Acta Geophysica

, Volume 66, Issue 4, pp 575–584 | Cite as

Stratigraphic absorption compensation based on multiscale shearlet transform

  • Chengming Liu
  • Deli Wang
  • Jialin Sun
  • Yibin Li
  • Fan Yang
Research Article - Applied Geophysics


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.


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


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.

A continuous affine system on \(L^{2} (R^{n} )\) can be written as follows:
$$\left\{ {\varvec{T}_{t} \varvec{D}_{\varvec{M}} \psi, \varvec{M} \in \varvec{G},t \in R^{n} } \right\}$$
where \(\psi \in L^{2} \left( {R^{2} } \right)\); \(\varvec{T}_{t}\) is a translation operator, defined as \(\varvec{T}_{t} f(x) = f(x - t)\), where \(t\) is the translation parameter; \(\varvec{D}_{\varvec{M}}\) is a dilation operator, given by \(\varvec{D}_{\varvec{M}} f(x) = \left| {\det \varvec{M}} \right|^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} f(\varvec{M}^{ - 1} x)\); and \(\varvec{G} \subset \varvec{G}L_{n} (R)\) is a set of matrices. When \(n = 2\), \(\varvec{G}\) represents a set of dilation matrices containing two parameters:
$$\varvec{G} = \{ \varvec{M}_{as} = \left( {\begin{array}{*{20}c} a & {\sqrt a s} \\ 0 & {\sqrt a } \\ \end{array} } \right):(a,s) \in R^{ + } \times R\}$$
It is worth noting that the dilation matrix \(\varvec{M}_{as}\) can be factorized into two matrices, namely, \(\varvec{M}_{as} = \varvec{S}_{s} \varvec{A}_{a}\), where \(S\) is the shear matrix and \(A\) is the scaling matrix.
$$\varvec{S}_{s} = \left( {\begin{array}{*{20}c} 1 & s \\ 0 & 1 \\ \end{array} } \right),\quad \varvec{A}_{a} = \left( {\begin{array}{*{20}c} a & 0 \\ 0 & {\sqrt a } \\ \end{array} } \right)$$
Here \(s\) is the shearing parameter and \(a\) is the scale parameter. In particular, for \(\xi = \left( {\xi_{1} ,\xi_{2} } \right) \in R^{2} ,\xi_{1} \ne 0\), \(\psi\) is given by
$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi } \left( \xi \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi } \left( {\xi_{1} ,\xi_{2} } \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{1} \left( {\xi_{1} } \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{2} \left( {\frac{{\xi_{2} }}{{\xi_{1} }}} \right)$$
where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{1}\) denotes the Fourier transform of \(\psi\) thus satisfying the Calderòn condition:
$$\int_{ 0}^{\infty } {\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{1} \left( {a\xi } \right)} \right|}^{2} \frac{{{\text{d}}a}}{a}\quad {\text{for}}\;\;{\text{a}} . {\text{e}} .\quad \xi \in R$$
And \(\left\| {\psi_{2} } \right\| = 1\). Equation (5) satisfies the conditions required for continuous real-valued wavelet transform under standard admissible conditions. With such an affine system, we have
$$\left\{ \psi_{a,s,t} (x) = a^{-3/4} \varphi (\varvec{M}_{as}^{ - 1} (x - t)):a \in R^{ + } ,s \in R,t \in R^{2} \right\}$$
Additionally, \(\psi_{1}\) represents the continuous wavelet \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{1} \in C^{\alpha } \left( R \right)\), and supp \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{1} \subset \left[ { - 2, - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right] \cup \left[ {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2},2} \right]\); \(\hat{\varphi }_{2} \in C^{\infty } (\hat{R})\), sup \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\psi }_{2} \subset \left[ { - 1,1} \right]\).
We consider that Eq. (4) satisfies the conditions required for shearlet transform and define the corresponding continuous shearlet transform as follows:
$$\left\{ \psi_{a,s,t} (x): = a^{-3/4} \psi (A_{a}^{ - 1} S_{s}^{ - 1} (x - t)):a \in R^{ + } ,s \in R,t \in R^{2} \right\}$$
where \(\psi_{a,s,t} (x)\) is a shearlet. The shearlet transform of the arbitrary square-integrable function f has the following form:
$$SH\left( f \right)\left( {a,s,m} \right): = \left\langle {f \cdot \varphi_{a,s,m} } \right\rangle = \int_{{{\mathbb{R}}^{2} }} {f\left( x \right)} \varphi_{a,s,m} \left( x \right)dx$$
Figure 1 illustrates the decomposition of a continuous shearlet in the frequency domain. It also reveals the roles of the parabolic scaling matrix \(\varvec{A}_{a}\) and the shear matrix \(\varvec{S}_{s}\) in construction. \(\varvec{A}_{a}\) controls the scale with two axial dilation factors, which ensures the lengthening and thinning of the support as \(a \to 0\). \(s\) determines the directionality of the shearlet.
Fig. 1

a Tiling of the frequency plane R2 induced by shearlets. b Frequency support of shearlet

