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H. J. He (2010)
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Mark Lulettgen, William Ka, A. Willsky, R. Tenney (1993)
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Jianlei Liu, K. Marfurt (2005)
Matching Pursuit Decomposition Using Morlet WaveletsSeg Technical Program Expanded Abstracts
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Yanghua Wang (2010)
Multichannel matching pursuit for seismic trace decompositionGeophysics, 75
Thang Nguyen, J. Castagna (2000)
Matching Pursuit of Two Dimensional Seismic Data And Its Filtering ApplicationSeg Technical Program Expanded Abstracts
Liu Qingyun (2004)
Time-frequency Analysis and Its Present SituationComputer Engineering
F. Zhang, Y. Y. Zhong, X. Y. Zhu (2006)
Application of time-frequency analysis to seismic wave spectrum researchSeismological and Geomagnetic Observation and Research, 27
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Jianlei Liu, Yafei Wu, D. Han, Xin‐Gong Li (2004)
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Guochang Liu, Sergey Fomel, Xiaohong Chen (2009)
Time-frequency characterization of seismic data using local attributesSeg Technical Program Expanded Abstracts, 28
Guochang Liu, Sergey Fomel, Xiaohong Chen (2011)
Time-frequency analysis of seismic data using local attributesGeophysics, 76
Hongcheng Fan, Qingfeng Meng, Youyun Zhang, Q. Gao, Fengni Wang (2009)
Matching pursuit based on nonparametric waveform estimationDigit. Signal Process., 19
Avijit Chakraborty, D. Okaya (1995)
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J. P. Castagna, S. J. Sun, R. W. Siegfried (2003)
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X. L. Du (2006)
409Nondestructive Testing, 28
Yang Sheng-li (2010)
Seismic Signal's MP Decomposition Based on Ricker WaveletComputer Knowledge and Technology
Yanghua Wang (2007)
Seismic time-frequency spectral decomposition by matching pursuitGeophysics, 72
Shen Yi (2006)
Denoising of Ultrasonic Testing Signals by Matching PursuitsNondestructive Testing
J. P. Castagna (2003)
120The Leading Edge, 22
F. C. Zhang, C. H. Li, X. Y. Yin (2010)
Seismic data fast matching pursuit based on dynamic matching wavelet libraryOil Geophysical Prospecting, 45
W. Zou, A. P. Chen, H. M. Gu (2004)
Joint time-frequency analysis and its application in seismic prospectingProgress in Exploration Geophysics, 27
effective tool for non-stationary signals, and has high time-frequency resolution and a transient structure with local self-adaption. We expand the time-frequency dictionary library with Ricker, Morlet, and mixed phase seismic wavelets, to make the method more suitable for seismic signal time-frequency decomposition. In this paper, we demonstrated the algorithm theory using synthetic seismic data, and tested the method using synthetic data with 25% noise. We compared the matching pursuit results of the time-frequency dictionaries. The results indicated that the dictionary which matched the signal characteristics better would obtain better results, and can reflect the information of seismic data effectively. Matching pursuit, seismic attenuation, wavelet transform, Wigner Ville distribution, time- Key words: frequency dictionary a transient structure with local self-adaption, and is used in 1 Introduction many areas (Du and Shen, 2006; He et al, 2010; Fan et al, Time-frequency analysis of seismic data is very important 2009; Wang, 2007; 2010; Zhang et al, 2010). It has the ability in processing and interpretation, such as measurements to extract more signal characteristics and is not affected by of seismic attenuation, detection of hydrocarbon, and the noise in signals. It can overcome the shortcomings of improvement of resolution. The short time Fourier transform traditional Fourier transform, windowed Fourier transform, (STFT) is used widely in time-frequency analysis, but due wavelet transform and S transform (Li, 2006; Liu et al, 2004b; to the limitation of Heisenberg’s uncertainty principle, it is Xu, 2000; Zhang et al, 2006; Zou et al, 2004). In the seismic difficult to consider the needs of both time and frequency exploration, the time and