On Backus average in modelling guided waves

We study the Backus (1962) average of a stack of layers overlying a halfspace to examine its applicability for the quasi-Rayleigh and Love wave dispersion curves. We choose these waves since both propagate in the same model. We compare these curves to values obtained for the stack of layers using the propagator matrix. In contrast to the propagator matrix, the Backus (1962) average is applicable only for thin layers or low frequencies. This is true for both a weakly inhomogeneous stack of layers resulting in a weakly anisotropic medium and a strongly inhomogeneous stack of alternating layers resulting in a strongly anisotropic medium. We also compare the strongly anisotropic and weakly anisotropic media, given by the Backus (1962) averages, to results obtained by the isotropic Voigt (1910) averages of these media. As expected, we find only a small difference between these results for weak anisotropy and a large difference for strong anisotropy. We perform the Backus (1962) average for a stack of alternating transversely isotropic layers that is strongly inhomogeneous to evaluate the dispersion relations for the resulting medium. We compare the resulting dispersion curves to values obtained using a propagator matrix for that stack of layers. Again, there is a good match only for thin layers or low frequencies. Finally, we perform the Backus (1962) average for a stack of nonalternating transversely isotropic layers that is strongly inhomogeneous, and evaluate the quasi-Rayleigh wave dispersion relations for the resulting transversely isotropic medium. We compare the resulting curves to values obtained using the propagator matrix for the stack of layers. In this case, the Backus (1962) average performs less well, but—for the fundamental mode—remains adequate for low frequencies or thin layers.