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Environ. Sci. Technol. 2010, 44, 4629–4634

Studying the Time Scale Dependence of Environmental Variables Predictability Using Fractal Analysis YUVAL* AND DAVID M. BRODAY Department of Civil and Environmental Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel

Received November 18, 2009. Revised manuscript received April 18, 2010. Accepted May 4, 2010.

Prediction of meteorological and air quality variables motivates a lot of research in the atmospheric sciences and exposure assessment communities. An interesting related issue regards the relative predictive power that can be expected at different time scales, and whether it vanishes altogether at certain ranges. An improved understanding of our predictive powers enables better environmental management and more efficient decision making processes. Fractal analysis is commonly used to characterize the self-affinity of time series. This work introduces the Continuous Wavelet Transform (CWT) fractal analysis method as a tool for assessing environmental time series predictability. The high temporal scale resolution of the CWT enables detailed information about the Hurst parameter, a common temporal fractality measure, and thus about time scale variations in predictability. We analyzed a few years records of half-hourly air pollution and meteorological time series from which the trivial seasonal and daily cycles were removed. We encountered a general trend of decreasing Hurst values from about 1.4 (good autocorrelation and predictability), in the subdaily time scale to 0.5 (which implies complete randomness) in the monthly to seasonal scales. The air pollutants predictability follows that of the meteorological variables in the short time scales but is better at longer scales.

1. Introduction Prediction of future weather and the associated air quality is an important scientific field. The quest for better forecasts prompts better understanding of the atmosphere dynamics and thermodynamics, and of the dispersion, advection, and chemical transformation of pollutants. Better forecasts are also important for practical purposes. In the short term they serve the public by providing information on future weather and air quality conditions. Predictions for the longer term are required for better planning and management. Interesting issue related to weather and air quality forecasts regards the predictability limits. How well meteorological and air quality variables can actually be predicted at different time scales? And, are there time scales in which prediction schemes are destined to fail because of random behavior of the variables at those scales? The answers lie with the correlation between the atmospheric system’s states, in other words, the interaction between the states and the length of the memory in the system. Random behavior of a variable at certain time scale * Corresponding author phone: +972 4 8292767; fax: +972 4 8295696; e-mail: [email protected]. 10.1021/es903495q

 2010 American Chemical Society

Published on Web 05/13/2010

means zero correlation and nil predictability. The stronger the correlation between the system states, the better are the prospects for an accurate prediction of its variables. The physical state of a system is described by a set of values of its physical variables. A place to start studying the correlation between the system’s states is a study of the variables’ time series. Many studies (e.g., refs 1-4) demonstrated the self-affine nature of meteorological and air quality variables. Self-affine series are characterized by a power spectrum that scales as a power law with frequency (5). The power law exponent is a good measure of the series’ autocorrelation, and thus their persistence and predictability. Numerous methods were suggested for estimating the strength of power law relationships in self-affine series (6). Among the ones applied for analyses of atmospheric variables one can find the rescaled Hurst analysis, originally introduced by ref (7) as an early attempt to characterize the Nile river flow fluctuations but that is still in use (e.g., ref (1)), and the power spectrum method (8). The frequent non-stationary nature encountered in data series lead to the use in more recent works of the Detrended Fluctuations Analysis (DFA; 2, 9, 10), and wavelet based methods (2, 11, 12). Each of the different fractal analysis methods provides a measure which characterizes the level of persistence in the data series. The many different measures calculated by the various methods are usually related by simple arithmetic relationship. In this work we refer to H, or the Hurst parameter. Values of H < 0.5 point to anti-persistency (i.e., lower than average value tends to be followed by a higher than the average value and vice versa) and are obtained in case of negative autocorrelations. Value of H ) 0.5 signify complete randomness. Cases with H > 0.5 are those with positive autocorrelation. They exhibit long memory that enables predictability. The larger is H the better is the predictability. Fractal analyses of meteorological data usually found power law relationships spanning surprisingly long-range, typically in the time scales from 10 days to decades (2, 13, 14). This finding points to a characteristic long memory in the meteorological variables beyond 10 days. However, (8, 15, 14), and (10) encountered cases of different governing power laws at different time scale ranges. Such temporal dependence of the fractal nature of a series was discussed from a theoretical point of view by (9) and (11), and studied using numerical experiments by (15). In general, long memory characteristics were used to gain insight into the nature of the atmosphereocean system, for example, typically longer memory is found in marine temperature data series than in terrestrial ones (13). The long memory characteristics are also very often suggested as benchmarks for testing simulations by weather and air pollution numerical models (4). In this work we propose as an additional role for fractal analysis the assessment of variables’ predictability at various time scales. As a suitable tool for that purpose we introduce the Continuous Wavelet Transform (CWT) fractal analysis. We show that the CWT method can accurately estimate the H value of a time series at any desired time scale resolution that the data’s temporal resolution warrants. Our implementation of the CWT uses a fast computer code that enables rapid analysis of very long series. Using data with sub-hourly resolution of a few environmental variables, we show variations in their series’ power laws between the time scales of a few hours to a few months. These variations in the power laws imply variability in the persistence of the series and in their potential predictability. VOL. 44, NO. 12, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 1. (a) log-log curve calculated for 8 years of half-hourly PM2.5 concentrations in Nave Shaanan, Haifa, using the DFA method. Linear fits to three segments of the curve are superimposed on it (see the text for more details). The scale is in hours. (b) Like (a) but for the DWT method. The scale is dyadic (i.e., powers of 2). (c) Like (a) but for the CWT method. The scale is in hours. (d) The H spectrum derived from the log-log curve in (a) and the H values derived from the linear fits. (e) Like (d) but for the DWT method. (f) Like (d) but for the CWT method.

