Spectral decomposition or time-frequency representation (TFR) is a powerful tool for analyzing time varying nature of seismic data. Mapping of a one dimensional seismic time trace to a two dimensional function of time and frequency reveals some characteristics of seismic signals that are not available in only time or frequency representations. This approach has been widely used in seismic exploration applications such as: denoising, direct hydrocarbon detection, seismic sequence analysis, reservoir characterization, and resolution enhancement. Obtaining a method with higher resolution in TFR and less computational cost is of fundamental significant in this field. Among different methods for spectral decomposition, the short time Fourier transform (STFT) or its squared modulus (the spectrogram), is the fastest one. However, it has limited resolution because of the uncertainty principle. The Wigner-Ville distribution (WVD) is another method that has superior TFR resolution, but its practical application is limited by undesirable cross terms because of its bilinear nature. The spectrogram obtained by the STFT method is related to
the WVD via a 2-D deconvolution. The Deconvolutive Short Time Fourier Transform Spectrogram (DSTFTS) is another method that has been introduced recently to increase the TFR resolution of the spectrogram by applying the Lucy-Richardson algorithm for 2-D deconvolution operation. However, as expected theoretically, the resolution of the DSTFTS is not very close to the resolution of the WVD.
In this paper, we first explain why the resolution of the deconvolutive spectrogram, introduced by Lu and Zhang (2009), is not as close as to that of the WVD, and then we introduce its modified version which effectively improves the TFR resolution. In the 2-D deconvolution process, the sampling interval of the input data must agree in both time and frequency. We have shown that the sampling interval of the WVD of the window function in the frequency direction is not equal to that of the spectrogram of the signal. A simple technique is proposed here to overcome this problem. The proposed modified deconvolutive spectrogram provides better results compared to those of Lu and Zhang (2009). The TFR resolution is very close to that of the WVD because of the correct implementation of the deconvolution process. As we performed deconvolution by the Lucy- Rchardson algorithm which is not an ideal and perfect process, our TFR is not the same as that of the WVD.
We have evaluated the performance of the modified deconvolutive spectrogram in spectral decomposition of a chirp signal; a synthetic signal consists of four Morlet wavelets and a real seismic trace. In comparison with the popular reassigned spectrogram method (Auger and Flandrin, 1995), the modified deconvolutive spectrogram effectively improves the TFR resolution and has no artifact and undesirable effects on TFR of any type of input signal. However, the reassigned spectrogram eliminates some components and disturbs the shape of the TFR of the complex input signals. These issues can make a method completely unusable.
We have shown that both the reassigned and the modified deconvolutive spectrograms have the same nature. The aim of the reassigned spectrogram is to improve the degree of the localization of the signal components by reallocating them over the time-frequency plane and applying a weighted integration. On the other hand, the aim of the modified deconvolutive spectrogram is to directly remove and compensate the damping effect of the window function by applying the 2-D deconvolution operation. From the utilized mathematical tools point of view, the 2-D deconvolution algorithms are more advanced and more reliable than the weighted integration (especially in case of signals with a complicated TFR such as seismic data).
Finally, we applied the method to detect low frequency shadow associated with a possible thin gas reservoir on a seismic section. The low frequency shadow has been used as a seismic indicator of a hydrocarbon accumulation. Several reasons have been proposed for this shadow, such as abnormally high attenuation in gas filled reservoir and some other mechanism in data processing. By spectral decomposition of all traces, a cube of data has been obtained from the seismic section. We have used single frequency seismic sections, extracted from the cube, for interpretation. According to the results, high amplitude energy on the 20 Hz single frequency seismic section has been disappeared on the 50 Hz single frequency seismic section. This high amplitude energy is a hydrocarbon indicator that exists beneath the reservoir. The superior resolution of the modified deconvolutive spectrogram resulted in a remarkably better localization of the reservoir. Therefore, the modified deconvolutive spectrogram is a fast and effective method for spectral decomposition of seismic data, especially when it is used for seismic attributes extraction.