Rayleigh-wave ellipticity in weakly heterogeneous layered media

We derive approximate expressions for the ellipticity (i.e. horizontal-to-vertical or vertical-to-horizontal ratio) of Rayleigh waves propagating in a layered medium. The approximation is based on the generalized energy equation for Rayleigh waves, which has been used previously to obtain perturbational results for ellipticity. For a medium with weakly heterogeneous layers, we obtain an approximation from the perturbational result by taking the background medium to be homogeneous. The generalized energy equation also requires an auxiliary function and we discuss how the various possible functions are related to the homogeneous Rayleigh-wave eigenfunction. The analysis reveals that, within the weak approximation, the product of ellipticity and squared phase velocity is linearly related to squared shear wave velocity in the subsurface. We show the accuracy of the approximation with a simple layer-over-half-space model and then demonstrate its utility in a linear inversion scheme for shear wave velocity.