Seismic tomography allows us to study the interior structure of the Earth on a range of lengthscales, with the method of waveform inversion being a promising and relatively recent development. In this work, we present new theoretical methods which may contribute to the study and solution of seismic tomography inverse problems. In particular, we set out the mathematical grounding for novel inversion frameworks which could be applied to waveform inversion.

In Chapter 2, we introduce an inversion framework for determining the topography of internal boundaries jointly with volumetric parameters, using waveform inversion. The approach makes use of a referential description of elasticity to encode boundary topography within a volumetric parameter, putting topography on an equal footing with other parameters in the joint inversion. We derive expressions for sensitivity kernels in this parameterisation using the adjoint method, and calculate examples on a 2D domain. A second-order adjoint method is used to derive expressions for the application of the Hessian to a model perturbation.

Chapter 3 sets out an approach for determining a suitable function space to use as a model space in geophysical inverse problems, including seismic tomography. In particular, we show that a Sobolev space is often a suitable choice, and allows us to specify a required degree of regularity for model parameters. We show that gradients in a Sobolev space can be calculated by a post-processing of a derivative, with the gradient being smoother than the derivative. The calculation of these gradients is demonstrated in different scenarios on spherical domains.

The final chapter presents a simple application of the Sobolev gradient method, to linearised ray tomography in 2D. This allows us to investigate the potential impacts of the choice of function space on tomography problems more generally. A particular advantage of the method used is that the minimum-norm solution to the least squares inverse problem with exact data can be calculated.