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.     

Shear-wave polarizations near the North Anatolian Fault – II. Interpretation in terms of crack-induced anisotropy

Summary. The polarizations of shear waves recorded by networks of digital three-component seismometers immediately above small earthquakes near the North Anatolian Fault in Turkey display shear-wave splitting on almost all shear-wave seismograms recorded within the shear-wave window. This splitting is incompatible with source radiation-patterns propagating through simple isotropic structures but is compatible with effective anisotropy of the internal structure of the rock along the ray paths. This paper interprets the phenomena in terms of widespread crack-induced anisotropy. Distributions of stress-induced cracks model many features of the observations, and synthetic polarization diagrams calculated for propagation through simulated cracked rock are similar to the observed patterns. This evidence for widespread crack-induced anisotropy lends strong support to the hypothesis of extensive-dilatancy anisotropy (EDA) suggested by laboratory experiments in subcritical crack-growth. The crucial evidence confirming some form of EDA would be observations of temporal changes in shear-wave splitting as the stress field alters the crack density and crack geometry. There is some weak evidence for such temporal changes at one site, but further analysis of suitable digital three-component seismometer networks in seismic areas is required to confirm EDA.     

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.     

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.     

The effects of crack anisotropy on the hypocentral locations of local earthquakes

Summary. Exact modelling of the earthquake location process in regions of dilatancy-anisotropy shows that failure to take account of the velocity anisotropy in the determination of local hypocentres can result in erroneous and misleading locations. In particular, the locations can indicate spurious migrations of foci from the true epicentral positions and the true depths of foci. These spurious locations may indicate planes of hypocentres deviating from the true fault plane. Observations of such phenomena have been noted several times in the literature. However, once the anisotropic model is known, a simple location program, incorporating the anisotropic velocity-variations, permits accurate location of local earthquakes using P and first S -wave arrival times.     

The basis for earthquake prediction

Summary. Recent advances in understanding the behaviour of shear waves propagating in the crust make the routine prediction of earthquakes seem practicable. Accumulating evidence suggests that most of the Earth's crust is pervaded by distributions of fluid-filled cracks and microcracks that are aligned by the contemporary stress-field so that the cracked rockmass is effectively anisotropic to seismic waves. This causes shear-waves to split, and shear-wave splitting is observed whenever shear-waves propagating along suitable raypaths in the crust are recorded by three-component instruments. These distributions of cracks are known as extensive-dilatancy anisotropy or EDA. Many characteristics of the crack- and stress-geometry can be monitored by analyzing shear-waves propagating through the cracked rockmass. Observations of temporal variations of the behaviour of shear-wave splitting in seismic gaps confirm these hypotheses, and suggest that stress changes before earthquakes may be monitored by analyzing shear-waves. In particular, monitoring earthquake preparation zones with three-component shear-wave vertical-seismic-profiles could lead to techniques for the routine prediction of earthquakes.     

Plane wave propagation in nonlinear-elastic anisotropic media

Summary. Kelvin-Christoffel equations describing plane wave propagation in anisotropic media are generalized to account for the effects of nonlinear elasticity. The polarization and waveform of nonlinear distortions of a transient plane wave are investigated by means of perturbation theory. Detailed analysis for an anisotropic medium with hexagonal symmetry shows that for "pure" shear-waves the polarization vector of the nonlinear component is always perpendicular to that of the linear wave. In the case of a high-amplitude excitation (for instance, in the vicinity of large earthquakes) the influence of nonlinearity may cause distortions of shear-wave polarization, which contains the most reliable information on the presence and characteristics of anisotropy. The solutions presented in this paper make it possible to solve reflection-transmission problems in nonlinear-elastic anisotropic media.     

