In this study, we use waveform tomography to image the upper mantle beneath the central and eastern Mediterranean, inverting for the seismic velocities vSH, vSV and vP as well as density ρ. Compared to the earlier, larger-scale European model of , our study provides a more local and zoomed-in model with clearer features that display stronger amplitudes. In the present study, we pay particular attention to developing a strategy that optimises the sensitivity to the deeper parts of the model domain by combining long-period surface wave data with a window selection strategy optimised for the selection of short-period body wave signals. This paper focuses on the technical construction of the model, discussing in detail the used methodology and the uncertainties and caveats of the methods and results. While we will highlight some key features of the resulting model, a detailed geological discussion will be the subject of a follow-up study.

After a brief introduction to the geological setting of the Mediterranean (Sect. ), we will focus on the methodological aspects of the waveform tomography executed, discussing the model domain and data selection (Sect. ) as well as the inversion setup (Sect. ). The latter includes a description of the effects of window selection on the inversion. In Sect. , we discuss the main features of the resulting model, after which Sect.  is dedicated to validity tests of the model and a discussion of the limitations of the methodology.

The chosen study area covers most of the central and eastern Mediterranean. This includes the tectonically interesting regions of the Italian peninsula, the Hellenic arc and Anatolia but also stretches towards the south to include the African coast (Fig. ). In order to avoid artificial reflections from the boundaries when simulating wave propagation within this model domain, a buffer zone with absorbing boundaries is implemented .

The inversion is carried out using data from around 80 earthquakes that occurred within the model domain between 1998 and 2017 (Fig. a, Table S1 in the Supplement). Most of the tectonic activity is in the north of the model domain, so in order to obtain a coverage that is as homogeneous as possible, events are initially selected manually from the Incorporated Research Institutions for Seismology (IRIS) Searchable Product Depository (SPUD) moment tensor catalogue (http://ds.iris.edu/spud/momenttensor, last access: July 2019). Additional events are then obtained using automatic event selection from the Large-scale Seismic Inversion Framework package LASIF;http://lasif.net, last access: March 2020.

All moment tensors are taken from the Global CMT Project https://www.globalcmt.org, last access: April 2020. Seismograms are downloaded automatically from the IRIS, ORFEUS, ETH, GFZ and KOERI data centres using the LASIF package. Since especially the depth and the moment tensor components linking horizontal and vertical motions can be poorly constrained when determined from long-period data e.g., we manually monitor the suitability of the earthquake data at all stages during the inversion. We excluded events that provided too few reliable measurements, e.g. because of cycle skipping or noise issues. In a few cases, we relocated or changed the timing of events where the (spatial) distribution of phase shifts indicated a clearly interpretable pattern (see Sect. S2 in the Supplement).

This approach results in around 80 events at the beginning of the inversion, corresponding to about 17 000 event–station pairs or 50 000 single-component channels, with a coverage that is excellent in the north but limited in the south (Fig. b). A table containing all earthquakes is available in Table S1.

Synthetic seismograms are computed by simulating seismic wave propagation in 3-D within the model domain. Here, we solve the following wave equation: 1ρ(x)u¨(x,t)-∇⋅σ(x,t)=f(x,t), where ρ denotes density, u is the displacement field (u¨ being its second time derivative the acceleration field), σ the stress field and f the forcing term. We use the SES3D wave propagation code and invert for a rheology that includes radial anisotropy for S-wave velocity (SH and SV). Anelasticity, kept fixed, is implemented using memory variables . The rheology relates stress σ to displacement u. There are therefore four inverted parameters: the seismic wave speeds (vSH, vSV, and vP) and density ρ.

In order to quantify the differences between observed and synthetic waveforms, we use the time–frequency phase misfit functional of : 2JTF,p(m):=∫t∫ωWp2(t,ω)ϕ(m,t,ω)-ϕobs(t,ω)2dtdω,Wp=log(1+|ũobs|)maxlog(1+|ũobs|), based on the phase shift (ϕ-ϕobs) for a given time window between observed and synthetic data from their time–frequency representations (calculated using the Gabor transform). A logarithmic time–frequency weighting function Wp determines the regions within that space that have sufficient amplitude such that meaningful measurements can be made, with ũ being the time–frequency representation of seismic signal u as calculated via the Gabor transform. Misfits for all windows, traces and events are summed to produce the total misfit for a given model.

