Velocity-Independent-Prestack Imaging

At this point in the analysis, application of velocity-independent DMO has produced a volume with normal moveout given by  

$$T^2 = T_0^2 + \frac{4((m - m_0)^2 + h^2)}{v^2}$$ (13)

Although one could, in principle, estimate v from this equation, the value would be relative to the normal from m+b and not to the desired (migrated) location, m₀. To see what must be accomplished to migrate the DMO volume let

 

h₀² = (m - m₀)² + h². (14)

The volume will be focused after points on the circle with center (m₀,0) and radius h₀ are mapped to the point (m₀,h₀). After focusing, the relation between T, T₀, h₀, and v is given by  

$$T^2 = \sqrt{T_0^2 + \frac{h_0^2}{v^2}}.$$ (15)

Velocity-independent focusing requires that m₀ and h₀ be expressed in terms of m and h. First, differentiate (13) with respect to m and h to get  

$$\frac{\partial T}{\partial m} = \frac{4(m - m_0)}{v^2T},$$ (16)

 

$$\frac{\partial T}{\partial h} = \frac{4h}{v^2T}.$$ (17)

Second, solve for the derivative of h with respect to m  

$$-\frac{\partial h}{\partial m} = \frac{\partial T/\partial m}{\partial T/\partial h} = \frac{(m - m_0)}{h}.$$ (18)

Finally, rewrite (18) as  

$$m_0 = m + h\frac{\partial h}{\partial m},$$ (19)

and substitute (19) into (14) so that  

$$h_0^2 = h^2 + h^2(\frac{\partial h}{\partial m})^2.$$ (20)

Equations (19) and (20) define PSI. As illustrated in Figure , they transform constant T slices from variables (m,h) to variables (m₀,h₀). Clearly, there is no velocity dependence, so PSI is also a velocity-independent process. As was the case for DMO, PSI can also be formulated Gardner (1993) as an (f,k) process. In this case the process is equivalent to (f,k) (Stolt) modeling rather than migration.

 

Fig4
Figure 4 Prestack imaging for constant slice T.