From ACH tomographic models to absolute velocity models

The ACH method, a widely used tomographic inverse method, is characterized by the use of relative residuals in order to avoid possible biases coming from outside the target volume. The ACH method thus does not really retrieve the 3-D structure of the target volume, but instead leads to velocity contrasts relative to the layer average of the velocity, this average value remaining unknown ( Aki et al. 1977 ). Two artefacts derive from this particularity: (1) velocity contrasts are known only in the horizontal direction and it is not possible, in a strict mathematical sense, to estimate the contrasts in the vertical direction with ACH alone; (2) negative anomalies are often interpreted as low velocities, whereas negative anomalies may correspond to high velocities if the average value of the corresponding layer is sufficiently high. The converse is true of positive anomalies. We show with synthetic data how these artefacts can affect the interpretation of tomographic images. We propose to correct the artefacts by reintroducing the 1-D regional average model, and show in synthetic experiments how effective this correction can be.
  The application of this procedure to data recorded in the Kunlun region shows that the retrieval of the absolute values of the 3-D velocity model is helpful for interpreting the tomographic images and better defining which features are anomalous.     

Imaging mantle transition zone thickness with SdS-SS finite-frequency sensitivity kernels

We invert differential SdS-SS traveltime residuals measured from stacked waveforms and finite-frequency sensitivity kernels for topography on the 410- and 660-km discontinuities. This approach yields higher resolution images of transition zone thickness than previous stacking methods, which simply average/smooth over topographic features. Apparent structure measured using simple stacking is highly dependent upon the bin size of each stack. By inverting for discontinuity topography with a variety of bin sizes, we can more accurately calculate the true structure. The inverted transition zone model is similar to simple stack models with an average thickness of 242 km, but the lateral variations in thickness are larger in amplitude and smaller in scale. Fast seismic velocities in 3-D mantle models such as SB4L18 correlate with areas of thicker transition zone. The elongated curvilinear regions of thickened transition zone that occur near subduction zones are narrow and high amplitude, which suggests relatively little lateral spreading and warming of subducted lithosphere within the transition zone. The anomalously thin transition zone regions are laterally narrow, and not broadly continuous. If these variations in transition zone thickness are interpreted as thermal in nature, then this model suggests significant temperature variations on small lateral scales.     

An alternative method for calculation of Rayleigh and Love wave phase velocities by using three-component records on a single circular array without a central station

A method for the computation of phase velocities of surface waves from microtremor waveforms is shown. The technique starts from simultaneous three-component records obtained in a circular array without a central station. Then, Fourier spectra of vertical, radial and tangential components of motion are calculated for each station and considered as complex-valuated functions of the azimuthal coordinate. A couple of intermediate real physical quantities, B and C , can be defined from the 0- and ±1-order coefficients of the Fourier series expansion of such functions. Finally, phase velocities of Rayleigh and Love waves can be retrieved from B and C by solving respective one-unknown equations. The basic assumption is the possibility of expanding the wavefield as a sum of plane surface waves with Rayleigh and Love wavenumbers being univocal functions of the circular frequency. The method is tested in synthetic ambient noise wavefields confirming its reliability and robustness for passive seismic surveying.     

Simultaneous estimation of terrestrial eigenvibrations

Summary. Recently, time domain methods have been shown to be advantageous in determining the quality of observations of terrestrial eigenvibrations. Complex demodulation has been used in estimating amplitude, period, decay constant Q and phase. Non-linear regression in the frequency domain yields formal uncertainties in each estimate. A crucial problem arises in isolating neighbouring modes of nearly the same frequency and comparable amplitude. Complex demodulation displays can be used to identify interferences such as beating. In such cases, the assumption of stationarity is violated in the regression analysis yielding biased results. Improved resolution has now been obtained by formulating the regression scheme to estimate simultaneously small sets of neighbouring spectral peaks so that stationarity is not violated. Under the assumption of stationarity this procedure becomes optimal in a statistical sense. A general analysis of variance serves to indicate the amount of correlation that exists between estimated parameters of neighbouring modes as well as the relative information that is supplied by each Fourier transform point to the overall system of equations. The covariance matrix indicates that there is no linear correlation between spectral peaks when their frequencies are separated by 5 per cent. There are, however, significant correlations when two frequencies differ by 0.5 per cent. The result is that overtone modes very near in frequency to fundamental modes can, under certain conditions, be resolved with a non-linear regression technique, although parameter uncertainties are underestimated in general.     

