A dictionary gives many definitions for the word
*run.*
They are related, but they are distinct.
Similarly,
the word
*migration*
in geophysical prospecting has about
four related but distinct meanings.
The simplest is like the meaning of the word
*move.*
When an object at some location in the (*x* , *z*)-plane
is found at a different location at a later time *t*,
then we say it
*moves.*
Analogously, when a wave arrival (often called an *event* )
at some location in the (*x* , *t*)-space of geophysical observations
is found at a different position for a different survey line
at a greater depth *z*, then we say it
*migrates.*

To see this more clearly,
imagine the four frames of Figure 3
being taken from a movie.
During the movie, the depth *z* changes
beginning at the beach (the earth's surface)
and going out to the storm barrier.
The frames are superimposed in Figure 4(left).

dcretard
Left shows a superposition of the hyperbolas
of Figure 3.
At the right the superposition incorporates a shift,
called retardation , to keep the hyperbola tops together.
Figure 4 |

Mainly what happens in the movie is that
the event migrates upward toward *t*=0.
To remove this dominating effect of vertical translation
we make another superposition,
keeping the hyperbola tops all in the same place.
Mathematically, the time *t* axis is replaced by a so-called
*retarded*
time axis *t* ' = *t* + *z*/*v*, shown in Figure 4(right).
The second, more precise definition of
*migration*
is the motion of an event in ( *x* , *t* ' )-space as *z* changes.
After removing the vertical shift,
the residual motion is mainly a shape change.
By this definition, hyperbola tops, or horizontal layers, do not migrate.

The hyperbolas in Figure 4 really extend to infinity,
but the drawing cuts each one off at a time equal times
its earliest arrival.
Thus the hyperbolas shown depict only rays
moving within 45 of the vertical.
It is good to remember this,
that the ratio of first arrival time on a hyperbola
to any other arrival time
gives the cosine of the angle of propagation.
The cutoff on each hyperbola is a ray at 45.Notice that the end points of the hyperbolas on the drawing
can be connected by a straight line.
Also, the slope at the end of each hyperbola is the same.
In physical space, the angle of any ray is
.For any plane wave
(or seismic event that is near a plane wave),
the slope is ,as you can see by considering a wavefront intercepting the earth's surface
at angle .So, energy moving on a straight line in physical (*x* , *z*)-space
migrates along a straight line in data (*x* , *t*)-space.
As *z* increases, the energy of all angles comes together to a focus.
The focus is the exploding reflector.
It is the gap in the barrier.
This third definition of migration is that it is the
process that somehow pushes observational data--wave
height as a function of *x* and *t* --from the beach to the barrier.
The third definition stresses not so much the motion itself,
but the transformation from the beginning point to the ending point.

To go further, a more general example is needed
than the storm barrier example.
The barrier example is confined to making Huygens sources
only at some particular *z*.
Sources are needed at other depths as well.
Then, given a wave-extrapolation process
to move data to increasing *z* values,
exploding-reflector images are constructed with

(2) |

t = 0 |
||

all | ||

Diffraction

is sometimes regarded as the natural process that creates
and enlarges hyperboloids.
*Migration*

is the computer process that does the reverse.

Another aspect of the use of the word
*migration*
arises
where the horizontal coordinate can be either shot-to-geophone midpoint *y*,
or offset *h*.
Hyperboloids can be downward continued
in both the (*y* , *t*)- and the (*h* , *t*)-plane.
In the (*y* , *t*)-plane this is called
*migration*
or
*imaging,*
and in the (*h* , *t*)-plane it is called
* focusing*
or

12/26/2000