Statistical Models and the Estimation of Low Flows

Statistical Models and the Estimation of Low Flows

Taha B.M.J. Ouarda, Christian Charron and André St-Hilaire

Abstract: The present paper provides a brief review of statistical models that are commonly used in the estimation of low flows both at sites with a reliable streamflow record and sites remote from data sources. Opportunities are identified for the regional estimation of low-flow characteristics at ungauged sites. The adaptation of the neighbourhood regionalization approach, commonly used in regional flood frequency analysis, can be extended to low-flow variables. Estimation approaches extending the usefulness of recession information in regional low-flow frequency analysis to ungauged sites using a canonical correlation analysis approach for the identification of hydrological neighbourhoods is described. The validity of recession parameters when estimated from very short hydrological data records is also discussed. Promising new directions for future research in the field of statistical low-flow frequency estimation are identified. Résumé : La présente communication offre un bref survol des modèles statistiques couramment utilisés pour l’estimation des basses eaux tant à des sites offrant un enregistrement fiable des débits d’un cours d’eau qu’à des sites éloignés des sources de données. Des possibilités d’estimation régionale des caractéristiques des basses eaux dans des sites non jaugés sont décrites. L’adaptation de l’approche de régionalisation par voisinage, couramment employée dans l’analyse régionale de la fréquence des crues, peut être étendue aux variables des basses eaux. Sont également décrites certaines approches d’estimation qui accroissent l’utilité des données sur les décrues dans l’analyse régionale de la fréquence des basses eaux pour des sites non jaugés, et ce, à l’aide d’une approche de l’analyse de corrélation canonique pour l’identification des voisinages hydrologiques. Il est aussi question de la validité des paramètres de décrue lorsque les estimations reposent sur des enregistrements de données hydrologiques s’étalant sur de très courtes périodes. De nouvelles orientations prometteuses pour les recherches futures dans le domaine de l’estimation statistique de la fréquence de l’étiage sont également dégagées.

Taha B.M.J. Ouarda1, Christian Charron1 and André St-Hilaire1 1

Canada Research Chair on the Estimation of Hydrological Variables Hydro-Québec/NSERC Chair in Statistical Hydrology, INRS-ETE, University of Québec, Quebec, QC G1K 9A9

Submitted April 2008; accepted May 2008. Written comments on this paper will be accepted until December 2008. Canadian Water Resources Journal Revue canadienne des ressources hydriques

Vol. 33(2): 195–206 (2008)

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Introduction The general complexity and diversity of low-flow processes and the multidisciplinary nature of low-flow studies (geologists, geomorphologists, climatologists, meteorologists, hydrologists, engineers, statisticians, soil scientists, geographers, and end users with different backgrounds) make it often difficult to find a common ground for exchange in scientific societies. The PUB Low-Flow Workshop, held in Quebec City, April 12-13, 2007, was intended to provide a forum for discussion to build up a common cross-disciplinary language and share methodologies. Such a forum is also important for the improvement of the general level of understanding of low-flow processes in natural and regulated river environments. The systematic crossdisciplinary summary, evaluation of current results and existing methodologies in this area provided by this special issue can guide scientists in new research directions that can lead to significant improvements in low-flow estimation capabilities. Low flows represent a crucial component of the natural river flow regimes. The spatial and temporal variability of river low-flow characteristics can be considerable. Smakhtin (2001) presented a comprehensive review of low-flow hydrology covering such issues as generating mechanisms, estimation methods and applications. Burn et al. (this issue) have summarized the processes and patterns of low flows in Canada, specifically. The availability of reliable lowflow occurrence and magnitude estimates is crucial for a wide array of engineering applications such as aquatic ecosystem modelling (Bradford and Heinonen, this issue), environmental impact analysis, water supply assessment for potable and irrigation purposes (Sellars et al., this issue), liquid waste effluent dilution, river navigation planning and water quality management. Where significant amounts of systematically observed flow data exist low-flow frequency analysis should be employed in decision-making because of potential economic impacts. A high economic value is associated with the activities of prediction and analysis of low-flows and the resulting long-term droughts. It should be mentioned that droughts have more severe consequences and are often more costly than flood events. Damage accounting to approximately US$40 billion occurred during the USA droughts of 1988-1989, while the financial cost of the floods in the Mississippi area in 1993 was in the range US$18-

