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Zeitschrift für angewandte
Mathematik und Physik
Journal of Applied Mathematics and
Physics / Journal de Mathématiques et
de Physique appliquées
ISSN 0044-2275
Volume 70
Number 3
Z. Angew. Math. Phys. (2019) 70:1-15
DOI 10.1007/s00033-019-1128-9
On the Backus average of a layered
medium with elasticity tensors in random
orientations
Len Bos, Michael A.Slawinski &
Theodore Stanoev
1 23
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Z. Angew. Math. Phys. (2019) 70:84
c
2019 Springer Nature Switzerland AG
https://doi.org/10.1007/s00033-019-1128-9
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
On the Backus average of a layered medium with elasticity tensors
in random orientations
Len Bos, Michael A. Slawinski and Theodore Stanoev
Abstract. As shown by Backus (J Geophys Res 67(11):4427–4440, 1962), the average of a stack of isotropic layers results in
a transversely isotropic medium. Herein, we consider a stack of layers consisting of randomly oriented anisotropic elasticity
tensors, which—one might reasonably expect—would result in an isotropic medium. However, we show—by means of a
fundamental symmetry of the Backus average—that the corresponding Backus average is only transversely isotropic and
not, in general, isotropic. In the process, we formulate, and use, an analogy between the Backus and Gazis et al. (Acta
Crystallogr 16(9):917–922, 1963) averages.
Mathematics Subject Classification. 86A15, 74Q20.
Keywords. Backus average, Elasticity theory, Quaternion rotation, Inhomogeneity, Anisotropy.
1. Introduction
In this paper, we consider a model of the Earth’s near surface used in applied seismology, namely that
of a stack of parallel layers. The properties of each layer are stated by its hyperelastic elasticity tensor.
We are interested in the so-called Backus [1] average, which replaces the stack of layers by a single
(approximately) equivalent medium and is commonly used in applied seismology. The behavior of such
a medium is analogous to the response of a stack of layers to a signal whose wavelength is much greater
than the thickness of individual layers. Herein, we discuss a basic mathematical property of this averaging
procedure. Specifically, we study the case for which the stack consists of the elasticity tensors that are
random rotations of a given generally anisotropic tensor, C.
We show that, in this situation, the arithmetic average of the rotated tensors results in the so-called
Gazis et al. [2] average, which is the isotropic tensor closest to C, according to the Frobenius norm,
as discussed in Sect. 2. Thus, the Gazis et al. average is the closest—in the Frobenius sense—isotropic
counterpart of C. In contrast, as we show in Theorem 5, the Backus average of the same randomly rotated
tensors is, in general, only transversely isotropic, and thus provides a transversely isotropic counterpart
for C.
Now for some details. Each layer is expressed by Hooke’s law,
σij =
3
k=1
3
=1
cijk εk ,i,j=1,2,3,(1)
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 202259.
0123456789().: V,-vol
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84 Page 2 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP
where the stress tensor, σij , is linearly related to the strain tensor,
εk := 1
2∂uk
∂x
+∂u
∂xk,k,=1,2,3,(2)
where uand xare the displacement and position vectors, respectively. The index symmetries of the
elasticity tensor, which has to be positive-definite as a consequence of hyperelasticity, are
cijk =ckij and cijk =cjik =cijk =cjik.(3)
The index symmetries for the former result from the existence of the strain-energy function whereas, for
the latter, from an assumption of the strong form of Newton’s third law and infinitesimal deformations [3,
Section 2.7]. Under the index symmetries, the elasticity tensor chas twenty-one linearly independent
components and can be written as (e.g., [4, expression (2.1)])
C=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
C11 C12 C13 √2C14 √2C15 √2C16
C12 C22 C23 √2C24 √2C25 √2C26
C13 C23 C33 √2C34 √2C35 √2C36
√2C14 √2C24 √2C34 2C44 2C45 2C46
√2C15 √2C25 √2C35 2C45 2C55 2C56
√2C16 √2C26 √2C36 2C46 2C56 2C66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,(4)
wherein the bijection of indices iand j(or kand )ofcijk to m(or n)ofCmn is given by m=
iδ
ij +(1−δij)(9−i−j) (e.g., [5, Section 3.2.4.2]). In addition, √2 and 2 ensure that the stress and
strain tensors have the same orthonormal basis, which is a convenient property in examining rotations
of C. Although the elasticity tensors cand Care different objects, they both exhibit proper tensorial
form and, hence, their pertinent properties are invariant under orthogonal transformation. Further, any
elasticity tensor of this form is also positive-definite (e.g., [6]).
