APPENDIX A - Derivation of formulas for FEAVO anomaly shapes

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 (), we first derive the shape of FEAVO effects in the offset domain.

 

vilus
Figure 7 Physical explanation for the expression of FEAVO anomalies in midpoint-offset 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 . Raypaths are in blue. The transmission anomaly is in B, at a depth of za. AE (of length f - full offset) is at the Earth's surface, C is on the reflector at the anomaly midpoint (ma), 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  (). For the zero-offset experiment, the focusing-generating anomaly affects only its own midpoint. For any other offsets, it affects two midpoints that grow increasingly distant with offset. In Figure , because the reflector is parallel to the surface,  

$$\left. {\begin{array} {*{20}c} {CD\vert\vert AE \Rightarrow... ...\kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$$ (51)

Applying the same reasoning to the left side of case B in Figure , we can write the equation for both slanted streaks at depth z as  

$$f = \frac{z}{{z - z_a }} \cdot 2\left\vert {m - m_a } \right\vert$$ (52)

Figure depicts parts of the corresponding surface for a 20m deep anomaly. Notice the arched form of the surface with the midpoint-depth vertical planes at maximum offset. This (with very different vertical scaling) is the ``bullet shape'' observed by () in a real dataset.

 

20_max500 Figure 9 Fragment of the surface described by equation , 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:  

$$\theta = A\hat DE \Rightarrow \tan \theta = \frac{AE}{ED} = \frac{f}{2z}.$$ (53)

Plugging in (),  

$$\left\vert {m - m_a } \right\vert = \left( {z - z_a} \right) \tan \theta$$ (54)

which can also be written as ().

 

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 - 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 . Panel 4 is in very good accord with panel 4 of Figure and with the analytical time delay (8.7 ms).