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Geometric Properties of Random Fields

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Book cover Random Fields for Spatial Data Modeling

Part of the book series: Advances in Geographic Information Science ((AGIS))

Abstract

This chapter deals with the main concepts and mathematical tools that help to describe and quantify the shapes of random fields. The geometry of Gaussian random functions is to a large extent determined by the mean and the two-point correlation functions. The classical text on the geometry of random fields is the book written by Robert Adler [10]. The basic elements of random field geometry are contained in the technical report by Abrahamsen [3]. A more recent and mathematically advanced book by Taylor and Adler exposes the geometry of random field using the language of manifolds [11].

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

Pythagoras

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Notes

  1. 1.

    More generally, it can be shown that the Gaussian covariance function admits derivatives of all orders at ∥r∥ = 0.

  2. 2.

    In [135] R 2(1) = ξ 1∕ξ 2 is used instead of R. The results from [135] are equivalent to those presented herein following the transformation R 2(1)↦1∕R.

  3. 3.

    Note the slight difference of notation in Λi,j compared to (5.27). For the second-order moments the indices i, j are sufficient instead of the vector (p 1, …, p d)⊤. We also drop the superscript (2) since the order of the moments is obvious.

  4. 4.

    There is nothing magical about 5%; a different level, such as the distance at half-maximum could be selected instead.

  5. 5.

    A unimodal function is a function with a single peak.

  6. 6.

    A “hyper-cube” is used in d > 3 dimensions; in d = 1 the “hyper-cubes” correspond to line segments and in d = 2 to squares.

  7. 7.

    Condensed matter physicists call this less-than-perfect correlation quasi-long-range order, to distinguish it from “true” long-range order. The latter implies that all the system variables have the same value. A typical example is a ferromagnetic system in which all the magnetic moments are aligned in the same direction.

  8. 8.

    The exponents β refer to the equations (4.14) and (4.15) respectively.

  9. 9.

    In this case we should speak of a spectral function, since the spectral density is not well-defined.

References

  1. Abrahamsen, P.: A Review of Gaussian Random Fields and Correlation Functions. Tech. Rep. TR 917, Norwegian Computing Center, Box 114, Blindern, N-0314, Oslo, Norway (1997)

    Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, DC, USA (1972)

    MATH  Google Scholar 

  3. Adler, R.J.: The Geometry of Random Fields. John Wiley & Sons, New York, NY, USA (1981)

    MATH  Google Scholar 

  4. Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer Science & Business Media, New York, NY, USA (2009)

    MATH  Google Scholar 

  5. Allard, D., Senoussi, R., Porcu, E.: Anisotropy models for spatial data. Math. Geosci. 48(3), 305–328 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Amit, D.J.: Field Theory, the Renormalization Group, and Critical Phenomena, 2nd edn. World Scientific, New York, NY, USA (1984)

    Google Scholar 

  7. Armstrong, M., Diamond, P.: Testing variograms for positive-definiteness. Math. Geol. 16(4), 407–421 (1984)

    Article  Google Scholar 

  8. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality – an explanation of 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)

    Article  ADS  Google Scholar 

  9. Ben-Avraham, D., Havlin, S.: Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge, UK (2000)

    Book  MATH  Google Scholar 

  10. Biermé, H., Richard, F.: Statistical tests of anisotropy for fractional Brownian texture. application to full-field digital mammography. J. Math. Imaging Vis. 36(3), 227–240 (2010)

    Google Scholar 

  11. Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Occam’s razor. In: Shavlik, J.W., Dieterich, T.G. (eds.) Information Processing Letters, pp. 377–380. Morgan Kaufmann, San Mateo, CA, USA (1987)

    Google Scholar 

  12. Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  13. Box, G.E.P., Jenkins, G.M., Reinsel, G.C., Ljung, C.M.: Time Series Analysis, 5th edn. Wiley, Hoboken, NJ, USA (2016)

    MATH  Google Scholar 

  14. Brockwell, P.G., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York, NY, USA (2006)

    MATH  Google Scholar 

  15. Bunde, A., Havlin, S.: Percolation I. In: Bunde, A., Havlin, S. (eds.) Fractals and Disordered Systems, pp. 59–113. Springer, London, UK (1991)

