Estimation of the propagation direction

The unit vectors ${\bf \hat{i}}$ and ${\bf \hat{r}}$ are estimated from the potential fields $\phi^s$ and $\phi^r$ using the following equations:

$$\begin{eqnarray} {\bf \hat{i}} & = & - \; \mbox{signum}({\partial \phi^s \over \... ...\over \partial t}) {\nabla \phi^r \over \mid \nabla \phi^r \mid}.\end{eqnarray}$$ (14)

I demonstrate below that these relations give the correct estimation of the propagation direction for the incident wavefield. Figure  shows the wavefield amplitude along the line defined by the gradient of the potential field.

 

wavedir Figure 11 A representation of the amplitude of potential field along the gradient direction.

The gradient of the potential field gives the direction, but not the sense of propagation. Any point A in the interval between points 1 and 2 in the figure will have a positive gradient and any point B in the interval between points 2 and 3 will have a negative gradient. Since the propagation direction is positive (to the right) point A will have a negative time derivative, while point B will have a positive time derivative. If the propagation direction was negative (to the left), then the gradients would remain unchanged, while the time derivatives would switch sign. As a result, the product of the gradient and the time derivative at any point will have the opposite sign of the propagation direction.

CORRELATION INTERVALS FOR THE PLANE-WAVE DECOMPOSITION

Figure  shows a snapshot of a wavefield at the particular time and location where a wavefront is being transmitted/converted, and reflected at an interface. No conversion is observed because only the P wave potential field is displayed. To implement the plane-wave decomposition imaging criterion requires the computation of the slant-stacks around all points in the grid for the upcoming and the downgoing wavefields, at regular angle intervals. These stacks correspond to a semi-plane-wave decomposition around each point because they are computed from (not across) the grid points.

 

wavesplit Figure 12 Snap-shot of a wavefield propagating through an interface. The incident and transmitted wavefronts propagate downward while the reflected wavefront propagates upward. All three wavefronts meet at the point of the interface where the partition takes place at that particular time.

Lets consider the two functions $\Phi^u(\theta)$ and $\Phi^d(\theta)$, corresponding, respectively, to the stacks of the upcoming and downgoing wavefields around the intersection point of the three wavefronts (incident, transmitted and reflected) in Figure . While $\Phi^u(\theta)$ will have only one maximum, in the direction tangent to the reflected wavefront at that point, $\Phi^d(\theta)$ will have two local maxima; one in the direction tangent to the incident wavefront and one in the direction tangent to the transmitted wavefront. An crucial step in this imaging criterion is the selection of the proper subdomain $\Theta^i$ of the distribution $\Phi^d(\theta)$ where the maximum associated with the incident wave is located.

First it is necessary to find the angle $\theta^u_{max}$ for which $\Phi^u(\theta)$ is maximum:

$$\Phi^u(\theta^u_{max}) = \max \left[\Phi^u(\theta)\right].$$

As explained below, the location of the subdomain $\Theta^i$, will depend on the quadrant where $\theta^u_{max}$ is located.

The following rules are applied in the analysis:

1.

The reflected wavefront can only propagate upward and the incident wavefront can only propagate downward.

2.

The reflected and incident wavefronts meet the interface at the same angle (Snell law).

3.

The reflecting interface is locally planar.

4.

The incident wavefront can only propagate towards the interface and the reflected wavefront can only propagate outwards from the interface.

Figure  shows four diagrams, each one corresponding to a different quadrant location for $\theta^u_{max}$. The dark bars in each diagram refers to the limit directions of the reflected wavefront (i.e., $\theta^u_{max}$) in each quadrant and the arrows indicate the only possible propagation direction for this wavefront.

 

fourquad
Figure 13 Diagrams representing the location of the reflected wavefront (dark bars) and its propagation direction (arrows). (a) The reflected wavefront is located in the first quadrant (-$\pi/2$). (b) The reflected wavefront is located in the second quadrant ($\pi/2$-$\pi$). (c) The reflected wavefront is located in the third quadrant ($\pi$-$3 \pi /2$). (d) The reflected wavefront is located in the fourth quadrant ($3 \pi /2$-$2 \pi$).

