An iterative, fast-sweeping-based eikonal solver for 3D tilted anisotropic media

Abstract

Computation of first-arrival traveltimes for quasi-P waves in the presence of anisotropy is important for high-end near-surface modeling, microseismic-source localization, and fractured-reservoir characterization — and it requires solving an anisotropic eikonal equation. Anisotropy deviating from elliptical anisotropy introduces higher order nonlinearity into the eikonal equation, which makes solving the eikonal equation a challenge. We addressed this challenge by iteratively solving a sequence of simpler tilted elliptically anisotropic eikonal equations. At each iteration, the source function was updated to capture the effects of the higher order nonlinear terms. We used Aitken’s extrapolation to speed up convergence rate of the iterative algorithm. The result is an algorithm for computing first-arrival traveltimes in tilted anisotropic media. We evaluated the applicability and usefulness of our method on tilted transversely isotropic media and tilted orthorhombic media. Our numerical tests determined that the proposed method matches the first arrivals obtained by wavefield extrapolation, even for strongly anisotropic and highly complex subsurface structures. Thus, for the cases where two-point ray tracing fails, our method can be a potential substitute for computing traveltimes. The approach presented here can be easily extended to compute first-arrival traveltimes for anisotropic media with lower symmetries, such as monoclinic or even the triclinic media.

Traveltime approximations and parameter estimation for orthorhombic media

“…Obtaining the roots of such a polynomial is very difficult. On the other hand, solving the eikonal equation for the simple ellipsoidal anisotropic (EA) medium, after a finite-difference discretization, is cheaper (Waheed et al, 2015b). In fact, by setting η 1 ¼ 0, η 2 ¼ 0, and χ ¼ ðv 2 ∕v 1 Þ, equation 1 reduces to the eikonal equation for an EA medium:…”

“…Traveltime contours for SEG/EAGE salt model described by its P-wave vertical velocity and the parameters η 1 ¼ 0.25, η 2 ¼ 0.2, δ 1 ¼ 0.2, δ 2 ¼ 0.15, and δ 3 ¼ 0.1. Solid contours are the reference traveltimes (Waheed et al, 2015b), and the dashed contours are (a) the ellipsoid background traveltimes (equation 5) and (b) the approximate traveltimes contours (equation 10). In both plots, the P-wave vertical velocity is shown in the background.…”

“…We show two examples that assess the accuracy of equation 10. The reference traveltime solutions are numerically obtained using the iterative approach of Waheed et al (2015b).…”

“…For instance, discretizing the eikonal equation describing a TI or an orthorhombic medium requires, respectively, finding the traveltime solutions of a quartic or a sextic polynomial at each computational step. To alleviate this difficulty, Ma and Alkhalifah (2013), , and Waheed et al (2015b) suggest computing solutions of the anisotropic eikonal equation by successively solving the simpler elliptic anisotropy eikonal equation, in which the additional nonlinearity of the more complex (i.e., TI) anisotropy is stored in the elliptic coefficients. Ray tracing for P-waves in orthorhombic media with varying tilt is proposed (Masmoudi and Pšenčík, 2014;Kainkaryam et al, 2015).…”

First-arrival traveltime tomography for anisotropic media using the adjoint-state method

“…Specifically, we use an iterative fast sweeping-based algorithm to solve the acoustic anisotropic eikonal equation. The details of the algorithm can be found in Waheed et al (2015b).…”