Remapping in $\omega_0$

As the variable $\omega$ is replaced with $\omega_0$ in the integral

$$\int d\omega p(\omega,k_y,k_h)= \int d\omega_0 {\left[{{d \omega} \over {d \omega_0}}\right]} p^*(\omega_0,k_y,k_h),$$

each value of $\omega_0$ in the integral needs to be associated with the appropriate value of the field $p(\omega,k_y,k_h)$ which is $p(\omega_0 \sqrt{1+{{v_h^2} \over {\omega_0^2-v_y^2}}},k_y,k_h)$.The new field $p^*(\omega_0,k_y,k_h)$ represents a remapping of the original field $p(\omega,k_y,k_h)$.Each value in the new field $p^*(\omega_0,k_y,k_h)$ with coordinates $(\omega_0,k_y,k_h)$ corresponds to the value in the field $p(\omega,k_y,k_h)$ with coordinates $(\omega={\omega_0 \sqrt{1+{{v_h^2} \over {\omega_0^2-v_y^2}}}} ,k_y,k_h)$.

The $\omega \rightarrow \omega_0$ remapping can be understood easier if one takes the particular case k_y=0. The change of variable in $\omega$ (5) becomes

$$\omega^2=\omega_0^2+v_h^2,$$

the Jacobian

$${\left[ {{d \omega} \over {d \omega_0}} \right]}= {{\mid {{2... ...over v} \mid} \over {\sqrt{{{4\omega_0^2} \over v^2}+k_h^2}}},$$

and the zero-offset migrated field becomes  

$$p_0(\omega_0,k_y=0)=\int dk_h {{\mid {{2\omega_0} \over v} ... ...v \over 2} {\sqrt{{{4\omega_0^2} \over v^2}+k_h^2}},k_y=0,k_h).$$ (10)

Compare equation (10) with the Gazdag (1978) and Stolt (1978) zero-offset migration equations:  

$$\begin{array} {lcl} p(z,k_y) & = & \displaystyle{ \int d\ome... ...n(\omega_0){v \over 2}\sqrt{\omega_0^2+k_y^2},k_y)}.\end{array}$$ (11)

Notice that the Stolt migration formula is comprised of an inverse Fourier transform and a remapping (interpolation). Equation (10) inside the integral in k_h has the same form as the remapping in the Stolt formula, save for a constant coefficient $v \over 2$. This similitude suggests the equivalent formulation:  

$$\begin{array} {lcl} p_0(t_0,k_y=0) & = & \displaystyle{ \int... ...mega)\sqrt{\omega^2-v_h^2}t_0} p(\omega,k_y=0,k_h)}.\end{array}$$ (12)

The validity of the new equation is verified by replacing in equation (12) the exponential expression

$$sign(\omega)\sqrt{\omega^2-v_h^2}$$

by a new variable $\omega_0$ to get back the inverse Fourier transform of the initial expression.

The integration in k_h represents the inverse Fourier transform at zero offset (h=0). However because equation (12) actually performs summation along hyperbolas in the spatial coordinate h, replacing the integral in k_h by the inverse Fourier transform will not return the constant-offset sections.

Applying the same technique to the case $k_y \neq 0$ I obtain  

$$p_0(t_0,k_y)= \int dk_h \int d\omega \; e^{-isign(\omega) {\... ...h^2)^2-4\omega^2v_y^2})\right]}^{1\over2}t_0} p(\omega,k_y,k_h)$$ (13)

which can be verified by substituting the exponential expression in $(\omega,v_y,v_h)$ with a new variable $\omega_0$defined as

$$\omega_0^2 = {1 \over 2} \left [ {\omega^2+v_y^2-v_h^2 + {\sqrt {(\omega^2+v_y^2-v_h^2)^2-4 \omega^2 v_y^2}}} \right ].$$ (14)