2. Materials and Methods 2.1. Data. We developed our methodology using half-hourly meteorological and air pollution time series observed at air quality monitoring stations in Israel. We examined records of 14 different variables from 38 monitoring stations. The examples shown in this work use the data from stations Nave Shaanan and Nesher, Haifa, which observed a large number of variables for up to 10 years. In all cases the daily and seasonal means were removed from the data to take off the trivial variability in the data series. This operation was carried out by subtracting from each datum the mean of its calendarian time point (i.e., the mean of points in the same calendarian day and the same hour in the day). Because of the relatively small number of years in the available time series (and thus a small number of time points to calculate each calendarian mean), the calculation included data of the relevant calendarian day and its adjacent six days (e.g., the calendarian mean of 15 January at 20:00 was calculated using the time points on January 12-18 at 20:00 in all the years in the study period). This results in 42-70 time points (depending on the variable) to calculate each mean. A seven days averaging period is short compared to the seasonal time scale so the accuracy of the seasonal means that we used was not compromised. 2.2. Fractal Analysis Methods. We briefly review here two state of the art fractal analysis methods and then elaborate on the details of the proposed method and how it can be 4630

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used for an estimation of a continuous fractal measure as function of the time scale. The Detrended Fluctuation Analysis (DFA) was introduced by (16) to quantify long-range power law correlation in nonstationary time series and is probably the most commonly used in many fields. A data series yi is integrated to yield xi i ) Σj)1 yj for i ) 1, · · · , N. The integrated series is divided into non-overlapping segments of length m and a polynomial trend xˆ is fit to the elements in each segment. The fluctuation of the detrended elements in each segment is given by F(m) m )1/mΣj)1 (xj - xˆj)2. For self-affine time series, the segments’ mean fluctuation is related to the scale of the segments’ length j by a power law F(m) ∼ m2H, and H can be found by a j regression of of log (F(m)) against log (m), using a number of division schemes with different segment length m. Figure j 1a shows the log-log curve of F(m) against the length scale m calculated for the 1999-2006 half-hourly PM2.5 (aerosols with effective diameter smaller than 2.5 µm) concentrations series (140 256 time points) observed in Nave Shaanan. The Discrete Wavelet Transform method (DWT; 11, 17) is a method to quantitate the long-range correlation in a time series which exploits the capability of the wavelet transform to deal with non-stationarities. The data series yi, i ) 1, · · · , N is passed through a quadrature mirror highpass and low-pass convolution filter pair which splits equally the frequency content in the series. The output of the low pass filter is then used as an input for the quadrature filter

