291 Int. J Sci. Emerging Tech
Vol‐5 No 5 May, 2013
Using Continuous Wavelet Transform and Short Time Fourier Transform as Spectral Decomposition Methods to Detect of Stratigraphic Channel in One of the Iranian South-West Oil Fields E. Shokrollahi#1, G. Zargar#2, M.A. Riahi*3 #
Petroleum exploration engineering department, Petroleum University of Technology, Abadan, Iran 1
[email protected] 2
*
[email protected]
(Institue of Geophysics,University of Tehran, Tehran, Iran) 3
[email protected]
Abstract--Classical seismic methods for characterization of hydrocarbon reservoirs have been used for decades. Stratigraphic events affect seismic sections in addition to structural events. Today the necessity of usage of seismic sections to determine reservoir extension and stratigraphic characteristics is increasing. Conventional seismic sections cannot display most of the important events in reservoir studies. The study of frequency content of seismic sections can provide better understanding to geoscientists, as well as interpreters in the field of petroleum engineering studies. An is frequency process is a powerful tool in terms of reservoir imaging. One of the methods that provides is frequency slice is Spectral Decomposition (SD). SD is a comprehensive method that eliminates most of limitations encountered in seismic data to reveal geological information. SD provides continuous analysis of amplitude, frequency, phase, and energy spectrum. Therefore, SD is applied to obtain an amplitude spectrum of frequency content of seismic trace which is attributed to the temporal center of the sample trace, so higher frequency resolution is obtained at lower frequencies and higher time resolution at higher frequencies. In this paper, Short Time Fourier Transform (STFT) and Continuous Wavelet Transform (CWT) as SD methods were used in order to detect stratigraphic channel at one of the Iranian South-West oil fields for which it has the potential to be considered as a reservoir. Keywords: Continuous Wavelet Transform (CWT), Short Time Fourier Transform (STFT), Spectral Decomposition (SD), stratigraphic channel, Sarvak.
1. Introduction Different methods have been introduced for SD of seismic data. The goal of all these methods is to decompose the seismic signal to its components in __________________________________________________________________________ International Journal of Science & Emerging Technologies IJSET, E‐ISSN: 2048 ‐ 8688 Copyright © ExcelingTech, Pub, UK (http://excelingtech.co.uk/)
order to obtain more geological information. Based on definition, SD refers to any method that produces continuous analysis of amplitude, frequency, phase, and energy spectrum. In this method, the output of each seismic trace is an amplitude spectrum of frequency content of input trace that refers to temporal center of sample. Therefore, the result of SD of a seismic trace is a time-frequency analysis window. One of the oldest methods in obtaining local spectrum is Discrete Fourier Transform (DFT) which is a classical frequency decomposition method. This transform calculates the relative intensity of each frequency component of the whole signal, but does not provide any information about frequency content changes with time. Therefore, DFT method is not suitable for non-stationary signals and cannot determine frequency changes with time (seismic waves are not stationary, i.e. Their frequency content varies with time). To resolve this problem of DFT, Short Time Fourier Transform (STFT) or Fast Fourier Transform (FFT) has been widely used to decompose nonstationary signals. The basic idea in STFT is to divide a non-stationary signal to small segments (which are considered as stationary parts) and calculate Fourier transform for each segment. To perform this division, a window function is chosen and multiplied by the signal. One of the setbacks of this method is the constant-length time interval obtained from window function. Short temporal window provides suitable time resolution but the resulting frequency resolution is low. If a larger temporal window is selected, frequency resolution will be improved but time resolution will be reduced. Therefore, if the goal is exact identification of separate time events, methods in which the window length is automatically adjusted to frequency should be used. Decomposition by Continuous Wavelet Transform (CWT) is a suitable method to overcome the resolution problems in decomposition by STFT method. In CWT the signal is multiplied by a function
292 Int. J Sci. Emerging Tech
such as window function in STFT in which the length of the window is not constant. The length of temporal window changes according to the required frequency resolution. This change in window length is the result of a window function called “wavelet function”. An important step in CWT method is selection of the wavelet function which is called Mother wavelet. Commonly used wavelets in CWT are Morlet wavelet, Gaussian wavelet and Mexican-Hat wavelet. Although there is no absolute optimal wavelet among the common wavelets, by considering the purpose and type of usage, specifications for suitable Mother wavelet must be considered to choose an optimal wavelet. As an example side lobes of Morlet wavelet reduce vertical resolution of CWT [1]. In this study, performance of CWT and STFT in mapping stratigraphic channel in one of the Iranian South-West oil fields using 3-D seismic data, is investigated.
