{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T23:52:01Z","timestamp":1772236321802,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,19]],"date-time":"2020-04-19T00:00:00Z","timestamp":1587254400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","award":["001"],"award-info":[{"award-number":["001"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["304421\/2015-4"],"award-info":[{"award-number":["304421\/2015-4"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["313431\/2018-3"],"award-info":[{"award-number":["313431\/2018-3"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2019s reflectivity. Our results show that the PSI based on Tsallis statistics outperforms the conventional PSI, especially in the non-Gaussian noisy-data case.<\/jats:p>","DOI":"10.3390\/e22040464","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T03:23:06Z","timestamp":1587439386000},"page":"464","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Tsallis Entropy, Likelihood, and the Robust Seismic Inversion"],"prefix":"10.3390","volume":"22","author":[{"given":"Igo Pedro","family":"de Lima","sequence":"first","affiliation":[{"name":"Programa de P\u00f3s-Gradua\u00e7\u00e3o em Ci\u00eancia e Engenharia de Petr\u00f3leo - Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6384-5049","authenticated-orcid":false,"given":"S\u00e9rgio Luiz E. F.","family":"da Silva","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Te\u00f3rica e Experimental, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1748-4040","authenticated-orcid":false,"given":"Gilberto","family":"Corso","sequence":"additional","affiliation":[{"name":"Programa de P\u00f3s-Gradua\u00e7\u00e3o em Ci\u00eancia e Engenharia de Petr\u00f3leo - Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil"},{"name":"Departamento de Biof\u00edsica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8462-4280","authenticated-orcid":false,"given":"Jo\u00e3o M.","family":"de Ara\u00fajo","sequence":"additional","affiliation":[{"name":"Programa de P\u00f3s-Gradua\u00e7\u00e3o em Ci\u00eancia e Engenharia de Petr\u00f3leo - Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil"},{"name":"Departamento de F\u00edsica Te\u00f3rica e Experimental, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. 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