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It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that both Frobenius norms result in similar isotropic counterparts, and the operator norm results in a counterpart with a slightly larger discrepancy. The reason for this discrepancy is rooted in the very definition of that norm, which is distinct from the Frobenius norms and which consists of the largest eigenvalue of the elasticity tensor. If we constrain the elasticity tensor to values expected for modelling physical materials, the three norms result in similar isotropic counterparts of a generally anisotropic tensor. To examine this important case and without loss of generality, we illustrate the isotropic counterparts by commencing from a transversely isotropic tensor obtained from a generally anisotropic one. Also, together with the three norms, we consider the L 2 slowness-curve fit. Upon this study, we infer that-for modelling physical materials-the isotropic counterparts are quite similar to each other, at least, sufficiently so that-for values obtained from empirical studies, such as seismic measurements-the differences among norms are within the range of expected measurement errors.
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Volume 11 ·2018 ·Pages 15–28
Effects of norms on general Hookean solids for their isotropic
counterparts
Tomasz Daneka·Andrea Noseworthy b·Michael A. Slawinski b
Communicated by Len Bos
Abstract
It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor,
it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a
given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the
Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that both
Frobenius norms result in similar isotropic counterparts, and the operator norm results in a counterpart
with a slightly larger discrepancy. The reason for this discrepancy is rooted in the very definition of that
norm, which is distinct from the Frobenius norms and which consists of the largest eigenvalue of the
elasticity tensor. If we constrain the elasticity tensor to values expected for modelling physical materials,
the three norms result in similar isotropic counterparts of a generally anisotropic tensor. To examine this
important case and without loss of generality, we illustrate the isotropic counterparts by commencing
from a transversely isotropic tensor obtained from a generally anisotropic one. Also, together with the
three norms, we consider the
L2
slowness-curve fit. Upon this study, we infer that—for modelling physical
materials—the isotropic counterparts are quite similar to each other, at least, sufficiently so that—for
values obtained from empirical studies, such as seismic measurements—the differences among norms
are within the range of expected measurement errors.
1 Introduction
The symmetry class of a Hookean solid is a property defined within its elasticity tensor. Such a solid, which is a mathematical
entity, might serve as an analogy—in the Platonic sense of mathematical physics—for physical properties of a given material. The
inference of properties of a physical material requires the interpretation of an elasticity tensor. Among these properties are its
symmetries. In particular, it is useful to compute an isotropic counterpart of the obtained tensor, which might be sufficiently
accurate for the modelling of materials, while offering a mathematical convenience. Furthermore, such a result can serve as a
starting model for more detailed study, especially if anisotropy does not significantly affect the wave propagation (e.g., Eken et
al. [
1
]). Regardless of the motivation, it is necessary to decide on an appropriate norm to compute such a counterpart. An insight
into such a decision is the crux of this paper.
An examination of several norms to obtain an isotropic counterpart is presented by Norris [
2
]. Herein, we numerically
compare isotropic counterparts of a generally anisotropic Hookean solid according to the Frobenius-36 norm and Frobenius-21
norm, to which we refer as
F36
and
F21
, respectively, as well as according to the operator norm, to which we refer as
λ
. Also,
for the case of reduction of a transversely isotropic tensor to its isotropic counterpart, we consider the
L2
slowness-curve fit.
Subsequently, to examine the importance in different results obtained with different norms, we use perturbation techniques to
examine the effect of errors on isotropic counterparts.
This paper is an examination of a reduction of an anisotropic tensor to isotropy. Reducing a generally anisotropic tensor—with
different norms—to a higher symmetry has been a research topic in which two of the authors have engaged for a decade.
Consequently, to avoid unnecessary repetitions, we provide references to papers that include proofs, explanations and descriptions
of methodologies: Bos and Slawinski [3], Bucataru and Slawinski [4], Danek et al. [5,6], Danek and Slawinski [7,8], Diner et
al. [
9
], Kochetov and Slawinski [
10
,
11
,
12
], Slawinski [
13
]. Also, these papers include references to many previous publications
of other authors upon which our discussions are based, such as the original work of Gazis et al. [14].
The purpose of this paper is the study of similarity and dissimilarity among different-norm counterparts for both general
Hookean solids and the ones commonly used in seismological models. The former case is restricted solely to the stability conditions
of the tensor; the latter is limited to values encountered in modelling terrestrial materials. An examination of results of reductions
under different norms has not been addressed in the any of the above-mentioned papers, except for a brief comment by Bos and
Slawinski [3]. Nor, as far as the authors are able to ascertain, has it been addressed anywhere else.
