We use satellite gravity data from the global gravity field model GOCO05S at the measurement height (225 km) of the GOCE satellite in its last mission phase . Often, gravity data are mathematically downward continued to a lower height, but this introduces omission errors , which we would like to avoid. In addition, the target density distributions are at such a great depth that downward continuation probably provides no additional sensitivity.

As a first step we calculate an ice-corrected Bouguer anomaly based on the topography of ETOPO1 . Onshore we assume a density of 2670 kg m-3, and offshore we use a correction density of 1770 kg m-3 based on a reference crustal density of 2800 kg m-3 and a water density of 1030 kg m-3. In addition, the influence of ice is removed assuming a density of 917 kg m-3.

Since we are interested in the mantle density structure, the second step is to account for the gravity effect of the crust and/or lithosphere. Two inherently different approaches exist to achieve this. The first option is to assume some form of isostatic compensation and use this to account for the crust and/or lithosphere. Alternatively, the gravity effect of crustal models, such as Crust1.0 or Litho1.0 can be calculated. The latter approach includes more geophysical data, but it also “propagates” errors of the crustal model into the mantle. In particular, all crustal models predict several kilometres of residual topography, even in the continents and are thus not nearly in isostatic balance. As a consequence, the topography would need to be supported dynamically, which should lead to higher free-air anomaly values than observed .

In light of this discrepancy, we explore two approaches. In the first approach, we assume that continental topography is compensated by crustal thickness variations, whereas ocean floor depth is compensated by variations in the mantle lithosphere density (the same isostatic model as in , with a compensation depth of 120 km). The isostatic residual is then 1isostaticresidual=FA-gtopo-giso, where FA is the free-air anomaly (Fig. a), gtopo is the gravity effect of topography and giso is the gravity effect of the isostatic compensation masses.

In the second approach, we use our recent crustal model based on kriging and assume a density contrast between crust and mantle of 400 kg m-3 to estimate the gravity effect of varying crustal thickness. This gives the crustal residual: 2crustalresidual=FA-gtopo-gcrust, where gcrust is the gravity response of the crustal thickness variations.

The isostatic residual is very similar to the free-air anomaly (Fig. b), except in areas of high topography. In contrast, the crustal residual has a much higher amplitudes and patterns (Fig. c). For comparison we also computed the residual (non-isostatic) topography of the kriging crustal model , which anti-correlates with the crustal residual on large scales (Fig. d).

Additional constraints are required to overcome the inherent non-uniqueness of a density inversion. We use information from whole mantle seismic tomographies in the form of vote maps to define volumes which could potentially contain density anomalies. Vote maps were first introduced by in the context of cluster analysis and then further developed by . A vote map is based on a collection of tomography models, and at each point it gives the number of tomographies that detect a significant anomaly at that location. The threshold that constitutes a significant anomaly is the standard deviation at that depth for each seismic tomography. Thus, even tomographies with vastly different amplitude ranges can be combined in a single vote map. We used the vote maps based on 17 S-wave tomography models as available on SubMachine at a resolution of 1∘ and 100 km depth spacing. There are two separate vote maps for positive and negative anomalies. Figure  shows an example of the vote maps at a depth of 2700 km, close to the core–mantle boundary.

The first step is to discretize the inversion problem by extracting regions of potential density anomalies from the vote map individually for each depth. Our underlying assumption is that density anomalies only occur in horizontally connected regions where at least N tomographies detect anomalous velocity. We choose N=10, because this is the cut-off that correlates with the strongest gradients of the vote maps. Each connected region is assigned a constant but unknown density anomaly. As a result of this process, we have a total of 1135 anomalous regions that are each described in terms of their depths, their pixels and unknown density anomaly values. These density anomalies are completely free parameters that will be adjusted during the inversion to fit the gravity field. Importantly, we impose no correlation between velocity and density.

Our inversion is a two-step approach. First, we forward calculate for each potential density anomaly region its characteristic gravity response. Second, the characteristic gravity responses of all of the potential density anomalies are placed into a matrix A and solved with a generalized Tikhonov regularization technique. The regularization parameters are chosen based on cross-validation.

The gravity effect of each potential density anomaly is estimated by approximating it as a collection of point masses. Each point mass represents a volume V, given by 3V=r2cos⁡(lat)(Δϕ)2Δz, where r is the radius, lat is the latitude, Δϕ is the angular size of each pixel (1∘) and Δz is the vertical resolution (100 km). The mass m is then found by multiplying the volume with the density anomaly ρ.