A rotation in R3, expressed conveniently in terms of quaternions, is given by
A=A(q)=⎡
⎣a2
0+a2
1−a2
2−a2
3−2a0a3+2a1a22a0a2+2a1a3
2a0a3+2a1a2a2
0−a2
1+a2
2−a2
3−2a0a1+2a2a3
−2a0a2+2a1a32a0a1+2a2a3a2
0−a2
1−a2
2+a2
3⎤
⎦,(5)
where q=[a0,a
1,a
2,a
3] is a unit quaternion. The corresponding rotation of the tensor (4)is,as
discussed by B´ona et al. [4, diagram (3.1)],
C=
AC
AT,(6)
where
Ais expression (A.1) in “Appendix A” applied to A(q). Throughout this article, we use notation
A
to denote the R6×6equivalent of A∈SO(3) and
Cto distinguish an entity that is the result of operation
AC
AT. Further information regarding
Acan also be found in B´ona et al. [4].
2. Backus and Gazis et al. averages
To examine the elasticity tensors, C∈R6×6, which are positive-definite, let us consider the space of all
matrices M:= R6×6. Its subspace of isotropic matrices is
Miso := {M∈M:
QM
QT=M, ∀Q∈SO(3)}.(7)
Miso is a linear space, since, as is easy to verify, if M1,M
2∈M
iso, then αM1+βM2∈M
iso,for
all α, β ∈R. Let us endow Mwith an inner product,
M1,M
2F:= tr M1MT
2=
6
i,j=1
(M1)ij (M2)ij ,(8)
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ZAMP On the Backus average of a layered medium Page 3 of 15 84
and the corresponding Frobenius norm,
M:= M1,M
2F.(9)
In such a context, Gazis et al. [2] prove the following theorem.
Theorem 1. The closest element—with respect to the Frobenius norm—to M∈Mfrom Miso is uniquely
given by
Miso :=
SO(3)
QM
QTdσ(Q),(10)
where dσ(Q)represents the Haar probability measure on SO(3) (see e.g., [7]).
Proof. It suffices to prove that
(M−Miso)⊥M
iso.(11)
To do so, we let N∈M
iso be arbitrary. Then, for any A∈SO(3),
M−Miso,NF=tr(M−Miso)NT
=tr⎛
⎜
⎝⎛
⎜
⎝M−
SO(3)
QM
QTdσ(Q)⎞
⎟
⎠NT⎞
⎟
⎠
=tr⎛
⎜
⎝
A⎛
⎜
⎝⎛
⎜
⎝M−
SO(3)
QM
QTdσ(Q)⎞
⎟
⎠NT⎞
⎟
⎠
AT⎞
⎟
⎠(as
Ais orthogonal)
=tr⎛
⎜
⎝
AMNT
AT−
A⎛
⎜
⎝
SO(3)
QM
QTdσ(Q)⎞
⎟
⎠
AT
ANT
AT⎞
⎟
⎠.(12)
But
A⎛
⎜
⎝
SO(3)
QM
QTdσ(Q)⎞
⎟
⎠
AT
=
SO(3)
A
QM
QT
ATdσ(Q) (by linearity)
=
SO(3)
A
QM
A
QT
dσ(Q)
=
SO(3)
AQM
AQT
dσ(Q) (by the properties of the tilde operation)
=
SO(3)
QM
QTdσ(Q) (by the invariance of the measure)
=Miso.(13)
Hence,
M−Miso,NF=tr
AMNT
AT−Miso
ANT
AT
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=tr
AM
AT
ANT
AT−Miso
ANT
AT
=tr
AM
ATNT−Miso NT
=tr
AM
ATNT−tr Miso NT,(14)
as by assumption, N∈M
iso.
Finally, integrating over A∈SO(3), we obtain
M−Miso,NF=tr⎛
⎜
⎝⎛
⎜
⎝
SO(3)
AM
ATdσ(A)⎞
⎟
⎠NT⎞
⎟
⎠−tr(Miso NT)
=tr(Miso NT)−tr(Miso NT)
=0,(15)
as required.