aDepartment of Geoinformatics and Applied Computer Science, AGH-University of Science and Technology, Kraków, Poland
bDepartment of Earth Sciences, Memorial University, St. John’s, Newfoundland,
A1B 3X5, Canada
Danek ·Noseworthy ·Slawinski 16
2 Elasticity tensors
For convenience of referring to standard expressions, let us state the following. A Hookean solid, ci j k`, is a mathematical object
defined by Hooke’s Law,
σi j =
3
X
k=1
3
X
`=1
ci jk`"k`,i,j=1, 2, 3, (1)
where
σi j
,
"k`
and
ci jk`
are the stress, strain and elasticity tensors, respectively. The components of the elasticity tensor can be
written—in Kelvin’s, as opposed to Voigt’s, notation (e.g., Chapman [15])—as a symmetric second-rank tensor in R6,
C=
c1111 c1122 c1133 p2c1123 p2c1113 p2c1112
c1122 c2222 c2233 p2c2223 p2c2213 p2c2212
c1133 c2233 c3333 p2c3323 p2c3313 p2c3312
p2c1123 p2c2223 p2c3323 2c2323 2c2313 2c2312
p2c1113 p2c2213 p2c3313 2c2313 2c1313 2c1312
p2c1112 p2c2212 p2c3312 2c2312 2c1312 2c1212
. (2)
If the elasticity tensor is transversely isotropic, its components can be written—in a coordinate system whose
x3
-axis coincides
with the rotation-symmetry axis—as
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 2cTI
2323 0 0
0 0 0 0 2cTI
2323 0
0 0 0 0 0 cTI
1111 cTI
1122
. (3)
If the elasticity tensor is isotropic, its components can be written—in any coordinate system—as
Ciso =
ciso
1111 ciso
1111 2ciso
2323 ciso
1111 2ciso
2323 000
ciso
1111 2ciso
2323 ciso
1111 ciso
1111 2ciso
2323 000
ciso
1111 2ciso
2323 ciso
1111 2ciso
2323 ciso
1111 000
0 0 0 2ciso
2323 0 0
0 0 0 0 2ciso
2323 0
0 0 0 0 0 2ciso
2323
, (4)
and expression (1) can be rewritten as
σi j =ciso
1111 δi j
3
X
k=1
"kk +2ciso
2323 "i j ,i,j=1,2, 3 .
3 Norms
To examine the closeness between elasticity tensors, as discussed by Bos and Slawinski [
3
]and by Danek et al. [
5
,
6
], we consider
several norms.
3.1 Frobenius norms
The Frobenius norm treats a matrix in
Rn×n
as a Euclidean vector in
Rn2
. In the case of a symmetric 6
×
6 matrix, where
Cmn =Cnm , we can choose either
||C||F36 =v
u
t
6
X
m=1
6
X
n=1
C2
mn ,
which uses the thirty-six components, including their coefficients of p2 and 2 , or
||C||F21 =v
u
t
6
X
m=1
m
X
n=1
C2
mn ,
which uses only the twenty-one independent components, including their coefficients of p2 and 2 .
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Danek ·Noseworthy ·Slawinski 17
3.2 Operator norm
As discussed by Bos and Slawinski [
3
], by treating a matrix as a vector, the Frobenius norms ignore the fact that a matrix is a
representation of a linear map from
Rn
to
Rn
. In view of equation (1), the elasticity tensor represents a linear map between the
strain tensor, whose components can be expressed as a symmetric 3
×
3 matrix, [
"k`
], and the stress tensor, whose components
can be expressed as a symmetric 3
×
3 matrix, [
σi j
]. The operator norm of the elasticity tensor considered as a mapping from
R3×3
to
R3×3
, where both the stress and strain tensors are endowed with the Frobenius norm,
F9
, is the operator norm of matrix
CR6×6.
Given a norm on Rn, the associated operator norm of matrix ARn×nis
kAk:=max
kxk=1kAxk.
An example of such a norm is the Euclidean operator norm, which—for symmetric matrices—becomes
kAk2:=max{|λ|:λan eigenvalue of A}.
The operator norm of an elasticity tensor—whose components in a given coordinate system can be expressed as a symmetric
6×6 matrix—is
||C||λ=max |λi|,
where λi∈ {λ1, . . . , λ6}is an eigenvalue of C.
4 Slowness-curve L2fit
In a manner similar to the
F36
norm,
F21
norm and operator norm, the slowness-curve
L2
fit can be used to find an isotropic
counterpart to an anisotropic Hookean solid. However, in contrast to these norms, which rely on finding the smallest distance
between tensors, it relies on finding the best fit of circles—according to a chosen criterion—to noncircular wavefronts.
To consider a generally anisotropic tensor, one would need to fit a sphere into a slowness surface. However, for the purpose
of insightful graphical illustrations, we consider a transversely isotropic tensor obtained from a generally anisotropic one, since
its directional dependence reduces to two dimensions.
For both the operator norm and fitting, would it be spheres or circles, we do not invoke explicit expressions for the components
of the closest elasticity tensor, as we do for the Frobenius norms, but instead we examine the effect of these components on
certain quantities. For the operator norm, this quantity consists of eigenvalues; for the fit, this quantity consists of wavefront
slownesses.
The direct results of the norms are the components of the corresponding isotropic tensors, and the wavefront-slowness circles
are their consequences. The direct result of the slowness-curve fit are slowness circles, and the components of the corresponding
isotropic tensor are their consequence.
