Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Wavelet-based multifractal analysis of earthquakes temporal distribution in Mammoth Mountain volcano, Mono County, Eastern California

  • 120 Accesses

  • 4 Citations


This paper presents a wavelet-based multifractal approach to characterize the statistical properties of temporal distribution of the 1982–2012 seismic activity in Mammoth Mountain volcano. The fractal analysis of time-occurrence series of seismicity has been carried out in relation to seismic swarm in association with magmatic intrusion happening beneath the volcano on 4 May 1989. We used the wavelet transform modulus maxima based multifractal formalism to get the multifractal characteristics of seismicity before, during, and after the unrest. The results revealed that the earthquake sequences across the study area show time-scaling features. It is clearly perceived that the multifractal characteristics are not constant in different periods and there are differences among the seismicity sequences. The attributes of singularity spectrum have been utilized to determine the complexity of seismicity for each period. Findings show that the temporal distribution of earthquakes for swarm period was simpler with respect to pre- and post-swarm periods.

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


  1. Arneodo, A., E. Bacry, and J.F. Muzy (1995), The thermodynamics of fractals revisited with wavelets, Physica A 213,1–2, 232–275, DOI: 10.1016/0378-4371(94)00163-N.

  2. Arneodo, A., Y. d’Aubenton-Carafa, E. Bacry, P.V. Graves, J.F. Muzy, and C. Thermes (1996), Wavelet based fractal analysis of DNA sequences, Physica D 96,1–4, 291–320, DOI: 10.1016/0167-2789(96)00029-2.

  3. Arneodo, A., B. Audit, E. Bacry, S. Manneville, J.F. Muzy, and S.G. Roux (1998), Thermodynamics of fractal signals based on wavelet analysis: application to fully developed turbulence data and DNA sequences, Physica A 254,1–2, 24–45, DOI: 10.1016/S0378-4371(98)00002-8.

  4. Arneodo, A., B. Audit, and P. Kestener (2007), Multi-fractal formalism based on the Continuous Wavelet transform, Scholarpedia 3,1–20, DOI: 10.4249/scholarpedia.4103.

  5. Bacry, E., J.F. Muzy, and A. Arneodo (1993), Singularity spectrum of fractal signals from wavelet analysis: exact results, J. Stat. Phys. 70,3–4, 635–674, DOI: 10.1007/BF01053588.

  6. Caruso, F., S. Vinciguerra, V. Latora, A. Rapisarda, and S. Malone (2006), Multifractal analysis of Mount St. Helens seismicity as a tool for identifying eruptive activity, Fractals 14,3, 179–186, DOI: 10.1142/S0218348X06003180.

  7. Castle, R.O., J.E. Estrem, and J.C. Savage (1984), Uplift across Long Valley Caldera, California, J. Geophys. Res. 89,B13, 11507–11516, DOI: 10.1029/JB089iB13p11507.

  8. Christiansen, L.B., S. Hurwitz, M.O. Saar, S.E. Ingebritsen, and P.A. Hsieh (2005), Seasonal seismicity at western United States volcanic centers, Earth Planet. Sci. Lett. 240,2, 307–321, DOI: 10.1016/j.epsl.2005.09.012.

  9. Crovelli, R.A., and C.C. Barton (1995), Fractals and the Pareto distribution applied to petroleum accumulation-size distributions. In: C.C. Barton and P.R. La Pointe (eds.), Fractals in Petroleum Geology and Earth Processes, Plenum Press, New York, 59–72, DOI: 10.1007/978-1-4615-1815-0_4.

  10. Currenti, G., C. Del Negro, and G. Nunnari (2005), Inverse modelling of volcanomagnetic fields using a genetic algorithm technique, Geophys. J. Int. 163,1, 403–418, DOI: 10.1111/j.1365-246x.2005.02730.x.

  11. de Souza, J., and S.P. Rostirolla (2011), A fast MATLAB program to estimate the multifractal spectrum of multidimensional data: Application to fractures, Comput. Geosci. 37,2, 241–249, DOI: 10.1016/j.cageo.2010.09.001.

  12. Enescu, B., K. Ito, and Z.R. Struzik (2006), Wavelet-based multiscale resolution analysis of real and simulated time-series of earthquakes, Geophys. J. Int. 164,1, 63–74, DOI: 10.1111/j.1365-246X.2005.02810.x.

