On mathematical modelling and parameter estimation of seismic media

Within the multidisciplinary field of geophysics, seismology is a branch that accounts for the generation and propagation of vibrations within the Earth, otherwise referred to as seismic waves. To study the mechanical properties of these waves, seismologists model the Earth’s subsurface as a continuous elastic medium. Such an approximation facilitates the interpretation of physical observations within a mathematical framework, which can be approached either as forward or inverse problem. To that end, this dissertation is comprised of two forward problems and one inverse problem. From a forward perspective, theoretical models are proposed based on a priori assumptions of mechanical properties of the subsurface, which can be quantified as changes in velocity with location (inhomogeneity) or direction (anisotropy). Two chapters of this dissertation reside within this context and are applied to homogeneous anisotropic media. In the first of these chapter, we determine the conditions for elliptical roots of the Christo!el equation in media that are the result of the Backus average. Within these conditions, we demonstrate that the slowness surfaces are nondetached. In the second chapter, we present a novel formulation for the purpose of forward modelling traveltimes. Through the Taylor expansion along vertical rays in a horizontally stratified Earth model, we obtain a homogeneous transversely isotropic medium within which the traveltimes are similar to the Fermat traveltimes of its constituent layers. From an inverse perspective, the parameters of the theoretical models are estimated so as to provide an agreement with physical observations. In this dissertation, one chapter resides within this context, where we perform an inversion on traveltime measurements acquired from a vertical seismic profile, otherwise referred to as field data. We implement a derivative-free approach to minimize the residual sum of squares between the measurements and the model. Since field data are not necessarily complete and can be subject to measurement errors, we conduct a simulation study on synthetically generated traveltimes to assess the accuracy of our estimates. Then, we apply our approach to the field data to estimate the inhomogeneity and anisotropy of the subsurface