ALBERT

Notes: We investigate surface wave propagation effects caused by P and S-wave velocity anisotropy and its orientation in the upper mantle. We compare coupled-mode synthetic seismograms, constructed from sums of free oscillations for different types of upper mantle models, in order to understand better the observable aspects of anisotropy and its fast direction in the earth. Our calculations suggest that upper mantle anisotropy, which is not transversely isotropic, of few per cent can generate significant waveform anomalies in long-period seismic records of 0–20 mHz. Those anomalies are termed ‘quasi-Love’ waves and are caused by spheroidal—toroidal free-oscillation coupling. Large quasi-Love waveforms are difficult to produce with levels of isotropic heterogeneity consistent with global tomographic models of the mantle. Coupled-mode waveform anomalies in long-period seismic records at a single station can be used to diagnose the presence of anisotropic structure in the upper mantle. Because 0Sl, —0Tl coupling pairs have weak sensitivity to aspherical wavenumbers s 〈l—l, and |l — increases with frequency for f 〉 4 mHz, smoother/rougher structure generates longer/shorter-period fundamental-branch waveform anomalies. Anisotropy in our models is assumed to have hexagonal symmetry, that is, a single axis of symmetry. However, the symmetry axis can be arbitrarily oriented. The waveform anomalies associated with spheroidal-toroidal coupling are sensitive to the orientation of the fast-velocity direction of upper mantle anisotropy. In particular, for the same level of velocity perturbation, coupled-mode waveform anomalies increase as the fast axis of anisotropic structure rotates from vertical to horizontal. The orientation of the fast axis can influence both phase shifts and coupled-mode waveforms of long-period surface waves. The coupled-mode waveform anomalies predicted by our theoretical calculations are observed in the data and seem to diagnose the presence of upper mantle anisotropy which is not transversely isotropic.

Notes: We have developed an expression for the first-order Born waveform perturbation for long-period seismic records on an aspherical earth, using an anelastic monopole reference model. We have derived both an explicitly anelastic first-order Born term u1(r, t) and an approximate, but much simpler, expression in which quasi-degenerate modal coupling is represented by secular terms in the time domain. We designate this expression the ‘strong’ Born approximation to identify the presence of ‘strong’ quasi-degenerate coupling. Earlier applications of Born theory to surface wave seismograms have typically neglected the coupling between different modal dispersion branches. The strong Born approximation is linear in the aspherical model, but the secular terms restrict its accuracy to ‘short’ times after event onset, typically fewer than 10 hr. Additional accuracy can be gained by treating the self-coupling of isolated multiplets explicitly by formally summing the Born series using a projection of the aspherical operator L1 onto the multiplet in question. We call this the ‘stronger’ Born approximation, which is a non-linear functional of the aspherical model. The stronger Born approximation is similar to, but formally less accurate than, the subspace projection algorithm. We verified the accuracy of synthetic seismograms calculated with the ‘strong’ and ‘stronger’ Born approximations against Galerkin coupled-mode synthetics using a zonally symmetric, weakly anisotropic upper mantle model. The Born seismograms replicate successfully the ‘quasi-Love’ waveforms that diagnose spheroidal-toroidal coupling, suggesting that the strong Born approximation is adequate to model the interaction of free-oscillation coupling partners that are closely spaced in the frequency domain, at least for short records. We performed a waveform inversion test using three-component synthetic seismograms from 297 source-receiver pairs. Once spheroidal—toroidal coupling is incorporated in the inverse formulation, the waveform perturbations associated with the anisotropic parameters are clearly distinct from waveform perturbations associated with the isotropic parameters. Although the model space used in this experiment is highly restricted compared to the real Earth, these results suggest that the trade off between anisotropy and isotropic lateral structure may be less problematic in a fully-coupled waveform inversion than in a tomographic inversion of surface wave phase delays.

Notes: We present an algorithm, based on the wavelet transform and multiple taper spectral analysis, for providing a low-variance spectrum estimate of a non-stationary data process. The‘multiwavelet’algorithm uses, within each frequency band, a number of mutually orthogonal Slepian wavelets, optimally concentrated in frequency. The sum of squared wavelet transforms with the Slepian wavelets results in a spectrum estimate that is both low-variance and resistant to broad-band bias. The multiwavelet algorithm is used to estimate the time-varying spectral density matrix S(f, t) for two or more time series, in particular for three-component seismic data. Coherent three-component motion is described by motion along a single trajectory, with appropriate projections onto the three component axes. This trajectory is found by applying a singular value decomposition (SVD) to a matrix M(f, t) of wavelet transform values. The normalized first singular value of the SVD determines whether a correlation among the three components of the seismogram is statistically significant. Where significant, coherent particle motion is reconstructed by a linear combination of the wavelets with coefficients specified by the first left-singular vector. The polarization of this motion with respect to the coordinate axes is given by the first right-singular vector. Where the wavelets are real-valued, the usefulness of this method is limited to cases in which the three components of the seismic record oscillate in phase with each other, as is often the case for seismic body waves. Elliptical polarization is handled by pairing even and odd Slepian wavelets into complex-valued wavelets, capable of detecting phase shifts between components. We demonstrate the mutiwavelet spectrum and polarization estimators on seismic data from a large shallow earthquake in the Solomon Islands, and from the recent deep earthquakes beneath Fiji (1994 March 9) and Bolivia (1994 June 9).

