Normal-mode representations of multiple-ray reflections in a spherical earth

Summary. In a spherically symmetric, isotropic earth model the duality between seismic rays and modes can be established completely by application of the principle of stationary phase to the summed normal-mode representation of the time signal. The requirement of stationary phase must be applied not only on the sum over the angular order but also over the radial-order summation.
This approach is illustrated by using asymptotic approximations to the equations for toroidal oscillations. In this way the travel-time formulae for rays from a surface source in a sufficiently smooth earth model are easily derived, and the distribution of modes between the different rays can be found. The result may be used as a selection criterion to reduce the number of modes that must be summed to construct synthetic seismograms at a certain distance and within a certain time window.