1-D results

Figure shows a representative shot gather. Realistic acquisition parameters were modeled as follows. The shot radiation pattern was modeled as $\cos\theta_s$ for a near-surface impulsive source, the receiver radiation pattern was modeled as $\cos^2\theta_r$ which emulates the receiver group and vertical component amplitude effect. The geometrical spreading was modeled proportional to r^-1.5 to simulate typical v(z) spreading. Each of 120 60-fold shots were modeled with a 120-trace cable, a minimum and maximum receiver offset of 0.3 and 3.3 km, and a 25 m receiver group spacing. Total trace length is 4.0 s at a 4 ms sample interval, and the data are bandpassed to the 10-50 Hz spectral range.

Figure shows the result of a standard prestack migration. The reflection amplitudes are meaningless because of the migration stacking bias along prestack hyperbolas with varying angle-dependent reflectivity amplitudes. The shallowest horizontal reflector is strongly positive due to the migration stacking bias for $\Theta \gt 15^{^{\circ}}$; the recording geometry gives an illumination aperture of 10-60. As the reflector depth increases, the recording geometry illumination aperture decreases also. This has the effect of decreasing the stacked reflector amplitudes with increasing depth, because the migration aperture stack is increasingly centered about both sides of the 15 polarity reversal, effectively summing to zero.

Figure shows the l₂ estimation of $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ for a common reflection point (CRP), or ``iso-x'' gather. This picture is the l₂ depth-migrated equivalent of an NMO and amplitude-corrected CMP gather. Note the correct polarity reversal with increasing offset. Figure is the associated $\Theta({\bf x};{\bf x}_h)$ reflection angle gather. Do not be alarmed by the apparent wild variation in this figure - it is an artifact of an artificial synthetic data set which contains a very sparse set of reflections. Elsewhere in this report, I show very stable reflection angle estimates for a real data pilot study over a producing gas reservoir in the Gulf of Mexico, using this same method. Figure is an overlay of the reflection angles (contoured in 5 increments), on top of the $\grave{P}\!\acute{P}$ estimates. Note that the 15 contour tracks the correct polarity crossing almost exactly, at each reflector depth.

Figure shows the l₂ estimation of $\grave{P}\!\acute{P}(\Theta)$ on the z = 1.0 km reflector at a common reflection point (CRP). Since the model is 1-D, these figures are independent of surface location, and depend only on source-receiver offset. The l₂ estimates (crosses) closely match the correct values (dots). There is an increasing error for angles beyond about 35 which is related to spatial aliasing of the shallow migration operator impulse response. I was unsuccessful at attempts to anti-alias this operator while simultaneously preserving wide-angle reflection amplitude estimates.

Figure shows a much improved l₂ estimate of $\grave{P}\!\acute{P}(\Theta)$ on the z = 1.5 km reflector. This depth is apparently below the spatial-aliasing limit, and so achieves accurate amplitude values for the entire 5-45 illumination aperture. Similarly, Figures to show very accurate amplitude recovery of the polarity reversal AVO curve over all recording geometry apertures. Note in particular that the l₂ nature of the inverse solution is compensating for finite recording aperture, since amplitudes are not degraded at the edges of the illumination cone (nearest and farthest angles). It is also worth mentioning that all these amplitude plots have been scaled to a single constant value. Since they all track the correct AVO curves in an absolute sense, the relative amplitude recovery from reflector to reflector, as well as in angle along a given reflector, is correct. Finally, note how the illumination aperture narrows from about 10-60 at 1 km depth, down to 5-25 at 3 km depth.

In summary, the l₂ theory derived here accurately estimates the theoretical angle-dependent reflectivity function for a 1-D reflector model in the presence of source/receiver radiation patterns, geometric spreading, high-frequency Q attenuation (not shown here), and limited data acquisition aperture. It may be worth noting that the so-called ``2.5-D'' amplitude correction to the recorded data is not required in the l₂ formulation, since data missing from the y-direction is just another form of finite recording aperture. However, the associated $\pi/4$ phase shift is required, since phase does not seem to be properly taken into account in the l₂ energy norm.

 

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Figure 2 Synthetic shot gather for five horizontal reflectors.

 

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Figure 3 Standard prestack migration and stack. Reflection amplitudes are meaningless due to migration stack bias along hyperbolic prestack angle-dependent reflectivity variation.

 

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Figure 4 Migrated estimate of $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ at a CRP or ``iso-x'' gather. Note correct AVO polarity reversal with offset.

 

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Figure 5 Migrated reflection angle estimates of $\Theta({\bf x};{\bf x}_h)$ for a CRP gather, contoured in 5 increments.

 

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Figure 6 Overlay of $\Theta({\bf x};{\bf x}_h)$ on $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$.

 

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Figure 7 $\grave{P}\!\acute{P}(\Theta)$ recovery on the horizontal reflector at 1.0 km depth.

 

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Figure 8 $\grave{P}\!\acute{P}(\Theta)$ recovery on the horizontal reflector at 1.5 km depth.

 

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Figure 9 $\grave{P}\!\acute{P}(\Theta)$ recovery on the horizontal reflector at 2.0 km depth.

 

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Figure 10 $\grave{P}\!\acute{P}(\Theta)$ recovery on the horizontal reflector at 2.5 km depth.

 

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Figure 11 $\grave{P}\!\acute{P}(\Theta)$ recovery on the horizontal reflector at 3.0 km depth.