Thickness estimates of the upper-mantle transition zone from bootstrapped velocity spectrum stacks of receiver functions

We modify the receiver-functions stacking technique known as velocity spectrum stacking (VSS) so as to estimate combinations of velocity model ( V_P and V_S ) and depth that stack the Ps conversion from upper-mantle discontinuities most coherently. We find that by estimating the differences in the depths to the 660 and 410 km discontinuities using velocities that maximize the stacked amplitudes of P410s and P660s phases we can estimate the thickness of the transition zone more accurately than the depths to either of these discontinuities. We present two examples indicating that the transition zone beneath Obninsk, Russia, is 252±6 km thick and that beneath Pasadena, California, is only 220±6 km thick.     

The shear-wave velocity structure in the upper mantle beneath Eurasia

We develop an approach that allows us to invert for the mantle velocity structure within a finely parametrized region as a perturbation with respect to a low-resolution, global tomographic model. We implement this technique to investigate the upper-mantle structure beneath Eurasia and present a new model of shear wave velocity, parametrized laterally using spherical splines with ∼2.9° spacing in Eurasia and ∼11.5° spacing elsewhere. The model is obtained from a combined data set of surface wave phase velocities, long-period waveforms and body-wave traveltimes. We identify many features as narrow as few hundred kilometres in diameter, such as subducting slabs in eastern Eurasia and slow-velocity anomalies beneath tectonically active regions. In contrast to regional studies in which these features have been identified, our model encompasses the structure of the entire Eurasian continent. Furthermore, including mantle- and body-wave waveforms helped us constrain structures at depths larger than 250 km, which are poorly resolved in earlier models. We find that up to +9 per cent faster-than-average anomalies within the uppermost ∼200 km of the mantle beneath cratons and some orogenic regions are separated by a sharp gradient zone from deeper, +1 to +2 per cent anomalies. We speculate that this gradient zone may represent a boundary separating the lithosphere from the continental root, which might be compositionally distinct from the overlying lithosphere and remain stable either due to its compositional buoyancy or due to higher viscosity compared with the suboceanic mantle. Our regional model of anisotropy is not significantly different from the global one.     

Regional tomographic inversion of the amplitude and phase of Rayleigh waves with 2-D sensitivity kernels

In this study, we test the adequacy of 2-D sensitivity kernels for fundamental-mode Rayleigh waves based on the single-scattering (Born) approximation to account for the effects of heterogeneous structure on the wavefield in a regional surface wave study. The calculated phase and amplitude data using the 2-D sensitivity kernels are compared to phase and amplitude data obtained from seismic waveforms synthesized by the pseudo-spectral method for plane Rayleigh waves propagating through heterogeneous structure. We find that the kernels can accurately predict the perturbation of the wavefield even when the size of anomaly is larger than one wavelength. The only exception is a systematic bias in the amplitude within the anomaly itself due to a site response.
An inversion method of surface wave tomography based on the sensitivity kernels is developed and applied to synthesized data obtained from a numerical simulation modelling Rayleigh wave propagation over checkerboard structure. By comparing recovered images to input structure, we illustrate that the method can almost completely recover anomalies within an array of stations when the size of the anomalies is larger than or close to one wavelength of the surface waves. Surface wave amplitude contains important information about Earth structure and should be inverted together with phase data in surface wave tomography.     

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.     

