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Consider a (nonlinear) function mapping
one element of the functional model space to
one element of the functional data space :
| |
(70) |

The tangent linear application to at point is
a linear operator defined by the expansion
| |
(71) |

where is a small perturbation in the model space.
The tangent linear application is also known under
the name of *Fréchet derivative* of
at point (109).
**fat3.sC
**

Figure 12 3D slowness model.

**fat3.fp3
**

Figure 13 3D sensitivity kernels for wave-equation MVA.
The frequency range is 1-16 Hz.
The kernels are complicated by the multipathing
occurring as waves propagate through the rough
salt body.
The image perturbation corresponds to a kinematic shift.

**fat3.fq3
**

Figure 14 3D sensitivity kernels for wave-equation MVA.
The frequency range is 1-16 Hz.
The kernels are complicated by the multipathing
occurring as waves propagate through the rough
salt body.
The image perturbation corresponds to an amplitude scaling.

**fat3.svty
**

Figure 15 Cross-section of 3D sensitivity kernels for wave-equation MVA.
The left panel corresponds to an image perturbation
produced a kinematic shift, while the right panel
corresponds to an image perturbation produced by
amplitude scaling.
The lowest sensitivity occurs in the center of the
kinematic kernel (left). In contrast, the maximum
sensitivity occurs in the center of the kernel (right).

frechet.exp can be written formally as

| |
(72) |

where
is a perturbation in the model space, and
is a perturbation in the image space.
If we denote by the *i*^{th} component of ,
and by an infinitesimal element of at location , we can write
| |
(73) |

*F*_{0}^{i} is, by definition, the integral kernel
of the Fréchet derivative ,*V* is the volume under investigation,
*dv* is a volume element of *V* and is the
integration variable over *V*.
The sensitivity kernel, a.k.a. *Fréchet derivative kernel* ,
*F*_{0}^{i} expresses the sensitivity of
to a perturbation of for
an arbitrary location in the volume *V*.
Sensitivity kernels occur in every inverse problem and have different meanings
depending of the physical quantities involved:

For **wideband traveltime tomography**
(104; 14; 3; 33; 55),
is represented by traveltime differences
between recorded and computed traveltimes in a reference medium.
The sensitivity kernels are infinitely-thin rays
computed by ray tracing in the background medium.
For **finite-frequency traveltime tomography**
(28; 52; 58; 76),
is represented by time shifts
measured by crosscorelation between the recorded wavefield
and a wavefield computed in a reference medium.
The sensitivity kernels are represented by hollow fat rays
(a.k.a. ``banana-doughnuts'') which depend on the background medium.
For **wave-equation tomography**
(114; 72; 73),
is represented by perturbations
between the recorded wavefield and the computed wavefield
in a reference medium.
The sensitivity kernels are represented by fat rays with similar forms
for either the Born or Rytov approximation.
For **wave-equation migration velocity analysis**
(10; 88; 89; 93),
is represented by image perturbations.
The sensitivity kernels are discussed in the following sections.

Wave-equation migration velocity analysis (WEMVA)
establishes a linear relation between
perturbations of the slowness model and
perturbations of migrated images . and correspond, respectively, to and
in linear.

Formally, we can write

| |
(74) |

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 frechet.exp.
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 the next examples, I 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 domain as follows:

| |
(75) |

and a purely dynamic type , implemented by
scaling the reference image with
an arbitrary number:
| |
(76) |

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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Stanford Exploration Project

11/4/2004