184 Seismic Imaging Figure 7.16 (top left) shows a VSP recorded in well Est 433 (Figure 7.3) as well as its frequency-wavenumber (f-k) diagram. It is composed of 221 levels, with a depth sampling of 5 m between 112 and 687 m, and a depth sampling of 10 m between 687 and 1,737 m. The source is a vibrator (sweep 14-140 Hz). The time sampling is 1ms and the recording length is 2 s after correlation. The down-going and up-going waves have been separated by f-k filtering. To obtain an estimate of the Q factor per layer, we used the fact that attenuation introduces dissipative dispersion, which can be measured from the frequency-dependent phase velocity of the VSP down-going wave (equation (7.6)). The picked times of the first pick of the first arrival have been used to compute the velocity log versus depth, and to measure the frequency log from the instantaneous frequency VSP section, using the analytic signal computed by the Hilbert transform. The 2 logs are shown in Figure 7.16 (top right). The down-going waves have been filtered with a low frequency band (8-28 Hz). The average value of the instantaneous frequency f2 of the filtered down-going wave is 21.5 Hz and the associated standard deviation is 2 Hz. A scan procedure in dΔt, equivalent to a velocity scan, was implemented to estimate the quantity ΔV and the Q factor (equation (7.6)), the search carried out in a realistic Q value domain, such as between 20 and 100 (Mari and Yven, 2018). If the f1/f2 ratio is close to a constant, the V/ΔV ratio is a linear function of Q. For Q ranging from 20 to 100, the V/ΔV ratio varies from 50 to 250 for an f1/f2 ratio of 3.5 (Figure 7.16, bottom left). Figure 7.16 (bottom left) shows, from top to bottom: the V/ΔV curve for an f1/f2 ratio of 3.5; the dΔt, ΔV and V/ΔV curves versus the level of the borehole geophone. The average value of the ΔV curve is 27 m/s and its associated standard deviation is 4 m/s. Figure 7.16 (bottom right, black curve) shows the computed Q factor log. Q values vary between 33 and 96. Due to standard deviation low values of the ΔV curve (4 m/s) and of the f2 frequency curve (2 Hz), the Q factors can be predicted using constant values for ΔV and f2. The predicted Q-factor curve is shown in Figure 7.16 (bottom right, red curve) as well as the associated relative uncertainty (10% on average) between the 2 Q-logs. The correlation coefficient between the 2 Q-logs is high (0.94). The correlation coefficient between Q factors and interval velocities is 0.69. The Q factor of the Cox (550-700 m) is estimated at 40, while the Q factors of the Oxfordian and Dogger limestone located directly above and below are higher, ranging from 65 to 90. The results have been validated on a second well, Est 412 (Figure 7.17), the predicted Q values being estimated from the law defined at well Est 433. The uncertainties between the 2 Q-logs are weak, ranging between 0.5 to 30%. Consequently, the predicted Q log can be applied to surface seismic data if interval velocities V ( f1) and instantaneous frequencies f1 are measured.
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