Seismic-waveform effects of conical points in gradually varying anisotropic media

Transcription

1 Geophys. J. Int. (1994) 118, Seismic-waveform effects of conical points in gradually varying anisotropic media G. Rumpker and C. J. Thomson Department of Geological Sciences, Queen s University, Kingston, Ontario, Canada, K7L 3N6 Accepted 1994 February 16. Received 1994 January 31; in original form 1993 September 24 SUMMARY We describe the effects of anisotropic slowness-surface conical points (acoustic axes) on quasi-shear wavefronts and waveforms in variable elastic media. Conical points have quite complicated geometrical consequences even for a point source in or a wave refracting into a homogeneous anisotropic medium. A hole develops in the fast quasi-shear wavefront and the swallowtail catastrophe plays an important role in the geometry of the slow quasi-shear front, which becomes folded with numerous self-intersections. The two fronts are joined along the rim of the hole. This geometry influences the waveforms, which show Hilbert-transform and diffraction effects. Therefore, standard ray theory is inapplicable even for a uniform medium and the Maslov method is needed to describe waveforms. The introduction of elastic gradients further complicates the geometry of the problem, because rays bend sharply as their slowness approaches that of the axis. An initially smooth, single-valued slow quasi-shear front will evolve in the gradient region into a front which is folded and multivalued and once again the swallowtail is important. However, in contrast to a homogeneous medium, no hole develops in the fast quasi-shear front and the slow and fast fronts separate completely. While such geometrical factors are included in the Maslov method, waveforms are also affected by coupling of the fast and slow waves on nearing the axis, where the rays and polarizations rotate most rapidly and their slownesses differ by very little. Numerical examples are presented for a cubic and an orthorhombic material. The differences between these two examples show that the fine structure of continuously varying internal conical refraction can vary considerably from material to material, though its basic principles are clearly defined. Waveforms are presented for a point source in a uniform medium and for fast and slow shear waves in a gradient, with and without coupling. Overall, we conclude that the wavefront-folding effects cause the most drastic waveform distortions. Coupling becomes most important when signals merge, as at cuspidal edges or at lower frequencies, since the net waveform could be altered significantly if its components are varied. Conical refraction complicates and yet could be decisive for the identification of seismic anisotropy and rock properties. Key words: anisotropy, body waves, catastrophe theory, seismic waveforms, shearwave coupling. 1 INTRODUCTION In an anisotropic elastic medium there are usually special propagation or slowness directions for which the two quasi-shear wavespeeds are the same. Such directions are called acoustic axes by analogy with the optical axes of crystallography (Fedorov 1968, p. 94). Examples arising in important Earth materials have been listed by Crampin & Yedlin (1981) and what they refer to as a point singularity of the slowness surface is of interest here. We follow Musgrave (1985) and others by referring to this feature as a slowness-surface conical point. From the above references it seems clear that conical points are very common and in this paper we study their effects on wave propagation in a smoothly depth-dependent inhomogeneous medium. These effects can be quite drastic and an 759

2 760 G. Riimpker and C. J. Thomson understanding of the waveform characteristics of such conical refraction should be helpful in both earthquake and controlled-source seismology. In the former, mantlemineral alignment and rock composition changes due to temperature, pressure and stress variations are the likely sources of gradual alterations in seismic anisotropy (especially in subduction zones). Nearer the surface, the closing of cracks as depth increases is an additional mechanism and, as for the other mechanisms, shear-wave data from around acoustic axes have an almost completely unexplored diagnostic potential. Such an investigation seems overdue, when one considers the great utility of axes for the classical optical identification of minerals in a laboratory. The effects of a conical point in a gradually varying medium are significantly different from those of conical refraction at a sharp boundary between two homogeneous media (described, e.g. by Musgrave 1970, Chapter 11). The optical version of the latter is used as a tool for the identification of crystals (see e.g. Born & Wolf 1980, Chapter 14). These differences lie in both the geometrical (or ray path) and the dynamic (or displacement) parts of the problem. Gradual variations in elastic properties cause the slowness directions of an axis to change gradually and this in turn causes rays passing near the axis to bend sharply. For the slow quasi-shear wave, this ray bending will cause an initially smooth, single-valued wavefront to fold and become multivalued, which in itself will have waveform implications. Furthermore, the two quasi-shear waves are coupled very strongly around a gradually changing acoustic axis and the frequency-dependent effects of this coupling on the waveforms should also be accounted for. A necessary preliminary step is to understand in some detail the wave-surface implications of slowness-surface conical points. The first to study this were Miller & Musgrave (1956) for the case of cubic Nickel. The excellent diagrams and comprehensive nature of their paper make it extremely useful. They considered the curvature properties?f the slowness surface around the acoustic axis and they listed precisely the resulting cusps and multiple intersections of the wave surface. Crampin (1981) presented wireframe diagrams of the wave surface around an axis of this type. Unfortunately, such diagrams are rather difficult to unravel, so that essential properties of the wave surface are obscured and their relationships to slowness surface properties are not revealed. Recently, Grechka & Obolentseva (1993) gave some details of the wave surface around an axis for the case of olivine (orthorhombic). They pointed out an interesting property of the locus of the cusps which must exist on the slow shear-wave sheet, though again the diagrams presented do not reveal the true solid form of the wave surface in 3-D*. In Section 2.1 we present a description of the wave surface that emphasizes the importance of swallowtails (in catastrophe-theory terminology). We describe the relationship of wave-surface swallowtails to slowness-surface features. It appears that although the number and arrangement of these swallowtails varies from material to material, they do form essential building blocks and once recognized they help to make the wave surface easier to understand. * This comment applies to the original submitted version of their paper. In Section 2.2 we progress from a point source to the case of an initially smooth single-valued curved wavefront, as might be incident from an isotropic region onto an anisotropic region. The slowness of the front encompasses the acoustic axis and the anisotropic region has depthdependent properties. The curvature of the initial wavefront naturally influences the final shape of the two quasi-shear wavefronts and the latter are certainly quite different to the wavefronts from a point source in a homogeneous medium (i.e. the wave surfaces). For example, there is no hole in the fast quasi-shear wavefront in the inhomogeneous medium. However, swallowtails do still develop. Various waveforms are presented in Section 3. Since the slow wavefront is generally multivalued, it is necessary to use Maslov slowness integrals (Kendall & Thomson 1993b) to represent the zeroth-order wavefield. The coalescing stationary points of these integrals describe the serious waveform distortions that arise near cusps such as those on the wave surface. The 2-D Maslov method also correctly describes the waveform effect of the conical point singularity on the Green s function (the so-called lid diffraction (Burridge 1967, Section 3.1 below). We should note here that Crampin (1991) has computed waveforms for a homogeneous orthorhombic material having a conical point. The source conditions were such that direct comparison between the exact Green s function and the computations was not permitted and the lid diffraction signal was not positively identified. The essential details of the wavesurface features due to the axis and their relationship to the waveforms were not considered. Although integrating over a range of slowness in the Maslov sense describes the effect of wavefront folding, a further step is required to include the effects of the quasi-shear-wave coupling caused by the gradient. Such coupling of the shear waves was described by Chapman & Shearer (1989) for a single Snell wave (uniform horizontal slowness) passing near a kiss singularity (in the terminology Crampin & Yedlin 1981). The case of a kiss singularity differs significantly from that of a conical point, because the group velocities on the two quasi-shear sheets are regular at a point of tangential touching whereas they are singular at a conical point. In Section 3.2 we review the basic equations for and describe the characteristics of the conical-point coupling, finishing with numerical examples for an arbitrary curved initial wavefront. 2 GEOMETRICAL ASPECTS OF THE PROBLEM Following Schoenberg & Protbio (1992) we use P (primary), S (secondary) and T (tertiary) to denote the three waves in an anistropic medium. We use Y to denote the three-sheeted slowness surface and W to denote the wave surface at a point in an anisotropic medium. The physical wavefront in our problem is referred to by WF. Befqre considering inhomogeneity and non-point-source initial conditions, it is helpful, even necessary, to describe some of the properties of Y and W associated with conical points, particularly for the T wave. In the vicinity of these points the S and T sheets of Y look like two identical cones placed tip to tip.