Since any elasticity tensor, C∈R6×6, is positive-definite, it follows that
Ciso :=
SO(3)
QC
QTdσ(Q) (16)
is both isotropic and positive-definite, since it is the sum of positive-definite matrices
QC
QT. Hence, Ciso
is the closest isotropic tensor to C, measured in the Frobenius norm.
If Qi∈SO(3), i=1, ..., n, is a sequence of random samples from SO(3), then the sample means
converge by the Law of Large Numbers to the true mean,
lim
n→∞
1
n
n
i=1
QiC
Qi
T=
SO(3)
QC
QTdσ(Q)=Ciso,(17)
which—in accordance with Theorem 1—is the Gazis et al. average of C.
This paper relies on replacing the arithmetic average in expression (17) by the Backus average, which
provides a single, homogeneous model that is long-wave equivalent to a thinly layered medium. We use
the coordinate x3to denote the depth of this medium. Thus, according to Backus [1], the average of the
function f(x3) of “width” is the moving average given by
f(x3):=
∞
−∞
w(ζ−x3)f(ζ)dζ, (18)
where the weight function, w(x3), centred and concentrated at x3= 0, and exhibits the following prop-
erties.
w(x3)0,w(±∞)=0,
∞
−∞
w(x3)dx3=1,
∞
−∞
x3w(x3)dx3=0,
∞
−∞
x2
3w(x3)dx3=()2.(19)
These properties define w(x3) as a probability-density function with mean zero and standard deviation ,
thus explaining the term “width” for . Typically, we would choose w(x3) to be an appropriate Gaussian
density or else a boxcar (uniform) density.
The Backus average is commonly used by Geophysics practitioners, especially in the context of seismic
exploration. Its theoretical properties have been studied, for example, in [8–11]. We rely on the formulation
derived in [10] and given in formulas (38)–(43).
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ZAMP On the Backus average of a layered medium Page 5 of 15 84
3. Block structure of C→
AC
AT
The action C→
AC
AThas a simple block structure that is exploited in Sect. 4.Toseethis,wecon-
sider q=[a0,0,0,a
3], with a0:= cos(θ/2), a3:= sin(θ/2); thus, in accordance with expression (5),
A=A(q)=⎡
⎣cos θ−sin θ0
sin θcos θ0
001
⎤
⎦(20)
and, in accordance with expression (A.1),
A=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
cos2θsin2θ00 0−1
√2sin (2 θ)
sin2θcos2θ00 0 1
√2sin (2 θ)
001000
000cosθsin θ0
000−sin θcos θ0
1
√2sin (2 θ)−1
√2sin (2 θ) 0 0 0 cos (2 θ)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(21)
For q=[0,a
1,a
2,0 ] , with a1:= cos(θ/2) and a2:= sin(θ/2),
A=A(q)=⎡
⎣cos θsin θ0
sin θ−cos θ0
00−1⎤
⎦(22)
and
A=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
cos2θsin2θ00 0 1
√2sin (2 θ)
sin2θcos2θ00 0−1
√2sin (2 θ)
001000
000cosθ−sin θ0
000−sin θ−cos θ0
1
√2sin (2 θ)−1
√2sin (2 θ)0 0 0 −cos (2 θ)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(23)
In both cases, permuting the rows and columns to the order ( 3 ,4,5,1,2,6 ) results in a diagonal block
structure for
A. For expression (20), we have
A→
A10
0
A2,(24)
where
A1=⎡
⎣10 0
0cosθsin θ
0−sin θcos θ⎤
⎦and
A2=⎡
⎢
⎢
⎢
⎣
cos2θsin2θ−1
√2sin (2 θ)
sin2θcos2θ1
√2sin (2 θ)
1
√2sin (2 θ)−1
√2sin (2 θ)cos(2θ)
⎤
⎥
⎥
⎥
⎦.(25)
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84 Page 6 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP
It can be readily verified that both
A1,
A2∈R3×3are rotation matrices; their determinants are + 1.
Similarly, for expression (22),
A→
A10
0
A2(26)
where, in this case, it can be checked that
A1,
A2∈R3×3are both reflections; their determinants are −1.
