The radiation modes of a vertically varying half-space: a new representation of the complete Green's function in terms of modes

Abstract

We prove that the complete Green's function for an isotropic vertically varying halfspace can be written in a dyadic form as a sum on the usual normal modes plus an integral over radiation modes. The radiation modes form the basis on which to represent the body waves, which are not trapped in the structure. The expressions f o r their eigenfunctions, the so-called improper eigenfunctions, are found for both the SH and the P-SV wave operators. We prove that they are orthogonal t o each other and to the normal-mode eigenfunctions. In a numerical example, we present improper eigenfunctions, and compare complete wavefield seismograms synthesized by the mode method and by the discrete wavenumber integration method.

Generalized eigenfunctions of layered elastic media and application to diffuse fields

“…The eigenvectors |e are solutions of the equation L |e = λ |e but do not belong to the domain of the operator, i.e., they are not solutions with finite energy. These eigenvectors are required to construct the complete modal solutions of the elastic wave equation 15 . Our work generalizes previous results obtained in the case of a homogeneous half-space 11,16 .…”

“…In the framework of the spectral theory presented in this paper, discrete, continuous or mixed spectra can be treated on the same footing. Other applications of the generalized normal mode expansion could be envisaged, such as the calculation of synthetic seismograms for geophysical applications 15 .…”

Finite element computation of trapped and leaky elastic waves in open stratified waveguides

“…From a mathematical point of view, leaky modes are complex poles that do not belong to the proper eigenvalue spectrum because such poles are located on a forbidden Riemann sheet [11,1]. These modes constitute a discrete set which is actually not part of the complete modal set constituted by the discrete set of trapped modes and the continuous spectrum of radiation modes representing the contribution of body waves [4,12,5,13]. However, leaky modes have the potential to give a good representation of the continuum over an area close to the core or the surface of the waveguide.…”

“…They constitute a continuous spectrum of standing waves [4,5]. This continuum is intrinsic to the unbounded nature of the cross-section of the waveguide.…”

A Review of the Discrete Wavenumber Method

“…The DWN method has been successfully tested against analytical solutions and other techniques (e.g., YAO and HARKRIDER, 1983;BEN-ZION and AKI, 1990) and has been extensively used to check the accuracy of other methods like finitedifferences, finite-elements, ray methods, mode summation, or pseudo-spectral techniques (e.g., STEPHEN et al, 1985;SAIKIA and HERRMANN, 1986;BEYDOUN and KEHO, 1987;MAUPIN, 1996;AOI and FUJIWARA, 1999;MOCZO et al, 1999).…”