Upper-mantle flow beneath French Polynesia from shear wave splitting

Upper-mantle flow beneath the South Pacific is investigated by analysing shear wave splitting parameters at eight permanent long-period and broad-band seismic stations and 10 broad-band stations deployed in French Polynesia from 2001 to 2005 in the framework of the Polynesian Lithosphere and Upper Mantle Experiment (PLUME). Despite the small number of events and the rather poor backazimuthal coverage due to the geographical distribution of the natural seismicity, upper-mantle seismic anisotropy has been detected at all stations except at Tahiti where two permanent stations with 15 yr of data show an apparent isotropy. The median value of fast polarization azimuths (N67.5°W) is parallel to the present Pacific absolute plate motion direction in French Polynesia (APM: N67°W). This suggests that the observed SKS fast polarization directions result mainly from olivine crystal preferred orientations produced by deformation in the sublithospheric mantle due to viscous entrainment by the moving Pacific Plate and preserved in the lithosphere as the plate cools. However, analysis of individual measurements highlights variations of splitting parameters with event backazimuth that imply an actual upper-mantle structure more complex than a single anisotropic layer with horizontal fast axis. A forward approach shows that a two-layer structure of anisotropy beneath French Polynesia better explains the splitting observations than a single anisotropic layer. Second-order variations in the measurements may also indicate the presence of small-scale lateral heterogeneities. The influence of plumes or fracture zones within the studied area does not appear to dominate the large-scale anisotropy pattern but may explain these second-order splitting variations across the network.     

Complex upper-mantle deformation beneath the North China Craton: implications for lithospheric thinning

The mechanism of lithospheric thinning of the North China Craton (NCC) remains controversial. To constrain the mechanism, this study investigated the upper-mantle deformation pattern of the craton by measuring shear wave splitting at the cratonic edge. The results, derived from data recorded at 47 stations, reveal a complex pattern of mantle deformation. Inside the eastern craton, the majority of fast direction trends SE–NW parallel to the tectonic extension direction accompanying with the lithospheric thinning. At the cratonic edge, 15 stations with only null splitting results indicate undetectable anisotropy beneath the stations. This may be due to upwelling or chaotic ascension of mantle flow. To the north, off the craton, large delay times and variation of splitting parameter with backazimuth are generated by the combination of lithospheric and asthenospheric anisotropy. Based on comparison of the splitting results and the predicted ones by the compelling models, it is likely that lithospheric delamination dominated the lithospheric thinning at the north edge of the NCC during the Mesozoic to Cenozoic.     

Three-dimensional image of the anisotropic bodies beneath central Honshu, Japan

Shear-wave splitting from local deep earthquakes is investigated to clarify the volume and the location of two anisotropic bodies in the mantle wedge beneath central Honshu, Japan. We observe a spatial variation in splitting parameters depending on the combination of sources and receivers, nearly N–S fast in the northern region, nearly E–W fast in the southern region and small time delays in the eastern region. Using forward modelling, two models with 30 and 10 per cent anisotropy are tested by means of a global search for the locations of anisotropic bodies with various volumes. The optimum model is obtained for 30 per cent anisotropy, which means a 5 per cent velocity difference between fast and slow polarized waves. The northern anisotropic body has a volume of 1.00° (longitude) × 0.5° (latitude) × 75 km (depth), with the orientation of the symmetry axis being N20°E. The southern anisotropic body has a volume of 1.25° × 1.25° × 100 km with the symmetry axis along N95°E. Our results show that the anisotropic bodies are located in low-velocity and low- Q regions of the mantle. This, together with petrological data and the location of volcanoes in the arc, suggests that the possible cause of the anisotropy is the preferred alignment of cracks filled with melt.     