frequency dictionaries of matching resolution. Liu et al (2011) proposed the local attributes pursuit are limited. Castagna et al used matching pursuit to time-frequency analysis method by shaping regularization process 2D seismic signals, and further used the method to on the basis of the conventional Fourier transform, and detect low-frequency shadows associated with hydrocarbons applied it successfully to detecting low-frequency shadows (Castagna et al, 2003; Nguyen and Castagna, 2000). Liu et al and ancient river sediments. Wavelet transform is also used used matching pursuit based on Morlet and Ricker wavelet widely in time-frequency analysis. Chakraborty and Okaya dictionaries to do time-frequency decomposition (Liu et (1995) compared the difference between wavelet transform al, 2004a; Liu and Marfurt, 2005). Wang and Yang (2010) and Fourier transform in time-frequency analysis, and they used the improved Ricker wavelet dictionary to sparsely pointed out that wavelet transform can improve the spectral decompose the actual seismic signals. However, most of resolution. Stockwell et al (1996) proposed the S transform. the actual seismic wavelets are mixed-phase, and matching This method can overcome the shortcomings of STFT, at the pursuit needs to scan the whole time and frequency domains same time it introduced multi-resolution analysis of wavelet of the signal, so the atom dictionary must be over-complete in transform. It has good time-frequency analysis capability, but order to decompose and reconstruct the signal best. Therefore, LWLVOLPLWHGLQSUDFWLFDODSSOLFDWLRQVGXHWRWKH¿[HGIRUPRI we should add some new dictionaries which are mixed-phase the basic wavelet function. to match the time and frequency characteristics of seismic Matching pursuit has high time-frequency resolution and signals better. 2 Matching pursuit * Corresponding author. email: 13626080@qq.com Received March 24, 2011 The principle of matching pursuit is to decompose any Pet.Sci.(2012)9:310-316 311 signal into a linear expansion of waveforms that belong to characteristics. a redundant dictionary of functions. These waveforms can Here, the function of seismic Ricker wavelet in ʌ gt match the signal structure best. The signal is decomposed into time domain is Rt ʌ gt ( g is the peak waveforms selected from a time-frequency atom dictionary, frequency), and the Wigner Ville distribution of it is which is obtained by extension, translation, and modulation ʌ gt of signal window functions. By adding the Wigner ʌ g Wt Z ʌ ʌʌ g t ʌ gt distribution to the selected atom dictionary, we can obtain (2) a time-frequency energy distribution. Because there are no ʌ ʌ ZZZ ʌ ʌ t interference terms in the distribution, we can obtain a clear picture in the time-frequency plane. ʌ gg ʌ ʌʌ g g The L (R) is the Hilbert space of complex valued function. The time-frequency atom dictionary is built by extending, The function of the Morlet wavelet is Z t translating, and modulating the signal window function it Z 2 I t ʌ , Z is the angular frequency, and such as g () t LR ( ). We suppose g(t) is real, continuously the Wigner Ville distribution of it is differentiable and 2 () , and suppose . So g 1 t 1 Z ZZ integration of g(t) is nonzero, and g (0)z 0. For any scale s t Z Z ZZ Wt ZZ t ZKLFKLVPRUHWKDQ]HURIUHTXHQF\PRGXODWLQJFRHI¿FLHQW ȟ DQGWUDQVODWLQJFRHI¿FLHQW uZHGH¿QH J (, su,[ ) , and let (3) The function of mixed-phase seismic wavelet (MSW) (Gao tu it[ and Yang, 2007) can change the phase and amplitude of the gt g (1) wavelet through regulating the four parameters so as to match the seismic signal. Most of the deconvolution processing We select an approximately countable subset of atoms is based on the seismic wavelet being time-invariant and [ gt ( )] J nN , because the time-frequency atom dictionary is its phase being zero. Actually the seismic wavelet is time- complete, matching pursuit will decompose every function variant, and most are mixed phase wavelets. So, using MSW ZKLFKVDWLV¿HV f () t LR ( ). After m iterations, a matching to build a mixed phase time-frequency dictionary is very pursuit decomposes the signal f. The number of iterations important for practical