Generalized orthogonality between rays and wavefronts in anisotropic inhomogeneous media

Abstract

We prove that in elastic anisotropic inhomogeneous media, rays and wavefronts are orthogonal to each other with respect to the metric induced by the phase-velocity function. The standard orthogonality of rays and wavefronts in elastic isotropic inhomogeneous media is a special case of this formulation.

Full text

Generalized orthogonality between rays and
wavefronts in anisotropic inhomogeneous
media
Ioan Bucataru∗Michael A. Slawinski†
Abstract
We prove that in elastic anisotropic inhomogeneous media, rays
and wavefronts are orthogonal to each other with respect to the metric
induced by the phase-velocity function. The standard orthogonality
of rays and wavefronts in elastic isotropic inhomogeneous media is a
special case of this formulation.
1 Introduction
It has been shown in several papers, e.g., Antonelli et al (2003), Bona and
Slawinski (2002) and Antonelli et al (2002) that Finsler geometry provides
a fruitful platform for the study of seismic ray theory in anisotropic inho-
mogeneous media. In this paper, we show that the Finsler metric, discussed
in the above papers, provides a natural context for the study of rays and
wavefronts in such media.
In general, in anisotropic media, rays and wavefronts are not orthogonal
to each other in the sense of Euclidean geometry. However, in this paper we
prove that rays and wavefronts are orthogonal to each other in the sense of
the geometry imposed by the properties of the medium, which are stated in
∗Dept. of Earth Sciences, Memorial University, St. John’s, Newfoundland A1B 3X7,
Canada. E-mail: nbucataru@esd.mun.ca, and Faculty of Mathematics, “Al.I.Cuza” Uni-
versity, Ia¸si, Romania. E-mail: bucataru@uaic.ro
†Dept. of Earth Sciences, Memorial University, St. John’s, Newfoundland A1B 3X7,
Canada. E-mail: mslawins@mun.ca
1

the context of a given phase-velocity function. Since this imposed geometry is
different from the Euclidean geometry we need to introduce its fundamental
concepts in the context of differential geometry.
We begin this paper by introducing the differential-geometry concepts
that are pertinent to ray theory in anisotropic inhomogeneous media. Therein,
we state definitions and equations that we need to investigate rays and wave-
fronts. Subsequently, we use our ray-theory Hamiltonian to obtain a general-
ized metric. Then, we show the generalized orthogonality between rays and
wavefronts in anisotropic inhomogeneous media. We conclude with a dis-
cussion of several related ray-theory properties. Orthogonality of geodesics
(rays) and geodesic balls (wavefronts) in the context of Finsler geometry is
known as Gauss Lemma, see Bao et al (2000). The proof we give here is dif-
ferent and uses Hamilton equations of rays. Moreover we prefer to work on
the cotangent space of a manifold rather then on the tangent space because
the eikonal equations are defined there and the Hamiltonian metric may not
be positive definite everywhere. This implies that the Legendre transforma-
tion may not be well defined somewhere and therefore we cannot use Euler
Lagrange equations to give the usual proof for the orthogonality.
2 Geometrical background
2.1 Geometric space
To obtain the results of this paper, in this section we define several necessary
entities and the spaces to which they belong.
The geometry of an elastic medium is the geometry of a triple (M, ρ, c),
where Mis a three-dimensional space, ρis the mass-density function, and
cis a fourth-rank tensor with particular properties. We note that — in a
mathematical context — Mis a manifold.
To study rays and wavefronts in the context of differential geometry, we
identify the physical space, M, of the elastic medium with an open subset of
the Euclidean three-dimensional space, where the coordinates are given by
functions x1,x2,x3. At every point xof M, we consider the differential of
functions ψat this point, namely,
dψ|x=
3
X
i=1
∂ψ
∂xidxi. (1)
2

