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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 *v*_{t} 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.

**imaset
**

Figure 1 The distorted images of the interface between two flat layers.
The star indicates the shot position and the dashed line represents
the receiver array. When *v*=0.9*v*_{t}, the distorted image bends upwards.
As *v* increases, the distorted image gradually moves to the left. Finally
when , the distorted image is completely migrated to the left
side of the shot.

As I mentioned earlier, is a Kirchhoff migration
operator. Its output is the superposition of a set of migration ellipses.
Therefore the image *I*(*t*_{0},*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*=0.9*v*_{t}, 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*.

**ellset
**

Figure 2 The distorted image of a reflector is defined by the envelopes of the
migration ellipses that are *v*-dependent. The star indicates the
position of the shot and the
dashed line represents the receiver array. (a) When , the
envelope of the migration ellipses is entirely on the left side of the
shot; (b) when *v*= 0.9*v*_{t}, the envelope of the migration ellipses is
entirely on the right side of the shot.

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** Up:** PRINCIPLES OF THE ALGORITHM
** Previous:** Kinematic operators
Stanford Exploration Project

1/13/1998