We investigate the problem of retrieving anisotropy as a function of depth in the mantle, from the observed azimuthal variations of Love and Rayleigh wave velocities. Following the approach of Smith and Dahlen, this azimuthal dependence is expressed in terms of a Fourier series of the azimuth &thgr;. For the most general case of anisotropy (provided it is small enough), some simple linear combinations of the elastic tensor coefficients are shown to describe the total effect of anisotropy (both polarization anisotropy and azimuthal anisotropy) on the propagation of surface waves. For the terms that do not depend on the azimuth the combinations are related to the elastic coefficients of a transversely isotropic mantle. For the azimuthal terms the relevant combinations are explicited. It is found that the partial derivatives of the azimuthal terms with respect to these combinations are easy to compute for they are proportional to the partial derivatives of a transversely isotropic modelin the case of a plane-layered model. In a first approximation the same property holds true for a spherical earth and we calculate from PREM all the partial derivatives needed for performing the inversion of the azimuthal anisotropy of surface waves in the period range 50--300 s. It is observed that very shallow anisotropy can be responsible for substantial azimuthal variations up to the longest periods. With this approach it is also easy to compute the azimuthal variations of surface wave velocities produced by any anisotropic model. When a Cij elastic tensor is chosen for the upper mantle, azimuthal variations up to 2% are obtained for Rayleigh waves. The azimuthal variations of Love wave velocities are very small. The 2&THgr;- term of the azimuthal variations of Rayleigh wave velocities is the dominant term. Its fast axis corresponds to the fast axis of P-waves.