applications. The wavelet function is is related to the decay rate of , and the norm of the WT t ( A,,WT , f ) stA ʌ ft , where parameters remainder decays exponentially. When the value of is DUHDPSOLWXGHFRHI¿FLHQWDWWHQXDWLRQIDFWRUSKDVHIDFWRUDQG less than the set accuracy, the decomposition will stop. peak frequency of the wavelet. Its Wigner Ville distribution is The Wigner distribution of f (t) is Wf (, t ZZ ) W [ f , f ](, t ). W tt TT Wt Z A ʌ ft Mm Eft ( ,ZZ )dtd f W We can obtain by the energy (4) ³³ ʌ f iZ m conservation. So Ef (, t Z ) can be considered as the energy ʌ f density of f in time-frequency plane (t, Ȧ ), and there are no crossing terms. The time-frequency energy distribution In these equations, t is time of the atom, and Z is frequency of the atom. We take the MSW dictionary as Ef (, t Z ) is a sum of single g(t) energy distribution (Mallat, an example, and decompose the signal which is generated 1993). by convoluting the mixed phase wavelet and reflection 3 Multi-wavelet time-frequency dictionary coefficient using the Gabor, Morlet and Ricker wavelet dictionaries and MSW dictionary respectively. Fig. 1(a) is construction the signal, (b), (c), (d) and (e) are time-frequency spectrums Though matching pursuit has high resolution of time which are the decomposition results of the signal using the and frequency, it also has different time and frequency Gabor, Morlet and Ricker wavelet dictionaries, and MSW characteristics in the processing of different signals. So the dictionary respectively. decomposition results will be more accurate when we select Because the signal has the same time-frequency the dictionary which best matches the signal. Based on the characteristics as the MSW dictionary, so the best Gabor dictionary, we select several wavelet functions to decomposition result should be achieved using the MSW expand the dictionary. These wavelet functions are selected dictionary in theory. Comparing (b), (c), (d) and (e), we can according to the characteristics of seismic signal, so they VHHWKDWDOORIWKHPFDQDFFXUDWHO\UHÀHFWWKHWLPHORFDWLRQV have the same time and frequency characteristics as the of signal components. (e) has the most concentrated time- seismic signal, and then the new dictionary can match frequency energy and the highest time resolution. The second the seismic signal better. We can use the Wigner Ville is (d), and its energy distribution is poorer than (e) from 500 distribution function of the wavelet to build a multi-wavelet ms to 600 ms. (b) and (c) have poorer time-frequency energy time-frequency dictionary. It can further improve the time- distribution and lower time resolution than (d) and (e). After frequency resolution of the signal, and the decomposed analysis, we conclude that in decomposition of the signal (a) atoms can maximally retain the original time-frequency we should select the MSW dictionary for the best result. f f f f fR fR 312 Pet.Sci.(2012)9:310-316 -1 -2 -3 0 100 200 300 400 500 600 700 800 900 1000 Time, ms (a) Original signal 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 900 1000 Time, ms (b) Gabor wavelet dictionary 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 900 1000 Time, ms (c) Morlet wavelet dictionary 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 900 1000 Time, ms (d) Ricker wavelet dictionary 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 900 1000 Time, ms (e) MSW dictionary Fig. 1 Comparison of time-frequency spectrums using different dictionaries. (a) synthetic seismic signal using mixed phase wavelet, (b) decomposition result using Gabor wavelet dictionary, (c) decomposition result using Morlet wavelet dictionary, (d) decomposition result using Ricker wavelet dictionary, (e) decomposition result using MSW dictionary Fig. 2 is the processing result of a simulated signal the reconstructed signal (blue) and original signal without using this method. (a) is the simulated signal which is noise (red). Their cross-correlation value is 0.86, and the -5 generated by mixed phase wavelet and its S/N value is average residual of every point is 1.8×10 . That is to say, two. We use the MSW dictionary to decompose this noisy the reconstructed signal through decomposition by the signal and choose first ten atoms which have stronger 06:GLFWLRQDU\FDQHIIHFWLYHO\UHÀHFWWKHRULJLQDOVLJQDO energy to reconstruct the signal. (b) is the comparison of without noise. 