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On the scaling behavior of rain event sequence recorded in Basilicata region (Southern Italy)
Journal of Hydrology 296 (2004) 234–240 www.elsevier.com/locate/jhydrol
On the scaling behavior of rain event sequence recorded in Basilicata region (Southern Italy) Luciano Telescaa,*, Gerardo Colangeloa, Vincenzo Lapennaa, Maria Macchiatob a
Istituto di Metodologie per l’Analisi Ambientale, Consiglio Nazionale delle Ricerche, Area della Ricerca di Potenza, Avanzate di Analisi Ambientale, C da S. Loja, Tito Scala (PZ) 85050, Italy b Dipartimento di Scienze Fisiche, Universita` ‘Federico II’, Naples, Italy Received 3 September 2003; revised 17 February 2004; accepted 25 March 2004
Abstract Rain events can be described in terms of point processes, analogously to other geophysical phenomena, such as earthquakes or avalanches. Power-law behaviors have been identified in the distribution of sizes, of rain event durations and drought durations, with scaling exponents 1.4, 2.4 and 1.3, respectively. Time-fractal approaches, like Fano Factor and Allan Factor, well suited to reveal scaling behavior in point processes, allow to detect power-law behavior in rainfall data, pointing out to the presence of clustering in the time distribution of the sequence of rain events. q 2004 Elsevier B.V. All rights reserved. Keywords: Fano factor; Allan factor; Basilicata rainfall
1. Introduction Rainfall is an end product of a number of complex atmospheric processes, which vary both in space and time (Baratta et al., 2003), and it can be considered as one of the dominant factor of the meteo-climatic features of an investigated area. The study of rainfall within a given catchment region can be utilized for several purposes, including hydrological structure design, flood prevention, hydrological engineering, reservoir safety assessment studies and land management (NERC Report, 1975; Lebel and Laborde, 1988; Anderson and Nadarajah, 1993; Anderson et al., 1994). * Corresponding author. Tel.: þ39-0971-427-206; fax: þ 390971-427-271. E-mail address: [email protected] (L. Telesca). 0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2004.03.024
Rainfall models have been developed, considering the daily rainfall within the season as a random event so that stochastic principles can be applied (Richardson, 1981). Rainfall has long been analyzed by means of standard statistics such as average value, variance, coefficient of variation and percentiles (Laughlin et al., 2003). The extreme complexity of atmospheric processes results from the coupling of several non-linear processes having completely different temporal and spatial characteristic scales. An event like rainfall combines, for instance, the O(1025 m) droplet condensation effect with the O(106 m) planetary transfer of air masses and moisture. Despite the richness of individual events, it has been more and more recognized that several properties of the statistical distributions of pertinent meteorological
L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
fields (temperature, air humidity,…) are independent of the particular (time or length) scale at which they are observed, i.e. they obey power law distributions (Andrade et al., 1998, and references therein). Nguyen et al. (1998), proposed a new method for estimating extreme rainfalls for partially gaged sites using the ‘scale-invariance’ (or ‘scaling’) concept, which is currently a popular tool in the modelling and analysis of various geophysical processes. The scale-invariance implies that statistical properties of extreme rainfall variables for different time scales are related to each other by a scale-changing operator involving only the scale ratio (Nguyen et al., 1998). Several works have been involved with the investigating the scaling properties of rainfall data. The robust method of the Detrended Fluctuation Analysis (DFA) has been applied by Rodriguez-Iturbe et al. (Matsoukas et al., 2000), who demonstrated that what controls the correlation structure is not the actual rainfall values but the pattern of alternating wet and dry spells. Bunde et al. (2002) analyzed four daily precipitation records (with length from 57 to 117 years) coming from several whether stations around the world. By means of the DFA, they found scaling regime with a slope of about 0.5, pointing to uncorrelated or weakly correlated behaviour at large time spans. Furthermore, the exponents did not show dependence on specific climatic or geographic conditions. Pakaj et al. (2002) found that the hourly rainfall distribution is well described by a scale invariant power law, with exponent increasing towards the higher latitudes; they also showed that the rainfall power dependence is universal, indicating that the underlying dynamical system may be best describable in terms of self organized criticality. The present study is based on the acknowledge of the event-like structure of the rain, as stressed in (Peters et al., 2002). Events are defined as a sequence of non-zero rain rates. The intervals of zero rain rates between events are called drought periods. P The size of a rain event is defined as M ¼ t qðtÞDt; with Dt ¼ 1 h, and it represents the accumulated water column during the event. This perspective is similar to that of other natural phenomena, as earthquakes and avalanches, which are described in terms of events. Peters et al. (2002) revealed that power laws describe the number of rain events versus size and number of droughts versus duration; furthermore, they showed
235
that the accumulated water column displays scale-less fluctuations. The novelty of the present work is also the investigation of the clustering structure in the time occurrence series of the rain events, defined as the starting time of the event.