Shear wave anisotropy in the crust around the San Andreas fault near Parkfield: spatial and temporal analysis

We systematically analysed shear wave splitting (SWS) for seismic data observed at a temporary array and two permanent networks around the San Andreas Fault (SAF) Observatory at Depth. The purpose was to investigate the spatial distribution of crustal shear wave anisotropy around the SAF in this segment and its temporal behaviour in relation to the occurrence of the 2004 Parkfield M 6.0 earthquake. The dense coverage of the networks, the accurate locations of earthquakes and the high-resolution velocity model provide a unique opportunity to investigate anisotropy in detail around the SAF zone. The results show that the primary fast polarization directions (PDs) in the region including the SAF zone and the northeast side of the fault are NW–SE, nearly parallel or subparallel to the SAF strike. Some measurements on the southwest side of the fault are oriented to the NNE–SSW direction, approximately parallel to the direction of local maximum horizontal compressive stress. There are also a few areas in which the observed fast PDs do not fit into this general pattern. The strong spatial variations in both the measured fast PDs and time delays reveal the extreme complexity of shear wave anisotropy in the area. The top 2–3 km of the crust appears to contribute the most to the observed time delays; however substantial anisotropy could extend to as deep as 7–8 km in the region. The average time delay in the region is about 0.06 s. We also analysed temporal patterns of SWS parameters in a nearly 4-yr period around the 2004 Parkfield main shock based on similar events. The results show that there are no appreciable precursory, coseismic, or post-seismic temporal changes of SWS in a region near the rupture of an M 6.0 earthquake, about 15 km away from its epicentre.     

Seismic P-wave anisotropy in the subcrustal lithosphere of north-west Australia

Summary. Reduced P_n travel times from the Archaean Pilbara Craton of north-west Australia show a strong correlation with azimuth, which could be used as evidence of anisotropy. However, the azimuthal correlation could also be explained by a southerly dip of between 1 and 2° on the crust–mantle boundary, although the models from several reversed seismic profiles across the craton suggest a smaller dip.
A time-term analysis of the P_n date yielded several models. The preferred solution, in which the dip on the crust–mantle boundary is similar to that in the models from the reversed profiles, has approximately 2 per cent anisotropy in the uppermost mantle, with the direction of maximum velocity 30° east of north. One possible cause of the anisotropy is that olivine crystals were aligned by syntectonic recrystallization and/or power law creep in the tensional environment caused at the base of the lithosphere by flexure during loading of the lithosphere by the strata of the Hamersley Basin which overlies the Pilbara Craton.
A seismic discontinuity occurs about 15 km below the crust–mantle boundary under the craton. A qualitative analysis of all available seismic data suggests that the velocity below the boundary is probably also anisotropic, with the direction of maximum velocity between north and 40° west of north. The direction of minimum velocity below the sub-Moho boundary correlates loosely with the direction of basement lineaments in the Proterozoic Capricorn Orogenic Belt to the south of the craton, suggesting that the anisotropy under the boundary may be younger than that immediately under the crust/mantle boundary. This is consistent with the notion that the Archaean lithosphere was thinner than the present lithosphere.     

Finite-frequency vectorial tomography: a new method for high-resolution imaging of upper mantle anisotropy

Amplitude measurements of the transverse component of SKS waves, the so-called splitting intensity, can be used to formulate a non-linear inverse problem to image the 3-D variations of upper mantle anisotropy. Assuming transverse isotropy (or hexagonal symmetry), one can parametrize anisotropy by two anisotropic parameters and two angles describing the orientation of the symmetry axis. These can also be written as two collinear pseudo-vectors. The tomographic process consists of retrieving the spatial distribution of these pseudo-vectors, and thus resembles surface wave vectorial tomography. Spatial resolution results from the sensitivity of low-frequency SKS waves to seismic anisotropy off the ray path. The expressions for the 3-D sensitivity kernels for splitting intensity are derived, including the near-field contributions, and validated by comparison with a full wave equation solution based upon the finite element method. These sensitivity kernels are valid for any orientation of the symmetry axis, and thus generalize previous results that were only valid for a horizontal symmetry axis. It is shown that both lateral and vertical subwavelength variations of anisotropy can be retrieved with a dense array of broad-band stations, even in the case of vertically propagating SKS waves.     

Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods

We investigate the sensitivity of finite-frequency body-wave observables to mantle anisotropy based upon kernels calculated by combining adjoint methods and spectral-element modelling of seismic wave propagation. Anisotropy is described by 21 density-normalized elastic parameters naturally involved in asymptotic wave propagation in weakly anisotropic media. In a 1-D reference model, body-wave sensitivity to anisotropy is characterized by 'banana–doughnut' kernels which exhibit large, path-dependent variations and even sign changes. P -wave traveltimes appear much more sensitive to certain azimuthally anisotropic parameters than to the usual isotropic parameters, suggesting that isotropic P -wave tomography could be significantly biased by coherent anisotropic structures, such as slabs. Because of shear-wave splitting, the common cross-correlation traveltime anomaly is not an appropriate observable for S waves propagating in anisotropic media. We propose two new observables for shear waves. The first observable is a generalized cross-correlation traveltime anomaly, and the second a generalized 'splitting intensity'. Like P waves, S waves analysed based upon these observables are generally sensitive to a large number of the 21 anisotropic parameters and show significant path-dependent variations. The specific path-geometry of SKS waves results in favourable properties for imaging based upon the splitting intensity, because it is sensitive to a smaller number of anisotropic parameters, and the region which is sampled is mainly limited to the upper mantle beneath the receiver.     

ScS wave splitting of deep earthquakes around Japan

ScS wave splitting of five deep earthquakes in subduction zones near Japan is investigated using horizontal seismograms recorded al JMA stations. For each earthquake, we clearly observe uniform ScS wave splitting in all stations over Japan, especially for the events located south of Honshu in 1982, 1984 and 1993. However, the directions of fast-polarized waves of these events differed by a maximum of about 50° from one another. The orientation of fast-polarized waves in the 1982 event was NNW-SSE; those in the two later events WNW-ESE. We also recognize this discrepancy in the results of the analysis of the 1971 Sea of Okhotsk event reported by Fukao (1984). The Sakhalin Islands event in 1990 reveals a linear particle motion without such a change in direction of the second arrivals, implying no anisotropy. These observations are interpreted as indicating an anisotropic region within the slab near the earthquake sources but not beneath the receivers, since the orientations of fast-polarized waves recorded at each station are not common to all the earthquakes. Furthermore, we consider that anisotropy exists non-uniformly within the slab. The event in 1982, which occurred in almost the same area as those in 1984 and 1993, showed a fast direction different from the events in 1984 and 1993. The 1982 event was 179 km deep, but the two later events were at 398 km and 360 km, respectively. The fast direction observed from the 1982 event is parallel to the fossil plate motion, whereas those from the events in 1984 and 1993 are parallel to the compression axis within the subducting slab. The depth of 400 km is a phase boundary, where olivine changes to β spinel. We consider that the most likely cause of the change in anisotropy direction is the re-orientation of crystals associated with the phase change of olivine to β spinel due to subduction of the slab.     

Teleseismic shear wave splitting measurements in noisyenvironments

High noise levels hamper teleseismic shear wave splitting measurements, which bandpass filtering does not always help. To investigate how robust splitting measurements are to noise, we analysed a set of synthetic records with known splitting parameters and added fixed levels of noise. In the presence of weak anisotropy, single-waveform splitting measurements are unreliable when operating with noisy data sets. A practical rule in terms of S/N ratio and splitting delay time parameters is that splitting is confidently detectable at S/N > 8, regardless of the wave's original polarization orientation. However, for the evidence of weak anisotropy to be detectable and measurable at an S/N value of 4, the backazimuth separation of the phases from the fast polarization direction needs to be higher than 20°. Stacks of individual measurements consistently yield reliable results down to S/N values of 4. Applying stacking to data from DSB (Dublin, Ireland), the fast polarization direction φ and lag time δt are 58° and 0.95  s. This orientation reflects surface trends of deformation in the area, as found elsewhere in the UK. Our result thus reinforces the proposed model that the detected anisotropy in the British Isles originates from lithospheric coherent deformation preserved from the last main tectonic episode.     

