Shallow layer correction for Spectral Element like methods

Today's numerical methods like the Spectral Element Method (SEM) allow accurate simulation of the whole seismic field in complex 3-D geological media. However, the accuracy of such a method requires physical discontinuities to be matched by mesh interfaces. In many realistic earth models, the design of such a mesh is difficult and quite ineffective in terms of numerical cost. In this paper, we address a limited aspect of this problem: an earth model with a thin shallow layer below the free surface in which the elastic and density properties are different from the rest of the medium and in which rapid vertical variations are allowed. We only consider here smooth lateral variations of the thickness and elastic properties of the shallow layer. In the limit of a shallow layer thickness very small compared to the smallest wavelength of the wavefield, by resorting to a second order matching asymptotic approximation, the thin layer can be replaced by a vertically smooth effective medium without discontinuities together with a specific Dirichlet to Neumann (DtN) surface boundary condition. Such a formulation allows to accurately take into account complex thin shallow structures within the SEM without the classical mesh design and time step constraints. Corrections at receivers and source—when the source is located within the thin shallow layer—have been also derived. Accuracy and efficiency of this formulation are assessed on academic tests. The stability and limitations of this formulation are also discussed.     

Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

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.     

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

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

Crustal structure of Iraq from receiver functions and surface wave dispersion: implications for understanding the deformation history of the Arabian–Eurasian collision

We report the crustal structure for two locations in Iraq estimated by joint inversion of P -wave receiver functions (RFs) and surface (Rayleigh) wave group velocity dispersion. RFs were computed from teleseismic recordings at two temporary broad-band seismic stations located in Mosul (MSL) in the Zagros Fold Belt and Baghdad (BHD) in the Mesopotamian Foredeep. Group velocity dispersion curves at the sites were derived from continental-scale tomography. The inversion results show that the crustal thicknesses are 39 km at MSL and 43 km at BHD. We observe a strong Ps _Moho at BHD consistent with a sharp Moho discontinuity. However, at MSL we observe a weak Ps _Moho suggesting a transitional Moho where crustal thickening is likely to be occurring in the deep crust. Both sites reveal low velocity surface layers consistent with sedimentary thickness of about 3 km at station MSL and 7 km at BHD and agreeing well with the previous reports. Ignoring the sediments, the crystalline crustal velocities and thicknesses are remarkably similar at both stations. The similarity of crustal structure suggests that the crust of the northeastern proto-Arabian Platform was uniform before subsidence and deposition of the sediments in the Cenozoic. If crystalline crustal structure is uniform across the northern Arabian Platform then crustal thickness variations in the Zagros Fold Belt and Thrust Zone should reveal the history of deformation and crustal shortening in the Arabian–Eurasian collision zone and not reflect pre-existing crustal thickness variations in the Arabian Plate.     

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.     

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.     

Surface wave array tomography in SE Tibet from ambient seismic noise and two-station analysis – II. Crustal and upper-mantle structure

We determine the 3-D shear wave speed variations in the crust and upper mantle in the southeastern borderland of the Tibetan Plateau, SW China, with data from 25 temporary broad-band stations and one permanent station. Interstation Rayleigh wave (phase velocity) dispersion curves were obtained at periods from 10 to 50 s from empirical Green's function (EGF) derived from (ambient noise) interferometry and from 20 to 150 s from traditional two-station (TS) analysis. Here, we use these measurements to construct phase velocity maps (from 10 to 150 s, using the average interstation dispersion from the EGF and TS methods between 20 and 50 s) and estimate from them (with the Neighbourhood Algorithm) the 3-D wave speed variations and their uncertainty. The crust structure, parametrized in three layers, can be well resolved with a horizontal resolution about of 100 km or less. Because of the possible effect of mechanically weak layers on regional deformation, of particular interest is the existence and geometry of low (shear) velocity layers (LVLs). In some regions prominent LVLs occur in the middle crust, in others they may appear in the lower crust. In some cases the lateral transition of shear wave speed coincides with major fault zones. The spatial variation in strength and depth of crustal LVLs suggests that the 3-D geometry of weak layers is complex and that unhindered crustal flow over large regions may not occur. Consideration of such complexity may be the key to a better understanding of relative block motion and patterns of seismicity.     

Seismic vertical array analysis for phase decomposition

We propose a vertical array analysis method that decomposes complex seismograms into body and surface wave time histories by using a velocity structure at the vertical array site. We assume that the vertical array records are the sum of vertically incident plane P and S waves, and laterally incident Love and Rayleigh waves. Each phase at the surface is related to that at a certain depth by the transfer function in the frequency domain; the transfer function is obtained by Haskell's matrix method, assuming a 1-D velocity structure. Decomposed P , S and surface waves at the surface are estimated from the vertical array records and the transfer functions by using a least-squares method in the frequency domain; their time histories are obtained by the inverse Fourier transform. We carried out numerical tests of this method based on synthetic vertical array records consisting of vertically incident plane P and S waves and laterally incident plane Love and Rayleigh waves. Perfect results of the decomposed P , S , Love and Rayleigh waves were obtained for synthetic records without noise. A test of the synthetic records in which a small amount of white noise was added yielded a reasonable result for the decomposed P , S and surface waves. We applied this method to real vertical array records from the Ashigara valley, a moderate-sized sedimentary valley. The array records from two earthquakes occurring at depths of 123 and 148 km near the array (epicentral distance of about 31 km) exhibited long-duration later phases. The analysis showed that duration of the decomposed S waves was a few seconds and that the decomposed surface waves appeared a few seconds after the direct S -wave arrival and had very long duration. This result indicated that the long-duration later phases were generated not by multireflected S waves, but by basin-induced surface waves.     

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.     

Surface wave tomography for azimuthal anisotropy in a strongly reduced parameter space

Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy.     

Traveltime measurements from noise correlation: stability and detection of instrumental time-shifts

We test the feasibility of using Green's functions extracted from records of ambient seismic noise to monitor temporal changes in the Earth crust properties by repeated measurements at regional distances. We use about 11 yr of continuous recordings to extract surface waves between three pairs of stations in California. The correlations are computed in a moving 1-month window and we analyse the temporal evolution of measured interstation traveltimes. The comparison of the arrival times in the positive and negative correlation time of Rayleigh and Love waves allows us to separate time-shifts associated with any form of physical change in the medium, those resulting from clock drift or other instrumental errors, and those due to change in the localization of the noise sources. This separation is based on the principle of time symmetry. When possible, we perform our analysis in two different period bands: 5–10 and 10–20 s. The results indicate that significant instrumental time errors (0.5 s) are present in the data. These time-shifts can be measured and tested by closure relation and finally corrected independently of any velocity model. The traveltime series show a periodic oscillation that we interpret as the signature of the seasonal variation of the region of origin of the seismic noise. Between 1999 and 2005, the final arrival time fluctuations have a variance of the order of 0.01 s. This allows us to measure interstation traveltimes with errors smaller than 0.3 per cent of the interstation traveltime and smaller than 1 per cent of the used wave period. This level of accuracy was not sufficient to detect clear physical variation of crustal velocity during the considered 11 yr between the three stations in California. Such changes may be more easily detectable when considering pairs of stations more closely located to each other and in the vicinity of tectonically active faults or volcanoes.