# Stratigraphic absorption compensation based on multiscale shearlet transform

## Abstract

Seismic waves propagating through viscoelastic media experience stratigraphic absorption and attenuation effects, which directly affect the imaging resolution in seismic exploration. Without stratigraphic absorption, the ratio of deep reflection energy to shallow reflection energy (attenuation ratio) is invariable at different frequencies. If a seismogram is decomposed into different frequency bands, these signals will show similar time–energy distributions. Therefore, the attenuation ratios should be similar across different frequency bands, except for frequency-variable weights. Nevertheless, the frequency-variable weights for different frequency bands can be obtained by benchmarking against the time–energy distributions of low-frequency information because the loss of low-frequency information is relatively insignificant. In this light, we obtained frequency-variable weights for different frequencies and established a stratal absorption compensation (SAC) model. The anisotropic basis of the shearlet enables nearly optimal representation of curved-shape seismic signals, and shearlets at different scales can represent signals for different frequency bands. Then, we combined the SAC model with the shearlet transform and established the new compensation method. As the signal and noise have different distributions in the shearlet domain, we selectively compensated the signals using a thresholding algorithm. Hence, it was possible to avoid noise enhancement. This is the prominent advantage of the proposed method over other compensation methods.

## Keywords

Shearlet transform Inverse*Q*filtering Stratal absorption compensation Time–frequency analysis

## Introduction

Seismic waves propagating outward from the shot point through viscoelastic media experience stratigraphic absorption and spherical divergence effects. Thus, the amplitude will attenuate with the increase in propagation distance and frequency, which can obscure fine structures located at considerable depths. Therefore, it is necessary to compensate for strata absorption to improve the resolution of signals from deep strata. Conventional stratal absorption compensation (SAC) methods fall under the following two main categories: quality factor-based (*Q* value) compensation methods and methods based on time–frequency analysis.

Methods based on the *Q* value require calculations of the quality factor in the time or frequency domains. Early compensation theories assumed wavelets to be time-invariant and considered travel time as the only factor related to amplitude attenuation. Thus, we just need to perform the spherical diffusion compensation and surface-consistent amplitude compensation with this approach. Based on the above theory, Zhou et al. (1994) made use of the main frequencies computed from reflection signals which were reflected by the two interfaces of a layer and slope of the spectrum, and were able to increase the stability of the spectral ratio method. Ling et al. (1997) put forward the one-dimensional (1D) elastic damping wave theory to improve the accuracy of sand-dune *Q* estimations in cases of low signal-to-noise ratio (SNR) data. Quan and Harris (1997) proposed an SAC method based on the central frequency shift. Scott et al. (2006) provided a *Q* estimation approach that uses the joint Gabor–Morlet time–frequency analysis method. Wang (2002, 2006) proposed a stabilized inverse *Q* filtering approach based on the wavefield extrapolation method and corrected the overcompensation for high-frequency noise. Ferber (2005) proposed a filter banks solution for absorption compensation, the key benefit of which is its applicability to cases of arbitrarily time-variant *Q*.

All the aforementioned compensation methods are based on accurate *Q* models, while the precision of *Q* models depends on accurate wavelet estimation. In noiseless or synthetic data, we can obtain accurate *Q* values as the wavelets are time invariant. However, *Q* values derived from field seismic data may not reflect actual stratal absorption, because field data have relatively low SNRs and wavelets are time variant as a result of the absorption by viscoelastic strata. Consequently, the application of inverse *Q* filtering is limited.

Compensation methods based on time–frequency analysis are more practical and applicable than methods based on the *Q* value, although they still require high SNR data. Zhao et al. (1994) proposed a new filter by differentiating the seismic trace. Their differential filter features an amplitude response proportional to the circular frequency, and it can be used to compensate for the high-frequency signals. Bai and Li (1999) developed an SAC technique using the short-time Fourier transform (STFT) by analyzing the time–frequency distribution characteristics of seismic reflection signals. Li et al. (2000), Liu et al. (2006), and Zhang et al. (2010) proposed absorption compensation methods based on the generalized S transform to overcome restrictions of the time window width in the STFT. With the rapid development of multiscale transform techniques, wavelet transform has been introduced to many SAC methods (Gao et al. 1996). Wang et al. (2013) presented an SAC method based on curvelet transform. However, only horizontal curvelet coefficients were used to compensate the poststack data and geological structures such as tilted uplifts or faults and vertical events were not well recognized. Nevertheless, the multidirectional curvelet transform (Candès and Donoho 2005a, b) was found to be still more efficient than the wavelet transform at improving the quality of seismic data.

