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Subsurface temperature measurements play a crucial role, for instance, in optimizing geothermal power plants and monitoring heat-storage systems. Previous studies have demonstrated that time-lapse variations in temperature can be correlated with variations in seismic wave speeds, offering the potential for temperature monitoring via seismic surveys. However, an apparent discrepancy has emerged between field and laboratory experiments. Field studies predominantly report positive correlations between temperature and seismic wave speeds, while laboratory experiments often show anticorrelations. This inconsistency underscores the need for a more comprehensive, physics-based understanding of temperature-induced wave speed changes. In this study, we strive to bridge the gap between field and laboratory findings by examining several mechanisms governing temperature-induced seismic wave speed changes, namely the intrinsic temperature dependency of elastic parameters and thermally induced elasticity. We present a physics-based modelling approach to identify the primary mechanisms responsible for temperature-induced seismic wave speed changes. By considering several end-member models, we find that intrinsic temperature dependency of elastic parameters (negative correlation) compete with thermal pressure effects (positive correlation). The precise initial and boundary conditions and physical parameters of the system under consideration will determine the weight of both effects. Temperature-related dilatation does not seem to play an important role. We apply our approach to loosely consolidated sediments in the shallow subsurface of the Groningen region, where subsurface temperature fluctuations are driven by seasonal atmospheric temperature fluctuations roughly between −5 and |$30\, ^\circ$| C. For these models, we predict seasonal temperature-induced changes in body-wave speeds of up to 8 per cent in the first few metres of the subsurface, high-frequency (above 2 Hz) surface wave phase velocity variations in the range of 1–2 per cent, and relative changes in site amplification on the order of 4 per cent. These findings contribute to a more comprehensive understanding of the intricate relationship between temperature and near-surface seismic properties, offering insights for applications as subsurface temperature monitoring systems. [ABSTRACT FROM AUTHOR]

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, Aïr, 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. Plain Language Summary: In this paper, we contribute to the knowledge about geological structures under the African continent. For that, we use publicly available seismograms (records of the ground motion at seismic stations) measured in Africa and the regions around it. These we process with an advanced imaging method called full‐waveform inversion (FWI). The result is a model of the speed of seismic waves propagating through the crust and upper mantle under Africa. We can use the model to generate maps of Earth's interior temperature anomalies in the given region. We can, for instance, clearly see hot structures beneath mountain ranges and volcanoes. Our FWI implementation benefits from several recent advances, allowing us to include a much higher number of seismograms than before, resulting in models with unprecedented resolution. We update the publicly available Collaborative Seismic Earth Model with those outputs. Finally, we propose a new strategy to assess the resolution and reliability of our resulting models. Key Points: We invert for crustal and mantle structure of the African Plate using recordings of 397 earthquakes down to periods of 35 sWe apply the dynamic mini‐batch approach to reduce computational costs and enable inclusion of more dataWe present a new approach to uncertainty analysis in which we perturb the final model and continue the inversion [ABSTRACT FROM AUTHOR]

This study introduces a technique for four‐dimensional pore pressure monitoring using passive image interferometry. Surface‐wave velocity changes as a function of frequency are directly linked to depth variations of pore pressure changes through sensitivity kernels. We demonstrate that these kernels can be used to invert time‐lapse seismic velocity changes, retrieved with passive image interferometry, for hydrological pore pressure variations as a function of time, depth, and region. This new approach is applied in the Groningen region of the Netherlands. We show good recovery of pore pressure variations in the upper 200 m of the subsurface from passive seismic velocity observations. This depth range is primarily limited by the reliable frequency range of the seismic data. Plain Language Summary: In this study, we develop a method for pore pressure monitoring using seismic ambient noise. We use passive image interferometry to estimate surface‐wave velocity changes as a function of frequency, and compute for surface‐wave velocities the sensitivity to pore pressure changes as a function of depth. These so‐called pore pressure sensitivity kernels are then used to invert surface‐wave velocity changes for pore pressure variations as a function of depth. By comparing different regions of Groningen, The Netherlands, we build a four‐dimensional pore pressure model for the shallowest 200 m of the subsurface. While the hydrological pore pressure variation can continue beyond 200 m depth, our method is limited by the shallow sensitivity and the frequency ranges for which seismic velocity measurements are possible. Key Points: Surface‐wave velocity changes are directly linked to pore pressure variations through sensitivity kernelsPore pressure sensitivity kernels enable an inversion of surface‐wave velocity changes for 4D pore pressure variationsThe shallow sensitivity to pore pressure changes in Groningen limits the method to the upper 200 m of the subsurface [ABSTRACT FROM AUTHOR]

