One-Dimensional Modeling of Multiple Scattering in the Upper Inner Core: Depth Extent of a Scattering Region in the Eastern Hemisphere

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1 Journal of Earth Science, Vol. 24, No. 5, p , October 2013 ISSN X Printed in China DOI: /s One-Dimensional Modeling of Multiple Scattering in the Upper Inner Core: Depth Extent of a Scattering Region in the Eastern Hemisphere Satoru Tanaka* Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokosuka , Japan ABSTRACT: Attenuation of PKP(DF) in the Eastern Hemisphere is examined in terms of multiple scattering to simultaneously explain a puzzling relationship, a relatively fast velocity anomaly corresponding to strong attenuation. Reflectivity synthetics with one-dimensional random velocity fluctuations are compared with observations of PKP(DF)/PKP(Cdiff) amplitude ratios and differential travel times of PKP(Cdiff)-PKP(DF) for the equatorial paths. A Gaussian distribution of P-wave velocity fluctuations with the standard deviations of 5%, 6%, and 7% in the uppermost 200 km of the inner core is superimposed on the velocity structure that is slightly faster than the typical structure in the Eastern Hemisphere, which is likely to explain both the travel time and amplitude data as far as only the one-dimensional structure is considered. Further examinations of the statistic characteristic of scatterer distribution in two and three-dimensions are required to obtain a realistic conclusion. KEY WORDS: seismology, the inner core, attenuation, scattering. INTRODUCTION The inner core, which is a very small body located at the center of the Earth, has a diameter of about 70% of the Moon s diameter and a volume equal to about 0.7% of the Earth s volume. However, the inner core plays important roles in Earth dynamics; for instance, it provides energy to the geodynamo through its growth (Lister and Buffett, 1995; Bragin- This study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (No ). *Corresponding author: stan@jamstec.go.jp China University of Geosciences and Springer-Verlag Berlin Heidelberg 2013 Manuscript received November 23, Manuscript accepted March 4, sky, 1963), and stability of the magnetic dipole field (Hollerbach and Jones, 1993). The seismological knowledge of the Earth s inner core is dramatically accumulated in recent decades, e.g., inner core anisotropy (Morelli et al., 1986; Woodhouse et al., 1986), inner core super-rotation (Song and Richards, 1996), and a hemispherical inner core (Tanaka and Hamaguchi, 1997). Based on such seismic observations, inner core dynamics have been discussed by many researchers (Deguen, 2012; Sumita and Bergman, 2009). The velocity and attenuation structures of the inner core show a puzzling relationship, such that relatively fast and slow velocity anomalies correspond to strong and weak attenuations, respectively (Souriau, 2007). It is well known that PKP(DF) phases traveling in a north-south direction are much faster than those

2 One-Dimensional Modeling of Multiple Scattering in the Upper Inner Core: Depth Extent of a Scattering Region 707 traveling in an east-west direction and that the amplitudes of PKP(DF) in a north-south are much smaller than those in an east-west as a result of anisotropy in inner core attenuation (Souriau and Romanowicz, 1996). This strange relationship has not been explained by any ordinary viscoelastic property in the mantle (Liu et al., 1976), but is explained by the effect of multipathing of PKP(DF) because of an abrupt increase in P-wave velocity in the north-south direction (Song and Helmberger, 1998). Furthermore, scattering attenuation in the inner core and a relaxation spectrum in which a lower frequency corner lies near the body wave frequency are suggested to explain