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GEOPHYSICS, VOL. 60, NO. 4 (JULY-AUGUST 1995); P. 1243-1248, 5 FIGS.

Short Note

Anisotropic parameters of layered media in termsof composite elastic properties

John Brittan*, Mike Warner*, and Gerhard Pratt*

INTRODUCTION

In elastic earth models, at a wide variety of scales, thesubsurface is a layered sequence of different constituentmedia. It is therefore important to understand the elasticproperties of such a sequence, in particular to determine theresponse of the layering to an investigating seismic wave. Itcan be shown that if the individual layer thicknesses aremuch less than the wavelength of a seismic wave passingthrough the stack, the wave will propagate as though it weretraversing a homogenous, anisotropic medium (Postma,1955). This property has been subjected to rigorous testingboth experimentally (Melia and Carlson, 1984) and numeri-cally (Carcione et al., 1991). The elastic properties of thisequivalent medium can be derived algebraically from the

elastic properties of the materials that compose the layers(Backus, 1962). The homogenous equivalent medium will betransversely isotropic (hereafter referred to as TI), the axisof symmetry lying perpendicular to the layering.

Thomsen (1986) has derived a set of parameters that canbe used to describe the transverse isotropy of a medium. Theparameters are defined in terms of components of the TImediums stiffness tensor. Thus, in principle, the Thomsenparameters of the equivalent medium of a stack of thin layerscan be derived analytically from the elastic properties of theconstituent layers. Obtaining the anisotropic parameters interms of the composite elastic properties enables a greaterunderstanding of the elastic properties of layered sequences,indicating to what degree particular combinations of materi-

als will appear anisotropic to seismic experiments. This maybe particularly relevant to studies of the subsurface wherethe seismic waves pass through what appear, at shorterwavelengths, to be highly layered sequences. Particularcombinations of materials may produce a highly anisotropicequivalent medium and consequently lead to an inaccuraterepresentation of the composition of the subsurface if theanisotropy is not taken into account. In this paper, we shall

investigate the simplest case, the elastic properties of alayered sequence of two isotropic materials using explicitderivations of the anisotropic parameters (we give equationsfor TI constituent layers in the Appendix).

ANISOTROPIC PARAMETERS OF A LAYERED SEQUENCE

The elasticity of any medium can be described by afourth-order stiffness tensor, the components of which canbe written as a 6 x 6 matrix with elements (following theVoigt recipe). For an isotropic medium there are only twoindependent nonzero elements, and These arerelated to the Lame parameters of the medium by

and

Backus (1962) showed that a stack composed of thin layersof two different homogenous materials behaved as a homog-enous TI medium with elastic parameters and Thethin layers can be isotropic or TI media. The case ofisotropic constituent layering will be discussed in detail here;the anisotropic parameters of a stack of two alternating TImedia are discussed in the Appendix.

The parameters and are defined, for a stack oftwo isotropic constituents, in terms of the respective Lameparameters and and the volume fraction ofmaterial 1 present in the stack, They are

(1)

= + (2)

R =

+ + +(3)

Manuscript received by the Editor June 1, 1994; revised manuscript received October 31, 1994.*Dept. of Geology, Imperial College, London SW7 2BP, United Kingdom. 1995, Society of Exploration Geophysicists. All rights reserved.

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Anisotropic Parameters of Layered Media 1245

rial density. All three terms are always positive. The relation-ship between the proportions of each material present and theanisotropy has been noted in laboratory analogs of wavepropagation in stratified media (Melia and Carlson, 1984).

The second section in equation (19) is simply the differ-ence between the shear moduli of the two composite mate-rials. If we define the layer with the greatest shear modulusas layer 1, this term is also positive at all times. Themagnitude of this term has a large influence on the magnitudeof To achieve large values of near-normal incidenceanisotropy, a large difference in shear moduli between thelayered materials is vital.

The third and final section of the equation holds the key tothe polarity of as well as being an influence on themagnitude of the parameters. The denominator term de-pends upon multiples of the materials volume fraction withits shear modulus. This implies that the maximum value ofdoes not necessarily occur when the layers are of equalthickness. Figure 1 shows the dependence of on thevolume fraction of material 1 for a weakly anisotropic TImedium. It can be noted that as the ratio of the two shearmoduli increases, the value of at which the maximumvalue of occurs increases. The denominator term isalways positive leaving the polarity of to be decided by theterm Thus for to have a negative polarity, the

FIG. Plot of the anisotropic parameter against the volumefraction of material 1 present in the layered stack Thevalues calculated are for standard Lame parameters = 1,

= 0.05, = 0.5 and for values of of 1, 3, and 5. Theskew of the curves is caused by the dependence of thedenominator term of equation (13) upon multiples of andthe materials shear moduli.

ratio of the Lame parameters must be smallest in themore rigid layer, i.e.,

This relationship is illustrated in Figure 2. The figure plotsagainst volume fraction for different ratios of the Lame

constants in each layer. The maximum values of (positiveand negative) occur when the difference in Lame ratios islargest, with the polarity of the parameter deriving from therelationship above. The maximum values can be increasedby enlarging the difference between the shear moduli of thetwo materials. Berryman (1979) showed that = 0 is attainedin the case where the shear moduli of the two layers areequal or in the case where the ratios of the two layersare equal. These results may also be seen from equation (19).It can be noted that 0 in both situations. This propertyis in accordance with earlier theoretical results (Postma,1955; Backus, 1962).

