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Imaging by wavefield extrapolation (WE) is based on recursive
continuation of the wavefields () from a given depth level
to the next, by means of an extrapolation operator ():
| |
(1) |

Here and hereafter, we use the following notation conventions:
stands for the linear operator
applied to *x*, and
stands for function *f* of argument *x*.
At any depth *z*,
the wavefield (), extrapolated through the background
medium characterized by the background velocity
(), interacts
with medium perturbations () and creates wavefield
perturbations ():

| |
(2) |

is a scattering operator relating slowness
perturbations to wavefield perturbations.
The total wavefield perturbation at depth is the
sum of the perturbation accumulated up to depth *z* from
all depths above (),
plus the scattered wavefield from depth ()extrapolated one depth step ():
| |
(3) |

We can use the recursive equation (3) to compute
a wavefield perturbation, given a precomputed
background wavefield and a slowness perturbation.
In a more compact notation, we can write
equation (3) as follows:

| |
(4) |

where and stand respectively
for the wavefield and slowness perturbations at all
depth levels, and , and are
respectively the wavefield extrapolation operator,
the scattering operator and the identity operator.
In our current implementation, refers to
a first-order Born scattering operator.
From the wavefield perturbation (),
we can compute an image perturbation () by applying
an imaging condition, .
For example, the imaging operator, () can be
a simple summation over frequencies.
If we accumulate all scattering, extrapolation
and imaging into a single operator
,we can write a simple linear expression relating
an image perturbation () to a
slowness perturbation ():

| |
(5) |

For wave-equation migration velocity analysis,
we use equation (5) to estimate
a perturbation of the slowness model from
a perturbation of the migrated image by minimizing
the objective function

| |
(6) |

where is a weighting operator related to the inverse
of the data covariance, indicating the reliability of the
data residuals.
Since, in most practical cases, the inversion problem
is not well conditioned, we need to add constraints on the
slowness model via a regularization operator.
In these situations, we use the modified objective
function
| |
(7) |

can be a regularization operator
which penalizes rough features of the model, and
is a scalar parameter which balances
the relative importance of the data residual,
, and the model residual,
.
An essential element of our velocity analysis method
is the image perturbation, . For the purposes
of the optimization problem in equation (7),
this is object is known and has to be precomputed,
together with the background wavefield used by
the operator .In the next section, we discuss how is estimated in practice.

** Next:** Image perturbations
** Up:** WEMVA theory
** Previous:** WEMVA theory
Stanford Exploration Project

5/23/2004