Single Pass Computation of First Seismic Wave Travel Time in Three Dimensional Heterogeneous Media With General Anisotropy

Abstract

We present a numerical method for computing the first arrival travel-times of seismic waves in media defined by a general Hooke tensor, in contrast with previous methods which are limited to a specific subclass of anisotropic media, such as "tilted transversally isotropic" (TTI) media or "tilted orthorhombic" (TOR) media [WYF15, LBBMV19]. Our method proceeds in a single pass over the discretized domain, similar to the fast marching method, whereas existing methods for these types of anisotropy require multiple iterations, similar to the fast sweeping method. We introduce a new source factorization model, making it possible to achieve third-order accuracy in smooth media. We also validate our solver by comparing it with the solution of the elastic wave equation in a 3D medium with general anisotropy.

Monotone Discretization of Anisotropic Differential Operators Using Voronoi’s First Reduction

“…Let us mention that, in a spirit similar to Theorem 1.3, a Lipschitz estimate for solutions of the discretized eikonal equation ( 47) is established in [15], using closely related techniques and starting from the bound max{0, δ e h u(x) ≤ max{0,…”

“…Therefore the error estimate ) with the optimal parameter choice ε = h 1 3 . The K-Lipschitz and ε-spanning properties lead to a Lipschitz estimate of the discrete solution u * h , namely |u * h (x) − u * h (y)| ≤ C|x − y|, for any x, y ∈ Ω h and any sufficiently small scale h > 0, see [15,Proposition 4.4]. This estimate rules out numerical instabilities such as checkerboards artifacts, and is also a necessary property if one wants to consider point source boundary conditions, so as to compute the geodesic distance from a single seed rather than from the domain's boundary.…”

“…The constants h 0 > 0 and n 0 only depend on (d, R, K, ε).Proof. A closely related argument appears in[15, Lemma 4.3], in the context of the discretization of a three-dimensional eikonal equation based on Selling's algorithm. Let h > 0 and x 0 ∈ T d h .…”

Monotone discretization of the Monge–Ampère equation of optimal transport

“…uniformly over x ∈ G h . Similarly, for any v ∈ V h and e ∈ v, we may assume, using ( 10) and ( 21), that h is small enough so that x ± he ∈ G h whenever x ∈ K ∩ G h , and then, using ( 14) and the same reasoning as above, (21) for the last equality,…”

Massively parallel computation of globally optimal shortest paths with curvature penalization

“…In that case, monotony holds by construction, whereas the causality property can be derived from a geometrical acuteness property of the discretization stencils. 23,[27][28][29][30] The fixed point of a monotonous operator Λ can be obtained using a variety of iterative methods such as References 17,19, or 9-12 on the GPU.…”