Angle-gathers

For reflection-angle gathers, the dispersion relation used to downward continue the wavefield is given by the DSR equation:

$$\begin{eqnarray} k_z &=& {k_{\rm zs}}+ {k_{\rm zr}}\nonumber \\ &=& {\rm SSR}{\vec{{k_m}}-\vec{k_h}} + {\rm SSR}{\vec{{k_m}}+\vec{k_h}}\end{eqnarray}$$ (22)

where s is the local slowness, ${k_{\rm zs}}$ and ${k_{\rm zr}}$are respectively the vertical wavenumber for the source and receiver components, and ${\bf {k}_{m}}$ and ${\bf {k}_{h}}$ are respectively the midpoint and offset wavenumbers.

 

ghmig
Figure 10 Reflection-angle gather implemented using Equation (20). We do not use the correct weighting of the transformation Jacobian, therefore the amplitudes are distorted.

 

gpmig
Figure 11 Offset ray-parameter gather implemented using Equation (21). We do not use the correct weighting of the transformation Jacobian, therefore the amplitudes are distorted.

The Jacobian for this transformation is thus the common prestack Stolt migration Jacobian:

$$\begin{eqnarray} \bold W_k_h &=& \left.\frac{d\omega}{dk_z}\right\vert _{{\bf {k... ...\rm zs}}} + \frac{\omega s}{{k_{\rm zr}}} \right ) \right ]^{-1}.\end{eqnarray}$$ (23)

As mentioned in the preceding sections, in the case of variable velocity media, those quantities are evaluated at the reflector location. For an arbitrary 2-D reflector geometry (Figure 1), we can rewrite Equation (23) as  

$$\bold W_k_h= \left [s\left (\frac{1}{\cos \left [\gamma-\alp... ...rac{1}{\cos \left [\gamma+\alpha\right ]}\right )\right ]^{-1},$$ (24)

where $\alpha$ is the structural dip angle, and $\gamma$ is the reflection angle.

For flat reflectors, defined by ${\bf {k}_{m}}=0$ and $\alpha=0$, the Jacobian takes the simple form  

$$\bold W_k_h= \frac{1}{2s}\cos \gamma.$$ (25)

 

chmig
Figure 12 Reflection-angle gather computed in the image space. The weighting factors restore correct amplitudes. Compare with the theoretical response in Figure 10.

 

cpmig
Figure 13 Offset ray-parameter gather computed in the data space. The weighting factors restore correct amplitudes.