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Low-frequency sparse double-constrained broadband seismic impedance inversion
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ScienceDirect Natural Gas Industry B 6 (2019) 556e563 www.keaipublishing.com/en/journals/natural-gas-industry-b/
Research Article
Low-frequency sparse double-constrained broadband seismic impedance inversion*,** Wen Xiaotao a,b, Yang Jixin a,b,*, Li Leihao a,b, He Jian a,b & Li Bo a,b b
a College of Geophysics, Chengdu University of Technology, Chengdu, Sichuan 610059, China State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Chengdu University of Technology, Chengdu, Sichuan 610059, China
Received 19 January 2019; accepted 25 May 2019 Available online 4 December 2019
Abstract The reflection coefficient solved by using the traditional impedance inversion method cannot reflect low-frequency information sufficiently and its continuity and resolution are limited, so the quality and resolution of impedance inversion are impacted seriously. In this paper, the advantages of rich low-frequency signals were discussed with the theoretical synthetic seismogram and wavelet simulation as the fulcrum. Then, the low-frequency sparse double-constrained reflection coefficient method was introduced to modify the sparse optimization item, and thus a new method was formed. Finally, based on the broadband data and non-broadband data of the actual work area, the reflection coefficient and impedance inversion solved by the traditional basis-pursuit reflection coefficient inversion were compared and then the calculation results of low-frequency sparse double-constrained reflection coefficient inversion were compared to verify the effect of the modified method. And the following research results were obtained. First, the broadband data with rich low-frequency information is less affected by the side lobe, and its seismic data resolution is higher, which is more favorable for the improvement of inversion accuracy and resolution. Second, in the new method, the L2 norm low-frequency model is added on the basis of the BPDN basis-pursuit denoising problem to constrain the residual, so as to realize the direct solution of the reflection coefficient with low-frequency information. Third, the reflection coefficient and wave impedance solved by using the new method have better continuity and resolution than those solved by using the traditional method and they are in good agreement with the well data. In conclusion, the new method achieves higher resolution on the impedance inversion of broadband data and non-broadband data and the accuracy of impedance inversion is increased, so it has higher application values in predicting the distribution of thin reservoirs. © 2019 Sichuan Petroleum Administration. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Compressed sensing; Low-frequency information; Impedance inversion; Broadband; Low-frequency constraint; Reflection coefficient; High resolution; South China Sea area
0. Introduction
*
Project supported by the National Natural Science Foundation of China “Fluid Identification and Fluid Mobility Prediction Based on FrequencyVariable Information” (No. 41774142) and “Extraction of FrequencyVariable Information and Its Application in Reservoir Prediction” (No. 4167040416). ** This is the English version of the originally published article in Natural Gas Industry (in Chinese), which can be found at https://doi.org/10.3787/j.issn. 1000-0976.2019.05.005. * Corresponding author. E-mail address: [email protected] (Yang JX). Peer review under responsibility of Sichuan Petroleum Administration.
Seismic low-frequency information can effectively broaden the reflected wave band, so the inversion accuracy of rich lowfrequency signals can be effectively improved [1e5]. Furthermore, the use of low-frequency information as background constraint for a model can obtain a relative reflection coefficient with higher continuity and resolution than the traditional inversion method, and the weave impedance derived from low-frequency sparse double-constrained reflection coefficient has a higher resolution.
https://doi.org/10.1016/j.ngib.2019.05.003 2352-8540/© 2019 Sichuan Petroleum Administration. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Wen XT et al. / Natural Gas Industry B 6 (2019) 556e563
Traditional wave impedance inversion methods usually invert the relative reflection coefficient through basis pursuit [6,7] or matching pursuit [8e13], and then uses the trace integral to obtain the wave impedance profile. Since they are only based on the original seismic records without making full use of the low-frequency information of seismic data, the reflection coefficient obtained cannot reflect sufficient lowfrequency information, with limited continuity and resolution. In this paper, based on the Basis Pursuit De-Noising (BPDN), the sparse optimization term was improved using the low-frequency sparse double-constrained inversion, and the residual term was added to the L2 norm low-frequency model, to realize the direct solution of reflection coefficient with low-frequency information. In recent years, offshore oil and gas resources have gradually become the focus of exploration [14e17]. The improved method was applied to the analysis of seismic data in an area of the South China Sea. Compared with the traditional basis-pursuit method, the lowfrequency sparse double-constrained method proposed is more effective in wave impedance inversion of broadband data and non-broadband data, with higher resolution and accuracy. 1. Analysis of low-frequency seismic signals According to the theory of seismic convolution model, the amplitude of seismic data is convoluted from reflection coefficient and wavelet [18]. Here, the convolution model is used to analyze the low-frequency absence in seismic data and illustrate the advantages of rich low-frequency signals. 1.1. Wavelet A seismic signal is convoluted from wavelet, and wavelet band has a great influence on seismic signal. A broadband signal is obtained by using the low-frequency broadening method to analyze the low-frequency characteristics of wavelet. Statistical wavelets are extracted from actual seismic traces, and then treated for low-frequency broadening. As shown in Fig. 1-a, the spectrum of broadband signals is higher in the low-frequency part than that of non-broadband signals. In Fig. 1-b, the main lobe amplitude of non-broadband wavelets is weaker than that of broadband wavelets, and the side lobes on both sides are weakened obviously after frequency broadening. Thus, rich low-frequency information can help strengthen the main lobes and weaken the side lobes, and
