Detection of Earth’s free oscillation and analysis of the non-synchronous oscillation phenomenon of normal modes

Abstract

Earth’s free oscillation can provide essential constraints for refining Earth models, inverting seismic source mechanisms, and studying the deep internal structure of the Earth. Large earthquakes can simultaneously excite numerous normal modes. Due to the Earth’s ellipticity, rotation, and internal heterogeneities, these normal modes undergo splitting, with the frequencies of singlets of normal modes becoming very close (only a few µHz apart). This imposes greater demands on the detection of normal modes. This paper introduces a novel method for normal mode detection based on the normal time–frequency transform (NTFT). Compared to classical FT spectrum methods and recent optimal sequence estimation (OSE), the proposed method not only detects more weak normal modes but also reveals the spatial distribution of the phase of each normal mode. Taking the detection of ₀S₂ as an example, the phase measurements of each singlet are spatially inconsistent. This phenomenon can provide prior information for other methods, such as product spectrum analysis (PSA), spherical harmonic stacking (SHS), multistation experiments (MSE), and OSE. Additionally, understanding the phase distribution patterns contributes to further study of geological structures, offering crucial foundational data and observational support.

Keywords

Normal time–frequency transform / Phase consistency / Normal modes / Mode splitting / Mode detection

References

L. Auer, L. Boschi, T.W. Becker, T. Nissen-Meyer, D. Giardini. Savani: A variable resolution whole-mantle model of anisotropic shear velocity variations based on multiple data sets. J. Geophys. Res. Solid Earth, 119 (4) (2014), pp. 3006-3034

D. Banka, D. Crossley. Noise levels of superconducting gravimeters at seismic frequencies. Geophys. J. Int., 139 (1) (1999), pp. 87-97

Boy, J. P., Barriot, J. P., Förste, C., Voigt, C., Wziontek, H., 2020. Achievements of the first 4 years of the International Geodynamics and Earth Tide Service (IGETS) 2015–2019. In Beyond 100: The Next Century in Geodesy. Proceedings of the IAG General Assembly, Montreal, Canada, 107-112.

J.P. Boy, S. Rosat, J. Hinderer, F. Littel. Superconducting gravimeter data from Strasbourg—level 1. GFZ Data Services (2017)

R. Buland, J. Berger, F. Gilbert. Observations from the IDA network of attenuation and splitting during a recent earthquake. Nature, 277 (5695) (1979), pp. 358-362

N. Courtier, B. Ducarme, J. Goodkind, J. Hinderer, Y. Imanishi, N. Seama, H. Sun, J. Merriam, B. Bengert, D.E. Smylie. Global superconducting gravimeter observations and the search for the translational modes of the inner core. Phys. Earth Planet. Inter., 117 (1–4) (2000), pp. 3-20

F.A. Dahlen. The normal modes of a rotating, elliptical Earth. Geophys. J. Int., 16 (4) (1968), pp. 329-367

F.A. Dahlen. The normal modes of a rotating, elliptical earth—II near-resonance multiplet coupling. Geophys. J. Int., 18 (4) (1969), pp. 397-436

A. Deuss, J. Ritsema, H. Van Heijst. Splitting function measurements for Earth's longest period normal modes using recent large earthquakes. Geophys. Res. Lett., 38 (4) (2011), p. L04303

A. Deuss, J. Ritsema, H. Van Heijst. A new catalogue of normal-mode splitting function measurements up to 10 mHz. Geophys. J. Int., 193 (2) (2013), pp. 920-937

H. Ding, B.F. Chao. Data stacking methods for the isolation of normal-mode singlets of Earth's free oscillation, extensions, comparisons, and applications. J. Geophys. Res.: Solid Earth, 120 (7) (2015), pp. 5034-5050

H. Ding, W.B. Shen. Search for the Slichter modes based on a new method: optimal sequence estimation. J. Geophys. Res.: Solid Earth, 118 (9) (2013), pp. 5018-5029

A.M. Dziewonski, D.L. Anderson. Preliminary reference Earth model. Phys. Earth Planet. Inter., 25 (4) (1981), pp. 297-356

R.J. Geller, S. Stein. Time-domain attenuation measurements for fundamental spheroidal modes (0S6 to 0S28) for the 1977 Indonesian earthquake. Bull. Seismolog. Soc. Am., 69 (6) (1979), pp. 1671-1691

F. Gilbert. The diagonal sum rule and averaged eigenfrequencies. Geophys. J. Int., 23 (1) (1971), pp. 119-123

F. Gilbert, A.M. Dziewonski. An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra. Philosophical Transactions of The Royal Society A Math. Phys Eng. Sci., 278 (1280) (1975), pp. 187-269

P. Koelemeijer, J. Ritsema, A. Deuss, H.J. Van Heijst. SP12RTS: A degree-12 model of shear- and compressional-wave velocity for Earth's mantle. Geophys. J. Int., 204 (2) (2016), pp. 1024-1039

L. Liu, H. Hsu. Inversion and normalization of time-frequency transform. 2011 International Conference on Multimedia Technology, (2011), pp. 2164-2168

L. Liu, H. Hsu. Inversion and normalization of time-frequency transform. Appl. Math. Inform. Sci., 6 (67) (2012), pp. 67-74

G. Masters, G. Laske, F. Gilbert. Matrix autoregressive analysis of free-oscillation coupling and splitting. Geophys. J. Int., 143 (2) (2000), pp. 478-489

C.S. Pedapudi, M.R. Katlamudi, S. Rosat. Observation of free oscillations after the 2010 Chile and 2011 Japan earthquakes by superconducting gravimeter in Kutch, Gujarat. India. Geodesy Geodynam., 14 (1) (2023), pp. 52-64

D.E. Smylie. The inner core translational triplet and the density near Earth’s center. Science, 255 (5052) (1992), pp. 1678-1682

S. Stein, E.A. Okal. Speed and size of the Sumatra earthquake. Nature, 434 (7033) (2005), pp. 581-582

X.Q. Su. Normal time-frequency transform theory and its applications in Earth science.. Ph.D. thesis, University of Chinese Academy of Sciences (2014), pp. 69-71

Van Camp, M., 1999. Measuring seismic normal modes with the GWR C021 superconducting gravimeter. Phys. Earth Planet. Interi. 116(1-4), 81-92.

R. Widmer-Schnidrig. Application of regionalized multiplet stripping to retrieval of aspherical structure constraints. Geophys. J. Int., 148 (2) (2002), pp. 201-213

J.H. Woodhouse, F.A. Dahlen. The effect of a general aspherical perturbation on the free oscillations of the Earth. Geophys. J. Royal Astron. Soc., 53 (2) (1978), pp. 335-354

Zhang, L. Y., 2018. Theoretical Simulation and Applications of Earth’s free oscillations. Ph.D. thesis, University of Chinese Academy of Sciences, 3 pp.