Skip to main content
SpringerLink
Log in

Fractal dimension modeling of seismology and earthquakes dynamics

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this study, we constructed the seismic wave equation in fractal dimensions based on the concept of product-like fractal measure introduced recently by Li and Ostoja-Starzewski in their formulation of anisotropic media. The solutions of the fractal seismic wave have proved the effect of fractal dimensions on the propagation of this type of wave in anisotropic media. The plane wave solutions were proved to be affected and deformed by fractal dimensions. It was observed that earthquakes in fractal dimension are characterized by a modified total energy which may be used to estimate the range of the fractal dimension. In particular, slow earthquakes with fractal dimension \(\alpha = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\) are characterized by a particular total energy \(E_{\alpha }\) which is \(E_{\alpha } \approx 10E_{c}\), \(E_{c}\) being the conventional total energy. Several additional points were discussed and analyzed.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. West, G.B.: The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Sci. 284, 1677–1679 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Varga, B.E., Gao, W., Laurik, K.L., Tatrai, E., Simo, M., Somfai, G.M., Cabrera DeBuc, D.: Investigating tissue optical properties and texture descriptors of the retina in patients with multiple sclerosis. PLoS ONE 10(11), e0143711 (2015)

    Article  Google Scholar 

  3. Ivanova, V. S., Bunin, I, J., Nosenko, V. I.: Fractal material science: a new direction in materials science, JOM50, 52–54. (1998)

  4. Carpinteri, A.: Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech. Mater. 18, 89–101 (1994)

    Article  Google Scholar 

  5. Agrisuelas, J., García-Jareño, J.J., Gimenez-Romero, D., Negrete, F., Vicente, F.: The fractal dimension as estimator of the fractional content of metal matrix composite materials. J. Solid State Electrochem. 13, 1599–1603 (2009)

    Article  Google Scholar 

  6. Balankin, A.S., Bugrimov, A.L.: A fractal theory of polymer plasticity. Polym. Sci. USSR 34, 246 (1992)

    Google Scholar 

  7. Balankin, A.S., Bugrimov, A.L.: Fractal theory of elasticity and rubber-like state of polymers. Polymer Sci. 34, 889 (1992)

    Google Scholar 

  8. Balankin, A.S., Tamayo, P.: Fractal solid mechanics. Rev. Mex. Phys. 40, 506 (1994)

    Google Scholar 

  9. Balankin, A.S.: Elastic behavior of materials with multifractal structure. Phys. Rev. B 53, 5438 (1996)

    Article  Google Scholar 

  10. Balankin, A.S.: The theory of multifractal elasticity: basic laws and constitutive equations. Rev. Mex. Phys. 42, 343 (1996)

    MathSciNet  MATH  Google Scholar 

  11. El-Nabulsi, R.A.: Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J. Stat. Phys. 172, 1617–1640 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. El-Nabulsi, R.A.: Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic Carrier concentrations. J. Phys. Chem. Sol. 127, 224–230 (2019)

    Article  Google Scholar 

  13. El-Nabulsi, R.A.: On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc. Roy. Soc. A476, 20190729 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. El-Nabulsi, R.A.: Inverse-power potentials with positive-bound energy spectrum from fractal, extended uncertainty principle and position-dependent mass arguments. Eur. Phys. J. P135, 683 (2020)

    Google Scholar 

  15. El-Nabulsi, R.A.: Dirac equation with position-dependent mass and Coulomb-like field in Hausdorff dimension. Few Body Syst. 61, 10 (2020)

    Article  Google Scholar 

  16. El-Nabulsi, R.A.: On generalized fractional spin, fractional angular momentum, fractional momentum operators and noncommutativity in quantum mechanics. Few Body Syst. 61, 1–13 (2020)

    Article  Google Scholar 

  17. Avron, J.E., Simon, B.: Almost periodic Hill’s equation and the rings of Saturn. Phys. Rev. Lett. 46, 1166–1168 (1981)

    Article  MathSciNet  Google Scholar 

  18. Lindner, J.F., Kohar, V., Kia, B., Hippke, M., Learned, J.G., Ditto, W.L.: Strange nonchaotic stars. Phys. Rev. Lett. 114(5), 054101 (2015)

