We carry out a semiclassical analysis of surface waves in Earth which is stratified near its boundary at some scale comparable to the wave length.

Propagation of such waves is governed by effective Hamiltonians which are non-homogeneous principal symbols of some pseudodifferential operators. Each Hamiltonian is identified with an eigenvalue in the discreet spectrum of a locally one-dimensional Schrödinger-like operator on the one hand, and generates a flow identified with surface wave bicharacteristics in the two-dimensional boundary on the other hand.

The eigenvalues exist under certain assumptions reflecting that wave speeds near the boundary are smaller than in the deep interior. This assumption is naturally satisfied in Earth’s crust and upper mantle.

Using the mentioned Hamiltonians, we obtain pseudodifferential surface wave equations. In the case of isotropic elasticity, the equations decouple into equations for Rayleigh and Love waves. In both cases, we perform a comprehensive analysis of the recovery of the S-wave speed from the semiclassical spectrum.

Our approach follows the ideas of Colin de Verdière pertaining to acoustic surface waves.