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High-resolution mantle circulation models (MCMs) together with thermodynamic mineralogical models make it possible to construct 3-D elastic mantle heterogeneity based on geodynamic considerations. Recently, we have shown that in the presence of a strong thermal gradient across D″ and corresponding large temperature variations in the lower mantle, the heterogeneity predicted from isochemical whole mantle flow agrees well with tomographic models in terms of magnitudes of S wave velocity (vs) variations. Here, we extend the comparison of geodynamic and tomographic structures by accounting explicitly for the limited resolving power of tomography. We focus on lateral variations in vs and use the resolution operator associated with S20RTS to modify our geodynamic models so that they reflect the long-wavelength (>1000 km) nature and the effects of heterogeneous data coverage and damping inherent to the tomographic inversion. Prior to the multiplication with , the geodynamic models need to be reparameterized onto the basis of S20RTS. The magnitude reduction introduced by this reparameterization is significant and needs careful assessment. We attempt a correction of the reparameterization effects and find that the inherent tomographic filtering alone then leads to a magnitude reduction by a factor of ∼2 in the lower mantle. Our tomographically filtered models with strong core heating agree well with S20RTS, which resolves maximum negative anomalies of around −1.5% in the lowermost mantle. Temperature variations on the order of +1000 K, corresponding to perturbations of around −3% in vs in the unfiltered model, would be seen as −1.5% when “imaged” with the data and damping of S20RTS. This supports our earlier finding that isochemical whole mantle flow with strong core heating and a pyrolite composition can be reconciled with tomography. In particular, the large lateral temperature variations associated with lower mantle plumes are able to account for the slow seismic anomalies in the large low-velocity zones under Africa and the Pacific. We also find that strong gradients in shear wave velocity of 2.25% per 50 km in our unfiltered models compare well with the sharp sides of the African superplume.
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In our earlier work [Schuberth et al., 2009], we did not account for the limited resolving power of tomography, which is known to be important in the analysis of geodynamic models. For example, Mégnin et al.  applied the “linear tomographic filter,” corresponding to model SAW12D [Li and Romanowicz, 1995] to mantle convection models. They showed that seismic data coverage significantly affects the spectral characteristics of their geodynamic models. Bunge and Davies  traced rays through mantle circulation models, and found a significant change in the pattern of heterogeneity in regions of poor ray coverage. Using the same approach, Davies and Bunge  observed a systematic bias toward negative synthetic traveltime residuals and explained this by most earthquakes being located in areas of fast seismic velocities (i.e., active subduction). Recently, Ritsema et al.  applied the resolution operator corresponding to the tomographic model S20RTS to isochemical and thermochemical models of mantle circulation and observed a substantial decrease in the total spectral power of heterogeneity and a change in geographic pattern.
The aforementioned studies clearly demonstrate the need to account for the limited resolving power of tomography. However, being based on simple depth-independent linear conversions between temperature and seismic velocities, the studies were limited in their ability to investigate the effects of tomographic resolution on the magnitudes of velocity anomalies. As noted before, our models benefit from recent progress in mineral physics, which now provides improved relations between thermal and elastic parameters based on the thermodynamically self-consistent treatment of mineral phase assemblages. By combining these advances with an analysis of the effects of limited tomographic resolution, a quantitative comparison of geodynamically predicted seismic heterogeneity with tomography can be attempted. In the present study, we therefore extend the assessment of our MCMs by “filtering” their shear wave velocity structures with the resolution operator of S20RTS.
We start this paper with a short description of the tomographic filtering process. This involves the transformation of our geodynamic models onto the parameterization of S20RTS, which has important implications for the following analysis of seismic heterogeneity. In the comparison of our tomographically filtered models with S20RTS we focus on statistical measures such as the spectral power, histograms and root-mean-square (RMS) profiles of heterogeneity.
The characteristics of our filtered models agree well with S20RTS. As expected, the magnitudes of seismic heterogeneity are significantly reduced during the filtering process. The amplitude reduction is due, in part, to the reparameterization and we explore possibilities to correct for this effect in a second step. Importantly, our MCM with strong core heating shows negative vs anomalies of around −1.5 to −2% in the lowermost mantle when only the effects of uneven data coverage and damping are considered. These values are fully compatible with the anomalies mapped by S20RTS. Finally, we note that the gradients of seismic heterogeneity in the original (unfiltered) version of this model compare well with those inferred from direct observations of traveltimes sampling the lower mantle under Africa [Ritsema et al., 1998; Ni et al., 2002].
