Table Of ContentOn commutativity of Backus and Gazis averages
David R. Dalton∗, Michael A. Slawinski †
6 January 12, 2016
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0
2
n
a
J Abstract
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1
We show that the Backus (1962) equivalent-medium average, which is an average over a spatial
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h variable, and the Gazis et al. (1963) effective-medium average, which is an average over a sym-
p
metrygroup,donotcommute,ingeneral. Theycommuteinspecialcases,whichweexemplify.
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e
g
1 Introduction
.
s
c
i
s Hookean solids are defined by their mechanical property relating linearly the stress tensor, σ, and
y
thestraintensor,ε,
h
p
[ 3 3
(cid:88)(cid:88)
σ = c ε , i,j = 1,2,3.
1 ij ijk(cid:96) k(cid:96)
v k=1 (cid:96)=1
9
6 Theelasticitytensor,c,belongstooneofeightmaterial-symmetryclassesshowninFigure1.
9
The Backus (1962) moving average allows us to quantify the response of a wave propagating
2
0 through a series of parallel layers whose thicknesses are much smaller than the wavelength. Each
.
1 layer is a Hookean solid exhibiting a given material symmetry with given elasticity parameters.
0 TheaverageisaHookeansolidwhoseelasticityparameters—and,hence,itsmaterialsymmetry—
6
1 allow us to model a long-wavelength response. This material symmetry of the resulting medium,
: towhichwereferasequivalent,isaconsequenceofsymmetriesexhibitedbytheaveragedlayers.
v
i The long-wave-equivalent medium to a stack of isotropic or transversely isotropic layers with
X
thicknesses much less than the signal wavelength was shown by Backus (1962) to be a homo-
r
a 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
∗DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected]
†DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected]
1
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
bothorthotropicandtrigonal,butorthotropicisnotasubgroupoftrigonalorvice-versa.
andapproximationsbehindtheBackus(1962)formulation,whichisderivedbyexpressingrapidly
varyingstressesandstrainsintermsofproductsofalgebraiccombinationsofrapidlyvaryingelas-
ticityparameterswithslowlyvaryingstressesandstrains. Theonlymathematicalapproximationin
the formulation is that the average of a product of a rapidly varying function and a slowly varying
functionisapproximatelyequaltotheproductoftheaveragesofthetwofunctions.
AccordingtoBackus(1962),theaverageoff(x )of“width”(cid:96)(cid:48) is
3
∞
(cid:90)
f(x ) := w(ζ −x )f(ζ)dζ, (1)
3 3
−∞
wherew(x )istheweightfunctionwiththefollowingproperties:
3
∞ ∞ ∞
(cid:90) (cid:90) (cid:90)
w(x ) (cid:62) 0, w(±∞) = 0, w(x )dx = 1, x w(x )dx = 0, x2w(x )dx = ((cid:96)(cid:48))2.
3 3 3 3 3 3 3 3 3
−∞ −∞ −∞
These properties define w(x ) as a probability-density function with mean 0 and standard devia-
3
tion(cid:96)(cid:48),explainingtheuseoftheterm“width”for(cid:96)(cid:48).
Gazisetal.(1963)averageallowsustoobtaintheclosestsymmetriccounterpart—intheFrobe-
niussense—ofachosenmaterialsymmetrytoagenerallyanisotropicHookeansolid. Theaverage
is a Hookean solid, to which we refer as effective, whose elasticity parameters correspond to the
symmetrychosenapriori.
2
Gazisaverageisaprojectiongivenby
(cid:90)
csym := (g ◦c)dµ(g), (2)
(cid:101)
Gsym
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; csym is the orthogonal projection of
(cid:101)
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,whichareallclassesexceptisotropyandtransverseisotropy.
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
differentbutknown—orientationofthecoordinatesystem.
Concludingthisintroduction,letusemphasizethatthefundamentaldistinctionbetweenthetwo
averagesistheirdomainofoperation. TheGazisetal.(1963)averageisanaverageoversymmetry
groupsatapointandtheBackus(1962)averageisaspatialaverageoveradistance. Bothaverages
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
mathematicaloperations.
2 Generally anisotropic layers and monoclinic medium
Letusconsiderastackofgenerallyanisotropiclayerstoobtainamonoclinicmedium. Toexamine
thecommutativitybetweentheBackusandGazisaverages,letusstudythefollowingdiagram,
B
aniso −−−→ aniso
G(cid:121) (cid:121)G (3)
mono −−−→ mono
B
andProposition1,below,
Proposition1. Ingeneral,theBackusandGazisaveragesdonotcommute.
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
averagesdonotcommute.