The gravity kernel K(P,Q) caused by a unit point mass located at P=(r,lat,long) on a measurement point Q=(ri,lati,longi) is given by 4K(P,Q)=Grs-rcos⁡γr2+rs2-2rrscos⁡γ3/2, where γ is the great-circle distance between the point mass and the measurement point, and G=6.67428×10-11 m3 (kg s2)-1 is the gravitational constant. The total gravity effect of each potential density anomaly is then found by summing over the gravity effect of all its pixels.

Next, the gravity response of all of the potential density anomalies are placed into a matrix A, where Aij is the gravity effect of potential density anomaly j on measurement point i, assuming a density of 1. Using Eq. (), Aij becomes 5Aij=∑k∈RjK(Pk,Qi)Vk, where Rj is the set of indices belonging to potential density anomaly j; P contains all subsurface points and Q all measurement points.

The unknown density values of the potential density anomalies are placed in a vector ρ. In principle, ρ=A-1g, where gi=g(Qi) is the measured gravity value at point Qi. However, in practice the solution will be unstable and non-unique, so that regularization is required. Here, we choose to minimize the following least-squares functional: 6χ(ρ)2=||Aρ-g||Σg-12+β||ρ||2+γ||Dρ||2. The three terms relate to the data misfit, the magnitude of density variations and the vertical derivative of density anomaly. ||⋅||M is the quadratic form related to matrix M, and ||⋅||=||⋅||I is the Euclidean norm. β and γ are the regularization parameters that enforce the minimization of absolute norm of the density anomalies and their vertical derivative respectively. In addition, the parameter β represents the ratio of data variance to model variance. D is a finite differencing matrix consisting of first-order forward vertical differences in density. Finally, Matrix Σg is an assumed correlation matrix of the gravity values. We use an isotropic correlation function with a correlation distance of 10∘, based on a semivariogram analysis . The main purpose of the matrix is to down-weigh observation points near the pole, which are otherwise over represented due to convergence near the pole.

The optimal ρ belonging to this formulation is 7ρ^=ATΣg-1A+βI+γDTD-1ATΣg-1g. The regularization parameters β and γ exert critical control over the resulting density structure and the achievable data fit. The most popular methods to determine such parameters are L curves and cross-validation . Here we use cross-validation to estimate the regularization parameter.

In k-fold cross-validation, the gravity data set is randomly split into k distinct index sets Ij. The first set I1=T then becomes the training set and the remaining sets are combined to form the validation set V. The training set is then used to solve the inversion problem giving a solution ρ1. Based on how well ρ1 predicts the data in T and V, the training and validation misfit is calculated. Each Ij becomes the training set once, and this is repeated for many random partitions of the data.

By design, the training misfit is minimal when β and γ are zero (if there are no numerical instabilities). However, small values of β and γ also risk overfitting the data. By carrying out cross-validation for all combinations of β and γ in a reasonable range of values, the optimal β and γ corresponding to the minimum validation misfit can be found.

Apart from constraining the regularization parameters, this procedure also gives a bootstrap estimate of the density structure. The collection of all the density anomaly estimates derived for all subsets of gravity data can be used to calculate a mean density anomaly value and its standard deviation for each potential anomaly, given certain regularization parameters. This is useful to determine how robust a single density anomaly is.

The surface and CMB deformations implied by the density model recovered by inversion are important to judge the quality of the results. The surface topography predicted by the density model should roughly agree with the residual topography, because topography is strongly sensitive to density. The magnitude of the CMB deformation is required to assess how much it affects the gravity field fitting.

We determine the deformation of the upper and lower domain boundary caused by density structure as a postprocessing step after inversion. We use both an isostatic and a simple dynamic formulation to determine this deformation.

Let tCMB(lat,long) be the CMB-topography as a function of longitude and latitude. Under the assumption of isostatic balance at the CMB, we find tCMB(lat,long)=-1ΔρCMB∫CMBCMB+Hδρ(r,lat,long)dr, where ΔρCMB is the density jump at the CMB, H is the height above the CMB where isostatic balance applies, and ρ‾ is the mean density between r=CMB and r=CMB+H.

