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Consider a (nonlinear) function mapping
one element of the functional model space to
one element of the functional data space :
| |
(1) |

The tangent linear application to at point is
a linear operator defined by the expansion
| |
(2) |

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 Tarantola (1987).
Equation (2) can be written formally as

| |
(3) |

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
| |
(4) |

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

- For
**wideband traveltime tomography**
Bishop et al. (1985); Kosloff et al. (1996); Stork (1992),
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**
Dahlen et al. (2000); Hung et al. (2000); Marquering et al. (1999); Rickett (2000),
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**
Pratt (1999); Woodward (1992),
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**
Biondi and Sava (1999); Sava and Biondi (2004a,b); Sava and Fomel (2002),
is represented by image perturbations.
The sensitivity kernels are discussed in the following sections.

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Stanford Exploration Project

5/23/2004