Global seismic tomography has greatly progressed in the past decades, with many global Earth models being produced by different research groups. Objective, statistical methods are crucial for the quantitative interpretation of the large amount of information encapsulated by the models and for unbiased model comparisons. Here we propose using a rotated version of principal component analysis (PCA) to compress the information in order to ease the geological interpretation and model comparison. The method generates between 7 and 15 principal components (PCs) for each of the seven tested global tomography models, capturing more than 97 % of the total variance of the model. Each PC consists of a vertical profile, with which a horizontal pattern is associated by projection. The depth profiles and the horizontal patterns enable examining the key characteristics of the main components of the models. Most of the information in the models is associated with a few features: large low-shear-velocity provinces (LLSVPs) in the lowermost mantle, subduction signals and low-velocity anomalies likely associated with mantle plumes in the upper and lower mantle, and ridges and cratons in the uppermost mantle. Importantly, all models highlight several independent components in the lower mantle that make between 36 % and 69 % of the total variance, depending on the model, which suggests that the lower mantle is more complex than traditionally assumed. Overall, we find that varimax PCA is a useful additional tool for the quantitative comparison and interpretation of tomography models.

Global seismic tomography has brought a new understanding of the current state of the mantle through the inversion of massive seismic datasets to build 3-D images of the Earth's interior of both isotropic and anisotropic structure, the latter being one of the most direct ways to constrain mantle flow

Statistical methods used in other disciplines to analyse and classify big data and models may be useful to further enhance the analysis of seismic tomography models by providing a common ground for comparison. For example, in recent years, clustering methods have been used to partition seismic tomography models into groups of similar velocity profiles, providing an objective way of comparing the models

Although the first PC, capturing the largest variance, often corresponds to an actual physical process, the others are increasingly difficult to interpret. The physical interpretation of the PCs and loads can be made easier by redistributing PCA components along other eigenvectors. We propose applying the varimax criterion

Varimax analysis has previously been successfully used in various applications, such as to analyse climate models, whereby the different models are projected on the same set of PCs, allowing a direct comparison in terms of captured variance and retrieved features

In Sect.

We use seven 3-D global seismic tomography models: (i) S20RTS

We obtained the global tomography models either directly from their authors or from the IRIS Earth model collaboration repository (REFS –

Global tomography models used in this study, including a short description of the data, parameterisation and the modelling approach used in their construction. All models were built using least-squares inversions with different regularisation choices.

Previous studies have compared global tomography models using

Considering the three-dimensional dataset

To make

A 2-D matrix (

In our case,

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 (

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

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

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

The associated horizontal structures,

Figure

Figure

As expected, the PCA profiles

On the other hand, the normalised varimax procedure recovers well-known structures. Within its first mode, we recover the tectonic patterns, and the LLSVPs gradually appear in the three last components (for a more detailed analysis, see the next section). Based on these comparisons, we find that the varimax PC method is useful to concentrate coherent information at different depths that is available in the seismic tomography models, without any preconception. The next sections will thus focus on the application of this method to the interpretation of the global tomography models considered in this study.

Results from the PC, varimax and

We use varimax PCA to compress the seven tomography models described in Sect.

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

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

This fulfils the first condition for the usefulness of the data compression mentioned above. In order to check the second condition previously mentioned, Fig.

Captured variance by the PC varimax method, applied to the isotropic part of the seven tomography models used in this study. The number of components is chosen such that during the PCA, we only keep the components explaining more than 1 % of the variance, which occurs after 7 (SAVANIi) to 15 (SGLOBE-ranii) components. The varimax PCs are sorted alphabetically from the shallowest one to the deepest ones.

Principal components for the different individual model varimax PCAs and the combined one (see Sect.

Figure

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.

We note in Fig.

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

The nine varimax components of the SEMUCB

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

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.

Depending on the region, the high-velocity craton signature should reach a maximum depth between 100 and 175 km

The low-velocity zones underneath the Tibetan Plateau

From

All models show a high-velocity zone between

In the East African Rift, the low-velocity anomaly aligned with the Afar Depression and the Main Ethiopian Rift in the uppermost mantle

All models show high-velocity subduction zones in the western Pacific, among others, notably underneath the Philippine Plate over two principal components with depths

More to the south (north of Papua New Guinea), the Caroline Ridge from

The Tonga–Kermadec subduction zone, located below the south Fiji Basin down to a depth of

All models evidence the LLSVPs, though they are less clear in some models, such as SAVANIi and S362WMANI

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.

