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Taking the one-dimensional *Z* transform of in the helical
coordinate system gives
| |
(1) |

Here, *Z*_{h} represents the unit delay operator in the sampled
(helical) coordinate system.
The summation in equation (1) can be split into two
components,
| |
(2) |

| (3) |

Ignoring boundary effects, a single unit delay in the helical
coordinate system is equivalent to a single unit delay on the
*x*-axis;
similarly, but irrespective of boundary conditions, *N*_{x} unit delays
in the helical coordinate system are equivalent to a single delay on
the *y*-axis.
This leads to the following definitions of *Z*_{h} and
*Z*_{h}^{Nx} in terms of delay operators, *Z*_{x} and *Z*_{y}, or
wavenumbers, *k*_{x} and *k*_{y}:

| |
(4) |

| (5) |

where and define the grid-spacings along the *x*
and *y*-axis respectively.
Substituting equations (4) and (5) into
equation (3) leaves

| |
(6) |

| (7) |

Equation (7) implies that, if we ignore boundary
effects, the one-dimensional FFT of in helical
coordinates is equivalent to its two-dimensional Fourier transform.

** Next:** Wavenumber in helical coordinates
** Up:** Theory
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Stanford Exploration Project

9/5/2000