On Backus average for generally anisotropic layers

0.15 MB·

by  Len Bos

On Backus average for generally anisotropic layers
Len Bos, David R. Dalton, Michael A. Slawinski, Theodore Stanoev
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0 June 22, 2016
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Abstract

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p Inthispaper,followingtheBackus(1962)approach,weexamineexpressionsforelasticityparam-
- eters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of
o
e thingenerally anisotropiclayers. Theseexpressionsreduce to theresults ofBackus (1962)forthe
g
caseofisotropicand transverselyisotropiclayers.
.
s
c In the over half-a-century since the publications of Backus (1962) there have been numerous
i
s publications applying and extending that formulation. However, neither George Backus nor the
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authors of the present paper are aware of further examinationsof the mathematical underpinnings
h
p oftheoriginalformulation;hence thispaper.
[
We prove that—within the long-wave approximation—if the thin layers obey stability condi-

2 tions then so does theequivalentmedium. We examine—withintheBackus-average context—the
v
approximationoftheaverageofaproductastheproductofaverages,whichunderliestheaveraging
7
6 process.
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In the presented examination we use the expression of Hooke’s law as a tensor equation; in
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0 other words, we use Kelvin’s—as opposed to Voigt’s—notation. In general, the tensorial notation
.
1 allowsustoconvenientlyexamineeffects dueto rotationsofcoordinatesystems.
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1 Introduction and historical background
:
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i
X
The study of properties of materials as a function of scale has occupied researchers for decades.
r
a Notably, the discipline of continuum mechanics originates, at least partially, from such a consid-
eration. Herein, we focus our attention on the effect of a series of thin and laterally homogeneous
layers on a long-wavelength wave. These layers are composed of generally anisotropic Hookean
solids.
∗DipartimentodiInformatica,Universita`diVerona,[email protected]
†DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected]
‡DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected]
§DepartmentofEarthSciences,MemorialUniversityofNewfoundland,[email protected]
1
Such a mathematicalformulation serves as a quantitativeanalogy for phenomenaexaminedin
seismology. The effect of seismic disturbances—whose wavelength is much greater than the size
of encountered inhomogeneities—is tantamount to the smearing of the mechanical properties of
such inhomogeneities. The mathematical analogy of thissmearing is expressed as averaging. The
result of this averaging is a homogeneous anisotropic medium to which we refer as anequivalent
medium.
We refer to the process of averaging as Backusaveraging, which is a common nomenclature
in seismology. However, several other researchers have contributed to the development of this
method.
Backus (1962) built on the work of Rudzki (1911), Riznichenko (1949), Thomson (1950),
Haskell (1953), White and Angona (1955), Postma (1955), Rytov (1956), Helbig (1958) and An-
