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![]() | Continuous time-varying Q-factor estimation method in the time-frequency domain | ![]() |
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The centroid frequency
with respect to the amplitude spectrum
can be defined as (Quan and Harris, 1997)
The variance
of the centroid frequency can be defined as
The time-frequency spectrum
replaces the Fourier amplitude spectrum
in equations 1 and equations 2, and the
instantaneous centroid frequency
and instantaneous variance
of the amplitude spectrum are defined as
The above equations show that the instantaneous centroid frequency and variance are calculated instantaneously using the amplitude spectrum information at a specific time. However, at times without effective spectrum information, reasonable results cannot be obtained using this calculation method.
Fomel (2007a) defined the local attributes of the seismic signals such as local frequency and local similarity using shaping regularization. In this paper, we use a similar method to define the local centroid frequency and local variance. Equation 3 shows that the instantaneous centroid frequency is a division regarding two integrals and can be expressed in linear algebraic notation as
The theory of shaping regularization comes from data smoothing. It has
fewer parameters and a faster convergence speed than the traditional
Tikhonov regularization method. When considering shaping
regularization, the shaping operator
can be defined as
The least-squares solution under the shaping regularization constraint can be obtained by substituting the above equation into equation 7
Similarly, the local variance
can be
calculated using the above method. When calculating the local centroid
frequency, only one smoothing parameter is needed to control the
locality and smoothness of the local centroid frequency. The local
centroid frequency is not calculated instantaneously using the
information at a specific time or calculated globally in a time window
but is calculated locally using the information around the time. Thus,
a relatively reasonable local centroid frequency can be continuously
and smoothly calculated at the time of missing information (such as
when the amplitude spectrum is zero). In this paper, we use the local
centroid frequency to estimate the continuous time-varying Q values of
the formation.
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![]() | Continuous time-varying Q-factor estimation method in the time-frequency domain | ![]() |
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