A finite difference scheme for solving the eikonal equation including surface topography

2009 ◽  
Author(s):  
Jianguo Sun ◽  
Zhangqing Sun ◽  
Fuxing Han
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Surface Topography ◽  
Finite Difference Scheme ◽  
Eikonal Equation

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 Microseismic Location Error Due to Eikonal Traveltime Calculation

2021 ◽  
Vol 11 (3) ◽  
pp. 982
Author(s):  
Dmitry Alexandrov ◽  
Umair bin Waheed ◽  
Leo Eisner
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Eikonal Equation ◽  
Velocity Model ◽  
Systematic Bias ◽  
Location Error ◽  
Fast Marching ◽  
Fast Sweeping ◽  
Microseismic Location

The accuracy of computed traveltimes in a velocity model plays a crucial role in localization of microseismic events. The conventional approach usually utilizes robust fast sweeping or fast marching methods to solve the eikonal equation numerically with a finite-difference scheme. These methods introduce traveltime errors that strongly depend on the direction of wave propagation. Such error results in moveout changes of the computed traveltimes and introduces significant location bias. The issue can be addressed by using a finite-difference scheme to solve the factored eikonal equation. This equation yields significantly more accurate traveltimes and therefore reduces location error, though the traveltimes computed with the factored eikonal equation still contain small errors with systematic bias. Alternatively, the traveltimes can be computed using a physics-informed neural network solver, which yields more randomized traveltimes and resulting location errors.

A finite difference scheme for solving the eikonal equation including surface topography

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. T53-T63 ◽  
Author(s):  
Jianguo Sun ◽  
Zhangqing Sun ◽  
Fuxing Han
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Eikonal Equation ◽  
Nonuniform Grid ◽  
Grid Spacing ◽  
Regular Grid ◽  
Grid Points ◽  
Upwind Finite Difference ◽  
Curved Interface

For solving the eikonal equation in the regions near the curved earth’s surface and the curved interface, we find a second order upwind finite difference scheme that uses nonuniform grid spacing in the regions near the earth’s surface and the interface, respectively. Specifically, in the direct neighborhood of the earth’s surface and of the considered interface, we replace the regular grid spacing in the vertical direction by the vertical distance between the surface (interface) point and the grid point under consideration. For the horizontal direction, however, only the regular grid points are used. As a result, the conventional upwind finite difference formulas are changed into the ones with nonuniform grid spacing. Furthermore, for capturing and propagating the local wavefront near the curved earth’s surface (interface), we adapt the fast marching method by introducing new point types, namely the surface point, the point above the surface, the interface point, and the point under the interface. If we use the scheme in a multistage fashion, we can compute not only the traveltimes of the first arrivals but also the traveltimes of the reflected and transmitted events. In comparison to the published schemes, our scheme has the following two advantages: (1) there is no need to construct a local unstructured grid for suturing the surface or the interface points to the neighboring regular grid points; (2) there is no need to make a local coordinate transform for capturing the local wavefront. Numerical results show that our scheme can treat the irregular region problem caused by the curved earth’s surface and by the curved interface with satisfactory effectiveness and flexibility.

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

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