2. Parameterization of Models and Tomographic Filtering
2.1. Tomographic Model S20RTS
S20RTS is a model of isotropic shear velocity perturbations from the (anisotropic) 1-D model PREM. The model is based on three different data types: normal mode splitting functions, Rayleigh wave phase velocities and body wave travel times [Ritsema et al., 2004]. It is parameterized laterally in spherical harmonics up to degree 20 and radially with 21 spline functions. The radial splines interpolate continuously across the 660 km discontinuity and their spacing is denser in the upper mantle where the resolving power of the data is higher. In total, S20RTS includes ∼10,000 parameters resulting in a lateral resolution of around 1000 km. After damping, about 3000 effective unknowns are resolved. In the inversion, it is assumed that P velocity anomalies are related to S velocity anomalies throughout the mantle by ∂ln vp = 0.5 ∂ln vs, a scaling based on globally recorded P and S wave traveltime residuals [Robertson and Woodhouse, 1995; Ritsema and van Heijst, 2002].
2.2. Mantle Circulation Models
We construct seismic mantle heterogeneity by postprocessing the present day temperature field obtained from mantle circulation models with one of the aforementioned thermodynamic mineralogical models, which includes all the effects of phase changes in a 5-oxide (CFMAS) pyrolitic composition [Stixrude and Lithgow-Bertelloni, 2005, 2007]. Apart from the isochemical, pyrolitic nature of the investigated models, two basic assumptions are made in the prediction of global mantle heterogeneity: (1) a large-scale flow structure related to past plate motion and (2) a radial viscosity profile that agrees with postglacial rebound and geoid observations. Details on the modeling and the characteristics of predicted vs heterogeneity are given by Schuberth et al. . We note that the flow calculations currently do not incorporate the dynamic effects of phase transformations. Furthermore, anelastic effects and the postperovskite phase are not included in the mineralogical model. In the following, we summarize the numerical parameterization of our MCMs, which is important in the context of tomographic filtering.
The MCMs are computed with the code TERRA [Bunge and Richards, 1996; Bunge et al., 1996, 1997] and are discretized with a mesh derived from the regular icosahedron, which provides almost equidistant grid spacing throughout the mantle. Horizontal grid spacing is around 30 km at the Earth's surface and, because of the sphericity of the model, about 15 km at the CMB. In radial direction, the model is discretized with 128 equidistant layers leading to ∼25 km vertical grid spacing. This fine discretization leads to a total of around 80 million grid points, which allows us to simulate large-scale mantle flow with Earth-like convective vigor and to employ a thermal Rayleigh number of ∼109 based on internal heating. Thus, we are able to resolve a characteristic thermal boundary layer thickness on the order of 100 km, comparable to that of oceanic lithosphere. The high-resolution mantle circulation models are implemented on 128 cores of a topical compute cluster dedicated to large-scale geophysical modeling [Oeser et al., 2006].
We focus our analysis of tomographic filtering effects on two of the four models described by Schuberth et al. , namely, M1 and M2. The models have the same viscosity stratification, but are end-member models in terms of bottom heating. The reference viscosity of the upper mantle is 1 × 1021 Pa s, while the viscosity in both the lithosphere and the lower mantle is 100 times larger. Model M1 is heated mainly from within (5% core heating), whereas model M2 has a strong component of bottom heating of around 35% of the surface heat flux. We note that in our study the term “lithosphere” is used only to denote the uppermost layer of our viscosity profile. As we do not account for the existence of chemical variations in the upper 100–150 km, we will not attempt a comparison of our models with tomography in this depth range.
2.3. Tomographic Filtering
We modify our theoretically predicted mantle heterogeneity using the resolution operator = G†G associated with S20RTS, where G is the operator of the seismic forward problem and G† its generalized inverse. Details of this procedure are given by Ritsema et al. . The geodynamic prediction of seismic heterogeneity is defined as the “true” model m, which is multiplied with to obtain a “filtered” representation m′ as if imaged by tomographic inversion:
Since includes all the effects on resolution due to limited data coverage and model regularization, this process is computationally easier and faster than ray tracing and inversion of synthetic traveltimes in case is explicitly available. The filtered geodynamic model can then be compared to the corresponding tomographic model. In this respect, it is important that the resolution operator is computed on the basis of the same damping parameter ε as the tomographic model. In the case of S20RTS, ε was chosen to be 0.035 [Ritsema et al., 2007], which we therefore adopt here as well.
