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On Choosing Effective Elasticity Tensors Using a Monte-Carlo Method


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A b s t r a c t A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its counterparts in different symmetry classes, as well as of their orientations and elasticity parameters.
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Acta Geophysica
vol. 63, no. 1, Feb. 2015, pp. 45-61
DOI: 10.2478/s11600-013-0197-y
Ownership: Institute of Geophysics, Polish Academy of Sciences;
© 2015 Danek and Slawinski. This is an open access article distributed under the Creative
Commons Attribution-NonCommercial-NoDerivs license,
On Choosing Effective Elasticity Tensors
Using a Monte-Carlo Method
Tomasz DANEK1,2 and Michael A. SLAWINSKI1
1Department of Earth Sciences, Memorial University of Newfoundland,
St. John’s, Canada; e-mail:
2Department of Geoinformatics and Applied Computer Science,
AGH – University of Science and Technology, Kraków, Poland
e-mail: (corresponding author)
A generally anisotropic elasticity tensor can be related to its closest
counterparts in various symmetry classes. We refer to these counterparts
as effective tensors in these classes. In finding effective tensors, we do
not assume a priori orientations of their symmetry planes and axes.
Knowledge of orientations of Hookean solids allows us to infer proper-
ties of materials represented by these solids. Obtaining orientations and
parameter values of effective tensors is a highly nonlinear process in-
volving finding absolute minima for orthogonal projections under all
three-dimensional rotations. Given the standard deviations of the compo-
nents of a generally anisotropic tensor, we examine the influence of
measurement errors on the properties of effective tensors. We use a glo-
bal optimization method to generate thousands of realizations of a gener-
ally anisotropic tensor, subject to errors. Using this optimization, we per-
form a Monte Carlo analysis of distances between that tensor and its
counterparts in different symmetry classes, as well as of their orientations
and elasticity parameters.
Key words: anisotropy, vertical seismic profile (VSP), inversion.
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The purpose of this paper is to examine a generally anisotropic elasticity ten-
sor, expressed in terms of twenty-one elasticity parameters and obtained
from vertical seismic profiling (VSP) measurements, to infer properties of
materials represented by this tensor. Beginning with a generally anisotropic
tensor, we are able to infer these properties by relating this tensor to its clos-
est counterparts in the sense of Frobenius norm, as defined by Gazis et al.
(1963), in all material symmetries of Hookean solids, as shown by Danek et
al. (2013). Herein, we focus our attention on examining symmetries used in
seismology: monoclinic, orthotropic, transversely isotropic, and isotropic
tensors. Following the definition and nomenclature of Kochetov and Slawin-
ski (2009a, b), we refer to these counterparts as effective tensors of these
symmetry classes. Note that we use generally anisotropic and orthotropic,
not triclinic, and orthorhombic crystals, respectively. The latter terms are as-
sociated with lattice symmetries of crystals, while in seismology we deal
with continua and their symmetries, which are symmetries of the elasticity
tensor. In such a case, orthotropic refers to three mutually orthogonal sym-
metry planes.
Consideration of a generally anisotropic tensor allows us to examine ef-
fective tensors belonging to distinct symmetry classes without a bias of prior
assumptions. In other words, the choice of the symmetry class model is
guided by the data from which the generally anisotropic tensor is derived.
Obtaining these orientations and parameter values is a mathematically
involved process. Explicit underpinnings of the methodology used in this
paper are presented by Danek et al. (2013), and the reader is referred to that
publication and references therein. Herein, we provide an overview to render
the present paper self-contained.
To infer information about materials examined through VSP measure-
ments, we consider relationships between the obtained tensor and its sym-
metric counterparts. Such a tensor was obtained by Dewangan and Grechka
(2003) from multi-component and multi-azimuth walkaway VSP data, and
such relationships are considered in terms of distance between tensors, as
proposed by Gazis et al. (1963). The concept of such a distance is discussed
by several researchers, including Norris (2006), Bόna (2009), and Kochetov
and Slawinski (2009a, b). The present work, which is formulated in the con-
text of a computationally efficient global optimization, allows us to obtain
thousands of solutions within a few hours on a multi-core CPU computer.