2.
Theory and Method
2.1
Spectral Decomposition Methods
Peyton et al. (1998) [2], partyka et al. (1999) [3], and Marfurt and Kirlin (2001) [4] are the prominent founders of this method. The basis of this method is selecting an optimal window length and making its convolution with Gaussian window, so that the Fourier Transform of this window can be calculated. Proper selection of the window is the main problem of this method. Based on the selected temporal window the STFT provides a time-frequency spectrum. In STFT, timefrequency resolution is fixed over the entire timefrequency space by preselecting a window length. Therefore, resolution in seismic data analysis becomes dependent on user-specified window length. Mathematically, STFT is the inner product of signal and a time shifted window in time τ and the frequency ω as shown in ‘Eq (1)’:
,
̅
Where ̅
(1)
is the complex conjugate of
Vol‐5 No 5 May, 2013
frequency covering length tends towards high frequencies and low frequencies respectively; therefore, by increasing frequency resolution, time resolution decreases consequently and vice versa [5]. As hydrocarbon reservoirs are detectable in low frequencies and thin layers with high temporal resolution are detectable in high frequencies this method is a proper method in reservoir characterization [6]. Wavelet transform was first used by Morlet et al. in 1980s for assessing of seismic data [7]. Wavelet with symbol (t) is usually defined as a function (t) ϵ L2(R) with zero mean which can be determined in two time and frequency domains. By translating this wavelet (t) a family of wavelets can be produced:
,
(2)
√
In which τϵR, σ ≠ 0, σ is scale parameter, τ Translation parameter and is complex conjugate of (t). As wavelets are usually normalized, a coefficient 1/√σ is used. CWT is defined as the inner product of the signal u (t) and a group of wavelets σ, τ (τ). Its formula is as:
2.1.1 Short Time Fourier Transform (STFT)
∞ ∞
.
, √
∞ ∞
In , scalogram).
,
,
(3)
which is the time-scale map (i.e. the
The scalogram cannot provide a direct and clear interpretation of frequency. For interpretation of the time-scale map in terms of a time-frequency map, a constant (c), which is obtained from the relation between scale and central frequency, must be calculated approximately as follows: (4)
The easiest step in conversion of scale to frequency is equating it with a scale-frequency map of wavelet. The useful wavelets commonly used in wavelet transform are Morlet, Gaussian and Mexican-Hat. In this article, these three wavelets are used in CWT analysis.
This method is a useful operator in studying and detection of thin layers.
2.2 Using CWT and STFT for Practical Study of the Field
2.1.2 Continuous Wavelet Transform (CWT)
In this paper, CWT and STFT are used as SD methods for detection of the channel in seismic data in one of the Iranian oil fields.
CWT in signal analysis is considered an alternative method for frequency distribution analysis of non-stationary time series. In CWT method by increasing or decreasing temporal covering length,
293 Int. J Sci. Emerging Tech
This oil field with a trend of S-N is located in the Dezful Embayment in the South-West of Iran. Dezful Embayment contains major oilfields of Iran, in other words, it is the center of South-West Iran oil province, which is geologically a part of Northern Arabian plate. This region, with a flat and even topography, is covered with a layer of quaternary alluvial sediments which has a thin thickness.
Vol‐5 No 5 May, 2013
Figure 3 shows the seismic amplitude response in time slice of 1800ms obtained from 3-D seismic cube of figure 1. This horizon slice is taken 20 ms under the Sarvak horizon shown in figure 2.