Theoretical predictions of mantle heterogeneity are usually taken from numerical simulations. As noted before, these are performed on grids with a large number of nodes to accommodate the vigorous convective regime of the Earth. Grid spacings have to be on the order of several tens of kilometers globally. This typically results in meshes with ∼107 degrees of freedom [e.g., Bunge et al., 2002; McNamara and Zhong, 2004]. In contrast, global tomographic models currently provide a maximum spatial resolution on the order of several hundred to thousand kilometers [Ritsema et al., 2004; Montelli et al., 2006; Simmons et al., 2007; Houser et al., 2008]. Geodynamic mantle structures thus have to be reparameterized onto the basis of the specific tomographic model being compared with, as resolution matrices are given in the corresponding parameterizations.
In our case, the reparameterization is carried out by expanding vs(r,θ,ϕ) from our MCMs into the parameterization of S20RTS:
Here, Fk(r) are the radial spline functions and Ylm(θ,ϕ) the spherical harmonics of degree l and order m. The coefficients aklm are determined by least squares fitting. We note that aliasing effects due to the expansion over a finite sum of global basis functions are in the case of our MCMs around 8 orders of magnitude smaller than the actual signal.
Then, the parameters of the adapted model, which defines the true structure m in equation (1), can be tomographically filtered by multiplication with . The reparameterization leads to a drastic reduction in the model dimensionality, which in turn results in a significant alteration of the model characteristics, as will be shown in section 3. We introduce the following notation for our MCMs: Mo denotes their original parameterization, Mr the reparameterization to the basis of S20RTS, and Mf denotes the reparameterized and filtered model (i.e., Mr multiplied with ). Changes in characteristics between the original model Mo and Mf are thus a combination of two separate effects. In the following, we use the terms “tomographic filtering” or “resolving power of tomography” only for the modification of models due to the multiplication with the resolution matrix.
3.1. Effects of Parameterization and Tomographic Filtering
Figure 1 shows the shear wave velocity structure of model M2 in the two parameterizations, and filtered to the resolution of S20RTS. In Figure 1a, M2 is plotted in its original parameterization on the numerical grid of the code TERRA (i.e., M2o). The Earth-like convective vigor leads to a narrow upper thermal boundary layer, around 100 km thick. Thin and elongated fast velocity structures at 340 km depth (e.g., all around the Pacific) correspond to cold, downwelling slabs located in regions of present day subduction. With increasing depth, their locations change according to earlier stages of subduction as given by the plate motion history. In the lowermost mantle, the downwelling material spreads laterally above the CMB and the strong lower thermal boundary layer leads to significant hot upwellings, as for example in the southeast Pacific.
In Figure 1b, M2 is plotted in the parameterization of S20RTS (i.e., M2r). Spatial resolution is significantly lower than in M2o owing to the restriction of model parameters to spherical harmonic degrees lower than 20. The reparameterization results in the spreading of anomalies and in a reduction of their amplitudes. Especially the thin slabs in the upper mantle are substantially broadened compared to the original parameterization. At 100 km depth, fast continental areas as well as slow mid-ocean ridge systems display lower values than in the original parameterization. The same is true in the lower mantle, where the magnitudes of the large slow anomalies located under the Pacific, Africa and the Indian Ocean are reduced.
Figure 1c shows M2f; that is, M2r filtered to the resolving power of S20RTS. The main effect of the filtering is to further reduce the amplitudes in the lower mantle, especially at 2800 km depth. There, the shape of the anomalies is not affected much in contrast to the midmantle. At 1450 km and 2100 km, some lateral smearing of heterogeneity can be observed. Interesting to note is that in certain regions of the upper mantle amplitudes appear to be enhanced owing to vertical smearing of structure from above or below. This is, for example, the case for the slow seismic velocities under the Pacific and the fast velocities corresponding to the Farallon slab under western North America at 340 km depth.
Irrespective of the general magnitude reduction, Figures 1b and 1c indicate that the pattern of seismic heterogeneity in the lowermost mantle is resolved quite well by S20RTS. This is probably related to the fact that the S20RTS data set includes a large number of traveltimes of S waves that have diffracted along the core, in addition to normal mode splitting coefficients [Ritsema et al., 2004], which optimizes the coverage in the lowermost mantle.
For comparison, S20RTS is shown in Figure 1d. The general character of our filtered model M2f agrees quite well with S20RTS. The poor geographic correlation, however, is primarily due to the unknown initial condition of the forward problem of mantle dynamics [Bunge et al., 2003], and uncertainties in the plate motion history [Bunge et al., 2002; McNamara and Zhong, 2005; Schuberth et al., 2009]. Thus, independently of the important effects of tomographic resolution, this limits the use of morphological considerations in the assessment of geodynamic models, unless better constraints, especially on the temperature variations in the past, become available.