Hence, we can infer properties of materials, together with reliability of such
inferences, by examining distributions, illustrated by histograms, of the elas-
ticity parameters of effective tensors and distributions of orientations of the-
se tensors. A computationally efficient scheme is crucial for generating such
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As discussed by Kochetov and Slawinski (2009a), the distance between
a generally anisotropic tensor and its counterpart belonging to a given sym-
metry class is obtained by finding the orientation that minimizes the dis-
tance. Performing a search under all orientations leads to a highly nonlinear
optimization problem, which commonly exhibits many local minima. In the
past, local-optimization methods have been used, which must be restricted to
the vicinity of the global minimum to avoid convergence to a local one.
A restriction based on visual examination of distance plots was proposed by
Kochetov and Slawinski (2009a, b). Such an examination, however, is prac-
tically impossible if we wish to perturb the generally anisotropic tensor thou-
sands of times to consider the effect of errors on the distribution of values
that describe properties of effective tensors. Herein, we address this problem
by applying a global optimization. Using this method, we can find effective
tensors by a Monte Carlo (MC) method (see, e.g., Tarantola 2005), and de-
termine distributions of their orientations and of their elasticity parameters.
These distributions arise from errors in which the original tensor is given.
That is, our inversion consists of distributions of values that describe proper-
ties and orientations of effective tensors. These distributions, which are akin
to error bars, allow us to gain an insight into the reliability of a given effec-
tive tensor in representing the generally anisotropic one.
This paper has a following layout. First, we review the concept of the ef-
fective elasticity tensor and describe the global optimization used in its
search. Then, using this optimization, we examine the generally anisotropic
tensor obtained by Dewangan and Grechka (2003) and discussed also in
Chapter 9 of Tsvankin and Grechka (2011). Next we analyze the sensitivity
of the solution to perturbation of elasticity tensor C using MC technique.
This way we can evaluate the reliability of the solutions.
For a fixed coordinate system, we can relate a general elasticity tensor, c, to
its counterpart, csym, which belongs to a particular symmetry class. Tensor
csym is the orthogonal projection of c, in the sense of the Frobenius inner
product (which is the sum of products of the corresponding components, aijkl
bijkl), onto the linear space containing all tensors of that symmetry class, as
described by Gazis et al. (1963).
The distance-squared between c and csym is:
2sym sym
sym .dcc cc=− = − (1)
The second equality is a consequence of the orthogonality of c and csym.
Components of tensor csym, and, hence, the value of distance obtained from
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Eq. 1, depend on the orientation of the coordinate system. To determine the
effective tensor without assuming an a priori orientation, we must minimize
dsym in Eq. 1 with respect to orientations; in other words, we have to perform
the minimization of dsym under all rotations. To find the solution of this high
dimensional minimization problem, taking into account an existence of mul-
tiple local minima, we formulate a metaheuristic global approach. In this ap-
proach, no prior knowledge about a solution is required. The only require-
ment is that – given two arbitrary points within that space – a candidate for
a solution can be chosen based on the difference of value of target function.
The manner in which candidates are selected depends on the choice of algo-
rithm. As a search strategy, we choose particle swarm optimization (PSO)
because of its simplicity and speed of computation. This search strategy was
formulated by Kennedy and Eberhart (1995) and used by Danek et al. (2013)
to find the closest tensor of a given symmetry class. Furthermore – unlike
other metaheuristics, say, genetic algorithms or simulated annealing – PSO
does not require algorithm-parameter tuning (see Donelli et al. 2006). PSO is
the stochastic technique that simulates social behavior of animals searching
for food, as exemplified by a swarm of fish, insects, etc. In the present case,
each particle represents a set of quaternion parameters in a four-dimensional
solution space. We choose this representation because quaternions are par-
ticularly convenient for describing three-dimensional rotations (see, e.g.,
Stillwell 2008). In particular, they are computationally more convenient than
the Euler angles. During the optimization process, each particle is “aware” of
three positions: its current position, xi, its best individual position, pi, and the
best position of the entire swarm, pg. Best positions are points in the solution
space for which a target function exhibits the lowest value obtained in all
previous iterations.
The amplitude of a jump from the previous to the current position is de-
fined by parameter vi, called velocity; its value depends on the difference be-
tween the best position of an individual particle and the best position found
so far by all particles. The canonical PSO formula is (Clerc and Kennedy
0, 0, ,
ii ii gi
=Φ− + Φ − Φ
Φ=Φ +Φ >
where U represents uniform distribution and is a component wise multi-
plication. Commonly, Φ, which is the sum of weights of a personal and
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swarm information, Φ1 and Φ2, respectively, is usually set to 4.1, which
means that a constant velocity multiplier, χ, is approximately 0.73 and U is a
random number between 0 and approximately 1.5, if both weights are equal.
