Comparison of spectral decomposition methods

John P. Castagna, University of Houston, and Shengjie Sun, Fusion Geophysical discuss a number of different methods for spectral decomposition before suggesting some improvements possible with their own variation of ‘matching pursuit’ decomposition. In seismic exploration, spectral decomposition refers to any method that produces a continuous time-frequency analysis of a seismic trace. Thus a frequency spectrum is output for each time sample of the seismic trace. Spectral decomposition has been used for a variety of applications including layer thickness determination (Partyka et al, 1999), stratigraphic visualization (Marfurt and Kirlin, 2001), and direct hydrocarbon detection (Castagna et al., 2003; Sinha et al., 2005). Spectral decomposition is a non-unique process, thus a single seismic trace can produce various time-frequency analyses. There are a variety of spectral decomposition methods. These include the DFT (discrete Fourier Transform), MEM (maximum entropy method), CWT (continuous wavelet transform), and MPD (matching pursuit decomposition). None of these methods are, strictly speaking, ‘right’ or ‘wrong’. Each method has its own advantages and disadvantages, and different applications require different methods. The DFT and MEM involve explicit use of windows, and the nature of the windowing has a profound effect on the temporal and spectral resolution of the output. In general, the DFT is preferred for evaluating the spectral characteristics of long windows containing many reflection events, with the spectra generally dominated by the spacing between events. The MEM is often difficult to parameterize and may produce unstable results. The CWT is equivalent to temporal narrow-band filtering of the seismic trace and has an advantage over the DFT for broad-band signals in that the window implicit in the wavelet dictionary is frequency dependent. The CWT has a great disadvantage, however, in that the wavelets utilized must be orthogonal. The commonly used Morlet wavelet, for example, has poor vertical resolution due to multiple side lobes. Furthermore, for typical seismic signals, the implicit frequency dependent windowing of the CWT is not particularly important, and experience has shown that a DFT with a Gaussian window of appropriate length produces almost the same result as a CWT with a Morlet wavelet. MPD (Mallat and Zhang, 1993) is a more computationally intensive process than the others, but, as will be shown in this paper, it has superior temporal and spectral resolution if a compact mother wavelet is utilized. Matching pursuit decomposition involves cross-correlation of a wavelet dictionary against the seismic trace. The projection of the best correlating wavelet on the seismic trace is then subtracted from that trace. The wavelet dictionary is then cross-correlated against the residual, and again the best correlating wavelet projection is subtracted. The process is repeated iteratively until the energy left in the residual falls below some acceptable threshold. As long as the wavelet dictionary meets simple admissibility conditions, the process will converge. Most importantly, the wavelets need not be orthogonal. The output of the process is a list of wavelets with their respective arrival times and amplitudes for each seismic trace. The inverse transform is accomplished simply by summing the wavelet list and the residual, thus reconstructing the original trace. The wavelet list is readily converted to a timefrequency analysis by superposition of the wavelet frequency spectra. Simple matching pursuit has difficulty in properly determining the precise arrival time of interfering wavelets – usually it will slightly misplace the wavelets which will also result in a slightly incorrect wavelet center frequency. Also, it can be seen that the process is path dependent: a slight change in the seismic trace may result in an entirely different order of subtraction. Thus, it may result in lateral instability of the non-uniqwue time-frequency analyses. Cross-correlation of the wavelet dictionary against the seismic trace is essentially a continuous wavelet transform, so it can be seen that the method involves iteratively performing hundreds, if not thousands, of wavelet transforms for each seismic trace. In this paper, we utilize a variation of matching pursuit called exponential pursuit decomposition (EPD). The method treats complex interference patterns as containing ‘gravity wells’ at the correct wavelet locations, and the selected wavelet location is iteratively attracted to the correct location. The profound advantage of EPD over other methods is that there is no windowing, and corresponding spectral smearing. The spectra for reflections from isolated interfaces that can be resolved by the method are the same as the seismic wavelet producing those reflections. The method can thus be used with confidence for direct hydrocarbon indication and stratigraphic visualization for thin beds. The classical Heisenberg Uncertainty Principle states that the product of temporal and frequency resolution is constant. One must normally pay the price of decreasing resolution in one domain, to increase resolution in the other. In EPD, there is no windowing and it is the bandwidth of the digital seismic data that limits resolution, not the windowing process. Thus, the Heisenberg Uncertainty Principle does not come into play. As a result, EPD provides better temporal AND spectral resolution than the other methods. In comparing spectral decomposition methods, it is important to keep in mind what the goal of the analysis is.