Distribution of seismic anisotropy in the subduction zone beneath the Wellington region, New Zealand

Shear wave splitting measurements from S arrivals of local earthquakes recorded at the Incorporated Research Institutions for Seismology (IRIS) broadband sensor SNZO are used to determine a basic anisotropic structure for the subduction zone in the Wellington region. With the use of high-frequency filters, fast anisotropic polarization ( φ ) and splitting time ( δt ) measurements typical of crustal anisotropy are evident, but the larger splitting expected from the mantle is often not resolved. The small splitting seen agrees well with the results of previous studies concerning shallow crustal anisotropy. With the use of lower-frequency filters, measurements more consistent with mantle anisotropy are made. Anisotropy of 4.4 ± 0.9 per cent with a fast polarization of 29° ± 38° is calculated for the subducting slab, from 20 to 70  km depth. Using this result in addition to the results of previous studies, a model is proposed. The model requires a frequency-dependent anisotropy of less than 1.4 per cent when measured with a period of ~2  s to be present in the sub-slab mantle.
Separate from this population, a band of events in northern Cook Strait with an 86° ± 10° fast polarization is seen. This is at about 40° from the strike of the Hikurangi margin, and suggests a source of shear strain 40° removed from that found in the majority of the region. The cause of this is probably a deformation in the subducting slab in this region, as it moves towards a greater incline to the south.     

Two-dimensional modelling of subduction zone anisotropy with application to southwestern Japan

We present a series of 2-D numerical models of viscous flow in the mantle wedge induced by a subducting lithospheric plate. We use a kinematically defined slab geometry approximating the subduction of the Philippine Sea plate beneath Eurasia. Through finite element modelling we explore the effects of different rheological and thermal constraints (e.g. a low-viscosity region in the wedge corner, power law versus Newtonian rheology, the inclusion of thermal buoyancy forces and a temperature-dependent viscosity law) on the velocity and finite strain field in the mantle wedge. From the numerical flow models we construct models of anisotropy in the wedge by calculating the evolution of the finite strain ellipse and combining its geometry with appropriate elastic constants for effective transversely isotropic mantle material. We then predict shear wave splitting for stations located above the model domain using expressions derived from anisotropic perturbation theory, and compare the predictions to ∼500 previously published shear wave splitting measurements from seventeen stations of the broad-band F-net array located in southwestern Japan. Although the use of different model parameters can have a substantial effect on the character of the finite strain field, the effect on the average predicted splitting parameters is small. However, the variations with backazimuth and ray parameter of individual splitting intensity measurements at a given station for different models are often different, and rigorous analysis of details in the splitting patterns allows us to discriminate among different rheological models for flow in the mantle wedge. The splitting observed in southwestern Japan agrees well with the predictions of trench-perpendicular flow in the mantle wedge along with B-type olivine fabric dominating in a region from the wedge corner to about 125 km from the trench.     

Teleseismic waveform modelling with a one-way wave equation

It is now widely accepted that elastic properties of the continental lithosphere and the underlying sublithospheric mantle are both anisotropic and laterally heterogeneous at a range of scales. To fully exploit modern three-component broad-band array data sets requires the use of comprehensive modelling tools. In this work, we investigate the use of a wide-angle, one-way wave equation to model variations in teleseismic 3-D waveforms due to 2-D elastic heterogeneity and anisotropy. The one-way operators are derived based on a high-frequency approximation of the square-root operator and include the effects of wave propagation as well as multiple scattering. Computational cost is reduced through a number of physically motivated approximations. We present synthetic results from simple 1-D (layer over a half-space) and 2-D (subduction zone) models that are compared with reference solutions. The algorithm is then used to model data from an array of broad-band seismograph stations deployed in northwestern Canada as part of the IRIS-PASSCAL/LITHOPROBE CANOE experiment. In this region radial-component receiver functions show a clear continental Moho and the presence of crustal material dipping into the mantle at the suture of two Palaeo-Proterozoic terranes. The geometry of the suture is better defined on the transverse component where subduction is associated with a ∼10 km thick layer exhibiting strong elastic anisotropy. The modelling reproduces the main features of the receiver functions, including the effects of anisotropy, heterogeneity and finite-frequency scattering.     