We convert the density difference associated with undulating topography into a surface density σ=tCMBΔρCMB and calculate its gravity response in spherical harmonics by g=∑l,mglmYlm=∑l,m4πGl+12l+1rCMBrsl+2σlmYlm, where σlm values are the spherical harmonic coefficients of the surface density, Ylm is a spherical harmonic function and rs=6371+225 km is the radius of the observations.

Isostatic surface topography is calculated in the same way using the upper 300 km of the domain and a density contrast of 2670 kg m-3 at the surface.

To determine the dynamic surface and CMB topography we use the propagator matrix approach of , with free slip boundary conditions on the surface and CMB. The viscosity is layered and increases by a factor of 100 at 700 km depth. This is a simplification of the real viscosity structure, which is additionally affected by lateral variations due to temperature and – possibly – compositional differences.

The resulting kernels show that for the assumed viscosity structure, the influence of density anomalies below 500 km on topography is limited, even at very long wavelengths (Fig. a). The CMB kernels decrease much more slowly, even with a layered viscosity (Fig. b) and even at degree 10.

A grid search of β and γ values between 10-3 and 103 using the cross-validation procedure leads to a minimum validation misfit of 5.32 mGal if β=0.03 and γ=1.0 (Fig. a). Although this minimum is not particularly well developed, we proceed with these values of β and γ, since neighbouring regularization settings give similar results. With these regularization parameters, the training misfit becomes 3.25 mGal, which is about 20 % of the rms of the input gravity field (96 % variance reduction).

The inversion is able to reasonably reproduce the main features of the isostatic residual gravity field (Fig. b). However, the amplitude of the residual field is systematically too small. A linear regression of the gravity field predicted from the model against the measured values gives a linear factor of 0.85, implying that the modelled anomalies are about 15 % too weak. This is probably a result of the regularization.

In the spectral domain, the predicted gravity field is able to reproduce the long-wavelength part up to spherical harmonic (SH) degree 10 (Fig. c). At shorter wavelengths, the predicted field is systematically too weak, going down to less than 10 % of the observed data at spherical harmonic degree 40. For comparison, the solution with no regularization (β=γ=0) is able to reproduce the spectrum up to spherical harmonic degree 25 perfectly. Thus, wavelengths shorter than SH degree 25 simply cannot be produced, due to the insufficient spectral content of the vote maps.

The inverted density anomalies show no clear correlation with sign of the velocity anomaly (Sect. ), which would suggest that many density anomalies are not caused only by temperature variations. The spatial distribution is highly complex, and not all density anomalies are easily related to expected features. However, some first-order trends can be identified (Fig. ).

The upper 100 km of the inversion results contain very little density variations (Fig. a). Some cratons (South America, South Africa, Eurasia) show a slight negative density anomaly (Fig. b). This is probably related to simplifications in our isostatic model and not representative of the actual density structure

At greater depths, subducted slabs would be expected in the density structure. This is overall reflected in our density inversion results, but not all slabs are resolved as positive density anomalies over their complete depth range. For example, the Andean slabs can be seen at a depth of 900 km as an anomaly of +6 kg m-3 (Fig. d) but not at 600 km depth.

The LLVPs have systematically a positive density anomaly of up to +10 kg m-3 (Fig. e and f). The top of the African LLVP appears at a depth of 1800 km and extends down to the CMB. The Pacific LLVP only appears at greater depths of 2200 km and is also continuous to the CMB. Both LLVPs are overlaid by slight negative density anomalies.

The preferred regularization parameters derived by the cross-validation procedure are similar to those for the isostatic residual, β=0.1 and γ=1.0 (Fig. a). With these values, a validation misfit of 34 mGal and a training misfit of 23 mGal are achieved, which corresponds to about 40 % of the input signal. Thus, the fit for the crustal residual is considerably worse than for the isostatic residual, both in an absolute and a relative sense. The predicted and input gravity data only agree qualitatively in some areas (Fig. b), and large residuals remain (Fig. d).

Overall the inverted density variations are much larger for the crustal residual than for the isostatic residual (Fig. ). In the upper 300 km or so, this is expected, since the lithospheric density structure has not been accounted for by the crustal model. The ocean cooling trend is not imaged well by the vote maps, and as a result, only the Pacific and Indian oceans' spreading ridges are resolved as crude negative density anomalies (Fig. a). Some cratons (North America, Eurasia) show strong positive density anomalies (30 kg m-3), but this neither is consistent vertically nor applies to most geographical regions (Fig. a and b). Thus, the ability to resolve mantle lithospheric sources is limited and probably contributes to the poor fit of the data.