The Turkish Dilatancy Project (TDP3): multidisciplinary studies of a potential earthquake source region

Summary. The section of the North Anatolian Fault lying near the city of Izmit, at the east of the Marmara Sea, has been identified as a seismic gap and the possible site of a future major earthquake. Previously published studies of records from an earthquake swarm within the gap (TDP1 and TDP2) provided the first evidence that shear-wave splitting occurs in earthquake source regions, a conclusion since verified by many studies at other locations. A third field study (TDP3) was mounted in the Izmit region during the summer of 1984. Observations were made over an eight-month period and included geomagnetic and geoelectric measurements in addition to a series of observations utilising dense arrays of three-component seismometers. Earthquake activity in the principal study area was monitored over a period of eight months. Records showed features similar to those observed in the earlier studies. In particular: (1) almost all shear waves emerging within the shear-wave window displayed shear-wave splitting; and (2) the polarizations of the first arriving (faster) split shear-waves showed sub-parallel alignments, characteristic of propagation through a distribution of parallel vertical cracks striking perpendicular to the minimum compressional stress.
These and other observations support the conclusion of earlier studies – that the upper crust is pervaded by distributions of micro-cracks aligned by stress, known as extensive-dilatancy anisotropy. A search for time dependence in shear-wave phenomena has revealed temporal variations in the delays between the split shear-waves throughout the course of the TDP3 study, but as yet this has not been correlated with specific earthquake activity.     

The variation of delays in stress-induced anisotropic polarization anomalies

Summary. Polarization anomalies in seismic shear wavetrains, diagnostic of propagation through anisotropic media, have now been observed in dilatancy zones in seismic regions. Stress-induced dilatancy will open cracks with preferred orientations, which will be effectively anisotropic to short-period seismic waves. The polarization anomalies are due to the shear waves splitting, in propagation through anisotropic media, into components with different polarizations and different velocities. This writes characteristic signatures into the shear wavetrains. The paper examines ways in which the differential shear-wave anisotropy (the delay between the split shear-waves) varies with direction by plotting stereograms of the relative delays, and their polarizations, for possible dilatancy symmetry-systems. It seems likely, that if sufficient observations of these anomalies can be obtained at each stage of the dilatancy episode, it will be possible to estimate the symmetry directions of the dilatancy and the geometry of the stress-induced crack-system, as well as monitoring the progress of the dilatancy episode.     

P-wave velocity anisotropy in crystalline rocks

Summary. Compressional wave velocities and anisotropy coefficients determined at high hydrostatic pressures are compiled from the data published for the main types of crystalline rocks. The crack-free elastic anisotropy of igneous crustal rocks is generally very low, between 1 and 3 per cent on average. The anisotropy of metamorphic rocks is higher (up to 22 per cent), but very variable. The average anisotropy coefficients in schists and amphibolites are about 10 per cent, in gneisses between 3 and 7 per cent, and in granulites less than 3 per cent. The average anisotropy of olivine ultramafites is between 7 and 12 per cent, whereas in pyroxenites and eclogites it is usually less than 4 per cent. A comparison of ranges of average velocities and average anisotropies for the individual rock groups suggests that, whereas in the crust the lateral velocity variations are mainly due to compositional changes, in the olivine of the uppermost mantle the velocity variations due to anisotropic structures could be of the same magnitude as the variations due to inhomogeneities.     

TRM deviations in anisotropic assemblages of multidomain magnetite

Anisotropic assemblages of multidomain magnetite particles develop an anisotropy of magnetic susceptibility (AMS), which in turn induces deviations of thermo-remanent magnetization (TRM) from the field direction. From the theories of multidomain TRM acquisition, it is shown that the TRM anisotropy tensor has its eigenvalue ratios ( T _i) related to the principal weak-field susceptibility ratios ( P _i) by the order of magnitude T _i≃ P ²_i. This relation has been experimentally verified on two sets of highly anisotropic rock samples. The exponent has been determined to be 1.94 in the samples from a Peruvian gabbro, and 1.81 in those from the granite of Flamanville (NW France). Accounting for experimental difficulties in determining the TRM anisotropy tensors, these exponents are judged to agree well with the expected one. It is therefore stressed that AMS measurements provide a good means of evaluating the magnetic field direction from deviated TRM directions, providing magnetic carriers are mainly multidomain magnetites.     

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.     