2. Methods In this article a rain event sequence is assumed to be a realization of a point process, completely defined by the set of event occurrence times. Fig. 1 shows an example of rain event. If the point process is Poissonian, the occurrence times are uncorrelated; for this memoryless process the interevent-interval probability density function pðtÞ behaves as a decreasing exponential function pðtÞ ¼ l e2lt ; for t $ 0; with l the mean rate of the process. If the point process is characterized by scaling behaviour, the interevent-interval probability density function pðtÞ generally decreases as a power-law function of the intervent time, pðtÞ ¼ kt2ð1þaÞ ; with a the so called fractal exponent (Thurner et al., 1997). The exponent a measures the strength of the clustering and represents the scaling coefficient of the decreasing power-law spectral density of the process Sðf Þ / f 2a (Lowen and Teich, 1993). The power spectral density furnishes the information about how the power of the process is concentrated at various frequency bands (Papoulis, 1990), and it gives information about the nature of the temporal fluctuations of the process. If a < 0; then the process is Poissonian, while if a – 0; the process present correlation structures and is characterized by memory effects. In this article, we analyze the rain event sequence, recorded at Matera station in Basilicata region (southern Italy) in order to detect the presence of 1=f a temporal fluctuations, by means of the Fano Factor (FF) and the Allan Factor (AF), that allow to estimate the fractal exponent a:
Fig. 1. A rain event is a sequence of hourly non-zero rain rate. In this example, 1’s indicate the presence of rain rate and the 0’s indicate the drought periods. The vertical arrows indicate the time occurrences of the rain events.
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A sequence of rain events can be mathematically expressed by a finite sum of Dirac’s delta functions centered on the occurrence times ti : yðtÞ ¼
n X
dðt 2 ti Þ:
ð1Þ
The AF is in relation with the variability of successive counts (Allan, 1966; Barnes and Allan, 1966), and is defined as the variance of successive counts for a specified counting time t divided by twice the mean number of events in that counting time:
i¼1
A representation of the point process is given by dividing the time axis into equally spaced contiguous counting windows of duration t; and producing a sequence of counts {Nk ðtÞ}; with Nk ðtÞ denoting the number of events in the k-th window: n ð tk X Nk ðtÞ ¼ dðt 2 tj Þ d t: ð2Þ tk21 j¼1
This sequence is a discrete-random process of nonnegative integers. An important feature of this representation is that it preserves the correspondence between the discrete time axis of the counting process {Nk } and the ‘real’ time axis of the underlying point process, and the correlation in the process {Nk } refers to correlation in the underlying point process (Thurner et al., 1997). With this representation two statistics can be used to reveal the presence of fractality in the process: the FF and the AF. The FF (Thurner et al., 1997) is a measure of correlation over different timescales. It is defined as the variance of the number of events in a specified counting time t divided by the mean number of events in that counting time; i.e. FFðtÞ ¼
kNk2 ðtÞl 2 kNk ðtÞl2 kNk ðtÞl
ð3Þ
where k l denotes the expectation value. The FF of a fractal point process with 0 , a , 1 varies as a function of counting time t as: a t FFðtÞ ¼ 1 þ ; ð4Þ t0 The monotonic power-law increase is representative of the presence of fluctuations on many timescales (Lowen and Teich, 1995); t0 is the fractal onset time, and marks the lower limit for significant scaling behaviour in the FF (Lowen and Teich, 1996), so that for t p t0 the clustering property becomes negligible. For Poisson processes the FF is always near unity for all counting times.