pair, and the process is repeated k times. This results in a filter bank of wavelet coefficients at k levels. For a self-affine series, the variance of the wavelet coefficients V(k) is scaled by the power law V(k) ∼ (2k)β, with H ) (β + 1)/2 (17). H can be found by a regression of log (V(k)) against log (2k). Figure 1b shows the log-log curve of V(k) against 2k calculated for the PM2.5 data from Nave Shaanan. Note that the abscissa is a non-dimensional coordinate, unlike the abscissa in Figure 1a which is in units of hours. In the following, we derive information in the temporal domain from the DWT log-log curve by looking at the dominant period in the Fourier transform of the DWT coefficients at each dyadic split of the power spectrum, and transforming this period to the time domain. The Continuous Wavelet Transform method (CWT) follows the same basic idea of the DWT method but uses the continuous wavelet transform. Consider the self-affine series yi, i ) 1, · · · , N. Its CWT coefficients are given by W(t, s) )

1 √s

∫

∞

-∞

( )

ˆt - t y(tˆ)ψ* dtˆ s

(1)

where s is the scale, ψ is a fast decaying continuous function in both time and frequency domains (usually referred to as the analyzing wavelet), and the * represents the complex conjugation operation. Using a range of s > 0 values results in a scalogram representation of the series y in the timescale space. Such a scalogram, along with examples of the CWT coefficients at four selected scales is shown in Figure S1 in the Supporting Information. For a self-affine time series the variance V(s) of the CWT coefficients W(t, s) relates to the scale by a power law V(s) ∼ sβ. The exponent β can be estimated as the slope of log (V(s)) against log (s) and the corresponding Hurst parameter H is given by H ) (β + 1)/2 (17). Equation 1 is a convolution of the input series y with the wavelet function ψ. An efficient computation of the convolution can be carried out in the frequency domain through multiplication of the Fourier transform of y by the Fourier transform representation of ψ at the correct frequency domain scale S ) πN/s. An inverse transform of the multiplication result yields the coefficients W(t) at scale s. We carried out this operation using the CWT transform Matlab function of the WaveLab850 software package (18). Computation of the log-log curve of a 170 000 time point series takes typically less than 30 s of CPU time on a single processor Pentium 4 PC. Using ψ which is the nth derivative of a fast decaying function ensures n vanishing moments and removal of polynomial trends of order n - 1 (19). We used the second derivative of the Gaussian as our analyzing wavelet, which ensures removal of linear trends from the data. Figure 1c shows the log-log curve of the variance of the CWT coefficients against the scale, calculated for the PM2.5 data from Nave Shaanan. Like in Figure 1a for the DFA method, the abscissa is in true temporal units. We compared the three methods described above for estimating the Hurst parameter of a self-affine time series. The comparison followed a similar comparison by (17) but we included the CWT method (not included by (17)) as one of the competitors, and we designed the synthetic self-affine series on which we tested the methods’ performance such that their statistical parameters simulate those of typical environmental variable series. The comparison established that the error of the DFA, DWT, and CWT methods in recovering the true H is almost always smaller than 5%. The CWT and DWT method have advantage over the DFA method in the very low H values. Details of the comparison scheme and its results are given in the Supporting Information. 2.3. Spectrum of Hurst Parameters in the Temporal Scale Domain. Log-log plots of the fractal analyses of 8 years of half-hourly PM2.5 time series using the DFA, DWT, and