The advantage of this misfit definition is that it combines the sensitivity to phase shifts from, for instance, a cross-correlation time shift misfit and to the shape of waveforms from an L2-norm misfit e.g.. It does not require the isolation of specific phases and is therefore specifically suitable for interfering phases. However, it is beneficial to isolate small- and large-amplitude signals in separate windows such that the information from small-amplitude signals is not suppressed by the weighting Wp (Sect. ).

The objective functional in Eq. () is minimised such that the synthetic seismograms for each consecutive model provide a better match with the observed data than the previous. We use a conjugate-gradient scheme to compute model updates. This algorithm makes use of the gradient of the current iteration's misfit with respect to the model parameters and a recursive term based on the previous iteration's descent direction.

The misfit gradient is constructed from sensitivity kernels obtained using the adjoint method e.g.. The raw gradients for each model parameter are preprocessed before a descent direction is computed in order to improve convergence properties of the gradients. Kernels for each event are clipped at the 99th percentile in order to avoid too-strong localisation of updates especially in the source region and then summed to produce the misfit gradient. The side and bottom edges are set to zero to remove potential boundary effects, and some smoothing is applied. This processing routine, fully described in Table S2, is based on experience from previous inversions and some initial experimentation. It is re-evaluated at several points during the inversion. The processed gradients are used to compute a descent direction using the conjugate-gradient scheme, and the step length is then determined using a quadratic interpolation between the current model and three test models made with steps of lengths of 5 %, 7 % and 10 % of the maximum gradient amplitude. The final model for an iteration is obtained from the computed descent direction and the step length thus obtained.

Because of the local nature of the gradient-based optimisation, the choice of initial model is of crucial importance: the closer the initial model is to the global minimiser, the more likely it is that it lies within the same “misfit valley”.

constructed a model of the European crust and upper mantle (with an embedded higher resolution model of Anatolia; ). These models were constructed using waveform tomography and already include considerable detail. The starting model for the current study is retrieved from these by way of the initial stages of the Collaborative Seismic Earth Model , which combines the local models with a smoother background of S20RTS anomalies . The transition between these is smooth, owing to the efforts of to combine regional seismic tomography models into a coherent and consistent global model. This gives a starting model with considerable detail in the northern parts of the model domain (resolution lengths of down to 25 km; see also Fig. 4 in ) but only very smooth structures in the southern parts (Fig. ).

The initial crust was implemented in the form of a velocity gradient, meaning that the initial model has no sharp crust–mantle discontinuity. This approach is widely used in regional- to global-scale tomography e.g., and it mostly serves two purposes. (1) The absence of a discontinuity facilitates the construction of a finite-element mesh, and in particular, it avoids the presence of very small elements that would require a very small numerical time step. (2) An initially smooth transition between crust and mantle allows the data to actually modify its sharpness, thereby avoiding artefacts that may result from an incorrect a priori implementation of the Mohorovičić discontinuity. Thus, in the final model, the velocity gradient from crust to mantle is as sharp as required by the data. These advantages are balanced by the disadvantage of somewhat limited interpretability of the final model in the sense that no sharp separation between crust and mantle exists.

To further mitigate the risk of descending towards (insignificant) local minima, we use a multi-scale approach . The lowest frequencies are inverted first, and as more of the data are explained by the model, higher frequencies are included in a stepwise manner. As a result of this approach, the large-scale structure within the model domain is obtained before small-scale details are filled in. This approach, a standard in waveform tomography e.g., mitigates the risk of cycle skips, spares computational resources and increases (at long periods) the proportion of the model domain that the data are sensitive to. This means that some of the long-wavelength structure of the southern, less-well-covered part of the model domain, can also be retrieved.

The used frequency bands are specified in Table . We start at long periods of 100–150 s to take full advantage of the comparatively minimal computational cost of such simulations and the broad sensitivity of such signals to almost the entire model space. Care is taken to avoid the introduction of cycle skips when new frequency content is introduced in the inversion. This is done by limiting the highest frequency to be less than 1.25 times the previous highest frequency. In each frequency band, ∼ 10–20 iterations are carried out.

