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Imaging seismic velocity of the Earth has been implemented widely for years. The majority of these studies are based on linear or non-linear methods that minimize the difference between seismic observations and predictions of these observations from simplified models of the Earth (tomographic models). Another family of methods, based on the work of Backus & Gilbert (1968), constrains Earth models by maximizing their resolution. A numerically tractable version of such linear local averaging methods, called SOLA, was recently been adapted to seismic tomography by Zaroli (2016). When correctly implemented, SOLA tends to reduce artifacts caused by uneven path coverage. It also provides information about model uncertainties and resolutions. We are the first to have applied the SOLA Backus-Gilbert method to group velocity dispersion tomography of the Northwest Iranian plateau. We used Rayleigh wave dispersion curves obtained from vertical component seismograms of local and regional M ≥ 4.5 earthquakes that occurred from 2010 to 2021. We also used cross-correlations of ambient seismic noise from January 2013 to the end of December 2015. We allowed the resolution to vary with location and adapted the target resolution based on the local path density. We included data uncertainties based on the location uncertainties of the earthquakes and on the energy in the dispersion curves at each period. We selected the trade-off parameter between model resolution and model uncertainties using a standard L-curve.We present group velocity maps at periods between 10 and 50 seconds as well as maps of model resolution lengths and uncertainties. We also present maps that mask regions where the anomalies are within the uncertainties to highlight the strongly anomalous regions. Our short-period maps reveal the relatively lower velocities in eastern Anatolia and western parts of NW Iran can be explained by partially melt zones in the crust, in accordance with the study of keshin (2003) who proposed extensive melting in the crust because of the interaction of hot asthenosphere with the Eastern Anatolian Accretionary Complex. Also, higher velocity anomalies along the Sanandaj-Srijan metamorphic zone (SSZ), can be related to the sedimentary and metamorphic Paleozoic-Cretaceous rocks. The low velocities observed along the Zagros fault thrust belt are also well correlated with high and shallow seismicity in this zone (Maggi et al 2000) which implies the presence of an upper crust tectonically very active.Our long-period maps reveal high-velocity anomalies beneath the Alborz and low-velocity zone in SSZ. The low-velocity anomalies are mainly due to a thin lithosphere or the absence of a lithospheric mantle, while high velocities can be related to the presence of a stable continental mantle lid or an oceanic-like lithosphere. Keywords: SOLA Backus-Gilbert, Group Velocity, Inverse theory, North-West of Iran, Tomography.

Linking geodynamic models to observations from seismology is essential for improving our understanding of the present-day thermodynamic state of the mantle. From the geodynamic perspective, 3D mantle circulation models (MCMs) yield physically relevant predictions of the global distribution of buoyancy forces, while complementing information is available from seismic data and tomography that can reveal the location and morphology of mantle heterogeneity. Investigating this powerful interplay in a fully synthetic framework has great potential. It allows us to make robust interpretations of mantle structure provided that quantitatively meaningful comparisons can be made. This especially relates to the magnitudes of heterogeneity that can not be effectively constrained by the individual modelling approaches.Following this general concept, there are two possible links: 1) synthetic seismic data can be predicted from the MCM and statistically be compared against observed data. 2) the MCM gets modified by a tomographic operator (informing us about spatially variable seismic resolution and, if applicable, model uncertainty), and subsequently this filtered version gets compared against the corresponding tomographic image from real observations.Here, we discuss these two strategies together based on observed data for S-wave cross-correlation traveltime residuals that have been applied to global seismic tomography. Taking the same set of source-receiver configurations, synthetic traveltime predictions are computed in a state-of-the-art MCM using ray theory (RT), paraxial finite-frequency kernels (FFK), as well as cross-correlation measurements on synthetic seismograms (SPECFEM). The latter requires computationally demanding 3D-wavefield simulations using SPECFEM3D_GLOBE for an earthquake catalog comprising over 4,200 teleseismic events.These data sets can be used for tomographic filtering by application of the generalized inverse operator of the actual tomographic model. Filtered MCMs derived from the differently predicted data sets appear largely similar on a global scale with regards to the shape and amplitudes of imaged mantle heterogeneity. This is observed despite the lack of more accurate wave physics in RT or FFK and possible measurement errors for the SPECFEM data that, although being computed in a synthetic case, can not be completely ruled out. Stronger differences between filtered models appear in regions of higher image resolution where model uncertainty by propagated data errors can play a more prominent role.We discuss the impact of the different filtering strategies by comparing filtered models to the original MCM and synthetic traveltime residuals to the underlying real observations. The results strongly highlight the need for incorporating both resolution and model uncertainty in combined tomographic-geodynamic studies.

