Aplinkos tyrimai, inžinerija ir vadyba, 2003.Nr.3(25), P.16-20 Environmental research, engineering and management, 2003.No.3(25), P.16-20 ISSN 1392-1649 The Hurst Phenomenon in Hydrology Gaudenta Sakalauskienė Institute of Mathematics and Informatics (received in July, 2003; accepted in October, 2003) Recent studies have shown that rescaled adjusted range analysis (R/S analysis) and fractional Gaussian noise (FGN) are useful in characterizing subsurface heterogeneities in addition to geophysical time series. Although those studies have led to a fairly good understanding of some aspects of R/S analysis and FGN, a comprehensive introduction to these stochastic, fractal functions is still lacking in the subsurface hydrology literature. In this paper we look for a possibility to use R/S analysis and fractional Gaussian noise in prediction of the Nemunas river flow using data on the average annual flow over the period of January 1812 December 2002. Key words: long-range dependence, R/S analysis, and hydrology. 1. Introduction The methods of time series analysis have been recognized as important tools for assisting in solving problems related to the management of water resources. R/S analysis has it roots in early work of the British hydrologist H. E. Hurst [1], who investigated dependence properties of such phenomena as levels of the river Nile. The Hurst constant H , as the index of dependence is often called, always lies between 0 and 1, and equals to 0.5 for processes that have independent increments. Particular interest focuses on the hypothesis that H > 0.5 indicates relatively longrange dependence. For example, Hurst observed that H = 0.91 in the case of the Nile data. This indicates a stretch of dependence that is well beyond what could be adequately explained assuming independent increments. Today the R/S analysis is mostly used for the hydrological studies, where H value is variously interpreted either as an indicator of dependence range or irregularity. Several authors have used R/S analysis to analyze data on hydrological parameters such as river flow, precipitation, temperature, etc [2-4]. R/S analysis was adapted to estimate natural annual flow for the Logan river for a 70-year period. The measure of long-term persistence (the Hurst coefficient), was approximately 0.8 and the lag-one correlation coefficient (measure of shorter-term persistence), was 0.4 [5]. The R/S analysis of Adibi and Collins [6] has been used to develop and evaluate long-term forecasts for the dry and wet period phenomenon. The most important statistical parameters of average annual flow in Lithuania were collected and researched by J. Jablonskis at the observatory of the Nemunas river near Smalininkai during the period from 1812 to 1991 [7]. This paper attempts to show that the Hurst phenomenon is essentially very simple to formulate, understand and reproduce in transformed time series. A mathematical formulation based on the relationship of the process variance with the temporal scale of the process is offered. In addition, it is demonstrated that the Hurst phenomenon could be used for analysis of fluctuations of hydrological processes at different time scales. This approach differs fundamentally from more traditional methods currently used for simulation, where the model is taken to be a relatively conventional discrete time series such as an autoregression or moving average, or autoregressive moving average. In this paper we look for a possibility to use R/S analysis and fractional Gaussian noise in prediction of the Nemunas river flow using data on the average annual flow over the period of January 1812 – December 2002. The Hurst Phenomenon in Hydrology 2. Hurst exponent power function, and is shown in Equation 5 [11]. Methods R* E n = cn H as n → ∞ . σˆ n 2.1. R/S analysis Hydrological processes such as rainfall, runoff, evaporation, etc are often modeled as stationary stochastic processes in discrete time. Let such a process be denoted as Z i with i = 1,2,..., denoting discrete time (e.g. years). The R/S analysis is the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. For an annual time series z1 , z 2 ,..., z n define the The expected value will be described by a power function with an exponent of 0.5 if the data set is a random walk. R* E n = cn 0.5 as n → ∞ . σˆ n (1) where z n is the mean of the n observations in the time series and s 0* = s n* = 0 . The adjusted range Rn* is defined as Rn* = M n* − mn* , (6) With this definition the Hurst exponent of 0.5 will correspond to a time series that is truly random (Normal distribution, Brown Noise, or Brownian motion). When Hurst exponent is within an interval of 0.5 < H ≤ 1 , it describes a dynamically persistent, or trend reinforcing series. A straight line with non-zero gradient will have a Hurst exponent of 1. Alternatively, the series have memory that increases with H . Importantly, the values at the beginning of the time series are as