2-D results

Realistic acquisition parameters were modeled as in the 1-D example. The shot radiation pattern was modeled as $\cos\theta_s$ for a near-surface impulsive source, the receiver radiation pattern was modeled as $\cos^2\theta_r$ to emulate 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 240 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 6.0 s at a 4 ms sample interval, and the data are bandpassed to the 10-50 Hz spectral range. Twice the number of shots and 2 extra seconds of recording time were required to capture the 30 degree dipping event in a comparative manner to the 1-D recording geometry.

Figure shows a representative shot gather. Note the crossing of events due to structural dip at the far offsets. This far offset event mixing is expected to cause some amplitude estimation problems, since the events never completely decouple. 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, and varying data aperture with dip. In fact, the horizontal reflector is strongly positive due to the migration stacking bias for $\Theta \gt 15^{^{\circ}}$, and the 30 dipping event is actually negative due to the data aperture bias at reflection angles less than 15.

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 DMO/NMO and amplitude-corrected CMP gather. Note the correct polarity reversal with increasing offset. This time, however, the 2-D dipping structure has caused the polarity crossing to be skewed to far offsets at depth, in contrast to the 1-D CRP of Figure . That is because far offset geometries tend to move the reflection point up dip on a dipping reflector, and narrow the aperture angle.

Figure is the associated $\Theta({\bf x};{\bf x}_h)$ reflection angle gather. 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, even though there is significant skew due to dip, especially on the 30dip at 3.0 km depth.

Figure shows the amplitude recovery with angle on the shallow horizontal reflector at 1.0 km depth. This is an interesting picture that can teach us a lot about practical amplitude recovery. At first, I thought that this plot should look identical to Figure , since they are both horizontal reflectors at 1.0 km depth. However, several factors come into play to degrade the far offset amplitude recovery. The first is spatial aliasing of the migration operator from the deeper, dipping reflections. As the dip increases, the spatial aliasing gets worse, and those aliased-operator artifacts can cause significant destructive interference and amplitude contamination on a completely different reflector (i.e., the shallow horizontal reflector). Second, the dip on the 15 and 30 events causes the three hyperbolas to overlap each other at far offsets (see Figure ). If you run a movie of all the shot gathers, you see that these three events never really decouple themselves very well at any point along the line. That means there is no easy way to distinguish their respective amplitude contributions at far offsets, and therefore the far offset amplitude estimates will be significantly in error. This effect would not be seen in a synthetic test involving only one dipping reflector, or non-overlapping reflection hyperbolas. This result provokes some sobering thoughts regarding the implications for AVO in 2-D dip structure using standard offset cables lengths.

As the dip increases, the illumination aperture narrows, and the amplitude estimate gets better because of the increased redundancy in the least-squares summation estimate. Also the effects of operator aliasing decrease with depth of target and smaller illumination angles. Figure shows reasonably good amplitude and angle estimation up to 30, and Figure shows excellent amplitude recovery on the 30 dipping event. Note that the range of $\Theta$ illumination on the deeper dipping reflector (2-22) is much less than the shallower horizontal reflector (5-45), as expected.