Modifications of basic equations

We can see that when we want to get back the initial data from the velocity panel, amplitudes at far offsets tend to resume their original values in a higher number of iterations than amplitudes at near offsets. This suggests that weighting depending on offset could speed the iterations.

Claerbout (1987) uses a weight that contains a square root of offset. It contains other factors such as time, slowness and count. Count does not play a significant role in most of the gather (see the section in this paper on count). I obtained the best results with this kind of weight when I used a square root of offset as weight. From Figure a we can see that this weight leads to the attenuation of near offsets. The result after one iteration may seem to look better with weighting than without weighting, but other iterations converge more slowly. Incorporating other factors into the weight led to the attenuation of amplitudes at high times.

Another possibility of how to give a higher weight to the far offsets is to stack all the amplitudes between two close hyperbolas. This is similar to the stochastic normal moveout (SMOC) (Jedlicka, 1989a), but there is no normalizing as in SMOC. It probably corresponds to the "antialias" operator in Claerbout (1987). There is a parameter describing how close the hyperbolas are. By playing with this parameter I obtained a result after one iteration (Figure b) with the same strength of amplitudes over all the offsets. Other iterations again converge more slowly and an event resembling a direct arrival appeared (Figure a). The velocity of the event is equal to the lowest velocity used in the inversion, which suggests that we have something to do with a computational artifact. A closer inspection of this event is in the section on count (Figure ). From this it follows that muting in NMO should attenuate this effect (Figure b).

Weighted modifications of the equations may serve for a quick look at the data after one iteration. For more iterations non-weighted equations give better results.