Inference from noisy data with an unknown number of discontinuities: ideas from outside the box

Abstract

The focus of this presentation is on new ways to extract information from complex datasets in situations where direct measurement is not possible. Such inverse problems are ubiquitous across the physical and mathematical sciences and are central to discovery of resources within the Earth upon which Australian society is dependent. A recurring problem is how to choose the number of unknowns with which to fit noisy data. If too few are chosen the data cannot be fit and if too many the inversion results contain unwarranted detail. Statistical methods are often used to find optimal numbers of unknowns, but these are based on simplistic assumptions and typically require multiple trial inversions to be performed with different numbers of variables. A new general approach recently applied to geophysical problems is to ask the data itself `How many unknowns should be used ?? While this may seem counter-intuitive at first sight it turns out to be entirely feasible. In effect the number of unknowns itself becomes an unknown. An extension of the basic approach also allows the level of noise on the data to also be included as an unknown. In this presentation we outline the central ideas, and illustrated through an example where a geophysical property varies only in 1-D (usually depth or time) and is constrained from surface measurements. Applications of the general approach are to airborne EM data, borehole geophysics, seismic interpretation and also palaeoclimate reconstructions