Born scattering of longperiod body waves

The Born approximation is applied to the modelling of the propagation of deeply turning longperiod body waves through heterogeneities in the lowermost mantle. We use an exact Green's function for a spherically symmetric earth model that also satisfies the appropriate boundary conditions at internal boundaries and the surface of the earth. The scattered displacement field is obtained by a numerical quadrature of the product of the Green's function, the exciting wavefield and structural perturbations. We study three examples: scattering of longperiod P waves from a plume rising from the coremantle boundary (CMB), generation of longperiod precursors to PKIKP by strong, localized scatterers at the CMB, and propagation of corediffracted P waves through largescale heterogeneities in D". The main results are as follows: (1) the signals scattered from a realistic plume are small with relative amplitudes of less than 2 per cent at a period of 20 s, rendering plume detection a fairly difficult task; (2) strong heterogeneities at the CMB of appropriate size may produce observable longperiod precursors to PKIKP in spite of the presence of a diffraction from the PKP B caustic; (3) corediffracted P  waves ( P _diff) are sensitive to structure in D" far off the geometrical ray path and also far beyond the entry and exit points of the ray into and out of D"; sensitivity kernels exhibit ringshaped patterns of alternating sign reminiscent of Fresnel zones; (4) P _diff also shows a nonnegligible sensitivity to shear wave velocity in D"; (5) down to periods of 40 s, the Born approximation is sufficiently accurate to allow waveform modelling of P _diff through largescale heterogeneities in D" of up to 5 per cent.     

Inference of core-mantle boundary topography from ISC PcP and PKP traveltimes

We invert ISC PcP and PKP absolute and differential traveltimes in an attempt to infer the long-wavelength topography of the core-mantle boundary (CMB). The data selection and processing methods are described and evaluated. These travel-time data are very noisy and the geographic distribution of the data is highly non-uniform, inhibiting reliable inference of CMB topography. Spatial averaging enhances the coherent component of the residual variance (related to heterogeneity), however, the random component of the variance is much larger than the coherent component. We show that for PcP data the coherent signal due to mantle heterogeneity overshadows that arising from the CMB, and that the effects of mantle heterogeneity are mapped into our inferred CMB solutions. The PcP data are not correlated across the spatial averaging bins and seem to have a strong bias due to small-scale structure and/or noise. The non-uniform geographic sampling of the data plays a role in the mapping of mantle heterogeneity onto the CMB. Spatial patterns of CMB models inferred from different phases do not agree. Amplitudes of seismically inferred CMB undulations vary greatly. The sensitivity of inferred CMB models to the processing, spatial averaging procedure, and inversion techniques are investigated. Topographic amplitudes increase strongly with increasing input residual variance. The power spectrum of inferred topography indicates that there are unmodelled heterogeneities that must be described with spherical harmonics of degree 6 and higher. Based on this work, we conclude that reliable inference of long-wavelength CMB topography is not likely with the current ISC data set or with a spherical harmonic expansion truncated to degree and order 6.     

Caustics in seismology

Summary. Catastrophe optics provides insights into the structure of the caustics produced by seismic events. Two examples are given. The first is the caustic formed on the surface of the Earth between 10° and 30° from the event by P rays that have reached the transition zone of the mantle. This is organized by one or more approximately circular cusp lines; these are generally below the surface, but because of lateral (regional) variations, they are puckered. Where they intersect the surface they give rise to characteristic beak-to-beak and lips patterns. The second example is the caustic structure produced by refraction at the core–mantle boundary. There is a four-cusped figure at the antipodal point, and, in addition, it is suggested that the main 143° PKP caustic is actually cusped, the cusps being smeared out when radially symmetric earth models are used. The cusps arise from bumps on the core–mantle boundary. In general, the caustic, and its accompanying diffraction structure, associated with a bump can be understood as an unfolding of the parabolic umbilic catastrophe.     

Electromagnetic core–mantle coupling—I. Explaining decadal changes in the length of day

Measured changes in the Earth's length of day on a decadal timescale are usually attributed to the exchange of angular momentum between the solid mantle and fluid core. One of several possible mechanisms for this exchange is electromagnetic coupling between the core and a weakly conducting mantle. This mechanism is included in recent numerical models of the geodynamo. The 'advective torque', associated with the mantle toroidal field produced by flux rearrangement at the core–mantle boundary (CMB), is likely to be an important part of the torque for matching variations in length of day. This can be calculated from a model of the fluid flow at the top of the outer core; however, results have generally shown little correspondence between the observed and calculated torques. There is a formal non-uniqueness in the determination of the flow from measurements of magnetic secular variation, and unfortunately the part of the flow contributing to the torque is precisely that which is not constrained by the data. Thus, the forward modelling approach is unlikely to be useful. Instead, we solve an inverse problem: assuming that mantle conductivity is concentrated in a thin layer at the CMB (perhaps D"), we seek flows that both explain the observed secular variation and generate the observed changes in length of day. We obtain flows that satisfy both constraints and are also almost steady and almost geostrophic, and therefore assert that electromagnetic coupling is capable of explaining the observed changes in length of day.     

