


 Seismic data analysis using local timefrequency decomposition  

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Geological events and geophysical data often exhibit fundamentally
nonstationary variations. Therefore, timefrequency characterization
of seismic traces is useful for geophysical data analysis. A widely
used method of timefrequency analysis is the shorttime Fourier
transform (STFT) (Allen, 1977). However, the window function limits
the timefrequency 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 timefrequency continuouswavelet transform (TFCWT) to
describe timefrequency map more accurately than the conventional
continuouswavelet 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 WignerVille distribution function (Wigner, 1932), which
achieves a higher resolution but is seldom used in practice because of
its wellknown drawbacks, such as interference and aliasing. For
this reason, Li and Zheng (2008) provided a smoothed WignerWille
distribution (SWVD) to reduce the interference caused by the
crossterm interference. The matching pursuit method is yet
another approach to representing the timefrequency 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 spectraldecomposition methods.
Liu et al. (2011,2009) recently proposed a new method of timevarying
frequency characterization of nonstationary seismic signals that is
based on regularized leastsquares inversion. In this paper, we
expand the method of Liu et al. (2011) by designing an invertible
nonstationary timefrequency
decomposition 
local timefrequency (LTF) decomposition
and its extensions  local timefrequencywavenumber (LTFK)
and local spacefrequencywavenumber (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
timefrequency, similar in properties to the S transform. The proposed
decompositions can provide local timefrequency
or spacewavenumber representations for common seismic dataprocessing
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 groundroll attenuation and
multicomponent image registration demonstrate that the method can be
effective in practical applications.



 Seismic data analysis using local timefrequency decomposition  

Next: Theory
Up: Liu and Fomel: Local
Previous: Liu and Fomel: Local
20130726