    Article  Google Scholar 

  19. Falconer, K.J.: Fractal Geometry-Mathematical Foundations and Applications. Wiley, New York (2003)

    Book  MATH  Google Scholar 

  20. Mandelbrot, B.B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1983)

    Book  Google Scholar 

  21. Jin, Y., Wu, Y., Li, H., Zhao, M., Pan, J.: Definition of fractal topography to essential understanding of scale-invariance. Sci. Rep. 7, 46672 (2017)

    Article  Google Scholar 

  22. Mainieri, R.: On the equality of Hausdorff and box counting dimensions, https://arxiv.org/abs/chao-dyn/9303007

  23. Schonwetter, M.: Fractal Dimensions in Classical and Quantum Mechanical Open Chaotic Systems, PhD Thesis, Angefertigt in der Arbeitsgruppe Dynamical Systems and Social Dynamics am Max-Planck-Institut fur Physik komplexer Systeme in Dresden. (2016)

  24. Chen, Y.: Equivalent relation between normalized spatial entropy and fractal dimension. Phys. A: Stat. Mech. Appl. 553, 124627 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bouda, M., Caplan, J.S., Saiers, J.E.: Box-counting dimension revisited: presenting an efficient method of minimizing quantization error and an assessment of the self-similarity of structural root systems. Front. Plant. Sci. 7, 149 (2016)

    Article  Google Scholar 

  26. Davey, K., Prosser, R.: Analytical solutions for heat transfer on fractal and pre-fractal domains. Appl. Math. Mod. 37, 554–569 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tarasov, V.E.: Continuous medium model for fractal media. Phys. Lett. A 336, 167–174 (2005)

    Article  MATH  Google Scholar 

  28. Tarasov, V.E.: Fractional hydrodynamic equations for fractal media. Ann. Phys. 318(2), 286–307 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Collins, J.C.: Renormalization. Cambridge University Press, Cambridge (1984)

    Book  MATH  Google Scholar 

  30. Demmie, P.N., Ostoja-Starzewski, M.: Waves in fractal media. J. Elasticity 104, 187 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ostoja-Starzewski, M.: Towards thermoelasticity of fractal media. J. Therm. Stress 30, 889 (2007)

    Article  MATH  Google Scholar 

  32. Ostoja-Starzewski, M., Li, J.: Towards thermoelasticity of fractal media. Z. Angew. Math. Phys. 60, 1 (2009)

    MathSciNet  Google Scholar 

  33. Li, J., Ostoja-Starzewski, M.: Micropolar continuum mechanics of fractal media. Int. J. Eng. Sci. 49, 1302 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ostoja-Starzewski, M.: Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech. 205, 161–170 (2009)

    Article  MATH  Google Scholar 

  35. Ostoja-Starzewski, M.: On turbulence in fractal porous media. Z. Angew. Math. Phys. 59(6), 1111–1117 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, J., Ostoja-Starzewski, M.: Fractal materials, beams and fracture mechanics. Z. Angew. Math. Phys. 60, 1–12 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, London (2009)

    Book  MATH  Google Scholar 

  38. Li, J., Ostoja-Starzewski, M.: Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A Math. Phys. Eng. Sci. 465, 2521 (2009). https://doi.org/10.1098/rspa.2010.0491

    Article  MathSciNet  MATH  Google Scholar 

  39. Ostoja-Starzewski, M., Li, J., Joumaa, H., Demmie, P.N.: From fractal media to continuum mechanics. Z. Angew. Math. Mech. 93, 1 (2013)

    MATH  Google Scholar 

  40. Li, J., Ostoja-Starzewski, M.: Fractal solids, product measures and continuum mechanics. In: Maugin, G.A., Metrikine, A.V. (eds.) Mechanics of Generalized Continua One Hundred Years after the Cosserats, pp. 315–323. Springer, Berlin (2010)

    Chapter  Google Scholar 

  41. El-Nabulsi, R.A.: Thermal transport equations in porous media from product-like fractal measure. J. Therm. Stress. 44, 899–912 (2021)