At each frequency band, the events to be included are re-evaluated, mainly for the benefit of reducing computational cost (see Table ). At low frequencies, events with few stations may provide valuable constraints on parts of the model domain that are otherwise poorly covered – several such events are located in northern Africa (Fig. ). These events are mostly discarded at higher frequencies, where the benefit of including the event becomes unfavourable compared to the computational cost.

In addition to the smoothing of the kernels as described in Sect. , we found it was necessary to apply additional processing to the total model update at the end of the iterations in some of the frequency bands. This mainly involved additional smoothing and removal of edge effects. All additional smoothing and damping is described in Table S2.

For the initial three frequency bands (down to periods of 65 s), the first 1200 s of data after the event origin time were evaluated. This is reduced in subsequent period bands, where the surface wave train becomes more compact (see Fig. ) and the computational cost per simulation increases (Table ). Ideally, complete seismograms are used as data in the inversion. However, due to potential source- and receiver-side issues, geometry and noise, some selection of the data needs to be performed . The initial window selection is carried out automatically using LASIF. For the first three frequency bands, windows are reviewed manually for all 50 000 seismograms; windows for later frequency bands are reviewed in part. During some initial experimentation, it was found that the inclusion of late arrivals could contribute to the formation of boundary artefacts, especially if the source or receiver are positioned close to the domain boundary. Such windows are therefore excluded.

Our window selection strategy is aimed to maximise sensitivity to deep structure. This is done by explicitly separating small- and large-amplitude signals into different windows, which allows us to make use of the (smaller-amplitude) body wave information. Figure  illustrates this. Two separate windows (A and B) reveal significant and complementary information in both wave packets, which is demonstrated by the deep sensitivity for vP for body wave window A. In the combined window (A + B), both the misfit and the vP sensitivity kernel are dominated by the large-amplitude surface wave signal, which is not significantly different from window B. Separating such windows accelerates the convergence of the body wave data and thus, given the finite total number iterations, allows us to better resolve deeper structure.

Figure displays the misfit development for all the frequency bands used in this study. The misfit reduction varies per frequency band but is on the order of 10 %–20 % within each band.

Comparing the initial and final models in the period band of 28–150 s, the misfit drops by 48 %. This is demonstrated in Figs.  and , where we compare observed traces for this period band with synthetics from the initial and final models using windows from the final iteration. Figure  demonstrates that the most important misfit improvements come from events clustering in the central Aegean part of the model domain and towards the southeast. The events with the largest initial misfit (which are the events with the largest number of windows) display the largest drop in misfit – both in absolute and in relative terms. Phase shifts per window become closer to zero for the final model.

Figure shows examples of seismogram fits for an event in the Aegean Sea. Especially at stations farther away from the source (e.g. stations 1 and 9), the difference between the synthetic seismograms for the initial and final models is striking.

After 100 iterations, the model has been updated considerably for all parameters down to the transition zone (Fig. ). Updates are strongest near the surface and decrease in strength with depth. P velocity is updated significantly less strongly than the other parameters and to shallower depths (Fig. a). Average shear velocities do not change significantly from the starting model (Fig. b), with average vSH slightly higher than vSV in the uppermost 100 km, as is reflected in the corresponding depth-averaged anisotropy vSH-vSVvS (Fig. c). Figure  shows how the Hellenic subduction zone becomes progressively more pronounced as iterations progress.

The final model is shown in Fig.  (compare to the starting model in Fig. ). In general, the S-velocity model has features that can be linked most coherently with our knowledge of the geological situation. In the upper ∼100 km, the continental areas have slow velocities, while the old oceanic lithosphere of the central/eastern Mediterranean is fast. In contrast to this, both the western Mediterranean and the Aegean Sea display low velocities, as expected from their history of young oceanic crust formation and recent or ongoing extension. Several high-velocity zones can be linked to areas of known current or previous subduction. These will be discussed in more detail below.

The picture is not so clear for P velocity and density. P velocity has generally increased in the upper ∼200 km of the model domain, with smaller amplitudes than in S velocity. Because for a given frequency, wavelengths are longer for P waves, the resulting model is smoother than the S-velocity model. The Hellenic subduction zone is visible down to ∼250 km. At first sight, there appears to be an anticorrelation between P and S velocity; however, upon closer inspection, this turns out to be not actually the case – the P-wave model is fast in general. The density model, in contrast, has much shorter spatial wavelengths. This is a result of the fact that it is mainly contrasts in density to which the waves are sensitive. As a result, the imaged structures are of stronger amplitude and tend to be more oscillatory. Although this reaches greater depths (even beyond the transition zone), the results are much less coherent. We will discuss this separately below.

