Abstract
The multi-fractal scaling properties of seismograms are investigated in order to quantify the complexity associated with high-frequency seismic signals. The third-order MDFA (MDFA3) method is capable of characterising the multi-fractality of earthquake records associated with frequency- and scale-dependent correlations of small and large fluctuations within seismogram. These correlations are related to changes in waveform properties and hence are a measure of the heterogeneities of the medium at different scales, sensed by direct and converted phases in a seismogram with different amplitudes and phases. The non-linear dependence of generalised Hurst and mass exponent with order q confirms the multi-fractal nature of earthquake records. Amongst different types of earthquakes analysed, the multi-fractal properties are more pronounced for signals with distinct P-, S- and coda waves. The multi-fractal singularity spectrum parameters (maximum, asymmetry and width) are used to measure the frequency-dependent complexity of seismograms. The degree of multi-fractality decreases with increasing frequency, and is generally more for the time period windowing dominant seismic phases in the seismogram. Significant difference in spectrum width between the original record and its randomly shuffled surrogates demonstrates that the multi-fractality in earthquake records is predominantly due to long-range correlation of small and large fluctuations within seismogram, although its origin due to broad probability distribution cannot be completely ruled out, based on the values of scaling exponent (H q ≈ 0.5) and their weak q-dependency for the surrogates.
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References
Ashkenazy Y, Ivanov PC, Havlin S, Peng CK, Goldberger AL, Stanley HE (2001) Magnitude and sign correlations in heart beat fluctuations. Phys Rev Lett 86:1900–1903
Ashkenazy Y, Havlin S, Ivanov PC, Peng CK, Schulte-Frohlinde V, Stanley HE (2003) Magnitude and sign scaling in power-law correlated time-series. Phys A 323:19–41
Buldyrev SV, Goldberger AL, Havlin S, Mantegna RN, Matsa ME, Peng CK, Simons M, Stanley HE (1995) Long-range correlation properties of coding and non-coding DNA sequences: GenBank analysis, Phys. Rev. E 51:5084–5091
Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter JH, Voigt K (2000) Corrrelated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736–3739
Chen Z, Ivanov PC, Hu K, Stanley HE (2002) Effect of non-stationarities on detrended fluctuation analysis. Phys Rev E 65:041107
Dimri VP (2005) Fractals in geophysics and seismology: an introduction. In: Dimri VP (ed) Fractal behaviour of the earth system. Springer, Berlin, pp 1–19
Godano C, Alonzo ML, Bottari A (1996) Multifractal analysis of the spatial distribution of earthquakes in southern Italy. Geophys J Int 125:901–911
Hirata T, Imoto M (1991) Multi-fractal analysis of spatial distribution of micro-earthquakes In the Kanto region. Geophys J Int 107:155–162
Hu K, Ivanov PC, Chen Z, Carpens P, Stanley HE (2001) Effects of trends on detrended fluctuation analysis. Phys Rev E 64:011114
Kagan YY, Knopoff L (1980) Spatial distribution of earthquakes: the two-point correlation function. Geophys J R Astron Soc 62:303–320
Kantelhardt JW, Zschiegner SA, Bunde EK, Havlin S, Bunde A, Stanley HE (2002) Multi-fractal detrended fluctuation analysis of non-stationary time series. Phys A 316:87–114
Kantelhardt JW, Konscienly-Bunde E, Rego HHA, Havlin S, Bunde A (2001) Detecting long-range correlations with detrended fluctuation analysis. Physica A 295:441–454
Kantelhardt JW, Rybski D, Zschiegner SA, Braun P, Bunde EK, Livina V et al (2003) Multifractality of river runoff and precipitation: comparison of fluctuation analysis and wavelet methods. Phys A 330:240–245
Lan T-H, Gao ZY, Abdalla Ahmed N, Cheng B, Wang S (2008) Detrended fluctuation analysis as a statistical method to study ion single channel signal. Cell Biol Int 32:247–252
