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All the derivations that follow are made under the assumption that the
velocity is constant (straight rays) and that all reflectors are flat
and horizontal.
In order to derive the shape of
FEAVO effects in the angle domain
(1),
we first derive the shape of FEAVO
effects in the offset domain.
vilus
Figure 7 Physical explanation for the
expression of FEAVO anomalies in midpointoffset space.
In the upper picture,
the blobs are transmission anomalies and the arrows are raypaths for
zero offset and 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}.
skema2
Figure 8 The right half
of case B in Figure 7. Raypaths are in blue. The
transmission anomaly is in B, at a depth of z_{a}. AE (of length f
 full offset) is at the Earth's surface, C is on the reflector at the
anomaly midpoint (m_{a}), D is on the reflector at midpoint m and
depth z. DE is perpendicular to the surface; BC is perpendicular to
the reflector.

 
Consider case B (the general case) in Figure 7
Vlad and Biondi (2002). For the zerooffset experiment, the
focusinggenerating anomaly affects only its own midpoint. For any
other offsets, it affects two midpoints that grow increasingly distant
with offset.
In Figure 8, because the reflector is parallel to the surface,
 
(2) 
Applying the same reasoning to the left
side of case B in Figure 7,
we can write the equation for both slanted streaks at
depth z
as
 
(3) 
Figure
9 depicts parts of the
corresponding surface for a 20m deep anomaly. Notice the arched form
of the surface with the midpointdepth vertical planes at maximum
offset. This (with very different vertical scaling) is the ``bullet
shape'' observed by Ottolini and Rocca (1982) in a real dataset.
20_max500
Figure 9 Fragment of the surface
described by equation 3, between 0 and 500m, for a
transmission anomaly 20 m deep. The shape resembles the bow
of an overturned boat.

 
The offset f can be easily replaced with the reflection angle in
this case because the reflector is flat:
 
(4) 
Plugging in (3),
 
(5) 
which can also be written as (1).
patpic
Figure 10 Left, from top to bottom:
1. Wavefield recorded 6 km deep after propagation through constant
velocity (background wavefield); 2. Quasi linearly scattered wavefield
(physically equivalent to the difference between the wavefield
propagated through the velocity model containing the slab  panel 6 of
Figure 6  and the background wavefield); Right, from
top to bottom: 3. Ratio between the maximum amplitudes in panel (1+2) and
panel 1, for each x location; 4. Difference between the times of the
maximum amplitudes in 1 and (1+2), for each x location. The wavefield was
propagated with the operator in equation 9. Panel 4 is in
very good accord with panel 4 of Figure 6 and with the
analytical time delay (8.7 ms).
Next: APPENDIX B  THE
Up: Vlad: Velocity estimation for
Previous: REFERENCES
Stanford Exploration Project
11/11/2002