Core-mantle boundary (CMB) Stoneley modes represent a unique class of normal modes with extremely strong sensitivity to wave speed and density variations in the D” region. We measure splitting functions of eight CMB Stoneley modes using modal spectra from 93 events with Mw> 7.4 between 1976 and 2011. The obtained splitting function maps correlate well with the predicted splitting calculated for S20RTS+Crust5.1 structure and the distribution of Sdiff and Pdiff travel time anomalies, suggesting that they are robust. We illustrate how our new CMB Stoneley mode splitting functions can be used to estimate density variations in the Earth's lowermost mantle.
 The D” region is the lowest 200–300 km of the mantle, atop the core-mantle boundary (CMB). D” is characterized by ultra-low-velocity zones (ULVZs), seismic discontinuities, anisotropy, CMB topography, and, most prominently, by large-low-shear-velocity provinces (LLSVPs) below Africa and the Pacific [e.g., Lay, 2007; Garnero and McNamara, 2008]. The LLSVPs extend hundreds of kilometers both laterally and vertically into the lower mantle [Ritsema et al., 1999]. To assess their effect on mantle dynamics, it is essential to have information on the density variations [Forte and Mitrovica, 2001].
 Observations of Earth's normal modes have the potential to constrain both wave speed and density variations in the mantle. Previous normal mode analyses suggest an anticorrelation between variations in the seismic shear velocity and density, particularly for the LLSVPs [e.g., Ishii and Tromp, 1999; Trampert et al., 2004; Mosca et al., 2012]. These results motivated the modeling of LLSVPs as long-lived “piles” of intrinsically dense material [e.g., Davaille, 1999; McNamara and Zhong, 2005]. However, it was debated whether some of these density models are robust, as they depend on the regularization and a priori constraints [Romanowicz, 2001; Kuo and Romanowicz, 2002], and the studied modes have sensitivity to both the upper and lower mantle [Resovsky and Ritzwoller, 1999].
 Here we revisit normal mode constraints on the density structure of the lower mantle by focusing on Stoneley modes [Stoneley, 1924]; a unique class of modes that are confined to solid-liquid interfaces such as the CMB (Figure 1). CMB Stoneley modes have extremely focused sensitivity to structures in D” and the outermost core and hence do not suffer from trade-offs with upper mantle structure. However, they have so far not been observed due to insufficient available data. We present, for the first time, splitting function measurements of CMB Stoneley modes, discuss the robustness of our measurements, and illustrate how they can be used in tomographic inversions to constrain density structures in the lowermost mantle.
2 Normal Modes
 Earth's normal modes are standing waves arising along the surface and radius of the Earth. They are observed as clear peaks in the amplitude spectra of several day long seismic recordings of large ( Mw>7.4) earthquakes. Modes only exist at discrete frequencies, due to the finite size of the Earth, and are characterized by their radial order n and angular order l. We focus here on spheroidal modesnSl which involve P-SV motion. Each normal mode consists of 2l+1 singlets with azimuthal order m in the range −l,...,l. These singlets are degenerate (i.e., have the same frequency) for a spherically symmetric, isotropic, nonrotating Earth model such as the Preliminary Reference Earth Model (PREM) [Dziewonski and Anderson, 1981]. Significant splitting of the singlets into different frequencies occurs by the rotation and ellipticity of the Earth and velocity and density heterogeneity, anisotropy, and topography on internal boundaries in the Earth.
 Normal mode splitting can be completely described using the splitting function approach introduced by Woodhouse and Giardini . Splitting function coefficients cst are linearly related to the perturbations of the reference Earth model according to:
where δmst(r) are the spherical harmonic coefficients of angular order s and azimuthal order t to describe Earth structure, including perturbations in S wave velocity (Vs), P wave velocity (Vp), density (ρ), and anisotropy. represent topography on discontinuities d, and Ks(r), are the associated sensitivity kernels [Woodhouse, 1980]. Splitting function maps F(θ,φ) are used to visualize splitting functions, i.e.,
where are the complex spherical harmonics of Edmonds . These maps show the local variation in splitting due to the underlying heterogeneity.