3.2. Effects on Spectral Characteristics of Heterogeneity
In Figure 2, we plot spectral heterogeneity maps (SHM) [Jordan et al., 1993], which are contour plots of spectral amplitude of heterogeneity versus depth. Shown are M1 and M2 in their original, reparameterized, and filtered representations (Figures 2a–2c and Figures 2d–2f, respectively). In addition, we show corresponding radial profiles of the root-mean-square power of the spherical harmonics expansion. Spectral power σl per degree l is computed at each depth level and for spherical harmonic degrees l = 1, …, 20 by [Dahlen and Tromp, 1998, B.8]
where alm and blm are the coefficients of the expansion and the RMS power δ up to degree lmax is given by
The spectral characteristics of both MCMs are quite similar with model M1 only showing less power overall. The strong degree two in the lower mantle is mainly a consequence of the long-wavelength structure of earlier stages of the plate motion history. Together with the viscosity stratification this generally leads to a red spectrum of the mantle [Bunge and Richards, 1996; Bunge et al., 1998, 2002].
In the original parameterization of M1 and M2, heterogeneity is strongest in the thermal boundary layers, where it is distributed over a wide range of spatial scales. In the midmantle, heterogeneity is weaker and concentrated in low degrees. Thin layers of increased heterogeneity in the upper mantle correspond to phase transformations at 410, 520 and 660 km depth.
Figures 2b and 2e show the SHMs of the reparameterized models M1r and M2r. Differences to the original models are small and derive from the difference in radial parameterization. The strong heterogeneity in the transition zone is smeared vertically and the distinct band of the 410 km discontinuity is smoothed away. Heterogeneity in the lowermost mantle is affected as well, which can also be seen from the RMS profiles.
The SHMs of the filtered models M1f and M2f are given in Figures 2c and 2f, respectively. Power is mainly reduced in high degrees, and the lowermost mantle is affected stronger than the rest. This is also visible from the RMS profiles. The band of high spectral power in the transition zone, which is more pronounced in M2r, is mostly filtered away and only low degrees (1–5) still show substantial power there.
3.3. Effects on Magnitudes of Heterogeneity
Figure 3 shows histograms of the lateral vs variations in our MCMs; that is, we contour the distribution of magnitudes of heterogeneity as a function of relative perturbation and depth. In analogy to the SHMs in Figure 2, the histograms are given for the original, reparameterized, and filtered versions of M1 and M2. First, we note that in the original parameterization (Figures 3a and 3d), largest amplitudes are located in the upper mantle and the transition zone with values of up to 5% and more. In the upper part of the lower mantle, heterogeneity is weaker and the distribution is asymmetric toward fast seismic anomalies (i.e., positive skewness). In M2, the skewness changes with depth being negative in the lowermost mantle as a consequence of the strong thermal gradient across D″. The most prominent feature in the original parameterization of M2 is the asymmetry of maxima in the lowermost mantle with values of −4% and +2.8%. In contrast, M1 with weak core heating only shows small negative anomalies of less than −1% in most of the lower mantle and a positive skewness throughout.
Figures 3b and 3e show the histograms of the reparameterized models M1r and M2r, and a comparison with Figures 3a and 3d illustrates that the reduction of magnitudes introduced in this step is depth-dependent. Amplitudes of heterogeneity are decreased stronger in the upper half than in the lower half of the mantle. This indicates that the short-scale variations in the upper mantle are not captured by the degree-20 parameterization. Amplitude reduction in the lower mantle is smaller owing to the predominance of long-wavelength structure there, which in turn is a consequence of the higher viscosity and the absence of major phase changes. Effects on positive and negative anomalies are different as can be seen, for example, from the change in skewness in the lower mantle. The same holds for the transition zone, where positive anomalies are affected stronger than negative perturbations, as is apparent from the fact that the positive anomalies at 660 km depth have all but disappeared.
As expected, the tomographic filtering further reduces the amplitudes of heterogeneity (see Figures 3c and 3f). Both models show similar magnitudes of positive anomalies in the lower mantle. Similar to the original parameterization, model M1f displays asymmetric distributions toward positive values in all of the lower mantle, only that maximum negative vs anomalies are around −0.5% now. The histogram of M2f is more symmetric and maximum anomalies in the lowermost mantle are around ±1.5% reaching up to ±2% close to the CMB.
