A review of the effects of anisotropic layering on the propagation of seismic waves

Transcription

1 GeophysJ. R. QStr. SOC. (1977) 49,9-27 A review of the effects of anisotropic layering on the propagation of seismic waves Stuart Crampin Institute of Geological Sciences, Murchison House, West Mains Road, Edinburgh EH9 3LA 1 Introduction Received 1976 May 10 Summary. This paper reviews recent work, much of it unpub~shed, on the effects of anisotropy on seismic waves, and lays the theoretical background for some of the other papers in this number of the Geophysical Journal. The propagation of both body and surface waves in anisotropic media is fundamentally different from their propagation in isotropic media, although the differences in behaviour may be comparatively subtle and difficult to observe. One of the most diagnostic of these anomalies, which has been observed on. some surface-wave trains, and should be evident in body-wave arrivals, is generalized, three-dimensional polarization, where the Rayleigh motion is coupled to the Love, and the P and SV motion is coupled to the SH. This coupling introduces polarization anomalies which may be used to investigate anisotropy within the Earth. A material displaying velocity anisotropy must have its effective elastic constants arranged in some form of crystalline symmetry. The behaviour of both body and surface waves in such anisotropic structures differs from that in isotropic structures, and the variation of velocity with direction is only one of the anomalies which may occur, where we use anomaly to mean differences in behaviour from that expected in isotropic material. Within an anisotropic material three body waves propagate in any direction, having different and varying velocity, and different and varying polarization. Away from directions of crystal symmetry there may be anomalous phases, body and surface waves will have anomalous polarizations, and energy propagation of body and surface waves will not be parallel to the propagation vector. It appears intuitively that many of the anomalies can be attributed to the subtle interplay of the three varying body waves, making the variations of these anomalies difficult to predict. Similarly, smd differences in the structure, such as the thickness of the layer, can make radical changes in the anomalous behaviour. In this paper, we shall describe the type of phenomena to be expected in seismic waves from the presence of a layer of anisotropy within the Earth. A more complete treatment of the mathematics for the general problem of a plane layered structure containing a layer of anisotropy can be found in Keith (1975) for body waves, and Crampin (1970) and Taylor & Crampin (1977) for surface waves.

2 10 S. earnpin It is worthwhile demonstrating why seismic waves behave differently in anisotropic media. 2 Bodywaves We shall consider plane waves propagating in the x1 direction, if necessary rotating the tensor cijkm of elastic constants so that this direction of propagation is preserved. We assume the periodic solution of the displacements of a seismic plane wave are Ui = aj exp [iw(r - pkxk/c)], j, k = I, 2,3 (1 1 where a is the polarization vector, and, for propagation in the x1 direction, p2 = p3= 0, and p1 = 1. Substituting ui in the equations of motion P a2u,lat2= CjkmnUm,nk, (2) we have three simultaneous equations which may be written as an eigenvalue problem for 5: (T- 5l)a = 0, where t = pc, and T is the 3 x 3 matrix (ciljl) for i, j = 1,2, 3. For an isotropic material, where the tensor of elastic constants is invariant with rotation, T is a diagonal matrix %) (4) In such cases the eigen equation (3) factorizes, and the well-known isotropic velocities can be written down immediately c = (Y = [(A + 2p)/p] 1 2 for the compressional velocity and the repeated root c = p = (p/p) 2 for the shear velocity. Similarly, for an anisotropic material such as orthorhombic olivine, when aligned along the a, b, and c axes (a propagating, say) the matrix T is diagonal. Equation (3) factorizes, and there are three independent body waves with polarizations parallel to the axis of symmetry, and having three different velocities. As soon as the direction of propagation moves away from an axis of symmetry the tensor transforms by (3) the off-diagonal elements of T are filled, and the eigen equation (3) no longer factorizes. There are three roots for c2, the three pairs of c roots map out three slowness surfaces, and the polarization vector for each root is no longer parallel to an axis. The matrix T is a symmetric positive-definite submatrix of the full symmetric positivedefinite tensor of elastic constants. Thus for propagation in any direction there always exists three real body waves with mutually orthogonal polarization, which in isotropic media are coincident with the dynamic axes formed by the wave front and propagation vector, but

