谱分解技术简介（英文版）

1、8 谱分解技术Recently, much attention has been given to spectral decomposition in exploration geophysics. The technique has been used for both bed thickness estimations and direct hydrocarbon indications. However, the spectral decomposition is not really a new invention of geophysicists, and its root is c

2、losely related to the so-called time-frequency representation of stationary and non-stationary time series that has been used in signal and imaging processing for over 50 years, though geophysicists have played a central role in its development in fact, Jean Morlet, a geophysicist, is one of the two

3、 “inventors of the continuous wavelet transform. The term “spectral decomposition was coined by geophysicist Greg Partyka of BP in 1999. The increase use of the spectral decomposition methods in exploration seismic in the last five years can be largely attributed to the pioneering work of Partyka (1

4、999) and Castagna et al (2003).Because of the Heisenberg uncertainty principle, which prevents us from achieving simultaneous localization in time and frequency, looking into the literature, we may find many techniques that can be used for the same purpose. In this study, we review six most commonly

5、 used time-frequency representation and spectral-decomposition techniques and compare them against a series of seismic signals. We aim to establish some practical guidelines about the choice of the methods. It is not surprised to find that all methods are correct, and there is no technique which can

6、 be claimed to be superior to others at all time and for all signals, though one technique may give better results (meaning better temporal AND frequency resolution) than others for certain kinds of signal or under certain conditions - hence everyone has his/her own preferred methods. 8.1 INTRODUCTI

7、ONThere is a growing interest in exploration geophysics in the application of spectral decomposition methods for bed thickness determination (Partyka 1999); stratigraphic visualization (Marfurt & Kirlin 2001); direct hydrocarbon (gas) detection (Castagna et al. 2003; Goloshubin et al. 2006); detecti

8、on of frequency-dependent seismic anisotropy (Liu et al. 2003); and study of the effects of attenuation/dispersion on seismic amplitudes-variations with offsets (Chapman et al. 2005). We shall see that the root of the spectral decomposition may be traced back to over 50 years, however, it is a relat

9、ively new technique and its popularity has increased in geophysics since the pioneering work of Partyka et al. (1999), Castagna et al. (2003), and more recently, Chapman et al. (2005), who have extended the application of spectral decomposition analysis to AVO studies in the framework of Rutherford

10、and Williams AVO analyses (Rutherford & Williams 1989). Seismic attributes from the time-frequency decomposition have also been proposed (see references in Section 6.4). According to Castagna & Sun (2006), spectral decomposition refers to any method that produces a continuous time-frequency analysis

11、 of a seismic trace. Thus a frequency spectrum is output for each time sample of the seismic trace. Spectral decomposition is a non-unique process, thus a single seismic trace can produce various time-frequency analyses. Despite the long historical root, exploration geophysicists have found many new

12、 applications of time-frequency representation and spectral decomposition methods in the last five years. Time-frequency analysis is a 2D representation of one-dimensional signal in time-frequency plane and it offers the ability to analyse relative long continuous segments of time series. The techni

13、que is simply a time-frequency analysis technique widely used in image processing, medical imaging, acoustics, electrical engineering, astrophysics, among others. One approach to time-frequency analysis is to represent the signal as a sum of waveforms with well-defined time-frequency properties. In

14、Fourier analysis, these waveforms are simply cosine and sine functions that provide good localization in frequency but not in time. To achieve a good localization both in time AND in frequency, there are a variety of time-frequency methods, including the short-time Fourier transform or windowed Four

15、ier Transform (WFT), maximum entropy method (MEM), Wigner-Ville transform (WVT) and various extensions, continuous wavelet transform (CWT), Stockwell or S-transform (SWT), and matching pursuit method (MPM). As we will see from this review, some of these techniques are not completed independent and a

16、re close relatives of each others. As Castagan & Sun (2006) correctly state that none of these methods are, strictly speaking, right or wrong. Each method has its own advantages and disadvantages, and different applications require different methods. Among the methods mentioned above, the WFT involv

17、es explicit use of fixed-length windows, and the nature of the windowing has a significant effect on the time and spectral resolution of the output. The MEM works in the similar way as the WFT, where the power spectrum of a signal is computed using the maximum entropy method instead of the conventio

