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

SUMMARY 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 for 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 to 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.