Jenö Gazdag

Migration of seismic data by phase shift plus interpolation

Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using l laterally uniform velocity fields. The intermediate result is l reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.

Modeling of the acoustic wave equation with transform methods

Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such er...

Time-differencing schemes and transform methods

Abstract In this paper, we study several time-differencing procedures for the numerical solution of partial differential equations. We find that “partially corrected” time-differencing schemes offer some advantages over single-step methods. Such differencing schemes consist of a predictor step and a corrector step, however, the time derivative is evaluated only once in the two steps. Partially corrected time-differencing schemes can be used with particular advantage in transform methods, where the numerical approximation of the derivative terms represents the major portion of computing. All differencing methods are tested on the nonlinear Vlasov-Poisson system of equations with two phase-space variables. In the case of the numerical example considered, the partially corrected schemes are only about 1% slower than the corresponding single-step procedures without correction.

NUMERICAL SOLUTION OF FISHER'S EQUATION

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fishers equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration. NONLINEAR DIFFUSION; EPIDEMIC WAVES; NUMERICAL SIMULATION

The recurrence of the initial state in the numerical solution of the Vlasov equation

Abstract The approximate recurrence of the initial state, observed recently in the numerical solution of Vlasovs equation by a finite-difference Eulerian model, is shown to be a property of three independent numerical methods. Some of the methods have exponentially growing modes (Dawsons beaming instabilities), and some others do not. The recurrence is in fact a manifestation of the finite velocity resolution of the numerical methods—a property which is independent of the approximation of a plasma by a finite number of electron beams. The recurrence is shown explicitly in the numerical simulation of Landau damping by three different methods: Fourier-Hermite, the recent variational method of Lewis, and the Eulerian finite-difference method.

Threshold conditions for electron trapping by nonlinear plasma waves

Numerical solutions of the nonlinear Vlasov equation show that the long time behavior of an electrostatic plasma wave depends critically on the value of the parameter γτ (γ is Landaus damping coefficient and τ is the electron bounce time), both for large and small initial wave amplitudes. When the initial wave amplitude is large, the initial nonlinear damping is stronger than Landau damping; for smaller initial amplitudes, the behavior is initially described by Landaus theory. For both types of problems if γτ > 0.5, the wave damps monotonically; if γτ < 0.5, the numerical results indicate that the plasma performs amplitude oscillations. When γτ ≈ 0.5, the plasma displays its critical behavior; here, after some initial damping the wave amplitude levels off at a constant value. The critical behavior observed for mild nonlinear problems is in good agreement with the experimental results of Franklin, Hamberger, and Smith. For a moderately strong nonlinear wave displaying the critical behavior, the wave perf...

Electrostatic Oscillations in Plasmas with Cutoff Distributions

When numerically solving the linear Vlasov equation with cutoff electron distributions, the well‐known Landau‐damped oscillation of the electric field was observed first but, after a short transition time, a constant‐amplitude harmonic oscillation with a frequency distinctly higher than Landaus frequency was found. This coexistence of two distinct modes of oscillation—one damped and one undamped—has not been observed before. The behavior is explained quantitatively by comparison of the dispersion equation solution with the results obtained from the numerical solution of Vlasovs equation.

Numerical solution of the Vlasov equation with the accurate space derivative method

Abstract A numerical procedure is described for the solution of the Vlasov-Poisson system of equations in two and three phase-space variables. In this approach the distribution function is represented over a computational mesh. Time integration is done by advancing the distribution in real phase space as in finite difference methods. However, the derivatives with respect to all the phase-space variables are computed by finite Fourier transform methods. Truncation errors are principally due to time discretization, and are controlled by the choice of the time step. As for the phase-space variables, the accuracy of the computation is determined by the harmonic content of the distribution function, since contributions due to coupling between Fourier modes are computed with high accuracy. The numerical method has been tested on linear and nonlinear problems, and our results agree remarkably well with those obtained from the Fourier-Hermite method. However, for comparable overall accuracy, the present method is about ten times more efficient (CPU time) than the Fourier-Hermite method in some of the examples discussed in this paper.

Concurrent computing by sequential staging of tasks

Described is a new approach to parallel formulation of scientific problems on shared-memory multiprocessors such as the IBM ES/3090 system. The class of problems considered is characterized by repetitive operations applied over the computational domain D. In each such operation, some fields of interest are extrapolated or advanced by an amount of Δτ. The integration variable τ may be time, distance, or iteration sequence number, depending on the problem under consideration. An extensively studied approach to parallel formulation of such computational problems is based on domain decomposition, which attempts to partition the domain of integration into many pieces, then construct the global solution from these local solutions. Thus, domain decomposition methods are confined to D alone at a single τ level. An inquiry into the possibilities of formulating parallel tasks in τ, or more significantly in the D × τ domain, opens up new horizons and untapped opportunities. The aim of this paper is to detail an approach to exploit this τ domain parallelism that will be referred to as sequential staging of tasks (SST). Concurrency is realized by means of ordering the tasks sequentially and executing them in a partially overlapped or pipelined manner. The SST approach can yield remarkable speedup for jobs requiring intensive paging I/O, even when a single processor is available for executing multiple tasks. Noteworthy features of the SST method are demonstrated and highlighted by using results obtained from computer experiments performed with a numerical solution method of the Poisson equation and migration of seismic reflection data.

Migration by phase shift; an algorithmic description for array processors

The phase shift method (Gazdag, 1978) is based on the solution, in the frequency domain, of an approximation (Claerbout, 1976) to the one‐way wave equation with initial conditions defined by a zero‐offset seismic section. Wave velocity is assumed to be constant within each layer of the section grid and is allowed to vary from layer to layer. Under these conditions, the equation written in the frequency domain reduces to a system of independent ordinary differential equations with initial values that can be solved analytically for each layer. The integration process simply amounts to multiplying the initial values by a complex number of unit modulus. The main advantages of this method are simplicity, stability, and high accuracy, since the precision of the Fourier approximation is limited only by the granularity of the seismic section. From a practical viewpoint of computer implementation, the phase shift method offers a great deal of flexibility. Some accuracy can be traded for speed, as needed, by exclud...