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![]() | Seismic data analysis using local time-frequency decomposition | ![]() |
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The Fourier series is by definition an expansion of a function in terms of
a sum of sines and cosines. Letting a causal signal,
, be in range of
, the Fourier series of the signal is given by
The notion of a Fourier series can also be extended to complex coefficients as follows:
Nonstationary regression allows the coefficients
to change with
. In the linear notation,
can be obtained by solving the
least-squares minimization problem
We use shaping regularization (Fomel, 2007b) instead of Tikhonov's regularization to constrain the least-squares inversion. Shaping is a general method for imposing constraints by explicit mapping the estimated model to the desired model, eg., smooth model. Instead of trying to find and specify an appropriate regularization operator, the user of the shaping-regularization algorithm specifies a shaping operator, which is often easier to design.
The absolute value of time-varying coefficients
provides a
time-frequency representation, and equation 2 provides the
inverse calculation. In the discrete form,
a range of frequencies can be decided by the Nyquist frequency
(Cohen, 1995) or by the user's assignment. In a somewhat different
approach, Liu et al. (2009) minimized the error between the input signal
and each frequency component independently. Their algorithm and the
proposed algorithm are equivalent when the decomposition is stationary
(or using a very large shaping radius), because they both reduce to
the regular Fourier transform. In the case of nonstationarity, their
approach does not guarantee invertability, because it processes each
frequency independently.
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![]() | Seismic data analysis using local time-frequency decomposition | ![]() |
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