A discontinuous Galerkin fast-sweeping eikonal solver for fast andaccurate traveltime computation in 3D tilted anisotropic media

Philippe Le Bouteiller1, Mondher Benjemaa2, Ludovic Métivier3, and Jean Virieux1

ABSTRACT

We tackle the challenging problem of efficient and accurateseismic traveltime computation in 3D anisotropic media byapplying the fast-sweeping method to a discontinuous Galer-kin (DG)-based eikonal solver. Using this method leads to astable and highly accurate scheme, which is faster than finite-difference schemes for a given precision, and with a low com-putational cost compared to the standard Runge-Kutta DGformulation. The integral formulation of the DG method alsomakes it easy to handle seismic anisotropy and complextopographies. Several numerical tests on complex models,such as the 3D SEG advanced modeling model, are givenas illustration, highlighting the efficiency and the accuracyof this new approach. In the near future, these results willbe used together with accurate solvers for seismic amplitudeand take-off angle computation to revisit asymptotic inversion(traveltime/slope tomography) and imaging approaches(quantitative migration involving amplitudes and angles).

INTRODUCTION

Asymptotic approaches based on traveltime and amplitude com-putation in the high-frequency approximation are widely used inmany seismic applications such as traveltime/slope tomography forinitial velocity model building (Le Meur, 1994; Hole and Zelt,1995; Billette and Lambaré, 1998; Leung and Qian, 2006; Tailland-ier et al., 2009; Lelièvre et al., 2011; Tavakoli F. et al., 2017) orray-based (quantitative) migration (Beylkin, 1985; Bleistein, 1987;Beylkin and Burridge, 1990; Jin et al., 1992; Gray and May, 1994;Lambaré et al., 2003; Operto et al., 2003). The advantage of a

slowly varying quantity (traveltime) compared with the highly os-cillating wavefield makes possible efficient decimations/interpola-tions over sparse grids for storage, which might be of great interestwhen working with large models and data sets (Mendes, 2000;Vanelle and Gajewski, 2002; Alkhalifah, 2011).In the high-frequency regime, we may consider the Lagrangian

framework of the ray theory to compute traveltimes by tracing thecharacteristics of the eikonal equation (Červený, 2001). Solving therelated ordinary differential equation with initial conditions (sourcelocation and shooting angle) is straightforward and easy to handle.However, when boundary conditions are considered (source andreceiver locations), the two-point ray tracing can be a quite challeng-ing task because of the nonuniform sampling of the medium byrays. Estimating traveltimes at a given point of the medium leadsto sophisticated interpolation/extrapolation techniques, especiallywhen considering shadow zones (Runborg, 2007). In addition, whenconsidering many source/receiver pairs, the ray-tracing approachmight become less efficient, as the computational cost scales asthe product between the number of sources and receivers. Alterna-tively, when one is only interested in first-arrival traveltimes, solvingthe eikonal equation within an Eulerian framework enables the com-putation of first-arrival traveltimes everywhere in the medium, thanksto the concept of viscosity solution (Crandall and Lions, 1983, 1984).Somehow, the geometric theory of diffraction is handled by the vis-cosity solution (Keller, 1962; Runborg, 2007). Moreover, the compu-tational cost is only proportional to the number of sources, whichmakes it very efficient for dense acquisition settings. The drawbackof the eikonal approach is that a nonlinear partial differential equation(PDE) has to be solved, which requires sophisticated numerical toolsto obtain reliable and accurate results.Vidale (1988) promotes an expanding box framework and triggers

an abundant literature about eikonal solvers. The original scheme ofVidale (1988) only computes traveltimes corresponding to outgoing

Manuscript received by the Editor 25 July 2018; revised manuscript received 12 October 2018; published ahead of production 26 December 2018; publishedonline 26 February 2019.

1Université Grenoble Alpes, ISTerre, Grenoble F-38000, France. E-mail: philippe.le-bouteiller@univ-grenoble-alpes.fr (corresponding author); jean.virieux@univ-grenoble-alpes.fr.

2University of Sfax, Laboratory of Stability and Control of Systems and Nonlinear PDE, Sfax 3029, Tunisia. E-mail: mondher.benjemaa@gmail.com.3Université Grenoble Alpes, ISTerre, Grenoble F-38000, France and Université Grenoble Alpes, LJK, CNRS, Grenoble F-38000, France. E-mail: ludovic

.metivier@univ-grenoble-alpes.fr.© 2019 Society of Exploration Geophysicists. All rights reserved.

C107

GEOPHYSICS, VOL. 84, NO. 2 (MARCH-APRIL 2019); P. C107–C118, 8 FIGS., 12 TABLES.10.1190/GEO2018-0555.1

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rays. In heterogeneous media, the first arrival is thus not guaranteedbecause some rays may go back into the expanding square, for in-stance, in the presence of high-velocity zones. Improvements of thistechnique are proposed by Podvin and Lecomte (1991), van Trier andSymes (1991), and Hole and Zelt (1995). Kim and Cook (1999)use (weighted) essentially nonoscillatory (ENO) schemes (Shu andOsher, 1988, 1989; Liu et al., 1994; Jiang and Shu, 1996; Jiangand Peng, 2000) for local discretization, while choosing the expandingbox framework (“Down’N’Out,” DNO) for computing the globalsolution. A postsweeping (PS) technique is added to retrieve the cau-sality and thus the first-arrival traveltimes, which leads to the ENO-DNO-PS algorithm. This is to some extent similar to the approachdeveloped by Hole and Zelt (1995) based on additional reverse propa-gation steps. Instead of relying on an arbitrary squared-box expansion,wavefront-tracking schemes are proposed to better fit causality by fol-lowing the expansion of the wavefront itself (Qin et al., 1992; Cao andGreenhalgh, 1994). Doing so, the PS technique is no longer necessary.In their concepts, ENO-DNO-PS schemes are close to fast-sweep-

