ALBERT

Description: SUMMARY Since its beginning in acoustics, the Time-Reversal method (hereafter referred as TR) has been explored by different studies to locate and characterize seismic sources in elastic media. But few authors have proposed an analytical analysis of the method, especially in the case of an elastic medium and for a finite body such as the Earth. In this paper, we use a normal mode approach (for general 3-D case and degenerate modes in 1-D reference model) to investigate the convergence properties of the TR method. We first investigate a three-point problem, with two fixed points which are the source and the receiver and a third one corresponding to a changing observation point. We extend the problem of a single channel TR experiment to a multiple channel and multiple station TR experiment. We show as well how this problem relates to the retrieval of Green’s function with a multiple source cross-correlation and also the differences between TR method and cross-correlation techniques. Since most of the noise sources are located close to the surface of the Earth, we show that the time derivative of the cross-correlation of long-period seismograms with multiple sources at the surface is different from the Green’s function. Next, we show the importance of a correct surface-area weighting of the signal resent by the stations according to a Voronoi tessellation of the Earth surface. We use arguments based on the stationary phase approximation to argue that phase-information is more important than amplitude information for getting a good focusing in TR experiment. Finally, by using linear relationships between the time-reversed displacement (resp. strain wavefields) and the components of a vector force source (resp. a moment tensor source), we show how to retrieve force (or moment tensor components) of any long period tectonic or environmental sources by time reversal.

Description: Azimuthal anisotropy is a powerful tool to reveal information about both the present structure and past evolution of the mantle. Anisotropic images of the upper mantle are usually obtained by analysing various types of seismic observables, such as surface wave dispersion curves or waveforms, SKS splitting data, or receiver functions. These different data types sample different volumes of the earth, they are sensitive to different length scales, and hence are associated with different levels of uncertainties. They are traditionally interpreted separately, and often result in incompatible models. We present a Bayesian inversion approach to jointly invert these different data types. Seismograms for SKS and P phases are directly inverted using a cross-convolution approach, thus avoiding intermediate processing steps, such as numerical deconvolution or computation of splitting parameters. Probabilistic 1-D profiles are obtained with a transdimensional Markov chain Monte Carlo scheme, in which the number of layers, as well as the presence or absence of anisotropy in each layer, are treated as unknown parameters. In this way, seismic anisotropy is only introduced if required by the data. The algorithm is used to resolve both isotropic and anisotropic layering down to a depth of 350 km beneath two seismic stations in North America in two different tectonic settings: the stable Canadian shield (station FFC) and the tectonically active southern Basin and Range Province (station TA-214A). In both cases, the lithosphere–asthenosphere boundary is clearly visible, and marked by a change in direction of the fast axis of anisotropy. Our study confirms that azimuthal anisotropy is a powerful tool for detecting layering in the upper mantle.

Description: 〈span〉〈div〉SUMMARY〈/div〉Accurate synthetic seismic wavefields can now be computed in 3-D earth models using the spectral element method (SEM), which helps improve resolution in full waveform global tomography. However, computational costs are still a challenge. These costs can be reduced by implementing a source stacking method, in which multiple earthquake sources are simultaneously triggered in only one teleseismic SEM simulation. One drawback of this approach is the perceived loss of resolution at depth, in particular because high-amplitude fundamental mode surface waves dominate the summed waveforms, without the possibility of windowing and weighting as in conventional waveform tomography.This can be addressed by redefining the cost-function and computing the cross-correlation wavefield between pairs of stations before each inversion iteration. While the Green’s function between the two stations is not reconstructed as well as in the case of ambient noise tomography, where sources are distributed more uniformly around the globe, this is not a drawback, since the same processing is applied to the 3-D synthetics and to the data, and the source parameters are known to a good approximation. By doing so, we can separate time windows with large energy arrivals corresponding to fundamental mode surface waves. This opens the possibility of designing a weighting scheme to bring out the contribution of overtones and body waves. It also makes it possible to balance the contributions of frequently sampled paths versus rarely sampled ones, as in more conventional tomography.Here we present the results of proof of concept testing of such an approach for a synthetic 3-component long period waveform data set (periods longer than 60 s), computed for 273 globally distributed events in a simple toy 3-D radially anisotropic upper mantle model which contains shear wave anomalies at different scales. We compare the results of inversion of 10 000 s long stacked time-series, starting from a 1-D model, using source stacked waveforms and station-pair cross-correlations of these stacked waveforms in the definition of the cost function. We compute the gradient and the Hessian using normal mode perturbation theory, which avoids the problem of cross-talk encountered when forming the gradient using an adjoint approach. We perform inversions with and without realistic noise added and show that the model can be recovered equally well using one or the other cost function.The proposed approach is computationally very efficient. While application to more realistic synthetic data sets is beyond the scope of this paper, as well as to real data, since that requires additional steps to account for such issues as missing data, we illustrate how this methodology can help inform first order questions such as model resolution in the presence of noise, and trade-offs between different physical parameters (anisotropy, attenuation, crustal structure, etc.) that would be computationally very costly to address adequately, when using conventional full waveform tomography based on single-event wavefield computations.〈/span〉

