Inversion of ray velocity and polarization for

elasticity tensor

Andrej Bona! and Ioan Bucataru" and Michael A. Slawinski!

Abstract

We construct a method for finding the elasticity parameters of an anisotropic ho-

mogeneous medium using only ray velocities and corresponding polarizations. We

use a linear relation between the ray velocities and wavefront slownesses, which de-

pends on the corresponding polarizations. Notably, this linear relation circumvents

the need to use explicitly the intrinsic relation between the wavefront slowness and

ray velocity, which – in general – is not solvable for the slownesses. We discuss

sensitivity of this method to the errors in measurements.

Key words: anisotropy; elasticity tensor; polarizations; traveltime; ray velocity

1 Introduction

In this paper, we discuss a method of using ray-velocity and polarization1

measurements to determine the complete set of the density-scaled elasticity2

1 !Department of Earth Sciences, Memorial University, St. John’s NL A1B 3X5,Canada, 1(709) 737 7541, mslawins@mun.ca2 "Faculty of Mathematics, ”Al. I. Cuza” University, Iasi 700506, Romania

Preprint submitted to Elsevier November 21, 2007

parameters that describe a Hookean solid. The problem of determining the3

twenty-one components of the elasticity tensor from wave propagation has4

been investigated by many researchers; among them, ?, ?, and ?. In all cases,5

the proposed methods relied on using polarizations and wavefront slownesses.6

We propose a method for finding these components using polarizations and7

ray velocities. To find the latter quantities, we consider traveltimes measured8

between a single point source and point receiver, which are directly related to9

the ray velocities. This method circumvents the need to measure the wavefront10

slownesses, which – in a seismological context – requires closely spaced sources11

or receivers.12

We also demonstrate that standard seismic measurements of polarizations and13

traveltimes allow us to obtain uniquely the density-scaled elasticity parame-14

ters. In view of the forward problem described in the next section and the15

inversion formulated in Section 3, we can infer that the relation of the elastic-16

ity parameters to the polarization and traveltime measurements is one-to-one17

in the context of the theory of elastodynamics.18

In Section 4, we discuss the error analysis for the proposed method and ex-19

emplify it with a numerical example.20

2

2 Waves and rays in Hookean solids21

A Hookean solid is fully described by its mass density and elasticity parame-22

ters, which are components of the elasticity tensor appearing in a constitutive23

equation. The constitutive equation of a Hookean solid is 324

!ij = cijkl"kl, i, j, k, l # {1, 2, 3} ,25

where !, c and " are stress, elasticity and strain tensors, respectively. Strain26

tensor is a symmetric second-rank tensor given by27

"ij =1

2

!#ui

#xj+

#uj

#xi

"

,28

where u and x are the displacement and position vectors, respectively. Elas-29

ticity tensor is a fourth-rank tensor that possesses the intrinsic symmetries,30

cijkl = cjikl = cklij, and is positive-definite, cijklviwjvkwl > 0, for nonzero31

vectors v and w. We will consider homogeneous media, where c and mass den-32

sity, $, do not depend on position. In such a case, the signal propagation is33

described by the following form of the elastodynamic equations:34

cijkl

$

#2uk

#xj#xl=

#2ui

#t2.35

These equations depend on the density-scaled elasticity tensor, a := c/$, whose36

components are the parameters that fully describe a medium in the context of37

this paper. The asymptotic solutions of these equations lead to the Christo!el38

3 We use repeated indices summation convention throughout this paper.

3

eikonal equation, its characteristic equations are48

49

x!i :=

dx!i

dt=

1

2

#H!

#pi(2)

p!i :=

dp!i

dt=$1

2

#H!