Seismic tomography provides valuable insights into the structure, composition and evolution of the mantle. However, the origin of structures like the Large-Low-Velocity-Provinces (LLVPs) in the lowermost mantle remains debated. Their velocity anomalies have been interpreted to be due to purely thermal or also compositional variations, with implications for mantle circulation, the evolution of the core and the Earth’s heat budget.To uniquely interpret seismic structures such as the LLVPs, it is crucial to constrain the relationships between different seismic observables, e.g. the ratio between shear-wave velocity (Vs) and compressional-wave velocity (Vp) variations. Joint inversions of seismic velocities have been performed, but their velocity amplitudes may be biased, uncertainties are typically not provided, and the resolution of Vs and Vp structures generally differs in existing models.To overcome these issues, we make use of the recently developed SOLA method (Zaroli, 2016), which is based on a Backus-Gilbert philosophy. Instead of finding a model with a particular data fit, we aim to construct model averages of the true Earth with uncertainties, whilst having a control on the model resolution. This direct control on resolution enables us to build Vs and Vp models that sample the same parts of the mantle, and therefore to robustly constrain the Vs/Vp ratio.Here, we test this philosophy by applying the SOLA method to normal modes. These free oscillations of the Earth are particularly useful to study the relationships between seismic velocities as they are directly sensitive to multiple physical parameters, including Vs, Vp as well as density. We illustrate our approach and discuss the trade-off between uncertainties and resolution using synthetic tests for both Vs and Vp, before showing real data inversions. Finally, we discuss the implications of our results for the Vs/Vp ratio in terms of mantle temperature and composition.

For a thorough understanding of the impact of mantle convection on vertical motions of the lithosphere, computational modeling plays a crucial role. Mantle circulation can be modeled by solving the equations of motion of a fluid using Earth-like input parameters assimilating plate motions at the surface in discrete steps through time. Thus, a realistic Earth model relies on the robustness of the inserted information. However, apart from the general difficulty of inferring deep Earth’s properties, also the plate tectonic model introduces uncertainty. Especially the linking of relative plate motions to absolute position relies on controversial assumptions such as fixity of structures in the mantle (e.g., plumes or Large-Low-Shear-Velocity Provinces) or the association between subducted plates at depth and high velocity regions in tomographic images. The latter specifically are restricted by non-uniqueness and the need to regularize the inversions, distorting structures and damping heterogeneity amplitudes.In order to infer secondary results from an MCM, it is thus important to validate the model against independent observations. Here, we employ Earth’s free oscillations that feature global sensitivity to 3-D structure for model assessment, complementing our earlier work using seismic body wave data. To this end, the temperature field of a published MCM is converted to seismic velocity with the help of a thermodynamic model of mantle mineralogy. An effective forward approach for the computation of normal mode data from synthetic Earth models is the calculation of splitting functions, describing the distortion of characteristic frequency peaks in the spectrum induced by even degree structural heterogeneity. A general problem is that the sensitivity of normal modes with depth often shows oscillatory behaviour preventing a straight forward relation of frequency shifts to structure in a certain depth range. This can be mitigated by combining kernels of several modes via a Backus-Gilbert approach to obtain focused sensitivity in pre-specified depth ranges of the mantle. For testing the significance of relevant model differences in splitting function data, geometrical alterations mimicking changes in the absolute reference frame and viscosity were applied to a pre-computed MCM. Current results indeed indicate that normal mode data are sensitive to such model changes within their respective uncertainty ranges.

SUMMARY Since most tomographic problems deal with imperfect data coverage and noisy data, an estimate of the seismic velocity in the Earth can only be a local average of the ‘true’ velocity with some attached uncertainty. We use the SOLA (subtractive optimally localized averages) method, a Backus–Gilbert-type method based on the resolution–uncertainty trade-off, to build a range of models of Rayleigh-wave velocities in the Pacific upper mantle. We choose one solution and show how to analyse the model using its resolution and uncertainties. We exploit the model statistics to evaluate the significance of deviations from a theoretical prediction: a half-space cooling model of the Pacific lithosphere. We investigate a slow-velocity anomaly located northeast of Hawaii, at about 200 km depth, and a pattern of alternatively slow- and fast-velocity bands, aligned approximately northwest to southeast, between 200 and 300 km depth. According to our resolution and uncertainty analyses, both features seem to be resolved.

