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On commutativity of Backus and Gazis averages
David R. Dalton∗
, Michael A. Slawinski †
January 12, 2016
Abstract
We show that the Backus (1962) equivalent-medium average, which is an average over a spatial
variable, and the Gazis et al. (1963) effective-medium average, which is an average over a sym-
metry group, do not commute, in general. They commute in special cases, which we exemplify.
1 Introduction
Hookean solids are defined by their mechanical property relating linearly the stress tensor, σ, and
the strain tensor, ε,
σij =
3
X
k=1
3
X
`=1
cijk` εk` , i, j = 1,2,3.
The elasticity tensor, c, belongs to one of eight material-symmetry classes shown in Figure 1.
The Backus (1962) moving average allows us to quantify the response of a wave propagating
through a series of parallel layers whose thicknesses are much smaller than the wavelength. Each
layer is a Hookean solid exhibiting a given material symmetry with given elasticity parameters.
The average is a Hookean solid whose elasticity parameters—and, hence, its material symmetry—
allow us to model a long-wavelength response. This material symmetry of the resulting medium,
to which we refer as equivalent, is a consequence of symmetries exhibited by the averaged layers.
The long-wave-equivalent medium to a stack of isotropic or transversely isotropic layers with
thicknesses much less than the signal wavelength was shown by Backus (1962) to be a homo-
geneous or nearly homogeneous transversely isotropic medium, where a nearly homogeneous
medium is a consequence of a moving average. Backus (1962) formulation is reviewed by Slaw-
inski (2016) and Bos et al. (2016), where formulations for generally anisotropic, monoclinic, and
orthotropic thin layers are also derived. Bos et al. (2016) examine the underlying assumptions
∗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:1601.02969v1 [physics.geo-ph] 12 Jan 2016
Figure 1: Order relation of material-symmetry classes of elasticity tensors: Arrows indicate subgroups in
this partial ordering. For instance, monoclinic is a subgroup of all nontrivial symmetries, in particular, of
both orthotropic and trigonal, but orthotropic is not a subgroup of trigonal or vice-versa.
and approximations behind the Backus (1962) formulation, which is derived by expressing rapidly
varying stresses and strains in terms of products of algebraic combinations of rapidly varying elas-
ticity parameters with slowly varying stresses and strains. The only mathematical approximation in
the formulation is that the average of a product of a rapidly varying function and a slowly varying
function is approximately equal to the product of the averages of the two functions.
According to Backus (1962), the average of f(x3)of “width” `0is
f(x3) :=
∞
Z
−∞
w(ζ−x3)f(ζ) dζ , (1)
where w(x3)is the weight function with the following properties:
w(x3)>0, w(±∞) = 0 ,
∞
Z
−∞
w(x3) dx3= 1 ,
∞
Z
−∞
x3w(x3) dx3= 0 ,
∞
Z
−∞
x2
3w(x3) dx3= (`0)2.
These properties define w(x3)as a probability-density function with mean 0and standard devia-
tion `0, explaining the use of the term “width” for `0.
Gazis et al. (1963) average allows us to obtain the closest symmetric counterpart—in the Frobe-
nius sense—of a chosen material symmetry to a generally anisotropic Hookean solid. The average
is a Hookean solid, to which we refer as effective, whose elasticity parameters correspond to the
symmetry chosen a priori.
2
Gazis average is a projection given by
ecsym := Z
Gsym
(g◦c) dµ(g),(2)
where the integration is over the symmetry group, Gsym , whose elements are g, with respect to
the invariant measure, µ, normalized so that µ(Gsym )=1;ecsym is the orthogonal projection of
c, in the sense of the Frobenius norm, on the linear space containing all tensors of that symmetry,
which are csym . Integral (2) reduces to a finite sum for the classes whose symmetry groups are
finite, which are all classes except isotropy and transverse isotropy.
The Gazis et al. (1963) approach is reviewed and extended by Danek et al. (2013, 2015) in
the context of random errors. Therein, elasticity tensors are not constrained to the same—or even
different but known—orientation of the coordinate system.
Concluding this introduction, let us emphasize that the fundamental distinction between the two
averages is their domain of operation. The Gazis et al. (1963) average is an average over symmetry
groups at a point and the Backus (1962) average is a spatial average over a distance. Both averages
can be used, separately or together, in quantitative seismology. Hence, an examination of their
commutativity might provide us with an insight into their physical meaning and into allowable
mathematical operations.
2 Generally anisotropic layers and monoclinic medium
Let us consider a stack of generally anisotropic layers to obtain a monoclinic medium. To examine
the commutativity between the Backus and Gazis averages, let us study the following diagram,
aniso B
−−−→ aniso
G
y
yG
mono −−−→
Bmono
(3)
and Proposition 1, below,
Proposition 1. In general, the Backus and Gazis averages do not commute.
