A conventional slant stack

Slant stacks are commonly applied to unsorted data, one shot at a time. This form is well suited to deconvolution of multiple reflections from flat reflectors. Such multiple reflections are periodic at zero-offset, but not at a single finite offset h.

A slant stack attempts to describe our recorded data as a sum of dipping lines. A dip p_s will measure the slope of time with offset holding a source position constant.  

$$p_s \equiv \left.{ \partial t \over \partial h }\right\vert _{s}$$ (1)

With ideal sampling and infinite offsets, this equation would describe a plane-wave source on the surface. A plane wave reflecting from flat reflectors would produce periodic multiples at any p_s. Predictive deconvolutions can detect this periodicity and remove multiple reflections.

The simplest slant-stack sums data over all lines within a feasible range of dips. Let $\tau_s$ be the intersection at zero offset of our imaginary plane wave in the shot gather.  

$$S(s,p_s , \tau_s ) \equiv \int d(s,r=s+h,t=\tau_s + p_s h ) dh$$ (2)

In practice the integral over offset h must be a discrete sum with a limited range of offsets.

The inverse of this transform looks much like another slant stack, with some adjustments of the spectrum. Papers are readily available to explain this inverse. I will concentrate instead on the conversion of one type of slant stack to another.