Tounderstandthiscorollary,weinvokethefollowinglemma,whoseproofisinAppendixA.1.
Lemma 1. For the effective monoclinic symmetry, the result of the Gazis average is tantamount
to replacing each c , in a generally anisotropic tensor, by its corresponding c of the mono-
ijk(cid:96) ijk(cid:96)
clinictensor,expressedinthenaturalcoordinatesystem,includingreplacementsoftheanisotropic-
tensorcomponentsbythezerosofthecorrespondingmonocliniccomponents.
3
LetusfirstexaminethecounterclockwisepathofDiagram3. Lemma1entailsacorollary.
Corollary2. Fortheeffectivemonoclinicsymmetry,givenagenerallyanisotropictensor,C,
C(cid:101)mono = Cmono; (4)
where C(cid:101)mono is the Gazis average of C, and Cmono is a monoclinic tensor whose nonzero entries
arethesameasforC.
AccordingtoCorollary2,theeffectivemonoclinictensorisobtainedsimplybysettingtozero—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,weobtain(Bosetal.,2015)
(cid:18) 1 (cid:19)−1 (cid:0)c2323(cid:1)
(cid:104)c (cid:105) = , (cid:104)c (cid:105) = D ,
3333 2323
c 2D
3333 2
(cid:0) (cid:1) (cid:0) (cid:1)
c1313 c2313
(cid:104)c (cid:105) = D , (cid:104)c (cid:105) = D ,
1313 2313
2D 2D
2 2
whereD ≡ 2(c c −c2 )andD ≡ (c /D)(c /D)−(c /D)2. Wealsoobtain
2323 1313 2313 2 1313 2323 2313
(cid:18) (cid:19)−1(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19)
1 c 1 c 1 c
1133 2233 3312
(cid:104)c (cid:105) = , (cid:104)c (cid:105) = , (cid:104)c (cid:105) = ,
1133 2233 3312
c c c c c c
3333 3333 3333 3333 3333 3333
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19)2
c2 1 c
(cid:104)c (cid:105) = c − 1133 + 1133 ,
1111 1111
c c c
3333 3333 3333
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19) (cid:18) (cid:19)
c c 1 c c
1133 2233 1133 2233
(cid:104)c (cid:105) = c − + ,
1122 1122
c c c c
3333 3333 3333 3333
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19)2
c2 1 c
(cid:104)c (cid:105) = c − 2233 + 2233 ,
2222 2222
c c c
3333 3333 3333
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19)2
c2 1 c
(cid:104)c (cid:105) = c − 3312 + 3312 ,
1212 1212
c c c
3333 3333 3333
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19) (cid:18) (cid:19)
c c 1 c c
3312 1133 1133 3312
(cid:104)c (cid:105) = c − +
1112 1112
c c c c
3333 3333 3333 3333
and
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19) (cid:18) (cid:19)
c c 1 c c
3312 2233 2233 3312
(cid:104)c (cid:105) = c − + ,
2212 2212
c c c c
3333 3333 3333 3333
where angle brackets denote the equivalent-medium elasticity parameters. The other equivalent-
mediumelasticityparametersarezero.
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,theBackusandGazisaveragesdonotcommute.
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. LetusconsiderthefollowingcorollaryofProposition1.
Corollary 3. For the monoclinic and orthotropic symmetries, the Backus and Gazis averages do
notcommute.
Tostudythiscorollary,letusconsiderthefollowingdiagram,
B
mono −−−→ mono
G(cid:121) (cid:121)G (5)
ortho −−−→ ortho
B
andthelemma,whoseproofisinAppendixA.2.
Lemma2. Fortheeffectiveorthotropicsymmetry,theresultoftheGazisaverageistantamountto
replacingeachc ,inagenerallyanisotropic—ormonoclinic—tensor,byitscorrespondingc
ijk(cid:96) ijk(cid:96)
of the orthotropic tensor, expressed in the natural coordinate system, including the replacements
bythecorrespondingzeros.
Lemma2entailsacorollary.
Corollary4. Fortheeffectiveorthotropicsymmetry,givenagenerallyanisotropic—ormonoclinic—
tensor,C,
C(cid:101)ortho = Cortho. (6)
whereC(cid:101)ortho istheGazisaverageofC,andCortho isanorthotropictensorwhosenonzeroentries
arethesameasforC.
