Complex inner core boundary from frequency characteristics of the reflection coefficients of PKiKP waves observed by Hi-net
© Tanaka and Tkalčić. 2015
Received: 21 April 2015
Accepted: 8 October 2015
Published: 4 November 2015
Frequency-dependent reflection coefficients of P waves at the inner core boundary (ICB) are estimated from the spectral ratios of PKiKP and PcP waves observed by the high-sensitivity seismograph network (Hi-net) in Japan. The corresponding PKiKP reflection locations at the ICB are distributed beneath the western Pacific. At frequencies where noise levels are sufficiently low, spectra of reflection coefficients show four distinct sets of characteristics: a flat spectrum, a spectrum with a significant spectral hole at approximately 1 or 3 Hz, a spectrum with a strong peak at approximately 2 or 3 Hz, and a spectrum containing both a sharp peak and a significant hole. The variety in observed spectra suggests complex lateral variations in ICB properties. To explain the measured differences in frequency characteristics of ICB reflection coefficients, we conduct 2D finite difference simulations of seismic wavefields near the ICB. The models tested in our simulations include a liquid layer and a solid layer above the ICB, as well as sinusoidal and spike-shaped ICB topography with varying heights and scale lengths. We find that the existence of a layer above the ICB can be excluded as a possible explanation for the observed spectra. Furthermore, we find that an ICB topographic model with wavelengths and heights of several kilometers is too extreme to explain our measurements. However, restricting the ICB topography to wavelengths and heights of 1.0–1.5 km can explain the observed frequency-related phenomena. The existence of laterally varying topography may be a sign of lateral variations in inner core solidification.
KeywordsInner core boundary Topography PKiKP Finite difference modeling
The inner core boundary (ICB) is one of the vital regions for understanding the Earth’s core dynamics (Loper and Roberts 1981; Loper 1983; Bergman and Fearn 1994; Shimizu et al. 2005; Deguen et al. 2007; Sumita and Bergman 2009; Deguen 2012). Seismological studies of the ICB and its inferred characteristics, such as the density jump between the inner and the outer cores, the shear-wave velocity at the top of the inner core, and scattering of seismic energy from small-scale topography at the inner core surface, are important in elucidating the growth mechanism of the inner core and the source of the geodynamo (Souriau 2007; Sumita and Bergman 2009; Deuss 2014; Tkalčić 2015).
The hypothesis of a hemispherical structure in the upper inner core (Tanaka and Hamaguchi 1997) has been widely accepted because it is supported by seismic observations of body waves and free oscillations (Creager 1999; Deuss et al. 2010). Hemispherical dichotomy is recognized as a global phenomenon near the ICB (Niu and Wen 2001; Waszek et al. 2011) and possibly near the center of the inner core (Lythgoe et al. 2014). To explain the hemispherical structure of the inner core, two models have been proposed: a large-scale asymmetric flow in the outer core (Sumita and Olson 1999; Aubert et al. 2008; Gubbins et al. 2011) and translational convection in the inner core (Alboussiere et al. 2010; Monnereau et al. 2010). The nature of these models is that they allow diametrically opposite scenarios of freezing and melting, i.e., one of the two hemispheres solidifies faster than the other. This has raised further questions about whether the surface of the inner core in the eastern hemisphere is melting or freezing. Although translational convection models predict that the eastern hemisphere is melting, outer-core large asymmetric flow models disagree about which hemisphere is melting, due to the problem setting, e.g., heat flux at the core–mantle boundary (CMB) and outer core dynamics.
The new era of modern instruments, dense networks, and improved global coverage enables observation of a large number of PKiKP and PcP phases on the same seismogram. For example, Koper et al. (2003) analyzed a significant number of PKiKP phases at shorter distances recorded by small aperture arrays of the International Monitoring System. This study was followed by a large number of new studies (Cao and Romanowicz 2004; Koper et al. 2004; Koper and Pyle 2004; Poupinet and Kennett 2004; Koper and Dombrovskaya 2005; Krasnoshchekov et al. 2005; Leyton et al. 2005; Kawakatsu 2006; Leyton and Koper 2007a, b; Peng et al. 2008).
