A divide-and-conquer algorithm for seismic data approximation by the Laguerre series

An algorithm of the Laguerre transform for approximating functions on large intervals is proposed. The idea of the considered approach is that the calculation of improper integrals of rapidly oscillating functions is replaced by a solution of an initial boundary value problem for the one-dimensional transport equation. It allows one to successfully avoid the problems associated with the stable implementation of the Laguerre transform. A divide-and-conquer algorithm based on shift operations made it possible to significantly reduce the computational cost of the proposed method. Numerical experiments have shown that the methods are economical in the number of operations, stable, and have satisfactory accuracy for seismic data approximation.

The Laguerre transform of a convolution product of vector-valued functions.

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Approximation by interpolation spectral subspaces of operators with discrete spectrum

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

The Laguerre spectral method as applied to numerical solution of a two−dimensional linear dynamic seismic problem for porous media

The initial boundary value problem of the dynamics of fluid saturated porous media, described by three elastic parameters in the reversible hydrodynamic approximation, is numerically solved. A linear two-dimensional problem as dynamic equations of porous media for components of velocities, stresses and pore pressure is considered. The equations of motion are based on conservation laws and are consistent with thermodynamic conditions. In this case, a medium is considered to be ideally isotropic (in the absence of energy dissipation) and twodimensional heterogeneous with respect to space. For a numerical solution of the dynamic problem of poroelasticity we use the Laguerre transform with respect to time and the finite difference technique with respect to spatial coordinates on the staggered grids with fourth order of accuracy. The description of numerical implementation of the algorithm offered is presented, and its characteristics are analyzed. Numerical results of the simulation of seismic wave fields for the test layered models have been obtained on the multiprocessor computer.

On the Numerical Solution of the Initial-Boundary Value Problem with Neumann Condition for the Wave Equation by the Use of the Laguerre Transform and Boundary Elements Method

We consider a numerical solution of the initial-boundary value problem for the homogeneous wave equation with the Neumann condition using the retarded double layer potential. For solving an equivalent time-dependent integral equation we combine the Laguerre transform (LT) in the time domain with the boundary elements method. After LT we obtain a sequence of boundary integral equations with the same integral operator and functions in right-hand side which are determined recurrently. An error analysis for the numerical solution in accordance with the parameter of boundary discretization is performed. The proposed approach is demonstrated on the numerical solution of the model problem in unbounded three-dimensional spatial domain.