The lateral variations of radial anisotropy, shown in Fig. , are dominated by structure with length scales of around 100–200 km. The maximum amplitudes of radial anisotropy reach nearly 15 %, already suggesting that intrinsic anisotropy (e.g. lattice-preferred orientation, or LPO, of olivine) is unlikely to be its only origin . Other contributions include the strong differential sensitivity of Love and Rayleigh waves and apparent anisotropy induced by sub-wavelength structure that cannot be tomographically resolved . On the one hand, radial anisotropy is well known to be required in order to jointly explain Love- and Rayleigh-wave dispersion. On the other hand, it can be shown both analytically and numerically that the inherent (LPO related) and apparent anisotropy cannot be distinguished from seismological observations . For the latter reason, we accept the required presence of radial anisotropy in our model but refrain from its geological interpretation.

Although computationally more expensive than ray-based tomography methods, the advantage of waveform tomography lies in its ability to make use of all and any part of seismograms that has a sufficient signal-to-noise ratio, allowing for a more complete extraction of all information that is carried by seismic waves. This allows us to take into account body and multimode surface waves, source effects and frequency dependence, and jointly invert for multiple parameters for the crust and upper mantle. This is done in a manner where the misfit is computed directly from the observed and simulated seismograms, thus providing a self-consistent inversion framework. This method is particularly powerful in tectonically active parts of the Earth such as the Mediterranean, where strong heterogeneity is unavoidably present – circumstances under which the assumptions underlying ray theory become invalid.

Nevertheless, it is important to understand that the results are unavoidably affected by observational and methodological errors. In the following paragraphs, we will discuss some of these issues.

Moment tensors used in this inversion are taken from the Global Centroid Moment Tensor catalogue using the point source approximation. These are not inverted for separately, as limited azimuthal coverage – in particular, near the model domain boundaries – may lead to a bias in the results, as well as potential overfitting. note that the model used for source inversions can have an imprint on tomographic results. Particularly in areas of limited coverage errors in the description of the source can have a large effect – this may be the case for anomaly E in Fig. . In order to minimise such effects, patterns in phase shift for each event are monitored manually and in some cases, source parameters were adjusted accordingly. In line with results from and , these adjustments are generally on the order of a few seconds or kilometres, and they correspond more closely to the reported International Seismological Centre (ISC-EHB) locations as well as results reported in regional tectonic studies such as . Examples are shown in Sect. S2. In a few cases, events were removed from the inversion altogether, as their waveforms showed a notable jump in complexity in the higher-frequency bands which may point to issues with the point source assumption or the source time function.

The wave propagation simulations are carried out on a regular mesh: a spherical chunk that includes no ellipticity, topography, ocean layer or explicitly meshed internal discontinuities. The neglect of topography can lead to small-scale artefacts near the surface, in particular if the topography is on the same length scale as the minimum wavelength . Similarly, internal discontinuities and the effect of the water layer are not modelled explicitly. Especially near the surface, the model thus has to be interpreted as an effective representation of the real Earth, valid for the frequencies used in the inversion . Sharp or small-scale layering and (apparent) anisotropy thus cannot be distinguished reliably.

Another source of error is the actual data used in the inversion. Such errors can pertain to the location, orientation and timing of the receivers, as well as instrument errors and errors in the reported response. Many of these issues are caught using the automated quality control carried out in our workflow, but subtle errors remain difficult to catch. In areas of dense data coverage, this will average out, but in the regions of poor coverage, these may still affect the inverted result.

Within the framework of the inversion and issues as outlined above, it is possible to analyse the trustworthiness (and limitations) of the model.

For this, we compute the misfit for the initial and final models for six earthquakes that were not part of the inversion (Table S1). We compare complete traces where no window selection has taken place in order to avoid preferential usage of data that “work well”. For this reason, and following and , we use for this a normalised L2 waveform difference misfit: 3JL2=∫tu(m,t)-uobs(t)2dt∫tu(m,t)2dt∫tuobs(t)2dt. Figure  shows the change in misfit for these events for all stations as well as example seismograms from events A–E (Table S1). The data fit improves both visually and quantitatively in the large majority of traces.