18、nal Fourier transform. The CWT is equivalent to temporal narrow-band filtering of the signal and has an advantage over the WFT for broadband signals in that the window is frequency dependent, i.e. varying window lengths. An alternative method, used in several motivating studies, is the S-transform i

19、nvented by Stockwell et al. (1996), which is an extension of the windowed Fourier transform which uses frequency-dependent scaling windows in analogy to the wavelet transform (however, it is not the wavelet transform because S-transform does not satisfy admissibility). This permits a frequency-depen

20、dent resolution with narrower windows at higher frequencies and wider windows at lower frequencies. Another powerful technique, MPM, developed by Mallat & Zhang (1993), is based on the wavelet transform, but uses a family of wavelets (“wavelet atom or wavelet dictionary) composed of Gabor functions

21、supplemented with a canonical basis of discrete Dirac functions and discrete Fourier basis of cosine and since functions. This dictionary provides a rich collection of atomic waveforms which are located on a much finer grid in the time-frequency space than wavelet and cosine packet tables. The MPM i

22、s a more computationally intensive process than the others, and Castagna & Sun (2006) claim that it has superior temporal and spectral resolution if a compact mother wavelet is utilized.In this paper, we provide a systematic overview of the most commonly used methods about time-frequency representat

23、ion of a time-variant signal, and compare them against some common signals. Our goal is to provide some practical guidelines to the choice of the methods in the context of exploration seismic applications. 8.2 FOURIER TRANSFORM AND HEISENBERG UNCERTAINTYWe start with the most basic technique for det

24、ermining the frequency distribution of a signal u(t), i.e. the Fourier transform. This is given by the familiar integral transform. (8-1)The Fourier transform is invertible, implying the signal u(t) can be recovered by. (8-2)The well-known property of the Fourier spectrum of a signal is that it is l

25、ocalized in frequency, but not in time. For example, an infinite length cosine signal will transformed to a delta pulse (single frequency). Figure 8-1 shows a seismic signal and its Fourier spectrum. We cannot tell the time corresponding to the frequency contents the arrival time of the low frequenc

26、y pulse, which is centred at 0.08 sec, cannot be revealed from the signals spectral plot. Note the multi-taper spectral algorithm of Park et al. (1987) may be used which can produce much smoother spectral than the conventional single taper FFT.Figure 8-1. A chirp-like guided wave from an inseam seis

27、mic experiment and its Fourier spectrum. Our aim is to obtain the time-dependent spectra of a signal. This can be achieved by taking the Fourier transform with a short moving window along the signal to obtain the time-frequency spectrum called spectrogram. Though the Fourier representation of a sign

28、al is unique, the time-frequency representation of this signal will not be unique. As a result, there are many methods that have been proposed to represent a transient signal aiming to achieve localized spectrum. The moving window Fourier transform or simply called windowed Fourier transform is one

29、of the simplest methods. Here we list six most popular methods that can usually be found in geophysical and other signal processing literature:Short-time Fourier transform or windowed Fourier transform (WFT)Maximum entropy method (MEM)Wigner-Ville transform (WVT)Continuous wavelet transform (CWT)Sto

30、ckwell or S-transform (SWT)Matching pursuit method (MPM)8.3 METHOD COMPARISON: TEST EXAMPLESWe test the methods in the previous sections on 3 signals: (a) a guided wave from an in-seam seismic experiment in a UK coal mine (after Liu et al. 1992); (b) a simple Ricker wavelet with a peak frequency of

31、50Hz, and (c) a synthetic time series containing a series of Ricker wavelets at different separations.8.3.1 Dispersive channel/guided wavesFigures 8-8 to 8-11 compare time-frequency spectrograms from various methods. We use this signal in our first test example because it shows obvious dispersive ch

32、aracteristics of guided or channel waves (which are very similar to chirp signals). In fact, a major application of time-frequency representation of time series signals in seismology has been the extraction of dispersion characteristics (velocity as a function of frequency) of surface waves generate

33、d from teleseismic events. The guided wave displayed in Figures 8-8 to 8-11 is very similar to a typical Rayleigh or Love surface waves from earthquake except that the frequency is significantly high. Comparing WFT and MEM in Figure 8-8, we can confidently say that both methods give similar results