Notes: We report a method to calculate surface waves in layered anisotropic media using a reflectivity algorithm. The method posits a finite stack of constant anisotropic layers over an isotropic half-space. The parametrization within each layer allows for anisotropy with an arbitrarily oriented axis of symmetry ŵ, with magnitude scaled by three constants related to wave-speed variations of P and S waves with respect to the azimuth relative to ŵ. Within each layer, eigensolutions to the equations of motion can be expressed as upgoing and downgoing oscillatory and/or evanescent plane waves, except possibly within narrow intervals of horizontal slowness p where the governing matrix becomes near-defective and an alternative set of eigensolutions is used to avoid numerical instability. Synthetic seismograms for simple anisotropic crustal models demonstrate that significant Love–Rayleigh coupling occurs between adjacent overtone branches, leading to significant ‘scattering’ in the absence of 3-D velocity structure. The dependence of this scattering on azimuth relative to ŵ can cause the Love-Rayleigh scattering to appear to be related to a spurious component of the source mechanism.

Notes: We have explored the long-period seismic response of several idealized isotropic and anisotropic earth models to determine the observable aspects of anisotropy in the Earth's upper mantle. The models chosen are zonally symmetric (i.e. constant with respect to longitude), with 3.0 per cent and ˜1.5 per cent peak-to-peak perturbations to shear and compressional velocities, respectively, to a depth of 216 km. Our calculations suggest that upper mantle azimuthal anisotropy of a few per cent, spatially coherent over the length scale of a typical ocean basin, is sufficient to generate significant waveform anomalies in long-period seismic records. These anomalies are termed ‘quasi-Love’ and ‘quasi-Rayleigh’ waves, and are best observed for shallow events near the Rayleigh and Love source-radiation minima, respectively. For three large, recent, shallow, predominantly strike-slip events, seismic observations at stations with auspicious source-receiver geometry show evidence for 4-7 mHz waveforms similar to those seen in synthetic experiments for ‘smooth’ anisotropic mantle structure. We infer the presence of smoothly varying upper mantle anisotropy after determining that other causative factors are less plausible. The waveforms observed could be explained by isotropic structures, but the velocity perturbations would need to be 〉20 per cent to explain the observations in the 4-7 mHz bandpass. ‘Rough’ anisotropic structure is also a possibility, but our calculations suggest that regional-scale anisotropy of a few per cent, if globally pervasive rather than confined to some regions (e.g. plate collision zones), would cause strong mixed-type coupling at frequencies f 〉 10 mHz. In the data we have analysed, the quasi-Love waveforms appear to be bandlimited to f≲ 10 - 15 mHz, depending on the source-receiver pair. Angular selection criteria for coupling between fundamental spheroidal and toroidal modes suggest that this frequency dependence is consistent with the presence of upper mantle anisotropy with angular degrees s≲ 10-15 and peak amplitude of a few per cent. However, the restriction of quasi-Love waveforms to Rayleigh wave minima restricts our resolution, for three predominantly strike-slip earthquakes, to six great circles, not one of which is free from apparent waveform anomalies.

Notes: We investigate normal mode multiplet coupling along the fundamental spheroidal mode branch using three different numerical methods. A suite of 300 synthetic vertical accelerograms computed using the great-circle approximation and the first-order subspace-projection algorithm are compared with accurate accelerograms computed using a Rayleigh-Ritz variational method. In each case, the accelerograms are computed by summing the responses of the hybrid multiplets 0S10–10S55; the resulting waveforms correspond to mantle Rayleigh waves with periods between 150 and 600 s. The great-circle approximation correctly represents all but 5 per cent of the waveform variance produced by the degree-8 model of upper mantle heterogeneity M84A, for each of the sequentially arriving wavegroups R1-R6. For a contrived model with a modest amount of additional heterogeneity up to degree 20, the relative great-circle error is substantially greater, approximately 30 per cent. The first-order subspace-projection algorithm requires an order of magnitude more computer time than the great-circle approximation; however it is much more accurate. For the first arriving wavegroup R1, it correctly represents all but 2 per cent of the waveform variance produced by model M84A; this relative error decreases for subsequent wavegroups to less than 1 per cent for R2 and less than 0.2 per cent for R3. For the contrived degree-20 model, the relative subspace-projection error is 6 per cent for R1, 1.5 per cent for R2 and 0.9 per cent for R3; the relative error is less for the later arriving wavegroups because of the increasing effect of the lateral heterogeneity as well as the more rapid attenuation of the higher frequency multiplets that are not as well modelled by the approximation.