Thus, in both cases,
A1and
A2are orthogonal matrices and
A∈R6×6is an orthogonal rotation matrix;
its determinant is +1.
In either case, the following lemma holds.
Lemma 2. Suppose that the rows and columns of Care permuted to the order (3,4,5,1,2,6) to have
the block structure
C→MB
KJ
,(27)
with M,B,K,J∈R3×3, and that the rows and columns of
Aare also so permuted. Then,
AC
AT→⎡
⎣
A1M
AT
1
A1B
AT
2
A2K
AT
1
A2J
AT
2⎤
⎦.(28)
Proof. Let P∈R6×6be the matrix obtained by permuting the rows of the identity to the order
(3,4,5,1,2,6 ). Our assumption is that
PCP
T=MB
KJ
.(29)
Then,
P
AC
ATPT=P
APTPCP
TP
ATPT
=
A10
0
A2MB
KJ
AT
10
0
AT
2
=⎡
⎣
A1M
AT
1
A1B
AT
2
A2K
AT
1
A2J
AT
2⎤
⎦,(30)
as required.
4. Fundamental symmetry of Backus average
Let us examine properties of the Backus average, which—for elasticity tensors, Ci—we denote by
(C1, ... , C
n).(31)
Theorem 3. For A∈R3×3, either a rotation about the x3-axis, which corresponds to form (20),or
a rotation-inversion about the x3-axis, which corresponds to form (22), and any elasticity tensor, C1,
... , C
n∈R6×6,
A(C1, ... , C
n)
AT=
AC
1
AT, ... ,
AC
n
AT,(32)
which is a symmetry condition. Conversely, if for an orthogonal matrix, A∈R3×3, we have equality (32),
for any collection of elasticity tensors, C1, ... , C
n∈R6×6, then Amust be of the form of expression (20)
or (22).
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Proof. As in Lemma 2, we permute the rows and columns of C∈R6×6to the order ( 3 ,4,5,1,2,6).
Thus, we have the block structure
C=MB
KJ
, M,B,K,J∈R3×3; (33)
herein, we use the notation of equations (5)–(9) of Bos et al. [10]. Also,
Ahas the block structure of
A=
A10
0
A2,
A1,
A2∈R3×3,(34)
and is orthogonal.
Let
C=
AC
AT=
A10
0
A2MB
KJ
A1T0
0
A2T=
M
B
K
J,(35)
where, by Lemma 2,
M=
A1M
A1T,
B=
A1B
A2T,
K=
A2K
A1T,
J=
A2J
A2T.(36)
In particular,
M−1=
A1M
A1T−1(37)
=
A1M−1
A1T.
The Backus-average equations, given by [10, Section 3], are
CBA =MBA BBA
KBA JBA ,(38)
where
MBA =M−1−1,(39)
BBA =M−1−1M−1B, (40)
KBA =KM
−1M−1−1,(41)
JBA =J−KM
−1B+KM
−1M−1−1M−1B,(42)
and where ◦denotes the arithmetic average of the expression ◦; for example,
M−1=1
n
n
i=1
M−1
i.(43)
Let MBA,BBA,KBA and JBA denote the associated sub-blocks of the Backus average of the
AC
j
AT.
Then,
A1MBA
A1T=
A1M−1−1
A1T
=
A1M−1
A1T−1
=
A1M−1
A1T−1
(by linearity)
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84 Page 8 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP
=
A1M
A1T−1−1
(by equation (37))
=
M−1−1
=
MBA ,(44)
A1BBA
A2T=
A1M−1−1M−1B
A2T
=
A1M−1−1
A1T
A1M−1B
A2T
=
MBA
A1M−1B
A2T(by the previous result)
=
MBA
A1M−1B
A2T(by linearity)
=
MBA
A1M−1
A1T
A1B
A2T
=
MBA
M−1
B
=
BBA ,(45)
A2KBA
A1T=
A2KM
−1M−1−1
A1T
=
A2KM
−1
A1T
A1M−1−1
A1T
=
A2KM
−1
A1T
MBA
=
K
M−1
MBA
=
KBA,(46)
and
A2JBA
A2T=
A2J−KM
−1B+KM
−1M−1−1M−1B
A2T
=
A2J
A2T−
A2KM
−1B
A2T+
A2KM
−1M−1−1M−1B
A2T
=
A2(J)
A2T−
A2(KM
−1B)
A2T+
A2KM
−1
A1T
A1M−1−1
A1T
A1M−1B
A2T
=
J−
A2K
A1T
A1M−1
A1T
A1B
A2T
+
A2KM
−1
A1T
A1M−1−1
A1T
A1M−1
A1T
A1B
A2T
=
J−
K
M−1
B+
K
M−1
MBA
M−1
B
=
JBA,(47)
which completes the proof of equality (32).