Heterogeneous structure around the Jemez volcanic field, New Mexico, USA, as inferred from the envelope inversion of active-experiment seismic data

We analyse active-experiment seismic data obtained by the 1993 Jemez Tomography Experiment (JTEX) programme to elucidate the heterogeneous structure of the Jemez volcanic field, which is located at the boundary between the Colorado Plateau and the Rio Grande Rift. Using a single isotropic scattering assumption, we first calculate the envelope Green's functions for the upper and lower crust and the uppermost mantle. By fitting the theoretical envelopes with the observed three-component data, we estimate depth-dependent features of the scattering coefficients around Valles Caldera. We estimate the ratios of scattering coefficients, rather than scattering coefficients themselves, because of the uncertainty of the seismic efficiency of the explosive sources and knowledge of absolute site-amplification factors. The strongest scattering coefficients are observed at a shallow depth beneath the Valles Caldera. This is considered to be related to the complex structure caused by two episodes of caldera formation and the ensuing resurgent uplift in the caldera, etc. The depth-dependent characters of the scattering coefficients for the Colorado Plateau and the Rio Grande Rift are similar to each other: a transparent upper crust and a heterogeneous lower crust (small and large scattering coefficients, respectively). However, the scattering coefficients beneath the Rio Grande Rift are several times larger than those beneath the Colorado Plateau. Depths of the lower crust and the Moho boundary beneath the Rio Grande Rift are shallower than those of the Colorado Plateau. From their geological settings and other geophysical results around the region, we infer that the larger scattering coefficients of the rift are associated with rift formation and volcanic activity, such as magma ascent from the upper mantle to the crust.     

Seismic Modelling of Lateral Heterogeneity At the Base of the Mantle

Summary. Results from several recent studies suggest that there are lateral heterogeneities of up to a few per cent in the lowermost 150–200 km of the mantle (Bullen's D " region). Inferred anomaly sizes span the range from less than 50 km to greater than 1000 km.
In this study differences in the velocity structure among regions at the base of the mantle were inferred from an analysis of amplitude ratios of PKPAB and PKPDF for given earthquake-station pairs at distances greater than 155° (Sacks, Snoke & Beach). We distinguish two kinds of regions: A (anomalous) regions in which the mean, median and spread in AB/DF amplitude ratios are significantly higher (> 50 per cent) than for a reference radial earth model and N (normal) regions in which the distribution of the amplitude ratios is as expected.
The AB branch has near-grazing incidence to the core and therefore maximum sensitivity to velocity structure compared to the near-normal incident DF phases. Using an iterative, forward-modelling approach, we have determined general characteristics of the velocity structure for regions at the base of the mantle which can produce amplitude-ratio distributions similar to those for an A region. Agreement between model and data is obtained over the period range from 0.5 s to greater than 10 s using a laterally heterogeneous model for the D " region. the model consists of cells which are 200 km in lateral extent with velocity variations of up to ±1 per cent. This structure is modulated by a region-wide (1000km) perturbation which increases smoothly from zero at the edges of the region to a negative 1 per cent at the centre. Small cells (∼40 km) cannot produce anomalously large amplitude, long-period AB arrivals, and larger cells (∼1000km) cannot match the observed scatter. the ∼200 km scale anomalies could be small-scale convection cells confined to the D " region.     

Elastic scattered waves from a continuous and heterogeneous layer

Elastic scattering from a continuous and laterally unbounded heterogeneous layer has been formulated using the Born approximation. A general solution of the scattered wave equation for the above-stated medium has been given in terms of a Fourier integral over plane waves. Far-field asymptotic expressions for weak elastic scattering by a finite, continuous and inhomogeneous layer have been presented which agree with earlier results. For perturbations of the two elastic parameters and the density having the same form of spatial variation, the spectrum of plane waves scattered from a heterogeneous layer is expressed as a product of an 'elastic scattering factor'and a 'distribution factor'. As in earlier results for small-scale heterogeneity, the scattering pattern depends on various combinations of perturbations of elastic parameters and density. In order to show the general characteristics of the elastic wave scattering, some scattering patterns have been given.     

