The V-stack criterion

It is possible to define an imaging criterion that does not produce undesirable false reflections when a discontinuous background model is used. Figure  shows the modeled wavefield and the reverse propagated recorded field at the same time. Not only the incident wavefield is reflected and transmitted (only the P wave component is shown) but also the backward-propagated recorded wavefield.

 

vstack
Figure 5 Right: The continuous thick line represents the modeled wavefield $\phi^s$ which is reflected and transmitted at the interface of a discontinuous background model. The dashed line represents the backward propagated recorded wavefield $\phi^r$ with an incident part A, which is also reflected B and transmitted C at the interface. Left: The overlapping region of both wavefields define a region with a characteristic ``V" shape.

The difference is that, because of its limited aperture, the reflections originating from the reverse propagation of recorded wavefield will be limited to a small region of space. As illustrated in the same figure, the product of the two wavefields has a peculiar V shape with vertex at the reflector. Figure  illustrates this concept for a synthetic shot profile.

 

vshape
Figure 6 Product of the modeled wavefield $\phi^s$ and the backward-propagated wavefield $\phi^r$ for a particular time t when the wavefields intercept two interfaces of the background model. Two V-shaped regions, whose vertex indicate the point of the interface where the wavefield partition takes place, can be observed in the figure

Application of this imaging criterion is not straightforward. The following steps can be used to implement the V-stack criterion:

• Define the product function $\Psi^{rs}$

  $$\Psi^{rs}(x,z,t;x_s) \; = \; \phi^r(x,z,t;x_s) \; \phi^s(x,z,t;x_s)$$

• Calculate the local semblance of $\Psi$ along all directions ($\theta$) around every point (x,z) of the profile x_s

  $$\mbox{Semb}[{\Psi}(x,z,t,\theta;x_s)] = {\left[ \int_{0}^{\D... ...i^{rs}(x + l \cos \theta, z + l \sin \theta,t,x_s)]^2 \; dl},$$

  where $\Delta l$ is the size of the local stacking segment.
• Find the two directions $\theta_s$ and $\theta_r$ for which $\mbox{Semb}[\Psi]$ is maximum.
• The estimation of the reflection coefficient is given by

$$R(x,z;x_s) = {\int \; [\int_{0}^{\Delta l} \phi_r(x + l \cos... ...phi^s(x + l \cos \theta_s, z + l \sin \theta_s) \; dl]^2 \; dt}$$

$$\mbox{where \hspace{2.0cm}} \beta \; = \; {\pi \; - \; \mid... ... \mid \over 2} \mbox{\hspace{2.0cm} is the angle of incidence.}$$

This criterion offers two advantages for the case of non-smooth backgrounds relative to the correlation criterion. It will not have spurious events caused by the overlap of secondary reflections, and the attribute is estimated from wavefield contributions away from the interface, where the interference with other modes is much weaker. These same advantages are present in the next criterion.