A theoretical and experimental comparison of the anisotropies of magnetic susceptibility and remanence in rocks and minerals

Summary. Susceptibility, thermo-remanent magnetization (TRM) and isothermal remanent magnetization (IRM) anisotropy ellipsoids have been determined for several rock samples. The results indicate that the ellipsoid of initial susceptibility is less anisotropic than the TRM and low field IRM ellipsoids which are found experimentally to be of identical shape. This suggests that palaeomagnetic data for anisotropic rocks may be corrected by using the anisotropy ellipsoid determined from magnetically non-destructive low field IRM measurements. Such IRM measurements can also be used to obtain anisotropy axes of samples which are inherently anisotropic but which have a susceptibility which is too weak to be accurately measured. The results for a series of artificial anisotropic samples containing magnetite particles of different sizes (in the range 0.2–90 μm) were very similar to those for the rocks. In contrast, a comparison of the susceptibility and IRM ellipsoids for anisotropic samples containing particles from a magnetic tape gave very different results in accordance with theory. Such results imply that susceptibility and IRM ellipsoids could be used to determine whether anisotropic rocks contain uniaxial single-domain particles (magnetization confined to the easy axis) or whether the particles are essentially multidomain.     

Correlation between seismic anisotropy and Bouguer gravity anomalies in Tibet and its implications for lithospheric structures

As a baseline measurement for understanding the Himalayan–Tibetan orogen, a product of continent–continent collision between India and Eurasia, we analyse digital seismic data in order to constrain the seismic anisotropy of the Indian shield. Based on spatially sparse data that are currently available in the public domain, there is little shear-wave birefringence for SKS phases under the Indian shield, even though it is part of a fast-moving plate in the hotspot frame of reference. If most of the northern Indian mantle has little transverse anisotropy, the onset of significant anisotropy under Tibet marks the northern terminus of intact Indian lithosphere that is thrusting under the Himalayan–Tibetan orogen. Beyond this terminus, tectonic fabric such as that associated with the deforming lithospheric mantle of Eurasia must be present in the upper mantle. Along the profile from Yadong to Golmud, the only profile in Tibet where a number of shear-wave birefringence data are available, the amount of birefringence shows two marked increases, near 30° and 33°N, between which a local high in Bouguer gravity anomaly is observed. Such a correlation between patterns of shear-wave birefringence and gravity anomalies is explained by the juxtaposition of Indian lithosphere against the overlying Eurasian lithosphere: while the Eurasian lithospheric mantle appears only to the north of 30°N, the Indian lithospheric mantle extends northwards to near 33°N.     

On the inversion of Love- and Rayleigh-wave dispersion and implications for Earth structure and anisotropy

Summary. Various factors can make it difficult to explain observations of Love- and Rayleigh-wave dispersion with the same relatively simple isotropic model. These factors include systematic errors which might occur in determinations of observed group and phase velocities, lateral variations in structure along the path of travel, and the attempt to explain observations with a model comprised of only a small number of thick layers. The last of these factors is illustrated by an inversion of dispersion data in the central United States where shear-wave anisotropy had previously been invoked as one way to explain incompatible Love- and Rayleigh-wave velocities. It is shown that the data can be satisfied equally well by an isotropic model consisting of several thin layers.
In cases where the incompatibility of Love- and Rayleigh-wave data might be produced by intrinsic anisotropy, it is necessary to invert those data using an anisotropic theory rather than by separate isotropic inversions of Love and Rayleigh waves. Inversions of fundamental-mode data for a region of the Pacific, assuming anisotropic media in which the layers are transversely isotropic with a vertical axis of symmetry, lead to models which are highly non-unique. Even if the inversions solve only for shear velocities in the litho-sphere and asthenosphere it is not possible, without supplementary information, to ascertain the depth interval over which anisotropy occurs or to determine the thickness of the lithosphere or asthenosphere with much precision.     

