Skip to main content
SpringerLink
Log in

Improvement of robustness and stability in estimating Rayleigh–Schuster’s hodograph parameters using different procedures of vector normalization

  • Published:
Seismic Instruments Aims and scope Submit manuscript

Abstract

This paper deals with Rayleigh–Schuster’s hodographs, intended for the detailed investigation of changes in the phase of quasiperiodic signals in time series. The hodographs are also known as the phasor-walkout method. A procedure of conditional vector normalization is proposed: it takes into account the vector amplitude for each period under consideration. The procedure considerably improves the robustness and stability of the hodograph approach to changes in the character of the processed data distributions and to various defects in the data. For example, when analyzing the earthquake catalog, the procedure strongly diminishes the influence of the event clustering caused, in particular, by swarms of earthquakes with comparable magnitude and the aftershock sequences of strong earthquakes. At the first stage, we calculate the vector sums (resulting vectors) for each period under investigation throughout the time series duration. For example, investigating diurnal periodicity of earthquakes, we first calculate the resulting vectors for each day of the observation. The further analysis of resulting vectors for each period throughout the time series duration can be performed with different procedures. We compare three procedures for normalization of the obtained resulting vectors, which are as follows: (1) th traditional one, preserving the real signal amplitude; (2) that with reduction of the obtained resulting vectors to the unit vector (phasor); and (3) that with conditional vector normalization, taking into account the amplitude of resulting vectors for each period throughout the time series duration. The third procedure diminishes the possible instability in some special distributions of the investigated data when the resulting vector for a period is close to zero. The procedures are compared using model signals and samples from real earthquake catalogs. All the procedures used give close results when processing random time series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Agayan, S.M., Bogoutdinov, Sh.R., Gvishiani, A.D., Graeva, E.M., Zlotniki, J., and Rodkin, M.V., A study of signal morphology on the basis of fuzzy logic algorithm, Geofiz. Issled., 2005, no. 1, pp. 143–155.

    Google Scholar 

  • Atlas vremennykh variatsii prirodnykh, antropogennykh i sotsial'nykh protsessov, vol. 1: Poryadok i khaos v litosfere i drugikh sferakh (Atlas of Temporal Variations in Natural, Anthropogenic, and Social Processes, vol. 1: Order and Chaos in the Lithosphere and Other Geospheres), Moscow: OIFZ RAN, 1994.

  • Atlas vremennykh variatsii prirodnykh, antropogennykh i sotsial'nykh protsessov, vol. 2: Tsiklicheskaya dinamika v prirode i obshchestve (Atlas of Temporal Variations in Natural, Anthropogenic, and Social Processes, vol. 2: Cyclic Dynamics I Nature and Society), Moscow: Nauchn. mir, 1998.

  • Atlas vremennykh variatsii prirodnykh, antropogennykh i sotsial'nykh protsessov, vol. 3: Prirodnye i sotsial'nye sfery kak chasti okruzhayushchei sredy i kak ob”ekty vozdeistvii (Atlas of Temporal Variations in Natural, Anthropogenic, and Social Processes, vol. 3: Natural and Social Media as Parts of the Environment and as Affected Objects), Moscow: Yanus-K, 2002.

  • Atlas vremennykh variatsii prirodnykh, antropogennykh i sotsial'nykh protsessov, vol. 4: Chelovek i tri okruzhayushchie ego sredy (Atlas of Temporal Variations in Natural, Anthropogenic, and Social Processes, vol. 4: Human and Three Media around Him), Moscow: OOO Svetoch Plyus, 2009.

  • Atlas vremennykh variatsii prirodnykh, antropogennykh i sotsial'nykh protsessov, vol. 5: Chelovek i tri okruzhayushchie ego sredy (Atlas of Temporal Variations in Natural, Anthropogenic, and Social Processes, vol. 5: Human and Three Media around Him), Moscow: Yanus-K, 2013.