this observation (Cormier and Li, 2002; Li and Cormier, 2002; Cormier et al., 1998). However, such a strange relationship also exists even for PKP(DF) propagating in an east-west direction, though the velocity difference is not as strong as in the case of inner core anisotropy. It has been observed that PKP(DF) phases propagating in the Eastern Hemisphere are slightly faster and smaller than those in the Western Hemisphere, and simple one-dimensional (1-D) velocity and Q structures have been proposed for each hemisphere (Tanaka, 2012; Yu and Wen, 2006; Cao and Romanowicz, 2004; Wen and Niu, 2002). Wen and Niu (2002) conducted two-dimensional (2-D) waveform modeling to examine scattering in the uppermost 100 km of the inner core as an alternative explanation of intrinsic attenuation by comparing waveforms of PKIKP and PKiKP at distances from 125 to 142. They found that the velocity fluctuations with a correlation scale length of 1 km and RMS P-wave velocity variations of 5% 9% in the 2-D modeling can explain the observed PKIKP/PKiKP amplitude ratios in the Eastern Hemisphere, whereas no scattering or a little fluctuation with the RMS less than 3% is required for the data in the Western Hemisphere. This explanation is consistent with the observation that PKiKP coda is strong in the Eastern Hemisphere but weak in the Western Hemisphere (Leyton and Koper, 2007). A geographical correlation among the thickness of the transition layer at the top of the inner core, the strength of PKIKP coda, and the hemispherical distribution of P-wave velocity and attenuation is tentatively explained by a characteristic direction and distribution of the texture in the uppermost 100 km of the inner core (Cormier, 2007). According to the hypothesis of lopsided growth of the inner core with translational convection, the hemispherical distribution of the grain sizes of hexagonal close-packed and body-centered cubic iron are inferred with the various combination of elastic coefficients determined by a grid search to explain both the velocity and apparent Q structures in the uppermost 90 km of the inner core (Monnereau et al., 2010), in which they use an analytical theory of multiple scattering (Calvet and Margerin, 2008). As mentioned above, the scattering properties of the hemispherical inner core have recently been considered but restricted in only the uppermost 100 km of the inner core. Actually, however, the hemispherical differences in velocity and attenuation structures extend deeper and vary with depth (Tanaka, 2012; Yu and Wen, 2006; Yu et al., 2005). Furthermore, Tanaka (2012) pointed out that his intrinsic attenuation modeling cannot fully explain the rapid variation found in the PKP(DF)/PKP(Cdiff) amplitude ratios as a function of epicentral distance. In this short note, multiple scattering in the 1-D modeling is used to numerically examine the radial structure through waveform simulation. DATA AND METHOD Tanaka (2012) examined PKP(DF) and PKP(Cdiff), ray paths of which are plotted in Fig. 1, to study the depth extent of a hemispherical inner core and collected more than 600 measurements of the travel time differences between PKP(Cdiff) and PKP(DF) and PKP(DF)/PKP(Cdiff) amplitudes ratios. He inferred the radial structures of P-wave velocity and Q in the uppermost approximately 500 km of the inner core for the Eastern and Western hemispheres by using bin-averaged data, for which best-fit models of radial structures are shown in Fig. 2. As Tanaka (2012) mentioned, the fit between observations and calculations of the amplitude ratios in the Eastern Hemisphere is insufficient. Here I attempt to explain his bin-averaged data of the Eastern Hemisphere by modifying only the velocity structure using the same Q structure as in the Western Hemisphere.