Observation and modelling of fault-zone fracture seismic anisotropy – II. P-wave polarization anomalies

Summary. In Part I of this paper we modelled shear-wave splitting observed in crystalline rock bordering an active, normal fault-zone at Oroville, California, with Červený's ray-tracing system applied to anisotropic heterogeneous media using Hudson's formulation of elastic constants for a medium containing aligned cracks. In Part II we use the ray-tracing results of Part I to quantitatively interpret P -wave polarization anomalies observed in the three-component seismograms recorded in the Oroville fault zone. We show that the eigenvectors of the first-order Christoffel tensor defined by the ray-tracing slowness vector and Hudson's first-order anisotropic corrections to the isotropic elastic tensor correctly account for P -wave first motion that deviates from the ray vector.     

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.     

Inversion of P-wave velocity anisotropy

Summary. We examine the way in which measurements of velocity anisotropy can add to our understanding of upper mantle structure. Measurements of P -wave velocity anisotropy in a single plane contain very little direct information about the anisotropic structure. A promising technique is to fit the observed velocity variation with a mixture of an assumed anisotropic constituent and a proportion of isotropic material. Using this technique, mixtures of orthorhombic and transversely isotropic olivine are obtained, which are in excellent agreement with observed velocity variations in the Pacific.     

Characterization of crack distribution: fabric analysis versus ultrasonic inversion

We analyse the relation between rock fabric, expressed by the preferred orientation of rock-forming minerals and microcracks, and elastic anisotropy of crystalline rock from the KTB pilot well. Detailed analyses of mineralogical composition, textures and microcrack fabrics were performed. In addition, ultrasonic velocity measurements of spherical samples in several directions were carried out at various confining pressures, and inverted in terms of the complete set of 21 elastic constants. By comparing the elastic tensors of the rocks at the final confining pressure (at which most of the microcracks are closed) with those at a lower pressure level, it is possible to separate the anisotropy induced by microcracks from that caused by mineral alignment. In contrast to previous work, no a priori knowledge of the type of anisotropy (triclinic, monoclinic, orthotropic etc.), or of the spatial orientation of the symmetry elements (planes, axes) of the cracked rock or of the intact rock is assumed. Furthermore, no restrictive assumptions on the orientation distribution function and the shape of the cracks are needed.
The results show that the elastic anisotropy characteristics, whether they are related to the microcracks or to the rock-forming minerals, are clearly correlated with the directly observed rock fabrics. We show that the symmetry directions of the mineral fabric and of microcrack fabric agree. A further result is that the microcrack-induced anisotropy dominates the other causes of anisotropy at confining pressures smaller than a few tens of megapascals, the situation being reversed at higher pressures. The laboratory data are quantitatively compared with sonic log data from the KTB well, showing the influence of pore fluids, effective pressure and crack density reduction on the anisotropy in situ .     

Global anisotropic phase velocity maps for higher mode Love and Rayleigh waves

It is well established that the Earth's uppermost mantle is anisotropic, but there are no clear observations of anisotropy in the deeper parts of the mantle. Surface waves are well suited to observe anisotropy since they carry information about both radial and azimuthal anisotropy. Fundamental mode surface waves, for commonly used periods up to 200 s, are sensitive to structure in the first few hundred kilometres, and therefore, do not provide information on anisotropy below. Higher mode surface waves have sensitivities that extend to and beyond the transition zone, and should thus give insight about azimuthal anisotropy at greater depths. We have measured higher mode Love and Rayleigh phase velocities using a model space search approach, which provides us with consistent relative uncertainties from measurement to measurement and from mode to mode. From these phase velocity measurements, we constructed global anisotropic phase velocity maps. Prior to inversion, we determine the optimum relative weighting for anisotropy. We present global azimuthal phase velocity maps for higher mode Rayleigh waves (up to the sixth higher mode) and Love waves (up to the fifth higher mode) with corresponding average model uncertainties. The anisotropy we derive is robust within the uncertainties for all modes. Given the ray theoretical sensitivity kernels of Rayleigh and Love wave modes, the source of anisotropy is complex, but mainly located in the asthenosphere and deeper. Our models show a good correspondence with other studies for the fundamental mode, but we have been able to achieve higher resolution.