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.     

On Love Waves across the Ocean

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Displacements of Love waves generated by a two-dimensional point source in a layered medium have been studied earlier by Sezawa & Sato by the method of successive reflections at the boundaries. In this paper the same problem has been worked out by using Green's function. The paper deals with the study of attenuation of Love waves of low periods in the coastal region. Experimental observations show that Love waves of smaller periods can be obtained only in the island observing stations. A slight intervention of the continental boundary is sufficient to attenuate lower period Love waves giving a hint thereby that attenuation of lower periods takes place perhaps at the continental margin. Taking a simplified configuration for the continental boundary and using Green's function technique, the displacement of Love waves due to a point source has been obtained and it has been shown that attenuation of Love waves of smaller periods takes place in the continental margin due to the slope of the boundary.     

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.     

A search for the oceanic Lg phase

Summary. The Lg phase has been shown previously to be a collection of higher-mode surface waves guided by the continental crust (Knopoff, Schwab & Kausel). A simple scaling between continental and oceanic crustal thicknesses suggests that a search for an oceanic Lg phase should be made in the period range from 1 to 2s. In a search for SH polarized Lg arrivals over oceanic paths, we found that in addition to the fundamental mode, seismo-grams at relatively short ranges in the Pacific showed the presence of only the first higher mode with group velocities on the steep portion of the dispersion curve rather than at the group velocity minimum as expected. Numerical model analysis indicates that, contrary to the continental case, there is no strong confluence of stationary phases of higher-mode crustal waves in the appropriate period range to produce Lg wave packets; this is due to small but significant differences in scaled crustal structures. Further, lateral variations in the thickness of oceanic sediments are sufficient to scatter most of the crustal surface-wave energy within a relatively short distance. Even were this thickness uniform, attenuation in the sediments would be strong enough to absorb the Lg stationary phases in a short distance.     

Surface-wave propagation in an ocean basin with an anisotropic upper mantle: observations of polarization anomalies

Summary. Observations of surface waves crossing ocean basins indicate that, at least in some cases, a better interpretation of what was previously considered to be the simultaneous arrival of fundamental Love and second Rayleigh modes, with similar group velocities, is the coupled motion of the second-generalized mode surface wave propagating in an anisotropic upper-mantle. This mode displays the polarization anomalies expected from anisotropic alignments caused by syntectonic recrystallization of olivine in a zone of horizontal shear. The polarity of the pattern of anomalies, found in waves crossing the Pacific Basin, suggests that the lithosphere is dragging the asthenosphere.     

Extension of the Thomson-Haskell method to non-homogeneous spherical layers

Summary. Amplitude spectra of Rayleigh and Love waves in a layered non-gravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson—Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20–300–s period and expressed in polynomials in the period.     

Guided waves in three-dimensional structures

A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal field: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in different directions in the horizontal plane. The influence of 3-D structure is included by allowing coupling between different modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial differential equations for the weight functions for different modal orders.
The representation of the guided wavefield requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as different polarization states of the modal spectrum. The coupling equations describe the interaction between the different polarizations induced by the presence of the 3-D structure.
The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or reflection of the surface wavefield can be included by allowing for transfer between modes travelling in different directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation.     

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.     

The upper mantle structure of the North Sea region from Rayleigh wave dispersion

Summary. The phase velocity dispersion of fundamental mode Rayleigh waves (period range 13–127 s) is determined by the interstation method for three profiles that traverse the North Sea region of northwest Europe. The resulting observations have been combined to produce a regional phase velocity curve with 95 per cent confidence intervals, which belongs to the aseismic continental platform category of Knopoff.
Inversions of the regional phase velocity curve by the'Hedgehog'method indicate that the North Sea region is characterized by an upper mantle low-velocity zone of S -wave velocity 4.35–4.45 km/s between depths of approximately 85–200 km.     

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.     

