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.

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.

BOAT-SAIL VORONOI DIAGRAM AND ITS APPLICATION

2009 ◽  
Vol 19 (05) ◽  
pp. 425-440 ◽  
Author(s):  
TETSUSHI NISHIDA ◽  
KOKICHI SUGIHARA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Voronoi Diagram ◽  
Water Surface ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Marching Method ◽  
Generalized Voronoi Diagram

A new generalized Voronoi diagram, called a boat-sail Voronoi diagram, is defined on the basis of the time necessary for a boat to reach on water surface with flow. A new concept called a boat-sail distance is introduced on the surface of water with flow, and it is used to define a generalized Voronoi diagram, in such a way that the water surface is partitioned into regions belonging to the nearest harbors with respect to this distance. The problem of computing this Voronoi diagram is reduced to a boundary value problem of a partial differential equation, and a numerical method for solving this problem is constructed. The method is a modification of a so-called fast marching method originally proposed for the eikonal equation. Computational experiments show the efficiency and the stableness of the proposal method. We also apply our equation to the shortest path problem and the simulation of the forest fire.

Multiple reflection and transmission phases in complex layered media using a multistage fast marching method

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1338-1350 ◽  
Author(s):  
N. Rawlinson ◽  
M. Sambridge
Keyword(s):  
Layered Media ◽  
Computational Domain ◽  
Grid Refinement ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Reflection And Transmission ◽  
Local Grid Refinement ◽  
Local Grid ◽  
Marching Method ◽  
Grid Based

Traditional grid‐based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only. However, later arrivals can be numerous and of greater amplitude, making them a potentially valuable resource for practical applications such as seismic imaging. The aim of this paper is to introduce a grid‐based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media. A finite‐difference eikonal solver known as the fast marching method (FMM) is used to propagate wavefronts from one interface to the next. By treating each layer that the wavefront enters as a separate computational domain, one obtains a refracted branch by reinitializing FMM in the adjacent layer and a reflected branch by reinitializing FMM in the incident layer. To improve accuracy, a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high. Several examples are presented which demonstrate the viability of the new method in highly complex layered media. Even in the presence of velocity variations as large as 8:1 and interfaces of high curvature, wavefronts composed of many reflection and transmission events are tracked rapidly and accurately. This is because the scheme retains the two desirable properties of a single‐stage FMM: computational speed and stability. Local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost.

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.

3D Facial Shape Reconstruction from a Single Frontal Image

2014 ◽  
Vol 989-994 ◽  
pp. 3544-3547
Author(s):  
Qian Ma
Keyword(s):  
Eikonal Equation ◽  
Shape From Shading ◽  
Shape Reconstruction ◽  
Fast Marching Method ◽  
Improved Method ◽  
Fast Marching ◽  
Facial Shape ◽  
Shape Invariant ◽  
Marching Method

In this paper, we propose an improved method for reconstruct 3D facial shape from a single frontal image. We use improved fast marching method to solve the Eikonal equation which can obtained from the method of shape from shading (SFS). In order to overcome the concave-convex ambiguity problems inherent to SFS, we find out the concave region and recover concave into the convex keeping the relative shape invariant to reconstruct the accuracy facial shape.

PyKonal: A Python Package for Solving the Eikonal Equation in Spherical and Cartesian Coordinates Using the Fast Marching Method

2020 ◽  
Vol 91 (4) ◽  
pp. 2378-2389
Author(s):  
Malcolm C. A. White ◽  
Hongjian Fang ◽  
Nori Nakata ◽  
Yehuda Ben-Zion
Keyword(s):  
Open Source ◽  
Eikonal Equation ◽  
Heterogeneous Media ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Cartesian Coordinates ◽  
Source Codes ◽  
High Level ◽  
Marching Method ◽  
Python Package

Abstract This article introduces PyKonal: a new open-source Python package for computing travel times and tracing ray paths in 2D or 3D heterogeneous media using the fast marching method for solving the eikonal equation in spherical and Cartesian coordinates. Compiled with the Cython compiler framework, PyKonal offers a Python application program interface (API) with execution speeds comparable to C or Fortran codes. Designed to be accurate, stable, fast, general, extensible, and easy to use, PyKonal offers low- and high-level API functions for full control and convenience, respectively. A scale-independent implementation allows problems to be solved at micro, local, regional, and global scales, and precision can be improved over existing open-source codes by combining different coordinate systems. The resulting code makes state-of-the-art computational capabilities accessible to novice programmers and is efficient enough for modern research problems in seismology.

scholarly journals Fast Marching Method Aplication for Forward Modelling of Seismic Wave Propagation

2018 ◽  
Vol 16 (3) ◽  
pp. 1
Author(s):  
Wahyu Srigutomo ◽  
Ghany Hanifan Muslim
Keyword(s):  
Wave Propagation ◽  
Viscosity Solution ◽  
Seismic Wave ◽  
Eikonal Equation ◽  
Heterogeneous Media ◽  
Seismic Wave Propagation ◽  
Time Travel ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Marching Method

One of the classical problem in seismology is to determine time travel and ray path of seismic wave betweentwo points at a given heterogeneous media. This problem is expressed by eikonal equation and can be seen as a propagation of a wavefront and interface evolution. One of methods to solve this problem is Fast Marching Method abbreviated as FMM. This method is used to produce entropy-satisfying viscosity solution of eikonal equation. FMM combines viscosity solution of Hamilton-Jacobi equation and Huygen's Principle that centered on min-heap data structure to determine the minimum value at every loop. In this study, FMM is applied to determine time travel and raypath of seismic wave. FMM also is used to determine the location of wavesource using simple algorithm. From our forward modeling schemes, it is found that FMM is an accurate, robust, and effcient method to simulate seismic wave propagation. For further study, FMM also can be used to be a part of passive seismic inverse scheme to locate hypocenter location.

A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions

2011 ◽  
Vol 33 (5) ◽  
pp. 2402-2420 ◽  
Author(s):  
Shahnawaz Ahmed ◽  
Stanley Bak ◽  
Joyce McLaughlin ◽  
Daniel Renzi
Keyword(s):  
Eikonal Equation ◽  
Two Dimensions ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Third Order ◽  
Marching Method
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