CWT methods are given in Figures 1a-1c, respectively. The curves in Figures 1b and 1c (DWT and CWT methods) are clearly concave. The dynamical ranges of the abscissa and ordinate values in the DFA (Figure 1a) results in an aspect ratio that visually may tempt one to consider it a straight line. However, a fitted line to the DFA log-log curve between the time scales of three hours and more than a year (≈ 9000 h) clearly shows that like in the case of the corresponding DWT and CWT curves, the residuals to the linear fit are correlated, which means that the log-log curve is not a straight line and thus the analyzed PM2.5 time series is not a monofractal. Our experience is that the DFA tends to produce log-log curves that may sometimes visually look linear even in cases where they are not straight. The DWT and CWT analyzes leave usually no such doubt. The H values derived from linear fits to the curves in Figures 1a-1c between the time scales of three hours and more than a year (≈ 9000 h) are similar, 0.75, 0.80, and 0.84, respectively. We also considered the PM2.5 time series as a doufractal and fitted two straight lines, at corresponding time scales, to segments in the log-log curves in Figures 1a-1c. The respectively corresponding H values are shown in Figures 1d-1f. The H values are (for DFA, DWT, and CWT, respectively) 0.95, 1.03, and 1.42 for the segments within scales of 3 h and a day, and 0.67, 0.63, and 0.65 for the segments spanning the time scales of about a week (168 h) to more than a season (≈ 2800 h). Only four points on the DFA and DWT log-log curves are available for calculation of H in the short time scales. Given that the beginning and end of the log-log curves are not reliable, the CWT value, computed while ignoring the first five points in the curve, is probably more accurate. The three methods thus yield similar estimates of H in the longer time scales when linear fits are imposed on the log-log curves. However, in the case of the curve produced by the CWT (Figure 1c), extracting H values using linear fits is neither the only, nor the best option. The fine scale resolution enables computation of a continuous spectrum of the Hurst parameter from the varying slope β(s) of CWT log-log curves. Such a spectrum is shown in Figure 1f. This curve is much finer and less jerky than the corresponding curves in Figures 1d and 1e using the DFA and the DWT, respectively. The H spectrum of the PM2.5 series and the corresponding curves of other environmental variables will be discussed in the results section. Note that suggesting to consider a continuous spectrum of H values implies multifractality. This multifractality must be distinguished from the more common multifractal analysis that considers H for different subsets of the data series with different magnitudes of the data Ho¨lder discontinuities (e.g., 20, 21).

3. Synthetic Data Example To aid in the interpretation of the real data results we constructed and analyzed a synthetic series of 175 200 time points. The synthetic record simulates hypothetical 10 years of half-hourly real data with H ) 1.4 at the short time scales (up to 96 time points) and H ) 0.7 at the longer time scales. The series is a superposition of two series. The first one was created by adjoining 3650 cycles of 48 time points, each of which was produced with H ) 1.4 by the method described in (17) for generating synthetic data with known H value (see also the Supporting Information for description of the method). This results in a series with H ) 1.4 and a dominant period of 48 time points. The first 480 time points in this series are shown in Figure 2a. The second series is composed of 1825 segments of 96 time points each. The data points within each segment are of equal value. The 1825 values of the segments are a series with H ) 0.7. The first 9600 time points of this series are shown in Figure 2b. The combined VOL. 44, NO. 12, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 2. (a) First 480 time points of a synthetic self-affine time series with 48 time points periodicity and H ) 1.4. (b) The first 9600 time points in a synthetic series composed of segments of 96 time points each. The segment values are a self-affine series with H ) 0.7. (c) The first 9600 time points of the superposition of the series in (a) and (b).

FIGURE 3. (a) CWT log-log curve for the 175 200 time points synthetic data series whose first 9600 time points are shown in Figure 2c. Linear fits to two segments of the curve are superimposed on it (see the text for more details). (b) The corresponding H values. The empty markers denote H values that should not be considered for predictability estimation. series is a simple superposition of the two components. Its first 9600 values are shown in Figure 2c. Figure 3a shows the log-log curve of the synthetic series, produced using the CWT method, with lines fitted to the segments of the log-log curve at time scales between 3-20 time points and between 600-6000 time points. Figure 3b shows the corresponding H spectrum, and the fixed H values that correspond to the fitted lines in Figure 3a. A kink in the log-log curve in Figure 3a can be seen around the scale of 50 time points. It is a manifestation of the lower variability at this time scale because of the incorporation of 48 time points periodicity in the data series. This kink results in a large convex feature in the H spectrum in Figure 3b, with a minimum around the 50 time points scale. Around the convex feature, from the minimal scale to a scale of 100 time points, the H spectrum is quite constant with H ) 1.35. Then the H 4632