3 Anisotropic layering and seismic wave propagation 11 c 0 0 Figure 1. Variation of the quasi-longitudinal (qp) and the two quasi-shear (qsh and qsv, specified as if the cut were horizontal) body-wave phase velocities for propagation in the three orthogonal planes of symmetry of crystalline olivine. The particle motion of the qp, qsh, and qsv waves is pure P, SH, and SV motion only for propagation along the axes of sagittal symmetry (after Crampin 1976). in anisotropic media, except along axis of symmetry, the orthogonal particle motion is not coincident with the dynamic axes. It is this departure from isotropic particle motion which forms one of the most distinctive features of wave propagation in anisotropic media. Such body waves are called quasi-compressional waves qp, and quasi-shear waves qs1 and qsz. In this paper and in Keith & Crampin (1977a, b and c), we have chosen to name the quasishear waves qsv and qsh according to the isotropic wave they most resemble. Fig. 1 shows the velocity anisotropies of the three body waves for propagation in the orthogonal planes of symmetry of olivine. The largest velocity anisotropies are 22 per cent for the quasi-compressional wave in the 001-plane and 14 per cent for the fastest quasishear wave (qsh) in the 010-plane, where we have taken the velocity anisotropy to be the percentage by which the minimum velocity is less than the maximum (1-min vei/max vel) x 100. In general the variation of the elastic constants which produce these velocity anisotropies will be reduced in proportion to the amount of isotropic or randomly orientated material mixed with the aligned olivine (the velocities vary as the square root of the appropriate elastic constants). Fig. 2 shows the velocity anisotropies of body waves in transversely isotropic olivine where the a axis (100) is the symmetry axis and the b and c axes are randomly orientated. This is a geophysically likely configuration in the oceanic upper mantle (Francis 1969). It is worthwhile adding an aside on these figures. The effects on both body (and surface waves) of a transversely isotropic olivine structure, where the a axis is horizontal and the b

4 12 S. Ckampin ia QP 9.a QP u W m 6.5 \ -i x z G.0 - P u a J W > 5.a G SV W > a QSH 3: 4.5 r 0 O m 4.0 a sa 60 sa Figure 2. Variation of body-wave velocities for propagation in three orthogonal planes of symmetry of transversely isotropic olivine. a (100) is axis of symmetry, and b and c axes are randomly orientated (after Crampin 1976). and c axes are randomly orientated, are very similar to those of crystalline olivine with either a and b axes or a and c axes in the horizontal plane. This is because the form and magnitude of the velocity variation in the (001)-plane is similar to that in the (010)-plane, and the velocity anisotropy in the (100)-plane is in any case considerably smaller than in the other two planes (Fig. 1). If olivine in the upper mantle is aligned with the a axis in the direction of spreading, as seems Ilkely, then the question of whether the b and c axes are randomly aligned is not of great significance for seismic-wave propagation. One further consequence of the variation of velocity with direction is that the propagation of the energy of a plane wave is no longer in the direction of the propagation vector. The kinematic group velocity is the gradient of the frequency with respect to the wave number UT = (aw/ak,, aw/akz, aw/ak 3), where for plane wave propagation in the x1 direction w = CK~. Thus the group velocity is ut = (c, ao/akz, aw/a~~), and the energy propagates along the propagation vector at the phase velocity c, with the addition of a component along the wave front. The absolute value of the group velocity is greater or equal to the phase velocity. The variations of particle motion and group velocity for different directions of propagation in the 001-plane of olivine are drawn in Fig. 3. The plots show the deviations of each 7.0 OSH C5V (6) (7)

5 p 3. s - w 'T THETF12 0 "T s Q N THETR' c ol. GROUP VELOC I S Y FlNCLES x3 Figure 3. Particle motion and goup velocity deviations for body waves propagating in the (001) plane of olivine for orientations between 0" (100) and 90' (010). The continuous curve is the quasi-longitudinal wave, the short dash is the faster of the two quasi-shear waves, and the long dash is the slower wave. The discontinuity at 10" shows only that the velocities of the two quasi-shear waves are coincident at this point, and marks a change in nomenclature, not a physical discontinuity (after Keith 1975).

6 14 S. Crampin body wave from the isotropic body wave it most resembles. Since olivine is orthorhombic, and there is a horizontal plane of symmetry, the polarization of the quasi-shear waves is close to that of isotropic SH and SV waves. The variations of motion in the other two planes of symmetry are similar in kind, but are smaller in amplitude by approximately 2/3 for the OlO-plane and 1/3 for the 100-plane. This figure demonstrates one of the most important characteristics of anisotropic body waves, when it is noted that the polarizations of the three body waves are fixed in relation to the planes of symmetry of the anisotropic media. If the solid is rotated about the direction of propagation so that the planes of symmetry passing through the propagation vector are no longer vertical and horizontal, the polarization of the corresponding shear waves will no longer bear any close resemblance to SV and SH waves. Thus the common understanding that SH and SV waves have different velocities in anisotropic material is only strictly true along certain directions. In all other directions there are two quasi-shear waves propagating with different velocities and orthogonal polarizations, but the polarizations may be intermediate between SH and SV. Consequently, both SH and SV waves will be generated at a horizontal interface, when a quasi-shear wave is refracted into an isotropic material. Pl PRRTICLE MOTiON RNGLES Figure 4. Particle motion in the (fl/2, - 1/2,0) plane of olivine, between 0" (001) and 90" (1/2,,/3/2, 0), where (d,, dz, d,) are direction cosines with respect to the a, b, c axes. Notation as in Fig. 3 (after Keith 1975).