This scheme guarantees convergences without particle velocity limitations.
The flowchart of the algorithm is presented in Fig. 1. Since the elasticity ten-
sor possesses index symmetries, cijkl = cjikl = cklij (see, e.g., Slawinski 2010),
we can write its components as entries of a symmetric 6 × 6 matrix. Hence,
Hooke’s law,
ij ijkl kl
kl c
= (3)
can be written in a manner that allows us for a convenient display of elastic-
ity parameters, namely,
Fig. 1. Flowchart describing the par-
ticle-swarm-optimization algorithm:
xi is the current particle position, pi
is its best position, and pg is the best
position for the entire swarm; see
text for details about the algorithm.
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11 1111 1122 1133 1123 1113
22 1122 2222 2233 2223 2213
33 1133 2233 3333
3323 3313
23 1123 2223 3323 2323 2313
13 1113 2213 3313 2313
12 1112 2212 3312 2312
ccc cc
ccc cc
ccc cc
1313 1311
1312 1212 12
22 2
⎤⎡ ⎤
⎥⎢ ⎥
⎥⎢ ⎥
⎥⎢ ⎥
⎥⎢ ⎥
⎥⎢ ⎥
⎥⎢ ⎥
⎥⎢ ⎥
⎦⎣ ⎦
which we refer to as the Kelvin notation, we shall denote the elasticity tensor
by C. Tensor C, which includes factors of 2 or 2 in its entries, allows us to
keep the same norm for both the strain and stress tensors, and as a conse-
quence allows us to conveniently examine rotations associated with symme-
try classes (see, e.g., Chapman 2004). Also, unlike the so-called Voigt
notation, Eq. 4 is a vector equation. Using the Kelvin notation, we can write
the squared distance between c and csym as
sym ,dCC=− (5)
which is equivalent to Eq. 1.
The crux for obtaining effective tensors by realizations of a generally anisot-
ropic tensor perturbed by errors relies on the aforementioned global optimi-
zation. We apply this method to the tensor obtained from VSP measurements
by Dewangan and Grechka (2003):
7.8195 3.4495 2.5667 2(0.1374) 2(0.0
3.4495 8.1284 2.3589 2(0.0812)
2.5667 2.3589 7.0908 2( 0.0092)
2(0.1374) 2(0.0812) 2( 0.0092) 2(1.6636)
2(0.0558) 2(0.0735) 2( 0.0286) 2( 0.0787)
2(0.1239) 2(0.1692) 2(0.1655) 2(0.1053)
558) 2(0.1239)
2(0.0735) 2(0.1692)
2( 0.0286) 2(0.1655)
2( 0.0787) 2(0.1053)
2(2.0660) 2( 0.1517)
2( 0.1517) 2(2.4270)
The components of this generally anisotropic tensor are the density-
scaled elasticity parameters; their units are km2/s2. Entries of matrix 6 were
obtained with the following standard deviations:
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0.1656 0.1122 0.1216 2(0.1176) 2(0.0774
0.1122 0.1862 0.1551 2(0.0797)
0.1216 0.1551 0.1439 2(0.0856)
2(0.1176) 2(0.0797) 2 (0.0856) 2(0.0714)
2(0.0774) 2(0.1137) 2 (0.0662) 2(0.0496)
2(0.0741) 2(0.0832) 2(0.1010) 2(0.0542)
2(0.1137) 2(0.0832)
2(0.0662) 2(0.1010)
2(0.0496) 2(0.0542)
2(0.0626) 2(0.0621)
2(0.0621) 2(0.0802)
Unlike matrixes 6 and 7 does not consist of components of a tensor; it
does not satisfy the conditions of tensorial transformations. Thus, S is fixed
in the coordinate system in which components 6 are expressed; it can be
used as a measure of errors in that system only.
In writing Eqs. 6 and 7, we do not imply that the number of decimal
points corresponds to the number of significant digits. We use more deci-
mals to examine numerical stability and to compare accurately our results
with those of Kochetov and Slawinski (2009a, b).
