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Geophys. J. R. astr. SOC.(1975) 40, 177-186. Distinctive Particle Motion of Surface Waves as a Diagnostic of Anisotropic Layering Stuart Crampin (Received 1974 October 21)* Summary Surface waves propagating along particular directions in anisotropic media may have one of three distinctive particle motions indicating orientations of planes of anisotropic symmetry with respect to the direction of propagation and the free surface. The particle motion of surface waves in a medium with a horizontal plane of symmetry has been observed on seismograms and used to indicate the presence of such anisotropy within the Upper Mantle. Introduction Aligned crystalline anisotropy patently exists on a small scale within the Earth. The presence of anisotropy on a larger scale is hard to establish because of the difficulty in finding a seismological parameter, which is sufficiently sensitive to anisotropy to be used to determine its scale and perhaps its nature. Seismic waves passing through an anisotropic region would be affected in many ways. Anomalies would be present in travel times, seismic velocities, the relationship between P, SV, and S H , body wave amplitudes, and surface wave dispersion, among others, but only in exceptional circumstances would the anomaly in these parameters be so marked that it could unequivocally be assigned to the presence of anisotropy, and not to some unknown inhomogeneity. In this paper we shall show that surface waves have distinctive particle motion when propagating in a structure, which contains a layer of anisotropic material with certain symmetry relations, and that t h s anomaly is sufficiently large to be used as a diagnostic for the presence of anisotropy. We consider the propagation of surface waves in media which contains a layer having crystalline anistropy. Except along directions of crystal symmetry, the three body waves in an anisotropic media have independent velocities, and particle motions which are a combination of linear oscillations in three dimensions. This phenomenon is mirrored by the surface waves, which do not have the degenerate form of two independent families of normal modes, but form one family of normal geneialized modes each with elliptical particle-motion in three dimensions (Crampin 1970, hereafter called Paper I). Along directions of symmetry, this generalized family degenerates into two independent families, one having the elliptical particle motion in the sagittal plane of the familiar Rayleigh wave, and the other the horizontally polarized transverse motion of the Love wave. Crystals have various types of symmetry, which considerably reduce the number of elastic constants below the maximum of 21. Nevertheless, analytical examination of the effects of anisotropy is confined to problems in unlayered homogeneous * Received in original form 1974 July 26. 177 C 178 S . Crampin Table 1 Relative phases of particle motion of surface waves (xl propagating, x3free surface) in anisotropic half-spaces for various symmetry conditions. Each component has oscillatory motion of the same frequency but dgerent phase. Crystal symmetry Phase differences between particle motion relative to phase u1 Planes of symmetry u1 1. No symmetry 2. x1 = 0 0 a B 0 +a12 +ff/2 =0 0 3. 4. XJ XI =0 u2 0 - a Type of particle motion u3 +,I2 B - 5. x1 = 0, x2 = 0, and xj = 0 Generalized Ti1ted-Rayleigh Inclined-Rayleigh Sloping-Rayleigh Love Rayleigh Love 6. Transversely isotropic 7. Isotropic Where a and /3 are constants not equal to 0 or f n / 2 , and ' - ' denotes no motion for that component. The word Ruyleigh refers to elliptical particle motion in a plane, where the plane is not necessarily the sagittal plane (see Fig. 1). (a) A b .... .............9. Propagation direction FIG. 1. Three types of particle motion characteristic of particular symmetry orientations: (a) Inclined-Rayleigh motion-horizontal plane of symmetry, (b) Tilted-Rayleigh motion-propagation at right angles to vertical plane of symmetry, and (c) Sloping-Rayleigh motion-propagation with the sagittal plane a plane of symmetry. Distinctive particle motion of sudace waves 179 Rotations about x-axis 4.0 '0 30' 60° 90' 4.0 0" (001) Angle from (001) 60' 90° Plane of cut (010) 30' (001) Plane of cut FIG. 2. Surface waves on an olivine half-space for three ranges of orientations. The upper figures show the variation with orientation of the phase velocities of the generalized surface wave, and the two slower body waves (the quasi-shear waves). The lower figures show the deviations of the particle motion from elliptic motion in the sagittal plane (0 in Fig. 1). The crosses and circles in the first two