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Introduction

Geological events and geophysical data often exhibit fundamentally nonstationary variations. Therefore, time-frequency characterization of seismic traces is useful for geophysical data analysis. A widely used method of time-frequency analysis is the short-time Fourier transform (STFT) (Allen, 1977). However, the window function limits the time-frequency resolution of STFT (Cohen, 1995). An alternative is the wavelet transform, which expands the signal in terms of wavelet functions that are localized in both time and frequency (Chakraborty and Okaya, 1995). However, because a wavelet family is built by restricting its frequency parameter to be inversely proportional to the scale, expansion coefficients in a wavelet frame may not provide precise enough estimates of the frequency content of waveforms, especially at high frequencies (Wang, 2007). Therefore, Sinha et al. (2009,2005) developed a time-frequency continuous-wavelet transform (TFCWT) to describe time-frequency map more accurately than the conventional continuous-wavelet transform (CWT). The S transform (Stockwell et al., 1996) is another generalization of STFT, which extends CWT and overcomes some of its disadvantages. Pinnegar and Mansinha (2003) developed a general version of the S transform by employing windows of arbitrary and varying shape. The clarity of the S transform is worse than the Wigner-Ville distribution function (Wigner, 1932), which achieves a higher resolution but is seldom used in practice because of its well-known drawbacks, such as interference and aliasing. For this reason, Li and Zheng (2008) provided a smoothed Wigner-Wille distribution (SWVD) to reduce the interference caused by the cross-term interference. The matching pursuit method is yet another approach to representing the time-frequency signature (Wang, 2010,2007; Liu and Marfurt, 2007). Matching pursuit involves several parameters and is a relatively expensive method. There are some other approaches to spectral decomposition. Castagna and Sun (2006) compare several different spectral-decomposition methods.

Liu et al. (2011,2009) recently proposed a new method of time-varying frequency characterization of nonstationary seismic signals that is based on regularized least-squares inversion. In this paper, we expand the method of Liu et al. (2011) by designing an invertible nonstationary time-frequency decomposition -- local time-frequency (LTF) decomposition and its extensions -- local time-frequency-wavenumber (LTFK) and local space-frequency-wavenumber (LXFK) decompositions. The key idea is to minimize the error between the input signal and all its Fourier components simultaneously using regularized nonstationary regression (Fomel, 2009) with control on time resolution. This approach is generic, in the sense that it is possible to combine other basis functions, eg., fractional splines, with regularization (Herrmann, 2001). Although there is an iterative inversion inside the algorithm, one can use LTF decomposition as an invertible "black box" transform from time to time-frequency, similar in properties to the S transform. The proposed decompositions can provide local time-frequency or space-wavenumber representations for common seismic data-processing tasks. We test the new method and compare it with the S transform by using a classical benchmark signal with two crossing chirps. The proposed LTF decomposition appears to provide higher resolution in both time and frequency when an appropriate parameters of the shaping regularization operator (Fomel, 2007b) are used to constrain the time resolution. Examples of ground-roll attenuation and multicomponent image registration demonstrate that the method can be effective in practical applications.


next up previous [pdf]

Next: Theory Up: Liu and Fomel: Local Previous: Liu and Fomel: Local

2013-07-26