Transdimensional approaches to geophysical inverse problems

Abstract

In geophysical inversion the model parameterisation, the number of unknown the level of smoothing and the required level of data fit are usually arbitrarily determined by the user prior to the inversion. These quantities are related to each other and define the formulation of the inverse problem; by definition they affect the final solution. They are often manually 'tuned' by means of subjective trial-and-error procedures, and this represents a recurring problem in geophysical inversion. In this thesis this issue is addressed by proposing an alternative inversion strategy. Different methodologies recently developed in the area of Bayesian statistics are combined to produce a general inversion algorithm, which lets the data themself formulate the inverse problem. This is done by treating the tunable quantities as unknowns to be constrained directly by the data. A major focus is on situations where data constrain a 2D spatially varying field, particularly seismic tomography. A variable parametrisation consisting of Voronoi cells with mobile geometry, shape and number, is treated as a set unknowns in the inversion. The reversible jump algorithm is used to sample the transdimensional model space within a Bayesian framework which avoids global damping procedures and the need to tune regularisation parameters. The method developed in this thesis is an ensemble inference approach, where many potential solutions are generated with variable numbers of cells. Information is extracted from the ensemble as a whole by performing Monte Carlo integration to obtain an expected Earth model. The inherent model averaging process naturally smooths out unwarranted structure in the Earth model, but maintains local discon{u00AC}tinuities if well constrained by the data. As a by product, uncertainty estimates are obtained for any point in the medium. In a transdimensional approach, the level of data uncertainty directly determines the model complexity needed to satisfy the data. Intriguingly, the Bayesian formulation can be extended to the case where data uncertainty is also uncertain. It is possible to estimate the level of data noise while at the same time controlling model complexity in an automated fashion. For 2D seismic tomography, this novel procedure gives promising results in situations where the ray coverage is far from ideal, as it performs better compared to standard methods that use regular parameterisations. The method is also applied to the inversion of three ambient noise datasets that span the Australian continent at different scales. A multiscale tomographic image of Rayleigh wave group velocity for the Australian continent is constructed. Experiments show that the procedure is particularly powerful when dealing with multiple data types that have different unknown levels of noise. Here it is possible to adjust the fit to different datasets and to provide a velocity map with a spatial resolution adapted to the quantity of informtation present in the data. Finally, two applications of ID problems are considered. The first is an application to a regression problem where the goal is to infer the position and number of abrupt changes in noisy geochemical records. The second is an application to receiver function waveform inversion, where both the magnitude and correlation of data noise are inverted for. Other fields where the general methodology can be applied are outlined. Show more