Puzzle in inverse problems: Tsallis noise and Tsallis norm

Abstract

Inverse problems are challenging in several ways, and we cite the non-linearity and the presence of non-Gaussian noise. Least squared is the standard method to construct a equivalent functional for optimization, which is equivalent to the L2 norm of the misfit. Alternative norms in the optimization process are an useful strategy in the inverse problem solution. Generalized statistics can be present at two sides of the inverse problem: in the noise that pollutes the data and in the norm used in the optimization algorithm. With help of a seismic problem, we polluted the signal with a q-exponential noise (using an exponent $q_{\text {noise}}$) and subsequently inverted the problem using a norm associated to a q-exponential (with an exponent $q_{\text {inv}}$). The same procedure was also applied to the simpler problem of a linear fitting. We tested the hypothesis of a relation between the exponents $q_{noise}$ and $q_{\text {inv}}$. The overall pattern observed is the following: inversion error are smaller for low $q_{\text {noise}}$ and high $q_{\text {inv}}$. In contrast, the worst inversion is found for high polluting noise (far from Gaussian noise) and for inversion with low $q_{\text {inv}}$ (close to the Gaussian case).

Graphic abstract