AFðtÞ ¼
kðNkþ1 ðtÞ 2 Nk ðtÞÞ2 l : 2kNk ðtÞl
ð5Þ
As the FF, the AF of a fractal point process varies with the counting time t with a power-law form:
t AFðtÞ ¼ 1 þ t1
a
ð6Þ
with the almost the same fractal exponent a; over a large range of counting times t (Thurner et al., 1997); t1 is the fractal onset time for the AF factor. As for FF, AF assumes values near unity for Poisson processes. In both the cases, the fractal exponent a < 0 for Poissonian processes.
3. Data The measuring station Matera, managed by Agenzia Lucana di Sviluppo e di Innovazione in Agricultura (ALSIA), is located in the Basilicata Region (southern Italy) (Fig. 2), and records hourly rainfall data since February 18, 1998 to February 14, 2002. The rain gauge (in accordance with the WMO recommendations) consists of a catchment area of 500 cm2 with an output magnetic reed switch and operating temperature between 2 20 and þ 50 8C; the sensitivity threshold is 0.2 mm and accuracy of 2%. Fig. 3 shows the size (a) and the duration (b) of the rain events versus the occurrence time.
4. Results Before applying the FF and AF statistics to the recorded rain events, we calculated the distribution of the event size M and the distribution of the event duration D: Fig. 4 shows the number density of the rain events NðMÞ versus event size M on a double logarithmic plot. The scaling regime extends over approximately two orders of magnitude
L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
Fig. 2. Location of the station Matera in Southern Italy.
Fig. 3. Size (a) and duration (b) of the sequence of rain events recorded at station Matera.
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Fig. 4. The distribution NðMÞ of the size of rain event sequence on a double logarithmic scale. The experimental data follow a power law /M 21:4 :
and the distribution follows a power law NðMÞ < M 2tM ;
tM < 1:4:
ð7Þ
This indicates the presence of scale invariance. This result is comparable with that shown by Andrade et al. (1998); they obtained for the Paris station a scaling relation with tM 1.1 up to approximately 10 mm size scale. Furthermore, Peters et al. (2002) obtained tM , 1:36: Fig. 5 shows the distribution of the duration TE of the rain events. The power law NðTE Þ < TE2tE ;
tE < 2:4
ð8Þ
Fig. 5. The distribution NðTE Þ of the duration of rain events on a double logarithmic scale. The experimental data follow the power law decay /TE22:4 :
Fig. 6. Relationship between event size M and event duration TE : A good positive correlation exists. The scaling exponent is approximately 1.8, this indicating that longer events tend to be more intense.
is very clearly visible. The value of the scaling exponent tE is larger than the exponent obtained by Peters (cond-mat/0204109), this is probably due to the different scaling regimes used to estimate the scaling coefficient, ranging in our case between 1 and 20 h. As observed by Peters and Christensen (2002), the exponent in the power law describing the duration of rain events to their frequency is different from that for the event sizes, indicating a non-trivial relationship between the duration and the size during a rain event. Fig. 6 shows this relationship, in which it seems that longer rain events are characterized by bigger size. It is interesting, however, that longer rain events are more intense (Peters and Christensen, 2002). This is reflected in the exponent of the power law indicated in Fig. 6 being approximately 1.8. Fig. 7 plots the distribution of the droughts; scale invariant behavior is shown in about two decades with scaling exponent tD < 1:3: Peters et al. (2002) obtained a comparable value tD < 1:42: We performed the clustering analysis of the timeoccurrences of the rain events by means the FF and the AF analyses. The results of the FF analysis are shown in Fig. 8. The FF curve increases with the counting time t; this indicating the presence of correlated temporal fluctuations. At low timescales the FF assumes an approximately constant unity value, denoting Poissonian temporal fluctuations involving
L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
Fig. 7. The distribution NðTD Þ of the duration of droughts on a double logarithmic scale. The experimental data follow the power law decay /TD21:3 :
these timescales. The value of the scaling exponent a has been estimated from the slope of the line fitting, by a least square method, the curve plotted in log – log scales over the linear range. The fractal exponent a , 0:32: Fig. 9 shows the results concerning the AF analysis. The linear range is clearly visible, over approximately the same timescales, with a-value consistent with the estimate based on the FF method. The AF curve appears more rough than the FF curve due to its definition involving the difference between
Fig. 8. Fano Factor analysis of the rain event sequence recorded at Matera station.
