The accuracy of computed traveltimes in a velocity model plays a crucial role in localization of microseismic events. The conventional approach usually utilizes robust fast sweeping or fast marching methods to solve the eikonal equation numerically with a finite-difference scheme. These methods introduce traveltime errors that strongly depend on the direction of wave propagation. Such error results in moveout changes of the computed traveltimes and introduces significant location bias. The issue can be addressed by using a finite-difference scheme to solve the factored eikonal equation. This equation yields significantly more accurate traveltimes and therefore reduces location error, though the traveltimes computed with the factored eikonal equation still contain small errors with systematic bias. Alternatively, the traveltimes can be computed using a physics-informed neural network solver, which yields more randomized traveltimes and resulting location errors.
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.
Fast and accurate traveltime computation for quasi-P waves in anisotropic media is an essential ingredient of many seismic processing and interpretation applications such as Kirchhoff modeling and migration, microseismic source localization, and traveltime tomography. Fast-sweeping methods are widely used for solving the anisotropic eikonal equation due to their flexibility in solving general equations compared to the fast-marching method. However, it has been observed that fast sweeping can be much less efficient than fast marching for models with curved characteristics and practical grid sizes. By representing a tilted transversely isotropic (TTI) equation as a sequence of elliptically isotropic (EI) eikonal equations, we determine that the fast-marching algorithm can be used to compute fast and accurate traveltimes for TTI media. The tilt angle is absorbed into the description of the effective EI model; therefore, the adopted approach does not compromise on the solution accuracy. Through tests on benchmark synthetic models, we test our fast-marching algorithm and discover considerable improvement in accuracy by using factorization and a second-order finite-difference stencil. The adopted methodology opens the door to the possibility of using the fast-marching algorithm for a wider class of anisotropic eikonal equations.
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.
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.
A comparison of fast marching, fast sweeping and fast iterative methods for the solution of the eikonal equation
