In the geometrical optics approximation the split eigenfrequencies Δω of an isolated multiplet on a laterally hetergeneous earth are certain eigencontours of the great circular average Δω¿ of the local eigenfrequency perturbation Δωlocal. The normal t¿ to a surface wave orbital plane, averaged over an orbit, precesses around the contours of Δω¿, and the quantization condition that selects the eigencontours is ⟨L¿z⟩=m, where L¿=(l+1/2)t¿ is the orbital angular momentum and ⟨L¿z⟩ is averaged over a precession cycle. All but one of the split eigenfrequencies in this approximation are degenerate doublets because of the even parity of Δω¿, and the spacing between adjacent doublets along a given branch of ⟨L¿z⟩ versus Δω¿ is △ωadjacent≈2&pgr;/T precess, where Tprecess is the precession period. There is a separate branch of the action ⟨L¿z⟩ versus Δω¿ associated with every local minimum or maximum of Δω¿ on the surface of the earth, and the resulting interleaving of doublets can lead to very irregular line spacing.

The doublets are split even on a nonrotating earth by tunneling between surface waves propagating in opposite directions about otherwise identical ray paths. Ordinarily, the tunneling time Ttunneling is much longer than Tprecess so the spacing △ωtunnel=2&pgr;/Ttunnel between the singlets in a quasi-degenerate doublet is extremely small. In general, the line spacing in a split multiplet is most complicated near the separatrix or slow-fast transition, since it is there that the precession period is the greatest and the tunneling time is the least.