A HIGH-ORDER FAST MARCHING SCHEME FOR THE LINEARIZED EIKONAL EQUATION

2001 ◽  
Vol 09 (03) ◽  
pp. 1095-1109 ◽  
Author(s):  
JONATHAN B. FRANKLIN ◽  
JERRY M. HARRIS
Keyword(s):  
Eikonal Equation ◽  
Evaluation Method ◽  
High Order ◽  
Fast Marching ◽  
Traveltime Tomography ◽  
First Order ◽  
Sequential Evaluation ◽  
Upwind Finite Difference ◽  
Coordinate Transforms ◽  
Marching Scheme

We present a high-order upwind finite-difference scheme for solving a useful family of first-order partial differential equations, of which the linearized eikonal equation is a member. Fast solutions of the linearized eikonal equation have applications in traveltime tomography and residual migration algorithms. The technique, besides being both accurate and stable, escapes aperture limitations inherent in static marching schemes. We use a time-sequential evaluation method similar to Sethian's Fast Marching strategy to insure causal operator evaluation. We apply our technique to several complex slowness distributions, including the Marmousi model. We also use an adaptation of our technique to compute Cartesian-to-Ray coordinate transforms for the same slowness models.

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. S211-S219 ◽  
Author(s):  
Siwei Li ◽  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Eikonal Equation ◽  
Order Partial Differential Equation ◽  
Fast Marching ◽  
First Order ◽  
Velocity Models ◽  
Kirchhoff Migration ◽  
Input Source ◽  
First Arrival ◽  
Synthetic Models

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.

Upwind finite‐difference calculation of traveltimes

Geophysics ◽  
1991 ◽  
Vol 56 (6) ◽  
pp. 812-821 ◽  
Author(s):  
J. van Trier ◽  
W. W. Symes
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Law ◽  
Finite Difference Scheme ◽  
Eikonal Equation ◽  
Upwind Schemes ◽  
Similar Scheme ◽  
Difference Calculation ◽  
Flow Variables ◽  
Upwind Finite Difference

Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite‐difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first‐arrival‐time field. The upwind finite‐difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first‐order upwind finite‐difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

scholarly journals Local Convergence for Multi-Step High Order Solvers under Weak Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 179
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Nonlinear Problems ◽  
Fréchet Derivative ◽  
High Order ◽  
Order Derivative ◽  
First Order ◽  
Lipschitz Constants ◽  
Convergence Study

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

Staggered-Grid High-Order Differencemethod for the First-Order Elastic Wave Equations

2000 ◽  
Vol 43 (3) ◽  
pp. 441-449 ◽  
Author(s):  
Liang-Guo DONG ◽  
Zai-Tian MA ◽  
Jing-Zhong CAO
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
High Order ◽  
Staggered Grid ◽  
First Order ◽  
Elastic Wave Equations
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