SLANT STACK AS GEOMETRY TRANSFORMATIONS

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

 

$${\bf d'}(x,t) = {\bf S}~ {\bf d}(p,\tau) .$$ (1)

Going from the x-t domain to the p-$\tau$ 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-$\tau$ domain

 

$${\bf d}(p,\tau) = ({\bf S^t~ S})^{-1}~ {\bf S^t}~ {\bf d}(x,t) .$$ (2)

In equation 1 and 2 we assumed that operator $\bf S$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 ${\bf d}(p,\tau)$ as a good representation of the data in the p-$\tau$ domain, we can now change the geometry. Having a p-$\tau$ data estimate, that is free of geometrical effects, is the key for this process. Of course, real p-$\tau$ data (noisy) are ``geometry free'' only in the least-squares sense. We would guess that changing the geometry of the transform $\bf S$, 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.