The geological studies of this oil field show that in the section related to Sarvak Formation, channel events commonly exist. Sarvak Formation stratigraphic ally locates in the second sequence of Middle cretaceous and has reservoir potential. Figure 1 shows 3-D seismic cube of the studied oil field and figure 2 shows the interpreted Sarvak horizon. The studied section by SD methods for detection of channel is beneath this horizon. Figure 3. Seismic amplitude response of time slice 1800 ms (20 ms below the Sarvak horizon) As it is seen, no specific geological event which causes changes in seisimc structure is detectable. CWT has been calculated as follows.CWT was calculated as isofrequency seismic sections based on all three Morlet,Gaussian and, Mexican-Hat wavelets at frequencies of 20, 30, and 40Hz and the results have been shown in figures 4 to 6 for a temporal depth of 1800ms. Figure 1. 3-D seismic volume of the studied oil field
Figure 2. Interpreted Sarvak horizon from 3-D seismic cube
At frequency of 20Hz in figure 4, both Morlet and Gaussian wavelets provide better resolutions compared to Mexican-Hat. At 30Hz frequency in figure 5, Morlet wavelet provides better resolution than Gaussian wavelet which in term provides better amplitude contrast than Mexican-Hat. In 40Hz frequency in figure 6, Gaussian wavelet provides relatively a good resolution, although Morlet wavelet does not provide relatively good resolution, it provides higher amplitude contrast compared to low resolution of Mexican-Hat wavelet. The minimum and maximum seismic amplitudes as a result of CWT method are shown in table 1. The obtained results from CWT are generally as follows: In 20 and 30Hz frequencies Morlet, Gaussian, and Mexican-Hat wavelets performed better respectively. In 30Hz frequencies Gaussian, Morlet and Mexican-Hat wavelets performed better respectively.
294 Int. J Sci. Emerging Tech
Vol‐5 No 5 May, 2013
(a) (a)
(b)
(b)
(c) Figure 4. Illustration of isofrequeny slices of 20Hz obtained from CWT method. (a) Morlet wavlet, (b) Gaussian wavelet, (c) Mexican-Hat wavelet
(c) Figure 5. Illustration of isofrequeny slices of 30Hz obtained from CWT method. (a) Morlet wavlet, (b) Gaussian wavelet, (c) Mexican-Hat wavelet
295 Int. J Sci. Emerging Tech
Vol‐5 No 5 May, 2013
(c)
(a)
Figure 6. Illustration of isofrequeny slices of 40Hz obtained from CWT method. (a) Morlet wavlet, (b) Gaussian wavelet, (c) Mexican-Hat wavelet
(b)
Table 1. Estimation of seismic amplitude of isofrequency sections obtained from CWT method Morlet Wavelet Max amplitude
GaussianWavelet
Mexican-Hat Wavelet
Min amplitude
Max amplitude
Min amplitude
Max amplitude
Min amplitude
20Hz isofrequency sections
0.557274
0.012638
0.166242
0.004634
0.342124
0.010825
30Hz isofrequency sections
0.646286
0.020745
0.245229
0.008189
0.37163
0.011042
40Hz isofrequency sections
0.603053
0.17234
0.288686
0.007864
0.352619
0.009143
296 Int. J Sci. Emerging Tech
Vol‐5 No 5 May, 2013
STFT is the second method of SD used in this study to obtain isofrequncy seismic sections.In the following STFT calculation is studied. STFT method was calculated as isofrequency seismic sections in frequencies of 15, 25, and 35Hz and in 10, 56, and 100ms time intervals and the results were shown in figures 7 to 9 for a temporal depth of 1800ms. Analyzing isofrequency images in figure 7 and 8 shows that isofrequency slices of the temporal window of 56 ms (figure 7-b and 8-b) illustrate the thickness and boundaries of the channel much better than the other two temporal windows of STFT method. Also 35Hz isofrequency image of 100ms temporal window shows the channel details better than 56 and 10ms temporal windows of STFT method in figure 9.
(a)
As shown in figures and observed in the maximum and minimum amplitude values of sections in table 2, isofrequency sections of STFT have a lower quality than CWT method and CWT displayed a better performance compared to STFT method in channel detection.
The obtained results from STFT are generally as follows: Comparison of the obtained results from different temporal windows in an isofrequency seismic section:
In 15 and 25Hz isofrequency sections, 56, 100, and 10ms time intervals performed better in channel detection respectively.
(b)
In 35Hz isofrequency sections, 100, 56, and 10ms time intervals performed better in channel detection respectively. Comparison of isofrequency seismic sections in equal temporal windows:
In 10ms temporal window, a better channel resolution is obtained in frequencies of 35, 25, and 15Hz respectively.
In 56ms temporal window, a better channel resolution is obtained in frequencies of 25, 15, and 35Hz respectively.
