### Free oscillations, surface waves and spectral theory

I revisited the construction and analysis of the relevant system of elasto-gravitational equations describing the free oscillations of the earth. The emphasis here is to reconcile (non-smooth, typically Lebesgue measurable) variations in material properties and the geodynamical processes in Earth’s interior with this construction. We adapted the weak formulation, in particular, with a view to solid-fluid boundaries, introduced appropriate function spaces and proved well-posedness using the theory of Hille-Yoshida and applying the Lumer-Phillip theorem. We also obtained energy estimates.

**Semi-classical analysis of surface waves**. We carried out an analysis of surface waves in a medium which is stratified near its boundary at some scale comparable to the wave length. Such a medium can also be thought of as a surficial layer (which can be thick) overlying a half space. We described how the (dispersive) propagation of such waves is governed by effective Hamiltonians on the boundary, and showed that the system is displayed by a space-adiabatic behavior. We obtained pseudodifferential surface wave equations and, via parametrix constructions, expansions of surface-wave amplitudes. These play a role, for example, in surface-wave tomography.

(Our analysis applies to the study of surface waves in Earth’s ‘near’ surface in the scaling regime mentioned above. The existence of such waves, that is, propagating wave solutions which decay exponentially away from the boundary of a homogeneous (elastic) half-space was first noted by Rayleigh. Rayleigh and (‘transverse’) Love waves can be identified with Earth’s free oscillation triples _{n}*S** _{l} *and

_{n}

*T*

*with*

_{l}*n*

*«*

*l*

*/*4. Love was the first to argue that surface-wave dispersion is responsible for the oscillatory character of the main shock of an earthquake tremor, following the ‘primary’ and ‘secondary’ arrivals.)

The Hamiltonians are pseudodifferential operators with non-trivial dispersion relations, that is, principal symbols which are non-homogeneous. We derived this with techniques from semi-classical analysis. Each Hamiltonian is identified with an eigenvalue of a locally one-dimensional Schrödinger problem on the one hand, and generates a flow identified with surface-wave bicharacteristics on the other hand. We showed that the eigenvalues exist under certain assumptions for the local profiles of coefficients in the boundary normal direction. The surface waves correspond with the discrete spectrum of the Schro¨dinger operator. The semi-classical asymptotics of the discrete spectrum provides a way to study the uniqueness and stability of the corresponding locally one-dimensional spectral inverse problems.

We used our results from semi-classical analysis to further develop a method, related to so-called eikonal tomography, for the recovery of anisotropic elastic parameters [preprint 9]. We first obtained a reconstruction of surface-wave bicharacteristics in the presence of azimuthal anisotropy enabled by appropriate geometric regularization from ambient noise generated dense array data. We then addressed the one-dimensional spectral inverse problems in the piecewise constant coefficients case. We applied our method to Rayleigh wave signals extracted from cross-correlations of ambient noise recorded at a seismograph network to constrain heterogeneity and anisotropy in the crust of SE Tibet.