an irregular surface (terrain) with TDAAPS is to give nodes below the boundary properties of rock (Bartel et al., 2000). This method is stable if three conditions are met: ...

... The original Beta Test plan called for a hill to be implemented as a transition over a single layer of nodes from the atmospheric conditions (C 342m/s and ρ 1.2Kg/m 3 ) to a hard, dense, rock-like material (C 3500m/s and ρ 1500Kg/m 3 ). We planned to use only a single transitional layer of density nodes to maintain stability (Bartel et al., 2000). However, once the test was underway we determined that " stair-step " diffractions from the surface of the hill were scattering far too much energy into the shadow zone behind the hill. ...

This document is intended to serve as a users guide for the time-domain atmospheric acoustic propagation suite (TDAAPS) program developed as part of the Department of Defense High-Performance Modernization Office (HPCMP) Common High-Performance Computing Scalable Software Initiative (CHSSI). TDAAPS performs staggered-grid finite-difference modeling of the acoustic velocity-pressure system with the incorporation of spatially inhomogeneous winds. Wherever practical the control structure of the codes are written in C++ using an object oriented design. Sections of code where a large number of calculations are required are written in C or F77 in order to enable better compiler optimization of these sections. The TDAAPS program conforms to a UNIX style calling interface. Most of the actions of the codes are controlled by adding flags to the invoking command line. This document presents a large number of examples and provides new users with the necessary background to perform acoustic modeling with TDAAPS.

... The simplest is the inclusion of a density contrast to mimic an air layer, at the expense of numerical accuracy. For elastic wave propagation, Bartel et al. (2000) find that some smoothing of the extreme density contrast is required for numerical stability. Boore (1972), Robertsson (1996), Mittet (2002), Bohlen and Saenger (2006), and Zeng et al. (2012) consider variants of the vacuum approach. ...

The presence of topography poses a challenge for seismic modeling with finite-difference codes. The representation of topography by means of an air layer or vacuum often leads to a substantial loss of numerical accuracy. A suitable modification of the finite-difference weights near the free surface can decrease that error. An existing approach requires extrapolation of interior solution values to the exterior while using the boundary condition at the free surface. However, schemes of this type occasionally become unstable and may be impossible to implement with highly irregular topography. One-dimensional extrapolation along coordinate lines results in a simple and efficient scheme. The stability of the 1D scheme is improved by ignoring the interior point nearest to the boundary during extrapolation in case its distance to the boundary is less than half a grid spacing. The generalization of the 1D scheme to more than one dimension requires a modification if the boundary intersects the finite-difference stencil on both sides of the central evaluation point and if there are not enough interior points to build the finite-difference stencil. Examples for the 2D constant-density acoustic case with a fourth-order finite-difference scheme demonstrate the method's capability. Because the 1D assumption is not valid in two dimensions if the boundary does not follow grid lines, the formal numerical accuracy is not always obtained, but the method can handle highly irregular topography. © 2017 Society of Exploration Geophysicists. All rights reserved.

... This can be amended by using very fine grids at the expense of computational time. A similar limitation is encountered when considering a heterogeneous medium that includes topography such as hills and mountains (Hestholm and Ruud, 1994;Bartel et al., 2000;Bohlen and Saenger, 2006). In contrast, finite-element methods (FEMs) easily handle unstructured meshes and spatial local refinement. ...

Modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources. Seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings. Finite elements enable accurate simulation of time-dependent wave propagation in heterogeneous media. They are more costly than finite-difference methods, but this is compensated by their superior accuracy if the finite-element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength, hence with the local wave velocity. However, 3D complex geologic settings often contain details on a very small scale compared to the dominant wavelength, requiring the mesh to contain elements that are smaller than dictated by the wavelength. Also, limitations of the mesh generation software may produce regions where the elements are much smaller than desired. In both cases, this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost. Local time stepping (LTS) can improve the computational efficiency and speed up the simulation. We evaluated a local formulation of an LTS scheme with second-order accuracy for the discontinuous Galerkin finite-element discretization of the wave equation. We tested the benefits of the scheme by considering a geologic model for a North-Sea-type example.

One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results.

The finite-difference method on rectangular meshes is widely used for time-domain modelling of the wave equation. It is relatively easy to implement high-order spatial discretization schemes and parallelization. Also, the method is computationally efficient. However, the use of finite elements on tetrahedral unstructured meshes is more accurate in complex geometries near sharp interfaces. We compared the standard eighth-order finite-difference method to fourth-order continuous mass-lumped finite elements in terms of accuracy and computational cost. The results show that, for simple models like a cube with constant density and velocity, the finite-difference method outperforms the finite-element method by at least an order of magnitude. Outside the application area of rectangular meshes, i.e., for a model with interior complexity and topography well described by tetrahedra, however, finite-element methods are about two orders of magnitude faster than finite-difference methods, for a given accuracy.

Three-dimensional (3D) elastic wave propagation modeling in the velocity-stress formulation using finite differences (FDs) have been investigated for a homogeneous medium covered by a representative, relatively steep surface topography consisting of a I D square root function. This scenario using various numerical implementations is explored. The behavior with regard to stability of long simulations is expected to be indicative of each numerical implementation's robustness for other types of topographies/media. Employing various combinations of the FD, order was found only to change the time of the first incidence of instability. On the other hand, nonequidistant grids in the horizontal and vertical directions are found to be extremely useful for long-term stability of 3D wave propagation modeling with our free-surface boundary condition for single-valued topographies. In particular dz greater than or equal to (3/2) A is found completely stable for all tested v(P)/v(S) ratios. Such relationships of using dz > dx are also favorable for more accurate Rayleigh-wave modeling. Setting the density equal to 1/10 of its interior value at one layer only at the surface is another simple means of achieving stability.

Stable and accurate numerical modeling of seismic wave propagation in the vicinity of high-contrast interfaces is achieved with straightforward modifications to the conventional, rectangular-staggered-grid, finite-difference (FD) method. Improvements in material parameter averaging and spatial differencing of wavefield variables yield high-quality synthetic seismic data.

I present a finite‐difference method for modeling P-SV wave propagation in heterogeneous media. This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid. The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson’s ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poisson’s ratio. Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid‐solid interface. Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results. The weathered‐layer and corner‐edge models show in seismograms the same converted phases obtained by previous authors. This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half‐space. The head wave preserves the correct amplitude. Finally, the corner‐edge model illustrates a more complex geometry for the liquid‐solid interface. As the Poisson’s ratio v increases from 0.25 to 0.5, the shear converted phases are removed from seismograms and from the time section of the wave field.

I describe the properties of a fourth-order accurate space, second-order accurate time, two-dimensional P-SV finite-difference scheme based on the Madariaga-Virieux 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 lay

Using a 3-D finite difference simulation, we calculated the effects of surface topography on the amplitudes and waveforms of seismic waves generated by a hypothetical nuclear explosion at the northern Novaya Zemlya test site (Matochkin Shar). The simulation shows substantial azimuthal variations in the amplitude of the downgoing P waveform at shallow depths beneath the source, caused by variations in the amplitude of pP. However, these azimuthal amplitude variations diminish as the wavefront propagates deeper in the crust. Based on these results, we estimate that the topographic scarp considered here produces a maximum azimuthal variation of +/-0.05 magnitude units for mb determined from teleseismic P-waves. The sloping topography causes substantial SH motion for azimuths along the strike of the scarp.