Travelling wave modes of a plane layered anelastic earth

Abstract

Abstract : Incorporation of attenuation into the normal mode sum representations of seismic signals is commonlyeffected by applying perturbation theory. This is fine for weak attenuation, but problematic forstronger attenuation. In this work modes of the anelastic medium are represented as complexsuperpositions of elastic eigenfunctions. For the P-SV system a generalized eigenvalue equation for thecomplex eigenwavenumbers and complex coefficients used to construct the anelastic eigenfunctions isderived. The generalized eigenvalue problem for the P-SV problem is exactly linear in theeigenwavenumber at the expense of doubling the dimension. The SH problem is exactly linear in thesquare of the eigenwavenumber. This is in contrast to a similar standing wave problem for the earthfree oscillations (Tromp and Dahlen,1990). Attenuation is commonly incorporated into syntheticseismogram calculations by introduction of complex frequency dependent elastic moduli. The modulidepend nonlinearly on the frequency. The independent variable in the standing wave free oscillationproblem is the frequency, which makes the eigenvalue problem nonlinear. The choice of the wavenumber asthe independent variable for the traveling wave problem leads to a linear problem.