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On Christoffel roots for nondetached slowness surfaces
Len Bos∗
, Michael A. Slawinski†
, Theodore Stanoev‡
Abstract
The only restriction on the values of the elasticity parameters is the stability condition.
Within this condition, we examine Christoffel equation for nondetached qP slowness surfaces in
transversely isotropic media. If the qP slowness surface is detached, each root of the solubility
condition corresponds to a distinct smooth wavefront. If the qP slowness surface is nondetached,
the roots are elliptical but do not correspond to distinct wavefronts; also, the qP and qSV
slowness surfaces are not smooth.
1 Introduction
Since the studies of Rudzki (1911)1, characterizing shapes of wavefronts in anisotropic media has
been of interest to seismologists. Postma (1955) derived a condition for elliptical velocity dependence
in homogeneous transversely isotropic media that is equivalent to alternating isotropic layers. This
condition was generalized by Berryman (1979) for “any horizontally stratified, homogeneous material
whose constituent layers are isotropic.” The proof for nonexistence of ellipticity of qP wavefronts
in media resulting from lamellation came from Helbig (1979), in response to Levin (1978). Shortly
thereafter, Helbig (1983, p. 826) stated the following. (1) The wavefront of qP waves is never an
ellipsoid; (2) the wavefront of qSV waves is never an ellipsoid; (3) the wavefront of SH waves is
always an oblate ellipsoid. Lamellation, which is described by Helbig (1979, 1983) as fine layering
on a scale small compared with the wavelength, is tantamount to using the Backus (1962) average;
throughout this paper, we use the methodology of the latter.
We consider the three roots of the solubility condition of the Christoffel equation to which we refer
as Christoffel roots. These roots correspond to the wavefront-slowness surfaces of the three waves
that propagate in an anisotropic Hookean solid. Herein, we examine transversely isotropic media
that results from the Backus average of isotropic layers. We derive the conditions under which the
spherical-coordinate plots of the three roots are ellipsoidal; we refer to such roots as elliptical. In
accordance with polar reciprocity, the ellipticity of wavefront slownesses is equivalent to ellipticity
of wavefronts.
As it turns out, a necessary condition for the ellipticity of roots is the nondetachment of the qP slow-
ness surface. Although the Hookean solids that represent most materials encountered in seismology
exhibit a detached qP slowness surface, the existence of both detached and nondetached slowness
surfaces is, indeed, permissible within the stability condition of the elasticity tensor (Bucataru and
Slawinski, 2009). Mathematically, this condition is the positive definiteness of the elasticity tensor.
∗Dipartimento di Informatica, Universit`a di Verona, Italy, leonardpeter.bos@univr.it
†Department of Earth Sciences, Memorial University of Newfoundland, mslawins@mac.com
‡Department of Earth Sciences, Memorial University of Newfoundland, theodore.stanoev@gmail.com
1This publication, which was presented to the Academy of Sciences at Cracow in 1911, has been translated with
comments by Klaus Helbig and Michael A. Slawinski; it appears as Rudzki (2003).
1
arXiv:1903.02514v1 [physics.geo-ph] 6 Mar 2019
2 Christoffel equation in Backus media
The existence of waves in anisotropic media is governed by the Christoffel equation; its solubility
condition is (e.g., Slawinski, 2015, Section 7.3)
det
3
X
j=1
3
X
`=1
cijk` pjp`−δik
= 0 , i, k = 1 ,2,3,
where cijk` is a density-scaled elasticity tensor and pis the wavefront-slowness vector. The three
roots of this bicubic equation can be stated as the expressions for the wavefront speeds of the qP ,
qSV and SH waves.
Let us consider a homogeneous transversely isotropic medium, whose elasticity parameters are
cTI =
cTI
1111 cTI
1122 cTI
1133 0 0 0
cTI
1122 cTI
1111 cTI
1133 0 0 0
cTI
1133 cTI
1133 cTI
3333 0 0 0
0 0 0 2 cTI
2323 0 0
0 0 0 0 2cTI
2323 0
0 0 0 0 0 cTI
1111 −cTI
1122
.(1)
Herein, superscript TI indicates transverse isotropy resulting from the Backus (1962) average. Using
n3= sin2ϑto express the wavefront orientation, we parameterize the expressions of the three
roots (e.g., Slawinski, 2015, equation (9.2.19), (9.2.20)) as
vqP,qS V =v
u
u
tcTI
3333 −cTI
1111(1 −n3) + cTI
1111 +cTI
2323 ±√∆
2ρ(2)
and
vSH =scTI
1212 n3+cTI
2323 (1 −n3)
ρ,(3)
where ∆ = a(n3)2+b n3+c, with
a=cTI
1111 + 2 cTI
1133 +cTI
3333cTI
1111 −2cTI
1133 −4cTI
2323 +cTI
3333,(4a)
b= 2 cTI
1111 cTI
2323 −2cTI
1111 cTI
3333 + 4 cTI
11332
+ 8 cTI
1133 cTI
2323 + 6 cTI
2323 cTI
3333 + 2 cTI
33332
,(4b)
c=cTI
2323 −cTI
33332
.(4c)
The reciprocal of root (3) is elliptical. The reciprocals of roots (2) are elliptical if and only if ∆
is a perfect square; in other words, if and only if the expression is pd+fsin2ϑ, where dand f
are nonzero real constants. This happens if and only if the discriminant of ∆ , which we denote by
Disc (∆) , is zero. In view of expressions (4a)–(4c),
Disc (∆) := 16 cTI
1133 +cTI
23232cTI
1111 cTI
2323 −cTI
3333+cTI
11332
+ 2 cTI
1133 cTI
2323 +cTI
2323 cTI
3333= 0 .