Tomographic-geodynamic model comparisons are a key component in studies of the present-day thermodynamic state of the mantle. A fundamental prerequisite for quantitatively meaningful comparisons is “tomographic filtering” of the geodynamic model. This means that geodynamically predicted mantle structures have to be modified to account for the spatially variable resolving power of tomographic images, i.e. to mimic the effects of uneven data coverage and regularization. Different approaches for tomographic filtering are available, but it is so far unclear which one will be the method of choice in the context of computationally demanding retrodictions of past mantle flow.Here, we investigate the impact of the possible filtering approaches in a fully synthetic framework. For the first time in a mantle circulation model (MCM), we simulate 3D-wavefields and seismograms for an entire tomographic earthquake catalogue with over 4,200 events using SPECFEM3D_GLOBE. We use both classic filtering with the resolution operator R, as well as the recently introduced “generalized inverse projection” (GIP; Freissler et al. 2020) to generate tomographically filtered versions of the MCM.In the GIP method, the generalized inverse operator of a given tomographic image is applied to synthetic seismic data predicted from the geodynamic model, as well as to potential data errors, to obtain the filtered MCM plus the propagated error. Important to note, the same generalized inverse operator is applied to an observed data set to build the tomographic model. A physically accurate prediction of synthetic data, here realized with the seismograms from numerical wave propagation, thus enables GIP filtering to consistently reproduce the tomographic imaging process. This is an important methodological advantage over classic filtering with R, where an unphysically reparametrized version of the MCM is filtered directly in model space and seismic data errors can not be considered.In our study, GIP-filtered models are computed with cross-correlation S-wave traveltime residuals from the synthetic seismograms, as well as with banana-doughnut kernel and ray-theoretical traveltime predictions. The differently filtered models are compared against each other using statistical measures. By taking the GIP-filtered model that is based on the 3D-wavefield simulations as a reference, we can quantify the impact of reparametrization in classic filtering versus the lack of exact wave physics when using less accurate methods for traveltime predictions in the GIP filtering. Additionally, all filtered models can be compared to the underlying original structure of the MCM.Detailed knowledge of tomographic filtering effects with different strategies is required prior to efforts on the associated uncertainty quantification in data-driven geodynamic retrodictions of mantle evolution.

Seismic tomography is a powerful tool to study the deep Earth, given the lack of direct observations. Seismic structures can be interpreted together with constraints from other disciplines, such as geodynamics and mineral physics, to provides valuable information about the structure, dynamics and evolution of the mantle. Nevertheless, a robust physical interpretation of seismic images remains challenging as tomographic models typically lack uncertainty information and may have biased amplitudes due to uneven data coverage and regularisation.We aim to build tomographic models of the mantle with associated uncertainties and unbiased amplitudes. For this, we use the SOLA method (Zaroli, 2016) applied to normal mode data, the Earth’s free oscillations. SOLA is based on a Backus-Gilbert approach, which explicitly constrains the amplitudes to be unbiased and inherently computes the model uncertainty and resolution. This approach enables us to perform meaningful physical interpretations of the imaged structures. By applying this method to normal modes, we obtain valuable insights on the long wavelength structure of the mantle. The use of normal modes also has several advantages: these data are sensitive to multiple parameters, including both Vs and Vp anisotropy as well as density, and they provide global data coverage.Here, we report on our progress towards a new 3-D mantle model based on the inversion of normal mode splitting function data. We discuss initial results from synthetic tests and isotropic inversions in terms of model estimates, uncertainties and resolution.

Tomographic models suffer from unevenly distributed noisy data and therefore have complicated resolution and uncertainties that can hinder their interpretation. Using linear Backus & Gilbert inversion, it is possible to obtain tomographic models with resolution and uncertainties in a single step. Using such a method, we aim to produce a three-dimensional tomographic model of the Pacific upper mantle from surface-wave data. To linearise the forward problem, we use finite-frequency theory to describe the sensitivity of surface-wave phase-delays to the three-dimensional shear-wave velocity. We build a data-base of phase-delay measurements for surface-waves that cross the Pacific Ocean. We estimate the data uncertainties caused by measurement errors using a multitaper technique and those caused by poor knowledge of the seismic source and crust by a Monte-Carlo method. Using the Backus & Gilbert approach, the phase-delay dataset, and the data uncertainty estimates, we obtain a model of the shear-wave velocity of the Pacific upper mantle together with its three-dimensional resolution and uncertainties. These allow us to discuss, using robust statistical arguments, the existence and the three-dimensional organisation of structures we expect to see in the Pacific upper mantle, such as plume-like upwellings or small-scale sub-lithospheric convections.