The set of the differentials at xof all functions on Mis a three-dimensional
vector space. We denote this space by T∗
xM, while we denote the collection
of T∗
xMat all points xby T∗M. We note that T∗Mis often called the
momentum space in classical mechanics, the phase space in Hamiltonian
mechanics and corresponds to the cotangent space in differential geometry.
Our geometry must be associated with T∗Msince the key entities that are
discussed below — such as the phase slowness, the Hamiltonian, Hamilton’s
equations and the Hamiltonian metric — are defined in this space.
Each element pof T∗
xMcan be expressed as p=p1dx1+p2dx2+p3dx3,
where — in view of expression (1) — we have
pi:= ∂ψ
∂xi,i∈ {1,2,3}. (2)
Then, the coordinates of T∗Mare denoted by (xi, pi), where i= 1,2,3.
Symbol := emphasizes that expression (2) is a definition not an equation.
Since the entities that we are about to define are, in general, not differ-
entiable at p= 0, we remove (x,0) from T∗M. We denote the remaining
space by g
T∗M. Furthermore, as these entities are homogeneous in p, if we
do not remove (x,0), these entities would be reduced to special cases that
would limit the generality of our subsequent formulation.
To give a seismological motivation, we note that expression (2) plays an
important role in ray theory. If we consider a particular function ψ(x), we
can write a moving wavefront as ψ(x) = t, where tdenotes time. At a given
instant, wavefronts are level sets of this particular function. Hence, in view
of expression (2) — in this case — pis normal to the wavefront. Since —
in this case — units of the piare the units of slowness, we may refer to p
as the phase-slowness vector. We note that in this context, the removal of
p= 0 has an immediate physical meaning since we remove the case that
would correspond to phase slowness that is equal to zero.
At this point, we also wish to emphasize the distinction between the
formulation of various entities in T∗Mand their meaning along particular
curves in this space that correspond to rays and wavefronts in ray theory.
Rigorous treatment of ray theory cannot be performed without global defini-
tions. In other words, while we are interested only in the properties of rays
and wavefronts, to study these properties, we require global definitions for
our Hamiltonian and its related entities.
Let us now return to triple (M, ρ, c). At each point xof M, the elasticity
tensor is a fourth-rank tensor whose components in our coordinate system
3

are cijkl (x), where i, j, k, l ∈ {1,2,3}. Using standard methods for solving
equations of motion in anisotropic inhomogeneous media, which employ an
asymptotic trial solution (e.g., B´ona and Slawinski, 2003), we obtain
£Γij (x,p)−δij ¤Aj(x) = 0,i, j ∈ {1,2,3}, (3)
where A(x) is the wavefront amplitude and Γ(x,p) is the matrix whose
entries are
Γij (x,p) =
3
X
k=1
3
X
l=1
cikjl (x)
ρ(x)pkpl,i, j ∈ {1,2,3}, (4)
with ρ(x) being mass density. Matrix Γij (x,p) is called Christoffel’s matrix.
Due to the properties of the elasticity tensor, cijk l (x), Christoffel’s matrix
is symmetric and positive-definite. Consequently, the three eigenvalues of
this matrix are real and positive. We denote them by Gα(x,p), where α∈
{1,2,3}. It can be shown that the eigenvalues, Gα, are homogeneous of degree
2 in p(e.g., ˇ
Cerven´y, 2001, p. 22). In other words, Gα(x, λp) = λ2Gα(x,p),
for any real number λ. Also, functions Gαare differentiable on g
T∗M. We
note that — in view of the homogeneity of Gα— including p= 0 in the
domain of differentiability would have limited Gαto a quadratic function
in p. Consequently, our subsequent formulation would have been limited to
elliptical anisotropy.
We wish to consider a given eigenvalue, Gα. Let us refer to it as Gand,
for convenience, let
H(x,p) := 1
2G(x,p), (5)
to which we shall refer as the ray-theory Hamiltonian. In view of the prop-
erties of G, we see that His also homogeneous of degree 2 in p. Since His
homogeneous of degree 2 in p, by factoring p2=p·p, we rewrite Hamiltonian
(5) as
H(x,p) = 1
2p2v2(x,p) , (6)
where, as we immediately see,
v2(x,p) := 2H(x,p)
p2. (7)
4