Herein, the best fit, in the L2sense, is the radius, r, that minimizes
S=
n
X
i=1
(siri)2, (5)
where
si
are
n
discretized values along the slowness curve, and
siri
is measured in the radial direction. Hence,
r
is the radius
of the slowness circle; it corresponds to isotropy.
5 Numerical results
5.1 Isotropic counterparts of general Hookean solids
In this section, we investigate isotropic counterparts derived using the three norms introduced in Section 3. We do so for the most
general Hookean solids. In other words, their material symmetry is a general anisotropy, and their parameter values are allowed
to vary randomly over a large range. The sole restriction on these values is the stability condition of the solid. This condition is
tantamount to the positive definiteness of matrix (2), which is equivalent to its eigenvalues being positive.
Consider Figures 1and 2. Both figures result from ten thousand repetitions of the tensor whose general form is stated
in expression (2). All histograms in this paper are generated by ten thousand repetitions, which are expressed by the total
height of all blocks; the interpretation of each histogram relies on the relative heights of these blocks. Herein, the values of
the density-scaled elasticity parameters, whose units are
km2/s2
, are allowed to vary randomly between 0 and 20 , on the main
diagonal of matrix (2), and between
10 and 10 , otherwise. The positive range, (0
,
20), along the main diagonal, is required by
the positive definiteness of the elasticity tensor.
The histograms in Figures 1illustrate the values of the elasticity parameters of a isotropic counterpart—according to the
F21
norm—and normalized by the values of the corresponding parameter according to the
F36
norm. If both norms result in the same
value of the parameter, their ratio is equal to unity. The histograms in Figures 2illustrate the values of the elasticity parameters
of an isotropic counterpart—according to the operator norm—and also normalized by the values of the corresponding parameter
according to the F36 norm.
Examining Figures 1and 2, we see that highest values of the histograms appear in the neighbourhood of unity. Thus, we
infer that the three norms result in similar isotropic counterparts of a general Hookean solid. Recognizing that the scale of the
horizontal axes is different for each plot, we see that the histograms of the operator norm are broader than for the
F21
norm. It
means that the F36 norm is more similar to the F21 norm than to the operator norm.
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Danek ·Noseworthy ·Slawinski 18
Figure 1: Deviation of ciso
1111 and ciso
2323 for general Hookean solid, according to F21 , and normalized by corresponding F36 parameters
Figure 2:
Deviation of
ciso
1111
and
ciso
2323
for general Hookean solid, according to operator norm, and normalized by corresponding
F36
parameters
To enquire into that behaviour, we examine Figure 3. On the horizontal axis is the ratio of the largest eigenvalue to the sum
of the remaining eigenvalues. The quantities of the vertical axis are the same as for the horizontal axis in Figure 2. As expected
from its definition, the operator norm, which is the value of the largest eigenvalue, is sensitive to that ratio. Quantitatively, the
correlation between the ratio and the breadth of the histogram for
ciso
1111
has the coefficient of 0
.
7210 , and for
ciso
2323
of 0
.
8578 ;
there is a significant correlation.
The similarity of the results obtained according to the F21 and F36 norms is a consequence of the fact that they differ by the
weight on the offdiagonal terms. These terms, however, tend to be smaller than the main-diagonal terms due to the requirement
of positive definiteness.
5.2 Isotropic counterparts of seismological Hookean solids
Let us consider a case of a Hookean solid that is pertinent to common terrestrial materials near the Earth’s surface. For that
purpose, we use the elasticity tensor obtained by Dewangan and Grechka [
16
]from measurements of vertical seismic profiling,
C=
7.8195 3.4495 2.5667 p2(0.1374)p2(0.0558)p2(0.1239)
3.4495 8.1284 2.3589 p2(0.0812)p2(0.0735)p2(0.1692)
2.5667 2.3589 7.0908 p2(0.0092)p2(0.0286)p2(0.1655)
p2(0.1374)p2(0.0812)p2(0.0092)2(1.6636)2(0.0787)2(0.1053)
p2(0.0558)p2(0.0735)p2(0.0286)2(0.0787)2(2.0660)2(0.1517)
p2(0.1239)p2(0.1692)p2(0.1655)2(0.1053)2(0.1517)2(2.4270)
. (6)
Figure 3: Relation between deviations illustrated in Figure 2and corresponding spreads of eigenvalues
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Danek ·Noseworthy ·Slawinski 19
Figure 4: Deviation of ciso
1111 and ciso
2323 for tensor (6), according to F21 , and normalized by corresponding F36 parameters
Figure 5: Deviation of ciso
1111 and ciso
2323 for tensor (6), according to operator norm, and normalized by corresponding F36 parameters
Figure 6: Relation between deviation illustrated in Figure 5and corresponding spreads of eigenvalues
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Danek ·Noseworthy ·Slawinski 20
Its components are density-scaled elasticity parameters; their units are
km2/s2
. Herein, the parameter values are more restricted
than for a general Hookean solid examined in Section 5.1.