557
enhance the main lobes of seismic signals and suppress the side lobes, so as to improve the inversion accuracy. 1.2. Convolution model The spectrum of seismic signals is affected by the wavelet spectrum. The seismic signal with rich low-frequency should have weaker side lobe and stronger main lobe in theory. Based on the simple regular convolution model and the complex irregular convolution model, the characteristics of lowfrequency absence and rich low-frequency were further demonstrated, and the performance of broadband in the resolution of geological model and seismic profile was illustrated using a simple fault convolution model. Fig. 2 shows a simple regular convolution model, which consists of 30 Hz Ricker wavelet convoluted with different reflection coefficients at different intervals and amplitudes. The model is simple and has regular distribution. The lowfrequency absence in the model shows that the lowfrequency part of the spectrum increases after the frequency broadening, which is higher than that before frequency broadening. Therefore, the original data is low-frequency absence compared with the broadband data after frequency broadening, while the data after frequency broadening show a lot of low-frequency parts in the spectrum. The main lobe amplitude of seismic wavelet enhances and side lobe amplitude decreases in the seismic signal. Fig. 2a shows the spectrum comparison of broadband and non-broadband in the simple regular convolution model. In order to analyze the influence of low-frequency information, it is only necessary to process the low-frequency part by frequency broadening. Fig. 2b shows the variation trend of main and side lobes in the simple convolution model, which is instructive for complex irregular model and actual seismic data. Fig. 3 shows a complex irregular convolution model, which is processed by frequency broadening and compared with lowfrequency absence features. The abundance of low-frequency information can affect the strength of main and side lobes in seismic signals, thus affecting the resolution and quality of seismic data. Although the model shown in Fig. 3 is relatively complex and distributed randomly, the strong amplitude of main lobe strengthening and the weak amplitude of side lobe weakening can still be seen after frequency broadening. The feature analysis of low-frequency information absence in simple regular and complex irregular convolution models
Fig. 1. Spectrograms of broadband wavelet and non-broadband wavelet.
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broadband fault model in Fig. 4b was obtained by convoluting the reflection coefficient of the fault (Fig. 4a) with nonbroadband wavelet. The broadband fault model in Fig. 4c was obtained by using the convolution of broadband wavelets. From the double-constrained synthetic seismograms in Fig. 4b and c, it can be seen that in the non-broadband double-constrained synthetic seismograms, the overall resolution of nonbroadband recordings is lower than that of broadband recordings due to the influence of strong side lobe amplitude and the weak main lobe amplitude. In broadband doubleconstrained synthetic seismograms, because the side lobe amplitude of wavelet becomes weaker and the main lobe amplitude becomes stronger, the recording resolution is improved, which is helpful to identify small faults. In summary, through numerical simulation using the wavelet and single-trace convolution model, as well as the fault convolution model, the broadband signals have more abundant low-frequency information for conventional signals. In wavelet and single-trace record, the main lobe amplitude of wavelet is enhanced, the side lobe amplitude is weakened, and the resolution is improved in the synthetic seismogram. 2. Low-frequency sparse double-constrained reflection coefficient method According to the Robinson seismic convolution model [18], each seismic signal is convoluted by seismic wavelet and corresponding reflection coefficient sequence. Therefore, the matrix equation is: Fig. 2. Comparison of broadband and non-broadband simple convolution models.
shows that the advantages of rich low-frequency information and broadband signal are strong main lobes and weak side lobes. The above two convolution models are all single-trace double-constrained synthetic seismograms. Through analysis, the advantages of broadband signals are shown as strong main lobes and weak side lobes. Theoretically, the advantages of broadband in seismic profiles with geological models should be reflected in the improvement of resolution. In order to demonstrate the superiority and high resolution of broadband signals in seismic profiles, based on the convolution model theory, we used the broadband and non-broadband wavelets in Fig. 1b to convolute fault models, as shown in Fig. 4. The non-
S ¼ WR
ð1Þ
where, S is the seismic records, W is the seismic wavelet matrix, and R is the reflection coefficient sequence. a R ¼ ½re ro ¼ Dm ð2Þ b According to the oddeeven decomposition theory of seismic signals, R is decomposed into Formula (2), and its decomposition diagram is shown in Fig. 5.where, D is the oddeeven pole library, m is the sparse solution, a and b are oddeeven pole coefficients. The even-pulse pairs (re) and oddpulse pairs (ro) are two functions of Formula (3), respectively. re ¼ dðtÞ þ dðt nDtÞ ð3Þ ro ¼ dðtÞ dðt nDtÞ
Fig. 3. Comparison of broadband and non-broadband complex convolution models.
