Fréchet derivatives of coupled seismograms with respect to an anelastic rotating earth

A theory, based on higher-order perturbations, is given and used to express the partial derivatives of the seismic waveform with respect to perturbations of density, anelasticity and anisotropy. Fréchet derivatives are expressed with respect to a general aspherical model by using modulation functions, which are already used for the computation of aspherical seismograms. A direct solution method (DSM) that optimizes other previously proposed DSMs is proposed for the computation of the modulation functions. Numerical tests point out significant differences between the general Fréchet derivatives and those used for more classical approaches (great-circle average or frozen-path approximation), as well as important focusing/defocusing effects. This theory will enable future imaging inversions of the small-scale heterogeneities of the Earth.