Hydrologic Time Series Analysis: Theory and Practice

Water is the most precious resource of the earth because no life is possible without water. It is essential for the survival and livelihood of every human. It also regulates ecosystems, grows our food and powers our industry. Hardly any economic activity can be sustained without water. Undoubtedly, water plays a vital role in our life. Different dimensions of water functions in society and nature are (Falkenmark and Rockstrom, 2004): (i) water as life-support and hence as a basic need and as a human and animal right; (ii) water as an economic commodity in some uses; (iii) water as an integral part of ecosystem (sustaining it and being sustained by it); (iv) water as a sacred resource; and (v) water as an inevitable component of cultures and civilizations. Thus, water is the key resource for the human/animal health, socio-economic development, and the survival of earth's ecosystems. On the other hand, natural ecosystems also play a crucial role in the availability and quality of water through their purifying and regulating services, thereby sustaining human development on the earth. In other words, water has social, economic and environmental values and is essential for sustainable development (Falkenmark and Rockstrom, 2004; UNESCO, 2003, 2009). In contrast with many other vital resources of the earth, there is no substitute for water in most activities and processes where it is needed!

Any hydrologic time series can be appropriately analyzed when knowledge about the basic statistical characteristics of the data series itself is first considered. Many time series analysis procedures are based on the assumptions that the time series possess certain characteristics which, in fact, are not true (Adeloye and Montaseri, 2002; Helsel and Hirsch, 2002; Rao et al., 2003). The results of such analyses based on false assumptions may provide incorrect and unreliable interpretations, or unnecessarily inconclusive. Therefore, it is essential to know about the common characteristics of hydrologic time series, which can help in selecting appropriate data analysis procedures for a given hydrologic time series.

Statistical methods are applied in all the phases of time series analysis from collecting data to evaluating results in hydrologic studies. Advances in computer technology has enabled most of the scientists/researchers to apply statistical analyses effectively; however, some of the researchers do not check parametric test assumptions, especially the normality assumption (Adeloye and Montaseri, 2002). Many methods of time series analysis depend on the basic assumption that data were sampled from a normal distribution (Madansky, 1988; USEPA, 1996; Thode, 2002). This assumption is very crucial for the reliability of results especially for parametric tests. These days many statistical software packages are available, which include several tests for checking the normality of time series data. However, the important point is to judge which test should be used under what condition (USEPA, 1996).

Natural time series, including hydrologic, climatic and environmental time series, which satisfy the assumptions of homogeneity, randomness, non- periodic, non-persistence and stationarity, seem to be the exception rather than the rule (Rao et al., 2003). In fact, for all water resources studies involving the use of hydrologic time series data, preliminary statistical analyses must always be carried out to confirm whether the hydrologic time series possess all the required assumptions/characteristics (Adeloye and Montaseri, 2002). Nevertheless, most time series analysis is performed using standard methods after relaxing the required conditions one way or another in the hope that the departure from these assumptions is not large enough to affect the analysis results (Rao et al., 2003). A comprehensive survey of the past studies on the hydrologic time series analysis (Machiwal and Jha, 2006) revealed that no studies considered all the aspects of time series analysis. Major work is reported dealing with only linear trend analysis, and the homogeneity, stationarity, periodicity, and persistence, which are equally important characteristics of the hydrologic time series, have been ignored. In most past studies on time series analysis, only regression and/or Kendall's rank correlation tests are applied for trend detection. Esterby (1996) and Hess et al. (2001) presented an overview of selected trend tests. Thus, very limited studies are reported to date concerning a detailed analysis of homogeneity, stationarity, periodicity and persistence in the hydrologic time series.

