A numerical-analytical algorithm for modeling of seismic and acoustic-gravity waves propagation is applied to a heterogeneous “Earth-Atmosphere" model. Seismic wave propagation in an elastic half-space is described by a system of first-order dynamic equations of elasticity theory. The propagation of acoustic-gravity waves in the atmosphere is described by the linearized Navier-Stokes equations with the wind. The algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform with respect to a spatial coordinate combined with a finite difference method for the reduced problem.

In this paper, the solution of two-dimensional Maxwell’s equations is considered using the Laguerre transform. Optimal parameters of the difference schemes for the equations are obtained and presented. Numerical values of these optimal parameters are given. Second-order difference schemes with the optimal parameters provide an accuracy of the solution of the equations that is comparable to the accuracy of the solution using fourth-order schemes. It is shown that, when using the Laguerre transform, the number of optimal parameters can be reduced compared to the Fourier transform. This reduction leads to a simplification of the difference scheme and a reduction in the amount of computation, i.e., to efficiency of the algorithm.

In this paper, we propose an analytical method of modeling seismic wave fields over a wide range of geophysical media: elastic, inelastic, anisotropic, anisotropic-inelastic, porous, random-inhomogeneous, etc., at very large distances. Since no finite-difference approximations are used, no grid dispersion occurs in computing wave fields for arbitrary media models and observation points. An analytical solution representation in the spectral domain makes it possible to analyze the wave field by parts, specifically, to obtain primary waves. A program of computing the wave fields has been developed, and a simulation of water waves and seismic “ringing” of the Moon has been carried out. The phenomenon of a monotonic shift of the resonance to the lower frequency area with increasing distance of recording is explained. This phenomenon was detected in some experiments with a seismic vibrator.

In [1], Mikhailenko proposed a method of solving dynamic problems of elasticity theory. The method is based on the Laguerre transform with respect to time. In this paper, we propose a modification of this approach, applying the Laguerre transform to a sequence of finite time intervals. The solution obtained at the end of one time interval is used as initial data for solving the problem on the next time interval. To implement the approach, four parameters are chosen: a scale factor to approximate the solution by Laguerre functions, an exponential coefficient of a weight function that is used for finding a solution on a finite time interval, the duration of this interval, and the number of projections of the Laguerre transform. A way to find parameters that provide stability of calculations is proposed. The effect of the parameters on the accuracy of calculations when using second- and fourth-order difference schemes is studied. It is shown that the approach makes it possible to obtain a high-accuracy solution on large time intervals.

A spectral method for solving the 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method is based on an expansion of the solution in terms of Laguerre functions in time. The operation of convolution of functions, which is part of the formulas describing the relaxation processes, is reduced to a sum of products of the harmonics. The Maxwell equations transform to a system of linear algebraic equations for the solution harmonics. In the algorithm, an inner parameter of the Laguerre transformis used. With large values of this parameter, the solution is shifted to high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution. Results of a comparison of the Laguerre method and a finite-difference method in accuracy both for a 2D medium structure and a layered medium are given. Results of a comparison of the spectral and finite-difference methods in efficiency for axial and plane geometries of the problem are presented.

В работе рассматривается численно-аналитический метод решения задачи о распространении сейсмических и акусто-гравитационных волн для неоднородной модели Земля-Атмосфера. Распространение сейсмических волн описывается системой динамических уравнений теории упругости первого порядка, распространение акусто-гравитационных волн в Атмосфере описываются линеаризированными уравнениями Навье-Стокса. Алгоритм основан на комплексировании инрегральных преобразований Лагерра по времени, конечных интегральных преобразований Бесселя по радиальной координате с конечно-разностным методом решения редуцированной задачи по вертикальной координате. В работе приведены примеры расчета сейсмических и акусто-гравитационных волн для неоднородной модели Земля-Атмосфера для различных положений источника.

We apply the spectral Laguerre transform in the time domain to solve 2D Maxwell's equations for propagation of electromagnetic waves in lossy anisotropic media. The new algorithm is simple and efficient as Maxwell's equations are reduced to a harmonic series of linear algebraic equations where the matrix is independent of the harmonic order and is the same for all harmonics.The efficiency of the algorithm is improved by fitting a specially introduced free parameter of the Laguerre transform. If it is large, the solution spectrum shifts toward higher harmonics which is formally equivalent to the case of ray approximation.The Laguerre solution is comparable with high-order accurate finite-difference time-domain (FDTD) solutions. The method is stable both in the region of the wavefield, where conductivity approaches zero and the spectral Fourier method is unstable, and in the high-conductivity region, where the explicit FDTD code requires a too small time step. (C) 2008, IGM, Siberian Branch of the RAS. Published by Elsevier B. V. All rights reserved.

