Next: Regularization of the angle Up: Sava & Fomel: Angle-gathers Previous: Introduction

# Equivalence to slant stacks

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