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The exploding reflector model provides the linear relation
between the subsurface image and the wavefield on the surface.
In forward modeling, the input is the subsurface image
and the output is the wavefield on the surface.
The migration is the conjugate operation to the forward modeling
(Claerbout, 1992a).
If the modeling operator is a unitary operator,
the migration operator, which is the conjugate of the modeling
operator, will backproject the wavefield
to the subsurface image, which results in the original input
of the modeling.
However, in reality the forward operator is the continuous wave equation
for a given earth model,
and the backprojection operator, which is migration operator,
is not the conjugate to the forward operator.
Instead we use an operator that is conjugate
to an approximated forward operator like
the Kirchhoff, phase shift, and finite-difference method.
This discrepancy along with the finite and the discrete limits
produces many artifacts on the image after
migration.
The forward modeling can be simply formulated as

| |
(1) |

where is the wave equation operator
that does the forward modeling,
is the continuous model space,
is the truncating and sampling operator
that simulates the finite and discrete seismic survey,
and is the data we obtain in reality.
In conventional migration, we approximate
the forward operator as
| |
(2) |

where represents an approximated
forward modeling operator.
Therefore, the conventional migration actually solve the problem as follows
| |
(3) |

Even though the sampling interval is small and the aperture is large,
which makes the operator identity matrix, some
artifacts might be appeared if the operator differs from *W*.
Thus, both the accurate approximation for the wave equation and
the wide aperture and the dense sampling are important to get a good image.
The other approach to getting a good image for a
given sampling interval and a given poor but cheap operator
might be to use the least-squares optimization technique to
find the best image.
To solve formally for the unknown in the least-squares sense, we solve the normal equation as follows:

| |
(4) |

This may not recover the original image because was generated by the operator which differs from the operator .If we assume that we have approximated the wave equation very accurately,
that is ,and the only problem that causes artifacts is due to the operator ,we are solving correct normal equation as
| |
(5) |

Usually the operator is too large in size to solve
the inversion of .An alternative might be an iterative inversion,
with the hope of a fast convergence,
such as the conjugate gradient method.
Fortunately, the initial guess for the conjugate gradient is
conventional migration, which is the image after applying the conjugate
operator of the forward operator.
It gets very close to the solution except for some artifacts.
We can expect a fast convergence and a close solution obtained
in a few iterations.
The object function we want to minimize using the conjugate gradient
can be formulated as follows :

| |
(6) |

where represents the data generated by the forward
operator and this operator could be any modeling operator
like the Kirchhoff, phase-shift, or finite-difference method.

** Next:** Kirchhoff migration in constant
** Up:** Ji: LS imaging, datuming
** Previous:** Introduction
Stanford Exploration Project

11/17/1997