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D. Machiwal, M. K. Jha1. IntroductionThe application of statistical hydrology in earlierdays was restricted to surface water problems, especiallyfor analyzing the hydrologic extremes suchas floods and droughts. However, during past threedecades, the statistical domain of hydrology hasbroadened to encompass the problems of both surfacewater and groundwater systems. With such abroad domain, statistics has emerged as a powerfultool for analyzing hydrologic time series. The mainaim of time series analysis is to detect and describequantitatively each of the generating processes underlyinga given sequence of observations (Shahinet al., 1993). In hydrology, time series analysis isused for building mathematical models to generatesynthetic hydrologic records, to forecast hydrologicevents, to detect trends and shifts in hydrologicrecords, and to fill in missing data and extend records(Salas, 1993).Most statistical analyses of hydrologic time seriesat the usual time scale encountered in waterresources studies are based on the following fundamentalassumptions: the series is homogenous,stationary, free from trends and shifts, non-periodicwith no persistence (Adeloye and Montaseri, 2002).Homogeneity implies that the data in the seriesbelong to one population, and hence have a timeinvariant mean. Non-homogeneity arises due tochanges in the method of data collection and/or theenvironment in which it is done (Fernando andJayawardena, 1994). Stationarity, on the otherhand, implies that the statistical parameters of theseries computed from different samples do notchange except due to sampling variations. A timeseries is said to be strictly stationary if its statisticalproperties do not vary with changes of time origin.A less strict type of stationarity (called weak stationarityor second-order stationarity) is that inwhich the first- and second-order moments dependonly on time differences (Chen and Rao, 2002). Innature, strictly stationary time series does not exist,and weakly stationary time series is practicallyconsidered as stationary time series.There are many different ways in which changesin hydro-meteorological series can take place. Achange can occur abruptly (step change) or gradually(trend) or may take more complex forms. Atime series is said to have trends, if there is a significantcorrelation (positive or negative) betweenthe observations and time. Trends and shifts in hydrologictime series are usually introduced due tonatural or artificial changes (Salas, 1993). Naturalchanges in hydrologic variables are usually gradualand are caused by a global or regional climatechange, which would be a representative of changesoccurring over the study area. Changes in monitoredvariables that may not be able to be extrapolatedover a study area could be caused by a gradualurbanization of the area surrounding the monitoringsite, changes in the method of measurement at themonitoring site, or by moving the monitoring siteeven a short distance away. The artificial change(or step change) is usually noted in the overall recordat a monitoring site, but this information is notalways presented with the sites’ data series. Thus,variables that appear to have a trend may actuallyjust represent a change in climatological conditionsnear the monitoring site. In such situations, theaffected climatological data should be modified sothat the values are better representative of the studyarea as a whole (Hameed et al., 1997). A key elementin this process is the ability to demonstratewhether a change or trend is present in the climatologicalrecord and to quantify this trend, if it ispresent.Periodicities in natural time series are generallydue to astronomical cycles such as earth’s rotationaround the sun (Kite, 1989). To identify and quantifythe periodicity in the hydrologic or climatologictime series, the time scale is to be considered lessthan a year (e.g., month or six months). In an annualtime series, periodicity effect is not discernibleand hence half-annual or monthly time series normallyencountered in hydrology can be used foranalyzing the periodicity. Lastly, the phenomenonof persistence is highly relevant to hydrologic timeseries. Persistence is the tendency for the magnitudeof an event to be dependent on the magnitudeof previous event(s), a memory effect, e.g., tendencyfor low streamflows to follow low streamflowsand that for high streamflows to follow highstreamflows; it can be considered synonymous withautocorrelation (O’Connel, 1977). Hurst (1951,1957) was the first person to describe persistencecomprehensively in his studies on a reservoir design.The phenomenon was defined in terms of aparameter called “Hurst’s coefficient”, the averagevalue of which is about 0.73 for very large samples.However, its theoretical value for an independentGaussian process to which hydrologic series areassimilated is 0.5 (Capodaglio and Moisello, 1990).If the theoretical and the observed values do notcorrespond, it is known as “Hurst’s phenomenon”.238
