Without loss of generality, we consider experiments with a receiver array spreading on the right side of the shot. (A split-spreading experiment can be decomposed into two one-side spreading experiments). Let us first look at a simple model that consists of a flat reflector with an overburden of a constant velocity medium. We want to investigate how the image of the reflector will be distorted as the parameter v in deviates from the true velocity vt of the medium.
Goldin (1982) proved that when v is equal to the crossing points of the caustics of rays and the image of the reflector are located right below the shot location. In Figure 1, a sequence of distorted images are drawn with . We see that when , the image of the reflector is completely migrated to the left side of the shot. Let us demonstrate this fact by graphics.
As I mentioned earlier, is a Kirchhoff migration operator. Its output is the superposition of a set of migration ellipses. Therefore the image I(t0,x) is constructed by the envelopes of these ellipses. Because the principle axes of migration ellipses depend on the parameter v, so do their envelopes. In Figure 2, I show a set of migration ellipses and their envelopes. When , the envelope of the migration ellipses appears entirely on the left side of the shot. In contrast, when v=0.9vt, the envelope of the migration ellipses appears entirely on the right side of the shot. In fact, it can be mathematically proved that, after the transformation with operator , the images of the events whose velocities are equal to will be completely separable from the images of the events whose velocities are greater than v.