    Article  Google Scholar 

  42. El-Nabulsi, R.A.: Superconductivity and nucleation from fractal anisotropy and product-like fractal measure. Proc. Roy. Soc. A477, 20210065 (2021)

    Article  MathSciNet  Google Scholar 

  43. El-Nabulsi, R.A.: Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal geometry. Phys. E: Low Dim. Syst. Nanostruct. 134, 114827 (2021)

    Article  Google Scholar 

  44. El-Nabulsi, R.A.: On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc. R. Soc. A476, 20190729 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. El-Nabulsi, R.A.: Quantization of Foster mesoscopic circuit and DC-pumped Josephson parametric amplifier from fractal measure arguments. Phys. E: Low Dim. Syst. Nanostruct. 133, 114845 (2021)

    Article  Google Scholar 

  46. El-Nabulsi, R.A.: Position-dependent mass fractal Schrödinger equation from fractal anisotropy and product-like fractal measure and its implications in quantum dots and nanocrystals. Opt. Quant. Elect. 53, 503 (2021)

    Article  Google Scholar 

  47. El-Nabulsi, R.A.: Fractal neutrons diffusion equation: uniformization of heat and fuel burn-up in nuclear reactor. Nucl. Eng. Des. 380, 111312 (2021)

    Article  Google Scholar 

  48. El-Nabulsi, R.A.: Fractal Pennes and Cattaneo-Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth. J. R. Soc. Interface 18, 20210564 (2021)

    Article  Google Scholar 

  49. El-Nabulsi, R.A., Anukool, W.: A mapping from Schrödinger equation to Navier-Stokes equations through the product-like fractal geometry, fractal time derivative operator and variable thermal conductivity. Acta Mech. 232, 5031–5039 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  50. El-Nabulsi, R.A., Anukool, W.: Quantum dots and cuboid quantum wells in fractal dimensions with position-dependent masses. Appl. Phys. A 127, 856 (2021)

    Article  Google Scholar 

  51. Malyarenko, A., Ostoja-Starzewski, M.: Fractal planetary rings: energy inequalities and random field model. Int. J. Mod. Phys. B 31, 1750236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. Mashayekhi, S., Miles, P., Hussaini, M.Y., Oates, W.S.: Fractional viscoelasticity in fractal and non-fractal media: theory, experimental validation, and uncertainty analysis. J. Mech. Phys. Solids 111, 134–156 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mashayekhi, S., Hussaini, M.Y., Oates, W.S.: A physical interpretation of fractional viscoelasticity based on the fractal structure of media: theory and experimental validation. J. Mech. Phys. Solids 128, 137–150 (2019)

    Article  MathSciNet  Google Scholar 

  54. Mashayekhi, S., Beerli, P.: Fractional coalescent. Proc. Nat. Acad. Sci. 116, 6244–6249 (2019)

    Article  Google Scholar 

  55. Mashayekhi, S., Sedaghat, S.: Fractional model of stem cell population dynamics. Chaos Solitons Fractals 146, 110919 (2021)

    Article  MathSciNet  Google Scholar 

  56. Oates, W., Stanisaukis, E., Pahari, B. R., Mashayekhi, S.: Entropy dynamics approach to fractional order mechanics with applications to elastomers. Behavior and Mechanics of Multifunctional Materials XV 11589, 1158905. (2021)

  57. El-Nabulsi, R.A.: Some geometrical aspects of nonconservative autonomous Hamiltonian dynamical systems. Int. J. Appl. Math. Stat. 5, 50–61 (2006)

    MathSciNet  MATH  Google Scholar 

  58. El-Nabulsi, R.A., Wu, G.-C.: Fractional complexified field theory from Saxena-Kumbhat fractional integral, fractional derivative of order () and dynamical fractional integral exponent. Afr. Diasp. J. Math. 13, 56–61 (2012)

    MathSciNet  MATH  Google Scholar 

  59. El-Nabulsi, R.A., Torres, D.F.M.: Fractional actionlike variational problems. J. Math. Phys. 49, 053521 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  60. El-Nabulsi, R.A.: Nonlinear wave equations from a non-local complex backward-forward derivative operator. Waves Comp. Rand. Med. 31, 1433–1442 (2021)