  13. Eneva, M. (1994), Monofractal or multifractal: a case study of spatial distribution of mining-induced seismic activity, Nonlin. Processes Geophys. 1,2/3, 182–190, DOI: 10.5194/npg-1-182-1994.

  14. Falconer, K. (2003), Fractal Geometry. Mathematical Foundations and Applications, 2nd ed., John Wiley & Sons Ltd, Chichester.

  15. Farge, M. (1992), Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24, 359–457, DOI: 10.1146/annurev.fl.24.010192.002143.

  16. Farrar, C.D., M.L. Sorey, W.C. Evans, J.F. Howle, B.D. Kerr, B.M. Kennedy, C.Y. King, and J.R. Southon (1995), Forest-killing diffuse CO2 emission at Mammoth Mountain as a sign of magmatic unrest, Nature 376,6542, 675–678, DOI: 10.1038/376675a0.

  17. Foulger, G.R., B.R. Julian, D.P. Hill, A.M. Pitt, P.E. Malin, and E. Shalev (2004), Non-double-couple microearthquakes at Long Valley caldera, California, provide evidence for hydraulic fracturing, J. Volcanol. Geoth. Res. 132,1, 45–71, DOI: 10.1016/S0377-0273(03)00420-7.

  18. Geilikman, M.B., T.V. Golubeva, and V.F. Pisarenko (1990), Multifractal patterns of seismicity, Earth Planet. Sci. Lett. 99,1–2, 127–132, DOI: 10.1016/0012-821X(90)90076-A.

  19. Goltz, C. (1997), Fractal and Chaotic Properties of Earthquakes, Lecture Notes in Earth Sciences, Vol. 77, Springer, Berlin Heidelberg, DOI: 10.1007/BFb0028316.

  20. Goupillaud, P., A. Grossmann, and J. Morlet (1984), Cycle-octave and related transforms in seismic signal analysis, Geoexploration 23,1, 85–102, DOI: 10.1016/0016-7142(84)90025-5.

  21. Grossmann, A., and J. Morlet (1984), Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15,4, 723–736, DOI: 10.1137/0515056.

  22. Hein, F.J. (1999), Mixed (“multi”) fractal analysis of Granite Wash fields/pools and structural lineaments, Peace River Arch area, northwestern Alberta, Canada; A potential approach for use in hydrocarbon exploration, Bull. Can. Petrol. Geol. 47,4, 556–572.

  23. Hill, D.P., R.A. Bailey, and A.S. Ryall (1985), Active tectonic and magmatic processes beneath Long Valley Caldera, eastern California: An overview, J. Geophys. Res. 90,B13, 11111–11120, DOI: 10.1029/JB090iB13p11111.

  24. Hill, D.P., W.L. Ellsworth, M.J.S. Johnston, J.O. Langbein, D.H. Oppenheimer, A.M. Pitt, P.A. Reasenberg, M.L. Sorey, and S.R. McNutt (1990), The 1989 earthquake swarm beneath Mammoth Mountain, California: An initial look at the 4 May through 30 September activity, Bull. Seismol. Soc. Am. 80,2, 325–339.

  25. Hirabayashi, T., K. Ito, and T. Yoshii (1992), Multifractal analysis of earthquakes, Pure Appl. Geophys. 138,4, 591–610, DOI: 10.1007/BF00876340.

  26. Jaffard, S. (1989), Hölder exponents at given points and wavelet coefficients, C. R. Acad. Sci. Paris Ser. I 308,4, 79–81.

  27. Jaffard, S. (1991), Pointwise smoothness, two-microlocalization and wavelet coefficients, Publ. Mat. 35,1, 155–168, DOI: 10.5565/PUBLMAT_35191_06.

  28. Jaffard, S. (1997a), Multifractal formalism for functions. Part I: Results valid for all functions, SIAM J. Math. Anal. 28,4, 944–970, DOI: 10.1137/S0036141095282991.

  29. Jaffard, S. (1997b), Multifractal formalism for functions. Part II: Self-similar functions, SIAM J. Math. Anal. 28,4, 971–998, DOI: 10.1137/S0036141095283005.

  30. Kagan, Y.Y., and D.D. Jackson (1991), Long-term earthquake clustering, Geophys. J. Int. 104,1, 117–133, DOI: 10.1111/j.1365-246X.1991.tb02498.x.

