The complex non-linear regime of the monthly rainfall in Catalonia (NE Spain) is analyzed by means of the reconstruction fractal theorem and the multifractal detrended fluctuation analysis algorithm. Areas with a notable degree of complex physical mechanisms are detected by using the concepts of persistence (Hurst exponent), complexity (embedding dimension), predictive uncertainty (Lyapunov exponents), loss of memory of the mechanism (Kolmogorov exponent), and the set of multifractal parameters (Hölder exponents, spectral asymmetry, spectral width, and complexity index). Besides these analyses permitting a detailed description of monthly rainfall pattern characteristics, the obtained results should also be relevant for new research studies concerning monthly amounts forecasting at a monthly scale. On one hand, the number of necessary monthly data for autoregressive processes could change with the complexity of the multifractal structure of the monthly rainfall regime. On the other hand, the discrepancies between real monthly amounts and those generated by some autoregressive algorithms could be related to some parameters of the reconstruction fractal theorem, such as the Lyapunov and Kolmogorov exponents.
REFERENCES
1
Burgueño
, A.
, Lana
, X.
, Serra
, C.
, and Martínez
, M. D.
, “Daily extreme temperature multifractals in Catalonia (NE Spain)
,” Phys. Lett. A
378
, 874
–885
(2014
). 2
Casas-Castillo
, M. C.
, Llabrés-Brustenga
, A.
, Rius
, A.
, Rodríguez-Solà
, R.
, and Navarro
, X.
, “A single scaling parameter as a first approximation to describe the rainfall pattern of a place: Application on Catalonia
,” Acta Geophys.
66
, 415
–424
(2018
). 3
Clavero
, P.
, Martín-Vide
, J.
, and Raso
, J.
, “Atles climàtic de catalunya: Termopluviometria” (Departament de Medi Ambient, Generalitat de Catalunya, 1996), pp. 1–42.4
Diks
, C.
, “Nonlinear time series analysis. Methods and applications
,” in Nonlinear Time Series and Chaos
, edited by H.
Tong
(World Scientific
, London
, 1999
), Vol. 4, p. 209
.5
Dimri
, V. P.
, “Fractals in geophysics and seismology: An introduction
,” in Fractal Behaviour of the Earth System
(Springer
, Berlin
, 2005
), pp. 1
–22
.6
Eckmann
, J. P.
, Oliffson
, S.
, Ruelle
, D.
, and Cilliberto
, S.
, “Lyapunov exponents from time series
,” Phys. Rev. A
34
, 4971
–4979
(1986
). 7
Enescu
, B.
, Ito
, K.
, Radulian
, M.
, Popescu
, E.
, and Bazacliu
, O.
, “Multifractal and chaotic analysis of Vrancea (Romania). Intermediate-depth earthquakes: Investigation of the temporal distribution of events
,” Pure Appl. Geophys.
162
, 249
–271
(2005
). 8
9
García-Marín
, A.
, Estévez
, J.
, Alcalá-Miras
, J. A.
, Morbidelli
, R.
, Flammini
, A.
, and Ayuso-Muñoz
, J. L.
, “Multifractal analysis to study break points in temperature data sets
,” Chaos
29
, 093116
(2019
). 10
García-Marín
, A.
, Estévez
, J.
, Jiménez-Hornero
, F. J.
, and Ayuso-Muñoz
, J. L.
, “Multifractal analysis of validated wind speed time series
,” Chaos
23
, 013133
(2013
). 11
Godano
, C.
, Alonzo
, M. L.
, and Bottari
, A.
, “Multifractal analysis of the spatial distribution of earthquakes in southern Italy
,” Geophys. J. Int.
125
, 901
–911
(1996
). 12
Goltz
, C.
, “Fractal and chaotic properties of earthquakes
,” in Lecture Notes in Earth Sciences
(Springer
, Berlin
, 1997
), Vol. 77, p. 178
.13
Grassberger
, P.
and Procaccia
, I.
, “Characterization of strange attractors
,” Phys. Rev. Lett.
50
, 346
–349
(1983a
). 14
Grassberger
, P.
and Procaccia
, I.
, “Estimation of the Kolmogorov entropy from a chaotic signal
,” Phys. Rev. A
28
, 448
–451
(1983b
).15
Herrera-Grimaldi
, P.
, García-Marín
, A.
, and Estévez
, J.
, “Multifractal analysis of diurnal temperature range over southern Spain using validated datasets
,” Chaos
29
, 063105
(2019
). 16
Hirabayashi
, T.
, Ito
, K.
, and Yoshii
, T.
, “Multifractal analysis of earthquakes
,” in Fractals and Chaos in the Earth Sciences
(Springer
, 1992
), pp. 591
–610
.17
Hirsch
, R. M.
, Helsel
, D. R.
, Cohn
, T. A.
, and Gilroy
, E. J.
, “Statistical analysis of hydrologic data
,” in Handbook of Hydrology
, edited by D. R.
Maidment
(McGraw-Hill
, New York
, 1992
), pp. 17.1
–17.55
.18
Kantelhardt
, J. W.
, Zschiegner
, S. A.
, Koscielny-Bunde
, A.
, Havlin
, S.
, Bunde
, A.
, and Stanley
, H. E.
, “Multifractal detrended fluctuation analysis of nonstationary time series
,” Phys. A Stat. Mech. Appl.