ing methods (FSMs) and wavefront-tracking schemes are similar tofast-marching methods (FMMs), two general classes of methodsdeveloped in the field of applied mathematics, which have foundapplications in many domains in the recent years. These methods relyon an ordering of the nodes. FMM belongs to single-pass algorithms,based on Dijsktra’s algorithm (Dijkstra, 1959), which considersthe propagation front and makes it evolve (Tsitsiklis, 1995; Sethian,1996, 1999). On the other hand, FSM belongs to multipass algorithmsrelying on global orderings of the nodes. All nodes are updated duringeach Gauss-Seidel iteration (sweep), following alternating orderings(Boué and Dupuis, 1999; Tsai et al., 2003; Kao et al., 2004; Zhao,2005; Luo and Zhao, 2016). The FMM and FSM have been inten-sively applied to solve the eikonal equation in a wide range of prob-lems. Extensive comparisons showing their numerical efficienciescan be found in Gremaud and Kuster (2006), and they highlight thatdetermining which strategy is the best is highly problem dependent.In the presence of anisotropy, the eikonal equation is more complex

and needs adequate numerical strategies. The ENO-DNO-PS princi-ples were extended to the anisotropic case in Dellinger and Symes(1997), Kim (1999), and Qian et al. (2001). Some extensions havealso been carried out concerning FMM (Cristiani, 2009). They arebased on approximations and are generally difficult to implement.Mir-ebeau (2014a) and Mirebeau (2014b) propose new approaches for theFMM for anisotropic eikonal. However, the most mature strategiesproposed so far in a geophysical context rely on FSM. The first ex-tensions have focused on elliptical anisotropy (Tsai et al., 2003; Qianet al., 2007a), which could be handled quite naturally because itamounts to a dilation in space. The general 2D tilted transverselyanisotropic (TTI) eikonal comprises spatial derivatives of the travel-time to the power of four, which is more challenging. Han et al. (2017)propose to solve the relatedquartic equationand to select theappropriateroot, yielding a high computational cost. Tavakoli F. et al. (2015),Waheed et al. (2015), and Waheed and Alkhalifah (2017) prefera fixed-point iteration technique to solve an elliptical equation ateach iterationwith a suitable right-hand side accounting for anellipticity.Most of the FSM extensions to anisotropy considered 2D prob-

lems only, except that by Waheed et al. (2015), who consider tiltedorthorhombic (TOR) media. Moreover, all the above-mentionedmethods are developed using finite-difference (FD) schemes, gen-erally of first order, or higher order at a high cost with noncompactstencils. The convergence order is in general less than one, or it is

equal to one if the source point is handled correctly by using thecelerity domain, the factorization or the perturbation methods,for instance (Pica, 1997; Zhang et al., 2005a; Fomel et al., 2009;Luo and Qian, 2011; Noble et al., 2014). This results in eikonalsolvers that are efficient and simple to implement, but with limitedaccuracy. Another strategy has been explored recently in Le Bou-teiller et al. (2018), using a discontinuous Galerkin (DG) finite-element discretization instead of an FD approach, to increase theconvergence order, and to obtain high accuracy on traveltimes andspatial derivatives in heterogeneous TTI media. Obtaining derivedquantities such as angles, amplitudes, or curvatures with high ac-curacy is crucial for tomographic/imaging methods. These quantitiesare based on first- and second-order derivatives of the traveltime, sothat second- or third-order schemes are required for traveltime com-putation. Another advantage of such a finite-element approach is thatthe integral finite-element formulation can be performed in complexgeometries, so that complex topographies can be precisely handled.This turns out to be of major interest when considering complex landtargets (Improta et al., 2002; Taillandier et al., 2009).Inspired by the work of Cheng and Shu (2007) and Cheng and

Wang (2014), Le Bouteiller et al. (2018) consider a numericalscheme applicable to generic time-dependent Hamilton-Jacobiequations, a class of equations to which eikonal belongs, and theymake use of the Runge-Kutta (RK) time integration with suitablesource condition until the steady state is reached, which corre-sponds to the sought traveltime solution. We shall refer to thismethod as the RK-DG solver. In terms of computational cost,the main bottleneck of this approach is the rather slow convergencein pseudotime to reach the static solution through all transient statesin every part of the medium, while the evolution of the front is local-ized. This results in an algorithmic complexity of O(dof3∕2) wherethe total number of degrees of freedom is denoted by #dof.It seems therefore natural to integrate the FSM approach in such a

formulation, with O(N) complexity, as an acceleration tool forreaching the steady state. Such an idea is already tested in an FDframework by Zhang et al. (2005b), using FSM as an accelerationloop over the time-marching procedure. Regarding DG approaches,a numerical strategy is developed in two dimensions to solve Ham-ilton-Jacobi equations with an FSM procedure over a DG discreti-zation by Li et al. (2008) and Zhang et al. (2011). Based on the DGsolver of Cheng and Shu (2007), this strategy exhibits some prac-tical limitations: An initial guess is needed and deduced from a pre-liminary FD computation; the local scheme requires a cumbersomeleast-squares L2 reconstruction of the solution’s derivatives at thecell interface. In addition, it is developed only for isotropic media,and the point-source singularity is not cured.We propose to integrate such FSM acceleration into the approach

developed by Le Bouteiller et al. (2018). The implementation ofFSM over this solver and the extension to three dimensions leadsto an accurate and efficient solver for 3D traveltime computation,which we refer to as the FSM-DG solver. Based on the state-of-the-art DG scheme of Cheng and Wang (2014), the L2 reconstruction ofspatial derivatives is avoided. Without any need of an initial solutionguess, features such as point-source singularity treatment, 2D-TTIand 3D-TOR anisotropy, and complex topographies are integratedin this new approach yielding highly accurate traveltime estimationwith a rather simple handling by potential users.The remainder of the paper is organized as follows. First,

the numerical method is detailed: The eikonal equation and its

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causality-consistent DG discretization are recalled, an FSM strategythat consists of a local solver and a global Gauss-Seidel-based strat-egy is presented, and a mesh deformation strategy for topographyhandling is exhibited. Second, the accuracy and the efficiency of theresulting FSM-DG solver are illustrated through various examplesin two dimensions and three dimensions: first in simple media forvalidation purpose, then in complex realistic media, such as the 3DSEAM model, with heterogeneities, topography, and anisotropy toillustrate the properties of the solver on challenging settings. In thelast example, a comparison is performed with a full-wavefield mod-eling. A conclusion closes this study.

