Imaging condition for the reflection coefficient

Migration includes two distinct steps: downward extrapolation in depth or backward extrapolation in time, and imaging for the desired attribute. When the desired attribute is the reflectivity, Claerbout's imaging principle (Claerbout, 1971) represents the basis for many imaging condition equations. According to this principle, a reflector exists at a point where the upcoming and the downgoing wavefields coincide in time and space.

Jacobs (1982) compared three different ways to implement this principle in pre-stack profile migration in the $\omega-x$ domain:  

$$c_1= \int U D^{\ast} \, d \omega, \mbox{\hspace{2.0cm}} c_2=... ...= \int U {D^{\ast} \over \mid D \mid^2 + \epsilon} \, d \omega.$$ (1)

His conclusion was that c₃, though theoretically more correct for estimating the reflection coefficient, was too noise sensitive to be used in that migration scheme. Hildebrand (1987) used a variation of this principle to image for the acoustic impedance rather than the reflection coefficient, using reverse-time migration to extrapolate the pressure and particle-velocity wavefields.

Most imaging methods described in the literature involve the correlation between scalar wavefields. An exception is the tensorial imaging condition formulated by Karrenbach (1991):

$$c_{jklm}= \int \partial_m (U_l) \partial_k (D_j) \, dt,$$

where c_jklm can be interpreted as the reflectivity function associated with the corresponding stiffness component.

Although this equation gives a more complete picture of the subsurface, I decided to focus attention on the P-P reflectivity by using a scalar imaging condition similar to c₃ in equation (1):

$$\begin{eqnarray} \Psi(x,z) & = & {\int u(x,z,t) w(x,z,t) \, dt \over E(x,z)}, \;... ... u(x,z,t) w(x,z,t) \, dt \over E_{cut}}, \;\;\; \mbox{elsewhere.}\end{eqnarray}$$ (2)

Equation (2) gives the correct estimation of the reflection coefficient in the regions well illuminated by the source ($E(x,z) \geq E_{cut}$) and a damped estimation in the dim regions.