In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves.

We present a comprehensive analysis of wavenumber resonances or leaking modes associated with the Rayleigh operator in a half space containing a heterogeneous slab, being motivated by seismology. To this end, we introduce Jost solutions on an appropriate Riemann surface, a boundary matrix and a reflection matrix in analogy to the studies of scattering resonances associated with the Schrödinger operator. We analyse their analytic properties and characterize the distribution of these wavenumber resonances. Furthermore, we show that the resonances appear as poles of the meromorphic continuation of the resolvent to the nonphysical sheets of the Riemann surface as expected.

Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. In the case of an 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 Lam e parameters from these data. We generalize spectral methods for Schrodinger operators to the Rayleigh problem, which is essentially not of Schrodinger type; and give comprehensive analysis of the wavenumber resonances, known in seismology as leaking modes.

This is joint work with Maarten V. de Hoop.

We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lamé parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and we point out parallels with a corresponding problem for the Schrödinger equation. The Jost function can be identified with spectral data. We derive a Gel’fand-Levitan type equation and obtain uniqueness with two distinct frequencies.

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.

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. In case of isotropic medium, the surface wave decouples up to principal parts, into Love and Rayleigh waves.

We present the conditional recovery of the Lamé parameters from spectral data, in two inverse problems approaches:- semiclassical techniques using the semiclassical spectra as the data;- exact methods for Sturm-Liouville operators, using the discrete and continuous spectra, or the Weyl function, as the data based on the solution of the Gel’fand-Levitan-Marchenko equation.

We conclude with discussion using resonances (leaking modes) as data.

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we focus on Love waves. Under certain generic conditions, we establish uniqueness and present a reconstruction scheme for theS-wavespeed with multiple wells from the semiclassical spectrum of these waves.

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we extend the treatment of Love waves [5] to Rayleigh waves. Under certain conditions, and assuming that the Poisson ratio is constant, we establish uniqueness and present a reconstruction scheme for the S-wave speed with multiple wells from the semiclassical spectrum of these waves.

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.

A simple mathematical model for free vibrations of an elastically clamped beam is suggested to interpret the results of the resonance frequency analysis developed for implant stability measurements in terms of the Implant Stability Quotient (ISQ) units. It is shown that the resonance frequency substantially depends on the lateral compliance of the implant/bone system. Based on the notion of the lateral stiffness of the implant/bone system, a new measure of the implant stability is introduced in the form similar to the ISQ scale and is called the Implant Clamping Quotient (ICQ), because it characterizes the jawbone’s clamp of the implant. By definition, the ICQ unit is equal to a percentage of the original scale for the lateral stiffness of the implant/bone system.