NUMERICAL METHOD

Eikonal equation

Hereafter, we adopt the dynamic formulation of the eikonal PDEin the Hamiltonian framework, which is written as

∂ξuðx; ξÞ þHðx;∇xuðx; ξÞÞ ¼ 0; (1)

where the spatial coordinates x span the space Rd with d ¼ 2 ord ¼ 3, and ξ denotes a pseudotime evolution parameter. In anisotropic medium, one can write

HISOðx;∇xuðx; ξÞÞ ¼ k∇xuðx; ξÞk −1

cðxÞ ; (2)

where the wave speed is denoted by cðxÞ. The stationary state ofHamilton-Jacobi equation 1 with the Hamiltonian of equation 2verifies the static eikonal equation H ¼ 0. At the stationary state,we have limξ→∞uðx; ξÞ ≡ TðxÞwhere TðxÞ is the traveltime field. Ina finite computational domain, we set the source boundary condi-tion to uðxs; ξÞ ¼ 0 at any pseudotime ξ at the source point xs, andsuch a stationary state is obtained at a finite pseudotime ξ⋆ once thesource information has been propagated from the source to the en-tire domain. Using this time-marching procedure to reach the steadystate is studied by Zhang et al. (2005b), whereas a formal link be-tween the static and the dynamic Hamilton-Jacobi equations is pro-posed by Osher (1993) through the level-set framework.Following Le Bouteiller et al. (2018), we write the Hamiltonian

for the 2D vertical transversely isotropic (VTI) case as

HVTI ¼ dðu;xÞ2 þ eðu;zÞ2 þ cðu;xÞ2ðu;zÞ2 − 1; (3)

where the derivatives of uðx; z; ξÞ with respect to x and z are, re-spectively, denoted by u;x and u;z, and with( c ¼ −2ðϵ − δÞV4

P;d ¼ ð1þ 2ϵÞV2

P;e ¼ V2

P;(4)

where the Thomsen’s parameters are denoted by ϵ and δ (Thomsen,1986) and the P-wave velocity along the vertical axis is denotedby VP. This derivation comes from Christoffel’s dispersion relationin an elastic medium (see, e.g., Červený, 2001; Slawinski, 2003),considering only the coupled P − SV propagation mode, under theacoustic approximation (Alkhalifah, 2000). The TTI case is re-trieved by applying the local rotation by the angle θðxÞ between thelocal rotation-symmetry axis and the vertical axis, yielding

HTTI¼dðu;x cosθþu;z sinθÞ2þeðu;z cosθ−u;x sinθÞ2þcðu;xcosθþu;z sinθÞ2ðu;zcosθ−u;x sinθÞ2−1: (5)

Similarly, the 3D orthorhombic (OR) Hamiltonian is writtenunder the acoustic approximation:

HOR¼aðu;xÞ2þbðu;yÞ2þcðu;zÞ2þdðu;xÞ2ðu;yÞ2þeðu;xÞ2ðu;zÞ2þfðu;yÞ2ðu;zÞ2þgðu;xÞ2ðu;yÞ2ðu;zÞ2−1; (6)

with

8>>>>>>>>>><>>>>>>>>>>:

a¼V2Pð1þ2ϵ2Þ;

b¼V2Pð1þ2ϵ1Þ;

c¼V2P;

d¼V4Pð1þ2ϵ2Þðð1þ2ϵ2Þð1þ2δÞ−ð1þ2ϵ1ÞÞ;

e¼−2ðϵ2−δ2ÞV4P;

f¼−2ðϵ1−δ1ÞV4P;

g¼−V6Pðð1þ2ϵ2Þ2ð1þ2δÞ−2ð1þ2ϵ2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ2δ2Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ2δ1Þ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ2δÞpþð1þ2δ2Þð1þ2δ1Þ−4ðϵ2−δ2Þðϵ1−δ1ÞÞ;

(7)

where anisotropic parameters ϵ2 and δ2 hold in the ½x; z� plane, aniso-tropic parameters ϵ1 and δ1 hold in the ½y; z� plane, and anisotropicparameter δ holds in the ½x; y� plane. This parameterization comesfrom Tsvankin (1997) and is consistent with an alternative parameter-ization proposed in Alkhalifah (2003) and used in Waheed et al.(2015). The TOR case is retrieved by applying the local 3 × 3 rotationoperator involving three rotation angles: the dip angle θðxÞ, the azi-muth angle ϕðxÞ, and the rotation angle ψðxÞ, which is the rotationangle of the elastic tensor in the rotated horizontal plane, correspond-ing to the crack orientation in this plane. The rotation operatorRTOR

is written as

RTOR ¼

0B@

cos ψ sin ψ 0

− sin ψ cos ψ 0

0 0 1

1CA0B@

cos θ 0 sin θ

0 1 0

− sin θ 0 cos θ

1CA

×

0B@

cos ϕ sin ϕ 0

− sin ϕ cos ϕ 0

0 0 1

1CA: (8)

Replacing quantities u;x, u;y, and u;z by their rotated expressions,obtained from equation 8, into the Hamiltonian of equation 6 yieldsthe TORHamiltonian. The substitution is as simple as for going fromexpression 3 to expression 5, although we do not explicitly write theTOR Hamiltonian here for the sake of concision.To increase the accuracy of the solver, we implement the point-

source factorization, as proposed by Pica (1997), Zhang et al. (2005a),Fomel et al. (2009), and Luo and Qian (2011). Le Bouteiller et al.(2018) extend the factorization principles to the DG discretizationand show that this makes possible to retrieve a second-order conver-gence when using approximations by polynomials of order one. Theadditive factorization embeds the source singularity inside a referencesolution u0ðxÞ such that

uðx; ξÞ ¼ u0ðxÞ þ τðx; ξÞ: (9)

We then plug expression 9 into Hamiltonian expressions 2, 3, and 6.We adopt the additive factorization because the Hamiltonian it yieldsdepends only on the gradient of the unknown τ. On the contrary, theHamiltonian obtained with a multiplicative strategy defined byuðx; ξÞ ¼ u0ðxÞτðx; ξÞ depends also on the unknown τ itself. The

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resulting equation would thus contain an additional term leading outof the frame of equation 1.Finally, the Hamiltonian obtained with the additive factorization is

plugged into the dynamic Hamilton-Jacobi equation 1 that we solvefor τðx; ξÞ. This leads to the factored Hamilton-Jacobi equation forthe isotropic case

∂ξτ þ k∇xu0 þ ∇xτk −1

c¼ 0: (10)

Factored equations for 2D-VTI and 3D-OR cases are given in Appen-dix A (equations A-1 and A-2, respectively), and tilted cases are re-trieved as described before. The reference solutions u0 are chosen tobe analytical solutions in isotropic, TTI, or TOR media with homo-geneous elastic parameters given by their values at the source point.