3.4. Quantitative Analysis of Reparameterization and Filtering Effects
To quantitatively analyze the effects of reparameterization and tomographic filtering, we first concentrate on root-mean-square amplitudes of heterogeneity. Figure 4a shows RMS profiles for vs variations in the three versions of M2 (i.e., the original, reparameterized, and filtered model). RMS values in the original parameterization are largest in the upper thermal boundary layer with values exceeding 2%. Between 300 and 1500 km depth, heterogeneity decreases to values of 0.5% and monotonically increases again from this midmantle minimum to values of up to 1.6% near the CMB. Strong variations of RMS in the upper mantle correspond to the phase transformations in the transition zone.
RMS amplitudes of the reparameterized model are lower than in the original parameterization. In addition, only one maximum is left in the transition zone as noted also for the SHMs in section 3.2. Between 800 km and 2000 km depth, RMS values are below 0.5% and reach ∼1.2% close to the CMB. The amplitude difference to the original parameterization is strongest in the transition zone and uppermost lower mantle and decreases with depth. As mentioned above, the depth dependence of this magnitude reduction relates to the different nature of heterogeneity between upper and lower mantle (i.e., short-scale variations in the upper mantle due to a complex set of phase changes, and predominantly long-wavelength features in the lower mantle resulting from the higher viscosity there).
After applying the tomographic filter, the RMS profile is nearly constant in most of the mantle with values close to 0.3%, but still shows a weak minimum at around 1500 km depth. A comparison with the profile of the reparameterized model shows that the reduction of RMS values is stronger in the lower half of the mantle indicating again that the resolving power of S20RTS is better in the upper mantle. There, mainly the local maximum in the transition zone is reduced. In the lowermost 300 km, the filtered model shows a strong increase in RMS values reaching ∼0.7% close to the CMB.
Figure 4b shows the total relative amplitude reduction resulting from the filtering procedure. In addition, we plot the separated contributions from the effects of reparameterization and tomographic filtering. The total relative amplitude reduction is almost constant with depth in the lower mantle and exceeds 0.5 nearly everywhere. Close to the CMB, about two thirds of the original amplitude is lost. Note that the profiles for the reparameterization and tomographic filtering effects cross at around 1500 km depth. Above this depth, the total reduction is dominated by the reparameterization while the effects of tomographic resolution are stronger below.
3.5. Correcting for the Effects of Reparameterization
As seen from Figures 2–4, the characteristics of the tomographically filtered models are significantly influenced by the effects of the reparameterization. A first-order correction of these effects could be attempted in a postprocessing step to the tomographic filtering due to the linear nature of ; that is, the heterogeneity of short-scale structure, which is lost in the degree-20 parameterization of the filtering procedure, can be considered as an additional contribution in the quantitative analysis. In the following, we try to do this by correcting for the total amplitude reduction resulting from the reparameterization. This allows estimates of the properties of our models as if only exposed to the limited resolving power of S20RTS. As will be discussed in section 4, the exact extent to which the reparameterization effects have to be accounted for depends on the amount of aliasing of short-scale heterogeneity into the long-wavelength structure of tomographic models. Thus, the correction attempted here provides an upper bound for the magnitudes of the tomographically filtered seismic heterogeneity derived from our MCMs.
The RMS values δf+ for the tomographic filter effect alone are obtained by computing the ratio of RMS values of the tomographically filtered model Mf with the reparameterized model Mr for each depth d. In this way, we isolate the component of amplitude reduction associated with . We then multiply this depth-dependent ratio with the RMS values of the original model δo:
In Figure 5, we show RMS profiles for our filtered models corrected in this manner, which we also denote M1f+ and M2f+. The RMS profile of S20RTS is plotted for comparison. The general characteristics of the original models, such as the large amplitudes close to the thermal boundary layers and peaks in the transition zone, are conserved when only accounting for the tomographic filtering; that is, without the bias due to the reparameterization. As in the original parameterization, model M2f+ with high CMB heat flux has larger values than M1f+ at all depths. Below 1500 km depth, the difference between M1f+ and M2f+ increases gradually. RMS values in M2f+ get larger below 2500 km, where model M1f+ only shows a moderate increase. Directly above the CMB models M1f+ and M2f+ display RMS amplitudes of 0.4% and 0.8%, respectively. Below 1500 km depth, M2f+ agrees remarkably well with S20RTS, while M1f+ shows values that are 20–30% lower than S20RTS there.
3.6. Correction of Spectral Characteristics
In a next step, we perform a similar correction as in section 3.5 for the spectral heterogeneity maps. In this case, we account for the loss of spectral power during the reparameterization for each spherical harmonic degree l, separately, again depending on depth d:
Figures 6a and 6b show the SHMs for the filtered and corrected models M1f+ and M2f+, respectively. The comparison with Figures 2a and 2d illustrates the now isolated effect of the limited tomographic resolution of S20RTS alone. Heterogeneity is mostly reduced in the high degrees of the lower mantle. Furthermore, the tomographic filtering results in vertical smearing of power from the transition zone into the upper part of the lower mantle.
