Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. T109-T115 ◽  
Author(s):  
Thomas Bohlen ◽  
Erik H. Saenger
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Rayleigh Waves ◽  
Complex Surface ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Image Method ◽  
The Earth ◽  
Grid Points

Heterogeneous finite-difference (FD) modeling assumes that the boundary conditions of the elastic wavefield between material discontinuities are implicitly fulfilled by the distribution of the elastic parameters on the numerical grid. It is widely applied to weak elastic contrasts between geologic formations inside the earth. We test the accuracy at the free surface of the earth. The accuracy for modeling Rayleigh waves using the conventional standard staggered-grid (SSG) and the rotated staggered grid (RSG) is investigated. The accuracy tests reveal that one cannot rely on conventional numerical dispersion discretization criteria. A higher sampling is necessary to obtain acceptable accuracy. In the case of planar free surfaces aligned with the grid, 15 to 30 grid points per minimum wavelength of the Rayleigh wave are required. The widely used explicit boundary condition, the so-called image method, produces similar accuracy and requires approximately half the sampling of the wavefield compared to heterogeneous free-surface modeling. For a free-surface not aligned with the grid (surface topography), the error increases significantly and varies with the dip angle of the interface. For an irregular interface, the RSG scheme is more accurate than the SSG scheme. The RSG scheme, however, requires 60 grid points per minimum wavelength to achieve good accuracy for all dip angles. The high computation requirements for 3D simulations on such fine grids limit the application of heterogenous modeling in the presence of complex surface topography.

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

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 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.

scholarly journals A Robust Optimal Finite Difference Scheme for the Three-Dimensional Helmholtz Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Dongsheng Cheng ◽  
Baowen Chen ◽  
Xiangling Chen
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Helmholtz Equation ◽  
Finite Difference Scheme ◽  
Three Dimensional ◽  
Weighted Average ◽  
Least Square ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Grid Points

We propose a robust optimal 27-point finite difference scheme for the Helmholtz equation in three-dimensional domain. In each direction, a special central difference scheme with 27 grid points is developed to approximate the second derivative operator. The 27 grid points are divided into four groups, and each group is involved in the difference scheme by the manner of weighted combination. As for the approximation of the zeroth-order term, we use the weighted average of all the 27 points, which are also divided into four groups. Finally, we obtain the optimal weights by minimizing the numerical dispersion with the least-square method. In comparison with the rotated difference scheme based on a staggered-grid method, the new scheme is simpler, more practical, and much more robust. It works efficiently even if the step sizes along different directions are not equal. However, rotated scheme fails in this situation. We also present the convergence analysis and dispersion analysis. Numerical examples demonstrate the effectiveness of the proposed scheme.

scholarly journals An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. T1-T9 ◽  
Author(s):  
Chong Zeng ◽  
Jianghai Xia ◽  
Richard D. Miller ◽  
Georgios P. Tsoflias
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Rayleigh Waves ◽  
Surface Boundary ◽  
Near Surface ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Grid Nodes

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM147-SM153 ◽  
Author(s):  
Yixian Xu ◽  
Jianghai Xia ◽  
Richard D. Miller
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Rayleigh Waves ◽  
Second Order ◽  
Body Waves ◽  
Staggered Grid ◽  
Difference Operators ◽  
Image Method ◽  
Elastic Boundary ◽  
Second Order In Time

The need for incorporating the traction-free condition at the air-earth boundary for finite-difference modeling of seismic wave propagation has been discussed widely. A new implementation has been developed for simulating elastic wave propagation in which the free-surface condition is replaced by an explicit acoustic-elastic boundary. Detailed comparisons of seismograms with different implementations for the air-earth boundary were undertaken using the (2,2) (the finite-difference operators are second order in time and space) and the (2,6) (second order in time and sixth order in space) standard staggered-grid (SSG) schemes. Methods used in these comparisons to define the air-earth boundary included the stress image method (SIM), the heterogeneous approach, the scheme of modifying material properties based on transversely isotropic medium approach, the acoustic-elastic boundary approach, and an analytical approach. The method proposed achieves the same or higher accuracy of modeled body waves relative to the SIM. Rayleigh waves calculated using the explicit acoustic-elastic boundary approach differ slightly from those calculated using the SIM. Numerical results indicate that when using the (2,2) SSG scheme for SIM and our new method, a spatial step of 16 points per minimum wavelength is sufficient to achieve 90% accuracy; 32 points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. When using the (2,6) SSG scheme for the two methods, a spatial step of eight points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. Our proposed method is physically reasonable and, based on dispersive analysis of simulated seismographs from a layered half-space model, is highly accurate. As a bonus, our proposed method is easy to program and slightly faster than the SIM.

Tensorial Elastodynamics for Anisotropic Media

Geophysics ◽  
2021 ◽  
pp. 1-60
Author(s):  
Tugrul Konuk ◽  
Jeffrey Shragge
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Anisotropic Media ◽  
Essential Elements ◽  
Constitutive Relationship ◽  
Staggered Grid ◽  
General Strategy ◽  
Reverse Time ◽  
Time Migration

Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse-time migration and full waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we propose a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic (TTI) or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid with a mimetic finite-difference (FSG+MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation where we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG+MFD numerical method has a computational complexity comparable to Cartesian scenarios using a similar FSG+MFD numerical approach. Numerical examples demonstrate that the proposed solution can simulate anisotropic elastodynamic field solutions on irregular geometries and is thus a reliable tool for anisotropic elastic modeling, imaging and inversion applications in generalized computational domains including handling free-surface topography.

An accuracy analysis of a Hamiltonian particle method with the staggered particles for seismic-wave modeling including surface topography

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. T189-T197 ◽  
Author(s):  
Junichi Takekawa ◽  
Hitoshi Mikada ◽  
Tada-nori Goto
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Surface Topography ◽  
Seismic Wave ◽  
Rayleigh Waves ◽  
Particle Method ◽  
Complex Surface ◽  
Seismic Wave Propagation ◽  
Numerical Error ◽  
Irregular Topography

A Hamiltonian particle method (HPM), which is one of the mesh-free methods, can simulate seismic wavefields for models including surface topography in a simple manner. Numerical error caused by a curved free surface or by particles not aligned with the surface is not obvious in HPM. In general, the accommodation of irregular free surfaces requires more grids or particles in a minimum wavelength for achieving sufficient accuracy in the simulation. We tested the accuracy of HPM with staggered particles for simulating seismic-wave propagation including the surface topography, and we established the relationship between desired accuracy and spatial resolution. We conducted numerical simulations for models with a planar free surface aligned with the regular particle alignment and a dipping free surface. Our accuracy tests revealed that the numerical error strongly depends on the dipping angle of the slope. We concluded that about 25 particles in a minimum wavelength are required to calculate Rayleigh waves propagating along the irregular topography with good accuracy. Finally, we simulated Rayleigh wave propagation along irregular topography using a layered model with a hill. HPM can reproduce not only surface-wave propagation but also the reflected and refracted waves. Our numerical results were in good agreement with those from a finite-element method. Our investigations indicated that HPM could be a solution to simulate Rayleigh waves in the presence of complex surface topography.

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