Function vis homogeneous of degree 0 in p. In other words, v(xi, λpi) =
v(xi, pi). This property means that the value of vdepends on direction of p
but not on the magnitude of p.
Expression (6) is the form of our Hamiltonian to be used in the remainder
of this paper. We note that our Hamiltonian is a product of two functions,
namely, p2and v2(x,p). The former function accounts for the homogeneity
of H, while the latter one, which is homogeneous of degree 0 in p, contains
all the information about the medium. For instance, if vdoes not depend on
p, the medium is isotropic.
Having defined the phase-slowness vector, p, function v, and the ray-
theory Hamiltonian, H, which are all associated with T∗M, we are now
ready to formulate the equations that will allow us to discuss wavefronts and
rays. At this point, we will focus on level sets on T∗Mthat correspond to
the wavefronts.
2.2 Wavefront and ray equations
To discuss the wavefronts, we invoke equation (3) and study the eigenvalues
that are associated with this equation. For nontrivial solvability of equation
(3), we require
det £Γij (x,p)−δij ¤= 0, (8)
which is an eigenvalue equation. Hence — in the context of equation (3) —
each of the three corresponding eigenvalues of matrix (4) gives us an equation,
which can be written as
G(x,p) = 1. (9)
This is an eikonal equation allowing us to describe the wavefronts. Following
expressions (5) and (6), we can restate eikonal equation (9) as
p2=1
v2(x,p). (10)
In view of equation (10) and since wavefronts are loci of constant phase, we
refer to v, defined in expression (7), as the phase-velocity function. Equations
(9) and (10) are valid for anisotropic inhomogeneous media since, considering
wavefronts and their normals, we see that xrefers to the dependence on
position and prefers to the dependence on direction.
5

We wish to explicitly state the eikonal equation as a differential equation.
In view of expression (2), we can write equation (10) as
3
X
i=1 µ∂ψ
∂xi¶2
=1
v2µxi,∂ψ
∂xi¶. (11)
This is a first-order nonlinear partial differential equation for function ψ. To
state equation (11) in terms of our Hamiltonian, H, in view of expression
(6), we can write
2H(x,p)≡2Hµxi,∂ψ
∂xi¶= 1, (12)
which is akin to equation (9) and is a standard form of the Hamilton-Jacobi
equation. The solution of equation (12) is function ψ, whose level sets are
the wavefronts.
Solving the eikonal equation by the method of characteristics (e.g., Courant
and Hilbert, 1989), we obtain Hamilton’s ray equations, namely,









dxi
dt=∂H
∂pi
dpi
dt=−∂H
∂xi
,i∈ {1,2,3}. (13)
Rays are curves whose components, [x1(t), x2(t), x3(t)], are solutions of
system (13).
Having formulated the equations describing wavefronts and rays, we are
now ready to study their geometrical relation, which is a function of a metric
that characterizes a given geometry. Thus, we begin by formulating pertinent
metrics.
2.3 Hamiltonian metric
In general, a given elastic medium exhibits two metric structures that are of
interest in our study. They are the Euclidean metric δij , where δij is Kro-
necker’s delta, and metric gij(x,p), to which we refer to as the Hamiltonian
metric. The Euclidean metric is intrinsically associated with the physical
6

