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# On Backus Average for Generally Anisotropic Layers

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## Abstract

In this paper, following the Backus (1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers. In over half-a-century since the publications of Backus (1962) there have been numerous publications applying and extending that formulation. However, neither George Backus nor the authors of the present paper are aware of further examinations of mathematical underpinnings of the original formulation; hence, this paper. We prove that---within the long-wave approximation---if the thin layers obey stability conditions then so does the equivalent medium. We examine---within the Backus-average context---the approximation of the average of a product as the product of averages, and express it as a proposition in terms of an upper bound. In the presented examination we use the expression of Hooke's law as a tensor equation; in other words, we use Kelvin's---as opposed to Voigt's---notation. In general, the tensorial notation allows us to examine conveniently effects due to rotations of coordinate systems.
J Elast (2017) 127:179–196
DOI 10.1007/s10659-016-9608-z
On Backus Average for Generally Anisotropic Layers
Len Bos1·David R. Dalton2·Michael A. Slawinski2·
Theodore Stanoev2
Received: 22 June 2016 / Published online: 7 November 2016
Abstract In this paper, following the Backus (in J. Geophys. Res. 67(11):4427–4440, 1962)
approach, we examine expressions for elasticity parameters of a homogeneous generally
anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic
layers. These expressions reduce to the results of Backus (1962) for the case of isotropic
and transversely isotropic layers.
In the over half-a-century since the publications of Backus (1962) there have been numer-
ous publications applying and extending that formulation. However, neither George Backus
nor the authors of the present paper are aware of further examinations of the mathematical
underpinnings of the original formulation; hence this paper.
We prove that—within the long-wave approximation—if the thin layers obey stability
conditions, then so does the equivalent medium. We examine—within the Backus-average
context—the approximation of the average of a product as the product of averages, which
underlies the averaging process.
In the presented examination we use the expression of Hooke’s law as a tensor equation;
in other words, we use Kelvin’s—as opposed to Voigt’s—notation. In general, the tensorial
notation allows us to conveniently examine effects due to rotations of coordinate systems.
Keywords Backus averaging ·General anisotropy ·Seismology ·Approximation ·Upper
bound ·Stability
BM.A. Slawinski
mslawins@mac.com
L. Bos
leonardpeter.bos@univr.it
D.R. Dalton
dalton.nﬂd@gmail.com
T. Stanoev
theodore.stanoev@gmail.com
1Dipartimento di Informatica, Università di Verona, Verona, Italy
2Department of Earth Sciences, Memorial University of Newfoundland, St. John’s, Newfoundland,
... Previous research for effective anisotropic properties of layered media was focused on the anisotropic velocity (e.g. Backus, 1962;Schoenberg and Muir, 1989;Bakulin, 2003;Bakulin and Grechka, 2003;Kumar, 2013;Bos et al., 2017), and much less attention has been devoted to layer-induced anisotropic attenuation. Carcione (1992) derived the effective anisotropic attenuation of isotropic layered media with interval isotropic attenuation. ...
... For simplicity, throughout the paper, we use a binary model with N = 2 and the anisotropic constituent is assumed to be one type of vertical transversely isotropic (VTI), horizontal transversely isotropic (HTI) or orthorhombic (ORT) medium in this stratified viscoelastic model. The effective stiffness coefficients of this stratified viscoelastic model are computed according to the Backus averaging/upscaling technique (Backus, 1962;Schoenberg and Muir, 1989;Kumar, 2013;Bos et al., 2017; see Appendix ) using ...
... The original Backus (1962) averaging technique was used to compute the effective properties for isotropic elastic layers in the long-wavelength limit. After years of research, the generalization of Backus averaging (or upscaling) is derived for generally anisotropic elastic layers, such as ORT, monoclinic and triclinic layers (Schoenberg and Muir, 1989;Kumar, 2013;Bos et al., 2017). The Backus averaging/upscaling technique also be applied to the viscoelastic layers in the long-wavelength limit (Carcione, 1992;, which means the dominant wavelength is much larger than the thickness of each layer. ...
