SEG/San Antonio 2007 Annual Meeting. Summary. Morlet wavelet transform

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1 Xiaogui Miao*, CGGVeritas, Calgary, Canada, Dragana Todorovic-Marinic and Tyler Klatt, Encana, Calgary Canada Summary Most geologic changes have a seismic response but sometimes this is expressed only in certain spectral ranges, buried within the broadband data. Spectral decomposition can be utilized to help interpretations or such cases. Compared with several dierent spectral decomposition technologies, the generalized S transorm is believed to be eicient and provides good temporal and spectral resolution. It has been used or heavy oil recovery and 4D SAGD monitoring cases and proved to be successul. The results correlate very well with well logs and provide useul inormation or seismic interpretations. Introduction Spectral decomposition is a novel technology developed in recent years. It has proved to be very useul or seismic data interpretation, because decomposing data into its spectral components reveals stratigraphic and structural details that are oten obscured in the broadband data. In a geologically complex area, a target event s variations in amplitude as a unction o requency can be traced more clearly when viewed in terms o a requency band. And lithology and luid driven spectral variations, such as peak requency shiting due to attenuation and absorption, can be better delineated. Thus, spectral decomposition provides a technique to help seismic interpretations. Popularly used spectral decomposition methods include Short windowed Fourier transorm (SWFT) (Partyka, 1999), Morlet wavelet based wavelet transorm (MWT) (X. Miao & W. Moon, 1994, Castagna, 2003), and Matching Pursuit Decomposition (MPD) (X. Miao & S. Cheadle 1998). SWFT involves explicit use o windows, which aects temporal and spectral resolution. Wave-packagelike spectral decomposition - even though it provides better spectral resolution - reduces temporal resolution, which is undesirable or thin bed interpretation. In this paper, in addition to the previously discussed MWT and MPD methods, we investigate a generalized S transorm (ST) based spectral decomposition method. We explore the merits and disadvantages o the methods, and apply them to the seismic data to show the interpretive beneits o spectral decomposition. Comparison o spectral decomposition methods Morlet wavelet transorm The Morlet wavelet is a modulated Gaussian unction, which is a non-orthogonal compactly supported complex wavelet, although it has side lobes. The decomposition o the MWT is represented in the scale (or voice) and time domain (X. Miao & W. Moon, 1994). Since scale is related to requency, we can convert the MWT representation into the requency-time domain and use it or spectral decomposition. The wavelet transorm based spectral decomposition has an implicitly deined analysis window, thus without the tapering eects inherent to the more commonly used short window Fourier method. S transorm The S transorm is proposed by Stockwell et. al. (1996) as an extension to the Morlet wavelet transorm. The mother wavelet or the S transorm is also a modulated Gaussian unction, but it keeps the modulation part with no scaling and no shiting. The S transorm is then deined as: s( t, ) D( t) g ( t -t )exp(-i2pt) dt. = - Where g (t) is a Gaussian unction given by g ( t) = exp( -( t) 2p With a small modiication to g (t) : 2 g ( t) = A exp( -a( t -b ) ), 2 ) and D(t) is the signal. the S transorm becomes the generalized S transorm (F. Tian et al. 2002). Here A, a and b are constants introduced to add a variety o orms in the mother wavelet so it can better correlate with signals. It directly decomposes signals into the requency and time domains. Matching pursuit decomposition The Matching Pursuit Decomposition technique is uniquely suited to providing high-resolution time-requency spectra. It inds a best matched wavelet rom a wavelet dictionary

2 a large collection o wavelets covering the ull ranges o time, requency, scale, and phase index - to represent each component o a signal, and so can enhance the spectral resolution (X. Miao and S. Cheadle, 1998) without side lobe eects. A successul spectral decomposition depends on its resolution as well as robustness. Among the above three algorithms, the MPD provides the highest temporal and spectral resolutions simultaneously, however is the most computationally expensive one and results sometimes is not unique due to the over redundancy o the wavelet library. Because o its eiciency and good localization to distinguish subtle changes, the MWT algorithm is still commonly used. Using the amplitude component can compensate its side lobe eect. The generalized S transorm shares most characteristics as the MWT, however gives higher spectral resolution due to variations o mother wavelet. We have compared all the methods using a synthetic signal and show the results in the Figure 1. First we created the synthetic signal. It consists o three wavelets with dierent requencies and temporal locations and two sinusoid waves at dierent starting and ending times. The requencies or these two sinusoid waves are 15 Hz and 25 Hz respectively. The Fourier spectrum o the synthetic signal is plotted on the top o Figure 1(b). Ater spectral decomposition, all the components in the synthetic trace are localized in the timerequency domain according to their spectral and temporal characteristics. It also shows that the MPD spectra (see Figure 1(d)) gives the highest resolution, and the generalized S transorm spectra (Figure 1(c)) gives the second highest resolution, however, it is much more eicient than the MPD method. MWT has slightly lower spectra resolution compared with the generalized S transorm method (Figure 1(b)). So in the ollowing applications we used generalized S transorm method or spectral decomposition. Because the wavelets in the above methods are complex, one can extract inormation about both the amplitude and phase o the signal being analyzed. The wavelet amplitude is helpul in analyzing attenuations and stratigraphic variation in time and requency simultaneously, while the wavelet phase is useul in locating discontinuities and identiying ractures. are developed. Some channels cut down into glauconitic sandstone, preserving patches o the sand bodies as isolated pools. The oil-water contact is dierent between separated pools, which perhaps indicate a more eective capture o hydrocarbon migration due to lithic channels. Interpretation becomes complicated; thereore we applied spectral decomposition to the seismic data. Figure 2 shows a result rom spectral decomposition. It is a horizon slice o the phase component rom an isorequency cube at 65 Hz. The color represents instantaneous phase changes. They clearly delineate discontinuities o stratigraphic variations. (a) Frequency (hz) Examples The irst example is rom a heavy oil recovery area. The target area has thick coals (4-5m), laterally extensive, draping over the sandy reservoirs and acting as partial seal. The sandstone up to 30 m thick penetrates with the oil pooling at the top o the reservoir. Within this upper Mannville sedimentary package, extensive lithic channels (b) 1438