Seismic tomography is essential for imaging the Earth’s interior in order to better understand the dynamic processes at work. However, robust physical interpretation of tomographic images remain difficult as the inverse problem is under-determined, model amplitudes are biased and uncertainties are usually not quantified.Commonly-used techniques, such as damped least-square inversions, break the non-uniqueness of the model solution by adding a subjective, ad hoc, regularization, which can lead to biased amplitudes and potential physical misinterpretations. The SOLA method (Zaroli, 2016; Zaroli et al., 2017), based on a Backus-Gilbert approach, removes the non-uniquess by averaging, rather than introducing a subjective regularization. The method explicitly constrains the amplitudes to be unbiased and the computation of the model resolution and uncertainty is inherent and efficient. Instead of aiming to minimize the data fit, the SOLA approach aims to minimize the size of the averaging volume and the associated uncertainties.We aim to build a new tomographic model of the Earth’s mantle using the SOLA method. We focus our observations on normal mode data, the standing waves of the Earth observed after very large earthquakes, which are not affected by an uneven data distribution. As normal modes are sensitive to multiple seismic parameters, we treat the sensitivity to different parameters as so called “3D noise” within the SOLA framework. We are specifically interested in constraining seismic anisotropy, which provides more direct information on mantle flow.Here, we report on some forward modelling results, fundamental to understanding normal mode sensitivity to seismic anisotropy at different depths and identifying which modes to focus on during inversions. We also show our initial work towards building a new tomography model, including the calculation of 3D noise and target kernels.

SUMMARY Tomographic-geodynamic model comparisons are a key component in studies of the present-day state and evolution of Earth’s mantle. To account for the limited seismic resolution, ‘tomographic filtering’ of the geodynamically predicted mantle structures is a standard processing step in this context. The filtered model provides valuable information on how heterogeneities are smeared and modified in amplitude given the available seismic data and underlying inversion strategy. An important aspect that has so far not been taken into account are the effects of data uncertainties. We present a new method for ‘tomographic filtering’ in which it is possible to include the effects of random and systematic errors in the seismic measurements and to analyse the associated uncertainties in the tomographic model space. The ‘imaged’ model is constructed by computing the generalized-inverse projection (GIP) of synthetic data calculated in an earth model of choice. An advantage of this approach is that a reparametrization onto the tomographic grid can be avoided, depending on how the synthetic data are calculated. To demonstrate the viability of the method, we compute traveltimes in an existing mantle circulation model (MCM), add specific realizations of random seismic ‘noise’ to the synthetic data and apply the generalized inverse operator of a recent Backus–Gilbert-type global S-wave tomography. GIP models based on different noise realizations show a significant variability of the shape and amplitude of seismic anomalies. This highlights the importance of interpreting tomographic images in a prudent and cautious manner. Systematic errors, such as event mislocation or imperfect crustal corrections, can be investigated by introducing an additional term to the noise component so that the resulting noise distributions are biased. In contrast to Gaussian zero-mean noise, this leads to a bias in model space; that is, the mean of all GIP realizations also is non-zero. Knowledge of the statistical properties of model uncertainties together with tomographic resolution is crucial for obtaining meaningful estimates of Earth’s present-day thermodynamic state. A practicable treatment of error propagation and uncertainty quantification will therefore be increasingly important, especially in view of geodynamic inversions that aim at ‘retrodicting’ past mantle evolution based on tomographic images.

