عنوان مقاله [English]
Spectral decomposition is a powerful tool for analysis of seismic data. Fourier transform determines the frequency contents of a signal. But for analysis of non-stationary signals, 1-D transform to frequency domain is not sufficient. In early years, transforming of seismic traces into time and frequency domain was done via windowed Fourier transform, called a Short Time Fourier Transform (STFT). In this method the resolution of the results in time-frequency domain was controlled by the width of the selected window. Continuous Wavelet transform (CWT) was a remedy to solve this problem by using the scaleable wavelets. The scale in CWT can be related to the frequency bandwidth of the wavelet. By converting the scale to frequency one can get the time-frequency map which is comparable to the time-frequency map obtained from STFT.
In this study, we applied CWT on a seismic section and extracted a single frequency seismic section from the resultant cube. The extracted sections were used as a seismic attribute to detect low frequency shadow of the hydrocarbon reserves over the study area as well as to analyze the existing thin layers on a synthetic seismic section.
Time-Frequency Analysis Methods, (a) STFT: The instantaneous frequency has often been considered as a way to introduce frequency dependence on time. If the signal is not narrow-band, however, the instantaneous frequency averages different spectral component in time. To become accurate in time, we therefore need a two dimensional time-frequency representation of the signal composed of spectral characteristics depending on time. By assuming that the signal is stationary when seen through a window of limited extent and moving the window along the signal in time, the Fourier transform of the windowed signals yields the Short Time Fourier Transform which is a two dimensional function in time-frequency domain.
(b) CWT: STFT requires a fixed time support. In practice, seismic data are non-stationary and using the STFT may not produce very reliable time-frequency map. Fixed window length and hence, fixed time-frequency resolution is a fundamental difficulty with the STFT for analyzing a non-stationary signal.
Continuous wavelet transform was introduced by Morlet et al. (1982). In CWT, time-frequency atoms are chosen in such a way that its time support changes for different frequencies honoring Heisenberg’s uncertainty principle (Mallat, 1999; Daubechies, 1992). In this study we used Morlet wavelet that provides an easy interpretation from scale to frequency (Torrence and Compo, 1998).
Single Frequency Seismic Section, Application 1: In this study we used a seismic section over one of the reservoirs in the South-West of Iran, where the reservoir interval is in sandstone in the Asmari Formation. Productivity is proven, and especially on its middle and upper part contains hydrocarbons.
Mapping of a seismic trace into the time-frequency domain produces a two dimensional data set by adding a frequency axis. In a similar way a two dimensional seismic section will generate a 3D data cube in which the third axis is frequency up to the Nyquist frequency. Sections of single frequency extracted from the cube are called single frequency seismic section (SFS). Comparison of different SFSs can be utilized to detect low frequency shadows caused by the presence of the hydrocarbon reservoirs. This method can potentially be utilized as a tool for direct hydrocarbon detection (Zabihi, 2006). We compared the single frequency seismic section of the STFT and CWT and discussed the differences between the results of the two methods. In a single frequency seismic section at frequency 15 Hz we found a low frequency anomaly below the reservoir which is a known phenomena. This anomaly disappeared at higher frequency single frequency seismic sections.
Application 2: Another application is thin layer analysis in a time-frequency domain. The idea is that tuning thickness is dependent on the dominant fre