To examine the influence of errors, we generate thousands of realizations
of tensor 6 with random perturbations whose standard deviations are given in
matrix 7. For each realization, using the PSO method, we obtain the effective
orthotropic tensor, whose natural orientation is illustrated in Fig. 2, and
Fig. 2. Four clusters of the effective orthotropic tensors: each black dot is the orien-
tation of the effective tensor corresponding to a realization of tensor 6 subject to er-
rors 7; gray points are projections of black points. The vertex of black lines is the
orientation of tensor 8, which results from tensor 6, without errors, and is the closest
to the original coordinate system. The axes denote the three Euler angles. Each clus-
ter corresponds to an equally valid natural coordinate system; their slightly different
appearances are a result of random perturbation.
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whose axes are the three Euler angles: the azimuth, defined by rotation about
the x3-axis, the tilt, by rotation about the x1-axis, and the bank, by rotation
about the new x3-axis. Values of these angles are obtained from the quater-
nions used in the PSO. This figure contains four clusters, which correspond
to orientations of natural coordinate systems of orthotropic tensors. Orienta-
tions of these systems differ in bank by π/2 or its multiple. None of these
systems has a privileged status; our choice is a matter of convenience.
Choosing effective tensor
Since a symmetry class has more than one natural coordinate system – each
associated with an aforementioned cluster – for our examinations, we choose
systems that are closest to the one in which components (Eq. 6) are ex-
pressed. Following the global optimization, components of the effective
orthotropic tensor – derived from tensor 6 without errors 7 – in the closest
natural coordinate system are
7.7740 3.3634 2.4276 0 0 0
3.3634 8.3762 2.4879 0 0 0
2.4276 2.4879 7.0810 0 0 0
0002(1.6497)0 0
0 0 0 0 2(2.0784) 0
0 0 0 0 0 2(2.3323)
and the azimuth, tilt, and bank are 2.4°, 2.6°, and 19.3°, respectively. Note
the similarity (expected) between tensors 8 and 6. Also to ensure consis-
tency, note that expression 25 in Kochetov and Slawinski (2009a) and ex-
pression 25 in Danek et al. (2013) describe the same effective tensor but
stated in a natural coordinate system that, relative to expression 8, is rotated
by π/2 about the new x3-axis.
According to the work of Dewangan and Grechka (2003) and Kochetov
and Slawinski (2009a), tensor 6 can be represented by its counterpart exhib-
iting orthotropic symmetry. Let us use our method to examine whether or not
a lesser or greater symmetry is a good representation of tensor 6 subject to
errors 7.
Using the aforementioned global optimization, we obtain the distribution
of shortest distances of tensor 6 to effective tensors of monoclinic, ortho-
tropic, transversely isotropic, and isotropic symmetries. We compare these
distributions to the distribution of the Frobenius norm of errors 7, which we
obtain by generating random realizations of a zero tensor with these errors.
This operation results in a distribution that is a square-root of the sum of
thirty-six squares of independent random variables, Mij, having a normal dis-
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tribution with the zero mean and standard deviations given in matrix 7. Even
though this distribution could be obtained analytically (see, e.g., Mathai and
Provost 1992), we use the same numerical method that we use to obtain the
distributions of distances of tensor 6 to effective tensors.
To clarify that the expected value of the norm of
11 16
61 66
(0, ) (0, )
(0, ) (0, )
where 2
(0, )
S are random variables and Sij are entries of matrix 7, is not
equal to zero, let us consider the variance
Var( ) : ,XEX EX
⎣⎦ (10)
where X is a random variable and E denotes the expected value. In our case,
E[X] = 0, so the expected value of the square of random variable is equal to
its variance. Hence, the expected value of the norm of matrix 9 is
= , (11)
which is the norm of matrix 7, namely, 0.7844. This value is in agreement
with the value obtained numerically: 0.7747, which corresponds to the loca-
tion of the apex of the black line in Fig. 3.
Fig. 3. Density distributions: the black line represents density of the norm of ma-
trix 9, whose mean value – by expression 11 – is equal to the norm of matrix 7. Pro-
ceeding from left to right, gray lines represent densities of distance distributions for
monoclinic, orthotropic, transversely isotropic, and isotropic symmetries, respectively.