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FIGURE 4. Spectrum of H values as a function of the temporal scale for the (a) temperature, (b) wind speed, (c) solar radiation, and (d) precipitation records from stations Nave Shaanan and Nesher, Haifa. The empty markers denote H values not considered for predictability estimation. The dotted line at H ) 0.5 denotes complete randomness and zero predictability. values diminish rapidly and are stabilized around H ) 0.7 between the scales of 600-6000 time points. The linear fit to the log-log curve between these scales results in H ) 0.69. Beyond the scale of 6000 time points the H spectrum fluctuates wildly. The synthetic series which we constructed is a simple multifractal, in fact a doufractal, with different long-range correlations at two different time scale ranges. This resulted in a log-log curve with two linear components and two corresponding relatively flat segments in the H spectrum. The CWT method quite accurately estimated the H values within the straight segments of the log-log curve. However, kinks in this curve result in prominent fluctuations in the H spectrum which complicate its interpretation. Whitcher and Jensen (11) noticed similar kinks at time scales when a sudden shift in the H value occurs. References (22) and (23), in which the effects of trends and non-stationarities on the log-log curves produced by the DFA method are studied, show many examples of such kinks (which they refer to as crossovers). We conclude that abrupt shifts in the log-log curves and the corresponding wild fluctuations in the H values must be considered with care. In the case of an expected periodicity like the one of 48 time points that we note in Figures 3a and 3b, the interpretation is obvious. However, the large fluctuations that typically appear toward the long time scales range are more difficult to explain. In any case, H values of such features should not be considered for predictability estimation. They were thus denoted in Figures 3, 4, and 5 with empty markers.

4. Real Data Examples Figure 4 shows the H spectra as a function of the temporal scale of ambient temperature, wind speed, solar radiation (SR), and precipitation, respectively. The temperature, wind speed, and precipitation data are from the monitoring station Nave Shaanan in Haifa, Israel. The SR data are from the

FIGURE 5. Spectrum of H values as a function of the temporal scale for the (a) SO2, (b) NO2, (c) O3, and (d) PM2.5 records from station Nave Shaanan, Haifa. The empty markers denote H values not considered for predictability estimation. The dotted line at H ) 0.5 denotes complete randomness and zero predictability. nearby station of Nesher. The scale range in the plots is from 1.5 h to about 1.5 years (13 000 h). Values beyond the time scale of a season (∼2000 h) cannot be reliably considered for predictability estimation and thus were denoted by empty markers. In all four plots of Figure 4 a general trend of decreasing H values from short to long time scales is evident. The temperature and wind speed plots (Figures 4a and 4b, respectively) show large H values above H ) 1.3 up to the scale of about 50 h, implying strong persistence and good predictability at the time scales of less than 2 days. A rapid decrease beyond that range suggests loss of persistence, and thus a loss of predictability. The H for temperature decreases smoothly and reaches a minimum of H ) 0.5 at a time scale of about 800 h (33 days). The H for the wind speed decrease more rapidly, reaches the H ) 0.5 value at time scale of about 180 h (7.5 days), and keeps staying at that level up to a scale of 2000 h (about 3 months). Recall that the value of H ) 0.5 implies random behavior and complete lack of predictive ability. Some long-range correlation and thus predictability is retained in the temperature data at almost all time scales. But the wind speed data seem to be unpredictable within the time scales of a week and a season. The H spectrum of SR (Figure 4c) is above H ) 1.0 only up to a few hours. Lower values around H ) 0.9 can be seen between time scales of 12 to 200 h where it drops to H ) 0.5. As the seasonal cycle was removed, the SR data variations result from varying cloudiness or dust levels. Our results point to three different regimes of these phenomena within the range of scales of less than a month. In the three H plots for the temperature, wind speed, and SR data a clear convex feature appears around 24 h. Recalling our synthetic data example, this feature is a manifestation of a 24 h periodicity. Its existence may seem surprising as we removed the daily cycle from our data. However, SR has an inherent residual daily cycle because of the diurnal periods of zero sun radiation during night hours. Residual periodicity may also occur in the temperature and wind speed records.