Except for their magnitudes, the spectral characteristics of M1f+ and M2f+ are very similar, in analogy to the original SHMs in Figures 2a and 2d. Both show a marked change in spectral pattern across the 660 km discontinuity, which is even more pronounced than in the unfiltered original models owing to the strong reduction of power in high degrees of the lower mantle. The characteristic difference between upper and lower mantle compares well with the change in spectral pattern seen in S20RTS (Figure 6c), and is a consequence of the phase changes in the transition zone together with the unresolved structure in the higher degrees of the lower mantle.
3.7. Correction of Histograms
We also attempt a first-order correction of the amplitude distributions in our MCMs. For this, we take the values of seismic heterogeneity from the filtered model and multiply them with the depth-dependent ratio of RMS values from the original model and the reparameterized model (which is similar to the operation in equation (5)). To account for possible asymmetries in the reparameterization, we perform this operation for positive and negative perturbations separately.
In Figure 7, we show histograms for the filtered and corrected models M1f+ and M2f+ together with the histogram of S20RTS. The main characteristics of the original models are conserved (e.g., largest amplitudes in the upper mantle, and in M2o a change in skewness from positive to negative throughout the lower mantle, which can also be seen in the histogram of S20RTS). Comparing the maximum values in Figures 7a and 7b with those in Figures 3a and 3d indicates that amplitudes of heterogeneity in our MCMs are lowered by almost a factor of 2 when affected by the limited tomographic resolution alone. The histogram of M2f+ (Figure 7b) compares well with S20RTS for both positive and negative amplitudes in most depth levels. This is markedly different for M1f+, which shows maximum negative anomalies of less than −0.5% in most of the lower mantle as a consequence of the lower core heating.
4.1. Lateral Temperature Variations and Magnitudes of Seismic Heterogeneity
An important result of our analysis is the change in the characteristics of our geodynamic models introduced by the reparameterization. The transformation of model parameters from the original high-resolution grid (∼25 km global grid spacing) onto the low-degree spherical harmonics basis of S20RTS is needed to perform the tomographic filtering and results in a significant “loss” of short-scale heterogeneity. The fact that heterogeneity in the upper mantle suffers more drastically from the long-wavelength reparameterization is related to velocity variations being present on very short length scales there (see SHMs in Figures 2a and 2d), which in turn is a consequence of the complex set of phase transformations in the transition zone [Schuberth et al., 2009]. Interestingly, the amplitude reduction associated with the reparameterization appears to be not only depth-dependent, but also different for positive and negative velocity variations, as can be seen from the change in skewness of the histograms in Figure 3. Most importantly, however, the amplitude reduction induced by the reparameterization is comparable in magnitude to the effects of uneven data coverage and damping.
The tomographic filtering is also depth-dependent, indicating a better ability of S20RTS to resolve velocity variations in the upper mantle due to fundamental mode and overtone surface wave constraints. Owing to their different depth dependencies of relative amplitude loss, the combination of reparameterization and subsequent multiplication with then leads to an almost constant total magnitude reduction in the lower mantle of around 60 percent of the original amplitudes (see Figure 4).
The loss of short-scale heterogeneity associated with the reparameterization in itself seems unphysical, as structural variations on length scales of around 100 km and less are expected to be present in the mantle (e.g., slabs). However, whether the reparameterization should be accounted for in comparisons of geodynamic and seismic models depends on the potential of short-scale structure to affect long-wavelength tomographic images. If this is the case, geodynamic models should be modified in such a way only, as to reflect the effects of limited resolving power alone.
In this respect, Mégnin et al.  have shown that aliasing of short-scale heterogeneity into the low degrees of tomographic models can be observed in the inversion of long-period S wave data for large-scale structure. In other words, velocity variations in the inverted long-wavelength structure may be larger than the actual large-scale structure in Earth due to short-scale structure “seen” by the seismic waves. This implies that the information on short-scale heterogeneity in the seismic data is not totally lost, as, for example, due to finite frequency effects such as wave front healing, but rather is mapped as additional contribution into the tomographic models. Put differently, the aliasing in tomography may counteract to some extent the reducing effect of limited tomographic resolution and result in a slight amplification of the long-wavelength anomalies.