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In Figure 3, the distribution of errors 7 overlaps not only with the distri-
bution of distances to the orthotropic symmetry but also to the monoclinic
and transversely isotropic symmetries. As suggested by Kochetov and
Slawinski (2009b), we view the overlap between the distribution of the dis-
tance from tensor 6 with its symmetric counterpart and with the distribution
of the Frobenius norm of errors 7 as an indication that a symmetric tensor
might represent tensor 6. Hence, the following question arises: could we
choose a tensor of monoclinic or transversely isotropic symmetry to repre-
sent tensor 6?
To address this question, we note that the natural-coordinate expressions
of the orthotropic, and higher, symmetries require c1112 = c2212 = c3313 = 0
(see, e.g., Slawinski 2010). For examination of this issue, we generate thou-
sands of realizations of tensor 6 subject to errors 7, and express them in the
orientations of their closest monoclinic counterparts. Since (0, 0, 0) is in the
center of the obtained cluster shown in Fig. 4, we conclude that tensor 6 with
errors 7 appears to be more symmetric than monoclinic.
Also, the natural-coordinate expressions of transverse isotropy require
c1111 = c2222, c1133 = c2233, and c2323 = c1313. To examine this issue, we gener-
ate thousands of realizations and express them in the orientations of their
closest transversely isotropic counterparts. Examining the left panels in
Fig. 4. Selected entries of realizations of tensor 6 subject to errors 7 in coordinate
systems whose orientations correspond to the closest monoclinic tensors. Note that –
with no constraints applied – the values of c1112, c2212, and c3313 are scattered around
zeros, which are highlighted by solid lines. From this pattern we infer that tensor 6
can be represented by an effective tensor whose symmetry is higher than monoclinic.
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Fig. 5. Selected entries of realizations of tensor 6 subject to errors 7 expressed in co-
ordinate systems whose orientations correspond to the closest transversely isotropic
tensors, and their marginal distributions. For picture clarity, only 1000 points are
shown on the left panel.
Fig. 5, we observe that obtained clusters are crossed by dashed lines showing
these equalities; however, from examination of the right panel, we see that
two-dimensional marginal distributions show that areas of the highest den-
sity are away from these equalities. We conclude that tensor 6 with errors 7
exhibits a lesser symmetry than transverse isotropy.
Thus, we choose the orthotropic symmetry to represent tensor 6.
Properties of chosen effective tensor
Having accepted that tensor 6 subject to errors 7 can be represented by an ef-
fective orthotropic tensor, let us examine its properties in the context of er-
rors. These properties are the elasticity parameters, whose distributions are
illustrated in Fig. 6, and orientation, whose distribution is illustrated in Fig. 7.
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Fig. 6. Histograms of density-scaled elasticity parameters of the effective orthotropic
tensors obtained from realizations of tensor 6, subject to errors 7, in natural coordi-
nates whose orientations are illustrated in Fig. 7. Solid lines correspond to the values
for the error-free case.
Furthermore, from the properties of this tensor we can infer properties of
materials examined by VSP measurements from which tensor 6 is obtained.
The behavior of the histograms in Fig. 6, including their unimodality and
relatively confined widths, suggests that the orthotropic symmetry contains
much information about tensor 6 subject to errors 7, which is consistent with
the symmetry choice discussed above. Examining the azimuth and tilt dis-
played in Fig. 7, we see that one of the symmetry planes of the effective or-
thotropic tensor is close to horizontal. The value of the bank indicates that
the axes of the system in which tensor 6 is expressed are oblique to natural
coordinates of the effective tensor. This information can be used to infer ori-
entations of layers and fractures in a manner akin to those examined by
Grechka and Kachanov (2006).
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Fig. 7. Histograms of the azimuth, tilt, bank (in degrees), and distance (in km2/s2),
from tensor 6 to effective orthotropic tensors obtained from realizations of tensor 6
subject to errors 7. The first three histograms are projections of the cluster in Fig. 2 –
whose bank is close to zero – to the three axes therein. Solid lines correspond to the
values for the error-free case.
To gain an insight into the strength of anisotropy under consideration,
following the formulation proposed by Tsvankin (1997) let us express the
pertinent components of tensor 8 in terms of the seven parameters that are
zero for the case of isotropy. Our results shown in Fig. 8 are consistent with
those presented by Dewangan and Grechka (2003) and further elaborated by
Tsvankin and Grechka (2011), Section 7.13.