    Chapter  MATH  Google Scholar 

  16. Chen, Z., Ivanov, P.C., Hu, K., Stanley, H.E.: Effect of nonstationarities on detrended fluctuation analysis. Phys. Rev. E 65(4), 041107 (2002)

    Article  ADS  Google Scholar 

  17. Chilès, J.P.: The generalized variogram. Tech. Rep. réf. R121030JCHI, École des Mines de Paris, Centre de Géosciences, Géostatistique (2012). http://www.cg.ensmp.fr/bibliotheque/public/CHILES_Rapport_02272.pdf

  18. Chilès, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty, 2nd edn. John Wiley & Sons, New York, NY, USA (2012)

    Book  MATH  Google Scholar 

  19. Chorti, A., Hristopulos, D.T.: Non-parametric identification of anisotropic (elliptic) correlations in spatially distributed data sets. IEEE Trans. Signal Process. 56(10), 4738–4751 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Christakos, G.: On the problem of permissible covariance and variogram models. Water Resour. Res. 20(2), 251–265 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  21. Christakos, G.: Random Field Models in Earth Sciences. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  22. Christakos, G., Hristopulos, D.T.: Spatiotemporal Environmental Health Modelling. Kluwer, Boston (1998)

    MATH  Google Scholar 

  23. Clauset, A., Shalizi, C., Newman, M.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Cramér, H., Leadbetter, M.R.: Stationary and Related Stochastic Processes. John Wiley & Sons, New York, NY, USA (1967)

    MATH  Google Scholar 

  25. Cressie, N.: Spatial Statistics. John Wiley & Sons, New York, NY, USA (1993)

    MATH  Google Scholar 

  26. Dagan, G., Neuman, S.P.: Subsurface Flow and Transport: A Stochastic Approach. Cambridge University Press, Cambridge, UK (2005)

    Google Scholar 

  27. Daley, R.: Atmospheric Data Analysis. Cambridge University Press, Cambridge, UK (1991)

    Google Scholar 

  28. Davies, S., Hall, P.: Fractal analysis of surface roughness by using spatial data. J. R. Stat. Soc. Ser. B (Stat Methodol.) 61(1), 3–37 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Davison, A.C., Gholamrezaee, M.M.: Geostatistics of extremes. Proc. R. Soc. A: Math. Phys. Eng. Sci. 468(2138), 581–608 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. De Micheaux, P.L., Liquet, B.: Understanding convergence concepts: a visual-minded and graphical simulation-based approach. Am. Stat. 63(2), 73–78 (2012)

    MathSciNet  Google Scholar 

  31. Di Federico, V., Neuman, S.P., Tartakovsky, D.M.: Anisotropy, lacunarity, and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms. Water Resour. Res. 35(10), 2891–2908 (1999)

    Article  ADS  Google Scholar 

  32. Embrechts, P., Maejima, M.: Selfsimilar Processes. Princeton University Press, Princeton, NJ, USA (2002)

    MATH  Google Scholar 

  33. Feder, J.: Fractals. Plenum Press, New York, NY, USA (1988)

    Book  MATH  Google Scholar 

  34. Fuglstad, G.A., Simpson, D., Lindgren, F., Rue, H.: Does non-stationary spatial data always require non-stationary random fields? Spat. Stat. 14(Part C), 505–531 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gelhar, L.W.: Stochastic Subsurface Hydrology. Prentice Hall, Englewood Cliffs, NJ (1993)

    Google Scholar 

  36. Gerlach, M., Altmann, E.G.: Testing statistical laws in complex systems. Phys. Rev. Lett. 122(16), 168301 (2019)

    Article  ADS  Google Scholar 

  37. Gneiting, T., Schlather, M.: Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46(2), 269–282 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Gneiting, T., Ševčíková, H., Percival, D.B., et al.: Estimators of fractal dimension: assessing the roughness of time series and spatial data. Stat. Sci. 27(2), 247–277 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gogolides, E., Constantoudis, V., Kokkoris, G., Kontziampasis, D., Tsougeni, K., Boulousis, G., Vlachopoulou, M., Tserepi, A.: Controlling roughness: from etching to nanotexturing and plasma-directed organization on organic and inorganic materials. J. Phys. D. Appl. Phys. 44(17), 174021 (2011)

