Enhancing the resolution of time–frequency spectrum using directional multichannel matching pursuit

Abstract

Matching pursuit is able to decompose signals adaptively into a series of wavelets and has been widely applied in signal processing of the geophysical fields. Single-channel matching pursuit could not take into account the lateral continuity of seismic traces, and the recent multichannel matching pursuit exploits the lateral coherence as a constraint, which helps to improve the stability of decomposition results. However, atoms searched by multichannel matching pursuit currently are just shared by lateral seismic traces at the same time slicers. The lack of directionality in multichannel search strategies leads to irrationality in dealing with large dip angle seismic traces. Considering that the waveforms of reflection events are relatively continuous and similar, an improved multichannel matching pursuit is proposed to realize the directional decomposition of adjacent signals. Based on the principle of seismic reflection events tracking and identification, directional multichannel decomposition of seismic traces is realized. The seismic channel to be decomposed is correlated with the time shift of the optimal atom determined by the previous seismic channel. The time position of the maximum correlation indicates the center time of the optimal atom. Optimal atoms identified by one iteration of multichannel decomposition have the same frequency and phase parameters, different center time and amplitude parameters. The center time of the optimal atoms is consistent with seismic reflection events. Tests illustrate that the algorithm can successfully reconstruct 2D seismic data without reducing accuracy. Besides, the application of field data is of great significance for reservoir exploration and hydrocarbon interpretation.

Introduction

The time–frequency analysis technique can convert the signal from the time domain to the time–frequency domain and can be used to extract more attribute characteristics of the signal, which plays an important role in field of geosciences (Hess-Nielsen and Wickerhauser 1996; Qian and Chen 1999). To estimate the time–frequency representations of the signal, a variety of methods including short-time Fourier transform (STFT), continuous wavelet transform (CWT), S transform (ST), matching pursuit (MP) are usually used (Duijndam and Schonewille 1999; Sinha et al. 2005; Stockwell et al. 2002; Mallat and Zhang 1993). Matching pursuit is widely applied in various subject areas for its superiority of resolution (Gribonval et al. 1996; Salomon and Ur 2004) and has been well discussed in signal processing of different fields (Durka et al. 2005).

The conventional matching pursuit is a greedy strategy to decompose a signal into a linear combination of time–frequency atoms chosen among a Gabor dictionary, which can adaptively sparsely represent signals and has high time–frequency resolution. However, the greedy global search strategy leads to low decomposition efficiency, which limits its widespread application for huge data processing (Lin et al. 2006). Therefore, various improved algorithms to deal with this problem are proposed. In order to reduce the computational cost of the algorithm, genetic algorithm is introduced to matching pursuit achieve this goal successfully (Stefanoiu and Lonescu 2003). Ricker or Morlet atomic libraries which are controlled by three parameters are usually constructed for seismic signal decomposition. The global search of all atoms in the atomic library results in extremely low computational efficiency. The instantaneous properties of the signal estimated by the Hilbert transform can be used to narrow the scanning range of the time–frequency atomic library, which improves the decomposition efficiency of the conventional MP algorithm (Liu et al. 2004; Liu and Marfurt 2005). Since the atoms in the dictionary are not orthogonal, the vertical projection of the signal residual at the selected atom is non-orthogonal, which makes each iteration not optimal but suboptimal. The proposed orthogonal matching pursuit helps to overcome this drawback, which can avoid the repeated selection of the optimal atoms by orthogonalizing the atoms in the dictionary (Tropp and Gilbert 2008; Miandji et al. 2017).

The methods above are usually single-channel decomposition strategies. The operation order of the single-channel matching pursuit has a great negative influence on the decomposition result. Slight changes in the seismic trace will result in changes in the order of matching pursuit, and the atoms obtained by the decomposition will be unstable (Jun et al. 2012). This leads to the lateral instability and non-uniqueness of the time–frequency analysis results, and it is difficult to ensure the spatial correlation of the decomposition result. Also, the decomposition is easily contaminated by the data noise.