For example, during clear winter periods the daylight hours are warmer than their calendarian mean because of enhanced solar radiation warming in the absence of clouds, while the nights are colder than their calendarian means because of enhanced infrared radiative cooling. Reference (24) also noted residual daily cycle in wind data stripped from diurnal periodic components. On the other hand, a strong manifestation of the yearly cycle that we noticed in the H plots of the series before deseasonalization (not shown) has indeed been removed and cannot be discerned in Figures 4a-4c. The H plot for the precipitation record is shown in Figure 4d. It depicts values around H ) 0.75 from the shortest time scales to about 150 h (6 days) and values around H ) 0.5 up to 1000 h (40 days). The H values for the precipitation record are generally low. A possible reason is that precipitation processes involve a combination of many variables of stochastic nature at several different time scales (e.g., convective turbulence, mesoscale eddies, synoptic cloud formation, etc.) that results in a more fluctuating behavior compared to the other meteorological variables. Similar to the wind speed case, the H ) 0.5 values beyond a time scale of a few days is probably a result of a lack of any correlation and predictability beyond the time scale of synoptic phenomena. At time scale longer than a season the H values in Figures 4a-4d do not seem to be reliable. More than 5-10 years of data that were available to us are probably needed to start noting the expected long memory effects of climatic patterns like the biennial oscillation or ENSO. Figures 5a-d show the H spectra of the SO2, NO2, O3, and PM2.5 time series in the Nave Shaanan monitoring station. Like in the case of the meteorological variables in Figure 4, there is a general trend of decreasing H values with scale in the plots for all four pollutants. The H curves of the three gaseous pollutants up to the daily time scale are similar to those of the wind speed and solar radiation, two meteorological variables with a major impact on air pollution dispersion and transformation. However, at the longer time scales the pollutants’ H values are generally higher than those of the meteorological variables. In the cases of NO2 and O3 the series do not reach the H ) 0.5 value, implying some predictability of these variables at all the time scales that can be resolved by the data. This is probably a result of slowly varying background values of these two pollutants. The large convex feature at 24 h scale in the SO2, NO2, and O3 plots is a results of the impact of the daily meteorological variations (breeze and sun radiation cycles) on the concentrations of these pollutants. In particular, for NO2 and O3 the meteorological effects are superimposed on the daily cycle of traffic, the dominant factor in the NOx/O3 cycle in the study study area. The H curve for NO2 (Figure 5b) exhibits also the effect of the 12 h cycle of traffic variations (morning and evening rush hours). In addition, H curves of NO2 and O3 (Figures 5b and 5c) display a second large convex feature at about 180 and 190 h, respectively. This is a results of the “weekend effect” on the variation in the concentrations of these pollutants. The lack of daily and weekly convex features in the H curve of the PM2.5 series (Figure 5d) points to a relatively small impact of the local traffic on the PM concentrations. This is in agreement with extensive literature (e.g., (25)) showing that a large proportion of the fine PM in the eastern Mediterranean is emitted by remote anthropogenic sources in eastern Europe and Turkey.

5. Discussion We have introduced the CWT fractal method as a tool for assessing the time-scale dependence of the predictability of environmental variables. The fine time scale resolution of the analysis enables studying subtle variation in the persistence and predictability of the variables. Large convex or concave fluctuations are real features that provide informaVOL. 44, NO. 12, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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tion about periodicities in the data series. However, care must be taken while interpreting such fluctuations at the large time scales. These fluctuations may be artifacts of insufficient sampling of long period phenomena. The method was demonstrated on time series of four meteorological variables and four air pollutants. We noted two or three distinct ranges of time scales with different predictability level in each of them. All the meteorological variables seem to behave in an unpredictable way at a certain time scale range. The gaseous air pollution series exhibit in the short time scales (up to 24 h) H values which resemble those of the wind speed and sun radiation. However, in the longer time scales they seem to possess longer memory than the meteorological variables. Naturally, the results we showed pertain to certain geographical location. Data observed at locations where the ambient conditions are different may result in different H curves. The chaotic nature of the atmosphere inherently limits the skills of predicting its variables. We can thus expect even the optimal prediction to depend to some degree on the length of time to which the statistical memory of the initial conditions is retained. The analysis described in this work provides a tool to measure the strength of that memory and gives some clues about the limits of predictability of several environmental variables at various time scales.

Acknowledgments Data for this study were kindly provided by the Haifa District Municipal Association for the Environment and by the Israel Electric Corporation. The authors would like to thank Dr. Rafi Linker for discussion regarding the scales transformation of the CWT method and two anonymous reviewers for their comments.

Supporting Information Available A plot demonstrating the graphical representation of a wavelet decomposition. Detailed description of the comparison between the fractal analysis methods, including a plot with the results and a discussion of their meaning. This material is available free of charge via the Internet at http:// pubs.acs.org.

Literature Cited (1) Rubalcaba, J. J. O. Fractal analysis of climatic Data: Annual precipitation records in Spain. Theor. Appl. Climatol. 1997, 56, 83–87. (2) Koscielny-Bunde, E.; Bunde, A.; Havlin, S.; Roman, H. E. Indication of a universal persistence law governing atmospheric variability. Phys. Rev. Lett. 1998, 81, 729–732. (3) Kira`ly, A.; Ja`nusi, L. M. Detrended fluctuation analysis of daily temperature records: Geographic dependence over Australia. Meteorol. Atmos. Phys. 2005, 88, 119–128.