At depths of 600 and 900 km, some subduction-related density anomalies can be seen, but overall correlation is poor. Only in the lowest mantle a more consistent picture emerges, in particular the LLVPs are resolved as positive density anomalies (+40 kg m-3). As in the case with isostatic residual, the African LLVP appears to extend further upwards as a positive density anomaly than the Pacific LLVP (Fig. e and f).

While the main features observed are the LLVPs, it is important to also scrutinize the results for the remaining mantle. A long-wavelength error in the upper-mantle density structures leads to an incorrect estimate of the density of the LLVPs. In the upper mantle above the transition zone (410–670 km depth) much of the seismic image can be linked to tectonic features: thick and cold lithospheric roots underlie the cratonic cores of the continents and can extend up to 300 km deep ; the oceans are characterized by a velocity trend related to cooling of the lithosphere ; and slabs formed during subduction sink through the mantle, sometimes stagnating at various depths and sometimes passing through the transition zone to the lowermost mantle (e.g. ).

The cratonic lithospheric mantle density is the result of a complex interaction between temperature, composition and mineralogy . The density increase due to low temperature is probably (partially) offset by its depleted composition (the isopycnic hypothesis; ). Indeed, a slightly positive density anomaly (+5 to 7.5 kg m-3) is recovered by the inversion with crustal residual in the cratonic cores of North America, central Europe, Siberia and East Antarctica. In contrast, the inversion with the isostatic residual does not show the cratons. The cratonic mantle keels affect surface topography and, hence, are removed by the isostatic correction.

The density of the oceanic mantle is expected to increase with age, following the cooling trend, which also explains seafloor subsidence. The inversion with crustal residual shows some decreased density associated with mid-oceanic ridges, but vote maps are unsuited to resolve the continuous gradient associated with the cooling of the plate. When using the isostatic residual, ocean floor subsidence and the cooling trend has already been accounted for during the isostatic correction and is thus not resolved by the inversion.

Subducted slabs are denser than surrounding mantle immediately after subduction. As the plate sinks, heat diffuses into the plate from the surrounding mantle, so that the plate slowly loses its negative buoyancy. How quickly a subducted plate thermally equilibrates depends on how effectively heat is transported to the slab by convection and conduction. Since some slabs seem to stagnate on mantle discontinuities, it is conceivable for a slab to loose all or most of its negative buoyancy. We compared our results with the slab depth contours from Slab 1.0 .

Out of the 12 slabs contained in Slab1.0, only five are detected as positive velocity anomalies based on the vote maps (Fig. a) and only at depths greater than 200 km or more. These are Izu–Bonin, Kermadec–Tonga, Kamchatka–Kurils–Japan, South America and Sumatra, according to the terminology of Slab1.0.

The reason might be that the tomography models are too heterogeneous in quality and resolution to resolve these relatively narrow features. Of course, vote maps are not the only way to extract features from seismic tomographies, and an alternative would be to use the “best” seismic tomography model available instead of a collection of models. But there are no clear criteria to decide which tomography result is the best, except that newer results are probably based on more and better data. Furthermore, there is some evidence that the differences between tomographic models partly reflect different regularization parameters, rather than different resolved features . Thus, vote maps only extract the features that are resolved by a majority of tomographic models

Still, out of the five detected slabs, four have positive density contrasts (+5 to +10 kg m-3) in the inversion with the isostatic residual (Fig. b). However, these density anomalies are not continuous over the entire depth range of the slabs and are limited to depths between 400 and 700 km. Similar results are obtained using the crustal residual as input, but the density anomalies are more sporadic and concentrated at a single depth of 400 km with stronger density variations of up to +30 kg m-3 (Fig. c).

Based on these results, we think that our density inversion gives acceptable results overall for most of the mantle. The binary classification scheme we use to extract regions from the vote maps leads to a very rough representation in the upper 300 km, which likely contributes to the poor data fit obtained using the crustal residual. In addition, we assume a constant density contrast over the vertical resolution of 100 km. Thus, the volume of relatively thin structures like slabs might be smeared out over a too large depth range.