3 2.1 9, W and conical points We describe two examples. The case of cubic nickel analysed by Miller & Musgrave (1956) and the case of orthorhombic olivine, which Grechka & Obolentseva (1993) have considered. The extent to which these cases are representative of all the possibilities is as yet unclear, but they do provide a great deal of insight. Cubic Nickel has an acoustic axis in the (1, 1, 1) direction and three-fold rotational symmetry about this axis. The properties of the T sheet of Y in the neighbourhood of this axis are summarized in Fig. 1. Note that points where the Gaussian curvature vanishes (called parabolic points) lie on a curve, denoted KO, which is open, not closed, around the acoustic axis, and that the curve KO also has cusps at the points K,. In Fig. 2 is sketched a portion of W corresponding to kth of 9 about the (1,1,1) axis, the rest of W being obtained by reflection and rotation operations from this portion. The cuspidal edge denoted C on W is associated with the parabolic points KO on 9. Note that the curve C merges smoothly at the point CC with the edge denoted I on the W portion shown and that points such as CC correspond to the cusps K, on the curve KO. The wave surface has a hyperbolic (negative Gaussian curvature) part associated with the slowness surface hyperbolic region and an elliptic (positive Gaussian curvature) portion associated with the elliptic part of the slowness surface. The curvature of W is singular at the cuspidal edge C, where these two portions meet. When the remaining five portions of W about the axis are generated, the net surface will have self-intersections that have been described by Miller & Musgrave (1956). There Pl J P3 t P2 Figure 1. Important slowness-surface properties for the T wave around the (1,1,1) acoustic axis in cubic Nickel (adapted from Miller & Musgrave 1956). The axis points out of the page at the centre and the slowness surface is locally a circular cone pointing inwards at that point. The thick line(s) KO show(s) the locus of points of zero Gaussian curvature. Stippling indicates negative Gaussian curvature. Note that KO is an open curve around the axis and that it is not smooth everywhere. Its cusps are denoted K,. Seismic conical refraction 761 R, Figure 2. A sketch of an elemental sixth of the T sheet of the wave-surface W corresponding to the hatched ith of the slowness surface portion in Fig. 1 (modified after Miller & Musgrave 1956). Parabolic points KO in Fig. 1 give rise to cuspidal edge C. This cuspidal edge and the bounding edge I of this wave surface kth meet tangentially at CC, corresponding to a cusp point K, in Fig. 1. Note that this sketch has the appearance of being more than bth of the local wave surface. This distortion is allowed simply for the convenience of showing the important types of overlap in Fig. 3. are essentially five different intersection types, which can be visualized from the three different types of joining of the iths of W. Namely, intersections arising from the joining of two iths along a common edge of type I in Fig. 2, intersections from a joining along a common edge of type I1 and intersections between iths of Y which are 2x13 radians apart around the axis. These intersections are shown in Fig. 3. We point out in particular the effect of joining along a common edge of type I. The point CC in Fig. 2 can then be seen to lie at the tip of a swallowtail catastrophe (see also Fig. 3d). According to Miller & Musgrave s (1956) Figs 11 and 12 and Table 5, the point CC corresponds to a cusp K, on the locus of parabolic points KO on the slow sheet of Y. However, we shall see below that it is possible to get swallowtails on W even when KO is a smooth curve. The sheet of W corresponding to the faster quasi-shear wave S is much less complicated and simply has a hole, which is circular in the cubic case (see Miller & Musgrave 1956, Fig. 9). The rim of the T sheet, denoted R in Fig. 2, fits precisely into this hole on the S sheet and the cone of group velocities on this common rim are just those allowed at the conical point on Y. The tangent to the wave surface is by definition continuous on passing across this rim from the faster to the slower sheet of W, though the continuity of the curvature of the wave-surface sheets is not so easily known. The wave-surface W relates directly to the WF for the problem of a point source in a homogeneous medium. For this problem, Burridge (1967) has found the leading form of a diffracted signal which lies in the plane of the rim R and

4 762 G. Rumpker and C. J. Thomson CL I Figure 3. When the various portions of W, such as that in Fig. 2, are joined together, points such as CC give rise to cusps on the joined cuspidal edges C and they lie at the tips of swallowtail catastrophes of the wave surface. (a) Joining along a common edge of type I in Fig. 2, giving the swallowtail. L denotes a line of self intersection of the T sheet of W. (b) Joining along a common edge of type I1 and (c) joining of two pieces 2n/3 radians apart around the axis, with self-intersections L, to L,. Part (d) shows a swallowtail catastrophe in isolation, with the conventional orientation rather than that which occurs in part (a). which has by definition the slowness of the conical point itself. His analysis begins with a representation for the Green s function as a slowness integral, which can be related to a Maslov integral straightforwardly. The necessity of considering a range of slowness around the singular point, rather than a single plane wave, is not surprising. It is also necessary to employ a spectrum of plane waves when considering signals with slowness close to those of the parabolic points KO on Y, corresponding to the cusps and swallowtails of W (Section 3.1). Orthorhombic Olivine has acoustic axes in the (p,, p3) plane, close to the p3 axis. Here we use the elastic coefficients of Kumazawa & Anderson (1969) (see Fig. 4). We have found by numerical means that the region of negative Gaussian curvature on the T sheet is closed around an axis for olivine, unlike the cubic case above. The locus of the parabolic points K,, is shown in Fig. 4(a). It is interesting that this curve is smooth and has no cusps like those denoted K, for the cubic case (Fig. 1). However, this does not necessarily imply that the corresponding cuspidal edges of W are also smooth curves in 3-D. Fig. 4(b) shows the numerically determined locus of group velocities for points along K, and we see that this locus is indeed cusped. This is more or less in agreement with the results of Grechka & Obolentseva (1993), who called this locus a J-curve (since they found it numerically from the vanishing of a geometrical spreading determinant denoted J). To understand how the curve C is cusped when the curve KO is smooth, one must consider the two principle curvatures of Y on travelling around KO. For this material, only one of the principle curvatures vanishes on K,, and in general the direction in Y corresponding to this vanishing curvature is not parallel to the direction of KO. However, at isolated points K, around KO the direction of the vanishing principle curvature is parallel to KO. The points K, correspond to the cusps CC on edges like C and the tips of swallowtail catastrophes on W. These features can be analysed via the Legendre transformation (Kendall & Thamson 1993b), which relates the slowness surface to the wave surface. In this case, it is appropriate to introduce a parameter along the path KO and to consider a 1-D Legendre transformation with regard to this parameter. The transformation will be unique with a unique inverse as long as the curvature of the slowness surface along the path K, is

