Parabolic shape of the multiples

Let's suppose the NMO-correction of a CMP gather is performed with the velocity curve of the primaries (velocity V_p). A multiple with RMS velocity V_m and 0-offset time t₀ will be moved to the position:

$$t(h)=t_0+\sqrt{t_0^2+{h^2\over V_m^2}}-\sqrt{t_0^2+{h^2\over V_p^2}}$$ (1)

Defining the residual slowness 1/V_r and the parameter p by:

$${1\over V_r^2}={1\over V_m^2}-{1\over V_p^2} \;, \mbox{\hspace{0.5cm}}p=1/(2t_0V_r^2) \;,$$

a series expansion of t(h) will give:

$$t(h)\simeq t_0+ph^2 \mbox{\hspace{0.5cm}}(\mbox{for } {h\over t_0V_r}\ll 1).$$ (2)

V_r² is guaranteed to be positive, since we expect the velocity of a multiple to be smaller than the velocity of a primary at the same 0-offset time. So, for ``not too large'' offsets, the shape of a multiple turns out to be parabolic after NMO-correction.