Full-waveform inversion based on Kaniadakis statistics

TL;DR: The results show that the κ-FWI outperforms the least-squares FWI, providing better parameter estimation of the subsurface, especially in situations where the seismic data are very noisy and with outliers, independently of theκ-parameter, important for the fast convergence of FWI.

Abstract: Full-waveform inversion (FWI) is a wave-equation-based methodology to estimate the subsurface physical parameters that honor the geologic structures. Classically, FWI is formulated as a local optimization problem, in which the misfit function, to be minimized, is based on the least-squares distance between the observed data and the modeled data (residuals or errors). From a probabilistic maximum-likelihood viewpoint, the minimization of the least-squares distance assumes a Gaussian distribution for the residuals, which obeys Gauss's error law. However, in real situations, the error is seldom Gaussian and therefore it is necessary to explore alternative misfit functions based on non-Gaussian error laws. In this way, starting from the $\ensuremath{\kappa}$-generalized exponential function, we propose a misfit function based on the $\ensuremath{\kappa}$-generalized Gaussian probability distribution, associated with the Kaniadakis statistics (or $\ensuremath{\kappa}$-statistics), which we call $\ensuremath{\kappa}$-FWI. In this study, we perform numerical simulations on a realistic acoustic velocity model, considering two noisy data scenarios. In the first one, we considered Gaussian noisy data, while in the second one, we considered realistic noisy data with outliers. The results show that the $\ensuremath{\kappa}$-FWI outperforms the least-squares FWI, providing better parameter estimation of the subsurface, especially in situations where the seismic data are very noisy and with outliers, independently of the $\ensuremath{\kappa}$-parameter. Although the $\ensuremath{\kappa}$-parameter does not affect the quality of the results, it is important for the fast convergence of FWI.