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where t denotes the propagation time of seismic wave, ms; n denotes the number of sampling points; Dt denotes the time sampling interval, ms. Substituting Formula (2) into Formula (1), we can obtain: S ¼ WDm
ð4Þ
Depending on the oddeeven decomposability and sparsity of reflection coefficients, Zhang et al. [6,7] introduced L1 norm and L2 norm as sparse constraints on inversion target, and obtained the inversion objective function in the form of basis-pursuit denoising problem, that is, m ¼ argkWDm Sk22 þ lkmk1
ð5Þ
min
Under the constrained inversion of Formula (5), the sparse solution (m) reflecting the sparse degree of reflection coefficient can be obtained. However, the residual term of the constrained optimization problem is only based on the original seismic signal and equally supplements all the band information of seismic signal. Because the medium- and highfrequency information may contain noise, we should not blindly pursue the high-frequency broadening. In view of the broadband characteristics of the existing seismic signals, we should focus on utilizing and developing the low-frequency advantages of broadband, and improving the accuracy and continuity of inversion according to the information of the actual work area [19]. The low-frequency information constructed from well data is used as the constraint reference of residual term. The m is mapped to the R reflection coefficient sequence through D, and then the R is integrally operated and the low-frequency filtered to reflect the low-frequency impedance information. The construction of low-frequency constraints is as follows. For the sampling point at time t, the relationship between Pwave impedance and reflection coefficient is: Fig. 4. Original reflection coefficients and models of faults.
1 Ip ðtÞ ln ¼ 2 Ip ðt0 Þ
Zt RðtÞdt t0
Fig. 5. Oddeeven decomposition diagram of reflection coefficients.
ð6Þ
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where, t0 is the time when seismic waves reach adjacent strata, ms, t0
ð7Þ
where, B denotes the integral matrix and x denotes the relative wave impedance sequence. B is expressed in the following forms: 3 2 1 61 1 7 7 ð8Þ B¼6 5 4« « 1 1 / 1 1 Ip ðtÞ xðtÞ ¼ ln 2 Ip ðt0 Þ
ð9Þ
According to Formula (2), Formula (7) can be rewritten as: BDm ¼ x
ð10Þ
The above is the relationship between inversion parameters and impedance sequence. In low-frequency constrained inversion, the impedance sequence on the right side of the equation needs to be transformed into a low-frequency impedance model derived from well data, and the two sides of Formula (10) need to be multiplied by a low-frequency filter operator L [20]. The construction form is shown in Reference [21]. An equivalent matrix relation is formed between inversion parameter m and the impedance model with lowfrequency: LBDm ¼ xL
ð11Þ
where, xL is an impedance model with low frequency. At this time, the low-frequency constraints are constructed, and then they are added to the target inversion function of BPDN in Formula (5). Thus, the low-frequency sparse double constraints are obtained as follows:
m ¼ argkWDm Sk22 þ lkmk1 þ kmLBDm xL k22
ð12Þ
min
Since the low-frequency constraints are similar to the residual constraints of seismic traces, they can be simplified further into: pffiffiffi pffiffiffi 2 m ¼ arg m L BD þ WD m S þ m x þ lkmk1 min
2
ð13Þ Based on the above formula, the solvable form of convex optimization problem with low-frequency sparse constraints is obtained. The improved low-frequency sparse doubleconstrained convex optimization is solved using compressed perception algorithm, that is, the sparse solution containing low-frequency information can be obtained. 3. Application Fig. 6 is a poststack well-tie seismic profile in an area of the South China Sea. The seismic data in this area are divided into poststack time data with conventional band (Fig. 6a) and poststack time data with broadband (Fig. 6b). Since the progress of acquisition technology has greatly promoted the advantages of low-frequency information of broadband data, the application effects of the traditional basis-pursuit reflection coefficient method and the low-frequency sparse doubleconstrained reflection coefficient method on conventional band data and broadband data were compared. First, compared with conventional data, broadband data can improve the accuracy of reservoir prediction and hydrocarbon detection because of the advantage of low-frequency information, and reflect the improved continuity and resolution on reflection coefficient and impedance profile. Second, the low-frequency sparse double-constrained reflection coefficient method has better low-frequency driving advantages than traditional basispursuit reflection coefficient method because of low-frequency constraints, and has better continuity and resolution in
Fig. 6. Post-stack time profile of an area in the South China Sea.
Wen XT et al. / Natural Gas Industry B 6 (2019) 556e563
reflection coefficient and impedance profile; it is in good agreement with well data. 3.1. Broadband data advantage In order to reflect the advantages of broad-band data, it is necessary to compare the reflection coefficient profiles and impedance profiles obtained by the traditional basis-pursuit method and the low-frequency sparse double-constrained method. Fig. 7a & b illustrate the conventional band and broadband reflection coefficients obtained by the traditional basis-pursuit method. It can be seen that the broadband reflection
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coefficients reflect more abundant layer information and higher continuity, and the overall information are more abundant and continuous. The trace integral of the reflection coefficients in Fig. 7a & b was calculated to determine the conventional band impedance and broadband impedance, as shown in Fig. 8a & b. In the white circle A, the impedance information in Fig. 8b has better continuity and resolution than that in Fig. 8-a, and better matches with well data. Fig. 7c & d illustrate the conventional band and broadband reflection coefficients obtained by the low-frequency sparse double-constrained method. Similarly, the broadband reflection coefficients have richer information and better continuity. Then the seismic impedance profiles are obtained from the
Fig. 7. Reflection coefficient profiles of conventional band and broadband by traditional basis-pursuit method and low-frequency sparse double-constrained method.
