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Estimating effective elasticity tensors from Christoffel equations
Abstract and Figures
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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... 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. The present work, which is formulated in the context of a computationally efficient global optimization, allows us to obtain thousands of solutions within a few hours on a multi-core CPU computer. ...
... As discussed by Kochetov and Slawinski (2009a), the distance between a generally anisotropic tensor and its counterpart belonging to a given symmetry class is obtained by finding the orientation that minimizes the distance. Performing a search under all orientations leads to a highly nonlinear optimization problem, which commonly exhibits many local minima. ...
... Note the similarity (expected) between tensors 8 and 6. Also to ensure consistency, note that expression 25 in Kochetov and Slawinski (2009a) and expression 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 x 3 -axis. According to the work of Dewangan and Grechka (2003) and Kochetov and Slawinski (2009a), tensor 6 can be represented by its counterpart exhibiting orthotropic symmetry. ...
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.
... Such a characterization was studied in [6], in which the use of some quantities, invariant with respect to SO(3) (such as the trace of the second-order symmetric Voigt and dilatation tensors), is proposed so as to make the minimization procedure easier. Numerical procedures for estimating such approximations and their reference frames (or "effective orientations") have been proposed and successfully applied in [21] and [20] (for transversely isotropic and orthotropic tensors, respectively); see also [19]. Note finally that the question of defining such approximations in the presence of measurement errors was recently addressed in [3]. ...
... The optimization problem (16) is solved under the following set of constraints: The set of constraints defined by Eqs. (17)(18)(19) basically corresponds to the one previously used and studied in [34] [35]. Eq. (17) is the classical normalization condition for the probability density function, while Eq. ...
... (18) means that the mean matrix is supposed to be known a priori. Eq. (19) implies the existence of the second-order moment of the inverse random matrix norm (see [34] [35]). Finally, the set of constraints defined by Eq. (20) allows one to partially prescribe the variances of m (m ≤ 6) selected random [25] for a discussion). ...
This study addresses the stochastic modeling of media whose elasticity ten-sor exhibits uncertainties on the material symmetry class to which it belongs. More specifically, we focus on the construction of a probabilistic model which allows realizations of random elasticity tensors to be simulated, under the constraint that the mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilis-tic model derived in [6] which allows the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. A new methodology and parameterization of the model are then defined. The proposed approach is exemplified considering the mean to transverse isotropy. The efficiency of the methodology is demonstrated by computing the mean distance of the random elasticity tensor to this material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemannian metric and the Euclidean projection, respec-tively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the nonparametric probabilistic approach (with anisotropic fluctuations) proposed in [12] is finally provided.
... Kullanılan bütün materyallerin maksimum ve ortalama ±CLV D, ±ISO sonuçları, maksimum ve ortalama dogrultu, dalım ve kayma açılarındaki degişim sonuçları, izotropik uzaya olan uzaklıkları (Kochetov 2009) ve Matlab tabanlı MTEX (Bachmann vd. 2010) Fay parametrelerindeki degişimler de non-DC yüzdeleri gibi bulunmuştur. ...
zet Anizotropik ortamdaki sismik kaynaklar izotropik or-tamdakilere göre daha karma¸sıkkarma¸sık moment tensörü ya-pısına sahiptirler. ˙ Izotropik ortamdaki kayma¸seklindekayma¸ kayma¸seklinde olu¸sanolu¸san depremler tamamen iki kuvvet çifti (DC) özelli-˘ gine sahip moment tensörler üretirler. Anizotropik or-tamdaki kayma¸seklindekayma¸ kayma¸seklinde olu¸sanolu¸san depremler (shear source) ise iki kuvvet çiftine ek olarak, izotropik (ISO) ve telafi edilmi¸sedilmi¸s do˘ grusal vektör dipolleri (CLVD) bile-¸ senlerini üretir. Moment tensörde bulunan DC, CLVD ve ISO yüzdeleri anizotropinin yönelimine ve büyüklü-˘ güne ba˘ glıdır. Bu çalı¸smadaçalı¸smada dü¸seydü¸sey atımlı fay türü seçilmi¸stirseçilmi¸stir. Bu