Wave Front Tracking Research Articles

Multiarrival wavefront tracking and its applications

Multivalued traveltimes have traditionally not been used in seismic imaging, with only a handful of notable exceptions in the field of exploration geophysics. For studies at local and regional scales (e.g., local earthquake/teleseismic tomography), the focus has largely been on first arrivals, with numerous ray‐ and grid‐based schemes developed for their calculation. However, later arrivals often contribute to the length and shape of a recorded wave train, particularly in regions of complex geology. These arrivals are likely to contain additional information about seismic structure, as their two point path differs from that of the first arrival; in particular, they are more amenable to sampling regions of lower velocity. In this work the wavefront construction principle is used as the basis of a new scheme for computing multivalued traveltimes that arise from smooth variations in both velocity structure and interface geometry. The idea is to represent the wavefront as a set of points in reduced phase space and use local ray tracing and interpolation to advance the wavefront in a series of time steps. The scheme is robust in the presence of strong velocity heterogeneity and interface curvature, with phases comprising multiple reflections, refractions, and triplications successfully tracked. Outside the field of exploration seismology wavefront construction techniques are rarely used, yet they hold great potential for addressing problems in other areas of seismology. This paper demonstrates the viability of the new wavefront construction scheme by applying it to a range of scenarios, including multiarrival body and surface wave tomography, teleseismic receiver function prediction using Gaussian beams, and the tracking of global phases such as PcP.

Nonclassical interactions portrait in a macroscopic pedestrian flow model

In this paper we describe the main characteristics of the macroscopic model for pedestrian flows introduced in [R.M. Colombo, M.D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci. 28 (13) (2005) 1553–1567] and recently sperimentally verified in [D. Helbing, A. Johansson, H.Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 75 (4) (2007) 046109]. After a detailed study of all the possible wave interactions, we prove the existence of a weighted total variation that does not increase after any interaction. This is the main ingredient used in [R.M. Colombo, M.D. Rosini, Existence of nonclassical Cauchy problem modeling pedestrian flows, technical report, Brescia Department of Mathematics, 2008] to tackle the Cauchy problem through wave front tracking, see [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl., vol. 20, Oxford Univ. Press, Oxford, 2000, The one-dimensional Cauchy problem; A. Bressan, The front tracking method for systems of conservation laws, in: C.M. Dafermos, E. Feireisl (Eds.), Handbook of Differential Equations; Evolutionary Equations, vol. 1, Elsevier, 2004, pp. 87–168; R.M. Colombo, Wave front tracking in systems of conservation laws, Appl. Math. 49 (6) (2004) 501–537]. From the mathematical point of view, this model is one of the few examples of conservation laws in which nonclassical solutions have a physical motivation, see [P.G. Lefloch, Hyperbolic Systems of Conservation Laws, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2002, The theory of classical and nonclassical shock waves], and an existence result is available.

A Tracking Algorithm for Car Paths on Road Networks

In this paper we introduce a computation algorithm to trace car paths on road networks, whose load evolution is modeled by conservation laws. This algorithm is composed of two parts: computation of solutions to conservation equations on each road and localization of car position resulting by interactions with waves produced on roads. Some applications and examples to describe the behavior of a driver traveling in a road network are shown. Moreover, a convergence result for wave front tracking approximate solutions, with BV initial data on a single road, is established.

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L 1 well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L 1 stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L 1 norm, and prove that the functional decreases in the flow direction. Then the L 1 stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.

Wavefront tracking within the stationary phase approximation

A physical optics approach to wavefront tracking is presented based on the stationary phase approximation. It allows for modeling transmission of radiation through a system of optical elements including coherence properties with an acceptable numerical effort. The code PHASE incorporating these ideas assumes small angles as it is the case for most synchrotron radiation and the free electron laser beam lines.

Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions

We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.

