The frequencies of Earth's normal modes are split by rotation, ellipticity, and internal structure of the Earth. Thus, models of mantle heterogeneity and discontinuity topography generate splitting that may be tested against observations. We insert maps of core-mantle boundary (CMB) topography, which are derived via either a purely seismic or a joint tomographic/geodynamic inversion of body waves data, on top of tomographic model S20RTS. We then calculate synthetic splitting functions for normal modes that have been shown to be sensitive to CMB topography and compare these to observed normal mode splitting data. The CMB topography maps obtained via geodynamically constrained tomography fit normal mode data better than purely seismic maps, in particular when the geodynamic constraint also accounts for the presence of post-perovskite in the D″ region. We test the significance of the reduction in misfit using the concept of observability which suggests that normal modes are able to observe the difference between the different CMB topography maps. In addition, we find that the statistical significance, assessed by checking what fraction of 1000 randomly generated CMB models achieve a comparatively good fit as the desired model, is higher than 92% for degree 2 and 98% for all degrees. In summary, we have identified a model of CMB topography that fits body wave data and improves, at least to some extent, the fit to normal mode data, and is coherent with the large-scale pattern of deep mantle heterogeneity expected on the basis of convection modeling.
 The core-mantle boundary (CMB) has a key role in the dynamics of deep mantle convection, the geodynamo and the evolution of the core, and has therefore been intensively studied from the beginning of the 1980s. Maps of CMB topography have been derived on the basis of seismological inversions of body wave traveltimes [e.g., Morelli and Dziewonski, 1987; Rodgers and Wahr, 1993; Obayashi and Fukao, 1997; Boschi and Dziewonski, 2000; Soldati et al., 2003] or normal mode splitting [Li et al., 1991b; Ishii and Tromp, 1999]. Alternatively, they are derived through a geodynamic approach [Forte et al., 1995; Lassak et al., 2007, 2010] which assumes that the CMB should be depressed/uplifted under relatively dense/light regions of the lowermost mantle. Despite the general agreement on the overall shape of CMB undulations (depression under the circum-Pacific Ring and elevation below the Central Pacific), the details of the peak-to-peak amplitude and pattern are still debated.
Soldati et al.  proposed a joint seismic-geodynamic technique to invert body wave data taking into account the coupling of CMB topography with mantle heterogeneity by viscous flow. The P wave velocity models of the mantle they obtained this way fit body wave data as well as the purely seismically derived ones and the corresponding maps of CMB topography are more stable (i.e., no observed discrepancy between maps based on PcP-PKP data sets) and have lower amplitude. In this formulation, the response of CMB topography to mantle density perturbations depends on the radial viscosity profile of the mantle. The perovskite to post-perovskite phase change in the D″ region [Murakami et al., 2004; Oganov and Ono, 2004] may be modeled as a reduction in mantle viscosity of three orders of magnitude [Yamazaki et al., 2006; Ammann et al., 2010; Cizkova et al., 2010]. The ellipticity of the resulting CMB maps can match the very long baseline interferometry data [Gwinn et al., 1986] only when the post-perovskite related low-viscosity jump is included. This partially confirms the findings by Nakagawa and Tackley  that geodynamic models with a low-viscosity post-perovskite layer can include high-density piles without producing unacceptably large CMB topography, and the analysis by Mosca et al.  whose probabilistic approach to joint body wave, surface wave, and normal mode tomography favor deep mantle chemical heterogeneity (including post-perovskite).
 Earth's normal modes have a completely different sensitivity than body waves to the lateral structure of both the mantle and CMB due to their global character. Normal mode-based CMB topography maps published so far [Li et al., 1991b; Ishii and Tromp, 1999] are poorly correlated with the results of body wave traveltime imaging [Morelli and Dziewonski, 1987; Sze and van der Hilst, 2003; Soldati et al., 2012]. In this contribution, we make use of existing normal mode data to investigate the influence of the different CMB topography maps of Soldati et al.  (Figure 1) and the effect of a low-viscosity post-perovskite layer on the splitting of Earth's free oscillations.
