S-transform with maximum energy concentration and its application to detect gas bearing zones and low-frequency shadows

Seismic attribute is a quantitative measure of an interested seismic characteristic. There are several seismic attributes. In recent years, time-frequency (TF) attributes have been developed which to reach them, TF analyzing of seismic data is required. A high resolution TF representation (TFR) can yield more accurate TF attributes. There are several TFR methods including short-time Fourier transform, wavelet transforms, S-transform, Wigner-Ville distribution, Hilbert-Huang transform and etc. In this paper, the S-transform is considered and an algorithm is proposed to improve its resolution. In the Fourier-based TFR methods, the width of the utilized window is the main factor affecting the resolution. The standard S-transform (SST) employs a Gaussian window which its standard deviation, controller the window width, changes inversely with frequency (Stockwell et al., 1996). It was an idea to use a frequency dependent window for TF decomposition. However, the TF resolution of SST is far from ideal; it demonstrates weak temporal resolution at low frequencies and weak spectral resolution at high frequency components. Later on, the generalized S-transform was proposed using an arbitrary window function whose shape is controlled by several free parameters (McFadden et a., 1999; Pinnegar and Mansinha, 2003). Another approach to improve the resolution of a TFR is based on energy concentration concept (Gholami, 2013; Djurovic et al., 2008). According this approach, in this paper, an algorithm is proposed to find the optimum windows for S-transform to get a TFR with maximum energy concentration. To reach this aim, an optimization problem is defined where an energy concentration measure (ECM) is employed to condition the windows so as the TFR would have the maximum energy concentration. Here, we utilize a Gaussian as the window function. Then different windows are constructed by a range of different values of standard deviations in a non-parametric form. Different TFRs are constructed by different windows. The optimum TFR is one with maximum energy concentration. The optimization is performed for each frequency component, individually, and hence, there would be an optimum window width for each frequency component. There are several ECMs which they are used in different applications (Hurley and Rickard, 2009). In this paper, we employ Modified Shannon Entropy as the ECM. As one knows, SST algorithm needs to be implemented in frequency domain (Stockwell et al., 1996). It is due to the dependency of the standard deviation of Gaussian window on the frequency. However, the proposed method of our paper can also be implemented in time domain where the optimum windows would be found, adaptively, for each time sample of the signal. We apply the proposed method to a synthetic signal to compare its performance with some other TF analysis methods in providing a well-concentrated TF map. The comparison of the results shows the superiority of the proposed method rather than STFT and SST. We also perform a quantitative experiment to evaluate the performance of the TFRs. The results confirm the best performance by the proposed method compared with STFT and SST. Then the proposed method is employed to detect gas bearing zones and low-frequency shadows on a seismic data set related to a gas reservoir of Iran. For this aim, some TF seismic attributes are extracted. The attributes include instantaneous amplitude, dominant instantaneous frequency, sweetness factor, single-frequency section and cumulative relative amplitude percentile (C80). The attributes are also extracted by SST to compare with those of the proposed method. The results show that the attributes obtained by the proposed method have more resolution; so that gas bearing zones and low-frequency shadows are better localized on the attribute sections obtained by the proposed method.