important to the dynamics as the most recently observed. When Hurst exponent is within an interval of 0 ≤ H < 0.5 , it describes an anti-persistent, or a mean reverting system. At the limit of zero the time series must change the direction of every sample as in the case of white noise. kth adjusted partial sum s k* by s k* = s k* −1 + z k − z n , (5) (2) where M n* = max(0, s1* , s 2* ,..., s n* ) is the adjusted surplus, mn* = min(0, s1* , s 2* ,..., s n* ) is the adjusted deficit. The rescaled adjusted range (RAR) is the given as Rn* = Rn* , σˆ n 2.2. The fractional Gaussian noise (3) The Fractional Gaussian Noise (FGN) was developed specifically to account for the Hurst phenomenon [9]. The connection with Hurst’s law is the parameter H in FGN, which is often estimated by the Hurst coefficient H in (4). To derive FGN first consider a Brownian motion or Wiener process that is denoted by B (t ) in continuous time where its increments B (t + u ) − B (t ) are Gaussian with mean zero and variance u and are independent for non-overlapping time intervals. Fractional Brownian motion is defined as where σˆ n is the sample standard deviation. By the studies of long term storage requirements on the Nile river Hurst stimulated in the RAR statistic in (3). On the basis of a study of 690 annual time series comprising stream flow, river or lake levels, precipitation, temperature, pressure and other geophysical processes, Hurst developed the empirical relationship Rn* t H n = , 2 ∫ B H (t ) − B H (0) = (t − s ) H −1 / 2 dB( s ) , (4) (7) 0 where H is the Hurst coefficient. Hurst found H to have an average value of 0.73 with a standard deviation of 0.09 from the historical records [8-10]. where t > 0 , 0 < H < 1 and B H (0) is the level of time t = 0. Discrete time fractional Gaussian noise (i.e., FGN) is then calculated as The Hurst exponent is estimated by calculating the average rescaled range over multiple regions of the data. In statistics, the average (mean) data set Z is sometimes written as the expected value E [Z ] . Using this notation, the expected value of R/S, is calculated over a set of regions converges on the z t = B H (t + 1) − B H (t ) , where t = 1,2,... . 17 (8) G. Sakalauskienė The sample mean and variance of FGN are consistent estimators of the true mean and variance and FGN is covariance stationary. The theoretical autocorrelation function (ACF) of FGN at lag k is given by [ 1 (k + 1) 2 H − 2k 2 H + (k − 1) 2 H 2 ρk = ] The data set used in this study consists of average time series of the Nemunas river flows near Smalininkai from January 1812 to December 2002. The data source is the Lithuanian Hydro meteorological Service at the Ministry of Environment. Annual data was used for the empirical study, 5 and 10 years moving average data on the Nemunas river flow. Fig. 1 shows estimated natural annual flow as well as the 5 and 10 years moving average flows of the Nemunas river near Smalininkai for the 190-year period. The proposed method for generating R/S analysis is demonstrated by transforming records with Hurst exponent equal to those of the historical standardized flow series at the Nemunas river. Fig. 2 shows R/S analysis for the Nemunas river near Smalininkai site. The measure of long-term persistence (the Hurst coefficient) is approximately 0.67 and the lag-one correlation coefficient (a measure of shorter-term persistence) is 0.29 (record length 190 years from 1812 to 2002). Moving average for the 5 and 10 years time series, Hurst coefficient are 0.80 and 0.87, lag-one correlation coefficient – 0.51 and 0.63, respectively. The first step of our empirical analysis was to test whether the Hurst coefficient of an asset was significantly different from 0.5 or not. A significant difference from 0.5 would indicate that Z i did not follow a Brownian motion. In order to test the null hypothesis that H = 0.5 against the alternative H ≠ 0.5 , i.e. (9) for 0 < H < 1 and k ≥ 1 . When a theoretical ACF is the able sum it must satisfy [12] ∞ ∑ ρk <∞. (10) k = −∞ The theoretical ACF for FGN in (9) is not the sum able for 0.5 < H ≤ 1 and therefore FGN is called a long memory process. The theoretical ACF of ARMA processes satisfy (10) and are referred to as short memory processes. When the parameter H lies in the interval 0 ≤ H < 0.5 , FGN also constitutes a short memory process. For many geophysical time series the estimates for H lie in the range 0.5 < H ≤ 1 . The statistical effect of past events on present behavior attenuates very slowly because FGN models possess long memory for H contained in this interval. 