Shear properties of mantle minerals are vital for interpreting seismic shear wave speeds and therefore inferring the composition and dynamics of a planetary interior. Shear wave speed and elastic tensor components, from which the shear modulus can be computed, are usually measured in the laboratory mimicking the Earth's (or a planet's) internal pressure and temperature conditions. A functional form that relates the shear modulus to pressure (and temperature) is fitted to the measurements and used to interpolate within and extrapolate beyond the range covered by the data. Assuming a functional form provides prior information, and the constraints on the predicted shear modulus and its uncertainties might depend largely on the assumed prior rather than the data. In the present study, we propose a data-driven approach in which we train a neural network to learn the relationship between the pressure, temperature and shear modulus from the experimental data without prescribing a functional form a priori. We present an application to MgO, but the same approach works for any other mineral if there are sufficient data to train a neural network. At low pressures, the shear modulus of MgO is well-constrained by the data. However, our results show that different experimental results are inconsistent even at room temperature, seen as multiple peaks and diverging trends in probability density functions predicted by the network. Furthermore, although an explicit finite-strain equation mostly agrees with the likelihood predicted by the neural network, there are regions where it diverges from the range given by the networks. In those regions, it is the prior assumption of the form of the equation that provides constraints on the shear modulus regardless of how the Earth behaves (or data behave). In situations where realistic uncertainties are not reported, one can become overconfident when interpreting seismic models based on those defined equations of state. In contrast, the trained neural network provides a reasonable approximation to experimental data and quantifies the uncertainty from experimental errors, interpolation uncertainty, data sparsity and inconsistencies from different experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider a simplified model of turbulence for large Reynolds numbers driven by a constant power energy input on large scales. In the statistical stationary regime, the behaviour of the kinetic energy is characterized by two well-defined phases: a laminar phase where the kinetic energy grows linearly for a (random) time tw followed by abrupt avalanche-like energy drops of sizes S due to strong intermittent fluctuations of energy dissipation. We study the probability distribution P[tw] and P[S] which both exhibit a quite well-defined scaling behaviour. Although tw and S are not statistically correlated, we suggest and numerically checked that their scaling properties are related based on a simple, but non-trivial, scaling argument. We propose that the same approach can be used for other systems showing avalanche-like behaviour such as amorphous solids and seismic events. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'. [ABSTRACT FROM AUTHOR]

We present a novel approach for imaging global mantle discontinuities based on full-waveform inversion (FWI). Over the past decades, extensive research has been done on imaging mantle discontinuities at approximately 400 and 670 km depth. Accurate knowledge of their topography can put strong constraints on thermal and compositional variations and hence geodynamic modelling. So far, however, there is little consensus on their topography. We present an approach based on adjoint tomography, which has the advantage that Fréchet derivatives for discontinuities and measurements, to be inverted for, are fully consistent. Rather than working with real data, we focus on synthetic tests, where the answer is known in order to be able to evaluate the performance of the developed method. All calculations are based on the community code SPECFEM3D_GLOBE. We generate data in fixed 1-D or 3-D elastic background models of mantle velocity. Our 'data' to be inverted contain topography along the 400 and 670 km mantle discontinuities. To investigate the approach, we perform several tests: (i) In a situation where we know the elastic background model 1-D or 3-D, we recover the target topography fast and accurately; (ii) The exact misfit is not of great importance here, except in terms of convergence speed, similar to a different inverse algorithm and (iii) In a situation where the background model is not known, the convergence is markedly slower, but there is reasonable convergence towards the correct target model of discontinuity topography. It has to be noted that our synthetic test is idealized and in a real data situation, the convergence to and uncertainty of the inferred model is bound to be larger. However, the use of data consistent with Fréchet kernels seems to pay off and might improve our consensus on the nature of mantle discontinuities. Our workflow could be incorporated in future FWI mantle models to adequately infer boundary interface topography. [ABSTRACT FROM AUTHOR]

Probabilistic seismic hazard analysis (PSHA) is the standard method used for designing earthquake‐resistant infrastructure. In recent years, several unexpected and destructive earthquakes have sparked criticism of the PSHA methodology. The seismological part of the problem is the true frequency‐magnitude distribution of regional seismicity. Two major models exist, the Gutenberg‐Richter (G‐R) and the Characteristic Earthquake (CE) model, but it is difficult to choose between them. That is because the instrumental, historical, and paleoseimological data available are limited in many regions of interest. Here, we demonstrate how a friction experiment on aggregates of glass beads can produce both regular (CE equivalent) and irregular (G‐R equivalent) stick‐slip. Using a new rotary shear apparatus we produced and analyzed large catalogs of acoustic emission (AE) events related to stick‐slip. The distributions of AE sizes, interevent times, and interevent distances were found to be sensitive to particle size and the applied normal stress, and, to a lesser degree, the stiffness of the loading apparatus. More importantly, the system spontaneously switched behavior for short periods of time. In the context of PSHA, if faults are able to switch behavior as our experimental system does, then justifying the choice of either the CE or the G‐R model is impossible based on existing observations. Key Points: A frictional interface can produce both characteristic and power‐law distributed instabilities in the laboratoryThe system can alternate between the two regimes on the flyIf these characteristics are applicable to natural faults, then seismic hazard analysis must take them into account [ABSTRACT FROM AUTHOR]