2 Theory and Data
 The occurrence of large earthquakes makes the Earth ring at discrete frequencies as first observed following the Chile earthquake of 1960 [Benioff et al., 1961]. Spheroidal modesnSl and toroidal modesnTl are characterized by their radial order n and angular order l, consisting of 2l+1 singlets with azimuthal order m. Modes are degenerated (i.e., all 2l+1 singlets have the same frequency) for a spherically symmetric, isotropic, nonrotating Earth model. In a more realistic Earth (e.g., aspherical), the effects of Earth's rotation, ellipticity and lateral heterogeneity, including discontinuity topography, remove the degeneracy of the modes (the 2l+1 singlets get different frequencies), an effect known as splitting.
 The splitting of a given mode can be described in terms of a splitting function [e.g., Woodhouse and Giardini, 1985], which represents the radially averaged structure as seen by the mode. Using perturbation theory, the splitting function coefficients cst are related linearly to the perturbations of the reference Earth model by
where mst represents the various model parameters (e.g., seismic velocities, density, and anisotropy), d represents the internal discontinuities with undulation δhst, both expanded in spherical harmonics of angular order s and azimuthal order t, and Ks(r), are the associated sensitivity kernels [Woodhouse, 1980]. Splitting function maps F(θ,φ) are used to visualize the splitting function coefficients:
where are the complex spherical harmonics of Edmonds . The values of the splitting function F(θ,φ) can be interpreted as the local radially averaged deviation from the degenerate frequency of the multiplet.
 In the recent years, the quantity and quality of normal mode data have improved substantially, due to the occurrence of several megathrust earthquakes such as the Sumatra events of 2004 (Mw =9.0) and 2005 (Mw =8.6) and the 2010 Chile event (Mw =8.8). We adopt the data set of splitting functions of normal modes obtained by Deuss et al.  after inversion of normal mode spectra from 91 large earthquakes in the time range 1976–2010. For the inversion of spectra, synthetic seismograms u(t) are computed by summation of normal modes using the method of Deuss and Woodhouse  and can be written as a harmonic function of time t:
where r and s are the source and receiver vector, respectively, and M is the matrix containing all the splitting function coefficients. Observed normal mode spectra are then fitted in an iterative way to obtain the optimum-splitting function coefficients. Since our focus is to distinguish the effect of different CMB topography maps on the normal modes, we only consider modes whose sensitivity kernels are most sensitive to CMB undulations, as demonstrated by Koelemeijer et al. . This results in a set of 16 normal modes from this data set with radial order n=0–5 for which the sensitivity kernels are given in Figure 2.
 We firstly compute synthetic splitting functions associated with mantle heterogeneity (no CMB topography), based on 3-D mantle model S20RTS [Ritsema et al., 1999], using crustal model Crust5.1 [Mooney et al., 1998] to correct for crustal thickness, surface topography, and sea level. P wave velocity and density heterogeneities are scaled to S wave velocity using δlnVp/ δlnVs=0.5 and δlnρ/ δlnVs=0.3, consistent with previous work [Karato, 1993; Li et al., 1991a]. The observed splitting functions were already corrected for rotation and ellipticity; hence, no correction is needed for the synthetic splitting functions.
 Using a scaled S model to account for P heterogeneity is preferable to using independent S and P models because: (i) modes are much more sensitive to S than P velocity; (ii) global S models are generally better resolved and more reliable than corresponding P models; (iii) implementation is simpler (no need to reparameterize a P model). In addition, Soldati et al.  show in their Figures 8, 9, and 14 that P-velocity mantle models associated to all the CMB models presented here differ only slightly: differences in the data fit found here can thus be ascribed to the important differences in CMB topography.