More specifically, Koper and Pyle (2004) measured PKiKP/PcP amplitude ratios from seismograms filtered between 1 and 3 Hz. Their analysis did not reveal any differences between the eastern and western hemispheres of the inner core. Using phases from a Mariana event observed in Japan, whose raypaths sampled the ICB beneath the western Pacific, Kawakatsu (2006) found little scattering energy in the PKiKP coda. However, Leyton and Koper (2007b) analyzed the coda of PKiKP and suggested the existence of small-scale heterogeneities in the uppermost inner core. This result indicates that the strong scattering region is located beneath the Pacific Ocean and Asia, which covers parts of the eastern and western hemispheres. On the basis of scattering properties and Q structure, Cormier (2007) inferred textural differences between the eastern and western hemispheres near the surface of the inner core, including vertically oriented structures in the eastern hemisphere.
Interestingly, the existence of a high-frequency PKiKP phase (up to 5 Hz) with steep incidence angles at the ICB was first observed by Poupinet and Kennett (2004) using phases recorded by narrow-aperture arrays and temporal broadband networks on the Australian continent, whose reflection points were in the eastern hemisphere. Recently, the same class of PKiKP waves was observed in the eastern hemisphere on Chinese and Japanese short-period and broadband stations (J-array) (Tkalčić et al., 2009, 2010) and by the high-sensitivity seismograph network (Hi-net) in Japan (Dai et al. 2012, Jiang and Zhao 2012).
To contribute to a better understanding of inner core dynamics and to constrain ICB structure in the eastern hemisphere, here, we collect an extensive dataset of PKiKP waves recorded by Hi-net in Japan. Our aim is to shift focus from analyzing a single value of the PKiKP/PcP amplitude ratio to evaluating its broad frequency characteristics, which is philosophically similar to how Cummins and Johnson (1988) evaluated pre-critical PKiKP waveforms and spectra by using a hybrid full wave-reflectivity algorithm. A dense configuration of borehole seismograms with high SNR observations of PKiKP waves over an unusually broad range of frequencies facilitates this new approach to estimate ICB properties. Thus, we examine data in the frequency domain and investigate possible broader implications for Earth’s core dynamics.
No. of used records
After applying the band-pass filter described above and retrieving record segments of ±10 s length around PKiKP arrivals, seismograms were divided into sub-groups comprising 70–230 stations with traces sorted by increasing epicentral distance. We then selected subsets of coherent waveforms by using the cross-correlation matrix method (Tkalčić et al. 2011), which retained 46–220 records per event. To find coherent PKiKP arrivals in each sub-group, we empirically determined the minimum percentage (τ) of all waveform pairs that should cross-correlate in such a way that the average cross-correlation coefficient equals or exceeds a threshold β. The algorithm calculates the cross-correlation coefficients for each pair of waveforms and counts the total percentage of pairs with cross-correlation coefficient exceeding β. For example, for β > 0.4, τ > 10 %, and 130 total waveforms, we found that 77 waveforms satisfied these criteria and were consequently selected as “mutually coherent” (Fig. 3c). Incoherent waveforms were not used in further analyses. Subsequently, we visually checked the waveforms that satisfied the above criteria to find possible PKiKP signals with high signal-to-noise ratios, which resulted in a station list of “good” sites for PKiKP observations.
To correct the amplitude ratios, we determined focal mechanisms using the program of Kikuchi and Kanamori (2003), rather than cataloged Global CMT solutions, USGS Moment tensor, and double-couple solutions. The broadband displacements of P and SH waves in the frequency range 0.002–1 Hz were used for the inversion. Furthermore, we compared the short-period (SP) P wave amplitudes (1–5 Hz) with the radiation pattern predicted from each focal mechanism solution. Although the observed SP amplitudes showed a high degree of scatter, we found that some events had clear energy even near the nodal plane and null axis, suggesting a smoothed radiation pattern. This may be due to scattering near the source region, as discussed in previous studies that determined magnitudes from SP data (e.g., (Schweitzer and Kværna 1999; Takemura et al. 2015). These results are summarized in the Remarks column of Table 1. The smoothed radiation pattern for the short period is a likely explanation for the observation of PcP and PKiKP in the events from Kuril (event 2) and Mindanao (event 11), in which the take-off azimuths and angles of PcP and PKiKP were located near the nodal plane.