    Article  ADS  Google Scholar 

  40. Gogolides, E., Constantoudis, V., Patsis, G.P., Tserepi, A.: A review of line edge roughness and surface nanotexture resulting from patterning processes. Microelectron. Eng. 83(4), 1067–1072 (2006)

    Article  Google Scholar 

  41. Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley, Reading, MA (1992)

    MATH  Google Scholar 

  42. Gouyet, J.F., Rosso, M., Sapoval, B.: Fractal surfaces and interfaces. In: Bunde, A., Havlin, S. (eds.) Fractals and Disordered Systems, pp. 263–301. Springer, London, UK (1996)

    Chapter  MATH  Google Scholar 

  43. Grohmann, C.H., Smith, M.J., Riccomini, C.: Multiscale analysis of topographic surface roughness in the Midland Valley, Scotland. IEEE Trans. Geosci. Remote Sens. 49(4), 1200–1213 (2011)

    Article  ADS  Google Scholar 

  44. Gu, G.F., Zhou, W.X.: Detrended fluctuation analysis for fractals and multifractals in higher dimensions. Phys. Rev. E 74(6), 061104 (2006)

    Article  ADS  Google Scholar 

  45. Gunning, J.: On the use of multivariate Lévy-stable random field models for geological heterogeneity. Math. Geol. 34(1), 43–62 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Heneghan, C., McDarby, G.: Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys. Rev. E 62(5), 6103 (2000)

    Article  ADS  Google Scholar 

  47. Hristopulos, D.T.: New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity. Stoch. Environ. Res. Risk Assess. 16(1), 43–62 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hristopulos, D.T.: Permissibility of fractal exponents and models of band-limited two-point functions for fGn and fBm random fields. Stoch. Environ. Res. Risk Assess. 17(3), 191–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. Hristopulos, D.T.: Covariance functions motivated by spatial random field models with local interactions. Stoch. Environ. Res. Risk Assess. 29(3), 739–754 (2015)

    Article  MathSciNet  Google Scholar 

  50. Hristopulos, D.T., Žukovič, M.: Relationships between correlation lengths and integral scales for covariance models with more than two parameters. Stoch. Environ. Res. Risk Assess. 25(1), 11–19 (2011)

    Article  MATH  Google Scholar 

  51. Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Stanley, H.E.: Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64(1), 011114 (2001)

    Article  ADS  Google Scholar 

  52. Hurst, H.E.: Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770–808 (1951)

    Google Scholar 

  53. Hurst, H.E., Black, R.P., Simaika, Y.: Long-term Storage: An Experimental Study. Constable, London, UK (1965)

    Google Scholar 

  54. Isichenko, M.B.: Percolation, statistical topography, and transport in random media. Rev. Mod. Phys. 64(4), 961–1043 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  55. Jackson, J.D.: Classical Electrodynamics, 3rd edn. John Wiley & Sons, New York, NY, USA (1998)

    MATH  Google Scholar 

  56. Jona-Lasinio, G.: Renormalization group and probability theory. Phys. Rep. 352(4–6), 439–458 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Kitanidis, P.K.: Statistical estimation of polynomial generalized covariance functions and hydrologic applications. Water Resour. Res. 19(2), 909–921 (1983)

    Article  ADS  Google Scholar 

  58. Kitanidis, P.K.: Parametric estimation of covariances of regionalized variables. Water Resour. Res. 23(4), 671–680 (1987)

    Google Scholar 

  59. Kitanidis, P.K.: Generalized covariance functions in estimation. Math. Geol. 25(5), 525–540 (1993)

    Article  Google Scholar 

  60. Kitanidis, P.K.: Introduction to Geostatistics: Applications to Hydrogeology. Cambridge University Press, Cambridge, UK (1997)

    Book  Google Scholar 

  61. Kitanidis, P.K.: Generalized covariance functions associated with the Laplace equation and their use in interpolation and inverse problems. Water Resour. Res. 35(5), 1361–1367 (1999)

    Article  ADS  Google Scholar 

  62. Kramer, P.R., Kurbanmuradov, O., Sabelfeld, K.: Comparative analysis of multiscale Gaussian random field simulation algorithms. J. Comput. Phys. 226(1), 897–924 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. Lantuéjoul, C.: Geostatistical Simulation: Models and Algorithms. Springer, Berlin, Germany (2002)