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Fig. 9. Allan Factor analysis of the rain event sequence recorded at Matera station.
successive counts, but the scaling behavior does not change.
5. Conclusions The 5-years long occurrence time sequence of rain events in the Matera site in Basilicata region has been studied in order to evidence the presence of fractality. Power-law behaviors have been evidenced in the event size distribution with exponent tM < 1:4; in the event duration distribution with scaling coefficient tE < 2:4; and in the drought duration distribution with scaling coefficient tD < 1:3: Concerning the clustering structure of the sequence of the time-occurrences of the rain events, the FF and AF statistics have displayed power law behaviors with exponents aFF < 0:32 and aAF < 0:39; respectively. The analysis performed on the time distribution of the rain events do not exclude that below approximately 1 day, the events could be poissonianly distributed; while for longer timescales up to almost 5 years the rain events are clusterized, denoting a clear time-correlation between them. The theory of self-organized criticality (SOC) has offered an innovative perspective of interpretation of the dynamics of many complex dissipative systems, which naturally evolve toward a statistically steady
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state, characterized by spatial and temporal fluctuations at all scales. The burst-like dynamics of rainfall data, whose statistics are characterized by scale-invariance, can be regarded as an instance of SOC. The SOC idea concerns with the tendency shown by many natural systems, such as the tectonic processes, driven by an energy input at a slow and constant rate, to enter states determined by scale free behavior (Peters and Christensen, 2002). Rainfalls present two separated timescale regimes: the regime of droughts, extending up to months, and that much shorter of the rain events (up to one day). As shown in (Peters and Christensen, 2002), ‘the atmosphere receives a slow and constant energy input from the Sun’s radiation. The absorbed energy evaporates water from the surface, which is intermediately stored in the atmosphere. During a rain shower, the water mass, slowly evaporated, is suddenly released with its original evaporation energy, and there is not constant light rainfall balancing the evaporated water mass immediately at every moment in time.’ Rain phenomena are ubiquitous. Since they are phenomena spanning on very broad temporal, spatial and intensity ranges, it is possible that local conditions such as total annual rainfall and orography become irrelevant to the scaling behaviour. This robustness with respect to changes in parameters is typical of critical systems and corroborates the idea of universality of scaling behaviour in rainfall, as expressed in Peters and Christensen (2002). The similarity among the values of the scaling exponents, qualifying and quantifying the scale-free behaviour of rainfall, in different regions of the globe confirm such universality.
Acknowledgements We gratefully acknowledge ALSIA, and in particular Dr Emanuele Scalcione, for providing the Matera rainfall data.
References Allan, D.W., 1966. Statistics of atomic frequency standards. Proc. IEEE 54, 221–230.
Anderson, C.W., Nadarajah, S., 1993. In: Barnett, V., Turkman, K. (Eds.), Statistics for the Environment, Wiley, Chichester, pp. 163–182. Anderson, C.W., Dwyer, I.J., Nadarajah, S., Reed, D.W., Tawn, J.A., 1994. Maximum reservoir water levels, Reservoir Safety and the Environment, Telford, London, pp. 200– 213. Andrade, R.F.S., Schellnhuber, H.J., Claussen, M., 1998. Analysis of rainfall records: possible relation to self-organized criticality. Physica A 254, 557–568. Baratta, D., Cicioni, G., Fasulli, F., Studer, L., 2003. Application of an ensemble technique based on singular spectrum analysis to daily rainfall forecasting. Neural Networks 16, 375– 387. Barnes, J.A., Allan, D.W., 1966. A statistical model of flicker noise. Proc. IEEE 54, 176 –178. Bunde, A., Eichner, J., Govindam, R., Havlin, S., Koscienly-Bunde, E., Rybski, D., Vjushin, D. Power law persistence in the atmosphere: an ideal test bed for climate models, arXiv:physics/ 0208019v1. Laughlin, G.P., Zuo, H., Walcott, J., Bugg, A.L., 2003. The rainfall reliability wizard—a new tool to rapidly analyse spatial rainfall reliability with examples. Environ. Mod. Software 18, 49–57. Lebel, T., Laborde, J.P., 1988. A geostatistical approach for areal rainfall statistics assessment. Stochast. Hydrol. Hydraul. 