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To show the converse claimed in the statement of Theorem 3, let us consider C1=Iand C2=2I.
Their Backus average is
B:=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
3
200000
03
20000
004
3000
0004
300
00004
30
000003
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(48)
Following rotation,
B=
AB
AT,(49)
where
Ais given in expression (A.1). It can be shown by direct calculation that the (3,3) entry of
Bis
B33 =4
31+2a2
1+a2
22a2
0+a2
42.(50)
Since C1and C2are multiples of the identity,
AC
1
AT=C1and
AC
2
AT=C2,(51)
and the Backus average of
AC
1
ATand
AC
2
ATequals the Backus average of C1and C2,whichis
matrix (48). Hence,
A(C1,C
2)
AT=
AC
1
AT,
AC
2
AT(52)
implies that, for expression (50),
4
31+2a2
1+a2
22a2
0+a2
32=4
3,(53)
which results in
2√2
3a2
1+a2
2a2
0+a2
4=0.(54)
Thus, either a1=a2=0ora0=a3= 0. This is a necessary condition for symmetry (32) to hold, as
claimed.
Remark 4. Theorem 3is formulated for general positive-definite matrices C∈R6×6, not all of which
represent elasticity tensors. However, expression (32) is continuous in the Ciand hence is true in general
only if it is also true for Ci,such as diagonal matrices, which are limits of elasticity tensors.
5. Backus average of randomly oriented tensors
In this section, we study the Backus average for a random orientations of a given tensor. As discussed in
Sect. 2, the arithmetic average of such orientations results in the Gazis et al. average, which is the closest
isotropic tensor with respect to the Frobenius norm. We see that—for the Backus average—the result is,
perhaps surprisingly, different.
Given an elasticity tensor, C∈R6×6, let us consider a sequence of its random rotations given by
Cj:=
QjC
QT
j,j=1, ... , n, (55)
where Qj∈R3×3are random matrices sampled from SO(3).
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The Cjare samples from some distribution and, hence, almost surely,
C:= lim
n→∞
1
n
n
j=1
Cj=μ(C),(56)
thetruemean,
μ(C)=
SO(3)
QC
QTdσ(Q),(57)
where dσ(Q) is Haar measure on SO(3). Note that μ(C) is just the Gazis et al. average of C, which—in
accordance with Theorem 1—is isotropic.
Similarly, for any expression X(C) of submatrices of C, which appear in the Backus-average formulas,
X:= lim
n→∞
1
n
n
j=1
Xj=μ(X).(58)
Hence, almost surely limn→∞ (C1, ... , C
n) equals the Backus-average formula with each expression X
replaced by
μ(X)=
SO(3)
X
QjC
QjTdσ(Q).(59)
Theorem 5. The limit B:= limn→∞ (C1, ... , C
n)exists almost surely, in which case it is transversely
isotropic. It is not, in general, isotropic.
Proof. Let A∈R3×3be either a rotation about the x3-axis, which corresponds to form (20), or a rotation-
inversion about the x3-axis, which corresponds to form (22). Then
Cj:=
AC
j
AT,j=1, ... , n,
=
A
QjC
QjT
AT
=
A
QjC
A
QjT
=
(AQj)C
(AQj)
T
(60)
by the properties of the tilde operation, are also random samples from the same distribution. Hence,
almost surely,
lim
n→∞
C1, ... ,
Cn= lim
n→∞ (C1, ... , C
n)=B, say.(61)
But by the symmetry property of the Backus average, Theorem 3,
C1, ... ,
Cn=
A(C1, ... , C
n)
AT.(62)
Thus
B=
AB
AT,(63)
which means that Bis invariant under a rotation of space by A. Consequently, Bis a transversely isotropic
tensor.
