On closest isotropic tensors and their norms

Abstract

An anisotropic elasticity tensor can be approximated by the closest tensor belonging to a higher symmetry class. The closeness of tensors depends on the choice of a criterion. We compare the closest isotropic tensors obtained using four approaches: the Frobenius 36-component norm, the Frobenius 21-component norm, the operator norm and the L2 slowness-curve fit. We find that the isotropic tensors are similar to each other within the range of expected measurement errors.

Full text

arXiv:submit/1533808 [physics.geo-ph] 13 Apr 2016
On closest isotropic tensors and their norms
Tomasz Danek∗
, Andrea Noseworthy†
, Michael A. Slawinski‡
April 13, 2016
Abstract
An anisotropic elasticity tensor can be approximated by the closest tensor belonging to a higher
symmetry class. The closeness of tensors depends on the choice of a criterion. We compare the closest
isotropic tensors obtained using four approaches: the Frobenius 36-component norm, the Frobenius 21-
component norm, the operator norm and the L2slowness-curve fit. We find that the isotropic tensors are
similar to each other within the range of expected measurement errors.
1 Introduction
For an elasticity tensor obtained from empirical information, the resulting symmetry class is explicitly the
property of a Hookean solid represented by that tensor, where this solid is a mathematical analogy of the
physical material in question. The inference of properties of that material requires further interpretation.
Among these properties there are symmetries of such a material, hence, it is useful to examine symmetries
of its models. In particular, it is useful to compute an isotropic counterpart of the obtained tensor. The
decision then lies in choosing an appropriate norm to compute the counterpart, hence the crux of this paper.
We compare isotropic counterparts according to the Frobenius-36 norm, Frobenius-21 norm, which we refer
to as F36 and F21 , respectively, as well as according to the operator norm and the L2slowness-curve fit,
which we refer to as λand L2, respectively.
2 Elasticity tensors
A Hookean solid, cijkℓ, is a mathematical object defined by Hooke’s Law,
σij =
3
X
k=1
3
X
ℓ=1
cijkℓεkℓ , i, j = 1,2,3,(1)
∗Department of Geoinformatics and Applied Computer Science, AGH-University of Science and Technology, Kraków, Poland,
Email: tdanek@agh.edu.pl
†Department of Earth Sciences, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada,
Email: anoseworthy@mun.ca
‡Department of Earth Sciences, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada,
Email: mslawins@mac.ca
1

where σij ,εkℓ and cijkℓ 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—as a symmetric second-rank
tensor in R6,
C=









c1111 c1122 c1133 √2c1123 √2c1113 √2c1112
c1122 c2222 c2233 √2c2223 √2c2213 √2c2212
c1133 c2233 c3333 √2c3323 √2c3313 √2c3312
√2c1123 √2c2223 √2c3323 2c2323 2c2313 2c2312
√2c1113 √2c2213 √2c3313 2c2313 2c1313 2c1312
√2c1112 √2c2212 √2c3312 2c2312 2c1312 2c1212









.(2)
For transverse isotropy, the components of Cbecome
CTI =








c1111 c1122 c1133 0 0 0
c1122 c2222 c2233 0 0 0
c1133 c2233 c3333 0 0 0
0 0 0 2c2323 0 0
0 0 0 0 2c2323 0
0 0 0 0 0 c1111 −c1122








.(3)
For isotropy, the components of Cbecome
Ciso =








c1111 c1111 −2c2323 c1111 −2c2323 0 0 0
c1111 −2c2323 c1111 c1111 −2c2323 0 0 0
c1111 −2c2323 c1111 −2c2323 c1111 0 0 0
0 0 0 2c2323 0 0
0 0 0 0 2c2323 0
0 0 0 0 0 2c2323