diagrams mark positions along directions of symmetry where the generalized wave degenerates into Rayleigh and Love type motion, respectively. QSH, QSV, QSl, and QS2 refer to quasi-SH, quasi-SY, and two undifferentiated quasi-shear waves, respectively. Table 2 Structural models for Figs 2-6. Olivine half-space Fig. 2(a) (010)cut, propagating over a range of directions between (001) and (100). Fig, 2(b) (100)-propagating for a range of cuts between (001) and (010). Fig. 2(c) x-propagating for a range of cuts between (001) and (100). Layered structure Figs 3-5 h 10 20 30 P 2.7 2.9 3 * 324 3.6 a B 3.4 6.6 3.8 olivine layer 9.0 5.4 5.8 The orientation of the olivine is given in the Figure captions. (100) 180 S. Crampin half-spaces, and frequently only to directions of symmetry within such a half space. The program used for the numerical analysis in this paper is based on the work in Paper I, and in Crampin & Taylor (1971, hereafter called Paper II), which has been further developed by Armstrong & Crampin (1972, 1973) to calculate surface wave motion in piezoelectric structures for microwave acoustic device applications. Surface-wave particle-motion in a homogeneous anisotropic half-space The particle motion of surface waves of a given frequency in a randomly orientated anisotropic crystalline half-space is oscillatory motion in three dimensions, in which each component has a different angle of phase. The orientation of the anisotropy with respect to the direction of propagation and the free surface has characteristic effects on the particle motion of the generalized normal-mode surface wave. It is found by numerical experiment (and can be shown theoretically, Crampin & Taylor, in preparation) that surface waves, propagating parallel or perpendicular to planes of symmetry, have components of displacement, which are in phase, out of phase by & n/2, or completely decoupled as the horizontally polarized transverse motion of Love waves is decoupled from Rayleigh wave motion in isotropic media. These phase differences persist in a layered structure if each layer contains similar symmetries. In particular, isotropic layers, which being wholly symmetric, are transparent to particle motion and do not affect the anisotropic phase differences. The effects on the particle motion of propagation parallel or perpendicular to planes of symmetry are summarized in Table 1. The terms inclined-Rayleigh, tilted-Rayleigh, and sloping-Rayleigh refer to motion of the types shown in Fig. 1. Fig. 2 shows some numerical evaluations of the deviation from isotropic Rayieigh wave motion of surface waves propagating on olivine half-spaces with three ranges of orientation. Olivine is orthorhombic with three mutually orthogonal planes of symmetry: (loo), (OlO), and (001). The elastic constants are taken from Verma (1960), and the orientations are given in Table 2. Fig. 2(a) shows the variation in motion over a range of directions when the free surface is a plane of symmetry. Fig. 2(b) shows the variation for propagation at right angles to a plane of symmetry as the plane is rotated. As the direction of propagation does not change, the velocities of the body waves are constant, and velocity of the generalized surface wave changes very little. Fig. 2(c) shows the variation of motion when the sagittal plane is a plane of symmetry as the plane is rotated. The generalized surface wave is independent of one of the two quasi-shear body waves (in this case the lowest), and this body wave identically satisfies the equations for Love waves over the whole range. In all three cases, the deviations from the isotropic Rayleigh-wave particlemotion are comparatively small, although no attempt was made to search for orientations where the particle motion would have larger deviations. Surface-wave particle-motion in layered earth models The layered models in Table 2 consist of a two-layered isotropic crust overlying an isotropic mantle with a layer of anisotropic olivine in the upper mantle beneath the crust. The earth models are not realistic but are intended to indicate that aligned anisotropy can have marked effects on particle motion at the surface, which are diEcult if not impossible to reproduce with any other type of non-isotropic phenomenon. Figs 3, 4 and 5 show the variation of the particle motion with period of the four gravest modes of generalized surface waves propagating in the layered structure Distinctive particle motion of surface waves I \ I I \ \/ 3, 1 2c 0 I I 181 I / / (D w 9( b H .-s - 6C (D n e n e Etn .L 0 w .-5 z .