Seismic wavefront tracking in 3D heterogeneous media: applications with multiple data classes

We demonstrate the potential of a recently developed grid-based eikonal solver for tracking phases comprising reflection branches, transmission branches, or a combination of these, in 3D heterogeneous layered media. The scheme is based on a multi-stage fast marching approach that reinitialises the wavefront from each interface it encounters as either a reflection or transmission. The use of spherical coordinates allows wavefronts and traveltimes to be computed at local, regional, and semi-global scales. Traveltime datasets for a large variety of seismic experiments can be predicted, including reflection, wide-angle reflection and refraction, local earthquake, and teleseismic.A series of examples are presented to demonstrate potential applications of the method. These include: (1) tracking active and passive source wavefronts in the presence of a complex subduction zone; (2) earthquake hypocentre relocation in a laterally heterogeneous 3D medium; (3) joint inversion of wide-angle and teleseismic datasets for P-wave velocity structure in the crust and upper mantle. Results from these numerical experiments show that the new scheme is highly flexible, robust and efficient, a combination seldom found in either grid- or ray-based traveltime solvers. The ability to track arrivals for multiple data classes such as wide-angle and teleseismic is of particular importance, given the recent momentum in the seismic imaging community towards combining active and passive source datasets in a single tomographic inversion.

Two-hydrophone localization of a click source in the presence of refraction

This paper introduces a ray-theoretic method for fast localization of an impulsive sound source in a general stratified ocean environment by measuring relative times of direct and surface-reflected arrivals at two hydrophones. The localization problem is solved through ray tracing and wave-front tracking in the time domain. This method is computationally efficient and allows for fast, near-real time estimation of the source location, which is a significant advantage for operational localization. The accuracy of the ray-theoretic approach is compared with that of the straight-line approximation. Further, localization errors associated with the angular and temporal discretization as well as with the variability of the hydrophone depths and the sound-speed profile are addressed.

Traffic Flow on a Road Network

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions. Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.

Source-Destination Flow on a Road Network

We construct a model of traffic flow with sources and destinations on a roads network. The model is based on a conservation law for the density of traffic and on semilinear equations for traffic-type functions, i.e. functions describing paths for cars. We propose a definition of solution at junctions, which depends on the traffic-type functions. Finally we prove, for every positive time T, existence of entropic solutions on the whole network for perturbations of constant initial data. Our method is based on the wave-front tracking approach.

Wave-front tracking for the equations of isentropic gas dynamics

We study the model equations of polytropic gas dynamics, which constitute a system of three hyperbolic conservation laws. Global in time BV solutions were obtained by Liu (Indiana Univ Math J 26(1):147–177, 1977) provided that \((\gamma - 1)\) times the total variation of the initial data is sufficiently small; here \(\gamma \) is the adiabatic coefficient. The aim of this paper is to give an alternative proof by exploiting the Dafermos–Bressan–Risebro wave-front tracking scheme. An original feature is the use of the path decomposition method to obtain pathwise estimates of the approximate solutions; these estimates show the decay properties of the solutions and play a crucial role in proving the stability of the wave-front tracking scheme.

Wave Front Tracking in Systems of Conservation Laws

This paper contains several recent results about nonlinear systems of hyperbolic conservation laws obtained through the technique of Wave Front Tracking.

Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.

BV estimates of Lax-Friedrichs’ scheme for a class of nonlinear hyperbolic conservation laws

We give uniform BV estimates and L-1-stability of Lax-Friedrichs' scheme for a class of n x n systems of strictly hyperbolic conservation laws whose integral curves of the eigenvector fields are straight lines, i.e., Temple class, under the assumption of small total variation. This implies that the approximate solutions generated via the Lax-Friedrichs' scheme converge to the solution given by the method of vanishing viscosity or the Godunov scheme, and then the Glimm scheme or the wave front tracking method.

