NMO for Multiple Reflections

In a horizontally-stratified, v(z) medium, multiple reflections can be treated as kinematically-equivalent primaries with the same source-receiver spacing but additional zero-offset traveltime $\tau^*$, as illustrated in Figure 1. We can write an extension to the NMO equation which flattens multiples to the zero-offset traveltime of the reflector of interest.  

$$t^2 = \sqrt{ (\tau+j\tau^{*})^2 + \frac{x^2}{V_{eff}^2} }$$ (1)

$j\tau^{*}$ is the two-way traveltime of a $j^{\mbox{th}}$-order multiple in the top layer. $V_{eff}(\tau)$ is the effective RMS velocity of the equivalent primary shown in the figure. For the simple case of constant velocity v^* in the multiple-generating layer,  

$$V_{eff}(\tau) = \frac{ \tau^{*} v^{*} + \tau V(\tau) }{ \tau^{*} + \tau }$$ (2)

So for the common case of relatively flat reflectors, v(z), and short offsets, equation (1) should do a reasonable job of flattening water-bottom multiples of any order to the $\tau$ of interest, assuming that we pick the water bottom ($\tau^*$) and that we know the seismic velocity of water.

 

schem Figure 1 Schematic for NMO of multiples. From the standpoint of NMO, multiples can be treated as pseudo-primaries with the same source-receiver spacing, but with extra zero-offset traveltime $\tau^*$, assuming that the velocity and time-thickness of the multiple layer are known.