Determining the optimal coefficients of the explicit finite-difference scheme using the Remez exchange algorithm

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. S137-S147 ◽  
Author(s):  
Zheng He ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Finite Difference ◽  
Simulated Annealing Algorithm ◽  
Least Squares Method ◽  
Computational Cost ◽  
Seismic Exploration ◽  
Traditional Methods ◽  
Exchange Algorithm ◽  
Optimal Coefficients ◽  
Constant Coefficients ◽  
Explicit Finite Difference

Explicit finite-difference (FD) schemes are widely used in the seismic exploration field due to their simplicity in implementation and low computational cost. However, they suffer from strong artifacts caused by using coarse grids for high-frequency applications. The optimization of constant coefficients is popular in reducing spatial dispersions, but current methods could not guarantee that the bandwidth of the tolerable dispersion error is the widest. We have applied the Remez exchange algorithm to optimize the constant coefficients of the explicit FD schemes, for conventional and staggered grids. The resulting dispersion errors are distributed alternately between the maxima and minima in the passband of the filter, which is consistent with the most important equal-ripple property of the error magnitude for the optimal solution according to the Chebyshev criterion. The Remez exchange algorithm can determine the optimal coefficients of the FD method with only a few iterations, and the resulting operator has a wider bandwidth compared with previous solutions. It can handle arbitrary orders without the influence of local minima. Its computational cost for solving the objective function is comparable to that of the least-squares method, but its bandwidth is wider. Its accuracy is also higher than that of the maximum norm solved by the simulated annealing algorithm, but its computational cost is much lower. Theoretically, the equal-ripple error can offer the widest bandwidth for suppressing numerical dispersions among all solutions obtained by the constant-coefficient optimization. In other words, we can obtain a smaller error limitation than traditional methods under the same bandwidth. This superiority over traditional methods is essential for reducing the total error accumulation, which is helpful to avoid rapid error accumulations especially for large-scale models and long-term problems.

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.

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.

scholarly journals Explicit finite difference solution for contaminant transport problems with constant and oscillating boundary conditions

2020 ◽  
Vol 24 (3 Part B) ◽  
pp. 2225-2231
Author(s):  
Svetislav Savovic ◽  
Alexandar Djordjevich
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Boundary Conditions ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Oscillating Boundary ◽  
Constant Coefficients ◽  
Explicit Finite Difference ◽  
Explicit Finite Difference Method

For constant and oscillating boundary conditions, the 1-D advection-diffusion equation with constant coefficients, which describes a contaminant flow, is solved by the explicit finite difference method in a semi-infinite medium. It is shown how far the periodicity of the oscillating boundary carries on until diminishing to below appreciable levels a specified distance away, which depends on the oscillation characteristics of the source. Results are tested against an analytical solution reported for a special case. The explicit finite difference method is shown to be effective for solving the advection-diffusion equation with constant coefficients in semi-infinite media with constant and oscillating boundary conditions.

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.

Globally optimized finite-difference extrapolator for strongly VTI media

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T125-T135 ◽  
Author(s):  
Jin-Hai Zhang ◽  
Zhen-Xing Yao
Keyword(s):  
Global Optimization ◽  
Finite Difference ◽  
Phase Angle ◽  
Simulated Annealing Algorithm ◽  
Taylor Series Expansion ◽  
Optimization Scheme ◽  
Fine Grid ◽  
Constant Coefficients ◽  
Anisotropy Parameters ◽  
Transversely Anisotropic

Implicit finite-difference (FD) migration is unconditionally stable and is popular in handling strong velocity variations, but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult. Traditional local optimizations generate the optimized coefficients for each pair of Thomsen anisotropy parameters independently, which can degrade results substantially for large anisotropy variations and lead to a huge table. We developed an implicit FD method using the analytic Taylor-series expansion and used a global optimization scheme to improve its accuracy for wide phase angles. We first extended the number of the constant coefficients; then we relaxed the coefficient of the time-delay extrapolation term by tuning a small factor such that error is less than 0.1%. Finally, we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0.5% simultaneously. The extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range. Compared with traditional local optimization, our scheme does not need any table and table lookup. For each order of the FD method, only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy. More importantly, our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters, no matter how strong the anisotropy is. For the globally optimized second-order FD method, the accurate phase angle is up to 58°, and the increase is about 18°–22°. For the globally optimized fourth-order FD method, the accurate phase angle is up to 77°, and the increase is about 22°–27°.

Global optimization of generalized nonhyperbolic moveout approximation for long-offset normal moveout

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. S141-S149 ◽  
Author(s):  
Hanjie Song ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Objective Function ◽  
Simulated Annealing Algorithm ◽  
Least Squares Method ◽  
Optimization Scheme ◽  
Direct Optimization ◽  
Fine Grid ◽  
Analytical Approximations ◽  
Long Offset ◽  
Constant Coefficients ◽  
Error Accumulation

The approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media. The generalized nonhyperbolic moveout approximation (GMA) brings considerable improvement in accuracy compared with known analytical approximations. However, this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters, which would degrade the inversion accuracy due to error accumulation in velocity analysis. We optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset-to-depth ratio (O/D) within practical ranges. Theoretical analyses and numerical experiments indicate that the traditional optimization scheme, using the two-norm objective function solved by the least-squares method, could not provide an error-constrained result; in addition, a direct optimization without constant-coefficient extension would not lead to a satisfactory accuracy improvement. We construct the objective function using the maximum norm and solve it by using a simulated annealing algorithm; in addition, we extend the total number of constant coefficients in the GMA to achieve additional significant improvements in the accuracy. We use a normalized traveltime and offset so that the optimized constant coefficients are independent of the model. The optimized constant coefficients are obtained over a fine grid of the anellipticity parameter (0–0.5) and the O/D (0–4) that covers most practical ranges. Our optimization scheme does not increase the computational complexity but can significantly improve the accuracy. The relative error after optimization is always below a given tolerable error threshold 0.01%, which is better than the original error 0.21% of GMA. Scanning of the velocity and anellipticity parameter indicates that the original GMA has relatively large errors; in contrast, the optimized GMA can obtain more accurate results, which are essential for flattening the moveout and helpful for reducing error accumulations.

Erratum: Explicit finite difference vectorial beam propagation method

1992 ◽  
Vol 28 (5) ◽  
pp. 517
Author(s):  
Y. Chung ◽  
N. Dagli ◽  
L. Thylé
Keyword(s):  
Finite Difference ◽  
Beam Propagation Method ◽  
Beam Propagation ◽  
Explicit Finite Difference

Thermal Modeling of Unlooped and Looped Pulsating Heat Pipes

2001 ◽  
Vol 123 (6) ◽  
pp. 1159-1172 ◽  
Author(s):  
Mohammad B. Shafii ◽  
Amir Faghri ◽  
Yuwen Zhang
Keyword(s):  
Heat Transfer ◽  
Finite Difference ◽  
Heat Pipes ◽  
Analytical Models ◽  
Charge Ratio ◽  
Sensible Heat ◽  
Governing Equations ◽  
Heating Wall ◽  
Heat Mode ◽  
Explicit Finite Difference

Analytical models for both unlooped and looped Pulsating Heat Pipes (PHPs) with multiple liquid slugs and vapor plugs are presented in this study. The governing equations are solved using an explicit finite difference scheme to predict the behavior of vapor plugs and liquid slugs. The results show that the effect of gravity on the performance of top heat mode unlooped PHP is insignificant. The effects of diameter, charge ratio, and heating wall temperature on the performance of looped and unlooped PHPs are also investigated. The results also show that heat transfer in both looped and unlooped PHPs is due mainly to the exchange of sensible heat.

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