Determination of the dynamical behavior of rainfalls by using a ...

Journal of the Korean Physical Society, Vol. 61, No. 4, August 2012, pp. 658∼661

Brief Reports

Determination of the Dynamical Behavior of Rainfalls by Using a Multifractal Detrended Fluctuation Analysis Seong Kyu Seo and Kyungsik Kim∗ Department of Physics, Pukyong National University, Busan 608-737, Korea

Ki-Ho Chang and Young-Jean Choi Hydrometeorological Resources Research Team, Meteorological Research Institute, Seoul 156-720, Korea

Keunyong Song Meteorological Industry Promotion Research Department, Korea Meteorological Industry Pomotion Aency, Seoul 110-101, Korea

Jong-Kil Park School of Environmental Sciences Engineering, Inje University, Gimhae 621-749, Korea (Received 17 May 2012, in final form 13 July 2012) We study the dynamical behavior of rainfalls in the complicated area of the Korean peninsula. Data for stations having no automatic weather system are simulated from those of ten stations with automatic weather stations via the ordinary kriging interpolation method. Using a multifractal detrended fluctuation analysis, we analyze all the stations from time series data for rainfalls. We are able to find forty high-ranked stations having no automatic weather system from the multifractal strength. In particular, our result may be useful and effective in determining the locations of new automatic weather stations and establishing the meteorological apparatus. PACS numbers: 05.10.-a, 05.45.Df, 89.75.-k, 89.90.+n Keywords: Multifractal detrended fluctuation analysis, Rainfall, Multifractal strength, Ordinary kriging interpolation DOI: 10.3938/jkps.61.658

The potential impact of complex systems has recently gained a great deal of interest and importance owing to new and practical computer-simulations for static and dynamical behaviors [1] in natural, social, engineering, and medical sciences. The methods and techniques used for complex systems have usefully contributed to research such as the intermittent nature of turbulence [2], various financial time-series [3], wavelet transform approaches [4,5], growing and non-growing networks [6–8], atmospheric phenomena of the climate change, seismic phenomena [9], etc. Among network theories, of interesting subjects are small-world and scale-free networks [6,10] that have led to many new insights into complex systems [11-13]. The last two decades have witnessed much effort and progress in efficient network theory for complex systems, as well as several practical attempts at simulating and analyzing the complex network of climate change.

The climate of each nation is really caused by several statistical factors [14–18] such as rainfall, temperature, humidity, and wind velocity. Atmospheric phenomena on the Korean peninsula, which is surrounded by the East, Yellow, and South Seas, is known to be mainly associated with the air-sea interaction under the influence of its seas in diverse aspects [19]. We present the multifractal property of stochastic elements based on the assumption that the process for meteorological variables can be modeled as a stochastic turbulent cascade process [20]. The cascade process in fully developed turbulence is well known to describe a breaking-up of eddies into smaller sub-eddies in terms of energy dissipation [21]. The multifractal formalism [22,23] is a widely useful technique for the delineation of fractal scaling properties in non-stationary time series. Complicated systems that consist of components with many interwoven fractal subsets of the time series have a multitude of scaling exponents. The generalized dimension and spectrum are introduced in order to extract the dynamic characteristics from the fractal structure in the

∗ E-mail: [email protected]; Tel: +82-51-620-6354; Fax: +82-51611-6357

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Determination of the Dynamical Behavior of Rainfalls by Using · · · – Seong Kyu Seo et al.

