Most people don't see slant stacks as geometry manipulation tool, but
rather as a tool for analyzing or filtering seismic data. However,
as long as one has a regular sampling in ray parameter, one can have
arbitrary irregular sampling in *x*, which makes slant stacks a convenient
process for manipulating source and receiver positions. In this paper
I will view slant stacks merely as a process for geometry transformation.
It turns out that these geometry transforms are closely related to
interpolation and extrapolation of data.

Given () data, we know exactly how to transform from *p*- domain
into the *x*-*t* domain. It is just a straight stack or plane wave
superposition. The operator denotes the stack:

(1) |

Going from the *x*-*t* domain to the *p*- domain represents some
difficulties, if we are concerned with the finiteness of the domain.
However, a least squares operation can be designed, which
transforms the data *d*(*x*,*t*) into the *p*- domain

(2) |

In equation 1 and 2 we assumed that operator incorporates in both instances the same geometry.
This does not have to be necessarily the case.
One can deliberately choose different geometries, which is reflected in
the different structure of the operators. Taking as
a good representation of the data in the *p*- domain, we can
now change the geometry. Having a *p*- data estimate, that is free
of geometrical effects, is the key for this process.
Of course, real *p*- data (noisy) are ``geometry free'' only
in the least-squares sense.
We would guess that changing
the geometry of the transform , on the way back,
leads us into the null
space of the operator. To see if this assertion is true I ran a few simple
synthetic tests.

- Geometry transform operator
- In terms of known and missing data
- Arbitrary (irregular) location shifting
- Interlacing

12/18/1997