In addition, we assess our model with what can be viewed as a “waveform tomography equivalent” for spike testing. While more sophisticated methods, such as resolution analysis through random probing do exist, spike tests do not rely on any assumptions about the shape of point spread functions which – especially in the case of density – may be strongly non-Gaussian. They thus give qualitative insight into smearing in a way that is visually easy to interpret.

As note, it is important in ray-theory travel-time tomography to use the ray paths of the actual model to assess recovery, smearing and trade-offs. To mimic this using the full wavefields computed from 3-D wave propagation, we add a grid of Gaussian anomalies onto one parameter in model 91 and use the real data to run five additional iterations in the final period band of 28–150 s. If the model is robust, the data will detect the (presumably) erroneous spikes, and updates will be generated that remove the anomalies. We do this for each parameter (vSV, vSH, vP and density) individually, thereby also checking trade-offs. Spikes are introduced in an alternating chequered pattern at different depth levels.

In Fig. , we compare, at a depth of 100 km, the input anomaly (top row) with the part of the model update that is a result of the addition of the spikes (bottom four rows). This is computed by subtracting the reference model update (without spikes) from the update after five iterations of the spiked model: uspike=(ms+5-ms)-(m96-m91).

Comparing Fig.  to Fig. , we immediately observe that those parts of the model domain that are well covered are indeed also the parts in which the spikes are most successfully removed. This is most visible for the first two rows, with spikes in vSH and vSV, respectively. These are also the parameters that are recovered best, with trade-offs to other parameters that are less significant. Recovery of vP is, in general, very limited. Trade-offs from vP to other parameters are partially stronger than the updates in vP itself. This is in line with our observations of the final model (Sect.  and Fig. ).

Recovery of density is also limited. Only in the central Aegean Sea is recovery of a level approaching that of vSV and vSH, and the negative spikes are surrounded by a positive “halo”. Furthermore, when comparing the different rows, we see that trade-offs to density from the other parameters are larger than the updates in density itself. While this is a test done over only a few iterations and it is possible that these trade-offs even out more as iterations progress, this does indicate that density is not a stable parameter.

The limited recovery of 3-D density structure is interesting in the context of previous numerical experiments, showing that density has a clear imprint on the seismic wavefield and may in principle be constrained , especially at longer periods . Furthermore, recent work in seismic exploration e.g. and regional seismology e.g. has been encouraging.

The fact that density cannot be constrained reliably in the present study can be attributed to two main factors. Firstly, as density affects the seismic wavefield mainly through reflection/backscattering, it is predominantly contrasts in density that are imaged. This results in a sensitivity to density at much shorter length scales than for S velocity. It is therefore likely that short-scale structure from other parameters is mapped into density – this is especially likely near sharp transitions such as the surface and the Mohorovičić discontinuity, and may be exacerbated by the lack of topography/bathymetry and the ocean layer in our mesh. In the context of exploration seismology, indirect sensitivity to density through impedance contrasts can be exploited much more effectively through the use of reflected waves.

Furthermore, variations in density have a smaller effect on the seismic wavefield than variations in seismic velocity . This means that in the presence of noise, inaccuracies in the modelling method as discussed above, or variations in parameters not included explicitly in the inversion (e.g. attenuation or further anisotropy), density is likely to act as an “inversion garbage bin” e.g.. For this reason, we presently do not further interpret density, although it is still important to include this parameter in the inversion.

The way ahead in waveform inversion for density structure may require significantly more effort than we, admittedly, anticipated at the beginning of this study. Other physical parameters that are at a similar level of importance as density will need to be included in order to avoid that the neglect of one parameter pollutes the image of another. Likely candidates for such parameters are attenuation and more complex forms of anisotropy. An even larger number of model parameters will unavoidably decrease resolution of individual parameters and increase the influence of subjective regularisation, e.g. damping and smoothing. This subjective imprint can only be avoided by adopting a probabilistic approach to waveform inversion that only incorporates prior knowledge that we actually do have. The result would be a posterior distribution that reflects our honest state of knowledge on 3-D density in the Earth. Thanks to increasing computational resources and methodological advances, probabilistic full-waveform based on Monte Carlo sampling is starting to be within reach e.g..