2
The solutions are
cTI
2323 =−cTI
1133 ,(5a)
cTI
2323 =−cTI
1133 and cTI
1111 =cTI
2323 ,(5b)
cTI
3333 =cTI
11332
+cTI
1111 cTI
2323 + 2 cTI
1133 cTI
2323
cTI
1111 −cTI
2323
; (5c)
solution (5b) is a special case of solution (5a) but we keep both for convenient referencing below.
For the Backus average, solution (5b) cannot be satisfied within the stability condition; it would
require Pn
i=1 (1/(c1111)i) = Pn
i=1 (1/(c2323)i),which is not allowed. Solution (5c) can be satisfied
if and only if c2323 is the same for all layers, which results in an isotropic average (Backus, 1962,
Section 6).
If we consider a stack of nisotropic layers, whose elasticity parameters are
c1111 ={(c1111)1, . . . , (c1111)n}and c2323 ={(c2323)1, . . . , (c2323)n},
the stability condition for each layer is (e.g., Slawinski, 2018, Exercise 5.3)
(c1111)i>4
3(c2323)i>0, i = 1 , . . . , n .
The Backus-average elasticity parameters are
cTI
1111 =1−2
nY2
n W −1+4
n(U−Z),(6a)
cTI
1122 =1−2
nY2
n W −1+2
n(U−2Z),(6b)
cTI
3333 =n W −1,(6c)
cTI
1133 =1−2
nYn W −1,(6d)
cTI
2323 =n V −1,(6e)
cTI
1212 =n−1U , (6f)
where
U:=
n
X
i=1
(c2323)i, V :=
n
X
i=1
1
(c2323)i
, W :=
n
X
i=1
1
(c1111)i
, Y :=
n
X
i=1
(c2323)i
(c1111)i
, Z :=
n
X
i=1
((c2323)i)2
(c1111)i
.
(7)
A standard form of these parameters is given by, for example, Slawinski (2018, Section 4.2.2); the
expressions, therein, and those of parameterizations (6), are equivalent to A,B,C,F,L,M
of Backus (1962, equations (13)), respectively. The stability of the Backus average is inherited from
the stability of the layers (Slawinski, 2018, Proposition 4.1); in other words, if the layers are stable,
so is the average.
3 Christoffel roots
Solutions (5a)–(5c) can be written in terms of parameterizations (6) as
64 (n(V+W)−2V Y )2n2−2n Y +U(W−V) + V Z −W Z +Y2
V3W3= 0 ,
3
whose solutions are
Y=n
21 + W V −1,(8a)
W=Vand Y=n , (8b)
Z=−n2+U V −U W + 2 n Y −Y2
V−W; (8c)
again, solution (8b) is a special case of solution (8a) but we keep both for convenient referencing
below. We proceed to prove the existence of solution (8a) for n>4 layers, followed by a numerical
example to illustrate the result.
3.1 Nondetachment
Within the constraints of the stability, both detachment and nondetachment are permitted. The
nondetachment of the qP slowness surface occurs if and only if cTI
2323 =−cTI
1133 . From solution (5a),
and its parameterization (8a), it follows that roots (2) are elliptical. Hence, we have the following
lemma.
Lemma 1. There exists a Backus average of at least four isotropic layers for which the Christoffel
roots are elliptical.
Proof. We fix a > 4
3and let x∈(0 ,1] so that
c1111 =na
x,a
x,a
x,a
xoand c2323 =x , 1
x,1
x,1
x.(9)
Following defintions (7), variables Y , V , and Wof solution (8a) become
Y(x) = x2
1
a+
n
X
i=2
(c2323)i
(c1111)i
=x2
a+
n
X
i=2
1/x
a/x =x2
a+n−1
a,
V(x) = 1
x+
n
X
i=2
1
(c2323)i
=1
x+
n
X
i=2
1
1/x =1+(n−1) x2
x,
W(x) =
n
X
i=1
1
(c1111)i
=
n
X
i=1
1
a/x =n x
a.