Proof. To prove this proposition and in view of Diagram 3, let us begin with the following corol-
lary.
Corollary 1. For the generally anisotropic and monoclinic symmetries, the Backus and Gazis
averages do not commute.
To understand this corollary, we invoke the following lemma, whose proof is in Appendix A.1.
Lemma 1. For the effective monoclinic symmetry, the result of the Gazis average is tantamount
to replacing each cijk` , in a generally anisotropic tensor, by its corresponding cijk` of the mono-
clinic tensor, expressed in the natural coordinate system, including replacements of the anisotropic-
tensor components by the zeros of the corresponding monoclinic components.
3
Let us first examine the counterclockwise path of Diagram 3. Lemma 1 entails a corollary.
Corollary 2. For the effective monoclinic symmetry, given a generally anisotropic tensor, C,
e
Cmono =Cmono ;(4)
where e
Cmono is the Gazis average of C, and Cmono is a monoclinic tensor whose nonzero entries
are the same as for C.
According to Corollary 2, the effective monoclinic tensor is obtained simply by setting to zero—in
the generally anisotropic tensor—the components that are zero for the monoclinic tensor. Then,
the second counterclockwise branch of Diagram 3 is performed as follows. Applying the Backus
average, we obtain (Bos et al., 2015)
hc3333i=1
c3333 −1
,hc2323i=c2323
D
2D2
,
hc1313i=c1313
D
2D2
,hc2313i=c2313
D
2D2
,
where D≡2(c2323c1313 −c2
2313)and D2≡(c1313 /D)(c2323/D)−(c2313/D)2. We also obtain
hc1133i=1
c3333 −1c1133
c3333 ,hc2233i=1
c3333 −1c2233
c3333 ,hc3312i=1
c3333 −1c3312
c3333 ,
hc1111i=c1111 −c2
1133
c3333 +1
c3333 −1c1133
c3333 2
,
hc1122i=c1122 −c1133 c2233
c3333 +1
c3333 −1c1133
c3333 c2233
c3333 ,
hc2222i=c2222 −c2
2233
c3333 +1
c3333 −1c2233
c3333 2
,
hc1212i=c1212 −c2
3312
c3333 +1
c3333 −1c3312
c3333 2
,
hc1112i=c1112 −c3312 c1133
c3333 +1
c3333 −1c1133
c3333 c3312
c3333
and
hc2212i=c2212 −c3312 c2233
c3333 +1
c3333 −1c2233
c3333 c3312
c3333 ,
where angle brackets denote the equivalent-medium elasticity parameters. The other equivalent-
medium elasticity parameters are zero.
4
Following the clockwise path of Diagram 3, the upper branch is derived in matrix form in
Bos et al. (2015). Then, from Bos et al. (2015) the result of the right-hand branch is derived by
setting entries in the generally anisotropic tensor that are zero for the monoclinic tensor to zero.
The nonzero entries, which are too complicated to display explicitly, are—in general—not the
same as the result of the counterclockwise path. Hence, for generally anisotropic and monoclinic
symmetries, the Backus and Gazis averages do not commute.
3 Higher symmetries
3.1 Monoclinic layers and orthotropic medium
Proposition 1 remains valid for layers exhibiting higher material symmetries, and simpler expres-
sions of the corresponding elasticity tensors allow us to examine special cases that result in com-
mutativity. Let us consider the following corollary of Proposition 1.
Corollary 3. For the monoclinic and orthotropic symmetries, the Backus and Gazis averages do
not commute.
To study this corollary, let us consider the following diagram,
mono B
−−−→ mono
G
y
yG
ortho −−−→
Bortho
(5)
and the lemma, whose proof is in Appendix A.2.
Lemma 2. For the effective orthotropic symmetry, the result of the Gazis average is tantamount to
replacing each cijk` , in a generally anisotropic—or monoclinic—tensor, by its corresponding cijk`
of the orthotropic tensor, expressed in the natural coordinate system, including the replacements
by the corresponding zeros.
Lemma 2 entails a corollary.
Corollary 4. For the effective orthotropic symmetry, given a generally anisotropic—or monoclinic—
tensor, C,
e
Cortho =Cortho .(6)
where e
Cortho is the Gazis average of C, and Cortho is an orthotropic tensor whose nonzero entries
are the same as for C.