Consider Figures 4and 5, which are generated using tensor (6), and its errors (Grechka, pers. comm., 2007),
±
0.1656 0.1122 0.1216 0.1176 0.0774 0.0741
0.1122 0.1862 0.1551 0.0797 0.1137 0.0832
0.1216 0.1551 0.1439 0.0856 0.0662 0.1010
0.1176 0.0797 0.0856 0.0714 0.0496 0.0542
0.0774 0.1137 0.0662 0.0496 0.0626 0.0621
0.0741 0.0832 0.1010 0.0542 0.0621 0.0802
, (7)
which are the estimates of the standard deviations corresponding to each component of expression (6), in the coordinate system
of data acquisition, since these values do not constitute components of a tensor. Either figure results from ten thousand repetitions.
The values of the elasticity parameters are allowed to vary randomly with a uniform distribution up to three standard deviations
stated in matrix (7). Figures 4and 5are analogous to Figures 1and 2, respectively, except that—by considering tensor (6) and
its standard deviations—the general Hookean solid is restricted to values that are representative for seismological models of the
Earth’s crust.
Comparing Figures 1and 4, and recognizing that the scale of the horizontal axes is different for each plot, we see that the
Frobenius norms produce isotropic counterparts for tensor (6) that are as close to one another as for the isotropic counterparts of
a general Hookean solid. This conclusion is confirmed quantitatively.
For the left plot of Figure 1, the mean value is 1
.
0071 , and for the left plot of Figure 4, the mean value is 1
.
0002 . For the
right plot of Figure 1, the mean value is 1
.
0011 , and for the right plot of Figure 4the mean value is 0
.
9882 . Thus, the values for
a seismological Hookean solid remain similarly close to unity as they do for a general Hookean solid, which is symptomatic of
both Frobenius norms resulting in similar isotropic counterparts for a wide range of generally anisotropic tensors.
Comparing Figures 2and 5, and recognizing that scales of the horizontal axes are different, we see that the operator norm
produces isotropic counterparts for tensor (6) whose scatter is narrower than for a general Hookean solid. Quantitatively, for the
left plot of Figure 2, the standard deviation is 0
.
1624 , and for the left plot of Figure 5, it is 0
.
019 ; for the right plots of these
figures, the corresponding standard deviations are 0.1373 and 0.0348 , respectively.
To explain that behaviour we see that, in contrast to a general Hookean solid and Figure 3, the correlation between the ratio
of the eigenvalues and the breadth of the histograms does not appear, as illustrated in Figure 6, where the quantities of the
vertical axis are the same as for the horizontal axis in Figure 5. We can infer that—in the case of a seismological Hookean solid
grounded in empirical information—the eigenvalues of the elasticity tensor are more similar to each other than in the case of a
general Hookean solid, whose elasticity parameters can assume any value as long as the tensor remains positive-definite.
5.3 Tensor CTI
a
To graphically illustrate the results for different norms and to examine further their properties, let us—without any significant
loss of generality—consider a transversely isotropic tensor. We choose a tensor computed by Danek et al. [
6
], which is the closest
counterpart, in the F36 sense, of tensor (6),
CTI
a=
8.0641 3.3720 2.4588 0 0 0
3.3720 8.0641 2.4588 0 0 0
2.4588 2.4588 7.0817 0 0 0
0 0 0 2(1.8625)0 0
0 0 0 0 2(1.8625)0
0 0 0 0 0 2(2.3460)
. (8)
The slowness curves for tensor (8) and its isotropic counterpart circles discussed in Sections 5.3.1,5.3.2 and 5.3.3, below, are
shown in Figure 7. Isotropic tensors examined in this section are counterparts of this tensor.
For the purpose of this section, the choice of norm to obtain tensor (8) from its generally anisotropic origins is insignificant,
provided the result is transversely isotropic. One could even begin with a generic transversely isotropic tensor, without relating it
to a generally anisotropic one. We derive the transversely isotropic tensor from a generally anisotropic one inferred from seismic
measurement in order to examine values commonly encountered in geophysics. Also, one could use this generally anisotropic
tensor itself and find its isotropic counterparts, as in Section 5.2. However, we choose to consider the isotropic counterparts of a
transversely isotropic tensor to be able to illustrate them graphically, as shown in Figures 711, below. To enhance the the details,
in each case, only the first quadrant is shown, since—due to symmetry—no further information is provided by the remaining
quadrants.
5.3.1 F36 norm
Let us consider the Frobenius norm for the thirty-six components. There are analytical formulæ to calculate—from a generally
anisotropic tensor—the two parameters of its closest isotropic tensor (Voigt, [
17
]). From a transversely isotropic tensor, these
parameters are
cisoF36
1111 =1
15 8cTI
1111 +4cTI
1133 +8cTI
2323 +3cTI
3333
and
cisoF36
2323 =1
30 7cTI
1111 5cTI
1122 4cTI
1133 +12cTI
2323 +2cTI
3333.