3 708 Satoru Tanaka 150 o o Cdiff DF BC Figure 1. The ray paths of PKP(DF), PKP(Cdiff) and PKP(BC) shown on a vertical cross section by solid lines, gray lines, and a broken line, respectively. The black star is the hypocenter. I model the PKP waveforms by using the reflectivity method (Kennett, 1988; Müller, 1985). The velocity, density and Q structures of ak135f (Montagner and Kennett, 1996) are adopted for the region above the inner core boundary (ICB); the P-wave velocity and Q structures below the ICB are based on those obtained by Tanaka (2012) shown in Fig. 2, the S-wave velocity and density structures are the same as ak135f. The structures are converted to a stack of flat layers with thickness of typically 1 km through earth-flattening approximation (Müller, 1985). The parameters for waveform calculation are the same as in Tanaka (2012), in which the sampling interval is 0.05 s, the frequency range is from 0.05 to 5 Hz, the range of the epicentral distance is from 149 to 160 in 0.2 increments, the focal depth is assumed to be 500 km. Application of the same band-pass filter as used in Tanaka (2012) results in the predominant frequency of approximately 0.5 Hz. To consider scattering, random velocity fluctuations are superimposed on a relevant velocity structure of the inner core. Here three cases are considered: (1) the absolute values of the fluctuations of P-wave velocity, (2) the positive and negative fluctuations of P-wave velocity, and (3) those of S-wave velocity. RESULTS Absolute Values of P-Wave Velocity Fluctuations I first assume that the anomalies of the travel times and amplitudes in the Eastern Hemisphere are caused by many inclusions with higher P-wave velocities in a matrix that has a slower velocity structure like that of the Western Hemisphere. Figure 3a shows the prepared P-wave velocity structures. The velocity Velocity (km/s) 1 000/Q Depth (km) Depth (km) Figure 2. Best-fit velocity structures in the inner core for the Eastern (thick black line) and Western (thick gray line) hemispheres, as inferred by Tanaka (2012). The broken line denotes the average of the P-wave velocity structure in the inner core. Radial structures of 1 000/Q for the Eastern (thick black line) and Western (thick gray line) hemispheres, as inferred by Tanaka (2012). The broken lines denote unreliable parts. fluctuations are generated by using a Gaussian distribution of random numbers with the standard deviations (SD) of 0.2%, 0.4%, 0.6%, 0.8%, 1.0%, 1.5%, and 2.0% at the ICB. The SD values decrease linearly with depth and become zero at 400 km below the ICB. The absolute values of the fluctuations are then finally adopted. They are superimposed on the P-wave velocity structure for the Western Hemisphere shown in Fig. 2a. Figures 3b and 3c present the comparisons of the

4 One-Dimensional Modeling of Multiple Scattering in the Upper Inner Core: Depth Extent of a Scattering Region 709 Velocity (km/s) Depth (km) (c) Figure 3. P-wave velocity profile around the inner core boundary. The positive velocity fluctuations with standard deviations of 0.2%, 0.4%, 0.6%, 0.8%, 1.0%, 1.5%, and 2.0%, which are represented by colors from red to blue, are superimposed on the velocity structure of the Western Hemisphere shown by the black line (same as the gray line in Fig. 2a). Differential travel times of PKP(Cdiff)-PKP(DF) as a function of epicentral distance. The black stars, triangles and circles are the distance bin-averaged observations for the Eastern Hemisphere, whole Earth, and Western Hemisphere, respectively. The colored lines are the differential travel times obtained from the synthetic waveforms for various velocity fluctuations. The colors correspond to the velocity fluctuations represented in. The gray broken lines are the differential travel times from the smoothed velocity structures by Tanaka (2012). (c) PKP(DF)/PKP (Cdiff) amplitude ratios as a function of epicentral distance. The black stars and circles are the distance bin-averaged observations for the Eastern and Western hemispheres, respectively. The colored lines are the amplitude ratios obtained from the synthetic waveforms for various velocity fluctuations. The colors correspond to the velocity fluctuations represented in. The red thick broken line is the theoretical amplitude ratio for the 4-layered Q model for the Eastern Hemisphere obtained by Tanaka (2012). observed differential travel times and amplitude ratios, respectively, with those obtained from the synthetic seismograms. The observed differential travel times of PKP(Cdiff)-PKP(DF) for the Eastern Hemisphere locate at those synthesized from the velocity fluctuations with the SDs of 1.0% and 1.5% at the ICB (Fig. 3b), which means that velocity structure of the Eastern Hemisphere can be explained by the superposition of the positive velocity fluctuations with the SDs of 1.0% to 1.5% on the structure of the Western Hemisphere. However, the PKP(DF)/PKP(Cdiff) amplitude ratios for the Eastern Hemisphere at epicentral distances less than 153 are not explained by those synthesized with any velocity fluctuations considered here (Fig. 3c). Fluctuations of P-Wave Velocity with Various Layer Thickness The above result would be almost equivalent to the case for smaller magnitude of the velocity fluctuations that are concentrated in the uppermost inner core and are superimposed on a fast reference velocity structure. Thus, I next assume that the anomalies in the Eastern Hemisphere are caused by positive and