  31. Kagan, Y., and L. Knopoff (1980), Spatial distribution of earthquakes: the two-point correlation function, Geophys. J. Roy. Astron. Soc. 62,2, 303–320, DOI: 10.1111/j.1365-246X.1980.tb04857.x.

  32. Kulkarni, O.C., R. Vigneshwar, V.K. Jayaraman, and B.D. Kulkarni (2005), Identification of coding and non-coding sequences using local Hölder exponent formalism, Bioinformatics 21,20, 3818–3823, DOI: 10.1093/bioinformatics/bti639.

  33. Langbein, J., D. Dzurisin, G. Marshall, R. Stein, and J. Rundle (1995), Shallow and peripheral volcanic sources of inflation revealed by modeling two-color geodimeter and levelling data from Long Valley Caldera, California, 1988–1992, J. Geophys. Res. 100,B7, 12487–12495, DOI: 10.1029/95JB01052.

  34. Mallat, S., and W.L. Hwang (1992), Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory 38,2, 617–643, DOI: 10.1109/18.119727.

  35. Mallat, S., and S. Zhong (1992), Characterization of signals from multiscale edges, IEEE Trans. Pattern Anal. Mach. Intellig. 14,7, 710–732, 10.1109/34.142909.

  36. Mandelbrot, B.B. (1989), Multifractal measures, especially for the geophysicist, Pure Appl. Geophys. 131,1/2, 5–42, DOI: 10.1007/BF00874478.

  37. Maruyama, F., K. Kai, and H. Morimoto (2011), Wavelet-based multifractal analysis of the El Niño/Southern Oscillation, the Indian Ocean dipole and the North Atlantic Oscillation, SOLA 7, 65–68, DOI: 10.2151/sola.2011-017.

  38. McAteer, R.T.J., C.A. Young, J. Ireland, and P.T. Gallagher (2007), The bursty nature of solar flare X-ray emission, Astrophys. J. 662,1, 691–700, DOI: 10.1086/518086.

  39. McKee, E.H. (1971), Tertiary igneous chronology of the Great Basin of western United States-Implications for tectonic models, Geol. Soc. Am. Bull. 82,12, 3497–3502, DOI: 10.1130/0016-7606(1971)82[3497:TICOTG]2.0.CO;2.

  40. McNutt, S.R. (2002), Volcano seismology and monitoring for eruptions. In: W.H.K. Lee, H. Kanamori, P.C. Jennings, and C. Kisslinger (eds.), International Handbook of Earthquake and Engineering Seismology, Vol. 81, Part A, Academic Press, Massachusetts, 383–406, DOI: 10.1016/S0074-6142(02) 80228-5.

  41. Muzy, J.F., E. Bacry, and A. Arneodo (1991), Wavelets and multifractal formalism for singular signals: Application to turbulence data, Phys. Rev. Lett. 67,25, 3515–3518, DOI: 10.1103/PhysRevLett.67.3515.

  42. Muzy, J.F., E. Bacry, and A. Arneodo (1993), Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method, Phys. Rev. E 47,2, 875–884, DOI: 10.1103/Phys RevE.47.875.

  43. Muzy, J.F., E. Bacry, and A. Arneodo (1994), The multifractal formalism revisited with wavelets, Int. J. Bifurcation Chaos 4,2, 245, DOI: 10.1142/S0218127494000204.

  44. Özger, M. (2011), Investigating the multifractal properties of significant wave height time series using a wavelet-based approach, J. Waterw. Port Coastal Ocean Eng. 137,1, 34–42, DOI: 10.1061/(ASCE)WW.1943-5460.0000062.

  45. Pastén, D., V. Muñoz, A. Cisternas, J. Rogan, and J.A. Valdivia (2011), Monofractal and multifractal analysis of the spatial distribution of earthquakes in the central zone of Chile, Phys. Rev. E 84,6, 66123-1–66123-11, DOI: 10.1103/PhysRevE.84.066123.

  46. Pitt, A.M., and D.P. Hill (1994), Long-period earthquakes in the Long Valley Caldera region, eastern California, Geophys. Res. Lett. 21,16, 1679–1682, DOI: 10.1029/94GL01371.

  47. Prejean, S., A. Stork, W. Ellsworth, D. Hill, and B. Julian (2003), High precision earthquake locations reveal seismogenic structure beneath Mammoth Mountain, California, Geophys. Res. Lett. 30,24, 2247, DOI: 10.1029/2003GL018334.