5 Seismic conical refraction 763 Figure 4. (a) The locus Ko of parabolic points of the T slowness sheet around an acoustic axis in orthorhombic olivine, projected onto the (p,, pz) plane and computed numerically by tracking the curve of zero Gaussian curvature. Note that KO is smooth. Points K, are where the vanishing principle curvature of the slowness surface lies in the plane tangential to K,,. (b) The corresponding group velocity trace, projected onto the (v,, v2) plane. Note the cusps. The elastic constants of Kumazawa & Anderson (1969) are used under conditions appropriate to a depth in the Earth of approximately 400 km (1700 C and 133 kb), following Kendall & Thomson (1993a). non-zero (Kendall & Thomson 1993b). Then to each point of KO there corresponds a unique point on a cuspidal edge C of a swallowtail. At points where the curvature of 9 vanishes in the direction of KO, this 1-D Legendre transformation breaks down and the cuspidal edge of W is itself cusped, giving a swallowtail tip CC. Fig. 5 shows the form of a typical fraction of the wave-surface W between two swallowtail tips, and Fig. 6 shows the joining of such portions. Three of the points K, lie close together in Fig. 4(a) and they give rise to three closely knit swallowtails with rather complicated multiple intersections that we do not try to show in detail (note the three-pointed star in Fig. 4b). Figure 5. A portion, roughly +, of the wave surface of olivine corresponding to the angular segment of the slowness surface between two points Koo in Fig Curved WF initial conditions and a depth-dependent medium The surfaces Y and W are clearly very important, especially for a point source in a homogeneous medium. More general situations are of interest here, though, and the following idealized model embodies the physical effects to be considered. A simple curved wavefront is incident from an isotropic region z < 0 and it propagates into an anisotropic region z > 0 where the wavespeeds increase smoothly with depth. The anisotropy is such that the acoustic axis lies in

6 764 G. Rumpker and C. J. Thomson Figure 6. Two types of joining of the wave-surface portions such as that in Fig. 5. (a) Joining along common edge of type I and (b) joining along common edge of type 11, showing the self-intersection lines 15. As there are six points K, in Fig. 4(a), three of them rather close together, the self intersections of Ware numerous and so we do not attempt a full 3-D picture. The principles are clear, however. the (x, z) plane and points downwards sufficiently to avoid turning rays in the gradient at the times of interest. An interface problem exists at z = 0, but this is not important for the WF evolution question. Even when we come to consider the waveforms later, we will simply assume initial displacements at z = 0 and concentrate on the subsequent propagation effects of the interacting gradient and acoustic axis. The initial time at z = 0 is given in the form T(x, YP 0) = w, Y) = p,,(x - x,) + $p:,(x - x,) + &y. (1) The reference x-slowness px, at the reference point (xr, y, = z = 0) is taken to be the same as the x component of slowness for the conical point, so that the incident WF spans a range of slownesses encompassing the acoustic axis?. t From now on we prefer to use Cartesians x, y, z and pr, p,,, pz, whereas in the last section we used p,, p2, p, to be consistent wlth previous, more crystallographic conventions. The WF reaches (x,, 0,O) at time zero simply for convenience. The curvature terms pi, and pi, are somewhat arbitrary, as is the velocity gradient. It is helpful to consider first the evolution of a single line in the incident wavefront (a so-called frontline ). We choose the frontline that passes through the plane z = 0 along a line y =constant. From (l), the out-of-plane component of slowness p,, is constant along, and therefore it characterizes, each frontline. It also remains constant along individual rays in this laterally homogeneous model. Fig. 7 shows the evolution of a typical frontline for both the S and T waves, along with the corresponding rays. It is easier to start with the rays and in order to understand them it is noted that the slowness surface shrinks as depth increases. Since horizontal ray slowness is conserved, the vertical slowness and the rayfgroup velocity vary in ways that can be inferred roughly from the slowness-surface sections shown in Fig. 7 (recall that group velocity is normal to the slowness surface). As a ray traverses the depth range that brings it closest to the acoustic axis, the group velocity rotates most rapidly and the ray bends most sharply. In the vertical plane, the S rays bend to the right and the Trays to the left. This ray bending becomes sharper as py decreases and the plane of the axis (p,, = 0) is approached. In addition, the depth range over which the rapidly rotating group velocity exists also decreases as the axis is approached. It is important to note that in general the group velocity has an out-of-plane component and therefore so does the distortion of the rays and frontlines. For S, this out-of plane component has the same sign as p,,. For T, however, the situation is more complicated. For py very close to zero, the out-of-plane group velocity near the conical point causes rays beginning at small positive values of y to stray across to negative y values (recall the wave-surface considerations of Section 2.1). On the other hand, for sufficiently large values of (pyl the rays are far from the hyperboik neighbourhood of the conical point on the slowness surface and the out-of-plane component of the group velocity has the same sign as py. Intermediate values of p,, are considered below. The effects of these ray bending considerations on the S frontlines are quite simple, but for the T wave they can result in the creation of a loop as shown most clearly in the out-of-plane part of Fig. 7. The vertical or in-plane projection of such a loop has the appearance of a triplication. For a given value of p,,, such a loop does not exist immediately at z = 0, but rather is created at some greater depth. At the moment of creation the frontline is no longer a smooth curve and it has a singular cusp. The argument by which this can be demonstrated is given momentarily. As p is decreased and the axis is approached, the loop first arises at ever closer depths to x = 0. The out-of-plane extent of a given loop is fixed by the deviation from the plane that the rays suffer as they pass near the conical point. The extent of a given loop in the x direction grows continuously with time, however, as more rays pass through the indentation in the T slowness surface (referring to Fig. 7, one can think of this as feeding a string from the left to the right so as to make the loop grow to the right). At a given depth, the x-direction extent is largest for the loop on frontline p,, = 0. However, this loop has no out-of-plane component for the model considered, since p,, = 0 is a mirror plane. Therefore, the frontline for pv = 0 is truly a

7 Seismic conical refraction 765 v y1 B g SLOWNESSES T (a) IN-PLANE GEOMETRY y-slowness ( s h ) 4.M loo. 10s x-slowness (skm) X oan) SLOWNESSES (b) OUT-OF-PL 1 ANE GEOMETRY m y-slowness ( S h ) I w y scale I loo x-slowness ( s h ) -18. x oun) s RAYS TRAYS s RAYS TRAYS g -12. t2 g -12 N loo. 10s loo X oun) x fkm) 90. 9s. loo s. loo v v. z SLOWNESSES (C) IN-PLANE GEOMETRY y-slowness ( s h ) 1 I N s. loo. 10s. 110 x-slowness (skm) x (km) S RAYS TRAYS -6. g -12. N loo ItO s. loo x (km) Figure 7. (a) Ray-tracing results from a depth-dependent medium for a single frontline in the incident wavefront, py = s km-. Clockwise from top left: the important parts of the S and T sheets of 9 in the plane of p,,; frontlines for S and T at three later times, projected into the vertical plane; rays for S and T projected into the vertical plane. Part (b) shows the out-of-plane deviations, which for the rays and frontlines are necessarily plotted in the plane with a separate scale bar. Part (c) shows the in-plane effects for p,, = s km-i, where the rays bend more sharply. x (km) triplication in the vertical plane, with cusps at each end. Note that this frontline remains continuous: no hole develops in the T (or for that matter the S) WF in the inhomogeneous medium, as it would in a homogeneous medium. By stacking together frontlines we may build up a 3-D picture of the WF at a given instant and a computer package such as Mathernatica can render the results as a surface plot. Such a plot is shown in Fig. 8 for the case of cubic symmetry in the anisotropic medium. As some details are obscured, Fig. 9 shows only that half of the WF which corresponds to py > 0 (and y > 0 at z = 0), viewed from both sides. Fig. 10 shows the T WF for the orthorhombic olivine, which reveals swallowtails rather clearly and suggests that the range of different shapes of WF could be rather large. Consider Fig. 9 as py decreases. For p,, greater than some

