Each hyperbola represents an equation with up to *nx* variables, where *nx*
is the number of offsets.
Up to *nv* hyperbolas may pass through every point of the gather, where
*nv* is the number of velocities.
So each point appears in up to *nv* equations.
If some hyperbolas are close to each other, it means that the corresponding
equations are almost identical.
From Figure we can see that this phenomenon happens with sampling in
velocity and alacrity domains. For the same number of velocities we may actually have
fewer independent equations than when sampling is done in the
sloth domain.
For sampling in the slowness domain, it seems that velocity analysis should
not be done up to infinite velocity (Figure c). If some
reasonable limit of velocity is chosen, then the operator is comparable
to the operator with evenly sampled sloth.
These results agree with the results obtained by use of
singular value decomposition.

This representation of can be also used for an explanation of why far offsets converge more slowly than near offsets in the conjugate gradient method (Figure and Figure d).

1/13/1998