Ray-parameter gathers

For the case of offset ray-parameter gathers, we can rewrite the DSR equation (22) as

$$\begin{eqnarray} k_z &=& {k_{\rm zs}}+ {k_{\rm zr}}\nonumber \\ &=& {\rm SSR}{\vec{{k_m}}-\omega\vec{p_h}} + {\rm SSR}{\vec{{k_m}}+\omega\vec{p_h}}.\end{eqnarray}$$ (26)

In this case, the imaging Jacobian becomes

$$\begin{eqnarray} \bold W_{p_h} &=& \left.\frac{d\omega}{dk_z}\right\vert _{{\bf ... ...vec{p_h}\right )\cdot {\bf {p_{h}}}}{4{k_{\rm zr}}} \right ]^{-1},\end{eqnarray}$$ (27)

which can be re-arranged as:

$$\begin{eqnarray} \bold W_{p_h} &=& \left [\left (s-\frac{{\bf {p_{h}}}\cdot{\bf... ...\rm zs}}} - \frac{\omega s}{{k_{\rm zr}}} \right ) \right ]^{-1}.\end{eqnarray}$$ (28)

For an arbitrary 2-D reflection geometry (Figure 1), we can write Equation (28) as  

$$\bold W_{p_h} = \left [\left (s-\frac{{\bf {p_{h}}}\cdot{\b... ...ac{1}{\cos \left [\gamma+\alpha\right ]}\right ) \right ]^{-1}.$$ (29)

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

$$\bold W_p_h= \frac{1}{2s} \frac{1}{\cos \gamma},$$ (30)

which is equivalent to the weighting factor introduced by Wapenaar et al. (1999).

For the case of flat reflectors, we also have  

$$\bold W_{p_h}=\frac{1}{4s^2} \frac{1}{\bold W_k_h}.$$ (31)

which explains the opposite behavior of the uncorrected migration amplitudes for reflection-angle gathers (Figure 10) and offset ray-parameter gathers (Figure 11).

After we apply the Jacobian weights, we obtain the corrected angle-gathers shown in Figures 12 and 13. As expected, the amplitudes are constant for the entire usable angular range.