In practice, hydrologists often deal with a limited amount of recorded data (i.e., a sample) while analyzing a hydrologic time series. This sample consists of a limited number of realizations of the population of same hydrologic process. When a hydrologic time series is characterized with statistical and probabilistic parameters, it represents a probability of occurrence of one of its possible stages. This probabilistic occurrence of the hydrologic time series is considered as one realization. All possible realizations of the hydrologic process constitute a population. The concept of terms sample and population has already been explained in Chapter 2. The main intent of the most hydrologic studies is to understand and quantitatively describe the population as well as the process that generates it based on a limited number of samples. Also, future predictions and/or simulations about the hydrologic time series can be made by applying statistical tools and techniques using probabilistic or stochastic models based on the historical data. When a hydrologic time series is analyzed in this manner, the technique is known as 'stochastic modelling" of time series and the parameters described with statistic and probabilistic terms are called 'stochastic parameters"

Time series analysis has been successfully applied in the fields like geology, ocean engineering, seismology, hydrology, climatology, etc. The hydrological and climatological time series studies have been carried out for analyzing the historic rainfall data (e.g., Henderson, 1989; De Michele et al., 1998; Mirza et al., 1998; Pagliara et al., 1998; Abaurrea and Cebrian, 2003; Pugacheva et al., 2003; Astel et al., 2004), streamflow data (Avinash and Ghanshyam, 1988; Capodaglio and Moisello, 1990; Radziejewski et al., 2000; Fanta et al., 2001; Adeloye and Montaseri, 2002; Chen and Rao, 2002), flood data (Grew and Werrity, 1995; Changnon and Kunkel, 1995; Westmacott and Burn, 1997; Robson et al., 1998; Reed et al., 1999; Lins and Slack, 1999; Loukas and Quick, 1996, 1999; Cayan et al., 1999; Jain and Lall, 2001; Douglas et al., 2000; Adamowski and Bocci, 2001; Zhang et al., 2001; Cunderlik and Burn, 2002), infiltration data (Schwankl et al., 2000), and surface water quality data (Jayawardena and Lai, 1989; Higashino et al., 1999) as well as for generating synthetic rainfall data in semi-arid regions (Janos et al., 1988), determining water consumption patterns (Maidment and Parzen, 1984), detecting trends in evapotranspiration and wind speed (Hameed et al., 1997; Raghuwanshi and Wallender, 1997), and for detecting climate change or variability (Kite, 1989; Khan, 2001).

The application of statistical hydrology in earlier days was restricted to surface water problems only, especially for analyzing the hydrologic extremes such as floods and droughts (McCuen, 2003). However, during past three decades, the statistical domain of hydrology has broadened to encompass the problems related to both surface water and groundwater resources (Shahin et al., 1993; Machiwal and Jha, 2006). With such a broad domain, time series analysis has emerged as a powerful tool for the efficient planning and management of scarce freshwater resources.

A comprehensive review on the applications of time series analysis in surface water hydrology, climatology and groundwater hydrology (Machiwal and Jha, 2006) revealed that although several studies deal with the application of time series analysis in surface water hydrology, the application of time series analysis in subsurface hydrology is greatly limited. In subsurface hydrology, time series analysis has been mostly used for detecting trends in groundwater quality (Loftis, 1996; Broers and van der Grift, 2004; Chang, 2008; Visser et al., 2009).

Streamflow statistics are extensively employed for the management and development of water resources. The magnitude and frequency of streamflows in the Susquehanna River Basin (SRB) are often used by the Susquehanna River Basin Commission (SRBC) and other agencies for the purposes of water resources planning and management (SRBC, 2006). For example, a wide range of streamflow statistics are used for consumptive water use mitigation, reservoir operation, and minimum release management. Water resources engineers and managers often implicitly assume that streamflow series are stationary over time when using streamflow data and statistics (SRBC, 2006; Zhang and Kroll, 2007a,b; Milly et al., 2008). This assumption may not be valid if the watershed under consideration is sensitive to human disturbance and/or climate change. More generally, climate variability, and change in population, land use and water use are implicated in the non- stationarity of streamflow series (Koutsoyiannis et al., 2009; Lins and Stakhiv, 1998; Milly et al., 2008). In a review of its consumptive use mitigation strategy, the SRBC examined the frequency and duration of consumptive use compensation releases from reservoirs located in the upper reaches of the SRB. It was evident that the number and frequency of 7-day-10-year low flow (Q₇₁₀) events had dropped substantially since around 1970. This suggests that the assumption of stationarity in the basin might be invalid. Therefore, an investigation of the assumption of streamflow stationarity in the SRB was of interest.

