Applications of chaos and fractals to geophysical inversion problems

Abstract

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.