space, while the Hamiltonian metric is induced by the phase-velocity func-
tion that corresponds to a given type of waves that propagate in the medium.
Since, in general, there are three distinct phase-velocity functions, which cor-
respond to the three types of waves that propagate in an anisotropic inhomo-
geneous medium, at every point of the medium there are three Hamiltonian
metrics.
Now, we wish to obtain the Hamiltonian metric that corresponds to the
properties of a given medium.
Consider the ray-theory Hamiltonian given by expression (6). Using this
Hamiltonian, we write a convenient metric (e.g., Rund, 1959) whose compo-
nents are given by
gij (x,p) := ∂2H
∂pi∂pj
(x,p) , i, j ∈ {1,2,3}. (14)
Inserting expression (6) into expression (14), we can obtain an explicit ex-
pression for the components of this metric, namely,
gij =v2δij + 2vµpi∂v
∂pj
+pj∂v
∂pi¶+p2µv∂2v
∂pi∂pj
+∂v
∂pi
∂v
∂pj¶,i, j ∈ {1,2,3}.
(15)
Thus, our Hamiltonian metric naturally emerges from the properties of a
given medium, which are contained in the phase-velocity function, v.
We require that Hbe regular, which means that the matrix with entries
given in expression (14) be nondegenerate on g
T∗M. In other words, det [gij ]6=
0. We note that, physically, det [gij ] = 0 corresponds to the inflection points
of a phase-slowness surface. In this paper, however, we do not study these
singular points. In view of expression (6), the assumption of differentiability
of His equivalent to the assumption of differentiability of v.
Examining expression (14), we observe the following properties of the
Hamiltonian metric (e.g., Miron, et al., 2001). Since His homogeneous of
degree 2 in the pi, it follows that the components of the Hamiltonian metric
are homogeneous of degree 0 in the pi. Furthermore, since det [gij ]6= 0, we
also have the inverse of the Hamiltonian metric, which we denote by gij (x,p);
in other words,
3
X
j=1
gij (x,p)gjk (x,p) = δk
i,i, k ∈ {1,2,3}. (16)
7

Examining expression (15), we recognize that we can also view this expres-
sion as the relation between the Hamiltonian metric, gij, and the Euclidean
metric, δij . We note that for an isotropic medium, where vis independent
of p, equation (15) reduces to gij =v2δij . This means that in an isotropic
medium the two metrics differ by a multiplicative scalar factor; in other
words, they are conformal to one another. This also justifies our choice of
metric (14).
Having formulated the Hamiltonian metric, we are now ready to complete
our study of a geometrical relation between rays and wavefronts, namely,
their orthogonality.
3 Orthogonality between rays and wavefronts
3.1 Euclidean and Hamiltonian gradients
To discuss orthogonality, we use the fact that a gradient of a function with
respect to a given metric is a vector that is orthogonal to the level sets of
this function with respect to this metric. For function ψon M, we can define
its gradient as being either the vector whose components are
(∇ψ)i=
3
X
j=1
δij ∂ψ
∂xj,i∈ {1,2,3}, (17)
with respect to the Euclidean metric, or as the vector whose components are
(∇gψ)i=
3
X
j=1
gij ∂ψ
∂xj,i∈ {1,2,3}, (18)
with respect to the Hamiltonian metric (e.g., Anastasiei, 1998). In other
words, expression (17) defines the Euclidean gradient, while expression (18)
defines the Hamiltonian gradient. Consequently, ∇ψis orthogonal to the
level sets of ψwith respect to the Euclidean metric, while ∇gψis orthogonal
to the level sets of ψwith respect to the Hamiltonian metric. For conciseness
of terminology, when dealing with the latter case, we will refer to it as the
Hamilton-orthogonality. In the next section, we show that ∇gψis Hamilton-
orthogonal to the level sets of ψ. Before showing this orthogonality, we wish
to show that gradients ∇ψand ∇gψare distinct from one another.
8