We define X(x) := n
21 + W(x)V(x)−1, which results in
X(x) = n
2 1 + n x
a1+(n−1) x2
x−1!=n
21 + 1
an x2
1+(n−1) x2.
As x→0+, we have
Y(0+)→n−1
a≈3
4(n−1) , X(0+)→n
2,
and, hence, Y(0+)> X(0+) for aclose to 4
3. Furthermore, as x→1−,
Y(1−)→1
a+n−1
aand X(1−)→n
21 + 1
a.
Thus, for n>4 , and aclose to, but greater than, 4
3,Y(1−)< X(1−) .
It follows form the Intermediate Value Theorem that there exists an x∈(0,1) for which Y(x) =
X(x),which completes the proof.
4
3.2 Numerical example
Let us consider a numerical example, where
x= 0.229852282578204 , a = 1.344000000000000 and Y=X= 2.271452434379770 ; (10)
indeed, there exists a Backus average of at least four isotropic layers, whose Christoffel roots are
elliptical.
Using results (10) with parameters (9), we obtain
c1111 ={5.8472 ,5.8472 ,5.8472 ,5.8472}and c2323 ={0.2299 ,4.3506 ,4.3506 ,4.3506};
the Backus-average parameters, following expressions (6), are
cTI
1111 = 3.6692 , cTI
1122 =−2.9717 , cTI
3333 = 5.8472 ,
cTI
1133 =−0.7936 , cTI
2323 = 0.7936 , cTI
1212 = 3.3204 .
(11)
The eigenvalues of tensor (1) with values (11) are λ1=λ2= 6.6409 , λ3= 6.0812 , λ4=λ5= 1.5873 ,
λ6= 0.4635 , which belong to a transversely isotropic tensor (B´ona et al., 2007a); since they are
positive, the stability condition of the average are satisfied. Also, Disc (∆) = 1.9883 ×10−12 , which
can be considered zero, as required. Consequently, equation (2) becomes
vqP,qS V =v
u
u
t4.4628 + 2.1781 (1 −n3)±q62.8718 (0.6373 −n3)2
2ρ.(12)
Recalling that n3= sin2θ, and since ρmust be a positive scalar quantity, we see that equation (12)
is pd+fsin2ϑ, as required. Letting ρ= 1 , the three roots are
vqP =v
u
u
tcTI
3333 −cTI
1111(1 −n3) + cTI
1111 +cTI
2323 +√∆
2ρ=p5.8472 + 5.0536 sin2ϑ(13a)
vqSV =v
u
u
tcTI
3333 −cTI
1111(1 −n3) + cTI
1111 +cTI
2323 −√∆
2ρ=p0.7936 −2.8756 sin2ϑ , (13b)
vSH =scTI
1212 n3+cTI
2323 (1 −n3)
ρ=p0.7936 + 2.5268 sin2ϑ . (13c)
The reciprocals of expressions (13) are ellipses, illustrated in Figure 1. Therein, the grey curve
represents 1/vSH ; the black curves represent 1/vqP and 1/vqSV .
3.3 Interpretation
Although values (11) do not represent typical Hookean solid used in seismology, they satisfy the
stability condition and can appear in computational searches. To gain an insight into the appear-
ance of slowness curves in Figure 1, let us examine an example using the density-scaled elasticity
parameters for Green-River shale (e.g., Slawinski (2015, Exercise 9.3); Thomsen (1986, Table 1)),
cTI
1111 = 13.55 , cTI
1133 = 1.47 , cTI
3333 = 9.74 , cTI
2323 = 2.81 , cTI
1212 = 3.81 ,(14)
5
Figure 1: Three Christoffel roots resulting in three slowness curves
(a) (b) (c)
Figure 2: Slowness curves, 1/vqP , 1/vqSV , 1/vSH , for modified values of elasticity parameters for
Green-River shale. As cTI
1133 → −cTI
2323 , the innermost curve ceases to be detached and smooth.
6
where each parameter is scaled by 106; herein, superscript TI refers to an intrinsically transversely
isotropic medium, as opposed to TI , which is a Backus average.