    Book  MATH  Google Scholar 

  64. Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge, UK (2010)

    Book  MATH  Google Scholar 

  65. Leach, R.: Characterisation of Areal Surface Texture. Springer, Berlin, Germany (2013)

    Book  Google Scholar 

  66. Lim, S., Teo, L.: Generalized Whittle-Matérn random field as a model of correlated fluctuations. J. Phys. A Math. Theor. 42(10), 105202 (2009)

    Article  ADS  MATH  Google Scholar 

  67. Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M.M., Ainsworth, M., Karniadakis, G.E.: What is the fractional laplacian? arXiv preprint arXiv:1801.09767 (2018)

    Google Scholar 

  68. Lodhia, A., Sheffield, S., Sun, X., Watson, S.S.: Fractional Gaussian fields: a survey. Probab. Surv. 13, 1–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  69. Lovejoy, S., Schertzer, D.: The Weather and Climate: Emergent Laws and Multifractal Cascades. Cambridge University Press, Cambridge, UK (2013)

    Book  MATH  Google Scholar 

  70. Løvsletten, O.: Consistency of detrended fluctuation analysis. Phys. Rev. E 96(1), 012141 (2017)

    Article  ADS  Google Scholar 

  71. Lu, S., Molz, F.J., Liu, H.H.: An efficient, three-dimensional, anisotropic, fractional Brownian motion and truncated fractional Lévy motion simulation algorithm based on successive random additions. Comput. Geosci. 29(1), 15–25 (2003)

    Article  ADS  Google Scholar 

  72. Makse, H.A., Davies, G.W., Havlin, S., Ivanov, P.C., King, P.R., Stanley, H.E.: Long-range correlations in permeability fluctuations in porous rock. Phys. Rev. E 54(4), 3129–3134 (1996)

    Article  ADS  Google Scholar 

  73. Mandelbrot, B.B.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. Mandelbrot, B.B.: The fractal Geometry of Nature. W. H. Freeman and Company, New York, NY, USA (1983)

    Book  Google Scholar 

  75. Mandelbrot, B.B.: Fractal analysis and synthesis of fracture surface roughness and related forms of complexity and disorder. Int. J. Fract. 138, 13–17 (2006)

    Article  MATH  Google Scholar 

  76. Mandelbrot, B.B., Passoja, D.E., Paullay, A.J.: Fractal character of fracture surfaces of metals. Nature 308, 721–722 (1984)

    Article  ADS  Google Scholar 

  77. Markovič, D., Gros, C.: Power laws and self-organized criticality in theory and nature. Phys. Rep. 536(2), 41–74 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  78. Martin, W., Flandrin, P.: Wigner-Ville spectral analysis of nonstationary processes. IEEE Trans. Acoust. Speech Signal Process. 33(6), 1461–1470 (1985)

    Article  Google Scholar 

  79. Matheron, G.: The theory of regionalized variables and its application. Tech. rep., Les Cahiers du Centre de Morphologie Mathematique de Fontainebleau, École Nationale Superieure des Mines de Paris (1971). http://cg.ensmp.fr/bibliotheque/cgi-bin/public/bibli_index.cgi#1971. [Online; accessed on 31 Oct 2018]

  80. Matheron, G.: The intrinsic random functions and their applications. J. Appl. Probab. 5(3), 439–468 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  81. McComb, W.D.: The Physics of Fluid Turbulence. Oxford University Press, Oxford, UK (1990)

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  83. Mirouze, I., Blockley, E., Lea, D., Martin, M., Bell, M.: A multiple length scale correlation operator for ocean data assimilation. Tellus A 68, 29744 (2016)

    Article  ADS  Google Scholar 

  84. Mishchenko, M.I., et al.: First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media. Phys. Rep. 632, 1–75 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  85. Mitchell, J.F.B., Lowe, J., Wood, R.A., Vellinga, M.: Extreme events due to human-induced climate change. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 364(1845), 2117–2133 (2006)

    Article  ADS  Google Scholar 

  86. Molz, F.J., Liu, H.H., Szulga, J.: Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: a review, presentation of fundamental properties, and extensions. Water Resour. Res. 33(10), 2273–2286 (1997)