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reflection coefficients in Fig. 7c & d. The conventional band impedance and broadband impedance are shown in Fig. 8c & d, respectively. In the white dotted line circles A and C, the broadband impedance has better continuity and more abundant information than the conventional band impedance, and it is in good agreement with well data. 3.2. Advantages of low-frequency sparse doubleconstrained method Fig. 7a & c are the seismic reflection coefficient profiles on conventional seismic data obtained by traditional basis-pursuit method and low-frequency sparse double-constrained method, respectively. It can be seen that the low-frequency sparse double-constrained method derives reflection coefficients with more information, better continuity and slightly higher resolution than the traditional basis-pursuit method. Trace integral was calculated for Fig. 7a & c to obtained the impedance, as
shown in Fig. 8a & c. In the white dotted line circles A and B, the low-frequency sparse double-constrained method derives reflection coefficients with much higher resolution and continuity at wells in Fig. 8c, but its features for low frequency are not exerted due to the lack of low-frequency information in conventional seismic data. Therefore, this method is applied to broadband seismic data with more abundant low-frequency information. Fig. 7b & d illustrate the seismic reflection coefficients on broadband seismic data obtained by the traditional basispursuit method and low-frequency sparse double-constrained method, respectively. It can be seen that the low-frequency sparse double-constrained method derives reflection coefficients with better continuity and higher resolution. Similarly, trace integral was calculated for Fig. 7b & d to obtain the impedance, as shown in Fig. 8b & d. In the white dotted line circles A, B and C, compared with traditional basis-pursuit method, the low-frequency sparse double-constrained
Fig. 8. Seismic impedance profiles of conventional band and broadband by traditional basis-pursuit method and low-frequency sparse double-constrained method.
Wen XT et al. / Natural Gas Industry B 6 (2019) 556e563
method in Fig. 8d shows better continuity, higher resolution and improved impedance information on conventional data, which means that this method can fully reflect the advantages of low-frequency in broadband data, with continuity maintained and also resolution improved, and is in good agreement with well data. 4. Conclusions 1) The lack or absence of low-frequency information was analyzed by frequency broadening, and numerical simulation was conducted using the single-trace synthetic seismograms of wavelet and convolution model and the fault convolution model. Results show that the broadband data with abundant low-frequency information have less side-lobe effect, so the resolution of seismic data is higher, facilitating the inversion accuracy. 2) The traditional basis-pursuit method does not fully reflect the low-frequency characteristics in the sparse expression of convex optimization problem, and the inversion results cannot better reflect the resolution, continuity and abundance of low-frequency information. 3) The low-frequency sparse double-constrained method introduces the low-frequency constraints into the convex optimization sparse problem, thus fully uncovering the low-frequency information in broadband data. This method can derive the impedance with better continuity and resolution than the traditional method, and it is in good agreement with well data. The proposed method can effectively predict thin layer distribution.
Conflicts of interest The authors declare that there is no conflicts of interest. References [1] Soubaras R & Whiting P. Variable depth streamer-The new broadband acquisition system. San Antonio: SEG Technical Program Expanded Abstracts; 2011. [2] Soubaras R & Dowle R. Variable depth streamer-A broadband marine solution. First Break 2010;28(12):89e96.
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[3] Zhao Renyong, Zhang Zhenbo & Xuan Yihua. Application of over/under streamer and over/under source seismic acquisition in the pearl river mouth basin. Oil Geophys Prospect 2011;46(4):517e21. [4] Yu Benshan & Sun Naida. Latest development of marine broadband seismic acquisition technology. Oil Forum 2015;(1):41e4. [5] Zhang Mugang, Luo Fei, Wang Changhui, Tao Zhifei & Liu Jinbao. Industrialization progress of wide-azimuth, broadband, and high-density (WBH) seismic acquisition technique. Nat Gas Ind 2017;37(11):1e8. [6] Zhang Rui, Sen MK & Srinivasasan S. A prestack basis pursuit seismic inversion. Geophysics 2013;78(1):R1e11. [7] Zhang Rui & Castagna J. Seismic sparse-layer reflectivity inversion using basis pursuit decomposition. Geophysics 2011;76(6):R147e58. [8] Mallat SG & Zhang Zhifeng. Matching pursuits with timeefrequency dictionaries. IEEE Trans Signal Process 1993;41(12):3397e415. [9] Nguyen T & Castagna J. Matching pursuit of two dimensional seismic data and its filtering application. San Antonio: SEG Technical Program Expanded Abstracts; 2000. [10] Wang Yanghua. Seismic timeefrequency spectral decomposition by matching pursuit. Geophysics 2007;72(1):13e20. [11] Liu Xiaolong, Liu Tianyou, Wang Hua & Zhao Shu’e. Spectrum imaging technique and its application based on matching pursuit algorithm. Oil Geophys Prospect 2010;45(6):850e5. [12] Yang Hao, Zheng Xiaodong, Ma Shufang & Li Jinsong. Thin-bed reflectivity inversion based on matching pursuit[C]//2011 SEG Annual Meeting. 