,(4)
and expression (1) can be written as
σij =c1111 δij
3
X
k=1
εkk + 2 c2323 εij , i, j = 1,2,3.
3 Norms
To examine the closeness between elasticity tensors, as discussed by Bos and Slawinski (2013) and by
Danek et al. (2013, 2015), we consider possible norms of tensor (2).
3.1 Frobenius norms
The Frobenius norm treats a matrix in Rn×nas 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
u
t
6
X
m=1
6
X
n=1
C2
mn ,
2

which uses the thirty-six components, including their coefficients of √2and 2, or
||C||F21 =v
u
u
t
6
X
m=1
m
X
n=1
C2
mn ,
which uses only the twenty-one independent components, including their coefficients of √2and 2.
3.2 Operator norm
The operator norm of a symmetric 6×6matrix is
||C||λ= max |λi|,
where λi∈ {λ1,...,λ6}, is an eigenvalue of C. As discussed by Bos and Slawinski (2013), such a norm of
cijkℓ results from equation (1) if σij and εkℓ are a priori endowed with a Frobenius norm. If cij kℓ is a priori
endowed with a Frobenius norm, its origin in the σij and εkℓ norm does not exhibit any standard form.
4 Slowness-curve L2fit
In a manner similar to the F36 norm, F21 norm and operator norm, the slowness-curve L2fit is used to find
an isotropic counterpart of 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.
In this approach, in a manner similar to the operator norm, we do not invoke explicit expressions for
the components of the closest elasticity tensor but we examine the effect of these components on certain
quantities. For the operator norm, this quantity consists of eigenvalues; for the slowness-curve 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.
The best fit, in the L2sense, is the radius, r, that minimizes
S=
n
X
i=1
(si−ri)2,(5)
where siare ndiscretized values along the slowness curve, and si−riis measured in the radial direction.
Hence, ris the radius of the slowness circle; it corresponds to isotropy.
5 Numerical results
5.1 Tensor C
In this section, we investigate isotropic counterparts for the three norms introduced in Section 3. For that
purpose, we use a transversely isotropic tensor derived from a generally anisotropic tensor obtained by
3

Dewangan and Grechka (2003),
C=









7.8195 3.4495 2.5667 √2(0.1374) √2(0.0558) √2(0.1239)
3.4495 8.1284 2.3589 √2(0.0812) √2(0.0735) √2(0.1692)
2.5667 2.3589 7.0908 √2(−0.0092) √2(0.0286) √2(0.1655)
√2(0.1374) √2(0.0812) √2(−0.0092) 2(1.6636) 2(−0.0787) 2(0.1053)
√2(0.0558) √2(0.0735) √2(0.0286) 2(−0.0787) 2(2.0660) 2(−0.1517)
√2(0.1239) √2(0.1692) √2(0.1655) 2(0.1053) 2(−0.1517) 2(2.4270)









.(6)
Its components are density-scaled elasticity parameters.
5.2 Tensor CTI
aand its isotropic counterparts
5.2.1 Tensor CTI
a
Let us consider a transversely isotropic tensor (Danek et al. 2013), which is the closest—in the F36 sense—
counterpart 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)








.(7)
Isotropic tensors discussed herein are counterparts of this tensor. The slowness curves for tensor (7) and its
isotropic counterpart circles discussed in Sections 5.2.2, 5.2.3 and 5.2.4, below, are shown in Figure 1; these
counterparts nearly coincide with each other.
5.2.2 F36 norm
Let us consider the Frobenius norm using the thirty-six components. There are analytical formulæ to
calculate—from a generally anisotropic tensor—the two parameters of its closest isotropic tensor (Voigt, 1910).
From a transversely isotropic tensor, these parameters are
cisoF36
1111 =1
15(8cTI
1111 + 4cTI
1133 + 8cTI
2323 + 3cTI
3333)
and
cisoF36
2323 =1
15(cTI
1111 −2cTI
1133 + 5cTI
1212 + 6cTI
2323 +cTI
3333).
Hence, the closest isotropic counterpart of tensor (7) 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)