Whole mantle SH velocity model constrained by waveform inversion based on three-dimensional Born kernels

A whole mantle SH velocity model is obtained by using a unique data set and techniques. Body and surface waveforms including major and multi-orbit phases are used as a data set and are inverted by using 3-D Born kernels. The resultant model, SH18CE, reveals the different natures of the two major upwelling systems: the strong low velocity anomalies beneath Africa extend for more than 1000 km from the core–mantle boundary (CMB), whereas those beneath the Pacific are restricted to 300–400 km from the CMB. The results also show the variable natures of stagnant slabs on the 670 discontinuity around Japan: the depths of the strongest high velocity anomalies within the stagnant slabs are different region by region, which is consistent with the detailed delay time tomography model in this area.     

Core precession: flow structures and energy

Experiments using a precessing liquid-filled oblate spheroid with ellipticity ( a − b )/ a =1/400 extend and clarify earlier research. They yield flow data useful for estimating flows in the Earth's liquid core. Observed flows illustrate and confirm a nearly rigid liquid sphere with retrograde drift and lagging a cavity (mantle) axis in precession. The similarities of the observed lag angle with that computed for a rigid sphere, and earlier energy dissipation research both support the use of a rigid sphere analytical model to predict energy dissipation and first-order flow within the core–mantle boundary (CMB). Second-order boundary layer and interior cylindrical flow structures also are photographed and measured. Interior flows are never turbulent or unstable at near-Earth parameters, although complex and transient flow patterns are observed within the boundary layer. Other mechanisms proposed to explain net heat loss from the Earth and maintenance of the geodynamo typically require acceptance of some critical but unproven premise. Precession and CMB configuration are known with certainty and precision. Analytical difficulties have been the obstacle. Experiments illustrate the consequences of precession and ellipticity, provide criteria for validating analytical and numerical models, and may yield direct knowledge of the Earth's deep interior with careful scaling.     

Seismic evidence for a sharp lithospheric base persisting to the lowermost mantle beneath the Caribbean

Broad-band data from South American earthquakes recorded by Californian seismic networks are analysed using a newly developed seismic wave migration method—the slowness backazimuth weighted migration (SBWM). Using the SBWM, out-of-plane seismic P -wave reflections have been observed. The reflection locations extend throughout the Earth's lower mantle, down to the core–mantle boundary (CMB) and coincide with the edges of tomographically mapped high seismic velocities. Modelling using synthetic seismograms suggests that a narrow (10–15 km) low- or high-velocity lamella with about 2 per cent velocity contrast can reproduce the observed reflected waveforms, but other explanations may exist. Considering the reflection locations and synthetic modelling, the observed out-of-plane energy is well explained by underside reflections off a sharp reflector at the base of the subducted lithosphere. We also detect weaker reflections corresponding to the tomographically mapped top of the slab, which may arise from the boundary between the Nazca plate and the overlying former basaltic oceanic crust. The joint interpretation of the waveform modelling and geodynamic considerations indicate mass flux of the former oceanic lithosphere and basaltic crust across the 660 km discontinuity, linking processes and structure at the top and bottom of the Earth's mantle, supporting the idea of whole mantle convection.     

Teleseismic waveform modelling with a one-way wave equation

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

New seismological constraints on differential rotation of the inner core from Novaya Zemlya events recorded at DRV, Antarctica

Novaya Zemlya nuclear test records at the seismic station DRV, Antarctica, are analysed in order to obtain further constraints on a possible differential rotation of the inner core with respect to the mantle. These data allow the sampling of the inner core along a nearly polar path in very stable conditions over more than two decades, from 1966 to 1990. The PKP (BC)– PKP (DF) traveltime residuals, which reflect the inner-core anisotropy and/or heterogeneities sampled along the path, exhibit a great stability through time. A computation of the residuals that are expected for various differential rotation rates and the same rotation axis as the mantle has been performed using the worldwide residual catalogue of Engdahl et al . (1997) for summary rays that include the time as an additional parameter in data stacking. Comparison of data and predictions shows that an eastward differential rotation with a rate as large as 3°  yr⁻¹, as suggested by some authors, is not possible, but an eastward rotation at 1°  yr⁻¹ or lower cannot be rejected.     

A model of time-periodic mantle flow

Summary. The instability of a layer consisting of a lighter viscous fluid on top of a heavier less viscous fluid is considered in the case when the heavy fluid is adiabatically stratified and the light fluid contains heat sources and possesses a lower heat conductivity. A perturbation in the thickness of the upper fluid layer causes horizontal temperature variations in the lower fluid. The motions induced by thermal buoyancy can interact with the distortion of the interface in such a way that the initial perturbation is reinforced in the form of an overstable oscillation. It is proposed that this mechanism is relevant to the problem of time-dependent flow in the Earth's mantle.     