Let us consider a monoclinic tensor and proceed counterclockwise along the first branch of Dia-
gram 5. Using the fact that the monoclinic symmetry is a special case of general anisotropy, we
invoke Corollary 4 to conclude that e
Cortho =Cortho , which is equivalent to setting c1112 ,c2212 ,
5
c3312 and c2313 to zero in the monoclinic tensor. We perform the upper branch of Diagram 5, which
is the averaging of a stack of monoclinic layers to get a monoclinic equivalent medium, as in the
case of the lower branch of Diagram 3. Thus, following the clockwise path, we obtain
c
1212 =c1212 −c2
3312
c3333 +1
c3333 −1c3312
c3333 2
,
c
1313 =c1313
D/(2D2), c
2323 =c2323
D/(2D2)
Following the counterclockwise path, we obtain
c
1212 =c1212 , c
1313 =1
c1313 −1
, c
2323 =1
c2323 −1
.
The other entries are the same for both paths.
In conclusion, the results of the clockwise and counterclockwise paths are the same if c2313 =
c3312 = 0 , which is a special case of monoclinic symmetry. Thus, the Backus average and Gazis
average commute for that case, but not in general.
3.2 Orthotropic layers and tetragonal medium
In a manner analogous to Diagram 5, but proceeding from the the upper-left-hand corner or-
thotropic tensor to lower-right-hand corner tetragonal tensor by the counterclockwise path,
ortho B
−−−→ ortho
G
y
yG
tetra −−−→
Btetra
(7)
we obtain
c
1111 =c1111 +c2222
2−c1111+c2222
22
c3333
+c1111 +c2222
2c3333 21
c3333 −1
.
Following the clockwise path, we obtain
c
1111 =c1111 +c2222
2−c2
1133 +c2
2233
2c3333
+1
2"c1133
c3333 2
+c2233
c3333 2#1
c3333 −1
.
These results are not equal to one another, unless c1133 =c2233 , which is a special case of or-
thotropic symmetry. Also c2323 must equal c1313 for c
2323 =c
2323. The other entries are the same
for both paths. Thus, the Backus average and Gazis average do commute for c1133 =c2233 and
c2323 =c1313 , which is a special case of orthotropic symmetry, but not in general.
6
Let us also consider the case of monoclinic layers and a tetragonal medium to examine the
process of combining the Gazis averages, which is tantamount to combining Diagrams (5) and (7),
mono B
−−−→ mono
G
y
yG
ortho −−−→
Bortho
G
y
yG
tetra −−−→
Btetra
(8)
In accordance with Proposition 1, there is—in general—no commutativity. However, the outcomes
are the same as for the corresponding steps in Sections 3.1 and 3.2. In general, for the Gazis
average, proceeding directly, aniso G
−→ iso , is tantamount to proceeding along arrows in Figure 1,
aniso G
−→ ··· G
−→ iso . No such combining of the Backus averages is possible, since, for each step,
layers become a homogeneous medium.
3.3 Transversely isotropic layers
Lack of commutativity can also be exemplified by the case of transversely isotropic layers. Follow-
ing the clockwise path of Diagram 5, the Backus average results in a transversely isotropic medium,
whose Gazis average—in accordance with Figure 1—is isotropic. Following the counterclockwise
path, Gazis average results in an isotropic medium, whose Backus average, however, is transverse
isotropy. Thus, not only the elasticity parameters, but even the resulting material-symmetry classes
differ.
Also, we could—in a manner analogous to the one illustrated in Diagram 8 —begin with gen-
erally anisotropic layers and obtain isotropy by the clockwise path and transverse isotropy by the
counterclockwise path, which again illustrates noncommutativity.
4 Discussion
Herein, we assume that all tensors are expressed in the same orientation of their coordinate sys-
tems. Otherwise, the process of averaging become more complicated, as discussed—for the
Gazis average—by Kochetov and Slawinski (2009a, 2009b) and as mentioned—for the Backus
average—by Bos et al. (2016).
Mathematically, the noncommutativity of two distinct averages is shown by Proposition 1, and
exemplified for several material symmetries.
We do not see a physical justification for special cases in which—given the same orientation
of coordinate systems—these averages commute. This behaviour might support the view that a
mathematical realm, which allows for fruitful analogies with the physical world, has no causal
connection with it.
7
Acknowledgments
We wish to acknowledge discussions with Theodore Stanoev. This 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 Council of Canada, grant
238416-2013.
References
Backus, G.E., Long-wave elastic anisotropy produced by horizontal layering, J. Geophys. Res.,67,
11, 4427–4440, 1962.
B´
ona, A., I. Bucataru and M.A. Slawinski, Space of SO(3)-orbits of elasticity tensors, Archives of
Mechanics,60, 2, 121–136, 2008
Bos, L, D.R. Dalton, M.A. Slawinski and T. Stanoev, On Backus average for generally anisotropic
layers, arXiv, 2016.
