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I now briefly discuss the 2-D pilot study I have conducted to indicate the practical utility of my prestack impedance inversion theory and algorithms.

Figure [*] shows two CMP gathers from a 1970's vintage 2-D marine survey. The gathers are from midpoint location 1 km and 2 km respectively, along the stacked section shown in Figure [*]. The CMP gathers exhibit a strong amplitude-offset (AVO) reflectivity response on the reflected event at about 2.3 seconds zero-offset time. Note the offset-variability in the raw AVO response. An advantage of prestack migration/inversion is that it uses an estimated variable velocity field to compensate for such wave-based focusing/defocusing effects on amplitudes in raw data.

The stacked section shows a few strong reflectors in the 2.3-2.5 second range, which are known to correspond to the reservoir interval of a producing gas field (Kjartansson, 1979). The offset stack across any AVO character will tend to invalidate the common assumption that the stacked section is an approximation of P impedance contrasts. This would have a negative impact on ``poststack impedance inversion'' for example.

Figure [*] shows the migration velocity semblance panels corresponding to the two CMP positions. The overlain black curves are the ``optimal'' migration velocities picked by a nonlinear Monte Carlo automatic search algorithm (Lumley, 1992b). After all 100 semblance scans are searched for the optimal migration velocity trajectory, the results are slightly smoothed and contoured in Figure [*]. The contours of the migration velocity field range from 1.6 km/s at the top to 2.3 km/s at the bottom, in 0.1 km/s intervals. The low velocity zone at the far left is apparently real, when the semblance scans at this location are examined. Also, note the low rms velocity zone at 2.3 seconds in the right scan, which will later be shown to correlate with an impedance anomaly indicative of increased gas saturation. This velocity model is used for the subsequent reflectivity estimation and impedance inversion.

Given 4800 input seismic traces, I perform 48 constant offset migrations ranging from offsets of 0.25 km to 2.65 km in 50 m increments. Figure [*] shows two Common Reflection Point (CRP) slices of the constant offset migration cube, each slice taken at a fixed midpoint position of 1 km and 2 km respectively, to correspond with the surface positions of the previous figures. This figure corresponds to the prestack-migrated equivalent of an NMO/DMO amplitude-corrected CMP gather in conventional AVO processing. The midpoint coordinate direction is perpendicular to the plane of the page. The top of the reservoir is at 2.3 seconds, and we are looking at how the migrated reflectivity estimate varies as a function of offset: $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$.At surface position 1 km, the top panel shows what appears to be a weak negative polarity reflection at the top of the reservoir, and the reflection strength approximately constant, although somewhat irregular, with increasing offset (angle). At the 2 km surface position (lower panel), the top of the reservoir appears to have changed to a somewhat stronger negative polarity and an increase in negative amplitude with increasing offset (angle). This difference suggests that the material properties in the reservoir have changed spatially from surface position 1 km to 2 km.

Figure [*] is the same plot as Figure [*], except the quantity being plotted is the incident reflection angle $\Theta({\bf x};{\bf x}_h)$ instead of the reflectivity itself. The contours range from reflection angles of $5^{^{\circ}}$ at near offsets to $30^{^{\circ}}$ at the farthest offsets, in increments of $5^{^{\circ}}$. Given the functions $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ and $\Theta({\bf x};{\bf x}_h)$ estimated from the data, the two can be combined to give the estimate of reflectivity as a function of reflection angle: $\grave{P}\!\acute{P}(\Theta({\bf x}))$.

For comparative purposes, I plot the prestack migrated stack in Figure [*]. This is obtained efficiently by merely summing the $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ gathers over the offset coordinate ${\bf x}_h$. It will be interesting to compare this image with the P and S impedance contrast maps.

Next, at each point in earth, I perform the linearized inversion of $\grave{P}\!\acute{P}(\Theta({\bf x}))$to estimate the elastic parameters $\{I_p,I_s,D\}$. The P impedance contrast section is shown in Figure [*], the S impedance contrast section is shown in Figure [*], and the density contrast section is shown in Figure [*]. As expected, the S/N ratio and stability of the inversion is good for P impedance, medium for S impedance, and very poor for density. Figure [*] justifies my claim that there is little or no useful independent information about subsurface density contrasts in standard acquisition geometry surface seismic data. Disregarding density, there appear to be some significant differences in information content comparing the P and S impedance contrast sections, especially near the 2 km midpoint location. Also, the migration stack of Figure [*] is most similar to the P impedance contrast section of Figure [*], as expected, but does have some orthogonal information content along the reservoir reflector at 2.3 seconds in the distance range of 1.2-1.6 km, for example.

A simple multiplication of the P and S sections produces an interesting P*S anomaly map, as shown in Figure [*]. In ``normal'' rock sequences, the P and S impedance contrasts are usually of the same polarity, and hence plot dark gray to black (blue) in Figure [*]. However, in a transition zone of changing rock properties, say from brine to gas-filled porespace, the P and S impedance contrasts may be of opposite polarity, and will plot as white (red) regions in Figure [*]. It is interesting to note that the white (red) anomalous regions of Figure [*] are well localized to the reflectors associated with the producing gas reservoir interval, and also importantly, nowhere else in the section.

This result indicates that the prestack impedance inversion theory and algorithms have been qualitatively tested and give reasonable physical results, since an AVO amplitude anomaly and a P*S impedance anomaly has been unambiguously detected by the method at the zone of a known gas producing reservoir interval.

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Next: CONCLUSIONS Up: Lumley: Prestack impedance inversion Previous: Impedance estimation
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