In 100ms temporal window, a better channel resolution is obtained in frequencies of 25, 35, and 15Hz respectively. (c) Figure 7. Illustration of isofrequeny slices of 15Hz obtained from STFT method. (a) 10 ms temporal window, (b) 56 ms temporal window, (c) 100 ms temporal window
297 Int. J Sci. Emerging Tech
Vol‐5 No 5 May, 2013
(a) (a)
(b) (b)
(c) (c) Figure 8. Illustration of isofrequeny slices of 25Hz obtained from STFT method. (a) 10 ms temporal window, (b) 56 ms temporal window, (c) 100 ms temporal window
Figure 9. Illustration of isofrequeny slices of 35Hz obtained from STFT method. (a) 10 ms temporal window, (b) 56 ms temporal window, (c) 100 ms temporal window
298 Int. J Sci. Emerging Tech
Vol‐5 No 5 May, 2013
Table 2. Estimation of seismic amplitude of isofrequency sections obtained from STFT method 10ms temporal window Max amplitude
56ms temporal window
100ms temporal window
Min amplitude
Max amplitude
Min amplitude
Max amplitude
Min amplitude
15Hz isofrequency sections
3.06994
0.09092
34.37968
1.485297
31.463892
25Hz isofrequency sections
4.39585
0.14683
52.11539
1.3913
52.763969
2.252071
35Hz isofrequency sections
5.84562
0.16486
58.22383
1.960338
82.458725
3.277157
1.115812
In figure 10, the position of the studied channel is shown in vertical seismic section. Vertical seismic section in figure 10-a is located where the channel main body is existed corresponding to Cross line number 5570 of seismic cube. Relative horizon to Sarvak Formation is characterized in figure. The oval shows the location of the main body of the channel. Figures 10-b shows 20Hz is frequency sections resulting from CWT method using Morlet wavelet, corresponding to time slice of 1800ms, in addition to the location of corresponding vertical section. (b) Figure 10. (a) Vertical seismic section corresponding to Crossline number 5570. Sarvak Horizon is specified in the figure. The oval shows the location of main branch of the channel. (b) 20Hz isofrequency section of CWT method, using Morlet wavelet corresponding to 1800ms time slice. Red line shows the location of vertical section shown in figure 10-a
3. Conclusion Spectral Decomposition is a powerful tool in detection of channels by using 3-D seismic data. (a)
Application of CWT and STFT as two effective SD-based methods is proper tools in imaging the buried channels which provide the possibility of comparison between the resulted is frequency slices for these methods. The best result in channel detection of all CWT is frequency sections is obtained from Morlet Mother Wavelet and in 20Hz section. The best result of all
299 Int. J Sci. Emerging Tech
STFT is frequency sections is obtained in 25Hz is frequency section of 56ms temporal window. Based on the obtained results of this paper, is frequency slices of CWT method relatively showed better quality than STFT method and detection of channel with more details is provided well by CWT method. This is because of constant Time-Frequency resolution in STFT method over the entire TimeFrequency space by preselecting a window function. Contrary to STFT in CWT method, the adaptive wavelet causes the production of optimum window length relative to frequency content of seismic data. In other words in CWT method by increasing or decreasing temporal covering length, frequency covering length tends towards high frequencies and low frequencies respectively.
References [1] Castagna, J.P., and Sun, S., 2006, Comparison of spectral decomposition methods: First Break, 24, 75-79.
Vol‐5 No 5 May, 2013
[2] Peyton, L., Bottjer, R., and Partyka, G., 1998, Interpretation of incised valleys using new 3-D seismic technique: A case history using spectral decomposition and coherency: The Leading Edge, 17, 1294-1298. [3] Partyka, G., Gridley, J., and Lopez, J., 1999, Interpretational applications of spectral decomposition in reservoir characterization. The Leading Edge, 18, 353–360. [4] Marfurt, K.J., and Kirlin, R.L., 2001, Narrow band spectral analysis and thin bed tuning: 66, 1274-1283. [5] Mallat, S., 1999, A wavelet tour of signal processing, 2nd ed.: Academic Press Inc. [6] Sinha, S.K., Routh, P.S., Anno, P.D., Phillips, C., and Castagna, J.P., 2005, Spectral Decomposition of seismic data with continuous-wavelet transform: Geophysics, 70, 19-25. [7] Grossmann, A., and Morlet, J., 1984, Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal., 15, 723-736,1984.
