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Fractals in Geophysics and Seismology: An Introduction

  • V.P. Dimri

Summary

Many aspects of nature are very much complex to understand and this has started a new science of geometrical complexity, known as ‘Fractal Geometry’. Various studies carried out across the globe reveal that many of the Earth’s processes satisfy fractal statistics, where examples range from the frequency-size statistics of earthquakes to the time series of the Earth’s magnetic field. The scaling property of fractal signal is very much appealing for descriptions of many geological features. Based on well-log measurements, Earth’s physical properties have been found to exhibit fractal behaviour. Many authors have incorporated this finding in various geophysical techniques to improve their interpretive utility. The aim of present chapter is to briefly discuss the fractal behaviour of the Earth system and the underlying mechanism by citing some examples from potential field and seismology.

Keywords

Tsunami Wave Geophysical Survey Aftershock Sequence Multifractal Analysis Self Organize Criticality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Abe K (1995) Modelling of runup heights of Hokkaido-Nansei-Oki tsunami of 12July 1993. Pure Appl Geophys 145:735–745Google Scholar
  2. Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A 38: 364–374CrossRefGoogle Scholar
  3. Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: An explanation of 1/f noise. Phys Rev Lett 59: 381–384CrossRefGoogle Scholar
  4. Bansal AR, Dimri VP (1999) Gravity evidence for mid crustal structure below Delhi fold belt and Bhilwara super group of western India. Geophys Res Lett 26: 2793–2795CrossRefGoogle Scholar
  5. Bansal AR, Dimri VP (2001) Depth estimation from the scaling power spectral density of nonstationary gravity profile. Pure Appl Geophys 158: 799–812Google Scholar
  6. Burrough MS, Tebbens SF (2005) Power-law scaling and probalistic forecasting of tsunami runup heights. Pure Appl Geophys 162:331–342Google Scholar
  7. Choi BH, Pelinovsky E, Ryabov I, Hong SJ (2002) Distribution functions of Tsunami wave heights. Natural Hazards 25:1–21CrossRefGoogle Scholar
  8. Christensen K, Danon L, Scanlon T, Bak P (2002) Unified scaling law for earthquakes. Proc Natl Acad Sci USA 99:2509–2513CrossRefGoogle Scholar
  9. Constable SC, Parker RL, Constable CG (1987) Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics 52: 289–300CrossRefGoogle Scholar
  10. De Rubies V, Dimitriu P, Papa Dimtriu E, Tosi P (1993) Recurrent patterns in the spatial behavior of Italian seismicity revealed by the fractal approach. Geophys Res Lett 20:1911–1914Google Scholar
  11. Dimri VP (1992) Deconvolution and inverse theory, Elsevier Science Publishers, Amsterdam London New York TokyoGoogle Scholar
  12. Dimri VP (1998) Fractal behavior and detectibility limits of geophysical surveys. Geophysics 63:1943–1946CrossRefGoogle Scholar
  13. Dimri VP (2000) Application of fractals in earth science. AA Balkema, USA and Oxford IBH Publishing Co New DelhiGoogle Scholar
  14. Dimri VP (2005) Tsunami wave modeling using fractals and finite element technique for irregular coastline and uneven bathymetries. Natl workshop on formulation of science plan for coastal hazard preparedness (18–19 Feb) NIO, Goa, IndiaGoogle Scholar
  15. Dimri VP, Nimisha V, Chattopadhyay S (2005) Fractal analysis of aftershock sequence of Bhuj earthquake — a wavelet based approach. Curr Sci (in press)Google Scholar
  16. Fedi M (2003) Global and local multiscale analysis of magnetic susceptibility data. Pure Appl Geophys 160: 2399–2417CrossRefGoogle Scholar
  17. Fedi M, Quarta T, Santis AD (1997) Inherent power-law behavior of magnetic field power spectra from a Spector and Grant ensemble. Geophysics 62:1143–1150Google Scholar
  18. Frisch U, Parisi G (1985) Fully developed turbulence and intermittency. In: Ghill M (ed) Turbulence and predictability in geophysical fluid dynamics and climate dynamics, North Holland, Amsterdam, pp 84Google Scholar