The correct implementation of the continuity conditions between different media is fundamental for the accuracy of any wave equation solver used in applications from seismic exploration to global seismology. Ideally, we would like to benchmark a code against an analytical Green's function. The latter, however, is rarely available for more complex media. Here, we provide a general framework through which wave equation solvers can be benchmarked by comparing plane wave simulations to transmission/reflection (R / T) coefficients from plane-wave analysis with exact boundary conditions (BCs). We show that this works well for a large range of incidence angles, but requires a lot of computational resources to simulate the plane waves. We further show that the accuracy of a numerical Green's function resulting from a point-source spherical-wave simulation can also be used for benchmarking. The data processing in that case is more involved than for the plane wave simulations and appears to be sufficiently accurate only below critical angles. Our approach applies to any wave equation solver, but we chose the poroelastic wave equation for illustration, mainly due to the difficulty of benchmarking poroelastic solvers, but also due to the growing interest in imaging in poroelastic media. Although we only use 2-D examples, our exact R / T approach can be extended to 3-D and various cases with different interface configurations in arbitrarily complex media, incorporating, for example, anisotropy, viscoelasticity, double porosities, partial saturation, two-phase fluids, the Biot/squirt flow and so on. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The method of ScS reverberation migration is based on a 'common reflection point' analysis of multiple ScS reflections in the mantle transition zone (MTZ). We examine whether ray-theoretical traveltimes, slownesses and reflection points are sufficiently accurate for estimating the thickness H of the MTZ, defined by the distance between the 410- and 660-km phase transitions. First, we analyse ScS reverberations generated by 35 earthquakes and recorded at hundreds of seismic stations from the combined Arrays in China, Hi-NET in Japan and the Global Seismic Network. This analysis suggests that H varies by about 30 km and therefore that dynamic processes have modified the large-scale structure of the MTZ in eastern Asia and the western Pacific region. Second, we apply the same procedure to spectral-element synthetics for PREM and two 3-D models. One 3-D model incorporates degree-20 topography on the 410 and 660 discontinuities, otherwise preserving the PREM velocity model. The other model incorporates the degree-20 velocity heterogeneity of S20RTS and leaves the 410 and 660 flat. To optimize reflection point coverage, our synthetics were computed assuming a homogeneous grid of stations using 16 events, four of which are fictional. The resolved image using PREM synthetics resembles the PREM structure and indicates that the migration approach is correct. However, ScS reverberations are not as strongly sensitive to H as predicted ray-theoretically because the migration of synthetics for a model with degree-20 topography on the 410 and 660: H varies by less than 5 km in the resolved image but 10 km in the original model. In addition, the relatively strong influence of whole-mantle shear-velocity heterogeneity is evident from the migration of synthetics for the S20RTS velocity model and the broad sensitivity kernels of ScS reverberations at a period of 15 s. A ray-theoretical approach to modelling long-period ScS traveltimes appears inaccurate, at least for continental-scale regions with relatively sparse earthquake coverage. Additional modelling and comparisons with SS precursor and receiver function results should rely on 3-D waveform simulations for a variety of structures and ultimately the implementation of full wave theory. [ABSTRACT FROM AUTHOR]

We analyze the sensitivity of PP precursor traveltimes that are often used to infer lateral variation in the depths of the 410‐ and 660‐km discontinuities in the mantle. Previous results were inconclusive due to complex wave phenomena, such as multiple energy conversions and focusing/defocusing, that hamper their interpretation. Using spectral‐element synthetics and Fréchet derivatives calculated with adjoint methods, we compute sensitivity kernels for volumetric and boundary parameters in a 1‐D model for representative epicentral distances of past studies, and a dominant period of 11–25 s. Our results indicate that the boundary sensitivity of PP precursors is low and that these phases are not coherently seen in exact synthetics. Our most important finding is the strong sensitivity to both shear and compressional wave speeds, indicating that wave interference and wave conversions are dominant. The PP precursor traveltimes appear more sensitive to structural parameters, that is, compressional and shear wave speed, than to the boundaries; therefore, they are unlikely sources for valuable insight into discontinuity topography. Plain Language Summary: Seismologists frequently use traveltimes of seismic waves to understand how the Earth's internal structure changes at various depths. This study presents pictorial examples of how traveltimes of seismic waves sense the depth variations inside the Earth. We focus on waves that reflect from specific depths, and they are often used to explain topography along interfaces at 410‐ and 660‐km depths. Along these depths, the ambient mantle conditions change significantly, indicating areas of major changes in material distribution, which enable vigorous movements within the Earth and cause geological phenomena observed at the surface. We aim at explaining why resulting models using these traveltimes are not in agreement. To achieve that, we use exact methods that give an accurate picture of the sensitivity of the traveltimes under investigation. Our contribution is important because we visualize many effects relating to how the waves travel, convert, and interact with each other. These effects, if not taken into account in an exact manner, could lead to inaccurate mapping of topography. Our explanations can help seismologists understand the reasons of disagreement between models produced using these traveltime data. We conclude that further interpretations relating to temperature and compositional variations should be considered with great care. Key Points: We present a sensitivity analysis of PP precursor traveltimes to clarify the reasons of their reported poor detectabilityThe Fréchet derivatives show that the sensitivity to topography is weak, but high to the compressional and shear wave structureThe usability of PP precursor traveltimes for imaging discontinuity topography is limited, given their weak sensitivity to boundary changes [ABSTRACT FROM AUTHOR]

When a continuum is subjected to an induced stress, the equations that govern seismic wave propagation are modified in two ways. First, the equation of conservation of linear momentum gains terms related to the induced deviatoric stress, and, second, the elastic constitutive relationship acquires terms linear in the induced stress. This continuum mechanics theory makes testable predictions with regard to stress‐induced changes in the elastic tensor. Specifically, it predicts that induced compression linearly affects the prestressed moduli with a slope determined by their local adiabatic pressure derivatives and that induced deviatoric stress produces anisotropic compressional and shear wave speeds. In this article we successfully compare such predictions against ab initio mineral physics calculations for NaCl and MgO. Key Points: We compare ab initio calculations of effects of induced stress on elastic parameters with predictions based on a continuum mechanics theoryWe find that the two methods are in good agreement, without the need of higher‐order theories of elasticityThe theory currently in use for accommodating the effects on nonhydrostatic prestress on seismic wave propagation needs to be modified [ABSTRACT FROM AUTHOR]