We converted the spectral ratios to reflection coefficients as a function of frequency. To the first order, geometric spreading is constant with respect to frequency under a ray-theoretical assumption. The reflection and transmission coefficients were then calculated for planar boundaries. The correction of the attenuation factor using t * gives a smooth exponential variation with frequency. Thus, these corrections will not result in any spectral holes or peaks.
To reduce the unwanted effect of the CMB on the PcP spectra, the spectral ratios of PcP/P were examined. Similar to the above procedure, we estimated the P wave reflection coefficients at the CMB as a function of frequency. The obtained reflection coefficients did not always coincide with theoretical values due to the uncertainty in focal mechanisms, large differences in P and PcP take-off angles, and other unknown causes. Thus, we corrected only the fluctuations in the reflection coefficients around the average values in the frequency range 1–3 Hz. The reflection coefficients at the ICB are multiplied by the fluctuations in reflection coefficients at the CMB, which can result in either amplifying an apparently small PKiKP/PcP due to a large peak in the PcP spectrum or reducing a large PKiKP/PcP due to a small peak in the PcP.
The frequency characteristics of the ICB reflection coefficients are quite complex, even in a narrow effective signal band (Fig. 8). Roughly speaking, peaks in reflection coefficients appear around 2 Hz (results 8, 9, and 10) and 3 Hz (result 3), and holes are observed around 1 Hz (results 2, 3, and 4) and 3 Hz (result 10). Although the discrimination is still qualitative, we recognize four general categories of frequency-dependent characteristics: (i) a flat variation, where fluctuations in the relative strengths of peaks and holes are between half and double those of the theoretical reflection coefficients (results 1, 5, 6, 7, and 11); (ii) a distinct single hole in each reflection coefficient spectrum (results 2 and 4); (iii) a strong single peak in each spectrum (results 8 and 9); and (iv) a strong peak and hole in the same spectrum (results 3 and 10).
Effects of the CMB
Regarding the effects of the CMB on PKiKP spectra during transmission through the CMB, we address this issue in the context of the results of previous studies. Using amplitude of precursors to PKIKP, Dai et al. (2012) and Yao and Wen (2014) showed that several regions exhibit weak scattering in the lowermost mantle beneath the southwestern Pacific. PKiKP phases from events that occurred in the Banda Sea (events 4 and 9), Sumatra (events 3 and 10), and the Philippines (event 11) enter a “normal” CMB. According to Hedlin and Shearer (2000), there is a relatively weak scattering region in the lowermost mantle beneath the Philippine Sea, which corresponds to the CMB entry points of PKiKP for events 1 and 5 and the CMB exit points for events 1, 3, 4, 5, 9, 10, and 11. Thus, we have reason to believe that the CMB effects on the estimated reflection coefficients for results 1, 2, 6, 8, 9, 10, and 11 will be negligible. However, a strong scattering area near the CMB exists beneath north Japan and the northwestern Pacific, which includes the PKiKP CMB entry and exit points for events 2, 6, 7, and 8. Thus, we cannot rule out the possibility that the frequency characteristics of calculated reflection coefficients for results 3, 4, 5, and 7 are CMB effects, e.g., the high-frequency components of PKiKP may be lost by scattering at the CMB.
Simulations for a thin layer above the ICB
Simulations for topography at the ICB
Earlier PKiKP and PcP amplitude ratio analyses made sporadic seismological observations of a lower density contrast at the ICB than predicted values for spherically symmetric Earth models, as low as 200–300 g/m3(Koper and Pyle 2004; Tkalčić et al. 2009). In addition, Gubbins et al. (2008) inferred a low density contrast from geodynamical considerations. These results indirectly support the existence of a dense layer at the top of the inner core (F-layer) to explain the smaller density differences between the outer and inner cores. However, there is no proof from seismology that such a layer is a global feature. Masters and Gubbins (2003) noted that even the relatively large density jump inferred from free oscillation analyses is consistent with an F-layer having a strong density gradient: the free oscillation data would average over thick layers below and above the ICB, while the body wave would be sensitive to variations across the ICB. A lower density contrast across the ICB would permit larger topography (Buffett 1997). Of particular interest is the possibility of sharp edges between the solidification and melting areas. Recent waveform modeling suggests significant topography (Dai et al. 2012); however, determination of the amplitude is a more difficult problem than determination of the wavelength.
