# Improving Seismic Inversion Robustness via Deformed Jackson Gaussian

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## Abstract

**:**

## 1. Introduction

## 2. Post-Stack Seismic Inversion

## 3. Likelihood Based on Jackson Statistics

## 4. Numerical Results

#### 4.1. Sensitivity to the Source Signature

#### 4.2. Sensitivity to Gaussian Noise

`NRMS`):

`NRMS`value close to zero means little error. The second measure used was the well-known Pearson’s coefficient (

`R`) [45], which is a similarity measure. It is worth remembering thata Pearson’s coefficient score varies between 0 and 1, with being 0 a perfect “uncorrelation” and 1 being a perfect correlation between two samples. The measures for the four Gaussian noise scenarios are summarized in Table 1.

#### 4.3. Sensitivity to Erratic Data (Outliers)

## 5. Final Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PSI | post-stack inversion |

CG | conjugate gradient |

NRMS | normalized root mean square |

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**Figure 1.**Probability plots of the $\overline{q}$-Gaussian distribution Equation (10) for typical $\overline{q}$-values, in which the black curve represents the standard Gaussian distribution ($\overline{q}\to 1.0$).

**Figure 2.**(

**a**) The 2D Marmousi acoustic impedance model considered as the true model. (

**b**) The initial model employed in the seismic inversion.

**Figure 3.**Source signatures used in this work. The solid black line represents the true seismic source (25 Hz Ricker wavelet). The dashed blue line, dotted red line, and solid green line represent the three incorrect source wavelets, respectively.

**Figure 4.**Models reconstructed for the correct source case using the PSI based on the (

**a**) conventional approach ($\overline{q}\to 1$) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.6$, (

**d**) $\overline{q}=0.4$, (

**e**) $\overline{q}=0.3$, (

**f**) $\overline{q}=0.1$, (

**g**) $\overline{q}=0.01$, or (

**h**) $\overline{q}=0.001$.

**Figure 5.**Models reconstructed for the incorrect source I case by using the PSI based on the (

**a**) conventional approach ($\overline{q}\to 1$) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.6$, (

**d**) $\overline{q}=0.4$, (

**e**) $\overline{q}=0.3$, (

**f**) $\overline{q}=0.1$, (

**g**) $\overline{q}=0.01$, or (

**h**) $\overline{q}=0.001$.

**Figure 6.**Models reconstructed for the incorrect source II case by using the PSI based on the (

**a**) conventional approach ($\overline{q}\to 1$) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.6$, (

**d**) $\overline{q}=0.4$, (

**e**) $\overline{q}=0.3$, (

**f**) $\overline{q}=0.1$, (

**g**) $\overline{q}=0.01$, or (

**h**) $\overline{q}=0.001$.

**Figure 7.**Models reconstructed for the incorrect source III case by using the PSI based on the (

**a**) conventional approach ($\overline{q}\to 1$) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.6$, (

**d**) $\overline{q}=0.4$, (

**e**) $\overline{q}=0.3$, (

**f**) $\overline{q}=0.1$, (

**g**) $\overline{q}=0.01$, or (

**h**) $\overline{q}=0.001$.

**Figure 8.**Vertical profiles of the true model (black curve), initial model (gray curve), and reconstructed models for the correct source in panel (

**a**) and incorrect sources I, II, and III in panels (

**b**–

**d**), respectively. In this figure, we depict the central vertical well at $Distance$ = 1.0 km.

**Figure 9.**Model reconstructed for the correct source case by using the PSI based on our proposal with: (

**a**) $\overline{q}={10}^{-3}$, (

**b**) $\overline{q}={10}^{-4}$, (

**c**) $\overline{q}={10}^{-5}$, and (

**d**) $\overline{q}={10}^{-6}$.

**Figure 10.**The black curves represent the observed data (waveforms) of the 51st seismic source ($Distance=1km$) contaminated by Gaussian noise with (

**a**) $SNR$ = 5 dB, (

**b**) $SNR$ = 10 dB, (

**c**) $SNR$ = 20 dB, and (

**d**) $SNR$ = 30 dB. The red curves represent the noiseless waveforms.

**Figure 11.**Models reconstructed for the Gaussian noise case, with $SNR=5\mathrm{dB}$, by using a PSI based on the (

**a**) conventional approach ($\overline{q}\to 1$) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.6$, (e) $\overline{q}=0.4$, (