  • Bachelier, L., Théorie de la Spéculation, Ann. Sci. Ec. Norm. Super. Ser., 1900, vol. 3, no. 17, pp. 21–86.

    Google Scholar 

  • Bartels, J., Random fluctuations, persistence, and quasipersistence in geophysical and cosmical periodicities, Terr. Magn. Atmos. Electr., 1935, vol. 40, no. 1, pp. 1–60.

    Article  Google Scholar 

  • Beeler, N. and Lockner, D., Why earthquakes correlate weakly with the solid Earth tides: Effects of periodic stress on the rate and probability of earthquake occurrence, J. Geophys. Res.: Solid Earth, 2003, vol. 108, no. B8. doi 10.1029/2001JB001518

    Google Scholar 

  • Bettinelli, P., Avouac, J.-P., Flouzat, M., Bollinger, L., Ramillien, G., Rajaure, S., and Sapkota, S., Seasonal variations of seismicity and geodetic strain in the Himalaya induced by surface hydrology, Earth Planet. Sci. Lett., 2008, vol. 266, pp. 332–344.

    Article  Google Scholar 

  • Cadicheanu, N., Van Ruymbeke, M., and Zhu, P., Tidal triggering evidence of intermediate depth earthquakes in the Vrancea zone (Romania), Nat. Hazards Earth Syst. Sci., 2007, vol. 7, no. 6, pp. 733–740.

    Article  Google Scholar 

  • Cochran, E.S., Vidale, J.E., and Tanaka, S., Earth tides can trigger shallow thrust fault earthquakes, Science, 2004, vol. 306, pp. 1164–1166.

    Article  Google Scholar 

  • Einstein, A., Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys., 1905, vol. 322, no. 8, pp. 549–560.

    Article  Google Scholar 

  • Fisher, N.I., Statistical Analysis of Circular Data, Cambridge: Cambridge Univ. Press, 1996.

    Google Scholar 

  • Gamburtsev, A.G., Aleksandrov, S.I., Oleinik, O.V., Gamburtseva, N.G., Ivanov-Kholodnyi, G.S., Novikov, A.M., Kutsenko, V.V., Novoselova, V.A., Lagunov, S.I., and Korsak, M.N., Comprehensive environmental monitoring and the Atlas of Temporal Variations, in Atlas vremennykh variatsii prirodnykh, antropogennykh i sotsial'nykh protsessov, vol. 3: Prirodnye i sotsial'nye sfery kak chasti okruzhayushchei sredy i kak ob”ekty vozdeistvii (Atlas of Temporal Variations in Natural, Anthropogenic, and Social Processes, vol. 3: Natural and Social Media as Parts of the Environment and as Affected Objects), Moscow: Yanus-K, 2002, pp. 160–169.

    Google Scholar 

  • Hartzell, S. and Heaton, T., The fortnightly tide and the tidal triggering of earthquakes, Bull. Seismol. Soc. Am., 1989, vol. 79, N 4, pp. 1282–1286.

    Google Scholar 

  • Hass, L., Ellis, M., and Rydelek, P.A., Minimum magnitude of completeness and rates of seismicity in the New Madrid region, Seismol. Res. Lett., 1992, vol. 63, pp. 395–405.

    Google Scholar 

  • Heaton, T.H., Tidal triggering of earthquakes, Geophys. J. R. Astr. Soc., 1975, vol. 43, pp. 307–326.

    Article  Google Scholar 

  • Heaton, T.H., Tidal triggering of earthquakes, Bull. Seismol. Soc. Am., 1982, vol. 72, no. 6, pp. 2181–2200.

    Google Scholar 

  • Kansowa, T. and Tatnall, A.R.L., A possible link between Earth tides and earthquakes, J. Environ. Sci., 2010, vol. 39. http://eprints.soton.ac.uk/153169

  • Lander, A.V., Levshin, A.L., and Pisarenko, V.F., on the spectral-temporal analysis of variations, Vychislit. Seismol., 1975, vol. 6, pp. 336–349.