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Danek ·Noseworthy ·Slawinski 21
Hence, the closest isotropic counterpart of tensor (8) is
CisoF36
a=
7.3662 2.9484 2.9484 0 0 0
2.9484 7.3662 2.9484 0 0 0
2.9484 2.9484 7.3662 0 0 0
0 0 0 2(2.2089)0 0
0 0 0 0 2(2.2089)0
0 0 0 0 0 2(2.2089)
. (9)
5.3.2 F21 norm
Let us consider the Frobenius norm for the twenty-one independent components. Following Slawinski [
13
], for a transversely
isotropic tensor, the analytical formulæ to calculate the two parameters of its closest isotropic tensor are
cisoF21
1111 =1
95cTI
1111 +2cTI
1133 +4cTI
2323 +2cTI
3333
and
cisoF21
2323 =1
18 4cTI
1111 3cTI
1122 2cTI
1133 +8cTI
2323 +cTI
3333.
Hence,
CisoF21
a=
7.4279 3.0716 3.0716 0 0 0
3.0716 7.4279 3.0716 0 0 0
3.0716 3.0716 7.4279 0 0 0
0 0 0 2(2.1781)0 0
0 0 0 0 2(2.1781)0
0 0 0 0 0 2(2.1781)
. (10)
5.3.3 λnorm
Unlike the Frobenius norms, the operator norm has no analytical formulæ for
cisoλ
1111
and
cisoλ
2323
. They must be obtained numerically.
The largest eigenvalues are obtained using a standard numerical procedure of the Singular Value Decomposition and then
optimized over a two-dimensional solution space using a similar procedure to the one described in Danek et al. [
6
]. For tensor (8),
we obtain
Cisoλ
a=
7.7562 3.0053 3.0053 0 0 0
3.0053 7.7562 3.0053 0 0 0
3.0053 3.0053 7.7562 0 0 0
0 0 0 2(2.3755)0 0
0 0 0 0 2(2.3755)0
0 0 0 0 0 2(2.3755)
. (11)
5.3.4 Distances among tensors
To gain insight into different isotropic counterparts of tensor (8), we calculate the
F36
distance between tensors (9) and (11),
which is 0
.
8993 . The
F36
distance between tensors (8) and (9) is 1
.
8461 . The
F36
distance between tensors (8) and (11) is
2
.
0535 , where we note that tensor (11) is the closest isotropic tensor according to the operator—not the
F36
—norm. Thus,
in spite of similarities between the isotropic tensors, the distance between them is large in comparison to their distances to
tensor (8).
This is an illustration of abstractness of the concept of distances in the space of elasticity tensors. A concrete evaluation is
provided by comparing the results obtained by minimizing these distances. Such results are tensors (9), (10), (11), and their
wavefront-slowness circles in Figure 7. This figure illustrates a similarity among these circles, which is a realm in which the
isotropic tensors can be compared. They can be compared within the slowness space.
5.4 Comparison of norms
Comparing tensors (9), (10) and (11), we see that the parameters of the closest isotropic tensor depend on the norm used. Given
two anisotropic tensors, we might be interested to know which of them is closer to isotropy. For a given norm, an answer is
obtained by a straightforward calculation. In general, for different norms, there is no absolute answer: the sequence in closeness
to isotropy can be reversed between two tensors; it depends on the norms.
5.4.1 F36 versus F21
Using a numerical search based on a simple random walk through a solution space with the target function being a difference
between the minimized
F21
distance and the maximized
F36
distance, an elasticity tensor is generated that is further from isotropy
than tensor (8) according to the
F36
norm, but closer to isotropy than tensor (8) according to the
F21
norm. The search results in
CTI
b=
7.3091 4.5882 2.9970 0 0 0
4.5882 7.3091 2.9970 0 0 0
2.9970 2.9970 6.6604 0 0 0
0 0 0 2(1.5631)0 0
0 0 0 0 2(1.5631)0
0 0 0 0 0 2(1.3605)
, (12)
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Danek ·Noseworthy ·Slawinski 22
Figure 7:
Slowness curves for tensor (8): solid lines rep-
resent the
qP
,
qSV
and
SH
waves; dashed lines represent
the
P
and
S
waves according to
F36
norm; dashed-dotted
lines represent the
P
and
S
waves according to
F21
norm;
the results of these norms almost coincide; dotted lines
represent the Pand Swaves according to λnorm.