In general, the limit tensor is not isotropic, as illustrated by the following example. Let
C= diag [ 1 ,1,1,1,0,0],(64)
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ZAMP On the Backus average of a layered medium Page 11 of 15 84
which, as stated in Remark 4, represents a limiting case of an elasticity tensor. Numerical evidence
strongly suggests that
B=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2
1
4
1
4000
1
4
1
2
1
4000
1
4
1
4
1
2000
000000
000000
000001
4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,(65)
which is not isotropic, since isotropic tensors have the form
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
xx−yx−y000
x−yx x−y000
x−yx−yx000
00 0y00
00 00y0
00 000y
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(66)
Although this is rather an artificial example, it could—with some computational effort—be “promoted”
to a legal proof. The conclusion is readily confirmed by the numerical examples presented in Sect. 6.
In fact, it is easy to identify the limiting matrix B; it is just the Backus-average expression (38), with
an expression X(C) replaced by the true mean
B=μ(X(C)) =
SO(3)
X
QC
QTdσ(Q).(67)
This limiting transversely isotropic tensor is of natural interest in its own right. It plays the role of the
Gazis et al. average in the context of the Backus average and is the subject of a forthcoming work.
6. Numerical example
Let us consider the elasticity tensor obtained by Dewangan and Grechka [12]; its components are estimated
from seismic measurements in New Mexico,
C=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
7.8195 3.4495 2.5667 √2(0.1374) √2(0.0558) √2(0.1239)
3.4495 8.1284 2.3589 √2(0.0812) √2(0.0735) √2(0.1692)
2.5667 2.3589 7.0908 √2(−0.0092) √2(0.0286) √2(0.1655)
√2(0.1374) √2(0.0812) √2(−0.0092) 2 (1.6636) 2 (−0.0787) 2 (0.1053)
√2(0.0558) √2(0.0735) √2(0.0286) 2 (−0.0787) 2 (2.0660) 2 (−0.1517)
√2(0.1239) √2(0.1692) √2(0.1655) 2 (0.1053) 2 (−0.1517) 2 (2.4270)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(68)
Using tensor (68), let us illustrate our theoretical results.
Consider a stack of equal-thickness layers, whose elasticity tensors are C. We rotate each Cusing
a random unit quaternion and perform the Backus average of the resulting stack of layers. Our main
result is that the Backus average—as the number of layers tends to infinity—tends almost surely to the
tensor Bgiven by expression (67).
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84 Page 12 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP
Fig. 1. Difference between tensors (69)and(70)
Indeed, using 107random layers as a proxy for infinity, the Backus average resulted in
B≡=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
7.3008 2.9373 2.9379 √2(0.0000) √2(0.0000) √2(0.0000)
2.9373 7.3010 2.9381 √2(0.0000) √2(0.0000) √2(0.0000)
2.9379 2.9381 7.2689 √2(0.0000) √2(−0.0001) √2(0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (2.1710) 2 (0.0000) 2 (0.0000)
√2(0.0000) √2(0.0000) √2(−0.0001) 2 (0.0000) 2 (2.1710) 2 (0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (0.0000) 2 (0.0000) 2 (2.1819)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(69)
To compare with the theoretical limit given by expression (67), we compute the triple integrals involved
numerically using Simpson’s and the trapezoidal rules. Effectively, the triple integral is replaced by a
weighted sum of the integrand evaluated at discrete points. The sums that approximate the integrals are
accumulated and are used in expressions (38).
Using the Simpson’s and the trapezoidal rules, with a sufficient number of subintervals, the limiting
Backus average is
B
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
7.3010 2.9373 2.9380 √2(0.0000) √2(0.0000) √2(0.0000)
2.9373 7.3010 2.9380 √2(0.0000) √2(0.0000) √2(0.0000)
2.9380 2.9380 7.2687 √2(0.0000) √2(0.0000) √2(0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (2.1711) 2 (0.0000) 2 (0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (0.0000) 2 (2.1711) 2 (0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (0.0000) 2 (0.0000) 2 (2.1818)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(70)
Notice that the components of expressions (69) and (70) are the same to the numerical accuracy displayed.