  48. Roux, S., J.F. Muzy, and A. Arneodo (1999), Detecting vorticity filaments using wavelet analysis: About the statistical contribution of vorticity filaments to intermittency in swirling turbulent flows, Eur. Phys. J. B 8,2, 301–322, DOI: 10.1007/s100510050694.

  49. Sadovskiy, M.A., T.V. Golubeva, V.F. Pisarenko, and M.G. Shnirman (1984), Characteristic dimensions of rock and hierarchical properties of seismicity, Izv. — Phys. Solid Earth 20, 87–95.

  50. Smalley Jr., R.F., J.-L. Chatelain, D.L. Turcotte, and R. Prévot (1987), A fractal approach to the clustering of earthquakes: Applications to the seismicity of the New Hebrides, Bull. Seismol. Soc. Am. 77,4, 1368–1381.

  51. Sorey, M.L., W.C. Evans, B.M. Kennedy, J. Rogie, and A. Cook (1999), Magmatic gas emissions from Mammoth Mountain, Mono County, California, Calif. Geol. 52,5, 4–16.

  52. Stanley, H.E., L.A.N. Amaral, A.L. Goldberger, S. Havlin, P.C. Ivanov, and C.-K. Peng (1999), Statistical physics and physiology: Monofractal and multifractal approaches, Physica A 270,1–2, 309–324, DOI: 10.1016/S0378-4371(99)00230-7.

  53. Telesca, L., V. Lapenna, and M. Macchiato (2004), Mono- and multi-fractal investigation of scaling properties in temporal patterns of seismic sequences, Chaos Soliton Fract. 19,1, 1–15, DOI: 10.1016/S0960-0779(03)00188-7.

  54. Toledo, B.A., A.C.L. Chian, E.L. Rempel, R.A. Miranda, P.R. Muñoz, and J.A. Valdivia (2013), Wavelet-based multifractal analysis of nonlinear time series: The earthquake-driven tsunami of 27 February 2010 in Chile, Phys. Rev. E 87,2, 22821-1–22821-11, DOI: 10.1103/PhysRevE.87.022821.

  55. Turcotte, D.L. (1992), Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge.

  56. Venugopal, V., S.G. Roux, E. Foufoula-Georgiou, and A. Arneodo (2006), Revisiting multifractality of high-resolution temporal rainfall using a waveletbased formalism, Water Resour. Res. 42,6, W06D14, DOI: 10.1029/2005WR004489.

  57. Vicsek, T. (1992), Fractal Growth Phenomena, 2nd ed., World Scientific Publ., Singapore.

  58. Wiemer, S. (2001), A software package to analyze seismicity: ZMAP, Seismol. Res. Lett. 72,3, 373–382, DOI: 10.1785/gssrl.72.3.373.

  59. Wiemer, S., S.R. McNutt, and M. Wyss (1998), Temporal and three-dimensional spatial analysis of the frequency-magnitude distribution near Long Valley Caldera, California, Geophys. J. Int. 134,2, 409–421, DOI: 10.1046/j.1365-246x.1998.00561.x.

  60. Zamani, A., and M. Agh-Atabai (2009), Temporal characteristics of seismicity in the Alborz and Zagros regions of Iran, using multifractal approach, J. Geodyn. 47,5, 271–279, DOI: 10.1016/j.jog.2009.01.003.

  61. Zamani, A., and M. Agh-Atabai (2011), Multifractal analysis of the spatial distribution of earthquake epicentres in the Zagross and Alborz-Kopeh Dagh regions of Iran, Iran. J. Sci. Technol. A1, 39–51.

  62. Zamani, A., J. Samiee, and J.F. Kirby (2013), Estimating the mechanical anisotropy of the Iranian lithosphere using the wavelet coherence method, Tectonophysics 601, 139–147, DOI: 10.1016/j.tecto.2013.05.005.

Download references

Author information

Correspondence to Amir Pirouz Kolahi Azar.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zamani, A., Kolahi Azar, A.P. & Safavi, A.A. Wavelet-based multifractal analysis of earthquakes temporal distribution in Mammoth Mountain volcano, Mono County, Eastern California. Acta Geophys. 62, 585–607 (2014). https://doi.org/10.2478/s11600-013-0184-3

Download citation

Key words

  • seismic hazards
  • natural disasters
  • earthquake swarm
  • volcano-tectonic seismicity
  • wavelet transform