Fast Inverse-Q Filtering in Time-frequency Domain

Local Time-frequency map can also used for time-varying Q-factor estimation and inverse-Q filtering. Wang et al. (2020) developed the local centroid frequency shift (LCFS) method for time-varying Q-estimation in time-frequency domain. The SLTFT method can provide accurate time-frequency spectrum for Q-factor estimation with high efficiency. We selected a 2D poststack seismic profile to perform numerical test (see Fig.10a). The seismic section has a time length of 4.5 s with a time sampling of 4 ms and contains 247 traces in total. We calculated the $t\textrm {-}f\textrm {-}x$ spectrum of the seismic section (see Fig.10b) by using the SLTFT ( $\varepsilon =0.985$). The spectrum shows that the frequency range is getting narrow as the seismic wave propagates. The local centroid frequency (LCF) can be calculated by using the $t\textrm {-}f\textrm {-}x$ map (Liu and Fomel, 2013; Wang et al., 2020), as is shown in Fig.10c. We used the LCFS method in time-frequency domain to estimate the time-varying equivalent Q-factors for each trace. Fig.11a-11c show the calculated Q-factors by using the S transform, the LTF decomposition and the SLTFT, respectively. The Q-factors obtained by the LTF decomposition and the SLTFT are reasonably distributed according to the LCF information (see Fig.10c), while the S-transform fails in providing an unbiased Q-factor map. The estimated Q-factors are used to perform inverse-Q filtering, and Fig.12a - 12c show the enhanced seismic sections corresponded to Fig.11a-11c, respectively. Because of its biased estimation of Q-factors, the S-transform exhibits limitations in effectively enhancing the resolution of deep reflection signals (around 3 $\sim$ 4 s) (see Fig.12a) when compared to the results of the LTF decomposition and the SLTFT (see Fig.12b and 12c), where the attenuation of the reflections is compensated well, and the original structural characteristics are reasonably preserved. Meanwhile, the proposed method effectively reduces more computational costs (see Table. 2) than the LTF decomposition. Additionally, it achieves storage cost reduction through flexible frequency down-sampling. Specifically, in the SLTFT, we utilize half the frequency samples compared to the S-transform, while maintaining a superior enhancing profile.

powdata,sltft0,ltft_cf
Figure 10. (a) 2D poststack seismic section, (b) the corresponding $t\textrm {-}f\textrm {-}x$ cube and (c) local centroid frequency.

st_eqvq,ltft_eqvq,sltft_eqvq
Figure 11. The estimated equivalent Q-factors by using (a) S-transform, (b) LTF decomposition and (c) SLTFT ( $\varepsilon =0.985$).

stresult,ltftresult,sltftresult
Figure 12. Enhanced seismic sections by using (a) S-transform, (b) LTF decomposition and (c) SLTFT ( $\varepsilon =0.985$).