Description: We present a method to construct non-stationary and anisotropic second-order random model realizations that can be used for numerical wave propagation simulations in various geometries. Models are generated directly from a given covariance matrix using its eigenvector decomposition (principal component or Karhunen-Loève method). Because this decomposition is very expensive computationally in 3-D, we use model symmetries to reduce the size of the covariance matrix to its non-stationary components. Stationary components can then be described through their power spectrum, such that models with axisymmetric or spherically symmetric statistics can be generated from a 1-D covariance matrix. We focus in particular on models with spherically symmetric statistics that are important to study wave propagation in the Earth. We use this method to show the influence of hypothetical small-scale structure in the Earth's mantle on the elastic wavefield. To this end, we extend tomographic models beyond their spatial resolution limit with different distributions of small-scale scatterers that generate a coda and attenuate direct waves (scattering attenuation). We observe that scattering attenuation of fundamental mode Rayleigh waves is small (0.5–2 per cent of the total attenuation), if the elastic mantle structure does not become significantly stronger at smaller scales. At the examined heterogeneity strengths, scattering attenuation scales linearly with the model variance. The long-period fundamental mode Rayleigh wave coda is difficult to measure because it is weak and overlaps with other signals. However, it can be shown that its intensity also scales linearly with model power, and that it depends strongly on the spherical geometry of the Earth. It can therefore be used to distinguish between models with different small-scale power. We show qualitatively that the coda generated by the type of random models we consider can explain observed scattered energy at long periods (100 s).

Description: SS precursor observations are a powerful tool to study the topography and character of transition zone discontinuities, especially in regions such as ocean basins where few seismic stations exist, precluding other high resolution approaches. Still, the available coverage is limited by the distribution of sources and stations, but also by the level of noise and by the fact that, in some distance ranges, interfering seismic phases mask the weak signal from the SS precursors. We introduce an array data processing tool, the local slant-stack filter, to address these challenges and clean up the otherwise noisy SS precursor record sections. We show that these filters are a powerful tool for extracting the weak yet coherent SS precursor signals while removing interfering seismic phases as well as random noise, yielding robust precursor traveltime measurements with spatial resolution higher than what can be achieved by the conventional common midpoint stacking method. The effectiveness of the filters are demonstrated by application to synthetic and real data. We systematically apply this filtering method to an SS precursor data set recorded by the U.S. Transportable Array that samples a vast region of the Pacific Ocean and its northwest margin, and present maps of 410 and 660 discontinuity topography. We discuss correlations observed between our discontinuity images and several fine-scale heterogeneities revealed by mantle shear wave tomography in the vicinity of Hawaii and the Pacific Superswell.

Description: In order to study fine scale structure of the Earth's deep interior, it is necessary to extract generally weak body wave phases from seismograms that interact with various discontinuities and heterogeneities. The recent deployment of large-scale dense arrays providing high-quality data, in combination with efficient seismic data processing techniques, may provide important and accurate observations over large portions of the globe poorly sampled until now. Major challenges are low signal-to-noise ratios (SNR) and interference with unwanted neighbouring phases. We address these problems by introducing scale-dependent slowness filters that preserve time-space resolution. We combine complex wavelet and slant-stack transforms to obtain the slant-stacklet transform. This is a redundant high-resolution directional wavelet transform with a direction (here slowness) resolution that can be adapted to the signal requirements. To illustrate this approach, we use this expansion to design coherence-driven filters that allow us to obtain clean PcP observations (a weak phase often hidden in the coda of the P wave), for events with magnitude M w  〉 5.4 and distances up to 80°. In this context, we then minimize a linear misfit between P and PcP waveforms to improve the quality of PcP–P traveltime measurements as compared to a standard cross-correlation method. This significantly increases both the quantity and the quality of PcP–P differential traveltime measurements available for the modelling of structure near the core–mantle boundary. The accuracy of our measurements is limited mainly by the highest frequencies of the signals used and the level of noise. We apply this methodology to two examples of high-quality data from dense arrays located in north America. While focusing here on body-wave separation, the tools we propose are more general and may contribute to enhancing seismic signal observations in global seismology in situations of low SNR and high signal interference.