We compare the recovered density values with velocity values from the SMEAN2 tomography, an update of SMEAN . SMEAN2 is itself an average of three tomography models and agrees reasonably well with the patterns of the vote maps. To correlate velocity and density, every potential density anomaly is assigned the average value of SMEAN2 over the area of the potential density anomaly.

Next, we calculated the distribution of the ρ/vs ratio at each depth. For this distribution, we used the variability in each density value that we calculated during the cross-validation procedure. In addition, we weighted the distribution by the area of each potential density anomaly. Two histograms were estimated for the fast and slow anomalies.

In the case of the isostatic residual, the median conversion factor changes significantly with depth (Fig. a). In the upper 300 km it is on average close to zero, starts to increase at depths greater than 300 km and reaches a maximum of ∼0.2 at 1000 km depth. Remarkably, the conversion factor then decreases rapidly with depth, becoming negative at around 1200 km depth. The distributions at the different depths are highly variable as well. At most depths, the scaling factor shows a multimodal distribution, so there is no single consistent velocity–density relation.

The velocity–density scaling relations we calculated could have the following causes. The small scaling values in the upper 300 km is due to the fact that we use the isostatic residual. The increased scaling factors in the transition zone depths could be related to undulations in the phase transitions depths, because the higher-pressure phases have both higher density and velocity. Thus, any undulation of the phase transition would strongly affect both density and velocity. The negative velocity-anomaly-to-density-anomaly ratios we calculated below 1200 km depth zone can only be explained by compositional variations. In Appendix A we calculate sensitivities of density and velocity to temperature, iron content and MORB (mid-oceanic ridge basalt) fraction changes. We find that the sensitivities to iron content and MORB fraction are fairly constant below the transition zone, whereas the temperature sensitivity slowly decreases with depth. Thus, the velocity and density values below the transition zone could be more indicative of compositional variations than above the transition zone.

The joint geodynamic–tomographic model Gypsum also recovered a depth-variable velocity anomaly to density anomaly scaling of similar magnitude as our results. In addition, they also obtained an increase towards a maximum at around 1000 km depth. However, while their results show a decrease below 1000 km, they did not obtain systematically negative scaling factors, because they regularized towards a thermal scaling factor, which is always positive.

However, the negative conversion factors might also be affected by unmodelled dynamic effects. Depending on the viscosity model, the geoid kernel can become negative at the longest wavelengths in the lower mantle . This is mainly due to the effect of dynamic topography at the surface and the CMB, which has the opposite sign of the density anomaly causing it. However, we do not think that this effect alone is sufficient to explain the sign change in the lower mantle. We use the gravity field, not the geoid, and the former is more sensitive to the dynamic surface topography, which is predicted to be small, based on our density model (see next section). Still, dynamic compensation of masses in the lower mantle on the CMB could bias our results in the direction of negative conversion factors (see also Sect. ).

In the case of the crustal residual, the increased overall density variation leads to higher velocity anomaly to density anomaly scaling values (Fig. b). However, the distribution of the scaling values are so broad that there is not a clear change in the scaling with depth.

We calculated the contribution of the mantle to surface topography. This allows one to assess the quality of our results, because the predicted mantle topography contribution should be zero for the isostatic residual and equal to the residual topography in the crustal residual case.

In the isostatic case, the upper mantle contributes less than 250 m to surface topography (Fig. a). Contrary to expectation, there is a slight positive topographic contribution in some regions with thick lithosphere (North America, eastern Australia, East Antarctica) due to negative density anomalies in the upper mantle. The likely explanation is that we only use the Moho depth to compensate topography, but in reality there is an additional contribution from the lithospheric mantle. Thus, our isostatic Moho depths is too shallow, making the isostatic correction too positive, which is then corrected by the density inversion.

These results are also confirmed by the dynamic topography calculation. With the two-layer viscosity model that we chose, the topography kernels are essentially zero except for the upper 300 km of the mantle. Hence, the predicted dynamic topography is very similar to the isostatic topography. Even at degree 2, where one would expect the strongest signal from dynamic topography, according to the kernels, there is hardly any topographic contribution, because the high-density LLVPs (the dominating degree 2 structure) are overlaid by negative density anomalies. The existence of a low-density layer on top of the LLVPs was also observed in a mantle convection simulation .

