On Backus average in modelling guided waves

Abstract

We examine the Backus [1] average of a stack of isotropic layers overlying an isotropic halfspace to examine its applicability for the quasi-Rayleigh and Love wave dispersion curves, both of which apply to the same model. We compare these curves to values obtained for the stack of layers using the propagator matrix. The Backus [1] average is applicable only for thin layers or low frequencies. This is true for both weakly inhomogeneous layers resulting in a weakly anisotropic medium and strongly inhomogeneous alternating layers resulting in a strongly anisotropic medium. We also compare the strongly anisotropic and weakly anisotropic media, given by the Backus [1] averages, to results obtained by the isotropic Voigt [2] 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 [1] average for a stack of alternating transversely isotropic layers that is strongly inhomogeneous to evaluate the dispersion curves for the resulting medium. We compare these 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 [1] average for a stack of nonalternating transversely isotropic layers that is strongly inhomogeneous, and evaluate the quasi-Rayleigh wave dispersion curves for the resulting transversely isotropic medium. We compare these curves to values obtained using the propagator matrix for the stack of layers. In this case, the Backus [1] average performs less well, but-for the fundamental mode-remains adequate for low frequencies or thin layers. IntroductionThis paper is an examination of the applicability of the Backus [1] average to guided-wave-dispersion modelling. We compare the dispersion curves of both Love and quasi-Rayleigh waves for the Backus [1] average of a stack of layers to the dispersion curves for these layers using the propagator-matrix method. The prefix distinguishes quasi-Rayleigh waves, which are guided waves, from classical Rayleigh waves, which propagate within a halfspace. We examine the effects of strength of inhomogeneity, anisotropy and layer thickness.The focus on examining both Love and quasi-Rayleigh waves is motivated by their existence in the same model. This is a consequence of compatibility of their wave equations and boundary conditions, as discussed by Dalton et al. [3].Similar work-for Love waves in a stack of alternating isotropic layers-was done by Anderson [4], in which the author drew on the results of Postma [5]. Herein, we broaden this work to quasi-Rayleigh waves and, drawing on the work of Backus [1], to a nonalternating stack of isotropic layers. We extend the scope to include Love waves for a stack of alternating transversely isotropic layers and quasi-Rayleigh waves for stacks of alternating and nonalternating transversely isotropic layers.We begin this paper by providing background information for the Backus [1] and Voigt [2] avera...