The Carnot-Carathéodory distance vis-à-vis the eikonal equation and the infinite Laplacian

2010 ◽  
Vol 42 (3) ◽  
pp. 395-404
Author(s):  
Thomas Bieske
Keyword(s):  
Eikonal Equation ◽  
Carathéodory Distance

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

Solution of the Eikonal equation by a finite‐difference method

1990 ◽  
Author(s):  
Fuhao Qin ◽  
Kim Bak Olsen ◽  
Yi Luo ◽  
Gerard T. Schuster
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Eikonal Equation

scholarly journals A new traveltime approximation for TI media

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. C37-C42 ◽  
Author(s):  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Power Series ◽  
Eikonal Equation ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Efficient Tool ◽  
Trade Off ◽  
Background Medium ◽  
The Common ◽  
Anisotropy Parameters ◽  
Ti Media

In a transversely isotropic (TI) medium, the trade-off between inhomogeneity and anisotropy can dramatically reduce our capability to estimate anisotropy parameters. By expanding the TI eikonal equation in power series in terms of the aneliptic parameter [Formula: see text], we derive an efficient tool to estimate (scan) for [Formula: see text] in a generally inhomogeneous, elliptically anisotropic background medium. For a homogeneous-tilted transversely isotropic medium, we obtain an analytic nonhyperbolic moveout equation that is accurate for large offsets. In the common case where we do not have well information and it is necessary to resolve the vertical velocity, the background medium can be assumed isotropic, and the traveltime equations becomes simpler. In all cases, the accuracy of this new TI traveltime equation exceeds previously published formulations and demonstrates how [Formula: see text] is better resolved at small offsets when the tilt is large.

Finite‐difference calculation of direct‐arrival traveltimes using the eikonal equation

Geophysics ◽  
2002 ◽  
Vol 67 (4) ◽  
pp. 1270-1274 ◽  
Author(s):  
Le‐Wei Mo ◽  
Jerry M. Harris
Keyword(s):  
Finite Difference ◽  
Ray Tracing ◽  
Finite Differences ◽  
Eikonal Equation ◽  
Velocity Model ◽  
Uniform Grid ◽  
Square Grid ◽  
Kirchhoff Migration ◽  
Difference Calculation

Traveltimes of direct arrivals are obtained by solving the eikonal equation using finite differences. A uniform square grid represents both the velocity model and the traveltime table. Wavefront discontinuities across a velocity interface at postcritical incidence and some insights in direct‐arrival ray tracing are incorporated into the traveltime computation so that the procedure is stable at precritical, critical, and postcritical incidence angles. The traveltimes can be used in Kirchhoff migration, tomography, and NMO corrections that require traveltimes of direct arrivals on a uniform grid.

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. S211-S219 ◽  
Author(s):  
Siwei Li ◽  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Eikonal Equation ◽  
Order Partial Differential Equation ◽  
Fast Marching ◽  
First Order ◽  
Velocity Models ◽  
Kirchhoff Migration ◽  
Input Source ◽  
First Arrival ◽  
Synthetic Models

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.

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