The inversion results with the crustal residual reproduce the main feature of the residual topography of our crustal model to first order. The isostatic topography contribution of the upper 300 km is considerable (up to 1.5 km). In the North American and Siberian craton, the isostatic topography is mainly negative due to the strong influence of thick, dense lithosphere, but in other areas (e.g. South Africa) the topographic contribution is negative. In the oceans, the expected signal from ocean floor spreading can only be seen in the Pacific. This is because the vote maps only reflect the oceanic cooling trends in a very crude way, such that only the broad ridges in the Pacific can be captured using this technique.

Still, there is qualitative agreement between the residual topography of the crustal model we used and the topography contribution from the upper mantle. The remaining misfit is due to an imperfect crustal model and the poor representation of the lithospheric mantle in the vote maps. In a recent model , it was found that topography and the gravity field can be explained by the crust and upper mantle (for wavelengths shorter than spherical harmonic degree 15), while still staying within the uncertainty in the global seismological model Litho1.0 . This suggests that when using a crustal model to account for the upper Earth, one should jointly consider the uncertainties in the crustal model, possibly even co-inverting a crustal model.

We did not explicitly include the gravity effect of CMB deformation in our inversion. To estimate the impact of CMB topography, we use the results based on the isostatic residual gravity. If the lower 800 km of the mantle is isostatically balanced on the CMB and assuming a density contrast of 4500 kg m-3 at the CMB, the inverted density structure would lead to CMB undulations of ±1200 m. The CMB topography is dominated by the depressions underneath Africa and the Pacific that are caused by the increased density of the LLVPs. At satellite altitude, these undulations would cause a gravity response of ±20 mGal. Due to the great depth of the source, the gravity response is anti-correlated with the proper gravity effect of the density anomalies in the lower 800 km of the mantle. Thus, the gravity effect of isostatic CMB deformations would reduce the gravity effect associated with lower-mantle density anomalies from ±30 to ±15 mGal but does not affect the spatial pattern. Hence, the actual density anomalies in the lower mantle would have to be twice as large to produce the same gravity effect as without CMB deformation. Therefore, we expect that the inversion underestimates the true density variations in the lowermost mantle.

These results also hold if dynamic topography is calculated using a two-layer viscosity model at least at the wavelengths where the gravity signal of the CMB would be visible (compare Figs. d and d). However, several studies have found that the convection dynamics in the lowest mantle might be more complicated. Viscosity probably varies laterally, because of strong temperature or composition differences . In addition, post-perovskite might have much lower viscosity than perovskite . In some scenarios the CMB deformation decorrelates from the density distribution in the lowest mantle. These findings imply that there is an additional source of uncertainty for the LLVP densities that we estimated.

Comparing our results with published CMB topography estimates based on seismological methods , we find that none of the models contain the first-order pattern of CMB depressions beneath Africa and the central Pacific, seen in our maps (Fig. c, d). In terms of the magnitude, the CMB topography based on the isostatic residual are much smaller than any seismic determination except Sze and van der Hilst (2003), whereas the crustal residual CMB topography has similar range as Doornbos and Hilton (1989) and Morelli and Dziewonski (1987).

The results obtained with isostatic and crustal residual disagree substantially. The correlation coefficient calculated for individual depth slices is typically less than 0.5 and even negative for some depths. The highest correlations are found in the depth ranges of 2500–2800 and 1500–2000 km and are mainly due to the influence of the LLVPs. In addition, the magnitude of the density anomalies is on average 4 times larger based on the crustal residual, which agrees with the relative magnitude of the input gravity fields.

Based on the fit to the data, the isostatic residual is preferred, because it achieves both higher absolute and relative data fit. However, the isostatic Moho depths disagree with the seismological determinations. Furthermore, the crustal residual lacks a correction for the oceanic cooling trend, which clearly contributes to the crustal residual. Thus, the better data fit preference for the isostatic residual is not as straightforward.

In any case, the isostatic and crustal residuals are extreme examples of how the crust and upper mantle can be accounted for in a gravity inversion. Our results clearly demonstrate that these different approaches lead to an enormous spread in terms of the recovered densities in the mantle. At the same time, signals from the deep Earth in terms of gravity or topography might affect modelling of the upper Earth. Commonly, high-pass filtering is used to remove the signal of the deep Earth , but this is clearly insufficient, due to the spectral overlap . Thus, a coupled approach that simultaneously considers the entire crust and mantle is required in order to properly model the gravity field and Earth's topography.