18e23 September 2011. San Antonio, Texas, USA. [13] Huang Handong, Guo Fei, Wang Jiabei & Ren Dunzhan. High precision seismic timeefrequency spectrum decomposition method and its application. Oil Geophys Prospect 2012;47(5):773e80. [14] Wan Yizhao, Wu Nengyou, Hu Gaowei, Xin Xin, Jin Guangrong, Liu Changling, et al. Reservoir stability in the process of natural gas hydrate production by depressurization in the Shenhu area of the South China Sea. Nat Gas Ind 2018;38(4):117e28. [15] Zhang Wei, Liang Jinqiang, He Jiaxiong, Cong Xiaorong, Su Pibo, Lin Lin, et al. Differences in natural gas hydrate migration and accumulation between GMGS1 and GMGS3 drilling areas in the Shenhu area, northern South China Sea. Nat Gas Ind 2018;38(3):138e49. [16] Zhang Houhe, Liu Peng, Liao Zongbao, He Shuanzhu & Zhu Xiaomin. Oil and gas exploration potential in beikang basin, Nansha Sea area. China Petrol Explor 2017;22(3):40e8. [17] Hu Xinghao. Challenges and measures in OBC seismic survey in Bohai Bay. China Petrol Explor 2017;22(6):112e7. [18] Robinson EA. Predictive decomposition of seismic traces. Geophysics 1957;22(4):767e78. [19] Yin Xingyao, Liu Xiaojing, Wu Guochen & Zong Zhaoyun. Basis pursuit inversion method under model constraint. Geophys Prospect Pet 2016;(1):115e22. [20] Zhang Fengqi, Wei Fuji & Wang Yanchun. Generalized linear AVO inversion with the priori constraint of trivariate Cauchy distribution based on Zoeppritz equation. Chin J Geophys 2013;56(6):2098e115. [21] Zhang Fengqi, Jin Zhijun, Sheng Xiujie & Liu Tao. AVA sparse layer inversion with the soft-low frequency constraint. Oil Geophys Prospect 2017;52(4):770e82.
ScienceDirect Natural Gas Industry B 6 (2019) 556e563 www.keaipublishing.com/en/journals/natural-gas-industry-b/
Research Article
Low-frequency sparse double-constrained broadband seismic impedance inversion*,** Wen Xiaotao a,b, Yang Jixin a,b,*, Li Leihao a,b, He Jian a,b & Li Bo a,b b
a College of Geophysics, Chengdu University of Technology, Chengdu, Sichuan 610059, China State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Chengdu University of Technology, Chengdu, Sichuan 610059, China
Received 19 January 2019; accepted 25 May 2019 Available online 4 December 2019
Abstract The reflection coefficient solved by using the traditional impedance inversion method cannot reflect low-frequency information sufficiently and its continuity and resolution are limited, so the quality and resolution of impedance inversion are impacted seriously. In this paper, the advantages of rich low-frequency signals were discussed with the theoretical synthetic seismogram and wavelet simulation as the fulcrum. Then, the low-frequency sparse double-constrained reflection coefficient method was introduced to modify the sparse optimization item, and thus a new method was formed. Finally, based on the broadband data and non-broadband data of the actual work area, the reflection coefficient and impedance inversion solved by the traditional basis-pursuit reflection coefficient inversion were compared and then the calculation results of low-frequency sparse double-constrained reflection coefficient inversion were compared to verify the effect of the modified method. And the following research results were obtained. First, the broadband data with rich low-frequency information is less affected by the side lobe, and its seismic data resolution is higher, which is more favorable for the improvement of inversion accuracy and resolution. Second, in the new method, the L2 norm low-frequency model is added on the basis of the BPDN basis-pursuit denoising problem to constrain the residual, so as to realize the direct solution of the reflection coefficient with low-frequency information. Third, the reflection coefficient and wave impedance solved by using the new method have better continuity and resolution than those solved by using the traditional method and they are in good agreement with the well data. In conclusion, the new method achieves higher resolution on the impedance inversion of broadband data and non-broadband data and the accuracy of impedance inversion is increased, so it has higher application values in predicting the distribution of thin reservoirs. © 2019 Sichuan Petroleum Administration. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Compressed sensing; Low-frequency information; Impedance inversion; Broadband; Low-frequency constraint; Reflection coefficient; High resolution; South China Sea area
0. Introduction
*
Project supported by the National Natural Science Foundation of China “Fluid Identification and Fluid Mobility Prediction Based on FrequencyVariable Information” (No. 41774142) and “Extraction of FrequencyVariable Information and Its Application in Reservoir Prediction” (No. 4167040416). ** This is the English version of the originally published article in Natural Gas Industry (in Chinese), which can be found at https://doi.org/10.3787/j.issn. 1000-0976.2019.05.005. * Corresponding author. E-mail address: [email protected] (Yang JX). Peer review under responsibility of Sichuan Petroleum Administration.