fay türleri enine izotropik, ortotropik ve monoklinik simetri sınıflarına sahip altı farklı materyalden olu¸sanolu¸san ortamlara uygulanmı¸stıruygulanmı¸stır. Anizotropik elastisite tensörü olası bütün açılarda döndürülmü¸sdöndürülmü¸s ve her döndürmede olu¸sanolu¸san yeni elastisite tensörü kullanılarak moment ten-sörler üretilmi¸stirüretilmi¸stir. Bu sayede anizotropinin yönelimi ile fayın yönelimi arasındaki açılar de˘ gi¸stirilmi¸sgi¸stirilmi¸gi¸stirilmi¸s ve sonuç-ları gözlenebilmi¸stirgözlenebilmi¸stir. Sonrasında bu moment tensörle-rin moment tensör çözümleri yapılmı¸syapılmı¸s ve ISO, CLVD ve DC bile¸senleribile¸senleri bulunmu¸sturbulunmu¸stur. DC ve CLVD bile¸senbile¸sen-lere ait grafikler çizdirilmi¸stirçizdirilmi¸stir. DC bile¸senibile¸seni kullanarak fay düzlemi oryantasyonu bulunmu¸sbulunmu¸s ve orjinal fay düz-lemi parameterlerinden olan sapmaları hesaplanmı¸stırhesaplanmı¸stır. Böylelikle anizotropik ortamın fay düzlemi parametre-leri üzerindeki etkisi bulunmu¸sturbulunmu¸stur. Tüm bu yüzdeler ve kayma miktarındaki de˘ gi¸simgi¸sim-ler anizotropi miktarına göre de˘ gi¸sece˘gi¸sece˘ gi için anizotropi miktarını da belirlenir. Verilen anizotropik materyalin en yakın izotropik uzaya ulan uzaklı˘ gı bulunmu¸sturbulunmu¸stur. P ve S dalgası anizotropileri bulunmu¸sbulunmu¸s ve P ve S dalgası azinotropi yüzdelerinin elastisite tensörlerinin izotro-pik uzaya ulan uzaklıkları ile ili¸skilendirilmi¸stirili¸skilendirilmi¸ili¸skilendirilmi¸stir. P dal-gası anizotropi yüzdesi ve elastisite tensörlerinin izot-ropik uzaya olan uzaklıklarının fay parametrelerindeki de˘ gi¸smdegi¸smde ve iki kuvvet çifti olmayan kayna˘ gın yüzdesi arasında bir ili¸skiili¸ski oldu˘ gu bulunmu¸sturbulunmu¸stur. Anahtar Kelimeler: deprem kayna˘ gı, anizotropik fay kayna˘ gı, fay paramet-leri Abstract Seismic sources in anisotropic medium have more complex moment tensor structures compared with the moment tensors of isotropic medium. Shear sources in an isotropic focal medium generate pure double-couple (DC) moment tensors. However in an anisotropic medium , shear sources can generate moment tensors with DC, compensated linear vector dipole (CLVD) and isot-ropic (ISO) components. The DC, CLVD and ISO percentages of a moment tensor depend on the magnitude and the orientation of the anisotropy. In this study, we choose dip-slip fault in a medium of different anisotropy classes; transversely isotropic, orthotropic and monoclinic. We rotated the anisotropic elasticity tensors of the medium for every possible orientation and evaluate the moment tensors of each cases. By rotating anisotropic elasticity tensors we can observe the effects of rotated source medium for fixed fault parameters. Then moment tensor decomposition is applied and DC, CLVD and ISO components are found. We plot the CLVD and ISO percentages of the moment tensors generated by anisotropy classes. By using the DC components , first we obtained fault plane orientation then we calculate the deviation from the original fault mechanism. Effects of anisotropy of the source region on calculated fault parameters are found. Distance from isotropic space of given anisotropic elasticity tensor and P/S wave velocity anisotropy percentages are measured. These percentages are proportional to the distance from isotropy. There is a correlation between distance to isotropy and P wave anisotropy with variation of fault plane parameters and percentages of non-DC components of earthquake source.
... The use is highlighted in the present article. 13 The modeling of the relaxation time would deserve to dedicate the whole article to this topic. Nevertheless the scope of the article on that matter is limited to justify at least partially the most common expression for the relaxation rate due to the Umklapp-Processes: 16,17 bT ...
... Such relations can be understood in terms of the concept of distance to a given symmetry class, which was proposed by Gazis et al. (10), and allow us to infer information about properties of the material represented by c, such as its layering or fractures. Following several papers, notably (11)(12)(13)(14), we consider the Frobenius norm to find the closest tensors that belong to particular symmetry classes. Such a tensor is referred to as the effective tensor of the given class. ...
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.
... in which (1 − L) ∈ R, [ Λ] ∈ M S n (R) and { τ k ∈ R} m k=1 are the Lagrange multipliers associated with constraints (30), (28) and (31) respectively, and k * 0 is the positive constant of normalization. It follows that [D] belongs to the ensemble SE ++ of positive-definite symmetric random matrices introduced in [34]. ...
