ETD Collection

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Subject: search.filters.subject.Inversion (Geophysics)

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    Applications of chaos and fractals to geophysical inversion problems
    (2019-10-22) Dias, Brandon
    In order to gain clear insight into the structure and composition of the Earth and its subsurface, geophysicists and geologists take readings of geophysics responses. Gravitational responses are among the most often recorded datasets and among the most used means of model analysis is the least-squares inversion process. Here; this is demonstrated using synthetic gravitational responses from buried sphere and cylinder models of different density contrasts to the background. The least-squares inversion attempts to utilize initial user chosen parameters to create models which correlate strongly with observed data and thus create potential geological models of the Earth’s subsurface or submerged geological structures. The inversion processes, misfit hyper-functions, basins of attraction and fractal dimensions are studied as functions of initial model parameters. We observe that the fractal dimension and basin of attraction vary with respect to observed model depths and positions. Additionally the fractal dimension is inversely proportional to the degree of damping of the least-squares inversion process. A potential problem with the least-squares inversion method is the possibility for solutions to tend to local minima. These may better fit an observed model response but may not provide a geologically viable option. The equation for the least-squares inversion, as applied to observed models, is altered to induce bifurcations and chaos within solutions. Chaos can be used to move the inversion process out of local minima on the misfit surface, potentially improving the fit of the model response to the data. Bifurcation diagrams are established and the periodicity analysed.
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    Aspects of the theory of inversion as applied to geophysical problems.
    (1997) Cooper, Gordon Robert John.
    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, in press). 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 leastsquares 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