Figure 8:
Slowness curves for tensor (12): solid lines rep-
resent the
qP
,
qSV
and
SH
waves; dashed lines represent
the
P
and
S
waves according to
F36
norm; dotted lines
represent the Pand Swaves according to F21 norm.
with its corresponding isotropic counterparts,
CisoF36
b=
6.8631 3.6422 3.6422 0 0 0
3.6422 6.8631 3.6422 0 0 0
3.6422 3.6422 6.8631 0 0 0
0 0 0 2(1.6104)0 0
0 0 0 0 2(1.6104)0
0 0 0 0 0 2(1.6104)
(13)
and
CisoF21
b=
6.9014 3.7188 3.7188 0 0 0
3.7188 6.9014 3.7188 0 0 0
3.7188 3.7188 6.9014 0 0 0
0 0 0 2(1.5913)0 0
0 0 0 0 2(1.5913)0
0 0 0 0 0 2(1.5913)
, (14)
respectively. The distances to isotropy for
CTI
a
and
CTI
b
—stated, respectively, in expressions (8) and (12)—using the
F36
and
F21 norms, are
da36 =1.8460 <db36 =2.0400 ,
da21 =1.6372 >db21 =1.5517 .
The slowness curves for tensor (12) and its isotropic counterparts are shown in Figure 8.
5.4.2 F36 versus λ
The second comparison is between the F36 norm and the λnorm. We obtain
CTI
bb =
6.8639 3.3046 2.8770 0 0 0
3.3046 6.8639 2.8770 0 0 0
2.8770 2.8770 8.3825 0 0 0
0 0 0 2(2.7744)0 0
0 0 0 0 2(2.7744)0
0 0 0 0 0 2(1.7797)
, (15)
which is further from isotropy according to the
F36
norm and closer to isotropy according to the
λ
norm. Its isotropic counterparts
in the sense of the F36 and λnorms are
CisoF36
bb =
7.5842 2.9125 2.9125 0 0 0
2.9125 7.5842 2.9125 0 0 0
2.9125 2.9125 7.5842 0 0 0
0 0 0 2(2.3358)0 0
0 0 0 0 2(2.3358)0
0 0 0 0 0 2(2.3358)
(16)
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Danek ·Noseworthy ·Slawinski 23
and
Cisoλ
bb =
7.4712 2.9171 2.9171 0 0 0
2.9171 7.4712 2.9171 0 0 0
2.9171 2.9171 7.4712 0 0 0
0 0 0 2(2.7704)0 0
0 0 0 0 2(2.7704)0
0 0 0 0 0 2(2.7704)
, (17)
respectively. The distances to isotropy for CT I
aand CTI
bb , using the F36 and λnorms, are
da36 =1.8460 <dbb36 =2.1825 ,
daλ=1.0259 >dbbλ=0.9947 .
The slowness curves for tensor (15) and its isotropic counterparts are shown in Figure 9.
5.4.3 F21 versus λ
The third comparison is between the F21 norm and the λnorm. The resulting tensor is
CTI
bbb =
4.5706 2.6852 2.9075 0 0 0
2.6852 4.5706 2.9075 0 0 0
2.9075 2.9075 5.2705 0 0 0
0 0 0 2(1.9145)0 0
0 0 0 0 2(1.9145)0
0 0 0 0 0 2(0.9427)
, (18)
with isotropic counterparts according to the F21 norm and the λnorm,
CisoF21
bbb =
5.2074 2.4297 2.4297 0 0 0
2.4297 5.2074 2.4297 0 0 0
2.4297 2.4297 5.2074 0 0 0
0 0 0 2(1.3889)0 0
0 0 0 0 2(1.3889)0
0 0 0 0 0 2(1.3889)
(19)
and
Cisoλ
bbb =
5.2926 2.4354 2.4354 0 0 0
2.4354 5.2926 2.4354 0 0 0
2.4354 2.4354 5.2926 0 0 0
0 0 0 2(1.4286)0 0
0 0 0 0 2(1.4286)0
0 0 0 0 0 2(1.4286)
, (20)
respectively. The distances to isotropy for both CT I
aand CTI
bbb using the F21 and λnorms are
da21 =1.6372 <dbbb21 =2.0842 ,
daλ=1.0259 >dbbbλ=0.9719.
The slowness curves for tensor (18) and its isotropic counterparts are shown in Figure 10.
5.5 Slowness-curve fit
Considering tensor (8) and applying a minimization for the
qP
wave, using formula (5), we find
S=0.0886
with
r=0.3770
.
Following the same procedure for the
qSV
and
SH
waves, we find
S=0.2973
, with
r=0.6832
, and
S=0.2169
, with
r=0.6831
,
respectively. Combining these results, we obtain
S=0.6029
, with
rP=0.3770
and
rS=0.6831
, which are the slownesses of
the
P
and
S
waves, respectively. Note that—since the slowness curves of the
qP
waves are detached from the curves for the
qSV
and SH waves—the value of rfor the Pwaves does not change by combining the results.
Since
vP
=
pc1111
and
vS
=
pc2323
are the
P
-wave and
S
-wave speeds, respectively, it follows that
c1111
=1
/r2
P
and
c2323
=1
/r2
S
.
Hence, we obtain
CisoL2
a=
7.0341 2.7485 2.7485 0 0 0
2.7485 7.0341 2.7485 0 0 0
2.7485 2.7485 7.0341 0 0 0
0 0 0 2(2.1428)0 0
0 0 0 0 2(2.1428)0
0 0 0 0 0 2(2.1428)
. (21)
The slowness curves for tensor (21) and its isotropic counterparts are shown in Figure 11.