**f**) $\overline{q}=0.3$, (

**g**) $\overline{q}=0.1$, or (

**h**) $\overline{q}=0.01$.

**Figure 12.**Model reconstructed for the Gaussian noise case, with $SNR=10\mathrm{dB}$, by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.6$, (

**e**) $\overline{q}=0.4$, (

**f**) $\overline{q}=0.3$, (

**g**) $\overline{q}=0.1$, or (

**h**) $\overline{q}=0.01$.

**Figure 13.**Model reconstructed for the Gaussian noise case, with $SNR=20\mathrm{dB}$, by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.6$, (

**e**) $\overline{q}=0.4$, (

**f**) $\overline{q}=0.3$, (

**g**) $\overline{q}=0.1$, or (

**h**) $\overline{q}=0.01$.

**Figure 14.**Model reconstructed for the Gaussian noise case, with $SNR=30\mathrm{dB}$, by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.9$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.6$, (

**e**) $\overline{q}=0.4$, (

**f**) $\overline{q}=0.3$, (

**g**) $\overline{q}=0.1$, or (

**h**) $\overline{q}=0.01$.

**Figure 15.**Vertical profiles of the true model (

**black curve**), initial model (

**gray curve**), and reconstructed models for the Gaussian noise case with (

**a**) $SNR$ = 5 dB, (

**b**) $SNR$ = 10 dB, (

**c**) $SNR$ = 20 dB, or (

**d**) $SNR$ = 30 dB. In this figure, we depict the central vertical well at $Distance$ = 1.0 km.

**Figure 16.**Vertical profiles of the true model (

**black curve**), initial model (

**gray curve**), and reconstructed models for the Gaussian noise case with (

**a**) $SNR$ = 5 dB, (

**b**) $SNR$ = 10 dB, (

**c**) $SNR$ = 20 dB, or (

**d**) $SNR$ = 30 dB. In this figure, we depict the central vertical well at $Distance$ = 180 m.

**Figure 17.**The black curves represent the observed data (

**waveforms**) of the 51st seismic source ($Distance$ = 1 km) contaminated by Gaussian noise and spiky-noise with (

**a**) $\%Spike=0.5$, (

**b**) $\%Spike=10$, (

**c**) $\%Spike=20$, or (

**d**) $\%Spike=50$. The red curves represent the noiseless waveforms.

**Figure 18.**The black curves represent the observed data (

**waveforms**) of the 10th seismic source ($Distance$ = 180 m) contaminated by Gaussian noise and spiky-noise with (

**a**) $\%Spike=0.5$, (

**b**) $\%Spike=10$, (

**c**) $\%Spike=20$, or (

**d**) $\%Spike=50$. The red curves represent the noiseless waveforms.

**Figure 19.**Model reconstructed for the $\%Spikes=0.5$ case by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.90$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.60$, (

**e**) $\overline{q}=0.40$, (

**f**) $\overline{q}=0.30$, (

**g**) $\overline{q}=0.10$, or (

**h**) $\overline{q}=0.01$.

**Figure 20.**Model reconstructed for the $\%Spikes=10$ case by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.90$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.60$, (

**e**) $\overline{q}=0.40$, (

**f**) $\overline{q}=0.30$, (

**g**) $\overline{q}=0.10$, or (

**h**) $\overline{q}=0.01$.

**Figure 21.**Model reconstructed for the $\%Spikes=20$ case by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.90$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.60$, (

