Robust approaches for inverse problems based on Tsallis and Kaniadakis generalised statistics

The inference of physical parameters from measured data is essential for describing and analysing several complex systems. In this regard, the inverse problem theory has been applied to solve this task based on the Boltzmann–Gibbs statistical mechanics by considering that the errors are Gaussian-like. However, in the non-Gaussian noise case, the classical inverse-problem approach estimates model parameters which may be inaccurate and grossly biased. In the case of extreme outliers in the data set, for example, the most common approach used is based on the l1-norm of the difference between modelled and observed data. This approach is famous for being robust because it is based on a long-tailed probability function: the Laplace distribution. However, l1-norm-based inverse problems equally weight noise and information, which limits its robustness to a large number of outliers in the data set. To mitigate the limitations of the l1-norm approach, we introduce in this work the inverse problem theory in the context of Tsallis and Kaniadakis generalised statistical mechanics, by generalising the Laplace distribution. Applications to a classical geophysical data-inversion problem demonstrate that inverse problems based on Tsallis q- and Kaniadakis κ-statistics outperform the classical and l1-norm-based frameworks. They also reveal that the best results are associated with the limit in which both generalised distributions are equivalent.