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).     

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

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

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

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

Anisotropy of heterogeneity scale lengths in the lower mantle from PKIKP precursors

The anisotropy of heterogeneity scale lengths in the lower mantle is investigated by modelling its effect on the high-frequency precursors of PKIKP scattered by the heterogeneities. Although models having either an isotropic or an anisotropic distribution of scale lengths can fit the observed coda shapes of short-period precursors, the frequency content of broad-band PKIKP precursors favours a dominantly isotropic distribution of scale lengths. Precursor coda shapes are consistent with 1 per cent fluctuations in P velocity in the wavenumber band 0.05–0.5  km⁻¹ extending to 1000  km above the core–mantle boundary, and with a D" region open to circulation throughout the lower mantle. The level of excitation of PKIKP precursors observed in the frequency band 0.02–2  Hz requires a power spectrum of heterogeneity that is nearly white or slowly increasing with wavenumber. Anisotropy of scale lengths may exist in a D" layer having larger horizontal than vertical scale lengths and produce little or no detectable effects on PKIKP precursors for P -velocity perturbations as high as 3 per cent when averaged over a vertical scale of several kilometres, and much higher when averaged over scales of hundreds of metres or less.     

Geomagnetic evidence for fluid upwelling at the core-mantle boundary

Summary. Previous studies, both geomagnetic and seismic, have been unable to show conclusively whether or not there is fluid upwelling at the core-mantle boundary. Here a new method is developed, in which an attempt is made to invert geomagnetic secular variation data measured at the Earth's surface for a frozen-flux purely toroidal core-mantle boundary (CMB) velocity field, under the assumption that the mantle is electrically insulating and flux is frozen in at the CMB. These data have previously been inverted for the core-mantle boundary radial secular variation, from which the appropriate fit between model and data is known. Two different main field models were used to assess the effect of uncertainty in its radial component at the CMB. The conclusions were the same in both cases: frozen-flux purely toroidal motions provide a poor fit. A statistical test allows very firm rejection of the hypothesis that the residuals are not significantly larger, whereas there is no statistical difference between the residuals of inversions for radial secular variation and frozen-flux velocity fields at the CMB if upwelling and down-welling is included. The inherent non-uniqueness in the velocity field obtained is not of concern, since only their statistical properties are utilized and no physical significance is attached to the flows obtained.     

t*— an unsuitable parameter to characterize anelastic attenuation in the Eastern Carpathians

Investigation of teleseismic P -wave recordings at a temporary network in the Eastern Carpathians, equipped with predominantly short-period sensors, is compared with synthetic modelling of anelastic attenuation of teleseismic waves in the upper mantle. Using the t * approach, we examine variations of amplitude decrease over frequency for teleseismic recordings in the frequency band 0.5–1.5 Hz. The results reveal a consistent pattern of increased t * values in the centre of the network, in the Vrancea region at the bend of the Carpathian Arc, although the magnitude of the observed variation in t * is much higher than expected. Synthetic t * parameter computations for the same event-receiver configurations reproduce the observed pattern in terms of relative variations. However, the amplitude of the synthetic t * values explains only 10–20 per cent of the observed variation in t *. t * is not a direct measure for anelastic attenuation but rather for a combination of anelastic and other attenuating effects such as scattering and amplitude fluctuation related to velocity inhomogeneities. If regional amplitude variations are solely attributed to anelastic attenuation, all other effects are mapped into Q . We discuss the role of anelastic attenuation and other effects in the case of the Eastern Carpathians and conclude that t * is an unsuitable parameter to characterize anelastic attenuation in the Eastern Carpathians.     

Postglacial rebound and lateral viscosity variations: a semi-analytical approach based on a spherical model with Maxwell rheology

A spectral method is employed to study the response to surface loads of a Maxwell earth including lateral viscosity variations. In particular, we focus on the effects of lithospheric cratons on the long-wavelength time-dependent displacement field for simple earth models. The viscosity contrast of the craton with respect to the surrounding mantle is kept fixed, whereas its thickness is allowed to vary. We show that the long-wavelength vertical displacement is not greatly affected by the presence of a lithospheric craton, while the tangential displacement is severely modified for the case of a homogeneous mantle. With increasing harmonic degree and thickness of the craton, the load-deformation coefficients deviate from those pertaining to a homogeneous mantle with a viscosity of 10²¹ Pa s. These deviations are particularly enhanced on timescales larger than a few hundred years. These findings indicate that the interpretation of the viscosity structure of the mantle inferred from postglacial rebound signatures based on radially stratified models is affected by the presence of lateral viscosity variations.