Residual-migration operator

Now we consider the image of a scatter in a homogeneous medium. Let the scatter have coordinates (x_t, z_t). Then, the angle of incident ray to the vertical is fixed.  

$$\alpha = \arctan {x_t-x_s \over z_t}.$$ (13)

The angle of the reflected ray can be any value between $-\pi$/2 and $\pi$/2. Therefore, equations (8) and (13) define the image of the scatter in the migrated shot profile. Figure 6 shows the two examples of the image of a scatter. We see that the image of the scatter is stretched out in both cases. When $\gamma < 1$, the image moves upwards and curves downwards; when $\gamma \gt 1$, the image moves downwards and curves upwards. Figure 6 also shows graphically that the curves defined by equations (8) and (13) are indeed the envelopes of migration ellipses.

 

envsca
Figure 6 When the reflected energy from a scatter in a homogeneous medium is migrated with a velocity of (a) -10% error; (b) +10% error, the image of the scatter, defined by the envelope of the migration ellipses, is stretched out and (a) curves downwards; (b) curves upwards.

For constant-velocity media, equations (8) and (13) actually define the kinematics of the summation operator that does residual migration for the migrated shot profiles. Figure 7 shows two sets of curves computed from these two equations. To revise distorted images on migrated shot profiles, we can sum the data samples along the curves defined by equations (8) and (13).

 

impres
Figure 7 Examples of the kinematics of the summation operator that does residual migration for the migrated shot profiles. The star indicates the shot position. The dots indicate the actual location of the scatters: (a) $\gamma = 0.9$; (b) $\gamma = 1.1$.