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OVERVIEW APPLICATIONS INTERACTIVE APPLETS HISTORY OF THE METHODS/FLOW CHART PUBLICATIONS EDUCATIONAL MATERIAL ACKNOWLEDGEMENTS ABOUT THE AUTHOR/CV Copyright: 1996, 1999, 2006 J.A. Sethian 
Applications to Seismology


AbstractWe present a fast algorithm for solving the eikonal equation in three dimensions, based on the Fast Marching Method (FMM). The algorithm is of 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 global 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 based on the Fast Marching Method (FMM), and follow with the numerical details. We then examples of traveltime propagation through the SEG/EAGE salt dome and analyze the errors in several 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 that it is the fastest of all possible consistent schemes to compute first arrivals. We suggest several modifications to the basic algorithm that can lead to a most energetic arrivals traveltime computation module.Download publications
AbstractWe present a fast, general computational technique for computing the phasespace solution of static HamiltonJacobi equations. Starting with the Liouville formulation of the characteristic equations, we derive ``Escape Equations'' which are static, timeindependent Eulerian PDEs. They represent all arrivals to the given boundary from all possible starting configurations. The solution is numerically constructed through a `onepass' formulation, building on ideas from semiLagrangian methods, Dijkstralike methods for the Eikonal equation, and Ordered Upwind Methods. To compute all possible trajectories corresponding to all possible boundary conditions, the technique is of computational order O(N \log N), where N is the total number of points in the computational phasespace domain; any particular set of boundary conditions is then extracted through rapid postprocessing. Suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance. As an application, we apply the technique to the problem of computing first, multiple, and most energetic arrivals to the Eikonal equation.
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