Migration in the reflection angle domain

Migration in the reflection angle domain is a subsurface-oriented technique. It is most easily understood as a Kirchhoff method. For each point in the subsurface (${\bf x}$), a ray couple can be created for a certain reflection angle ($\theta$) and a certain dip angle ($\phi$) as shown in Figure 1. To create a CIG, simply sum over all of the dip angles:

$$a({\bf x},\theta) = \int w(\phi,\theta,{\bf x}) d(s,r,t) d\phi$$ (1)

where $a({\bf x},\theta)$ is the RAD CIGs, $w(\phi,\theta,{\bf x})$is an appropriate weighting function, and d(s,r,t) is the data in source, receiver and time space. From this, it is clear that each event in a RAD CIG is created by one and only one ray couple. This means there is no multipathing. Therefore, we will have fewer artifacts, especially in complex areas.

 

rangle Figure 1 Geometry of the reflection angle

In this paper, we actually use a wave-equation method to create the RAD CIG. We begin by using the Double Square Root (DSR) equation to downward continue the data (represented as the wavefield recorded at the surface $P\left(\omega,{\bf m},h_x;z=0\right)$ where $\omega$ indicates the frequency domain, ${\bf m}$ is the midpoint location, h_x is offset, and z is depth), then we slant stack (transforming the data into vertical traveltime space $\tau$)at each depth level and image at zero time:

$$\begin{eqnarray} P\left(\omega,{\bf m},h_x;z=0\right) & \stackrel{\rm DSR}{\Long... ...maging}{\Longrightarrow} & P\left(\tau=0,{\bf m},p_{hx};z\right). \end{eqnarray}$$ (2) (3) (4)

The actual CIG is extracted at a fixed midpoint. The resulting CIG has an axis that is in offset ray parameter (p_hx) rather than reflection angle ($\theta$). However, the offset ray parameter is related to the reflection angle by:

$${\partial t \over \partial h}=p_{hx}=\frac{2 \sin \theta \cos\phi}{V\left(z,{\bf m}\right)}.$$ (5)

Sava and Fomel (2000) describe an alternative method of obtaining RAD CIGs in the Fourier domain.

The advantages of the RAD in helping to reduce illumination problems are not immediately obvious. Imagine upward continuing a wavefield from a reflector below a complex structure such as a salt dome. As the wavefield passes through the complex structure, the energy gets spread out unevenly over a large distance. In the offset and shot domains, this distribution can lead to the loss of energy that is present but doesn't conform to the regular surface-oriented geometry. The subsurface-oriented reflection angle domain does not have this difficulty. Unfortunately, there is another problem. Our acquisition geometry is always finite, therefore we can never record the entire wavefield at the surface. When we downward continue the finite recorded wavefield, the "missing" data creates a model null space. This null space cannot be filled with migration alone, so we turn to inversion.