Main IG, Burton PW (1984) Information theory and the earthquake-magnitude distribution. Bull Seismol Soc Am 74:1409–1426
Padhy S, Mishra OP, Subhadra N, Dimri VP, Singh OP, Chakrabortty GK (2014) Effects of errors and biases on the scaling of earthquake spatial pattern: application to the 2004 Sumatra-Andaman sequence. Nat Hazards 1–22
Paladin G, Vulpiani A (1987) Anomalous scaling laws in multifractal objects. Phys Rep 156:147
Rundle JB (1989) Derivation of the complete Gutenberg-Richter frequency-magnitude relation using the principle of scale invariance. J Geophys Res 94:12337–12342
Scafetta N, Griffin L, West BJ (2003) Hölder exponent spectra for human gait. Phys A 328:561–583
Shang P, Lu Y, Kamae S (2008) Detecting long-range correlations of traffic time series with multifractal detrended fluctuation analysis. Chaos Solitons Fractals 36:82–90
Shimizu Y, Thurner S, Ehrenberger K (2002) Multifractal spectra as a measure of complexity in human posture. Fractals 10:103–116
Smalley RF, Chatelain JL, Turcotte DL, Prevot R (1987) A fractal approach to the clustering of earthquakes: applications to the seismicity of the New Hebrides. Bull Seismol Soc Am 77:1368–1381
Sornette D (2004) Critical phenomena in natural sciences, 2nd edn. Springer, Berlin
Talkner P, Weber RO (2000) Power spectrum and detrended fluctuation analysis: application to daily temperatures. Phys Rev E 62:150–160
Tang YJ, Chang YF, Liou TS, Chen CC, Wu YM (2012) Evolution of the temporal multi-fractal scaling properties of the Chiayi earthquake (ML = 6.4), Taiwan. Tectonophysics 546–547:1–9
Telesca L, Cuomo V, Lapenna V, Macchiato M (2001) Identifying space-time clustering properties of the 1983–1997 Irpinia-Basilicata (southern Italy) seismicity. Tectonophysics 330:93–102
Telesca L, Lapenna V, Macchiato M (2004) Mono- and multi-fractal investigation of scaling properties in temporal patterns of seismic sequences. Chaos Solitons Fractals 19:1–15
Telesca L, Lovallo M, Chamoli A, Dimri VP, Srivastava K (2013) Fisher-Shannon analysis of seismograms of tsunamigenic and non-tsunamigenic earthquakes. Phys A 392:3424–3429
Telesca L, Lovallo M, Alcaz V, Ilies I (2014a) Investigating the inner time properties of seismograms by using the Fisher information measure. Phys A 409:154–161
Telesca L, Lovallo M, Martì Molist J, Lopez Moreno C, Abella Mendelez R (2014b) Using the Fisher-Shannon method to characterize continuous seismic signal during volcanic eruptions: application to 2011–2012 El Hierro (Canary Islands) eruption. Terra Nova 26:425–429
Telesca L, Lovallo M, Molist JM, Moreno CL, Melendez RA (2015) Multi-fractal investigation of continuous seismic signal recorded at El Hierro volcano (Canary Islands) during the 2011–2012 pre- and eruptive phases. Tectonophysics 642:71–77
Vjushin D, Govindan RB, Monetti RA, Havlin S, Bunde A (2001) Scaling analysis of trends using DFA. Phys A 302:234–243
Yuan Y, Zhuang X-T, Jin X (2009) Measuring multi-fractality of stock price fluctuation using multi-fractal detrended fluctuation analysis. Phys A 388:2189–2197
Acknowledgements
The author (SP) sincerely thanks anonymous reviewers and Prof. Vijay P. Dimri for their helpful review that improved the clarity of this work. SP acknowledges Prof. Dimri for his kind invitation to contribute this work as one of the chapters of the book to be published by Springer. The Director, CSIR-NGRI is thanked for his kind permission to publish this work.
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Padhy, S. (2016). The Multi-fractal Scaling Behavior of Seismograms Based on the Detrended Fluctuation Analysis. In: Dimri, V. (eds) Fractal Solutions for Understanding Complex Systems in Earth Sciences. Springer Earth System Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-24675-8_7
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