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Danek ·Noseworthy ·Slawinski 24
Figure 9:
Slowness curves for tensor (15): solid lines rep-
resent the
qP
,
qSV
and
SH
waves; dotted lines represent
its
P
and
S
waves according to
F36
norm; dashed lines
represent its Pand Swaves according to λnorm.
Figure 10:
Slowness curves for tensor (18): solid lines
represent the
qP
,
qSV
and
SH
waves; dotted lines repres-
ent its
P
and
S
waves according to
F21
norm; dashed lines
represent its Pand Swaves according to λnorm.
Figure 11:
Slowness curves for tensor (21): solid lines represent the
qP
,
qSV
and
SH
waves; dotted lines represent its
P
and
S
waves according
to the slowness-curve L2fit.
5.6 Thomsen parameters
To consider the empirical importance of the numerical study presented in Section 5, we examine whether or not tensors (8), (12),
(15) and (18) might be representative of seismic media. Herein, we show that these tensors exhibit the strength of anisotropy
that is consistent with cases of interest to geophysicists. To show this consistency, we calculate the Thomsen [18]parameters,
α=qcTI
3333 ,
β=qcTI
2323 ,
γ=cTI
1212 cTI
2323
2cTI
2323
,
δ=(cTI
1133 +cTI
2323)2(cTI
3333 cTI
2323)2
2cTI
3333(cTI
3333 cTI
2323),
ε=cTI
1111 cTI
3333
2cTI
3333
.
The values of these parameters for tensors (8), (12), (15) and (18) are shown in Table 1. Comparing results of this table to data
of Auld [19]and Thomsen [18], we see that these tensors represent common geological materials.
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Danek ·Noseworthy ·Slawinski 25
Tensor α β γ δ ε
CTI
a2.6612 1.2986 0.1956 -0.1561 0.0694
CTI
b2.5808 1.2503 -0.6483 -0.0764 0.0487
CTI
bb 2.2958 1.3837 -0.2538 0.3389 -0.6640
CTI
bbb 2.8953 1.6657 -0.1793 0.0052 -0.0906
Table 1: Thomsen parameters for tensors (8), (12), (15) and (18)
5.7 Isotropic counterparts as functions of anisotropy strength
Let us examine the effect of the three norms and the slowness-curve fit as a function of the strength of anisotropy. We take
Thomsen’s parameter εto quantify this strength.
For this examination, we choose tensor (9), and vary the value of
c1111
to obtain the values of
ε
between
0
.
4 and 1 . This
range covers both weak and strong anisotropy. The
P
-wave slowness values of the isotropic counterparts of the norms and the fit
are shown in Figure 12.
As expected, for isotropy,
ε
=0 , the
P
-wave-slowness values coincide for the three norms and the slowness-curve fit. The
common value is 1
/ÆcTI
1111
=1
/ÆcTI
3333
=0
.
3684 . Also, both Frobenius-norm values,
F21
and
F36
, and the operator-norm
values,
λ
, are similar to each other for the entire range of
ε
. Their differences are negligible in the context of measurement
errors. However, the values for the slowness-curve fit are similar to the values for the norms only in the vicinity of
ε
=0 . For the
fit, the behaviour of the P-wave slowness as a function of εis different, as illustrated by the shape of the graphs.
Figure 12: P
-wave slowness of isotropic counterparts as function of
ε
of transversely isotropic tensor;
F21
(light gray),
F36
(dark gray),
λ
(black);
and L2(dotted)
5.8 Error propagation
Components of an anisotropic tensor obtained from experimental measurements exhibit uncertainties due to measurement errors.
These uncertainties propagate to its symmetric counterparts.
In-depth studies of probability laws for the stiffness components was a subject of a paper by Guilleminot and Soize [
20
]. In
general, the offdiagonal terms may be assumed to be Gaussian but the diagonal ones are Gamma-distribution random variables.
The statistical dependence structure for the six strongest symmetry classes, namely, isotropic, TI, cubic, tetragonal, trigonal and
orthotropic, is presented in Table 1 of Guilleminot and Soize [
20
]. From the point of view of seismic observations, this problem
was analyzed by Rusmanugroho and McMechan [
21
]. In this case, normality—expressed as a large-shape parameter of the
Gamma-distribution variables—and the independence assumptions are good analogies for observations, even though certain
components, such as
c1212
and
c1223
, have the values of the crosscorrelation matrix significantly higher than others due to the
relation between their horizontal and vertical stress and horizontally polarized strain. These assumptions, namely, independence
of components and normality of their distributions, are crucial in the approach of Danek et al. [
5
]. They are also—at least
partially—required to obtain matrix (7) through numerical simulations performed by Dewangan and Grechka [16].
