Plane-wave decomposition criterion

The relevance of obtaining an estimation of the several angle-dependent reflection coefficients is the prospect of later using these information in a Zoeppritz-based elastic inversion. However, since the Zoeppritz equations represent the plane-wave response of a planar interface, it is necessary that the reflectivity estimation also represent the in situ plane-wave response of the interface.

From the previous criteria, only the V-stack represents a reasonable approximation to this requirement. To obtain an estimation completely consistent with the plane-wave response it necessary to perform a complete plane-wave decomposition of the upward and downward propagating wavefields at each point of the model, for all time steps. Although this criterion can be applied to general RTM schemes, it is particularly appropriate for the one-experiment approach described in Cunha (1992), where a single wavefield (with combined information form the modeled and recorded wavefields) is backward propagated in time using a discontinuous background model.

The implementation of the plane-wave decomposition criterion for the one-experiment RTM scheme is outlined below

• First, it is necessary to separate the downgoing $\phi^d(x,z,t;x_s)$ and upcoming $\phi^u(x,z,t;x_s)$ components of the wavefield potential, using the propagation-direction unit vector ${\bf \hat{k}}(x,z,t;x_s) \;\;=\;\; (k_x,k_z)$ Appendix A describes how to obtain the propagation direction from the field potentials. The downgoing field (within the forward time reference system) includes the incident and the transmitted wavefields, while the upcoming field contains only the reflections. Figure  shows the estimated propagation directions overlain on the wavefield and Figure  shows the resulting upcoming and downgoing separated wavefields.
• The second step involves the computation of the stacking power of the two wavefields ($\phi^d$ and $\phi^u$) along the directions represented by the angle $\theta$ around each point of the model As in the V-stack case, the stacking power here is computed outward from the image point (stacking trajectories are like a clock's hand), instead of across it. This implies that $\Phi^d(\theta) \neq \Phi^d(\theta + \pi)$. As discussed in appendix B the point of a given interface that intersects the downgoing wavefield at time t (reflection point) will have two directions $\theta$ (almost opposite) for which the $\Phi^d$ has a local maximum, corresponding to the incident and transmitted waves, but only one direction for which the $\Phi^u$ is maximum.
   
  
  wdir
  Figure 7 Wavefield partition at an interface. The incident and transmitted wavefields propagate downward while the reflected wavefield propagates upward. The overlaid arrows indicate the local propagation direction estimated from the wavefield as described in appendix A.
  
   
  
  wavesep
  Figure 8 Separation of the upward-propagating and downward-propagating parts of a wavefield using the local propagation direction. (a) The downward-propagating part corresponding to the incident and transmitted wavefields. (b) The upward-propagating part corresponding to the reflected wavefields.
  

• Evaluate the angle $\theta^u_{max}$ for which $\Phi^u$ is maximum. For reasons discussed in appendix B this angle determines the subdomain $\Delta \theta(\theta^u_{max})$ in which the correlation between the two stacking power functions is to be evaluated.
• Next, the crosscorrelation $\cal C$ of the two stacking powers, and the autocorrelation $\cal A$ of the stacking power of the downgoing wavefield are evaluated, in the interval $\Delta \theta(\theta^u_{max})$ for points $\;\;\; (x,z,t;x_s) \;\;\;$ where $\;\;\; - {\pi \over 2} \; \leq \; \theta^u_{max} \; \leq \; {\pi \over 2},$ and for points $\;\;\; (x,z,t;x_s) \;\;\;$ where $\;\;\; {\pi \over 2} \; \leq \; \theta^u_{max} \; \leq \; {3 \pi \over 2},$where $\beta$ is the reflection angle.
• The estimated attribute is given by

  $$\begin{eqnarray} R(x,z,\beta;x_s) & = & {\int {\cal C}(x,z,t,\beta;x_s) \; dt \o... ...x_s) \; dt \over E_{cut}(x,z,\beta;x_s)}, \;\;\; \mbox{elsewhere,}\end{eqnarray}$$ (12)

  where

  $$E(x,z,\beta;x_s) \; = \; \int {\cal A}(x,z,t,\beta;x_s) \; dt.$$

 

mig1time
Figure 9 PP reflectivity for a small (6 time steps) time window of the wavefield in Figure , for different values of the angle of incidence $\beta$.

Figure  shows the twelve images obtained over a small time window centered at the time-frame of Figure . Each of these images correspond to $R(x,z,\beta;x_s)$ at a different local angle of incidence $\beta$. The ray-theoretical values for the angle of incidence of the wavefronts that reach the two interfaces at that particular time are: 48 degrees for upper interface and 30.5 degrees for the bottom interface. From the figure it is clear that the point of the upper interface that is imaged at this time has maximum energy at the frames corresponding to 45 and 52.5 degrees, while for the bottom interface the maximum occurs at 30 and 37.5 degrees. Though special care will be required to avoid the undesirable focusing at small angles that we observe in the figure, it is encouraging to see that the location of the main maxima are consistent with the expected values.