phase space of a dynamical system [24]. The multifractal detrended fluctuation analysis is one of the main analysis tools for detecting and analyzing fractal scaling properties in non-stationary time series. This analysis is considered as an extension of the conventional detrended fluctuation analysis [25], which is a scaling analysis technique for a non-stationary time series [26–28]. Several published papers [29–31] have recently proposed an alternative approach based on a generalization of the detrended fluctuation analysis method. The alternative approach was basically devised for the analysis of non-stationary time series by incorporating with a detrending procedure. In this paper, our purpose is to study the dynamical behavior of rainfalls in the Andong area on the Korean peninsula. Data for the variability analysis are timeseries data on the summertime rainfall for five years. The method is simulated from the automatic weather system data of ten stations by using the ordinary kriging interpolation [32–34]. We analyze the time series data of stations having no automatic weather system by using the simulated multifractal detrended fluctuation analysis. We focus on the multifractal strength via the multifractal detrended fluctuation analysis of the time series of the rainfall in the climate change. Firstly, the difference of the rainfall between time t and t + τ is defined by xi (τ ) = Si (t + τ ) − Si (t), i = 1, . . . , N , where Si (t) is the i-th rainfall at time t. For the time series of rainfalls N , the profile Y (i) is defined by Y (i) =

i


[xk − < x >] ,

i = 1, . . . , N,

(1)

k=1

where xk (= xk (τ )) is the k-th time series for the difference between two rainfalls, and xk − < x > is the deviation for the difference of two rainfalls. The variances are given by F 2 (v, s) =

 Fq (s) ≡

2N q/2 1
s  2 F (v, s) 2Ns v=1

1/q ∼ sh(q) ,

(4)

where h(q) is called the generalized Hurst exponent. When q = 2, this is the Hurst exponent describing the rainfall feature of the time series. The relation between the Renyi exponent τq [29] and the generalized Hurst exponent hq is analytically obtained as τq = qhq − 1.

(5)

The Lipschitz-H¨older exponent αq [35] is related to the singularity spectrum fα [36] as αq = hq + qdhq /dq

(6)

fα = q[αq − hq ] + 1

(7)

and

via a Legendre transform, where the partition function Zβ scales as ω τβ , and ω and β mean the length unit and the order, respectively. τ β is known as the Renyi exponent [3]. The generalized dimension Dq is defined as Dq = τq /(q − 1) = (qhq − 1)/(q − 1),

(8)

which is used interchangeably with the Renyi exponent. The multifractal strength ∆α [15] can be quantified by using the difference between the maximum and the minimum values of αq , which is given by (9)

(2)

and F 2 (v, s) =

to scan the degree of contribution of a certain segment with size s to fluctuations of a time series, we define the q-th order fluctuation function [29] as

∆α = αmax − αmin ,

s

1
{Y [(v − 1)s + i] − yv (i)}2 s i=1

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s

1
{Y [N − (v − Ns )s + i] − yv (i)}2 , (3) s i=1

where Yv (i) is the fitting polynomial described by the local trend, and s is the number of time series, and v = 1, . . . , Ns for each segment of the game v. Ns ≡ int(N/s) is the number of segments with size s over a time series, and N is the length of the rainfall series. The length N of the time series is often not a multiple of the time scale s, i.e., N = Ns s + N  . In order to include the information contained in the remaining part N  , after the time series is split into Ns intervals of length s starting from the front in Eq. (2), we split the time series into Ns intervals of length s starting from the back in Eq. (3). In order

where ∆α is an important parameter describing the width of the multifractal spectrum in the rainfall. The larger the ∆α is, the stronger the multifractality is. Hence, for the time series of the rainfall, we can estimate the multifractal strength in terms of the range of αq in order to identify the multifractal properties of the rainfall. As unobserved stations, we choose 63-by 63 (3,969) stations in the Andong area of the Korean peninsula in order to use measurements of rainfalls. The kriging interpolation allows one to tackle problems of making estimates for unsampled locations. The kriging algorithm accounts for data solely being continuously estimated and for incorporating secondary information. From the linear regression paradigm, three of its most important variants are the simple kriging, the ordinary kriging, and the kriging with a trend model. The multivariate colocated cokriging primarily describes several parameters,

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Journal of the Korean Physical Society, Vol. 61, No. 4, August 2012 Table 1. Values of the multifractal strength ∆α in the 10 high-ranked stations. Number 1 2 3 4 5 6 7 8 9 10

Fig. 1. (Color online) Plot of 63-by-63 (3,969) stations having no automatic weather system for the rainfall calculated by using the ordinary kriging interpolation on the 15th July of 2009. The color bar on the right side denotes the height, where the unit is meters above sea level.

