FEAVO effects in the data

The first sign of FEAVO that one may encounter in a dataset are subvertical streaks of alternating high and low amplitudes in constant-offset sections (left panel in Figure ). At a closer inspection, the affected areas show traveltime departures from hyperbolicity as small as 2-3 ms Carazzone et al. (1984) and as large as 20 ms Kjartansson (1979). An illustration of these effects is presented in the middle panel of Figure . The amplitudes in these areas may be easily three times larger than those in unaffected areas White et al. (1988). The effects may be frequency-dependent and distort the wavelet Stephens and Sheng (1985); Vlad and Biondi (2002).

Needed: modeling of purely-velocity and purely-absorption FEAVO, in order to investigate whether the frequency-dependent effects can serve to discriminate between FEAVO caused by absorption and that caused by velocity.

While the above-described effects are visible and are what started FEAVO research in the seventies, FEAVO's certain ``signature'' in the data domain (before migration) is the ``Kjartansson V's''. These shapes appear if we window a prestack 2-D line to roughly include the areas with anomalous amplitudes, then take the absolute value of each sample, stack the prestack dataset along the time axis and display the resulting midpoint-offset plot with an appropriate gain. ``V'' shapes become visible (right panel in Figure ). This may not occur if the background velocity in the medium varies so strongly with midpoint as to distort these shapes too much.

 

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Figure 3 FEAVO in the data domain (before migration). Grand Isle dataset, also used by Vlad and Biondi (2002) Left: vertical amplitude streaks in constant-offset section. Center: Milisecond-sized departures from hyperbolicity. Most visible at 2.35s. Right: Kjartansson V's in the midpoint-offset space, after stacking the unsigned values along the time axis.

The heuristic used by Kjartansson (1979) to explain the formation of the ``V'' shapes is presented in Figure .

 

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Figure 4 The physical explanation for the expression of FEAVO anomalies in midpoint-offset space Kjartansson ``V''s. In the upper picture, the blobs are transmission anomalies and the arrows are raypaths for the zero offset and for the maximum offset recordings. For case A (anomaly on the reflector), only a single midpoint is affected, for all offsets. Case C (anomaly at the surface), is actually a static: its ``footprint'' is a pair of streaks slanting 45^o from the offset axis. Case B (in between) gives a pair of streaks with angles smaller than 45^o. From Vlad and Biondi (2002).

The ``V'' shapes are the result of stacking along the time axis surfaces which in constant velocity are described by  

$$h = \frac{t}{{t - t_a }} \cdot \left\vert {m - m_a } \right\vert$$ (1)

A form of this equation is given by Rocca and Toldi (1982), with a simpler proof of another form in Vlad (2002). h is the half offset, t is traveltime, m is midpoint, t_a and m_a are the location of the heterogeneity that causes the focusing. Figure gives a better view of the path of the FEAVO effects through the prestack dataset. The shape resembles the bow of a capsized boat. Its slope becomes asymptotically vertical with time and the opening of the ``V''s becomes asymptotically $45^\circ$as the traveltime to the focusing source becomes negligible. It is visible now why stacking along the time axis a window in the middle of the prestack data volume would produce a ``V''.

 

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Figure 5 Path of FEAVO effects through the prestack data volume in constant velocity, flat reflectors. The source of focusing is at midpoint 0 and traveltime 20ms. Two views are provided for a better perception of the tridimensional surface.

A subtle, little-studied aspect of FEAVO effects is the distribution of the anomalous amplitudes when the anomaly paths described above intersect reflectors, for which I will use the name ``FEAVO microstructure''. Since an absorption-free, velocity-only ``lens'' conserves energy, any increase in amplitudes would have to be bordered by one or two shadow zones, and a decrease - by two illuminated zones. Finite-frequency wave theory predicts this for absorption-free media and ultrasonic experiments confirm it, as illustrated by Figure 4 of Spetzler et al. (2004). I am not aware of any equivalent studies for absorption. The existence or not of shadow/hightlight border zones for absorption is important because it may offer an avenue of discriminating between absorption-caused FEAVO and velocity-caused FEAVO, a key issue when trying to remove focusing with methods based on the physics of the phenomena (not just image processing).

Needed: A theoretic/numeric study of the magnitude of the shadow/highlight efects bordering FEAVO with parameters likely to be encountered in real exploration surveys, including absorption.

The polarity of the FEAVO on a reflector depends on the polarity of the velocity anomaly causing the FEAVO (negative or positive with respect to the background) and on the polarity of the reflector itself. The rightmost ``V'' in the bottom panel of Figure illustrates the dependence of the polarity of the ``V''s on the sign of the velocity anomaly. This will have important consequences on the choice of FEAVO removal strategies, treated in a separate section towards the end of this paper.