Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. The propagation of such waves is governed by effective Hamiltonians on the boundary with a space-adiabatic behavior.
Effective Hamiltonians of surface waves correspond to eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities.Using these Hamiltonians, we obtain pseudodifferential surface wave equations. Then we carry out the semiclassical construction of general surface waveparametrices. In the process, we introduce locally Schrödinger-like operators in the boundary normal coordinate and their eigenvalues signifying effective Hamiltonians in the boundary(tangential) coordinates describing surface-wave propagation.
In case of isotropic medium the surface wave decouples up to principal parts into Love and Rayleigh waves associated to scalar and matrix spectral problems, respectively.
Since the mathematical features (such as spectrum, resonances) of these problems can be extracted from the seismograms, we are interested in recovering the Lamé parameters from these data. Plan of lectures:
1) Semiclassical description of surface waves.
2) Semiclassical inverse spectral problems for Love and Rayleigh waves: conditional recovery of S-wave speed using the semiclassical spectra as the data.
3) Recovery of S-and P-wave speeds from the discrete and continuous spectra using the exact methods for Sturm-Liouville operators: Gel'fand-Levitan-Marchenko approach.
4) Leaking modes as analogues of scattering resonances.
The material of these lectures is result of collaboration with Maarten V. de Hoop at Rice University and his Geo-Mathematical Imaging Group.