    Google Scholar 

  • Li, Q. and Xu, G.-M., Tidal triggering of earthquakes in Longmen Shan region: The relation to the fold belt and basin structures, Earth, Planets Space, 2012, vol. 64, pp. 771–776. doi: 10.5047/eps.2011.06.037

    Article  Google Scholar 

  • Li, Q. and Xu, G.-M., Precursory pattern of tidal triggering of earthquakes in six regions of China: The possible relation to the crustal heterogeneity, Nat. Hazards Earth Syst. Sci., 2013, vol. 13, pp. 2605–2618. doi: 10.5194/nhess-13-2605

    Article  Google Scholar 

  • Lindberg, C.R. and Park, J., Multiple-taper spectral analy-2013 sis of terrestrial free oscillations: Part II, Geophys. J. R. Astron. Soc., 1987, vol. 91, no. 3, pp. 795–836.

    Article  Google Scholar 

  • Lockner, D. and Beeler, N., Premonitory slip and tidal triggering of earthquakes, J. Geophys. Res.: Solid Earth, 1999, vol. 104, no. B9, pp. 20133–20151. doi 10.1029/1999JB900205

    Article  Google Scholar 

  • Mardia, K.V., Statistics of Directional Data, London: Academic Press, 1972.

  • Mardia, K.V. and Jupp, P.E., Directional Statistics, Chichester: Wiley, 2000.

    Google Scholar 

  • McMurry, H., Periodicity of deep-focus earthquakes, Bull. Seismol. Soc. Am., 1941, vol. 31, no. 1, pp. 33–59.

    Google Scholar 

  • Park, J., Lindberg, C.R., and Thomson, D.J., Multiple taper spectral analysis of terrestrial free oscillations: Part I, Geophys. J. R. Astron. Soc., 1987a, vol. 91, no. 3, pp. 755–794.

    Article  Google Scholar 

  • Park, J., Lindberg, C.R., and Vernon III, F.L., Multitaper spectral analysis of high frequency seismograms, J. Geophys. Res., B, 1987b, vol. 92, no. 12, pp. 12675–12784.

    Article  Google Scholar 

  • Pearson, K., The problem of the random walk, Nature, 1905, vol. 72, no. 1865, p. 294.

    Article  Google Scholar 

  • Pearson, K. and Blakeman J., Mathematical Contributions to the Theory of Evolution—XV. A Mathematical Theory of Random Walk, London: London Univ., 1906.

    Google Scholar 

  • Rayleigh, Lord, On the result of a large number of vibrations of the same pitch and arbitrary phase, Philos. Mag., Ser. 5, 1880, vol. 10, pp. 73–78.

    Article  Google Scholar 

  • Rayleigh, Lord, On the problem of random vibrations and of random flights in one, two or three dimensions, Philos. Mag., Ser. 6, 1919a, vol. 37, pp. 321–347.

    Article  Google Scholar 

  • Rayleigh, Lord, On the resultant of a number of unit vibrations whose phases are at random over a range not limited to an integral number of periods, Philos. Mag., Ser. 6, 1919b, vol. 37, pp. 498–515.

    Article  Google Scholar 

  • Rydelek, P. and Hass, L., On estimating the amount of blasts in seismic catalogs with Schuster's method, Bull. Seismol. Soc. Am., 1994, vol. 84, no. 4, pp. 1256–1259.

    Google Scholar 

  • Rydelek, P.A. and Sacks, I.S., Comment on “Seismicity and detection/location threshold in the Southern Great Basin seismic network” by Joan Gomberg, J. Geophys. Res.: Solid Earth, 1992, vol. 97, no. B11, pp. 15361–15362. doi: 10.1029/92JB00604

    Article  Google Scholar 

  • Rydelek, P.A. and Sacks, I.S., Testing the completeness of earthquake catalogues and the hypothesis of self-similarity, Nature, 1989, vol. 337, pp. 251–253. doi: 10.1038/337251a0.