Seismic body waves in anisotropic media: synthetic seismograms

Summary. Synthetic seismograms and particle motion diagrams are computed for simple, layered Earth models containing an anisotropic layer. The presence of anisotropy couples the P, SV and SH wave motion so that P waves incident on the anisotropic layer from below produce P, SV and small-amplitude SH waves at the surface both the P velocity and the amplitudes of the converted phases vary with azimuth. Significant SH amplitudes may be generated even when the wavelength of the P wave is much greater than the thickness of the anisotropic layer. Incident SV or SH waves may each generate large amplitudes of both SV and SH motion. This strong coupling is largely independent of the degree of velocity anisotropy of the medium. The arrivals from short-period S waves exhibit S-wave splitting, but arrivals from longer period S waves superpose into a modified waveform. This strong coupling does not allow the arrival of separate phases with pure SV and SH polarization except along directions of symmetry where the motion decouples.     

Seismic anisotropy as measured under high-pressure, high-temperature conditions

Summary. Four types of crustal and upper-mantle rocks have been used for the investigation of seismic P -wave velocities in three mutually perpendicular directions. Hydrostatic pressure, up to 6 kbar and temperatures up to 500°C were applied to the samples. Measurements of the ultrasonic P -wave travel times and velocities were carried out along two geotherms. All rock types show an anisotropic behaviour which is caused by the orientation of certain minerals. The anisotropy is not dependent on temperature and pressure. Gneiss and peridotite have 5–6 per cent anisotropy whereas granite and a metagabbro show values of only 2–3 per cent. The smallest velocity is always in the z direction, perpendicular to a schistocity or foliation. It is shown that the data agree with those of field observation. We conclude that anisotropy caused by preferred orientation of minerals must be expected in the whole lithosphere. Additional effects of layering, of cracks, and of nonhydrostatic stresses are estimated.     

Shear-wave anisotropy: spatial and temporal variations in time delays at Parkfield, Central California

Shear-wave splitting is analysed on data recorded by the High Resolution Seismic Network (HRSN) at Parkfield on the San Andreas fault, Central California, during the three-year period 1988-1990. Shear-wave polarizations either side of the fault are generally aligned in directions consistent with the regional horizontal maximum compressive stress, at some 70° to the fault strike, whereas at station MM in the immediate fault zone, shear-wave polarizations are aligned approximately parallel to the fault. Normalized time delays at this station are found to be about twice as large as those in the rock mass either side. This suggests that fluid-filled cracks and fractures within the fault zone are elastically or seismically different from those in the surrounding rocks, and that the alignment of fault-parallel shear-wave polarizations are associated with some fault-specific phenomenon.
Temporal variations in time delays between the two split shear-waves before and after a M_L = 4 earthquake can be identified at two stations with sufficient data: MM within the fault zone and VC outside the immediate fault zone. Time delays between faster and slower split shear waves increase before the M_L = 4 earthquake and decrease near the time of the event. The temporal variations are statistically significant at 68 per cent confidence levels. Earthquake doublets and multiplets also show similar temporal variations, consistent with those predicted by anisotropic poroelasticity theory for stress modifications to the microcrack geometry pervading the rock mass. This study is broadly consistent with the behaviour observed before three other earthquakes, suggesting that the build-up of stress before earthquakes may be monitored and interpreted by the analysis of shear-wave splitting.     

Anisotropy in multi-offset deep-crustal seismic experiments

Modelling of deep-seismic wide-angle data commonly assumes that the Earth is heterogeneous and isotropic. It is important to know the magnitudes of errors that may be introduced by isotropic-based wide-angle models when the Earth is anisotropic. It is equally important to find ways of detecting anisotropy and determining its properties.
  This paper explores the errors introduced by interpreting anisotropic seismic data with isotropic models. Errors in P -wave reflector depths are dependent on the magnitude of the velocity anisotropy and the direction of the fast axis. The interpreted, isotropic, model velocity function is found to correspond closely to the horizontal velocity of the anisotropic medium. An additional observed parameter is the time mismatch , which we define to be the difference between the vertical two-way traveltime to a reflector and the time-converted wide-angle position of the reflector. The magnitude of the time mismatch is typically <1.0  s (when the whole crust is anisotropic) and is found to be closely related to the magnitude and sign of the anisotropic anellipticity. The relationships are extendible to more complicated models, including those with vertical velocity gradients, crustal zonation, and lower symmetry orders.
  A time mismatch may be symptomatic of the presence of anisotropy. We illustrate the observation of a time mismatch for a real multi-offset seismic data set collected north of Scotland and discuss the implications for crustal anisotropy in that region.     