One possibility of most simplified geometry is that a sinusoidal topographic structure might develop at the largest scales (λ = 10–100 km) (Buffett 1997). At these length scales, there is an inverse relationship between relaxation time scale and wavelength (Turcotte and Schubert 2002). In addition, the time scale required for topography to relax varies inversely with density contrast and linearly with viscosity. As the viscosity of the outer core is effectively zero (de Wijs et al. 1998), the rate of relaxation is thus entirely controlled by deformation in the inner core.
On the other hand, spike-shaped topographic structures might develop as a result of dendritic growth, likely at smaller scales (λ = 10–several 100 m) (Bruce Buffett, pers. comm.). Such topography could be relaxed through melting and freezing (thermal relaxation). As the temperature gradients are steeper at short wavelengths, this can drive the heat flow needed to melt or freeze.
The resultant characteristics are slightly different from those produced by the sinusoidal topographies, even though their structural dimensions are the same. The frequencies of the distinct spectral peaks decrease with increasing wavelength and height (Fig. 14b–d), whereas no distinct peaks are observed for the topography with λ = H = 0.5 km. The spiky topography with λ = H = 1.0 km results in a distinct peak around 1.7 Hz for θ = 10°–20° and a large hole around 1.5 Hz for θ = 30° (Fig. 14b). The topography with λ = 1.5 and H = 1.5 km yields a distinct single peak at roughly f = 1.2 Hz. There are several spectral holes for θ = 10°–20° and peaks for θ = 30° at f > 2 Hz (Fig. 14c). Of all the test cases with spiky topography, the case with λ = H = 1.0 km most closely matches the observed spectral ratios.
In summary, our observations and numerical simulations suggest that the inner core boundary is a sharp boundary without transitional layers. The hypothesis of melt at the surface of the inner core in the quasi-eastern hemisphere is not supported by our simulations. The most likely scenario to explain some of the observed spectral characteristics is the existence of topography at the ICB; however, more than one class of topography must be invoked to explain all observations. We therefore conclude that the topography characteristics of the ICB vary laterally. These variations may result from lateral variations of inner core solidification. If solidification is dynamically driven from top to bottom, its geographical pattern will be controlled by the pattern of outer core convection (Bergman et al. 2002; Aubert et al. 2008; Gubbins et al. 2011). If the solidification is instead driven from the bottom up, the pattern will be affected by variations in inner core convection (Deguen and Cardin 2011). Furthermore, small-scale variations in topographic characteristics suggest small-scale convection in a mushy zone at the ICB (Bergman and Fearn 1994; Deguen et al. 2007). While we cannot distinguish between these hypotheses in the present study, largely due to the fact that we sample only sparse and limited areas of the ICB, further observations of PKiKP and PcP will improve our understanding of large-scale ICB structure and dynamics.
Frequency characteristics of ICB reflection coefficients were investigated for the area around Japan using Hi-net vertical component seismograms. We found four patterns in the frequency-dependent behavior of reflection coefficients: (a) a nearly flat spectrum (little variation), (b) a significant hole at a frequency of approximately 1 or 3 Hz, (c) a peak at a frequency of approximately 2 or 3 Hz, and (d) the existence of a hole and a peak. The variety in observed spectra reflects the complex nature of the ICB. To interpret these observations, we conducted 2D finite difference simulations. Since we tested only limited cases with planar geometry, further simulations are required. Our modeling results suggest that holes and peaks in the spectra of reflection coefficients can be qualitatively explained by a sinusoidal or spike-like topography at the ICB, with wavelengths and heights ~1–1.5 km, whereas a liquid or solid layer overlying the ICB does not reproduce any of the observed spectral features.
inner core boundary
The authors are grateful to the National Research Institute for Earth Science and Disaster Prevention (NIED), Japan, for providing high-quality seismograms recorded by the high-sensitivity seismograph network (Hi-net). The authors thank two anonymous reviewers who give valuable comments. S.T. was supported in part by MEXT KAKENHI grant number 15H05832. H.T. was supported by the Japan Society for the Promotion of Science and the Research School of Earth Science, The Australian National University during his stay in Japan. Most of the figures were drawn using GMT (Wessel and Smith 1998).
