Finite difference method

Abstract: The first‐arrival quasi‐P wave traveltime field in an anisotropic elastic solid solves a first‐order nonlinear partial differential equation, the q P eikonal equation. The difficulty in solving this eikonal equation by a finite‐difference method is that for anisotropic media the ray (group) velocity direction is not the same as the direction of the traveltime gradient, so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field. However, establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty. Furthermore, the solution of the paraxial q P eikonal equation, an evolution equation in depth, gives the first‐arrival traveltime along downward propagating rays. A second‐order upwind finite‐difference scheme solves this paraxial eikonal equation in O(N) floating point operations, where N is the number of grid points. Numerical experiments using 2‐D and 3‐D transversely is...

Abstract: A Chebyshev finite difference method has been proposed in order to solve linear and nonlinear second-order Fredholm integro-differential equations. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a nonuniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique.

Abstract: [1] High-frequency seismograms mainly consist of incoherently scattered waves. Although their phases are more or less random, their envelopes show smooth and stable variations depending on frequency and distance. Envelope modeling can thus be used to infer stochastic parameters of the heterogeneous Earth medium. Radiative transfer theory (RTT) describes energy transport through a random heterogeneous medium neglecting phase information and has been frequently used to simulate observed mean square (MS) envelopes of high-frequency waves. The radiative transfer equations can be numerically solved by Monte Carlo simulations. So far, mostly isotropic scattering and acoustic approximations have been used. Here we present an extension of the Monte Carlo method to the full elastic case including P, S, and conversion scattering where the single scattering events are described by angular-dependent scattering coefficients in random media which follow from the Born approximation. In order to validate the method, the simulated envelopes are compared to average envelopes obtained by full waveform modeling with a finite difference method in two-dimensional random media with Gaussian and exponential correlation functions. Envelope shapes agree remarkably well for both short and long lapse times and for a broad range of scattering parameters. We conclude that the use of Born scattering coefficients in RTT does not pose severe limits on its validity range. Even in the strong forward scattering regime, envelope broadening and peak amplitude delays can be successfully modeled if one includes the wandering effect as obtained from the parabolic wave equation and Markov approximation into RTT.

Abstract: An analysis is made of the flow of an electrically conducting fluid in a channel with constrictions in the presence of a uniform transverse magnetic field. A solution technique for governing magnetohydrodynamic (MHD) equations in primitive variable formulation is developed. A coordinate stretching is used to map the long irregular geometry into a finite computational domain. The governing equations are discretized using finite difference approximations and the well-known staggered grid of Harlow and Welch is used. Pressure Poisson equation and pressure-velocity correction formulas are derived and solved numerically