On Backus Average for Generally Anisotropic Layers

Bos, Len; Dalton, David R.; Slawiński, Michael A.; Stanoev, Theodore

doi:10.1007/s10659-016-9608-z

Cited by 12 publications

(30 citation statements)

References 25 publications

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“…The case of g ≈ 0 might also occur for anisotropic layers discussed by Bos et al [2]. For such cases, there are more expressions analogous to the fractional term in expression (3.2), as exemplified for orthotropic layers by Slawinski [5,Exercise 4.6].…”

Section: Discussionmentioning

confidence: 96%

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Statistical and Numerical Considerations of Backus-Average Product Approximation

Bos

Danek²,

Slawiński

et al. 2017

J Elast

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This version contains corrections of typographical errors in Bos, L., Danek, T., Slawinski, M.A., Stanoev, T. (2018) Statistical and numerical considerations of Backus-average product approximation. Journal of Elasticity 132(1), 141-159.Abstract. In this paper, we examine the applicability of the approximation, f g ≈ f g , within Backus [1] averaging. This approximation is a crucial step in the method proposed by Backus [1], which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions.Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al.[2], we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions.Even though, according to the upper bound, Backus [1] averaging might not appear as a viable approach, we show that-for cases representative of physical scenarios modelled by such an averaging-the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy.In particular, we examine a special case for which the approximation results in spurious values. However, such a case-though physically realizable-is not likely to appear in seismology, where Backus [1] averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus [1] averaging is also considered.

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Section: Discussionmentioning

confidence: 96%

“…If g = 0 , this error becomes 100% ; hence, the case of g = 0 is of concern, and we discuss it in Section 3.6. To obtain expression (1.4), for a fixed value of x , we set W (ζ) := w(ζ − x) , as discussed by Bos et al [2,Appendix C]. Then, W 0 and ∞ −∞ W (ζ) dζ = 1 .…”

Section: Introductionmentioning

confidence: 99%

Statistical and Numerical Considerations of Backus-Average Product Approximation

Bos

Danek²,

Slawiński

et al. 2017

J Elast

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“…Backus [1] shows that-for thicknesses of individual layers that are much less than the wavelength-waves travelling through parallel isotropic layers behave as if they were travelling through a single transversely isotropic medium. An examination and extension of the Backus [1] average, as well as its limitations, are discussed by Bos et al [7,8].…”

Section: Introductionmentioning

confidence: 99%

On Backus average in modelling guided waves

Dalton

Meehan

Slawiński

2019

Journal of Applied Geophysics

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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...

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“…Let us mention only a few of them: Postma (), Backus (), Schoenberg and Muir (), Berryman, Grechka and Berge (), Kumar () or Bos et al . (). Postma () pointed out that an inhomogeneous medium consisting of fine, isotropic layers can be treated as a homogeneous, transversely isotropic (TI) medium.…”

Section: Introductionmentioning

confidence: 99%

“…Schoenberg and Muir (1989) showed a method of calculating the elasticity coefficients for the media equivalent to layers exhibiting lower symmetries than the TI symmetry. Kumar (2013) and Bos et al (2017) presented analytical solutions of Schoenberg and Muir (1989) approach for layers with anisotropies up to monoclinic and up to generally anisotropic, respectively. Herein, we only examine equivalent TI media obtained upon Backus averaging of isotropic layers.…”

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confidence: 99%

On possible fluid detection in equivalent transversely isotropic media

Adamus

2019

Geophysical Prospecting

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We consider a transversely isotropic medium that is long‐wave equivalent to a stack of thin, parallel, isotropic layers and is obtained using the Backus average. In such media, we analyse the relations among anisotropy parameters; Thomsen parameters, ε and δ, and a new parameter ϕ. We discuss the last parameter and show its essential properties; it equals 0 in the case of isotropy of equivalent medium and/or constant Lamé coefficient λ in layers. The second property occurs to make ϕ sensitive to variations of λ in thin‐bedded sequences. According to Gassmann, in isotropic media the variation of fluid content affects only the Lamé coefficient λ, not μ; thus, the sensitivity to changes of λ is an essential property in the context of possible detection of fluids. We show algebraically and numerically that ϕ is more sensitive to these variations than ε or δ. Nevertheless, each of these parameters is dependent on the changes of μ; to understand this influence, we exhibit comprehensive tables that illustrate the behaviour of anisotropy parameters with respect to specific variations of λ and μ. The changes of μ in layers can be presented by the Thomsen parameter γ that depends on them solely. Hence, knowing the values of elasticity coefficients of equivalent transversely isotropic medium, we may compute ϕ and γ, and based on the aforementioned tables, we predict the expected variation of λ; in this way, we propose a new method of possible fluid detection. Also, we show that the prior approach of possible detection of fluids, proposed by Berryman et al., may be unreliable in specific cases. To establish our results, we use the Monte Carlo method; for the range and chosen variations of Lamé coefficients λ and μ – relevant to sandstones – we generate these coefficients in thin layers and, after the averaging process, we obtain an equivalent transversely isotropic medium. We repeat that process numerous times to get many equivalent transversely isotropic media, and – for each of them – we compute their anisotropy parameters. We illustrate ϕ, ε and δ in the form of cross‐plots that are relevant to the chosen variations of λ and μ. Additionally, we present a table with the computed ranges of anisotropy parameters that correspond to different variations of Lamé coefficients.

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On Backus Average for Generally Anisotropic Layers

Cited by 12 publications

References 25 publications

Statistical and Numerical Considerations of Backus-Average Product Approximation

Statistical and Numerical Considerations of Backus-Average Product Approximation

On Backus average in modelling guided waves

On possible fluid detection in equivalent transversely isotropic media

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