316
, 87
–114
(2002
). 19
Kaplan
, J. K.
and Yorke
, J. A.
, “Chaotic behaviour of multidimensional difference equations
,” in Functional Difference Equations and Approximation of Fixed Points
, edited by H. O.
Walter
and H. O.
Peitgen
(Springer
, Berlin
, 1979
), Vol. 730
, pp. 204
–227
.20
21
Koscielny-Bunde
, E.
, Bunde
, S.
, Havlin
, S.
, Roman
, H. E.
, Goldreich
, Y.
, and Schellnhuber
, H. J.
, “Indication of a universal persistence law governing atmospheric variability
,” Phys. Rev. Lett.
81
, 729
–732
(1998
). 22
Koscielny-Bunde
, E.
, Kantelhardt
, J. W.
, Braund
, P.
, Bunde
, A.
, and Havlin
, S.
, “Long-term persistence and multifractality of river runoff records: Detrended fluctuation studies
,” J. Hydrol.
322
, 120
–137
(2006
). 23
Lana
, X.
, Burgueño
, A.
, Martínez
, M. D.
, and Serra
, C.
, “Complexity and predictability of the monthly western Mediterranean oscillation index
,” Int. J. Climatol.
36
, 2435
–2450
(2016
). 24
Lana
, X.
, Burgueño
, A.
, Serra
, C.
, and Martínez
, M. D.
, “Fractal structure and predictive strategy of the daily extreme temperature residual at fabra observatory (NE Spain, years 1917-2005)
,” Theor. Appl. Climatol.
121
, 225
–241
(2015
). 25
Lana
, X.
, Burgueño
, A.
, Serra
, C.
, and Martínez
, M. D.
, “Monthly rain amounts at fabra observatory (Barcelona, NE Spain): Fractal structure, autoregressive processes and correlation with monthly western Mediterranean oscillation index
,” Int. J. Climatol.
37
, 1557
–1577
(2017
). 26
Lana
, X.
, Martinez
, M. D.
, Serra
, C.
, and Burgueno
, A.
, “Complex behaviour and predictability of the European dry spell regimes
,” Nonlinear Processes Geophys.
17
, 499
–512
(2010
). 27
Llabrés-Brustenga
, A.
, Rius
, A.
, Rodríguez-Solà
, R.
, Casas-Castillo
, M. C.
, and Redaño
, A.
, “Quality control process of the daily rainfall series available in catalonia from 1855 to the present
,” Theor. Appl. Climatol.
137
, 2715
–2729
(2019
). 28
Mandelbrot
, B. B.
, The Fractal Geometry of Nature
(WH Freeman and Co.
, New York
, 1983
), revised and enlarged edition, p. 495
.29
Mc Night
, T. L.
and Hess
, D.
, “Climate zone and types
,” in Physical Geography. A Landscape Appreciation
(Prentice Hall
, Upper Saddle River
, NJ
, 2000
), p. 200
.30
Movahed
, M. S.
and Hermanis
, E.
, “Fractal analysis of river flow fluctuations
,” Phys. A
387
, 915
–932
(2008
). 31
Muzy
, J. F.
, Bacry
, E.
, and Arneodo
, A.
, “The multifractal formalism revisited with wavelets
,” Int. J. Bifurc. Chaos
4
, 245
–302
(1994
). 32
Ozturk
, S.
, “Statistical correlation between b-value and fractal dimension regarding Turkish epicenter distribution
,” Earth Sci. Res. J.
16
(2
), 103
–108
(2012
).33
Press
, W. H.
, Teukolsky
, S. A.
, Vetterling
, W. T.
, and Flannery
, B. P.
, “Interpolation by kriging
,” in Numerical Recipes: The Art of Scientific Computing
, 3rd ed. (Cambridge University Press
, New York
, 2007
), Sec. 3.7.4
.34
Rodríguez
, R.
, Casas
, M. C.
, and Redaño
, A.
, “Multifractal analysis of the rainfall time distribution on the metropolitan area of Barcelona (Spain)
,” Meteorol. Atmos. Phys.
121
, 181
–187
(2013
). 35
Shimizu
,Y.
, Thurner
,S.
, and Ehrenberger
,K.
, “Multifractal spectra as a measure of complexity human posture
,” Fractals
10
, 104
–116
(2002
), SMC Meteorological Service of Catalonia, Environmental Department, www.meteocat, Government of the Generalitat de Catalunya
(2019
).36
Stein
, M. L.
, Statistical Interpolation of Spatial Data: Some Theory for Kriging
(Springer
, New York
, 1999
).37
Stoop
, F.
and Meier
, P. F.
, “Evaluation of Lyapunov exponents and scaling functions from time series
,” J. Opt. Soc. Am. B
5
, 1037
–1045
(1998
). 38
Talkner
, P.
and Weber
, R. O.
, “Power spectrum and detrended fluctuation analysis: Application to daily temperatures
,” Phys. Rev. E
62
(1
), 150
–160
(2000
). 39
Turcotte
, D. L.
, Fractal and Chaos in Geology and Geophysics
, 2nd ed. (Cambridge University Press
, Cambridge
, 1997
), p. 398
.40
Wiggins
, S.
, “Introduction to applied nonlinear dynamical systems and chaos
,” in Texts in Applied Mathematics
, 2nd ed. (Springer
, New York
, 2003
), Vol. 2, p. 844
.© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