The occurrence of structural aliasing is expected from theoretical considerations [Dahlen, 2004]: In the case of perfect illumination of a given velocity perturbation by waves with finite frequency, the anomaly recovered with ray tomography will have the same volume-integrated total perturbation. The recovered anomaly, however, will be broader because of lateral smearing and will have a smaller maximum amplitude. This “mass conservation” of seismic anomalies suggests that only uneven data coverage and the associated need to damp inversions are the reason for tomographically resolved amplitudes being on average (in a root-mean-square sense) smaller than in the Earth.
As for the geodynamic models, the volume integral of the velocity perturbations is not conserved during reparameterization and we have therefore attempted to correct for the associated magnitude reduction. Taking advantage of the linearity of the resolution operator, it is possible to consider the short-scale structure lost in the long-wavelength parameterization as additional contribution in the statistical characteristics of the tomographically filtered MCMs (see sections 3.5–3.7). This way, an approximation of the magnitude reduction of seismic heterogeneity due to the limited tomographic resolving power alone is provided.
After correcting for reparameterization induced changes in the characteristics of our tomographically filtered geodynamic models, we find that shear wave velocity anomalies are reduced by a factor of around 2 in the bottom 500 km of the mantle (i.e., from the tomographic filtering alone). More specifically, large negative anomalies of −3 to −4% in the lowermost mantle, corresponding to plume excess temperatures of +1000–1500 K, would be mapped as −1.5 to −2%, and are therefore in good agreement with the values of S20RTS. We note that large lateral temperature variations in the lower mantle can be anticipated from a number of geodynamic considerations on plume excess temperatures, including the adiabatic ascent of plumes in the subadiabatic mantle [Bunge, 2005] and the fact that the adiabatic gradient is itself temperature dependent, getting steeper with increasing temperature [e.g., Piazzoni et al., 2007].
The corrected histogram of our strongly bottom heated model M2 shows slightly larger maximum velocity anomalies than S20RTS, especially for positive values (compare Figures 7b and 7c). This may show that accounting for the total amplitude reduction of the reparameterization overestimates the aliasing effect in tomography. In fact, it is likely that the actual structure that should be compared with the tomographic model lies somewhere between models M2f and M2f+. In this case, the temperature variations may not fully account for the seismic anomalies seen in S20RTS. However, as mentioned in section 2, the mineralogical model used here does not incorporate the effects of anelasticity on seismic velocities. Including these effects would result in a higher sensitivity of vs to temperature by up to 20–30% compared to the purely elastic case [e.g., Goes et al., 2004; Brodholt et al., 2007]. Thus, in the more realistic case of anelasticity, the thermal anomalies in M2 are still likely to explain most of the seismic signal in the large low-velocity zones even if no structural aliasing was present in tomographic inversions (compare M2f in Figure 3f with S20RTS in Figure 7c).
We do not preclude compositional variations in Earth's mantle, but the tomographic filtering of our MCMs shows that it is quite possible to explain seismic heterogeneity by temperature variations alone. This finding may place limits on the potential role of chemical heterogeneity, as the large plume temperatures anticipated in the lower mantle already account for the seismic signal in the large low-velocity zones. An increase in the iron content, for example, added on to the large thermal anomalies, is expected to further enhance the seismic anomalies [Wang and Weidner, 1996; Jackson, 1998] and will therefore possibly overpredict the magnitudes of velocity variations. Thus, our study lends further support to the notion that global mantle circulation is predominantly thermally driven and has a strong core heating component.
4.2. Horizontal Gradients of Thermal and Seismic Heterogeneity
Sharp horizontal gradients in seismic velocities have sometimes been advanced as an argument for chemical variations [Ni et al., 2002; Brodholt et al., 2007]. Evidence for strong gradients in shear wave velocity, which may be related to anomalously warm buoyant mantle due to continental insulation [Anderson, 1982; Phillips and Bunge, 2005], comes from the observation of rapid variations of body wave traveltimes over a small azimuthal range for raypaths turning in the deep mantle under Africa [e.g., Ritsema et al., 1998]. On the basis of this direct observation, Ni et al.  estimated that shear wave velocities in the lower mantle vary by up to 3% over a distance of 50 km (i.e., a gradient of relative perturbations of ∼6 × 10−7 m−1).
To investigate the abruptness with which velocity variations change laterally in isochemical whole mantle flow, we can directly examine our unfiltered models, as we do not need to account for tomographic resolution effects in the comparison to the direct observation of rapid azimuthal traveltime variations.