8 766 G. Riimpker and C. J. Thomson T WAVEFRONT: CUBIC CASE Figure 8. A perspective of the WF of T after propagation for some time into the cubic material. The depth increases upwards and so we are observing the front from the as-yet undisturbed region ahead of it. special value, say pyc, the frontlines do have a component in the negative direction, but have not yet formed loops. Frontline pyc is the one for which at this time instant a loop is about to appear. The value of pyc increases as time increases. However, pyc must reach some limiting value, as it has already been noted that for large lp,l the out-of-plane group velocity will not have a negative y component even at the initial depth z = 0. The looped frontlines for py <pyc have created what appears to be a swallowtail singularity CC, on the wavefront. It might not seem obvious that the edges C1 and C, are cusped, not rounded. To confirm that they are cusped, it need only be remembered that between any two arbitrarily close points on the WF the slowness changes by an arbitrarily small amount. This is an inherent property of the ray tracing and it remains true for points on different branches of the WF (i.e. on passing over an edge). If the edges were rounded, then on passing from one branch to the other the WF normal and the slowness would deviate greatly from the values at nearby points on each branch. Hence the edges must be cusped. Similar reasoning can be used to recognize that a single frontline must be cusped at the moment a loop is created, even though the normal to a line is not completely unique. It is tempting to try to understand the formation of the swallowtail in terms of the combined curvature matrices of the slowness surface and the initial WF, as an extension of what was done in Section 2.1 for the wave-surface W. However, this is beyond the scope of the present discussion and we simply accept the numerical results. Indeed, even these numerical results have limited resolution. The edge C2 does not appear to end at the mid-plane, but travels a certain distance into the region y > 0. It presumably terminates at another swallowtail CC,, assuming normal catastrophe-theory arguments apply. It is interesting to note the hyperbolic regions directed towards the p, and pz axes in Fig. 1 and to speculate on a connection between these and CC2. More diagrams and perhaps finer ray tracing would be needed to clarify this and other features, but again we choose not to pursue these questions for now. For smallish py and initial value of y at z = 0, the effect of the negative y component of group velocity is to cause some loops to pass over to the negative side of the vertical plane y = 0. On the other hand, for py = 0 the frontline remains entirely within the plane y = 0. There is a trade-off between the magnitude of the negative group-velocity component and the positive initial value of y at z = 0, so that for each x at the later time there is some py for which the distance travelled into y < 0 is a maximum. Hence a new edge E is created and by the previous arguments it too must be cuspidal. The points where this edge meets the mid-plane are special and the edge E appears to be kinked there (when joined to the other half of the front). The rays at these points have passed right through the axis.. When the two halves of the wavefront are put together, the fact that both sides of the plane y = 0 are occupied means that there must be further self-intersections. At points such as P, for example, there are five geometrical arrivals (i.e. an integral representation of the wave functions must have five stationary phase points). As an edge E is (a) T WAVEFRONT: CUBIC CASE (b) T WAVEFRONT: CUBIC CASE Figure 9. The half of the WF for T which began from y > 0 at z = 0. Views from (a) behind and (b) ahead of the front. See text for discussion.

9 Seismic conical refraction 767 T WAVEFRONT: ORTHORHOMBIC CASE Figure 10. Same as Fig. 8, but for the orthorhombic olivine. Note that two swallowtails are clearly visible and contrast this overall shape with that in Fig. 8. approached two of these must merge together. As an edge like C, is approached two others merge. Three merge at CCI. At infinitely small times after T = 0, the bubble on the T front is infinitely small. It grows from zero, at first about equally rapidly in the x and y directions. The extent in the y direction is eventually limited, though, by the degree to which rays pass out of the (x, z) plane as they travel close to the conical point. The rays corresponding to the leading face of the bubble have relatively large and roughly constant separation, in comparison to the geometrical spreading on the other parts of the WF. (This can be deduced from the S grid on the WF, which began as a regular rectangular grid at x = 0.) The S front in the inhomogeneous medium is much simpler and so it is not shown. This front remains single valued, though the rays and frontlines deviate as indicated in Fig. 7. One can infer that the geometrical spreading is again large in the central region corresponding to rays that have passed close to the axis. It is important to note that the S and T fronts separate completely. There is no point of continuous contact. Also, at very high frequency the effects of shear-wave coupling are important in two isolated localized regions. For the T front this is around the mid-plane at the rightmost point where it is intersected by edge E, i.e. where T rays are currently closest to the axis and changing direction most rapidly. Similarly, for the S front near a mid-point at left, where S rays come nearest to the axis (see the rays in Fig. 7). Of course, at lower frequencies these regions overlap and the space between the two fronts is disturbed in a complicated way. 2.3 Polarization around a conical point Figure 11 shows the polarization vectors of the S and T sheets around a conical-point axis in slowness space. The polarizations rotate rapidly in this space and the free eigenvectors have a line of discontinuity on each sheet, because they rotate through n radians on circling the axis once. This discontinuity does not imply that displacements can be discontinuous in physical problems. The modulation effect of given initial conditions will always be so as to smooth out the discontinuity (by multiplication by zero). Furthermore, from the previous sections it is clear that initially close rays in slowness space can travel far away from each other in position space, so that practical observations T Px Figure 11. Polarizations around the conical point for cubic nickel.