The curves of 1/vqP , 1/vqSV and 1/vS H with values (14) are illustrated in Figure 2(a); such curves
are typical in seismology. The progression of Figures 2(a)–(c), however, illustrates an important
property. For detached qP slowness surfaces, the expressions for the qP ,qSV and SH wavefront
speeds, indeed, correspond to distinct smooth wavefronts. However, if cTI
1133 =−cTI
2323 , the qP and
qSV slowness surfaces lose their smoothness. Also, their expressions—not only their curves—become
connected with one another; neither root corresponds to a distinct slowness curve nor does a given
curve result from a single root.
For each slowness surface, the criterion of belonging to a particular wave on either side of an intersec-
tion is not its belonging to a single root but the orientation of the corresponding eigenvectors, which
are the displacement vectors of a given wave (B´ona et al., 2007b). As can be readily shown, for the
innermost surface, the displacement vector is normal to it along the rotation-symmetry axis and in
the plane perpendicular to it; hence, it corresponds to the qP wave. Figure 2 supports heuristically
a rigorous statement based on the eigendecomposition theorem.
4 Ellipticity condition
According to Thomsen (1986), the ellipticity condition is ε=δ, where
ε=cTI
1111 −cTI
3333
2cTI
3333
and δ=cTI
1133 +cTI
23232−cTI
3333 −cTI
23232
2cTI
3333 cTI
3333 −cTI
2323,(15)
for either TI or TI . However, equations (13a)–(13c) lead to ellipsoidal forms, even though, therein,
ε=−0.2363 6=−0.4445 = δ. To avoid this discrepancy, we state the following proposition with a
qualifier.
Proposition 2. The detached qP slowness surface is ellipsoidal if and only if ε=δ.
Proof. Following expressions (15), ε=δif and only if
cTI
2323 =−cTI
1133 and cTI
1111 =cTI
2323 or cTI
3333 =cTI
11332+cTI
1111 cTI
2323 + 2 cTI
1133 cTI
2323
cTI
1111 −cTI
2323
,
which are solutions (5b) and (5c) , respectively. Solution (5a), cTI
2323 =−cTI
1133 , is—in general—the
condition for nondetachment, and—for the Backus average—is the condition for elliptical roots. For
the Backus average, solution (5b) is not allowed within the stability condition and solution (5c)
results in an isotropic average, hence, circular roots. Thus, the ellipticity condition, ε=δ, is valid
for detached qP slowness surfaces only.
To gain an insight into Proposition 2, we modify parameters for Green-River shale—illustrated in
plot 3(a)—by applying expression (5c) to obtain an ellipsoidal qP slowness surface illustrated in
plot 3(b). Applying subsequently expression (5a), we obtain three elliptical Christoffel roots, where
ε6=δ, as expected in view of expressions (13a) and (13b). The innermost slowness surface is neither
detached nor elliptical, as illustrated in plot 3(c).
Let us apply these expressions in the opposite order. Using expression (5a), we obtain the result
illustrated in plots 2(c) and 4(b). Applying subsequently expression (5c), we obtain three elliptical
slowness surfaces, illustrated in plot 4(c). The stability conditions are satisfied but δis indeterminate.
7
(a) (b) (c)
Figure 3: Slowness curves, 1/vqP , 1/vqSV , 1/vSH , for modified values of elasticity parameters for
Green-River shale. Plot (b) corresponds to expression (5c); plot (c) corresponds to expression (5c)
followed by expression (5a).
(a) (b) (c)
Figure 4: Slowness curves, 1/vqP , 1/vqSV , 1/vSH , for modified values of elasticity parameters for
Green-River shale. Plot (b) corresponds to modifications by expression (5a); plot (c) is modified by
expression (5a) followed by expression (5c).
8
However, if we let cTI
2323 ≈ −cTI
1133 , as opposed to cTI
2323 =−cTI
1133 , we obtain δ=ε, as a consequence
of detachment.
In contrast to these results, the Backus average—for either order of expressions (5a) and (5c)—does
not satisfy the stability condition.
5 Conclusion
The only restriction on the values of the elasticity parameters is the stability condition. Within this
condition, we examine properties of the Christoffel roots for nondetached qP slowness surfaces in
transversely isotropic media. The qP slowness surface is detached if and only if cTI
2323 6=−cTI
1133 .
Under such a condition, each root corresponds to a distinct smooth wavefront. The qP slowness
surface is nondetached if and only if cTI
2323 =−cTI
1133 . Under such conditions, the roots are elliptical
but do not correspond to distinct wavefronts; also, the qP and qSV slowness surfaces are not smooth.
Acknowledgements
The authors wish to acknowledge Ran Bachrach for fruitful discussions leading them to revisit the
classic theorem and statements of Helbig (1979, 1983), and Elena Patarini for her graphical support.
This research was performed in the context of The Geomechanics Project supported by Husky En-
ergy. Also, this research was partially supported by the Natural Sciences and Engineering Research
Council of Canada, grant 202259.
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