DG discretization

The DG spatial discretization of equation 1 is proposed by Chengand Wang (2014) and further adapted to the factored TTI eikonal byLe Bouteiller et al. (2018). We present the scheme in its most generalformulation, which handles unstructured polygonal (two dimensions)and polyhedral (three dimensions) meshes. Note that the rectangular/cuboid Cartesian formulation simplifies the discretization of themedium following a natural ordering along the x-, y-, and z-axes,which will be useful for the FSM algorithm, as we shall see. The spaceΩ is partitioned into n elements denoted byKi; i ¼ 1; : : : ; n. For eachelement Ki, we choose a local approximation space Pi spanned by abasis of shape functions ϕj

i ðxÞ. In practice, we consider polynomialspaces Pk containing all polynomials of degree at most k. In ournumerical tests, we used P1 and P2 spaces. We define nKi

to be theoutward unit normal to the boundary of the element Ki. At elementinterfaces, traces v�h and jumps ½vh� of any numerical quantity vhdefined inside two neighboring elements are given, respectively, by

v�h ðxÞ ¼ limϵ↓0

vhðx� ϵnKiÞ;

½vh�ðxÞ ¼ vþh ðxÞ − v−h ðxÞ: (11)

With these expressions, as well as their spatial derivatives and theirprojections on boundaries, different key quantitiesF ;G;K are definedat the boundary between two elements to build a causality consistentflux estimation, which is an essential ingredient of the DG formu-lation. These quantities depend nonlinearly on the solution itself andon the local values of the Hamiltonian function. For the sake of con-cision, the expressions F ;G;K are given in Appendix B.The weak formulation of equation 1 can be stated as follows:

Finduhð:;ξÞ∈fv∶vjKi∈Pi; ∀ i∈f1; :::;ngg ∀ξ≥0such thatZ

Ki

ð∂ξuhðx;ξÞþHðx;∇xuhðx;ξÞÞÞviðxÞdx

þZ∂Ki

Fðx;∇xu�h ðxÞÞ½uh�ðx;ξÞv−i ðxÞds

−CΔKi

XSji∈∂Ki

1

ΔSji

ZSji

Gðx;∇xu�h ðxÞÞ½∇xuh ·nKi�ðx;ξÞv−i ðxÞds

−2CΔKi

XS̄ji∈∂Ki

1

ΔS̄ji

ZS̄ji

Kðx;∇xu�h ðxÞÞð∇xu−h ðx;ξÞ ·nKiÞv−i ðxÞds¼0;

for each i∈f1; :::;ng and for any test functionvi∈Pi; (12)

where ΔKi (respectively ΔSji ) is the volume of the element Ki (re-spectively, the area of the face j of element Ki). The set ∂Ki denotesthe internal faces of element Ki, which are shared with other ele-ments. The set ∂Ki denotes the external faces of element Ki, whichare part of the domain boundary ∂Ω. The test functions vi are shapefunctions as usual for Galerkin approaches (Zienkewicz and Mor-gan, 1983). The first term of scheme 12 ensures consistency, em-bedding a weak formulation of the Hamilton-Jacobi equation. Thesecond term determines the information flow direction and acts asan upwind flux term. It also captures potential shocks, keeping thesmallest traveltime next to triplications of the wavefield. The thirdterm treats so-called rarefaction situations in which noncausal en-tropy violations may occur in the traveltime solution. The scalingfactor C is chosen empirically: Numerous observations have shownthat 0.25 gives stable schemes in practice (Cheng and Wang, 2014).The fourth term, acting on external edges only, is added to enforcesuitable radiative boundary conditions (Le Bouteiller et al., 2018).For further technical knowledge about setting up the RK-DGimplementation, we refer the reader to Cockburn and Shu (1998).

Local solver

Instead of solving scheme 12 over the whole domain in a time-marching approach as proposed by Le Bouteiller et al. (2018), inthis study, we evolve elements one by one to a temporary local sta-tionary solution with a local iterative strategy. In other words, givencurrent solutions in its neighbors, solution inside an element evolvesuntil its local steady state is reached, before considering the nextelement. Therefore, the local solver consists of solving scheme 12repeatedly for a given element Ki. Formally, we introduce the var-iable ξi as a local pseudotime variable, which is no longer global.The local integration, which can be considered as a local fixed-pointprocedure, is performed in ξi with an explicit Euler method. Usingan explicit time integration is advantageous because scheme 12 ishighly nonlinear: The numerical fluxes depend on the solution itselfin a nonlinear way. Moreover, we verify that it is not necessary touse higher order time schemes such as second-order Runge-Kuttamethod in this local solver. The implementation of the local inte-gration is straightforwardly derived from the RK-DG formulation,taking care of restricting the computation to the current element.Iterations are performed until a convergence criterion is reached,which we refer to as the real quantity local_conv.However, in some transient configurations, such convergence

may not be reached due to the nonlinearity of the problem and cau-sality considerations. When the upwind flux is initially oriented to-ward a given direction at an edge of a cell (e.g., an initial orientationgiven by a direct wave), the cell could be in a configuration suchthat changing its orientation to another direction (e.g., a diving wavecoming through a higher velocity zone) turns out to be impossibleduring the fixed-point local procedure. Once the local solver hasreached a given number max_local_iter of local iterations withoutlocal convergence, we have designed a specific procedure, that wecall the degenerate local solver, which is activated for overcomingsuch a situation. This degenerate solver detects, among the four(two dimensions) or six (three dimensions) neighbors of the currentelement, the one with the lowest traveltime values (in practice, wecompare the maximum values at edges). Once the fastest edge hasbeen retrieved, we simulate a (nonphysical) plane wave comingfrom this edge and traveling through the element at the local wavespeed. This allows us to reconstruct a temporary solution with a

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correct orientation of fluxes and respecting the velocity inside theelement. Such a solution will not be the final solution after thesweeping strategy we describe in the next paragraph. This tempo-rary simple estimation enables various branches of the final solutionto propagate along the current sweep and eventually to keep thefastest one at each location. We proceed with sweeps until this de-generate solver is not activated anymore: This is one of the criteriarequired for global convergence.The degenerate local solver we have designed is a key procedure,

which unlocks several crucial issues for developing an efficientFSM-DG method. First, it avoids to be trapped in a wrong causalitysetting, as described above. Second, as a consequence, it exemptsfrom the need of a good initial solution, which was a practical limi-tation in Zhang et al. (2011). Finally, it exempts from a severeconstraint on the Hamiltonian: the RK-DG scheme used in Le Bou-teiller et al. (2018) required a Lipschitz continuous Hamiltonian todefine a suitable CFL condition, which would not depend on thesolution. This limitation prompted the authors to use a complicated2D TTI Hamiltonian to ensure stability in all the elements at everytimestep. In our FSM-DG method, if an instability raises during thelocal integration, then the degenerate solver acts as a posttreatmentlimiter, by replacing the solution with an approximate one respect-ing the local causality. Therefore, we are able to use standard aniso-tropic Hamiltonians of equations 3 and 6.

