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# Numerical examination of commutativity between Backus and Gazis et al. averages

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## Abstract

Dalton and Slawinski (2016) show that, in general, the Backus (1962) average and the Gazis et al. (1963) average do not commute. Herein, we examine the extent of this noncommutativity. We illustrate numerically that the extent of noncommutativity is a function of the strength of anisotropy. The averages nearly commute in the case of weak anisotropy.
Numerical examination of commutativity between
Backus and Gazis et al. averages
David R. Dalton
, Michael A. Slawinski
September 5, 2016
Abstract
Dalton and Slawinski (2016) show that, in general, the Backus (1962) average and the Gazis
et al. (1963) average do not commute. Herein, we examine the extent of this noncommutativ-
ity. We illustrate numerically that the extent of noncommutativity is a function of the strength of
anisotropy. The averages nearly commute in the case of weak anisotropy.
1 Introduction
Dalton and Slawinski (2016) show that—in general—the Backus (1962) average, which is an aver-
age over a spatial variable, and the Gazis et al. (1963) average, which is an average over a symmetry
group, do not commute. These averages result in the so-called equivalent and effective media, re-
spectively. In this paper, using the monoclinic and orthotropic symmetries, we numerically study
the extent of the lack of commutativity between these averages. Also, we examine the effect of the
strength of the anisotropy on noncommutativity. We consider the following diagram.
mono B
mono
G
y
y
G
ortho
Bortho
.(1)
Herein, Band Gstand for the Backus (1962) average and the Gazis et al. (1963) average, respec-
tively. The upper left-hand corner of Diagram 1 is a series of parallel monoclinic layers. The lower
right-hand corner is a single orthotropic medium. The intermediate clockwise result is a single
monoclinic tensor: an equivalent medium; the intermediate counterclockwise result is a series of
parallel orthotropic layers: effective media.
Department of Earth Sciences, Memorial University of Newfoundland, dalton.nfld@gmail.com
Department of Earth Sciences, Memorial University of Newfoundland, mslawins@mac.com
1
arXiv:1609.01034v1 [physics.geo-ph] 5 Sep 2016
Given monoclinic tensors, cijk , in the upper left-hand corner of Diagram 1 and following the
clockwise path, we have , according to Dalton and Slawinski (2016) and Bos et al. (2016),
c
1212 =c1212 c2
3312
c3333 +1
c3333
1c3312
c3333 2
,
c
1313 =c1313
D/(2D2), c
2323 =c2323
D/(2D2),
where D2(c2323c1313 c2
2313)and D2(c1313 /D)(c2323/D)(c2313 /D)2;c
ijk are the elasticity
parameters of the orthotropic tensor in the lower right-hand corner. Following the counterclock-
wise path, we have
c
1212 =c1212 , c
1313 =1
c1313
1
, c
2323 =1
c2323
1
.
The other parameters are the same for both paths.
As stated by Dalton and Slawinski (2016), the results of the clockwise and counterclockwise
paths are the same for all elasticity parameters if c2313 =c3312 = 0 , which is a special case
of monoclinic symmetry. For that case, the Backus (1962) average and the Gazis et al. (1963)
average commute.
2 Numerical testing
Even though, in general, the Backus (1962) average and the Gazis et al. (1963) average do not
commute, it is important to consider the extent of their noncommutativity. We wish to enquire
to what extent—in the context of a continuum-mechanics model and unavoidable measurement
errors—the averages could be considered as approximately commutative.
To do so, we numerically examine two cases. In one case, we begin—in the upper left-hand
corner of Diagram 1—with ten strongly anisotropic layers. In the other case, we begin with ten
weakly anisotropic layers.
Elasticity parameters for the strongly anisotropic layers are derived by random variation from
the H002 sanidine alkali feldspar given in Waeselmann et al. (2016), but with the x3-axis perpen-
dicular to the symmetry plane rather than the x2-axis, used by Waeselmann et al. (2016). These
parameters are given in Table 1.
2
Table 1: Ten strongly anisotropic monoclinic tensors. The elasticity parameters are density-scaled;
their units are 106m2/s2.
layer c1111 c1122 c1133 c1112 c2222 c2233 c2212 c3333 c3312 c2323 c2313 c1313 c1212
1 23.9 11.6 12.2 1.53 71.4 6.64 2.94 52.0 -2.89 8.00 -6.79 8.21 4.54
2 33.5 8.24 12.2 -0.98 66.9 5.65 2.02 82.3 -1.12 6.35 -5.16 17.4 7.36
3 33.2 9.79 16.9 0.57 62.1 6.19 3.81 83.4 -7.34 10.2 -2.33 16.6 4.72
4 38.1 8.33 12.2 1.51 55.0 4.87 3.11 56.8 -1.43 4.10 -0.20 8.25 11.2
5 37.4 11.5 14.4 -0.79 72.6 3.93 3.00 76.5 -6.07 9.58 -4.38 14.8 8.70
6 38.4 10.7 17.1 1.55 63.8 7.11 1.99 55.2 -0.98 9.66 -6.85 11.1 11.4
7 29.2 11.4 11.7 0.59 59.5 5.23 3.74 82.7 -3.81 10.1 -5.09 9.78 6.89
8 31.9 9.03 19.1 -0.07 71.6 4.18 1.98 70.4 -0.25 4.84 -0.33 8.21 10.9
9 37.5 10.5 19.4 0.37 76.7 5.02 3.57 76.7 -0.16 7.84 -1.62 13.8 10.7
10 36.0 9.65 18.9 -0.43 73.1 3.94 2.53 60.4 -7.20 5.44 -2.20 9.25 5.20
Weakly anisotropic layers are derived from the strongly anisotropic ones by keeping c1111 and
c2323 , which are the two distinct elasticity parameters of isotropy, approximately the same as for
the corresponding strongly anisotropic layers, and by varying other parameters away from isotropy.