7 Anisotropic layering and seismic wave propagation 15 Fig. 4 shows similar variations for an off-symmetry plane, which illustrates a frequent situation, where quasi-shear waves have particle motion which smoothly changes between qsh and qsv as the direction of propagation in the material varies. This causes problems of nomenclature if we try to identify the quasi-transverse waves with either qsh or qsv. These deviations from isotropic motion result in fundamental changes in the propagation, reflection, and refraction of body waves. The requirement for the continuity of displacements and stresses across a (horizontal) interface between two media results in the continuity of the sum of the particle motions. Consequently, the independence of SH motion from P and SV motion is lost. The passage of a P wave through a plane horizontal layer of anisotropy can give rise to reflected and refracted SH waves as well as P and SV, because the continuity of the displacements at each interface in general require all three possible wave types to match any incident wave. Similarly, the passage of an SH wave can give rise to P and SV waves as well as SH. As a consequence of such behaviour, anomalous phase arrivals can be expected from anisotropic media. Fig. 5 contrasts the effects of propagation through an anisotropic layer with propagation through an isotropic layer. The representation is symbolic only as plane waves cannot propagate in anisotropic media unless there is a source supplying energy parallel to the wave front (Lighthill 1960) and the representation of a plane wave by a ray is particularly unrealistic in anisotropic media. However, the use of a ray notation is convenient to illustrate a point: the energy of a ray, such as those shown in the figure, propagates in the group direction not along the propagation direction, and will not in general be in the plane of incidence unless the incident planes is also a plane of elastic symmetry. Thus, while the behaviour at the interfaces is determined by the phase velocity, the passage of the energy through the layer is determined by the group velocity. The propagation of energy within the layer will deviate away from the plane of incidence, but will return to propagating parallel to the incident plane on entering an isotropic media again if all the interfaces are plane and parallel. 3 Surface waves We shall also demonstrate why surface waves behave differently in anisotropic material. Consider propagation in the x,-direction in a halfspace with a surface xg = 0. A surface wave (propagation vector coplanar but gro vectors dtvergen (coplanar) (coplanar) Figure 5. Comparison of rays incident on (a) an isotropic layer an anisotropic layer. This is a symbolic representation only.

8 16 S. Gampin travelling through an elastic media with a horizontal phase velocity c, decomposes into inhomogeneous plane wave components: ui = 2 f(n) ai(n) exp [iw(t - pk(n)xl/c)], j, k = 1,2,3 n=l where f(n) are excitation factors for each component, a(n) is the polarization vector, and pk (n)/c is the slowness. For propagation in the x1 direction at a given phase velocity, p2 = 0 and p1 = 1, and we drop the subscript on ps Substituting the displacement into the equations of motion (2), we have three simultaneous equations which may be written (Taylor & Crampin 1977) Fa = (Rp' t Sp + T - Zpc') a = 0, (9) where det (F) = 0 is the slowness equation, R and V are the matrices (ci3j3) and (Cj3jl) respectively, for i, j = 1,2,3. S = V t VT, and T is defined in equation (3). Equation (9) is another way of expressing the sixth-order polynomial slowness equation of Synge (1957). Taylor & Crampin show that the complete solution for surface waves in plane layered media can be specified in terms of the R, V, and T matrices. For propagation in an isotropic material, we have The SH motion then factorizes in equation (9) with a slowness for the Love wave decomposition of p/c = f - 11 '"/c, The remaining equation can be solved for p giving the Rayleigh wave decompositions p/c = f [(c/a)' - 11 '"/c for the longitudinal wave, and a repeat of the shear root for the SV wave, but with an orthogonal polarization vector. Across plane, parallel interfaces in the structure, the displacements, and normal, transverse, and tangential stresses are continuous. These are real function; of p for normal modes, and for isotropic media, where p is real or imaginary, the resulting equations are either pure real or pure imaginary. Thus, surface wave motion in isotropic media decouples into two independent families of Love and Rayleigh waves, and computations require only real arithmetic. For propagation in anisotropic material, the R, I/ and T matrices are filled, equation (9) does not factorize (unless x2 = 0 is a plane of symmetry), the p-roots and continuity equations are complex, and computations require complex arithmetic. (The conditions for x2 = 0 to be a plane of symmetry is that the elastic constants ciklm vanish when one or three of the suffices are equal to 2.) When there is no factorization, the sagittal plane and transverse components are coupled, each plane wave component has generalized particle motion, and one family of Generalized surface modes propagate. Alternate Generalized modes being, in general, the equivalent of the Rayleigh and Love modes, but with three-dimensional particle motion. The dispersion and other characteristics of these Generalized modes vary with direction. Along directions possessing sagittal symmetry, the Generalized family separates into two independent families having Rayleigh- and Love-type polarization. This similarity with isotropic motion is misleading, as the propagation is considerably more complicated in anisotropic media. Crampin (1976) discussed the complications of Rayleigh wave equations for propagation in directions of sagittal symmetry in orthorhombic olivine and in transversely isotropic media with horizontal axis of symmetry. The degree of complication of the wave equation can be indicated by the number of different elastic constants that the equation