5 710 negative velocity fluctuations with three different thicknesses of a fluctuated layer on a fast velocity structure like that of the Eastern Hemisphere. Figure 4 shows the prepared P-wave velocity structures. The velocity fluctuations are generated using a Gaussian distribution of random numbers with the SDs of 1%, 2%, 3%, 4%, 5%, 6%, and 7% throughout the uppermost 100 km (Fig. 4a), 200 km (Fig. 4b), and 300 km (Fig. 4c) of the inner core. They are superimposed on a fast P-wave velocity structure, in which P-wave velocity at the ICB is 0.60% faster than the average inner core structure shown in Fig. 2a. The difference between the fast velocity structure and the average decreases linearly, and coincidence with the average inner core structure occurs at 500 km below the ICB. This structure is slightly faster than that of the Eastern Hemisphere shown in Fig. 2a, in which P-wave velocity at the ICB is 0.50% faster than the average and linearly decreases to coincide with the average at 400 km below the ICB. Figure 5 shows comparisons of the observed differential travel times and amplitude ratios with those obtained from the synthetic seismograms for different thickness of the layer having the velocity fluctuations. In general, the larger velocity fluctuation results in slower and smaller PKP(DF). Thus the magnitudes of the RMS velocity fluctuations and the superimposed velocity structure create a trade-off to obtain the best-fit travel times. In the configuration described above, the observed differential travel times of PKP (Cdiff)-PKP(DF) show good coincidence with those synthesized with the SD of 7% for thickness of 100 km (Fig. 5a), the SDs of 5% to 6% for the thickness of 200 km (Fig. 5c), and the SDs of 3% to 4% for the thickness of 300 km (Fig. 5e). However, the reduction of the amplitude ratio for the thicknesses of 100 and 300 km is insufficient at distances from 151 to 152 for all the velocity fluctuations considered here (Figs. 5b, 5f). Furthermore, the reduction for the thickness of 300 km is too great at distances around 155 (Fig. 5f). The observed amplitude ratios for the whole distance range are well explained by those synthesized for the fluctuations with the SDs of 5%, 6%, and 7% and the thickness of 200 km (Fig. 5d). Compared to the amplitude ratios by the 4-layered Q structure, presented by the red broken line, the scattering reproduces better Velocity (km/s) Velocity(km/s) Velocity (km/s) Satoru Tanaka (c) Depth (km) Figure 4. P-wave velocity profiles around the inner core boundary. The positive and negative velocity fluctuations with standard deviations of 1%, 2%, 3%, 4%, 5%, 6%, and 7% in the uppermost 100 km, 200 km, and (c) 300 km of the inner core, represented by colors from red to blue, are superimposed on a tentative reference velocity structure in which the P-velocity at the ICB is 0.60% faster than the average structure and the transition thickness is 500 km (thick black line). The thin gray line is that of the average structure by Tanaka (2012) (same as the thin broken line in Fig. 2a).

6 One-Dimensional Modeling of Multiple Scattering in the Upper Inner Core: Depth Extent of a Scattering Region 711 profiles as a function of epicentral distance, especially for the rapid increase of the amplitude ratios at distances from 151 to 155 (Fig. 5d). Therefore it is likely that the thickness of the layer having velocity fluctuations is 200 km. However, note that some negative peaks of velocity fluctuations for the SD of 5%, 6%, 7% fall below the P-wave velocity virtually extended from that in the outer core, corresponding to the P-wave velocity of liquid iron, which is not realistic. Thickness=100 km Thickness=100 km (c) Thickness=200 km (d) Thickness=200 km (e) Thickness=300 km (f) Thickness=300 km Figure 5. Comparison of the observed differential travel times, (c), and (e) and amplitude ratios, (d), and (f) with those obtained from the synthetic waveforms for the considered thickness of 100 km,, 200 km (c), (d), and 300 km (e), (f). The symbols are the same as in Fig. 3b for, (c), and (e) and as in Fig. 3c for, (d), and (f) except that the positive and negative P-wave velocity fluctuations with standard deviations of 1.0% to 7.0% are represented by colors from red to blue.

