Angle-domain CIG by wave-equation migration

In the previous sections we discussed the advantages of angle-domain CIGs over offset-domain CIGs when a complex velocity function induces multipathing and event triplication. In this section we show how to extract angle-domain CIGs from downward-continued prestack data.

Recorded 3-D seismic data can be organized as a function of midpoint coordinates (${\bf m}$)and offset coordinates (${\bf h}$). Prestack data are efficiently downward continued using the DSR equation in the frequency ($\omega$) domain. Furthermore, since we either use 2-D downward continuation or 3-D common-azimuth downward continuation, the offset space is restricted to the in-line offset h_x, and thus we express the recorded wavefield as $P\left(\omega,{\bf m},{h_x};z=0\right)$,where z is depth and z=0 indicates data recorded at the surface.

The prestack wavefield at depth is obtained by downward continuing the recorded data using the DSR, and is imaged by extracting the values at zero time

$$\begin{eqnarray} P\left(\omega,{\bf m},{h_x};z=0\right) & \stackrel{{\rm DSR}}{\... ...rel{Imaging}{\Longrightarrow} & P\left(t=0,{\bf m},{h_x};z\right) \end{eqnarray}$$ (1) (2)

The downward-continuation process focuses the wavefield towards zero offset (left panel in Figure 5) and if the continuation velocity is correct, a migrated image can be obtained by extracting the value of the wavefield at zero offset. However, the zero-offset wavefield has limited diagnostic information for velocity updating, and no information on the amplitude of the reflections versus reflection angle (AVA). We therefore perform a slant stack along the offset axis before imaging and obtain an image as a function of the offset ray parameter p_hx, as

$$\begin{eqnarray} P\left(\omega,{\bf m},{h_x};z=0\right) & \stackrel{{\rm DSR}}{\... ...g}{\Longrightarrow} & P\left(\tau=0,{\bf m},{p_{hx}};z\right). \ \end{eqnarray}$$ (3) (4) (5)

Angle-domain CIGs are subsets of $P\left(\tau=0,{\bf m},{p_{hx}};z\right)$ at fixed midpoint location. The right panel in Figure 5 shows the angle-domain CIG gather corresponding to the downward-continued offset gather shown in the left panel. Notice that because in downward-continued offset gathers the energy is concentrated around zero offset, the slant stack decomposition does not suffer from the usual artifacts caused by the boundary conditions.

Strictly speaking, the CIG gathers obtained by the proposed procedure are function of the offset ray parameters p_hx and not of the aperture angle $\theta$.However, p_hx is linked to $\theta$ by the following simple trigonometric relationship  

$${\partial t \over \partial h}={p_{hx}}=\frac{2 \sin \theta \cos\phi}{V\left(z,{\bf m}\right)} ,$$ (6)

where $\phi$ is the geological dip along the in-line direction and $V\left(z,{\bf m}\right)$ is the velocity function.

 

AVO-hydrate-off-angle
Figure 5 Left: Offset panel after downward continuation. Right: Angle-domain CIG