Considering the three-dimensional dataset D(λi,ϕi,zj), the k-means algorithm defines k clusters, corresponding to the k sets of horizontal positions (λi,ϕi) closest to their average zi profiles. The algorithm is based on an iterative procedure. At the first iteration, it randomly chooses k horizontal positions used as cluster centres. Each point of the dataset is then associated with the cluster centre to which it is the closest. The average radial profile of the points attributed to each given cluster is computed and used as the new cluster centre for the next iteration. This is repeated until convergence is achieved.

To make k-means and PCA representations somewhat comparable, we use the clusters as horizontal patterns and the average vertical profiles of each cluster as the PCs. By construction, the variance captured by the k-means is noticeably smaller than that for the other methods, since it is not meant to propose a compressed representation of the dataset but rather to separate the dataset into subsets, which results in an important loss of information.

A 2-D matrix (Fj,k, j=1, 2, …, J; k=1, 2, …, K) is transformed by the PCA into a sum of components, with each component being composed of a load αn,j and an eigenvector An,k. 1Fj,k=∑n=1N=min(J,K)αn,jAn,k

In our case, Fj,k corresponds to the velocity anomaly at horizontal position pj=(λj,ϕj) and depth zk. The An,k are the eigenvectors, or principal components (PCs), of the covariance matrix. These PCs are orthogonal vertical structures representing the covariance between the slices of the model. It has large positive values if the horizontal structures from two layers are positively correlated, zero values if the structures are not correlated, and large negative values if they are anticorrelated. The loading patterns αn,j are also orthogonal to each other, and each αi,j results from the projection of the dataset on the PC i, capturing the horizontal structure i associated with the vertical anomaly profile Ai,k. The loads take continuous positive and negative values. Here, those patterns correspond to horizontal maps showing where each vertical structure is more or less important in the model.

The components are ordered by decreasing eigenvalue, as the variance captured by each PC is directly proportional to the eigenvalue of the PC. Due to their orthogonality and to the mathematical properties of the transformation, the variance captured by each PC drops rapidly with the order so that a small number of independent components (Ñ≪N=29) is often sufficient to capture most of the information, allowing an efficient compression of the dataset.

Unlike clustering methods, which are binary in that any horizontal location only belongs to one cluster, PCA computes the amplitude of the contribution from each principal component for every horizontal location, providing a compressed reconstruction of the dataset.

The first PC corresponds to the dominant covariance, which might be physically associated with a global phenomena – in our case, a structure that would develop on the whole mantle depth – or a more local feature, i.e. associated with a limited depth range. But this covariance structure might also correlate with other features from other depths, which will also be retrieved in the first PC. The second PC being orthogonal to the first, some of the physics might have been subtracted by the computation of the first PC, and it is even more so for the following principal components.

Used on space–time datasets, PCA often produces artefacts from the domain geometry : the topographies of the PCs are primarily determined by the shape of the domain and not by the covariation among the data. In other words, different correlation functions on a geometrically shaped domain have similar load patterns in a predictable sequence, which do not reflect the underlying covariation. This is the case of square domains found, for example, on meteorological maps; in our case, the unrotated PCs show a harmonic-like progression, with one maximum for the first PC, two for the second, and so on. In contrast, the varimax rotation captures distinct, well-defined depth domains in the mantle, which are easier to interpret physically. Hence, the physical interpretation of the PCs and load can be made easier by redistributing PCA components along other eigenvectors, thereby maximising a functional of the loads that favours some physical properties that appear physically meaningful . There are several possible redistribution options for the PCs . Among those, varimax rotation favours PCs with large values concentrated on the smallest possible subset of depths and preserves the orthogonality of the PCs.

Varimax rotation corresponds to a rotation on the basis of the same information space as that generated by the PCA; consequently, the total variance captured by Ñ components is exactly the same for both types of PCs but distributed differently between the components, with a slower variance decrease for the varimax rotated solution. Considering the variance captured by the PCA components, the number Ñ of PCs to keep is selected to meet a given criterion: the total variance captured, a fixed number of components or the minimum variance captured by a PC kept. Then we define new Ñ components so that 2Fj,k≃∑n=1Ñαn,jAn,k=∑n=1Ñβn,jBn,k.

βn,j represents the new horizontal structures, and the new PCs, Bn,k, are chosen to maximise a given objective function V, defined by the sum of the values of the objective functions Vn computed over each PC: 3V=∑nÑVnBn,k, with 4Vn=1K∑k=1KBn,ksk4-1K2∑k=1KBn,ksk2, where sk represents normalisation factors, with sk=1, for all k in the case of varimax rotation.

