Optimizing the finite-difference implementation of three-dimensional free-surface boundary in frequency-domain modeling of elastic waves

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. T363-T379
Author(s):  
Jian Cao ◽  
Jing-Bo Chen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Frequency Domain ◽  
Computational Cost ◽  
Good Choice ◽  
Weighted Averaging ◽  
Seismic Exploration ◽  
Surface Boundary ◽  
Domain Modeling ◽  
Near Surface

The problem of modeling seismic wave propagation for multiple sources, such as in the solution of gradient-based elastic full-waveform inversion, is an important topic in seismic exploration. The frequency-domain finite-difference (FD) method is a good choice for this purpose, mainly because of its simple discretization and high computational efficiency. However, when it comes to modeling the complete elastic wavefields, this approach has limited surface-wave accuracy because, when modeling with the strong form of the wave equation, it is not always easy to implement an accurate stress-free boundary condition. Although a denser spatial sampling is helpful for overcoming this problem, the additional discrete points will significantly increase the computational cost in the resolution of its resulting discrete system, especially in 3D problems. Furthermore, sometimes, when modeling with optimized schemes, an inconsistency in the computation precision between the regions at the free surface and inside the model volume would happen and introduce numerical artifacts. To overcome these issues, we have considered optimizing the FD implementation of the free-surface boundary. In our method, the problem was formulated in terms of a novel system of partial differential equations satisfied at the free surface, and the weighted-averaging strategy was introduced to optimize its discretization. With this approach, we can impose FD schemes for the free surface and internal region consistently and improve their discretization precision simultaneously. Benchmark tests for Lamb’s problem indicate that the proposed free-surface implementation contributes to improving the simulation accuracy on surface waves, without increasing the number of grid points per wavelength. This reveals the potential of developing optimized schemes in the free-surface implementation. In particular, through the successful introduction of weighting coefficients, this free-surface FD implementation enables adaptation to the variation of Poisson’s ratio, which is very useful for modeling in heterogeneous near-surface weathered zones.

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.

An explicit method to calculate implicit spatial finite differences

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Computational Cost ◽  
New Method ◽  
Seismic Exploration ◽  
Explicit Method ◽  
Lu Decomposition ◽  
Implicit Methods ◽  
Band Matrices ◽  
Explicit Finite Difference

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

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.

An implicit finite-difference operator for the Helmholtz equation

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T97-T107 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Difference Operator ◽  
Computational Cost ◽  
Finite Difference Discretization ◽  
Implicit Finite Difference ◽  
Explicit Finite Difference ◽  
The One ◽  
Difference Discretization

We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly.

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