9 Anisotropic layering and seismic wave propagation 17 Table 1. The number of independent elastic constants in surface wave equations for propagation in symmetry directions. The number of constants is a guide to the complexity of the equations. Material symmetry Symmetry relations relative to x, propagation, with boundaries x, = constant Number of independent elastic constants in surface wave equations for directions of sagittal symmetry: Rayleigh-type particle motion Love-type particle motion Isotropic Transversely isotropic Orthorhombic General anisotropic Spherical symmetry i x, axis of symmetry x, axis of symmetry x, axis of symmetry x,, x2, x, planes of symmetry x, plane of symmetry 2 2 (identical with isotropic equations) (identical with isotropic equations) 2 2 contains. Table 1 lists the constants for some symmetry conditions. Clearly, it is difficult or impossible to approximate to propagation in anisotropic structures by using isotropic models even when there is sagittal symmetry, except for directions perpendicular to axes of transverse isotropy, where it is possible to use isotropic results if allowance is made for different SH and SV velocities. The single family of Generalized modes has further complexities. In a direction of sagittal symmetry, the phase dispersion curves of modes from the Rayleigh- and Love-type families may cross each other, as is frequently the case with Second Rayleigh (2R) and Second Love (2L) modes in Earth structures. Away from sagittal symmetry, the two families coalesce, and the dispersion curves of the two modes can no longer cross. Instead, the modes approach each other in a pinch, and at the pinch exchange particle motion and dispersion characteristics. So that as a Generalized mode passes a pinch it will change from resembling a Rayleigh mode (say) to resembling a Love mode. Fig. 6 shows the dispersion of modes at a pinch. These pinches are one further phenomena which may disturb the regular wave trains of higher modes in Earth structures. Crampin (1 975) calculated the particle motion of surface waves propagating in particular symmetry directions in anisotropic media and showed that propagation in some directions showed particle motion anomalies diagnostic of the symmetry (Fig. 7). Computations show that a layer of anisotropy in the upper mantle would have a major effect on the particle motion of the Third Generalized mode (3G), the equivalent of the 2R mode in isotropic models. An example of the dispersion of a Generalized mode for a simplified Earth structure is shown in Fig. 8 for a 30-km thick layer of olivine in the upper mantle, in four directions of propagation. We see that even such a thick layer of strongly anisotropic olivine only produces maximum velocity anisotropies of about 3,4 and 5 per cent for the equivalents of the Fundamental Rayleigh mode, Fundamental Love mode, and 2R mode, respectively. The behaviour of the velocity anisotropy is different for alternate modes. The quasi-rayleigh modes show a largely monotonic variation of velocity for directions between two planes of symmetry. The quasi-love modes show an increase and a decrease for the same range of propagation directions. The behaviour of Love modes are closely related in this example to the qsh body waves, which also show an increase and decrease in velocity over the same range of directions. 2 3

10 18 S. Crampin Frequency (MHz) Figure 6. Phase velocity dispersion of the first six generalized normal mode surface waves propagating in two directions in 30 p of zinc oside, on 10 p silicon, overlayering a sapphire half-space. The projection of the group velocity on the sagittal plane is shown for the fundamental mode in the 0" direction (after Crampin & Taylor 1971). Figure 7. Three types of particle motion characteristic of particular symmetry directions: (a) Inclined- Rayieigh motion - horizontal plane of Tilted-Rayleigh motion - propagation at right angles to a vertical plane of symmetry, and (c) Sloping-Rayleigh motion - sagittal plane of symmetry (after Crampin 1975).