This transformation corresponds to a rotation of the PCs because the subspace generated by the transformation – or the reconstructed model – is the same as with the non-rotated PCA limited to Ñ components. In our case, it limits the vertical extension of the PCs; i.e. each PC shows large values on only a few depths and/or slices.

The associated horizontal structures, βn,j, are recomputed by projection of the tomography model on the rotated vertical profile. This rotation conserves the orthogonality of the eigenvector, which is not the case for all the possible rotations, but the horizontal loads are often not orthogonal anymore . The total variance captured by the rotated PCs is the same as that from unrotated PCs, but the decrease in the variance is slower than that from the original decomposition.

We use varimax PCA to compress the seven tomography models described in Sect.  into a set of components, keeping only the most important ones, as explained below. Each component is composed of a vertical profile obtained directly from the varimax process and an associated horizontal pattern, which is computed by projecting the model on the profile. Such data compression is useful to compare the models if three major conditions hold. First, a subset of components must capture most of the variance of the signal, with the number of components being significantly smaller than the original number of depth splines and boxes, and enhance the signal-to-noise ratio. Secondly, the relevant structures in the mantle, which will be used for comparison and for geological interpretation, should not be distorted by the compression; i.e. their shape and position must remain unchanged. Third, the power spectral densities should not be altered by the compression process.

In order to facilitate the comparison of the horizontal structures in the models, we label the varimax components obtained from the varimax PCA using capital letters in alphabetical order from components sensitive to shallow mantle structure to components sensitive to the lowermost mantle structure. Figure  shows the variance captured by the varimax PCA applied to the different models. The varimax analysis is performed by considering all the components capturing more than 1 % of the variance of the signal in the classical PCA. Keeping only the components with variance above 1 % limits the number of maps, facilitating the quest for relevant information. Tests with 5 % and 10 % thresholds showed that some important information is lost. When using a 10 % threshold, all the models are represented by three components only, which misrepresent known structures such as ridges or subduction zones. Moreover, the depth distributions of the corresponding PCAs become quite broad and imprecise, each stretching over more than 1000 km depth ranges. In the 5 % threshold case, the models are represented by four (SAVANIi) to six components (SEMUCB-WM1i, SEISGLOB2). Again, information is lost concerning e.g. ridges or subduction zones, and the depth information is spread over a depth range greater than 500 km for all components and for all models.

The simplification brought by the varimax method is particularly efficient for tomography models with weak regularisation, such as SGLOBE-ranii, wherein short-scale structure is likely mixed with noise. Figure  shows that the number of varimax components ranges from 7 for model SAVANIi to 15 for SGLOBE-ranii, with the total variance captured by all these principal components always exceeding 97.3 % (see also Table , which summarises the components kept in the varimax analysis). The number of varimax components required by each model depends on the details of the model's construction, such as the data used (Table 1), and, importantly, on subjective choices made, such as the level of regularisation used. Increasing the strength of regularisation reduces the model's effective number of free parameters and hence the number of varimax components required by the model. As the SAVANI model only needs seven varimax PCA components, the shallowest component concentrates a lot of information (26 % of the variance) that is spread into more components for the other models. Table  shows that the number of PCs needed to explain 97.3 % or more of the total information in the tomography models is always smaller than the number of splines or layers used in the models' original depth parameterisation, with 29 % to 75 % fewer PCs than depth splines and layers.

This fulfils the first condition for the usefulness of the data compression mentioned above. In order to check the second condition previously mentioned, Fig.  compares six examples of depth slices in the original SEMUCB-WM1i model with those obtained from the model's reconstruction using varimax analysis. The differences between the original and reconstructed models are small and random, highlighting the fact that the compression process does not distort the model's features. The residual signal is the part of the models that is not covariant vertically. We also verified that there are only very small, random differences between the power spectra of the original and the reconstructed models (see Figs.  and in the Appendix). Hence, the third condition of usefulness of the data compression used in this study is also satisfied.

Figure  shows the varimax PCs for all the tomography models used in this study, together with the spline functions or the variable thickness depth layers used in the models' parameterisation. The vertical profiles differ from one model to the other in both numbers of components and the depths of their maxima. For example, as shown in Fig. , SAVANIi requires 7 components, while SGLOBE-ranii needs 15 components. We re-emphasise that the number of principal components obtained for a given model reflects their amount of independent information, which in turn depends heavily on choices made during the model's construction, such as regularisation and the amount and type of data used.

