Tsallis Entropy, Likelihood, and the Robust Seismic Inversion

The nonextensive statistical mechanics proposed by Tsallis have been successfully used to model and analyze many complex phenomena. Here, we study the role of the generalized Tsallis statistics on the inverse problem theory. Most inverse problems are formulated as an optimisation problem that aims to estimate the physical parameters of a system from indirect and partial observations. In the conventional approach, the misfit function that is to be minimized is based on the least-squares distance between the observed data and the modelled data (residuals or errors), in which the residuals are assumed to follow a Gaussian distribution. However, in many real situations, the error is typically non-Gaussian, and therefore this technique tends to fail. This problem has motivated us to study misfit functions based on non-Gaussian statistics. In this work, we derive a misfit function based on the q-Gaussian distribution associated with the maximum entropy principle in the Tsallis formalism. We tested our method in a typical geophysical data inverse problem, called post-stack inversion (PSI), in which the physical parameters to be estimated are the Earth's reflectivity. Our results show that the PSI based on Tsallis statistics outperforms the conventional PSI, especially in the non-Gaussian noisy-data case.