Seismic anisotropy in the core–mantle transition zone

Split S waves observed at Hockley, Texas from events in the Tonga–Fiji region of the southwest Pacific show predominantly vertically polarized shear-wave ( SV  ) energy arriving earlier than horizontally polarized ( SH ) energy for rays propagating horizontally through D" . After corrections are made for the effects of upper-mantle anisotropy beneath Hockley, a time lag of 1.5 to 2.0  s remains for the furthest events (93.9°–100.6° ), while the time lags of the nearer observations (90.5°–92.9° ) nearly disappear. At closer distances, the S waves from these same events do not penetrate as deeply into the lower mantle, and are not split. These observations suggest that a patch of D" beneath the central Pacific is anisotropic, while the mantle immediately above the patch is isotropic. The thickness of the anisotropic zone appears to be of the order of 100–200  km.
  Observations of shear-wave splitting have previously been made for paths that traverse D" under the Caribbean and under Alaska. SH leads SV , the reverse of the Hockley observations, but in these areas the fact that SV  leads SH in the HKT data shown here suggests a different sort of anisotropy under the central Pacific from that under Alaska and the Caribbean. The case of SH travelling faster than SV  is consistent with transverse isotropy with a vertical axis of symmetry (VTI) and does not require variations with azimuth. The case of SV  leading SH is consistent with transverse isotropy with a horizontal axis of symmetry (HTI), an azimuthally anisotropic medium, and with a VTI medium formed by a hexagonal crystal. Given that (Mg,Fe)SiO₃ perovskite appears unlikely to form anisotropic fabrics on a large scale, the presence of anisotropy may point to chemical heterogeneity in the lowermost mantle, possibly due to mantle–core interactions.     

Modelling the compliance of crustal rock—I. Response of shear-wave splitting to differential stress

We show that seismic shear waves may be used to monitor the in situ stress state of deep inaccessible rocks in the crust. The most widespread manifestation of the stress-related behaviour of seismic waves is the shear-wave splitting (shear-wave birefringence) observed in almost all rocks, where the polarizations of the leading split shear waves are usually subparallel to the direction of the local maximum horizontal stress. It has been recognized that such shear-wave splitting is typically the result of propagation through distributions of stress-aligned fluid-filled microcracks and pores, known as extensive-dilatancy anisotropy or EDA. This paper provides a quantitative basis for the EDA hypothesis. We model the evolution of anisotropic distributions of microcracks in triaxial differential stress, where the driving mechanism is fluid migration along pressure gradients between neighbouring microcracks and pores at different orientations to the stress field. This leads to a non-linear anisotropic poroelasticity (APE) model for the stress-sensitive behaviour of fluid-saturated microcracked rocks. A companion paper shows that APE modelling matches a range of observed phenomena and is a good approximation to the equation of state of a stressed fluid-saturated rock mass.     

P-wave traveltime and polarization tomography of VSP data

The presence of anisotropy requires that tomographic methods be generalized to account for anisotropy. This generalization allows geological structure to be correctly imaged and allows the anisotropic parameters to be estimated. Use of isotropic inversion for imaging anisotropic structures gives systematic trends in the traveltime and polarization residuals. However, due to the limited directional coverage, the traveltimes along may not be sufficient to study the anisotropic properties of the structure. Polarizations can provide independent information on the structure. Traveltime and polarization inversion are applied to synthetic examples simulating VSP experiments. Transverse isotropy and 1-D structure are assumed. Plots of traveltime and polarization residuals are an important tool to detect the anomalies due to the presence of anisotropy. For receivers located in anisotropic layers, polarization residuals display consistent anomalies of several degrees. The synthetic examples show that even the simple 1-D problem is difficult, when using direct arrivals only. Large a posteriori errors in anisotropic parameters are obtained by traveltime inversion in layers where available incidence angles are less than 45°. Resolution of the tomographic image of VSP data is greatly improved by a combination of traveltime and polarization information. In order to obtain accurate inversion results, the measurement error of polarization data should be kept to within a few degrees.     

Reflection coefficients for weak anisotropic media

The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy.
Classical perturbation theory is applied on a 6times6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S -wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6times6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lamé parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.