  19. Grant FS (1985) Aeromagnetics, geology and ore environments 1. Magnetite in igneous, sedimentary and metamorphic rocks: An overview. Geoexploration 23:303–333Google Scholar
  20. Gregotski ME, Jensen OG, Arkani-Hamed J (1991) Fractal stochastic modeling of aeromagnetic data. Geophysics 56:1706–1715CrossRefGoogle Scholar
  21. Grossmann A, Morlet J (1984) Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J Math Anal 15:723–736CrossRefGoogle Scholar
  22. Gupta HK (2005) A Note on the 26 December 2004 tsunami in the Indian ocean. J Geol Soc India 65: 247–248Google Scholar
  23. Hanken H (1983) Advanced synergetics: Instability hierarchies of self-organizing systems and devices, Springer, Berlin Heidelberg New YorkGoogle Scholar
  24. Hirabayashi T, Ito K, Yoshi (1992) Multifractal analysis of earthquakes. Pure Appl Geophys 138: 591–610CrossRefGoogle Scholar
  25. Hirata T, Imoto M (1991) Multifractal analysis of spatial distribution of micro earthquakes in the Kanto region. Geophys J Int 107:155–162Google Scholar
  26. Hirata T, Sato T, Ito K (1987) Fractal structure of spatial distribution of microfracturing in rock. Geophys J R A Soc 90:369–374Google Scholar
  27. Kagan YY, Knopoff L(1980) Spatial distribution of earthquakes. The two point correlation function. Geophys J R A Soc 62:303–320Google Scholar
  28. Kanamori H, Anderson DL(1975) Theoretical basis of some empirical relations in seismology. Bull Seis Soc America 65:1073–1095Google Scholar
  29. Korvin G (1992) Fractal models in the earth sciences. Elsevier Science Publishers, Amsterdam London New York TokyoGoogle Scholar
  30. Kowalik Z (2003) Basic relations between tsunamis calculation and their physics— II. Sci Tsunami Haz 21: 154–173Google Scholar
  31. Kowalik Z, Murty TS (1993a) Numerical modeling of ocean dynamics. World Sci Publ, Singapore New Jersey London Hong KongGoogle Scholar
  32. Kowalik Z, Murty TS (1993b) Numerical simulation of two-dimensional tsunami runup. Marine Geodesy 16: 87–100CrossRefGoogle Scholar
  33. Lovejoy S, Schertzer S, Ladoy P (1986) Fractal characterization of homogeneous geophysical measuring network. Nature 319:43–44CrossRefGoogle Scholar
  34. Malamud BD, Turcotte DL (1999) Self affine time series I: generation and analysis. In: Dmowska R, Saltzman B (ed.) Advances in Geophysics: Long Range Persistence in Geophysical Time Series, vol 40. Academic Press, San Diego, pp 1–87Google Scholar
  35. Mandal P, Mabawonku AO, Dimri VP (2005) Self-organized fractal seismicity of reservoir triggered earthquakes in the Koyna-Warna seismic zone, Western India. Pure Appl Geophys 162: 73–90Google Scholar
  36. Mandelbrot BB (1983) The Fractal Geometry of Nature. WH Freeman & Company, New YorkGoogle Scholar
  37. Mandelbrot BB (1989) Mulifractal measures, especially for geophysicists. In: Scholz CH, Mandelbrot BB (eds) Fractals in geology and geophysics, Berkhäuser Verlag, Basel, pp 5–42Google Scholar
  38. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM rev.10:422–437CrossRefGoogle Scholar
  39. Maus S (1999) Variogram analysis of magnetic and gravity data. Geophysics 64:776–784Google Scholar
  40. Maus S, Dimri VP (1994) Fractal properties of potential fields caused by fractal sources. Geophys Res Lett 21: 891–894CrossRefGoogle Scholar
  41. Maus S, Dimri VP (1995) Potential field power spectrum inversion for scaling geology. J Geophys Res 100: 12605–12616CrossRefGoogle Scholar
  42. Maus S, Dimri V (1996) Depth estimation from the scaling power spectrum of potential fields? Geophys J Int 124: 113–120Google Scholar
  43. Maus S, Gordon D, Fairhead JD(1997) Curie-depth estimation using a self-similar magnetization model. Geophys J Int 129:163–168Google Scholar
  44. Mofjeld HO, Gonzalez FI, Newman JC (1999) Tsunami prediction in coastal regions. In: Mooers CNK (ed) Coastal ocean prediction, Am Geohys Union, Washington DC, pp 353–375Google Scholar
  45. Moharir PS (2000) Multifractals. In: Dimri VP (ed) Application of fractals in earth sciences AA Balkema, USA Oxford IBH Pub Co New Delhi, pp 46–57Google Scholar