10 168 G. Riimpker and C. J. Thomson of small-scale rapid polarization changes are not necessarily implied by Fig. 11. It is interesting to note that Al'shits & Lothe (1979a,b) have explained how a kiss singularity can degenerate into neighbouring conical points when a triclinic perturbation is added to a medium of higher symmetry. The.line of free-polarization discontinuity naturally joins the two conical points in the perturbed state. 3 WAVEFORMS Ordinary ray theory assumes that there are no rapid variations in WF curvature and so its single-plane-wave approximation is inappropriate, even as a first guess which ignores coupling. First-order Maslov integrals will be valid at the cusps and even the swallowtails of the wave surface. In order to incorporate the effects of a conical point, however, second-order (double) Maslov integrals are needed. It is convenient once again to consider first the point source in a homogeneous medium, following on from Section 2.1. We show in Section 3.1 seismograms for the different kinds of intersections and examples of waveforms in the neighbourhood of swallowtail tips and cusps. The significance of the lid-diffraction signal (Burridge 1967) is compared to that of the geometrical arrivals. In Sections we show examples of waveforms on the wavefront propagating through the depth-dependent medium. First we present Maslov seismograms with the coupling ignored, which shows the 'geometrical' waveform effects of the T-front edges, for example. Then the coupling due to conical points on the slowness surface is investigated for Snell waves in the depth-dependent medium. Finally, the effect of coupling is included in the waveform calculations using 2-D Maslov integrals, and its significance is assessed in comparison with the geometrical folding effects Waveforms on W We have already noted that the wave-surface W relates directly to the wavefronts from a point source in a homogeneous anisotropic medium. The corresponding displacement field can be expressed exactly as an integral over the slowness-surface Y for the medium (see Lighthill 1960; Burridge 1967) and this integral relates directly to a 2-D-Maslov slowness integral. Explicitly, the displacement-field dv) (where (v) indicates the wavetypes P, S or T) due to an impulsive point-source sb(x)b(t) can be given in the form where $j")(px, p,) = fg'" : g'"] - S, g(v)(px, p,) is the polarization vector, (:) signifies the dyadic outer product and v!y)(p,, p,) is the z component of the group velocity. Note that we have chosen to represent the waves as an integral over p, and py and that this parameterization of 3' is useful only for a certain range of directions in space. We shall rotate the (3) coordinate system so that the new z direction lies along the acoustic axis (the (1, 1, 1) direction in the cubic case). Then the important area of 2-D integration is given by the real part of the slowness surface p!y)(p,,p,) in the neighbourhood of the origin px =py = 0 of the new coordinates. The main contributions to the 2-D slownessintegral in (2) are from stationary points of O(")(px, p,). At these values of (px, p,) the normal to the slowness surface (i.e. the group velocity) is parallel to the receiver direction. The stationary points are the extrema-in elliptic regionsand the saddle points-in hyperbolic region-f the 13(~)(p,, p,) surface or, equivalently, of the slowness-surface pz(px, p,) (note that the difference between these two types of surface is just the linear function p,x +pyy). The relationship between these features on the slowness surface and corresponding waveforms has been shown by Buchwald (1959) and Burridge (1967) by stationary phase analysis of (2). In the high-frequency limit, maxima and minima of O(")(px, p,) are related to positive and negative d pulses, whereas saddle points relate to Hilbert-transformed 6 pulses. A higher-order phase approximation is necessary for parabolic points on O(v)(px, p,), corresponding to cusps on the wave surface. The waveforms at the cusps can be described in terms of Airy functions (Buchwald 1959). Examples of Maslov theory swallowtail waveforms in ocean acoustics have been calculated by Brown (1986). These asymptotic and earlier numerical results provide some guidelines for the interpretation of the signals to be presented Numerical examples For the numerical evaluation of (2), the 2-D-slowness integration is performed first using the trapezoidal rule followed by the frequency integration using the FFT. At this stage the frequency integral may be evaluated analytically first (the Chapman 1978, method), though later the frequency dependence of the amplitudes rules this out (Section 3.3). Some care must be exercised with the 2-D slowness integration in order to include all geometrical arrivals of interest and to avoid end-point contributions in the time-window chosen. The rapidly varying eigenvectors very near the conical point also require special attention. Figure 12 shows contours of the vertical slowness ~:~)(p,, p,) and the corresponding displacement iiht)(px, p,) for a point source s = (1,1,0) (in the rotated coordinates) in the neighbourhood of the conical point for cubic nickel. The slowness surface shows the three-fold rotational symmetry around the acoustic axis. In addition to the conical point at the origin, three maxima and three saddle points can be identified. Hence, for a receiver on the acoustic axis x = y = 0, z > 0 and at high frequencies, the seismogram will contain six geometrical T-wave arrivals at two different arrival times and a precursory diffraction signal due to the conical point (the lid). As the receiver moves off the acoustic axis, the six arrivals will become nondegenerate.and eventually some of the stationary points will merge and annihilate each other. In order to understand how the six possible arrivals relate to W, it is best to begin with Fig. 3(c). The heavy line CL is a radial line from the origin. It intersects the two wave-surface portions shown four times, near the two

11 Seismic conical refraction 769 I::::::,... _.g t t ! PE (s/w Figure 12. (a) The vertical slowness pir) as a function of the horizontal slownesses px, py in the neighbourhood of the acoustic axis for cubic nickel. The conical point CP at the origin is surrounded by three maxima MA and three saddle points SP of the surface ptr). (b) The corresponding displacement qt) for a point source s= (1,l. 0) (see eq. 2). overlapping cuspidal edges. This gives four arrivals. Then, referring to Fig. 3(a), we recognize that all cuspidal edges are preceded by an arrival on a hyperbolic part of the wavefront, and from 3(b) we see that all cuspidal edges are followed by an elliptical portion of the wavefront. This gives a total of six arrivals. In Fig. 13 the corresponding T-wave seismogram is shown for a range of receiver positions along x =Om with z = 1000 m. Geometrical traveltimes are superposed. The theoretical arrival time of the 'lid' signal is shown with a dashed line. Scalar waveforms are shown: the magnitudes of the displacements are plotted so as to show all important signals on one set of traces. At positions y = 0 m only two separate arrivals can be identified. The waveform of the first corresponds to a Hilbert-transformed 6-function and the waveform of the second to a 6 function, respectively. A pulsewidth of At = s is chosen to ensure sufficient separation of the geometrical arrivals. On varying y we see that each of the separate arrivals at y =Om is a superposition of three geometrical arrivals with waveforms of the same type. Therefore, three hyperbolic and three elliptic portions of the T wavefront intersect here. On close comparison with Fig. 12, we find that the amplitude variations correspond to the amplitudes of the displacement vectors in the area of horizontal slownesses that contributes most to the slowness integration (i.e. the stationary points of e'73(px, Py)). An intersection of a hyperbolic and an elliptic section can