Fast-sweeping algorithm

The global solution is obtained by applying the local solversuccessively to all the elements in a block Gauss-Seidel approach:Nonlinear equations are solved element by element, each elementrepresenting several unknowns (degrees of freedom). As in Zhao(2005), the alternating sweepings follow the four (two dimensions)or eight (three dimensions) natural orderings of the structured Car-tesian mesh. For unstructured grids, it could be possible to precom-pute specific orderings of the elements for sweeping: We have notimplemented such a strategy because of the additional implemen-tation complexity (Qian et al., 2007a, 2007b). For Cartesian grid,the natural directions allow to sample efficiently the characteristicsof the eikonal equation.A boundary condition is set at the source: Inside the source

element, the numerical solution τ is set to zero and does not evolve,so that the traveltime solution equals the reference solution insidethis source element.

Initialization

Initialization steps are defined for the local solver procedure aswell as for the sweeping procedure. Regarding the sweeping, the firstfour (two dimensions) or eight (three dimensions) sweeps are per-formed from the source element toward the boundaries, respectively,in the four (two dimensions) or eight (three dimensions) quarters ofthe domain defined by the horizontal and vertical axes aligned withthe source point. We have found this to be the best initialization tooptimize the number of sweeps needed to reach the convergence. Atthe local scale, the first time the local solver is called for a givenelement, the degenerate solver is executed at first, then the local iter-ative procedure occurs. The element is tagged as updated afterward.When executing the local solver, only the at-least-once updatedneighbors are considered. At edges where a neighbor has not been

Algorithm 1. Sweeping.

1: procedure SWEEPING

2: call INIT_SWEEP

3: for k ¼ 1 → max iter do

4: select case modðk − 1; 4Þ þ 1

5: case(1) call SWEEP1

6: case(2) call SWEEP2

7: case(3) call SWEEP3

8: case(4) call SWEEP4

9: if kðu − uoldÞ∕uoldk < global conv then return

10: procedure SWEEP1

11: for i ¼ 1 → Nx do

12: for j ¼ 1 → Nz do

13: call LOCAL_SOLVER(i; j)

14: procedure SWEEP2

15: for i ¼ Nx → 1 do

16: for j ¼ 1 → Nz do

17: call LOCAL_SOLVER(i; j)

18: procedure SWEEP3

19: for i ¼ Nx → 1 do

20: for j ¼ Nz → 1 do

21: call LOCAL_SOLVER(i; j)

22: procedure SWEEP4

23: for i ¼ 1 → Nx do

24: for j ¼ Nz → 1 do

25: call LOCAL_SOLVER(i; j)

Algorithm 2. Init sweeping.

1: procedure INIT_SWEEP

2: updated(:,:)=False

3: for i ¼ ixs → Nx do

4: for j ¼ izs → Nz do

5: call LOCAL_SOLVER(i; j)

6: updatedði; jÞ =True7: for i ¼ ixs → 1 do

8: for j ¼ izs → Nz do

9: call LOCAL_SOLVER(i; j)

10: updatedði; jÞ =True11: for i ¼ ixs → 1 do

12: for j ¼ izs → 1 do

13: call LOCAL_SOLVER(i; j)

14: updatedði; jÞ =True15: for i ¼ 1 → ixs do

16: for j ¼ izs → 1 do

17: call LOCAL_SOLVER(i; j)

18: updatedði; jÞ =True

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updated yet, a boundary condition is applied, which is the same as atthe domain boundaries (the fourth term in scheme 12).For sake of conciseness, the algorithms are presented in two di-

mensions. The block Gauss-Seidel procedure and its initializationare detailed in Algorithms 1 and 2. The structure of the local solveris presented in Algorithm 3, and the degenerate solver is detailed inAlgorithm 4. The extension to higher dimensions of these algo-rithms is straightforward by sweeping in the additional dimensions.

Mesh deformation for topography

Finite-element methods are able to handle complex geometries ina natural way, thanks to the integral formulation allowing deformed

elements. Scheme 12 is written in a general formulation in whichcells could be of any polygonal type. For example, we could design2D unstructured triangle meshes, as shown in Le Bouteiller et al.(2018), case study 4. Instead, at the exploration scale, verticallydeformed Cartesian grids are very attractive for their simplicity(Hestholm, 1999; Tarrass et al., 2011; Trinh et al., 2018). The mainadvantages are that there is no mesh generation step to be performedby the user: The spatial position of each element can be accesseddirectly by its indices in x, y, and z. Moreover, the grid allows us tokeep the natural ordering of elements when performing the FSMalgorithm, yielding an optimal efficiency for the Gauss-Seidel pro-cedure. The topography variation is described by a gradual verticaldeformation of elements, keeping a constant number of elements inthe x-, y-, and z-directions. After interpolating the topography z⋆

over the ðx; yÞ nodes, the explicit mapping between a reference unitcube and each deformed element is established. The Gauss pointsused for computing integrals in scheme 12 are defined in the refer-ence cube, and the quantities in the physical space are estimated atthese Gauss points using the mapping, the local Jacobian, and chainrules for the spatial derivatives. The quantities needed at Gausspoints are precomputed prior to solving eikonal itself, and only oncefor a given deformed grid. Figure 1 shows an example of a verticallydeformed mesh in two dimensions.

NUMERICAL RESULTS

All of the 2D and 3D numerical experiments are performed on alaptop computer using a single core of an Intel Core i7-4600U com-puter processing unit with a frequency of 2.10 GHz, and 8 GB ofDDR3 SDRAM. The computations are done in double precision.