The main objective of studies on analysis of trends is to ascertain how the statistical characteristics (e.g., mean and variance) of hydrological variables change over time at a given location or at a number of locations in a watershed/ region. From the historical perspective, much of the earlier studies on temporal trends in time series of hydrological variables were focussed on water quality related parameters. Most of the earlier studies, reported during 1970s and 1980s, have been reviewed and documented in the work of Helsel and Hirsch (1992) and Hipel and McLeod (1994). Quite recently, interest in the investigation of trends in time series of hydrological variables has increased enormously and numerous studies have been undertaken in different parts of the world. It is difficult to present an exhaustive account of these studies in this chapter and therefore only some of these studies are listed here: Chiew and McMahon (1993), Yulianti and Burn (1998), Lins and Slack (1999), Douglas et al. (2000), Yue et al. (2002b), Robson (2002), Xiong and Shenglian (2004), Hannaford and Marsh (2006), Dixon et al. (2006), Fu et al. (2007), Khaliq et al. (2008, 2009a, 2009b) and Khaliq and Gachon (2010) for trends in streamflows (e.g. mean annual, low and high flows); Hisdal et al. (2001) for trends in hydrological droughts; Suppiah and Hennessy (1998), Haylock and Nicholls (2000), Kunkel et al. (2003), Krishnamurthy et al. (2009) and Kumar et al. (2010) for trends in precipitation related variables (e.g., annual or seasonal total precipitation, frequency and magnitude of extreme events and dry days). Scientific research on the identification of trends in time series of hydrological variables is still continuing, however using improved approaches and with an enhanced focus on the interpretation of trends. It is important to note that the majority of the studies on trends over the last two decades were driven mainly by concerns of climate change and less due to the influence of other factors like agricultural and industrial developments that could also influence time evolution of hydrological variables.

Scientific literature and successive assessment reports of the Intergovernmental Panel on Climate Change (IPCC, 2001; IPCC, 2007; Min et al., 2011) show that the net anthropogenic radiative forcing causes the global warming and intensification of hydrological cycle with consequent increase in the occurrence of extreme weather events. To trace the future of water resources under climate change, climate research uses simulation models well known as general circulation models (GCMs) for forecasting (Koutsoyiannis and Montanari, 2007). Trend analysis of paleoclimatic observation has been an important tool to test the presence of a systematic component (i.e., signal) against the background of natural variability and randomness (i.e. noise) of the instrumental record of hydroclimatic time series (e.g., Zhang et al., 2001; Bhutiyani et al., 2007; Wilson et al., 2010). Huntington (2006) reported that trends in hydrologic variables are consistent with an intensification of the water cycle. However, substantial uncertainty in trends exists due to regional differences of response variables and unavailability of datasets.

Understanding trends and variations of current and historical hydroclimatic variables is pertinent to the future development and sustainable management of water resources of a particular region. Information regarding hydroclimatological issues is important within the context of global warming, water and energy cycles and the increasing demand for water due to population and economic growth (Sankarasubramanian and Vogel, 2003; Oguntunde et al., 2006). Changes in the climate system and land cover have been widely accepted to have important consequences for regional to global water resources management and conservation. The extent to which human alteration of earth’s environment affects the global hydrologic cycle is still largely unknown (Szilagyi, 2001). Valuable historical records of hydrologic patterns over complex drainage basins help to understand anthropogenic and climatic effects on large-scale terrestrial ecosystems (Vörösmarty and Sahagian, 2000). One of the very important necessities of research into climate change is to analyze and detect historical changes in the climatic system (Houghton et al., 1996). Rainfall is a principal element of the hydrological cycle, hence understanding its behaviour may be of profound social and economic significance. Detection of trends and oscillations in the rainfall time series yields important information for understanding the climate. However, rainfall changes are particularly hard to gauge, because rainfall is not uniform and varies considerably from place to place and time to time, even on small scales.