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Limitations to published formulas are identified for depth-dependent stresses.A corrected perturbation theory is derived for general anisotropy.Explicit expressions are given for an isotropic half space.Validation with an exact method is provided.Examples include depth-dependent third order elastic constants.This article revisits the evaluation by a perturbation theory of the modification of the Rayleigh wave velocity under a static loading varying with depth. Two derivations, that have been exposed in the past and presented as comparable, are questioned. A new derivation of the perturbation formula is given by adapting Auld’s approach. Validation with exact calculations is provided. The examples cover depth-varying static stress as well as depth-varying third order elastic properties.