Effects of slight anisotropy on surface waves

We present a complete ray theory for the calculation of surface-wave observables from anisotropic phase-velocity maps. Starting with the surface-wave dispersion relation in an anisotropic earth model, we derive practical dynamical ray-tracing equations. These equations allow calculation of the observables phase, arrival-angle and amplitude in a ray theoretical framework. Using perturbation theory, we also obtain approximate expressions for these observables. We assess the accuracy of the first-order approximations by using both theories to make predictions on a sample anisotropic phase-velocity map. A comparison of the two methods illustrates the size and type of errors which are introduced by perturbation theory. Perturbation theory phase and arrival-angle predictions agree well with the exact calculation, but amplitude predictions are poor. Many previous studies have modelled surface-wave propagation using only isotropic structure, not allowing for anisotropy. We present hypothetical examples to simulate isotropic modelling of surface waves which pass through anisotropic material. Synthetic data sets of phase and arrival angle are produced by ray tracing with exact ray theory on anisotropic phase-velocity maps. The isotropic models obtained by inverting synthetic anisotropic phase data sets produce deceptively high variance reductions because the effects of anisotropy are mapped into short-wavelength isotropic structure. Inversion of synthetic arrival-angle data sets for isotropic models results in poor variance reductions and poor recovery of the isotropic part of the anisotropic input map. Therefore, successful anisotropic phase-velocity inversions of real data require the inclusion of both phase and arrival-angle measurements.     

Controls on Rayleigh wave amplitudes: attenuation and focusing

A large data set of amplitude measurements of minor and major arc Rayleigh waves in the period range 73–171 s is collected. By comparing these amplitudes with the amplitudes of synthetic waveforms calculated by mode summation, maps of lateral variations in the apparent attenuation structure of the Earth are constructed. An existing formalism for predicting the effects of focusing is employed to calculate amplitude perturbations for the same data set. These perturbations are used to construct 'pseudo‐attenuation' maps and these results are compared with the apparent attenuation maps calculated from the data. It is shown that variations in Rayleigh wave amplitude perturbations in the Earth are dominated by attenuation at long wavelengths (below about degree 8) and by elastic structure at shorter wavelengths. It is also shown that the linear approximation for focusing is successful at predicting Rayleigh wave amplitudes using existing phase velocity maps. These results indicate that future attempts to model the velocity structure of the Earth would be assisted by incorporating amplitude data and by jointly inverting for Q structure.     

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.     

Upper mantle anisotropy: evidence from free oscillations

Summary Isotropic earth models are unable to provide uniform fits to the gross Earth normal mode data set or, in many cases, to regional Love-and Rayleigh-wave data. Anisotropic inversion provides a good fit to the data and indicates that the upper 200km of the mantle is anisotropic. The nature and magnitude of the required anisotropy, moreover, is similar to that found in body wave studies and in studies of ultramafic samples from the upper mantle. Pronounced upper mantle low-velocity zones are characteristic of models resulting from isotropic inversion of global or regional data sets. Anisotropic models have more nearly constant velocities in the upper mantle.
Normal mode partial (Frediét) derivatives are calculated for a transversely isotropic earth model with a radial axis of symmetry. For this type of anisotropy there are five elastic constant. The two shear-type moduli can be determined from the toroidal modes. Spheroidal and Rayleigh modes are sensitive to all five elastic constants but are mainly controlled by the two compressional-type moduli, one of the shear-type moduli and the remaining, mixed-mode, modulus. The lack of sensitivity of Rayleigh waves to compressional wave velocities is a characteristic only of the isotropic case. The partial derivatives of the horizontal and vertical components of the compressional velocity are nearly equal and opposite in the region of the mantle where the shear velocity sensitivity is the greatest. The net compressional wave partial derivative, at depth, is therefore very small for isotropic perturbations. Compressional wave anisotropy, however, has a significant effect on Rayleigh-wave dispersion. Once it has been established that transverse anisotropy is important it is necessary to invert for all five elastic constants. If the azimuthal effect has not been averaged out a more general anisotropy may have to be allowed for.     