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- Alboussiere T, Deguen R, Melzani M (2010) Melting-induced stratification above the Earth’s inner core due to convective translation. Nature 466:744–9View ArticleGoogle Scholar
- Aubert J, Amit H, Hulot G, Olson P (2008) Thermochemical flows couple the Earth’s inner core growth to mantle heterogeneity. Nature 454:758–62View ArticleGoogle Scholar
- Bergman MI, Fearn DR (1994) Chimneys on the Earth’s inner-outer core boundary. Geophys Res Lett 21:477–80View ArticleGoogle Scholar
- Bergman MI, Cole DM, Jones JR (2002) Preferred crystal orientations due to melt convection during directional solidification. J. Geophys. Res., 107: doi:10.1029/2001JB000601
- Buchbinder GGR (1972) Travel times and velocities in the outer core from PmKP. Earth Planet Sci Lett 14:161–8View ArticleGoogle Scholar
- Buffett BA (1997) Geodynamic estimates of the viscosity of the Earth’s inner core. Nature 388:571–3View ArticleGoogle Scholar
- Cao A, Romanowicz B (2004) Constraints on density and shear velocity contrast at the inner core boundary. Geophys J Int 157:1146–51View ArticleGoogle Scholar
- Cormier VF (2007) Texture of the uppermost inner core from forward- and back-scattered seismic waves. Earth Planet Sci Lett 258:442–53View ArticleGoogle Scholar
- Creager KC (1999) Large-scale variations in inner core anisotropy. J Geophys Res 104:23127–39View ArticleGoogle Scholar
- Cummins P, Johnson L (1988) Synthetic seismograms for an inner core transition of finite thickness. Geophys J 94:21–34Google Scholar
- Dai Z, Wang W, Wen L (2012) Irregular topography at the Earth’s inner core boundary. Proc Natl Acad Sci 109:7654–8View ArticleGoogle Scholar
- de Wijs GA, Kresse G, Vocadlo L, Dobson DP, Alfè D, Gillan M, Price GD (1998) The viscosity of liquid iron at the physical conditions of the Earth’s core. Nature 392:805–7View ArticleGoogle Scholar
- Deguen R (2012) Structure and dynamics of Earth’s inner core. Earth Planet Sci Lett 333–334:211–25View ArticleGoogle Scholar
- Deguen R, Cardin P (2011) Thermochemical convection in Earth’s inner core. Geophys J Int 187:1101–18View ArticleGoogle Scholar
- Deguen R, Alboussiere T, Brito D (2007) On the existence and structure of a mush at the inner core boundary of the Earth. Phys Earth Planet Inter 164:36–49View ArticleGoogle Scholar
- Deuss A (2014) Heterogeneity and anisotropy of Earth’s inner core. Ann Rev Earth Planet Sci 42:103–26View ArticleGoogle Scholar
- Deuss A, Irving JCE, Woodhouse JH (2010) Regional variation of inner core anisotropy from seismic normal mode observations. Science 328:1018–20View ArticleGoogle Scholar
- Engdahl ER, Flinn EA, Romney CF (1970) Seiemic waves reflected from the Earth’s inner core. Nature 228:852–3View ArticleGoogle Scholar
- Engdahl ER, Flinn EA, Massé RP (1974) Differential PKiKP travel times and the radius of the inner core. Geophys J R Astron Soc 39:457–63View ArticleGoogle Scholar
- Gubbins D, Masters G, Nimmo F (2008) A thermochemical boundary layer at the base of Earth’s outer core and independent estimate of core heat flux. Geophys J Int 174:1007–18View ArticleGoogle Scholar
- Gubbins D, Sreenivasan B, Mound J, Rost S (2011) Melting of the Earth’s inner core. Nature 473:361–3View ArticleGoogle Scholar
- Hedlin MAH, Shearer PM (2000) An analysis of large-scale variations in small-scale mantle heterogeneity using global seismographic network recordings of precursors to PKP. J Geophys Res 105:13655–73View ArticleGoogle Scholar