    Article  MathSciNet  Google Scholar 

  61. Yu, L., Zou, Z.: The Fractal Dimensionality of Seismic Wave. In: Yuan, Y., Cui, J., Mang, H.A. (eds.) Computational Structural Engineering. Springer, Dordrecht (2009). https://doi.org/10.1007/978-90-481-2822-8_33

    Chapter  Google Scholar 

  62. Olami, Z., Feder, H.J.S., Christensen, K.: Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett. 68, 1244 (1992)

    Article  Google Scholar 

  63. Angulo-Brown, F., Ramirez-Guzman, A.H., Yepez, E., Rudoif-Navarro, A., Pavia-Miller, C.G.: Fractal geometry and seismicity in the Mexican Subduction Zone. Geofisica Internat. 37, 29–33 (1998)

    Google Scholar 

  64. Sunmonu, L.A., Dimri, V.P.: Fractal geometry and Seismicity of Koyna-Warna, India. Pure Appl. Geophys. 157, 1393 (2000)

    Article  Google Scholar 

  65. Idziak, A., Teper, L.: Fractal dimension of faults network in the upper Silesian coal basin (Poland): preliminary studies. Pure Appl. Geophys. 147, 239 (1996)

    Article  Google Scholar 

  66. Sukmono, S., Zen, M.T., Kadir, W.G.A., Hendrajjya, L., Santoso, D., Dubois, J.: Fractal pattern of the Sumatra active fault system and its geodynamical implications. J. Geodynam. 22, 1 (1996)

    Article  Google Scholar 

  67. Sukmono, S., Zen, M.T., Hendrajjya, L., Kadir, W.G.A., Santoso, D., Dubois, J.: Fractal pattern of the Sumatra fault seismicity and its application to earthquake prediction. Bull. Seismol. Soc. Amer. 87, 1685 (1997)

    Google Scholar 

  68. Roy, P.N.S., Ram, A.: Fractal dimensions of blocks using a box-counting technique for the 2001 Bhuj Earthquake, Gujarat, India. Pure Appl. Geophys. 162, 531 (2005)

    Article  Google Scholar 

  69. Nanjo, K., Nagahama, H.: Spatial distribution of aftershocks and the fractal structure of active fault systems. Pure Appl. Geophys. 157, 575 (2000)

    Article  Google Scholar 

  70. Chakrabarti, B.K., Stinchcombe, R.B.: Stick-slip statistics for two fractal surfaces: a model for earthquakes. Phys. A 270, 27 (1999)

    Article  Google Scholar 

  71. Bhattacharya, P., Chakrabarti, B.K., Kamal, S.D.: Fractal models of earthquake dynamics. In: Schuster, H.G. (ed.) Reviews of Nonlinear Dynamics and Complexity, pp. 107–158. Wiley - VCH Verlag GmbH & Co. KGaA, Weinheim (2009)

    Google Scholar 

  72. Bhattacharya, P., Chakrabarti, B. K., Kamal, S.D.: A fractal model of earthquake occurrence: Theory, simulations and comparisons with the aftershock data, J. Phys.: Conf. Ser. 319, 012004, (2011)

  73. Chen, C.-C., Wang, W.-C., Chang, Y.-F., Wu, Y.-M., Lee, Y.-H.: A correlation between the b-value and the fractal dimension from the aftershock sequence of the 1999 Chi-Chi, Taiwan, earthquake. Geophys. J. Int. 167, 1215–1219 (2006)

    Article  Google Scholar 

  74. Henderson, J.R., Barton, D.J., Foulger, G.R.: Fractal clustering of induced seismicity in the Geysers geothermal area, California. Geophys. J. Int. 139, 317–324 (1999)

    Article  Google Scholar 

  75. Hirata, T.: A correlation between the b-value and the fractal dimension of earthquakes. J. Geophys. Res. 94, 7507–7514 (1989)