Vertical gradient of velocity

We first exhibit the efficiency of the FSM-DG technique in a 2Disotropic case, in which the velocity of the medium varies linearlywith depth. In a 4 × 4 km square, the velocity varies from a valueof 2 km∕s at the surface to a value of 4 km∕s at depth. The pointsource is located at the surface with coordinates xs ¼ 2 km andzs ¼ 0 km. The knowledge of the exact solution enables error com-putation for traveltime as well as its spatial derivatives (Fomel et al.,2009). A refinement study is carried out, and the results are shownin Tables 1 and 2: L2 errors are exhibited with respect to the numberof degrees of freedom when using P1 polynomial approximations,

together with a comparison of CPU times be-tween the RK-DG and the FSM-DG techniques.We perform similar experiments using a fast-sweeping FD solver from Noble et al. (2014); re-sults are given in Table 3. All of these results arecompared in Figure 2.The convergence of the schemes are high-

lighted in Figure 2a. The FD method exhibitsa first-order convergence: When the spatial dis-cretization step is divided by two, the error is alsodivided by two. Note that RK-DG and FSM-DGmethods yield the same error because they yieldthe same final state in a given discretization afterintegration in ξ. As expected, the higher slope ofdecrease of the error with respect to the number ofdegrees of freedom (dof) highlights the second-order convergence of the P1 DG approximation.

Algorithm 3. Local solver.

1: procedure LOCAL_SOLVER(i; j)

2: for n ¼ 1 → max_local_iter do

3: inside = HJ_integral(i; j)

4: forðk; lÞ ∈ neighborsði; jÞ do5: if updatedðk; lÞ then6: edges←edges + flux_integral(i; j; k; l)

7: else

8: edges←edges + boundary_integral(i; j; k; l)

9: uij ¼ uijold þ Δtðinside þ edgesÞ10: if kðuij − uijoldÞ∕uijoldk < local conv then return

11: uijold ¼ uij

12: call DEGENERATE_SOLVER(i; j)

Algorithm 4. Degenerate solver.

1: procedure DEGENERATE_SOLVER(i; j)

2: tmin ¼ 100000

3: for ðk; lÞ ∈ neighborsði; jÞ do tmin ¼ minðtmin;maxðukledgeÞÞ4: uij ¼ DG projectionðtmin þ distðx; edge minÞ∕minKij

ðcðxÞÞÞ

Figure 1. Vertically deformed mesh for a cross section extracted from SEAM II Foot-hills benchmark model. The topography (the blue line) has been interpolated over 101equally distributed points in the x-direction. The mesh is 100 × 40 elements.

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Coherently, the x-derivative of the solution exhibits a first-orderconvergence (Tables 1 and 2). This result was already exhibitedin Le Bouteiller et al. (2018). In terms of computational cost,the analysis of the slopes in Figure 2b shows that the RK-DGmethod has a computational complexity in Oðdof3∕2Þ, comparedwith the linear complexity (OðdofÞ) of the FSM-FD method. Thisis why the RK-DG method is not efficient. However, the new FSM-DG algorithm constitutes a huge improvement because it exhibits a

Table 1. The RK-DG results from the first numerical examplein two dimensions.

RK-DG

N #dof Error Order dx error Order CPU (s)

21 1323 1.74E-03 — 1.13E-02 — 0.36

41 5043 4.66E-04 1.97 5.51E-03 1.08 1.6

81 19,683 1.21E-04 1.98 2.71E-03 1.04 11

161 77,763 3.07E-05 1.99 1.34E-03 1.02 78

321 309,123 7.74E-06 2.00 6.69E-04 1.01 600

641 1,232,643 1.94E-06 2.00 3.34E-04 1.00 4900

Note: Number of elements along one direction (N), number of degrees of freedom(#dof), L2 error of the solution and its derivative along the x-direction, convergenceorders, and CPU times, for P1 polynomial approximation.

Table 2. The FSM-DG results from the first numericalexample in two dimensions.

FSM-DG

N #dof Error Order dxerror

Order CPU(s)

CPUratio

21 1323 1.73E-03 — 1.18E-02 — 0.21 1.7

41 5043 4.65E-04 1.96 5.60E-03 1.11 0.43 3.7

81 19,683 1.21E-04 1.98 2.73E-03 1.06 0.68 16

161 77,763 3.07E-05 1.99 1.35E-03 1.03 2.0 39

321 309,123 7.74E-06 2.00 6.70E-04 1.01 6.6 91

641 1,232,643 1.95E-06 1.99 3.35E-04 1.00 26 188

Note: Number of elements along one direction (N), number of degrees of freedom(#dof), L2 error of the solution and its derivative along the x-direction, convergenceorders, CPU times, and CPU ratio between RK-DG and FSM-DG, for P1 polynomialapproximation. Please note the slow increase of the CPU time with respect to thenumber of degrees of freedom (see Figure 2b).

Table 3. The FSM-FD results from the first numerical examplein two dimensions.

FSM-FD

N #dof Error Order CPU (s)

72 5184 6.08E-03 — 0.01

140 19,600 2.91E-03 1.06 0.05

278 77,284 1.46E-03 0.99 0.18

556 309,136 7.32E-04 1.00 0.76

1110 1,232,100 3.67E-04 1.00 3.1

2220 4,928,400 1.84E-04 1.00 12

4440 19,713,600 9.42E-05 0.97 50

Note: Number of elements along one direction (N), number of degrees of freedom(#dof), L2 error of the solution, convergence order, and CPU times.

Figure 2. Comparison of FD and DG methods with respect to accu-racy (L2 error), CPU time, and number of degrees of freedom. (a) TheDG scheme enables high-order approximation, yielding a lower errorfor a fixed number of degrees of freedom compared with the FDmethod. (b) Although the DGmethod implies a higher computationalburden for a fixed number of degrees of freedom compared with theFD method, the fast-sweeping algorithm applied to the DG method(FSM-DG) exhibits a linear complexity, which is more efficient thanthe RK-DGmethod. (c) Finally, the resulting efficiency is higher withthe FSM-DG method than with the FD method: Reaching a fixedlevel of error is done with a lower CPU time.