- Jiang G, Zhao D (2012) Observation of high-frequency PKiKP in Japan: insight into fine structure of inner core boundary. J Asian Earth Sci 59:167–84View ArticleGoogle Scholar
- Kawakatsu H (2006) Sharp and seismically transparent inner core boundary region revealed by an entire network observation of near-vertical PKiKP. Earth Planets Space 58:855–63View ArticleGoogle Scholar
- Kennett BLN, Engdahl ER, Buland R (1995) Constraints on seismic velocities in the Earth from travel-times. Geophys J Int 122:108–24View ArticleGoogle Scholar
- Kikuchi M, Kanamori H (2003), Note on teleseismic body-wave inversion program, http://www.eri.u-tokyo.ac.jp/ETAL/KIKUCHI/
- Koper KD, Dombrovskaya M (2005) Seismic properties of the inner core boundary from PKiKP/P amplitude ratios. Earth Planet Sci Lett 237:680–94View ArticleGoogle Scholar
- Koper KD, Pyle ML (2004) Observations of PKiKP/PcP amplitude ratios and implications for Earth structure at the boundaries of the liquid core. J. Geophys. Res., 109: doi:10.1029/2003JB002750
- Koper KD, Pyle ML, Franks JM (2003) Constraints on aspherical core structure from PKiKP-PcP differential travel times. J Geophys Res 108:2168, doi:2110.1029/2002JB001995View ArticleGoogle Scholar
- Koper KD, Franks JM, Dombrovskaya M (2004) Evidence for small-scale heterogeneity in Earth’s inner core from a global study of PKiKP coda waves. Earth Planet Sci Lett 228:227–41View ArticleGoogle Scholar
- Krasnoshchekov DN, Kaazik PB, Ovtchinnikov VM (2005) Seismological evidence for mosaic structure of the surface of the Earth’s inner core. Nature 435:483–7View ArticleGoogle Scholar
- Larsen SC, Shultz CA (1995), E3D:2D/3D Elastic finite-difference wave propagation code, Lawrence Livermore National Laboratory,Livermore, CA, USA. 1–18.
- Lay T, Wallace T (1995) Modern global seismology. Academic, San DiegoGoogle Scholar
- Leyton F, Koper KD (2007a) Using PKiKP coda to determine inner core structure: 1. Synthesis of coda envelopes using single-scattering theories. J. Geophys. Res., 112: doi:10.1029/2006JB004369.
- Leyton F, Koper KD (2007b) Using PKiKP coda to determine inner core structure: 2. Determination of QC. J. Geophys. Res., 112: doi:10.1029/2006JB004370.
- Leyton F, Koper KD, Zhu L, Dombrovskaya M (2005) On the lack of seismic discontinuities within the inner core. Geophys J Int 162:779–86View ArticleGoogle Scholar
- Loper DE (1983) Structure of the inner core boundary. Geophys Astrophys Fluid Dyn 25:139–55View ArticleGoogle Scholar
- Loper DE, Roberts PH (1981) A study of conditions at the inner core boundary of the Earth. Phys Earth Planet Inter 24:302–7View ArticleGoogle Scholar
- Lythgoe KH, Deuss A, Rudge JF, Neufeld JA (2014) Earth’s inner core: Innermost inner core or hemispherical variations? Earth Planet. Sci Lett 385:181–9Google Scholar
- Masters G, Gubbins D (2003) On the resolution of density within the Earth. Phys Earth Planet Inter 140:159–67View ArticleGoogle Scholar
- Monnereau M, Calvet M, Margerin L, Souriau A (2010) Lopsided growth of Earth’s inner core. Science 328:1014–7View ArticleGoogle Scholar
- Niu FL, Wen LX (2001) Hemispherical variations in seismic velocity at the top of the Earth’s inner core. Nature 410:1081–4View ArticleGoogle Scholar
- Okada Y, Kasahara K, Hori S, Obara K, Sekiguchi S, Fujiwara H, Yamamoto A (2004) Recent progress of seismic observation networks in Japan –Hi-net, F-net, K-NET and KiK-net–. Earth Planets Space, 56: xv-xxviii
- Peng ZG, Koper KD, Vidale JE, Leyton F, Shearer P (2008) Inner-core fine-scale structure from scattered waves recorded by LASA. J. Geophys. Res., 113: doi:10.1029/2007jb005412