Figure 2 shows a set of shearlets at different scales along the same direction. With the increase in scale, the shearlets become elongated. They are directional and band-passed in nature with increasing central frequencies from left to right; thus, they be used successfully to represent high-frequency components and provide sparse representations with higher precision (Lakshman et al. 2015).
Fig. 2

Examples of shearlet elements for three scales in the same direction (top row: frequency domain, bottom row: spatial domain)

Figure 3 illustrates the distribution of shearlet coefficients of synthetic seismic data along different directions at a given scale. Figure 3a shows the synthetic seismogram and Fig. 3b–d shows the shearlet coefficients along three directions at the third scale. Figure 4 shows the shearlet coefficients at the second, third, and fourth scales along the same direction. Large-scale shearlets can effectively capture high-frequency information along certain directions, especially fine events. The comparison of frequency spectra at the three scales demonstrate that shearlets at different scales can effectively represent information for different frequency bands.
Fig. 3

Distribution of shearlet coefficients along different directions at a given scale: a synthetic seismogram, b upper left 45°, c horizontal, and d upper right 45°

Fig. 4

Examples of shearlet coefficients at different scales along a given direction and the corresponding frequency spectra: a second scale, b third scale, c fourth scale, df frequency-wavenumber spectrum for (ac)

Tables 1 and 2 compare the number of coefficients, redundancy, and computational costs of the shearlet and curvelet transforms in representing the test seismic data respectively. The test data are shown in Fig. 3a, the size being \(501 \times 361\). As shown in Table 1, the curvelet coefficients at different scales do not have a uniform size and, thus, a fine-to-coarse scale relationship follows the many-to-one correspondence. In contrast, the size of shearlet coefficients remains uniform across different scales. We will show that this feature makes the SAC method based on the shearlet transform more effective than the curvelet method.
Table 1

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


501 × 361

501 × 361

501 × 361

501 × 361

501 × 361

Table 2

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


Number of coefficients


Forward transform time (s)

Inverse transform time (s)











The implementation of curvelet-generated angles at each scale is 1, 16, 32, 32, and 64, and the corresponding values for the shearlet transform are 1, 8, 8, 16, and 16

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

Seismic records should undergone time-varying low-pass filtering with increase in the propagation distance and frequency. We usually describe this process as 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:
$$A_{k} (\omega ,T_{k} )= A_{0} (\omega ,0 ) {\text{e}}^{{ - \omega T_{k} /Q_{\text{eq}} (T_{k} )}}$$
$$Q_{\text{eq}} (T_{k} )= \frac{{T_{k} }}{{\int_{0}^{{T_{k} }} {\frac{\tau }{Q (\tau )}{\text{d}}\tau } }}$$
where \(Q_{\text{eq}} \left( {T_{k} } \right)\) is the equivalent 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:
$$\hat{x} (\omega )= \sum\limits_{0}^{K} {r_{k} A_{k} (\omega ,T_{k} ) {\text{e}}^{{ - j\omega T_{k} }} \hat{\omega }} (\omega )$$
Assuming that the strata are sufficiently thick and waves reflected from the adjacent reflecting surfaces do not superpose in the seismic records, we have attenuation ratios at \(T_{k}\):
$$a (\omega ,T_{k} )= \frac{{\left| {x (\omega ,T_{k} )} \right|}}{{\left| {x (\omega ,0 )} \right|}} = \frac{{\left| {r_{k} A_{0} (\omega ,0 )A_{k} (\omega ,T_{k} )\hat{\omega } (\omega )} \right|}}{{\left| {A_{0} (\omega ,0 )\hat{\omega } (\omega )} \right|}} = \left| {r_{k} A_{k} (\omega ,T_{k} )} \right|$$
where \(A_{k} (\omega ,T_{k} )= {\text{e}}^{{ - \omega T_{k} Q_{eq}^{ - 1} (T_{k} )}}\) is the amplitude of reflected waves at \(T_{k}\). Thus the frequency-variable weights at any frequency \(\omega\) can be obtained through dividing \(a (\omega ,T_{k} )\) by \(a (\omega_{0} ,T_{k} )\):
$$C (\omega ,T_{k} )= \frac{{a (\omega ,T_{k} )}}{{a (\omega_{0} ,T_{k} )}} = \frac{{{\text{e}}^{{ - \omega T_{k} Q_{eq}^{ - 1} (T_{k} )}} }}{{{\text{e}}^{{ - \omega_{0} T_{k} Q_{eq}^{ - 1} (T_{k} )}} }}$$

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.

