Synthetic examples

Figure  shows a slice through the broad-band impulse response of the 45$^\circ$ equation. As with the 2-D 45$^\circ$ equation, evanescent energy at high dip appears as noise, and takes the form of a cardioid. This is never a problem on field data, and has been removed from the depth-slices shown in Figures  and .

Figure  compares the impulse responses of the 45$^\circ$ equation obtained with the x-y splitting approximation [panel (a)] and the helical factorization methodology described in this chapter [panel (b)]. Implicit migration with the full Laplacian, instead of a splitting approximation, produces an impulse response that is azimuthally isotropic without the need for any phase corrections.

Figure  shows the effects of the different boundary conditions on the two spatial axes. The fast spatial axis (top and bottom of Figure) have helical boundary conditions, and show wrap-around. The slow spatial axis (left and right of Figure) has a zero-value boundary condition, and hence is reflective.

For the examples in this chapter, we set the `one-sixth' parameter Claerbout (1985), $\beta_{1/6}=0.125$, and used the isotropic nine-point Laplacian from equation () that corresponds to $\gamma = 2/3$ in equation ().

 

3Dcardioid
Figure 1 Vertical slice through broad-band impulse response of 45$^\circ$ wave equation, showing the distinctive cardioid.

 

3Dtimeslice
Figure 2 Depth-slice of centered impulse response corresponding to a dip of 50$^\circ$. Panel (a) shows the result of employing an x-y splitting approximation, and panel (b) shows the result of the helical factorization. Note the azimuthally isotropic nature of panel (b). Evanescent energy has been removed by dip-filtering prior to migration.

 

3Dboundary Figure 3 Depth-slice of offset impulse response corresponding to a dip of 45$^\circ$. Note the helical boundary conditions on the fast spatial axis.