Fig. 2. (Color online) Plot of 40 high-ranked stations (blue squares) obtained from the multifractal strength ∆α by using a multifractal detrended fluctuation analysis, where the red circles denote the 10 stations having an automatic weather system.

including the height correction. However, the kriging interpolation method can be used to calculate as some important properties such as weights and variances. The ordinary kriging interpolation [34] is one of the methods with the least errors among various interpolations, and it interpolates known sample data in the light of the distance and the degree of change. We apply this inter-

Latitude 36.77710 36.77710 37.02940 36.82210 36.85820 36.86720 36.88520 36.62380 36.88520 36.88520

Longitude 129.00900 128.98600 128.82900 128.86300 128.84000 128.87400 128.85200 128.59300 128.91900 128.89700

∆α 3.743 3.399 3.388 3.111 3.019 3.016 2.994 2.952 2.938 2.929

polation to the rainfall calculation for stations having no automatic weather system. The data that we used for the variability analysis are hourly time series of the summertime rainfall for five years from Jan 2005 to Dec 2009 from the ten stations with existing automatic weather system. We analyzed time series data of 3,969 stations without automatic weather systems by using the ordinary kriging interpolation. Using a multifractal detrended fluctuation analysis, we identified the 10 high-ranked stations from the multifractal strength ∆α, which indicates the intensity of multifractals. Figure 1 is a plot of the rainfall calculated by using the ordinary kriging interpolation on the 15th July of 2009 for 63-by-63 (3,969) stations having no automatic weather systems. The color bar on the right side represents the height, where the unit is meters above sea level. Figure 2 shows the 40 high-ranked stations having no automatic weather systems that were obtained from the values of the multifractal strength ∆α by using the multifractal detrended fluctuation analysis. Here, the blue squares denote the 40 high-ranked stations, and the red circles denote the 10 stations with automatic weather systems. Table 1 summarizes the values of the multifractal strength for the 10 stations having no automatic weather systems. These values were obtained from an ordinary kriging interpolation based on data from stations with automatic weather systems. The location observing a large rainfall change has a strong multifractal feature rather than a fractal property. It is proper to establish a new automatic weather station at a location observing a large rainfall change. In conclusion, we have studied the multifractal features of rainfalls for the summer season in the Andong area of the Korean peninsula. We found the 10 stations (among the 40 stations) based on the multifractal strength ∆α of rainfalls obtained by using a multifractal detrended fluctuation analysis, as shown in Fig. 2. The larger the value of the multifractal strength ∆α is, the stronger the observed multifractal property is (or the

Determination of the Dynamical Behavior of Rainfalls by Using · · · – Seong Kyu Seo et al.

more its volatility changes). When the rainfall volatility for all the stations having no automatic weather system is considered in the Andong area, the 10 high-ranked stations are found to be mostly concentrated in regions along a river. The 1st, 2nd, and 5th – 7th regions have been well known to be the regions with automatic weather stations having localized heavy rainfalls year by year. Furthermore, if meteorological quantities are to be measured, our results should be particularly useful and effective in determining new automatic weather stations and establishing the meteorological apparatus. According to increased correlation between meteorological variables and elevation, a much improved result may be obtained. We anticipate that our capability can be extended to a study of using the multivariate colocated cokriging to map meteorological variables for cases having no automatic weather system. For the sake of a more detailed investigation of the multifractality, we need to extend our research to other regions of our peninsular in the future. This multifractal detrended fluctuation analysis formalism can be extended to encompass both the discrimination and the characterization of rainfall, temperature, wind velocity, and humidity.

ACKNOWLEDGMENTS This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea Ministry of Education, Science and Technology (MEST) (No. 2011-0029809), by Center for Atmospheric Sciences and Earthquake Research (CATER 2012-6110), and by the “Hydrometeorology research for the test-bed region” of the National Institute of Meteorological Research (NIMR) funded by the Korea Meteorological Administration (KMA).

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