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 C(cid:101)ortho = Cortho, which is equivalent to setting c , c ,
1112 2212
5
c andc tozerointhemonoclinictensor. WeperformtheupperbranchofDiagram5,which
3312 2313
is the averaging of a stack of monoclinic layers to get a monoclinic equivalent medium, as in the
caseofthelowerbranchofDiagram3. Thus,followingtheclockwisepath,weobtain
(cid:18) (cid:19) (cid:18) (cid:19)−1(cid:18) (cid:19)2
c2 1 c
c(cid:8) = c − 3312 + 3312 ,
1212 1212 c c c
3333 3333 3333
(cid:16)c (cid:17) (cid:16)c (cid:17)
(cid:8) 1313 (cid:8) 2323
c = /(2D ), c = /(2D )
1313 D 2 2323 D 2
Followingthecounterclockwisepath,weobtain
(cid:18) (cid:19)−1 (cid:18) (cid:19)−1
1 1
(cid:9) (cid:9) (cid:9)
c = c , c = , c = .
1212 1212 1313 c 2323 c
1313 2323
Theotherentriesarethesameforbothpaths.
In conclusion, the results of the clockwise and counterclockwise paths are the same if c =
2313
c = 0, which is a special case of monoclinic symmetry. Thus, the Backus average and Gazis
3312
averagecommuteforthatcase,butnotingeneral.
3.2 Orthotropic layers and tetragonal medium
In a manner analogous to Diagram 5, but proceeding from the the upper-left-hand corner or-
thotropictensortolower-right-handcornertetragonaltensorbythecounterclockwisepath,
B
ortho −−−→ ortho
G(cid:121) (cid:121)G (7)
tetra −−−→ tetra
B
weobtain
c +c (cid:0)c1111+c2222(cid:1)2 (cid:18)c +c (cid:19)2(cid:18) 1 (cid:19)−1
c(cid:9) = 1111 2222 − 2 + 1111 2222 .
1111 2 c 2c c
3333 3333 3333
Followingtheclockwisepath,weobtain
(cid:34)(cid:18) (cid:19)2 (cid:18) (cid:19)2(cid:35)(cid:18) (cid:19)−1
c +c c2 +c2 1 c c 1
c(cid:8) = 1111 2222 − 1133 2233 + 1133 + 2233 .
1111 2 2c 2 c c c
3333 3333 3333 3333
These results are not equal to one another, unless c = c , which is a special case of or-
1133 2233
(cid:8) (cid:9)
thotropic symmetry. Also c must equal c for c = c . The other entries are the same
2323 1313 2323 2323
for both paths. Thus, the Backus average and Gazis average do commute for c = c and
1133 2233
c = c ,whichisaspecialcaseoforthotropicsymmetry,butnotingeneral.
2323 1313
6
Let us also consider the case of monoclinic layers and a tetragonal medium to examine the
processofcombiningtheGazisaverages,whichistantamounttocombiningDiagrams(5)and(7),
B
mono −−−→ mono
G(cid:121) (cid:121)G
ortho −−−→ ortho (8)
B
G(cid:121) (cid:121)G
tetra −−−→ tetra
B
InaccordancewithProposition1,thereis—ingeneral—nocommutativity. However,theoutcomes
are the same as for the corresponding steps in Sections 3.1 and 3.2. In general, for the Gazis
G
average, proceeding directly, aniso −→ iso, is tantamount to proceeding along arrows in Figure 1,
G G
aniso −→ ··· −→ iso. No such combining of the Backus averages is possible, since, for each step,
layersbecomeahomogeneousmedium.
3.3 Transversely isotropic layers
Lackofcommutativitycanalsobeexemplifiedbythecaseoftransverselyisotropiclayers. Follow-
ingtheclockwisepathofDiagram5,theBackusaverageresultsinatransverselyisotropicmedium,
whoseGazisaverage—inaccordancewithFigure1—isisotropic. Followingthecounterclockwise
path, Gazis average results in an isotropic medium, whose Backus average, however, is transverse
isotropy. Thus,notonlytheelasticityparameters,buteventheresultingmaterial-symmetryclasses
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
counterclockwisepath,whichagainillustratesnoncommutativity.
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—byBosetal. (2016).
Mathematically,thenoncommutativityoftwodistinctaveragesisshownbyProposition1,and
exemplifiedforseveralmaterialsymmetries.
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
connectionwithit.
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
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Mechanics,60,2,121–136,2008
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layers,arXiv,2016.
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Appendix A
Appendix A.1
LetusproveLemma1.
8
Proof. Fordiscretesymmetries,wecanwriteintegral(2)asasum,
1 (cid:16) (cid:17)
C(cid:101)sym = A˜symCA˜symT +...+A˜symCA˜symT , (9)
n 1 1 n n
where C(cid:101)sym is expressed in Kelvin’s notation, in view of Thomson (1890, p. 110) as discussed in
Chapman(2004,Section4.4.2).