The horizontal patterns obtained from the varimax PCA also show distinct features, but this is not always in the same way as for the vertical profiles (Figs. , –). S20RTS shows sharp vertical profiles (Fig. ) and even contains one more PC (13) in the upper mantle than the updated S40RTS model (12, Fig. ), but the horizontal patterns are smoother. This is likely due to the fact that the latter model is constrained by about 10 times more data and used a different level of regularisation . SGLOBE-ranii, SEMUCB-WM1i and SEISGLOB2 (Figs. , and ) depict sharper horizontal patterns than SAVANIi, S362WMANI+Mi and S20RTS (Figs. , and ). SEMUCB-WM1i and SAVANIi (Figs.  and ) show vertical profiles concentrated closer to the surface, but their horizontal patterns are different. The PC B of SAVANIi (∼200 km of depth; Fig. ) corresponds to low-velocity anomalies underneath all oceans, which is also the case for S20RTS and S362WMANI+Mi (Figs.  and ), but not for SGLOBE-ranii and SEISGLOB2 (Figs.  and ). Such upper mantle low velocities beneath the oceans also appear in the models S40RTS and SEMUCB-WM1i, but with a higher level of detail (Figs.  and ).

We note in Fig.  that the vertical components obtained from the varimax analysis do not fully correspond to the depth parameterisations used to build the models, especially above the 660 km discontinuity, where there are 2.3 to 3.3 fewer PCs than original spline functions or boxes. This implies that the PCs do not simply reflect the model parameterisation and inform us about the independence between the slices reconstructed from the model. The PCA objectifies the number of splines and boxes consistent with the amount of information present in the model. In the upper mantle, for all models, we end up with three to four PCs. In the lower mantle, the correspondence differs from one model to the other, whereas we observe three categories.

SGLOBE-rani. The PCA reproduces the original spline functions quite well except the two deepest ones, which are recovered into one PCA. This is probably due to the relatively weak regularisation used .

This shows that overall the tomography models do not have a strong imprint of the depth regularisation used in their construction. This is especially true above the 600 km discontinuity or in the whole mantle for the SAVANIi, S362WMANI+Mi and SEMUCB-WM1i models, for which the information is recovered by fewer PCs than the number of spline functions or boxes.

One of the most striking differences between the models is the way the signal is distributed between 500 and 1500 km of depth. In this region the different tomography models require between two (SAVANIi D–E) and seven PCs (three for S362WMANI+Mi C–E and SEMUCB-WM1i D–F; five for SEISGLOB2 D–H; six for S20RTS D–I and S40RTS C–H; seven for SGLOBE-ranii D–J). The observed variability in the number of required PCs likely reflects the level of model regularisation used in the construction of the various tomography models as well as the amount and variety of data used (e.g. if the model does not include constraints from overtones, structures in the transition zone and mid-mantle might not be well-retrieved). Moreover, the treatment of discontinuity topography may also matter because neglecting this topography could map directly into isotropic wave-speed variations in the mid-mantle. The PC E of SAVANIi (∼1200 km of depth) is dominated by low-velocity anomalies and shows a substantially different pattern than e.g. the PC F of SEMUCB-WM1i and the PC G of SEISGLOB2 (∼1300 km of depth), which depict alternating low- and high-velocity zones. The principal components G (∼1000 km of depth) and H (∼1100 km of depth) of SGLOBE-ranii present mostly low-velocity anomalies, which are similar to the components F of S40RTS and G of S20RTS, both at ∼1100 km of depth, but in these two latter models we also observe a high-velocity anomaly under the northwestern Indian ocean. In SAVANIi, this is also observed on its PC E (∼1200 km of depth), which, is however, much more broadly distributed at depth. These differences between the models reflect the high level of uncertainty for this part of the mantle, which is likely due to its limited data coverage.

A fully detailed geological and geophysical discussion of the models is beyond the scope of this study and has already been performed in many previous studies (see e.g. ). Varimax PCA recovers all the features discussed in these studies. Table  compares how key Earth structures are captured by the varimax analysis for the isotropic part of the seven tomography models used in this study (Figs. , –). The table covers ridges, rifts, plateaus (low-velocity anomalies, red) and cratons (high-velocity anomalies, blue) at depths of 50–300 km, which corresponds to the heterosphere , subducted slabs (high-velocity anomalies) at 300–1300 km of depth, and the LLSVPs and the Perm low-velocity zone in the lowermost mantle (red anomalies).