The SAC method based on the shearlet transform involves the following steps:
  1. (a)

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

  2. (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 \(X_{j,l}\).

  3. (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.

  4. (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.

  5. (e)

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

  6. (f)

    Apply the inverse shearlet transform to the weighted shearlet coefficients.


Numerical examples

To test the validity of the proposed method, both curvelets and shearlets were processed on the prestack data shown in Fig. 3a. Both methods boosted the weak events in the deep area and improved the resolution. From Fig. 5, we can find two main problems caused by the curvelet method. The first problem is that the curvelets exhibit unreliable presentations of large curvature events as indicated by the arrow at 0.3 s. Another is that the distortion appeared in the fine structure. Figure 5d–f shows zoomed-in plots of Fig. 5a–c; micro-stratigraphic boundaries of many small geological features such as pinch-outs and faults, which became distorted (circled) after curvelet compensation. In contrast, the micro-structures were improved with the shearlet compensation. The reason for this is that the curvelets cannot provide a uniform size of coefficients at different scales and, thus, the frequency-variable weights for high-frequency bands are a kind of approximation. Therefore, the frequency-variable weights of curvelets are less accurate than those of the shearlets (Fig. 6).
Fig. 5

Examples of the prestack seismic profile a before compensation; b after compensation with curvelets, c after compensation with shearlets; df zoomed-in plots of (a), (b), and (c) respectively

Fig. 6

Frequency-wavenumber spectra of prestack seismic data before and after compensation

In the field data test, we selected field marine data with target reservoirs ranging from 0.8 to 1.2 s. Because of absorption, the seismic events appeared obscure and discontinuous below 0.8 s. The profile can be divided into shallow, middle, and deep parts, as shown in Fig. 7a–c, respectively. Then, we decomposed the data into different frequency bands and directions using the shearlet transform. Four scales of shearlet transform were applied to the numerical examples in this study. Additional scales can be used to obtain more refined frequency bands, but doing so will significantly increase the computing burden. The compensation results for shallow signals are insignificant because shallow stratal absorption does not lead to much energy loss in each frequency band. The resolution of seismic events from the middle strata was significantly improved after compensation, especially for those under 0.8 s, which were originally discontinuous but became smooth and continuous. The frequency spectrum (> 20 Hz) was clearly widened by the shearlet method. The most significant effect was observed in the deep area, where high-frequency components suffered substantial losses. We can see that the fine structures that were originally obscured could be identified easily without increasing noise after compensation. In particular, the uplift around 1.25 s (circled) presented reasonable resolution after compensation. The comparison of frequency spectra also indicates that the proposed method effectively broadens frequencies higher than 30 Hz. The numerical results clearly demonstrate that this algorithm is robust and effective at compensating high-frequency signals and improving the resolution.
Fig. 7

Compensation for the poststack seismic profile: ac signals from shallow, middle, and deep strata before compensation. df signals from shallow, middle, and deep strata after compensation, and gi comparison of the frequency spectra before and after compensation

To further evaluate the compensation effects for poststack data for different frequency bands, frequency scans of the profile before and after compensation were performed (Fig. 8). As the figure shows, the compensation effect was not apparent for frequencies lower than 30 Hz, possibly because of the less loss of low-frequency signals during propagation. For frequencies higher than 30 Hz, signals from strata deeper than the location at 0.9 s were significantly strengthened after compensation and the seismic data contained abundant high-frequency information. Moreover, the profile after compensation exhibited excellent event consistency without ambiguities.
Fig. 8

Frequency scanning before and after compensation


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.



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).


