In his pioneering 1961 paper on seismic anisotropy in a layered earth, Don L. Anderson (hereafter referred to as DLA) introduced a parameter often referred to in global seismology as η without providing any reasoning. This note hopes to clarify the significance of η in the context of the dependence of body wave velocities in a transversely isotropic system on the angle of incidence, and also its relation with the other well-known anisotropic parameters introduced by Leon Thomsen in 1986

Kawakatsu, H

Montagner, J-P

Song, TRA

UCL Discovery

On DLA's Hitoshi Kawakatsua, Jean-Paul Montagnerb and Teh-Ru Alex SongcaEarthquake Research Institute, The University of Tokyo,1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032 JAPAN (hitosi@eri.u-tokyo.ac.jp)bInstitut de Physique du Globe de Paris, Universite Paris-Diderot,1 rue Jussieu 75238-Paris cedex 05 FRANCE (jpm@ipgp.fr)cDepartment of Earth Sciences, University College London,Gower Street, WC1E 6BT, London, United Kingdom (tehrusong@gmail.com)January 29, 2015SUMMARY1In his pioneering paper on seismic anisotropy in a layered earth, Anderson (1961) introduced a2parameter often referred in global seismology as  without providing any reasoning. This note3hopes to clarify the signicance of  in the context of the dependence of bodywave velocities in4a transversely isotropic system on the angle of incidence, and also its relation with the other5well-known anisotropic parameters introduced by Thomsen (1986).6Key words: Seismic anisotropy, transverse isotropy, radial anisotropy.7Introduction8To describe a radially anisotropy (transversely isotropy with a vertical symmetry axis, VTI)9system, we employ the Love's original notation (Love, 1927), where stress and strain tensors10are related by11 2666666411223323311237777775 =26666664A H FH A FF F CLLN3777777526666664e11e22e332e232e312e1237777775 (1)where H = A  2N . There are ve independent parameters, A; C; F; L; N , to describe this12system, while there are two, ; , for the isotropic case, for which A = C =  + 2, F = ,13L = N = . For convenience, Anderson (1961) introduced the following \anisotropy factors":14' = C=A = 2V =2H (2) = (A H)=2L = N=L = 2H=2V (3) = (A  2L)=F; (4)which are all equal to 1 for isotropic case (V =pC=, H =pA=, H =pN=, V =15 pL=, where  gives the density.).161While both ' and  have simple meanings (degree of anisotropy in P- and S-wave, re-17spectively), the physical meaning of  is not so trivial. Takeuchi and Saito (1972), in their18monograph on seismic surface waves, reversed the order of the denominator and numerator in19the denition of  as,20 = F=(A  2L); (5)without commenting on the physical meaning either. As the expression of Takeuchi and Saito21(1972) is now commonly used in the global seismological community, we will use this notation22and denote it as DLA = F=(A 2L) in the following. In his text book, Anderson (1989) called23this DLA \the fth parameter required to fully describe transverse isotropy". In Dziewonski24and Anderson (1981), by showing examples, the eect of DLA on the incident angle dependence25of the phase velocity of P and S waves is discussed, and we generally think that DLA controls,26to some extent, the incidence angle dependence of those bodywaves, as well as those of Rayleigh27waves.28The purpose of this short note is provide simple theoretical background to how DLA aects29the bodywave propagation.30Incidence angle dependence of bodywaves31By solving an eigenvalue problem of an appropriate Christoel matrix, the incident angle, 32dependence of bodywave phase velocities can be obtained as33v2P () =(L+ C) + (A  C) sin2  +pS2(6)v2SV () =(L+ C) + (A  C) sin2   pS2(7)v2SH() = L+ (N   L) sin2 ; (8)where vP , vSV , and vSH denote phase velocities of pseudo- P, SV and SH waves respectively,34and35S = f(A  L) sin2    (C   L) cos2 g2 + (F + L)2 sin2 2 (9)= f(A  L) sin2  + (C   L) cos2 g2 + f(F + L)2   (C   L)(A  L)g sin2 2 (10)= f(C   L) + (A  C) sin2 g2 + f(F + L)2   (C   L)(A  L)g sin2 2 (11)= (C   L)2 + (A  C)(A+ C   2L) sin2  + f(F + L)2   (A+ C2  L)2g sin2 2: (12)When the condition36(F + L)2 = (C   L)(A  L) (13)is satised, equation (11) will be S = f(C   L) + (A  C) sin2 g2, andv2P () = C + (A  C) sin2  (14)v2SV () = L (15)v2SH() = L+ (N   L) sin2 : (16)The condition (13) is called by Thomsen (1986) the elliptic condition, since, in the absence37of the sin2 2 term, the forms of the wave velocity surfaces as a function of incidence angle 38are elliptical with only a sin2  dependence. When condition (13) is not satised the presence39of the sin2 2 term means that the wavesurfaces can be either convex or concave. (The convex-40ity/concavity of the P velocity is in the opposite sense to that of the SV velocity. This is an41explicit consequence of the presence of thepS term in (6) and (7) with opposite signs.)422Thus if we were to introduce an additional parameter to characterize the incidence angle43dependence of bodywaves, one reasonable choice may be44 =F + L(A  L)1=2(C   L)1=2 ; (17)and  = 1 for the isotropic case.45Further considering46(A  L)(C   L) =A+ C2  L2 A  C22;470 =F + LA+C2   L(18)may be another possibility that might make sense by looking at equation (12).48One of the good points of DLA is that it is simple and depends on just A and not C.49Assuming P-wave anisotropy is small, if we substitute A+C2 in (18) by A, we get5000 =F + LA  L (19)It is instructive how these parameters ('s) behave when both P- and S-wave anisotropy is51absent (i.e., A = C and L = N). When these conditions are satised,52v2P () =(L+A) +pS2v2SV () =(L+A) pS2v2SH() = L;and53S = f(A  L)g2 + f(F + L)2   (A  L)2g sin2 2;and sin2  dependence disappears. In this case, , 0 , and 00 reduce to the same form. Also,54it is easy to see that DLA = 1 gives the elliptic condition, and so in this sense, DLA   155becomes a measure of a departure from the elliptic condition to dictate the convex/concave56pattern.57For more general case,  = DLA   1 is small for weak anisotropy,58 = DLA   1 = F  A+ 2LA  2L : (20)Similarly5900 = 00   1 = F  A+ 2LA  L = A  2LA  L ; (21)and as long as A L > A 2L > 0 is satised, 00 has the same sign as , and  > 00, indicating6000 is also small. So in this respect, if anisotropy is weak (especially in P), DLA might be a61good proxy for  whose departure from unity provides a measures of the deviation from elliptic62anisotropy and dictates the convex/concave pattern of the incidence angle dependence of vP63and vSV .643Thomsen's parameters65Thomsen (1986) introduced three parameters for VTI system, now referred to as Thomsen's66parameters, and they are dened as67" =A  C2C=12(' 1   1) (22) =N   L2L=12(   1) (23) =(F + L)2   (C   L)22C(C   L) ; (24)which are all small for weak anisotropy. While " and  are directly related to ' and  respec-68tively as shown above and thus to P- and S-wave anisotropy,  was introduced such that it69dominates vP in the case of near vertical incidence as in reection proling.70Considering that  = " is their condition for elliptical anisotropy, examination of "  leads71to72"   = A  C2C  (F + L)2   (C   L)22C(C   L) (25)=(A  L)(C   L)  (F + L)22C(C   L) (26)= 1  2 A  L2C; (27)and we now see the connection between Thomsen's  and  introduced here. If DLA were a73proxy of  for weak anisotropy, we might be able to say that a connection between DLA and74Thomsen's  is established.75For weak anisotropy, the incidence angle dependence of bodywaves are, according to Thom-76sen (1986),77vP () = H 1 +  sin2  cos2  + " sin4 (28)vSV () = V1 +2H2V("  ) sin2  cos2 (29)vSH() = V (1 +  sin2 ); (30)and when the elliptic condition is satised78vP () = H(1 + " sin2 )vSV () = VvSH() = V (1 +  sin2 );which show simple incidence angle dependences.79(28)(29)(30) may be expressed in