Hans Tieman spoke to the Stanford Exploration Project on Jan 23, 1996 about depth imaging with slant slacks. Among his techniques is a clever method of converting slant stacks of midpoint gathers into equivalent slant stacks of source gathers. A source gather best represents a physical experiment that can be modeled easily by wave-equation methods. Midpoint gathers, however, better include the coherence of steep reflections and better avoid aliasing. A conversion takes advantage of the best of both domains.

For a given source, we have a limited range of receivers (perhaps 3–5 kilometers), and vice versa. Receiver positions are often sampled two or four times as densely as source positions. In marine data, both are relatively evenly sampled, but a spatial Fourier transform must pay attention to aliasing or edge effects from the short span. Land data will be much more arbitrarily sampled.

Define the coordinates of full offset and midpoint . Resorted data can be written as . The well-sampled midpoint coordinate covers the entire span of the survey.

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

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 . 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 be the intersection at zero offset of our imaginary plane wave in the shot gather.

In practice the integral over offset 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.

The second step uses the Fourier transform of a delta function . The third uses the behavior of a delta function in an integral .

If the data are first sorted by midpoint and half-offset , 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:

where The Fourier version of a common-midpoint slant stack can be derived exactly as before. Let be the Fourier frequency of : Unfortunately, this slant stack does not correspond to any single seismic experiment, and wave-equation modeling is much more awkward.

and

To place the second integral (11) in the form of the first (10), we should change the variables of integration from and to and . (The Jacobian of this transformation is .) Substituting we get

Thus, a two-dimensional stretch of the midpoint-gather transform becomes equivalent to the source-gather transform. For a given dip over offset in a midpoint gather , we can identify a dip over midpoint

(13)

The adjustment of
subtracts
half of this midpoint dip from the offset dip.
With a careful application of the chain rule, and carefully
distinguishing partial derivatives, we could arrive at the
same result
(14)