As shown in Fig. 8, distributions of several among the aforementioned
parameters contain zero. Nevertheless, absolute average values of δ(1) and γ(2)
are close to 0.2, which suggests that anisotropy is not weak. Moreover, the
shear-wave-splitting coefficient, (c1313 c2323)/(2c2323), which is important in
fracture detection, is about 0.12; again, it is similar to 0.1 obtained by
Tsvankin and Grechka (2011). This value is relatively large since, typically,
the observed splitting coefficients are less than 0.05 (Tsvankin 2013, pers.
Note that values of i in δ(i), ε(i), and γ(i), in this figure, are interchanged
with respect to values in Kochetov and Slawinski (2009b) because the coor-
dinate systems differ by π/2 about the new x3-axis. Similarly, the interchange
in Dewangan and Grechka (2003) and Tsvankin and Grechka (2011) is
a consequence of coordinate systems belonging to different clusters shown in
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Fig. 8. Histograms of the seven elasticity parameters of orthotropic symmetry,
whose values are zero in the case of isotropy. Solid lines correspond to the values
for the error-free case.
Fig. 2. None of the clusters is privileged, as long as all results are expressed
with respect to the same system.
The presented method allows us to infer from seismic measurements infor-
mation about materials represented by a generally anisotropic tensor. This
method extends the approach introduced by Kochetov and Slawinski (2009a)
in two important ways.
First, as discussed by Danek et al. (2013), it invokes a global optimiza-
tion method, which allows us to directly consider tensors of all symmetry
classes, regardless of their orientation being described by two or three Euler
angles. We note that, as discussed by Kochetov and Slawinski (2009a, b),
constraining the local search to obtain absolute minima requires an examina-
tion that is possible only for tensors whose orientations are described by only
two Euler angles; such tensors are either monoclinic or transversely iso-
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Second, this direct approach allows us to perturb the values of the origi-
nal tensor thousands of times to obtain estimates of its effective tensors in
the presence of errors. Hence, by using the Monte-Carlo approach, we can
estimate ranges of values for elasticity parameters and orientations of effec-
tive tensors for cases such as that represented by tensor 6 with errors 7.
Within the assumption of a normal distribution of errors, one could also
examine the best fit in terms of likelihood by including errors in the distance
function, as considered by Bόna (2009). We use the same errors to perturb
the original tensor and – while remaining within coordinate-invariant defini-
tion of distance – obtain distributions of solutions, within whose range we
would find the effective tensor obtained by the approach of Bόna (2009).
In this study, we confirm and further quantify conclusions obtained orig-
inally by Dewangan and Grechka (2003) and examined also by Kochetov
and Slawinski (2009a) about the symmetry, orientation, and component val-
ues of tensor 8. In particular, we conclude that tensor 6 with errors 7 is con-
sistent – in the Monte-Carlo sense – with the orthotropic symmetry class.
Also the results of this paper are consistent with comments of Grechka and
Kachanov (2006), according to whom orthotropy might suffice for many
scenarios encountered in exploration seismology.
Ac kn ow le dg me nt s. This work was inspired by collaboration with
the late Albert Tarantola. Also, the authors acknowledge discussions with,
and fruitful suggestions of Misha Kochetov, Ken Larner, Daniel Peter, Mi-
chael Rochester and Ilya Tsvankin, editorial help of David Dalton and
graphic support of Elena Patarini. TD received funding from the Atlantic In-
novation Fund and the Research and Development Corporation of New-
foundland and Labrador through the High Performance Computing for
Geophysical Applications Project and from Polish National Science Center
through grant number 2011/01/B/ST10/07305. MS’s research was partially
supported by the Discovery Grant of The Natural Sciences and Engineering
Research Council of Canada. This research was performed in the context of
The Geomechanics Project supported by Husky Energy.
Bόna, A. (2009), Symmetry characterization and measurement errors of elasticity
tensors, Geophysics 74, 5, 75-78, DOI: 10.1190/1.3184013.
Chapman, C. (2004), Fundamentals of Seismic Wave Propagation, Cambridge Uni-
versity Press, Cambridge.
Author copy
Clerc, M., and J. Kennedy (2002), The particle swarm – explosion, stability, and
convergence in a multidimensional complex space, IEEE Trans. Evolut.
Comp. 6, 1, 58-73, DOI: 10.1109/4235.985692.