- Poupinet G, Kennett BLN (2004) On the observation of high frequency PKiKP and its coda in Australia. Phys Earth Planet Inter 146:497–511View ArticleGoogle Scholar
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1988) Numerical recipes in C: the art of scientific computing. Cambridge University Press, CambridgeGoogle Scholar
- Rodgers A, Tkalčić H, McAllen D (2006) Seismic ground motion and site response in Las Vegas Valley, Nevada from NTS explosions and earthquake data. Pure Appl Geophys 163:55–80View ArticleGoogle Scholar
- Schweitzer J, Kværna T (1999) Influence of source radiation patterns on globally observed short-period magnitude estimates (mb). Bull Seism Soc Am 89:342–7Google Scholar
- Shearer P, Masters G (1990) The density and shear velocity contrast at the inner core boundary. Geophys J Int 102:491–8View ArticleGoogle Scholar
- Shimizu H, Poirier JP, Le Mouël JL (2005) On crystallization at the inner core boundary. Phys Earth Planet Inter 151:37–51View ArticleGoogle Scholar
- Souriau A (2007) Deep Earth structure—the Earth’s cores. In: Romanowicz B, Dziewonski AM (eds) Treatise on geophysics, vol. 1, seismology and structure of the Earth. Elsevier, Amsterdam, pp 655–93View ArticleGoogle Scholar
- Souriau A, Souriau M (1989) Ellipticity and density at the inner core boundary from subcritical PKiKP and PcP data. Geophys J Int 98:39–54View ArticleGoogle Scholar
- Sumita I, Bergman M (2009) Inner-core dynamics. In: Olson P (ed) Treatise on geophysics, vol. 8, core dynamics. Elsevier, Amsterdam, pp 299–318Google Scholar
- Sumita I, Olson P (1999) A laboratory model for convection in Earth’s core driven by a thermally heterogeneous mantle. Science 286:1547–9View ArticleGoogle Scholar
- Takemura S, Furumura T, Maeda T (2015) Scattering of high-frequency seismic waves caused by irregular surface topography and small-scale velocity inhomogeneity. Geophys J Int 201:459–74View ArticleGoogle Scholar
- Tanaka S, Hamaguchi H (1997) Degree one heterogeneity and hemispherical variation of anisotropy in the inner core from PKP(BC)-PKP(DF) times. J Geophys Res 102:2925–38View ArticleGoogle Scholar
- Tkalčić H (2015) Complex inner core of the Earth: the last frontier of global seismology. Rev. Geophys., 53: doi:10.1002/2014RG000469
- Tkalčić H, Kennett BLN, Cormier VF (2009) On the inner-outer core density contrast from PKiKP/PcP amplitude ratios and uncertainties caused by seismic noise. Geophys J Int 179:425–43View ArticleGoogle Scholar
- Tkalčić H, Cormier VF, Kennett BLN, He K (2010) Steep reflections from the earth’s core reveal small-scale heterogeneity in the upper mantle. Phys Earth Planet Inter 178:80–91View ArticleGoogle Scholar
- Tkalčić H, Chen Y, Liu R, Huang Z, Sun L, Chan W (2011) Multistep modelling of teleseismic receiver functions combined with constraints from seismic tomography: crustal structure beneath southeast China. Geophys J Int 187:303–26View ArticleGoogle Scholar
- Turcotte D, Schubert G (2002) Geodynamics, 2nd edn. Cambridge University Press, CambridgeView ArticleGoogle Scholar
- Waszek L, Irving J, Deuss A (2011) Reconciling the hemispherical structure of Earth’s inner core with its super-rotation. Nat Geosci 4:264–7View ArticleGoogle Scholar
- Wessel P, Smith WHF (1998) New improved version of generic mapping tools released. EOS Trans Am Geophys Un 79:579View ArticleGoogle Scholar
- Yao J, Wen L (2014) Seismic structure and ultra-low velocity zones at the base of the Earth’s mantle beneath Southeast Asia. Phys Earth Planet Inter 233:103–11View ArticleGoogle Scholar