Waveform inversion of mantle Love waves:the Born seismogram approach

Summary. Normal mode theory, extended to the slightly laterally heterogeneous earth by the first-order Born approximation, is applied to the waveform inversion of mantle Love wave (200–500 s) for the Earth's lateral heterogeneity at l = 2 and a spherically symmétric anelasticity ( Q _μ) structure. The data are from the Global Digital Seismograph Network (GDSN). The l =2 pattern is very similar to the results of other studies that used either different méthods, such as phase velocity measurements and multiplet location measurements, or a different data set, such as mantle Rayleigh waves from different instruments. The results are carefully analysed for variance reduction and are most naturally explained by heterogeneity in the upper 420 km. Because of the poor resolution of the data set for the deep interior, however, a fairly large heterogeneity in the transition zones, of the order of up to 3.5 per cent in shear wave velocity, is allowed. It is noteworthy that Love waves of this period range cannot constrain the structure below 420 km and thus any model presented by similar studies below this depth are likely to be constrained by Rayleigh waves (spheroidal modes) only.
The calculated modal Q values for the obtained Q _μ model fall within the error bars of the observations. The result demonstrates the discrepancy of Rayleigh wave Q and Love wave Q and indicates that care must be taken when both Rayleigh and Love wave data, including amplitude information, are inverted simultaneously.
Anomalous amplitude inversions of G2 and G3, for example, are observed for some source-receiver pairs. This is due to multipathing effects. One example near the epicentral region, which is modelled by the obtained l = 2 heterogeneity, is shown.     

Generation of complete theoretical seismograms for SH—

Summary. We report the initial results of our attempts to obtain theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram, including both body waves and surface waves, can be generated for a spherical, anelastic earth by simple inverse Fourier transformation of the sum of the propagating fundamental and higher-mode surface waves. The key to success in reproducing the WWSSN records involves the number of modes, and the minimum period used in these computations; here we use eight modes and a minimum period of 2 s. Efficient computational algorithms make it possible to handle up to 2000 frequency points for each mode; approximately 200 layers are used to model the radial heterogeneity of the earth; attenuation is treated exactly. Examples are given of the SH theoretical seismograms resulting from dislocation sources buried at various depths in the Earth.     

Transitional continental–oceanic structure beneath the Norwegian Sea from inversion of surface wave group velocity data

We have analysed the fundamental mode of Love and Rayleigh waves generated by 12 earthquakes located in the mid-Atlantic ridge and Jan Mayen fracture zone. Using the multiple filter analysis technique, we isolated the Rayleigh and Love wave group velocities for periods between 10 and 50  s. The surface wave propagation paths were divided into five groups, and average group velocities calculated for each group. The average group velocities were inverted and produced shear wave velocity models that correspond to a quasi-continental oceanic structure in the Greenland–Norwegian Sea region. Although resolution is poor at shallow depth, we obtained crustal thickness values of about 18  km in the Norwegian Sea area and 9  km in the region between Svalbard and Iceland. The abnormally thick crust in the Norwegian Sea area is ascribed to magmatic underplating and the thermal blanketing effect of sedimentary layers. Maximum crustal shear velocities vary between 3.5 and 3.9  km  s⁻¹ for most paths. An average lithospheric thickness of 60  km was observed, which is lower than expected for oceanic-type structure of similar age. We also observed low shear wave velocities in the lower crust and upper mantle. We suggest that high heat flow extending to depths of about 30  km beneath the surface can account for the thin lithosphere and observed low velocities. Anisotropy coefficients of 1–5 per cent in the shallow layers and >7 per cent in the upper mantle point to the existence of polarization anisotropy in the region.     

Rayleigh Wave Spectra and Group Velocity Minima, and the resonance of P waves in layered structures

Summary. A connection is established between the group velocity of Rayleigh waves, the spectral amplitudes of surface waves generated by a source, and the resonance of vertically travelling P waves. It implies that a minimum in a group velocity curve is reflected in the spectral amplitudes as a maximum. That this is so, appears to have been first noticed by Longuet-Higgins in a study of microseisms. Also when a sharp impedance contrast occurs in a plane-layered model of the crust, the group velocity minimum in the fundamental mode occurs close to a period equal to four times the travel time of P -waves from the surface to the interface. More than one such contrast gives rise in general to more than one minimum. Similar relations hold for the higher modes.     

