Fast-phase space computation of multiple arrivals S. Fomel and J. A. Sethian* Department of Mathematics, University of California, Berkeley, CA 94720 Edited by Cathleen S. Morawetz, New York University, New York, NY, and approved January 28, 2002 (received for review September 7, 2001) We present a fast, general computational technique for computing of angle-gather migration (2, S. Brandsberg-Dahl, M.V. de Hoop the phase-space solution of static Hamilton–Jacobi equations. Start- & B. Ursin, unpublished work). In contrast with our approach, ing with the Liouville formulation of the characteristic equations, we we note that to obtain the exit time and position from each of derive ‘‘Escape Equations’’ which are static, time-independent Eule- the N2 interior grid points using a Lagrangian approach would rian PDEs. They represent all arrivals to the given boundary from all require integration from each of N3 starting values; a typical possible starting configurations. The solution is numerically con- integration would require N steps, giving an operation count of structed through a ‘‘one-pass’’ formulation, building on ideas from N4. Analogously, for a problem in three-dimensional physical semi-Lagrangian methods, Dijkstra-like methods for the Eikonal space with N points on each edge of a computational cube in equation, and Ordered Upwind Methods. To compute all possible physical space, we find all possible exit trajectories for all trajectories corresponding to all possible boundary conditions, the possible boundary conditions in O(N5 log N). technique is of computational order O(N log N), where N is the total In the case where particular boundary conditions are known number of points in the computational phase-space domain; any in advance, computational speedup is possible; this is discussed particular set of boundary conditions then is extracted through rapid after the algorithm is introduced. post-processing. Suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in Formulation of Problem advance. As an application, we apply the technique to the problem of Consider the static Hamilton–Jacobi equation H(x, ٌu) ϭ 0. A computing first, multiple, and most energetic arrivals to the Eikonal nonlinear Hamiltonian H may not have a unique solution, even with equation. smooth boundary data and smooth H. A particular, viscosity-type solution can be selected (3, 4), corresponding to the earliest arrival e present a fast, general computational technique for from the given boundary. Fast algorithms for computing these Wcomputing phase-space solutions of static Hamilton– viscosity-satisfying first-arrival solutions have been developed in Jacobi equations. We derive a set of ‘‘Escape Equations’’ that are recent years. Tsitsiklis developed the first such method (5) for static, time-independent Eulerian partial differential equations solving the Eikonal equation in a Dijkstra-like setting based on an which represent all arrivals to the given boundary from all optimal control perspective; Fast Marching Methods (6, 7) take a possible starting configurations. Following the strategy proposed finite difference perspective to obtain higher order schemes and in (1) we solve these Escape Equations by systematically con- schemes on unstructured meshes (see also ref. 8 for comparison of structing space marching the solution in increasing order, using a similar algorithm with a volume-of-fluid approach); and Sethian a ‘‘one-pass’’ formulation. This means that the solution at each and Vladimirsky (ref. 9, and J. Sethian & A. Vladimirsky, unpub- point in the computational mesh is computed only k times, where lished work) developed so-called ‘‘Ordered Upwind Methods’’ to k does not depend on the number of points in the mesh. compute solutions of general convex static Hamilton–Jacobi equa- The algorithm combines ideas of semi-Lagrangian methods, tions which arise in anisotropic front propagation and optimal Dijkstra-like methods for the Eikonal equation, and Ordered control. These first arrivals are of considerable importance in a Upwind Methods. The method is unconditionally stable, with no large collection of problems, such as computing seismic travel times time-step restriction, and can be made higher-order accurate. (10); see refs. 7 and 11 for reviews. APPLIED We demonstrate the applicability of this technique by computing However, later arrivals may carry additional valuable infor- MATHEMATICS multiple arrivals to the Eikonal equation in a variety of settings. mation, and it is often desirable to compute all possible solutions. The methods presented here are efficient. The Escape Equa- For example, in geophysical simulations, first arrivals may not tions are posed time-independent Eulerian equations in phase correspond to the most energetic arrivals, and this can cause space, whose solution gives the exit time and location for all problems in seismic imaging (12, 13). possible trajectories, starting from all interior