3 Methods and Data
 Splitting functions are measured from the inversion of spectra observed for large earthquakes. We make use of a recent normal mode spectra data set of 92 events with Mw> 7.4 for the period 1976–2011 [Deuss et al., 2011, 2013], with the addition of the 2011 Tohoku event (Mw= 9.0). Following Deuss et al. , we measure the splitting functions using nonlinear iterative least squares inversion [Tarantola and Valette, 1982], starting from PREM or predictions for mantle and crust structure. Cross validation is used to determine the errors of our measured coefficients.
 Measuring CMB Stoneley modes is complicated as they generally overlap in frequency with a (high-amplitude) fundamental mode with n = 0. Hence, we must invert for their splitting functions jointly as previously done for1S14 by Resovsky and Ritzwoller . We also measure the fundamental mode separately to verify that we improve the misfit by including the CMB Stoneley mode. We account for the coupling between fundamental spheroidal and toroidal modes due to Earth's ellipticity and rotation. The misfit is smaller when the CMB Stoneley mode is added (Table 1), which is also visible for individual spectra (Figure 2).
Table 1. Misfit for the Measured Splitting Functionsa
aPREM denotes the misfit including only ellipticity and rotation, and S20 denotes the misfit for S20RTS+Crust5.1 synthetics. The final misfit is given for the measurement without (mst i) and with (mst s) the CMB Stoneley mode (denoted by‘s’). The number of spectra (Ns) and events (Ne) is shown. Bold modes correspond to new modes, and modes in brackets are included for rotation and ellipticity coupling.
 We compare our measurements to predictions for mantle and crust structure, calculated using mantle Vs model S20RTS [Ritsema et al., 1999]. We assume scaling factors of the form Rp = δlnVp/δlnVs = 0.5 and Rρ = δlnρ/δlnVs = 0.3, consistent with previous work [Karato, 1993; Li et al., 1991]. The contributions of crustal thickness, surface topography, and water level are calculated using model Crust5.1 [Mooney et al., 1998].
4.1 Splitting Function Observations
 We have made splitting function measurements of 23 modes in total, including eight CMB Stoneley modes and four other new modes along the same overtone branches (1S15,1S16,2S14 and3S25). In addition, we have measured the associated fundamental modes up to0S26.
 The observed splitting function maps (Figure 3) show the “Ring around the Pacific” pattern of high frequencies and pronounced low frequencies at the LLSVPs. Within the “Ring” structure, isolated patches of elevated frequencies are identified, particularly underneath Southeastern Asia and South America. The splitting function maps resemble the predictions for S20RTS+Crust5.1 structure closely. However, individual coefficients such as the c20 differ substantially from the predictions (Figure S1 in the supporting information). In addition, the misfit is significantly lower for our measurements (Table 1). We verify using F-test statistics that the misfit reduction due to including the CMB Stoneley mode is significant (90% confidence level). Details on the misfit calculation and F-test can be found in the supplementary online material. Corresponding center frequencies and quality factors for our measurements are in Table 2, and our splitting function coefficients can be found online (Table S1).
Table 2. Normal Mode Center Frequencies in μHz and Quality Factors Q for the Modes Measured in This Study Compared With PREM Valuesa
aBold modes correspond to new modes.
2345.64 ± 0.41
405 ± 33
2552.55 ± 0.09
374 ± 15
2764.32 ± 0.11
331 ± 4
2973.73 ± 0.15
288 ± 5
3168.96 ± 0.10
207 ± 2
3337.58 ± 0.08
164 ± 1
3062.25 ± 0.07
182 ± 2
3238.69 ± 0.03
247 ± 2
3440.80 ± 0.14
334 ± 5
5397.21 ± 0.20
325 ± 5
5427.09 ± 0.15
238 ± 3
5620.73 ± 0.24
431 ± 12
4.2 Comparison to Sdiff and Pdiff Data
 CMB Stoneley modes have similar sensitivity to waves diffracting around the core such as the Sdiff and Pdiff phase. We use the travel time anomaly data set from Ritsema and Van Heijst  obtained for events between 1980 and 2000 with Mb > 5.9. The binned data show good coverage in the northern hemisphere but less in the southern hemisphere (Figures 4a–4c). Again, we observe a characteristic “Ring around the Pacific” pattern, becoming even clearer in the even spherical harmonic expansion of the diffracted wave data (Figures 4b–4d), though, some of the southern hemisphere structure could be due to inherent symmetry of the even degree expansion.