12 770 G. Riimpker and C. J. Thomson * I -I T - WAVE SEISMOGRAM X=Om show geometrical traveltimes for four more ranges of : I \ I Figure 13. T-wave seismograms for receiver positions along x = 0 m with z = loo0 m. The dashed line marks the theoretical arrival time of the lid-diffraction signal due to the conical point. In addition to Fig. 13, T-wave seismograms are shown for a range of receiver positions along y = 0 m in Fig. 15. Here, only two geometrical arrivals separate for receiver positions off the symmetry axis at x = 0 m. The cusp near x = +300 m, therefore, results from four degenerate geometrical arrivals, related to two hyperbolic and two elliptic wavefront sections. These wavefront sections all meet at the intersecting cusps shown in Fig. 3(c). The early arrival with relatively large amplitude near x = -300 m relates to a swallowtail tip (see Section 2. l), where two hyperbolic and one elliptic wavefront sections merge together. This is shown in detail in Fig. 16 for nine receiver positions in the neighbourhood of the swallowtail tip at y = 0 m, x = -300 m (Fig. 16d). The approximate receiver positions on the wavefronts are indicated in Fig. 3(d). On moving off the tip to x = -200 m (Fig. 16a), hyperbolic and elliptic sections of the wavefront separate. The receiver off the symmetry plane at y = 50 m (Fig. 16b) shows a further separation of the two hyperbolic wavefront sections. Finally, at y = 100 m (Fig. 16c), hyperbolic and elliptic sections of the wavefront merge to form one of the cuspidal edges related to the swallowtail structure. In addition to the seismograms in Figs 13, 15 and 16, we receiver positions in Fig. 17. A comparison with the results at x = 0 m and x = 100 m shows that the wavefront section corresponding to the early arrivals at x = 200 m is only apparently detached from the subsequent parts of the whole T wavefront. The results for y = 100m and y = 200m elucidate the four separate geometrical arrivals that form the intersecting cusps at y = 0 m with x = 300 m. be found near y = f130m and there are four cusps (near y = f270 m and y = f600 m) where hyperbolic and elliptic sections merge in correspondence to the parabolic points on q. These cuspidal features are accompanied by diffraction arrivals, here only visible near y = f300 m at t = 0.36 s. Burridge (1967) shows that the lid diffraction signal forms a plane circular wavefront with radius of 469m (at z = 1OOOm). On the rim of this circular wavefront the lid signal merges with a geometrical arrival. The form of this diffraction signal due to the conical point on Y is asymptotically a step function. This related precursory signal cannot be identified in Fig. 13 because it is negligibly small compared to the geometrical arrivals shown. Therefore, in Fig. 14 we show enlarged seismogram traces for a range of receiver positions along y 2 0 m. Here, the contribution of the S-wave slowness sheet is also included. The amplitude scale is chosen so that the second geometrical arrival at y =Om (see Fig. 13) is equal to 1. The numerically calculated seismogram is compared with the asymptotic form of the lid-diffraction signal. For receiver positions close the the centre of the lid (y = 0 m) the numerically calculated and asymptotic solution agree in step size. The amplitude offset for the numerical solution results from the acausal parts of the following geometrical arrivals (Hilberttransformed 6 functions). The disagreement between the two solutions increases for receiver positions closer to the rim of the lid at y = 469 m, where, as Burridge (1967) points out, the step approximation breaks down and a higher-order asymptotic solution is necessary. 3.2 Depth-dependent medium Theory The basic equations for Maslov seismograms in a depth-dependent anisotropic medium are presented here in a form suitable for the later inclusion of coupling. The Fourier-transformed equation of motion and the stressstrain relation can be written in the matrix form -- ef(z)- iwa(p,, p,,, z)f(z), dz where f = (6, 6)T is a six-vector of transformed displacement and vertical stresses and A(p,, p,,, z) is the system matrix for the medium (Woodhouse 1974; Fryer & Frazer 1984). An approximate fundamental matrix F of the system (4) is the leading-order term of the WKBJ expansion (Chapman 1981) F(z) = D(z) exp {iwt(z)}, (5) where A is the diagonal matrix of eigenvalues of A and D is the matrix of eigenvectors. The six eigenvalues correspond to the allowed vertical slowness p p) of the three downgoing (v = P+, S+, T+) and three upgoing (v = P-, S-, T-) waves. A particular solution of (4) can be written in the (4)

13 LID - WAVEFORMS Seismic conical refraction 771 O.l Q) a 5 2 O*O Figure 14. S- and T-wave seismograms showing the lid diffraction at six receiver positions from the centre of the lid wavefront to just beyond its rim. The numerical results (solid line) are compared with Burridge s (1967) analytical solution (dashed line). See text for discussion. form f(z) = D(z) exp {iwt(z)}c, (7) where c is a vector of constants determined by initial conditions. The Maslov (or WKBJ) seismogram is then given by a back-transformation of the displacement components of f into the (x, t) domain: solved properly. Although this leads to interesting questions about how the conical point affects, say, the reflected wave, such an investigation is not within the scope of the present paper. Instead we simply assume a specified total displacement at z =0, which is resolved into the three downgoing waves at that depth: x exp C WPP + PyY - t)} dpx dp, d o (8) (Chapman 1978). The scale factor (iw/2n) * for each slowness integration ensures that the stationary-phasehrstmotion evaluation of (8) formally agrees with the ray-theory result (Thomson & Chapman 1985). The vector of constants in (7) takes the form c= (E~+, E ~ + ct+,, 0, 0, 0), all upgoing reflections from the gradient being neglected at z = 0. Naturally the waveform results depend on the excitation functions ep+, etc., and we cannot explore all possible values. Hopefully our examples will nevertheless be instructive and representative Initial conditions Numerical examples: coupling neglected Returning to the example of Section 2.2, we note that the reflection/transmission problem at z = 0 should really be In this section we present the waveforms corresponding to the wavefront examples shown in Section 2.2.

14 ~ 772 C. Riimpker and C. J. Thomson s X 0.60, I T - WAVE SEISMOGRAM Y=Om 1 1 I Figure 15. Same as Fig. 13, but for receiver positions along y =Om. The early arrival with large amplitude near x = -300 m lies very close to the tip of a swallowtail catastrophe of the wave surface. Figure 18 shows the phase function O(")(px, py, z) = Jrpp)(px, p,,, z) dz +pxx +pvy at a depth of z = 24 km with x = 104 km and y = 0 km for the S wave and the T wave. Whereas 0(') has one maximum, five stationary points can be identified for 0(? two maxima, two saddle points and one minimum. This leads to maximum number of six distinct geometrical arrivals for the corresponding S- and 7'-wave seismogram. The initial (z = 0 km) and momentary (z = 24 km) position of the conical point on the slowness surface is projected onto the plane of horizontal slownesses. Rays with horizontal slowness p,, = 0 and pf)'px 'pp' therefore have passed through the conical point at some depth. In Fig. 19 the corresponding displacement vectors at a depth of z = 24 km for an initially horizontally polarized displacement field 6(px, py, 0) = (1, 1,O) are shown. Initial and momentary position of the conical point are again marked. We already note here that rays (with py = 0 s km-') that pass the conical point at some depth undergo a rotation in displacement of 90" leading to the strong coupling between the S and T waves. This rotational effect decreases for rays with larger values of Ipy(. Figure 20 (dashed lines) presents two sets of (scalar) S- and T-wave seismograms for receiver positions along a line (a) y = 1 km and (b) x = 103 km, respectively. Waveforms that include the effects of coupling are also shown in Fig. 20 (solid lines) and will be discussed in detail in Section The pulsewidth in these examples is At = s. In Fig. 20(a) we chose receiver positions slightly out of the symmetry plane y = 0 km to show the maximum number of six distinct geometrical arrivals in one set of seismograms. The corresponding section through the T wavefront is approximately indicated in Fig. 8. The precursory S-wave arrival is followed by up to five distinct geometrical T-wave arrivals. A triplication is built by two saddle-point arrivals that intersect near x = 104 km and the arrival due to the minimum of OCn. The later maximum amval with relatively small amplitude forms a cusp with one of the saddle-point arrivals near x = 110 km. This cusp corresponds to the cusp C, in Fig. 9(a). We will see later (Fig. 21b) that there is a related cusp C1 near x = 80 km. The relatively large amplitude near x = 112 km is related to the neighbourhood of the point CC, (see Figs 9a-b), which probably corresponds to a swallowtail tip of the T wavefronts close to the plane y = 0 km, where a saddle-point signal merges with two maximum signals. There is an additional (non-geometrical) signal at x = 110 km with t = 8.66 s following a diffraction signal off the cusp and interfering with the geometrical amval at t=8.67s. It is probably related to strong bending of this section of the T wavefront associated with the suspected small-scale swallowtails near x = 112 km. In Fig. 20(b) the S wavefront is followed by an elliptic and a hyperbolic section of the T wavefront apparently disconnected from the following hyperbolic and elliptic sections that form a triplication. There is a total number of four cuspidal edges near y = f 7 km and y = f 6 km. The latter correspond to cusp(s) C, (see Fig. 9a). To further elucidate the T-wavefront structure, traveltimes are calculated for six ranges of receiver positions with x = 24 km. Fig. 21 shows the traveltimes for positions on (y = 0) and slightly off (y >0) the symmetry plane. Comparison with Fig. 20(a) shows that in the symmetry plane (Fig. 21a) two elliptic wavefront sections intersect each other. The separation of these sections is shown in parts (b) and (c) of Fig. 21. The arrow in Fig. 21(a) relates to the point CC2 but because of the limited resolution it is not possible to decide whether there is a swallow tip in the symmetry plane or possibly two tips on either side. Another swallowtail (in fact two tip to tip) can be found near x=75 km, where the cusp C, terminates in a new catastrophe that we have not further explored. Traveltimes for perpendicular cross-sections (Fig. 21d-e) show how the precursory wavefront sections are attached to the following parts of the wavefront (x = 104km and x = 105 km). The swallowtail tips CC, are marked at x = 105 km and they are connected with point(s) CC, along the cusplines C,. 3.3 Quasi-shear-wave coupling Basic theory Coupling between the quasi-shear waves occurs whenever the medium is inhomogeneous and the qs-wave polarizations twist along the curved ray paths. It will be strongest when the slownesses are approximately equal, as they are near the conical point. The frequency-dependent energy transfer of quasi-shear waves in a hexagonally symmetric, depth-dependent medium has been investigated by Chapman & Shearer (1989) for the so-called intersection