Our unfiltered MCM M2o (Figure 1a) shows slow anomalies in the lowermost mantle directly bounded by fast material. The resulting gradients are around 4.5 × 10−7 m−1 or 2.25% over 50 km, comparable to the values suggested by Ni et al. . These gradients in shear wave velocity relate to thermal gradients of ∼0.015–0.018 K m−1 or around 750–900 K per 50 km. Such strong thermal variations in isochemical mantle circulation models are entirely expected, as heat transport inside the Earth is dominated by advection (typical Peclet numbers of around 10–100) so that sharp gradients are preserved.
4.3. Current Limitations in the Comparison of Geodynamic and Tomographic Seismic Heterogeneity
In the present study, we concentrated on variations in shear wave velocity and the comparison to S20RTS. It would be helpful to also compare our MCMs to other tomographic S wave models, as well as to compare the additionally predicted vp heterogeneity to tomographic P wave models. However, resolution matrices are currently available for only a few tomographic models. In fact, for models with a large number of parameters in the inversion, the construction of is computationally not feasible (G. Nolet, personal communication, 2008). A further complication arises from the varying degrees of approximations concerning the forward problem of seismology used in different tomographic studies. Furthermore, direct comparison of S and P wave models may be biased owing to the different frequency content of the respective data sets. The approach taken here thus seems currently not well suited to address the question of potential anticorrelation of vs with bulk sound velocity vϕ, or density, in the lowermost mantle.
With respect to structural aliasing in tomographic inversions, further studies are needed to better quantify the extent to which this is affecting the long-wavelength tomographic images. It also remains to be seen whether the rise of multifrequency traveltime and amplitude tomography [Sigloch and Nolet, 2006; Sigloch et al., 2008], or alternatively inversions based on the adjoint method [e.g., Fichtner et al., 2006a, 2006b], will result in better resolution of the geometry, amplitude and especially the gradients of seismic heterogeneity.
Our conclusions are furthermore limited in that the mineralogical models currently do not include the postperovskite phase. Recent studies point to the possibility that the anticorrelation of lower mantle shear and bulk sound velocity variations may be caused by phase heterogeneity associated with postperovskite [Hutko et al., 2008; Hernlund and Houser, 2008]. However, postperovskite is likely to occur only in cold regions of the lowermost mantle [Tateno et al., 2009] and therefore our results on high plume excess temperatures will probably not be affected much.
We have quantitatively related lateral temperature variations to magnitudes of seismic heterogeneity taking into account their complex relation through mantle mineralogy and the effects of limited tomographic resolution. In our analysis, we concentrated on isochemical whole mantle circulation to study the effects of bottom heating without any complications due to compositional variations. We have focused on models with strong core heating, as a number of studies have argued for a large thermal gradient across D″ on the order of 1000 K and a high core heat flow.
The tomographic filtering of seismic velocity structures involves the reparameterization of our geodynamic models onto the long-wavelength basis of S20RTS. The drastic decrease in model dimensionality from 8 × 107 to 104 results in an artificial reduction of the magnitude of seismic heterogeneity. On the basis of the linearity of the resolution operator, we have shown that the short-scale heterogeneity lost in the reparameterization can be considered as an additional contribution to the statistical characteristics of the filtered MCMs. With this correction, we arrive at a consistent comparison with tomography, as it is likely that seismic inversions inherently incorporate information on short-scale variations due to structural aliasing.
We find a good agreement of the tomographically filtered and reparameterization corrected MCMs with S20RTS. However, only the MCM with strong core heating shows magnitudes of slow seismic anomalies in the lowermost mantle that are compatible with the values seen in S20RTS. Shear wave velocity perturbations of −3 to −4% in this model are reduced by a factor of 2 when accounting for uneven data coverage and damping. Furthermore, thermal gradients of around 750–900 K per 50 km in the corresponding unfiltered model result in shear wave velocity gradients of ∼2.25% over 50 km, which are in good agreement with the study of Ni et al.  for the sharp sides of the African superplume. Our results thus confirm the conclusion of Schuberth et al.  that isochemical whole mantle circulation with substantial CMB heat flow of 9–12 TW and a pyrolite composition is consistent with a number of seismic observations on the lower mantle.
We are grateful to Saskia Goes and an anonymous reviewer for detailed and constructive comments that helped to improve the manuscript. The mantle circulation models were partly computed on the supercomputing facilities of the Leibniz-Rechenzentrum (LRZ) Munich, Germany. We thank the staff of the LRZ for their support. Figure 1 was generated with GMT4.0 [Wessel and Smith, 1991]. This research was funded by the International Graduate School THESIS within the Elite-Network of Bavaria (BSAS) and the German Science Foundation (DFG) under grant BU 2010/3-1 (H.P.B.). J.R. was supported by NSF grant EAR-0609763.