Broadly speaking, the LLVPs could originate from any combination of increased temperature and compositional variation. After their first discovery, the LLVP were initially considered as a purely thermal feature: two “superplumes” that rise from the core–mantle boundary . However, even if the LLVPs are purely due to temperature increase, a more likely explanation is that they are swarms of smaller plumes that are smeared due to the limited resolution of seismic tomographies , with an additional influence from the stability field of post-perovskite .

In contrast to this isochemical view, some authors have proposed that the LLVPs are chemically distinct from the normal (pyrolitic) mantle. The chemical distinctiveness of the LLVP can either be accumulated over time or be a primitive reservoir that separated early in Earth's history . A likely process for accumulation is the separation of oceanic crust from subducted lithosphere . However, these two hypotheses are not mutually exclusive, and a small amount of subducted oceanic lithosphere might be incorporated in pre-existing structure, as proposed in the basal melange (BAM) hypothesis . If the LLVPs have a compositional component, they have to be intrinsically denser than surrounding pyrolitic mantle to have remained near the core–mantle boundary, despite very high temperatures. If the LLVPs are indeed old features, this would also explain the proposed spatial correlation between plume generation and the edges of the LLVPs over the last 200 million years .

Our inversion results indicate a positive density anomaly for the LLVPs both using isostatic and crustal residual. This would rule out a purely thermal origin of the LLVPs, since such a scenario would lead to negative density anomalies.

In order to test different scenarios using our inversion results, we make use of the petrological database of . We construct a simple adiabatic model (see Appendix A) based on a pyrolitic composition for the mantle that serves as reference model. The adiabatic model is entirely self-consistent, and the only free parameter is the temperature at the top of the model, which we adjusted in order to fit the surface wave dispersion curves from PREM . Unlike previous thermochemical interpretations for the lower mantle (e.g. ), this corrects for the bias between the petrophysical database and PREM. Next, we applied first-order perturbations to the model and obtained sensitivities of shear wave velocity and density to changes in temperature, iron content and fraction of mid-ocean ridge basalt (MORB).

We assume a negative S-wave velocity deviation of 2 % for the LLVPs, since most mantle tomographies display roughly this amount of slowness. Our inversion results would place the density anomaly of the LLVP between 0.1 % (isostatic residual) and 0.3 % (crustal residual), but due to possible isostatic compensation at the CMB, (Sect. ), the density anomalies could be twice as large (up to 0.6 %).

A temperature increase of 670 K leads to the required velocity reduction; however it would also entail a density change of -1 %, which would be incompatible with our findings. Likewise, a 2.6 % increase in iron content (without temperature change) fits the velocity reduction but leads to a density increase of 1.6 %. To fit both velocity and our lowest density estimate, a temperature increase of 380 K and an iron increase of 1.1 % is required, while our highest density estimate would require changes of 260 K and 1.6 % FeO respectively.

Adding a MORB fraction leads to higher required temperatures, because MORB is slightly faster than pyrolitic mantle at the depths of the LLVPs, according to our petrophysical calculations. Our lowest density estimate (+0.1 %) would require a temperature change of +870 K and a MORB fraction of 40 %, and the highest density estimate (+0.6 %) requires +960 K and 58 % MORB. also found high excess temperatures (+1500 K) were necessary to explain the shear wave velocity and bulk sound speed if the LLVPs are MORB-enriched. However, our interpretation is dependent on the petrological database of , which is based on sparse measurements at extremely high pressures. Recent measurements of calcium–perovskite (Ca–Pv) imply that seismic velocities in MORB-enriched zones would be lower than thought previously. Thus, less extreme temperatures would be required to reconcile increased density and reduced velocity. Furthermore, the chemical composition of MORB is highly variable, especially its iron content , which could also strongly affect the impact of mixing a MORB fraction into a pyrolitic mantle.

Based on our results it is difficult to express preference for MORB or iron enrichment. Since MORB is introduced to the mantle by subduction, plate reconstructions place some constraints on the amount of MORB produced. estimate that the total amount of basalt input into the mantle corresponds to about 10 % of the volume of the mantle. As an absolute upper limit, assuming the LLVPs have a height of 800 km and cover 25 % of the core surface area, the volume would be around 4 % of the total mantle. Thus, to reach a MORB fraction of 60 % in the LLVPs, about a fourth of the total generated subducted basalt would need to accumulate.