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(4) Varotsos, C.; Ondov, J.; Efstathiou, M. Scaling properties of air pollution in Athens, Greece and Baltimore, Maryland. Atmos. Environ. 2005, 39, 4041–4047. (5) Malamud, B. D.; Turcotte, D. L. Self-affine time series: measures of weak and strong persistence. J. Stat. Plan. Infer. 1999, 80, 173–196. (6) Beran, J. 1994. Statistics for Long-Memory Processes; Chapman and Hall: New York, 1994. (7) Hurst, H. E. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civil Eng. 1951, 116, 770–808. (8) Pelletier, J. D. Analysis and modeling of the natural variability of climate. J. Climate 1997, 10, 1331–1342. (9) Talkner, P.; Weber, R. O. Power spectrum and detrended fluctuation analysis: Application to daily temperatures. Phys. Rev. E 2000, 62, 150–160. (10) Varotsos, C.; Kirk-Davidoff, D. Long-memory processes in ozone and temperature variations at the region 60°S-60°N. Atmos. Chem. Phys. 2006, 6, 4093–4100. (11) Whitcher, B.; Jensen, M. J. Wavelet estimation of a local long memory parameter. Exp. Geophys. 2000, 31, 94–103. (12) Whitcher, B.; Byers, S. D.; Guttorp, P.; Percival, D. B. Testing for homogeneity of variance in time series: Long memory, wavelets, and the Nile River. Water Resour. Res. 2002, 38, 1054. 12: 1-16. (13) Eichner, J. F.; Koscielny-Bunde, E.; Bunde, A.; Havlin, S.; Schellnhuber, H.-J. Power-law persistence and trends in the atmosphere: A detailed study of long temperature records. Phys. Rev. E 2003, 68, 0461331-5. (14) Monetti, R. A.; Havlin, S.; Bunde, A. Long-term persistence in the sea surface temperature fluctuations. Phys. A (Amsterdam, Neth.) 2003, 320, 581–589. (15) Matsoukas, C.; Islam, S.; Rodriguez-Iturbe, I. Detrended fluctuation analysis of rainfall and streamflow time series. J. Geophys. Res. 2000, 105 (D23), 165–172. (16) Peng, C.-K.; Buldyrev, S. V.; Havlin, S.; Simons, M.; Stanely, H. E.; Goldberger, A. L. Mosaic organization of DNA nucleotides. Phys. Rev. E 1994, 49, 1685–1689. (17) McSharry, P. E.; Malamud, B. D. Quantifying self-similarity in cardiac inter-beat interval time series. Comput. Cardiol. 2005, 32, 459–462. (18) WaveLab850, Matlab software package; http://www-stat. stanford.edu/wavelab. (19) Mallat, S. A Wavelet Tour of Signal Processing; Academic Press: San Diego, CA, 1999. (20) Muzy, J. F.; Bacry, E.; Arneodo, A. Wavelets and multifractal formalism for singular signals: Application to turbulence data. Phys. Rev. Lett. 1991, 67, 3515–3518. (21) Turiel, A.; Pe´rez-Vicente, C. J.; Grazzini, J. Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study. J. Comput. Phys. 2006, 216, 362–390. (22) Hu, K.; Ivanov, P. Ch.; Chen, Z.; Carpena, P.; Stanley, H. E. Effect of trends on detrended fluctuation analysis. Phys. Rev. E 2002, 64, 0111141-11. (23) Chen, Z.; Ivanov, P. Ch.; Hu, K.; Stanley, H. E. Effect of nonstationarities on detrended fluctuation analysis. Phys. Rev. E 2002, 65, 0411071-11. (24) Harris, R. I. The macrometeorological spectrum-a preliminary study. J. Wind Eng. Ind. Aerodyn. 2008, 96, 2294–2307. (25) Erel, Y.; Kalderon-Asael, B.; Dayan, U.; Sandler, A. European atmospheric pollution imported by cooler air masses to the eastern Mediterranean during the summer. Environ. Sci. Technol. 2007, 41, 5198–6203.

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