# High-Resolution Seismic Data Deconvolution by A0 Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

**Assertion 1.**

## 2. What Is Special about the Algorithm $\mathit{A}\mathit{0}$

**E–step:**

**M–step:**

## 3. Example of A0 Applied to Seismic Tasks

- model 1—using the Ricker wavelet with a dominant frequency of 30 Hz,
- model 2—using a wavelet estimated from the seismic data spectrum,
- model 3—using a wavelet estimated from the spectrum of seismic data, with the addition of phase rotation of 90 degrees.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SP&ARCM | Searching for Positions and Amplitudes of the Reflection Coefficients of the Medium |

OMP | Orthogonal Matching Pursuit |

DE | Differential Evolution |

BP | Basic Pursuit |

MP | Matching Pursuit |

RMSE | Root Mean Square Error |

## References

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**Figure 2.**Dependence of the proportion of linearly independent rows on the size of the matrix ${W}_{K}^{\left(N\right)}$.

**Figure 3.**Dependencies of inverse convergence for a seismic trace with $N=200$ for a different number of reflecting coefficients K.

**Figure 7.**Dependence of the error (distance) between vectors and for the case with one reflecting plane.

**Figure 8.**The result of the application of the algorithm A0 on the acoustic model of the well: DT is the curve of the acoustic logging; RHOB—density curve; Rc initial—reflection coefficient curve; Rc recover—restored curve of reflection coefficients; error—relative error values (color settings: blue—error less than 10%, green—error 10–20%, red—error more than 20%); Synthetic is a model wave field.

**Figure 9.**Wavelets used for the formation of acoustic models in the well: blue—Ricker wavelet with a dominant frequency of 30 Hz (model 1); green—statistical pulse from seismic data (model 2); red—statistical pulse with a phase rotation of 90 degrees (model 3).

**Figure 10.**The result of applying the A0 algorithm on the wave field along a well with different fractions of the noises: Rc init is the reflection coefficient curve; Rc recover—the restored curve of the reflection coefficients without noise; Rc recover 5%—the restored curve of reflection coefficients at a noise content of 5%; Rc recover 10%—restored curve of reflection coefficients at a noise content of 10%; error Seis, error per 5, error per 10—the corresponding relative mean square error curves (color settings: blue—error less than 10%, green—error 10–20%, red—error more than 20%).

**Figure 11.**(

**top**) picture is the acoustic impedance model for three layers; (

**below**) cuts along the lines I-I’ and II-II’.

**Figure 13.**(

**left**), cuts of the reflection coefficients along the lines I-I’ and II-II’, obtained as a result of applying the algorithm; (

**right**) the model sections of the reflection coefficients.

**Figure 14.**(

**left**): model section of the reflection coefficients, (

**center**): model wave field; (

**right**): the result of the algorithm $\mathit{A}\mathit{0}$.

**Figure 15.**(

**left**): the seismic section with reflective coefficients in well, (

**right**): the result of the algorithm $\mathit{A}\mathit{0}$ work.

Model | Correlation Coefficient | Relative Standard Deviation, % |
---|---|---|

1 | 0.93 | 4.33 |

2 | 0.94 | 4.24 |

3 | 0.03 | 20.43 |

The Noise Fraction,% | Correlation Coefficient | Relative Standard Deviation, % |
---|---|---|

0 | 0.94 | 4.24 |

5 | 0.85 | 17.16 |

10 | 0.70 | 24.93 |

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## Share and Cite

**MDPI and ACS Style**

Krasnov, F.; Butorin, A.
High-Resolution Seismic Data Deconvolution by *A0* Algorithm. *Geosciences* **2018**, *8*, 497.
https://doi.org/10.3390/geosciences8120497

**AMA Style**

Krasnov F, Butorin A.
High-Resolution Seismic Data Deconvolution by *A0* Algorithm. *Geosciences*. 2018; 8(12):497.
https://doi.org/10.3390/geosciences8120497

**Chicago/Turabian Style**

Krasnov, Fedor, and Alexander Butorin.
2018. "High-Resolution Seismic Data Deconvolution by *A0* Algorithm" *Geosciences* 8, no. 12: 497.
https://doi.org/10.3390/geosciences8120497