Parallel Algorithms for Multifractal Analysis of River Networks

  • Leonardo PrimaveraEmail author
  • Emilia Florio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11973)


The dynamical properties of many natural phenomena can be related to their support fractal dimension. A relevant example is the connection between flood peaks produced in a river basin, as observed in flood hydrographs, and the multi-fractal spectrum of the river itself, according to the Multifractal Instantaneous Unit Hydrograph (MIUH) theory. Typically, the multifractal analysis of river networks is carried out by sampling large collections of points belonging to the river basin and analyzing the fractal dimensions and the Lipschitz-Hölder exponents of singularities through numerical procedures which involve different degrees of accuracy in the assessment of such quantities through different methods (box-counting techniques, the generalized correlation integral method by Pawelzik and Schuster (1987), the fixed-mass algorithms by Badii and Politi (1985), being some relevant examples). However, the higher accuracy in the determination of the fractal dimensions requires considerably higher computational times. For this reason, we recently developed a parallel version of some of the cited multifractal methods described above by using the MPI parallel library, by reaching almost optimal speed-ups in the computations. This will supply a tool for the assessment of the fractal dimensions of river networks (as well as of several other natural phenomena whose embedding dimension is 2 or 3) on massively parallel clusters or multi-core workstations.


Multifractal dimension River networks Parallel algorithms 


  1. 1.
    Badii, R., Politi, A.: Hausdorff dimension and uniformity factor of strange attractors. Phys. Rev. Lett. 52, 1661–1664 (1984)MathSciNetCrossRefGoogle Scholar
  2. 2.
    De Bartolo, S.G., Ambrosio, L., Primavera, L., Veltri, M.: Descrittori frattali e caratteri morfometrici nella risposta idrologica. In: Caroni, E., Fiorotto, V., Mancinelli, A., Salandin, P. (eds.) La Difesa Idraulica del Territorio, pp. 47–60. Tergeste, Trieste (2003)Google Scholar
  3. 3.
    De Bartolo, S.G., Gabriele, S., Gaudio, R.: Multifractal behaviour of river networks. Hydrol. Earth Syst. Sci. 4, 105–112 (2000)CrossRefGoogle Scholar
  4. 4.
    De Bartolo, S.G., Veltri, M., Primavera, L.: Estimated generalized dimensions of river networks. J. Hydrol. 322, 181–191 (2006)CrossRefGoogle Scholar
  5. 5.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)CrossRefGoogle Scholar
  6. 6.
    Florio, E., Maierù, L.: The scientific knowledge of book XVI De subtilitate by G. Cardano used in the Trattato sulla divinatione naturale cosmologica by P.A. Foscarini. Open J. Human. 1, 385–422 (2019)Google Scholar
  7. 7.
    Gaudio, R., De Bartolo, S.G., Primavera, L., Veltri, M., Gabriele, S.: Procedures in multifractal analysis of river networks, vol. 286, pp. 228–237. IAHS Publication (2004)Google Scholar
  8. 8.
    Grassberger, P.: Generalized dimensions of strange attractors. Phys. Lett. A 97, 227–230 (1983)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50, 346–349 (1983)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mandelbrot, B.B.: Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In: Rosenblatt, M., Van Atta, C. (eds.) Statistical Models and Turbulence. LNP, vol. 12, pp. 333–351. Springer, Berlin (1972). Scholar
  11. 11.
    Pawelzik, K., Schuster, H.G.: Generalized dimensions and entropies from a measured time series. Phys. Rev. A 35, 481–484 (1987)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità della CalabriaRendeItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità della CalabriaRendeItaly

Personalised recommendations