Attenuation of truncation effect

In Jedlicka, 1989, I introduced a velocity analysis using stochastic normal moveout (SMOC). Let us briefly review the basics. A sampling interval $\Delta m$ is chosen in the sloth domain, but instead of a sum along hyperbolas, a sum is made from all amplitudes lying between adjacent hyperbolas. In other words a histogram is created (Figure ). This operation is equivalent to the filtering of $(\tau,m)$-space with a boxcar filter over the sloth domain. A spectrum of the triangular filter

$${\Huge \bf \triangle = \Box * \Box}$$ (11)

has lower sidelobes than the spectrum of the boxcar filter and therefore it causes less overlapping (Figure ). Furthermore it can be viewed as an approximation to a Gaussian. This is the initial idea of Francis Muir that led to the concept of SMOC.

From Figure b we can see that the truncation effect is strongly attenuated. For comparison, a velocity analysis panel shown in Figure c was filtered over the sloth domain; the result should be equivalent to the velocity stack produced with SMOC. However, the result shown in Figure a is not so clean. This comparison shows the advantage of using SMOC over the filtering in the sloth domain. The length of the filter should not be chosen to be too small, otherwise the truncation effect is not attenuated (Figure c). Choosing the length of the filter to be too big leads to a fuzzy appearance of the velocity analysis panel.