Seismic low-frequency information can effectively broaden the reflected wave band, so the inversion accuracy of rich lowfrequency signals can be effectively improved [1e5]. Furthermore, the use of low-frequency information as background constraint for a model can obtain a relative reflection coefficient with higher continuity and resolution than the traditional inversion method, and the weave impedance derived from low-frequency sparse double-constrained reflection coefficient has a higher resolution.
https://doi.org/10.1016/j.ngib.2019.05.003 2352-8540/© 2019 Sichuan Petroleum Administration. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Wen XT et al. / Natural Gas Industry B 6 (2019) 556e563
Traditional wave impedance inversion methods usually invert the relative reflection coefficient through basis pursuit [6,7] or matching pursuit [8e13], and then uses the trace integral to obtain the wave impedance profile. Since they are only based on the original seismic records without making full use of the low-frequency information of seismic data, the reflection coefficient obtained cannot reflect sufficient lowfrequency information, with limited continuity and resolution. In this paper, based on the Basis Pursuit De-Noising (BPDN), the sparse optimization term was improved using the low-frequency sparse double-constrained inversion, and the residual term was added to the L2 norm low-frequency model, to realize the direct solution of reflection coefficient with low-frequency information. In recent years, offshore oil and gas resources have gradually become the focus of exploration [14e17]. The improved method was applied to the analysis of seismic data in an area of the South China Sea. Compared with the traditional basis-pursuit method, the lowfrequency sparse double-constrained method proposed is more effective in wave impedance inversion of broadband data and non-broadband data, with higher resolution and accuracy. 1. Analysis of low-frequency seismic signals According to the theory of seismic convolution model, the amplitude of seismic data is convoluted from reflection coefficient and wavelet [18]. Here, the convolution model is used to analyze the low-frequency absence in seismic data and illustrate the advantages of rich low-frequency signals. 1.1. Wavelet A seismic signal is convoluted from wavelet, and wavelet band has a great influence on seismic signal. A broadband signal is obtained by using the low-frequency broadening method to analyze the low-frequency characteristics of wavelet. Statistical wavelets are extracted from actual seismic traces, and then treated for low-frequency broadening. As shown in Fig. 1-a, the spectrum of broadband signals is higher in the low-frequency part than that of non-broadband signals. In Fig. 1-b, the main lobe amplitude of non-broadband wavelets is weaker than that of broadband wavelets, and the side lobes on both sides are weakened obviously after frequency broadening. Thus, rich low-frequency information can help strengthen the main lobes and weaken the side lobes, and
557
enhance the main lobes of seismic signals and suppress the side lobes, so as to improve the inversion accuracy. 1.2. Convolution model The spectrum of seismic signals is affected by the wavelet spectrum. The seismic signal with rich low-frequency should have weaker side lobe and stronger main lobe in theory. Based on the simple regular convolution model and the complex irregular convolution model, the characteristics of lowfrequency absence and rich low-frequency were further demonstrated, and the performance of broadband in the resolution of geological model and seismic profile was illustrated using a simple fault convolution model. Fig. 2 shows a simple regular convolution model, which consists of 30 Hz Ricker wavelet convoluted with different reflection coefficients at different intervals and amplitudes. The model is simple and has regular distribution. The lowfrequency absence in the model shows that the lowfrequency part of the spectrum increases after the frequency broadening, which is higher than that before frequency broadening. Therefore, the original data is low-frequency absence compared with the broadband data after frequency broadening, while the data after frequency broadening show a lot of low-frequency parts in the spectrum. The main lobe amplitude of seismic wavelet enhances and side lobe amplitude decreases in the seismic signal. Fig. 2a shows the spectrum comparison of broadband and non-broadband in the simple regular convolution model. In order to analyze the influence of low-frequency information, it is only necessary to process the low-frequency part by frequency broadening. Fig. 2b shows the variation trend of main and side lobes in the simple convolution model, which is instructive for complex irregular model and actual seismic data. Fig. 3 shows a complex irregular convolution model, which is processed by frequency broadening and compared with lowfrequency absence features. The abundance of low-frequency information can affect the strength of main and side lobes in seismic signals, thus affecting the resolution and quality of seismic data. Although the model shown in Fig. 3 is relatively complex and distributed randomly, the strong amplitude of main lobe strengthening and the weak amplitude of side lobe weakening can still be seen after frequency broadening. The feature analysis of low-frequency information absence in simple regular and complex irregular convolution models
Fig. 1. Spectrograms of broadband wavelet and non-broadband wavelet.
