Geometrical representation of ${\bf H^TH}$

In Appendix B an expression for the operator ${\bf H^TH}$ is derived. It says that the value of the operator ${\bf H^TH}$ at a point (t,x) is equal to a weighted sum of all values lying on the hyperbolas passing through the point (t,x). The hyperbolas correspond to the velocities used in velocity analysis. Operators for various kinds of sampling are shown in Figure . From these figures we can judge which sampling to choose.

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 ${\bf H^TH}$ 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).