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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.
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,k6dRd×dd×d,
with Cik Rd×dand (Cik)j`:=ci jk`. Thus, matrix (2) can be written as
ˆptCik ˆp16i,k6d=(A p)tCik (A p)16i,k6d=ptAtCik Ap16i,k6dRd×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
AtptAtCik Ap16i,k6dARd×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,
AO(3), not only ASO(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 cosθsinθ
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
00.018424 0.949406
and
ˇ
Γ:=
3
j=1
3
`=1
ˇci jk`pjp`=
0.192934 0 0
0 0.562667 0.299407
00.299407 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
ΓI] = 0.027013 =det[ˇ
ΓI]. In general, the two determinants are equal
to one another. Hence, if det[ˆ
ΓI] = 0 , so does det[ˇ
ΓI], 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[ΓI] = 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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"The author dedicates this book to readers who are concerned with finding out the status of concepts, statements and hypotheses, and with clarifying and rearranging them in a logical order. It is thus not intended to teach tools and techniques of the trade, but to discuss the foundations on which seismology — and in a larger sense, the theory of wave propagation in solids — is built. A key question is: why and to what degree can a theory developed for an elastic continuum be used to investigate the propagation of waves in the Earth, which is neither a continuum nor fully elastic. But the scrutiny of the foundations goes much deeper: material symmetry, effective tensors, equivalent media; the influence (or, rather, the lack thereof) of gravitational and thermal effects and the rotation of the Earth, are discussed ab initio. The variational principles of Fermat and Hamilton and their consequences for the propagation of elastic waves, causality, Noether's theorem and its consequences on conservation of energy and conservation of linear momentum are but a few topics that are investigated in the process to establish seismology as a science and to investigate its relation to subjects like realism and empiricism in natural sciences, to the nature of explanations and predictions, and to experimental verification and refutation. A study of this book will help the reader to firmly put seismology in the larger context of mathematical physics and to understand its range of validity. I would not like to miss it from my bookshelf." Klaus Helbig
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The present book — which is the second, and significantly extended, edition of the textbook originally published by Elsevier Science — emphasizes the interdependence of mathematical formulation and physical meaning in the description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray theory to explain phenomena resulting from the propagation of seismic waves. The book is divided into three main sections: Elastic Continua, Waves and Rays and Variational Formulation of Rays. There is also a fourth part, which consists of appendices. In Elastic Continua, we use continuum mechanics to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such a material. In Waves and Rays, we use these equations to identify the types of body waves propagating in elastic continua as well as to express their velocities and displacements in terms of the properties of these continua. To solve the equations of motion in anisotropic inhomogeneous continua, we invoke the concept of a ray. In Variational Formulation of Rays, we show that, in elastic continua, a ray is tantamount to a trajectory along which a seismic signal propagates in accordance with the variational principle of stationary traveltime. Consequently, many seismic problems in elastic continua can be conveniently formulated and solved using the calculus of variations. In the Appendices, we describe two mathematical concepts that are used in the book; namely, homogeneity of a function and Legendre's transformation. This section also contains a list of symbols.
Article
Based on the rotation of a slowness surface in anisotropic media, we have derived a set of mapping operators that establishes a point-to-point correspondence for the traveltime and relative-geometric-spreading surfaces between these calculated in nonrotated and rotated media. The mapping approach allows one to efficiently obtain the aforementioned surfaces in a rotated anisotropic medium from precomputed surfaces in the nonrotated medium. The process consists of two steps: calculation of a necessary kinematic attribute in a nonrotated, e.g., orthorhombic (ORT), medium, and subsequent mapping of the obtained values to a transformed, e.g., rotated ORT, medium. The operators we obtained are applicable to anisotropic media of any type; they are 3D and are expressed through a general form of the transformation matrix. The mapping equations can be used to develop moveout and relative-geometric-spreading approximations in rotated anisotropic media from existing approximations in nonrotated media. Although our operators are derived in case of a homogeneous medium and for a one-way propagation only, we discuss their extension to vertically heterogeneous media and to reflected (and converted) waves.
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Traveltime parameters, defined through the series coefficients of the traveltime squared as a function of the horizontal offset projections, play an important role in moveout approximations and corrections, and in model parameter inversion. We evaluate an approach to derive the traveltime parameters in a single homogeneous anisotropic layer of tilted orthorhombic symmetry for one- and two-way traveling waves. The approach allows us to obtain the traveltime parameters of pure and converted modes. We use numerical models to illustrate the dependence of the high-order traveltime parameters on the Euler angles and the anisotropy parameters. The traveltime parameter inversion is a strongly ill-posed problem in anisotropic media, and improvements due to inclusion of the high-order traveltime parameters can sufficiently reduce the space of equivalent kinematic models. We perform a numerical model parameter inversion using the concept of artificial neural networks to demonstrate the accuracy improvements due to inclusion of the highorder traveltime parameters over the inversion of the second- order coefficient, conventionally known as normal moveout velocity, only. We demonstrate algebraically and numerically that the presented approach to calculate the traveltime parameters is easily extended to multilayered media. It can be used for Dix-type inversion to obtain the interval medium parameters.
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Article
The present book-which is the third, significantly revised edition of the textbook originally published by Elsevier Science-emphasizes the interdependence of mathematical formulation and physical meaning in the description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray theory to explain phenomena resulting from the propagation of seismic waves. The book is divided into three main sections: Elastic Continua, Waves and Rays and Variational Formulation of Rays. There is also a fourth part, which consists of appendices. In Elastic Continua, we use continuum mechanics to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such a material. In Waves and Rays, we use these equations to identify the types of body waves propagating in elastic continua as well as to express their velocities and displacements in terms of the properties of these continua. To solve the equations of motion in anisotropic inhomogeneous continua, we invoke the concept of a ray. In Variational Formulation of Rays, we show that, in elastic continua, a ray is tantamount to a trajectory along which a seismic signal propagates in accordance with the variational principle of stationary traveltime. Consequently, many seismic problems in elastic continua can be conveniently formulated and solved using the calculus of variations. In the Appendices, we describe two mathematical concepts that are used in the book; namely, homogeneity of a function and Legendre’s transformation. This section also contains a list of symbols. © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
Chapter
The seismic ray method plays an important role in seismology, seismic exploration, and in the interpretation of seismic measurements. Seismic Ray Theory presents the most comprehensive treatment of the method available. Many new concepts that extend the possibilities and increase the method's efficiency are included. The book has a tutorial character: derivations start with a relatively simple problem, in which the main ideas are easier to explain, and then advance to more complex problems. Most of the derived equations are expressed in algorithmic form and may be used directly for computer programming. This book will prove to be an invaluable advanced text and reference in all academic institutions in which seismology is taught or researched.