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 $\kappa$-generalized exponential function, we propose a misfit function based on the $\kappa$-generalized Gaussian probability distribution, associated with the Kaniadakis statistics (or $\kappa$-statistics), which we call $\kappa$-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 $\kappa$-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 $\kappa$-parameter. Although the $\kappa$-parameter does not affect the quality of the results, it is important for the fast convergence of FWI.

Full-waveform inversion (FWI) is a powerful seismic imaging methodology to estimate geophysical parameters that honour the recorded waveforms (observed data), and it is conventionally formulated as a least-squares optimization problem. Despite many successful applications, least-squares FWI suffers from cycle skipping issues. Optimal transport (OT) based FWI has been demonstrated to be a useful strategy for mitigating cycle skipping. In this work, we introduce a new Wasserstein metric based on q-statistics in the context of the OT distance. In this sense, instead of the data themselves, we consider the graph of the seismic data, which are positive and normalized quantities similar to probability functions. By assuming that the difference between the graphs of the modelled and observed data obeys the q-statistics, we introduce a robust q-generalized graph-space OT objective function in the FWI context namely q-GSOT-FWI, in which the standard GSOT-FWI based on l₂-norm is a particular case. To demonstrate how the q-GSOT-FWI deals with cycle skipping, we present two numerical examples involving 2-D acoustic wave-equation modelling. First, we investigate the convexity of q-GSOT objective function regarding different time-shifts, and, secondly, we present a Brazilian pre-salt synthetic case study, from a crude initial model which generates significant cycle-skipping seismic data. The results reveal that the q-GSOT-FWI is a powerful strategy to circumvent cycle skipping issues in FWI, in which our objective function proposal presents a smoother topography with a wider attraction valley to the optimal minimum. They also show that q-statistics leads to a significant improvement of FWI objective function convergence, generating higher resolution acoustic models than classical approaches. In addition, our proposal reduces the computational cost of calculating the transport plan as the q-value increases.

In an attempt to overcome the difficulties of the full waveform inversion (FWI), several alternative objective functions have been proposed over the last few years. Many of them are based on the assumption that the residuals (differences between modelled and observed seismic data) follow specific probability distributions when, in fact, the true probability distribution is unknown. This leads FWI to converge to an incorrect probability distribution if the assumed probability distribution is different from the real one and, consequently it may lead the FWI to achieve biased models of the subsurface. In this work, we propose an objective function which does not force the residuals to follow a specific probability distribution. Instead, we propose to use the non-parametric kernel density estimation technique (KDE) (which imposes the least possible assumptions about the residuals) to explore the probability distribution that may be more suitable. As evidenced by the results obtained in a synthetic model and in a typical P-wave velocity model of the Brazilian pre-salt fields, the proposed FWI reveals a greater potential to overcome more adverse situations (such as cycle-skipping) and also a lower sensitivity to noise in the observed data than conventional L²- and L¹-norm objective functions and thus making it possible to obtain more accurate models of the subsurface. This greater potential is also illustrated by the smoother and less sinuous shape of the proposed objective function with fewer local minima compared with the conventional objective functions.

The detection of information-bearing signals in a time series is very important for describing and analyzing a wide variety of complex physical systems. However, identifying events in low signal-to-noise ratio circumstances remains a challenge once heavy data preprocessing is usually required. In this work, we propose a robust methodology based on the instantaneous-spectral Shannon entropy for capturing microseismic events in noisy environments without the requirement of data preprocessing. We call our proposal the instantaneous spectral entropy detection (ISED) method. We tested the ISED in a couple of real empirical seismic records to detect microseismic events. Our methodology detects microseismic patterns even without any preprocessing, in contrast to usual methods in the literature which need appreciable data preprocessing.

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.