A simple method of representing azimuthal anisotropy on a sphere

We describe a method of expressing azimuthally anisotropic surface wave velocities on the Earth using a local and smooth spherical-spline parametrization. Anisotropy in the Earth leads to azimuthally varying Love and Rayleigh wave velocities that can be expressed as (cos 2ζ, sin 2ζ) and (cos 4ζ, sin 4ζ) perturbations to the isotropic velocities, where ζ is the direction of surface-wave propagation. The strength of the perturbations varies laterally, and a current goal of seismic tomography is the detailed global mapping of these variations. Several parametrizations have previously been used to describe azimuthally varying velocities. The representation proposed here uses spherical splines and is designed to describe smooth variations in both the strength and geometry of azimuthal anisotropy. The method builds on a simple geometrical approximation for the local azimuth of propagation expressed at the defining spline knot points. It avoids the singularities at the poles that result when azimuthal variations are parametrized using traditional scalar spherical harmonics. Compared with a generalized spherical-harmonic expansion of the tensor fields that represent 2ζ and 4ζ azimuthal variations smoothly on a sphere, the new method offers the advantages of local geographical support and simplicity of implementation.     

Evidence for anisotropy in the upper mantle beneath Eurasia from the polarization of higher mode seismic surface waves

Summary. Analysis of NORSAR records and a number of Soviet microfilms reveals second-mode surface Caves propagating along paths covering a large part of Eurasia. These second modes in the 6–15-s period band are frequently disturbed by other surface-wave modes and by body-wave arrivals. However, in all cases, where the modes appear to be undisturbed and show normal dispersion, the Second Rayleigh modes have a slowly varying phase difference with the Second Love modes. This coupling has the particle motion of Inclined Rayleigh waves characteristic of surface-wave propagation in anisotropic media, where the anisotropy possesses a horizontal plane of symmetry. Numerical examination of surface wave propagating in Earth models, with an anisotropic layer in the upper mantle, demonstrate that comparatively small thicknesses of material with weak velocity anisotropy can produce large deviations in the polarizations of Inclined Rayleigh Second modes. In many structures, these inclinations are very sensitive to small changes in anisotropic orientation and to small changes in the surrounding isotropic structure. It is suggested that examination of second mode inclination anomalies of second mode surface waves may be a powerful technique for examining the detailed anisotropic structure of the upper mantle.     

On crustal corrections in surface wave tomography

Mantle models from surface waves rely on good crustal corrections. We investigated how far ray theoretical and finite frequency approximations can predict crustal corrections for fundamental mode surface waves. Using a spectral element method, we calculated synthetic seismograms in transversely isotropic PREM and in the 3-D crustal model Crust2.0 on top of PREM, and measured the corresponding time-shifts as a function of period. We then applied phase corrections to the PREM seismograms using ray theory and finite frequency theory with exact local phase velocity perturbations from Crust2.0 and looked at the residual time-shifts. After crustal corrections, residuals fall within the uncertainty of measured phase velocities for periods longer than 60 and 80 s for Rayleigh and Love waves, respectively. Rayleigh and Love waves are affected in a highly non-linear way by the crustal type. Oceanic crust affects Love waves stronger, while Rayleigh waves change most in continental crust. As a consequence, we find that the imperfect crustal corrections could have a large impact on our inferences of radial anisotropy. If we want to map anisotropy correctly, we should invert simultaneously for mantle and crust. The latter can only be achieved by using perturbation theory from a good 3-D starting model, or implementing full non-linearity from a 1-D starting model.     

Surface wave tomography of the western United States from ambient seismic noise: Rayleigh and Love wave phase velocity maps

We present the results of Rayleigh wave and Love wave phase velocity tomography in the western United States using ambient seismic noise observed at over 250 broad-band stations from the EarthScope/USArray Transportable Array and regional networks. All available three-component time-series for the 12-month span between 2005 November 1 and 2006 October 31 have been cross-correlated to yield estimated empirical Rayleigh and Love wave Green's functions. The Love wave signals were observed with higher average signal-to-noise ratio (SNR) than Rayleigh wave signals and hence cannot be fully explained by the scattering of Rayleigh waves. Phase velocity dispersion curves for both Rayleigh and Love waves between 5 and 40 speriod were measured for each interstation path by applying frequency–time analysis. The average uncertainty and systematic bias of the measurements are estimated using a method based on analysing thousands of nearly linearly aligned station-triplets. We find that empirical Green's functions can be estimated accurately from the negative time derivative of the symmetric component ambient noise cross-correlation without explicit knowledge of the source distribution. The average traveltime uncertainty is less than 1 s at periods shorter than 24 s. We present Rayleigh and Love wave phase speed maps at periods of 8, 12, 16,and 20 s. The maps show clear correlations with major geological structures and qualitative agreement with previous results based on Rayleigh wave group speeds.     