3. Results and discussion The Nemunas river is a major source of surface water for all Lithuanian territory. The Nemunas river basin accounts for 72% of all Lithuanian surface water. The total surface of the Nemunas river basin is 96100 km2 (97900 km2 with the river delta). The Nemunas river near Smalininkai hydrology monitoring station was established in October 1811. The monitoring station is 111 km away from the river mouth and it measures water flow from an area of 81190 km2. So, the Nemunas river flow in this monitoring station is the best indicator of changes in Lithuanian hydrology. m /s h1 : H ≠ 0.5 , we approximated the distribution of Hˆ − H conditional on the null hypothesis, and calculated the p-values. We studied the assets for which the estimated Hurst coefficient H was significantly different from 0.5. Hence the Nemunas river flow near Smalininkai time series (both annual and 5 and 10 moving average) is a long memory process. Annual value Average, 10 years 3 900 h0 : H = 0.5 Average, 5 years 800 700 600 500 400 1999 1988 1977 1966 1955 1944 1933 1922 1911 1900 1889 1878 1867 1856 1845 1834 1823 1812 300 Year Fig. 1. The Nemunas river annual, 5 and 10 years moving average flow time series in 1812 - 2002 18 The Hurst Phenomenon in Hydrology H=0.67 H=0.80 H=0.85 2000 1500 1000 500 Fig. 2. 1999 1988 1977 1966 1955 1944 1933 1922 1911 1900 1889 1878 1867 1856 1845 1834 1823 -500 1812 0 Year Plot of a transformed time series generated using the statistics of standardized discharge at the * Nemunas river ( s k ) 6. As a result of the normalization process, every realization curve contains at least on minimum or one maximum global point (i.e. it has at least one half cycle if a full cycle incorporates wet-dry-wetdry periods). It was possible to analyze the time series of the Nemunas river flow near Smalininkai by employing R/S analysis and predict that durations of wet and dry periods are 27 and 5 years respectively. 7. 8. 9. 10. 4. References 1. Hurst H. E. Long – term storage capacity of reservoirs. // Transactions of the American Society of Civil Engineering Nr. 116. 1951, P. 770-799. Bellin A., Pannone M., Fiori A., Rinaldo A. On transport in porous formations characterized by heterogeneity of evolving scales. // Water Resource Research. Vol 32. No. 12 3485. Koutsoyiannis, D. The Hurst phenomenon and fractional Gaussian noise made easy. // Hydrological Sciences Journal. Nr. 47(4) 2002, P. 573-595. Koutsoyiannis, D. Climate change, the Hurst phenomenon, and hydrological statistics. // Hydrological Sciences Journal. Nr. 48(1) 2003, P. 3-24. Burgers Stephen J. Some aspects of Hydrologic variability. // Managing water Resources in the West Under Conditions of climate uncertainly. 1991, P. 275-280. 2. 3. 4. 5. 11. 12. Adibi M. and Collins M. A. Anticipatory conservation based drought management of water supply reservoirs. P. 7. http://www.eng.warwick.ac.uk. Jablonskis J. Run-off of the Nemunas during the 180 years. // Power engineering. Nr. 4. ISSN 02357208. Vilnius: Academia, 1994. P. 19-32 (in Lithuanian). Kottegoda N. T. Stochastic water resources technology. Halsted Press, New, 1980. P. 11-47. Anderson O. D. Time series. 1980, P. 73-103. Hall P. and et. all. Semiparametric bootstrap approach to hypothesis tests and confidence intervals for the Hurst coefficients. P. 22. http://netec.mcc.ac.uk/WoPEc/data/Papers//wophu mbsf1999-62.html. Bras R.L. and Rodriguez-Iturbe I. Random function and hydrology, Addison-Wesley, USA. 1985, P. 221. Billinger D. R. Time series data analysis and theory. 1975, P. 10-35. Dr. Gaudenta Sakalauskienė, researcher at the Institute of Mathematics and Informatics. Research interests: water pollution, water quality modeling, and time series analysis. Address: Akademijos str. 4, LT-2005 Vilnius, Lithuania Tel.: +370 5 2722554 Fax.: +370 5 2729209 E - mail: [email protected] 19 G. Sakalauskienė Hursto reiškinys hidrologijoje Gaudenta Sakalauskienė Matematikos ir informatikos institutas (gauta 2003 m. liepos mėn.; atiduota spaudai 2003 m. spalio mėn.) Straipsnyje išsamiai išdėstyta R/S statistikos ir fraktalinio Brauno judėjimo teorija bei jų pritaikymo galimybės hidrologijoje. Nemuno upės hidrologinėje stotyje ties Smalininkais nepertraukiamai atliekami vandens lygio, vandens nuotėkio stebėjimai, matavimai ir kaupiami duomenys nuo 1812 m. Gautų matavimų duomenys sudaro ilgalaikę laiko eilutę. Sukauptų duomenų eilutės yra unikalios ir istorine, ir moksline prasme. Turimoms metinėms vandens nuotėkio laiko eilutėms nuo 1812 iki 2002 metų apdoroti buvo panaudota procesų teorija, t. y. R/S statistikos ir fraktalinio Brauno judėjimo teorija. Pritaikę minėtą procesų teoriją Nemuno ties Smalininkais metiniams bei 5 ir 10 slankių vidurkių nuotėkio laiko eilutėms gauta, kad visos trys nagrinėtos laiko eilutės atspindi ilgalaikę priklausomybę visam nagrinėtam laikotarpiui (Hursto koeficientas: 0,67, 0,8 ir 0,85, atitinkamai su tikimybe 0,05). Be to, šią procesų teoriją labai sėkmingai galima taikyti hidrologinėms laiko eilutėms, norint įvertinti sausringus ir labai drėgnus meteorologinius laikotarpius. Taigi, analizuojant Nemuno nuotėkio eilutę, pastebėta, kad labai drėgni laikotarpiai truko 27 metus, o sausringi – tik 5 metus, ir tai leistų tiksliau atlikti vandens nuotėkio prognozę. 20
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