terms of 2 and 4 to make the incidence angle dependence80more explicit:81vP () = Hh1 +"2(1  cos 2)  !2(1  cos 4)i(31)vSV () = V1 +2H2V!2(1  cos 4)(32)vSH() = Vh1 +2(1  cos 2)i; (33)where ! = ("   )=4 is introduced. These equations show that ("   ) dictates the con-82vex/concave nature (i.e, cos 4 dependence) of vP and vSV .834DLA and  for weakly anisotropic models84To nish up this short note, we compare distributions of -related parameters for some of85weakly anisotropic cases.86Millefeuille (isotropic layers) case87In the rst example, we present a series of VTI models constructed by the Backus averaging88(Backus, 1962) of a stack of two kinds of homogeneous isotropic layers: soft layers embedded in89a background solid matrix (e.g., Kawakatsu et al., 2009). We parameterize (i) the proportional90reduction of rigidity of soft layers to the background by a (0  a  1), (ii) the proportional91reduction of the bulk modulus by a=2, and (iii) the volume fraction of soft layers by f (0 92f  1). Both a and f are varied in intervals of 0:05. Figure 1(a) compares  with DLA93(blue circles) or 0 (magenta crosses). While  and 0 give almost the same values, DLA940.4 0.6 0.8 1 1.2 1.4 1.60.40.60.811.21.41.6Millefeuilleηκη DLA , η κʹ0.40.60.8 1 1.2 1.4 1.60.40.60.811.21.41.6General TIηκη DLA  , η κʹ0.40.60.8 1 1.2 1.4 1.60.40.60.811.21.41.6A−type olivineηκη DLA , η κʹ(a) (b)(c)Figure 1: Comparison of -related parameters for various weakly anisotropic models. (a)Millefeuille case, (b) general TI case, and (c) rotated A-type olivine case. Green asteriskscorrespond to  vs: DLA for (a) a = 0:9, f = 0:01, (b) peak-to-peak anisotropy for both Pand S waves is 1:5% with DLA = 0:9; 1:0, and (c) the A-type olivine fabric case whose fast axislies in the horizontal plane, all for which examples of incident angle dependency of bodywavesare shown in Figure 2.5gives slightly smaller values. As   1 is guaranteed (Berryman, 1979), all values appear95generally less than 1. Although DLA in this case slightly deviates from , nearly one-to-one96correspondence may be observed to make DLA a reasonable proxy to .97General case98For a more general case, we construct a series of VTI models which have a maximum of 5%99anisotropy in both V;H and V;H , and 0:5 < DLA < 1:5 (Figure 1(b)). While  and 0 give100almost the same values, DLA deviates signicantly from the corresponding .101A-type olivine case102As a third example, we construct a series of VTI models by azimuthal averaging (Montagner103and Nataf, 1986; Montagner and Anderson, 1989) of an arbitrarily rotated A-type olivine fabric104(Jung et al., 2006) (Figure 1(c)) (rotation is done with a 30-degree interval for each Euler angle).105In a similar way to the preceding cases,  and 0 have almost the same values, but DLA106deviates from corresponding .107Examples of the incidence angle dependence of representative VTI models (denoted by108green asterisks in Figure 1) are shown in Figure 2. Note that the convex pattern of vSV109velocity occurs when  < 1.110Discussion111The incidence angle dependence of bodywave phase velocities in a radially anisotropic system112has not been discussed much in the geophysical literature as it is a dicult eect to observe.113In the laboratory, on the other hand, the simple sin  and sin 2 dependence (e.g., (6) and (11))114has been used to obtain the fth elastic constant from measurement along the angle 45 degrees115from the symmetric axes (e.g., Christensen and Crosson, 1968; Anderson, 1966). Song and116Kawakatsu (2012, 2013) recently suggested that such incident angle dependency in the Earth117may be constrained at subduction zones where the dip of the lithosphere/asthenosphere changes118along with the subduction, aecting the eective incidence angle of teleseismic bodywaves to119the system. If such analyses can be made generally, the new parameter  (or 0) might be120a