    Article  Google Scholar 

  76. Legrand, D.: Fractal dimensions of small, intermediate, and large earthquakes. Bull. Seism. Soc. Am. 92, 3318–3320 (2002)

    Article  Google Scholar 

  77. Mandal, P., Mabawonku, A.O., Dimri, V.P.: Self-organized fractal seismicity of reservoir triggered earthquakes in the Koyna-Warna seismic zone, western India. Pure Appl. Geophys. 162, 73–90 (2005)

    Article  Google Scholar 

  78. Yin, L., Li, X., Zheng, W., Yin, Z., Song, L., Ge, L., Zeng, Q.: Fractal dimension analysis for seismicity spatial and temporal distribution in the circum-Pacific seismic belt. J. Earth Syst. Sci. 128, 22 (2019)

    Article  Google Scholar 

  79. Nakaya, S.: Fractal properties of seismicity in regions affected by large, shallow earthquakes in western Japan: Implications for fault formation processes based on a binary fractal fracture network model. J. Geophys. Res. Sol Earth 110, 1–15 (2005)

    Article  Google Scholar 

  80. Wang, J.H., Lee, C.W.: Multifractal measures of earthquakes in west Taiwan. Pure Appl. Geophys. 146, 131–145 (1996)

    Article  Google Scholar 

  81. Chen, S.J., David, H., Ma, L., Wang, L.F.: Research on the multifractal characteristics of the temporal-spatial distribution of earthquakes over New Zealand area. Acta Seismol. Sin. 16, 312–322 (2003)

    Article  Google Scholar 

  82. Pastén, D., Muñoz, V., Cisternas, A., Rogan, J., Valdivia, J.A.: Monofractal and multifractal analysis of the spatial distribution of earthquakes in the central zone of Chile. Phys. Rev. E 84, 066123 (2011)

    Article  Google Scholar 

  83. Hui, C., Cheng, C., Ning, L., Yang, J.: Multifractal characteristics of seismogenic systems and b values in the Taiwan seismic region. Int. J. Geo-Inform. 9, 384 (2020)

    Article  Google Scholar 

  84. Balcerak, E.: Seismic quiescence before the 2003 Tokachi-oki earthquake. Eos Trans. Am. Geophys. Union 93, 16 (2012)

    Google Scholar 

  85. Dimri, V.P.: Fractals in Geophysics and Seismology: An Introduction. In: Dimri, V.P. (ed.) Fractal Behaviour of the Earth System. Springer, Berlin, Heidelberg (2005). https://doi.org/10.1007/3-540-26536-8_1

    Chapter  Google Scholar 

  86. Gutenberg, B., Richter, C.F.: Frequency of earthquakes in California. Bull. Seismol. Soc. Am. 34, 185–188 (1944)

    Article  Google Scholar 

  87. Dahlen, F.A., Tromp, J.: Theoretical Global Seismology. Princeton, New Jersey, Princeton University Press, USA (1998)

    Google Scholar 

  88. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  89. Zimbardo, G., Malara, F., Perri, S.: Energetic particle superdiffusion in solar system plasmas: which fractional transport equation? Symmetry 13, 2638 (2021)

    Article  Google Scholar 

  90. Failla, G., Zingales, M.: Advances materials modeling via fractional calculus: challenges and perspectives. Phil. Trans. A: Math. Phys. Eng. Sci. 378, 20200050 (2020)

    Article  Google Scholar 

  91. Caputo, M.: Methods of fluids in porous media with memory. Water Res. Res. 36, 693–705 (2000)

    Article  Google Scholar 

  92. Ge, J., Everett, M.E., Weiss, C.J.: Fractional diffusion analysis of the electromagnetic field in fractured media-Part 2; 3D approach. Geophys. 80, E175–E185 (2015)

    Article  Google Scholar 

  93. Weiss, C.J., von Bloemen Waanders, B.G., Antil, H.: Fractional operators applied to geophysical electromagnetics. Geophys. J. Int. 220, 1242–1259 (2020)

    Google Scholar 

  94. Balankin, A.S.: Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems. Chaos Solitons Fractals 132, 109572 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  95. Tarasov, V.E.: Wave equation for fractal solid string. Mod. Phys. Lett. B 19(15), 721–728 (2005)