In seismic tomography we map the wave speed structure inside the Earth, but we ultimately seek to interpret those images in terms of physical parameters. This is challenging because many parameters can trade‐off with each other to produce a given wave speed. The problem is compounded by the convention of mapping seismic structures as perturbations relative to a 1‐D reference model, rather than absolute wave speeds. Using a full waveform tomography model of Europe as a case study, we quantify the extent to which thermochemical and dynamic properties can be constrained using only S wave speed, expressed in absolute values. The wave speed distributions of this tomography model are compared with 4 million thermochemical models, whose seismic properties are computed via thermodynamic modeling. These models sample the full range of realistic mantle compositions, including variable water and melt contents, and mineral intrinsic anelasticity is taken into account. Intrinsic anelasticity causes waves to travel more slowly at higher temperatures, leading to seismic attenuation, but the sensitivity of the wave speed reduction to temperature is, in turn, controlled by the wave frequency. Global studies of surface waves indicate an anticorrelation between S wave speed and attenuation. We therefore only retain thermochemical models satisfying this anticorrelation. Our study indicates that the frequency dependence of anelasticity, α, depends on temperature or rheology, with α ≈ 0.1 being most appropriate in cold or lithospheric mantle and α ≈ 0.3 in warmer regions (i.e., the asthenosphere). Additionally, the slowest regions require specific compositions and/or a velocity‐weakening mechanism, such as partial melting, elastically accommodated grain boundary sliding, or water. Key Points: Knowing the S wave speed precisely with a loose constraint on mantle attenuation places strong constraints on the temperatureThe frequency dependence of anelasticity α is variable: ~0.1 in cold/lithospheric mantle and ~0.3 in warm/asthenospheric mantleEither partial melting or elastically accommodated grain boundary sliding is required to explain the slowest wave speeds at 130 km [ABSTRACT FROM AUTHOR]

We apply Hamilton's principle to obtain the exact equations of motion for an elastic planet that is rotating, self-gravitating and comprises both fluid and solid regions. This variational problem is complicated by the occurrence of tangential slip at fluid--solid boundaries, but we show how this can be accommodated both directly and using the method of Lagrange multipliers. A novelty of our approach is that the planet's motion is described relative to an arbitrary reference configuration, with this generality offering advantages for numerical calculations. In particular, aspherical topography on the free surface or internal boundaries of the planet's equilibrium configuration can be converted exactly into effective volumetric heterogeneities within a geometrically spherical reference body by applying a suitable particle relabelling transformation. The theory is then specialized to consider the linearized motion of a planet about a steadily rotating equilibrium configuration, with these results having applications to normal mode coupling calculations used within studies of long-period seismology, tidal deformation and related fields. In particular, we explain how our new theory will, for the first time, allow aspherical boundary topography to be incorporated exactly within such coupling calculations. [ABSTRACT FROM AUTHOR]

We present a general concept for evolutionary, collaborative, multiscale inversion of geophysical data, specifically applied to the construction of a first-generation Collaborative Seismic Earth Model. This is intended to address the limited resources of individual researchers and the often limited use of previously accumulated knowledge. Model evolution rests on a Bayesian updating scheme, simplified into a deterministic method that honors today's computational restrictions. The scheme is able to harness distributed human and computing power. It furthermore handles conflicting updates, as well as variable parameterizations of different model refinements or different inversion techniques. The first-generation Collaborative Seismic Earth Model comprises 12 refinements from full seismic waveform inversion, ranging from regional crustal- to continental-scale models. A global full-waveform inversion ensures that regional refinements translate into whole-Earth structure. [ABSTRACT FROM AUTHOR]

We demonstrate how effects of induced stress may be incorporated in seismic modelling and inversion. Our approach is motivated by the accommodation of pre-stress in global seismology. Induced stress modifies both the equation of motion and the constitutive relationship. The theory predicts that induced pressure linearly affects the unstressed isotropic moduli with a slope determined by their adiabatic pressure derivatives. The induced deviatoric stress produces anisotropic compressional and shear wave speeds; the latter result in shear wave splitting. For forward modelling purposes, we determine the weak form of the equation of motion under induced stress. In the context of the inverse problem, we determine induced stress sensitivity kernels, which may be used for adjoint tomography. The theory is illustrated by considering 2-D propagation of SH waves and related Fréchet derivatives based on a spectral-element method. [ABSTRACT FROM AUTHOR]

Using spectral-element simulations of wave propagation, we investigated the sensitivity of seismic waveforms, recorded on transverse components, to upper-mantle discontinuities in 1-D and 3-D background models. These sensitivity kernels, or Fréchet derivatives, illustrate the spatial sensitivity to model parameters, of which those for shear wave speed and the surface topography of internal boundaries are discussed in this paper. We focus on the boundaries at 400 and 670 km depth of the mantle transition zone. SS precursors have frequently been used to infer the topography of upper-mantle discontinuities. These seismic phases are underside reflections off these boundaries and are usually analysed in the distance range of 110°-160°. This distance range is chosen to minimize the interference from other waves. We show sensitivity kernels for consecutive time windows at three characteristic epicentral distances within the 110°-160° range. The sensitivity kernels are computed with the adjoint method using synthetic data. From our simulations we can draw three main conclusions: (i) The exact Fréchet derivatives showthat in all timewindows, and also in those centred on the SS precursors, there is interference from other waves. This explains the difficulty reported in the literature to correct for 3-D shear wave speed perturbations, even if the 3-D structure is perfectly known. (ii) All studies attempting to map the topography of the 400 and 670 km discontinuities to date assume that the traveltimes of SS precursors can be linearly decomposed into a 3-D elastic structure and a topography part. We recently showed that such a linear decomposition is not possible for SS precursors, and the sensitivity kernels presented in this paper explain why. (iii) In agreement with previous work, we showthat other parts of the seismograms have greater sensitivity to upper-mantle discontinuities than SS precursors, especially multiply bouncing S waves exploiting the S-wave triplications due to the mantle transition zone. These phases can potentially improve the inference of global topographic variations of the upper-mantle discontinuities in the context of full waveform inversion in a joint inversion for (an)elastic parameters and topography. [ABSTRACT FROM AUTHOR]