Danek, T., M. Kochetov, and M.A. Slawinski (2013), Uncertainty analysis of effec-
tive elasticity tensors using quaternion-based global optimization and
Monte-Carlo method, Q. J. Mech. Appl. Math. 66, 2, 253-272, DOI:
Dewangan, P., and V. Grechka (2003), Inversion of multicomponent, multiazimuth,
walkawayVSP data for the stiffness tensor, Geophysics 68, 3, 1022-1031,
DOI: 10.1190/1.1581073.
Donelli, M., G. Franceschini, A. Martini, and A. Massa (2006), An integrated mul-
tiscaling strategy based on a particle swarm algorithm for inverse scattering
problems, IEEE Trans. Geosci. Remote Sens. 44, 2, 298-312, DOI:
10.1109/TGRS. 2005.861412.
Gazis, D.C., I. Tadjbakhsh, and R.A. Toupin (1963), The elastic tensor of given
symmetry nearest to an anisotropic elastic tensor, Acta Crystallogr. 16, 9,
917-922, DOI: 10.1107/S0365110X63002449.
Grechka, V., and M. Kachanov (2006), Seismic characterization of multiple fracture
sets: Does orthotropy suffice? Geophysics 71, 3, D93-D105, DOI: 10.1190/
Kennedy, J., and R. Eberhart (1995), Particle swarm optimization. In: Proc. IEEE
Int. Conf. Neural Networks, 27 November – 1 December 1995, Perth, Aus-
tralia, 1942-1948.
Kochetov, M., and M.A. Slawinski (2009a), Estimating effective elasticity tensors
from Christoffel equations, Geophysics 74, 5, 67-73, DOI: 10.1190/
Kochetov, M., and M.A. Slawinski (2009b), On obtaining effective orthotropic elas-
ticity tensors, Q. J. Mech. Appl. Math. 62, 2, 149-166, DOI: 10.1093/
Mathai, A.M., and S.B. Provost (1992), Quadratic Forms in Random Variables:
Theory and Applications, Statistics: Textbooks and Monographs, Vol. 126,
Dekker, New York.
Norris, A.N. (2006), The isotropic material closest to a given anisotropic material,
J. Mech. Mater. Struct. 1, 2, 223-238, DOI: 10.2140/jomms.2006.1.223.
Slawinski, M.A. (2010), Waves and Rays in Elastic Continua, World Scientific
Publ., Singapore.
Stillwell, J. (2008), Naive Lie Theory. Undergraduate Texts in Mathematics,
Springer, New York, DOI: 10.1007/978-0-387-78214-0.
Tarantola, A. (2005), Inverse Problem Theory and Methods for Model Parameter
Estimation, SIAM, Philadelphia.
Tsvankin, I. (1997), Anisotropic parameters and P-wave velocity for orthorhombic
media, Geophysics 62, 4, 1292-1309, DOI: 10.1190/1.1444231.
Author copy
Tsvankin, I., and V. Grechka (2011), Seismology of Azimuthally Anisotropic Media
and Seismic Fracture Characterization, Geophysical References Series,
Society of Exploration Geophysicists, DOI: 10.1190/1.9781560802839.
Received 30 July 2013
Received in revised form 7 October 2013
Accepted 14 October 2013
Author copy
... One can distinguish a group of techniques, which are based on the separation of an additive part of the elasticity tensor (e.g., by way projecting), possessing one or another type of symmetry [34][35][36][37]. In this case, the identification problem is reduced to determination of the symmetric part, which is the closest in some metric to a given tensor, as e.g., in [38][39][40][41][42][43][44][45]. Note that depending on the choice of the metric, the separated parts may have special properties. ...
... There may exist multiple local and global minimums, so that the direct use of determined numerical methods is obstructed. However, for the majority of practical applications, there is no need for an exact approximation of the solutions to a global minimum, so that the optimization problems can be solved by using the heuristic algorithms, e.g., the particle swarm optimization method [71] employed in [39,41]. It seems plausible to use as a criterion of assigning a material by its properties to one or another class a small value of the residual caused by the related approximation in its constitutive equation. ...
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The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.
... Thus, one should treat the tensor as a set of normal distributions of entries, not just their strict values, because the entries are burdened with measurement errors. The matrix of standard deviations of tensor entries is given below (Danek & Slawinski 2015): ...
... 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 consider error propagation from tensor (6) to tensor (8). ...
