Layer-induced elastic anisotropy Part 2: inversion of compound parameters to constituent parameters

Abstract

was shown that a medium consisting of a periodic sequence of layers is, in the long-wavelength approximation, equivalent to a homogeneous compound medium with elastic parameters that are generalized averages of the constituents' stiffnesses. Though the matrix-algorithm described in Part 1 works with anisotropic constituents, the most interesting application is to layer sequences with isotropic constituents, i.e., to transversely isotropic (TI) compound media. Part 2 discusses the possibility to obtain information about the (thin-layer) constituents from the properties of the compound medium. Though every periodic sequence of isotropic layers results in a TI medium, the reverse is not true: there are TI media that cannot be "modeled" by a periodic sequence of isotropic layers. Those that can be modeled can be inverted to layer sequences that result in precisely the observed anisotropy. This inversion is not unique, but it constrains the possibilities. The critical tool to determine the possibility of modeling a TI medium is the concept of stability. Unstable compound media -that release energy on being deformed -would not exist. However, for inversion we must insist that not only the compound medium, but also the potential constituents are stable. In preparing a catalog that covers all possible media, instability is the boundary beyond which the calculation becomes meaningless. Inversion means to determine possible causes of the observed anisotropy, ideally the elastic parameters of the constituents and their contribution to the compound medium. This is possible, though under several restrictions: Not all TI media are long-wave equivalent to a periodically layered sequence of isotropic layers. Those that are can be "modeled" by a variety of layer sequences. Every TI medium that can be modeled at all can be modeled by as few as three layers, but the set of all models is a threeparametric manifold. If a TI medium can be modeled by two constituents only, this can be done only in one way, unless the constituents have the same ratio of S-to P-velocity. In that case, the set of possible models forms a one-parametric manifold.

On Backus Average for Generally Anisotropic Layers

“…The subject of Backus average was examined by several researchers, among them, Helbig and Schoenberg (1987), Schoenberg and Muir (1989), Berryman (1997), Helbig (1998Helbig ( , 2000, Carcione et al (2012), Kumar (2013), Brisco (2014), and Danek and Slawinski (2016). However, further venues of investigation remain open.…”