7 712 Satoru Tanaka Velocity (km/s) Depth (km) Vp Vs (c) Figure 6. P- and S-wave velocity profiles around the inner core boundary. The positive and negative velocity fluctuations with standard deviations of 2%, 4%, 6%, 8%, 10%, 15%, and 20%, which are represented by colors from red to blue, are superimposed on the S-wave velocity structure (labeled Vs). The black line (labeled Vp) is a tentative reference P-wave velocity profile in which P-wave velocity at the ICB is 0.70% faster than the average structure obtained by Tanaka (2012) and the transition thickness is 400 km. The differential travel times of PKP(Cdiff)-PKP(DF) as a function of epicentral distance. (c) The PKP(DF)/PKP(Cdiff) amplitude ratios as a function of epicentral distance. Symbols in and (c) are the same as in Figs. 3b and 3c except for the meaning of the color for the S-wave velocity fluctuations with standard deviations of 2%, 4%, 6%, 8%, 10%, 15%, and 20%. Fluctuations of S-Wave Velocity I now examine velocity fluctuations in S-wave velocity in the inner core. The fluctuations of S-wave velocity are generated using a Gaussian distribution of random numbers with the SDs of 2%, 4%, 6%, 8%, 10%, 15%, and 20% throughout the uppermost 200 km of the inner core (Fig. 6a). As multiple scattering creates delayed PKIKP phases even though only S-wave velocity is fluctuated, the P-wave velocity in the upper inner core is assumed to be 0.7% faster than the average at the ICB with transition thickness of 400 km (Fig. 6a). This is slightly faster than the best-fit P-wave velocity structure for the Eastern Hemisphere (Fig. 2a). The resultant travel time residuals suggest that the S-wave velocity fluctuations with the SDs of 10% 15% can explain the data (Fig. 6b); however, the reduction of the amplitude ratios is insufficient at distances less than 152 and 153 for the SD of 15% and 10%, respectively, and the reduction is too great at distance greater than 152 for the SD of 15% (Fig. 6c). The velocity fluctuations with the SD of 20% make both the differential travel times and the amplitude ratios much smaller than the observation of the Eastern Hemisphere in the entire distance range considered in this study (Figs. 6b and 6c). Although it is likely that both P- and S-wave velocity fluctuations exist, the effects of S-wave velocity fluctuations would be negligible if the percentages of P- and S-wave velocity fluctuations are almost the same.

8 One-Dimensional Modeling of Multiple Scattering in the Upper Inner Core: Depth Extent of a Scattering Region 713 SD=5% SD=5% (c) (e) SD=6% SD=7% (d) (f) SD=6% SD=7% Figure 7. Comparison of the observed differential travel times, (c), and (e) and amplitude ratios, (d), and (f) with those obtained from the synthetic waveforms for 10 seeds and three standard deviations considered in the generation of the P-wave velocity fluctuations in the uppermost 200 km of the inner core. The velocity structures are considered for fluctuation with the three standard deviations of 5% and, 6% (c) and (d), and 7% (e) and (f). The black stars, triangles and circles are the distance bin-averaged observations for the Eastern Hemisphere, whole Earth, and Western Hemisphere, respectively. The white and gray circles are original data for the Eastern and Western hemispheres, respectively. The red lines are the theoretical curves of the travel times and amplitudes for the seed of 1 appeared in Fig. 5; the other colors correspond to another nine seeds. The gray broken lines in, (c), and (e) are the differential travel times from the smoothed velocity structure model of Tanaka (2012).

