Fig 1.
Workflow of the computational experiments.
Fig 2.
η-Gaussian probability distribution.
The black dashed line represents the conventional curves.
Fig 3.
(a)-(c) objective functions and (d)-(f) influence functions generalized based on Rényi statistic.
The black dashed line represents the conventional curves.
Fig 4.
q-Gaussian probability distribution.
The black dashed line represents the conventional curves.
Fig 5.
(a)-(c) objective functions and (d)-(f) generalized influence functions based on Tsallis statistic.
The black dashed line represents the conventional curves.
Fig 6.
κ-Gaussian probability distribution.
The black dashed line represents the conventional curves.
Fig 7.
(a)-(b) objective functions and (c)-(d) generalized influence functions based on Kaniadakis statistic.
The black dashed line represents the conventional curves.
Fig 8.
Fit of the estimated lines using the objective functions of (a) Rényi, (b) Tsallis and (c) Kaniadakis.
The dashed line indicates an ideal line, and the points highlighted in red indicate the inserted outliers.
Fig 9.
Relation between estimated parameters and entropic indexes.
The zoom in window in (a) and (b) emphasizes the regions 0.5 ≤ η < 1/3, 2.5 ≤ q < 3 and 0.5 ≤ κ < 2/3.
Fig 10.
The geophysical model employed to illustrate the inversion methodology.
In (a) the synthetic acoustic impedance model called Marmousi2. In (b) we show the initial model employed in the inversion methodology.
Fig 11.
The noiseless seismic data in (a). The same data contaminated with white-noise (signal-to-noise ratio SNR = 80) and spike noise with (b) 0.5%, (c) 5%, and (d) 80%. The black line in panels (e) represents a single seismic trace from the middle of panel (a). The same trace contaminated by noise is represented in (f) spikes (25%).
Fig 12.
Acoustic impedance model recovered for an observed data contaminated with white noise (SNR = 80) and spike noise (0.5%) using objective function (a) conventional (b) Rényi with η = 0.3334; (c) Tsallis with q = 2.9999; and (d) Kaniadakis with κ = 0.6666.
Fig 13.
Acoustic impedance model recovered for an observed data contaminated with white noise (SNR = 80) and spike noise (80%) using objective function (a) conventional (b) Rényi with η = 0.3334; (c) Tsallis with q = 2.9999; and (d) Kaniadakis with κ = 0.6666.
Fig 14.
Heat-map representing the correlation between synthetic and recovered impedance models for the objective functions: (a) Rényi, (b) Tsallis and (c) Kaniadakis.
The white markings indicate points such that R = 0.9 (strong correlation).
Table 1.
Summary of smallest Mean Absolute Error (MAE) and symmetric Mean Absolute Percentage Error (sMAPE) values for objective functions based on Rényi, Tsallis and Kaniadakis frameworks.
%Sp indicates the percentage of spikes, while ζ represents a normalization constant for the MAE to present this error metric best. The maximum MAE value is ζ = 4, 020.91, which happened in the numerical test with %Sp = 80 using the conventional objective function.