3‐D eikonal solvers: First‐arrival traveltimes

Geophysics ◽  
2002 ◽  
Vol 67 (4) ◽  
pp. 1225-1231 ◽  
Author(s):  
Seongjai Kim
Keyword(s):  
Grid Point ◽  
Level Set Methods ◽  
Second Order ◽  
Normal Velocity ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Maximum Angle ◽  
First Arrival ◽  
Grid Points ◽  
Marching Method

The article is concerned with the development and comparison of three different algorithms for the computation of first‐arrival traveltimes: the fast marching method (FMM), the group marching method (GMM), and a second‐order finite‐difference eikonal solver. GMM is introduced as a variant of FMM. It proceeds the solution by advancing a selected group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. The second‐order eikonal solver studied in the article is an expanding‐box, essentially nonoscillatory scheme for which the stability is enforced by the introduction of a down ‘n’ out marching and a post‐sweeping iteration. Techniques such as the maximum angle condition, the average normal velocity, and cache‐based implementation are introduced for the algorithms to improve the numerical accuracy and efficiency. The algorithms are implemented for solving the eikonal equation in 3‐D isotropic media, and their performances are compared. GMM is numerically verified to be faster than FMM. However, the second‐order algorithm turns out to be superior to these first‐order level‐set methods in both accuracy and efficiency; the incorporation of average normal velocity improves accuracy dramatically for the second‐order scheme.

scholarly journals On correctness of first and second order fast marching method

2011 ◽  
Vol 1 (2) ◽  
Author(s):  
Christoph Lass
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unique Solution ◽  
Nonlinear Partial Differential Equations ◽  
Second Order ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Order Case ◽  
Upwind Discretization ◽  
Marching Method

AbstractIn this article we will discuss the Fast Marching Method which was introduced by James A. Sethian to solve some types of nonlinear partial differential equations efficiently. We will show that this method yields the unique solution to an upwind discretization. Furthermore we will present the correct algorithm for the second order case where existence and unicity of the solution will be proven as well.

On: “3-D traveltime compuation using the fast marching method” (J. A. Sethian and A. M. Popovici, GEOPHYSICS, 64, 516–523).

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 682-682
Author(s):  
Fuhao Qin
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Marching Method ◽  
Difference Solution

The Sethian and Popovici paper “3-D traveltime computation using the fast marching method” that appeared in Geophysics, Vol. 64, 516–523, discussed a method to solve the eikonal equation for first arrival traveltimes which was called the “fast marching” method. The method, as the authors demonstrated, is very fast and stable. However, their method is very similar to the method discussed by F. Qin et al. (1992), entitled “Finite difference solution of the eikonal equation along expanding wavefronts,” Geophysics, Vol. 57, 478–487. F. Qin et al. first proposed the “expanding wavefront” method for solving eikonal equation in the 60th Ann. Internat. Mtg. of the SEG in 1990.

A Second Order Multi-Stencil Fast Marching Method With a Non-Constant Local Cost Model

2019 ◽  
Vol 28 (4) ◽  
pp. 1967-1979
Author(s):  
Susana Merino-Caviedes ◽  
Lucilio Cordero-Grande ◽  
Maria Teresa Perez ◽  
Pablo Casaseca-de-la-Higuera ◽  
Marcos Martin-Fernandez ◽  
...  
Keyword(s):  
Cost Model ◽  
Second Order ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Marching Method

scholarly journals Segmentation of Image Using Watershed and Fast Level set methods

2012 ◽  
pp. 38-43
Author(s):  
Minal M. Purani ◽  
Shobha Krishnan
Keyword(s):  
Medical Imaging ◽  
Level Set Method ◽  
Level Set ◽  
Medical Images ◽  
Level Set Methods ◽  
Initial Boundary ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Watershed Transform ◽  
Marching Method

Technology is proliferating. Many methods are used for medical imaging .The important methods used here are fast marching and level set in comparison with the watershed transform .Since watershed algorithm was applied to an image has over clusters in segmentation . Both methods are applied to segment the medical images. First, fast marching method is used to extract the rough contours. Then level set method is utilized to finely tune the initial boundary. Moreover, Traditional fast marching method was modified by the use of watershed transform. The method is feasible in medical imaging and deserves further research. It could be used to segment the white matter, brain tumor and other small and simple structured organs in CT and MR images. In the future, we will integrate level set method with statistical shape analysis to make it applicable to more kinds of medical images and have better robustness to noise.

3-D traveltime computation using the fast marching method

Geophysics ◽  
1999 ◽  
Vol 64 (2) ◽  
pp. 516-523 ◽  
Author(s):  
James A. Sethian ◽  
A. Mihai Popovici
Keyword(s):  
Point Source ◽  
Eikonal Equation ◽  
Initial Conditions ◽  
Plane Waves ◽  
Three Dimensions ◽  
Computational Domain ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Grid Points ◽  
Marching Method

We present a fast algorithm for solving the eikonal equation in three dimensions, based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagation wavefronts. We begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details. We then show examples of traveltime propagation through the SEG/EAGE salt model using point‐source and plane‐wave initial conditions and analyze the error in constant velocity media. The algorithm allows for any shape of the initial wavefront. While a point source is the most commonly used initial condition, initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another. The algorithm presented here is designed for computing first‐arrival traveltimes. Nonetheless, since it exploits the fast marching method for solving the eikonal equation, we believe it is the fastest of all possible consistent schemes to compute first arrivals.

A Generalized Fast Marching Method on Unstructured Triangular Meshes

2013 ◽  
Vol 51 (6) ◽  
pp. 2999-3035 ◽  
Author(s):  
E. Carlini ◽  
M. Falcone ◽  
Ph. Hoch
Keyword(s):  
Fast Marching Method ◽  
Triangular Meshes ◽  
Fast Marching ◽  
Marching Method
Sign in / Sign up

Export Citation Format

Share Document