#xi,

which are the Hamilton equations. The solutions of these equations are rays.50

In homogeneous media, the second equation reduces to p = 0, which implies51

that the rays are straight lines. Equation (2) defines the ray velocity, which in52

the case of straight rays is given by the vector connecting the source and the53

receiver divided by the traveltime; both the source and receiver locations are54

known from the experimental setup, and the traveltime is a key measurement.55

3 Inversion for density-scaled elasticity parameters56

In this section, we derive expressions that allow us to obtain the density-57

scaled elasticity parameters using the measured traveltimes and corresponding58

polarizations. The relationship between the ray velocity, x, and wavefront59

slowness, p, is given by equation (2). Using this equation, it is, in general,60

impossible to express p as an explicit function of x and a. In our formulation,61

we circumvent this problem by including the measured polarization, A, and62

expressing p in terms of x, a and A.63

To do so, we consider the Christo!el equation (??),64

aijklpjplAk (p) = Ai (p) ,65

4

Figure 1. Synthetic measurements of ray velocity and corresponding polarizationsshowing qP and qS waves in the case of 3000 measurements.

which describes polarization A as an eigenvector of the Christo!el matrix,66

". The Christo!el equation restricts the eigenvalues to the slowness surfaces67

given by68

H! (p) = 1.69

Since H! (p) is an eigenvalue of " (p) with corresponding unitary eigenvector70

A! (p), we write71

"ik (p) A!k (p) = H! (p) A!

i (p) ; (3)72

multiplying by A!i (p), we write73

aijklpjplA!i (p) A!

k (p) = H! (p) , (4)74

5

since |A! (p)| = 1. Using expression (4) in equation (2), we write75

x!i = ajiklplA

!j (p) A!

k (p) + amjklpjpl#A!

m (p)

#piA!

k (p) .76

Using equation (3), we get77

x!i = ajiklplA

!j (p) A!

k (p) . (5)78

This equation can be found, among others, in the classic book of ?, pp. 150-79

151. Equation (5) can be written as80

x! = " (A! (p)) p.81

Since x and A are obtained from the measurements, equations (5) form a82

linear system for p. The solution exists since the determinant of this system,83

det#ajiklA!

j A!k

$, is nonzero due to the positive definiteness of the Christo!el84

matrix. The solution is85

p! = "#1 (A!) x!. (6)86

This equation expresses the wavefront-slowness vector as a function that is87

linear in the ray velocity and quadratic in the polarization. The dependence88

on the density-scaled elasticity tensor is contained in ". For a wavefront given89

by a fixed %, we obtain the measurements of x and A that result in p. In view90

of multiple arrivals of the same wave, from now on, the index % distinguishes91

among di!erent wavefront arrivals for a single source-receiver pair, and not92

among the three waves, as was the case above. Still, to obtain all twenty-one93

6

parameters of elasticity, we need to consider the measurements correspond-94

ing to the three waves since, for a given wave, some of the components of a95

might not appear in equations (8), as exemplified by transverse isotropy: the96

expressions for the traveltime and the polarization of the fastest wave do not97

contain a1212 even though the transversely isotropic medium is described by98

a1111, a1133, a3333, a2323 and a1212, see, e.g., ?, pp. 229-232.99

Combining expression (6) with the Christo!el equation (??), we obtain a as100

an implicit function of x and A, which we write as101

"#"#1 (A!) x!

$A! = A!. (7)102

Writing the inverse of " using its minors, we can rewrite expression (7) as103

aijklMinor (" (A!))jm x!mMinor (" (A!))ln x!

nA!k = (det " (A!))2 A!

i , (8)104

where i # {1, 2, 3}. For a given % and a fixed source-receiver pair, this is an105

implicit system of three equations for a. These equations are polynomials in106

components of a, A and x; they contain only the fifth and the sixth powers107

of the components of a, the thirteenth powers of components of A, and the108

second powers of the components of x multiplied by the ninth powers of the109

components of A.110

We can view each of the three equations (8) as an equation for a hypersurface111

in the twenty-one dimensional space of density-scaled elasticity parameters.112

We need to obtain twenty-one such hypersurfaces. From each measurement,113

7

we obtain three hypersurfaces. If we measure all three waves for a given source-114

receiver pair, we obtain at least nine hypersurfaces; more than nine if there115

are multiple arrivals of the same wave. In an ideal case with no measurement116

errors, all hypersurfaces intersect at the point corresponding to the density-117

scaled elasticity tensor that describes the medium. For the intersection of these118