derson(1961)toshowthatahomogeneoustransverselyisotropicmediumwithaverticalsymmetry
axiscouldbealong-waveequivalenttoastackofthinisotropicortransverselyisotropiclayers. In
other words, the Backus average of thin layers appears—at the scale of a long wavelength—as a
homogeneoustransverselyisotropicmedium.
Inthispaper,wediscussthemathematicalunderpinningsoftheBackus(1962)formulation. To
doso,weconsiderahomogeneousgenerallyanisotropicmediumthatisalong-waveequivalentto
a stack of thin generally anisotropic layers. The cases discussed explicitly by Backus (1962) are
special cases ofthisgeneral formulation.
2 Averaging Method
2.1 Assumptions
We assume the lateral homogeneity of Hookean solids consisting of a series of layers that are
parallel to the x x -plane and have an infinite lateral extent. We subject this series to the same
1 2
traction above and below, independent of time or lateral position. It follows that the stress tensor
components σ , where i 1,2,3 , are constant throughout the strained medium, due to the
i3
∈ { }
requirement of equality of traction across interfaces (e.g., Slawinski (2015), pp. 430–432), and to
thedefinitionofthestresstensor,
3
T = σ n , i 1,2,3 ,
i ij j
∈ { }
j=1
X
where T is traction and n is the unit normal to the interface. No such equality is imposed on
the other three components of this symmetric tensor; σ , σ and σ can vary wildly along the
11 12 22
x -axisdueto changesofelasticpropertiesfrom layertolayer.
3
Furthermore, regarding the strain tensor, we invoke the kinematic boundary conditions that
require no slippage or separation between layers; in other words, the corresponding components
of the displacement vector, u , u and u , must be equal to one another across the interface (e.g.,
1 2 3
Slawinski(2015), pp.429–430).
2
Theseconditionsaresatisfiedifuiscontinuous. Furthermore,forparallellayers,itsderivatives
withrespecttox andx ,evaluatedalongthex -axis,remainsmall. However,itsderivativeswith
1 2 3
respect tox , evaluatedalong thataxis,can varywildly.
3
Thereason forthedifferingbehaviourofthederivativesresideswithinHooke’slaw,
3 3
σ = c ε , i,j 1,2,3 , (1)
ij ijkℓ kℓ
∈ { }
k=1 ℓ=1
XX
where
1 ∂u ∂u
k ℓ
ε := + , k,ℓ 1,2,3 . (2)
kℓ
2 ∂x ∂x ∈ { }
ℓ k
(cid:18) (cid:19)
Within each layer, derivatives are linear functions of the stress tensor. The derivatives with re-
spect to x and x remain within a given layer; hence, the linear relation remains constant. The
1 2
derivativeswithrespect to x exhibitchangesdueto differentpropertiesofthelayers.
3
In view of definition (2), ε , ε and ε vary slowly along the x -axis. On the other hand,
11 12 22 3
ε , ε and ε can vary wildlyalong thataxis.
13 23 33
Herein, we assume that the elasticity parameters are expressed with respect to the same coor-
dinatesystemforalllayers. However,thisaprioriassumptioncanbereadilyremovedbyrotating,
ifnecessary,thecoordinatesystemstoexpressthem inthesameorientation.
2.2 Definitions
FollowingthedefinitionproposedbyBackus(1962),theaverageofthefunctionf(x )of“width”ℓ
3 ′
isthemovingaveragegivenby
∞
f(x ) := w(ζ x )f(ζ)dζ, (3)
3 3
−
Z
−∞
wheretheweightfunction,w(x ),isanapproximateidentity,whichisanapproximateDiracdelta
3
thatacts likethedeltacentred at x = 0, withthefollowingproperties:
3
∞ ∞ ∞
w(x ) > 0, w( ) = 0, w(x )dx = 1, x w(x )dx = 0, x2w(x )dx = (ℓ)2.
3 ±∞ 3 3 3 3 3 3 3 3 ′
Z Z Z
−∞ −∞ −∞
These properties define w(x ) as a probability-density function with mean 0 and standard devia-