Partial derivatives of surface wave phase velocities for flat anisotropic models

Summary. Surface wave behaviour in flat anisotropic structures is first illustrated by performing an exact computation on a simple two-layer model. The variational procedure of Smith & Dahlen is then used to compute the partial derivatives of surface wave phase velocities with respect to the elastic parameters in more realistic earth models. Linear relationships between the partial derivatives for a general anisotropic structure and those for a transversely isotropic structure are derived. When considering waves propagating in a fixed direction, there are only four independent derivatives for Rayleigh waves, and two for Love waves. To avoid the lack of resolution in an inverse method, we propose to use physically constrained models. These results are illustrated by using a model with hexagonal symmetry and a symmetry axis oriented either vertically or horizontally. Quasi-Love- and quasi-Rayleigh-wave partial derivatives are computed for both axis orientations. Modes up to the second overtone and periods ranging between 45 and 130 s have been considered. Finally, anomalies of phase velocity are computed in an oceanic model made of 1/6 oriented olivine crystals with horizontal or vertical preferred orientations of the a -axis.     

Length scales, patterns and origin of azimuthal seismic anisotropy in the upper mantle as mapped by Rayleigh waves

We measure the degree of consistency between published models of azimuthal seismic anisotropy from surface waves, focusing on Rayleigh wave phase-velocity models. Some models agree up to wavelengths of ∼2000 km, albeit at small values of linear correlation coefficients. Others are, however, not well correlated at all, also with regard to isotropic structure. This points to differences in the underlying data sets and inversion strategies, particularly the relative 'damping' of mapped isotropic versus anisotropic anomalies. Yet, there is more agreement between published models than commonly held, encouraging further analysis. Employing a generalized spherical harmonic representation, we analyse power spectra of orientational (2Ψ) anisotropic heterogeneity from seismology. We find that the anisotropic component of some models is characterized by stronger short-wavelength power than the associated isotropic structure. This spectral signal is consistent with predictions from new geodynamic models, based on olivine texturing in mantle flow. The flow models are also successful in predicting some of the seismologically mapped patterns. We substantiate earlier findings that flow computations significantly outperform models of fast azimuths based on absolute plate velocities. Moreover, further evidence for the importance of active upwellings and downwellings as inferred from seismic tomography is presented. Deterministic estimates of expected anisotropic structure based on mantle flow computations such as ours can help guide future seismologic inversions, particularly in oceanic plate regions. We propose to consider such a priori information when addressing open questions about the averaging properties and resolution of surface and body wave based estimates of anisotropy.     

Wavefield smoothing and the effect of rough velocity perturbations on arrival times and amplitudes

Geometric ray theory is an extremely efficient tool for modelling wave propagation through heterogeneous media. Its use is, however, only justified when the inhomogeneity satisfies certain smoothness criteria. These criteria are often not satisfied, for example in wave propagation through turbulent media. In this paper, the effect of velocity perturbations on the phase and amplitude of transient wavefields is investigated for the situation that the velocity perturbation is not necessarily smooth enough to justify the use of ray theory. It is shown that the phase and amplitude perturbations of transient arrivals can to first order be written as weighted averages of the velocity perturbation over the first Fresnel zone. The resulting averaging integrals are derived for a homogeneous reference medium as well as for inhomogeneous reference media where the equations of dynamic ray tracing need to be invoked. The use of the averaging integrals is illustrated with a numerical example. This example also shows that the derived averaging integrals form a useful starting point for further approximations. The fact that the delay time due to the velocity perturbation can be expressed as a weighted average over the first Fresnel zone explains the success of tomographic inversions schemes that are based on ray theory in situations where ray theory is strictly not justified; in that situation one merely collapses the true sensitivity function over the first Fresnel zone to a line integral along a geometric ray.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

Quasi-isotropic approximation of ray theory for anisotropic media

Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails.     