Article
An important cause of seismic anisotropic attenuation is the interbedding of thin viscoelastic layers. However, much less attention has been devoted to layer‐induced anisotropic attenuation. Here, we derive a group of unified weighted average forms for effective attenuation from a binary isotropic, transversely isotropic‐ and orthorhombic‐layered medium in the zero‐frequency limit by using the Backus averaging/upscaling method and analyse the influence of interval parameters on effective attenuation. Besides the corresponding interval attenuation and the real part of stiffness, the contrast in the real part of the complex stiffness is also a key factor influencing effective attenuation. A simple linear approximation can be obtained to calculate effective attenuation if the contrast in the real part of stiffness is very small. In a viscoelastic medium, attenuation anisotropy and velocity anisotropy may have different orientations of symmetry planes, and the symmetry class of the former is not lower than that of the latter. We define a group of more general attenuation‐anisotropy parameters to characterize not only the anisotropic attenuation with different symmetry classes from the anisotropic velocity but also the elastic case. Numerical tests reveal the influence of interval attenuation anisotropy, interval velocity anisotropy and the contrast in the real part of stiffness on effective attenuation anisotropy. Types of effective attenuation anisotropy for interval orthorhombic attenuation and interval transversely isotropic attenuation with a vertical symmetry (vertical transversely isotropic attenuation) are controlled only by the interval attenuation anisotropy. A type of effective attenuation anisotropy for interval TI attenuation with a horizontal symmetry (horizontal transversely isotropic attenuation) is controlled by the interval attenuation anisotropy and the contrast in the real part of stiffness. The type of effective attenuation anisotropy for interval isotropic attenuation is controlled by all three factors. The magnitude of effective attenuation anisotropy is positively correlated with the contrast in the real part of the stiffness. Effective attenuation even in isotropic layers with identical isotropic attenuation is anisotropic if the contrast in the real part of stiffness is non‐zero. In addition, if the contrast in the real part of stiffness is very small, a simple linear approximation also can be performed to calculate effective attenuation‐anisotropy parameters for interval anisotropic attenuation.
... The Backus average is commonly used by Geophysics practitioners, especially in the context of seismic exploration. Its theoretical properties have been studied, for example, in [8][9][10][11]. We rely on the formulation derived in [10] and given in formulas (38)-(43). ...
... Its theoretical properties have been studied, for example, in [8][9][10][11]. We rely on the formulation derived in [10] and given in formulas (38)-(43). ...
... herein, we use the notation of equations (5)-(9) of Bos et al. [10]. Also, A has the block structure of ...
Article
Full-text available
As shown by Backus (J Geophys Res 67(11):4427–4440, 1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of randomly oriented anisotropic elasticity tensors, which—one might reasonably expect—would result in an isotropic medium. However, we show—by means of a fundamental symmetry of the Backus average—that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, an analogy between the Backus and Gazis et al. (Acta Crystallogr 16(9):917–922, 1963) averages.
... The problem of fine, parallel layering and its long-wave equivalent medium approximation has been treated by a number of authors. Among many of them there are: Postma (1955), Backus (1962), Helbig and Schoenberg (1987), Schoenberg and Muir (1989), Berryman et al. (1999), and Bos et al. (2017). One of the first authors who stated and derived that the layered medium may be viewed as transversely isotropic (TI), was Postma (1955). ...
... Seven years later, fundamental work of Backus (1962) provided us an elegant formula of the TI medium, long-wave equivalent to isotropic or TI layers of different thicknesses, with no assumption of periodicity. Aforementioned formula was extended to lower symmetry classes-including generally anisotropic case-by Bos et al. (2017). However, in this paper, we do not consider lower symmetry classes than the TI one, only Backus formula for isotropic layers is used. ...
Preprint
Full-text available
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 author's method \citep{Adamus}, using anisotropy parameter $\varphi$. Both methods are based on detecting variations of the Lam\'e 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 Lam\'e 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 Lam\'e 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.
... herein, we use the notation of equations (5)-(9) of Bos et al. (2017). Also, A has the block structure of ...
... The Backus-average equations are given by (Bos et al., 2017) ...