Waveform modelling of teleseismic S, Sp, SsPmP, and shear-coupled PL waves for crust- and upper-mantle velocity structure beneath Africa

We describe a waveform modelling technique and demonstrate its application to determine the crust- and upper-mantle velocity structure beneath Africa. Our technique uses a parallelized reflectivity method to compute synthetic seismograms and fits the observed waveforms by a global optimization technique based on a Very Fast Simulated Annealing (VFSA). We match the S , Sp, SsPmP and shear-coupled PL phases in seismograms of deep (200–800 km), moderate-to-large magnitude (5.5–7.0) earthquakes recorded teleseismically at permanent broad-band seismic stations in Africa. Using our technique we produce P - and S -wave velocity models of crust and upper mantle beneath Africa. Additionally, our use of the shear-coupled PL phase, wherever observed, improves the constraints for lower crust- and upper-mantle velocity structure beneath the corresponding seismic stations. Our technique retains the advantages of receiver function methods, uses a different part of the seismogram, is sensitive to both P - and S -wave velocities directly, and obtains helpful constraints in model parameters in the vicinity of the Moho. The resulting range of crustal thicknesses beneath Africa (21–46 km) indicates that the crust is thicker in south Africa, thinner in east Africa and intermediate in north and west Africa. Crustal P - (4.7–8 km s⁻¹) and S -wave velocities (2.5–4.7  km s⁻¹) obtained in this study show that in some parts of the models, these are slower in east Africa and faster in north, west and south Africa. Anomalous crustal low-velocity zones are also observed in the models for seismic stations in the cratonic regions of north, west and south Africa. Overall, the results of our study are consistent with earlier models and regional tectonics of Africa.     

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.     

Resolution potential of surface wave phase velocity measurements at small arrays

The deployment of temporary arrays of broadband seismological stations over dedicated targets is common practice. Measurement of surface wave phase velocity across a small array and its depth-inversion gives us information about the structure below the array which is complementary to the information obtained from body-wave analysis. The question is however: what do we actually measure when the array is much smaller than the wave length, and how does the measured phase velocity relates to the real structure below the array? We quantify this relationship by performing a series of numerical simulations of surface wave propagation in 3-D structures and by measuring the apparent phase velocity across the array on the synthetics. A principal conclusion is that heterogeneities located outside the array can map in a complex way onto the phase velocities measured by the array. In order to minimize this effect, it is necessary to have a large number of events and to average measurements from events well-distributed in backazimuth. A second observation is that the period of the wave has a remarkably small influence on the lateral resolution of the measurement, which is dominantly controlled by the size of the array. We analyse if the artefacts created by heterogeneities can be mistaken for azimuthal variations caused by anisotropy. We also show that if the amplitude of the surface waves can be measured precisely enough, phase velocities can be corrected and the artefacts which occur due to reflections and diffractions in 3-D structures greatly reduced.     

Shear and compressional velocity models of the mantle from cluster analysis of long-period waveforms

We present a new technique for the efficient measurement of the traveltimes of long period body wave phases. The technique is based on the fact that all arrivals of a particular seismic phase are remarkably similar in shape for a single event. This allows the application of cross-correlation techniques that are usually used in a regional context to measure precise global differential times. The analysis is enhanced by the inclusion of a clustering algorithm that automatically clusters waveforms by their degree of similarity. This allows the algorithm to discriminate against unusual or distorted waveforms and makes for an extremely efficient measurement technique.
This technique can be applied to any seismic phase that is observed over a reasonably large distance range. Here, we present the results of applying the algorithm to the long-period channels of all data archived at the IRIS DMC from 1976 to 2005 for the seismic phases S and P (from 23° to 100°) and SS and PP (from 50° to 170°). The resulting large data sets are inverted along with existing surface wave and updated differential traveltime measurements for new mantle models of S and P velocity. The resolution of the new model is enhanced, particularly, in the mid-mantle where SS and PP turn. We find that slow anomalies in the central Pacific and Africa extend from the core–mantle boundary to the upper mantle, but their direct connection to surface hotspots is beyond our resolution. Furthermore, we find that fast anomalies that are likely associated with subducting slabs disappear between 1700 and 2500 km, and thus are not continuous features from the upper to lower mantle despite our extensive coverage and high resolution of the mid-mantle.     

