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.

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.

A modified-boundary condition algorithm for efficient 3D forward modeling of CSEM data

2021 ◽  
Author(s):  
Rahul Dehiya
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Source Term ◽  
Forward Modeling ◽  
Field Vector ◽  
Fine Grid ◽  
Boundary Field ◽  
Radiation Boundary ◽  
3D Forward Modeling ◽  
Modeling Algorithm

<p>I present a newly developed 3D forward modeling algorithm for controlled-source electromagnetic data. The algorithm is based on the finite-difference method, where the source term vector is redefined by combining a modified boundary condition vector and source term vector. The forward modeling scheme includes a two-step modeling approach that exploits the smoothness of the electromagnetic field. The first step involves a coarse grid finite-difference modeling and the computation of a modified boundary field vector called radiation boundary field vector. In the second step, a relatively fine grid modeling is performed using radiation boundary conditions. The fine grid discretization does not include stretched grid and air medium. The proposed algorithm derives computational efficiency from a stretch-free discretization, air-free computational domain, and a better initial guess for an iterative solver. The numerical accuracy and efficiency of the algorithm are demonstrated using synthetic experiments. Numerical tests indicate that the developed algorithm is one order faster than the finite-difference modeling algorithm in most of the cases analyzed during the study. The radiation boundary method concept is very general; hence, it can be implemented in other numerical schemes such as finite-element algorithms.</p>

A Singular Perturbation Problem and A Neutral Differential-Difference Equation

1974 ◽  
Vol 17 (1) ◽  
pp. 77-83
Author(s):  
Edward Moore
Keyword(s):  
Difference Equation ◽  
Taylor Series Expansion ◽  
Linear Difference Equation ◽  
Constant Matrix ◽  
Perturbation Problem ◽  
Constant Coefficients ◽  
Explicit Formulae ◽  
Differential Difference Equation ◽  
The Difference ◽  
Image Position

Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients(1.1)where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients(1.2)(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

scholarly journals A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. T191-T207
Author(s):  
Xingguo Huang ◽  
Hui Sun ◽  
Zhangqing Sun ◽  
Nuno Vieira da Silva
Keyword(s):  
Point Source ◽  
Eikonal Equation ◽  
Taylor Series Expansion ◽  
Transversely Isotropic ◽  
Complex Point ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Complex Point Source ◽  
Complex Eikonal

The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.

scholarly journals Exact Finite-Difference Schemes ford-Dimensional Linear Stochastic Systems with Constant Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Peng Jiang ◽  
Xiaofeng Ju ◽  
Dan Liu ◽  
Shaoqun Fan
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Symplectic Structure ◽  
Stochastic Systems ◽  
First Integral ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Linear Stochastic Systems ◽  
Constant Coefficients

The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

Optimal Circumferential Placement of Cylindrical Thermocouple Probes for Reduction of Excitation Forces

1992 ◽  
Author(s):  
Eben C. Cobb ◽  
Tsu-Chien Cheu ◽  
Jay Hoffman
Keyword(s):  
Sensitivity Analysis ◽  
Linear Programming ◽  
Finite Difference ◽  
Design Methodology ◽  
Turbine Stage ◽  
Linear Programming Method ◽  
Optimization Scheme ◽  
Forcing Function ◽  
Finite Difference Formulation ◽  
Critical Excitation

This paper presents a design methodology to determine the optimal circumferential placement of cylindrical probes upstream of a turbine stage for reduced excitation forces. The potential flow forcing function generated by the probes is characterized by means of a Fourier analysis. A finite difference formulation is used to evaluate the sensitivity of the forcing function to the probe positions. An optimization scheme, based on the linear programming method, uses the sensitivity analysis results to reposition the probes such that the Fourier amplitudes of critical excitation orders are reduced. The results for an example design situation are presented.

scholarly journals A Dynamic Model of Rescuer Parameters for Optimizing Blood Gas Delivery during Cardiopulmonary Resuscitation

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Ali Jalali ◽  
Allan F. Simpao ◽  
Jorge A. Gálvez ◽  
Robert A. Berg ◽  
Vinay M. Nadkarni ◽  
...  
Keyword(s):  
Global Optimization ◽  
Cardiopulmonary Resuscitation ◽  
Optimization Technique ◽  
Performance Measure ◽  
Sequential Optimization ◽  
Blood Gas ◽  
Optimization Scheme ◽  
Total Blood ◽  
Active Research

Introduction. The quality of cardiopulmonary resuscitation (CPR) has been shown to impact patient outcomes. However, post-CPR morbidity and mortality remain high, and CPR optimization is an area of active research. One approach to optimizing CPR involves establishing reliable CPR performance measures and then modifying CPR parameters, such as compressions and ventilator breaths, to enhance these measures. We aimed to define a reliable CPR performance measure, optimize the CPR performance based on the defined measure and design a dynamically optimized scheme that varies CPR parameters to optimize CPR performance. Materials and Methods. We selected total blood gas delivery (systemic oxygen delivery and carbon dioxide delivery to the lungs) as an objective function for maximization. CPR parameters were divided into three categories: rescuer dependent, patient dependent, and constant parameters. Two optimization schemes were developed using simulated annealing method: a global optimization scheme and a sequential optimization scheme. Results and Discussion. Variations of CPR parameters over CPR sequences (cycles) were analyzed. Across all patient groups, the sequential optimization scheme resulted in significant enhancement in the effectiveness of the CPR procedure when compared to the global optimization scheme. Conclusions. Our study illustrates the potential benefit of considering dynamic changes in rescuer-dependent parameters during CPR in order to improve performance. The advantage of the sequential optimization technique stemmed from its dynamically adapting effect. Our CPR optimization findings suggest that as CPR progresses, the compression to ventilation ratio should decrease, and the sequential optimization technique can potentially improve CPR performance. Validation in vivo is needed before implementing these changes in actual practice.

Sign in / Sign up

Export Citation Format

Share Document