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

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.

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

Geophysics ◽

10.1190/1.1444764 ◽

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.

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Improved Characteristic Fast Marching Method for the Generalized Eikonal Equation in a Moving Medium

Journal of Scientific Computing ◽

10.1007/s10915-019-01093-z ◽

2019 ◽

Vol 81 (3) ◽

pp. 2484-2502

Author(s):

Myong-Song Ho ◽

Ju-Hyok Ri ◽

Sin-Bom Kim

Keyword(s):

Eikonal Equation ◽

Moving Medium ◽

Fast Marching Method ◽

Fast Marching ◽

Marching Method

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3D Facial Shape Reconstruction from a Single Frontal Image

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.989-994.3544 ◽

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.

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

Seismological Research Letters ◽

10.1785/0220190318 ◽

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.

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Fast Marching Method Aplication for Forward Modelling of Seismic Wave Propagation

Jurnal Geofisika ◽

10.36435/jgf.v16i3.107 ◽

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.

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3-D traveltime computation using the fast marching method

Geophysics ◽

10.1190/1.1444558 ◽

1999 ◽

Vol 64 (2) ◽

pp. 516-523 ◽

Cited By ~ 268

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.

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Computing Pressure Front Propagation Using the Diffusive-Time-of-Flight in Structured and Unstructured Grid Systems via the Fast-Marching Method

SPE Journal ◽

10.2118/201771-pa ◽

2021 ◽

pp. 1-21

Author(s):

Hongquan Chen ◽

Tsubasa Onishi ◽

Jaeyoung Park ◽

Akhil Datta-Gupta

Keyword(s):

Finite Difference ◽

Analytical Solutions ◽

Eikonal Equation ◽

Front Propagation ◽

Fast Marching Method ◽

Fast Marching ◽

Reservoir Properties ◽

Grid Systems ◽

Marching Method ◽

Diffusive Time

Summary Diffusive-time-of-flight (DTOF), representing the travel time of pressure front propagation, has found many applications in unconventional reservoir performance analysis. The computation of DTOF typically involves upwind finite difference of the Eikonal equation and solution using the fast-marching method (FMM). However, the application of the finite difference-based FMM to irregular grid systems remains a challenge. In this paper, we present a novel and robust method for solving the Eikonal equation using finite volume discretization and the FMM. The implementation is first validated with analytical solutions using isotropic and anisotropic models with homogeneous reservoir properties. Consistent DTOF distributions are obtained between the proposed approach and the analytical solutions. Next, the implementation is applied to unconventional reservoirs with hydraulic and natural fractures. Our approach relies on cell volumes and connections (transmissibilities) rather than the grid geometry, and thus can be easily applied to complex grid systems. For illustrative purposes, we present applications of the proposed method to embedded discrete fracture models (EDFMs), dual-porositydual-permeability models (DPDK), and unstructured perpendicular-bisectional (PEBI) grids with heterogeneous reservoir properties. Visualization of the DTOF provides flow diagnostics, such as evolution of the drainage volume of the wells and well interactions. The novelty of the proposed approach is its broad applicability to arbitrary grid systems and ease of implementation in commercial reservoir simulators. This makes the approach well-suited for field applications with complex grid geometry and complex well architecture.

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