High-resolution structures of the Landers fault zone inferred from aftershock waveform data

High-frequency body waves recorded by a temporary seismic array across the surface rupture trace of the 1992 Landers, California, earthquake were used to determine fault-zone structures down to the seismogenic depth. We first developed a technique to use generalized ray theory to compute synthetic seismograms for arbitrarily oriented tabular low-velocity fault-zone models. We then generated synthetic waveform record sections of a linear array across a vertical fault zone. They show that both arrival times and waveforms of P and S waves vary systematically across the fault due to transmissions and reflections from boundaries of the low-velocity fault zone. The waveform characteristics and arrival-time patterns in the record sections allow us to locate the boundaries of the fault zone and to determine its P - and S -wave velocities independently as well as its depth extent. Therefore, the trade-off between the fault-zone width and velocities can be avoided. Applying the method to the Landers waveform data reveals a low-velocity zone with a width of 270–360 m and a 35–60 per cent reduction in P and S velocities relative to the host rock. The analysis suggests that the low-velocity zone extends to a depth of ∼7 km. The western boundary of the low-velocity zone coincides with the observed main surface rupture trace.     

Computation of Large Anisotropic Seismic Heterogeneities (CLASH)

A general tomographic technique is designed in order (i) to operate in anisotropic media; (ii) to account for the uneven seismic sampling and (iii) to handle massive data sets in a reasonable computing time. One modus operandi to compute a 3-D body wave velocity model relies on surface wave phase velocity measurements. An intermediate step, shared by other approaches, consists in translating, for each period of a given mode branch, the phase velocities integrated along ray paths into local velocity perturbations. To this end, we develop a method, which accounts for the azimuthal anisotropy in its comprehensive form. The weakly non-linear forward problem allows to use a conjugate gradient optimization. The Earth's surface is regularly discretized and the partial derivatives are assigned to the individual grid points. Possible lack of lateral resolution, due to the inescapable uneven ray path coverage, is taken into account through the a priori covariances on parameters with laterally variable correlation lengths. This method allows to efficiently separate the 2ψ and the 4ψ anisotropic effects from the isotropic perturbations. Fundamental mode and overtone phase velocity maps, derived with real Rayleigh wave data sets, are presented and compared with previous maps. The isotropic models concur well with the results of Trampert & Woodhouse. Large 4ψ heterogeneities are located in the tectonically active regions and over the continental lithospheres such as North America, Antarctica or Australia. At various periods, a significant 4ψ signature is correlated with the Hawaii hotspot track. Finally, concurring with the conclusions of Trampert & Woodhouse, our phase velocity maps show that Rayleigh wave data sets do need both 2ψ and 4ψ anisotropic terms.     

Ocean-bottom seismograph observations on the Mid-Atlantic Ridge near 45° N

Summary. Rayleigh-wave phase velocities at very long periods (185–290 s) are investigated and regionalized, taking into account the lateral heterogeneities within ocean plates revealed by earlier studies at shorter periods. The two-station method is applied to a few 'pure-age' oceanic paths, and is shown to be compatible with the average Earth model C2 (Anderson & Hart 1976) below depths of 180 km. Under this assumed oceanic model, regionalized for age above 180 km, continental velocities are then derived from a set of experimental great-circle values, both new or taken from previously published studies. The results basically agree with earlier studies (Dziewonski 1970; Kanamori 1970), although they exhibit less scatter than Kanamori's model. Results are successfully checked against a set of values derived by the two-station method from a pure continental path.
Although the shield velocities are substantially different from the mean oceanic ones, they still fall within the range of variation of oceanic velocities with the age of the plate. This makes velocities derived theoretically from Jordan's (1975a, b) models of deep continent—ocean lateral heterogeneities, inconsistent with the present set of experimental data. Finally, we show that Dziewonski's (1971) model S2 reconciles all experimental seismic data relative to shields, without being significantly different from oceanic models below 240 km.     

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.     

Upper mantle shear structure of North America

Summary. The waveforms and travel times of S and SS phases in the range 10°–60° have been used to derive upper mantle shear velocity structures for two distinct tectonic provinces in North America. Data from earthquakes on the East Pacific Rise recorded at stations in western North America were used to derive a tectonic upper mantle model. Events on the north-west coast of North America and earthquakes off the coast of Greenland provided the data to investigate the upper mantle under the Canadian shield. All branches from the triplications due to velocity jumps near 400 and 660 km were observed in both areas. Using synthetic seismograms to model these observations placed tight constraints on heterogeneity in the upper mantle and on the details of its structure. SS–S travel-time differences of 30 s along with consistent differences in waveforms between the two data sets require substantial heterogeneity to at least 350 km depth. Velocities in the upper 170 km of the shield are about 10 per cent higher than in the tectonic area. At 250 km depth the shield velocities are still greater by about 4.5 per cent and they gradually merge near 400 km. Below 400 km no evidence for heterogeneity was found. The two models both have first-order discontinuities of 4.5 per cent at 405 km and 7.5 per cent at 695 km. Both models also have lids with lower velocities beneath. In the western model the lid is very thin and of relatively low velocity. In the shield the lid is 170 km thick with very high elocity (4.78 km s^-1); below it the velocity decreases to about 4.65 km s^-1. Aside from these features the models are relatively smooth, the major difference between them being a larger gradient in the tectonic region from 200 to 400 km.     