3
tionℓ , explainingtheuseoftheterm“width”forℓ .
′ ′
To understand the effect of such averaging, which is tantamount to smoothing by a wave, we
mayconsideritseffect on thepurefrequency,f(x ) = exp( iωx ),
3 3
−
∞ ∞ ∞
f(x ) = w(ζ x )f(ζ)dζ = w(ζ x )exp( ιωζ)dζ = w(u)exp( ιω(u+x ))du,
3 3 3 3
− − − −
Z Z Z
−∞ −∞ −∞
3
whereu := ζ x and ι := √ 1; itfollowsthat
3
− −
∞
f(x ) = exp( ιωx ) w(u)exp( ιωu)du = exp( ιωx )w(ω),
3 3 3
− − −
Z
−∞
b
wherew(ω)istheFouriertransformofw(x ).
3
If, in addition, w(x ) is an even function, then w(ω) is real-valued and we may think of f(x )
3 3
asthepurefrequency,exp( ιωx ),whose“amplitude”isw(ω). TheclassicalRiemann-Lebesgue
b 3
−
Lemmaimpliesthatthisamplitudetendstozero as thefrequencygoes toinfinity. Toexaminethis
b
decay ofamplitude,we mayconsidera commonchoiceforw(x ), namely,theGaussiandensity,
b 3
1 x2
w(x ) = exp 3 .
3 ℓ√2π −2(ℓ)2
′ (cid:18) ′ (cid:19)
As iswellknown,inthiscase,
(ωℓ)2
′
w(ω) = exp ,
− 2
(cid:18) (cid:19)
which is a multiple of the Gaussian density with standard deviation 1/ℓ . In particular, one notes
b ′
thefastdecay, as theproductωℓ increases.
′
Perhaps itisuseful tolookat anotherexample. Consider
1
w(x ) = I ,
3 2√3ℓ′ [−√3ℓ′,√3ℓ′]
whichistheuniformdensityontheinterval[ √3ℓ,√3ℓ],andwhichsatisfiesthedefiningprop-
′ ′
−
erties ofw(x ), as required. ItsFouriertransformis
3
sin(√3ωℓ)
′
w(ω) = ,
√3ωℓ
′
and, as expected, this amplitudetends to zero as ω , but at a much slower rate than in the
b → ±∞
Gaussiancase; herein, thedecay rateisorder 1/(ωℓ).
′
2.3 Properties
To perform the averaging,we useits linearity,according to which theaverageof asum is thesum
oftheaverages,f +g = f +g. Also,weusethefollowinglemma
Lemma 1. Theaverageof thederivativeisthederivativeoftheaverage,
∂f ∂
= f , i 1,2,3 .
∂x ∂x ∈ { }
i i
This lemma is proved in AppendixA. In AppendixB, we prove the lemma that ensures that the
averageofHookean solidsresultsina Hookeansolid,whichis
Lemma 2. Iftheindividuallayerssatisfythestabilitycondition,sodoestheirequivalentmedium.
TheproofofthislemmainvokesLemma3, below.
4
2.4 Approximations
In AppendixC, westateandprovearesult thatmaybeparaphrased as
Lemma3. Iff(x )isnearlyconstantalongx andg(x )doesnotvaryexcessively,thenfg f g.
3 3 3
≈
Anapproximation—withinthephysicalrealm—isourapplyingthestatic-casepropertiestoex-
aminewavepropagation,whichisadynamicprocess. AsstatedinSection2.1,inthecaseofstatic
equilibrium, σ , where i 1,2,3 , are constant. We consider that these stress-tensor compo-
i3
∈ { }
nents remain nearly constant along the x -axis, for the farfield and long-wavelength phenomena.
3
As suggested by Backus (1962), the concept of a long wavelength can be quantified as κℓ 1,
′
≪
where κ is the wave number. Similarly, we consider that ε , ε and ε remain slowly varying
11 12 22
alongthat axis.
Also, we assume that waves propagate perpendicularly, or nearly so, to the interfaces. Oth-
erwise, due to inhomogeneity between layers, the proportion of distance travelled in each layer
is a function of the source-receiver offset, which—in principle—entails that averaging requires
differentweightsforeach layerdependingontheoffset(Daltonand Slawinski,2016).
3 Equivalent-medium elasticity parameters
Considertheconstitutiveequationforagenerally anisotropicHookeansolid,
σ c c c √2c √2c √2c ε
11 1111 1122 1133 1123 1113 1112 11
σ c c c √2c √2c √2c ε
22  1122 2222 2233 2223 2213 2212  22
   