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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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Geophysics—similarly to astrophysics—relies on remote sensing. Inferring material properties of the Earth’s interior is akin to inferring the composition of a distant star. In both cases, scientists rely on matching theoretical predictions or explanations with observations. Notably, obtaining a sample of a material from the interior of our planet might not be less difficult than obtaining a sample from a distant celestial object. To infer the presence and orientations of subsurface fractures, seismologists might use directional properties of Hookean solids. In other words—using such a solid as a mathematical model— seismologists match its quantitative predictions with observations.
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We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular sym-metry that corresponds to measurable traveltime and polar-ization quantities. These quantities — the wavefront-slow-ness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization prob-lem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of partic-ular symmetry without assuming its orientation is challeng-ing. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we dis-tinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.
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The present book — which is the second, and significantly extended, edition of the textbook originally published by Elsevier Science — emphasizes the interdependence of mathematical formulation and physical meaning in the description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray theory to explain phenomena resulting from the propagation of seismic waves. The book is divided into three main sections: Elastic Continua, Waves and Rays and Variational Formulation of Rays. There is also a fourth part, which consists of appendices. In Elastic Continua, we use continuum mechanics to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such a material. In Waves and Rays, we use these equations to identify the types of body waves propagating in elastic continua as well as to express their velocities and displacements in terms of the properties of these continua. To solve the equations of motion in anisotropic inhomogeneous continua, we invoke the concept of a ray. In Variational Formulation of Rays, we show that, in elastic continua, a ray is tantamount to a trajectory along which a seismic signal propagates in accordance with the variational principle of stationary traveltime. Consequently, many seismic problems in elastic continua can be conveniently formulated and solved using the calculus of variations. In the Appendices, we describe two mathematical concepts that are used in the book; namely, homogeneity of a function and Legendre's transformation. This section also contains a list of symbols.
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We consider the problem of representing a generally anisotropic elasticity tensor, which might be obtained from physical measurements, by a tensor belonging to a chosen material symmetry class, so-called ‘effective tensor'. Following previous works on the subject, we define this effective tensor as the solution of a global optimization problem for the Frobenius distance function. For all nontrivial symmetry classes, except isotropy, this problem is nonlinear, since it involves all orientations of the symmetry groups. We solve the problem using a metaheuristic method called particle-swarm optimization and employ quaternions to parametrize rotations in 3-space to improve computational efficiency. One advantage of this approach over previously used plot-guided local methods and exhaustive grid searches is that it allows us to solve a large number of instances of the problem in a reasonable time. As an application, we can use Monte-Carlo method to analyze the uncertainty of the orientation and elasticity parameters of the effective tensor resulting from the uncertainty of the given tensor, which may be caused, for example, by measurement errors.
It is often desirable to approximate a full anisotropic tensor, given by 21 independent parameters, by one with a higher symmetry. If one considers measurement errors of an elasticity tensor, the standard approaches of finding the best approximation by a higher symmetric tensor do not produce the most likely tensor. To find such a tensor, I replace the distance metric used in previous studies with one based on probability distribution functions of the errors of the measured quantities. In the case of normally distributed errors, the most likely tensor with higher symmetries coincides with the closest higher symmetric tensor, using a deviation-scaled Euclidean metric. © 2009 Society of Exploration Geophysicists. All rights reserved.
Geometry of complex numbers and quaternions.- Groups.- Generalized rotation groups.- The exponential map.- The tangent space.- Structure of Lie algebras.- The matrix logarithm.- Topology.- Simply connected Lie groups.
Presenting a comprehensive introduction to the propagation of high-frequency body-waves in elastodynamics, this volume develops the theory of seismic wave propagation in acoustic, elastic and anisotropic media to allow seismic waves to be modelled in complex, realistic three-dimensional Earth models. The book is a text for graduate courses in theoretical seismology, and a reference for all academic and industrial seismologists using numerical modelling methods. Exercises and suggestions for further reading are included in each chapter.
The book provides an up-to-date description of the methods used for fitting experimental data, or to estimate model parameters, and to unify these methods into the Inverse Problem Theory. The first part of the book deals with problems and describes Maximum likelihood, Monte Carlo, Least squares, and Least absolute values methods. The second part deals with inverse problems involving functions. Theoretical concepts are emphasized, and the author has all the useful formulas listed, with many special cases included. The book serves as a reference manual.