Slant stacking and linear moveout

Looking on profiles or gathers for events of some particular stepout p=dt/dx amounts to scanning hyperbolic events to find the places where they are tangent to a straight line of slope p. The search and analysis will be easier if the data is replotted with linear moveout--that is, if energy located at offset x=g-s and time t in the (x,t)-plane is moved to time $\tau = t-px$ in the $(x, \tau )$-plane. This process is depicted in Figure 5.

 

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Figure 5 Linear moveout converts the task of identifying tangencies to constructed parallel lines to the task of locating the tops of convex events.

The linear moveout converts all events stepping out at a rate p in (x,t)-space to ``horizontal'' events in $(x, \tau )$-space. The presence of horizontal timing lines facilitates the search for and the identification and measurement of the locations of the events.

After linear moveout $\tau = t-px$,the components in the data that have Snell parameters near p are slowly variable along the x'-axis. To extract them, apply a low-pass filter on the x'-axis, and do so for each value of $\tau$.The limiting case of low-frequency filtering is extracting the mean. This leads to the idea of slant stack.

To slant stack , do linear moveout with $\tau = t-px$, then sum over x'. This is the same as summing along slanted lines in (t,x)-space. In either case, the entire gather $\ P ( x , t ) \$ gets converted to a single trace that is a function of $\tau$.

Slant stack assumes that the sum over observed offsets is an adequate representation of integration over all offset. The (slanted) integral over offset will receive its major contribution from the zone in which the path of integration becomes tangent to the hyperboloidal arrivals. On the other hand, the contribution to the integral is vanishingly small when the arrival-time curve crosses the integration curve. The reason is that propagating waves have no zero-frequency component.

The strength of an arrival depends on the length of the zone of tangency. The Fresnel definition of the length of the zone of tangency is based on a half-wavelength condition. In an earth of constant velocity (but many flat layers) the width of the tangency zone would broaden with time as the hyperbolas flatten. This increase goes as $\sqrt{t}$, which accounts for half the spherical-divergence correction. In other words, slant stacking takes us from two dimensions to one, but a $\sqrt{t}$ remains to correct the conical wavefront of three dimensions to the plane wave of two.