9 714 DISCUSSION AND CONCLUDING REMARKS I use pseudorandom numbers generated by a computer in the previous sections. The statistical characteristic of these pseudorandom numbers depends on a calculation algorithm and adopted constants. I use the linear congruential method as a uniform random number generator, which is expressed by r i =mod(αr i 1 +β, m) (1) where r i is the numerical sequence for a pseudorandom number; α, β and m are fixed at 1 229, , and (Watanabe et al., 1989), respectively; r 0 is an initial value called a seed for which I have so far used a value of 1 (Knuth, 1997; Press et al., 1988). Subsequently the Box-Muller method is used for the conversion to a Gaussian distribution (Knuth, 1997; Press et al., 1988). Here I additionally examine the effects of seeds ranging from 1 to 10 on the travel times and amplitudes. The thickness of 200 km is adopted for the fluctuated layer because it is most suitable, as noted in the previous section. The reference velocity structure is the same as those described in the section above. Figure 7 shows the results for the P-wave velocity fluctuations of the SDs of 5% (Figs. 7a and 7b), 6% (Figs. 7c and 7d), and 7% (Figs. 7e and 7f). The differential travel times for the larger SDs generally tend to be reduced especially at shorter epicentral distances, which can be compensated by changing the reference radial velocity structure in the uppermost inner core. The variations in the travel times with the seeds seem to correspond to the range of the actual data distribution in the Eastern Hemisphere (Figs. 7a, 7c, and 7e). Although Tkalčić (2010) suggested that the scatter of PKP(BC)-PKP(DF) travel time data would be partly influenced by heterogeneity at the base of the mantle, my result suggests that a statistical nature of the velocity perturbation in the inner core can also affect the variation in the travel times as well as the magnitudes of the velocity fluctuations. Furthermore, the variations in the amplitude ratios due to the different seeds at distances smaller than 154 are consistent with the range of the actual data distribution in the Eastern Hemisphere (Figs. 7b, 7d and 7f). However, those at distances greater than 155 become narrow, and cannot explain the actual data distribution. In summary, the variation in seeds can explain many aspects of the actual data of the travel times and amplitudes, except for the scattering of the amplitudes Satoru Tanaka that occurs at larger distances. From the 1-D velocity structures that I have examined, a Gaussian distribution of P-wave velocity fluctuations with the standard deviations of 5% 7% in the uppermost 200 km of the inner core likely explains the observed travel times and amplitudes. However, I have not considered the typical autocorrelation functions and power spectral densities of the velocity fluctuation that are usually used in studies of scattering in the crust and lithosphere in which the correlation length is also assigned (Sato and Fehler, 1997). Furthermore, numerical experiments for the distribution of scatterers in the upper mantle suggest that the magnitude of the velocity perturbations in 2-D are smaller (almost half) than that in 1-D experiments to explain the same amount of coda waves (Ryberg et al., 2000; Tittgemeyer et al., 1996). Thus, further work toward constructing the 2-D and three-dimensional velocity structures is required, as is consideration for the statistic nature of scatterer distribution. ACKNOWLEDGMENTS I thank Profs. Sidao Ni, Marc Monnereau, David A Yuen, Vernon Cormier, and Heping Sun for organizing a nice workshop of core dynamics and providing me an opportunity to write this paper. Invaluable comments by two anonymous reviewers and Vernon Cormier greatly improved the manuscript. REFERENCES CITED Braginsky, S. J., Structure of the F Layer and Reasons for Convection in the Earth s Core. Dokl. Akad. Nauk SSSR, 149: 8 10 Calvet, M., Margerin, L., Constraints on Grain Size and Stable Iron Phases in the Uppermost Inner Core from Multiple Scattering Modeling of Seismic Velocity and Attenuation. Earth Planet. Sci. Lett., 267(1 2): Cao, A., Romanowicz, B., Hemispherical Transition of Seismic Attenuation at the Top of the Earth s Inner Core. Earth Planet. Sci. Lett., 228(3 4): Cormier, V. F., Li, X., Choy, G. L., Seismic Attenuation of the Inner Core: Viscoelastic or Stratigraphic. Geophys. Res. Lett., 25(21): Cormier, V. F., Li, X., Frequency-Dependent Seismic Attenuation in the Inner Core 2. A Scattering and Fabric Interpretation. J. Geophys. Res., 107(B12): ESE 14-1 ESE 14-15,

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