To write the elements of the monoclinic symmetry group as 6×6 matrices, we must consider
orthogonal transformations in R3. Transformation A ∈ SO(3) of c corresponds to transforma-
ijk(cid:96)
tionofC givenby
√ √ √
A2 A2 A2 2A A 2A A 2A A
11 12 13 √ 12 13 √ 11 13 √ 11 12
A2 A2 A2 2A A 2A A 2A A
21 22 23 √ 22 23 √ 21 23 √ 21 22
A˜ = √ A231 √ A232 √ A233 2A32A33 2A31A33 2A31A32 ,
√2A21A31 √2A22A32 √2A23A33 A23A32+A22A33 A23A31+A21A33 A22A31+A21A32
2A A 2A A 2A A A A +A A A A +A A A A +A A
√ 11 31 √ 12 32 √ 13 33 13 32 12 33 13 31 11 33 12 31 11 32
2A A 2A A 2A A A A +A A A A +A A A A +A A
11 21 12 22 13 23 13 22 12 23 13 21 11 23 12 21 11 22
(10)
whichisanorthogonalmatrix,A˜ ∈ SO(6)(Slawinski(2015),Section5.2.5).1
Therequiredsymmetry-groupelementsare
1 0 0 0 0 0
0 1 0 0 0 0
1 0 0
Amono = 0 1 0 (cid:55)→ 0 0 1 0 0 0 = A˜mono
1 0 0 0 1 0 0 1
0 0 1
0 0 0 0 1 0
0 0 0 0 0 1
1 0 0 0 0 0
0 1 0 0 0 0
−1 0 0
Amono = 0 −1 0 (cid:55)→ 0 0 1 0 0 0 = A˜mono.
2 0 0 0 −1 0 0 2
0 0 1
0 0 0 0 −1 0
0 0 0 0 0 1
Forthemonocliniccase,expression(9)canbestatedexplicitlyas
(cid:16) (cid:17) (cid:16) (cid:17)T (cid:16) (cid:17) (cid:16) (cid:17)T
A˜mono C A˜mono + A˜mono C A˜mono
1 1 2 2
C(cid:101)mono = .
2
1Readersinterestedinformulationofmatrix(10)mightrefertoBo´naetal.(2008).
9
Performingmatrixoperations,weobtain
√
c c c 0 0 2c
1111 1122 1133 √ 1112
c1122 c2222 c2233 0 0 √2c2212
c c c 0 0 2c
C(cid:101)mono = 1133 2233 3333 3312 , (11)
0 0 0 2c 2c 0
2323 2313
0 0 0 2c 2c 0
√ √ √ 2313 1313
2c 2c 2c 0 0 2c
1112 2212 3312 1212
which exhibits the form of the monoclinic tensor in its natural coordinate system. In other words,
C(cid:101)mono = Cmono,inaccordancewithCorollary2.
Appendix A.2
LetusproveLemma2.
Fororthotropicsymmetry,A˜ortho = A˜mono andA˜ortho = A˜mono and
1 1 2 2
1 0 0 0 0 0
0 1 0 0 0 0
−1 0 0
Aortho = 0 1 0 (cid:55)→ 0 0 1 0 0 0 = A˜ortho,
3 0 0 0 −1 0 0 3
0 0 −1
0 0 0 0 1 0
0 0 0 0 0 −1
1 0 0 0 0 0
0 1 0 0 0 0
1 0 0
Aortho = 0 −1 0 (cid:55)→ 0 0 1 0 0 0 = A˜ortho.
4 0 0 0 1 0 0 4
0 0 −1
0 0 0 0 −1 0
0 0 0 0 0 −1
Fortheorthotropiccase,expression(9)canbestatedexplicitlyas
(cid:16) (cid:17) (cid:16) (cid:17)T (cid:16) (cid:17) (cid:16) (cid:17)T (cid:16) (cid:17) (cid:16) (cid:17)T (cid:16) (cid:17) (cid:16) (cid:17)T
A˜ortho C A˜ortho + A˜ortho C A˜ortho + A˜ortho C A˜ortho + A˜ortho C A˜ortho
1 1 2 2 3 3 4 4
C(cid:101)ortho = .
4
Performingmatrixoperations,weobtain
c c c 0 0 0
1111 1122 1133
c1122 c2222 c2233 0 0 0
C(cid:101)ortho = c1133 c2233 c3333 0 0 0 , (12)
0 0 0 2c 0 0
2323
0 0 0 0 2c1313 0
0 0 0 0 0 2c
1212
which exhibits the form of the orthotropic tensor in its natural coordinate system. In other words,
C(cid:101)ortho = Cortho,inaccordancewithCorollary4.
10