 We subsequently compute the splitting due to heterogeneous CMB topography, based on the CMB topography models obtained by Soldati et al.  applying alternatively a classic tomography approach (T) [e.g., Morelli and Dziewonski, 1987; Boschi and Dziewonski, 2000] and the joint tomography-geodynamic (TG) approach summarized in section 1 (Figure 1). The geodynamic “regularisation” is based on different rheology profiles. Model cmbTG is based on the radial viscosity profile proposed by Mitrovica and Forte . Models cmbTGppv200 and cmbTGppv250 are derived using a modified profile that accounts for a post-perovskite layer via a reduction of viscosity of three orders of magnitude [Cizkova et al., 2010] in the lowermost 200 km and 250 km of the mantle, respectively.
 In Figure 3, we present the splitting coefficients predictions for the c20 and Im(c22) coefficients associated with the 16 modes used in this study. Instead of plotting the actual value of the prediction, we plot the difference from the S20RTS+Crust5.1 prediction to be able to see the variations. We choose to show these coefficients as they are important contributors to the Pacific “Ring-of-fire” pattern observed in lowermost mantle tomography models. Values close to zero indicate small variations from S20RTS+Crust5.1. The data are fitted better by the CMB topography maps than S20RTS+Crust5.1 when the predictions plot on the same side of the horizontal line as the data.
 For some modes, the predictions with added CMB topography move toward the observed splitting coefficients as is the case for the fundamental modes0S4-0S9 (Im(c22) coefficient) and the first-order overtones1S8-1S10 and1S14 (c20 coefficient). However, in other cases, such as for the third-order overtones3S8-3S9 (both coefficients), the predictions for CMB topography move away from the data. Generally, these trends depend on the mode and coefficient in question. For some modes, we have observations up to structural degree t=12, hence a total number of 80 coefficients. Therefore, looking at individual coefficients does not give a straightforward way of determining the best CMB topography map. In addition, Koelemeijer et al.  demonstrated that the splitting functions can be also influenced by other structures, such as anisotropy and ultra low-velocity zones, which are not accounted for here.
 We calculate misfit values between the observed splitting functions and the predictions for the different CMB topography maps to quantitatively assess the best CMB topography map. To this purpose, we define the L1 and L2 norms for structural degree s as
where are the predicted splitting function coefficients for the CMB topography maps on top of S20RTS+Crust5.1 using equation (1), and are the coefficients of the observed splitting functions. a is either one or two for the L1 and L2 norms, respectively. In this definition, the misfit is not normalized as otherwise small coefficients (which contain little information about structure) dominate the misfit. We also normalize the misfit by the associated uncertainties in the data to give more weight to coefficients with small measurement uncertainties. These uncertainties were estimated using the maximum range in observed coefficients in 10 cross-validation runs, with different events being left out in different runs [Deuss et al., 2012].
 The misfit to the data is summarized in Table 1 for different structural degrees and all degrees for which we have data (up to s=12) together. Degree 2 misfits have the highest absolute values, as these coefficients have the largest amplitude in the data. Except at degree 6, the misfit values for all geodynamically constrained tomography maps are lower than the misfits for the pure seismic one. On top of that, the misfit values for cmbTGppv200 and cmbTGppv250 are very similar but always lower than the values for cmbTG. This suggests that the addition of a low-viscosity post-perovskite layer fits the observations better, although the thickness of the layer is less important. For degrees 2 and 4, the cmbTGppv250 produces the smallest misfit out of the different CMB topography maps, whereas for degree 6, the cmbTGppv200 has a lower misfit.
Table 1. Misfit of the Synthetic Splitting Coefficients in μHz Based on Different CMB Topography Maps on Top of S20RTS+Crust5.1 to the Observed Splitting Function Dataa
aThe misfit is computed using the L2 norm (equation (4)) for structural degree 2, 4, 6, and all degrees up to s=12 together. In bold, we indicate the lowest misfit value in each case.