    Article  MATH  Google Scholar 

  96. Carpinteri, A., Mainardi, F.: Fractals and fractional calculus in continuum mechanics. Springer, Wien (1997). https://doi.org/10.1007/978-3-7091-2664-6

    Book  MATH  Google Scholar 

  97. S. Sidhardh, S. Patnaik, F. Semperlotti, Thermodynamics of fractional-order nonlocal continua and its application to the thermoelastic response of beams, http://arxiv.org/abs/2003.10215

  98. Cusimano, N., Bueno-Orovio, A., Turner, I., Burrage, K.: On the order of the fractional Laplacian in determining the spatio- temporal evolution of a space-fractional model of cardiac electrophysiology. PLoS ONE 10, e0143938 (2015)

    Article  Google Scholar 

  99. Ostoja-Starzewski, M.: Electromagnetism on anisotropic fractal media. Z. Angew. Math. Phys. 64, 381–390 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  100. Billingham, J., King, A.C.: Wave Motion. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

  101. Bedford, A., Drumheller, D.S.: Introduction to Elastic Wave Propagation, 1st edn. Wiley, New York (1994)

    MATH  Google Scholar 

  102. Aster, R.: The Seismic Eave Equation. Lectures given at Colorado State University, Warner College of Natural Resources (2011)

    Google Scholar 

  103. Abramowitz, M., Stegun, I.A. (eds.), Exponential Integral and Related Functions, Ch. 5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 227–233 (1972)

  104. Cerveny, V., Psencik, I.: Time-averaged and time-dependent energy-related quantities of harmonic waves in inhomogeneous viscoelastic anisotropic media. Geophys. J. Int. 170, 1253–1261 (2007)

    Article  Google Scholar 

  105. Bai, Q., Bao, W.-B., Jin, S.-J.: Estimation of time-dependent power spectral density of seismic wave based on wavelet transform. J. Shenyang Univ. Tech. 32, 342–348 (2010)

    Google Scholar 

  106. Masuda, K., Ide, S., Ohta, K., Matsuzawa, T.: Bridging the gap between low-frequency and very-low-frequency earthquakes. Earth Planets Space 72, 47 (2020)

    Article  Google Scholar 

  107. Kinjo, A., Nakamura, M.: Low-frequency earthquakes along the Rkykyu islands triggered by teleseismic earthquakes. Earth Planets Space 73, 118 (2021)

    Article  Google Scholar 

  108. Molchan, G., Kronrod, T.: The fractal description of seismicity. Geophys. J. Int. 179, 1787–1799 (2009)

    Article  Google Scholar 

  109. Rundle, J.B., Turcotte, D.L., Shcherbakov, R., Klein, W., Sammis, C.: Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems. Rev. Geophys. 41, 1019 (2003)

    Article  Google Scholar 

  110. Abe, Y., Kato, N.: Complex earthquake cycle simulations using a two-degree-of-freedom spring-block model with a rate- and state-friction law. Pure Appl. Geophys. 170, 745–765 (2013)

    Article  Google Scholar 

  111. Huang, J., Turcotte, D.L.: Evidence for chaotic fault interactions in the seismicity of the San Andreas fault and Nankai trough. Nature 348, 234–236 (1990)

    Article  Google Scholar 

  112. Huang, J., Turcotte, D.L.: Chaotic seismic faulting with a mass-spring model and velocity-weakening friction. Pure Appl. Geophys. 138, 569–589 (1992)

    Article  Google Scholar 

  113. Goebel, T.H.W., Kwiatek, G., Becker, T.W., Brodsky, E.E., Dresen, G.: What allows seismic events to grow big?: Insights from b-value and fault roughness analysis in laboratory stick-slip experiments. Geology 45, 815–818 (2017)

    Article  Google Scholar 

  114. Scholz, C.H.: Earthquakes and friction laws. Nature 391, 37–42 (1998)

    Article  Google Scholar 

  115. Dixiong, Y., Changgeng, Z.: Fractal characterization and frequency properties of near-fault ground motions. Earthq. Eng. Eng. Vib. 12, 503–518 (2013)

