ALBERT

Description: We present a new conceptual approach to scattering-integral-based seismic full waveform inversion (FWI) that allows a flexible, extendable, modular and both computationally and storage-efficient numerical implementation. To achieve maximum modularity and extendability, interactions between the three fundamental steps carried out sequentially in each iteration of the inversion procedure, namely, solving the forward problem, computing waveform sensitivity kernels and deriving a model update, are kept at an absolute minimum and are implemented by dedicated interfaces. To realize storage efficiency and maximum flexibility, the spatial discretization of the inverted earth model is allowed to be completely independent of the spatial discretization employed by the forward solver. For computational efficiency reasons, the inversion is done in the frequency domain. The benefits of our approach are as follows: (1) Each of the three stages of an iteration is realized by a stand-alone software program. In this way, we avoid the monolithic, unflexible and hard-to-modify codes that have often been written for solving inverse problems. (2) The solution of the forward problem, required for kernel computation, can be obtained by any wave propagation modelling code giving users maximum flexibility in choosing the forward modelling method. Both time-domain and frequency-domain approaches can be used. (3) Forward solvers typically demand spatial discretizations that are significantly denser than actually desired for the inverted model. Exploiting this fact by pre-integrating the kernels allows a dramatic reduction of disk space and makes kernel storage feasible. No assumptions are made on the spatial discretization scheme employed by the forward solver. (4) In addition, working in the frequency domain effectively reduces the amount of data, the number of kernels to be computed and the number of equations to be solved. (5) Updating the model by solving a large equation system can be done using different mathematical approaches. Since kernels are stored on disk, it can be repeated many times for different regularization parameters without need to solve the forward problem, making the approach accessible to Occam's method. Changes of choice of misfit functional, weighting of data and selection of data subsets are still possible at this stage. We have coded our approach to FWI into a program package called ASKI (Analysis of Sensitivity and Kernel Inversion) which can be applied to inverse problems at various spatial scales in both Cartesian and spherical geometries. It is written in modern FORTRAN language using object-oriented concepts that reflect the modular structure of the inversion procedure. We validate our FWI method by a small-scale synthetic study and present first results of its application to high-quality seismological data acquired in the southern Aegean.

Description: SUMMARY We present a new, S -velocity model of the European upper mantle, constrained by inversions of seismic waveforms from broad-band stations in Europe and surrounding regions. We collected seismograms for the years 1990–2007 from all permanent stations in Europe for which data were available. In addition, we incorporated data from temporary experiments. Automated multimode inversion of surface and S -wave forms was applied to extract structural information from the seismograms, in the form of linear equations with uncorrelated uncertainties. The equations were then solved for seismic velocity perturbations in the crust and mantle with respect to a 3-D reference model with a realistic crust. We present two versions of the model: one for the entire European upper mantle and another, with the highest resolution, focused on the upper 200 km of the mantle beneath western and central Europe and the circum Mediterranean. The mantle lithosphere and asthenosphere are well resolved by both models. Major features of the lithosphere–asthenosphere system in Europe and the Mediterranean are indentified. The highest velocities in the mantle lithosphere of the East European Craton (EEC) are found at about 150 km depth. There are no indications for a deep cratonic root below about 330 km depth. Lateral variations within the cratonic mantle lithosphere are resolved as well. The locations of kimberlites correlate with reduced S -wave velocities in the shallow cratonic mantle lithosphere. This anomaly is present in regions of both Proterozoic and Archean crust, pointing to an alteration of the mantle lithosphere after the formation of the craton. Strong lateral changes in S -wave velocity are found at the northwestern margin of the EEC and may indicate erosion of cratonic mantle lithosphere beneath the Scandes by hot asthenosphere. The mantle lithosphere beneath western Europe and between the Tornquist–Teisseyre Zone and the Elbe Line shows moderately high velocities and is of an intermediate character, between cratonic lithosphere and the thin lithosphere of central Europe. In central Europe, Caledonian and Variscian sutures are not associated with strong lateral changes in the lithosphere–asthenosphere system. Cenozoic anorogenic intraplate volcanism in central Europe and the circum Mediterranean is found in regions of shallow asthenosphere and close to changes in the depth of the lithosphere–asthenosphere boundary. Very low velocities at shallow upper-mantle depths are present from eastern Turkey towards the Dead Sea transform fault system and Sinai, beneath locations of recent volcanism. Low-velocity anomalies extending vertically from shallow upper mantle down to the transition zone are found beneath the Massif Central, Sinai and the Dead Sea, the Canary Islands and Iceland.