For a comprehensive link between seismic tomography and geodynamic models, uncertainties in the seismic model space play a non-negligible role. More specifically, knowledge of the tomographic uncertainties is important for obtaining meaningful estimates of the present-day thermodynamic state of Earth's mantle, which form the basis of retrodictions of past mantle evolution using the geodynamic adjoint method. A standard tool in tomographic-geodynamic model comparisons nowadays is tomographic filtering of mantle circulation models using the resolution operator R associated with the particular seismic inversion of interest. However, in this classical approach it is not possible to consider tomographic uncertainties and their impact on the geodynamic interpretation. Here, we present a new method for 'filtering' synthetic Earth models, which makes use of the generalised inverse operator G†, instead of using R. In our case, G† is taken from a recent global SOLA Backus–Gilbert S-wave tomography. In contrast to classical tomographic filtering, the 'imaged' model is constructed by computing the Generalised-Inverse Projection (GIP) of synthetic data calculated in an Earth model of choice. This way, it is possible to include the effects of noise in the seismic data and thus to analyse uncertainties in the resulting model parameters. In order to demonstrate the viability of the method, we compute a set of travel times in an existing mantle circulation model, add specific realisations of Gaussian, zero-mean seismic noise to the synthetic data and apply G†. Our results show that the resulting GIP model without noise is equivalent to the mean model of all GIP realisations from the suite of synthetic 'noisy' data and also closely resembles the model tomographically filtered using R. Most important, GIP models that include noise in the data show a significant variability of the shape and amplitude of seismic anomalies in the mantle. The significant differences between the various GIP realisations highlight the importance of interpreting and assessing tomographic images in a prudent and cautious manner. With the GIP approach, we can moreover investigate the effect of systematic errors in the data, which we demonstrate by adding an extra term to the noise component that aims at mimicking the effects of uncertain crustal corrections. In our presentation, we will finally discuss ways to construct the model covariance matrix based on the GIP approach and point out possible research directions on how to make use of this information in future geodynamic modelling efforts.

3D-printing techniques allow us to visualise geophysical concepts that are difficult to grasp, making them perfect for incorporation into teaching and outreach packages. Abstract models, often represented as 2D coloured maps, become more tactile when represented as 3D physical objects. In addition, new questions tend to be asked and different features noticed when handling such objects, while they also make outreach and education more inclusive to the visually impaired.Some of our most effective models are simply exaggerated planetary topography in 3D, including Earth, Mars and the Moon. The resulting globes provide a powerful way to explain the importance of plate tectonics in shaping a planet and linking surface features to deeper dynamic processes. In addition, we have developed a simple method for portraying abstract global models by 3D printing globes of surface topography, representing the parameter of interest as additional, exaggerated long-wavelength topography. This workflow has been applied to models of dynamic topography, the geoid and seismic tomography. In analogy to Russian nesting dolls, the resulting “seismic matryoshkas” have multiple layers that can be removed by the audience to explore the structures present deep within our planet and learn about the ongoing dynamic processes.While these 3D objects are easily printed on a cheap (

Globe representation of the Earth has a long history in pedagogy and outreach. To help people realize global processes, these representations allow the conception and the manipulation of global fields and planetary geography. The realization of a physical representation of such global fields is demanding. 3D printing allows representing well scalar data at a fixed time, via for example the deformation of elevation maps. We propose here an alternative allowing to represent easily dynamic fields, and reproducing in a simple principle the effect obtained by the first astronauts visualizing planet Earth as a "pale blue dot". To that effect, we use virtual reality and represent mobile fields on a globe, associated with a physical object permitting spatial manipulation. The open software Unity, common in videogame conception and development, and the library Vuforia, allowing virtual reality, are utilized for the development. The fields represented are associated with the solid earth, and with oceanic and atmospheric dynamics: Seismic velocity fields, global seismicity catalogs, geoid, geothermal gradient, or oceanic and atmospheric currents. The software is can be easily deployed on tablets and phones, complementing printed images.

This proof-of-concept study presents a parameter-free, linear Backus–Gilbert inversion scheme, tractable for seismic tomography problems. It leads to efficient computations of unbiased tomographic images, accompanied by meaningful resolution and uncertainty infor- mations. Moreover, as there is no need to parametrize the model space in this parameter-free approach, it enables numerically accurate data sensitivity kernels to be effectively exploited in tomographic inversions. This is a major benefit over discrete tomographic methods, for which data sensitivity kernels are often inaccurate, as they are projected on a given model parametrization prior to be exploited in the inversion, and these parametrizations are usually coarse to limit the number of parameters and keep tractable the problems of model estima- tion and/or appraisal. Therefore, this new tomographic scheme fuels great hopes on better constraining multiscale seismic heterogeneities in the Earth’s interior by exploiting accurate data sensitivity kernels, that is, taking full advantage of known wave-propagation physics, and enabling quantitative appraisals of tomographic features. As a remark, since its computational cost grows as a function of the total number of data squared, it may be better suited to han- dle moderate-size data sets, typically encountered in regional-scale tomography. Theoretical developments are illustrated within a finite-frequency physical framework. A set of 27 070 teleseismic S -wave time residuals is inverted, with focus on imaging and appraising shear- wave velocity anomalies lying in the mantle below Southeast Asia, in the 350–1410 km depth range.