15 SWALLOWTAIL - WAVEFORMS Seismic conical refraction 773 Y=Om Y=50m Y=100m X=-200m 1.0 (a) I I I U I I I o - ls0 X=-400m (g) I 1 I r( F 4 Q) a I I singularities, where the two slowness sheets cross, and for kiss singularities, where the slowness sheets touch tangentially. Their derivation of the basic quasi-shear-wave coupling equations applies here too, although coupling of qs-waves near conical points differs in several respects. First we follow Chapman & Shearer (1989) and consider coupling for a single ray with constant horizontal slowness (i.e. a Snell wavefront). Later we will extend these calculations to a range of rays and corresponding horizontal slownesses (i.e. a generally curved wavefront) to show the effects of coupling on the waveform examples shown in Section A more complete solution of the system (4) can be usefully written in the form f(z) = D(z) exp {ioz(z)}r(z), ( 10) where the elements of r(z) are the continuously changing scalar amplitudes of the up- and downgoing waves. It is worth noting that we ought really to write r(px,py, z, o) and it is the non-trivial frequency dependence of r which makes the calculations so time consuming.

16 774 G. Riimpker and C. J. Thomson T - WAVE TRAVELTIMES s W 0.70 I * Y 2 & 0 I< x=2oom I I W x 0.00 Y Q) Y=lm I I I Figure 17. T-wave traveltimes due to a point source at the origin for six ranges of receiver positions at z = 1000 m. After substitution of ansatz (10) into the differential system (4), we obtain given by with the anti-symmetric coupling matrix C, given by For our numerical examples we consider only the coupling between the two downgoing quasi-shear waves and ignore the coupling to upgoing waves and P waves. This is justified by the relatively large difference in slowness between these other wavetypes, which means they are weakly coupled to the S+ and T+ waves of interest (see Chapman & Shearer 1989, for a more complete explanation of this). A reduced version of system (11) for downgoing S and T waves is then where the anti-symmetry of the coupling matrix has been used. Eq. (13) describes the frequency-dependent coupling between the two quasi-shear waves. The solution for r with given initial conditions at z = 0 is found straightforwardly by the Runge-Kutta method. The final waveforms are then constructed from (10) and (8) Snell wave coupling near conical points Figure 22 shows polarizations for S and T waves with horizontal slownesses close to the conical point at depth z = 0. Consider now a downgoing Snell wave (S or T) with

17 S Seismic conical refraction 775 T ! ds) (s) Pz (S/W P, (s/km) Figure 18. The phase O(") for the S wave and the T wave as a function of the horizontal slownesses px, py at a depth z = 24 km with x = 104 km and y = 0 km in a cubic material. The initial position (right 0) and the momentary position (left 0) of the conical point are projected onto the (Pm P,) plane. constant horizontal slowness py = 0 and px <p$)). At some We show two examples of the coupling for Snell waves depth, say z = Eo, we have px =pf' and the slownesses for with (a) p,, = s km-' and (b) p = s km-', the two qs wavetypes will be the same whereas the ly. respectively, whereas px = s km- is constant. The polarization of the Snell wave undergoes a rotation of 90". lines in Fig. 22 show the approximate change in polarization The coupling at this depth will be singular and it must be with depth for these two cases. Numerical results show that approached as a limit, which is fortunately quite obvious. the slowness of the Snell waves, in these examples, comes S 0.02 I I 0.02 I i T Pz (4kI-n) Pz (s/km) Figure 19. The displacement E@') for the S wave and the T wave as a function of the horizontal slownesses px, pv at a depth z = 24 km. The initial displacement d = (1, 1,O) is horizontally polarized. The initial position (right 0) and the momentary position (left 0 ) of the conical point are projected onto the (px, p,) plane.

18 G. Rumpker and C. J. Thornon 116. (a) S - T SEISMOGRAM Y=lkm (b) S - T SEISMOGRAM X=103km Transformed time (s) -LO Figure 20. S- and 7 -wave seismograms for receiver positions along (a) y = 1 km and (b) x = 103 km at a depth of z = 24 km. Approximate lines of receiver positions for the T wave are shown in Fig. 8. Results without the effects of coupling bnetween the quasi-shear waves (dashed line) are compared with results that include coupling (solid line). A pulsewidth of Ar = s has been chosen. For reasons of presentation a transformed or reduced time t = t - a(x - xo) with xo = 104 km and LY = 0.12 s km- is used in (a). close to the slowness of the conical point at a depth of about z = 4 km, where the coupling coefficients are largest and the slowness differences come close to zero. In Fig. 23 the resulting S- and T-wave (scalar) amplitudes for an incident T wave are shown at three depths. The pulsewidth in these examples is Ar = s and a depth increment of Az/2 = 0.04 km has been chosen for the coupling calculations using the Runge-Kutta method. The resulting T- and S-wave pulses (a) show that the relatively weak coupling in this case leads to a broad S pulse of small amplitude. The amplitude of the incident T wave is only slightly decreased at z = 7.52 km though the pulse is deformed with a precursory indentation. For a smaller value of p,, (b), the initial T wave has lost about half of its energy to the secondary S wave at depth 4.00 km and the energy transfer is almost complete at depth 7.52km. Since the velocity difference between the two wavetypes is only small, the separation between the two pulses is not very pronounced Effects of coupling for a curved wavefront We now proceed to investigate the effects coupling for the curved wavefront examples in Section 3.2.3, Seismograms that include the effects of coupling are shown in Figs 20(a)-(b) (solid line) and are compared with the results without coupling (dashed line). Unlike for the Snell wave coupling examples, the results presented here include the effects of both coupling from the S to the T wavefront and vice versa. This is in agreement with the physical problem stated in Section 2.2, since both qs waves will be excited in the anisotropic medium at the initial depth z = 0. The seismograms in Fig. 20(a) show that the coupling only affects the S wave and the early T wave arrivals in the time window between 8.40 s and 8.55 s. The later arriving saddle-point signals are only marginally and the maximum signals are not at all affected by the coupling. The coupling causes a deformation of the S-wave pulses; the amplitude is slightly reduced and enlargement shows that the onset of the S-wave pulse is more gradual. The situation for the T wavefront is more complicated as a result of the interference of the geometrical arrivals with each other. At the receiver position x = 104 km, the first minimum arrival is well separated from the following parts of the wavefront. Like for the S wave, the amplitude of the first T-wave pulse is slightly reduced. On the other hand, at receiver positions off the cusps near x =96 km and x = 114 km, amplitudes are much larger when coupling is taken into account. The cancellation of the geometrical arrivals that merge at the cusp seems to be disturbed by the inclusion of coupling. The results of Section 2.2 suggest that the strong effect near x = 114 km results from coupling of the T wave to the S wave, whereas near x = 96 km coupling from S to T is most prominent. The results for a perpendicular range of receiver positions in Fig. 20(b) show again that only the S wave and the first