Lets consider each of the four possibilities:

• a) Reflected wavefront in the first quadrant
  If the reflected wavefront is horizontal then the incident wavefront must be restricted to the second quadrant, otherwise one of the above stated rules would be violated. If the reflected wavefront is vertical then the incident wavefront must be restricted to the second or third quadrants.

  $$0 \le \theta^u_{max} \le {\pi \over 2} \;\;\; \longrightarrow \;\;\; {\pi \over 2} \le \theta^d \le {3 \pi \over 2}$$

• b) Reflected wavefront in the second quadrant
  If the reflected wavefront is horizontal then the incident wavefront must be restricted to the first quadrant. If the reflected wavefront is vertical then the incident wavefront must be restricted to the first or fourth quadrants.

  $${\pi \over 2} \le \theta^u_{max} \le \pi \;\;\; \longrightarrow \;\;\; -{\pi \over 2} \le \theta^d \le {\pi \over 2}$$

• c) Reflected wavefront in the third quadrant
  If the reflected wavefront is horizontal then the incident wavefront must be restricted to the first quadrant. If the reflected wavefront is vertical then the incident wavefront must be restricted to the first or fourth quadrants.

  $$\pi \le \theta^u_{max} \le {3 \pi \over 2} \;\;\; \longrightarrow \;\;\; -{\pi \over 2} \le \theta^d \le {\pi \over 2}$$

• d) Reflected wavefront in the fourth quadrant
  If the reflected wavefront is horizontal then the incident wavefront must be restricted to the second quadrant. If the reflected wavefront is vertical then the incident wavefront must be restricted to the second or third quadrants.

  $$-{\pi \over 2} \le \theta^u_{max} \le 0 \;\;\; \longrightarrow \;\;\; {\pi \over 2} \le \theta^d \le {3 \pi \over 2}$$

The above relations can be summarized as

$$-{\pi \over 2} \le \theta^u_{max} \le {\pi \over 2} \;\;\; \... ...htarrow \;\;\; {\pi \over 2} \le \theta^d \le {3 \pi \over 2}$$

$${\pi \over 2} \le \theta^u_{max} \le {3 \pi \over 2} \;\;\; ... ...htarrow \;\;\; -{\pi \over 2} \le \theta^d \le {\pi \over 2}.$$

Figure  is a graphical representation of theses relations. When the reflected wavefront is located in the first or fourth quadrants (dark bar in -a) the incident wavefront must be in the second or third quadrants (white bar). When the reflected wavefront is located in the second or third quadrants (dark bar in -b) the incident wavefront must be in the first or fourth quadrants (white bar). As indicated in the figure the angle between the direction of the two wavefronts is equal to twice the angle of incidence.

 

twoquad
Figure 14 Diagrams showing the relation between the reflected (dark bar) and transmitted (white bar) wavefronts. (a) If the reflected wavefront is located in the first or fourth quadrants the transmitted wavefront must be located in the second or third quadrants. (b) If the reflected wavefront is located in the second or third quadrants the transmitted wavefront must be located in the first or fourth quadrants. $\beta$ is the angle of incidence.

Using the above relations, the correlation between the two distributions $\Phi^u(\theta)$ and $\Phi^d(\theta)$ is will be given by

$${\cal C}(\beta) \;=\; \sum_{\theta=\pi/2}^{3\pi/2} \left[ \P... ...ace{.5cm}} {-\pi \over 2} \le \theta^u_{max} \le {\pi \over 2},$$

and

$${\cal C}(\beta) \;=\; \sum_{\theta=-\pi/2}^{\pi/2} \left[ \P... ...ce{.5cm}} {\pi \over 2} \le \theta^u_{max} \le {3 \pi \over 2},$$

and the autocorrelation of $\Phi^d(\theta)$ by

$${\cal A}(\beta) \;=\; \sum_{\theta=\pi/2}^{3\pi/2} \left[ \P... ...ace{.5cm}} {-\pi \over 2} \le \theta^u_{max} \le {\pi \over 2},$$

and

$${\cal A}(\beta) \;=\; \sum_{\theta=-\pi/2}^{\pi/2} \left[ \P... ...ce{.5cm}} {\pi \over 2} \le \theta^u_{max} \le {3 \pi \over 2}.$$

The reflectivity estimation for that particular time would be given by the ratio ${\cal C}/{\cal A}$.