To make full use of the spatial continuity of seismic data, the multichannel matching pursuit (MCMP) exploits lateral coherence as a constraint to improve the uniqueness of the solution (Wang 2010). The genetic algorithm is introduced to the multichannel matching pursuit to reduce the complexity of the algorithm (Jun et al. 2012). Multichannel matching pursuit keeps better stability, but there are some limitations. Some improved algorithms are proposed to optimize the decomposition strategy of the algorithm (Ning and Wang 2014).

We know from sedimentology that the layered lithologic body is mainly distributed underground, and the seismic waves of adjacent seismic traces show better horizontal continuity and spatial coherence in the same seismic horizon (Yuan et al. 2015). The optimal atom searched in the multichannel matching pursuit best matches the average of all seismic traces, rather than the best match for each channel. The algorithm does not take the information of seismic reflection events into account, and the matching optimization direction does not match the direction of the seismic event. Therefore, in this study, we propose an improved directional multichannel matching pursuit algorithm, which can make full use of the lateral continuity of the seismic data and can directionally search for multichannel atoms. The proposed multichannel matching pursuit has been proved to enhance the resolution of the time–frequency spectrum profile, which help to guide reservoir prediction and hydrocarbon detection.

Methods

Single-channel matching pursuit theory

In matching pursuit, the signal can be expressed in a linear combination of multiple atoms obtained by decomposition. Given a seismic trace \( f(t) \), we select a parent function according to the characteristics of the signal to create a redundant dictionary \( D = \left\{ {g_{\gamma } \left( t \right)} \right\} \), where \( g_{\gamma } \left( t \right) \) is an atom defined by parameter group \( \gamma = \left( {u,\omega ,\varphi } \right) \) and \( \left\| {g_{\gamma } \left( t \right)} \right\| = 1 \), where \( u \), \( \omega \) and \( \varphi \) represent time, frequency and phase, respectively. The signal is decomposed as follows.

Firstly, the atom which matches the signal best is selected from the over-complete atomic library and satisfies the following equation (Mallat and Zhang 1993):

$$ \left| {\left\langle {f\left( t \right),g_{{\gamma_{1} }} \left( t \right)} \right\rangle } \right| = \sup_{{i \in \left( {1,2, \ldots k} \right)}} \left| {\left\langle {f\left( t \right),g_{{\gamma_{i} }} \left( t \right)} \right\rangle } \right| $$
(1)

where \( g_{{\gamma_{1} }} \left( t \right) \) is the atom that best matches the signal at the first iteration. \( g_{{\gamma_{i} }} \left( t \right) \) presents each atom in the dictionary. sup is an upper bound operator. k is the number of atoms in the redundant dictionary.

The signal can be expressed as follows:

$$ f\left( t \right) = \left\langle {f\left( t \right),g_{{\gamma_{1} }} \left( t \right)} \right\rangle g_{{\gamma_{1} }} \left( t \right) + R_{1} f\left( t \right) $$
(2)

where R is the residual operator. \( R_{1} f\left( t \right) \) is the residual component of the signal after the first iteration. Continuously performing such an iterative process, the nth decomposition process is:

$$ R_{n - 1} f\left( t \right) = \left\langle {R_{n - 1} f\left( t \right),g_{{\gamma_{n} }} \left( t \right)} \right\rangle g_{{\gamma_{n} }} \left( t \right) + R_{n} f\left( t \right) $$
(3)

where \( R_{n - 1} f\left( t \right) \) and \( R_{n} f\left( t \right) \) are the residual signal at \( n - 1{\text{th}} \) and nth step, respectively. \( g_{{\gamma_{n} }} \left( t \right) \) is the atom that fits to \( R_{n - 1} f\left( t \right) \) best at the nth iteration, and \( g_{{\gamma_{n} }} \left( t \right) \) satisfies the following equation:

$$ \left| {\left\langle {R_{n - 1} f\left( t \right),g_{{\gamma_{n} }} \left( t \right)} \right\rangle } \right| = \sup_{{i \in \left( {1,2, \ldots k} \right)}} \left| {R_{n - 1} f\left( t \right),g_{{\gamma_{i} }} \left( i \right)} \right|. $$
(4)

