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Wave-equation migration velocity analysis (WEMVA)
is based on a linear relation established between
perturbations of the slowness model and
perturbations of migrated images . and correspond, respectively, to and
in equation (3).
Formally, we can write

| |
(5) |

where is the linear first-order Born
wave-equation MVA operator.
The operator incorporates all first-order scattering
and extrapolation effects for media of arbitrary complexity.
The major difference between WEMVA and
wave-equation tomography is that is formulated
in the image space for the former as opposed to the
data space for the later. Thus, with WEMVA we are able
to exploit the power of residual migration in perturbing
migrated images - a goal which is much harder to achieve
in the space of the recorded data.
By construction, the linear operator depends on
the wavefield computed by extrapolation of the surface
data using the background slowness,
which corresponds to in equation (2).
Thus, the operator depends directly on the type of
recorded data and its frequency content,
and it also depends on the background slowness model.
Thus, the main elements that control the shape
of the sensitivity kernels are

- the frequency content of the background wavefield,
- the type of source from which we generate the background
wavefield (e.g. point source, plane wave), and
- the type of perturbation introduced in the image space,
which for this problem corresponds to the data space.

In our examples, we define two types of image perturbations:
a purely kinematic type , implemented simply as a derivative
of the image with respect to depth, which can be implemented as
a multiplication in the depth wavenumber domain
Sava and Biondi (2004a,b) as follows:

| |
(6) |

and a purely dynamic type , implemented by
scaling the reference image with
an arbitrary number:
| |
(7) |

In both cases, the perturbations are limited to
a small portion of the image.
The main difference between and is given by the phase-shift between the
two image perturbations.

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** Up:** Theory
** Previous:** Fréchet derivative integral kernels
Stanford Exploration Project

5/23/2004