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In this paper, we discuss five parameters that indicate the inhomogeneity of a stack of parallel isotropic layers. We show that, in certain situations, they provide further insight into the intrinsic inhomogeneity of a Backus medium, as compared to the Thomsen parameters. Additionally, we show that the Backus average of isotropic layers is isotropic if and only if $gamma=0$. This is in contrast to parameters $delta$ and $epsilon$, whose zero values do not imply isotropy.
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In general, the Backus average of an inhomogeneous stack of isotropic layers is a transversely isotropic medium. Herein, we examine a relation between this inhomogeneity and the strength of resulting anisotropy, and show that, in general, they are proportional to one another. There is an important case, however, in which the Backus average of isotropic layers results in an isotropic -- as opposed to a transversely isotropic -- medium. We show that it is a consequence of the same rigidity of layers, regardless of their compressibility. Thus, in general, the strength of anisotropy of the Backus average increases with the degree of inhomogeneity among layers, except for the case in which all layers exhibit the same rigidity.
Elastic anisotropy might be a combined effect of the intrinsic anisotropy and the anisotropy induced by thin-layering. The Backus average, a useful mathematical tool, allows us to describe such an effect quantitatively. The results are meaningful only if the underlying physical assumptions are obeyed, such as static equilibrium of the material. We focus on the only mathematical assumption of the Backus average, namely, product approximation. It states that the average of the product of a varying function with nearly-constant function is approximately equal to the product of the averages of those functions. We discuss particular, problematic case for which the aforementioned assumption is inaccurate. Further, we focus on the seismological context. We examine numerically if the inaccuracy affects the wave propagation in a homogenous medium -- obtained using the Backus average -- equivalent to thin layers. We take into consideration various material symmetries, including orthotropic, cubic, and others. We show that the problematic case of product approximation is strictly related to the negative Poissons ratio of constituent layers. Therefore, we discuss the laboratory and well-log cases in which such a ratio has been noticed. Upon thorough literature review, it occurs that examples of so-called auxetic materials (media that have negative Poissons ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poissons ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. Except for the main objectives, we also show that the averaging of cubic layers results in an equivalent medium with tetragonal (not cubic) symmetry. Additionally, we present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.
Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution $P(k)sim k^{-gamma}$, where the degree exponent $gamma$ describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various $gamma in (2,1+frac{ln 3}{ln 2}]$, with an aim to explore the impacts of structure heterogeneity on APL and RWs. We show that the degree exponent $gamma$ has no effect on APL $d$ of RSFTs: In the full range of $gamma$, $d$ behaves as a logarithmic scaling with the number of network nodes $N$ (i.e. $d sim ln N$), which is in sharp contrast to the well-known double logarithmic scaling ($d sim ln ln N$) previously obtained for uncorrelated scale-free networks with $2 leq gamma <3$. In addition, we present that some scaling efficiency exponents of random walks are reliant on degree exponent $gamma$.
We consider a long-wave transversely isotropic (TI) medium equivalent to a series of finely parallel-layered isotropic layers, obtained using the citet{Backus} average. In such a TI equivalent medium, we verify the citet{Berrymanetal} method of indicating fluids and the authors method citep{Adamus}, using anisotropy parameter $varphi$. Both methods are based on detecting variations of the Lame parameter, $lambda$, in a series of thin isotropic layers, and we treat these variations as potential change of the fluid content. To verify these methods, we use Monte Carlo (MC) simulations; for certain range of Lame parameters $lambda$ and $mu$---relevant to particular type of rocks---we generate numerous combinations of these parameters in thin layers and, after the averaging process, we obtain their TI media counterparts. Subsequently, for each of the aforementioned media, we compute $varphi$ and citet{Thomsen} parameters $epsilon$ and $delta$. We exhibit $varphi$, $epsilon$ and $delta$ in a form of cross-plots and distributions that are relevant to chosen range of $lambda$ and $mu$. We repeat that process for various ranges of Lame parameters. Additionally, to support the MC simulations, we consider several numerical examples of growing $lambda$, by using scale factors. As a result of the thorough analysis of the relations among $varphi$, $epsilon$ and $delta$, we find eleven fluid detectors that compose a new fluid detection method. Based on these detectors, we show the quantified pattern of indicating change of the fluid content.
In this paper, we consider a long-wave equivalent medium to a finely parallel-layered inhomogeneous medium, obtained using the Backus average. Following the work of Postma and Backus, we show explicitly the derivations of the conditions to obtain the equivalent isotropic medium. We demonstrate that there cannot exist a transversely isotropic (TI) equivalent medium with the coefficients $c^{overline{rm TI}}_{1212} eq c^{overline{rm TI}}_{2323}$, $c^{overline{rm TI}}_{1111} = c^{overline{rm TI}}_{3333}$ and $c^{overline{rm TI}}_{1122} = c^{overline{rm TI}}_{1133}$. Moreover, we consider a new parameter, $varphi$, indicating the anisotropy of the equivalent medium, and we show its range and properties. Subsequently, we compare $varphi$ to the Thomsen parameters, emphasizing its usefulness as a supportive parameter showing the anisotropy of the equivalent medium or as an alternative parameter to $delta$. We argue with certain Berryman et al. considerations regarding the properties of the anisotropy parameters $epsilon$ and $delta$. Additionally, we show an alternative way---to the one mentioned by Berryman et al.---of indicating changing fluid content in layered Earth.
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