3 can see excellent alignment o the known thickness o net pay with this seismically-predicted attribute. It also indicates that using high-resolution spectral decomposition can detect thin beds such as coals without the ringing eects o traditional narrow pass ilters. Another excellent application is using the tuning cube or interpretation (Partyka, 1999). A tuning cube containing amplitude spectra or a target zone can be created along a horizon. It is used to analyze requency dependent stratigraphic and structural eatures as well as to identiy the dominant requency at each cmp location or the examined horizon. Dominant requency can be calibrated with thickness rom well log inormation, so that the tuning cube portrays estimated thickness o target zones. Figure 3(a) shows an example o using tuning cube to estimate sand channel thickness. It is the tuning cube at requency 60 Hz or a 20 ms window centered on the target coal horizon. Pink and red colors correspond to higher amplitudes. Green color represents lower amplitudes. Figure 3(b) is the estimated relative thickness o the sand channels using the tuning cube. Pink and red colors represent thicker sands. It corresponds well with the known well log inormation. (d) Figure 1. (a) A synthetic trace composed o three wavelets with dierent requencies and temporal locations and two sinusoid waves on the let and the synthetic trace is on the right. The time-requency spectra o the synthetic trace decomposed respectively by MWT (b), generalized ST (c) and MPD (d). The horizontal axis is requency; the vertical axis is time in seconds. On the top o 1(b) is the Fourier spectrum o the synthetic trace. By overlaying with the known thickness o the net pay estimated rom well logs - shown in white contours - one We have also applied spectral decomposition technology successully to the SAGD monitoring or heavy oil recovery. Ater conventional 4D processing to both base and repeated seismic surveys, amplitude decrease and time sage in the target zone due to recovery are observed. However, detailed steaming activities are not easily monitored in the broad band data, because the dimensions o wells and steaming aected area are only sensitive to certain wavelengths. While it is diicult to identiy locations o horizontal wells with the broadband data, they are more readily distinguished by means o iso-requency cubes generated by the spectral decomposition. The development o heating lows along the well paths causes requency dependent spectral attenuations; these can be very well traced in these iso-requency cubes, so that the steaming quantities are better monitored. As well, 4D dierences along the horizontal well paths due to steaming recovery are more readily distinguished with iso-requency cubes. Results will be shown in the presentation. Conclusions Among the spectral decomposition methods, the generalized S transorm proved to be valuable in terms o temporal and spectral resolution as well as eiciency. Examples show that spectral decomposition has provided valuable attributes or seismic interpretation and the results correlate well with the well log inormation. 1439

4 Acknowledgements The authors greatly appreciate Encana Corporation or allowing us to present these results, and Bashir Durrani o CGGVeritas or helping process the data, and Bogden Batlai o Hampson-Russell or providing processed well log inormation. Figure 2. Phase component o the iso-requency cube at 65Hz (in color). The white contours represent the known thickness o net pay estimated rom well logs, which correspond very well with the iso-requency phase map. (a) (b) Figure 3. (a) Tuning cube at requency 60 Hz or a 20ms window centered on a coal horizon. (b) Relative thickness o sand channels, estimated rom calibration o dominant requency o the tuning cube with well log inormation. 1440

5 EDITED REFERENCES Note: This reerence list is a copy-edited version o the reerence list submitted by the author. Reerence lists or the 2007 SEG Technical Program Expanded Abstracts have been copy edited so that reerences provided with the online metadata or each paper will achieve a high degree o linking to cited sources that appear on the Web. REFERENCES Castagna, J. P., S. Sun, and R. Seigried, 2003, Instantaneous spectral analysis: Detection o low-requency shadows associated with hydrocarbons: The Leading Edge, 22, Miao, X., and S. Cheadle, 1998, High resolution seismic data analysis by wavelet transorm and matching pursuit decomposition: Geo-Triad, CSEG, CSPG and CWLS Joint convention, Miao, X., and W. Moon, 1994, Application o wavelet transorm in seismic data processing: 64th Annual International Meeting, SEG, Expanded Abstracts, Partka, G. A., J. M. Gridley, and J. Lopez, 1999, Interpretational applications o spectral decomposition in reservior characterization: The Leading Edge, 18, Stockwell, R. G., L. Mansinha, and R. P. Lowe, 1996, Localization o complex spectrum: The S-transorm: IEEE Transactions on Signal Processing, Tian, F., S. Chen, E. Zhang, J. Gao, W. Chen, Z. Zhang, and Y. Li, 2002, Generalized S transorm and its applications or analysis seimsic thin beds: 72nd Annual International Meeting, SEG, Expanded Abstracts,