11 Anisotropic layering and seismic wave propagation 19

12 20?! I L ' I First modes 0. - g S. Oampin 00 60' A- - J - I Second modes- I I I I I - Third modes - 70 Fourth / / / I CP. =--- Zero vertical displacement Figure 9. Inclination of horizontal particle motion - angle e in Fig. 7(a) - at the surface to the propagation vector corresponding to the phase velocities of Fig. 8. The inclination is measured x, towards x2 for retrograde particle motion - the dashed line represents prograde motion (after Crampin & Taylor 1971). I I I (5) - 5. J 70 0' Fourth O * modes B 5. I Figure 10. Deviation of group direction, measured x, towards x,, for the non-symmetry propagation directions in Fig. 8 (after Crampin & Taylor 1971). The particle motion of the first four Generalized mode surface waves for the same structure are shown in Fig. 9. As the anisotropy has a horizontal plane of symmetry, the motion is of Inclined-Rayleigh type, where the inclination is the angle 0 in Fig. 7(a). The anomalies for the first, second, and fourth modes are comparatively small, but the particle motion of

13 Anisotropic layering and seismic wave propagation 21 the 3G mode (equivalent to 2R) swings through 180" in just a few seconds change of period. It would be expected that the third mode shows the greatest anomaly as it has the largest proportion of its energy propagating at the depth of the anisotropic layer. The group velocity of surface waves in anisotropic media diverges from the phase velocity. For normal modes in the surface wave guide am/& = 0, and equation (6) for the group velocity becomes The component of group velocity in the direction of the propagation vector may be obtained, as in isotropic media, by differentiation of the phase dispersion. However, there is now a component parallel to the wave front. Thus the group direction will not be in the propagation direction unless the sagittal plane is also a plane of elastic symmetry. Evaluation of group direction can only be found by numerical analysis of the phase velocity, consequently, the group velocity at any period in a particular direction can only be found by a process of trial and error by choosing the phase velocities in adjacent directions. The group velocity is of major physical significance, as in the absence of refracting boundaries energy propagates in a straight line at the group velocity, with divergent phase propagation directions. The difficulty of obtaining the true group velocity is unlikely to prove a serious problem as the deviation is usually small; in an olivine half-space the deviation is less than 7", and for the Earth models so far examined the deviation is less than 5". Hence, the magnitude of the group velocity is little affected by.the group deviation, although no search has yet been made for particularly divergent structures. The variation of group direction for the same structures as Figs 8 and 9 is shown in Fig. 10. The deviations are all small. An example of the difficulty of predicting anisotropic phenomena is the difference between the variation for the first and second modes: there is a change in the polarity of the deviation of the equivalent Love mode just as there is a change in the rate of change of the dispersion deviation, but the more difficult result is that the deviation of the equivalent of the Love mode is as large at 70-s period as the maximum deviation of the Rayleigh equivalent at 224 period. 4 Body waves propagation in weakly anisotropic material The weaker the anisotropy the closer, in general, the various body and surface wave characteristics resemble those in isotropic material. In particular, the anomalies become more difficult to observe accurately, and more easily mistaken for structural inhomogeneities. Small deviations of surface-wave polarizations from isotropic standards, for example, could be interpreted as lateral refractions. There is, however, an anomaly of quasi-transverse waves, which may be important as it depends on the extent and orientation of the material traversed rather than the degree of anisotropy. The presence of an elastic constant in the off-diagonal elements of matrix T prevents equation (3) factorizing, and consequently three different body waves propagate. In particular, there are two quasi-transverse waves, which will have nearly identical velocities if the anisotropy is weak, but however weak the anisotropy, the two body waves will have orthogonal polarization vectors with fked orientations relative to the symmetry. The fixed polarization for each quasi-transverse wave varies only with the propagation direction in the anisotropic media. Two orthogonal quasi-shear waves with fixed polarization and similar velocities will approximate to isotropic waves of any given transverse particle motion (SV or SH, say) by combining in the correct proportions to give the required vectorial sum of their polarization

