Extracting FEAVO effects in angle gathers

Extracting the FEAVO anomalies assumes that the long spatial wavelength velocity model is good enough that the reflectors are flat in angle gathers, and only the amplitudes remain to be fixed. Since the FEAVO effects are expressed both in the midpoint-angle domain and in the angle-domain common image gathers, their separation must proceed in a synergistic fashion. This involves the entire data volume. For the simplest case (constant velocity, flat reflectors), the FEAVO effects generated by a velocity anomaly at depth z_a and midpoint m_a will be distributed in the depth(z)-midpoint(m)-angle($\theta$) space along a surface described by:  

$$z = z_a + \left\vert {m - m_a } \right\vert \cot \theta.$$ (50)

The derivation is laid out in Appendix B, and the shape of the surface is shown in Figure .

 

ang3d_20_500_pi4 Figure 3 Shape of the FEAVO ``footprint'' in the depth-midpoint-angle space due to a velocity anomaly 20 m deep in an otherwise constant velocity medium with flat reflectors. For a better 3D visual understanding, the shape resembles the bow of a flipped boat.

Even for a v(z) case (Grand Isle dataset), the shape of the anomaly will not be very different, especially for a limited angle range (due to a finite range of offsets). Figure  shows the dips of the FEAVO effects are confined to a limited range.

 

ang20_100_pi4 Figure 4 Midpoint-angle contour map of FEAVO effects generated by a velocity anomaly 20m deep. The angle range is wide (up to $45^\circ$), which is wider than the range recorded in most of the real data sets. Therefore, it is unlikely that curvature of the anomalies be observed in real data panels.

Therefore, as a first measure for separating them, we can apply an appropriate f-k dip filter to the midpoint-angle slices (Figure ). This eliminates the largest part of the petrophysical AVO. There is, however, no guarantee that the remaining energy within the plausible FEAVO dip range does actually belong to FEAVO. I will have to separate the signal from noise in the manner of ():

1.

For each point in the depth-midpoint section, consider that it ``houses'' an anomaly and precompute the FEAVO-effect surface that depends on the known long spatial wavelength velocity field.

2.

Sum (or compute a semblance-like operator) along the precomputed surfaces to obtain a depth-midpoint ``anomaly map,'' taking care to distinguish between FEAVO caused by absorption and that caused by velocity.

3.

Filter the image based on its statistical properties, so only the most focused points remain.

4.

Spread the filtered image along the precomputed surfaces back into the depth-midpoint-angle space. Alternately, focusing could be done using the downward continuation operator itself.