The midpoint gather

We should prefer a transform of a single source gather because these gathers correspond to a physical experiment that can be modeled easily by wave-equation methods. Unfortunately, reflections from dipping layers and point scatters may have a very complicated expression in a source gather. We may be obliged to use many dips to capture their coherence. Worse, many reflections will have minimum times at finite offset, and a slant stack will alias some of their energy.

If the data are first sorted by midpoint y and half-offset h, then reflections from dipping lines and from points will still remain symmetric about zero offset. A slant stack of a midpoint gather will better capture the coherence of the reflections:  

$$Y(y,p_y, \tau_y ) \equiv \int d ( s=y-h/2, r=y+h/2, t=\tau_y + p_y h ) dh$$ (7)

where  

$$p_y = \left. {\partial t \over \partial h } \right\vert _y .$$ (8)

The Fourier version of a common-midpoint slant stack can be derived exactly as before. Let f_y be the Fourier frequency of $\tau_y$: 

$$\tilde Y (y,p_y ,f_y ) = \int \exp( i 2 \pi f_y p_y h ) \tilde d (s=y-h/2,r=y+h/2,f=f_y ) dh .$$ (9)

Unfortunately, this slant stack does not correspond to any single seismic experiment, and wave-equation modeling is much more awkward.