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linear complexity in OðdofÞ, as the FSM-FD scheme. Finally, theanalysis of Figure 2c underlines that, for reaching low levels of er-ror, the most efficient algorithm is the new FSM-DG. For a givenlevel of error, this algorithm is much faster than FSM-FD because itneeds fewer degrees of freedom to reach the same accuracy.In terms of memory requirements, in this 2D case, the FSM-FD

code requires approximately 315 MB of memory for the case inwhich N ¼ 4440. This amount is obtained by summing the sizes

of the arrays allocated for the traveltime and for the velocity model.It corresponds to 16 bytes per degree of freedom. In our currentimplementation, the DG codes require approximately 115 MB ofmemory for the case in which N ¼ 641. Because there are three

Table 4. The FSM-DG results from the first numericalexample in three dimensions.

FSM-DG

N #dof Error Order dx error Order CPU (s)

11 5324 1.41E-03 — 2.06E-02 — 2.4

21 37,044 3.73E-04 2.05 1.11E-02 0.96 4.3

41 275,684 9.59E-05 2.03 5.76E-03 0.98 15

81 2,125,764 2.43E-05 2.01 2.93E-03 0.99 133

161 16,693,124 6.07E-06 2.02 1.48E-03 0.99 1150

Note: Number of elements along one direction (N), number of degrees of freedom(#dof), L2 error of the solution and its derivative along the x-direction, convergenceorders, and CPU times, for P1 polynomial approximation.

Figure 3. The FSM-DG results from the first numerical example inthree dimensions: L2 error and CPU time with respect to the numberof degrees of freedom. The curved shape for small numbers of de-grees of freedom when the CPU time is lower than 10 s is explainedby initialization steps in the code that do not depend on the size of themesh, such as model reading and source handling, for instance.

Figure 4. Traveltime isocontours computed in the homogeneousTOR model with (blue) and without (red) the topography. Panelsat z ¼ 945 m (left), x ¼ 1150 m (right), and y ¼ 50 m (bottom).Please note that the thick zone is the contour of the topography:The blue solution is not built above the topography, whereas thered solution assumes a flat topography at the top of the domain.

Figure 5. The 30 × 70 × 20 vertically deformed mesh built for theSEAM II model.

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degrees of freedom per element, it corresponds to 90 bytes per de-gree of freedom. The difference with the FD code comes from addi-tional arrays allocated in memory for the values of the referencesolution u0 and its spatial derivatives at all the Gauss points. Thisis designed as such to optimize the CPU time. Alternatively, thesequantities could be computed on the fly. Note that the FSM-DG andRK-DG codes have the same memory requirements.Keeping the vertical gradient of velocity, we now perform similar

simulations on a 3D 4 × 4 × 4 km cube with the FSM-DG method.The point source is located at the surface with coordinates xs ¼ 2 km,ys ¼ 2 km, and zs ¼ 0 km. The results obtained with a P1 approxi-mation are detailed in Table 4 and highlighted in Figure 3. Asexpected, we retrieve a second-order convergence of the 3DDG discretization (Figure 3a). Moreover, the FSM-DG method still

exhibits a linear complexity (Figure 3b), which makes it very efficientin three dimensions as well.In terms of memory requirements, in the 3D case with N ¼ 161,

the FSM-DG codes require approximately 3500 MB of memory,yielding 200 bytes per degree of freedom. Here again, the valuesof the reference solution u0 and its spatial derivatives at all the Gausspoints are stored in memory.

Homogeneous 3D TOR medium with complextopography

In this example, we consider a homogeneous 3D TOR medium,with anisotropic parameters chosen as follows:

8>>>>>>>><>>>>>>>>:

VP ¼ 2000 m∕s;ϵ2 ¼ 0.2; δ2 ¼ 0.1;ϵ1 ¼ 0.4; δ1 ¼ 0.3;δ ¼ 0.1;ϕ ¼ 30°;θ ¼ 45°;ψ ¼ −15°:

(13)

We define a physical domain with a complextopography using a part of the SEAM II model(Regone et al., 2017). We consider a domain ofsize 3 km along the x-axis, 7 km along the y-axis,and 2 km along the z-axis. This domain is re-stricted in the z-direction by a topography pro-vided with the model, which we amplify by a

Figure 6. The VP model extracted from the SEAM II model, with traveltime isocontourssuperimposed.

Figure 7. Displacement field in the x-direction from SEM46, an elas-tic spectral-element solver, with an impulse source. The traveltimeisocontours are superimposed. Snapshots in the plane x ¼ 1500 mand corresponding isocontours at time 0.6 (top) and 0.9 (bottom)s. One could notice the numerical noise occurring ahead of the front,which comes from the wave-propagation solution, and which is vis-ible due to the saturation of the plot.

Figure 8. Displacement field in the x-direction from SEM46, an elas-tic spectral-element solver, with an impulse source. The traveltimeisocontours are superimposed. Snapshots in the plane z ¼ 500 m andcorresponding isocontours at time 0.6 (top) and 0.9 (bottom) s. Thetopographical contour is expressed by the white zone where no sol-ution is computed. Here again, one could notice the numerical noiseoccurring ahead of the front.

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factor two to clearly exhibit its imprint on the traveltimes. We builda vertically-deformed mesh from a regular Cartesian grid to followthis topography. The mesh spacing before deformation is 100 m inthe three directions. Traveltimes are computed with and without thetopography, for a point source located at x ¼ 1150 m, y ¼ 50 m,and z ¼ 945 m. Both results are superimposed in Figure 4. The im-print of the topography is clearly visible in the near-surface areasand might be understood by applying the Huygens principle at thebottom points of the topography, from which the upper parts of thedomain are illuminated. This example shows the importance of tak-ing care of topography, and the good behavior of the FSM-DGmethod when doing so. Let us emphasize that not only the travel-times, but also the spatial derivatives, are modified by the topogra-phy, which indicates the front propagation direction. Thesederivatives are of major interest when considering subsequent am-plitude or angle estimations.

