Equivalence to slant stacks

The Fourier-domain stretch represented by equation (1) is equivalent to a slant stack in the $z-\vec h$ domain. Indeed, we can convert an image gather in the offset-domain (${\bf H}$) to one in the angle-domain (${\bf A}$), using a slant-stack equation of the form  

$${\bf A}\left (z,\vec \mu\right ) = \int \H{z+\vec \mu\cdot\vec h}{\vec h} d\vec h,$$ (2)

where $\vec \mu$ is a vector describing the direction of the stack.

Fourier transforming equation (2) over the depth axis, we obtain

$$\underline {\bf A}\left (k_z,\vec \mu\right ) = \int\left [\int \H{z+\vec \mu\cdot\vec h}{\vec h} d\vec h\right ]e^{i k_zz} dz$$

where the underline stands for a 1-D Fourier transform. We can continue by writing the equation

$$\underline {\bf A}\left (k_z,\vec \mu\right ) = \int \int \H... ...vec \mu\cdot\vec h\right )-ik_z\vec \mu\cdot\vec h} d\vec hdz,$$

where we can re-arrange the terms as

$$\underline {\bf A}\left (k_z,\vec \mu\right ) = \int \left [... ...t\vec h\right )} dz\right ]e^{-ik_z\vec \mu\cdot\vec h}d\vec h,$$

which highlights the relation between the 1-D Fourier-transformed angle-domain and offset-domain representation of the seismic images:

$$\underline {\bf A}\left (k_z,\vec \mu\right ) = \int \underl... ...}\left (k_z,\vec h\right ) e^{-ik_z\vec \mu\cdot\vec h}d\vec h.$$

We recognize on the right-hand side of the previous equation additional Fourier transforms over the offset axes, and therefore we can write

$$\underline {\bf A}\left (k_z,\vec \mu\right ) = \underline{\underline{\underline {\bf H}}} \left (k_z,-\vec \mu k_z\right ),$$

where the triple underline stands for the 3-D Fourier transform of the offset-domain common-image gather. Finally, defining $-\vec \mu k_z=\vec{k_h}$, 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,  

$$\underline {\bf A}\left (k_z,\vec \mu\right ) = \underline{\underline{\underline {\bf H}}} \left (k_z,\vec{k_h}\right ),$$ (3)

subject to the stretch of the offset axis according to the simple law  

$$\vec \mu= -\frac{\vec{k_h}}{k_z}.$$ (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

$$\begin{array} {l} \gamma_x= -\tan^{-1} \left (\frac{{k_h}_x... ...a_y= -\tan^{-1} \left (\frac{{k_h}_y}{k_z} \right ),\end{array}$$

or compute one angle corresponding to the entire offset vector:

$$\gamma= -\tan^{-1} \left (\frac{\vert\vec{k_h}\vert}{k_z} \right ).$$