28 billion (Demuth, 2005). On the other hand, where the amount of available local observed streamflow records is meager, regional estimation techniques can be utilized. The purpose of the present paper is to provide an overview of existing statistical models that are commonly used in the estimation of low flows, to present some recent results dealing with the estimation of low-flow characteristics at sites remote from data sources, and to propose some potential directions for improvements in regional estimation methods. The remainder of the paper is organized as follows. A general introduction to local and regional statistical models for the estimation of low flows is presented in the following section, followed by some recent results concerning the adaptation of the neighbourhood regionalization approach to low-flow variables. The usefulness of recession information in regional lowflow frequency analysis is then addressed, followed by non-stationary low-flow frequency analysis. Finally, some conclusions are presented and promising new research directions in statistical low-flow estimation are suggested.

Statistical Models for the Estimation of Low Flows A number of low-flow statistics can be used to characterize the low-flow regime at a given location. Among the most commonly used are the quantiles Q d,y of the lowest mean discharge over a consecutive d-day period corresponding to a recurrence interval (return period) of y-years. In Canada, the indices Q 7,10 , Q 7,2 and Q 30,5 are the most widely used characteristics for the analysis of water supply systems during droughts and for the study of the waste assimilative capacity of streams. Other characteristics, such as the one-day, or three-day 20-year low-flow quantiles can also be used for specific purposes. These statistics can be estimated locally at sites with a significant amount of observed data, or regionally at ungauged sites. In Canada and in other northern countries characterised by cold climates, the streams and the soil can freeze completely during the winter period. This leads to a winter low-flow season. Summer can also correspond to a low-flow season as the snowmelt processes were already completed during the spring season, and periods of low liquid precipitations can © 2008 Canadian Water Resources Association

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be common in some regions (Burn et al., this issue). Therefore, it is often necessary to consider separately the summer and winter low-flow seasons as they are characterised by completely different phenomena and low-flow data can be represented by different statistical distributions. Local Low-Flow Frequency Analysis

Extreme value frequency analysis is a predictive statistical tool commonly used in hydrology to make inference concerning the probability of occurrence of extreme events such as floods and low flows. A series of observed annual or seasonal (winter and summer seasons) flow minima is used in the case of local lowflow frequency analysis. The first step in a low-flow frequency analysis consists of verifying the hypotheses of homogeneity, stationarity and independence. In the second step, a statistical distribution representing the relationship between the magnitudes of the events and the exceedance probabilities is fitted to the observed low-flow data, and the parameters of the probability distribution are estimated. In low-flow frequency analysis, the most commonly used distributions are the Weibull, Gumbel, Log-Normal, Gamma, Pearson Type III and Log-Pearson Type III distributions. Tasker (1987) has shown that the use of these hypothetical distributions performs better in terms of mean square error than the Log-Boughton method, which is based upon a curve fit through plotted points of the observed data (plotting positions) (Loganathan et al., 1985). Several methods for the estimation of the distribution parameters can be considered (such as the method of moments, the method of probability weighted moments, or the maximum likelihood method), depending on the distribution being adopted. The testing of the adequacy of the fit and the selection of the distribution that best fits the lowflow data sample can be carried out using either the Akaike (1974) or the Bayesian information criteria (Schwartz, 1978). Parsimony considerations, related to the number of parameters in the distribution, can become important when selecting the distribution that best fits the low-flow data. In the third step of a low-flow frequency analysis, the computation of quantiles corresponding to different return periods can be carried out. The fitted probability

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distribution, obtained in step 2, can be used to predict the probability of non-exceedance for a specified magnitude or, equivalently, the magnitude associated with a specific non-exceedance probability. The availability of the fitted probability distribution allows extrapolation beyond the range of the probabilities of the observed data series, which is limited by the length of the data record. Non-parametric approaches, such as the one using an Epanechnikov kernel (NP) can also be used in lowflow frequency analysis. The use of non-parametric methods for the estimation of quantiles allows one to eliminate the effect of the subjectivity due to the choice of distribution. However, while these non-parametric methods enjoy a strong descriptive power, they suffer from the lack of predictive capability (extrapolation beyond the range of available data). The treatment of zeros in low-flow frequency analysis can be performed either by adopting a conditional probability adjustment procedure (Stedinger et al., 1993), or by considering them as censored values (Durrans et al., 1999). Regional Estimation of Low Flow Characteristics at Ungauged Basins

