NMO DEFINED

In a continuous world, we could define NMO precisely as a mapping transformation. If M(t) is a continuous zero-offset trace, and D(t) is a trace at some non-zero offset x, then conventional NMO with a constant velocity v is defined by the relationship  

$$M(t) = D(\sqrt{t^2+\frac{x^2}{v^2}})\;.$$ (1)

If the continuous dot products in the model and data space have the usual L₂ form

$$\begin{eqnarray} \left(M_1 (t),M_2 (t)\right) & = & \int M_1 (t)\,M_2 (t)\,dt\;, \\ \left(D_1 (t),D_2 (t)\right) & = & \int D_1 (t)\,D_2 (t)\,dt\;,\end{eqnarray}$$ (2) (3)

the adjoint of operator (1) is simply  

$$D(t) = \frac{t}{\sqrt{t^2-\frac{x^2}{v^2}}}\,M(\sqrt{t^2-\frac{x^2}{v^2}})\;,$$ (4)

where the scaling factor appears as the result of the time coordinate stretching Crawley (1995). A generic inverse NMO operator can be defined accordingly as  

$$D(t) = W(t)\,M\left(\tau^{-1}(t)\right)\;,$$ (5)

where the function $\tau$ defines the time coordinate transformation, and W describes the amplitude scaling. The continuous adjoint of (5) has the form  

$$M(t) = \left\vert\frac{d \tau^(t)}{dt}\right\vert\,W(\tau (t))\, D\left(\tau(t)\right)\;.$$ (6)

The adjoint operators (5) and (6) have quite different appearance in the discrete word of digital signal processing. After the stretch $\tau (t)$ or $\tau^{-1} (t)$ the signal transforms from a regular grid to an irregular distribution on the time axis. After it is interpolated back to the regular grid, discretization errors enter the computation, and we discover that operators (5) and (6) are no longer adjoint to the machine precision. This fact splits the implementation of the NMO transform into two branches: stretching and squeezing, or, as Claerbout 1995a calls them, `` push'' and ``pull''. In some applications, only one of them is available. This happens, for example, when the transform function $\tau (t)$ doesn't have an easily computed inverse. In other applications, both approaches are possible, which opens interesting possibilities for their comparison.