In this work, we address the stochastic modeling of apparent elasticity tensors, for which both material symmetry and stochastic boundedness constraints have to be taken into account, in addition to the classical constraint of invertibility. We first introduce a stochastic measure of anisotropy, which is defined using metrics in the set of elasticity tensors and used for quantitatively characterizing the fulfillment of material symmetry constraints. After having defined a numerical approximation for the stochastic boundedness constraint, we then propose a methodology allowing one to unify maximum entropy based models that have been previously derived by considering some of these constraints and which consists in constructing a probabilistic model for an auxiliary random variable. The latter can be interpreted as a stochastic compliance tensor, for which the available information to be used in the maximum entropy formulation can be readily deduced from the one considered for the elasticity tensor. A numerical illustration of the approach to an elastic microstructure is finally provided.
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.
The problem of reducing a fully anisotropic (Triclinic) stiffness tensor comprised of 21 distinct components to a higher symmetry form having fewer distinct elements is of interest in many geophysical applications, where assuming a particular type of symmetry is often necessary in order to sufficiently reduce computational complexity to allow practical solutions. In addition, recent advances in the upscaling of realistically large field-scale discrete fracture networks (DFN) have led to the need for an efficient way to derive a very large number of nearest medium approximations to the Triclinic stiffness tensors obtained. Owing to rotational symmetries and non-linearity, the problem of efficiently finding such approximations is generally non-trivial, as optimal orientations are intrinsically non-unique. The algorithm proposed by Dellinger (2005) computes nearest Orthotropic and Transverse Isotropic approximations for a given stiffness tensor, using the Federov norm as an objective function to iteratively minimize the fit error. Although this method is appropriate for computing solutions to single problem instances, the implementation is too inefficient for production situations, where a very large number of invocations of the algorithm is required. The enhanced algorithm proposed here is accurate, efficient and general, allowing nearest medium approximations to be determined for arbitrary symmetry types, including Isotropic, Cubic, Transverse Isotropic, Orthotropic and Monoclinic.
Inversion of 3D, 9C wide azimuth vertical seismic profiling (VSP) data from the Weyburn Field for 21 independent elastic tensor elements was performed based on the Christoffel equation, using slowness and polarization vectors measured from field data. To check the ability of the resulting elastic tensor to account for the observed data, simulation of the 3C particle velocity seismograms was done using eighth-order, staggered-grid, finite-differencing with the elastic tensor as input. The inversion and forward modeling results were consistent with the anisotropic symmetry of the Weyburn Field being orthorhombic. It was dominated by a very strong, tranverse isotropy with a vertical symmetry axis, superimposed with minor near-vertical fractures with azimuth similar to 55 degrees from the inline direction. The predicted synthetic seismograms were very similar to the field VSP data. The examples defined and provided a validation of a complete workflow to recover an elastic tensor from 9C data. The number and values of the nonzero tensor elements identified the anisotropic symmetry present in the neighborhood of a 3C borehole geophone. Computation of parameter correlation matrices allowed evaluation of solution quality through relative parameter independence.
Inversion of phase slowness and polarization vectors measured from multicomponent vertical seismic profile data can yield estimates of all 21 density-normalized elastic moduli for anisotropic elastic media in the neighborhood of each 3C geophone. Synthetic test data are produced by direct evaluation of the Christoffel equation, and by finite-difference solution of the elastodynamic equations. Incompleteness of the data, with respect to illumination (polar and azimuth angle) apertures (qP and/or qS) wave types, wave-propagation directions, and the amount of data (e.g., with or without horizontal slowness components), produces solutions with variations in quality, as revealed by the distribution of model parameter correlations. In a good solution, with all parameters well constrained by the data, the correlation matrix is diagonally dominant. qP-only and qS-only solutions typically produce complementary distributions in their correlation matrices, as they are orthogonal in their sampling of the medium with respect to polarization. The elastic moduli become less independent as the data apertures decrease. If the other input data are relatively complete, the horizontal components of the slowness vector are not needed as the information they contain is redundant. The main consequence of omitting horizontal slowness components is slower convergence. When modest amounts of random noise are added to the slowness and polarization data, in otherwise adequately sampled apertures, the solution is still very close to the correct model, but with larger residual variance.
Vertical seismic profiling (VSP), an established technique, can be used for estimating in‐situ anisotropy that might provide valuable information for characterization of reservoir lithology, fractures, and fluids. The P‐wave slowness components, conventionally measured in multiazimuth, walkaway VSP surveys, allow one to reconstruct some portion of the corresponding slowness surface. A major limitation of this technique is that the P‐wave slowness surface alone does not constrain a number of stiffness coefficients that may be crucial for inferring certain rock properties. Those stiffnesses can be obtained only by combining the measurements of P‐waves with those of S (or PS) modes.