**e**) $\overline{q}=0.40$, (

**f**) $\overline{q}=0.30$, (

**g**) $\overline{q}=0.10$, or (

**h**) $\overline{q}=0.01$.

**Figure 22.**Model reconstructed for the $\%Spikes=50$ case by using the PSI based on the (

**a**) conventional approach (

**$\overline{q}\to 1$**) or our proposal with: (

**b**) $\overline{q}=0.90$, (

**c**) $\overline{q}=0.75$, (

**d**) $\overline{q}=0.60$, (

**e**) $\overline{q}=0.40$, (

**f**) $\overline{q}=0.30$, (

**g**) $\overline{q}=0.10$, or (

**h**) $\overline{q}=0.01$.

**Figure 23.**Vertical profiles of the true model (

**black curve**), initial model (

**gray curve**), and reconstructed models for the spiky-noise cases with (

**a**) $\%Spikes=0.5$, (

**b**) $\%Spikes=10$, (

**c**) $\%Spikes=20$, and (

**d**) $\%Spikes=50$. In this figure, we depict the central vertical well at $Distance$ = 1.0 km.

**Figure 24.**Vertical profiles of the true model (

**black curve**), initial model (

**gray curve**), and reconstructed models for the spiky-noise cases with (

**a**) $\%Spikes=0.5$, (

**b**) $\%Spikes=10$, (

**c**) $\%Spikes=20$, and (

**d**) $\%Spikes=50$. In this figure, we depict the central vertical well at $Distance$ = 180 m.

**Figure 25.**A heatmap as a graphical representation of Pearson’s R for the numerical experiments carried out in this work, in which the white curve indicates the

`R`= 0.8 case.

**Table 1.**Main statistics of the PSI results from datasets contaminated by Gaussian noise with different SNRs.

Strategy | SNR = 5 dB | SNR = 10 dB | SNR = 20 dB | SNR = 30 dB | ||||
---|---|---|---|---|---|---|---|---|

NRMS | R | NRMS | R | NRMS | R | NRMS | R | |

Conventional PSI ($\overline{q}\to 1.0$) | 0.6144 | 0.4342 | 0.3678 | 0.7587 | 0.1889 | 0.9290 | 0.1019 | 0.9789 |

Our proposal ($\overline{q}=0.90$) | 0.4784 | 0.6287 | 0.2779 | 0.8528 | 0.1589 | 0.9486 | 0.1004 | 0.9791 |

Our proposal ($\overline{q}=0.75$) | 0.4662 | 0.6435 | 0.2788 | 0.8520 | 0.1588 | 0.9487 | 0.1003 | 0.9791 |

Our proposal ($\overline{q}=0.60$) | 0.4068 | 0.7077 | 0.2787 | 0.8520 | 0.1586 | 0.9488 | 0.1003 | 0.9792 |

Our proposal ($\overline{q}=0.40$) | 0.3594 | 0.7542 | 0.2770 | 0.8536 | 0.1588 | 0.9487 | 0.1004 | 0.9791 |

Our proposal ($\overline{q}=0.30$) | 0.3162 | 0.8013 | 0.2746 | 0.8549 | 0.1591 | 0.9485 | 0.1004 | 0.9791 |

Our proposal ($\overline{q}=0.10$) | 0.3176 | 0.7957 | 0.2666 | 0.8623 | 0.1589 | 0.9486 | 0.1003 | 0.9791 |

Our proposal ($\overline{q}=0.01$) | 0.2947 | 0.8234 | 0.2566 | 0.8698 | 0.1590 | 0.9486 | 0.1004 | 0.9791 |

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## Share and Cite

**MDPI and ACS Style**

Silva, S.A.; da Silva, S.L.E.F.; de Souza, R.F.; Marinho, A.A.; de Araújo, J.M.; Bezerra, C.G.
Improving Seismic Inversion Robustness via Deformed Jackson Gaussian. *Entropy* **2021**, *23*, 1081.
https://doi.org/10.3390/e23081081

**AMA Style**

Silva SA, da Silva SLEF, de Souza RF, Marinho AA, de Araújo JM, Bezerra CG.
Improving Seismic Inversion Robustness via Deformed Jackson Gaussian. *Entropy*. 2021; 23(8):1081.
https://doi.org/10.3390/e23081081

**Chicago/Turabian Style**

Silva, Suzane A., Sérgio Luiz E. F. da Silva, Renato F. de Souza, Andre A. Marinho, João M. de Araújo, and Claudionor G. Bezerra.
2021. "Improving Seismic Inversion Robustness via Deformed Jackson Gaussian" *Entropy* 23, no. 8: 1081.
https://doi.org/10.3390/e23081081