0.5 -0.5 -1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 (a) Time, ms Reconstructed signal (blue) 0.5 Original signal without noise (red) -0.5 -1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 (b) Time, ms Fig. 2 (a) The original signal which is generated by mixed phase wavelet (S/N ratio is 2), (b) comparison of the original signal without noise (red) and reconstructed signal using the MSW dictionary (blue) Frequency, Hz Frequency, Hz Frequency, Hz Frequency, Hz Amplitude Amplitude Amplitude Pet.Sci.(2012)9:310-316 313 the quality factor increases to 30 evenly. Both ends are the 4 Applications tight sandstone, and the quality factor is 150. The upper and In order to verify the applicability of this method in lower formations are mudstone, the velocity is 2400 m/s, analysis of seismic wave attenuation, we built a seismic and the quality factor is 150. We add 25% Gaussian random record model which is generated by the mixed phase wavelet noise in the model (the noise frequency band is 0-120 Hz). (Fig. 3). The reflection layer F is equivalent to the reflection near There is a sand formation in this model, whose thickness the bottom of the sand, and there are 50 seismic traces. No is 10 m, and velocity is 3200 m/s. The quality factor of the denoising method is used in calculating the seismic attributes traces from 11 to 25 which indicate the gas-containing sand of the model. body varies from 30 to 5 evenly. From trace 25 to trace 40, 0 5 10 15 20 25 30 35 40 45 50 CDP trace number Fig. 3 Synthetic seismic record with a sand formation about 10 m thick CMP )LJD E DQGF DUHWLPHIUHTXHQF\HQHUJ\SUR¿OHV 0 5 10 15 20 25 30 35 40 45 obtained from matching pursuit decomposition using the Morlet wavelet, Ricker wavelet, and MSW dictionaries of the signal in Fig. 3 respectively. From Fig. 4(a) and (b), we can see that the overall signal energy changes irregularly because the time-frequency dictionary does not match the wavelet RIWKHVHLVPLFGDWDDQGWKHLQÀXHQFHRIQRLVHLVVLJQL¿FDQW The time resolution is low, and the sand layer in Figs. 4(a) and (b) is thicker than that in the true model. It is difficult to recognize the actual information of the sand layer. From Fig. 4(c), we can see that the top and bottom interfaces of (b) CMP the sand layer are clear, and the signal energy decays near 0 5 10 15 20 25 30 35 40 45 WKHUHÀHFWLRQOD\HU)ZKLFKLQGLFDWHVJDVFRQWDLQLQJ8VLQJ the matching pursuit method, we can obtain a high time- CMP 0 5 10 15 20 25 30 35 40 45 (c) Fig. 4 (a) 0DWFKLQJSXUVXLWWLPHIUHTXHQF\J\HQHUSUR¿OHXVLQJWKH0RUOHW ZDYHOHWE GLFWLRQDU\PDWFKLQJSXUVXLWWLPHIUHTXHQF\J\HQHUSUR¿OHXVLQJ the Ricker wavelet dictionary, (c) matching pursuit time-frequency energy (a) GLFWLRQDU\SUR¿OHXVLQJWKH06: Time, ms 500 400 300 200 100 0 Time, ms 0 0.2 0.4 0.6 0.8 1 Time, ms Time, ms 500 400 300 200 100 0 500 400 300 200 100 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 314 Pet.Sci.(2012)9:310-316 IUHTXHQF\UHVROXWLRQDQGWKHVDQGOD\HULQ)LJF UHÀHFWV Blue: MSW 1.9 Green: Ricker its actual position in the model. For this example, it is feasible Red: Morlet 1.8 to identify a 10 m-thick layer using the wavelet whose 1.7 1.6 frequency is 30 Hz. 1.5 Fig. 5 is comparison of the change rate of matching 1.4 SXUVXLWWLPHIUHTXHQF\J\HQHUDWWKHUHÀHFWLRQOD\HU)XVLQJ 1.3 MSW dictionary, Ricker wavelet dictionary, and Morlet 1.2 1.1 wavelet dictionary respectively. The change rate is the ratio of time-frequency energy of every trace to that of tight 0.9 sandstone. The blue one represents the energy change rate 0.8 using MSW dictionary, the green one represents the energy 0.7 0.6 change rate using Ricker wavelet dictionary, and the red 0.5 one represents the energy change rate using Morlet wavelet 0.4 dictionary. Near the bottom of the gas-containing sand, the 0.3 0.2 J\FKDQJHGLFWLRQDU\XVLQJHQHULV06:PRUHVLJQL¿FDQWWKDQ 0.1 that using other dictionaries. Therefore, when the sand layer is 0 5 10 15 20 25 30 35 40 45 50 10 m