Description: The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s perspective on mathematical language may appeal also to those who are not friends of or experts on Wittgenstein’s later philosophy of mathematics.

Description: SUMMARY We present an algorithm for automated S -phase arrival time determination of local, regional and teleseismic events based on autoregressive (AR) prediction of the waveform. The waveforms of the horizontal components are predicted using a scalar AR model for multicomponent recordings. The AR coefficients are estimated in a short moving window using a least-squares approach minimizing the forward prediction error. Synthetic tests with single-component data show that the least-squares approach yields similar or even better results than the Yule–Walker and Burg’s algorithms. We discuss the choice of the AR model and show that the corresponding prediction error of the AR model, applied to both horizontal components, is sufficient to detect instantaneous changes in amplitude, frequency, phase and polarization. The rms prediction error of both horizontal components defines the characteristic function, to which an algorithm for the estimation of the arrival time is applied. The proposed algorithm also accounts for automatic quality assessment of the estimated S -onset times. Four quality criteria are used to define the weight of the automatically estimated S -arrival time. They are based on two different estimations of the slope of the characteristic function and on two signal-to-noise ratios (SNRs). The proposed algorithm is applied to a large data set recorded by a dense regional seismic network in the southern Aegean. The data set contains recordings of local and regional crustal as well as intermediate deep earthquakes. The reliability and the robustness of the picking algorithm is tested by comparing more than 2600 manual S readings, serving as reference picks, with the corresponding automatically derived S -onset times. We find an average deviation from the reference picks of 0.5 s ± 0.8 s. If only excellent automatic picks are considered, the average difference from the reference picks is reduced to −0.057 s ± 0.12 s. The proposed automatic quality weighting scheme yields similar weights for the individual S onsets as the ones set by the analysts. The presented algorithm works reliably and robust even when applied to a data set with heterogeneous SNRs. Furthermore, the proposed method may be suitable for the implementation in an earthquake early-warning system as additional, accurate S -wave arrival time estimates stabilize the location, especially the determination of the event depth.

Description: 〈span〉〈div〉Summary〈/div〉An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as two-dimensional surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1D by comparison to an analytical solution and in 2D by comparing the results of a simulation valid in 1D and 2D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1D and 2D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.〈/span〉

Description: 〈span〉〈div〉SUMMARY〈/div〉An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.〈/span〉

Description: We infer seismic azimuthal anisotropy from ambient-noise-derived Rayleigh waves in the wider Vienna Basin region. Cross-correlations of the ambient seismic field are computed for 1953 station pairs and periods from 5 to 25? s to measure the directional dependence of interstation Rayleigh-wave group velocities. We perform the analysis for each period on the whole data set, as well as in overlapping 2°-cells to regionalize the measurements, to study expected effects from isotropic structure, and isotropic–anisotropic trade-offs. To extract azimuthal anisotropy that relates to the anisotropic structure of the Earth, we analyse the group velocity residuals after isotropic inversion. The periods discussed in this study (5–20? s) are sensitive to crustal structure, and they allow us to gain insight into two distinct mechanisms that result in fast orientations. At shallow crustal depths, fast orientations in the Eastern Alps are S/N to SSW/NNE, roughly normal to the Alps. This effect is most likely due to the formation of cracks aligned with the present-day stress-field. At greater depths, fast orientations rotate towards NE, almost parallel to the major fault systems that accommodated the lateral extrusion of blocks in the Miocene. This is coherent with the alignment of crystal grains during crustal deformation occurring along the fault systems and the lateral extrusion of the central part of the Eastern Alps.