14 22 S. CLampin VERTICAL RADIAL TRANSVERSE INCIDENT P WAVE Figure 11. Synthetic seismograms for a 4-s damped-sine wave P-pulse, incident at 45", and transmitted at a range of azimuths through the structure: h P ff P (km) (g/cc) (km/s) Otm/s) Isotropic Anisotropic (001)-cut of 20 per cent transversely isotropic olivine plus 80 per cent isotropic (01 = 7.5, p = 4.3), azimuths measured from top every 15" from 100 to 010 (from top of figure) Isotropic The structure is x, propagating with xj into halfspace. Surface efi'ects are not taken into account. vectors. The two quasi-transverse waves will then constructively interfere to produce the given particle motion and destructively interfere to destroy its transverse. However, the velocities of the interfering quasi-transverse waves will not be identical. Neither the constructive nor the destructive interference will be complete. There will be introduction of motion transverse to the primary motion, and such anomalies in body wave particle motion will exist however weak the anisotropy. The magnitude of the anomaly depends on the direction and distance travelled through the anisotropic material rather than the degree of velocity anisotropy. The delay of the particle motion anomaly relative to the initial onset of the transverse arrival will give some indication of the extent of the anisotropy. Figs 11 and 12 illustrate such an anomaly in synthetic seismograms calculated by the procedure outlined in Keith & Crampin (1977b). Fig. 11 shows synthetic seismograms from a 4-s P-pulse transmitted from a range of azimuths through a 30-km thick layer of weakly anisotropic material having a 4 per cent Fwave velocity anisotropy in the (001)-plane. n e anomalous SH component is negligible. Fig. 12 shows seismograms for the same structure but with the anisotropic layer now 100 km thick. In the middle of the range of azimuth, the SH pulse is approximately a third of the radial amplitude and is a distinctive arrival some 14s after the onset of the initial P wave. The 14 s being the S-P delay from the bottom of the anisotropic layer. In this example the anisotropy causes a variation of less than a second in

15 Anisotropic layering and seismic wave propagation VERTICAL RADiAL TRANSVERSE I--- INCIDENT P WAVE Figure 12. As for Fig. 11 but with a 100-km thick anisotropic layer. the arrival time of a teleseismic P wave travelling through the layer, which is unlikely to be detected. Thinner layers of the same weakly anisotropic material would produce comparable results for shorter period P arrivals. 5 The variation of body wave velocity with azimuth in a weakly anisotropic material The velocities of the three body waves propagating in the x1 direction in an anisotropic material can be obtained from the eigen value problem (T - pc21)a = 0 of equation (3). If the anisotropy is weak the off-diagonal terms of the T matrix are small and the body wave velocities are given by Pc: = Cllll +x, Pc; = C212l + Y, and Pc?3 = c z, where cl, c2 and c3 are the qp, 4SH and 4SV velocities, respectively, and X, Y and 2 are terms of second order in the off-diagonal elements, which are zero in directions of sagittal symmetry when x3 = constant is also a plane of symmetry. The variation of velocity with azimuth is obtained by rotating the elastic tensor about the vertical x3 axis. An element in the new coordinate system after a rotation of 8 can be expressed in terms of the original elements and multiples of 8. The variation of the qp velocity is given by PC: = (3cllll + 2(c cI2,,) + 3C2222) '/s +(cl,ll-~2222) ~ ~ 0 +(c2111 ~ 2 +clz2,)sin (c c c1212) + cz2zz) '18 cos 48 + (c Cl22Z) 4i sin 48, (12) (1 1)

16 24 S. Crampin the variation of the qsh velocity by PCi = (CIIII -- ~(CIIZZ - ~CIZIZ) -h C2222) '/8 (- CIIII -k 2(c1122 i- 2C1212) - C2222) '/8 COS 48 + (- c c1222) % sin 48 (13) and the variation of the 4SVvelocity by PC; + ~2323) % + ( c - c233) ~ ~ % cos ~ 28 t ~ c2313 sin 28. Equation (12) was first derived by Backus (1965) and was used successfully by Backus and subsequently other authors to describe the variation of P,, velocities. If the direction from which the azimuth 8 is measured has sagittal symmetry, the elastic constants cjjkm are zero when either one or three of the subscripts is equal to 2, and equations (12), (13) and (14) can be further reduced to (14) 4P: pc:=al + B1 cos28 + C, cos48, (15) qsh: pci= A2 t B2 cos 48, (1 6) 4SV: pc~=a3+b3cos28, (17) where Cl = - B2. The only symmetry restriction on equations (15), (16) and (17) is that the azimuth is measured from a direction having sagittal symmetry, whch, since the C, coefficient is usually smaller than B1, can be identified from observations by being a maximum or minimum of P-wave velocity. Equations (15), (16) and (17) transform to the form of (12), (13) and (14), respectively, on substitution of 8 +a, where cy is the angular separation of the origin from the direction of sagittal symmetry. In experiments to determine anisotropic structure in the Earth from observations of 4P velocities in refraction experiments, the question of whether it is better to solve for five variables in equation (12), or choose a direction of symmetry and solve the reduced equation (15) for three variables must be decided on the amount and distribution of data available. The various coefficients of terms in equations (15), (16) and (17) contain only a limited number of elastic constants and there is not enough information to distinguish between olivine and transversely isotropic olivine (with horizontal symmetry axis). Consequently, the velocity variation of one or all three of the body waves over the surface of the Earth cannot distinguish between a transversely isotropic and an orthorhombic material. Although the relations (15), (16) and (17) have only been proved for weakly anisotropic propagation, they approximately describe strongly anisotropic media such as olivine. The velocity anisotropy in Figs 1 and 2 demonstrates (1) 4P varying as cos28, but having a minimum at about 20" from 001 in the 010-cut (Fig. I(c)) from a small cos48 term. (2) 4SH varying as cos 48, with repeated values every 90" of azimuth. The qsh wave largely, but not completely, controls the velocity variation of the odd-order Generalized surface waves (the quasi-love modes), which are also nearly repeated every 90". These results are important for observations of velocity anisotropy for both Love and SH waves. (3) 4SV varying with the same sign as 4P but with smaller amplitude and no secondary minima. We see from these results that the variation of the qsh wave is quite distinct from the qsv wave. However, the two quasi-shear roots are analytically continuous for if we follow