    Article  Google Scholar 

  116. Huang, J., Turcotte, D.L.: Are earthquakes an example of deterministic chaos. Geophys. Res. Lett. 17, 223–226 (1990)

    Article  Google Scholar 

  117. Keith, C.M., Crampin, S.: Seismic body waves in anisotropic media: synthetic seismograms. Geophys. J. Int. 49, 225–243 (1977)

    Article  Google Scholar 

  118. Fryer, G.J., Neil Frazer, L.: Seismic waves in stratified anisotropic media. Geophys. J. R. Astro. Soc. 78, 691–710 (1984)

    Article  MATH  Google Scholar 

  119. El-Nabulsi, R.A., Anukool, W.: Fractal dimensions in fluid mechanics and their effects on the Rayleigh problem, the Burger’s vortex and the Kelvin-Helmholtz instability. Acta Mech. 233, 363–381 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  120. Chakraborty, S., Chattopadhyay, S.: Exploring the Indian summer monsoon rainfall through multifractal detrended fluctuation analysis and the principle of entropy maximization. Earth Sci. Inform. 14, 1571–1577 (2021)

    Article  Google Scholar 

  121. Chattopadhyay, G., Chattopadhyay, S., Midya, S.K.: Fuzzy binary relation based elucidation of air quality over a highly polluted urban region of India. Earth Sci. Inform. 14, 1625–1631 (2021)

    Article  Google Scholar 

  122. Chakraborty, S., Chattopadhyay, S.: A probe into the behaviour of total ozone time series through multifractal detrended fluctuation analysis. Theor. Appl. Climat. (2022). https://doi.org/10.1007/s00704-022-03967-z

    Article  Google Scholar 

  123. Chattopadhyay, G., Chattopadhyay, S.: A probe into the chaotic nature of total ozone time series by correlation dimension method. Soft Comput. 12, 1007–1012 (2008)

    Article  Google Scholar 

  124. Chakraborty, S., Chattopadhyay, S.: A time-domain approach to the total ozone time series and a test of its predictability within a univariate framework. Remote Sens. Lett. 12, 20–29 (2021)

    Article  Google Scholar 

  125. El-Nabulsi, R.A.: Geostrophic flow and wind-driven ocean currents depending on the spatial dimensionality of the medium. Pure Appl. Geophys. 176, 2739–2750 (2019)

    Article  Google Scholar 

  126. El-Nabulsi, R.A., Anukool, W.: Ocean-atmosphere dynamics and Rossby waves in fractal anisotropic media. Meteor. Atmosph. Phys. (2022). https://doi.org/10.1007/s00703-022-00867-9

    Article  MATH  Google Scholar 

  127. Petrov, L.B., Golubev, V.I., Petrukhin, VYu., Nikitin, I.S.: Simulation of seismic waves in anisotropic media. Doklady Math. 103, 146–150 (2021)

    Article  MATH  Google Scholar 

  128. Pailoplee, S., Choowong, M.: Earthquake frequency-magnitude distribution and fractal dimension in mainland Southeast Asia. Earth Planets Space 66, 8 (2014)

    Article  Google Scholar 

  129. Lenhardt, W.A.: Fractal Concepts and their Applications to Earthquakes in Austria. In: Lehner, F.K., Urai, J.L. (eds.) Aspects of Tectonic Faulting. Springer, Berlin, Heidelberg (2000). https://doi.org/10.1007/978-3-642-59617-9_4

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their useful comments and valuable suggestions.

Funding

The authors would like to thank Chiang Mai University for funding this research.

Author information

Authors and Affiliations

  1. Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  2. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  3. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

Authors
  1. Rami Ahmad El-Nabulsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Waranont Anukool
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rami Ahmad El-Nabulsi.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The authors confirm the absence of sharing data.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Nabulsi, R.A., Anukool, W. Fractal dimension modeling of seismology and earthquakes dynamics. Acta Mech 233, 2107–2122 (2022). https://doi.org/10.1007/s00707-022-03213-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03213-7

Search

Navigation