In this study, we developed a compensation method based on shearlet transform and combined it with the SAC model proposed by Bai and Li (1999). It is noteworthy that our scheme does not need as many assumptions as Bai’s method (here, the strata are sufficiently thick and waves reflected from the adjacent reflecting surfaces do not superpose in the seismic records). The paper is arranged as follows. First, an overview of the shearlet transform and its characteristics are presented. Then, we show how we combined the shearlet transform with the SAC model. Finally, the validity of the proposed method is verified through testing on both prestack and poststack field data.

## Introduction to the shearlet transform

Over the past two decades, wavelet transform has achieved tremendous success in digital signal processing, owing to their efficiency in work involving the capturing of singularities and analyzing piecewise smooth signals. However, the application of wavelets to natural images does not provide satisfactory results, because natural images, unlike digital signals, are normally composed of anisotropic features such as “lines” and “curves”. In the due course of time, multidirectional wavelet transforms emerged and developments in this area have been rapid. Seismic data, which consist primarily of curved-shape elements, can be perfectly characterized using multiscale and multidirectional sparse transforms. Curvelet transform, a multiscale mathematical transform that can provide excellent sparse representations of 2D images, has been widely applied to seismic data processing (Herrmann 2007; Herrmann et al. 2008). However, the basis functions of curvelets derived by using rotations do not allow for unitary treatment of continuous and discrete systems.

Shearlets (Guo et al. 2004a, b, 2006; Häuser 2012), which have similar sparse properties, form an affine-like system and have simpler mathematical structures. As such, 2D shearlet systems are carefully designed to efficiently encode such anisotropic features. The shearlet transform uses shearing instead of rotations (curvelet transform) and provides unitary treatment of continuous and discrete transforms. This allows for optimal sparse representations of piecewise smooth images.

*f*has the following form:

Comparison of the shearlet and curvelet transforms in representing the test seismic data

Scale 1 | Scale 2 | Scale 3 | Scale 4 | Scale 5 | |
---|---|---|---|---|---|

Curvelet (horizontal/vertical) | 41 × 31 | 31 × 30/42 × 23 | 63 × 31/42 × 45 | 126 × 61/84 × 90 | 251 × 61/84 × 180 |

Shearlet | 501 × 361 | 501 × 361 | 501 × 361 | 501 × 361 | 501 × 361 |

Number of coefficients, redundancy, and computational burden of shearlet and curvelet coefficients

Number of coefficients | Redundancy | Forward transform time (s) | Inverse transform time (s) | |
---|---|---|---|---|

Curvelet | 1,307,287 | 7.2281 | 0.4041 | 0.5570 |

Shearlet | 8,862,189 | 49 | 0.5472 | 0.6231 |

From Table 2, we can see the shearlet transform exhibits seven times the redundancy of the curvelet transform and the computing time for shearlet is just over that of curvelet owing to its special mathematical structure. For the example presented here, the compensation method requires only one forward transform and one inverse transform with a relatively low computation burden.

## Stratal absorption compensation model

*Q*-filtering. Let \(r_{k} \left( {k = 0,1 \ldots k} \right)\) be the reflection sequences, which have been subjected to different

*Q*filtering and let \(T_{k}\) represent the double travel time. Then, the amplitude response of the

*Q*-filter (Saatcilar and Coruh 1999) can be obtained through the following equations:

*Q*value at time \(T_{k}\); \(\omega\) is the frequency, and \(A_{0} \left( {\omega ,0} \right)\) is the initial equivalent amplitude. If we only consider the amplitude attenuation, then the reflection record \(\hat{x} (\omega )\) can be written as follows:

Equation (12) shows that the attenuation ratio is only time related and the frequency-variable weights of any frequency \(\omega\) can be derived by the reciprocal of a benchmark frequency \(\omega_{0}\) from Eq. (13). In other words, for a given frequency, seismic waves exhibit uniform waveforms at different times, but with amplitude differences in frequency-variable weights. As the loss of low-frequency information is relatively insignificant, frequency-variable weights for different frequency bands can be obtained by benchmarking against the time–energy distributions for the low-frequency bands. Then, the high-frequency records can be compensated by the reciprocal of the frequency-variable weights.

Traditional SAC methods based on time–frequency analysis are susceptible to noise. Nevertheless, this problem can be overcome by the remarkable directionality of the shearlet transform. In the shearlet domain, effective signals appear as high-amplitude and concentrated coefficients, whereas shearlet coefficients of noises have low amplitudes and relatively dispersed distributions. Thus noises can be separated from signals by thresholding (Liu et al. 2014) and excluded from the calculation of attenuation ratios. Doing so avoids noise enhancement during signal compensation.

