High frequency elastic wave inversion

This thesis is concerned with the problem of high frequency elastic wave inversion. This is the problem of determining sharp, localised changes in the properties of materials beneath the surface of the earth using only measurements of reflected seismic waves taken at or near the surface. The central objective of this thesis is to construct multiparameter inversion operators which map data from surface wave measurements into accurate estimates for the high frequency perturbations in the density, ρ, and in the 21 independent Hooke’s tensor components, cijkl, of subsurface anisotropic inclusions. Using results from the field of microlocal analysis of Fourier Integral Operators, it is shown that asymptotically valid inversion operators exist which can invert all 22 independent elastic parameter perturbations directly, without relying on statistical estimates. To gather the required data, the technique of using ensembles of linked seismic experiments is introduced and extensively analysed in the context of a standard linearised single scattering model for elastic waves based on the Born approximation. This technique builds on work by Burridge and others in [10:Burridge R. ; De Hoop M. V. ; Miller D. ; Spencer C. ; 1998], and by Nolan in [17:Clifford J. Nolan 1997][49:Nolan and Ryan 2007]. In addition, a fundamental theoretical analysis of the seismic inversion problem is carried out. By analysing important components of the seismic forward problem, a theoretical framework is introduced which allows the determination of whether or not multiparameter inversion is possible, and specifically with what types of seismic ensembles and elastic wave modes. In particular, this framework will show under which circumstances multiparameter inversion is not possible, both for the case of point anisotropic inclusions, and in particular for larger volume inclusions. It is shown that these results can be extended to inversion in multiple type of seismic backgrounds. To complement the theoretical results, an application of evolutionary algorithm is presented which is used to find practical invertible seismic ensembles which allow inversion to be carried out feasibly. The thesis also presents a introductory overview of the techniques of Fourier Integral Operators and microlocal analysis, which are used to construct later inversion models. Other more elaborate mathematical techniques used in the thesis are also introduced or expanded on in the appendices for the benefit of the general reader. Finally, supporting lemmas in the appendices introduce a new method for determining the component dependencies of linear elastic materials.