  1. Bai H, Li K (1999) Stratigraphic absorption based on time–frequency analysis. Oil Geophys Prospect 34(6):642–648 (in Chinese) Google Scholar
  2. Candès EJ, Donoho DL (2005a) Continuous curvelet transform: I. Resolution of the wavefront set. Appl Comput Harmonic Anal 19(2):162–197CrossRefGoogle Scholar
  3. Candès EJ, Donoho DL (2005b) Continuous curvelet transform: II. Discretization and frames. Appl Comput Harmonic Anal 19(2):198–222CrossRefGoogle Scholar
  4. Ferber R (2005) A filter bank solution to absorption simulation and compensation. SEG Tech Program Expand Abstr 24(1):2668Google Scholar
  5. Gao J, Ling Y, Zhou X, Mou YG (1996) Compensation for spherical divergence and absorption in time and frequency domain. Oil Geophys Prospect 31(6):856–866 (in Chinese) Google Scholar
  6. Guo K, Lim W, Labate D, Weiss G, Wilson E (2004a) Wavelets with composite dilations. Electron Res Announc Am Math Soc 10:78–87CrossRefGoogle Scholar
  7. Guo K, Lim W, Labate D, Weiss G, Wilson E (2004b) The theory of wavelets with composite dilations. In: Heil C (ed) Harmonic analysis and applications. Birkhäuser, Boston, pp 231–250Google Scholar
  8. Guo K, Lim W, Labate D, Weiss G, Wilson E (2006) Wavelets with composite dilations and their MRA properties. Appl Comput Harmonic Anal 20:231–249CrossRefGoogle Scholar
  9. Häuser S (2012) Fast finite Shearlet transform. arXiv:1202.1773
  10. Herrmann FJ (2007) Recent developments in curvelet-based seismic processing. Seismic Laboratory for Imaging and Modeling, European Association of Geoscientists and Engineers, London. CrossRefGoogle Scholar
  11. Herrmann FJ, Moghaddam P, Stolk CC (2008) Sparsity- and continuity-promoting seismic image recovery with curvelet frames. Appl Comput Harmonic Analsis 24(2):150–173CrossRefGoogle Scholar
  12. Lakshman H, Lim WQ, Schwarz H et al (2015) Image interpolation using shearlet based iterative refinement. Sig Process Image Commun 36:83–94CrossRefGoogle Scholar
  13. Li K, Li Y, Zhang X (2000) A method to compensate earth filtering based on wavelet packet. Chin J Geophys 43(4):542–549 (in Chinese) CrossRefGoogle Scholar
  14. Ling Y, Gao J, Zhang R (1997) Sand-dune Q absorption compensation based on 1-D elastic damping wave theory. Oil Geophys Prospect 32(6):795–803 (in Chinese) Google Scholar
  15. Liu X, Nian J, Liu H (2006) Generalized S-transform based on compensation for stratigraphic absorption of seismic attenuation. Geophys Prospect Petrol 45(1):9–14Google Scholar
  16. Liu C, Wang D, Wang T et al (2014) Random seismic noise attenuation based on the shearlet transform. Acta Petrol Sin 35(4):692–699 (in Chinese) Google Scholar
  17. Quan YL, Harris JM (1997) Seismic attenuation tomography using the frequency shift method. Geophysics 62(3):895–905CrossRefGoogle Scholar
  18. Saatcilar R, Coruh C (1999) Seismic Q estimations for lithological interpretation. SEG Tech Program Expand Abstr 14(1):1366Google Scholar
  19. Scott S, Taner MT, Treitel S (2006) Q estimation using Gabor–Morlet joint time–frequency analysis. In: SEG expanded abstracts, pp 2610–2614.
  20. Wang Y (2002) A stable and efficient approach of inverse Q filtering. Geophysics 67(2):657CrossRefGoogle Scholar
  21. Wang Y (2006) Inverse Q-filter for seismic resolution enhancement. Geophysics 71(3):v5l–v61CrossRefGoogle Scholar
  22. Wang D, Sun J, Meng D et al (2013) Absorption compensation based on curvelet transform. J Seism Explor 22(1):19–32Google Scholar
  23. Zhang G, Xiong X, Rong JJ, Cai ZD (2010) Stratum absorption and attenuation compensation based on improved generalized S transform. Oil Geophys Prospect 45(4):512–515 (in Chinese) Google Scholar
  24. Zhao S, Wang X, Wang J (1994) A simple frequency compensation method. Oil Geophys Prospect 29(2):231–235 (in Chinese) Google Scholar
  25. Zhou H, Qu G, Yang B (1994) New methods for calculating Q value using spectrum of seismic record. J Jilin Univ (Earth Sci Edn) 4:461–467 (in Chinese) Google Scholar

Copyright information

© Institute of Geophysics, Polish Academy of Sciences & Polish Academy of Sciences 2018

Authors and Affiliations

  • Chengming Liu
    • 1
  • Deli Wang
    • 1
  • Jialin Sun
    • 2
  • Yibin Li
    • 3
  • Fan Yang
    • 1
  1. 1.College of Geo Exploration Science and TechnologyJilin UniversityChangchunPeople’s Republic of China
  2. 2.China National Offshore Oil CorporationTianjinPeople’s Republic of China
  3. 3.China National Offshore Oil CorporationZhanjiangPeople’s Republic of China

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