-- 3c 0 - a I I I I I I -30 1 10 1 I 30 Period in seconds 20 I 40 FIG.3. The variation with period of the deviations from isotropic Rayleigh particlemotion of generalized waves, having inclined-Rayleigh particle motion, propagating in the layered structure of Table 2, when the olivine has a horizontal plane of symmetry. The olivine layer is (010) cut and propagation is at 30" (solid line) and 60" (dashed line) measured from (001) towards (100). when the olivine layer is 30 km thick. These four modes are equivalent to the fundamental and second modes of Rayleigh and Love waves in an isotropic structure. The correspondence between the modes in an anisotropic structure and those in an isotropic structure is not always simple: there may be occasions when two modes pinch together (Paper 11) and the modes exchange characteristics, and the orthogonal nature of the particle motion of Rayleigh and Love modes in an isotropic structure does not exist between the diffeient normal modes in an anisotropic structure. 182 S. Crampin I I I 1 I I I 12c I mode4 9c -m 5 c) 6( r .E" . " I c I I I r 0 ._ L 0 c ._ I I I c c E g ln I I I 3c I P c I I c 30 10 1 1 I 20 30 40 Period in seconds FIG.4. The variation with period of the deviations of generalized waves, having tilted-Rayleigh particle-motion, propagating in the layered structure at rightangles to a plane of symmetry. The Olivine layer is (100)-propagating at cuts of 30" (solid line) and 60" (dashed line) measured from (001) towards (010). The orientations of the Olivine layers in the models used for Figs 3, 4 and 5 are two directions or cuts within the ranges of orientations used for Fig. 2(a), 2(b) and 2(c), respectively, and the waves have the correspondingly inclined-, tilted-, and slopingRayleigh particle-motion. The size of the deviations in Figs 3 and 4 are generally similar. The deviations of the first and second generalized modes peak near a period of 30 s, with a maximum deviation of 20" or less. The higher modes, in particular the third mode (usually Distinctive particle motion of surface waves -10 30 40 183 50 60 50 60 70 I -10- -20Y) 7 -30- 2 -40- : al a 0 -50m -60- L -L I 10 20 30 2 40 I . I 70 PerirJd in seconds FIG.5. The variation with period of the deviations of generalized waves, having sloping-Rayleigh particle-motion, propagating in the layered structure, when the sagittal plane is a plane of symmetry. The olivine layer is x-propagating at cuts of 30" (solid line) and 60" (dashed line) measured from (001) towards (100). equivalent to the second Rayleigh mode in isotropic media), show a much larger range of deviations. There are two contributing factors to the large deviations: (1) The variation with depth of the particle motion of third modes has a large lobe of amplitude below the crust near the depth of the olivine layer (the exact depth of the lobe depending on the period) so that most of the energy of the third mode travels in the anisotropic layer. Consequently, the anisotropy usually has a large effect on this mode. (2) In these anisotropic earth models, the third and four generalized modes have similar dispersion (as have the corresponding modes, the second Rayleigh and second Love modes, in an isotropic earth model). Pinching occurs when two modes approach each other in period and velocity. The closeness of the third- and fourth-mode dispersion curves makes them liable to pinching, and each pinch will result in a rapid transition in properties, in particular the particle motion may show a rapid change. These factors result in the plane elliptical particle motion of the third and fourth modes changing in just a few seconds period, from being nearly in the sagittal plane to being nearly in the plane of the wavefront (inclined-Rayleigh motion of Fig. 3), or nearly in the horizontal plane (tilted-Rayleigh motion of Fig. 4). The sloping-Rayleigh wave motion in Fig. 5 behaves rather differently from the motion in the previous two figures. Rapid deviations due to pinching are not possible, as, for this orientation of olivine, the generalized family of modes has degenerated into the two independent families of the sloping-Rayleigh waves and the Love waves (see Table 1, Symmetry type 4), and pinching usually occurs only between adjacent pairs of generalized modes. Thus, in Fig. 5, the second sloping-Rayleigh mode shows little deviation, and the elliptical motion of the fundamental mode shows, particularly for the 60" cut, a change at 40-s period from being vertically elongated to being horizontally elongated, which is a possible fluctuation of isotropic Rayleigh motion. Fig. 6 shows the deviations in the inclined-Rayleigh motion as the thickness of the olivine layer is varied from 30 to 5 km with the orientation kept constant. The S. Crampin propagating with the same structure and orientation as for Fig. 3, but fo; a range