hypersurfaces to be zero-dimensional, it is necessary that the normals to these119

hypersurfaces be linearly independent.120

To find the density-scaled elasticity parameters from equation (8) we can use121

regression analysis. In the next section, we discuss the relation between the122

number of measurements and the accuracy of results. A large number of mea-123

surements also ensures that we have enough equations to determine uniquely124

the density-scaled elasticity tensor.125

4 Stability analysis: Numerical example126

In this section, we discuss the stability of the proposed method; in particular,127

we discuss sensitivity of the elasticity parameters to the ray-velocity and polar-128

ization measurement errors. To do so, we consider the following density-scaled129

8

elasticity tensor. 4130

%

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&'

4.00 2.06 2.10 $0.05 0.01 $0.02

3.83 1.96 0.12 $0.05 0.13

3.96 0.11 0.03 $0.09

1.00 0.11 $0.07

S Y M 0.88 0.01

1.11

(

)))))))))))))))))))))))))))))))))))))*

+km2

s2

,

(9)131

The entries of this matrix are related to the components of the elasticity132

tensor using the so-called Voigt notation, e.g., ?, pp. 65-66. We concisely write133

equation (8) as134

F (a, A, x) = 0.135

The errors #a are, to the first order, related to the errors #A and #x by the136

following linear relation.137

#F

#a#a = $#F

#A#A$ #F

#x#x. (10)138

For an overdetermined system given by many measurements, the smallest er-139

rors in a result from taking the pseudoinverse, (#F/#a)†, of the rectangular140

matrix #F/#a and applying it to the vector on the right-hand side of equation141

9

(10), in other words,142

#a =

!#F

#a

"† +

$#F

#A#A$ #F

#x#x

,

. (11)143

Evaluating equation (11) using synthetic measurements corresponding to the144

density-scaled elasticity tensor (9), we obtain the values of linear dependence145

of #a on errors #A and #x. In Figure 1 we show 3000 synthetic measurements146

of ray velocities and corresponding polarizations used in these computations.147

Figures 2 and 3 show the error dependence for the sets of 30 and 3000 mea-148

surements of both ray velocities and corresponding polarizations. For example,149

Figure 2 tells us that the error in each polarization measurement is magnified150

about nine times in the error of a16.151

Let us examine the sensitivity to errors in the the ray-velocity measurements152

and polarizations. In Figures 2 and 3, we show this sensitivity for the cases of 30153

and 3000 measurements, respectively. Examining these figures, we observe that154

this sensitivity is small, and decreases with the number, n, of measurements155

proportionally to%

n, as expected in view of the standard deviation of the156

sample mean.157

We see that the sensitivity of the density-scaled elasticity parameters to the158

errors in polarization measurement is several times greater than the sensi-159

tivity to the errors in ray-velocity measurements. To obtain the absolute160

errors in these parameters, we have to find the absolute errors in the ray-161

4 These values were used by ?.

10

Figure 2. Multiplicative factors for error sensitivity of inversion for density-scaledelasticity parameters, a, to the ray-velocity measurements – marked by empty tri-angles – and to the polarization measurements – marked by solid triangles – in thecase of 30 measurements.

velocity and polarization measurements. The errors in ray velocity depend,162

to the first order, on errors in traveltime and trajectory measurements as163

#x & x (#x/x$#t/t). Considering the ray velocity of about 2akm/s, source-164

receiver distance of 1akm, error in position of ±1am and error in traveltime165

of ±2ams, we see that the error in ray velocity is in the order of ±10am/s,166

which translates to the error of about ±0.02akm2/s2 in a11, for the case of 30167

measurements. The errors in polarization measurements of 10$ is an error in168

one of the components of the normalized polarization vector of ±0.015, and169

translates to the error in a11 of about ±0.21akm2/s2, for the case of 30 mea-170

11

Figure 3. Multiplicative factors for error sensitivity of inversion for the density-s-caled elasticity parameters, a, to the ray-velocity measurements – marked by emptytriangles – and to the polarization measurements – marked by solid triangles – inthe case of 3000 measurements.

