Multifractal Structures in Temperature and Humidity

$Multifractal Structures in Temperature and Humidity$

Journal of the Korean Physical Society, Vol. 57, No. 2, August 2010, pp. 296∼299 Brief Reports

Dong-In Lee and Cheol-Hwan You Department of Environmental Atmospheric Sciences, Pukyong National University, Busan 608-737

Jae-Won Jung and Ki-Ho Chang National Institute of Meteorological Research, Seoul 156-720

Soo Yong Kim Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701

Kyungsik Kim∗ Department of Physics, Pukyong National University, Busan 608-737

(Received 6 January 2010)

The multifractal structure of the temperature and the humidity is investigated in seven cities of Korea. For our cases, we estimate the generalized Hurst exponent, the Renyi exponent, and the singularity spectrum from tick data of the temperature and the humidity. We mainly discuss the diﬀerent values of the scaling exponent characterizing the multifractality from the hourly data for thirty years. To analyze the multifractality of the temperature and the humidity, we compare the multifractal properties of seven cities (Gwangju, Mokpo, Busan, Seoul, Cholwon, Gangrung, and Sokcho) from our results and discuss the unusual statistical behavior of each city.

PACS numbers: 05.45.-a, 05.10.-a, 05.40.-a, 89.75.-k Keywords: Generalized Hurst exponent, Renyi exponent, Singularity spectrum, Temperature, Humidity DOI: 10.3938/jkps.57.296

After the pioneering work of Mandelbrot [1], many in- a dynamical system [2]. An alternative approach based vestigations have extended a fractal nature to find in sci- on a generalization of the detrended fluctuation analysis entific models, such as the coast line, mountains, ocean method [11,12] has been proposed recently. This treated tomography, cloud drifts, rainfall, deposition models, the analysis of nonstationary time series by incorporat- and so on. Since the considerable attention [2-5] has ing a detrending procedure. The multifractal detrended focused on the fractal nature, several studies have unan- fluctuation analysis can be considered to be an exten- imously indicated the presence of a fractal nature and sion of the conventional detrended fluctuation analysis the possibility of multi-scaling in atmospheric phenom- [13], which is a scaling analysis technique of a time series ena. The multifractals can be modeled as a stochastic [14-16]. turbulent cascade process [6], and the cascade process The purpose of this letter is to analyze and simulate in fully developed turbulence has shown the breaking-up the multifractals of the temperature and the humidity in of eddies into the smaller sub-eddies due to the energy seven cities, i.e., Gwangju, Mokpo, Busan, Seoul, Chol- dissipation [7]. won, Gangrung, and Sokcho, of Korea. We mainly es- Furthermore, the multifracal formalism is a widely timate the generalized Hurst exponent, Renyi exponent, used technique for the delineation of fractal scaling prop- and singularity spectrum for tick data of the temperature erties in non-stationary time series [8-10]. Complicated and the humidity. systems that consists of components with many interwo- To analyze the differences in the temperature and the ven fractal subsets of the time series have multitudes in humidity, we present the detrended fluctuation analysis scaling exponents. The generalized dimension and the method in the following steps: spectrum can be extended to describe the dynamic characteristics from the fractal structure in the phase space of (1) Consider a correlated time series, x(i), for i = 1, 2, ..., N, with N being the total number of a corre- ∗Corresponding author; E-mail: [email protected]; Tel: +82-51- lated time series. The time series x(i) is integrated Pk 629-5562; Fax: +82-51-629-5549 to obtain z(k) = i=1(x(i)− < x >), where < x > means the average value of x(i); -296- Multifractal Structures in Temperature and Humidity – Dong-In Lee et al. -297-

Fig. 1. Generalized Hurst exponent of the temperature for Fig. 2. Plot of the Renyi exponent τ(q) of the temperature four seasons in seven cities (Gwangju, Mokpo, Busan, Seoul, for four seasons in seven cities. Cholwon, Gangrung, and Sokcho).

(2) z(k) is divided into boxes of equal size s;

(3) For each box, z(k) is fitted using a polynomial function. The y coordinate of the fitting line in each box is denoted by zn(k), and we denote the algorithm as the detrended fluctuation analysis;

(4) The integrated series z(k) is detrended by subtract- ing the local trend zn(k) within each box;

2 (5) The variance, Fν (s), for the integrated and detrended series is calculated as

s 1 X F 2(s) = {z[(ν − 1)s + i] − z (i)}2 (1) ν s ν i=1 Fig. 3. (Color online) Plot of the singularity spectrum f(α) of the temperature for four seasons in seven cities. for each segment ν, ν = 1, 2, ..., Ns, where Ns ≡ int(N/s) is the number of segments with size s over a time series. In order to describe the degree of contribution of a cer- and tain segment with size s to fluctuations of a time series, the q−th order fluctuation function can be represented f(α) = q[α − h(q)] + 1 (5) in terms of via a Legendre transform, where the partition function 1/q τ(β) " 2Ns # Z(β) scales as , where means the length unit, and 1 X F (s) = (F 2(s))q/2 ∼ sh(q). (2) τ(β) is known as the Renyi exponent [4]. The generalized q 2N ν s ν=1 dimension D(q) is defined as τ(q) qh(q) − 1 The relation between the Renyi exponent τ(q) and the D(q) ≡ = , (6) multifractal scaling exponent h(q) is obtained as q − 1 q − 1 which is used interchangeably with the Renyi exponent. τ(q) = qh(q) − 1. (3) The strength of the multifractality [17] is presented as The Lipschitz-Hölder exponent α and the singularity ∆α = αmax − αmin, (7) spectrum f(α) are, respectively, calculated as where this can be quantified by the difference between α = h(q) + qh0(q) (4) the maximum and the minimum values of α. For a -298- Journal of the Korean Physical Society, Vol. 57, No. 2, August 2010