σ c c c √2c √2c √2c ε
33 = 1133 2233 3333 3323 3313 3312 33 , (4)
 √2σ   √2c √2c √2c 2c 2c 2c  √2ε 
 23   1123 2223 3323 2323 2313 2312  23 
 √2σ13   √2c1113 √2c2213 √2c3313 2c2313 2c1313 2c1312  √2ε13 
 √2σ12   √2c1112 √2c2212 √2c3312 2c2312 2c1312 2c1212  √2ε12 
 
   
 
where theelasticitytensor, whosecomponentsconstitutethe6 6 matrix,C, is positive-definite.
×
Thisexpressionisequivalentto thecanonicalform ofHooke’slawstated inexpression(1). In ex-
pression(4),theelasticitytensor,c ,which initscanonical formisafourth-rank tensorinthree
ijkℓ
dimensions, is expressed as a second-rank tensor in six dimensions, and equations (4) constitute
tensor equations (e.g., Chapman (2004, Section 4.4.2) and Slawinski (2015, Section 5.2.5)). This
formulation is referred to as Kelvin’s notation. A common notation, known as Voigt’s notation,
does notconstituteatensorequation.
To apply the averaging process for a stack of generally anisotropic layers, we express equa-
tions (4) in such a manner that the left-hand sides of each equation consist of rapidly varying
stresses or strains and the right-hand sides consist of algebraic combinations of rapidly varying
layer-elasticityparameters multipliedbyslowlyvaryingstressesorstrains.
5
First, considertheequationsforσ , σ and σ , whichcan bewrittenas
33 23 13
σ =c ε +c ε +c ε +√2c √2ε +√2c √2ε +√2c √2ε
33 1133 11 2233 22 3333 33 3323 23 3313 13 3312 12
√2σ =√2c ε +√2c ε +√2c ε +2c √2ε +2c √2ε +2c √2ε
23 1123 11 2223 22 3323 33 2323 23 2313 13 2312 12
√2σ =√2c ε +√2c ε +√2c ε +2c √2ε +2c √2ε +2c √2ε ,
13 1113 11 2213 22 3313 33 2313 23 1313 13 1312 12
whichthen can bewrittenas thematrixequation,
c √2c √2c ε σ c ε c ε √2c √2ε
3333 3323 3313 33 33 1133 11 2233 22 3312 12
− − −
√2c 2c 2c √2ε =√2σ √2c ε √2c ε 2c √2ε 
3323 2323 2313 23 23 1123 11 2223 22 2312 12
− − −
√2c3313 2c2313 2c1313 √2ε13 √2σ13 √2c1113ε11 √2c2213ε22 2c1312√2ε12
    − − − 
    
M E A
σ c c √2c ε
| {z }| {z } | 33 1133 {z2233 3312 11}
=√2σ23 √2c1123 √2c2223 2c2312  ε22  .
−
√2σ13 √2c1113 √2c2213 2c1312 √2ε12
   
 
   
G B F
(5)
| {z } | {z }| {z }
M is invertible, since it is positive-definite and, hence, its determinant is strictly positive. This
positivedefiniteness follows from the positivedefiniteness of C, given in expression (4), for x
R3 0 and y := [0,0,xt,0]t, xtMx = ytCy > 0 as y = 0. This follows only if C is in Kelvi∈n
\{ } 6
notation, and allows us to conclude that—since the positive definiteness is the sole constraint on
the values of elasticity parameters—the Backus average is allowed for any sequence of layers
composedofHookeansolids.
Notably,determinantsofM , inexpression(5),differby afactoroffourbetweenVoigt’snota-
tionandKelvin’snotation,usedherein. Thefinalexpressionsfortheequivalentmedium,however,
appearto bethesameforbothnotations.
Multiplyingbothsides ofequation(5) byM 1, weexpresstherapidly varyingE as
−
E = M 1A = M 1(G BF) = M 1G (M 1B)F , (6)
− − − −
− −
whichmeans that
M 1G = E +(M 1B)F ,
− −
and can beaveraged toget
M 1G E +(M 1B)F ,
− −
≈
and, hence, effectively,
1 1 1
G = (M 1)− E +(M 1B)F = (M 1)− E +(M 1)− (M 1B)F . (7)
− − − − −
h i
6
Comparingexpression(7)withthepattern ofthecorrespondingthreelinesofC in expression(4),
weobtainformulæfortheequivalent-mediumelasticityparameters.
To obtain theremainingformulæ, letus examinetheequations fortherapidlyvaryingσ , σ
11 22
and σ ,which, fromequation(4), can bewritten as
12
σ c c √2c ε c √2c √2c ε
11 1111 1122 1112 11 1133 1123 1113 33
 σ22  =  c1122 c2222 √2c2212 ε22 + c2233 √2c2223 √2c2213√2ε23 .
√2σ12 √2c1112 √2c2212 2c1212 √2ε12 √2c3312 2c2312 2c1312 √2ε13
    
   
    
