Representation of a velocity analysis panel

The CMP gather G can be viewed as a sum of gathers G_tx, each containing only one non-zero amplitude:

$$G = \sum_{t,x} A(t,x) G_{tx},$$ (4)

where G_tx is a gather containing a spike at (t,x) and A(t,x) is the amplitude of a trace with offset x at time t.

Because the image of G_tx is a $(\tau,m)$-space containing a parabola, we can write

$$V = \sum_{t,x} A(t,x) V_{tx},$$ (5)

i.e., the $(\tau,m)$-space can be viewed as a sum of panels, each containing one parabola. For a constant offset x, the resulting sets of parabolas are shown in Figure .

Let us denote as V_x the panel that we get for one offset x:

$$V_x = \sum_t A(t,x) V_{tx}.$$ (6)

V_x is an image of a trace with offset x. Finally we have

$$V=\sum_x V_x.$$ (7)

V_x consists of horizontal lines for x=0. For increasing offset V_x consists of parabolas with increasing slopes (Figure ).