Table 1. Multifractal strength ∆α shown for the temper- the multifractal property is strongly observed while in ature in seven cities. Seoul it is weakly observed. Hence, Busan has no monofractal property, indicating that its characteristic of the Cities ∆α αmax αmin humidity is quite different from that of the other cities. Gwangju 1.263 1.830 0.570 In conclusion, we have studied the multifractal de- Mokpo 1.275 1.861 0.590 trended fluctuation analysis of the temperature and the Busan 1.223 1.828 0.600 humidity for four seasons in seven cities of Korea. In all Seoul 1.252 1.820 0.570 seasons of the seven cities, the multifractal strength of Cholwon 1.255 1.808 0.550 the temperature has similar values, significantly different Gangrung 1.212 1.815 0.600 from the feature of the rainfall [12] and the humidity. In particular, for the humidity, the multifractal strength of Sokcho 1.213 1.863 0.650 Busan (Seoul) is larger (smaller) than those for the other cities. Table 2. Multifractal strength, ∆α, for the humidity in We will analyze regions of other countries for the sake seven cities. of more detailed study of multifractal characteristics. In the future, we expect the multifractal detrended fluctu- Cities ∆α αmax αmin ation analyses to be used to both discriminate and char- Gwangju 0.353 1.167 0.814 acterize for various meteorological phenomena of other Mokpo 0.299 1.148 0.849 countries, as has been done for the financial market, the Busan 0.598 1.542 0.944 complex heartbeat dynamics, and the brain and neuron Seoul 0.333 1.128 0.795 networks. Cholwon 0.453 1.310 0.857 Gangrung 0.538 1.407 0.869 Sokcho 0.591 1.493 0.902 ACKNOWLEDGMENTS

This work was supported by the Korea Re- search Foundation(KRF) grant funded by the Korea monofractal time series, the generalized Hurst exponent government(MEST)(No.2009-0074635) and by the Ko- h(q) [15] is expected to be independent of q, so that from rea Foundation for International Cooperation of Science Eqs. (4) and (5) α is constant over all q. We estimate the & Technology (KICOS) through a grant provided by the strength of multifractality of the time series in terms of Korean Ministry of Education, Science & Technology the range of α in order to identify its multifractal prop- (MEST) in 2009 (No.K20607010000). erties. We analyze the temperature and the humidity time series for four seasons (spring, summer, autumn, and winter) in seven cities on the Korean peninsula. Fig- REFERENCES ure 1 shows the temperature tick data for seven cities (Gwangju, Mokpo, Busan, Seoul, Cholwon, Gangrung, [1] B. B. Mandelbrot, The Fractal Geometry of Nature and Sokcho), recorded from January of 1977 to Decem- (Freeman, San Francisco, 1983). ber of 2006, where the time lag is one hour. [2] S.-G. Lee and S. B. Lee, J. Korean Phys. Soc. 52, 209 From Eq. (2), we find the generalized Hurst exponent (2008). h(q) as a function of q. τ(q) is obtained after substituting [3] J. S. Kim, B. Kahng, D. Kim and K.-I. Goh, J. Korean Phys. Soc. 52, 350 (2008). h(q) into Eq. (3). As shown in Figs. 1 and 2, the gen- [4] K. E. Lee and J. W. Lee, J. Korean Phys. Soc. 50, 178 eralized Hurst exponent and the Renyi exponent can be (2007). found for the temperature of the seven cities (Gwangju, [5] S.-H. Kim, J. Korean Phys. Soc. 54, 809 (2009). Mokpo, Busan, Seoul, Cholwon, Gangrung, and Sokcho) [6] G. Parasi and U. Frisch, in Proceedings of the Interna- during four seasons. tional School on Turbulence and Predictability in Geo- Let us show the multifractal strength ∆α from now on. physical Fluid Dynamics and Climate Dynamics, edited The multifractal strength of the temperature in Table 1 by P. Huerre and P. Coullet (Plenum, New York, 1988). can be obtained from the result shown in Fig. 3. Among [7] J. F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. Lett the seven cities we addressed, the multifractal strength 67, 3515 (1991). of Mokpo is largest, that is, ∆α = 1.275. This means [8] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. I. Shraiman, Phys. Rev. A 33, 1141 (1986). for the temperature that Mokpo is the city in which the [9] T. Vicsek, F ractal Growth P henomena (World Scien- multifractal property is strongly observed. As calculated tific, Singapore, 1988). in Table 2, the case of the humidity is different from that [10] G. Paladin and A. Vulpiani, Phys. Rep. 156, 147 (1987). of the temperature. That is, the multifractal strength [11] J. W. Kantelhardt, S. A. Zschiegner, E. K. Bunde, S. ∆α of the humidity in Busan is 0.598 while that in Seoul Havlin, A. Bunde and H. E. Stanley, Physica A 316, 87 is 0.333. This means that Busan is the city in which (2002). Multifractal Structures in Temperature and Humidity – Dong-In Lee et al. -299-

[12] D. I. Lee, C. H. You, S. Y. Kim, I. H. Park and K. Kim, Pecknold, J. Geophys. Res. 101, 427 (1996). J. Korean Phys. Soc. 55, 2344 (2009). [16] P. Norouzzadeh and B. Rahmani, Physica A 367, 328 [13] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian and J. (2006) D. Farmer, Physica D 58, 77 (1992). [17] Z.-Q. Jiang and W.-X. Zhou, Physica A 387, 3605 [14] K. Hu, P. Ch. Ivanov, Z. Chen, P. Carpena and H. E. (1984); ibid, 4881 (2008). Stanley, Phys. Rev. E 64, 011114 (2001). [15] Y. Tessier, S. Lovejoy, P. Hubert, D. Schertzer and S.