    Article  Google Scholar 

  • Rydelek, P.A., Sacks, I.S., and Scarpa, R., On tidal triggering of earthquakes at Campi Flegrei, Italy, Geophys. J. Int., 1992, vol. 109, pp. 125–137.

    Article  Google Scholar 

  • Savage, H.M. and Marone, C., Effects Effects of shear velocity oscillations on stick-slip behavior in laboratory experiments, J. Geophys. Res.: Solid Earth, 2007, 112, B02301. doi: 10.1029/2005JB004238.

    Google Scholar 

  • Schuster, A., On lunar and solar periodicities of earthquakes, Proc. R. Soc. London, 1897, vol. 61, pp. 455–465.

    Article  Google Scholar 

  • Shirley, J.H., Lunar periodicity in great earthquakes, 1950–1965, Gerlands Beitr. Geophys., 1986, vol. 95, pp. 509–515.

    Google Scholar 

  • Shlien, S., Earthquake-tide correlation, Geophys. J. R. Astron. Soc., 1972, vol. 28, pp. 27–34.

    Article  Google Scholar 

  • Sidorin, A.Ya., Are the Earth's resistivity seasonal variations linked to changes in seismic rate?, Geofiz. Issled., 2005a, no. 2, pp. 83–92.

    Google Scholar 

  • Sidorin, A.Ya., Midday effect in the time series of earthquakes and seismic noise, Dokl. Earth Sci., 2005b, vol. 403, no. 5, pp. 771–776.

    Google Scholar 

  • Sidorin, A.Ya., A new insight into the problem of annual periodicity of the electric resistivity of the Earth's crust in earthquake prediction research, Seism. Instrum., 2007, vol. 43, no. 1, pp. 11–20. doi 10.3103/S0747923907010021

    Article  Google Scholar 

  • Sidorin, A.Ya., On application of the Rayleigh–Schuster method in studies of the periodicity of earthquakes, Seism. Prib., 2009, vol. 45, no. 3, pp. 29–40.

    Google Scholar 

  • Sidorin, A.Ya., Diurnal periodicity of earthquakes and its seasonal variations, Seism. Instrum., 2010, vol. 46, no. 3, pp. 293–305. doi 10.3103/S0747923910030114

    Article  Google Scholar 

  • Sidorin, A.Ya., Relationship between the parameters of diurnal periodicity phase diagrams and recurrence graph for the earthquakes in the Gram Test Area, Vopr. Inzh. Seismol., 2011, vol. 38, no. 4, pp. 59–82.

    Google Scholar 

  • Sidorin, A.Ya., On the causes of unordinary periodicity of earthquakes, Seism. Instrum., 2012a, vol. 48, no. 2, pp. 196–207. doi 10.3103/S0747923912020090

    Article  Google Scholar 

  • Sidorin, A.Ya., Changes in the diurnal periodicity of Southern California earthquakes in the neighborhood of the catalog magnitude of completeness, Seism. Instrum., 2012b, vol. 48, no. 4, pp. 342–350. doi 10.3103/S0747923912040044

    Article  Google Scholar 

  • Sidorin, A.Ya., Nonuniform intra-annual distribution of strong earthquakes in the Garm region, Izv., Atmos. Ocean. Phys., 2013a, vol. 40, no. 7, pp. 780–784. doi 10.1134/S0001433813070050

    Google Scholar 

  • Sidorin, A.Ya., Differences in diurnal phase diagrams of series of earthquakes with different energies, Seism. Prib., 2013b, vol. 49, no. 2, pp. 71–84.

    Google Scholar 

  • Sidorin, A.Ya., Influence of aftershocks on parameters of diurnal periodicity of earthquakes, Vopr. Inzh. Seismol., 2013c, vol. 40, no. 2, pp. 68–78.