International audience; The appraisal of tomographic models, of fundamental importance towards better understanding the Earth's interior, consists in analysing their resolution and covariance. The discrete theory of Backus–Gilbert, solving all at once the linear problems of model estimation and appraisal, aims at evaluating weighted averages of the true model parameters. Contrary to damped least-squares techniques, one key advantage of Backus–Gilbert inversion is that no subjective regularization is needed to remove the non-uniqueness of the model solution. Indeed, it is often possible to identify unique linear combinations of the parameters even when the parameters themselves are not uniquely defined. In other words, the non-uniqueness can be broken by averaging rather than regularizing. Over the past few decades, many authors have considered that, in addition to a high computational cost, it could be a clumsy affair in the presence of data errors to practically implement the Backus–Gilbert approach to large-scale tomographic applications. In this study, we introduce and adapt to seismic tomography the Subtractive Optimally Localized Averages (SOLA) method, an alternative Backus–Gilbert formulation which retains all its advantages, but is more computationally efficient and versatile in the explicit construction of averaging kernels. As a leitmotiv, we focus on global-scale S-wave tomography and show that the SOLA method can successfully be applied to large-scale, linear and discrete tomographic problems.

We investigate the influence of crust on time residual measurements made by cross-correlation in the 10–51 s filtering period range on a global scale, considering two crustal models: CRUST2.0 and CRUST1.0. This study highlights, in a quantitative way, crust-related time corrections. One part of this correction is directly linked to the body wave traveltime through the crust as predicted by the ray theory, whereas a second part is related to interferences with multiple crustal reflections. This second component, called finite-frequency (FF) crustal correction, is frequency-dependent unlike the ray-theory based correction. We show that if this frequency-dependent crust-related correction is not taken into account in cross-correlation measurements, it may lead to a dispersive effect in S-wave delay-times that could ultimately bias tomographic models. On average, this FF correction increases with the filtering period. Comparisons between the two crustal models highlight the significant dispersive effect of the crust, which has complex patterns depending on geological contexts, with an important role of the sediment thickness. Although ray crustal corrections remain important, FF crustal effects may lead to a bias in measurements if not properly taken into account; on average they may reach 0.9–1.6 s for CRUST2.0 and 0.5–1.6 s for CRUST1.0, for period ranging from 10 to 51 s, respectively.

International audience; Inverse problems occur in a wide range of scientific applications, such as in the fields of signal processing, medical imaging, or geophysics. This work aims to present to the field practitioners, in an accessible and concise way, several established and newer cutting-edge computational methods used in the field of inverse problems—and when and how these techniques should be employed.

In this study, we confirm the existence of a change in the shear velocity spectrum around 1000 km depth based on a new shear velocity tomographic model of the Earth’s mantle, SEISGLOB2. This model is based on Rayleigh surface wave phase velocities, self- and crosscoupling structure coefficients of spheroidal normal modes and body wave traveltimes which are, for the first time, combined in a tomographic inversion. SEISGLOB2 is developed up to spherical harmonic degree 40 and in 21 radial spline functions. The spectrum of SEISGLOB2 is the flattest (i.e. richest in ‘short’ wavelengths corresponding to spherical harmonic degrees greater than 10) around 1000 km depth and this flattening occurs between 670 and 1500 km depth.We also confirmvarious changes in the continuity of slabs and mantle plumes all around 1000 km depth where we also observed the upper boundary of Large Low Shear Velocity Provinces. The existence of a flatter spectrum, richer in short-wavelength heterogeneities, in a region of the mid-mantle can have great impacts on our understanding of the mantle dynamics and should thus be better understood in the future. Although a viscosity increase, a phase change or a compositional change can all concur to induce this change of pattern, its precise origin is still very uncertain.