In general, hydrometric gauging records are not available at the site of interest. Where these records are available, they may be of short length, leading to high uncertainties in the selection of the probability distribution and the estimation of the parameters of the selected model. When the observed streamflow records are unavailable or inadequate for a proper local frequency analysis, other approaches must be used. Regional frequency analysis is one of the most commonly used tools for the estimation of extreme hydrological events (floods, droughts) at sites where little or no reliable data are available (Vogel and Kroll, 1990; Tucci et al., 1995; Durrans and Tomic, 1996; Hamza et al., 2001; Ouarda et al., 2005; McLean and Watt, 2005; Laaha and Bloschl, 2006). In general, a regional frequency analysis procedure is composed of two main steps: the identification of groups of hydrologically homogeneous basins (or “regions”) and the application of a regional estimation method within each delineated region. It is possible to define the homogeneous regions in a variety of manners: as geographically contiguous regions, as geographically non-contiguous regions, or as © 2008 Canadian Water Resources Association

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hydrological neighbourhoods (or “regions of influence” as in Burn (1990)). The delineation of homogeneous hydrological regions can be carried out using a variety of methods, most of which are based on multivariate statistical tools, such as hierarchical cluster analysis or principal component analysis (Ouarda et al., 2008). The “neighbourhood” approach provides additional flexibility in comparison to the other two types of regions. It assumes that each target-site has its own homogeneous region or neighbourhood. This approach will be discussed in the following section. The second step of the regional frequency analysis procedure consists in transferring the information from the gauged sites of the region into the ungauged target site. Regressive models represent the most commonly used approaches for regional estimation. Pilon (1990) extended the index-flood method, used in regional flood frequency analysis, to low-flow analysis when the regional distribution is assumed to be the threeparameter Weibull. The performance of the approach was demonstrated by its application to the southern part of the province of Ontario. Other approaches that can be used for the estimation of low-flow characteristics at sites where inadequate discharge records are available include the drainage area ratio method, spatial interpolation techniques, and the use of regional prediction curves. The drainage area ratio method is a linear drainagearea discharge relationship that allows the estimation of low-flow quantiles at an ungauged basin on the basis of low-flow quantile values at a nearby station and the ratio of the areas of the two basins. Spatial interpolation and regional mapping techniques, such as the development of flow contour maps, assume the existence of a well defined relation between the flow field and a set of explanatory physiographic variables. Regional prediction curves can be established at ungauged sites by standardizing flows at gauged sites by a scale index and by combining the information into a single regional curve. It is also possible to synthetically generate a large number of single-site or multi-site streamflow time series based on other gauging station records and then proceed with the estimation of the low-flow characteristics.

Practical Tools for Regional Low-Flow Frequency Analysis

While all available frequency analysis software is suitable for a local low-flow estimation study, very few general ready-to-use tools are available for practitioners to carry out regional low-flow frequency analysis projects. On the other hand, simple regional regression equations have been identified for different regions in several parts of the world. These regional regression equations represent simple practical tools with very limited general interest and whose performance can be improved by adopting more advanced regional estimation approaches. The best known ready-to-use tool for regional lowflow frequency analysis is probably the software Low Flows 2000 (Young et al., 2003). It has been adopted by the Environment Agency (EA) and the Scottish Environment Protection Agency for estimating low flows in ungauged catchments. Low Flows 2000 allows the user to estimate annual and monthly flow duration curves, and to carry out a complete regional frequency analysis. Low Flows 2000 also facilitates the estimation of the Base Flow Index for ungauged sites within the UK. The REGIONS software (Ouarda et al., 2002) represents a general tool for regional frequency analysis of extreme streamflow events on both tails of the distribution. The low-flow module integrates several approaches for the delineation of homogeneous regions, for regional information transfer and for the combination of local and regional information. It also includes modules for regional frequency analysis in both stationary and non-stationary frameworks. In the province of Quebec, the software ARIDE (Ouarda et al., 2005) combines hydrological, statistical and GIS tools so that a regional low-flow frequency analysis can be performed in any site on the territory of the province. The adoption of recently developed regional estimation approaches by the practical engineering community goes through the development of general user friendly tools that allow familiarizing the hydrologic community with the potential of these approaches and providing the framework for a simple and efficient technological transfer.