.(8)
4

5.2.3 F21 norm
Let us consider the Frobenius norm using the twenty-one independent components. The analytical formulæ
to calculate the two parameters of its closest isotropic tensor are (Slawinski, 2016)
cisoF21
1111 =1
9(−cTI
1122 + 2(3cTI
2222 +cTI
2233 + 2cTI
2323 +cTI
3333))
and
cisoF21
2323 =1
18(−5cTI
1122 + 6cTI
2222 −2cTI
2233 + 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)








.(9)
5.2.4 λnorm
Unlike the Frobenius norms, the operator norm has no analytical formulæ for cisoλ
1111 and cisoλ
2323 . They must
be obtained numerically. For tensor (7), 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)








.(10)
5.2.5 Distances among tensors
To gain insight into different isotropic counterparts of tensor (7), we calculate the F36 distance between
tensors (8) and (10), which is 0.8993 . The F36 distance between tensors (7) and (8) is 1.8461 . The F36
distance between tensors (7) and (10) is 2.0535 , where we note that tensor (10) 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 (7).
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 (8), (9), (10), and their wavefront-slowness circles in Figure 1. 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.
5.3 Comparison of norms
Comparing tensors (8), (9) and (10), 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, a unique answer is obtained by a straightforward calculation. For different
norms, there is no unique answer: the sequence in closeness to isotropy can be reversed between two tensors.
5

Figure 1: Slowness curves for tensor (7):
solid lines represent the qP ,qSV and SH
waves; dashed lines represent the Pand
Swaves according to F36 norm; dashed-
dotted lines represent the Pand Swaves
according to F21 norm; the results of these
norms almost coincide; dotted lines rep-
resent the Pand Swaves according to
λnorm.
Figure 2: Slowness curves for tensor (11):
solid lines represent the qP ,qSV and SH
waves; dashed lines represent the Pand S
waves according to F36 norm; dotted lines
represent the Pand Swaves according to
F21 norm.
5.3.1 F36 versus F21
Using a numerical search, an elasticity tensor is generated that is further from isotropy than tensor (7)
according to the F36 norm, but closer to isotropy than tensor (7) 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)








,(11)
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)








(12)
6

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)








,(13)
respectively. The distances to isotropy for CTI
aand CTI
b, using the F36 and F21 norms, are
da21 = 1.6372 > db21 = 1.5517 ,
da36 = 1.8460 < db36 = 2.0400 .
The slowness curves for tensor (11) and its isotropic counterparts are shown in Figure 2.
5.3.2 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)








,(14)
which is further from isotropy according to the F36 norm and closer to isotropy according to the λnorm. 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)








(15)
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)








,(16)
respectively. The distances to isotropy for CTI
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 (14) and its isotropic counterparts are shown in Figure 3.
7

5.3.3 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)








,(17)
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)








(18)
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)








,(19)
respectively. The distances to isotropy for both CTI
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 (17) and its isotropic counterparts are shown in Figure 4.
5.4 Slowness-curve fit
Considering tensor (7) 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 Pand Swaves, 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=√c1111 and vS=√c2323 are the P-wave and S-wave speeds, respectively, it follows that
8

Figure 3: Slowness curves for tensor (14):
solid lines represent the qP ,qSV and SH
waves; dotted lines represent its Pand S
waves according to F36 norm; dashed lines
represent its Pand Swaves according to
λnorm.
Figure 4: Slowness curves for tensor (17):
solid lines represent the qP ,qSV and SH
waves; dotted lines represent its Pand S
waves according to F21 norm; dashed lines
represent its Pand Swaves according to
λnorm.
c1111 = 1/r2
Pand 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)








.(20)
The slowness curves for tensor (20) and its isotropic counterparts are shown in Figure 5.
5.5 Thomsen parameters
Tensors (7), (11), (14) and (17) exhibit the strength of anisotropy that is consistent with cases of interest to
geophysicists. To show this consistency, we calculate the Thomsen (1986) parameters using the following
formulæ,
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),
ǫ=cTI
1111 −cTI
3333
2cTI
3333
.
9

