de Lima Igo Pedro, da Silva Sérgio Luiz E F, Corso Gilberto, de Araújo João M
Programa de Pós-Graduação em Ciência e Engenharia de Petróleo - Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil.
Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil.
Entropy (Basel). 2020 Apr 19;22(4):464. doi: 10.3390/e22040464.
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 -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.
Tsallis提出的非广延统计力学已成功用于对许多复杂现象进行建模和分析。在此，我们研究广义Tsallis统计在反问题理论中的作用。大多数反问题被表述为一个优化问题，旨在从间接和部分观测中估计系统的物理参数。在传统方法中，要最小化的失配函数基于观测数据与建模数据（残差或误差）之间的最小二乘距离，其中假设残差服从高斯分布。然而，在许多实际情况中，误差通常是非高斯的，因此这种技术往往会失效。这个问题促使我们研究基于非高斯统计的失配函数。在这项工作中，我们基于与Tsallis形式体系中的最大熵原理相关的-高斯分布推导了一个失配函数。我们在一个典型的地球物理数据反问题，即叠后反演（PSI）中测试了我们的方法，其中要估计的物理参数是地球的反射率。我们的结果表明，基于Tsallis统计的PSI优于传统的PSI，特别是在非高斯噪声数据的情况下。