Variance [%] obtained from the individual varimax analysis of each model. In parentheses, the number of components capturing more than 1 % of the variance is shown. The last column provides the variance captured by 12 components for the combined analysis of the seven models, as discussed in Sect.

Examples of key geophysical patterns recovered in the mantle by the varimax analysis (see Figs.

The horizontal patterns associated with each PC result from the projection of the tomography model on the varimax PCs, which differ from one model to the other. As suggested by

PC F (maximum at 1000 km of depth) of the combined analysis of the isotropic models. On the right are the principal components or vertical profiles, and on the left are the associated horizontal structures. The other components are shown in the Appendix in Figs.

Components A–D (at

Most of the patterns described in Table

As shown in the previous section, all models display several independent components in the lower mantle, making between 36 % (SAVANIi) and 69 % (SEISGLOB2) of the total components, depending on the model. This highlights complexity in the lower mantle and supports recent studies suggesting that the region of the lower mantle above the lowermost D

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

The two LLSVPs appear on the deepest PC J of SGLOBE-rania and SEMUCB

For PC B (with a maximum at

Components B (maximum depth 100 km; Fig.

Global seismic tomography models typically involve thousands to tens of thousands of parameters, which can be cumbersome to handle and difficult to interpret. This is also true for model comparison; we lack a common basis for comparing models built with different parameterisations.
In this study we used a rotated version of principal component analysis to compress the information and ease the geological interpretation and model comparison. The varimax PC analysis results in a separation of the information into different components associated with depth distributions, which are linked to a horizontal pattern obtained by orthogonal projection. We tested the analysis on seven global tomography models: S20RTS, S40RTS, SEISGLOB2, SEMUCB

We found that by using the varimax method we reduced the number of independent depth components needed to describe more than 97 % of the total information in the tomography models by 29 % to 75 %.
We note that the scale of heterogeneity is not relevant for the varimax PCA method, which is only based on the vertical covariance. Considering the low amount of variance lost in the reconstruction (e.g. Fig. 4) and the spectrum shown in the Appendix, we capture most of the information, and we do not change the spectrum of the signal. Thus, the method is valid for any scale, as long as the signal is robust. In the varimax comparison, what is called noise is not the small-scale features, but rather the part of the models that is not covariant vertically. Hence, the varimax analysis simplifies the number of patterns that needs to be analysed without any significant loss of information; by ensuring the orthogonality of the depth components, it eased the detection and comparison of the relevant information. Overall, the large majority of depth components and horizontal maps obtained from the varimax analysis are different from the original parameterisations used for building the models. This is especially true above the 600 km discontinuity and in the whole mantle for the SAVANIi, S362WMANI

Being data-based, the varimax method is neutral with respect to the assumptions made in the model's construction. The combined multi-model varimax analysis allows the comparison of the various tomography models on a neutral set of modes determined by the level of compression fixed by the user. Based on the vertical consistency between the various tomography models, it provides a set of data-based vertical distribution functions. Those functions represent the information present in the PC-based reconstruction and how the models relate to each other.
It is fast and simple to implement, and, as we maximise the captured variance, the level of compression is lower for a given number of components and depths than would be required by other methods such as

When applying the varimax analysis to isotropic tomography models, we found that the most important elements of the models contributing to most of the information are (i) large low-shear-velocity provinces (LLSVPs) in the lowermost mantle, (ii) subducted slabs and low-velocity anomalies probably associated with mantle plumes in the upper and lower mantle, and (iii) ridges and cratons in the uppermost upper mantle. The analysis highlights several independent components in the lower mantle that make between 36 % and 69 % of the total components depending on the model, which supports recent studies suggesting that the lower mantle is more complex than previously thought. The reasons for this complexity remain a very active field of research. On the other hand, we find limited agreement between the radial anisotropy structure of the models, with common features mainly in the asthenosphere and to some extent in the lower mantle beneath the Pacific and beneath subduction zones.

