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It is shown that the wavenumber dcrivative of the dispersion relation, dω(k)/dk, is a Fréchet differentiable gross Earth datum. The differential kernel functions are given explicitly as expressions quadratic in the normal mode eigenfunctions and their wavenumber derivatives. Explicit expressions for the latter are given as particular solutions to the inhomogeneous propagator equations and as expansions in terms of eigenfunctions. Both spherical stratification and plane stratification are considered in detail.
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Freeman Gilbert (1976) studied this question.
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