An inverse solution to the ID wave equation is obtained using the spectral Laguerre transform to find the distribution of wave velocities at some point of the medium. The problem is solved as optimization in which the function of Laguerre harmonics is minimized by the conjugate gradient or Newton's algorithms. Reported are velocities of a wave defined by a stepwise constant function. The accuracy of the inverse Solution for the Laguerre harmonics is investigated against the approximation accuracy in the boundary problem. The accuracy and efficiency of the Laguerre method are compared to those in the Fourier method. (c) 2007, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved.

We model propagation of electromagnetic waves in frequency-dependent media applying the Laguerre transform in time domain. The new algorithm is fourth-order accurate in space and computationally efficient. Maxwell's equations are reduced to a harmonic series of linear algebraic equations in which only the right side depends on the harmonic number and the inverse matrix is the same for all harmonics. The efficiency of computation for the algebraic equations is improved by fitting the free parameter of the Laguerre transform. The value of this parameter is easy to find and is likewise the same for all harmonics. The Laguerre scheme provides a better accuracy than the second-order accurate finite-difference solution at large path lengths. The method is stable both in the region of the wavefield where conductivity approaches zero and the spectral Fourier method is unstable, and in the high-conductivity region where the explicit FDTD code requires a too small time step.

One-dimensional Maxwell's equations with relaxation are solved using the Laguerre time series. Space derivatives are obtained by difference approximation of the fourth order of accuracy. The equations reduce to a harmonic series of linear algebraic equations, in which each harmonic is obtained by multiplication of the matrix by the right term. The matrix is universal and is the same for all harmonics. The algorithm shows a high efficiency.With the optimum choice of parameters, the Laguerre transform is advantageous over the finite-difference and Fourier methods. It is more precise than the second order approximation of the finite-difference solution, and uncertainty grows at a lower rate.The method is applicable to two- and three-dimensional Maxwell's equations, without significant limitations.

The wave propagation in real media can be described within the theory of linear viscoelasticity. The presence of convolutional integral in Boltzmann's superposition principle poses the main difficulties in implementing the direct numerical methods in time domain. The paper presents a new algorithm, based on the application of the spectral Laguerre method for the approximation of temporal derivatives and convolution as applied to the problem of seismic wave propagation in the heterogeneous viscoelastic medium. Examples of the calculation of synthetic seismograms for different models of viscoelastic media are presented.

Seismic wave propagation in a viscoelastic media can be described by a system of integro‐differential equations. The solution of such equations requires special methods when using finite‐difference techniques in the time domain. In the frequency domain, the integral terms are represented by complex elastic parameters. This paper presents an efficient algorithm for viscoelastic modelling based on the integral Laguerre transform for the approximation of temporal derivatives and for the calculation of convolution integrals. For the calculation of spatial derivatives, it is possible to use various methods: finite‐difference and finite‐element techniques, spectral and pseudo‐spectral methods. We then obtain a system of algebraic equations with a matrix independent of the parameter m , i.e. the degree of the Laguerre polynomials. In this case, only the right‐hand side of the system has recurrent dependence on the parameter m , which is an analogue of the temporal frequency in the frequency domain. The obtained system with a large number of right‐hand sides can be solved using fast methods, where the matrix is transformed only once, as opposed to the frequency‐domain approach, when the matrix is transformed for each temporal frequency.

P005 NUMERICAL VISCOELASTIC MODELING BY SPECTRAL LAGUERRE METHOD 1 Summary The wave propagation in real media can be described within the theory of linear viscoelasticity. The presence of a convolutional integral in Boltzmann's superposition principle involves the main difficulties in implementing the direct numerical methods in the time domain. The paper presents an efficient algorithm based on the application of the spectral Laguerre method for approximation of temporal derivatives and convolution as applied to the problem of seismic wave propagation in the heterogeneous viscoelastic medium. Introduction The paper presents the new efficient algorithm based on the application of the spectral

The paper presents some efficient algorithms based on the application of the integral Laguerre transform for approximation of temporal derivatives. Some specific features of employing this algorithm for the first and the second order equations with respect to time are considered. A few examples of calculation of seismic fields for the layered medium model with drastically contrast elastic parameters and for the 2-D heterogeneous medium model are presented.

The paper presents an efficient algorithm based on the combination of the integral Laguerre transforms for the temporal derivatives with the Legendre transforms and finite difference method for the spatial variables. Several examples of synthetic seismograms computed for the SH waves propagating in the radial- heterogeneous spherical Earth are presented.

The wave propagation in real media can be described within the framework of the theory of linear viscoelasticity. The presence of convolutional integral in Boltzmann's superposition principle poses the main difficulties in implementing the direct numerical methods in time domain. The paper presents an efficient algorithm, based on the application of the spectral Laguerre method for approximation of temporal derivatives as applied to the problem of seismic wave propagation in the heterogeneous viscoelastic medium.

This paper presents some efficient algorithms based on the spectral Laguerre approximations of temporal derivatives for time dependent problems.