Here, we extend the idea of Horne and Leaney, who proved the feasibility of joint inversion of the slowness and polarization vectors of P‐ and SV‐waves for parameters of transversely isotropic media with a vertical symmetry axis (VTI symmetry). We show that there is no need to assume a priori VTI symmetry or any other specific type of anisotropy. Given a sufficient polar and azimuthal coverage of the data, the polarizations and slownesses of P and two split shear (S1 and S2) waves are sufficient for estimating all 21 elastic stiffness coefficients c ij that characterize the most general triclinic anisotropy. The inverted stiffnesses themselves indicate whether or not the data can be described by a higher‐symmetry model.
We discuss three different scenarios of inverting noise‐contaminated data. First, we assume that the layers are horizontal and laterally homogeneous so that the horizontal slownesses measured at the surface are preserved at the receiver locations. This leads to a linear inversion scheme for the elastic stiffness tensor c. Second, if the S‐wave horizontal slowness at the receiver location is unknown, the elastic tensor c can be estimated in a nonlinear fashion simultaneously with obtaining the horizontal slowness components of S‐waves. The third scenario includes the nonlinear inversion for c using only the vertical slowness components and the polarization vectors of P‐ and S‐waves. We find the inversion to be stable and robust for the first and second scenarios. In contrast, errors in the estimated stiffnesses increase substantially when the horizontal slowness components of both P‐ and S‐waves are unknown. We apply our methodology to a multiazimuth, multicomponent VSP data set acquired in Vacuum field, New Mexico, and show that the medium at the receiver level can be approximated by an azimuthally rotated orthorhombic model.
We consider the problem of finding the transversely isotropic elasticity tensor closest to a given elasticity tensor with respect to the Frobenius norm. A similar problem was considered by other authors and solved analytically assuming a fixed orientation of the natural coordinate system of the transversely isotropic tensor. In this paper we formulate a method for finding the optimal orientation of the coordinate system—the one that produces the shortest distance. The optimization problem reduces to finding the absolute maximum of a homogeneous eighth-degree polynomial on a two-dimensional sphere. This formulation allows us a convenient visualization of local extrema, and enables us to find the closest transversely isotropic tensor numerically.
An insightful, structurally appealing and potentially utilitarian formulation of the anisotropic form of the linear Hooke's law due to Lord Kelvin was independently rediscovered by Rychlewski (1984, Prikl. Mat. Mekh.48, 303) and Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math.43, 14). The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor, considered as a second-rank tensor in six-dimensional space, are called eigentensors when projected back into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is therefore six. The concept of an eigentensor was introduced by Kelvin (1856, Phil. Trans. R. Soc.166, 481) who called eigentensors “the principal types of stress or of strain”. Kelvin determined the eigentensors for many elastic symmetries and gave a concise summary of his results in the 9th edition of the Encyclopaedia Britannica (1878). The eigentensors for a linear isotropic elastic material are familiar. They are the deviatoric second-rank tensor and a tensor proportional to the unit tensor, the spherical, hydrostatic or dilatational part of the tensor. Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math.43, 14) give explicit forms of the eigentensors for all of the linear elastic symmetries except monoclinic and triclinic symmetry. We discuss two approaches for the determination of eigentensors and illustrate these approaches by partially determining the eigentensors for monoclinic symmetry. With the nature of the eigentensors for monoclinic symmetry known, a rather complete table of the structural properties of all linear elastic symmetries can be constructed. The purpose of this communication is to give the most specifically detailed presentation of the eigenvalues and eigentensors of the Kelvin formulation to date.
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 Cowin–Mehrabadi theorem is generalized to allow less restrictive and more flexible conditions for locating a symmetry plane in an anisotropic elastic material. The generalized theorems are then employed to prove that the number of linear elastic symmetries is eight. The proof starts by imposing a symmetry plane to a triclinic material and, after new elastic symmetries are found, another symmetry plane is imposed. This process exhausts all possibility of elastic symmetries, and shows that there are only eight elastic symmetries. At each stage when a new symmetry plane is added, explicit results are obtained for the locations of the new symmetry plane that lead to a new elastic symmetry. It takes as few as three, and at most five, symmetry planes to reduce a triclinic material (which has no symmetry plane) to an isotropic material for which any plane is a symmetry plane.
Harmonic and Cartan decompositions are used to prove that there are eight symmetry classes of elasticity tensors. Recent results in apparent contradiction with this conclusion are discussed in a short history of the problem.