Choices such as data types and amounts, as well as the strength of regularisation used in the construction of tomography models are probably key controls on the number of varimax components required by each model. Hence, the PCA-based model compression preserves the impact of the choices made in the construction of the tomography models and facilitates their interpretation in terms of geophysical objects. Future work will expand this analysis for the interpretation and comparison of local and regional models, which tend to use more diverse underlying datasets than in global models and have highly variable spatial resolutions. Moreover, we will also focus on comparisons with other sources of information, such as geodynamical models, gravity anomalies, magnetic anomalies and heat fluxes.

Depth slices of the isotropic and anisotropic models used in this study at the depths of 100, 200, 300, 400, 600, 800, 1000, 1400, 2000 and 2900 km. The isotropic

Examples of six depth slices in the original SEMUCB

PSDs of the original models.

PSDs of the reconstructed models from the varimax modes. The number of modes is given in Table

The 15 varimax components of the isotropic part of the SGLOBE-rani model. On the right are the principal components or vertical profiles, and on the left are the associated horizontal structures.

The seven varimax components of the isotropic part of the SAVANI model.

The 13 varimax components of the S20RTS model.

The 12 varimax components of the S40RTS model.

The eight varimax components of the isotropic part of the S362WMANI

The 13 varimax components of the isotropic part of the SEISGlob2 model.

PC A (maximum at 50 km of depth) of the combined analysis of the isotropic parts of the models. On the right are the principal components or vertical profiles, and on the left are the associated horizontal structures.

PC B (maximum at 200 km of depth) of the combined analysis of the isotropic parts of the models.

PC C (maximum at 300 km of depth) of the combined analysis of the isotropic parts of the models.

PC D (maximum at 600 km of depth) of the combined analysis of the isotropic parts of the models.

PC E (maximum at 800 km of depth) of the combined analysis of the isotropic parts of the models.

PC G (maximum at 1200 km of depth) of the combined analysis of the isotropic parts of the models.

PC H (maximum at 1400 km of depth) of the combined analysis of the isotropic parts of the models.

PC I (maximum at 1700 km of depth) of the combined analysis of the isotropic parts of the models.

PC J (maximum at 2000 km of depth) of the combined analysis of the isotropic parts of the models.

PC K (maximum at 2300 km of depth) of the combined analysis of the isotropic parts of the models.

PC L (maximum at 2800 km of depth) of the combined analysis of the isotropic parts of the models.

PC A (maximum at 50 km of depth) of the combined analysis of the anisotropic parts of the models. On the right are the principal components or vertical profiles, and on the left are the associated horizontal structures.

PC B (maximum at 100 km of depth) of the combined analysis of the anisotropic parts of the models.

PC C (maximum at 200 km of depth) of the combined analysis of the anisotropic models.

PC D (maximum at 400 km of depth) of the combined analysis of the anisotropic parts of the models.

PC E (maximum at 600 km of depth) of the combined analysis of the anisotropic parts of the models.

PC F (maximum at 800 km of depth) of the combined analysis of the anisotropic parts of the models.

PC G (maximum at 1300 km of depth) of the combined analysis of the anisotropic parts of the models.

PC H (maximum at 1900 km of depth) of the combined analysis of the anisotropic parts of the models.

PC I (maximum at 2300 km of depth) of the combined analysis of the anisotropic parts of the models.

PC J (maximum at 2800 km of depth) of the combined analysis of the anisotropic parts of the models.

A Python function for computing PCA and varimax PCA
is provided at

OdV and AG performed the formal analysis, OdV and MVC developed and adapted the methodologies used, OdV made the visualisation, and all the authors contributed to the investigation and validation of the results.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Ana M. G. Ferreira is grateful for support from NERC grant NE/N011791/1. We greatly thank Lapo Boschi, Barbara Romanowicz and Raj Moulik for providing us with files containing their model parameterisations. The reviewers of the previous version and of this version of the paper are gratefully acknowledged for their insightful comments.

This research has been supported by the CNES as a methodological case study for the exploitation of geodetic space missions.

This paper was edited by Juliane Dannberg and reviewed by two anonymous referees.