Time series analysis of hydrologic data for water resources planning and management: a reviewAll the stochastic models proposed to representhydrologic time series have attempted to reproduceor incorporate the persistence phenomenon; however,with the time series records commonly availablein hydrology, it is virtually impossible to identifyany long-term persistence in the data (Capodaglioand Moisello, 1990).Although some researchers have presented reviewson trend detection and estimation techniquesin the past (Esterby, 1996; Hess et al., 2001), acomprehensive review encompassing the applicationpotential of time series analysis techniques inhydrology and climatology is lacking. Esterby(1996) reviewed some parametric and nonparametrictrend detection methods by confining their applicationto water-quality time series, while Hess etal. (2001) presented an overview of six linear trenddetection methods used for environmental timeseries. Thus, the earlier reviews focused on a singlecharacteristic (i.e., trend) of the time series, with anemphasis on some favored techniques only. At thedawn of 21st century, there is a need to highlightthe role of time series analysis in water resourcesplanning and management amidst emerging newissues of sustainable water management. Therefore,in the present study, an attempt has been made topresent a review of the application of time seriesanalysis techniques in hydrology and climatology.There has been a dramatic increase in the applicationof time series analysis in hydrology during pastfour or five decades. However, the present reviewmainly includes the research findings reported duringpast ten years bearing in mind the size of a reviewpaper.2. Overview of the tests for determining varioustime series characteristics2.1 Homogeneity or consistencyAs mentioned earlier, homogeneity implies thatall the collected data belong to the same statisticalpopulation having a time invariant mean. Therefore,the tests to check the homogeneity of dataseries are based on evaluating the significance ofchanges in the mean value. The features of threehomogeneity tests viz., the von Neumann Test,Cumulative Deviations, and the Bayesian Test arediscussed in Buishand (1982) and Jayawardena andLau (1990). In these studies, all the three tests havebeen used as test statistics.Buishand (1982, 1984) presents detailed methodologyfor the above-mentioned three homogeneitytests, which serves as major guidelines for thesetests. Kanji (2001) in his excellent collection of 100statistical tests, have reported various homogeneitytests for multiple comparisons (e.g., Tukey, Link-Wallace, Dunnett, Bartlett, and Hartley tests).However, it has major drawbacks that the objectivesof tests are not clear and that original referencesare missing. On the other hand, some researchers(e.g., Radziejewski et al., 2002) have consideredonly a few homogeneity tests in the list oftrend detection tests. Such a reference may createconfusion about the general perception of homogeneityand trend for the researchers having no accessto other references. In the hydrologic time seriesanalysis, multiple comparison tests are still contemporary,while these tests are considered classical inthe geotechnical field (e.g., Phoon et al., 2003).2.2 StationarityA time series is said to be strictly stationary, ifits statistical properties do not vary with changes oftime origin. That is, if two non-overlapping timeintervals are selected from a given time series, thenthe two subseries will look almost the same. In fact,both the subseries will differ from one another, butwill be scattered around the same mean value.Therefore, a stationary time series cannot have anytrend or periodic component. This is the reason thatsometimes trend and periodicity tests are used tocheck the stationarity of hydrologic time series.There are two general approaches for testing stationarity:parametric and nonparametric. A detailedreview of the literature indicated that the parametricapproach is usually used by the researchers workingin the time domain, such as economists, who makecertain assumptions about the nature of their data.On the other hand, the nonparametric approach ismore commonly used by the researchers working inthe frequency domain, such as electrical engineers,who often treat the system as a "black box" andcannot make any basic assumptions about the natureof the system. In hydrology, however, bothparametric and nonparametric approaches are used.Furthermore, nonparametric tests are not based onthe knowledge or assumption that the population isnormally distributed (Bethea and Rhinehart, 1991).Therefore, the nonparametric tests are more widelyapplicable than parametric tests. However the nonparametrictests are reported to be less powerfulthan parametric tests. To arrive at the same conclusionwith the same confidence level, the nonpara-239
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