    Article  ADS  Google Scholar 

  87. Montgomery, D.C., Jennings, C.L., Kulahci, M.: Introduction to Time Series Analysis and Forecasting. John Wiley & Sons, Hoboken, NJ, USA (2008)

    MATH  Google Scholar 

  88. Nan, T., Neuman, S.P., Riva, M., Guadagnini, A.: Analyzing randomly fluctuating hierarchical variables and extremes. In: Cushman, J.H., Tartakovsky, D.M. (eds.) The Handbook of Groundwater Engineering, 3rd edn., pp. 443–456. CRC Press, Taylor & Francis Group, Boca Raton, FL, USA (2016)

    Google Scholar 

  89. Neuman, S.P.: Generalized scaling of permeabilities: validation and effect of support scale. Geophys. Res. Lett. 21(5), 349–352 (1994)

    Article  ADS  Google Scholar 

  90. Neuman, S.P., Guadagnini, A., Riva, M., Siena, M.: Recent advances in statistical and scaling analysis of earth and environmental variables. In: Mishra, P.K., Kuhlman, K.L. (eds.) Advances in Hydrogeology, pp. 1–25. Springer, New York, NY, USA (2013)

    Google Scholar 

  91. O’Malley, D., Cushman, J.H.: Anomalous dispersion. In: Cushman, J.H., Tartakovsky, D.M. (eds.) The Handbook of Groundwater Engineering, 3rd edn., pp. 497–505. CRC Press, Taylor & Francis Group, Boca Raton, FL, USA (2016)

    Google Scholar 

  92. Painter, S.: Random fractal models of heterogeneity: the Lévy-stable approach. Math. Geol. 27(7), 813–830 (1995)

    Article  Google Scholar 

  93. Painter, S.: Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations. Water Resour. Res. 32(5), 1183–1195 (1996)

    Article  ADS  Google Scholar 

  94. Painter, S.: Stochastic interpolation of aquifer properties using fractional Lévy motion. Water Resour. Res. 32(5), 1323–1332 (1996)

    Article  ADS  Google Scholar 

  95. Palma, W.: Long-memory Time Series: Theory and Methods, vol. 662. John Wiley & Sons, Hoboken, NJ, USA (2007)

    Book  MATH  Google Scholar 

  96. Papoulis, A., Pillai, S.U.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York, NY, USA (2002)

    Google Scholar 

  97. Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49(2), 1685 (1994)

    Article  ADS  Google Scholar 

  98. Potthoff, J.: Sample properties of random fields III: Differentiability. Commun. Stoch. Anal. 4(3), 335–353 (2010)

    MathSciNet  MATH  Google Scholar 

  99. Robert, C.Y.: Some new classes of stationary max-stable random fields. Statist. Probab. Lett. 83(6), 1496–1503 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  100. Røislien, J., Omre, H.: T-distributed random fields: a parametric model for heavy-tailed well-log data. Math. Geol. 38(7), 821–849 (2006)

    Article  Google Scholar 

  101. Rosenburg, M.A., Aharonson, O., Head, J.W., Kreslavsky, M.A., Mazarico, E., Neumann, G.A., Smith, D.E., Torrence, M.H., Zuber, M.T.: Global surface slopes and roughness of the moon from the lunar orbiter laser altimeter. J. Geophys. Res. Planets 116(E2), E02001 (2011)

    Article  ADS  Google Scholar 

  102. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam, Netherlands (1993)

    MATH  Google Scholar 

  103. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall/CRC (2000)

    Google Scholar 

  104. Scheuerer, M.: Regularity of the sample paths of a general second order random field. Stoch. Process. Appl. 120(10), 1879–1897 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  105. Scheuerer, M., Schlather, M.: Covariance models for divergence-free and curl-free random vector fields. Stoch. Model. 28(3), 433–451 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  106. Schlather, M.: Models for stationary max-stable random fields. Extremes 5(1), 33–44 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  107. Schröder, K.P., Connon Smith, R.: Distant future of the Sun and Earth revisited. Mon. Not. R. Astron. Soc. 386(1), 155–163 (2008)