Existence result for a class of nonconservative and nonstrictly hyperbolic systems

We consider the class of nonconservative hyperbolic systems $\partial_t u+A(u) \partial_x u =0,\quad\partial_t v + A(u) \partial_x v =0,$ where $u=u(x,t),\quad v=v(x,t)\in\mathbb R^N$ are the unknowns and $A(u)$ is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat (Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995), 483--548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows: (i) We solve the Riemann problem based on a prescribed family of paths. (ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations $u^h$, $v^h$. (iii) We justify rigorously the passage to the limit in the nonconservative product $A(u^h) \partial_x v^h$, based on the local uniform convergence properties of $u^h$, by extending an argument due to LeFloch and Liu (Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), 261--280). Our results provide a generalization to the existence theorem established earlier in the scalar case ($N=1$) by the second author (An existence and uniqueness result for two nonstrictly hyperbolic systems, IMA Volumes in Math. and its Appl. 27, Nonlinear evolution equations that change type, ed. B.L. Keyfitz and M. Shearer, Springer Verlag, 1990, pp. 126--138.)

A Wavefront Tracking Algorithm for N×N Nongenuinely Nonlinear Conservation Laws

We introduce a wavefront tracking algorithm for N×N hyperbolic systems of conservation lawsut+F(u)x=0, that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue λi(u) of DF(u) in the direction of the ith right eigenvector ri(u), vanishes on a single (N−1)-dimensional hypersurface in the u-space, transversal to the field ri(u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc.30 (1981), no. 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S.A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax.

Stability of $L^\infty$ Solutions for Hyperbolic Systems with Coinciding Shocks and Rarefactions

We consider a hyperbolic system of conservation laws \[ \left\{ \begin{array}{c} u_t + f(u)_x = 0, u(0,\cdot) = u_0, \end{array} \right. \] where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates w, we prove that there exists a semigroup of solutions $u(t) = \mathcal{S}_t u_0$, defined on initial data $u_0 \in L^\infty$. The semigroup $\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{{\rm loc}}$ topology. Moreover, $\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.

Partition of Waves in a System of Conservation Laws

The aim of this paper is to give a simple proof of the classical Liu estimate on the decay of positive waves in a solution of a n×n system of conservation laws. In the first part, we transcribe the wave partition technique introduced in Comm. Math. Phys.57 (1977), 135–148 (by means of the Glimm scheme) to the case of approximate solutions constructed by the wave front tracking scheme. Then, we use a decoupling argument on the characteristic speeds to establish the desired estimate.

3-D traveltimes and amplitudes by gridded rays

Seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium. They can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite‐difference methods. Unfortunately, this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used. We propose a new method for computing traveltimes and amplitudes in 3-D media that is simple, fast, unconditionally stable, and robust. Defining the slowness vector as [Formula: see text] and assuming an isotropic medium, the ray velocity v is related to the slowness vector by the relation [Formula: see text]. Rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane. The components of the slowness vector are then interpolated to gridpoints on this next horizontal plane. This is termed regridding; the process of downward propagation of rays, one vertical grid step at a time, is continued until some prescribed depth is reached. Computation of amplitudes is achieved using a method similar to that for obtaining the zero‐order approximation in asymptotic ray theory. We show comparisons with a full‐wave method on readily accessible 3-D velocity models.

To: “An analytical expression for the gravity field of a polyhedral body with linearly varying density” R. O. Hansen (GEOPHYSICS, 64, 75–77)

The domain of applicability of the expressions derived in this paper is stated incorrectly. The second‐last sentence of the section “Derivation of the Field Equations” should read “Using the numerical techniques discussed by Pohanka for circumventing this problem, an expression applicable everywhere in space can be obtained.” The following sentence is incorrect and should be deleted. I thank Dr. Marion Ivan for point this error out to me. After acceptance of the paper, an article covering substantially the same material, though with a somewhat different point of view, appeared. The article is: Pohanka, V., 1998, Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density: Geophysical Prospecting, 46, 391–404. Were I aware of Dr. Pohanka’s paper, it would of course have been referenced in mine. To: “Finite‐difference solution of the eikonal equation using an efficient, first‐arrival, wavefront tracking scheme” S. Cao and S. Greenhalgh (Geophysics, 59, 635) Equation 8a is in error. The parenthetical expressions on the left should be squared to make it dimensionally correct.