H J F K E
(8)
| {z } | {z }| {z } | {z }| {z }
Notethat K = Bt. Substitutingexpression(6) forE, weget
H = JF +KM 1(G BF) = JF +KM 1G KM 1BF .
− − −
− −
Averaging,we get
H J F +KM 1G KM 1BF
− −
≈ −
1
=(J KM 1B)F +KM 1 (M 1)− E +(M 1B)F
− − − −
−
n 1 h io 1
= J KM 1B +KM 1(M 1)− (M 1B) F +KM 1(M 1)− E. (9)
− − − − − −
−
h i
Comparingequation(9)withthepatternofthecorrespondingthreelinesinequation(4),weobtain
formulæfortheremainingequivalent-mediumparameters.
Wedonotlistindetailtheformulæforthetwenty-oneequivalent-mediumelasticityparameters
of a generally anisotropic solid, since just one such parameter takes about half-a-dozen pages.
However, a symbolic-calculation software can be used to obtain those parameters. In Section 4,
we usethe monoclinicsymmetryto exemplifythe process and listin detail theresultingformulæ,
and wealsosummarizetheresultsfororthotropicsymmetry.
TheresultsofthissectionaresimilartotheresultsofSchoenbergandMuir(1989),Helbigand
Schoenberg (1987, Appendix), Helbig (1998), Carcione et al. (2012) and Kumar (2013), except
that the tensorial form of equation (4) requires factors of 2 and √2 in several entries of M , B,
J and K. This notation allows for a convenient study of rotations, which arise in the study of
elasticitytensorsexpressedincoordinatesystemsofarbitrary orientations.
4 Reduction to higher symmetries
4.1 Monoclinic symmetry
Let us reducetheexpressionsderivedforgeneral anisotropyto highermaterialsymmetries. To do
so,let usfirst considerthecase ofmonocliniclayers.
7
Thecomponentsofa monoclinictensorcan bewrittenin amatrixform as
c c c 0 0 √2c
1111 1122 1133 1112
c c c 0 0 √2c
1122 2222 2233 2212
 
c c c 0 0 √2c
Cmono = 1133 2233 3333 3312 ; (10)
 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 
 
this expression corresponds to the coordinate system whose x -axis is normal to the symmetry
3
plane. Insertingthesecomponentsintoexpression(5), wewrite
1
0 0
c 0 0
3333 c
3333
 c c 
M =  0 2c2323 2c2313 , M−1 = 0 1313 2313 ,
 D − D 
 0 2c2313 2c1313  0 c2313 c2323 
 
  − D D
 
 
whereD 2(c c c2 ). Then, wehave
≡ 2323 1313 − 2313
1
1 −
0 0
1
c
0 0 (cid:18) 3333(cid:19) 
c c c
 3333  2323 2313
c c 1  
M−1 =  0 1D313 − 2D313 , (M−1)− =  0 (cid:16) DD (cid:17) (cid:16) DD (cid:17) ,
   2 2 
 c c   c c 
 0 2313 2323   2313 1313 
   
 − D D   0 D D 
   (cid:16) D (cid:17) (cid:16) D (cid:17)
 2 2 
 
2
whereD (c /D)(c /D) (c /D) . Wealsohave
2 1313 2323 2313
≡ −
c c √2c
1133 2233 3312
B =  0 0 0  ,
 0 0 0 
 
 
whichleads to
c c √2c c c √2c
1133 2233 3312 1133 2233 3312
c c c c c c
 3333 3333 3333   3333 3333 3333 
M 1B = , M 1B = .
− −
0 0 0  0 0 0 
 
 
 
 
 0 0 0   0 0 0 
 
 
 
 
8
Furthermore,
1 1 1
1 − c 1 − c 1 − √2c
1133 2233 3312
1 (cid:18)c3333(cid:19) (cid:18)c3333(cid:19) (cid:18)c3333(cid:19) (cid:18)c3333(cid:19) (cid:18)c3333(cid:19) c3333 !
(M 1)− (M 1B) = .
− −
 0 0 0 
 