Depending on the region, the high-velocity craton signature should reach a maximum depth between 100 and 175 km , but tomography models often show a deeper signature, likely due to smearing effects. The model SGLOBE-ranii is the most consistent with this depth limit, as most of the cratons are concentrated in the second PC (∼100 km of depth; PC B in Fig. ), separating them from the cold oceanic crust. This is possibly due to the huge set of data sensitive to the upper mantle used in SGLOBE-ranii's construction, including massive sets of both phase- and group-velocity measurements. Beneath Africa and the Baltic region, a high-velocity zone remains visible on the third PC (∼300 km of depth; PC C in Fig. ). For the other models, the craton signatures extend from 50 to ∼200–300 km of depth.

The low-velocity zones underneath the Tibetan Plateau and Hangai Dome, southwest of Lake Baikal , are recovered by the first PC of all models with a maximum at 50 km, but with different shapes. These zones are smaller in SEMUCB-WM1i (Fig. ) than in SGLOBE-ranii and SAVANIi (Figs.  and ), and the Tibetan Plateau extends more to the south in SAVANIi and S362WMANI+Mi (Figs.  and ), being subdivided into three small zones in S40RTS and SEMUCB-WM1i (Figs.  and ).

From ∼ 150 km to ∼ 800 km of depth, global tomography models often show a low-velocity anomaly beneath the Pacific e.g.. For SAVANIi, at ∼ 200 km of depth, it is difficult to distinguish between the Pacific and other oceans. Its PC C, with a maximum at ∼ 400 km of depth, is confined to the central and western Pacific, while the PC C of S362WMANI+Mi at ∼ 500 km of depth is more to the southwest, resembling the PC D of S20RTS (∼ 600 km of depth), D of S40RTS (∼ 700 km of depth), E of SGLOBE-ranii (∼ 600 km of depth), D of SEMUCB-WM1i (∼ 600 km of depth) and D of SEISGLOB2 (∼ 700 km of depth).

All models show a high-velocity zone between ∼ 300 and 700 km of depth beneath the central Atlantic and along the Atlantic coasts of South America and Africa, which is similar to that from Fig. 9 of and already discussed in . Nevertheless, this zone appears less clearly in SEMUCB-WM1i (Fig.  C–D) and especially SGLOBE-ranii (Fig.  C–D), where it is mixed with low-velocity patches. This anomaly is a region with long transform faults, high gravity, anomalous ocean depth and low melt production. It is thought to be the region of the Atlantic that formed during the final stages of the opening of the Atlantic because it was presumably the strongest part of the Pangean continent .

In the East African Rift, the low-velocity anomaly aligned with the Afar Depression and the Main Ethiopian Rift in the uppermost mantle appears in all the models from the surface to the LLSVP, with narrower contours in S40RTS, SEMUCB-WM1i and SGLOBE-ranii than for the other models. This is consistent with the presence of one or multiple mantle plumes in the region, as proposed in previous studies (e.g. ).

All models show high-velocity subduction zones in the western Pacific, among others, notably underneath the Philippine Plate over two principal components with depths ∼ 400–800 km. This complex system mixes different subduction zones . The Izu–Bonin slab subducts westward down to ∼ 870 km of depth and is connected in the upper ∼ 300–400 km of depth to the Marianas to the south, which plunges vertically down to ∼ 1200 km of depth (components C–E in Fig. , C–G in Fig. , C–D in Fig. , D–F in Fig. , C–D in Fig. , C–D in Fig.  and C–F in Fig. ).

More to the south (north of Papua New Guinea), the Caroline Ridge from ∼ 475 to 750 km presents high-velocity anomalies. West of those zones, Manila and Sangihe present high-velocity anomalies. Even more west, Banda, Sumatra and Burma also present high-velocity anomalies. This is recovered by SGLOBE-ranii (components D–E in Fig. ), SEMUCB-WM1i (component D in Fig. ) and SEISGLOB2 (component D in Fig. ), but it is broader, especially to the north, in S20RTS (components D–E in Fig. ), S40RTS (component C in Fig. ), SAVANIi (components C–D in Fig. ) and S362WMANI+Mi (component C in Fig. ).

The Tonga–Kermadec subduction zone, located below the south Fiji Basin down to a depth of ∼ 1300 km in the lower mantle , is recovered by all models but is less clear in S362WMANI+Mi (Fig. ). Conversely, SGLOBE-ranii, S40RTS, SEISGLOB2 and to a lesser extent SEMUCB-WM1i show a narrow arc-shaped signature of this zone. On the other hand, it is difficult to assess a maximum depth of this subduction zone in SGLOBE-ranii.