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broadband fault model in Fig. 4b was obtained by convoluting the reflection coefficient of the fault (Fig. 4a) with nonbroadband wavelet. The broadband fault model in Fig. 4c was obtained by using the convolution of broadband wavelets. From the double-constrained synthetic seismograms in Fig. 4b and c, it can be seen that in the non-broadband double-constrained synthetic seismograms, the overall resolution of nonbroadband recordings is lower than that of broadband recordings due to the influence of strong side lobe amplitude and the weak main lobe amplitude. In broadband doubleconstrained synthetic seismograms, because the side lobe amplitude of wavelet becomes weaker and the main lobe amplitude becomes stronger, the recording resolution is improved, which is helpful to identify small faults. In summary, through numerical simulation using the wavelet and single-trace convolution model, as well as the fault convolution model, the broadband signals have more abundant low-frequency information for conventional signals. In wavelet and single-trace record, the main lobe amplitude of wavelet is enhanced, the side lobe amplitude is weakened, and the resolution is improved in the synthetic seismogram. 2. Low-frequency sparse double-constrained reflection coefficient method According to the Robinson seismic convolution model [18], each seismic signal is convoluted by seismic wavelet and corresponding reflection coefficient sequence. Therefore, the matrix equation is: Fig. 2. Comparison of broadband and non-broadband simple convolution models.
shows that the advantages of rich low-frequency information and broadband signal are strong main lobes and weak side lobes. The above two convolution models are all single-trace double-constrained synthetic seismograms. Through analysis, the advantages of broadband signals are shown as strong main lobes and weak side lobes. Theoretically, the advantages of broadband in seismic profiles with geological models should be reflected in the improvement of resolution. In order to demonstrate the superiority and high resolution of broadband signals in seismic profiles, based on the convolution model theory, we used the broadband and non-broadband wavelets in Fig. 1b to convolute fault models, as shown in Fig. 4. The non-
S ¼ WR
ð1Þ
where, S is the seismic records, W is the seismic wavelet matrix, and R is the reflection coefficient sequence. a R ¼ ½re ro ¼ Dm ð2Þ b According to the oddeeven decomposition theory of seismic signals, R is decomposed into Formula (2), and its decomposition diagram is shown in Fig. 5.where, D is the oddeeven pole library, m is the sparse solution, a and b are oddeeven pole coefficients. The even-pulse pairs (re) and oddpulse pairs (ro) are two functions of Formula (3), respectively. re ¼ dðtÞ þ dðt nDtÞ ð3Þ ro ¼ dðtÞ dðt nDtÞ
Fig. 3. Comparison of broadband and non-broadband complex convolution models.
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where t denotes the propagation time of seismic wave, ms; n denotes the number of sampling points; Dt denotes the time sampling interval, ms. Substituting Formula (2) into Formula (1), we can obtain: S ¼ WDm
ð4Þ
Depending on the oddeeven decomposability and sparsity of reflection coefficients, Zhang et al. [6,7] introduced L1 norm and L2 norm as sparse constraints on inversion target, and obtained the inversion objective function in the form of basis-pursuit denoising problem, that is, m ¼ argkWDm Sk22 þ lkmk1
ð5Þ
min
Under the constrained inversion of Formula (5), the sparse solution (m) reflecting the sparse degree of reflection coefficient can be obtained. However, the residual term of the constrained optimization problem is only based on the original seismic signal and equally supplements all the band information of seismic signal. Because the medium- and highfrequency information may contain noise, we should not blindly pursue the high-frequency broadening. In view of the broadband characteristics of the existing seismic signals, we should focus on utilizing and developing the low-frequency advantages of broadband, and improving the accuracy and continuity of inversion according to the information of the actual work area [19]. The low-frequency information constructed from well data is used as the constraint reference of residual term. The m is mapped to the R reflection coefficient sequence through D, and then the R is integrally operated and the low-frequency filtered to reflect the low-frequency impedance information. The construction of low-frequency constraints is as follows. For the sampling point at time t, the relationship between Pwave impedance and reflection coefficient is: Fig. 4. Original reflection coefficients and models of faults.
1 Ip ðtÞ ln ¼ 2 Ip ðt0 Þ
Zt RðtÞdt t0
Fig. 5. Oddeeven decomposition diagram of reflection coefficients.
ð6Þ
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where, t0 is the time when seismic waves reach adjacent strata, ms, t0
ð7Þ
where, B denotes the integral matrix and x denotes the relative wave impedance sequence. B is expressed in the following forms: 3 2 1 61 1 7 7 ð8Þ B¼6 5 4« « 1 1 / 1 1 Ip ðtÞ xðtÞ ¼ ln 2 Ip ðt0 Þ
ð9Þ
According to Formula (2), Formula (7) can be rewritten as: BDm ¼ x
ð10Þ
The above is the relationship between inversion parameters and impedance sequence. In low-frequency constrained inversion, the impedance sequence on the right side of the equation needs to be transformed into a low-frequency impedance model derived from well data, and the two sides of Formula (10) need to be multiplied by a low-frequency filter operator L [20]. The construction form is shown in Reference [21]. An equivalent matrix relation is formed between inversion parameter m and the impedance model with lowfrequency: LBDm ¼ xL
ð11Þ
where, xL is an impedance model with low frequency. At this time, the low-frequency constraints are constructed, and then they are added to the target inversion function of BPDN in Formula (5). Thus, the low-frequency sparse double constraints are obtained as follows:
m ¼ argkWDm Sk22 þ lkmk1 þ kmLBDm xL k22
ð12Þ
min
Since the low-frequency constraints are similar to the residual constraints of seismic traces, they can be simplified further into: pffiffiffi pffiffiffi 2 m ¼ arg m L BD þ WD m S þ m x þ lkmk1 min
2
ð13Þ Based on the above formula, the solvable form of convex optimization problem with low-frequency sparse constraints is obtained. The improved low-frequency sparse doubleconstrained convex optimization is solved using compressed perception algorithm, that is, the sparse solution containing low-frequency information can be obtained. 3. Application Fig. 6 is a poststack well-tie seismic profile in an area of the South China Sea. The seismic data in this area are divided into poststack time data with conventional band (Fig. 6a) and poststack time data with broadband (Fig. 6b). Since the progress of acquisition technology has greatly promoted the advantages of low-frequency information of broadband data, the application effects of the traditional basis-pursuit reflection coefficient method and the low-frequency sparse doubleconstrained reflection coefficient method on conventional band data and broadband data were compared. First, compared with conventional data, broadband data can improve the accuracy of reservoir prediction and hydrocarbon detection because of the advantage of low-frequency information, and reflect the improved continuity and resolution on reflection coefficient and impedance profile. Second, the low-frequency sparse double-constrained reflection coefficient method has better low-frequency driving advantages than traditional basispursuit reflection coefficient method because of low-frequency constraints, and has better continuity and resolution in
Fig. 6. Post-stack time profile of an area in the South China Sea.