Geophysical tomographic studies traditionally exploit linear, damped least-squares inversion methods. We demonstrate that the resulting models can be locally biased toward lower or higher amplitudes in regions of poor data illumination, potentially causing physical misinterpretations. For example, we show that global model S40RTS is locally biased toward higher amplitudes below isolated receivers where ray paths are quasi vertical, such as on Hawaii. This leads to questions on the apparent low-velocity structure interpreted as the Hawaii hotspot. We prove that a linear Backus–Gilbert inversion scheme can bring the Earth's interior into focus through unbiased tomographic lenses, as its model estimates are constrained to be averages over the true model. It also efficiently computes the full generalized inverse required to infer both model resolution and its covariance, enabling quantitative interpretations of tomographic models.

The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite precision arithmetic) the exact solution after a finite number of iterations. It is thus well suited for many types of inverse problems. On the other hand, the method requires the computation of the gradient. Here difficulty can arise, since the functional of interest to the given inverse problem may not be differentiable. In this paper, we review two approaches to deal with this situation: iteratively reweighted least squares and convolution smoothing. We apply the methods to a more generalized, two parameter penalty functional. We show advantages of the proposed algorithms using examples from a geotomographical application and for synthetically constructed multi-scale reconstruction and regularization parameter estimation., arXiv admin note: text overlap with arXiv:1408.6795

SUMMARY Current interpretations of seismic observations typically argue for significant chemical heterogeneity being present in the two large low shear velocity provinces under Africa and the Pacific. Recently, however, it has been suggested that large lateral temperature variations in the lowermost mantle resulting from a strong thermal gradient across D″ may provide an alternative explanation. In case of a high heat flux from the core into the mantle, the magnitude of shear wave velocity variations in tomographic models can be reconciled with isochemical whole mantle flow and a pyrolite composition. So far, the hypothesis of strong core heating has been tested in a consistent manner only against tomographic S-wave velocity models, but not against P-wave velocity models. Here, we explore a new approach to assess geodynamic models and test the assumption of isochemical whole mantle flow with strong core heating directly against the statistics of observed traveltime variations of both P and S waves. Using a spectral element method, we simulate 3-D global wave propagation for periods down to 10 s in synthetic 3-D elastic structures derived from a geodynamic model. Seismic heterogeneity is predicted by converting the temperature field of a high-resolution mantle circulation model (MCM) into seismic velocities using thermodynamic models of mantle mineralogy. Being based on forward modelling only, this approach avoids the problems of limited resolution and non-uniqueness inherent in tomographic inversions while taking all possible finite-frequency effects into account. Capturing the correct physics of wave propagation allows for a consistent test of the assumption of high core heat flow against seismic data. The statistics of long-period body wave traveltime observations show a markedly different behaviour for P and S waves: the standard deviation of P-wave delay times stays almost constant with turning depth, whereas that of the S-wave delay times increases strongly throughout the mantle. Surprisingly, synthetic traveltime variations computed for the isochemical MCM reproduce these different trends. This is not expected from a ray-theoretical point of view and highlights the importance of finite-frequency effects. Most importantly, the large lateral temperature variations in the lower mantle related to strong core heating are able to explain most of the standard deviation of observed P- and S-wave delay times. This is a strong indication that seismic heterogeneity in the lower mantle is likely dominated by thermal variations on the length scales relevant for long-period body waves.

SUMMARY We present a globally distributed data set of ∼400 000 frequency-dependent SH-wave traveltimes. An automated technique is used to measure teleseismic S, ScS and SS traveltimes at several periods ranging from 10 to 51 s. The targeted seismic phases are first extracted from the observed and synthetic seismograms using an automated time window algorithm. Traveltimes are then measured at several periods, by cross-correlation between the selected observed and synthetic filtered waveforms. Frequency-dependent effects due to crustal reverberations beneath each receiver are handled by incorporating crustal phases into WKBJ synthetic waveforms. After correction for physical dispersion due to intrinsic anelastic processes, we observe a residual traveltime dispersion on the order of 1–2 s in the period range of analysis. This dispersion occurs differently for S, ScS and SS, which is presumably related to their differing paths through the Earth. We find that: (1) Wavefront-healing phenomenon is observed for S and to a lesser extent SS waves having passed through very low velocity anomalies. (2) A preferred sampling of high velocity scatterers located at the CMB may explain our observation that ScS waves travel faster at low-frequency than at high-frequency. (3) A frequency-dependent attenuation q(ω) ∝q0×ω−α, with α∼ 0.2, is compatible with the globally averaged dispersion observed for S waves.