Image of a point

A velocity analysis panel is obtained by transformation of a CMP gather. In this paper an operator of summation will be used. This operator ${\bf S}$ transforms (t,x)-space into $(\tau,m)$-space; i.e., it is an operator of transformation from the time-offset domain into the time-sloth domain. We are interested in the image in $(\tau,m)$-space of a point in (t,x)-space.

Let a gather G_tx contain only one non-zero amplitude that is located at time t of a trace with the offset x (Fig. a). Let us apply the operator ${\bf S}$ on the gather G_tx. The operator ${\bf S}$ gives non-zero results only for hyperbolas passing through the point (T,X) (Fig. ). Each hyperbola is determined by the time $\tau$ and the sloth m, which satisfy the equation

$$T^2 = \tau^2 + X^2m,$$ (1)

which can be written

$$\tau ^2 = -X^2 m + T^2.$$ (2)

This is the equation of a parabola, so we have

$${\bf S}(point) = parabola.$$ (3)

A parabola corresponding to the point (T,X) is shown in Figure b.