    Google Scholar 

  • Sidorin, A.Ya., Features of the diurnal periodicity of earthquakes in Japan, Seism. Instrum., 2014, vol. 50, no. 3, pp. 246–267. doi 10.3103/S0747923914030098

    Article  Google Scholar 

  • Sidorin, A.Ya., A weekend effect in the seismicity of Japan, Izv., Atmos. Oceanic Phys., 2015a, vol. 50, no. 8, pp. 824–838. doi 10.1134/S0001433814080076

    Article  Google Scholar 

  • Sidorin, A.Ya., Search for technogenic effects in the earthquake catalog of the Garm Test Area, Geofiz. Protsessy Biosfera, 2015b, vol. 14, no. 1, pp. 80–94.

    Google Scholar 

  • Smoluchowski, M., von, Unregelmassigkeiten in der Verteilung vop Gasmolekeln und deren Einlus auf Entropie und Zustandg-Gleichung, in Boltzmann–Festschrift, Leipzig, 1904, pp. 626–633.

    Google Scholar 

  • Smoluchowski, M., von, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. Phys., 1906, vol. 326, no. 14, pp. 756–780.

    Article  Google Scholar 

  • Stroup, D.F., Bohnenstiehl, D.R., Tolstoy, M., Waldhauser, F., and Weekly, R.T., Pulse of the seafloor: Tidal triggering of microearthquakes at 9°50' N East Pacific Rise, Geophys. Res. Lett., 2007, vol. 34, p. L15301. doi 10.1029/2007GL030088

    Article  Google Scholar 

  • Tanaka, S., Ohtake, M., and Sato, H., Evidence for tidal triggering of earthquakes as revealed from statistical analysis of global data, J. Geophys. Res.: Solid Earth, 2002, vol. 107, no. B10, p. 2211. doi 10.1029/ 2001JB001577

    Google Scholar 

  • Tanaka, S., Sato, H., Matsumura, S., and Ohtake, M., Tidal triggering of earthquakes in the subducting Philippine Sea plate beneath the locked zone of the plate interface in the Tokai region, Japan, Tectonophysics, 2006, vol. 417, pp. 69–80.

    Article  Google Scholar 

  • Tsuruoka, H., Ohtake, M., and Sato, H., Statistical test of the tidal triggering of earthquakes: Contribution of the ocean tide loading effect, Geophys. J. Int., 1995, vol. 122, pp. 183–194.

    Article  Google Scholar 

  • Ulbrich, U., Ahorner, L., and Ebel, A., Statistical investigations on diurnal and annual periodicity and on tidal triggering of local earthquakes in Central Europe, J. Geophys., 1987, vol. 61, pp. 150–157.

    Google Scholar 

  • Wilcock, W.S.D., Tidal triggering of microearthquakes on the Juan de Fuca Ridge, Geophys. Res. Lett., 2001, vol. 28, pp. 3999–4002.

    Article  Google Scholar 

  • Young, D. and Zürn, W., Tidal triggering of earthquakes in Swabian Jura?, J. Geophys., 1979, vol. 45, N 2, pp. 171–182.

    Google Scholar 

  • Zürn, W. and Rydelek, P., Revisiting the phasor-walkout method for detailed investigation of harmonic signals in time series, Surv. Geophys., 1994, vol. 15, no. 4, pp. 409–431.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, str. 1, Moscow, 123242, Russia

    A. V. Desherevskii & A. Ya. Sidorin

Authors
  1. A. V. Desherevskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Ya. Sidorin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Desherevskii.

Additional information

Original Russian Text © A.V. Desherevskii, A.Ya. Sidorin, 2015, published in Seismicheskie Pribory, 2015, Vol. 51, No. 2, pp. 56–80.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Desherevskii, A.V., Sidorin, A.Y. Improvement of robustness and stability in estimating Rayleigh–Schuster’s hodograph parameters using different procedures of vector normalization. Seism. Instr. 52, 79–97 (2016). https://doi.org/10.3103/S0747923916010047

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0747923916010047

Keywords

Search

Navigation