We propose a new approach for generating synthetic earthquakes based on the physics of soft glasses. The continuum approach produces yield-stress materials based on Lattice-Boltzmann simulations. We show that if the material is stimulated below yield stress, plastic events occur, which have strong similarities to seismic events. Based on a suitable definition of displacement in the continuum, we showthat the plastic events obey a Gutenberg-Richter lawwith exponents similar to those for real earthquakes. We also find that the average acceleration, energy release, stress drop and interoccurrence times scale with the same exponent. Furthermore, choosing a suitable definition for aftershocks, we show that they follow Omori's law. Finally, the far field power spectra of elastic waves generated by these plastic events decay as ω-2 similar to those observed for seismic waves. Our approach is fully self-consistent and all quantities can be calculated at all scales without the need of ad hoc friction or statistical assumptions. We therefore suggest that our approach may lead to new insights into the physics connecting the micro- and macroscales of earthquakes. [ABSTRACT FROM AUTHOR]

Owing to the increasing availability of computational resources, in recent years the probabilistic solution of non-linear, geophysical inverse problems by means of sampling methods has become increasingly feasible. Nevertheless, we still face situations in which a Monte Carlo approach is not practical. This is particularly true in cases where the evaluation of the forward problem is computationally intensive or where inversions have to be carried out repeatedly or in a timely manner, as in natural hazards monitoring tasks such as earthquake early warning. Here, we present an alternative to Monte Carlo sampling, in which inferences are entirely based on a set of prior samples--that is, samples that have been obtained independent of a particular observed datum. This has the advantage that the computationally expensive sampling stage becomes separated from the inversion stage, and the set of prior samples--once obtained--can be reused for repeated evaluations of the inverse mapping without additional computational effort. This property is useful if the problem is such that repeated inversions of independent data have to be carried out. We formulate the inverse problem in a Bayesian framework and present a practical way to make posterior inferences based on a set of prior samples. We compare the prior sampling based approach to a Markov Chain Monte Carlo approach that samples from the posterior probability distribution. We show results for both a toy example, and a realistic seismological source parameter estimation problem. We find that the posterior uncertainty estimates obtained based on prior sampling can be considered conservative estimates of the uncertainties obtained by directly sampling from the posterior distribution. [ABSTRACT FROM AUTHOR]

Using the spectral-element method, we explored the effect of topography of upper-mantle discontinuities on the traveltimes of SS precursors recorded on transverse component seismograms. The latter are routinely used to infer the topography of mantle transition zone discontinuities. The step from precursory traveltimes to topographic changes is mainly done using linearised ray theory, or sometimes using finite-frequency kernels. We simulated exact seismograms in 1-D and 3-D elastic models of the mantle. In a second simulation, we added topography to the discontinuities. We compared the waveforms obtained with and without topography by cross correlation of the SS precursors. Since we did not add noise, the precursors are visible in individual seismograms without the need of stacking. The resulting time anomalies were then converted into topographic variations and compared to the original topographic models. Based on the correlation between initial and inferred models, and provided that ray coverage is good, we found that linearised ray theory gives a relatively good idea on the location of the uplifts and depressions of the discontinuities. It seriously underestimates the amplitude of the topographic variations by a factor ranging between 2 and 7. Real data depend on the 3-D elastic structure and the topography. All studies to date correct for the 3-D elastic effects assuming that the traveltimes can be linearly decomposed into a structure and a discontinuity part. We found a strong non-linearity in this decomposition which cannot be modelled without a fully non-linear inversion for elastic structure and discontinuities simultaneously. [ABSTRACT FROM AUTHOR]

Whenever a geophysical image is to be constructed, a variety of choices must be made. Some, such as those governing data selection and processing, or model parametrization, are somewhat arbitrary: there may be little reason to prefer one choice over another. Others, such as defining the theoretical framework within which the data are to be explained, may be more straightforward: typically, an 'exact' theory exists, but various approximations may need to be adopted in order to make the imaging problem computationally tractable. Differences between any two images of the same system can be explained in terms of differences between these choices. Understanding the impact of each particular decision is essential if images are to be interpreted properly—but little progress has been made towards a quantitative treatment of this effect. In this paper, we consider a general linearized inverse problem, applicable to a wide range of imaging situations. We write down an expression for the difference between two images produced using similar inversion strategies, but where different choices have been made. This provides a framework within which inversion algorithms may be analysed, and allows us to consider how image effects may arise. In this paper, we take a general view, and do not specialize our discussion to any specific imaging problem or setup (beyond the restrictions implied by the use of linearized inversion techniques). In particular, we look at the concept of 'hybrid inversion', in which highly accurate synthetic data (typically the result of an expensive numerical simulation) is combined with an inverse operator constructed based on theoretical approximations. It is generally supposed that this offers the benefits of using the more complete theory, without the full computational costs.We argue that the inverse operator is as important as the forward calculation in determining the accuracy of results. We illustrate this using a simple example, based on imaging the density structure of a vibrating string. [ABSTRACT FROM AUTHOR]

The robust and automated determination of earthquake source parameters on a global and regional scale is important for many applications in seismology. We present a novel probabilistic method to invert a wide variety of (waveform) data for point-source parameters in real time using pattern recognition. Inferences are made in the form of marginal probability density functions for point-source parameters and incorporate realistic posterior uncertainty estimates. The neural-network-based method is calibrated using samples from the prior distribution, which are synthetic data vectors, and corresponding sources located in a predefined monitoring volume. Once a set of trained neural networks is available, inversions are fast with very moderate demands on computational resources: an inversion takes less than a second on a standard desktop computer. Uncertainties in the layered Earth model are taken into account in the Bayesian framework and increase the robustness of the results with respect to neglected 3D heterogeneities. Moreover, we find that the method is very robust with respect to perturbations such as observational noise and missing data and therefore is potentially well suited for automated and real-time tasks, such as earthquake monitoring and early warning. We demonstrate the method by means of synthetic tests and by inverting an observed high-rate Global Positioning System displacement dataset for the 2010 Mw 7.2 El Mayor-Cucapah event. Our results are compatible with published point-source estimates for this event within the respective uncertainty bounds. [ABSTRACT FROM AUTHOR]

