Asymptotic Structure in Torsional Free Oscillations of the Earth-I. Overtone Structure

By means of a Liouville transformation, the torsional eigenfrequencies nσl of a spherically symmetric, non-rotating, elastic and isotropic (SNREI) Earth can be defined as the eigenvalues of a second order differential equation with no first derivative term. From this equation, an asymptotic formula for nσl for fixed l can be derived in terms of the radial travel time for shear waves as well as other structural parameters for the mantle. This formula has a number of geophysical applications which yield new constraints on the rigidity structure of the mantle. In particular, the asymptotic structure of torsional free oscillation data contains information about the size, sharpness and position of discontinuities in the upper mantle.