Continuous time-varying Q-factor estimation method in the time-frequency domain

Estimation of local centroid frequency

To verify the feasibility of calculating the local centroid frequency using the LTFT method, a nonstationary signal (Figure 1b) is generated by convolving the Ricker wavelet with a random reflection coefficient (Figure 1a). The dominant frequency of the signal is a function varying with time $f_d(t)=100-75t^2$ . Figure 2a shows a time-frequency spectrum consisting of the Ricker wavelet’s frequency spectrum (the dominant frequency is $f_d(t)=100-75t^2$ ). According to the dominant frequency of the Ricker wavelet, we can calculate the theoretical centroid frequency (black line in Figure 2a) of the Ricker wavelet $f_c(t)=2f_d(t)/\surd{\pi}$ (Hu et al., 2013). Figures 2b and 2c show the time-frequency spectrum of the synthetic signal obtained using the LTFT and S-transform, respectively. We estimated the local centroid frequency from these two time-frequency spectra, as shown in Figure 3 (the blue line is estimated from the LTFT, and the purple line is estimated from the S-transform). Compared with the theoretical centroid frequency (black line in Figures 2a and 3), the local centroid frequency obtained using the LTFT method is closer to the theoretical curve, so the LTFT analysis method is selected for calculating the local centroid frequency and time-varying Q-factors.

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Figure 1. Theoretical model. Random reflectivity series (a), synthetic nonstationary signal (b).

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Figure 2. Time-frequency spectrum. Theoretical time-frequency spectrum (the black line represents the theoretical centroid frequency) (a), time-frequency spectrum of the LTFT (b), time-frequency spectrum of the S-transform (c).

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Figure 3. Local centroid frequency estimation (the black line represents the theoretical centroid frequency, the blue line is estimated using the LTFT method, and the purple line is estimated using the S-transform).

Continuous time-varying Q-factor estimation method in the time-frequency domain