3D complex SEAM II model

In this example, our FSM-DG scheme is applied onto the samepart of the isotropic SEAM II model (Regone et al., 2017). Here, weconsider the original topography as provided with the model (noamplification factor). To do so, we build a vertically-deformed meshas in the previous example: The original mesh step is 100 m inthe three directions, and the resulting deformed mesh is shown inFigure 5. The P-wave velocity model, shown in Figure 6, exhibitscomplex velocity structures with layering and faulting, as well asnear-surface low-velocity areas which are known to impair imagingand inversion results if not carefully considered. We use the P-wavevelocity of the model to compute first-arrival traveltimes for asource located at x ¼ 375 m, y ¼ 350 m, and z ¼ 528 m. The trav-eltime isocontours are superimposed over the P-wave velocitymodel in Figure 6. To illustrate the quality of the result, we performa computation using a spectral-element solver of the elastic-waveequation applied to the same model. For this computation, we usethe code SEM46 (Trinh et al., 2017) with an impulse source in a60 × 140 × 40 mesh. In Figures 7 and 8, the traveltime isocontoursobtained by the FSM-DG method are superimposed over the dis-placement field in the x-direction obtained with the SEM46 code,in the vertical and horizontal planes, highlighting an almost perfectagreement between the first-arrival traveltimes and the wavefront,although they are based on different equations (the eikonal equationversus the elastic-wave equation) and different numerical methods(the DG method versus the spectral element method), respectively.In the presence of a nonflat topography, the memory requirements

are higher than in the Cartesian 3D case. The last example requiresapproximately 180 MB of memory, corresponding to 1000 bytes perdegree of freedom. This is mainly due to the values of the derivativesof the basis functions that we keep in memory at each Gauss point.Alternatively, these values could be computed on the fly.

CONCLUSION

An FSM algorithm has been introduced in a DG approach for solv-ing the 3D eikonal equation. The DG approach had been previouslyintroduced in a 2D framework, using a Runge-Kutta solver, respon-sible for a high computational complexity in Oðdof3∕2Þ. The newFSM algorithm provides a significant increase of efficiency, makingit possible to reach a linear complexity as for the FSM-FD approach,while benefiting for the high accuracy and higher-order convergence

rate associated with DG approach. For this reason, higher accuracyfor the traveltime solution and its spatial derivatives is obtained com-pared with FD methods, while complex structures are handled in astable and accurate way, thanks to the FE properties. This is illus-trated by the use of deformed Cartesian grid for handling topography.The 2D and 3D implementations are performed, with TTI and TORanisotropy, thanks to the general Hamiltonian formulation of the DGscheme. Even more general Hamiltonians could be considered in thefuture, accounting, for instance, for triclinic anisotropy.We may now use these results together with accurate solvers for

seismic amplitudes and take-off angle computations to revisit asymp-totic inversion (traveltime/slope tomography) and imaging approaches(quantitative migration using amplitudes and angles).

ACKNOWLEDGMENTS

This study was partially funded by the SEISCOPE consortium(http://seiscope2.osug.fr), sponsored by AKERBP, CGG, CHEV-RON, EXXON-MOBIL, JGI, PETROBRAS, SCHLUMBERGER,SHELL, SINOPEC, STATOIL, and TOTAL. This study was grantedaccess to the HPC resources of the Froggy platform of the CIMENTinfrastructure (https://ciment.ujf-grenoble.fr), which is supportedby the Rhône-Alpes region (grant no. CPER07_13 CIRA), theOSUG@2020 labex (reference ANR10 LABX56), and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the programmeInvestissements d’Avenir supervised by the Agence Nationale pourla Recherche, and the HPC resources of CINES/IDRIS/TGCC underthe allocation 046091 made by GENCI.

APPENDIX A

FACTORED HAMILTON-JACOBI EQUATIONS

Plugging expression 9 into 2D-VTI Hamiltonian expression 3,and finally, into the dynamic Hamilton-Jacobi equation 1 that wesolve for τðx; ξÞ, leads to the factored 2D-VTI Hamilton-Jacobiequation:

∂ξτ þ dðu0;x þ τ;xÞ2 þ eðu0;z þ τ;zÞ2þ cðu0;x þ τ;xÞ2dðu0;z þ τ;zÞ2 − 1 ¼ 0: (A-1)

Now, using the 3D-OR Hamiltonian of equation 6, the factored3D-OR Hamilton-Jacobi equation writes

∂ξτ þ aðu0;x þ τ;xÞ2 þ bðu0;y þ τ;yÞ2 þ cðu0;z þ τ;zÞ2þ dðu0;x þ τ;xÞ2ðu0;y þ τ;yÞ2 þ eðu0;x þ τ;xÞ2ðu0;z þ τ;zÞ2þ fðu0;y þ τ;yÞ2ðu0;z þ τ;zÞ2þ gðu0;x þ τ;xÞ2ðu0;y þ τ;yÞ2ðu0;z þ τ;zÞ2 − 1 ¼ 0: (A-2)

APPENDIX B

DETAILED DG SCHEME

At an interface of element Ki, we define a 2C vector by the ex-pression:

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∇xu�hKi¼

� ð∇xuh · nKiÞ�

∇xuh · tKi

�: (B-1)

The first component ð∇xuh · nKiÞ� is the projection onto the normal

nKi, of the gradient of the numerical solution computed inside the

Ki cell (−), or inside its neighbor (þ). The second component∇xuh · tKi

holds for the mean of the projections onto the tangentialvector tKi

of the gradient of the numerical solution computed insidethe Ki cell and inside its corresponding neighbor.We also introduce the following quantities:

H�Ki

¼ Hðx�;∇xu�hKiÞ;

HnKi¼ ∇∇uH · nKi

;

H�nKi

¼ HnKiðx�;∇xu�hKi

Þ;

~HnKiðxÞ ¼

� HþKi−H−

Ki½∇xuh·nKi

�ðxÞ ; if ½∇xuh · nKi�ðxÞ ≠ 0;

12ðHþ

nKiþH−

nKiÞ; otherwise;

δnKiðxÞ ¼ maxð0; ~HnKi

ðxÞ −H−nKi

;HþnKi

− ~HnKiðxÞÞ;

χnKiðxÞ ¼ maxðδnKi

ðxÞ; j ~HnKiðxÞjÞ: (B-2)

Quantities F ;G;K introduced in scheme 12 are now written as

F ¼ minð ~HnKiðxÞ; 0Þ;

G ¼ χnKiðxÞ − j ~HnKi

ðxÞj;K ¼ minðH−

nKiðxÞ; 0Þ: (B-3)

The key quantity for preserving causality is ~HnK , referred to asthe Roe speed: Its sign specifies the information flow direction at aninterface between two cells. Thanks to the min operator in F , whencomputing the second integral of scheme 12, the downwind cellreceives information from the upwind cell, whereas the upwind cellis not affected by the downwind cell information. The continuitybetween elements is thus weakly enforced in an upwind manner.

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