19 115.0 T - WAVE TRAVELTIMES 777 s v Y a l(d) I I I rce) (r) I Figure 21. T-wave traveltimes for six ranges of receiver positions at z = 24 km. The arrows mark the tips of four swallowtail catastrophes. A T k3 9 n < F W Pz ( S / W Pz Wkm) Figure 22. Initial (z = 0 km) polarizations for the S and T wave as a function of the horizontal slownesses p,, p,. The position of the conical point ( 0) is projected onto the (px, p,,) plane. The dashed lines mark the change in polarization with depth for Snell wave coupling examples in Fig. 23.

20 778 G. Riimpker and C. J. Thomson (4 Z4.48km Z=7.52km c/l I I I Y I I I E _ g 0.0 < z4.ookm P- J \ L !$ 0.0 El 53 d Z=7.52km E E Figure 23. S and T Snell-wave seismograms at three depths for an incident T Snell wave for (a) py = s km-' and (b) py = O.OOO1 S km-' with px =0.1227s km-'. T-wave arrival are affected by the coupling. There is also a Comparison with the displacement (Fig. 19) and the gradual decrease of the coupling effects from y = 0 to larger (Fig. 18) shows that the rotation in values of Iyl, because rays that contribute to the pulses displacement and therefore the coupling is strongest for rays around y = 0 pass by the conical point closest (see Section with horizontal slownesses corresponding to the maximum 2.2). of O(') and the minimum of 0(7) respectively. This agrees

21 Seismic conical refraction 779 (a) S - T SEISMOGRAM (b) S - T SEISMOGRAM 116. A Y=lh :: X=103km Transformed time (s) Figure 24. Same as Fig. 20, but with a pulsewidth of Af = 0.1 s. with the results shown here, where the S-wave and the first T-wave signal are affected most. A pulsewidth of AT =0.018s been chosen for the calculation of seismograms in Fig. 20 to ensure a good separation of the individual pulses of the S and T wavefronts. Examples of seismograms for a lower frequency range resulting in a pulsewidth At = 0.1 s are presented in Fig. 24. As before, seismograms are shown with and without coupling effects included. Note that the time axis has also been lengthened. For these lower frequencies, the S wavefront interferes with the early T-wavefront sections and the pulses of the T wavefront cannot be separated, which leads to an underestimation of the total number of distinct arrivals. Sharp features, such as cuspidal edges, are smoothed out in both Figs 24(a) and (b). Neglecting coupling effects in Fig. 24(a) does not cause the waveforms to change considerably, but amplitude variations are less extreme near x = 96 km and x = 114 km without coupling. The effects of coupling for receivers with offset in the y direction (Fig. 24b) are relatively insignificant. Generally, as a result of the pulse broadening due to coupling, the onset of the S-wave pulse starts about 0.1 s earlier. 4 DISCUSSION In this paper we applied the 2-D-Maslov method to seismograms in homogeneous and depthdependent media. The Maslov method allows the calculation of waveforms near cusps and swallowtails that are related to the parabolic points of the slowness surface. A restriction to relatively smooth areas of the wavefront, as for example in Gajewski (1993), is therefore not necessary. We have numerically verified Burridge s (1967) asymptotic solution for the waveform of the lid diffraction signal in a homogeneous cubic medium. In addition, waveforms have been calculated near the rim of this plane wavefront, where the diffraction interferes with geometrical arrivals. Near the centre of the lid, we found its amplitude to be negligibly small compared to neighbouring geometrical arrivals and therefore the use of this diffraction effect as a diagnostic tool seems limited. The properties of the slowness surface in the neighbourhood of the conical point cause swallowtail structures on and multiple self-intersections of the T wavefront for both the homogeneous and the depth-dependent medium. The S wavefront in our examples remains smooth (as does the P wavefront). The examples for a cubic and an orthorhombic medium show that the overall features of the T wavefronts are fairly similar. A more systematic investigation of the relation between global slowness-surface characteristics and wavefront evolution is desirable. The numerical waveform examples related to the swallowtail structures indicate strong effects, i.e. pulse broadening and deformation, sometimes large amplitudes and Hilbert-transform type waveforms, which may serve to identify these complicated wavefront features. The evolution of the wavefronts in the depth-dependent medium depends on the initial conditions chosen, but we regard the slightly curved incident wavefront as a fairly general example. Different choices for the initial polarizations will not greatly affect the waveforms for distinct arrivals, only their relative amplitudes, but it would lead to significant modifications of net waveforms when these arrivals merge, such as at cusps or at lower frequencies.

22 780 G. Rumpker and C. J. Thomson Nevertheless, we hope that the examples shown give an impression of the possible waveform variability. Differences between plane and curved wavefront-coupling results can be considerable. The Snell-wave examples depend strongly on the horizontal y slowness chosen, i.e. the distance to the plane that contains the conical point. In this plane the energy transfer is complete, but the effects for the generally curved wavefront are less extreme (for the frequency range chosen) due to neighbouring plane waves that contribute to the result on integration. We found that coupling effects are most pronounced near cuspidal edges where the (geometrical) arrivals interact. Instead of diffractions off the cusps (without coupling) the theoretical seismograms exhibit Hilbert-transform-type waveforms with amplitudes comparable to those of the geometrical arrivals. Overall, though, coupling effects on waveforms are less drastic than those of multiple arrivals caused by wavefront folding. Since our results show that shear waves, incident onto an anisotropic region, generally split or evolve into more than two phases if the initial wavefront is sufficiently curved (for example near the source), at teleseismic receivers the effect might be mistaken as a complicated source mechanism or an effect due to reverberations. We note too, that dimples or indentations on any sheet of the slowness surface also imply curvature sign changes and therefore they, as well as conical points, can cause wavefronts to bifurcate in an inhomogeneous medium. Lastly, it is appropriate to mention that the elastic-wave effects described in this paper should be of interest to laboratory experimentalists as well as seismologists. ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Bullard Laboratories of Cambridge University (BL), Schlumberger Cambridge Research (SCR) and the Seismologisches Zentralobservatorium in Erlangen (GRF), where parts of this paper were written. Computations were performed at BL and SCR. We are indebted to Dave Lyness for providing access to computer systems at BL and also housing for one of us (GR)! We thank Carl Spencer for his help with the Mathernatica graphics. Conversations with Bob Burridge, Chris Chapman, John Hudson and others proved very helpful. GR is supported by Queen s University Graduate Scholarships. Additional funding for CJT was provided by SCR and the German Academic Exchange Service (DAAD) and work on this paper continued while CJT held a Visiting Fellowship at the Research School of Earth Sciences of The Australian National University. Ongoing support for CJT is provided by an NSERC (Canada) Operating Grant. 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