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** Up:** Sava & Fomel: Angle-gathers
** Previous:** Introduction

The Fourier-domain stretch represented by equation (1) is equivalent to a slant stack in the domain. Indeed, we can convert an image gather in the offset-domain () to one in the angle-domain (), using a slant-stack equation of the form
| |
(2) |

where is a vector describing the direction of the stack.
Fourier transforming equation (2) over the depth
axis, we obtain

where the underline stands for a 1-D Fourier transform. We can continue by writing the equation
where we can re-arrange the terms as
which highlights the relation between the 1-D Fourier-transformed angle-domain and offset-domain representation of the seismic images:
We recognize on the right-hand side of the previous equation additional Fourier transforms over the offset axes, and therefore we can write
where the triple underline stands for the 3-D Fourier transform of the
offset-domain common-image gather. Finally, defining , we
can conclude that the 1-D Fourier transforms of angle-domain gathers
are equivalent to the 3-D Fourier transforms of the offset-domain
gathers,
| |
(3) |

subject to the stretch of the offset axis according to the simple law
| |
(4) |

We can recognize in equation (4) the fundamental relation between the reflection angle and the Fourier-domain quantities that are evaluated in wave-equation migration. This equation also shows that the angles evaluated by (1) are indeed equivalent to slant stacks on offset-domain
common-image gathers. Therefore, we could either compute angles for each of the two offset axes with the equations
or compute one angle corresponding to the entire offset vector:

** Next:** Regularization of the angle
** Up:** Sava & Fomel: Angle-gathers
** Previous:** Introduction
Stanford Exploration Project

4/27/2000