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reflection coefficient and impedance profile; it is in good agreement with well data. 3.1. Broadband data advantage In order to reflect the advantages of broad-band data, it is necessary to compare the reflection coefficient profiles and impedance profiles obtained by the traditional basis-pursuit method and the low-frequency sparse double-constrained method. Fig. 7a & b illustrate the conventional band and broadband reflection coefficients obtained by the traditional basis-pursuit method. It can be seen that the broadband reflection
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coefficients reflect more abundant layer information and higher continuity, and the overall information are more abundant and continuous. The trace integral of the reflection coefficients in Fig. 7a & b was calculated to determine the conventional band impedance and broadband impedance, as shown in Fig. 8a & b. In the white circle A, the impedance information in Fig. 8b has better continuity and resolution than that in Fig. 8-a, and better matches with well data. Fig. 7c & d illustrate the conventional band and broadband reflection coefficients obtained by the low-frequency sparse double-constrained method. Similarly, the broadband reflection coefficients have richer information and better continuity. Then the seismic impedance profiles are obtained from the
Fig. 7. Reflection coefficient profiles of conventional band and broadband by traditional basis-pursuit method and low-frequency sparse double-constrained method.
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reflection coefficients in Fig. 7c & d. The conventional band impedance and broadband impedance are shown in Fig. 8c & d, respectively. In the white dotted line circles A and C, the broadband impedance has better continuity and more abundant information than the conventional band impedance, and it is in good agreement with well data. 3.2. Advantages of low-frequency sparse doubleconstrained method Fig. 7a & c are the seismic reflection coefficient profiles on conventional seismic data obtained by traditional basis-pursuit method and low-frequency sparse double-constrained method, respectively. It can be seen that the low-frequency sparse double-constrained method derives reflection coefficients with more information, better continuity and slightly higher resolution than the traditional basis-pursuit method. Trace integral was calculated for Fig. 7a & c to obtained the impedance, as
shown in Fig. 8a & c. In the white dotted line circles A and B, the low-frequency sparse double-constrained method derives reflection coefficients with much higher resolution and continuity at wells in Fig. 8c, but its features for low frequency are not exerted due to the lack of low-frequency information in conventional seismic data. Therefore, this method is applied to broadband seismic data with more abundant low-frequency information. Fig. 7b & d illustrate the seismic reflection coefficients on broadband seismic data obtained by the traditional basispursuit method and low-frequency sparse double-constrained method, respectively. It can be seen that the low-frequency sparse double-constrained method derives reflection coefficients with better continuity and higher resolution. Similarly, trace integral was calculated for Fig. 7b & d to obtain the impedance, as shown in Fig. 8b & d. In the white dotted line circles A, B and C, compared with traditional basis-pursuit method, the low-frequency sparse double-constrained
Fig. 8. Seismic impedance profiles of conventional band and broadband by traditional basis-pursuit method and low-frequency sparse double-constrained method.
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method in Fig. 8d shows better continuity, higher resolution and improved impedance information on conventional data, which means that this method can fully reflect the advantages of low-frequency in broadband data, with continuity maintained and also resolution improved, and is in good agreement with well data. 4. Conclusions 1) The lack or absence of low-frequency information was analyzed by frequency broadening, and numerical simulation was conducted using the single-trace synthetic seismograms of wavelet and convolution model and the fault convolution model. Results show that the broadband data with abundant low-frequency information have less side-lobe effect, so the resolution of seismic data is higher, facilitating the inversion accuracy. 2) The traditional basis-pursuit method does not fully reflect the low-frequency characteristics in the sparse expression of convex optimization problem, and the inversion results cannot better reflect the resolution, continuity and abundance of low-frequency information. 3) The low-frequency sparse double-constrained method introduces the low-frequency constraints into the convex optimization sparse problem, thus fully uncovering the low-frequency information in broadband data. This method can derive the impedance with better continuity and resolution than the traditional method, and it is in good agreement with well data. The proposed method can effectively predict thin layer distribution.
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