All models evidence the LLSVPs, though they are less clear in some models, such as SAVANIi and S362WMANI+Mi (Fig. ), and they appear quite patchy in S40RTS and SGLOBE-ranii (Figs.  and ). All the models show low-velocity anomalies spreading from the core–mantle boundary (CMB), where the LLSVPs are clearly visible, to about 1500 km of depth, where low-velocity structures are less coherent (for example, components G–I in Fig. ). All together, the components encompassing the LLSVPs capture 11 % (SAVANIi) to 29 % (S40RTS) of the models' information. The Perm anomaly is recovered in all models for the two deepest components, apart for SAVANIi, wherein it is recovered by the last PC only. This is because this PC is quite broad, extending from ∼ 2000 km of depth to the CMB. These depths are consistent with e.g. the findings of and , which estimate that LLSVPs spread up to ∼ 500 km above the CMB. Note that it is difficult to estimate the top of the LLSVPs in S362WMANI+Mi, as there are persistent low-velocity zones beneath e.g. eastern Europe up to the PC E centred at ∼ 1300 km of depth (Fig. )

Our analysis allows determining the importance of the various elements of the models. For example, for all models, principal components with maxima in their varimax PCs below 1700 km of depth and dominated by LLSVPs explain 11 % (SAVANIi) to 24 % (SEISGLOB2) of the models' information. On the other hand, principal components with maxima in the top 300 km dominated by ridges, rifts and cratons explain 22 % (SGLOBE-ranii) to 45 % (SAVANIi) of the models' information.

In addition to isotropic shear-wave-speed anomalies, four of the models considered in our study also include radial anisotropy perturbations: that is, speed differences between vertically and horizontally polarised shear waves in SGLOBE-rania, SEMUCB-WM1a, SAVANIa and S362WMANI+Ma. The seismic imaging of anisotropy is more challenging than that of isotropic structure because the sensitivity of seismic data to anisotropy is weaker e.g.. Moreover, it has been shown that if crustal effects are not properly modelled, this can lead to substantial errors in the estimated mantle anisotropy . These difficulties are at least partly responsible for the strong differences between existing radial anisotropy models (e.g. Fig. b). Varimax PC analysis is thus a natural candidate to analyse and compare the models since it enhances their robust information. Figures  and – show the vertical profiles (varimax PCs) and horizontal patterns from the combined varimax analysis on the anisotropic part of the four radially anisotropic models, for which 10 components capture more than 1 % of the variance. As expected, there is poorer agreement between the radial anisotropy structure in the models than between the isotropic structure discussed in the previous sections, though some common features can be identified.

The two LLSVPs appear on the deepest PC J of SGLOBE-rania and SEMUCB-WM1a (Fig. ), which captures 10 % of the models' information. The Pacific LLSVP barely appears in the last PC for SAVANIa and S362WMANI+Ma. However, as explained in Sect. 3, e.g. , , and showed that the signature of LLSVPs in radial anisotropy models is an artefact due to leakage of isotropic structure into artificial anisotropic structure in the lowermost mantle.

For PC B (with a maximum at ∼ 100 km of depth; Fig. ), a positive zone appears beneath the Pacific and the Nazca plates in SGLOBE-rania, SAVANIa and to a lesser extent SEMUCB-WM1a, while no clear pattern is evidenced in S362WMANI+Ma. The same holds true for mid-ocean ridges. A positive anomaly is observed on PC C under the Pacific Plate for all models, with a maximum around a depth of 200 km (Fig. ). A broad positive radial anisotropy anomaly beneath the Pacific at these upper mantle depths has been well-documented in previous studies and may be due to horizontal mantle flow in the region and/or thin layers of partial melt in the asthenosphere (e.g. ).

Components B (maximum depth 100 km; Fig. ), D (maximum depth 400 km; Fig. ) and especially C (Fig. ) evidence subduction patterns in SGLOBE-rania (Alaska, Izu–Bonin, Fiji–Tonga–Kermadec). This is also observed, but less clearly, for SAVANIa and S362WMANI+Ma on PC C. We also distinguish subduction signatures deeper in the mantle on PC F along Cascadia, Central and South America, Tonga, the western Pacific, and the north of the Mediterranean Sea (Fig. ). An overall red negative anisotropy anomaly for the first (PC A) is common to SEMUCB-WM1a and SAVANIa (Fig. ). SGLOBE-rani shows such a red anomaly only under the oceans, which is probably due to the different way the crust is treated in this model, with crustal thickness perturbations being jointly inverted along with isotropic and anisotropic structure .