After nth iterations, the signal becomes:

$$ f\left( t \right) = \sum\limits_{i = 1}^{n} {\left\langle {R_{i - 1} f\left( t \right),g_{{\gamma_{i} }} \left( t \right)} \right\rangle } g_{{\gamma_{i} }} \left( t \right) + R_{i} f\left( t \right) $$
(5)

here \( R_{0} f\left( t \right) = f\left( t \right) \). As discussed above, the signal can be expressed as the sum of the linear combination of all atoms and the residual signal.

At each step of decomposition, the atom extracted from the redundant dictionary satisfies Eq. (4). When the number of iterations reaches the preset value or the signal residual energy is less than a certain threshold, the decomposition process is completed. Although the matching pursuit can be used to yield a sparse representation model of the signal, this algorithm has the main disadvantage of a large amount of calculation (Masood and Al-Naffouri 2013).

Directional multichannel matching pursuit theory

Given M seismic traces \( \left\{ {f_{1} \left( t \right),f_{2} \left( t \right), \ldots ,f_{M} \left( t \right)} \right\} \), we start the first iterative decomposition from the first seismic trace \( f_{1} \left( t \right) \). Similar to the strategy of searching for the optimal atom in the single-channel matching pursuit algorithm, we decompose the signal from the time position of the maximum envelope amplitude. Utilize the transient characteristics of the time position to build the dynamic atomic library, and search for the atom \( g_{{\gamma_{1} }} \left( t \right) \) that best matches the signal at the first iteration. The instantaneous phase \( \varphi_{1} \), instantaneous frequency \( \omega_{ 1} \) and central time \( u_{1} \) parameters of the optimal atom are recorded. Then estimate the amplitude of the optimal atom corresponding to the first trace according to the following equation:

$$ a_{ 1} = \frac{{\left| {\left\langle {f_{ 1} \left( t \right),g_{{\gamma_{ 1} }} \left( t \right)} \right\rangle } \right|}}{{\left\| {g_{{\gamma_{ 1} }} \left( t \right)} \right\|^{2} }}. $$
(6)

The signal residual can be expressed as follows:

$$ Rf_{ 1} \left( t \right) = f_{ 1} \left( t \right) - a_{ 1} g_{{\gamma_{ 1} }} \left( t \right) $$
(7)

where R represents residual signal. Considering that the same interface’s event of reflected waves often has similar waveform characteristics, we use the similarity of seismic wavelet waveform to identify the seismic event and realize the fast decomposition of adjacent seismic traces. The search strategy is shown in Fig. 1.

Fig. 1
figure1

Directional multichannel search schematic

The red wavelet is the optimal atom determined by the first search. The green dot is the central time position \( u_{1} \) of the optimal atom. For the search of the next seismic trace, we only need to select the appropriate delay time. After the delay of the optimal wavelet determined by the search of the previous channel, we can do correlation analysis with the cableway to be inspected to obtain the maximum correlation coefficient, which shows that the optimal wavelet is the most similar to the seismic channel to be retrieved after the time delay. The correlation coefficient is calculated as follows:

$$ g_{{\gamma_{2} }} \left( t \right) = \arg \hbox{max} \left| {\left\langle {f_{2} \left( t \right),g_{{\gamma_{1} }} \left( {t - \tau } \right)} \right\rangle } \right| $$
(8)

where \( \tau \) is the time length of sliding up and down of the optimal wavelet in correlation calculation. As shown in Fig. 1, the purple shadow area represents the size of the sliding window, that is, first determine the time position of the maximum instantaneous amplitude closest to the center time of the optimal atom obtained from the previous seismic channel search, and the time position of the maximum instantaneous amplitude closest to the top and bottom of this position is the upper and lower limit of the sliding window.

