A slowness matching Eulerian method for multivalued solutions of Eikonal equations

Traveltime, or geodesic distance, is locally the solution of the eikonal equation of geometric optics. However traveltime between sufficiently distant points is generically multivalued. Finite difference eikonal solvers approximate only the viscosity solution, which is the smallest value of the (mul...

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Bibliographic Details
Published in:Journal of scientific computing 2003-12, Vol.19 (1-3), p.501-526
Main Authors: SYMES, William W, JIANLIANG QIAN
Format: Article
Language:English
Subjects:
Algorithms
Differential geometry
Eikonal equation
Exact sciences and technology
Finite difference method
Geodesy
Geometrical optics
Matching
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Partial differential equations, boundary value problems
Sciences and techniques of general use
Solvers
Stitches
Viscosity
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Summary:Traveltime, or geodesic distance, is locally the solution of the eikonal equation of geometric optics. However traveltime between sufficiently distant points is generically multivalued. Finite difference eikonal solvers approximate only the viscosity solution, which is the smallest value of the (multivalued) traveltime (“first arrival time”). The slowness matching method stitches together local single-valued eikonal solutions, approximated by a finite difference eikonal solver, to approximate all values of the traveltime. In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation, so that the eikonal equation may be viewed as an evolution equation in one of the spatial directions. This paraxial assumption simplifies both the efficient computation of local traveltime fields and their combination into global multivalued traveltime fields via the slowness matching algorithm. The cost of slowness matching is on the same order as that of a finite difference solver used to compute the viscosity solution, when traveltimes from many point sources are required as is typical in seismic applications. Adaptive gridding near the source point and a formally third order scheme for the paraxial eikonal combine to give second order convergence of the traveltime branches.
ISSN:0885-7474
1573-7691
DOI:10.1023/A:1025380731197