In expression (15), we obtained the relation between the Euclidean and
Hamiltonian metrics. Using this relation we can also derive an analytical
expression relating the components of the two corresponding gradients, as
follows. In view of expressions (15), (17) and (18), we can write
(∇gψ)i=v2(∇ψ)i+vp2∂v
∂pi
,i∈ {1,2,3}. (19)
Thus, in general, the two gradients are distinct from one another.
3.2 Hamilton-orthogonality of ∇gψand wavefronts
Herein, we rigorously show that vector ∇gψis Hamilton-orthogonal to the
wavefronts. This is equivalent to saying that ∇gψis Hamilton-orthogonal to
any curve that belongs to a level set of ψ.
Consider such a curve described by our coordinate functions as
ψ(x(t)) = C, (20)
where Cdenotes a constant. Taking the derivative of both sides of equation
(20) with respect to t, we obtain
3
X
i=1
∂ψ
∂xi
dxi
dt= 0. (21)
To state expression (21) in terms of the Hamiltonian gradient, using ex-
pression (16), we can rewrite expression (21) as
3
X
i=1
3
X
j=1
3
X
k=1
gji gjk ∂ψ
∂xk
dxi
dt= 0. (22)
Recognizing that P3
k=1 gjk ∂ψ/∂xkis expression (18), we can write expression
(22) as
3
X
i=1
3
X
j=1
gji (∇gψ)jdxi
dt= 0. (23)
Equation (23) states that the scalar product — with respect to the Hamil-
tonian metric — of the Hamiltonian gradient and the vector tangent to the
wavefront vanishes. Thus, the Hamiltonian gradient is Hamilton-orthogonal
to the wavefront.
9

3.3 Hamilton-orthogonality of rays and wavefronts
To show the Hamilton-orthogonality of rays and wavefronts, it now suffices
to show that the vector tangent to the ray coincides with the Hamiltonian-
gradient vector ∇gψ.
Consider a ray described in our coordinates as a curve given by x(t).
The vector tangent to this ray can be written as dx/dt. In view of the first
equation of system (13), we can write the components of dx/dtas
dxi
dt=∂H
∂pi
,i∈ {1,2,3}. (24)
We want to show that ∂H/∂piare the components of the Hamiltonian gra-
dient of ψ.
Since His homogeneous of degree 2 in the pi, by Euler’s homogeneous-
function theorem, we can write
3
X
i=1
∂H
∂pi
pi= 2H. (25)
Furthermore, since His homogeneous of degree 2 in the pi, it follows that
∂H/∂piis homogeneous of degree 1 in the pi. Following Euler’s homogeneous-
function theorem, we obtain
3
X
j=1
∂
∂pjµ∂H
∂pi¶pj=∂H
∂pi
,i∈ {1,2,3}. (26)
Multiplying both sides of equation (26) by piand summing over i, we get
3
X
i=1
3
X
j=1
∂2H
∂pi∂pj
pipj=
3
X
i=1
∂H
∂pi
pi. (27)
Following expressions (14) and (25), we can rewrite equation (27) as
2H(x,p) =
3
X
i=1
3
X
j=1
gij (x,p)pipj. (28)
Taking partial derivatives of equation (28) with respect to pi, we obtain
2∂H
∂pi
= 2
3
X
j=1
gij pj+
3
X
j=1
3
X
k=1
∂gkj
∂pi
pkpj,i∈ {1,2,3}. (29)
10

