Statistical and Numerical Considerations of Backus-Average Product Approximation

Abstract

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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“…31a). As a model benchmark I used Backus averaged (Backus 1962) welllog data from well L1 (averaging for f = 200 Hz using Seismic Unix) (Liner and Fei 2007), which is a reasonable upscaled approximation of a sonic log data (Maxwell et al 2010;Bos et al 2018). The benchmark model has an accuracy of 17 m of average mislocation.…”

On problematic case of product approximation in Backus average

“…As discussed by Bos et al (2018), the assumption of product approximation may be inaccurate only in the case of g ≈ 0. Since, in such a situation, the relative error,…”

On possible fluid detection in equivalent transversely isotropic media

“…However, taking into consideration that the values of λ that are close to 0 might cause the issue within Backus averaging (Bos et al . ; Kudela and Stanoev ), we set the ranges to be and . These ranges might also correspond to shales and other sedimentary rocks.…”

“…Based on works of Castagna and Smith (1994) and Wanniarachichi et al (2017), the approximate ranges of these coefficients are λ ∈ [−3 , 20] GPa and μ ∈ [1 , 30] GPa, where GPa are gigapascals. However, taking into consideration that the values of λ that are close to 0 might cause the issue within Backus averaging (Bos et al 2018;Kudela and Stanoev 2018), we set the ranges to be λ ∈ [3 , 20] GPa and μ ∈ [1 , 30] GPa. These ranges might also correspond to shales and other sedimentary rocks.…”