  46. Naidu P (1968) Spectrum of the potential field due to randomly distributed sources, Geophysics 33: 337–345Google Scholar
  47. Omori (1894) On aftershocks. Rep Imp Earthq Investig Comm 2:103–139 (in Japanese)Google Scholar
  48. Ouchi T, Uekawa T (1986) Statistical analysis of the spatial distribution of earthquakes before and after large earthquakes. Phys Earth Planet Int 44: 211–225Google Scholar
  49. Pelinovsky EN (1989) Tsunami climbing a beach and Tsunami zonation. Sci Tsunami Haz 7:117–123Google Scholar
  50. Peterson I (1984) Ants in the Labyrinth and other fractal excursions. Science News 125: 42–43Google Scholar
  51. Pilkington M (1997) 3-D magnetic imaging using conjugate gradients. Geophysics 62:1132–1142CrossRefGoogle Scholar
  52. Pilkington M, Todoeschuck JP (1990) Stochastic inversion for scaling geology. Geophys J Int 102:205–217Google Scholar
  53. Pilkington M, Todoeschuck JP (1991) Naturally smooth inversions with a priori information from well logs. Geophysics 56:1811–1818CrossRefGoogle Scholar
  54. Pilkington M, Todoeschuck JP (1993) Fractal magnetization of continental crust. Geophys Res Lett 20: 627–630Google Scholar
  55. Pilkington M, Todoeschuck JP (1995) Scaling nature of crustal susceptibilities. Geophys Res Lett 22: 779–782CrossRefGoogle Scholar
  56. Pilkington M, Todoeschuck JP (2004) Power-law scaling behavior of crustal density and gravity. Geophys Res Lett 31: L09606, doi: 10.1029/2004GL019883CrossRefGoogle Scholar
  57. Pilkington M, Gregotski ME, Todoeschuck JP (1994) Using fractal crustal magnetization models in magnetic interpretation. Geophys Prosp 42:677–692Google Scholar
  58. Quarta T, Fedi M, Santis AD (2000) Source ambiguity from an estimation of the scaling exponent of potential field power spectra. Geophys J Int 140: 311–323CrossRefGoogle Scholar
  59. Raju PS, Raghavan RV, Umadevi E, Shashidar D, Sarma ANS, Gurunath D, Satyanarayana HVS, Rao TS, Naik RTB, Rao NPC, Kamuruddin Md, Gowri Shankar U, Gogi NK, Baruah BC, Bora NK Kousalya M, Sekhar M, Dimri VP(2005) A note on 26 December,2004 Great Sumatra Earthquake. J Geol Soc India 65: 249–251Google Scholar
  60. Ravi Prakash M, Dimri VP (2000) Distribution of the aftershock sequence of Latur earthquake in time and space by fractal approach. J Geol Soc India 55: 167–174Google Scholar
  61. Saggaf M, Toksoz M (1999) An analysis of deconvolution: Modeling reflectivity by fractionally integrated noise. Geophysics 64:1093–1107CrossRefGoogle Scholar
  62. Spector A, Grant FS (1970) Statistical models for interpreting magnetic data. Geophysics 35:293–302Google Scholar
  63. Sunmonu LA, Dimri VP, Ravi Prakash M, Bansal AR (2001), Multifractal approach of the time series of M ≥ 7 earthquakes in Himalayan region and its vicinity during 1985–1995. J Geol Soc India 58: 163–169Google Scholar
  64. Thorarinsson, F, Magnusson SG (1990) Bouguer density determination by fractal analysis. Geophysics 55: 932–935CrossRefGoogle Scholar
  65. Todoeschuck JP, Jensen OG (1988) Joseph geology and scaling deconvolution. Geophysics 53:1410–1411CrossRefGoogle Scholar
  66. Todoeschuck JP, Jensen OG (1989) Scaling geology and seismic deconvolution. Pure Appl. Geophys 131: 273–288CrossRefGoogle Scholar
  67. Toverud T, Dimri VP, Ursin B (2001) Comparison of deconvolution methods for scaling reflectivity. Jour Geophy 22:117–123Google Scholar
  68. Tubman KM, Crane SD (1995) Vertical versus horizontal well log variability and application to fractal reservoir modeling. In: Barton CC, LaPointe PR (eds) Fractals in Petroleum Geology and Earth Sciences, Plenum press, New York, pp 279–294Google Scholar
  69. Turcotte DL (1989) Fractals in geology and geophysics. In: Scholz CH, Mandelbrot BB (eds) Frcatals in Geophysics, Berkhäuser Verlag, Basel, pp 171–196Google Scholar
  70. Turcotte DL (1992) Fractals and chaos in geology and geophysics. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • V.P. Dimri
    • 1
  1. 1.National Geophysical Research InstituteHyderabadIndia

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