Inverse theory provides an important tool that the geophysicist can use to explore the structure of the Earth. This thesis examines several new approaches to the inverse problem, and suggests ways of improving the conventional least-squares technique. Non least-squares inversion was applied to borehole temperature data from South Africa, and when the norm of the inversion was controlled by the statistics of the misfit, it reduced by over 50% the number of iterations required for the inversion to converge upon a solution. Various damping schemes were also examined, and the use of the misfit in controlling the damping is shown to provide the best solution of those studied (Cooper and Jones, Geophysics March/April 1998). Improvements to the efficiency of the inverse process were also achieved by the fitting of parabolic forms to portions of the misfit surface, using both the misfit value and the gradient of the surface, for gravity data. The presence of nearby minima other than the one that the inversion has just converged to can also be detected in this manner.
The set of initial models that converged to a particular solution using least-squares inversion was studied for magnetic data, and it was noted to have a fractal nature. The fractal dimension of the set was found to be inversely proportional to the damping of the inverse problem.
The inverse process was pushed into a chaotic state by the modification of the least- squares inversion equation. The chaotic state was studied, and exploited to ensure that a broad range of possible solutions was examined by the inversion, and to escape from local minima when these were encountered. Chaotic inversion can be used as an alternative to the Monte Carlo technique, over which it has some advantages.
Finally, the project involved the writing of a large quantity of geophysical software using object-oriented techniques, the structure of which is described in detail.
The last two decades has seen the emergence of new phenomena being observed in all areas of nonlinear dynamics. Principal among these has been chaotic vibrations; random-like motions produced from completely deterministic systems. It has long been known that this phenomena exists in low order systems such as simple one degree-of-freedom mechanical problems, however little research has been directed towards more complex mechanical systems such as multi-body systems.
This PhD research project concerns an investigation to identify and control possible chaotic instabilities occurring in non-linear mechanical systems involved with the dynamics of rotating multi-body systems. In particular, results for a three dimensional rigid body with internal moving parts and energy dissipation are presented. This is analogous to a nutation damper system used for stability in dual-spin spacecraft. Numerical and analytical results are presented, identifying chaotic phenomena in this model when the rotor is subjected to a sinusoidally varying torque for a range of forcing amplitude and frequency. Such a torque, in practice, may arise under malfunction of the control system or from an unbalanced rotor. The motion is studied using the techniques of time history, phase space, frequency spectrum, Poincare map analysis and Lyapunov characteristic exponents analysis.
Three recently developed methods of feedback control are then introduced to eliminate the chaotic instabilities in the dual-spin spacecraft system and numerical simulations show the effectiveness of each method. The first two methods are model independent techniques designed to eliminate chaotic instabilities in any nonlinear system. A third, model dependent, method has also been derived from total system energy considerations and is shown to be more effective. Similar results for a rotating body with energy dissipation and a torsional driveline incorporating a Hookes (Universal) Joint are also developed using the same techniques.
MSc (in Mathematics of Nonlinear Models, joint with Edinburgh University):
Source: Chris Eilbeck (email@example.com).
A listing of Recent Theses at Heriot-Watt University also appeared in UK Nonlinear News 10, November 1997. ((Contact to shortly change to Jonathan Sherratt (firstname.lastname@example.org.))