 The same test conducted on the basis of tomographic model S40RTS [Ritsema et al., 2011] (derived inverting also the normal mode splitting observations from Deuss et al. ) instead of S20RTS obtained better fit to the data but gives analogous conclusions: misfit values are lowest for the cmbTGppv CMB topography maps for both the L1 and L2 norms as demonstrated in Table 2 for all degrees together. This table also gives the misfit values when we use the associated uncertainties to normalize the misfit. In this case, the cmbTGppv maps perform best except when the L1 norm is used with S20RTS as basis. However, in this case, the difference with the cmbTGppv250 map is less than a percent and not significant.
Table 2. Not Normalized and Uncertainty Normalized Misfit of the Synthetic Splitting Coefficients for Different CMB Topography Maps on Top of S20RTS or S40RTS to the Observed Splitting Function Dataa
Not Normalized [μHz]
Error Normalized [-]
aThe misfit is computed using the L1 and L2 norm (equation (4)) for all degrees up to s=12 together. In bold, we indicate the lowest misfit value in each case.
 In the remainder of this letter, we choose to display the not normalized misfit as we found that misfit values using the data or uncertainties as normalization are dominated by few coefficients with anomalous small values.
4 Robustness Estimate
 Generally, the difference in misfit between the cmbT and cmbTGppv maps is similar in magnitude to the difference in misfit between the S20RTS and S40RTS predictions. This indicates that these results are robust; yet, a more quantitative estimate of their significance is necessary.
4.1 Random Model Test
 To test the significance of the misfit values, we generate 1000 random models of CMB topography with the same harmonic spectrum as our preferred maps but with different geographic patterns. Given that the CMB topography maps differ quite significantly in amplitude, we use two different models as our basis for the random models. We use primarily the cmbTGppv200 map which has a low amplitude and gives us a conservative estimate. A larger amplitude map such as for the cmbTG model is expected to give higher confidence numbers as the normal modes favor low amplitude topography maps. We compute the splitting functions for these models and calculate the misfit of these predictions with the data for the same selection of modes. In this way, we are testing how likely it is that we improve the fit “by chance” rather than physical reasons. For example, if 100 out of 1000 models have a lower misfit than the real model, then this indicates that the misfit of the real model is 90% significant. By restricting ourselves to models with the same amplitude spectrum as the real model, we are also conservative in the sampling of the null space.
 Figure 4 shows the histograms of the L2 norm obtained for degrees 2, 4, 6, and all degrees together for random models based on the cmbTGppv200 map. For degrees 2 and 4, the cmbTGppv maps have misfits very close together at a significance level of 92% and 75%, respectively. However, for degree 6, the CMB topography models generally have a higher misfit than the random models. When all degrees are considered together, the cmbTGppv200 model has the lowest misfit, at a 99.7% significance level.
 The bimodal shape characterizing the degree 2 distribution of misfit achieved by the random CMB models may be an effect of the harmonic spectrum, which is strongly dominated by degree 2. Depending on the (random) sign of the l=2, m=0 coefficient, two classes of models emerge: one resembling the real world, with elevated topography under South Africa and the Pacific; the other anticorrelated, and hence inconsistent with the data. This bimodal distribution is also evident in the misfit for all degrees, which is dominated by degree 2. We report in Table 3 the relative number of random CMB topography models based on both the cmbTGppv200 and cmbTG maps which fit the data better than all the different models of mantle/CMB topography employed here. We can see that especially for degrees 2 and 4, the reduction in misfit for the cmbTGppv models is significant. For each degree, the significance numbers based on the two different CMB topography maps agree. Hence, we can confidently use the more conservative significance estimates based on model cmbTGppv200.
Table 3. Percentage of Random Models Based on Model cmbTGppv200 Obtaining a Better Fit to the Data than the Model Indicated in the First Columna
aSimilarly, the value in brackets indicates the number of random models in this case based on model cmbTG having a better fit. In bold, we indicate the lowest misfit value in each case.