EPSILON

For a seismic wave passing through a TI medium, theparameter dominates the P-wave velocity at propagationangles nearly perpendicular to the symmetry axis. The valueof represents the percentage difference between the hori-zontal and vertical compressional-wave velocities. It canalso be used with 6 to relate group and phase velocitieswithin TI media.

FIG. 2. Plot of the anisotropic parameter against the volumefraction of material 1 present in the layered stack Thevalues are calculated for standard Lame parameters = 1,

= 1, = 0.5 and for values of of and3


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Anisotropic Parameters of Layered Media 1247

FIG. 5. Plot of the anisotropic parameter against thevolume fraction of material present in the layered stack

The values are calculated for standard Lame parameters= 0.5 and for values of of 0.75, 1 and 2.

is the SH-wave velocity perpendicular to theaxis of symmetry, and is the S-wave velocity parallel tothe axis of symmetry. It can also be used to calculate the

normal moveout ofSHarrivals and to relate SHgroup andphase angles.

The relationship between and the Lame parameters ofthe component layers [equation (17)] indicates that is afunction of only the volume fraction and the shear moduliof the two layers. The first part of the equation shows that,

similar to the maximum value of occurs when the layershave equal thickness = 0.5). The actual magnitude ofdepends, as do the magnitudes of and on the differencebetween the shear moduli of the layers. This relationship isillustrated in Figure 5. One important point to be noted isthat by equation (21), is always positive. In equation (23)above, this suggests that for a two-layer sequence thevelocity of the SH-wave perpendicular to the axis of sym-metry is always greater than or equal to the velocity of theS-wave parallel to the axis of symmetry.

CONCLUSIONS

In the case of a sequence of layers of two materials, theparameters describing the anisotropy of the equivalent ho-mogenous medium can be derived from the elastic parame-ters of the original materials. To produce a highly anisotropicequivalent medium from a stack of two isotropic layers, thelayers must have significantly different shear moduli and

ratios.

ACKNOWLEDGMENTS

The authors would like to thank Paul Williamson and LeonThomsen for their useful comments. Financial support forJohn Brittan was provided by a Shell International Petro-leum Company scholarship.

REFERENCES

Backus, G. E., 1962, Long-wave elastic anisotropyhorizontal layering: J. Geophys. Res., 67, 4427-4440.

reduced by

Berryman, J. G., 1979, Long-wave elastic anisotropy in transverselyisotropic media: Geophysics, 44, 896-917.

Carcione, J. M., Kosloff, D., and Behle, A., 1991, Long-waveanisotropy in stratified media: A numerical test: Geophysics, 56,245-254.

Chapman, C. H., and Pratt, R. G., 1992, Traveltime tomography inanisotropic media-I. Theory: Geophys. J. Int., 109, 1-19.

Helbig, K., 1979, Discussion on The reflection, refraction and

diffraction of waves in media with elliptical velocity dependenceb

Melia, P. J., and Carlson, R. L., 1984, An experimental test ofF. K. Levin: Geophysics, 44, 987-990.

P-wave anisotropy in stratified media: Geophysics, 49, 364-378.Postma, G. W., 1955, Wave propagation in a stratified medium:

Geophysics, 20, 3-17.Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51,

19541966.

APPENDIX

ELASTIC PROPERTIES FOR TI COMPOSITE MEDIA

A stack composed of alternating thin layers of two TImedia may be approximated by an equivalent homogenousTI medium. The anisotropic parameters of the equivalentmedium can be found in terms of the elasticity-matrixcomponents of the individual layers. Using the same nota-tion as that used in Backus (1962), each layer has anelasticity matrix


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Denoting the matrix components for each medium by asubscript, the anisotropic parameters of the equivalent me-dium may be found using equation (9) of Backus andequations (13), (15), and (16) of this paper. When simplified,the anisotropic parameters and are given by

+

+

+ +

where is the volume fraction of material 1 present in the

stack and,

=

(A-1)

(A-2)

(A-3)

In the case of isotropic layering, equations (A-l)-(A-3)c on ve rg e t o their respective isotropic equivalent[equations (19), (20), and (21)].

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