17 Anisotropic layering and seismic wave propagation 25 the variation of any one quasi-shear root around the cubic corner formed by the junction of the three planes of symmetry of olivine in Fig. 1 (say) the two roots cross at least once, and on return to the starting axis we have transferred to the other branch. We may note that the form of equation (12), five terms in even powers of 8, is not unexpected. It is the first five terms of a Fourier Series expansion of a function which repeats every 180". The expansion is of c2, rather than c, because the coefficients can then be written explicitly. Similarly the Fourier expansion of the azimuthal variation of a Generalized surface wave of mode m, and frequency w, in a plane layered structure having symmetry which repeats every 180, may be written m c(m, w) = A(m, w) + C [ B (m, ~ w) cos2n0 + C, (m, w) sin2n01, n=l where 8 is measured from an arbitrary origin. If we omit terms in higher powers than 4, we have the equation of Smith & Dahlen (1973) 2 c(m, w) = A(m, w) t C [B,(m, a) cos 2nd t Cn(m, w) sin2ne], (19) n=l which holds for approximately the same conditions and order of accuracy as equation (12). By symmetry considerations, we can immediately write the reduced form of (19) as c(m, w) =A (m, w) + Bl(m, w) cos 28 + B,(m, w) cos 48, (20) where 8 is now measured from a direction having sagittal symmetry. 6 Discussion Recently an increasing amount of direct evidence has been gathered that the sub-crustal lithosphere is anisotropic. Certainly all the circumstantial evidence points to aligned anisotropy in the upper mantle, and the vast majority of hand sized rock specimens show some aligned anisotropy. Immediately beneath the Moho, where brittle fracture can still take place and the rocks are still crystalline, the lithosphere is created under stress beneath oceanic ridges, or is subjected to the enormous and enduring forces of plate motion beneath continents. In such situations there must be very special reasons if anisotropy is not to exist, and as yet none have been suggested. The counter-claim that no very large scale velocity anisotropies have yet been established can be met by the numerical examples of Crampin & Taylor (1971) for surface waves, and Keith (1975) for teleseismic body waves, which show that even for a 30 km thick layer of olivine in the upper mantle the velocity anomalies at the surface are generally small and could be otherwise explained. In refraction and reflection experiments which are the most direct measures of velocity in the Earth, a comparatively thin isotropic refractor above the anisotropic layer could effectively hide any velocity anomaly. Quite the most distinctive phenomena associated with the presence of anisotropy, are anomalies in polarization. These include anomalous body-wave conversions P and SV to SH, and SH to P and SV, which must exist at the Moho if velocity anisotropy exists, and coupling between Rayleigh and Love modes. Surface-wave phenomena are particularly important, because higher modes in particular frequency ranges are the only seismic waves which effectively sample some parts of the upper mantle. In view of the magnitude of these polarization anomalies, which certainly exist, if the upper mantle is anisotropic, why have they not been noticed? Observations of polarization require analysis of rotated, calibrated, digitized (or preferably digital), three-component