Let us examine the error propagation between the transversely isotropic tensor and its isotropic counterparts. Apart from
inferring the stability of these counterparts, such an examination allows us to gain an insight into a range of tensors whose
values are pertinent to seismological studies. Even though our conclusions stem from a single transversely isotropic tensor, the
perturbation of its components is akin to considering a multitude of such tensors.
The standard deviations of components of tensor (6) are given in expression (7). Since these values do not constitute
components of a tensor—and, hence, are valid only in the coordinate system of measurements—there is a need for a simulation to
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Danek ·Noseworthy ·Slawinski 26
consider error propagation from tensor (6) to tensor (8). Probability distributions of the values of the components of tensor (8)—
obtained by a Monte-Carlo simulation (Danek et al. 2013)—are shown in Figures 13,14,15,16,17. Therein, different histograms
have different horizontal scales.
The probability distributions of the two parameters for its isotropic
F36
counterpart are obtained in the same manner; they
are shown in Figure 18. Their mean values are given in tensor (9). In the same figure, we show the probability distributions of
parameters for the F21 and λcounterparts, whose mean values are given in tensors (10) and (11).
Examining Figure 18, we infer that, for tensors commonly encountered in seismology, their isotropic counterparts obtained
with distinct norms might be similar to each other within a typical range of measurement errors. For both Frobenius norms,
probability distributions of the corresponding parameters are very similar to one another.
Independently of the parameter values, in Figure 18, we observe their distributions. The distributions for the operator norm
are different than for the Frobenius norms. Also, within the operator norm, there is a significant difference between the
c1111
and
c2323
distributions. This is a consequence of properties of the operator norm, where only the largest among the six eigenvalues is
taken into consideration.
7.50 8.50
Figure 13: c1111 of tensor (8)
3.00 3.50
Figure 14: c1122 of tensor (8)
2.00 2.80
Figure 15: c1133 of tensor (8)
1.70 2.00
Figure 16: c2323 of tensor (8)
6.50 7.50
Figure 17: c3333 of tensor (8)
6 Discussions and conclusions
The essence of this paper consists of Section 5, in particular, Sections 5.1 and 5.2, therein, as well as Figures 16, in which we
examine the isotropic counterparts of generally anisotropic elasticity tensors as a function of different norms. Subsequently, we
examine consequences of the choice of a norm in reducing a typical tensor obtained from measurements, subject to experimental
errors, to its isotropic counterparts. For a general tensor, restricted solely to its stability conditions, the two Frobenius norms
result in isotropic counterparts that are sufficiently close to one another that one might neglect their difference within the context
of experimental errors. In seismological practice—and perhaps for many naturally occurring materials—one might also ignore the
differences between the two Frobenius norms and the operator norm. The similarity of results for the three norms is emphasized
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Danek ·Noseworthy ·Slawinski 27
Figure 18: c1111 (left panel) and c2323 (right panel) of F21 (light gray), F36 (dark gray) and λ(black) isotropic counterparts of tensor (8)
in Sections 5.3,5.4 and 5.6, in which—given tensor (8), which represents a typical material—we can find another transversely
isotropic tensor representative of common materials such that one of them is closer to isotropy according to one norm and the
other one closer to isotropy according to another norm.
Thus, as discussed in Sections 5.1,5.2,5.7 and 5.8, the differences among the results of the three norms—and, perhaps,
also the slowness-curve fit—might not be crucial within the context of typical materials and measurement errors. Thus, for an
important range of Hookean solids, the choice of the norm might be of secondary importance, and, pragmatically, we might
choose a Frobenius norm, since it offers analytical formulæ for an isotropic counterpart.
Acknowledgments
We wish to acknowledge the inspiring and fruitful environment of the Dolomites Research Weeks on Approximation in Alba
di Canazei where—within the section of Approximations in Seismology—substantial parts of this research were accomplished.
Also, we wish to acknowledge discussions with Len Bos, David Dalton, Michael Rochester and Theodore Stanoev, as well as the
editorial support of David Dalton and graphical support of Elena Patarini.
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 238416-2013, the Polish National
Science Center under contract No. DEC-2013/11/B/ST10/0472, and by AGH - University of Science and Technology as a part of
the statutory project No. 11.11.140.613.
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Dolomites Research Notes on Approximation ISSN 2035-6803
... Let us consider a stack of alternating transversely isotropic layers, given in Table 3, where the parameters of the odd-numbered layers are from tensor C TI a of Danek et al. [19] and the parameters of the even-numbered layers are twice those of the tensor C TI bb of Danek et al. [19]. We chose these parameters to illustrate varying levels of anisotropy, quantified by parameters (12), (13) and (14). ...
... Let us consider a stack of alternating transversely isotropic layers, given in Table 3, where the parameters of the odd-numbered layers are from tensor C TI a of Danek et al. [19] and the parameters of the even-numbered layers are twice those of the tensor C TI bb of Danek et al. [19]. We chose these parameters to illustrate varying levels of anisotropy, quantified by parameters (12), (13) and (14). ...
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