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Multivariate Modelling of Low Flows at Ungauged Basins

While traditional frequency analysis has focused on univariate local and regional extremes, in recent years an increase in the use of multivariate local methods has been observed. However, the development of multivariate regional tools has not yet received sufficient attention. Even less attention has been devoted specifically to the regional multivariate modelling of low-flow characteristics. Low-flow events are described by multivariate characteristics (such as duration and volume). It is therefore important to jointly consider all these characteristics for a rational multivariate local or regional low-flow frequency analysis. The pioneering work dealing with multivariate modelling of low-flow characteristics was presented by Yevjevich (1967) who proposed the “theory of runs” which considers events for which discharge remains below a given threshold level for a specific duration. This theory has been used in a variety of studies dealing with low-flow and drought analysis and management (e.g., Burn and DeWit, 1996; El-Jabi et al., 1997). A detailed description of the theory of runs and its application to the local analysis of low-flow events is presented in Zelenhasic and Salvai (1987). A low-flow event can be described by its “duration” Di, its “volume” V i (also called “severity” representing the total water deficit below the threshold Q R) and “magnitude” Mi represented by the ratio of the volume Vi over the duration Di. Other characteristics of interest are the “intensity” Ii, represented by the lowest streamflow level below Q R that is reached during the low-flow, and the starting date of the low-flow event (see Figure 1). Recently, Burn and Farid (2007) and Ouarda (2007) proposed multivariate regional procedures for the joint estimation of several low-flow characteristics that are based on the use of copulas. Multidimensional copulas (Nelsen, 1999) have a strong potential of use for the regional multivariate modelling of low-flow characteristics. Indeed, they overcome all the major limitations of classical bivariate and multivariate distributions (limited form of marginal distribution, the same marginal distribution for all variables, difficulties in fitting the variables and in estimating the parameters, the multivariate case is practically impossible except for the normal distribution).

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Chebana and Ouarda (2007) proposed a Lmoment multivariate homogeneity test based on the use of L-co-moments to define the statistics and copula models to describe the statistical behaviour of dependent variables. Chebana and Ouarda (2007) showed the importance of simultaneously treating all variables and identifying a single homogeneous region that takes into consideration the dependence between the various variables. While the usefulness of the methodology was illustrated on flood events, the multivariate homogeneity test has a strong potential in regional multivariate low-flow frequency analysis. The development of this test represents a first step towards a rational and general framework for regional multivariate frequency analysis of extreme hydrological events. Other steps should follow.

Usefulness of the Neighbourhood Approach in Regional Low-Flow Frequency Analysis GREHYS (1996) presented an inter-comparison study of various regional flood estimation procedures obtained by coupling several methods for delineating homogeneous regions and for regional estimation. The study concluded that the neighbourhood approach (Burn, 1990; Ouarda et al., 2001) for the delineation of groups of hydrologically homogeneous basins is superior to the approaches based on building a fixed set of homogeneous regions. Ouarda et al. (2005) carried out a regional low-flow frequency analysis in which five different approaches were compared on the basis of their application to the regional estimation of summer and winter low-flow characteristics of the hydrometric station network of southern Quebec. All approaches are based on the use of catchment physiographic and meteorological attributes as surrogates to catchment hydrologic properties. The delineation of homogeneous regions was carried out using the following methods: determination of regions based on the L-moment homogeneity test (LM), hierarchical cluster analysis (HC), region of influence (ROI), canonical correlation analysis (CCA), and spatial interpolation (SI). For each site and within each homogeneous region, a multiple linear regression model is constructed from physiographic and meteorological catchment attributes to obtain a regional estimate of low-flow characteristics. A jackknife method was used © 2008 Canadian Water Resources Association

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Figure 1. Multivariate characterization of a low-flow event.

