On orthogonal transformations of the Christoffel equations

Abstract

We prove the equivalence—under rotations of distinct terms—of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to \({{\mathbb {R}}}^3\), nor is it limited to the elasticity tensor with its index symmetries. Furthermore, the proposition is valid for orthogonal transformations, not only for rotations. The sought equivalence is a corollary of that proposition.

Full text

GEM - International Journal on Geomathematics
On orthogonal transformations of the Christoffel equations
--Manuscript Draft--
Manuscript Number: IJGE-D-19-00027
Full Title: On orthogonal transformations of the Christoffel equations
Article Type: Original Paper
Corresponding Author: Theodore Stanoev
Memorial University of Newfoundland
CANADA
Corresponding Author Secondary
Information:
Corresponding Author's Institution: Memorial University of Newfoundland
Corresponding Author's Secondary
Institution:
First Author: Len Bos
First Author Secondary Information:
Order of Authors: Len Bos
Michael A. Slawinski
Theodore Stanoev
Maurizio Vianello
Order of Authors Secondary Information:
Funding Information: Natural Sciences and Engineering
Research Council of Canada
(202259)
Dr. Michael A. Slawinski
Abstract: We prove the equivalence---under rotations of distinct terms---of different forms of a
determinantal equation that appears in the studies of wave propagation in Hookean
solids, in the context of the Christoffel equations.To do so, we prove a general
proposition that is not limited to~${\mathbb R}^3$\,, nor is it limited to the elasticity
tensor with its index symmetries.Furthermore, the proposition is valid for orthogonal
transformations, not only for rotations. The sought equivalence is a corollary of that
proposition.
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International Journal on Geomathematics manuscript No.
(will be inserted by the editor)
On orthogonal transformations of the Christoffel equations
Len Bos ·Michael A. Slawinski ·Theodore
Stanoev ·Maurizio Vianello
Received: date / Accepted: date
Abstract We prove the equivalence—under rotations of distinct terms—of different forms
of a determinantal equation that appears in the studies of wave propagation in Hookean
solids, in the context of the Christoffel equations. To do so, we prove a general proposition
that is not limited to R3, nor is it limited to the elasticity tensor with its index symmetries.
Furthermore, the proposition is valid for orthogonal transformations, not only for rotations.
The sought equivalence is a corollary of that proposition.
Keywords Orthogonal transformation ·Christoffel equation ·Tensor algebra ·Algebraic
formulation
1 Introduction
In this paper, we prove analytically a conjecture used in Ivanov and Stovas (2016, 2017,
2019), which is based on numerical considerations for rotations of slowness surfaces in
anisotropic media. The conjecture is essential for their proposed technique of obtaining a
set of mapping operators that establish point-to-point correspondences for traveltimes and
relative-geometric-spreading surfaces between those calculated in nonrotated and rotated
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.
Len Bos
Dipartimento di Informatica, Universit`
a di Verona, Italy
E-mail: leonardpeter.bos@univr.it
Michael A. Slawinski
Department of Earth Sciences, Memorial University of Newfoundland, Canada
E-mail: mslawins@mac.com
Theodore Stanoev
Department of Earth Sciences, Memorial University of Newfoundland, Canada
E-mail: theodore.stanoev@gmail.com
Maurizio Vianello
Dipartimento di Matematica, Politecnico di Milano, Italy
E-mail: maurizio.vianello@polimi.it
Manuscript Click here to download Manuscript BSSV_GEM_2019.tex
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2 Len Bos et al.
media. Thus, this proof results in mathematical rigor to substantiate relevant techniques of
