# Time-varying wavelet estimation and its applications in deconvolution and seismic inversion

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## Abstract

Wavelet holds an essential role in seismic data processing and characterization, for examples deconvolution and seismic inversion. Unfortunately, wavelet is an unknown data. Several existing methods attempt to estimate and extract the wavelet from seismic data. However, the methods give only a single wavelet from one seismic trace. When seismic data are non-stationer, single wavelet usage will cause a problem, that is raising the error. This paper proposes a time-varying wavelet estimation method to accommodate this problem. It uses matrix diagonalization to estimate a set of wavelets. Next, the time-varying wavelet is applied to deconvolution and seismic inversion. The experiment shows that time-varying wavelet improves the results in both deconvolution and seismic inversion. The errors decreased and spectrum bandwidth broadened.

## Keywords

Time-varying wavelet Time-varying deconvolution Time-varying seismic inversion## Introduction

Wavelet estimation holds an important process in seismic processing and inversion. Several methods were proposed to obtain the best-estimated wavelet (Ricker 1953; Walden and White 1998; Cui and Margrave 2014). In the prior publications, seismic frequency analysis is required to extract a wavelet because seismic spectrum represents the wavelet spectrum. The simple method of wavelet estimation is by using the mathematical equation, for example, Ricker’s wavelet (Ricker 1953) and bandpass or Ormsby wavelet (Ryan 1994) equations. However, these methods achieve a good result when the shape of seismic spectrum is similar to the mathematical wavelet spectrum. Besides the mathematical equation, there are wide-known methods: statistical wavelet estimation and spectral smoothing (Cui and Margrave 2014). These two methods are more flexible to any seismic conditions.

All listed wavelet estimation methods above are aimed for a stationary signal. In reality, our seismic data are non-stationary. When the seismic data are non-stationary, the explained methods will produce the most optimum single wavelet. This single wavelet may have no negative effect on the next processing stage if the non-stationarity of the signal is weak. However, if the non-stationarity is strong enough, it may cause several problems, for example, the raise of higher error in the next processing stage. Prior researches in time-varying wavelets were published to solve this problem, especially in deconvolution, either explicitly (van der Baan 2008) or implicitly (Clarke 1968; Margrave et al. 2011). In this paper, it will be introduced a proposed method of wavelet estimation in the time domain. This method estimates a set of wavelets which varies over time. Therefore, it can compensate for the non-stationarity of the signal. Moreover, it will be explained applications of this proposed method not only in deconvolution but also in seismic inversion.

## Theory and methods

### Square root of a matrix

Then, \({\mathbf{B}}\) is called the square root of \({\mathbf{A}}\). To get \({\mathbf{B}}\) is not as directly as by taking a square root of the matrix elements of \({\mathbf{A}}\). There is a method which uses matrix diagonalization to extract \({\mathbf{B}}\) when \({\mathbf{A}}\) is the only known information (Levinger 1980).

### Time-varying wavelet estimation

#### Time-varying wavelet estimation

#### Time-varying wavelet’s applications

*Time*-

*varying deconvolution*For time-varying deconvolution, a time-varying wavelet is aimed to build deconvolution operators, so we do not need to have wavelet’s phase information. The deconvolution operators are intended to transform the wavelets to be spike series (represented by identity matrix). It is defined as

*Model*-

*based inversion*Model-based inversion is one of seismic inversion methods. This inversion results P-Impedance as the final result. Model-based inversion is started from a low-frequency model. Then, the model is perturbed each iteration to get the final inversion result. The process of model-based inversion is based on a mathematical model in the following equation (Hampson-Russel Software 1999; Hampson et al. 2005),

## Results and discussion

Pre-whitening effect to RMS error and coefficient correlation

Pre-whitening, \(\lambda\) | RMS error | Coeff. correlation |
---|---|---|

0.1 | 0.0616 | 0.7245 |

0.2 | 0.0643 | 0.7343 |

0.3 | 0.0665 | 0.7279 |

## Conclusion

Square of a matrix by using matrix diagonalization could be used to estimate time-varying wavelets from a signal. This concept is based on wavelet estimation by using autocorrelation. Time-varying wavelet can be applied in geophysical data processing for examples deconvolution and inversion. The time-varying wavelet is used to build a set of deconvolution operators to perform deconvolution. Then, the operators are applied to the input signal. As a result of deconvolution, this method enhances the non-stationer signal properly because the deconvolution operators change over time. Besides deconvolution, time-varying wavelets improve seismic inversion. A set of time-varying wavelet is used as a replacement of single wavelet in the inversion process. In the experiment, inversion error decreases by approximately 17%.

## Notes

### Acknowledgements

The author would like to thank Adhitya Ryan Ramadhani, M.Sc. (Mechanical Engineering) and Dr. Ida Herawati (Geophysical Engineering) for proofreading this article.

### Funding

This research did not receive any grant from funding agencies in the public, commercial, or not-for-profit sectors.

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## Copyright information

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