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We present a seismic model of the Africa Plate, constructed with the technique of full-waveform inversion. The purpose of our model is to serve as a foundation for quantitative geodynamic and geochemical interpretation, earthquake-induced ground motion predictions, and earthquake source inversion. Starting from the first-generation Collaborative Seismic Earth Model, we invert seismograms filtered to a minimum period of 35 s and compute gradients of the misfit function with respect to the model parameters using the adjoint state method. We use dynamically changing mini-batches of the complete data set to compute approximate gradients at each iteration. This approach has three significant advantages: (a) it reduces computational costs for model updates and the inversion, (b) it enables the use of larger datasets without increasing iteration costs, and (c) it makes it trivial to assimilate new data since we can extend the data set without changing the misfit function. We invert data from 397 unique earthquakes and 184,356 unique source-receiver pairs. We clearly image tectonic features such as the Afar triple junction and low-velocity zones below the Hoggar, Air, and Tibesti Mountains, pronounced more than in earlier works. Finally, we introduce a new strategy to assess model uncertainty. We deliberately perturb the final model, perform additional mini-batch iterations, and compare the result with the original final model. This test uses actual seismic data instead of artificially generated synthetic data and requires no assumptions about the linearity of the inverse problem., Journal of Geophysical Research: Solid Earth, 128 (6), ISSN:2169-9313, ISSN:0148-0227, ISSN:2169-9356

We present an accelerated full-waveform inversion based on dynamic mini-batch optimization, which naturally exploits redundancies in observed data from different sources. The method rests on the selection of quasi-random subsets (mini-batches) of sources, used to approximate the misfit and the gradient of the complete data set. The size of the mini-batch is dynamically controlled by the desired quality of the gradient approximation. Within each mini-batch, redundancy is minimized by selecting sources with the largest angular differences between their respective gradients, and spatial coverage is maximized by selecting candidate events with Mitchell’s best-candidate algorithm. Information from sources not included in a specific mini-batch is incorporated into each gradient calculation through a quasi-Newton approximation of the Hessian, and a consistent misfit measure is achieved through the inclusion of a control group of sources. By design, the dynamic mini-batch approach has several main advantages: (1) The use of mini-batches with adaptive size ensures that an optimally small number of sources is used in each iteration, thus potentially leading to significant computational savings; (2) curvature information is accumulated and exploited during the inversion, using a randomized quasi-Newton method; (3) new data can be incorporated without the need to re-invert the complete data set, thereby enabling an evolutionary mode of full-waveform inversion. We illustrate our method using synthetic and real-data inversions for upper-mantle structure beneath the African Plate. In these specific examples, the dynamic mini-batch approach requires around 20 per cent of the computational resources in order to achieve data and model misfits that are comparable to those achieved by a standard full-waveform inversion where all sources are used in each iteration., Geophysical Journal International, 221 (2), ISSN:0956-540X, ISSN:1365-246X

In this paper, we present a series of mathematical abstractions for seismologically relevant wave equations discretized using finite-element methods, and demonstrate how these abstractions can be implemented efficiently in computer code. Our motivation is to mitigate the combinatorial complexity present when considering geophysical waveform modelling and inversion, where a variety of spatial discretizations, material models, and boundary conditions must be considered simultaneously. We accomplish this goal by first considering three distinct classes of abstract mathematical models: (1) those representing the physics of an underlying wave equation, (2) those describing the discretization of the chosen equation onto a finite-dimensional basis and (3) those describing any spatial transforms. A full representation of the discrete wave equation can then be constructed using a hierarchical nesting of models from each class. Additionally, each class is functionally orthogonal to the others, and with certain restrictions models within one class can be interchanged independently from changes in another. We then show how this recasting of the relevant equations can be implemented concisely in computer software using an abstract object-oriented design, and discuss how recent developments in the numerical and computational sciences can be naturally incorporated. This builds to a set of results where we demonstrate how the developments presented can lead to an implementation capable of multiphysics waveform simulations in completely unstructured domains, on both hypercubical and simplical spectral-element meshes, in both two and three dimensions, while remaining concise, efficient and maintainable., Geophysical Journal International, 216 (3), ISSN:0956-540X, ISSN:1365-246X

The sources of ambient seismic noise are extensively studied both to better understand their influence on ambient noise tomography and related techniques, and to infer constraints on their excitation mechanisms. Here we develop a gradient-based inversion method to infer the space-dependent and time-varying source power spectral density of the Earth's hum from cross correlations of continuous seismic data. The precomputation of wavefields using spectral elements allows us to account for both finite-frequency sensitivity and for three-dimensional Earth structure. Although similar methods have been proposed previously, they have not yet been applied to data to the best of our knowledge. We apply this method to image the seasonally varying sources of Earth's hum during North and South Hemisphere winter. The resulting models suggest that hum sources are localized, persistent features that occur at Pacific coasts or shelves and in the North Atlantic during North Hemisphere winter, as well as South Pacific coasts and several distinct locations in the Southern Ocean in South Hemisphere winter. The contribution of pelagic sources from the central North Pacific cannot be constrained. Besides improving the accuracy of noise source locations through the incorporation of finite-frequency effects and 3-D Earth structure, this method may be used in future cross-correlation waveform inversion studies to provide initial source models and source model updates. ISSN:2169-9313 ISSN:0148-0227 ISSN:2169-9356

