Modern visualization can be formulated as inversion problems that aim to obtain structural information about a complex medium through wave excitations. However, without numerically efficient forward calculations, even state-of-the-art inversion procedures are too computationally intensive to implement. We adapt a method previously used to treat transport in electronic waveguides to describe acoustic wave motion in complex media with high gains in computational time. The method consists of describing the system as if it was made of disconnected parts that are patched together. By expressing the system in this manner, wave-propagation calculations that otherwise would involve a very large matrix can be done with considerably smaller matrices instead. In particular, by treating one of such patches as a target whose parameters are changeable, we are able to implement target-oriented optimization in which the model parameters can be continuously refined until the ideal result is reproduced. The so-called Patched Green's function (PGF) approach is mathematically exact and involves no approximations, thus improving the computational cost without compromising accuracy. Given the generality of our method, it can be applied to a wide variety of inversion problems. Here we apply it to the case of seismic modeling where acoustic waves are used to map the earth subsurface in order to identify and explore mineral resources. The technique is tested with realistic seismic models and compared to standard calculation methods. The reduction in computational complexity is remarkable and paves the way to treating larger systems with increasing accuracy levels.

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.

The estimation of physical parameters from data analysis is a crucial point 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). We show the effectiveness of our proposal 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.