A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation.

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.

Generalized statistics: Applications to data inverse problems with outlier-resistance.

The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of Rényi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyze the outlier-resistance of each proposal through the so-called influence function. In this way, we formulate inverse problems by constructing objective functions linked to the maximum likelihood estimators. To demonstrate the robustness of the generalized methodologies, we consider an important geophysical inverse problem with high noisy data with spikes. The results reveal that the best data inversion performance occurs when the entropic index from each generalized statistic is associated with objective functions proportional to the inverse of the error amplitude. We argue that in such a limit the three approaches are resistant to outliers and are also equivalent, which suggests a lower computational cost for the inversion process due to the reduction of numerical simulations to be performed and the fast convergence of the optimization process.

Full-waveform inversion based on generalized Rényi entropy using patched Green's function techniques.

The estimation of physical parameters from data analyses is a crucial process for the description and modeling of many complex systems. Based on Rényi α-Gaussian distribution and patched Green's function (PGF) techniques, we propose a robust framework for data inversion using a wave-equation based methodology named full-waveform inversion (FWI). From the assumption that the residual seismic data (the difference between the modeled and observed data) obeys the Rényi α-Gaussian probability distribution, we introduce an outlier-resistant criterion to deal with erratic measures in the FWI context, in which the classical FWI based on l2-norm is a particular case. The new misfit function arises from the probabilistic maximum-likelihood method associated with the α-Gaussian distribution. The PGF technique works on the forward modeling process by dividing the computational domain into outside target area and target area, where the wave equation is solved only once on the outside target (before FWI). During the FWI processing, Green's functions related only to the target area are computed instead of the entire computational domain, saving computational efforts. We show the effectiveness of our proposed approach by considering two distinct realistic P-wave velocity models, in which the first one is inspired in the Kwanza Basin in Angola and the second in a region of great economic interest in the Brazilian pre-salt field. We call our proposal by the abbreviation α-PGF-FWI. The results reveal that the α-PGF-FWI is robust against additive Gaussian noise and non-Gaussian noise with outliers in the limit α → 2/3, being α the Rényi entropic index.

κ-statistics approach to optimal transport waveform inversion.

Extracting physical parameters that cannot be directly measured from an observed data set remains a great challenge in several fields of science and physics. In many of these problems, the construction of a physical model from waveforms is hampered by the phase ambiguity of the recorded wave fronts. In this work, we present an approach for mitigating the effect of phase ambiguity in waveform-driven issues. Our proposal combines the optimal transport theory with the κ-statistical thermodynamics approach. We construct an energy function from the most probable state of a system described by a finite-variance κ-Gaussian distribution to introduce an optimal transport metric. We demonstrate that our proposal outperforms the classical frameworks by considering a nonlinear geophysical data-driven problem based on a wave equation numerical solution. The κ-generalized optimal transport metric is easily adapted to various inverse problems, from estimating power-law exponents to machine learning approaches in quantum mechanics.

Microseismic event detection in noisy environments with instantaneous spectral Shannon entropy.

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.

Robust parameter estimation based on the generalized log-likelihood in the context of Sharma-Taneja-Mittal measure.

The problem of obtaining physical parameters that cannot be directly measured from observed data arises in several scientific fields. In the classic approach, the well-known maximum likelihood estimation associated with a Gaussian distribution is employed to obtain the model parameters of a complex system. Although this approach is quite popular in statistical physics, only a handful of spurious observations (outliers) make this approach ineffective, violating the Gauss-Markov theorem. In this work, starting from the generalized logarithmic function associated to the Sharma-Taneja-Mittal (STM) information measure, we propose an outlier-resistant approach based on the generalized log-likelihood estimation. In particular, our proposal deforms the Gaussian distribution based on a two-parameter generalization of the ordinary logarithmic function. We have tested the effectiveness of our proposal considering a classic geophysical inverse problem with a very noisy data set. The results show that the task of obtaining physical parameters based on the STM measure from noisy data with several outliers outperforms the classic approach, and therefore, our proposal is a useful tool for statistical physics, information theory, and statistical inference problems.

Improving Seismic Inversion Robustness via Deformed Jackson Gaussian.

The seismic data inversion from observations contaminated by spurious measures (outliers) remains a significant challenge for the industrial and scientific communities. This difficulty is due to slow processing work to mitigate the influence of the outliers. In this work, we introduce a robust formulation to mitigate the influence of spurious measurements in the seismic inversion process. In this regard, we put forth an outlier-resistant seismic inversion methodology for model estimation based on the deformed Jackson Gaussian distribution. To demonstrate the effectiveness of our proposal, we investigated a classic geophysical data-inverse problem in three different scenarios: (i) in the first one, we analyzed the sensitivity of the seismic inversion to incorrect seismic sources; (ii) in the second one, we considered a dataset polluted by Gaussian errors with different noise intensities; and (iii) in the last one we considered a dataset contaminated by many outliers. The results reveal that the deformed Jackson Gaussian outperforms the classical approach, which is based on the standard Gaussian distribution.

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.

An objective function for full-waveform inversion based on frequency-dependent offset-preconditioning.

Full-waveform inversion (FWI) is a powerful technique to obtain high-resolution subsurface models, from seismic data. However, FWI is an ill-posed problem, which means that the solution is not unique, and therefore the expert use of the information is required to mitigate the FWI ill-posedness, especially when wide-aperture seismic acquisitions are considered. In this way, we investigate the multiscale frequency-domain FWI by using a weighting operator according to the distances between each source-receiver pair. In this work, we propose a weighting operator that acts on the data misfit as preconditioning of the objective function that depends on the source-receiver distance (offset) and the frequency used during the inversion. The proposed operator emphasizes information from long offsets, especially at low frequencies, and as a consequence improves the update of deep geological structures. To demonstrate the effectiveness of our proposal, we perform numerical simulations on 2D acoustic Marmousi2 case study, which is widely used in seismic imaging tests, considering three different scenarios. In the first two ones, we have used an acquisition geometry with a maximum offset of 4 and 8 km, respectively. In the last one, we have considered all-offsets. The results show that our proposal outperforms similar strategies, for all scenarios, providing more reliable quantitative subsurface models. In fact, our inversion result has the lowest error and the highest similarity to the true model than similar approaches.

Full-waveform inversion based on Kaniadakis statistics.

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