Crustal and uppermost mantle structure in southern Africa revealed from ambient noise and teleseismic tomography

Rayleigh wave phase velocity maps in southern Africa are obtained at periods from 6 to 40 s using seismic ambient noise tomography applied to data from the Southern Africa Seismic Experiment (SASE) deployed between 1997 and 1999. These phase velocity maps are combined with those from 45 to 143 s period which were determined previously using a two-plane-wave method by Li & Burke. In the period range of overlap (25–40 s), the ambient noise and two-plane-wave methods yield similar phase velocity maps. Dispersion curves from 6 to 143 s period were used to estimate the 3-D shear wave structure of the crust and uppermost mantle on an 1°× 1° grid beneath southern Africa to a depth of about 100 km. Average shear wave velocity in the crust is found to vary from 3.6 km s^–1 at 0–10 km depths to 3.86 km s^–1 from 20 to 40 km, and velocity anomalies in these layers correlate with known tectonic features. Shear wave velocity in the lower crust is on average low in the Kaapvaal and Zimbabwe cratons and higher in the surrounding Proterozoic terranes, such as the Limpopo and the Namaqua-Natal belts, which suggests that the lower crust underlying the Archean cratons is probably less mafic than beneath the Proterozoic terranes. Crustal thickness estimates agree well with a previous receiver function study of Nair et al. . Archean crust is relatively thin and light and underlain by a fast uppermost mantle, whereas the Proterozoic crust is thick and dense with a slower underlying mantle. These observations are consistent with the southern African Archean cratons having been formed by the accretion of island arcs with the convective removal of the dense lower crust, if the foundering process became less vigorous in arc environments during the Proterozoic.     

Computation of qP-wave rays, traveltimes and slowness vectors in orthorhombic media

The eikonal equation is the equation of the phase slowness surface for isotropic and anisotropic media. In general anisotropic media, there is no simple explicit expression for the phase slowness surface. An approximate expression of the eikonal equation may be obtained in weakly anisotropic media. In orthorhombic media, the approximate eikonal equation of the qP wave is the sum of an ellipsoidal form and a more complicated term. The ellipsoidal form corresponds to what we call ellipsoidal anisotropy. Ray equations written in the Hamiltonian formulation are characteristics of the eikonal equation. Ray perturbation theory may be used to compute changes in ray paths and physical attributes (traveltime, polarization, amplitude) due to changes in the medium with respect to a reference medium. Examples obtained in homogeneous orthorhombic media show that a reference medium with ellipsoidal anisotropy is a better choice to develop the perturbation approach than an isotropic reference medium. Models with strong anisotropy can be considered. The comparison with results obtained by an exact ray program shows a relative traveltime error of less than 0.5 per cent for a model with relatively strong anisotropy. We propose a finite element approach in which the medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, using a perturbation approach, analytical expressions for rays and traveltimes are obtained Ray tracing reduces to connecting these analytical solutions at the vertices of the cells.     

Ambient noise Rayleigh wave tomography of New Zealand

We present the first New Zealand-wide study of surface wave dispersion, using ambient noise observed at 42 broad-band stations in the national seismic network (GeoNet) and the Global Seismic Network (GSN). Year-long vertical-component time-series recorded between 2005 April 1 and 2006 March 31 have been correlated with one another to yield estimated fundamental mode Rayleigh wave Green's functions. We filter these Green's functions to compute Rayleigh wave group dispersion curves at periods of 5–50 s, using a phase-matched filter, frequency–time analysis technique. The uncertainties of the measurements are estimated based on the temporal variation of the dispersion curves revealed by 12 overlapping 3-month stacks. After selecting the highest quality dispersion curve measurements, we compute group velocity maps from 7 to 25 s period. These maps, and 1-D shear wave velocity models at four selected locations, exhibit clear correlations with major geological structures, including the Taranaki and Canterbury Basins, the Hikurangi accretionary prism, and previously reported basement terrane boundaries.     