Crosshole seismic tomography has been used by Vale to provide geophysical images of mineralized massive sulphides in the Eastern Deeps deposit at Voisey's Bay. High resolution seismic images are presented by applying acoustic waveform tomography to these data. In waveform tomography, an initial model is required which predicts the first arrival traveltimes to within a half-cycle of the lowest frequency in the data. Because seismic velocity anisotropy can be significant, the initial model must quantify the anisotropy, as well as the velocity, in order to meet the half-cycle criterion. In our case study, difficulties were encountered in generating an accurate anisotropy model through traveltime tomography, and the starting model for waveform tomography failed the half-cycle criterion. We formulate a new, semi-global approach for finding the best-fit 1-D elliptical anisotropy model using simulated annealing, and successfully apply this technique to the Vosiey's Bay dataset, as well as synthetically generated datasets.

В развитие методологии стохастической границы введено понятие достижимого производственного потенциала, обобщающее понятие граничного производственного потенциала. Построена модель достижимого производственного потенциала, учитывающая возможности управления факторами неэффективности и затраты на управление. Получена оценка условной технической эффективности производства относительно достижимого производственного потенциала. Предложены формальное описание мероприятия, направленного на управление факторами неэффективности производства, способы оценивания его технической и экономической эффективности. Приводятся результаты экспериментальных расчетов., In development of the stochastic frontier methodology the introduc a concept of achievable production potential generalizing that of the frontier production potential. A model of frontier production potential was built taking in account possibilities of management of factors of inefficiency and the related costs. An estimate of the conditional technical efficiency of production relative to the production potential was obtained. A formal description of measures, directed at management of factors of production inefficiency, and methods of estimation of their technical and economic efficiency are offered. Finally, we present the results of experimental calculations are given.

The sources of ambient seismic noise are extensively studied both to better understand their influence on ambient noise tomography and related techniques, and to infer constraints on their excitation mechanisms. Here we develop a gradient-based inversion method to infer the space-dependent and time-varying source power spectral density of the Earth's hum from cross correlations of continuous seismic data. The precomputation of wavefields using spectral elements allows us to account for both finite-frequency sensitivity and for three-dimensional Earth structure. Although similar methods have been proposed previously, they have not yet been applied to data to the best of our knowledge. We apply this method to image the seasonally varying sources of Earth's hum during North and South Hemisphere winter. The resulting models suggest that hum sources are localized, persistent features that occur at Pacific coasts or shelves and in the North Atlantic during North Hemisphere winter, as well as South Pacific coasts and several distinct locations in the Southern Ocean in South Hemisphere winter. The contribution of pelagic sources from the central North Pacific cannot be constrained. Besides improving the accuracy of noise source locations through the incorporation of finite-frequency effects and 3-D Earth structure, this method may be used in future cross-correlation waveform inversion studies to provide initial source models and source model updates., Journal of Geophysical Research: Solid Earth, 122 (11), ISSN:2169-9313, ISSN:0148-0227, ISSN:2169-9356

In recent years, seismic station coverage in polar regions has been increasing steadily, providing new insight into the deep structure and dynamics of these remote parts of the globe. Numerical seismic wave propagation through polar regions is complicated by the presence of ice sheets. At periods relevant for regional-scale waveform tomography, the ice acts as a thin layer that, for an exact solution, needs to be meshed by elements that are small compared to the minimum wavelength. As a consequence, computational costs may become prohibitively high. In this study, we investigate the effects of various simplifications of the mesh using spectral-element simulations of regional seismic wave propagation, with propagation distances of several hundred kilometres. Our goals are to quantify errors in phase and amplitude, and to identify simplifications that significantly reduce computational costs while producing numerical errors that are tolerable given a concrete application, dataset and frequency band. Using the example of western Greenland, we specifically consider meshes where 1) the ice sheet is replaced by bedrock, 2) where the ice sheet is removed but bedrock topography is honored, and 3) where any topography and the ice sheet are ignored. For periods of 10 s we find that the group velocity of fundamental-mode surface waves can be off by more than 2 % when the ice sheet topography is ignored (2 and 3). The actual elastic properties of the ice play a comparatively smaller role. Furthermore, the ice sheet-related traveltime errors diminish with increasing period, whereas the topography-related errors are more frequency-independent. This suggests (1) that regional-scale topography with height much less than a wavelength should be modelled also in ice-free regions, and (2) that computational-costs may be reduced substantially by replacing ice with bedrock in the simulations, while fully honoring topography. Amplitude errors in the fundamental-mode surface waves are equivalent to variations in attenuation that easily exceed 100 %, regardless of the type of simplification. It follows that the ice sheet should be meshed accurately when amplitude information is used to infer regional-scale Earth structure.