 The expanded Sdiff and Pdiff data and the CMB Stoneley mode splitting functions of2S25 and 3S26have a strong resemblance. The correlation between the modes and Sdiff data is typically 0.99 and 0.75 for degrees 2 and 4, respectively, whereas the correlation to Pdiff data is lower at 0.98 and 0.50. This similarity strengthens our confidence in our CMB Stoneley mode measurements. The additional advantage of the normal modes is that they automatically provide coverage in the southern hemisphere.
4.3 Sensitivity to Density
 The sensitivity kernels of the CMB Stoneley modes (Figure 1) show a strong sensitivity to Vp at the CMB, whereas the sensitivity to Vs and ρ is similar and peaks in the D” above the CMB. Thus, Stoneley modes are useful to constrain Rρ in the D” which plays an important role in determining the nature of the LLSVPs. We calculate splitting function synthetics using S20RTS and Crust5.1 in which we vary Rp and Rρ between −1 and 2 for a 300 km thick D” layer. We compute the misfit between the observed and calculated splitting function coefficients for individual modes.
 Contour plots of the misfit for s = 2 are shown for mode1S10 (previously observed, e.g., Resovsky and Ritzwoller  and Deuss et al. ) and CMB Stoneley mode3S26 (Figure 5).1S10 can be used to put some constraints on Rp but cannot constrain density variations even though the sensitivity in D” is nonzero. However,3S26 has a strong sensitivity to both Rp and Rρ, and the same is observed for other CMB Stoneley modes. Assuming values for Rp of 0.5 and 0.25 as a range of possible values [Karato, 1993; Ritsema and Van Heijst, 2002], we observe for3S26 a best fitting Rρof 0.1 and 0.5, respectively, close to the range of ratios that would be consistent with purely thermal variations [e.g., Karato, 1993; Mosca et al., 2012]. This suggests that the anticorrelation between density and shear wave velocity might not be required by our new data. However, without good constraints on Rp, we cannot constrain Rρ accurately. In addition, Rpand Rρ trade-off with other structures in D” such as CMB topography and ULVZs [Koelemeijer et al., 2012], and therefore, a proper inversion is required to draw any firm conclusions regarding thermal versus thermochemical LLSVPs.
5 Concluding Remarks
 Using a data set of 93 large earthquakes, we make robust splitting function observations of eight CMB Stoneley modes. Their splitting function maps correlate well with expanded Sdiff and Pdiff data suggesting they are robust. We demonstrate the sensitivity of the Stoneley modes to density variations in the lowermost mantle and illustrate the trade-off with P wave velocity structure. This trade-off can be partially removed when we consider thinner layers (100 km thick) due to the nature of the sensitivity kernels (Figure 1). In addition, a large number of P wave sensitive normal mode observations is available [Deuss et al., 2013], and body wave data also provide constraints on Rp. Therefore, when our new measurements are included with these in tomographic inversions, they will help to provide tighter constraints on the density variations in the lowermost mantle.
 We thank the Editor (Michael Wyssession), Caroline Beghein, and Joseph Resovsky for their detailed comments, which greatly improved the manuscript. Data were provided by the IRIS/DMC. PJK and AD are funded by the European Research Council under the European Community's 7th Framework Programme (FP7/2007-2013)/ERC grant agreement 204995. PJK is also supported by the Nahum Scholarship in Physics and a Graduate Studentship, both from Pembroke College, Cambridge. AD is also funded by a Philip Leverhulme Prize, and JR is supported by NSF grant EAR-1014749. We would like to thank Anna Mäkinen for advice on the F-test statistics. Figures have been produced using the GMT software [Wessel and Smith, 1998].
 The Editor thanks Caroline Beghein and an anonymous reviewer for their assistance in evaluating this paper.