Surface-wave polarization data and global anisotropic structure

In the past few years, seismic tomography has begun to provide detailed images of seismic velocity in the Earth's interior which, for the first time, give direct observational constraints on the mechanisms of heat and mass transfer. The study of surface waves has led to quite detailed maps of upper-mantle structure, and the current global models agree reasonably well down to wavelengths of approximately 2000 km. Usually, the models contain only elastic isotropic structure, which provides an excellent fit to the data in most cases. For example, the variance reduction for minor and major arc phase data in the frequency range 7–15 mHz is typically 65–92 per cent and the data are fit to within 1–2 standard deviations. The fit to great-circle phase data, which are not subject to bias from unknown source or instrument effects, is even better. However, there is clear evidence for seismic anisotropy in various places on the globe. This study demonstrates how much (or little) the fit to the data is improved by including anisotropy in the modelling process. It also illuminates some of the trade-offs between isotropic and anisotropic structure and gives an estimate of how much bias is introduced by neglecting anisotropy. Finally, we show that the addition of polarization data has the potential for improving recovery of anisotropic structure by diminishing the trade-offs between isotropic and anisotropic effects.     

Investigation of 3D wavefields of reflected shear-waves and converted waves: mathematical modelling and reflection data processing

Summary. Seismic investigations using shear-wave and converted wave techniques show that very often reflected PS - and SS -waves have anomalous polarizations ( accessory components ). This phenomenon cannot be explained in terms of isotropic models with dipping boundaries. Computations of synthetic seismograms of reflected PS - and SS -waves were made for different models of transversely isotropic media with dipping anisotropic symmetry axes not normal to the boundaries. Synthetic seismograms were computed by ray techniques using an optimization algorithm to construct all rays arriving at a given receiver. These computations indicate that accessory components arise when the medium above the boundary is anisotropic, where they are caused by the constructive interference of qSV - and qSH -waves. If a low-velocity layer is present, displacement vectors of both waves have horizontal projections which are approximately orthogonal. The algorithm for wave separation is presented and some results of its use are given.     

Three-dimensional Rayleigh hysteresis of oriented core samples from the German Continental Deep Drilling Program: susceptibility tensor, Rayleigh tensor, three-dimensional Rayleigh law

Rayleigh hysteresis, as defined by the well-known Rayleigh relations, has been observed not only when magnetization of pyrrhotite-bearing KTB-samples is measured in parallel to a weak dc magnetic field, but also in experiments where field and measuring directions have been adjusted strictly perpendicularly to each other. Nine-tupels of independent Rayleigh hysteresis loops could thus be compiled. Their characteristic coefficients X ^ijk of initial susceptibility together with the Rayleigh loss coefficients α^jk have been proved to determine completely the samples' weak-field magnetic anisotropy. Interpreting the coefficient matrices ( X ^ijk) and (α^jk) as the tensor of initial susceptibility and the Rayleigh tensor, respectively, generalization of the isotropic Rayleigh relations in terms of corresponding tensor relationships has been suggested for the anisotropic case. Application to the KTB samples showed 3-D Rayleigh hysteresis measurements to be an excellent tool for rock magnetic analysis in terms of ore content and crystalline texture. In particular, a magnetocrystalline double texture of the basal planes of pyrrhotite precipitates and their [1120] directions of easy magnetization have been clearly detected. Surprisingly, the welt-known theorem α= const. X ²_I, formulated by Néel (1942) for the isotropic case, has been found to hold true even in tensor generalization (α_jk) = const ( X ²_jk). To reach sufficient resolution for the measurements performed, a sensitive vibrating coil magnetometer (VCM) has been developed.     

Global multiresolution models of surface wave propagation: comparing equivalently regularized Born and ray theoretical solutions

I invert a large set of teleseismic phase-anomaly observations, to derive tomographic maps of fundamental-mode surface wave phase velocity, first via ray theory, then accounting for finite-frequency effects through scattering theory, in the far-field approximation and neglecting mode coupling. I make use of a multiple-resolution pixel parametrization which, in the assumption of sufficient data coverage, should be adequate to represent strongly oscillatory Fréchet kernels. The parametrization is finer over North America, a region particularly well covered by the data. For each surface-wave mode where phase-anomaly observations are available, I derive a wide spectrum of plausible, differently damped solutions; I then conduct a trade-off analysis, and select as optimal solution model the one associated with the point of maximum curvature on the trade-off curve. I repeat this exercise in both theoretical frameworks, to find that selected scattering and ray theoretical phase-velocity maps are coincident in pattern, and differ only slightly in amplitude.