 We have used the misfit and the random model test to show that the reduction in misfit is significant. However, we also want to know whether the normal mode data used here are sensitive to the differences between these CMB topography models. To this purpose, we use the concept of observability [Koelemeijer et al., 2012] defined as follows:
where and are the splitting function coefficients for two models A and B, and are the associated uncertainties in the data [Deuss et al., 2012]. The observability provides us with a quantitative measure to assess whether small variations in synthetic splitting functions are significant enough with respect to the data limitations. Observability values larger than one indicate that the difference between two models is larger than the uncertainty in the data, and hence, it is significant. As a rule of thumb, if more than half of the Os values are above one, then we would call the difference between those models observable under the current data uncertainties. We calculate the observability of the CMB topography maps with respect to the S20RTS+Crust5.1 predictions and between the different CMB topography maps. In Figure 5, we show the degree 2 observability (O2) values for the selection of normal modes.
 The observability values summed for each mode (Figure 5a) indicate that the CMB topography maps cmbT and cmbTG are the easiest to observe (12 above versus 4 below 1), whereas when a low-viscosity layer is added (cmbTGppv200 and cmbTGppv250), the observability is reduced (6 above versus 10 below 1). The same is true when looking at the observability for individual coefficients (Figures 5b–5e), where the signal due to the cmbTGppv models is mainly within the uncertainty of the data (20–30 above versus 60–50 below 1). However, these are observability values with respect to the S20RTS+Crust5.1 predictions, whereas we are interested in the observability between different CMB topography models. In that case, the observability values are in fact larger; the difference between the cmbT and cmbTG maps are close to the data uncertainties (30 above versus 50 below 1), but the difference between the cmbT and cmbTGppv250 maps is more observable (40 above versus 40 below 1). This implies that the normal modes are able to observe the difference between these kind of models, although being close to the data uncertainties and hence that the difference in misfit we observe is significant.
 The peak-to-peak amplitude of the CMB topography maps decreases with increasing complexity from 12.4 km for cmbT, 8.7 km for cmbTG, 3.5 km for cmbTGppv200 to 2.1 km for cmbTGppv250. One may think that the better fit we obtain for the cmbTGppv maps is purely due to the amplitude of the topography maps with the normal modes preferring smaller amplitudes. Given that the amplitude of the topography is of more influence than its pattern [Koelemeijer et al., 2012], it is difficult to assess whether this is the case. The random test models show that even when we have the same amplitude spectrum, the misfit of the desired maps are significantly lower than those of the random maps, at least for degrees 2 and 4. Additionally, both the cmbTGppv maps fit the requirement of a peak-to-peak amplitudes less than 5 km, and as a result, they are our preferred maps on the basis of all these considerations.
 We calculate synthetic splitting functions for different CMB topography maps on top of the 3-D mantle S wave model S20RTS and crustal model Crust5.1. The maps of CMB topography employed (retrieved via purely seismic or geodynamically constrained tomographic inversions of P wave data) generally fit the normal mode splitting observations. Introducing the geodynamic term in the CMB topography maps, constrained by the mechanical coupling with lowermost mantle structure, helps with respect to pure tomography. Accounting for the presence of post-perovskite in the form of a low-viscosity layer at the base of the mantle reduces the misfit further. As the improvements in misfit are small, we assess the significance of these results using a random model test and the concept of observability. The random model test demonstrates that the reduction in misfit of the cmbTGppv maps is about 92% significant for degree 2 and about 98% significant for all degrees together. The observability values indicate that some of the maps are within the uncertainty of the data. However, normal modes are able to observe the differences between the different CMB topography maps, implying that the reduction in misfit is significant enough. We can thus state that our CMB tomography maps cmbTGppv significantly improve the fit to the normal mode observations while at the same time being geodynamically consistent. This implies that joint seismic-geodynamic inversions and the incorporation of a low-viscosity layer are important steps toward a more realistic modeling of the Earth's lowermost mantle.
 We thank Mark Panning and an anonymous reviewer for helpful reviews of this manuscript. AD and PJK are funded by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 204995. AD was supported by a Philip Leverhulme Prize, and PJK was also supported by the Nahum Scholarship in Physics and a Graduate Studentship, both from Pembroke College, Cambridge. Figures have been produced using the GMT software [Wessel and Smith, 1998].