The optimal atoms obtained in one iteration of multichannel search have the same frequency and phase parameters, different center time and amplitude parameters. Therefore, it is only necessary to record the center time and amplitude parameters of the optimal atom. For the next seismic channel to be decomposed, we need to update the center time of the optimal atom and determine the sliding window and then calculate the correlation coefficient between the atom and the seismic channel to be decomposed. The amplitude of the adjacent wavelets on the same reflection event changes gradually. When the amplitudes of the adjacent seismic channels change greatly, it is considered that there are discontinuous layers, faults or pinch-out points, etc., and then, the current multichannel decomposition is stopped. Stop the first iteration of the multichannel search until all seismic traces have completed or the amplitude ratio does not satisfy the following formula:

$$ \alpha_{1} \ge \frac{{a_{k + 1} }}{{a_{k} }} \ge \alpha_{2} $$
(9)

where \( \alpha_{ 1} \) and \( \alpha_{ 2} \) are the multichannel search threshold to ensure the existence of the fault and the discontinuous layer in the reconstructed seismic traces. After testing, we think that amplitude ratio \( \alpha_{1} = 1.5,\;\alpha_{2} = 0.5 \) can get better decomposition results. It is not perfect to use only the amplitude information as the parameter indicating the optimal atom, so it is possible to constrain the lateral search direction by combining coherence and phase parameters in the future.

Results

Synthetic examples

Considering the wide application of Ricker wavelet in modeling, processing, inversion and interpretation of seismic data (Yuan et al. 2019), we use Ricker wavelet dictionary to decompose and reconstruct model data and field data. To verify the feasibility of the directional multichannel matching pursuit method, we build a geological model of multilayer sand bodies to get the synthetic seismic data. As shown in Fig. 2a, the synthetic seismic data consist of 700 traces with the vertical length of 800 ms. The dominant frequency of the zero-phase Ricker wavelet is 30 Hz, and the vertical sampling interval is 2 ms. We apply the proposed multichannel matching pursuit to decompose and reconstruct the theoretical data.

Fig. 2
figure2

Estimated results of theoretical data using the proposed multichannel matching pursuit. a A synthetic 2D seismic profile. The matching atoms (b) and the corresponding residual (c) identified by the third iteration, respectively. The matching atoms (d) and the corresponding residual (e) identified by the sixth iteration, respectively. f The reconstructed seismic profile. g The final residual

In order to better understand the decomposition process of the algorithm, we demonstrate the iterative decomposition process in detail. Figure 2b, d shows the matching atoms identified by the third and sixth iteration of the first seismic trace using the proposed multichannel matching pursuit, respectively. Figure 2c, e displays the corresponding residual seismic profiles, respectively. For the third iteration of the first trace, we first scan the dynamic atomic library to determine parameters of the optimal atom, and then, we just need to calculate the time shift correlation between this atom and the next seismic channel. Continuously update the center time and amplitude parameters of the optimal atom. Figure 2b shows the optimal atoms determined by the above steps. The optimal atoms shared by lateral seismic traces have different time parameters, and the directions are consistent with the seismic events. The atoms estimated by the multichannel search are best matched to each seismic trace, not the best match to the average of all the seismic traces. Therefore, the proposed multichannel decomposition strategy makes the signal residuals get the fastest convergence and decline. We set the amplitude ratio as the criterion to cease the multichannel search, which can guarantee the accuracy of the reconstructed seismic traces.

We can see from Fig. 2d that the horizon continuity of estimated atoms is preserved well, and the atoms determined by multichannel search keep high consistency with seismic reflection events. The matching atoms shown in Fig. 2d only need to search the atomic library six times, while the single-channel matching pursuit algorithm needs to search the atomic library once to determine each optimal atom. The proposed algorithm greatly reduces the time cost of searching the optimal atom. Figure 2f shows the final result of the reconstructed profile after 120 iterations. The value of the seismic residual shown in Fig. 2g is close to zero, and the reconstructed seismic profile shares the same characteristics with the theoretical data as shown in Fig. 2a.

