CSP migration, 3D case

To develop true-amplitude formulas for the CSP case, we shall use the simple fact that if $A_E ({\bf r})$ is the amplitude of the discontinuity received as a result of wave-field continuation with the amplitude-equivalent operator ${\bf P}_E^{(-)} (v)$ with the kernel

$$w_E ({\bf r},r) = {{\cos \theta} \over {v ({\bf r})}} \sqrt{{v(r)} \over {v({\bf r})}} {1 \over {\sqrt{J({\bf r},r)}}}$$

and if the kernel of the operator ${\bf P}_w^{(-)} (v)$ is

$$w ({\bf r},r) = w_E ({\bf r},r) \cdot \Psi ({\bf r},r),$$

then as a result of applying the operator ${\bf P}_w^{(-)} (v)$ we receive the amplitude  

$$A ({\bf r}) = A_E ({\bf r}) \cdot \Psi ({\bf r},r^{*}),$$ (117)

(note that the star means that r^* belongs to the same ray of eiconal $\tau^{(-)}$ as r).

But the operator ${\bf P}_E^{(-)} (v)$ acts inversely (in the sense of q-equivalence). This means that at the point ${\bf r}$ on the reflector,

$$A_E ({\bf r}) = {{c \cdot R} \over {\sqrt{J(s,{\bf r}^{(-)})}}} \sqrt{{v(s)} \over {v({\bf r})}}$$

where ${\bf r}^{(-)}$ means the point ${\bf r}$ before the reflection.

According to equation (117),

$$A ({\bf r}) = {{c \cdot R} \over {\sqrt{J(s,{\bf r}^{(-)})}}} \sqrt{{v(s)} \over {v({\bf r})}} \Psi ({\bf r},r^{*}).$$

Now we apply the condition

$$A ({\bf r}) = c \cdot R$$

and receive

$$\Psi ({\bf r},r^{*}) = \sqrt{ { {v({\bf r})} \over {v(s)}} J (s,{\bf r})}.$$

So the kernel we are looking for is

$$w_{tr} ({\bf r},r) = {{\cos \theta} \over {v({\bf r})}} \sqrt{ { {v(r)} \over {v(s)}} {{J (s,{\bf r})} \over {J ({\bf r},r)}}}$$

or in homogeneous media

$$w_{tr} ({\bf r},r) = {{\cos \theta} \over {v}} \sqrt{ { {{(s... ...-x)}^2 + z^2}} } = {{\cos \theta} \over v} {\rho_0 \over \rho}.$$

For land surveys, in the general case we have

$$\stackrel{\sim}{w}_{tr} = {1 \over {v({\bf r})}} \sqrt{ { {v(r)} \over {v(s)}} {{J (s,{\bf r})} \over {J ({\bf r},r)}}}$$

and in homogeneous media

$$\stackrel{\sim}{w}_{tr} = \sqrt{ { {{(s-x)}^2 + z^2} \over {{(r-x)}^2 + z^2}} } = {\rho_0 \over \rho}.$$