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Extensive and nonextensive statistics in seismic inversion

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  • da Silva, Sérgio Luiz Eduardo Ferreira
  • dos Santos Lima, Gustavo Zampier
  • de Araújo, João Medeiros
  • Corso, Gilberto

Abstract

Seismic inversion is a central procedure for estimating subsurface physical parameters from observed data. In geophysical applications, the seismic inversion is usually formulated as an optimisation problem that aims to minimise the difference between modelled and observed data through the Gauss’ error law, which is linked to the extensive Boltzmann–Gibbs (BG) statistics. However, this approach is known to be sensitive to non-Gaussian errors, especially to outliers in the data set. Therefore, error laws determined by non-Gaussian statistics are essential for robust seismic inversion. In this way, we present a comparative study of seismic inversions using the extensive statistics of Rényi and also on the nonextensive statistics of Tsallis and Kaniadakis. In particular, we consider a classical seismic inversion problem so-called Post-Stack Inversion (PSI), which analyses the interaction between seismic waves and subsurface reflectivity to infer geological structures. Considering a realistic subsurface reflectivity model taking into account spike-noisy data, the numerical results show that the PSI based on generalised statistics outperforms the PSI based on standard BG statistics. We note that, the best results are for the Tsallis case in which the Lévy–Gnedenko central-limit theorem is valid (q>53).

Suggested Citation

  • da Silva, Sérgio Luiz Eduardo Ferreira & dos Santos Lima, Gustavo Zampier & de Araújo, João Medeiros & Corso, Gilberto, 2021. "Extensive and nonextensive statistics in seismic inversion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    • Handle: RePEc:eee:phsmap:v:563:y:2021:i:c:s0378437120307949
    DOI: 10.1016/j.physa.2020.125496
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    References listed on IDEAS

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    1. Enrico Moretto & Sara Pasquali & Barbara Trivellato, 2017. "A non-Gaussian option pricing model based on Kaniadakis exponential deformation," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(10), pages 1-10, October.
    2. Kaniadakis, G., 2001. "Non-linear kinetics underlying generalized statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(3), pages 405-425.
    3. da Silva, Sérgio Luiz Eduardo Ferreira & da Costa, Carlos A.N. & Carvalho, Pedro Tiago C. & de Araújo, João Medeiros & dos Santos Lucena, Liacir & Corso, Gilberto, 2020. "Robust full-waveform inversion using q-statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 548(C).
    4. Tsallis, Constantino & Mendes, RenioS. & Plastino, A.R., 1998. "The role of constraints within generalized nonextensive statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 534-554.
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    Cited by:

    1. da Silva, Sérgio Luiz E.F. & Silva, R. & dos Santos Lima, Gustavo Z. & de Araújo, João M. & Corso, Gilberto, 2022. "An outlier-resistant κ-generalized approach for robust physical parameter estimation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    2. Adson Alexandre Quirino da Silveira & Renato Ferreira Souza & Jonathas da Silva Maciel & Jessica Lia Santos da Costa & Daniel Teixeira dos Santos & João Medeiros Araujo & Sérgio Luiz E. F. da Silva & , 2023. "Puzzle in inverse problems: Tsallis noise and Tsallis norm," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(3), pages 1-7, March.
    3. da Silva, Sérgio Luiz Eduardo Ferreira, 2021. "Newton’s cooling law in generalised statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    4. da Silva, Sérgio Luiz E.F., 2021. "κ-generalised Gutenberg–Richter law and the self-similarity of earthquakes," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

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