useful tool in global seismology to characterize VTI (radially anisotropic) systems. How121-related parameters might be constrained from Rayleigh wave dispersion needs also to be122understood (e.g., Anderson, 1966).123Acknowledgments124This work was conducted while the rst author was a visiting professor at IPGP in 2014. He125thanks the members of IPGP, especially those of the seismology section for hospitality. We126thank Brian Kennett and Thorsten Becker for their excellent and prompt reviews, and the127editor, Gillian Foulger for her judgement. It is unfortunate that Professor Anderson's decease128robbed us of the opportunity to discuss with him directly how he came up with his .1296References130Anderson, D.L., 1961. Elastic wave propagation in layered anisotropic media. J. Geophys. Res.1319, 2953{2963.132Anderson, D.L., 1966. Recent evidence concerning the structure and composition of the Earth's133mantle, In Physics and Chemistry of the Earth. Pergamon, Oxford. volume 6. pp. 1{131.134Anderson, D.L., 1989. Theory of the Earth. Blackwell, Boston.135Backus, G.E., 1962. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys.136Res. 67, 4427{4440.137Berryman, J.G., 1979. Long-wave elastic anisotropy in transversely isotropic media. Geophysics13844, 896{917.139Christensen, N., Crosson, R.S., 1968. Seismic anisotropy in the upper mantle. Tectonophysics1406, 93{107.141Dziewonski, A.M., Anderson, D.L., 1981. Preliminary reference Earth model. Phys. Earth142Planet. Inter. 25, 297{356.143Jung, H., Katayama, I., Jian, Z., Hiraga, T., Karato, S., 2006. Eect of water and stress on144the lattice-preferred orientation of olivine. Tectonophysics 421, 1{22.145Kawakatsu, H., Kumar, P., Takei, Y., Shinohara, M., Kanazawa, T., Araki, E., Suyehiro, K.,1462009. Seismic Evidence for Sharp Lithosphere-Asthenosphere Boundaries of Oceanic Plates.147Science 324, 499{502.148Love, A.E.H., 1927. A treatise on the mathematical theory of elasticity. Dover, New York.149Montagner, J.P., Anderson, D. L., 1989. Constraints on elastic combinations inferred from150petrological models. Phys. Earth Planet. Inter. 54, 82{105.151Montagner, J.P., Nataf, H. C., 1986. On the inversion of the azimuthal anisotropy of surface152waves. J. Geophys. Res. 91, 511{520.153Song, T.R.A., Kawakatsu, H., 2012. Subduction of oceanic asthenosphere: evidence from154sub-slab seismic anisotropy. Geophy. Res. Lett. 39, L17301. doi:10.1029/2012GL052639.155Song, T.R.A., Kawakatsu, H., 2013. Subduction of oceanic asthenosphere: A Critical Appraisal156in Central Alaska. Earth Planetary Science Lett. 367, 82{94.157Takeuchi, H., Saito, M., 1972. Seismic surface waves. In Seismology: Surface Waves and Earth158Oscillations, Methods in Computational Physics. Academic Press, New York. volume 11. pp.159217{295.160Thomsen, L., 1986. Weak elastic anisotropy. Geophysics 51, 1954{1966.16170 20 40 60 800.9511.05MF, a =0.9, f =0.01, Rp=0.54%,Rs=3.9%,ηDLA=0.889,ηκ=0.941incidence angle [deg]Scaled Velocity0 20 40 60 800.9511.05GE, Rp=3%,Rs=3%,ηDLA=0.9,1,ηκ=0.983,1.04incidence angle [deg]Scaled Velocity0 20 40 60 800.9511.05Averaged A−type, Rp=4%,Rs=1.7%,ηDLA=0.9,ηκ=0.997incidence angle [deg]Scaled VelocityFigure 2: Examples of the incidence angle dependency of bodywaves for the VTI modelsrepresented by asterisks in Figure 1. Blue (and magenta in middle) solid lines, red (and cyan)dash-dot lines, and green dashed lines are respectively for vP , vSV , and vSH . Phase velocitiesare scaled by those of corresponding reference isotropic models. (Top), (middle), and (bottom)correspond to the models in (a), (b) and (c) in Figure 1. In the middle panel, vP and vSVshown by magenta and cyan lines are for DLA = 1,  = 1:04 case, and vSH behaves the samefor two cases.8

On DLA's η

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On DLA's η

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