These parameters are given in Table 2.
Table 2: Ten weakly anisotropic monoclinic tensors. The elasticity parameters are density-scaled;
their units are 106m2/s2.
layer c1111 c1122 c1133 c1112 c2222 c2233 c2212 c3333 c3312 c2323 c2313 c1313 c1212
1 24 9 9 0.2 29 7 0.3 27 -0.3 8 -1 8.2 7
2 34 15 18 -0.1 38 14 0.2 39 -0.1 6 -1 7.5 6.5
3 33 12 14 0.06 37 10 0.4 38 -0.7 10 -0.5 12 8.5
4 38 20 22 0.15 40 15 0.3 41 -0.1 4 -0.2 5 6
5 37 14 16 -0.08 42 10 0.3 41 -0.6 10 -0.8 11 9
6 38 15 18 0.16 41 14 0.2 40 -0.1 10 -1 10.5 11
7 29 9.5 9.5 0.06 32 8 0.4 34 -0.4 10 -0.8 10 9
8 32 15 19.5 -0.01 36 13 0.2 36 -0.03 5 -0.3 6 6
9 38 16 20 0.04 43 14 0.4 42 -0.02 8 -0.4 9 9
10 36 18 23 -0.04 40 15 0.3 39 -0.7 5 -0.5 6 5
Assuming that all layers have the same thickness, we use an arithmetic average for the Backus (1962)
averaging; for instance,
c1212 =1
10
10
X
i=1
ci
1212 .
The results of the clockwise and counterclockwise paths for the three elasticity parameters that
differ from each other are given in Table 3. It appears that the averages nearly commute for the case
of weak anisotropy. Hence, we might conclude that the extent of noncommutativity is a function
of the strength of anisotropy.
3
Table 3: Comparison of numerical results.
anisotropy c
1212 c
1212 c
1313 c
1313 c
2323 c
2323
strong 8.06 8.16 9.13 10.84 6.36 6.90
weak 7.70 7.70 7.88 7.87 6.82 6.81
3 Strength of anisotropy
To quantify the strength of anisotropy, we invoke the concept of distance in the space of elasticity
tensors. In particular, we consider the closest isotropic tensor according to the Frobenius norm, as
formulated by Voigt (1910). Examining one layer from the upper left-hand corner of Diagram 1,
we denote its weakly anisotropic tensor as cwand its strongly anisotropic tensor as cs.
Using explicit expressions of Slawinski (2016), we ﬁnd that the elasticity parameters of the
closest isotropic tensor, cisow, to cwis cisow
1111 = 25.52 and cisow
2323 = 8.307 . The Frobenius distance
from cwto cisowis 6.328 .
The closest isotropic tensor, cisos, to csis cisos
1111 = 39.08 and cisos
2323 = 11.94 . The distance from
csto cisosis 49.16 .
Thus, as required, cs, which represents strong anisotropy, is much further from isotropy than
cw, which represents weak anisotropy.
4 Discussion
Dalton and Slawinski (2016) show that—in general—the Backus (1962) average and the Gazis
et al. (1963) average do not commute. Herein, using the the case of monoclinic and orthotropic
symmetries, we numerically show that noncommutativity is a function of the strength of anisotropy.
For weak anisotropy, which is a common case of seismological studies, the averages appear to
nearly commute.
In our future work, we will consider aspects of the approximation theory to rigorously examine
the commutativity issues between these averages. Also, in such an examination, we will invoke
Acknowledgments
This numerical examination was motivated by a fruitful discussion with Robert Sarracino. The
research was performed in the context of The Geomechanics Project supported by Husky Energy.
Also, this research was partially supported by the Natural Sciences and Engineering Research
4
References
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Bos, L, D.R. Dalton, M.A. Slawinski and T. Stanoev, On Backus average for generally anisotropic
layers, arXiv [physics.geo-ph], 1601.02967, 2016.
Dalton, D. R. and M.A. Slawinski, On commutativity of Backus average and Gazis et al. average,
arXiv [physics.geo-ph], 1601.02969, 2016.
Gazis, D.C., I. Tadjbakhsh and R.A. Toupin, The elastic tensor of given symmetry nearest to an
anisotropic elastic tensor, Acta Crystallographica,16, 9, 917–922, 1963.
Slawinski, M.A., Waves and rays in seismology: Answers to unasked questions, World Scientiﬁc,
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Voigt, W., Lehrbuch der Kristallphysik, Teubner, Leipzig, 1910.
Waeselmann, N., J.M. Brown, R.J. Angel, N. Ross, J. Zhao and W. Kamensky, The elastic tensor
of monoclinic alkali feldspars, American Mineralogist,101, 1228–1231, 2016.
5
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On commutativity of Backus average and Gazis et al. average, arXiv [physics.geo
• D R Dalton
• M A Slawinski
Dalton, D. R. and M.A. Slawinski, On commutativity of Backus average and Gazis et al. average, arXiv [physics.geo-ph], 1601.02969, 2016.