2, 245 –261. Lowen, S.B., Teich, M.C., 1993. Estimating the dimension of a fractal point process. SPIE Chaos Biol. Med., 2036. Lowen, S.B., Teich, M.C., 1995. Estimation and simulation of fractal stochastic point processes. Fractals 3, 183–210. Lowen, S.B., Teich, M.C., 1996. The Periodogram and Allan Variance reveal fractal exponents greater than unity in auditorynerve spike trains. J. Acoust. Soc. Am. 99, 3585–3591. Matsoukas, C., Islam, S., Rodriguez-Iturbe, I., 2000. Detrended Fluctuation Analysis of rainfall and streamflow time series. J. Geophys. Res. 105, 29165–29172. Natural Environmental Research Council. Flood Studies Report. Natural Environmental Research Council, London, 1975. Nguyen, V.T.V., Nguyen, T.D., Wang, H., 1998. Regional estimation of short duration rainfall extremes. Wat. Sci. Tech. 37, 15–19. Pankaj, J., Suman, J., Gauher, S. Universality and scale invariance in hourly rainfall, airXiv:physics/0202007v1. Papoulis, A., 1990. Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York. Peters, O., Christensen, K., 2002. Rain: relaxation in the sky. Phys. Rev. E 66, 036120. Peters, O., Hertlein, C., Christensen, K., 2002. A complexity view of rainfall. Phys. Rev. Lett. 88, 018701. Richardson, C.W., 1981. Stochastic simulation of daily precipitation, temperature and solar radiation. Water Resour. Res. 17, 182 –190. Thurner, S., Lowen, S.B., Feurstein, M.C., Heneghan, C., Feichtinger, H.G., Teich, M.C., 1997. Analysis, synthesis, and estimation of fractal-rate stochastic point processes. Fractals 5, 565 –596.
On the scaling behavior of rain event sequence recorded in Basilicata region (Southern Italy) Luciano Telescaa,*, Gerardo Colangeloa, Vincenzo Lapennaa, Maria Macchiatob a
Istituto di Metodologie per l’Analisi Ambientale, Consiglio Nazionale delle Ricerche, Area della Ricerca di Potenza, Avanzate di Analisi Ambientale, C da S. Loja, Tito Scala (PZ) 85050, Italy b Dipartimento di Scienze Fisiche, Universita` ‘Federico II’, Naples, Italy Received 3 September 2003; revised 17 February 2004; accepted 25 March 2004
Abstract Rain events can be described in terms of point processes, analogously to other geophysical phenomena, such as earthquakes or avalanches. Power-law behaviors have been identified in the distribution of sizes, of rain event durations and drought durations, with scaling exponents 1.4, 2.4 and 1.3, respectively. Time-fractal approaches, like Fano Factor and Allan Factor, well suited to reveal scaling behavior in point processes, allow to detect power-law behavior in rainfall data, pointing out to the presence of clustering in the time distribution of the sequence of rain events. q 2004 Elsevier B.V. All rights reserved. Keywords: Fano factor; Allan factor; Basilicata rainfall
1. Introduction Rainfall is an end product of a number of complex atmospheric processes, which vary both in space and time (Baratta et al., 2003), and it can be considered as one of the dominant factor of the meteo-climatic features of an investigated area. The study of rainfall within a given catchment region can be utilized for several purposes, including hydrological structure design, flood prevention, hydrological engineering, reservoir safety assessment studies and land management (NERC Report, 1975; Lebel and Laborde, 1988; Anderson and Nadarajah, 1993; Anderson et al., 1994). * Corresponding author. Tel.: þ39-0971-427-206; fax: þ 390971-427-271. E-mail address: [email protected] (L. Telesca). 0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2004.03.024
Rainfall models have been developed, considering the daily rainfall within the season as a random event so that stochastic principles can be applied (Richardson, 1981). Rainfall has long been analyzed by means of standard statistics such as average value, variance, coefficient of variation and percentiles (Laughlin et al., 2003). The extreme complexity of atmospheric processes results from the coupling of several non-linear processes having completely different temporal and spatial characteristic scales. An event like rainfall combines, for instance, the O(1025 m) droplet condensation effect with the O(106 m) planetary transfer of air masses and moisture. Despite the richness of individual events, it has been more and more recognized that several properties of the statistical distributions of pertinent meteorological
L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