Figure 5: Slowness curves for tensor (20): solid lines represent the qP ,qSV and SH waves; dotted lines
represent its Pand Swaves according to the slowness-curve L2fit.
The values of these parameters for tensors (7), (11), (14) and (17) are shown in Table 1. Comparing results
of this table to data of Auld (1973) and Thomsen (1986), we see that these tensors can represent common
geological materials.
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 (7), (11), (14) and (17)
5.6 Error propagation
Components of an anisotropic tensor obtained from experimental measurements exhibit uncertainties due to
measurement errors. These uncertainties propagate to its symmetric counterparts. The standard deviations
of components of tensor (6) are (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








.(21)
These values do not constitute components of a tensor. They are valid only in the coordinate system of
measurements; rotation is not allowed. Hence, to consider error propagation from tensor (6) to tensor (7),
there is a need for a simulation. Probability distributions about the mean values of the components of
10

tensor (7)—obtained by a Monte-Carlo simulation (Danek et al. 2013)—are shown in Figures 6, 7, 8, 9, 10.
Different histograms have different horizontal scales.
The probability distributions of the two parameters for its isotropic F36 counterpart are obtained in the
same manner and shown in Figures 11 and 12. Their mean values are given in tensor (8). The proba-
bility distributions of parameters for its F21 counterpart are shown in Figures 13 and 14. The probability
distributions of parameters for its λcounterpart are shown in Figures 15 and 16.
7.50 8.50
Figure 6: c1111 of tensor (7)
3.00 3.50
Figure 7: c1122 of tensor (7)
2.00 2.80
Figure 8: c1133 of tensor (7)
1.70 2.00
Figure 9: c2323 of tensor (7)
6.50 7.50
Figure 10: c3333 of
tensor (7)
11

7.10 7.60
Figure 11: c1111 of tensor (8)
2.10 2.35
Figure 12: c2323 of tensor (8)
7.20 7.70
Figure 13: c1111 of tensor (9)
2.05 2.30
Figure 14: c2323 of tensor (9)
6 Discussions and conclusions
In Section 3, we consider several types of norms for obtaining—for a transversely anisotropic tensor—its
closest isotropic counterpart. We examine the Frobenius norms and the operator norm. In Section 4, we
consider the slowness-curve L2fit to obtain such a counterpart.
The closeness to isotropy is norm-dependent. Yet, the order of several tensors according to their close-
ness to isotropy for a particular norm remains the same for all norms. However, as shown in Sections 5.2, 5.3
and 5.5, given tensor (7) we can find another tensor—representative of common geological materials—such
that one of them is closer to isotropy according to one norm and the other closer to isotropy according to
another norm.
In view of Section 5.6, we conclude that the results of the three norms and the slowness-curve fit are so
similar to each other that their corresponding values might be indistinguishable in the context of measure-
ment errors. Thus, the choice of the closeness criterion might be of secondary importance. Pragmatically,
12

7.50 8.00
Figure 15: c1111 of tensor (10)
2.20 2.60
Figure 16: c2323 of tensor (10)
we might choose a Frobenius norm, since it offers analytical formulæ to obtain an isotropic counterpart.
Both Frobenius norms result in similar counterparts, since they differ only by a weight doubling of the off-
diagonal components, whose values are small. Also, in view of this similarity, the preference in closeness
to isotropy for the pairs of tensors discussed in Section 5.3 might be indistinguishable.
Performing a simple error-propagation analysis, we observe that—for Frobenius norms—probability
distributions of the corresponding parameters are very similar to one another. For the operator norm, how-
ever, the c2323 distributions differ significantly. This result might be a consequence of the properties of the
operator norm, where only the largest among six eigenvalues is taken into consideration.
Acknowledgments
We wish to acknowledge discussions with David Dalton, Michael G. Rochester and Theodore Stanoev, as
well as graphical support by Elena Patarini. This research was performed in the context of The Geomechan-
ics Project supported by Husky Energy. Also, this research was partially supported by the Natural Sciences
and Engineering Research Council of Canada, grant 238416-2013, and by the Polish National Science Cen-
ter under contract No. DEC-2013/11/B/ST10/0472.
13

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