seismological modelling.
The existence and properties of three waves that propagate in a Hookean solid are a
consequence of the Christoffel equations (e.g., Slawinski, 2015, Chapter 9), whose solubility
condition is
det"3
∑
j=1
3
∑
`=1
ci jk`pjp`−δik #=0,i,k=1,2,3,(1)
which is a cubic polynomial equation in p2, whose roots are the eikonal equations (e.g.,
Slawinski, 2015, Section 7.3). The matrix in condition (1),
"3
∑
j=1
3
∑
`=1
ci jk`pjp`#∈R3×3,
is commonly referred to as the Christoffel matrix, where ci jk`is a density-normalized elas-
ticity tensor and pis the wavefront-slowness vector.
Studies of Hookean solids by Ivanov and Stovas (2016, equations (7)–(12)), Ivanov and
Stovas (2017, equations (10), (11)) and Ivanov and Stovas (2019, equations (A.3), (A.5),
(A.6)) invoke a property that we state as Corollary 1, which is a consequence of Proposi-
tion 1. Ivanov and Stovas (2016, 2017, 2019) verify the equivalence of the equations given
in Corollary 1, without a general proof, hence, this paper.
The purpose of this paper is to prove Proposition 1 and, hence, Corollary 1. In doing so,
we gain an insight into a tensor-algebra property that results in this corollary. The equiva-
lence of the aforementioned equations is a consequence of two orthogonal transformations
of ci jk`and pithat result in two matrices that are similar to one another.
2 Proposition and its corollary
Proposition 1 Consider a tensor, ci jk`, in Rd. Also, consider a vector, pi, in Rd, and an
orthogonal transformation, A ∈Rd×d.It follows that matrices
"d
∑
j=1
d
∑
`=1
ci jk`ˆpjˆp`#16i,k6d
∈Rd×d(2)
and "d
∑
j=1
d
∑
`=1
ˇci jk`pjp`#16i,k6d
∈Rd×d,(3)
where t are similar to one another and, consequently, have the same spectra.
Proof The fourth-rank tensor, ci jk`, in Rdcan be viewed as a d×dmatrix, whose entries
are d×dmatrices,
C= [Cik]16i,k6d∈Rd×dd×d,
with Cik ∈Rd×dand (Cik)j`:=ci jk`. Thus, matrix (2) can be written as
ˆptCik ˆp16i,k6d=(A p)tCik (A p)16i,k6d=ptAtCik Ap16i,k6d∈Rd×d,
where tdenotes the transpose.
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On orthogonal transformations of the Christoffel equations 3
We claim that matrix (3) can be written as
AtptAtCik Ap16i,k6dA∈Rd×d.(4)
To see this, we let matrix (4) be
M=AtX A,
where X:= [ pt(AtCik A)p]16i,k6d, to write
Mik =
d
∑
m=1
d
∑
o=1
(At)im Xmo Aok =
d
∑
m=1
d
∑
o=1
Ami Xmo Aok .
Defining Y:=AtCmo A, we have
Xmo =
d
∑
j=1
d
∑
`=1
Yj`pjp`,
where
Yj`=
d
∑
n=1
d
∑
q=1
(At)jn (Cmo)nq Aq`=
d
∑
n=1
d
∑
q=1
An j Aq`cmnoq .
Hence,
Xmo =
d
∑
j=1
d
∑
`=1 d
∑
n=1
d
∑
q=1
An j Aq`cmnoq !pjp`
and, in turn,
Mik =
d
∑
j=1
d
∑
`=1 d
∑
m=1
d
∑
n=1
d
∑
o=1
d
∑
q=1
Ami An j Aok Aq`cmnoq !pjp`=
d
∑
j=1
d
∑
`=1
ˇci jk`pjp`,
i,k=1, ... , d,
which is matrix (3), as required.
Corollary 1 From Proposition 1—and the aforementioned fact that the similar matrices
share the same spectrum, as well as the fact that the similarity of matrices is not affected by
subtracting from them the identity matrices—it follows that
det"3
∑
j=1
3
∑
`=1
ci jk`ˆpjˆp`−δik #=det "3
∑
j=1
3
∑
`=1
ˇci jk`pjp`−δik#,i,k=1,2,3,
and, hence, equations
det"3
∑
j=1
3
∑
`=1
ci jk`ˆpjˆp`−δik #=0,i,k=1,2,3,(5)
and
det"3
∑
j=1
3
∑
`=1
ˇci jk`pjp`−δik#=0,i,k=1,2,3,(6)
are equivalent to one another.
Corollary 1 is valid even without requiring the index symmetries of Hookean solids. Also,
A∈O(3), not only A∈SO(3), which is more general than the property invoked by Ivanov
and Stovas (2016, 2017).
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4 Len Bos et al.
3 Numerical example
Consider an orthotropic tensor (Ivanov and Stovas, 2016, Table 2), whose components are
c1111 =6.3,c2222 =6.9,c3333 =5.4,
c1122 =c2211 =2.7,c1133 =c3311 =2.2,c2233 =c3322 =2.4,
c1212 =c2112 =c2121 =c1221 =1.5,c1313 =c3113 =c3131 =c1331 =0.8,
c2323 =c3223 =c3232 =c2332 =1.0.
Also, consider vector p=h0,0,q1
c3333 i. Rotating this vector by
A=