Description: Resolving the topography of the core–mantle boundary (CMB) and the structure and composition of the D '' region is key to improving our understanding of the interaction between the Earth's mantle and core. Observations of traveltimes and amplitudes of short-period teleseismic body waves sensitive to lowermost mantle provide essential constraints on the properties of this region. Major challenges are low signal-to-noise ratio of the target phases and interference with other mantle phases. In a previous paper (Part I), we introduced the slant-stacklet transform to enhance the signal of the core-reflected ( PcP ) phase and to isolate it from stronger signals in the coda of the P wave. Then we minimized a linear misfit between P and PcP waveforms to improve the quality of PcP–P traveltime difference measurements as compared to standard cross-correlation methods. This method significantly increases the quantity and the quality of PcP–P traveltime observations available for the modelling of structure near the CMB. Here we illustrate our approach in a series of regional studies of the CMB and D '' using PcP–P observations with unprecedented resolution from high-quality dense arrays located in North America and Japan for events with magnitude M w 〉5.4 and distances up to 80 $\deg$ . In this process, we carefully analyse various sources of errors and show that mantle heterogeneity is the most significant. We find and correct bias due to mantle heterogeneities that is as large as 1 s in traveltime, comparable to the largest lateral PcP–P traveltime variations observed. We illustrate the importance of accurate mantle corrections and the need for higher resolution mantle models for future studies. After optimal mantle corrections, the main signal left is relatively long wavelength in the regions sampled, except at the border of the Pacific large-low shear velocity province (LLSVP). We detect the northwest border of the Pacific LLSVP in the western Pacific from array observations in Japan, and observe higher than average P velocities, or depressed CMB, in Central America, and slightly lower than average P velocities under Alaska/western Canada.

Description: We present evidence for the presence of complex anisotropy in the lowermost mantle from 3-D waveform modelling of observed core-diffracted shear waves that sample the southern edge of the African Large Low Shear Velocity Province (LLSVP). The anomalously strong amplitude of the SV component for the shear core-diffracted phase at large distances indicates the presence of anisotropy. We measure shear wave splitting parameters to determine which part of the elastic tensor is constrained by this particular data set. The modelling is performed using the spectral element method. The anisotropy is strong outside the LLSVP, weakens or rotates close to its boundary, and appears to be absent inside the LLSVP. The presence of the LLSVP margin may cause flow in the mantle to change direction. The occurrence of strong anisotropy in the region of fast seismic velocities is compatible with lattice-preferred orientation in post-perovskite due to accommodation of flow through dislocation creep.

Description: A general approach for constructing numerical equivalents of time-reversal mirrors is introduced. These numerical mirrors can be used to regenerate an original wavefield locally within a confined volume of arbitrary shape. Though time-reversal mirrors were originally designed to reproduce a time-reversed version of an original wavefield, the proposed method is independent of the time direction and can be used to regenerate a wavefield going either forward in time or backward in time. Applications to computational seismology and tomographic imaging of such local wavefield reconstructions are discussed. The key idea of the method is to directly express the source terms constituting the time-reversal mirror by introducing a spatial window function into the wave equation. The method is usable with any numerical method based on the discrete form of the wave equation, for example, with finite difference (FD) methods and with finite/spectral elements methods. The obtained mirrors are perfect in the sense that no additional error is introduced into the reconstructed wavefields apart from rounding errors that are inherent in floating-point computations. They are fully transparent as they do not interact with waves that are not part of the original wavefield and are permeable to these. We establish a link between some hybrid methods introduced in seismology, such as wave-injection, and the proposed time-reversal mirrors. Numerical examples based on FD and spectral elements methods in the acoustic, the elastic and the visco-elastic cases are presented. They demonstrate the accuracy of the method and illustrate some possible applications. An alternative implementation of the time-reversal mirrors based on the discretization of the surface integrals in the representation theorem is also introduced. Though it is out of the scope of the paper, the proposed method also apply to numerical schemes for modelling of other types of waves such as electro-magnetic waves.