Impulse response

A slice through the broad-band impulse response of the 45$^\circ$equation is shown in Figure . As with the 2-D implementation of the 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-slice shown in Figure . 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 paper, we set the `one-sixth' parameter (), $\beta_{1/6}=0.125$, and used the isotropic nine-point Laplacian from equation ().

 

3Dtimeslice Figure 2 Depth-slice of centered impulse response corresponding to a dip of 45$^\circ$. Note the azimuthally isotropic nature of the full implicit migration. 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.