Asymptotic eigenfrequencies of the Earth's normal modes

We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A.     

Multiplet-Multiplet Coupling Due to Lateral Heterogeneity: Asymptotic Effects On the Amplitude and Frequency of the Earth's Normal Modes

Summary. Starting with the first-order formulation of quasi-degenerate splitting theory for the normal modes of a laterally heterogeneous earth, we have obtained an asymptotic expression for the coupling terms corresponding to neighbouring multiplets along the same dispersion branch as the mode considered, valid to order 1/ℓ, where ℓ is the angular order of this mode (ℓ≥ 1).
We show that, to order zero, these coupling terms introduce a small shift in epicentral distance into the expression for the long period seismogram obtained by normal mode summation. This shift depends on the difference between the great circle and the minor arc averages of the local frequency. the coupling terms thus permit us to reconcile results obtained by normal-mode summation and by a propagating wave approach, as far as the dependence on structure of the phase of surface waves is concerned.
To order 1/ℓ, the coupling terms result in a perturbation in the amplitude of the mode considered, which depends on spatial derivatives of the local frequency and thus on the structure in the vicinity of the source station great circle path. We show that this term is equivalent to that which is found using ray perturbation methods for propagating surface waves. We compare and discuss the assumptions underlying both approaches and illustrate, by an example, the potential of the asymptotic normal-mode formulation for improved modelling of lateral heterogeneity in the earth.     

B-polarization induction in two thin half-sheets coupled to the mantle by a conducting crust—II. Solution by the mode-matching method, the fields above the conductors, and example results

The mode-matching method is used to obtain an exact analytical solution to the problem of B -polarization induction in two adjacent thin half-sheets, lying on a conducting layer that is terminated by a perfect conductor at finite depth. These components of the model represent, respectively, the Earth's conducting surface layers, crust, and mantle. In dimensionless variables, the model has three independent parameters, these being the two thin-sheet conductances and the layer thickness. The mode-matching solution obtained in this paper is shown to be identical lo that derived via the Wiener-Hopf method in a companion paper (Dawson 1996), and so provides additional verification of that solution. As was shown in the companion paper, the solution for the present model contains, as special limiting cases, those for three models considered earlier by various authors. The second part of the present paper addresses the solutions for the electric fields in the non-conducting half-space above the conductors, which represents the atmosphere. In the final part, sample numerical calculations are presented to illustrate the solution.