of thicknesses of the olivine layer: 30 km, solid line; 20 km, dashed line; 10 km, dash/dot line; and 5 km, dotted line. The Olivine layer is (010) cut and 45" propagating (001 towards 100). The deviations for mode 3 have not been plotted for periods less than 9.5 s, as they show rapid fluctuations which could be followed only with difficulty. deviations of Mode 1 are reduced as the thickness of the olivine layer decreases, but even a thickness of only 5 km produces a peak deviation of some 13" at 23-s period. The particle motion of Mode 2 is little effected by the layer of anisotropy and the peak deviation does not exceed 7". The fourth mode also shows little effect. The third mode shows rapid fluctuations for the whole range of thicknesses of the olivine layer; in particular there is still rapid change at the lowest thickness of 5 km. The Distinctive particle motion of surface waves 185 deviation at this thickness ranges over 7112 in 6-s period, which includes a section between 10- and 14-s period where the deviation only varies between 35" and 50". Conclusions The figures show that for some orientations even quite small thicknesses of anisotropy in the upper mantle produce marked effects on the particle motion of surface waves, which are characteristic not only of anisotropy but also of the orientations of some of the planes of symmetry. If the upper mantle, or more particularly, a continental plate, contains any crystalline material, the long-persisting horizontal stresses required to move plates will be sufficient to align crystals over a large area of the plate. The forces likely to align crystals will be the stress causing the plate movement acting in a horizontal plane, and vertical forces such as gravity, and forces associated with differentiating processes, resulting in two orthogonal planes of symmetry: horizontal and vertical. Thus, of the orientations resulting in the three distinctive particle motions discussed here, only orientations producing inclinedRayleigh motion appears to be possible in the Earth, except perhaps on a limited local scale. The effects on particle motion of anisotropy in the upper mantle are most marked for the motion of the third generalized mode (second Rayleigh mode), because these modes have most of their energy travelling in the upper mantle, and because the third and fourth modes have similar dispersion making mode pinching liable to occur in earth models containing anistiopy. Paper I1 presented calculations of several surface-wave parameters for similar structures. From Paper I1 and other numerical examinations of the effects of anisotropy, it appears that particle motion is probably the most sensitive observable parameter for the unambiguous recognition of anisotropy in the upper mantle at a single station. This distinction belongs to particle motion because one of the major effects of anisotropy on elastic wave propagation in a homogeneous medium is the loss of independence of P and S waves and the consequent breakdown in polarity between motion in the sagittal plane and SH motion characteristic of horizontally layered isotropic media. Inclined-Rayleigh particle motion has been observed by Crampin (1966, 1967) for Second Rayleigh/Second Love modes (Third Generalized mode) over many paths across the Russian shield. The observations show many different angles of inclination as we would expect from Fig 3 and 6. The persistence of the inclined motion over long paths, and the known sensitivity of higher modes to changes in structure, with consequent attenuation, suggest that the only explanation for the inclined particle motion is the presence of anisotropy in the upper mantle. Acknowledgments 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. Global Seismology Unit, Imtitute of Geological Sciences, Edinburgh. References Armstrong, G. A. & Crampin, S . , 1972. Piezo-electric surface-wave calculations in multilayered anisotropic media, Electron. Lett,, 8, 521-522. Armstrong, G. A. & Crampin, S . , 1973. Preferential excitation of 2nd-mode piezoelectric surface waves in Zinc-oxide-layered substrates, Electron. Lett., 9, 322-323, 186 S. Crampin Crampin, S., 1966. Higher modes of seismic surface wave: Propagation in Eurasia, Bull. seism. SOC.Am., 56, 1227-1239. Crampin, S., 1967. Coupled Rayleigh-Love second modes, Geophys. J. R. astr. SOC., 12, 229-235. Crampin, S., 1970. The dispersion of surface waves in multilayered anisotropic media, Geophys. J. R., astr. SOC.,21, 387-402. Crampin, S. & Taylor, D. B., 1971. The propagation of surface waves in anisotropic media, Geophys. J . R. astr. SOC.,25,7147. Verma, R. K., 1960. Elasticity of some high density crystals, J. geophys. Res., 65, 757-766.