Inverse problems, non-uniqueness and symmetries in geology

Most research in geology requires solving inverse problems. A geological inverse problem could be, for example, to extrapolate conditions in the past given limited observations today, or to unravel properties of the Earth’s interior given incomplete measurements gathered at the surface. Unfortunately, inverse problems usually have non-unique solutions. However, these solutions are often linked by quite simple relationships even when problems are non-linear. These relationships express the symmetry for a problem since predictions are unaltered given proper changes in a solution obeying the rules. Symmetries are powerful tools since they enable an existing solution to be directly transformed into an alternative solution. This property leads to a novel inversion procedure based on numerical models, i.e. find a simple solution that produce the desired model and then have all additional solutions derived from the initial solution. Calculation of multiple solutions allows properties common to all solutions to be deduced and hence allows end-member hypotheses to be examined. Hence, it is possible to determine what is physically reasonable and what is not for an inverse problem, even though a unique solution is not available. The principle of symmetry is quite general and can be widely applied in various fields of geology. Synthetic and real-data examples presented in this thesis cover sedimentology, thermochronology and geophysics.