Variable-velocity Jacobians

The transformation Jacobians, given by Equation (23) for reflection-angle gathers and by Equation (28) for offset ray-parameter gathers, are strictly speaking valid only for constant velocity media. In the case of variable velocity, downward continuation is implemented in the mixt $\omega-k$ and $\omega-x$ domain. The dispersion relation is approximated using an expansion of the DSR equation. In one of the most usual forms, the dispersion relation can be written as

$$\begin{eqnarray} k_z&=& {k_{\rm zs}}+ {k_{\rm zr}} \nonumber \\ {k_{\rm zs}}&=... ...\left. \frac{dk_z}{d s}\right\vert _{s=so}\left (s_r-s_0\right ). \end{eqnarray}$$ (39)

Equations (39) give the effective dispersion relation for which we need to implement the Jacobian weighting.

Appendix A outlines the derivation for the expresion of the Jacobian in variable velocity media. For the case of migration with output in offset, the Jacobian expression is:  

$$\bold W_k_h= \left [ \frac{\omega s_0}{{k_{\rm zs_0}}} s_0+... ...{k_z}_0}\right ]^2 \right )\left (s_r-s_0\right ) \right ]^{-1}$$ (40)

where $\left (\ss-s_0\right )$ and $\left (s_r-s_0\right )$ are respectively the slowness perturbations at the source and receiver. In constant velocity media, $\left (\ss-s_0\right )=0$ and $\left (s_r-s_0\right )=0$, Equation (40) obviously takes the same form as Equation (23) we derived earlier for the case of constant velocity.

For the expressions of all Jacobians, we need to compute the quantity $\omega s/k_z$ which may lead to numerical instability when k_z approaches zero. A simple way so stabilize the Jacobian, which we have used for the current examples, is to add a small positive quantity to the denominator, and compute $\omega s/\left (k_z+ \epsilon \right )$.

Another approach, which we have not tested yet, but which appears to have the potential to be more accurate, is to write

$$\begin{eqnarray} \frac{\omega s}{k_z} &=& \frac{\omega s}{{\rm SSR}{\vec{k}}} \... ...}{\sqrt{1-\frac{{\bf m}{\vec{k}}^2}{\left [2\omega s\right ]^2}}}.\end{eqnarray}$$ (41)

For small values of $x={\bf m}{\vec{k}} / \left [2\omega s\right ]$ we can compute $\omega s/k_z$ using the following Taylor series expansion:  

$$\frac{1}{\sqrt{1-x^2}} \approx 1 + \frac{1}{2}x^2 + \frac{3}{8}x^4 + \frac{5}{16}x^6 + \frac{35}{128}x^8 + \dots$$ (42)

This expansion is very similar to the transformation used by Huang et al. (1999) for the extended Local Born Fourier migration method, or by Sava and Biondi (2000) in wave-equation MVA. This interesting possibility, however, awaits future research.