The SAC model proposed above is constructed using the following two assumptions: the strata are sufficiently thick and the waves reflected from adjacent reflecting surfaces do not superpose. The results shown in Figs. 3 and 4 lead to the conclusion that overlap events can be recognized along different scales and directions in the shearlet domain. By exploiting the identification of shearlet orientation, the proposed method can be applied to the SAC model even in cases where the abovementioned assumptions are not met.

## Implementation of the SAC model based on the shearlet transform

Based on the theoretical SAC model presented above, we derive frequency-variable weights of high-frequency bands, which can be obtained by benchmarking low-frequency signals. The multiscale shearlet transform is applied to decompose a seismic record into signals for different directional and frequency bands \(x_{j,l}\), where \(j\) and \(l\) represent the scale and shearing parameters, respectively. Coefficients \(X_{j,l}\) (free of noise) are first derived by thresholding \(x_{j,l}\) and, then, the attenuation ratios \(a_{j,l}\) are determined by pointwise recursion \(X_{j,l}\) instead of \(x_{j,l}\). Subsequently, the original shearlet coefficients \(x_{j,l}\) are weighted by the reciprocal of \(C_{j,l}\) (frequency-variable weights) to remove the effects of stratal absorption. Finally, the inverse shearlet transform of the weighted \(x_{j,l}\) is performed, which gives the seismic record after the SAC. After estimating frequency-variable weights, we invert for all shearlet coefficients except those representing signals. This is because high-frequency signals usually have low amplitudes and we cannot guarantee that all the signals are reserved after thresholding. By inverting all shearlet coefficients, we can ensure high fidelity without loss of signals. In addition, at least noise was not compensated for, although we cannot guarantee the compensation of all the signals.

- (a)
Apply the forward shearlet transform to the seismic record to obtain the shearlet coefficients \(x_{j,l}\).

- (b)
Suppress random noise along different scales and directions by thresholding the shearlet coefficients (Liu et al. 2014) in order to enable directional compensation for effective signals. The shearlet coefficients after noise suppression are defined as \(X_{j,l}\).

- (c)
Calculate the attenuation ratios of deep reflection energy to shallow reflection energy for each scale and each direction by pointwise recursion using \(a_{j,l} \left( {\omega_{m} ,T_{k} } \right) = \frac{{X_{j,l} \left( {\omega_{m} ,T_{k} } \right)}}{{X_{j,l} \left( {\omega_{m} ,T_{f} } \right)}}\), where \(T_{k}\) is the sampling time, and \(T_{f}\) denotes the first-arriving-time in the prestack data or shallow strong reflections in the poststack data chosen as per experience.

- (d)
Calculate the frequency-variable weights for different frequency bands using \(C_{j,l} \left( {\omega_{m} ,T_{k} } \right) = \frac{{a_{j,l} \left( {\omega_{m} ,T_{k} } \right)}}{{a_{2,l} \left( {\omega_{0} ,T_{k} } \right)}}\), where \(j = 2\) represents the reference scale, and \(\omega_{0}\) is the reference low frequency.

- (e)
Weight the original shearlet coefficients \(x_{j,l} \left( {\omega_{m} ,T_{k} } \right)\) by the reciprocal of \(C_{j,l} \left( {\omega_{m} ,T_{k} } \right)\).

- (f)
Apply the inverse shearlet transform to the weighted shearlet coefficients.

## Numerical examples

## Conclusions

In this study, we introduced a multiscale and multidirectional shearlet transform and establish a new algorithm combined with the absorption compensation model. The sparsity of shearlets enables the calculation of compensation weights for effective signals only. Hence, it is possible to avoid noise enhancement during signal compensation. The multi-direction of the shearlets facilitates the efficient identification of seismic events. Therefore, the proposed model is not restricted by the assumption of a sufficiently thick adjacent reflector. Moreover, owing to the directionality of the shearlets, the proposed absorption compensation method is applicable to both prestack and poststack seismic data. We calculated the attenuation ratios of signals for different frequency bands on the basis of the low frequency and compensated the high-frequency components only. The numerical examples showed that the proposed method has wide ranging applicability for practical applications.

## Notes

### Acknowledgements

We thank the ShearLab for sharing their codes available on the web. This research is supported by the Major Projects of the National Science and Technology of China (Grant No. 2016ZX05026-002-003), National Natural Science Foundation of China (No. 41374108), and Graduate Innovation Fund of Jilin University (No. 2017090).

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