Deconvolution of teleseismic recordings for mantle structure

Delineation of detailed mantle structure frequently requires the separation of source signature and structural response from seismograms recorded at teleseismic distances. This deconvolution problem can be posed in a log-spectral domain where the operation of time-domain convolution is reduced to an additive form. The introduction of multiple events recorded at many stations leads to a system of consistency equations that must be honoured by both the source time functions and the impulse responses associated with propagation paths between sources and receivers. The system is inherently singular, and stabilization is accomplished through the supply of an initial estimate of the source time function. Although alternative choices exist, an effective estimate is derived from the eigenimage associated with the largest eigenvalue in a singular-value decomposition of the suite of aligned seismograms corresponding to a given event. The relation of the deconvolution scheme to simultaneous least-squares deconvolution is examined. Application of the methodology to broadband teleseismic P waveforms recorded on the Canadian National Seismograph Network demonstrates the retrieval of effective Green's functions including secondary phases associated with upper-mantle structure.     

Methodologies for estimating t*(f) from short-period body waves and regional variations of t*(f) in the United States

Summary. In this paper we discuss some aspects of estimating t * from short-period body waves and present some limits on t* (f) models for the central and south-western United States (CUS and SWUS). We find that for short-period data, with frequencies above 1 or 2 Hz, while the average spectral shape is stable, the smaller details of the spectra are not; thus, only an average t *, and not a frequency-dependent t *, can be derived from such information. Also, amplitudes are extremely variable for short-period data, and thus a great deal of data from many stations and azimuths must be used when amplitudes are included in attenuation studies.
The predictions of three pairs of models for t* (f) in the central and south-western United States are compared with time domain observations of amplitudes and waveforms and frequency domain observations of spectral slopes to put bounds on the attenuation under the different parts of the country. A model with the t * values of the CUS and SWUS converging at low frequencies and differing slightly at high frequencies matches the spectral domain characteristics, but not the time domain amplitudes and waveforms of short-period body waves. A model with t * curves converging at low frequencies, but diverging strongly at high frequencies, matches the time domain observations, but not the spectral shapes. A model with nearly-parallel t* (f) curves for the central and south-western United States satisfies both the time and frequency domain observations.
We conclude that use of both time and frequency domain information is essential in determining t* (f) models. For the central and south-western United States, a model with nearly-parallel t* (f) curves, where Δ t *∼ 0.2 s, satisfies both kinds of data in the 0.3–2 Hz frequency range.     

Magnitude and Q determinations in southern Africa using Lg wave amplitudes

b
The amplitude of vertical, short period (1 s) Lg -waves from 575 shallow earthquakes recorded within the distance range 0|Mo-30|Mo by the Rhodesian seismograph network during the period 1968–77 are analysed to separate the effects of earthquake size, epicentral distance and station structure.
When corrected for geometrical spreading and Airy phase dispersion the decay of amplitude with distance yields an estimate of anelastic attenuation of 0.160 deg^-1 which gives an average value of Q (the specific quality factor) of 603 |Mp 50 for propagation paths that lie along and across the East African Rift System. Inversion of the amplitude—distance curve gives the calibration or distance normalizing function. Thereby the amplitude of Lg can be used to provide an estimate of the size of small, local earthquakes in terms of the teleseismic body wave magnitude m_b (after Henderson). The station effects of the six seismograph stations making up the network all lie within |Mp0.1 magnitude units. Since three of the stations lie on the Rhodesian craton while the remaining three lie on Precambrian mobile belts adjacent to the craton, the Precambrian basement geology does not significantly affect the amplitude of Lg