To test the stability of the proposed method, we add random noise to get the model data shown in Fig. 3a with signal-to-noise ratio of 2:1. Figure 3b, c displays the matching wavelets and the corresponding residual profile identified by the sixth iteration, respectively. We can see that the decomposition iteration process of the proposed method is relatively stable although in the presence of random noise. The matching wavelets shown in Fig. 3b keep good lateral continuity. Figure 3d, e is the final results of the reconstructed profile and the corresponding residual profile. We can see that the reconstructed profile contains less noise than the noisy synthetic data in Fig. 3a. The residual profile shown in Fig. 3e does not contain valid reflection information and keeps a high similarity with the random noise profile shown in Fig. 3f, which proves that the proposed directional multichannel matching pursuit has a certain anti-noise ability.

Fig. 3
figure3

Estimated results of noisy theoretical data using the proposed multichannel matching pursuit. a A noisy synthetic 2D seismic profile with a signal-to-noise ratio of 2:1. The matching atoms (b) and the corresponding residual (c) identified by the sixth iteration, respectively. d The reconstructed seismic profile. e The final residual. f The added random noise

In addition, we built a complex geological model to verify the applicability of the proposed method. The model data shown in Fig. 4a is composed of 28 Hz Ricker wavelet, and Fig. 4b, c displays the matching wavelets and the corresponding residual identified by the 12th iteration, respectively. We can see that even for complex geological model data, the proposed method can still identify multichannel matching wavelets along the direction of the seismic reflection events. The matching wavelets shown in Fig. 4b are obtained by only iteratively decomposing the first seismic trace for 12 times. The reconstructed profile in Fig. 4d obtained by 430 iterations keeps high consistency with the original profile in Fig. 4a. The residual profile in Fig. 4e does not contain effective reflection information, which proves that the proposed matching pursuit can realize the directional decomposition of the synthetic data.

Fig. 4
figure4

Estimated results of complex theoretical data using the proposed multichannel matching pursuit. a A synthetic 2D seismic profile. The matching atoms (b) and the corresponding residual (c) identified by the 12th iteration, respectively. d The reconstructed seismic profile. e The final residual

Field examples

Furthermore, the proposed directional multichannel matching pursuit is applied to the field data as displayed in Fig. 5a to test the stability and reliability. The data consist of 450 traces with a longitudinal time of 500 ms. The vertical sampling interval is 1 ms. Figure 5b, d shows the matching wavelets identified by the 5th and 30th iteration of the first seismic trace using the proposed multichannel matching pursuit, respectively. Figure 5c, e displays the corresponding residual seismic profiles. Although we only decompose the first seismic trace, we can accurately estimate the optimal atoms of adjacent seismic traces shown in Fig. 5b, d. The matching wavelets between adjacent seismic traces keep better horizon continuity. We can see that the center time of the extracted optimal atoms is different, which coincides with the trend of stratigraphic structure. We can see from Fig. 5d that the proposed algorithm can adaptively break a multichannel search at the location of the pinch out, ensuring the accuracy of reconstructing seismic traces. The algorithm can estimate many matching wavelets by iterating a few times, which can make full use of the lateral continuity of seismic traces.

Fig. 5
figure5

The iterative decomposition process of field data using the proposed multichannel matching pursuit. a A seismic profile extracted from a 3D seismic cube. The matching atoms (b) and the corresponding residual (c) identified by the fifth iteration, respectively. The matching atoms (d) and the corresponding residual (e) identified by the 30th iteration, respectively. f The reconstructed seismic profile. g The final residual

Figures 5f shows the reconstructed field seismic profile obtained by 870 iterations, which accurately resembles the original seismic section shown in Fig. 5a. We can see that the weak seismic signals can also be better identified. The corresponding decomposition residual is displayed in Figs. 5g. The mismatch value is around zero, which demonstrates the high decomposition and reconstruction precision of the proposed multichannel matching pursuit in the field data.

