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## Fréchet derivative integral kernels

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 ith component of , and by an infinitesimal element of at location , we can write
 (4)
G0i 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 , G0i 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