fields (temperature, air humidity,…) are independent of the particular (time or length) scale at which they are observed, i.e. they obey power law distributions (Andrade et al., 1998, and references therein). Nguyen et al. (1998), proposed a new method for estimating extreme rainfalls for partially gaged sites using the ‘scale-invariance’ (or ‘scaling’) concept, which is currently a popular tool in the modelling and analysis of various geophysical processes. The scale-invariance implies that statistical properties of extreme rainfall variables for different time scales are related to each other by a scale-changing operator involving only the scale ratio (Nguyen et al., 1998). Several works have been involved with the investigating the scaling properties of rainfall data. The robust method of the Detrended Fluctuation Analysis (DFA) has been applied by Rodriguez-Iturbe et al. (Matsoukas et al., 2000), who demonstrated that what controls the correlation structure is not the actual rainfall values but the pattern of alternating wet and dry spells. Bunde et al. (2002) analyzed four daily precipitation records (with length from 57 to 117 years) coming from several whether stations around the world. By means of the DFA, they found scaling regime with a slope of about 0.5, pointing to uncorrelated or weakly correlated behaviour at large time spans. Furthermore, the exponents did not show dependence on specific climatic or geographic conditions. Pakaj et al. (2002) found that the hourly rainfall distribution is well described by a scale invariant power law, with exponent increasing towards the higher latitudes; they also showed that the rainfall power dependence is universal, indicating that the underlying dynamical system may be best describable in terms of self organized criticality. The present study is based on the acknowledge of the event-like structure of the rain, as stressed in (Peters et al., 2002). Events are defined as a sequence of non-zero rain rates. The intervals of zero rain rates between events are called drought periods. P The size of a rain event is defined as M ¼ t qðtÞDt; with Dt ¼ 1 h, and it represents the accumulated water column during the event. This perspective is similar to that of other natural phenomena, as earthquakes and avalanches, which are described in terms of events. Peters et al. (2002) revealed that power laws describe the number of rain events versus size and number of droughts versus duration; furthermore, they showed
235
that the accumulated water column displays scale-less fluctuations. The novelty of the present work is also the investigation of the clustering structure in the time occurrence series of the rain events, defined as the starting time of the event.
2. Methods In this article a rain event sequence is assumed to be a realization of a point process, completely defined by the set of event occurrence times. Fig. 1 shows an example of rain event. If the point process is Poissonian, the occurrence times are uncorrelated; for this memoryless process the interevent-interval probability density function pðtÞ behaves as a decreasing exponential function pðtÞ ¼ l e2lt ; for t $ 0; with l the mean rate of the process. If the point process is characterized by scaling behaviour, the interevent-interval probability density function pðtÞ generally decreases as a power-law function of the intervent time, pðtÞ ¼ kt2ð1þaÞ ; with a the so called fractal exponent (Thurner et al., 1997). The exponent a measures the strength of the clustering and represents the scaling coefficient of the decreasing power-law spectral density of the process Sðf Þ / f 2a (Lowen and Teich, 1993). The power spectral density furnishes the information about how the power of the process is concentrated at various frequency bands (Papoulis, 1990), and it gives information about the nature of the temporal fluctuations of the process. If a < 0; then the process is Poissonian, while if a – 0; the process present correlation structures and is characterized by memory effects. In this article, we analyze the rain event sequence, recorded at Matera station in Basilicata region (southern Italy) in order to detect the presence of 1=f a temporal fluctuations, by means of the Fano Factor (FF) and the Allan Factor (AF), that allow to estimate the fractal exponent a:
Fig. 1. A rain event is a sequence of hourly non-zero rain rate. In this example, 1’s indicate the presence of rain rate and the 0’s indicate the drought periods. The vertical arrows indicate the time occurrences of the rain events.
