In the research of the Geomechanics Project, the focus is on seismic ray theory. The choice of that method over numerical methods has been largely arbitrary, though full numerical methods may require more computation time and storage than ray methods and may be less able to model propagation of singularities. So far, the attention of the Geomechanics Project has been mostly on traveltimes and ray trajectories or wavefronts but not on the displacement amplitude. My study will centre on modelling that amplitude which is described in the context of ray theory by the transport equation. I plan to further review existing approaches to amplitude modelling in the literature including those in chapters 5-6 of Cervený (2001), and in Aki and Richards (2002) and Kennett (2001). Then I will try to devise a new approach, based on Fourier-integral operators (FIOs), to study the equations of motion. I expect to show that FIOs extend the results of the classical transport equation. As a preliminary to solution of the equations of motion I will apply the method of characteristics to the equations of motion and then demonstrate that a principal-symbol formulation yields the same characteristics. That formulation is linked to, and should facilitate development of, a full analytic FIO solution for simple models and/or approximations. But, for a general case, FIO reduction of the equations of motion may be only to a point where some numerical work is still required but much less than if the original equations were numerically solved. In order to check FIO solutions I will have to perform one full finite element or finite difference numerical solution for at least one test model.