Fréchet derivatives and single scattering theory

Despite their obvious importance in geophysical inversion schemes, Fréchet derivatives have previously only been studied for very simple earth models. Although these derivatives can be interpreted mathematically as sensitivity functions, the basis for this assertion is not well understood in terms of physical laws. To some extent these limitations are the result of the conventional method used to derive the derivatives. By writing the equation(s) for a linear field (potential, diffusive, wave) in an arbitrarily perturbed medium as an integral rather than differential equation, one sees that the Fréchet derivatives or related sensitivity functions for any region in a model are just an integral over Green's functions. However, Fréchet differentiability must be established and the proof of this is shown to depend on the finiteness of the fields at the boundaries of the perturbed volume. If the scattering representation is Fréchet differentiale, the Fréchet derivatives are easily given a physical interpretation in terms of scattering theory. The method is illustrated by deriving an analytical expression for the DC resistivity Fréchet derivative in a uniform halfspace and the result is compared to previously published work.