18 26 s.c).ampin recordings. As it is only recently that the significance of body-wave polarization studies for investigation of anisotropy has been recognized little has yet been attempted. Refraction experiments frequently use only vertical instrumentation, and even with adequate instrumentation, the effects will be difficult to observe, as any anomaly existing at the Moho will be fiitered by a largely isotropic, and inhomogeneous crust before it can be observed. Thus, transverse coupling of a short-period P wave emerging from the Moho will be separated by at least a 4 or 5-s S-P delay before the surface is reached. Long-period P waves probably provide a less ambiguous anomaly. To find an anomaly in surface wave particle motion, we must examine a mode which has a large amount of energy travelling in the anisotropic layer. The equivalent of the 2R and 2L are such modes for anisotropy in the continental upper mantle. Unfortunately, there are comparatively few observations of these modes in the period range of interest (6-15 s) as they are attenuated by inhomogeneities along the path. Crampin (1967) observed such coupled motion from a Ryukyu Islands earthquake at a number of stations across Asia, but at that time the numerical calculations were not yet developed. A further study of higher mode polarization at NORSAR and at Soviet stations in Eurasia (Crampin & King 1977), now suggests aligned anisotropy beneath much of Eurasia. At the present stage in the investigation of anisotropy within the Earth, the importance of polarization studies is two-fold: diagnostic and confirmatory. Comparatively simple analysis of a few three-component records can lead to a strong indication of the presence of anisotropy, and any studies or suggestions of velocity anisotropy should be confirmed by the presence of appropriate particle motion anomalies. We can expect polarization observations to provide a sensitive test for any proposed anisotropic model, and, in particular, indicate directions of symmetry. These symmetry directions will be the result of present or fossil stress patterns within the upper mantle, and will have implications for plate motions and tectonic activity. The delays of the converted SH arrivals for incident P waves will be a powerful tool for examining the extent of the anisotropy. The example shown in Fig. 12 of anomalies from a teleseismic P wave at a range of azimuths through a weakly anisotropy layer indicates the symmetry directions by the absence of the SH anomaly at some azimuths, and extent of the anisotropy by the S-P delay from the lower interface. Acknowledgments I have benefited, over the years, by many discussions with collaborators in anisotropic investigations, particularly with G. Alisdair Armstrong, Microwave Electronic Systems Limited, Colum M. Keith, Edinburgh University (now at Department of Energy, Mines and Resources, Ottawa), David W. King, NORSAR (now at Sydney University), David B. Taylor, Edinburgh Regional Computing Centre, and more recently by discussions with David Bamford, Edinburgh University. I am especially indebted to Colum Keith for permission to use his program for calculations for Figs 11 and 12. This work was undertaken as part of the research programme of the Institute of Geological Sciences and is published with approval of the Director, IGS. References Backus, G. E., Possible forms of seismic anisotropy of the uppermost mantle under oceans, J. geophys. Res. 70, Crampin, S., Coupled Rayleigh-Love second modes, Geophys. J. R. astr. Soc., 12, Crampin, S., The dispersion of surface waves in multikdyered anisotropic media, Geophys. J. R. asfr. SOC., 21, Crampin, S., Distinctive particle motion of surface waves as a diagnostic of anisotropic layering, Geophys. J. R. astr. Soc, 40,

19 Anisotropic layering and seismic wave propagation 27 Crampin, S., A comment on The early structural evolution and anisotropy of the oceanic Upper Mantle, Geophys. J. R. astr. Soc., 46, Crampin, S. & Taylor, D. B., The propagation of surface waves in anisotropic media, Geophys. J. R. astr. Soc., 25, Crampin, S. & King, D. W., Evidence for anisotropy in the upper mantle beneath Eurasia from the polarization of higher mode seismic surface waves, Geophys. J. R. astr. SOC., 49, Francis, T. J. G., Generation of seismic anisotropy in the Upper Mantle along the mid-oceanic ridges. Nature, 221, Keith, C. M., Propagation of seismic body waves in layered anisotropic structures, PhD dissertation, Edinburgh University. Keith, C. M. & Crampin, S., 1977a. Seismic body waves in anisotropic media: reflection and refraction at a plane interface, Geophys. J. R. astr. Soc., 49, Keith, C. M. & Crampin, S., 1977b. Seismic body waves in anisotropic media: propagation through a layer, Geophys. J. R. astr. Soc., 49, Keith, C. M. & Crampin, S., 1977c. Seismic body waves in anisotropic media: synthetic seismograms, Geophys. J. R. astr. Soc., 49, Lighthill, M. J., On magneto-hydrodynamic waves and other anisotropic wave motions, Phil. Trans R. SOC. Lond. A, 252, Smith, M. L. & Dahlen, F. A., The azimuthal dependence of Love and Rayleigh wave propagation in a slightly anisotropic medium, J. geophys. Res., 78, Synge, J. L., Elastic waves in anisotropic media, J. marh. Phys., 35, Taylor, D. B. & Crampin, S., Surface waves in anisotropic media: propagation in a homogeneous halfspace, Phil. Trans. R. Soc. Lond., in press.