1 0 0
0 cosθ−sinθ
0 sinθcosθ
,(7)
with an arbitrary angle of θ=π/5 , and the tensor by At, we obtain
ˆ
Γ:=
3
∑
j=1
3
∑
`=1
ci jk`ˆpjˆp`=

0.192934 0 0
0 0.331749 −0.018424
0−0.018424 0.949406 

and
ˇ
Γ:=
3
∑
j=1
3
∑
`=1
ˇci jk`pjp`=

0.192934 0 0
0 0.562667 −0.299407
0−0.299407 0.718488 
,
respectively. The eigenvalues of these matrices are the same, λ1=0.949955 , λ2=0.33120
and λ3=0.192934 , as required for similar matrices. Their corresponding eigenvectors are
related by transformation (7).
Herein, det[ˆ
Γ−I] = −0.027013 =det[ˇ
Γ−I]. In general, the two determinants are equal
to one another. Hence, if det[ˆ
Γ−I] = 0 , so does det[ˇ
Γ−I], and vice versa.
The equivalence of equations (5) and (6) does not imply their equivalence to
det"3
∑
j=1
3
∑
`=1
ci jk`pjp`−δik #=0,i,k=1,2,3.
The eigenvalues of
Γ:=
3
∑
j=1
3
∑
`=1
ci jk`pjp`=

0.148148 0 0
0 0.185185 0
0 0 1 

are λ1=0.148148 , λ2=0.185185 and λ3=1 , which are distinct from the eigenvalues
of ˆ
Γand ˇ
Γ. Herein—in view of pand ci jk `representing, respectively, the slowness vector
along the x3-axis and its corresponding elasticity tensor—det[Γ−I] = 0 , which results in
the eikonal equations. We emphasize, however, that Proposition 1 and Corollary 1 are valid
for arbitrary vectors and fourth-rank tensors, even though, in this example, they are related
by the Christoffel equations.
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On orthogonal transformations of the Christoffel equations 5
4 Conclusion
The corollary proven in this article is necessary to address problems of quantitative seis-
mology (Ivanov and Stovas, 2016, 2017, 2019). As shown in the proof of the proposition, it
relies on certain intricacies of tensor algebra, which, intrinsically, do not exhibit any physical
meaning but are pertinent to mathematical operations in examining geophysical concepts.
In summary, the proof contributes the bridging of geoscience and mathematics through an
examination of anisotropy, within the context of elasticity theory, for techniques in seismo-
logical modelling.
Acknowledgements The authors wish to acknowledge Yuriy Ivanov for presenting and clarifying aspects
of this problem, Igor Ravve, an editor for Geophysics, for encouraging the submission of the proof to peer
review, Sandra Forte for fruitful discussions, and David Dalton for insightful proofreading.
Conflict of interest
The authors declare that they have no conflict of interest.
References
Ivanov Y, Stovas A (2016) Normal moveout velocity ellipse in tilted orthorhombic media.
Geophysics 81(6):319–336
Ivanov Y, Stovas A (2017) Traveltime parameters in tilted orthorhombic medium. Geo-
physics 82(6):187–200
Ivanov Y, Stovas A (2019) 3D mapping of kinematic attributes in anisotropic media. Geo-
physics 84(3):C159–C170
Slawinski MA (2015) Waves and rays in elastic continua, 3rd edn. World Scientific
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