The determination of earthquake source parameters is an important task in seismology. For many applications, it is also valuable to understand the uncertainties associated with these determinations, and this is particularly true in the context of earthquake early warning (EEW) and hazard mitigation. In this paper, we develop a framework for probabilistic moment tensor point source inversions in near real time. Our methodology allows us to find an approximation to p(m|d), the conditional probability of source models (m) given observations (d). This is obtained by smoothly interpolating a set of random prior samples, using Mixture Density Networks (MDNs)—a class of neural networks which output the parameters of a Gaussian mixture model. By combining multiple networks as ‘committees’, we are able to obtain a significant improvement in performance over that of a single MDN. Once a committee has been constructed, new observations can be inverted within milliseconds on a standard desktop computer. The method is therefore well suited for use in situations such as EEW, where inversions must be performed routinely and rapidly for a fixed station geometry. To demonstrate the method, we invert regional static GPS displacement data for the 2010 MW 7.2 El Mayor Cucapah earthquake in Baja California to obtain estimates of magnitude, centroid location and depth and focal mechanism. We investigate the extent to which we can constrain moment tensor point sources with static displacement observations under realistic conditions. Our inversion results agree well with published point source solutions for this event, once the uncertainty bounds of each are taken into account. [ABSTRACT FROM AUTHOR]

How do body-wave traveltimes constrain the Earth's radial (1-D) seismic structure? Existing 1-D seismological models underpin 3-D seismic tomography and earthquake location algorithms. It is therefore crucial to assess the quality of such 1-D models, yet quantifying uncertainties in seismological models is challenging and thus often ignored. Ideally, quality assessment should be an integral part of the inverse method. Our aim in this study is twofold: (i) we show how to solve a general Bayesian non-linear inverse problem and quantify model uncertainties, and (ii) we investigate the constraint on spherically symmetric P-wave velocity (VP) structure provided by body-wave traveltimes from the EHB bulletin (phases Pn, P, PP and PKP). Our approach is based on artificial neural networks, which are very common in pattern recognition problems and can be used to approximate an arbitrary function. We use a Mixture Density Network to obtain 1-D marginal posterior probability density functions (pdfs), which provide a quantitative description of our knowledge on the individual Earth parameters. No linearization or model damping is required, which allows us to infer a model which is constrained purely by the data.We present 1-D marginal posterior pdfs for the 22 VP parameters and seven discontinuity depths in our model. P-wave velocities in the inner core, outer core and lower mantle are resolved well, with standard deviations of ∼0.2 to 1 per cent with respect to the mean of the posterior pdfs. The maximum likelihoods of VP are in general similar to the corresponding ak135 values, which lie within one or two standard deviations from the posterior means, thus providing an independent validation of ak135 in this part of the radial model. Conversely, the data contain little or no information on P-wave velocity in the D′′ layer, the upper mantle and the homogeneous crustal layers. Further, the data do not constrain the depth of the discontinuities in our model. Using additional phases available in the ISC bulletin, such as PcP, PKKP and the converted phases SP and ScP, may enhance the resolvability of these parameters. Finally, we show how the method can be extended to obtain a posterior pdf for a multidimensional model space. This enables us to investigate correlations between model parameters. [ABSTRACT FROM AUTHOR]

We develop and apply a full waveform inversion method that incorporates seismic data on a wide range of spatio-temporal scales, thereby constraining the details of both crustal and upper-mantle structure. This is intended to further our understanding of crust–mantle interactions that shape the nature of plate tectonics, and to be a step towards improved tomographic models of strongly scale-dependent earth properties, such as attenuation and anisotropy.The inversion for detailed regional earth structure consistently embedded within a large-scale model requires locally refined numerical meshes that allow us to (1) model regional wave propagation at high frequencies, and (2) capture the inferred fine-scale heterogeneities. The smallest local grid spacing sets the upper bound of the largest possible time step used to iteratively advance the seismic wave field. This limitation leads to extreme computational costs in the presence of fine-scale structure, and it inhibits the construction of full waveform tomographic models that describe earth structure on multiple scales. To reduce computational requirements to a feasible level, we design a multigrid approach based on the decomposition of a multiscale earth model with widely varying grid spacings into a family of single-scale models where the grid spacing is approximately uniform. Each of the single-scale models contains a tractable number of grid points, which ensures computational efficiency. The multi-to-single-scale decomposition is the foundation of iterative, gradient-based optimization schemes that simultaneously and consistently invert data on all scales for one multi-scale model.We demonstrate the applicability of our method in a full waveform inversion for Eurasia, with a special focus on Anatolia where coverage is particularly dense. Continental-scale structure is constrained by complete seismic waveforms in the 30–200 s period range. In addition to the well-known structural elements of the Eurasian mantle, our model reveals a variety of subtle features, such as the Armorican Massif, the Rhine Graben and the Massif Central. Anatolia is covered by waveforms with 8–200 s period, meaning that the details of both crustal and mantle structure are resolved consistently. The final model contains numerous previously undiscovered structures, including the extension-related updoming of lower-crustal material beneath the Menderes Massif in western Anatolia.Furthermore, the final model for the Anatolian region confirms estimates of crustal depth from receiver function analysis, and it accurately explains cross-correlations of ambient seismic noise at 10 s period that have not been used in the tomographic inversion. This provides strong independent evidence that detailed 3-D structure is well resolved. [ABSTRACT FROM AUTHOR]

