Fréchet derivative integral kernels

Consider a (nonlinear) function $\bf{g}$ mapping one element of the functional model space ${\bf m}$ to one element of the functional data space $\d$: 

$$\d = \bf{g}\left({\bf m}\right)\;.$$ (1)

The tangent linear application to $\bf{g}$ at point ${\bf m}={\bf m}_0$ is a linear operator ${\bf G}_0$ defined by the expansion  

$$\bf{g}\left({\bf m}_0+\delta {\bf m}\right)= \bf{g}\left({\bf m}_0\right)+ {\bf G}_0\delta {\bf m}+ \dots \;,$$ (2)

where $\delta {\bf m}$ is a small perturbation in the model space. The tangent linear application ${\bf G}_0$ is also known under the name of Fréchet derivative of $\bf{g}$ at point ${\bf m}_0$ Tarantola (1987).

Equation (2) can be written formally as  

$$\delta \d= {\bf G}_0\delta {\bf m}\;,$$ (3)

where $\delta {\bf m}$ is a perturbation in the model space, and $\delta \d$ is a perturbation in the image space. If we denote by $\delta d^i$ the i^th component of $\delta \d$, and by $\delta m\left({\bf x}\right)$ an infinitesimal element of $\delta {\bf m}$at location ${\bf x}$, we can write  

$$\delta d^i= \int_\mathcal VG_0^i\left({\bf x}\right)\; \delta m\left({\bf x}\right)\; d v\left({\bf x}\right)\;.$$ (4)

G₀ⁱ is, by definition, the integral kernel of the Fréchet derivative ${\bf G}_0$,$\mathcal V$ is the volume under investigation, d v is a volume element of $\mathcal V$ and ${\bf x}$ is the integration variable over $\mathcal V$.The sensitivity kernel, a.k.a. Fréchet derivative kernel , G₀ⁱ expresses the sensitivity of $\delta d^i$ to a perturbation of $\delta m\left({\bf x}\right)$ for an arbitrary location ${\bf x}$ in the volume $\mathcal V$.

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), $\delta \d$ 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), $\delta \d$ 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), $\delta \d$ 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), $\delta \d$ is represented by image perturbations. The sensitivity kernels are discussed in the following sections.