It has been known for over 50 years that seismic anisotropy must be included in a realistic analysis of most seismic data. The evidence for this consists of the observed dependency in many contexts (reviewed briefly here) of seismic velocity upon angle of propagation and upon angle of S-wave polarization. Despite this well-established understanding, many current investigations continue to employ less realistic isotropic assumptions. One result is the appearance of artifacts which can be interpreted in terms of details of Earth structure rather than of the restrictive assumptions in the analysis.

The reason for this neglect of anisotropy is presumably the greater algebraic complexity and the larger number of free parameters of anisotropic seismics. However, the seismic anisotropy in the Earth is usually weak, and the equations for weak anisotropy are only marginally more complex than for isotropy. Further, the additional parameters are commonly required to describe the data. Moreover, the parameters of weak anisotropy defined below (combinations of the anisotropic elastic moduli) are less subject to compounding of uncertainty and to spatial resolution issues than are the individual anisotropic moduli themselves. Hence inversions should seek to fit data with these parameters, rather than with those individual moduli. We briefly review the theory for weak anisotropy and present new equations for the weakly anisotropic velocities of surface waves. The analysis offers new insights on some well-known results found by previous investigations, for example the “Rayleigh wave–Love wave inconsistency”, including the facts that Raleigh wave velocities depend not only on the horizontal SV velocity but also on the anisotropy, and Love wave velocities depend not only on the horizontal SH velocity but also on the anisotropy.