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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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84 Page 4 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP

=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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ZAMP On the Backus average of a layered medium Page 7 of 15 84

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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ZAMP On the Backus average of a layered medium Page 9 of 15 84

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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84 Page 10 of 15 L. Bos, M. A. Slawinski, and T. Stanoev ZAMP

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)

Author's personal copy

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)

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[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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