More generally, the rate of convergence in terms of the number of layers is illustrated in Fig. 1, where the
horizontal axis is the number of layers and the vertical axis is the maximum componentwise difference
between the two tensors. As indicated by the figure, for all intents and purposes, the convergence occurs
quite rapidly.
Expression (70) is transversely isotropic, as expected from Theorem 5, and in accordance with B´ona
et al. [6, Section 4.3], since its four distinct eigenvalues are
λ1=13.1658 ,λ
2=4.3412 ,λ
3=4.3636 ,λ
4=4.3421 ,(71)
with multiplicities of m1=m2=1andm3=m4= 2. The eigenvalues of expression (69)arein
agreement—up to 10−3—with eigenvalues (71) and their multiplicities. Furthermore, in accordance with
Theorem 5, in the limit, the distance to the closest isotropic tensor for expression (70)is0.0326 =0;
thus the distance does not reduce to zero.
Expressions (69) and (70) are transversely isotropic, which is the main conclusion of this work, even
though, for numerical modeling, one might view them as isotropic. This is indicated by Thomsen [13]
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ZAMP On the Backus average of a layered medium Page 13 of 15 84
parameters, which for tensor (70)are
γ=2.4768 ×10−3,δ=1.5816 ×10−3,=2.2219 ×10−3; (72)
values much less than unity indicate very weak anisotropy.
We emphasize, as previously remarked, that the limit of the arithmetic average of random layers
results in the classical Gazis average, which—in this case—is the closest isotropic tensor to the initial C.
Tensor (67) is therefore a “Backus average” analogue of the Gazis average.
7. Conclusions and future work
Examining the Backus average of a stack of layers consisting of randomly oriented anisotropic elasticity
tensors, we show that—in the limit—this average results in a homogeneous transversely isotropic medium,
as stated by Theorems 3and 5. In other words, the randomness within layers does not result in a medium
lacking a directional pattern. Both the isotropic layers, as shown by Backus [1], and randomly oriented
anisotropic layers, as shown herein, result in the average that is transversely isotropic, as a consequence
of inhomogeneity among parallel layers. This property is discussed by Adamus et al. [14], and herein it
is illustrated in “Appendix B”.
In the limit, the transversely isotropic tensor is the Backus counterpart of the Gazis et al. average.
Indeed, the arithmetic average of randomized layers of an elasticity tensor produces the Gazis et al.
average and is its closest isotropic tensor, according to the Frobenius norm. On the other hand, the
Backus average of the layers resulting from a randomization of the same tensor produces the transversely
isotropic tensor given in expression (67). This tensor and its properties are the subject of a forthcoming
paper.
Acknowledgements
We wish to acknowledge discussions with Michael G. Rochester, the proofreading of David R. Dalton and
Filip P. Adamus, as well as the graphic support of Elena Patarini.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps
and institutional affiliations.
Appendix A: Rotations by unit quaternions
The R6equivalent for A∈SO(3), which is a rotation of the tensor (4), is (e.g., [5, equation (3.42)]),
˜
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 A23 A31 +A21A33 A22A31 +A21 A32
√2A11A31 √2A12 A32 √2A13A33 A13 A32 +A12A33 A13 A31 +A11A33 A12A31 +A11 A32
√2A11A21 √2A12 A22 √2A13A23 A13 A22 +A12A23 A13 A21 +A11A23 A12A21 +A11 A22
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(A.1)
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84 Page 14 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP
8. Appendix B: Alternating layers
Consider a randomly generated elasticity tensor,
C=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
