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On commutativity of Backus and Gazis averages David R. Dalton∗, Michael A. Slawinski † 6 January 12, 2016 1 0 2 n a J Abstract 2 1 We show that the Backus (1962) equivalent-medium average, which is an average over a spatial ] h variable, and the Gazis et al. (1963) effective-medium average, which is an average over a sym- p metrygroup,donotcommute,ingeneral. Theycommuteinspecialcases,whichweexemplify. - o 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 corollary. 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 commutativity. 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 generally 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 systems. 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 Backus,G.E.,Long-waveelasticanisotropyproducedbyhorizontallayering,J.Geophys.Res.,67, 11,4427–4440,1962. Bońa,A.,I.BucataruandM.A.Slawinski,SpaceofSO(3)-orbitsofelasticitytensors,Archivesof Mechanics,60,2,121–136,2008 Bos,L,D.R.Dalton,M.A.SlawinskiandT.Stanoev,OnBackusaverageforgenerallyanisotropic layers,arXiv,2016. Chapman,C.H.,Fundamentalsofseismicwavepropagation,CambridgeUniversityPress,2004. Danek, T., M. Kochetov and M.A. Slawinski, Uncertainty analysis of effective elasticity tensors usingquaternion-basedglobaloptimizationandMonte-Carlomethod,TheQuarterlyJournal ofMechanicsandAppliedMathematics,66,2,pp.253–272,2013. Danek, T., M. Kochetov and M.A. Slawinski, Effective elasticity tensors in the context of random errors,JournalofElasticity,2015. Gazis, D.C., I. Tadjbakhsh and R.A. Toupin, The elastic tensor of given symmetry nearest to an anisotropicelastictensor,ActaCrystallographica,16,9,917–922,1963. Kochetov,M.andM.A.Slawinski,Onobtainingeffectiveorthotropicelasticitytensors,TheQuar- terlyJournalofMechanicsandAppliedMathematics,62,2,pp.149-166,2009a. Kochetov, M. and M.A. Slawinski, On obtaining effective transversely isotropic elasticity tensors, JournalofElasticity,94,1-13.,2009b. Slawinski, M.A. Wavefronts and rays in seismology: Answers to unasked questions, World Scien- tific,2016. Slawinski,M.A.,Wavesandraysinelasticcontinua,WorldScientific,2015. Thomson, W., Mathematical and physical papers: Elasticity, heat, electromagnetism, Cambridge UniversityPress,1890 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)mightrefertoBońaetal.(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,Aõrtho = A˜mono andAõrtho = 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  = Aõrtho, 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  = Aõrtho. 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 Aõrtho C Aõrtho + Aõrtho C Aõrtho + Aõrtho C Aõrtho + Aõrtho C Aõrtho 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