We propose a simple method where the inversion of synthetic data, corresponding to a zero-mean random input vector, is used to infer the average horizontal and vertical resolution lengths of tomographic models. The method works well if the resolution operator has a diagonally dominant structure. This assumption, although often verified in seismic tomography, can be tested by simply cross-correlating the input with the output of the synthetic simulation. The method is as efficient as a single checkerboard test, but reveals more easily interpretable information. [ABSTRACT FROM AUTHOR]

Abstract: The centroid-moment-tensor (CMT) algorithm provides a straightforward, rapid method for the determination of seismic source parameters from waveform data. As such, it has found widespread application, and catalogues of CMT solutions – particularly the catalogue maintained by the Global CMT Project – are routinely used by geoscientists. However, there have been few attempts to quantify the uncertainties associated with any given CMT determination: whilst catalogues typically quote a ‘standard error’ for each source parameter, these are generally accepted to significantly underestimate the true scale of uncertainty, as all systematic effects are ignored. This prevents users of source parameters from properly assessing possible impacts of this uncertainty upon their own analysis. The CMT algorithm determines the best-fitting source parameters within a particular modelling framework, but any deficiencies in this framework may lead to systematic errors. As a result, the minimum-misfit source may not be equivalent to the ‘true’ source. We suggest a pragmatic solution to uncertainty assessment, based on accepting that any ‘low-misfit’ source may be a plausible model for a given event. The definition of ‘low-misfit’ should be based upon an assessment of the scale of potential systematic effects. We set out how this can be used to estimate the range of values that each parameter might take, by considering the curvature of the misfit function as minimised by the CMT algorithm. This approach is computationally efficient, with cost similar to that of performing an additional iteration during CMT inversion for each source parameter to be considered. The source inversion process is sensitive to the various choices that must be made regarding dataset, earth model and inversion strategy, and for best results, uncertainty assessment should be performed using the same choices. Unfortunately, this information is rarely available when sources are obtained from catalogues. As already indicated by Valentine and Woodhouse (2010), researchers conducting comparisons between data and synthetic waveforms must ensure that their approach to forward-modelling is consistent with the source parameters used; in practice, this suggests that they should consider performing their own source inversions. However, it is possible to obtain rough estimates of uncertainty using only forward-modelling. [Copyright &y& Elsevier]

SUMMARY In a synthetic tomographic experiment, we succeeded to recover an idealized narrow mantle plume reaching deep into the lower mantle by using a misfit based on the instantaneous phase difference. A misfit based on simple cross-correlation traveltime shifts leaves the lower mantle part of the plume largely unresolved, despite the use of finite-frequency sensitivity kernels. The time-continuous and amplitude-independent instantaneous phase misfit allows us to measure the interaction between direct and diffracted waves as a function of time, which is difficult to capture by simple cross-correlation traveltime measurements. The diffracted waves arriving later than the main phase are essential to improve the tomographic result. The measurement of diffracted waves yields the necessary information to recover the plume correctly even in the lower mantle. The instantaneous phase measurement is ideal to capture this interaction, but other time- or frequency-dependent measurements may give similar results. We also investigated the effect of wavefront healing on cross-correlation traveltime shifts for a range of differently sized idealized mantle plumes. We confirm that wavefront healing severely reduces traveltime shifts when the plume conduit is considerably thinner than the width of the first Fresnel zone. For plume conduits with a diameter on the order of 100 km, even traveltime shifts measured at periods as short as T= 1 s are affected. [ABSTRACT FROM AUTHOR]

SUMMARY What makes a seismogram look like a seismogram? Seismic data sets generally contain waveforms sharing some set of visual characteristics and features-indeed, seismologists routinely exploit this when performing quality control 'by hand'. Understanding and harnessing these characteristics offers the prospect of a deeper understanding of seismic waveforms, and opens up many potential new techniques for processing and working with data. In addition, the fact that certain features are shared between waveforms suggests that it may be possible to transform the data away from the time domain, and represent the same information using fewer parameters. If so, this would be a significant step towards making fully non-linear tomographic inversions computationally tractable. Hinton & Salakhutdinov showed that a particular class of neural network, termed 'autoencoder networks', may be used to find lower-dimensional encodings of complex binary data sets. Here, we adapt their work to the continuous case to allow the use of autoencoders for seismic waveforms, and offer a demonstration in which we compress 512-point waveforms to 32-element encodings. We also demonstrate that the mapping from data to encoding space, and its inverse, are well behaved, as required for many applications. Finally, we sketch a number of potential applications of the technique, which we hope will be of practical interest across all seismological disciplines, and beyond. [ABSTRACT FROM AUTHOR]