thick, with 25% noise in the records, the time-frequency CDP trace number energy of matching pursuit using MSW dictionary which matches the characteristics of the signal is the most sensitive Fig. 5 Comparison of change rate of matching pursuit time-frequency to the quality factor. In other words, when the formation HQHUJ\DWWKHUHÀHFWLRQOD\HU)XVLQJWKH06:GLFWLRQDU\5LFKHUZDYHOHW thickness is 10 m, noise will not affect the time-frequency dictionary, and Morlet wavelet dictionary, respectively 1000m 2000m 3000m 4000m 5000m -5000m -5000m 5000:1 Ratio of velocity: 2:1 -5390 3400m/s 25m Thickness of basal conglomerate 25m -5500 4146m/s -5590 Thickness of coalbed 6m 4197m/s 4485m/s -5625 8m J x-5 oil reservoir 2 3917m/s Dry sand area -5685 -5680 1800m/s Reservoir area -5785 Reservoir area 4045m/s Reservoir area 4454m/s J s sand group 1 21 4454m/s 4475m/s 14m -5947 Dry sand area 4496m/s 4475m/s Reservoir area -6020 -6025 J s sand group 1 22 Dry sand area J b top of coalbed 1 1 Velocity of dry sand 4200m/s 4571m/s -6320 2500m/s -6400 (a) Line of dry sand Top line Line of thin coalbed Bottom line J s J s 1 22 1 21 Dry sand 2 J s 1 22 dry sand body 1 No.1 sand body J x-5 (b) Fig. 6 (a) Geological cross-section of physical model, (b) No.1 sand body shape Variation rate of time frequency energy T240 T220 T200 T180 T160 T140 T120 T100 T80 T60 T40 T20 Pet.Sci.(2012)9:310-316 315 energy using appropriate matching pursuit dictionary. The thin. In this area, when frequency is 22 Hz, the thickness of reason is that when calculating the time-frequency energy, the sand layer can be recognized. (c) is the single frequency the dictionary can best match the signal, and the noise is slice of No.1 sand body (50 Hz), where the red area looks not calculated in the reconstruction. We can say this method like a circle. There is a gap in the southwest, which means the improves the S/N ratio. layer is thick in the middle and thin in the surroundings, and Fig. 6(a) is the geological cross-section of a physical the shape of the sand body is approximately oval. (d) is the model, (b) is the shape of No.1 sand body, which is in the peak amplitude of No.1 sand body. The boundary of the red middle of J s DUHDFDQUHÀHFWWKHVKDSHRI1RVDQG)URPERG\WKHVLQJOH 1 22 of the physical model. Fig. 8(a) is the single frequency slice frequency slices with different frequency, we can estimate of No.1 sand body (22 Hz). The red area is in the middle, the thickness, location and shape of the sand, and especially which means the area is thick, and when the frequency is identify the thin layer. However, a higher frequency does not 22 Hz the tuning energy is a maximum. (b) is the single produce a clearer geological body distribution, so we should frequency slice of No.1 sand body (30 Hz). In this figure select an appropriate frequency to analyze the geological the red area is small, which means the middle of the layer is body. T50 T100 T150 T200 900 900 1000 1000 1100 1100 No.1 sand body 1200 1200 1300 1300 1400 1400 470m LINE 73 PH MODELO1.sdv 9 tr/cm 11 IPS E 0LJUDWLRQSUR¿OHRISK\VLFDOPRGHO Fig. 7 T20 T40 T60 T80 T100 T120 T140 T160 T180 T200 T220 T240 T20 T40 T60 T80 T100 T120 T140 T160 T180 T200 T220 T240 L160 L160 L160 L160 L140 L140 L140 L140 L120 L120 L120 L120 L100 L100 L100 L100 L80 L80 L80 L80 L60 L60 L60 L60 L40 L40 L40 L40 PH MOD MEM Amp.bri 500m TIME 22 PH MOD MEM Amp.bri 500m TIME 30 W E W E (a) (b) T20 T40 T60 T80 T100 T120 T140 T160 T180 T200 T220 T240 L160 L160 L160 L160 L140 L140 L140 L140 L120 L120 L120 L120 L100 L100 L100 L100 L80 L80 L80 L80 L60 L60 L60 L60 L40 L40 L40 L40 PH MOD MEM Amp.bri 500m TIME 50 (c) (d) Fig. 8 (a) Single frequency layer slice of No.1 sand body (22 Hz), (b) single frequency layer slice of No.1 sand body (30 Hz), (c) single frequency layer slice of No.1 sand body (50 Hz), (d) peak amplitude of No.1 sand body VDQGERG\JURXS)LJLVWKHPLJUDWLRQSUR¿OH 316 Pet.Sci.(2012)9:310-316 Geophysical and Geochemical Exploration. 2010. 32(6): 641-644 (in 5 Conclusions Chinese) Li H B. 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Petroleum Science – Springer Journals
Published: Aug 17, 2012
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