Multichannel adaptive deconvolution based on streaming prediction-error filter

1D attenuation model

We start with a 1D synthetic example with the quality factor $Q$ attenuation according to the modified Kolsky model (Wang, 2008; Wang and Guo, 2004). In this model, the phase velocity is defined as

$$\frac{1}{v(\omega)}=\frac{1}{v_r}(1-\frac{1}{\pi Q_r}\ln{\left\ve... ...approx\frac{1}{v_r} {\left\vert \frac{\omega}{\omega_h} \right\vert}^{-\gamma},$$ (16)

where $\gamma=\frac{1}{\pi Q_r}$ , $Q_r$ and $v_r$ are the quality factor and phase velocity at a reference frequency $\omega_r$ (the dominant frequency in genernal), $\omega_h$ is the tuning frequency. We generate a time-varying trace (figure 4b), where the dominant frequency of the unattenuated minimum-phase wavelet is 40 Hz, the time interval is 1 ms, and the Q value is 30. Figure 4a shows the actual reflectivity. For comparison, we use the traditional predictive deconvolution to squeeze all wavelets (figure 5a), the filter length $N$ is ten and the prewhitening factor is 0.0001. The traditional method produces a reasonable result at the high-frequency locations; however, the predictive deconvolution still loses part of the original amplitudes. We design the streaming PEF deconvolution with a constant prediction step ($N=3$ , $\epsilon_t=1.5$ , and $\alpha=1$ ) to further handle the variability of wavelet (figure 5c). The streaming PEF residual visually shows a result similar to the traditional method, however, a close-up comparison between the traditional deconvolution (figure 5b) and the proposed deconvolution (figure 5d) shows an obvious resolution difference, which proves better nonstationary characteristics of the streaming PEF.

Next, we improve the adaptive deconvolution result by involving the time-varying prediction step, and the result is shown in figure 6. Figures 6a and  6b show the decay of local frequency and the time-varying prediction step by using equation 14 where $b=0.06$ , respectively. Figure 6c shows that the proposed method keeps the relative amplitude relationship without auto gain correction (AGC) and the time resolution is reasonably enhanced. Figure 6d shows amplitude spectrum of the synthetic data before and after deconvolution, where the grey line is the original synthetic data and the black line is the deconvolution result. It can be seen from figure 6d that the amplitude spectrum broadens after the deconvolution.

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Figure 4. A synthetic seismic trace example. The reflectivity (a), synthetic trace with Q attenuation (b).

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Figure 5. Deconvolution results by using different methods. Traditional predictive deconvolution (a), local display of (a) (b), streaming PEF deconvolution (c), local display of figure (c) (d).

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Figure 6. Deconvolution by using streaming PEF with time-varying prediction steps. Local frequency (a), time-varying prediction step (b), the deconvolution result with the proposed method (solid line), which is compared with the original trace (dotted line) (c), amplitude spectrum (The grey line is the original data, and the black line is the deconvolution result) (d).

Multichannel adaptive deconvolution based on streaming prediction-error filter