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Eikonal Equation (eikonal + equation)
Selected AbstractsA CFL-like constraint for the fast marching method in inhomogeneous chemical kineticsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2008Ramón Escobedo Abstract Level sets and fast marching methods are a widely used technique for problems with moving interfaces. Chemical kinetics has been recently added to this family, for the description of reaction paths and chemical waves in homogeneous media, in which the velocity of the interface is described by a given field. A more general framework must consider variable velocities due to inhomogeneities induced by chemical changes. In this case, a constraint must be satisfied for the correct use of fast marching method. We deduce an analytical expression of this constraint when the Godunov scheme is used to solve the Eikonal equation, and we present numerical simulations of a case which must be enforced to obey the constraint. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008 [source] Light propagation in multi-step index optical fibresLASER & PHOTONICS REVIEWS, Issue 3 2008J. Zubia Abstract This paper reviews the theoretical analysis of light propagation we have carried out on multimode multi-step index (MSI) optical fibres. Starting from the Eikonal equation, we derive the analytical expressions that allow calculating the ray trajectories inside these fibres. We also analyse the effects of leaky rays on the transmission properties of MSI fibres. For this purpose, a single analytical expression for the evaluation of the ray power transmission coefficient is calculated. Afterwards, we investigate the effects of extrinsic and intrinsic coupling losses on the performance of MSI fibres, providing analytical expressions to calculate the coupling loss and, also, determining the most critical parameters. Finally, we carry out a comprehensive numerical analysis of the fibre bandwidth under different source configurations. [source] A Maslov-propagator seismogram for weakly anisotropic mediaGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2002Georg Rümpker Summary We introduce a formalism to calculate shear-wave seismograms for weakly-anisotropic and inhomogeneous media. The method is based on the combination of the forward-propagator method, which accounts for shear-wave interaction along a single reference ray, and the Maslov ray-summation, which incorporates amplitude and phase information from neighbouring rays to account for waveform and diffraction effects at caustics and in shadow regions. The approach is based on the assumption that the multiply split shear waves, on the way to a given receiver, travel along a common ray path that can by obtained from ray tracing in an isotropic reference medium (i.e. the common-ray approximation). The forward propagator and the Maslov amplitude are expressed with respect to radial and transverse coordinates (perpendicular to the ray propagation direction) that are defined uniquely by the initial conditions. Local polarizations and slownesses of the fast and slow shear-waves in the direction of propagation are obtained from the eikonal equation. The Maslov-propagator phase is given by the average shear-wave traveltime along the reference ray. Phase advances and delays of individual shear wave components are accounted for by the propagator. The geometrical-spreading information required for the Maslov integration is supplied by dynamic ray tracing in the isotropic reference medium. In the high-frequency limit effective phase functions are defined to assess the validity of the Maslov propagator phase information. For a homogeneous isotropic reference medium, we find good agreement with exact Maslov phase functions for anisotropic perturbations of up to 20 per cent. As a numerical application we consider effects of inhomogeneous anisotropy in a shear-wave cross-hole survey. The variations of the transversely-isotropic medium require 2-D slowness integrals. The method can handle discontinuities of the fast polarization along the ray path and also for neighbouring rays which is important for the slowness integration. Smooth transitions between isotropic and anisotropic regions along the ray path can be accounted for without the need to switch between numerical formulations. [source] A hybrid fast algorithm for first arrivals tomographyGEOPHYSICAL PROSPECTING, Issue 5 2009Manuela Mendes ABSTRACT A hybrid algorithm, combining Monte-Carlo optimization with simultaneous iterative reconstructive technique (SIRT) tomography, is used to invert first arrival traveltimes from seismic data for building a velocity model. Stochastic algorithms may localize a point around the global minimum of the misfit function but are not suitable for identifying the precise solution. On the other hand, a tomographic model reconstruction, based on a local linearization, will only be successful if an initial model already close to the best solution is available. To overcome these problems, in the method proposed here, a first model obtained using a classical Monte Carlo-based optimization is used as a good initial guess for starting the local search with the SIRT tomographic reconstruction. In the forward problem, the first-break times are calculated by solving the eikonal equation through a velocity model with a fast finite-difference method instead of the traditional slow ray-tracing technique. In addition, for the SIRT tomography the seismic energy from sources to receivers is propagated by applying a fast Fresnel volume approach which when combined with turning rays can handle models with both positive and negative velocity gradients. The performance of this two-step optimization scheme has been tested on synthetic and field data for building a geologically plausible velocity model. This is an efficient and fast search mechanism, which permits insertion of geophysical, geological and geodynamic a priori constraints into the grid model and ray path is completed avoided. Extension of the technique to 3D data and also to the solution of ,static correction' problems is easily feasible. [source] Traveltime computation with the linearized eikonal equation for anisotropic mediaGEOPHYSICAL PROSPECTING, Issue 4 2002Tariq Alkhalifah A linearized eikonal equation is developed for transversely isotropic (TI) media with a vertical symmetry axis (VTI). It is linear with respect to perturbations in the horizontal velocity or the anisotropy parameter ,. An iterative linearization of the eikonal equation is used as the basis for an algorithm of finite-difference traveltime computations. A practical implementation of this iterative technique is to start with a background model that consists of an elliptically anisotropic, inhomogeneous medium, since traveltimes for this type of medium can be calculated efficiently using eikonal solvers, such as the fast marching method. This constrains the perturbation to changes in the anisotropy parameter , (the parameter most responsible for imaging improvements in anisotropic media). The iterative implementation includes repetitive calculation of , from traveltimes, which is then used to evaluate the perturbation needed for the next round of traveltime calculations using the linearized eikonal equation. Unlike isotropic media, interpolation is needed to estimate , in areas where the traveltime field is independent of ,, such as areas where the wave propagates vertically. Typically, two to three iterations can give sufficient accuracy in traveltimes for imaging applications. The cost of each iteration is slightly less than the cost of a typical eikonal solver. However, this method will ultimately provide traveltime solutions for VTI media. The main limitation of the method is that some smoothness of the medium is required for the iterative implementation to work, especially since we evaluate derivatives of the traveltime field as part of the iterative approach. If a single perturbation is sufficient for the traveltime calculation, which may be the case for weak anisotropy, no smoothness of the medium is necessary. Numerical tests demonstrate the robustness and efficiency of this approach. [source] A multigrid upwind strategy for accelerating steady-state computations of waves propagating with curvature-dependent speedsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2002Jonathan Rochez Abstract A multigrid strategy using upwind finite differencing is developed for accelerating the steady state computations of waves, [14] propagating with curvature-dependent speeds. This will allow the rapid computation of a "burn table." In a high explosive material, a burn table will allow the elimination of solving chemical reaction ODEs by feeding in source terms to the reactive flow equations for solution of the system of ignition of the high explosive material. Standard iterative methods show a quick reduction of the residual followed by a slow final convergence to the solution at high iterations. Such systems, including a nonlinear system such as this, are excellent choices for the use of multigrid methods to speed up convergence. Numerical steady-state solutions to the eikonal equation on several test grids are conducted. Results are presented for these cases in 2D and a cubic grid in 3D using a Runge-Kutta time iteration for the smoothing operator until steady state is reached. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 179,192, 2002; DOI 10.1002/num.1002 [source] | ||||||||||