Article
Full-text available
As shown by Backus (1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of a randomly oriented anisotropic elasticity tensor, which-one might expect-would result in an isotropic medium. However, we show-by means of a fundamental symmetry of the Backus average-that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, a relationship between the Backus and Gazis et al. (1963) averages.
... The Backus average is a method that produces a homogenous medium that is long-wave equivalent to an inhomogeneous stack of thin layers. Notwithstanding the ubiquitous acceptance of the Backus average, it has been the topic of recent study for Adamus et al. (2018) and Bos et al. (2017aBos et al. ( ,b, 2018 as well as Dalton and Slawinski (2016) and Dalton et al. (2018). While the mathematical underpinnings of the Backus approach are analyzed by Bos et al. (2017a), there may exist a possible issue with the sole mathematical approximation used by Backus (Bos et al., 2017b). ...
... Notwithstanding the ubiquitous acceptance of the Backus average, it has been the topic of recent study for Adamus et al. (2018) and Bos et al. (2017aBos et al. ( ,b, 2018 as well as Dalton and Slawinski (2016) and Dalton et al. (2018). While the mathematical underpinnings of the Backus approach are analyzed by Bos et al. (2017a), there may exist a possible issue with the sole mathematical approximation used by Backus (Bos et al., 2017b). ...
Article
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In this paper, we continue the study of Bos et al. (2017b) regarding statistical and numerical considerations of the Backus (1962) product approximation. While the approximation is typically quite good for seismological scenarios, Bos et al. (2017b) demonstrate a physical scenario that could, in spite of the stability conditions for isotropic media, lead to an issue within the Backus average. Using the Preliminary Reference Earth Model of Dziewo\'nski and Anderson (1981), we investigate whether this issue is likely to occur in the context of seismology.
... For more than a half-century, the researchers have taken the product assumption for granted. Bos et al. [5] are the first authors to discuss its validity in the context of the Backus average. A year later, Bos et al. [6] find and examine statistically a particular case for which the product approximation results in spurious values. ...
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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 a nearly constant function is approximately equal to the product of the averages of those functions. We analyse particular problematic case for which the aforementioned assumption is inaccurate. Furthermore, we focus on the seismological context. We examine the inaccuracy’s effect on the wave propagation in a homogenous medium—obtained using the Backus average—equivalent to thin layers. Numerical simulations indicate clearly that the product approximation inaccuracy has negligible effect on wave propagation; irrespective of layers’ symmetries. To give the results a practical focus, we show that the problematic case of product approximation is strictly related to the negative Poisson’s ratio of constituents layers. 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 Poisson’s ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poisson’s ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. In addition to the main objectives, we also show that the averaging of cubic layers results in an equivalent medium with tetragonal (not cubic) symmetry. We present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.
... For more than a half-century, the researchers take the product assumption for granted. Bos et al. (2017) are the first authors to discuss its validity in the context of the Backus average. A year later, Bos et al. (2018) find and examine statistically a particular case for which the product approximation results in spurious values. ...
Preprint
Full-text available
Seismic anisotropy is often 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 the latter one quantitatively. The results are meaningful only if the underlying physical assumptions are obeyed, such as the low frequency of the propagating wave. In this paper, however, 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 examine numerically if this 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 Poisson's 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 rocks (rocks that have negative Poisson's ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poisson's ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. Except for the main objectives of the paper, we additionally show that the averaging of cubic layers results in an equivalent medium having tetragonal (not cubic) symmetry. Also, we present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.
... The problem of fine, parallel layering and its long-wave equivalent medium approximation has been treated by a number of authors, among many of them are: Postma (1955), Backus (1962), Helbig and Schoenberg (1987), Schoenberg and Muir (1989), Berryman et al. (1999), and Bos et al. (2017). In the work of Postma, the periodic, isotropic, two-layered medium (PITL) was considered; the formula for the equivalent homogeneous transversely isotropic (TI) medium was shown. ...
Preprint
Full-text available
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 $c^{\overline{\rm TI}}_{1212} \neq c^{\overline{\rm TI}}_{2323}$, $c^{\overline{\rm TI}}_{1111} = c^{\overline{\rm TI}}_{3333}$ 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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