    Article  ADS  Google Scholar 

  108. Schwartz, L.M.: Mathematics for the Physical Sciences. Dover, Mineola, NY, USA (2008)

    MATH  Google Scholar 

  109. Serinaldi, F., Kilsby, C.G.: Stationarity is undead: uncertainty dominates the distribution of extremes. Adv. Water Resour. 77, 17–36 (2015)

    Article  ADS  Google Scholar 

  110. Shumway, R.H., Stoffer, D.S.: Time Series Analysis and its Applications. Springer Science & Business Media, New York, NY, USA (2000)

    Book  MATH  Google Scholar 

  111. Sornette, D.: Critical Phenomena in Natural Sciences. Springer, Berlin, Germany (2004)

    MATH  Google Scholar 

  112. Stauffer, D., Aharony, A.: Introduction to Percolation Theory. Taylor and Francis, London, UK (1994)

    MATH  Google Scholar 

  113. Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York, NY, USA (1999)

    Book  MATH  Google Scholar 

  114. Stein, M.L.: Local stationarity and simulation of self-affine intrinsic random functions. IEEE Trans. Inf. Theory 47(4), 1385–1390 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  115. Strogatz, S.H.: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, Boca Raton, FL (2018)

    Book  MATH  Google Scholar 

  116. Swerling, P.: Statistical properties of the contours of random surfaces. IRE Trans. Inf. Theory IT-8(4), 315–321 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  117. Tessier, Y., Lovejoy, S., Schertzer, D.: Universal multifractals: theory and observations for rain and clouds. J. Appl. Meteorol. 32(2), 223–250 (1993)

    Article  ADS  Google Scholar 

  118. Vanmarcke, E.H.: Properties of spectral moments with applications to random vibration. J .Eng. Mech. Div. 98(2), 425–446 (1972)

    Google Scholar 

  119. Vourdas, N., Tserepi, A., Gogolides, E.: Nanotextured super-hydrophobic transparent poly (methyl methacrylate) surfaces using high-density plasma processing. Nanotechnology 18(12), 125304 (2007)

    Article  ADS  Google Scholar 

  120. Wang, F., Fan, Q., Stanley, H.E.: Multiscale multifractal detrended-fluctuation analysis of two-dimensional surfaces. Phys. Rev. E 93(4), 042213 (2016)

    Article  ADS  Google Scholar 

  121. Weron, R.: Estimating long-range dependence: finite sample properties and confidence intervals. Physica A: Stat. Mech. Appl. 312(1), 285–299 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  122. West, G.: Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies. Penguin, New York, NY, USA (2017)

    Google Scholar 

  123. Whitehouse, D.J.: Handbook of Surface and Nanometrology. CRC Press, Boca Raton, FL, USA (2010)

    Book  Google Scholar 

  124. Yaglom, A.M.: Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2(3), 273–320 (1957)

    Article  Google Scholar 

  125. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions, vol. I. Springer, New York, NY, USA (1987)

    Book  MATH  Google Scholar 

  126. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Supplementary Notes and References, vol. II. Springer, New York, NY, USA (1987)

    Google Scholar 

  127. Yaremchuk, M., Sentchev, A.: Multi-scale correlation functions associated with polynomials of the diffusion operator. Q. J. R. Meteorol. Soc. 138(668), 1948–1953 (2012)

    Article  ADS  Google Scholar 

  128. Yokota, Y., Gwinner, K., Oberst, J., Haruyama, J., Matsunaga, T., Morota, T., Noda, H., Araki, H., Ohtake, M., Yamamoto, S., Gläser, P., Ishihara, Y., Honda, C., Hirata, N., Demura, H.: Variation of the lunar highland surface roughness at baseline 0.15–100 km and the relationship to relative age. Geophys. Res. Lett. 41(5), 1444–1451 (2014)

    Article  ADS  Google Scholar 

  129. Young, I.R., Ribal, A.: Multiplatform evaluation of global trends in wind speed and wave height. Science 364(6440), 548–552 (2019)

    Article  ADS  Google Scholar 

  130. Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, New York, NY, USA (2001)

    MATH  Google Scholar 

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  1. Technical University of Crete, Chania, Greece

    Dionissios T. Hristopulos

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Hristopulos, D.T. (2020). Geometric Properties of Random Fields. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_5

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