Theor Appl Climatol (2010) 102:75–85 DOI 10.1007/s00704-009-0242-6
ORIGINAL PAPER
Nonlinear dynamics of meteorological variables: multifractality and chaotic invariants in daily records from Pastaza, Ecuador Humberto Millán & Aleksandar Kalauzi & Milena Cukic & Riccardo Biondi
Received: 14 April 2009 / Accepted: 28 November 2009 / Published online: 29 December 2009 # Springer-Verlag 2009
Abstract Weather represents the daily state of the atmosphere. It is usually considered as a chaotic nonlinear dynamical system. The objectives of the present study were (1) to investigate multifractal meteorological trends and rhythms at the Amazonian area of Ecuador and (2) to estimate some nonlinear invariants for describing the meteorological dynamics. Six meteorological variables were considered in the study. Datasets were collected on a daily basis from January 1st 2001 to January 1st 2005 (1,460 observations). Based on a new multifractal method, we found interesting fractal rhythms and trends of antipersistence patterns (Fractal Dimension >1.5). Nonlinear time series analyses rendered Lyapunov exponent spectra
H. Millán (*) Department of Physics and Chemistry, University of Granma, Apdo. 21, 85100 Bayamo, Granma, Cuba e-mail:
[email protected] A. Kalauzi Department for Life Sciences, Institute for Multidisciplinary Research, Kneza Viseslava 1, Belgrade, Serbia e-mail:
[email protected] M. Cukic Laboratory for Neurophysiology, Institute for Medical Research, Clinical Center of Serbia, Dr Subotica 6, Belgrade, Serbia e-mail:
[email protected] R. Biondi Dipartimento di Ingegneria Elettronica e dell’Informazione, University of Perugia, Perugia, Italy e-mail:
[email protected]
containing more than one positive Lyapunov exponent in some cases. This sort of hyperchaotic structures could explain, to some extent, larger fractal dimension values as the Kaplan–Yorke dimension was also in most cases larger than two. The maximum prediction time ranged from ξ= 1.69 days (approximately 41 h) for E/P ratio to ξ= 14.71 days for evaporation. Nonlinear dynamics analyses could be combined with multifractal studies for describing the time evolution of meteorological variables.
1 Introduction Climate was defined as a set of atmospheric states of a dynamical, chaotic system showing deterministic variability (Lorenz 1993). That is, a set of atmospheric states characterizing a given area (Barry and Chorley 1985). In a more precise way, there are differences between weather, meteorology, and climate in terms of time scales for definitions. There exist many computational, numerical, and statistical methods for analyzing time series from meteorological and/or hydrological variables. However, most of them are linear by its nature. Classical methods (e.g., linear univariate or multivariate statistics) do not permit in many cases to extract relevant information as nonlinear components remain hidden. Analyzing time series imply decomposition into trend and oscillation (Balling et al. 1998), but in recent years, nonlinear approaches such as monofractal, multifractal and chaos methods have been introduced for investigating time series derived from physical processes (Deidda 2000; Valdez-Cepeda et al. 2003; Kurnaz 2004). In most published research on this subject, the underlying hypothesis is that an entire climatic or meteorological signal can be described with only one fractal parameter. Further-
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more, almost all estimated fractal dimensions are ≤1.5 which represent a persistent (< sign) or a purely Brownian (= sign) behavior of the studied process. Some symbolic and numerical methods derived from chaos theory have been also incorporated for processing atmospheric signals (Washington 2000; Gutiérrez 2004; Gutiérrez et al. 2006; Chaudhuri 2006; Primo et al. 2007). Ferguson and Messier (1996) combined fractal and chaos methods for investigating the impact of climate variability on ecological systems. Sivakumar (2004) presented a comprehensive revision on the application of chaos methods to geophysical events. In general, there is little information on the use of new multifractal methods with climatic or meteorological data (Mandelbrot 1989, 2003). Recently, Kalauzi et al. (2006) and Millán et al. (2008) incorporated the consecutive differences method developed by Kalauzi et al. (2005) for analyzing meteorological records from the Amazonian area of Ecuador. Those investigations were based on short time series from mean monthly observations over 31 years (N= 372 months). However, larger time series collected at shorter time intervals (e.g., on a daily basis) could reveal fine structures and chaotic invariants useful for a better understanding of meteorological dynamics. The objectives of the present study were (1) to investigate multifractal meteorological trends and rhythms at the Amazonian area of Ecuador and (2) to estimate some nonlinear invariants for describing the meteorological dynamics.
with stable slopes and Y-intercepts proportional to signal FD. One basic rationale underlying the consecutive finite difference scheme is based on comparing the empirical time series with a theoretical complex signal such as Weierstrass functions which usually simulate natural signals (e.g., meteorological time series). To search for relations between Yint and FD, it is then used a family of Weierstrass functions: X WHg ðtÞ ¼ g iH cos 2p i t ð3Þ i
with the parameters H (01.5) might be also associated to the underlying chaotic structure of each meteorological variable. In general, this interplay between multifractality and chaoticity needs to be addressed in future works for a better understanding of the underlying climate dynamics. In addition, it is possible the existence of time scales where 1/f noise is the dominant phenomenon driving the rainfall pattern (FD=2, Hurst exponent, H=0). This is a widespread natural phenomenon which has been interpreted within a self-organized criticality perspective (Bak et al. 1987). The 1/f noise time scales include components of all durations. We have advanced two possible explanations for those FD values around, in some cases over, the classical fractal limit (FD=2) even though we are well aware that atmosphere represents a complex system with
many active degrees of freedom. From our point of view, the present study is consistent with the concept of climate advocated by Lorenz (1993) as we found positive Lyapunov exponents, a test of chaotic dynamics, and deterministic components (Δ values).
5 Conclusions We have used a new multifractal method and nonlinear time series analysis with some raw meteorological variables collected from the Amazonian area of Ecuador. In all cases, fractal rhythms followed almost the same oscillation patterns as those corresponding to meteorological dynamics. Even though deterministic and random patterns were preserved, each spectrum was composed by fractal dimen-
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sion values larger than 1.5 and very near to 2 or over that value for the case of rainfall sequence. This accounts for some sort of flicker noise within the considered time scale. We estimated some nonlinear parameters for interpreting, to some extent, the computed multifractal spectra. The nonlinear time series analysis detected Kaplan–Yorke dimensions larger than 2 for most time series and positive Lyapunov exponents for the six meteorological variables. Some meteorological variables can include a large deterministic component (e.g., relative humidity and relative sunshine duration) with a large complex dynamics (e.g., positive Lyapunov exponents). Future works using fractal approaches need to include nonlinear time series analyses to improve future interpretations.
Acknowledgments This work was supported by the Ministry of Science and Environmental Protection of the Republic of Serbia (projects 143045 and 143027). We thank National Institute of Meteorology and Hydraulic Resources (I.N.A.M.H.I, Ecuador) for access to the climatic database. The present investigation was conducted while the first author served as an invited professor of Environmental Physics at Amazonian State University (Ecuador).
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