Now, we will show that the double summation is identically zero. Recall-
ing expression (14), we can write
∂gkj
∂pi
=1
2
∂3H
∂pi∂pk∂pj
=∂gij
∂pk
,i, j, k ∈ {1,2,3}, (30)
where the second equality results from the equality of the mixed partial
derivatives. Using expression (30), we can rewrite the double summation as
3
X
j=1
3
X
k=1
∂gkj
∂pi
pkpj=
3
X
j=1 Ã3
X
k=1
∂gij
∂pk
pk!pj,i∈ {1,2,3}. (31)
Since gij is homogeneous of degree 0 in the pi, by Euler’s homogeneous-
function theorem, the term in parentheses vanishes. Consequently, the double
summation is identically zero and expression (29) is reduced to
∂H
∂pi
=
3
X
j=1
gij pj,i∈ {1,2,3}. (32)
Using expression (32), we can now write equation (24) as
dxi
dt=
3
X
j=1
gij pj,i∈ {1,2,3}. (33)
In view of expressions (2) and (18), we can rewrite equation (33) as
dxi
dt= (∇gψ)i,i∈ {1,2,3}. (34)
Equation (34) states that the vector tangent to the ray coincides with the
Hamiltonian gradient of ψ, whose level sets are the wavefronts. Hence, the
proof of the statement that rays and wavefronts are Hamilton-orthogonal to
each other is complete.
4 Discussion and conclusions
This paper proves the Hamilton-orthogonality between rays and wavefronts
in anisotropic inhomogeneous media. In other words, the Euclidean orthog-
onality associated with isotropic media is extended to the anisotropic ones.
11

This can be explained in the following way. The Hamiltonian metric, which
is derived from the angle-dependent phase velocity, contains this angular de-
pendence. This means that the anisotropy has been accounted for by the
metric itself.
In addition, this study allows us to illustrate the following ray-theory
properties.
We recall that the Hamiltonian metric is expressed in terms of the phase-
velocity function, as shown in expression (15). It is also possible to obtain
the phase-velocity function from the Hamiltonian metric alone. To see this,
consider expression (28). Dividing both sides of equation (28) by p2and in
view of expression (6), we obtain
v2(x,p) =
3
X
i=1
3
X
j=1
gij (x,p)pi
p
pj
p=
3
X
i=1
3
X
j=1
gij (x,n)ninj. (35)
In expression (35), p=√p·pand ni=pi/p.
We note that the fact that rays and wavefronts are Hamilton-orthogonal
to each other means that each elementary wavefront generated by a point
source is a unit ball of the geometry induced by the phase-velocity function.
In other words — in the context of the Hamiltonian metric — each elementary
wavefront can be viewed as a sphere with unit radius. To see this, we can
use expression (28) to rewrite Hamilton-Jacobi equation (12) as
3
X
i=1
3
X
j=1
gij µxk,∂ψ
∂xk¶∂ψ
∂xi
∂ψ
∂xj= 1. (36)
Equation (36) can be expressed in terms of the Hamiltonian gradient, as fol-
lows. The squared magnitude of the Hamiltonian gradient ∇gψwith respect
to the Hamiltonian metric is
(∇gψ)2=
3
X
i=1
3
X
j=1
3
X
k=1
3
X
l=1
gklgki ∂ψ
∂xiglj ∂ψ
∂xj=
3
X
i=1
3
X
j=1
gij ∂ψ
∂xi
∂ψ
∂xj. (37)
In view of expression (37), we can concisely write equation (36) as
(∇gψ)2= 1, (38)
which is a unit ball. Notably, this is the indicatrix of the Finsler geometry
(e.g., Bao et al., 2000), and we see that the geodesics are orthogonal to
indicatrices.
12

The results we have shown include, in particular, the case of isotropy. If
a medium is isotropic, then the phase-velocity function depends on position
only, namely, v=v(x). In such a case, expression (19) can be written as
∇gψ=v2∇ψ. (39)
Also, in such a case, expression (15) reduces to
gij =v2δij ,i, j ∈ {1,2,3}, (40)
which means that, for isotropy, the Hamiltonian and Euclidean metrics are
conformal to one another. From equations (39) and (40), we see that, in
isotropic media, the Hamilton-orthogonality reduces to the standard Eu-
clidean orthogonality.
Acknowledgements
We would like to acknowledge the valuable editorial work of Cathy Beveridge,
the fruitful collaboration with Drs. Andrej B´ona and Michael Rochester.
This study has been done in the context of The Geomechanics Project.
Hence, the authors wish to acknowledge the support of EnCana Energy,
Husky Energy and Talisman Energy.
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