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There are two basic tasks to be done:
(1) make data from a model, and (2) make models from data.
The latter is often called **imaging**.

Imagine the earth had just one point reflector at (*x*_{0},*z*_{0}).
This reflector explodes at *t*=0.
The data at *z*=0 is a function of location *x* and travel time *t*
would be an impulsive signal along
the hyperbolic trajectory *t*^{2} = ((*x*-*x*_{0})^{2}+*z*_{0}^{2})/*v*^{2}.
Imagine the earth had more point reflectors in it.
The data would then be a superposition of more hyperbolic arrivals.
A dipping bed could be represented as points along a line in (*x*,*z*).
The data from that dipping bed must be
a superposition of many hyperbolic arrivals.

Now let us take the opposite point of view:
we have data and we want to compute a model.
This is called ``**migration**''.
Conceptually,
the simplest approach to migration is also based on the idea of
an impulse response.
Suppose we recorded data that was zero everywhere
except at one point (*x*_{0},*t*_{0}).
Then the earth model should be a spherical mirror
centered at (*x*_{0},*z*_{0}) because this model produces the required data,
namely, no received signal except when the sender-receiver pair
are in the center of the semicircle.

This observation plus the superposition principle
suggests an algorithm for making earth images:
For each location (*x*,*t*) on the data mesh *d*(*x*,*t*) add in
a semicircular mirror of strength *d* into the model *m*(*x*,*z*).
You need to add in a semicircle for every value of (*t*,*x*).
Notice again we use the same equation *t*^{2} = (*x*^{2}+*z*^{2})/*v*^{2}.
This equation is the ``conic section''.
A slice at constant *t* is a circle in (*x*,*z*).
A slice at constant *z* is a hyperbola in (*x*,*t*).

Examples are shown in Figure 2.
Points making up a line reflector diffract to a line reflection, and
how points making up a line reflection migrate to a line reflector.

**dip
**

Figure 2
Left is a superposition of many hyperbolas.
The top of each hyperbola lies along a straight line.
That line is like a reflector, but instead of using a continuous line,
it is a sequence of points.
Constructive interference gives an apparent reflection off to the side.
Right shows a superposition of semicircles.
The bottom of each semicircle lies along a line that
could be the line of an observed plane wave.
Instead the plane wave is broken into point arrivals,
each being interpreted as coming from a semicircular mirror.
Adding the mirrors yields a more steeply dipping reflector.

Besides the semicircle superposition migration method,
there is another migration method
that produces a similar result conceptually,
but it has more desirable results numerically.
This is the ``**adjoint modeling**'' idea.
In it, we sum the data over a hyperbolic
trajectory to find a value in model space that is located
at the apex of the hyperbola.
This is also called the ``**pull**'' method of migration.

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Stanford Exploration Project

3/1/2001