On high-frequency spheroidal modes and the structure of the upper mantle

Summary. The asymptotic properties of spheroidal mode dispersion at high frequency for fixed phase velocity are related to the intercept times τ_β( p ) for P and S waves. If the mode eigenfrequency and the ratio of horizontal to vertical displacement at the surface for the mode are known τ_α( p ) and τ_β( p ) may be separately estimated. If discontinuities exist in the velocity model then 'solotone' effects occur, in frequency at fixed slowness, and in τ_α( p ), τ_β( p ) estimated from the mode dispersion as a function of slowness. The coupling of P and S waves in the spheroidal modes means that the interaction of P waves with upper-mantle discontinuities affects also the estimates of the S wave τ_β( p ) values for which the corresponding turning points lie in the lower mantle. The asymptotic formalism also shows that sharp pulses formed by superposition of spheroidal modes correspond to multiple PS reflections.
A study of τ_α( p ), τ_β( p ) estimates derived from spheroidal modes with periods from 45–50s, calculated for model 1066B, shows that even in the presence of strong upper-mantle discontinuities the errors in intercept time are only about one-tenth of a period. The asymptotic properties may there-for provide a useful means of estimating intercept times from modes with a few seconds period as a supplement to travel-time methods.     

Some remarks about the degree-one deformation of the Earth

The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the M_o mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity.
The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale).     

A parabolic approximation for surface waves

Summary. A parabolic approximation to the equation of motion of elastic waves as a sum of surface modes and discovering a parabolic approximation be applied directly to surface waves. The approximation depends on the material properties varying slowly within a wavelength, whereas surface waves may travel in a surface wave guide whose depth is of the same order of magnitude as a wavelength. This difficulty is overcome by representing the waves as a sum of surface modes and discovering a parabolic approximation for the amplitudes as a function of position on the surface. The theory is applicable to the propagation of Love or Rayleigh waves in a structure which is vertically stratified in an arbitrary way, but varies slowly in any horizontal direction.     

In search of an elusive Antarctic circumpolar wave in sea ice extents: 1978–1996

For ease in discerning an Antarctic circumpolar wave in the perimeter of the ice pack, we construct a time series of the sea ice extents (essentially the area within the ice perimeter) in 1-degree longitudinal sectors for the period 1978–1996, as observed with the multichannel microwave imagers on board the NASA Nimbus 7 and the DOD (Dept. of Defense) DMSP (Defense Meteorological Satellite Program) F8. F11, and F13 satellites. After converting the time series into complex numbers by means of a Hilbert transform, we decompose the time series of the 360 sectors into its complex principal components (CPCs), effectively separating the spatial and temporal values. Then we decompose the real and imaginary parts of the temporal portions of the first three CPCs (complex principal compenents) by Empirical Mode Decomposition into their intrinsic modes, each representing a narrow frequency band, resulting in a collection of three CPCs for each intrinsic mode. Finally, we reconstruct the data in two different ways. First, we low-pass filter the data by combining all of the intrinsic modes of each CPC with periods longer than two years, which we designate as low-pass filtered. Next, we select the intrinsic mode of each CPC with periods of approximately four years, which we designate the quasiquadrennial (QQ) modes. The low-pass filtered time series shows eastward propagating azimuthal motion in the Ross and Weddell Seas, but no clearly circumpolar motion. The QQ time series, on the other hand, clearly shows castward propagating circumpolar waves, but with occasional retrograde motion to the west.     

Surface-wave study of the Labrador Sea earthquake 1971 December

Source parameters of an earthquake that occurred on the continental slope off the coast of Labrador, the first earthquake known to have occurred in this region, are determined by using body-and surface-wave data. The focal mechanism is of the dip slip type and the focal depth is 16 km. The local deviatric stress field applied to the region is estimated to be compressional oriented normal to the North American continent. The observed surface-wave spectra are characterized by a sharp fall-off near the period T = 15–20 s. This observed fall-off is attributed to thick sediment at the continental slope. Consequently, when an earthquake is not located within a continental area, the source parameter determination using short-period surface waves is restricted in application.     

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.