34、in terms of temporal and frequency resolution. The resolution from WVT results shown in Figure 8-9 is clearly higher than in Figure 8-8, but they are very noisy due to the spurious peaks in the WVT method as discussed in Section 6.3.3. Note that spurious peaks appear along frequency axis for the Tim

35、e-domain WVT, and spurious peaks appear along tme axis for the frequency domain WVT. Nevertheless the time-frequency resolution is quite good. The SWT and Gabor transform give slightly better resolution for this particular signal than WFT or MEM and could be better if parameters k in SWT is adjust (

36、there is no automatic role to set this parameter and it has to be adjusted by users). In Figure 8-11, we display scalogram for CWT instead of spectrogram, i.e. we have not converted the a to frequency f. It is clear that CWT resolution is similar to the Gabor.Figure 8-8. Comparison of spectrograms g

37、enerated using WFT and MEM for the chirp-like guided wave in Figure 1. A Blackman window with a length of 32 samples is used (for MEM the parameter M=20).Figure 8-11. Comparison of spectrograms generated using CWT displayed in scale and WFT with the Gaussian window function (the Gabor transform). No

38、tice scale is inversely proportional to frequency.8.3.2 A single Ricker wavelet frequency resolutionRicker wavelet is not the ideal wavelet to test spectral decomposition method because of its wide spectral band. For example, a Ricker wavelet with a peak frequency of 20Hz will have a bandwidth of al

39、so around 20Hz the dominant frequency will span from 10 to 30Hz frequency band is defined at the drop of 30 dB in frequency peaks. A Ricker wavelet with a peak frequency of 50 Hz has energy between 20 to 100 Hz. That really means that the Ricker wavelet is only located in time, but not in frequency.

40、 Nevertheless, because of its popularity in seismic exploration, we have adapted this wavelet for comparison of different time-frequency representation.Figure 8-11 compares the spectrograms for a Ricker wavelet with a peak frequency of 50Hz from WVT time-domain and WVT frequency domain. For comparis

41、on, we also show the spectrograms from the WFT (with a Blackman window function) and SWT. WVT spectrograms do give good resolution in time, and the frequency resolution is comparable to WFT and SWT. The FFT spectrum is shown on top of the plots, indicating that the lack of frequency resolution is du

42、e to the spectral characteristics of the Ricker wavelet itself.In Figure 8-12, we compare the Gabor transform implemented by the multi-filtering techniques and the CWT with a Morlet wavelet. The Gabor spectrogram is the very similar to the CWT result shown in the previous Figure. However, it seems t

43、hat the CWT scalogram shows that the scale range of this Ricker wavelet is between 1 and 2.6. Figure 8-12. Comparison of spectrograms for a synthetic trace with a single Ricker wavelet with a peak frequency of 50 Hz: Four methods are used: WVT time-domain and frequency domain, WFT with a Blackman wi

44、ndow and SWT.Figure 8-12. Continued from previous figure. WFT with the Gaussian window function (Gabor transform) and CWT displayed in scale rather than frequency.Note that for smooth spectra, such as the synthetic Ricker wavelets, the difference between MEM and WFT spectrograms are small and thats

45、why we have not shown any results for MEM.8.3.3 Multiple Ricker wavelets time resolutionIn the next test example, we create a synthetic seismic trace containing nine Ricker wavelets to form three groups: the first group contains three wavelets at 0.5, 0.515, 0.53 sec with peak frequencies of 40, 50,

46、 60Hz. The second group also contains three wavelets at 0.8, 0.825, 0.85 sec with peak frequencies of 40, 50, 60Hz. The third group contains three wavelets at1.3, 1.35, 1.4 sec with peak frequencies of 40, 50 and 60 Hz. Total number of samples is 1024 with a sample rate of 2ms. The amplitudes of all

47、 wavelets are set to 1. Various spectrograms are shown in Figures 8-13 and 8-14. Just look at these figures, we cannot immediately judge which one gives better time resolution because they are all similar. Actually, in each time-frequency representation methods presented in Section 6.3 have different parameters to adjust (briefly summarized in Appendix). We can in theory achieve similar time and frequency re