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L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
A sequence of rain events can be mathematically expressed by a finite sum of Dirac’s delta functions centered on the occurrence times ti : yðtÞ ¼
n X
dðt 2 ti Þ:
ð1Þ
The AF is in relation with the variability of successive counts (Allan, 1966; Barnes and Allan, 1966), and is defined as the variance of successive counts for a specified counting time t divided by twice the mean number of events in that counting time:
i¼1
A representation of the point process is given by dividing the time axis into equally spaced contiguous counting windows of duration t; and producing a sequence of counts {Nk ðtÞ}; with Nk ðtÞ denoting the number of events in the k-th window: n ð tk X Nk ðtÞ ¼ dðt 2 tj Þ d t: ð2Þ tk21 j¼1
This sequence is a discrete-random process of nonnegative integers. An important feature of this representation is that it preserves the correspondence between the discrete time axis of the counting process {Nk } and the ‘real’ time axis of the underlying point process, and the correlation in the process {Nk } refers to correlation in the underlying point process (Thurner et al., 1997). With this representation two statistics can be used to reveal the presence of fractality in the process: the FF and the AF. The FF (Thurner et al., 1997) is a measure of correlation over different timescales. It is defined as the variance of the number of events in a specified counting time t divided by the mean number of events in that counting time; i.e. FFðtÞ ¼
kNk2 ðtÞl 2 kNk ðtÞl2 kNk ðtÞl
ð3Þ
where k l denotes the expectation value. The FF of a fractal point process with 0 , a , 1 varies as a function of counting time t as: a t FFðtÞ ¼ 1 þ ; ð4Þ t0 The monotonic power-law increase is representative of the presence of fluctuations on many timescales (Lowen and Teich, 1995); t0 is the fractal onset time, and marks the lower limit for significant scaling behaviour in the FF (Lowen and Teich, 1996), so that for t p t0 the clustering property becomes negligible. For Poisson processes the FF is always near unity for all counting times.
AFðtÞ ¼
kðNkþ1 ðtÞ 2 Nk ðtÞÞ2 l : 2kNk ðtÞl
ð5Þ
As the FF, the AF of a fractal point process varies with the counting time t with a power-law form:
t AFðtÞ ¼ 1 þ t1
a
ð6Þ
with the almost the same fractal exponent a; over a large range of counting times t (Thurner et al., 1997); t1 is the fractal onset time for the AF factor. As for FF, AF assumes values near unity for Poisson processes. In both the cases, the fractal exponent a < 0 for Poissonian processes.
3. Data The measuring station Matera, managed by Agenzia Lucana di Sviluppo e di Innovazione in Agricultura (ALSIA), is located in the Basilicata Region (southern Italy) (Fig. 2), and records hourly rainfall data since February 18, 1998 to February 14, 2002. The rain gauge (in accordance with the WMO recommendations) consists of a catchment area of 500 cm2 with an output magnetic reed switch and operating temperature between 2 20 and þ 50 8C; the sensitivity threshold is 0.2 mm and accuracy of 2%. Fig. 3 shows the size (a) and the duration (b) of the rain events versus the occurrence time.
4. Results Before applying the FF and AF statistics to the recorded rain events, we calculated the distribution of the event size M and the distribution of the event duration D: Fig. 4 shows the number density of the rain events NðMÞ versus event size M on a double logarithmic plot. The scaling regime extends over approximately two orders of magnitude
L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
Fig. 2. Location of the station Matera in Southern Italy.
Fig. 3. Size (a) and duration (b) of the sequence of rain events recorded at station Matera.
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Fig. 4. The distribution NðMÞ of the size of rain event sequence on a double logarithmic scale. The experimental data follow a power law /M 21:4 :
and the distribution follows a power law NðMÞ < M 2tM ;
tM < 1:4:
ð7Þ
This indicates the presence of scale invariance. This result is comparable with that shown by Andrade et al. (1998); they obtained for the Paris station a scaling relation with tM 1.1 up to approximately 10 mm size scale. Furthermore, Peters et al. (2002) obtained tM , 1:36: Fig. 5 shows the distribution of the duration TE of the rain events. The power law NðTE Þ < TE2tE ;
tE < 2:4
ð8Þ
Fig. 5. The distribution NðTE Þ of the duration of rain events on a double logarithmic scale. The experimental data follow the power law decay /TE22:4 :
Fig. 6. Relationship between event size M and event duration TE : A good positive correlation exists. The scaling exponent is approximately 1.8, this indicating that longer events tend to be more intense.