SUMMARY We propose a new method for the quantitative resolution analysis in full seismic waveform inversion that overcomes the limitations of classical synthetic inversions while being computationally more efficient and applicable to any misfit measure. The method rests on (1) the local quadratic approximation of the misfit functional in the vicinity of an optimal earth model, (2) the parametrization of the Hessian in terms of a parent function and its successive derivatives and (3) the computation of the space-dependent parameters via Fourier transforms of the Hessian, calculated with the help of adjoint techniques. In the simplest case of a Gaussian approximation, we can infer rigorously defined 3-D distributions of direction-dependent resolution lengths and the image distortion introduced by the tomographic method. We illustrate these concepts with a realistic full waveform inversion for upper-mantle structure beneath Europe. As a corollary to the method for resolution analysis, we propose several improvements to full waveform inversion techniques. These include a pre-conditioner for optimization schemes of the conjugate-gradient type, a new family of Newton-like methods, an approach to adaptive parametrization independent from ray theory and a strategy for objective functional design that aims at maximizing resolution. The computational requirements of our approach are less than for a typical synthetic inversion, but yield a much more complete picture of resolution and trade-offs. While the examples presented in this paper are rather specific, the underlying idea is very general. It allows for problem-dependent variations of the theme and for adaptations to exploration scenarios and other wave-equation-based tomography techniques that employ, for instance, georadar or microwave data. [ABSTRACT FROM AUTHOR]

We present an extension of the adjoint method that allows us to compute the second derivatives of seismic data functionals with respect to Earth model parameters. This work is intended to serve as a technical prelude to the implementation of Newton-like optimization schemes and the development of quantitative resolution analyses in time-domain full seismic waveform inversion. The Hessian operator H applied to a model perturbation can be expressed in terms of four different wavefields. The forward field excited by the regular source, the adjoint field excited by the adjoint source located at the receiver position, the scattered forward field and the scattered adjoint field . The formalism naturally leads to the notion of Hessian kernels, which are the volumetric densities of . The Hessian kernels appear as the superposition of (1) a first-order influence zone that represents the approximate Hessian, and (2) second-order influence zones that represent second-order scattering. To aid in the development of physical intuition we provide several examples of Hessian kernels for finite-frequency traveltime measurements on both surface and body waves. As expected, second-order scattering is efficient only when at least one of the model perturbations is located within the first Fresnel zone of the Fréchet kernel. Second-order effects from density heterogeneities are generally negligible in transmission tomography, provided that the Earth model is parameterized in terms of density and seismic wave speeds. With a realistic full waveform inversion for European upper-mantle structure, we demonstrate that significant differences can exist between the approximate Hessian and the full Hessian-despite the near-optimality of the tomographic model. These differences are largest for the off-diagonal elements, meaning that the approximate Hessian can lead to erroneous inferences concerning parameter trade-offs. The full Hessian, in contrast, allows us to correctly account for the effect of non-linearity on model resolution. [ABSTRACT FROM AUTHOR]

Resolution in seismic tomography intimately depends on data coverage, with different parts of seismograms sensitive to different parts of Earth's structure. In classical seismic tomography, the usable amount of data is often restricted because of approximations to the wave equation. 3-D numerical simulations of wave propagation provide new opportunities for increasing the amount of usable data in seismograms by choosing appropriate misfit functions which have direct control on Fréchet derivatives. We propose new misfit functions for full waveform tomography based on instantaneous phase differences and envelope ratios between observed and synthetic seismograms. The aim is to extract as much information as possible from a single seismogram. Using the properties of the Hilbert transform, we separate phase and amplitude information in the time domain. To gain insight in the advantages and disadvantages of chosen misfit functions, we make qualitative comparisons of the corresponding finite-frequency adjoint sensitivity kernels with those from commonly used misfit functions based on cross-correlation traveltime, amplitude and waveform differences. The major advantages of our misfit functions are: (1) working in the Hilbert domain reduces non-linear behaviour of waveforms due to interaction of phase and amplitude information, and (2) we show with noise-free synthetic seismograms that it is possible to use a complete seismogram without losing information from low-amplitude phases. Complementary to instantaneous phase measurements, envelope measurements provide a way of using amplitude information of waveforms, which may also easily be extended to constrain anelastic properties. The properties of the kernels allow us to simplify the tomography problem by separating elastic and anelastic inversions. First indications are that the kernels remain well behaved in the presence of noise. [ABSTRACT FROM AUTHOR]

Wave front healing, in which diffractions interfere with directly travelling waves causing a reduction in recorded traveltime delays, has been postulated to cause a bias towards faster estimated earth models. This paper reviews the theory from the mathematical physics community that explains the properties of diffractions and applies it to a suite of increasingly complicated numerical examples. We focus in particular on the elastic case and on the differences between P and S healing. We find that rather than introducing a systemic fast bias, wave front healing gives a more complicated bias in the results of traveltime tomography, with fast anomalies even manifesting themselves as slow anomalies in some situations. Of particular interest, we find that a negative correlation between the bulk and shear or compressional velocities may result to a large extend from healing. [ABSTRACT FROM AUTHOR]

We investigated the agreement between real seismograms and those predicted by long wavelength mantle models by looking at phase and amplitude differences. We computed full synthetic seismograms using a spectral element method together with 3-D mantle models and the appropriate crustal model on top. We selected differently damped mantle models to see the effect of regularization on the computed seismograms. To check the phase agreement, we measured time-shifts between the real and synthetic surface waves and body wave phases such as P, S and SS using a cross-correlation technique. We also compared the amplitudes of the real and synthetic seismograms to understand how well the models explain not only the phases, but the whole waveforms. 3-D mantle models improve the phase agreement of surface waves in particular. The remaining misfit, however, is still so large that we cannot distinguish between different tomographic models. We suggest that this is mainly due to an imperfect modelling of the crust, and/or source location if body waves are included, which have to be addressed in future inversions. Amplitude mismatches are large, regardless which 3-D mantle model is used. We observe that 3-D scattering or focusing/defocusing effects can only explain half of the surface wave amplitudes whereas body wave amplitudes are dominated by scattering effects. 1-D Q models, particularly in the crust and upper-mantle, strongly affect surface wave amplitudes and have to be modelled properly. [ABSTRACT FROM AUTHOR]