Chapman, C. H., Fundamentals of seismic wave propagation, Cambridge University Press, 2004.
Danek, T., M. Kochetov and M.A. Slawinski, Uncertainty analysis of effective elasticity tensors
using quaternion-based global optimization and Monte-Carlo method, The Quarterly Journal
of Mechanics and Applied Mathematics,66, 2, pp. 253–272, 2013.
Danek, T., M. Kochetov and M.A. Slawinski, Effective elasticity tensors in the context of random
errors, Journal of Elasticity, 2015.
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.
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terly Journal of Mechanics and Applied Mathematics,62, 2, pp. 149-166, 2009a.
Kochetov, M. and M.A. Slawinski, On obtaining effective transversely isotropic elasticity tensors,
Journal of Elasticity,94, 1-13., 2009b.
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tific, 2016.
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University Press, 1890
Appendix A
Appendix A.1
Let us prove Lemma 1.
8
Proof. For discrete symmetries, we can write integral (2) as a sum,
e
Csym =1
n˜
Asym
1C˜
Asym
1
T+. . . +˜
Asym
nC˜
Asym
n
T,(9)
where e
Csym is expressed in Kelvin’s notation, in view of Thomson (1890, p. 110) as discussed in
Chapman (2004, Section 4.4.2).
To write the elements of the monoclinic symmetry group as 6×6matrices, we must consider
orthogonal transformations in R3. Transformation A∈SO(3) of cij k` corresponds to transforma-
tion of Cgiven by
˜
A=
A2
11 A2
12 A2
13 √2A12A13 √2A11 A13 √2A11A12
A2
21 A2
22 A2
23 √2A22A23 √2A21 A23 √2A21A22
A2
31 A2
32 A2
33 √2A32A33 √2A31 A33 √2A31A32
√2A21A31 √2A22 A32 √2A23A33 A23 A32 +A22A33 A23A31 +A21 A33 A22A31 +A21 A32
√2A11A31 √2A12 A32 √2A13A33 A13 A32 +A12A33 A13A31 +A11 A33 A12A31 +A11 A32
√2A11A21 √2A12 A22 √2A13A23 A13 A22 +A12A23 A13A21 +A11 A23 A12A21 +A11 A22
,
(10)
which is an orthogonal matrix, ˜
A∈SO(6) (Slawinski (2015), Section 5.2.5).1
The required symmetry-group elements are
Amono
1=
1 0 0
0 1 0
0 0 1
7→
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
=˜
Amono
1
Amono
2=
−100
0−1 0
0 0 1
7→
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 −100
0 0 0 0 −1 0
0 0 0 0 0 1
=˜
Amono
2.
For the monoclinic case, expression (9) can be stated explicitly as
e
Cmono =˜
Amono
1C˜
Amono
1T
+˜
Amono
2C˜
Amono
2T
2.
1Readers interested in formulation of matrix (10) might refer to B´
ona et al. (2008).
9
Performing matrix operations, we obtain
e
Cmono =
c1111 c1122 c1133 0 0 √2c1112
c1122 c2222 c2233 0 0 √2c2212
c1133 c2233 c3333 0 0 √2c3312
0 0 0 2c2323 2c2313 0
0 0 0 2c2313 2c1313 0
√2c1112 √2c2212 √2c3312 0 0 2c1212
,(11)
which exhibits the form of the monoclinic tensor in its natural coordinate system. In other words,
e
Cmono =Cmono , in accordance with Corollary 2.
Appendix A.2
Let us prove Lemma 2.
For orthotropic symmetry, ˜
Aortho
1=˜
Amono
1and ˜
Aortho
2=˜
Amono
2and
Aortho
3=
−1 0 0
010
0 0 −1
7→
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 −1 0 0
0 0 0 0 1 0
0 0 0 0 0 −1
=˜
Aortho
3,
Aortho
4=
1 0 0
0−1 0
0 0 −1
7→
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 −1 0
0 0 0 0 0 −1
=˜
Aortho
4.
For the orthotropic case, expression (9) can be stated explicitly as
e
Cortho =˜
Aortho
1C˜
Aortho
1T
+˜
Aortho
2C˜
Aortho
2T
+˜
Aortho
3C˜
Aortho
3T
+˜
Aortho
4C˜
Aortho
4T
4.
Performing matrix operations, we obtain
e
Cortho =
c1111 c1122 c1133 000
c1122 c2222 c2233 000
c1133 c2233 c3333 000
0 0 0 2c2323 0 0
0 0 0 0 2c1313 0
0 0 0 0 0 2c1212
,(12)
which exhibits the form of the orthotropic tensor in its natural coordinate system. In other words,
e
Cortho =Cortho , in accordance with Corollary 4.
10