is very clearly visible. The value of the scaling exponent tE is larger than the exponent obtained by Peters (cond-mat/0204109), this is probably due to the different scaling regimes used to estimate the scaling coefficient, ranging in our case between 1 and 20 h. As observed by Peters and Christensen (2002), the exponent in the power law describing the duration of rain events to their frequency is different from that for the event sizes, indicating a non-trivial relationship between the duration and the size during a rain event. Fig. 6 shows this relationship, in which it seems that longer rain events are characterized by bigger size. It is interesting, however, that longer rain events are more intense (Peters and Christensen, 2002). This is reflected in the exponent of the power law indicated in Fig. 6 being approximately 1.8. Fig. 7 plots the distribution of the droughts; scale invariant behavior is shown in about two decades with scaling exponent tD < 1:3: Peters et al. (2002) obtained a comparable value tD < 1:42: We performed the clustering analysis of the timeoccurrences of the rain events by means the FF and the AF analyses. The results of the FF analysis are shown in Fig. 8. The FF curve increases with the counting time t; this indicating the presence of correlated temporal fluctuations. At low timescales the FF assumes an approximately constant unity value, denoting Poissonian temporal fluctuations involving
L. Telesca et al. / Journal of Hydrology 296 (2004) 234–240
Fig. 7. The distribution NðTD Þ of the duration of droughts on a double logarithmic scale. The experimental data follow the power law decay /TD21:3 :
these timescales. The value of the scaling exponent a has been estimated from the slope of the line fitting, by a least square method, the curve plotted in log – log scales over the linear range. The fractal exponent a , 0:32: Fig. 9 shows the results concerning the AF analysis. The linear range is clearly visible, over approximately the same timescales, with a-value consistent with the estimate based on the FF method. The AF curve appears more rough than the FF curve due to its definition involving the difference between
Fig. 8. Fano Factor analysis of the rain event sequence recorded at Matera station.
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Fig. 9. Allan Factor analysis of the rain event sequence recorded at Matera station.
successive counts, but the scaling behavior does not change.
5. Conclusions The 5-years long occurrence time sequence of rain events in the Matera site in Basilicata region has been studied in order to evidence the presence of fractality. Power-law behaviors have been evidenced in the event size distribution with exponent tM < 1:4; in the event duration distribution with scaling coefficient tE < 2:4; and in the drought duration distribution with scaling coefficient tD < 1:3: Concerning the clustering structure of the sequence of the time-occurrences of the rain events, the FF and AF statistics have displayed power law behaviors with exponents aFF < 0:32 and aAF < 0:39; respectively. The analysis performed on the time distribution of the rain events do not exclude that below approximately 1 day, the events could be poissonianly distributed; while for longer timescales up to almost 5 years the rain events are clusterized, denoting a clear time-correlation between them. The theory of self-organized criticality (SOC) has offered an innovative perspective of interpretation of the dynamics of many complex dissipative systems, which naturally evolve toward a statistically steady
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state, characterized by spatial and temporal fluctuations at all scales. The burst-like dynamics of rainfall data, whose statistics are characterized by scale-invariance, can be regarded as an instance of SOC. The SOC idea concerns with the tendency shown by many natural systems, such as the tectonic processes, driven by an energy input at a slow and constant rate, to enter states determined by scale free behavior (Peters and Christensen, 2002). Rainfalls present two separated timescale regimes: the regime of droughts, extending up to months, and that much shorter of the rain events (up to one day). As shown in (Peters and Christensen, 2002), ‘the atmosphere receives a slow and constant energy input from the Sun’s radiation. The absorbed energy evaporates water from the surface, which is intermediately stored in the atmosphere. During a rain shower, the water mass, slowly evaporated, is suddenly released with its original evaporation energy, and there is not constant light rainfall balancing the evaporated water mass immediately at every moment in time.’ Rain phenomena are ubiquitous. Since they are phenomena spanning on very broad temporal, spatial and intensity ranges, it is possible that local conditions such as total annual rainfall and orography become irrelevant to the scaling behaviour. This robustness with respect to changes in parameters is typical of critical systems and corroborates the idea of universality of scaling behaviour in rainfall, as expressed in Peters and Christensen (2002). The similarity among the values of the scaling exponents, qualifying and quantifying the scale-free behaviour of rainfall, in different regions of the globe confirm such universality.
Acknowledgements We gratefully acknowledge ALSIA, and in particular Dr Emanuele Scalcione, for providing the Matera rainfall data.
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