surements. In other words, using our methods we would be able to determine171

a11 as 4.00 ± 0.23akm2/s2. For the case of 3000 measurements, we list all the172

12

parameters with corresponding errors, namely,173

%

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&'

4.00 ± 0.03 2.06 ± 0.03 2.10 ± 0.03 $0.05 ± 0.02 0.01 ± 0.02 $0.02 ± 0.02

3.83 ± 0.03 1.96 ± 0.03 0.12 ± 0.02 $0.05 ± 0.02 0.13 ± 0.02

3.96 ± 0.04 0.11 ± 0.02 0.03 ± 0.02 $0.09 ± 0.02

1.00 ± 0.02 0.11 ± 0.01 $0.07 ± 0.01

S Y M 0.88 ± 0.01 0.01 ± 0.01

1.11 ± 0.02

(

)))))))))))))))))))))))))))))))))))))*

+km2

s2

,

.(12)174

An n-fold increase of measurements results in approximately an%

n-fold de-175

crease in errors. Had the errors been an order of magnitude greater, as would176

have been the case of fewer measurements, we could characterise this tensor177

as isotropic with the density-scaled Lame parameters of approximately & = 2178

and µ = 1.179

5 Conclusions and discussions180

We have proposed a method for obtaining the density-scaled elasticity pa-181

rameters from seismic measurements. These parameters are related explicitly182

to the wavefront slowness and the polarization via the Christo!el equations.183

However, considering a point source and a point receiver, we deal directly184

13

with the ray velocity rather than the wavefront slowness. These two quan-185

tities are related to one another by equation (2). In general, this equation186

cannot be solved explicitly for slowness. Moreover, at the inflection points of187

the slowness surfaces, it cannot be solved at all due to the vanishing of the188

Hessian. To circumvent this di$culty, we have expressed wavefront slowness189

as a linear function of ray velocity by including the measured polarization as190

another argument. For each measurement, we combined this expression with191

the Christo!el equation (??) to obtain a system of three implicit nonlinear192

equations for the density-scaled elasticity parameters that define three hyper-193

surfaces in the twenty-one dimensional space. To uniquely determine these194

parameters, we need su$cient number of measurements to obtain twenty-one195

equations that define hypersurfaces with linearly independent normals. This196

set of equations can be solved by regression analysis. The error analysis of197

the regression approach shows robustness of the inversion and the expected198

improvement with the number of measurements.199

The key practical limitation of the proposed method is the assumption of ho-200

mogeneity. This limitation could be removed by considering an approach akin201

to layer stripping, which is a subject of future work. The numerical example202

shows that we can obtain good estimates of the density-scaled elasticity pa-203

rameters given the number of measurements of the order of thousands. This204

requirement can be accommodated by the modern advances in data acquisi-205

tion. The theoretical usefulness of the proposed method lies in the derivation of206

14

equation (8), which allows us to obtain the components of the elasticity tensor207

in an arbitrary coordinate system that are required to begin the coordinate-free208

characterization proposed by ?. Except for the isotropic case, the parameters209

obtained from measurements are not generally expressed in the natural coor-210

dinates. To determine the symmetry class of the corresponding Hookean solid211

and the orientation of its symmetry axes, we have to proceed with further212

analysis, such as the coordinate-free characterization. In view of experimental213

errors, we can also consider the possible symmetries of the material that are214

within the range of the errors by studying the method proposed by ?.215

Using the knowledge of the wavefront slownesses and the associated polariza-216

tions, ? derives the explicit conditions for the measurement locations to ensure217

a unique inversion. Deriving such conditions for our method, which uses the218

knowledge of ray velocities and polarizations, is a subject of future work. How-219

ever, the very concept of regression analysis requires the use of a large number220

of measurements to obtain an accurate and unique solution.221

Acknowledgements222

We wish to acknowledge insightful comments of Vladimir Grechka and an-223

other, anonymous, reviewer, as well as the editorial work of Cathy Beveridge.224

This research was conducted within The Geomechanics Project. The research225

of A.B. and M.A.S. was supported also by NSERC.226

15