 
 0 0 0 
 
 
Then, ifwewriteequation(7)as
σ ε
33 33 ε
11
√2σ  = (M 1)−1√2ε +(M 1)−1(M 1B) ε22
23 − 23 − −  
√2ε
√2σ13 √2ε13  12
 
   
and compareittoequation(4), weobtain
c
1 2323
1 −
D
c = , c = ,
h 3333i c h 2323i (cid:16)2D (cid:17)
(cid:18) 3333(cid:19) 2
c c
1313 2313
D D
c = , c = ,
h 1313i (cid:16)2D (cid:17) h 2313i (cid:16)2D (cid:17)
2 2
1 1 1
1 − c 1 − c 1 − c
1133 2233 3312
c = , c = , c = ,
1133 2233 3312
h i c c h i c c h i c c
(cid:18) 3333(cid:19) (cid:18) 3333(cid:19) (cid:18) 3333(cid:19) (cid:18) 3333(cid:19) (cid:18) 3333(cid:19) (cid:18) 3333(cid:19)
whereanglebrackets denotetheequivalent-mediumelasticityparameters.
To calculate the remaining equivalent elasticity parameters from equation (9), we insert com-
ponents(10)intoexpression(8)to write
c c √2c c c √2c
1111 1122 1112 1111 1122 1112
J =  c c √2c  , J =  c c √2c  ,
1122 2222 2212 1122 2222 2212
√2c √2c 2c  √2c √2c 2c 
 1112 2212 1212   1112 2212 1212 
   
c
c 1133
1133 0 0
0 0
c
c 0 0 c3333  (cid:18) 3333(cid:19) 
1133
 
K = √c22c2333312 00 00 , KM−1 = √2cc23c2333333312 00 00 , KM−1 =  (cid:18)cc32c323333(cid:19) 0 0 ,
   c3333  √2(cid:18)c33333132(cid:19) 0 0
 
9
c2 c c c c
1133 1133 2233 √2 3312 1133
c c c
3333 3333 3333
 
c c c2 c c
KM−1B =  1133 2233 2233 √2 3312 2233 ,
 c c c 
 3333 3333 3333 
 
 c c c c c2 
√2 3312 1133 √2 3312 2233 2 3312 
 
c c c
 3333 3333 3333 
 
c2 c c c c
1133 1133 2233 √2 3312 1133
c c c
 (cid:18) 3333(cid:19) (cid:18) 3333 (cid:19) (cid:18) 3333 (cid:19)
c c c2 c c
KM−1B =  1133 2233 2233 √2 3312 2233  ,
 (cid:18) c3333 (cid:19) (cid:18)c3333(cid:19) (cid:18) c3333 (cid:19)
 
 c c c c c2 
√2 3312 1133 √2 3312 2233 2 3312 
 c c c 
 (cid:18) 3333 (cid:19) (cid:18) 3333 (cid:19) (cid:18) 3333(cid:19) 
 
1
c 1 −
1133
0 0
c c
 (cid:18) 3333(cid:19)(cid:18) 3333(cid:19) 
1
1  c 1 − 
KM−1(M−1)− =  c2233 c 0 0 ,
 (cid:18) 3333(cid:19)(cid:18) 3333(cid:19) 
 
 1 
 c 1 − 
√2 3312 0 0
 c c 
 (cid:18) 3333(cid:19)(cid:18) 3333(cid:19) 
 
1
KM 1(M 1)− M 1B
− − −
1 −1 c 2 1 −1 c c 1 −1 c c
1133 1133 2233 √2 1133 3312
c c c c c c c c
3333 3333 3333 3333 3333 3333 3333 3333
 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
−1 −1 2 −1
1 c c 1 c 1 c c
=  1133 2233 2233 √2 2233 3312 .
 c c c c c c c c 
3333 3333 3333 3333 3333 3333 3333 3333
 (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
 
 1 −1 c c 1 −1 c c 1 −1 c 2 
√2 1133 3312 √2 2233 3312 2 3312 
 c c c c c c c c 
 (cid:18) 3333(cid:19) (cid:18) 3333(cid:19)(cid:18) 3333(cid:19) (cid:18) 3333(cid:19) (cid:18) 3333(cid:19)(cid:18) 3333(cid:19) (cid:18) 3333(cid:19) (cid:18) 3333(cid:19) 
 
Then, ifwewriteequation(9)as
σ ε ε
11 11 33
1 1
 σ  = J KM 1B +KM 1(M 1)− (M 1B)  ε +KM 1(M 1)− √2ε 
22 − − − − 22 − − 23
−
√2σ  h i√2ε  √2ε 
 12  12  13
     
10