14.5739 6.3696 2.9020 √2(9.4209) √2(3.8313) √2(3.5851)
6.3696 10.7276 6.2052 √2(4.0375) √2(5.1333) √2(6.0745)
2.9020 6.2052 11.4284 √2(1.9261) √2(9.8216) √2(1.3827)
√2(9.4209) √2(4.0375) √2(1.9261) 2 (13.9034) 2 (0.2395) 2 (2.0118)
√2(3.8313) √2(5.1333) √2(9.8216) 2 (0.2395) 2 (10.7353) 2 (0.0414)
√2(3.5851) √2(6.0745) √2(1.3827) 2 (2.0118) 2 (0.0414) 2 (9.0713)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,(B.2)
whose eigenvalues are
λ1=34.0318 ,λ
2=18.1961 ,λ
3=10.4521 ,λ
4=4.8941 ,λ
5=2.2737 ,λ
6=0.5921.(B.3)
The Backus average of 107alternating layers composed of randomly oriented tensors (68) and (B.2)is
B
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
8.4711 1.1917 1.2572 √2(0.0000) √2(0.0000) √2(0.0000)
1.1917 8.4710 1.2570 √2(0.0000) √2(0.0000) √2(0.0000)
1.2572 1.2570 6.6648 √2(−0.0001) √2(0.0000) √2(0.0000)
√2(0.0000) √2(0.0000) √2(−0.0001) 2 (2.8440) 2 (0.0000) 2 (0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (0.0000) 2 (2.8440) 2 (0.0000)
√2(0.0000) √2(0.0000) √2(0.0000) 2 (0.0000) 2 (0.0000) 2 (3.6340)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(B.4)
Its eigenvalues show that this is a transversely isotropic tensor,
λ1=10.4892 ,λ
2=5.8384 ,λ
3=7.2794 ,λ
4=7.2793 ,λ
5=5.6880 ,λ
6=5.6878.(B.5)
Its Thomsen parameters,
γ=0.1400 ,δ=0.0433 ,=0.1353 ,(B.6)
indicate greater anisotropy than for tensor (70), as expected. In other words, an emphasis of a pattern
of inhomogeneity results in an increase of anisotropy.
References
[1] Backus, G.E.: Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res. 67(11), 4427–4440 (1962)
[2] Gazis, D.C., Tadjbakhsh, I., Toupin, R.A.: The elastic tensor of given symmetry nearest to an anisotropic elastic tensor.
Acta Crystallogr. 16(9), 917–922 (1963)
[3] Slawinski, M.A.: Waves and Rays in Elastic Continua, 3rd edn. World Scientific, Singapore (2015)
[4] B´ona, A., Bucataru, I., Slawinski, M.A.: Space of SO(3)-orbits of elasticity tensors. Arch. Mech. 60(2), 123–138 (2008)
[5] Slawinski, M.A.: Waves and Rays in Seismology: Answers to Unasked Questions, 2nd edn. World Scientific, Singapore
(2018)
[6] B´ona, A., Bucataru, I., Slawinski, M.A.: Coordinate-free characterization of the symmetry classes of elasticity tensors.
J. Elast. 87(2–3), 109–132 (2007)
[7] Haar Measure. https://en.wikipedia.org/wiki/Haar measure. Accessed 19 Apr 2019
[8] Kumar, D.: Applying Backus averaging for deriving seismic anisotropy of a long-wavelength equivalent medium from
well-log data. J. Geophys. Eng. 10(5), 055001 (2013)
[9] Dalton, D.R., Slawinski, M.A.: Numerical examination of commutativity between Backus and Gazis et al. averages.
arXiv (2016)
[10] Bos, L., Dalton, D.R., Slawinski, M.A., Stanoev, T.: On Backus average for generally anisotropic layers. J. Elast. 127(2),
179–196 (2017)
[11] Bos, L., Danek, T., Slawinski, M.A., Stanoev, T.: Statistical and numerical considerations of Backus-average product
approximation. J. Elast. 132(1), 141–159 (2018)
Author's personal copy
ZAMP On the Backus average of a layered medium Page 15 of 15 84
[12] Dewangan, P., Grechka, V.: Inversion of multicomponent, multiazimuth, walkaway VSP data for the stiffness tensor.
Geophysics 68(3), 1022–1031 (2003)
[13] Thomsen, L.: Weak elastic aniostropy. Geophysics 51(10), 1954–1966 (1986)
[14] Adamus, F.P., Slawinski, M.A., Stanoev, T.: On effects of inhomogeneity on anisotropy in Backus average.
arXiv:1802.04075 [physics.geo-ph] (2018)
Len Bos
Dipartimento di Informatica
Universit`adiVerona
Ver o n a
Italy
e-mail: leonardpeter.bos@univr.it
Michael A. Slawinski and Theodore Stanoev
Department of Earth Sciences
Memorial University of Newfoundland
St. John’s
Canada
e-mail: mslawins@mac.com
Theodore Stanoev
e-mail: theodore.stanoev@gmail.com
(Received: October 5, 2018; revised: April 24, 2019)
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