points, initialized There are two approaches to multiple arrivals. in all directions. The computational speed depends on whether Y The first is the Lagrangian (ray tracing) approach (14, 15) and one wants to, in fact, obtain results for all possible boundary its variations (16, 17). Here, the phase space characteristic conditions, or, in fact, only for a particular subset of possibilities. equations are integrated, often from a source point, resulting To illustrate, consider a two-dimensional problem consisting in a Lagrangian structure which fans out over the domain. This of a region and its boundary; we discretize the region with a is a valuable and common approach; however, it can face square mesh with N points on each side. Thus, the physical space difficulties either in low-ray density zones where there are very corresponding to the interior consists of N2 points, with N points few rays or near caustics where rays cross. In addition, the use on the boundary (we ignore constants). of irregular computational grid is often inconvenient. In the most general form of boundary conditions, such as those Y A different approach is to work with an Eulerian description which occur in applications such as tomography and seismic of the problem, in either the physical domain or phase space, migration, one needs to solve multiple boundary problems with and attempt to extract multiple arrivals. In recent years, this -H(x, ٌu) ϭ 0 and the point-source boundary condition u(x) ϭ has led to many fascinating and clever Eulerian partial dif 0 for x ϭ s with a set of sources s distributed on the surface of ferential equation based approaches to computing multiple the observational domain. In this case, the solutions span arrivals, including slowness-matching algorithms (18), dy- three-dimensional space, composed of x and s. Because of our namic surface extension algorithms (19) and its modification use of a fast-ordering scheme, we can find all possible exit times and locations for all possible trajectories in O(N3 log N). One can use the output of such computation either for extracting multiple This paper was submitted directly (Track II) to the PNAS office. arrivals for a particular set of sources or directly, as in the method *To whom reprint requests should be addressed. E-mail: [email protected]. www.pnas.org͞cgi͞doi͞10.1073͞pnas.102476599 PNAS ͉ May 28, 2002 ͉ vol. 99 ͉ no. 11 ͉ 7329–7334 Downloaded by guest on September 28, 2021 (20), segment projection methods, and ‘‘big-ray tracing’’ (21) collide, the solution in the physical space becomes multi-valued, and (see also ref. 22). We note that the regularity of the phase interpolating it onto a regular x grid presents a difficult computa- space has been utilized previously in theoretical studies on the tional problem (26). asymptotic wave propagation (23, 24). Liouville Formulation of Phase-Space Solution. We now convert the As an example, consider a one-dimensional closed curve bound- phase-space approach into a set of Liouville equations; these have ing a region in the plane, and suppose one has a collection of sources been used extensively in different applications by Chorin, Hald, and located along the entire boundary; the goal is to consider a front Kuperfman (27, 28). Eqs. 3 and 4 form a system of coupled ordinary propagating inwards from this boundary. The Lagrangian approach differential equations, starting with a particular set of initial con- is to work in phase space and discretize this boundary into a set of ditions. The Liouville equation is a partial differential equation for marker points, whose motion is determined by solving the charac- the same solution with the differentiation performed with respect teristic equations. The curve then evolves in three-dimensional to the initial conditions; it describes the local change in the solution phase space, and the projection of the curve back into two dimen- in response to changes in the initial conditions. sional physical space produces the multiple arrivals. An Eulerian To simplify notation, let us denote the phase-space vector (x, p), formulation of this same approach was pursued by Engquist, by y, the right-hand side of system given in Eq. 3 by vector function Runborg, and Tornberg (35) by using the Vlasov equation to R(y), and the right-hand side of Eq. 4 by the function r(y). In this describe the motion of this curve. In their Segment Projection Method, the curve moving in three-dimensional phase space is notation, the Hamilton–Jacobi system takes the form viewed from several different coordinate systems, so that it always Ѩy͑y , ␴͒ Ѩu͑y , ␴͒ 0 ϭ ͑ ͒ 0 ϭ ͑ ͒ remains locally a graph. A version of this approach using two-level Ѩ␴ R y ; Ѩ␴ r y , [5] set functions was performed by Osher, Cheng, Kang, Shim, and Tsai (unpublished work). and is initialized at ␴ ϭ 0asy ϭ y and u ϭ 0.† The approach presented in this paper computes the solution 0 In the Appendix, we show that the solution of system Eq.