To verify the accuracy of the algorithm for data reconstruction, we randomly extract several seismic traces from the reconstructed section and the original section shown in Fig. 5a, f, respectively. Figure 6a–d displays the decomposition results of the 30th, 110th, 260th and 410th trace (red means reconstruction signal, black means the original signal, green means residual signal) with the proposed multichannel matching pursuit, respectively. We can see from Fig. 6 that the green curve is close to zero and the red curve is highly consistent with the black line, which proves that the reconstruction accuracy of the algorithm is preserved well and the algorithm can be used for signal processing.

Fig. 6
figure6

Reconstruction precision of different trace a the 30th trace; b the 110th trace; c the 260th trace; d the 410th trace (red lines represent reconstructed signals with the proposed matching pursuit, black lines represent original signals, green lines represent residual signals)

And then, the proposed multichannel matching pursuit is applied to generate time–frequency spectrum for the oil and gas reservoir. Spectral decomposition has been widely used in the field of geophysics, which can convert the seismic traces from the time domain to the time–frequency domain. It can display more characteristics of the target reservoir and play an important role in oil and gas exploration. Figure 7a shows the seismic profile across three wells. The black seismic event at about 40 ms is the gas reservoir, and the black axis at 80 ms is the oil reservoir.

Fig. 7
figure7

Time–frequency decomposition to detect reservoir. a A seismic line across three wells. b 30 Hz and c 45 Hz spectral components estimated by proposed multichannel matching pursuit. d 30 Hz and e 45 Hz spectral components estimated by single-channel matching pursuit

Figure 7b, c shows the 30-Hz spectral profiles estimated by the proposed multichannel matching pursuit, respectively. We can see that the spectral components exhibit strong amplitudes at the gas and oil layers, and the effective time–frequency information gradually decreases with the increase in frequency. Figure 7d, e is the frequency profiles of 45-Hz obtained by single-channel matching pursuit. Through comparison, we find that the formation continuity of the proposed matching pursuit is better than the single-channel matching pursuit, which may be due to the fact that the proposed method searches for multichannel optimal atoms along the seismic reflection events. The frequency profiles shown in Fig. 7d, e are difficult for us to distinguish the boundary between layers. The proposed algorithm has significantly improved time and frequency resolution relative to the single-channel matching pursuit. Therefore, the directional multichannel matching pursuit method can achieve better application results in time–frequency analysis.

Conclusion

A novel multichannel matching pursuit is proposed in this study, which could realize the directional multichannel decomposition of seismic traces. The optimal atoms estimated by the proposed multichannel search are not at the same time slicers, which is consistent with the direction of seismic reflection events. The synthetic test and field data case illustrate the feasibility and stability of the directional multichannel matching pursuit. The application of time–frequency analysis contributes to the identification of oil and gas reservoir. The formation continuity and time–frequency resolution of spectrum profile are inferior to the constant spectrum profile generated by single-channel matching pursuit. The comparisons in Fig. 7 illustrate the superiority of our methodology in time–frequency analysis. The application example shows that the proposed multichannel matching pursuit can be better used for geophysical exploration.

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Acknowledgements

We would like to acknowledge the sponsorship of National Grand Project for Science and Technology (2016ZX05024-004), SINOPEC Project (P18026), and Science Foundation from SINOPEC Key Laboratory of Geophysics (wtyjy-wx2017-01-07).

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Correspondence to Xingyao Yin.

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Xu, L., Yin, X. & Li, K. Enhancing the resolution of time–frequency spectrum using directional multichannel matching pursuit. Acta Geophys. 68, 1643–1652 (2020). https://doi.org/10.1007/s11600-020-00490-5

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Keywords

  • Matching pursuit (MP)
  • Reflection event
  • Time–frequency analysis
  • Directional multichannel decomposition
  • Optimal atom