Zero-offset migration

There is a remarkable relationship between the amplitude $B({\bf r})$ of a discontinuity on zero-offset migrated sections and the amplitude $A({\bf r})$ of the corresponding discontinuity on a CSP section for the point R that belongs to the same normal ray:

$$A ({\bf r}) = 4 B ({\bf r}).$$

This means that we can use the formulas for CSP patterns at s=r; the constant factor of 4 doesn't play any role. If we take into account the equality

$$v(r) \cdot \sqrt{J(r,{\bf r})} = v({\bf r}) \sqrt{J({\bf r},r)},$$

we receive

$$w_{tr} = {{const} \over {v(r)}} \cdot \cos \theta$$

and

$$\stackrel{\sim}{w}_{tr} = {{const} \over {v(r)}}.$$

The operator ${\bf P}_{w_{tr}}^{(-)} ({v \over 2})$ is not unique. If the medium is homogeneous then there is a whole family of operators ${\bf P}_{{\overline{w}}_{tr}}^{(-)} ({v \over 2}) \cdot t^m (m=0,1,2,\ldots)$that acts according to the formula:

$$U^{(-)} = {\bf P}_{{\overline{w}}_{m}}^{(-)} ({v \over 2}) [ t^m U_0],$$

and

$${\overline{w}}_m = {{\cos \theta} \over {\rho^{m}}}.$$

In the 2D case the amplitude of the image depends on the radius of transverse curvature of the reflector. But if the reflector is planar or cylindrical, we have a family of true-amplitude operators of the same type as

$${\overline{w}}_m = {{\cos \theta} \over {\rho^{m}}} \sqrt{\rho}.$$