for the assessment of the performance of each of the approaches. The results clearly indicated that the two approaches that fall under the neighbourhood category (ROI and CCA), outperform all the other methods. While the approach based on CCA led to best results for summer low-flows, the ROI method was shown to be superior to the other approaches for winter lowflows. The approach based on the test of homogeneity using L-moments often failed to meet the homogeneity criteria and could not even lead to the definition of homogeneous regions for information transfer. The results of this study demonstrated that the neighbourhood approach (CCA, ROI) for the delineation of homogeneous regions can be applied successfully to low-flow regionalization and leads to improvements over other traditional approaches commonly used in regional low-flow estimation. The use of other approaches that do not rely on the definition of a fixed set of homogeneous regions, such as artificial neural networks, should also be investigated for the regional estimation of low-flow characteristics at ungauged basins.

Statistical Use of Recession Curves in Regional Low-Flow Frequency Analysis For the aim of regional low-flow estimation, regressive models are often used at target sites. Independent variables are usually related to catchment and climatic characteristics. Those related to the geology of the basin have often been reported to have primordial influence on low-flow behaviour (Bingham, 1986; Tallaksen, 1989; Smakhtin, 2001). However, the practical problem is related to the best way to represent this influence numerically (Tallaksen, 1995). In many studies, baseflow recession characteristics have been used in regressive regional models to account for the effect of geology (Bingham, 1986; Tallaksen, 1989; Arihood and Glatfelter, 1991;Vogel and Kroll, 1992; 1996; Demuth and Hagemann, 1994; Kroll et al., 2004). A recession is defined by the gradual depletion of streamflow discharge by groundwater and soil discharge into the stream, and by evapotranspiration during dry weather periods (Figure 2). During these periods, the streamflow is composed principally of the baseflow, which is the portion of the flow that comes © 2008 Canadian Water Resources Association

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Figure 2. Hydrograph of the 022003 station on the Rimouski River (dashed lines are recession segments).

from groundwater and other delayed sources (Hall, 1986). Because the most important part of baseflow usually comes from groundwater storage, recession flows depend primarily on catchment geology (Smakhtin, 2001). Based on this assumption, recession characteristics have often been used as surrogates for geological characteristics. Recession characteristics are generally based on the relation between storage and outflow. The relationship between the storage S and the outflow Q of a penetrating stream channel into an unconfined aquifer can be expressed by S = aQb

(1)

where a and b are constants (Hall, 1986). The constant a is called the recession coefficient. The specific case when b = 1 corresponds to the linear reservoir model. The outflow then becomes the simple exponential function Qt = Q 0e-at, where Qt is the outflow at time t and Q 0 is the initial outflow. For the ease of analysis, many authors made the assumption that the reservoir model is linear (Bingham, 1986; Arihood

and Glatfelter, 1991; Vogel and Kroll, 1992). However, many authors indicated that a non-linear relationship is more realistic and concluded that the value of the parameter b is around 0.5 in most cases (Brutsaert and Nieber, 1977; Wittenberg, 1994; Chapman, 2003). Recession variables used in the context of regional estimation have never been defined by considering the non-linear reservoir model. One opportunity would be to include a recession parameter from the non-linear reservoir model in a regional model. Performances obtained with this parameter could be compared with those obtained when including a recession variable from the linear reservoir model. In addition, local hydrological data are needed to estimate a recession characteristic; consequently it cannot be estimated for ungauged catchments. Very limited attempts have been made to obtain recession characteristics estimates with partial record stream gauges (Eng and Milly, 2007). Furthermore, no study provided a statistical evaluation of the usefulness of the method with recession variables obtained from linear and non-linear reservoir models. Another approach would be to investigate the validity of the © 2008 Canadian Water Resources Association

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method by estimating recession parameters with very short hydrological data records (ranging from one to five years). These parameters, estimated with partial data, could then be included in regressive models to estimate quantiles at gauged stations and performance determined. Thus informative estimates of the recession characteristics at partially gauged sites could be obtained with only a few years of data. For the purposes of regional low-flow frequency analysis, the recession parameter from the non-linear reservoir model has better prediction power than the one from the linear model. Another conclusion is that it is then possible to combine local hydrological information with regional data in a regional model that includes a recession variable.

A Comment on Non-Stationarity and LowFlow Estimation Streamflows are sensitive to changes in climatic inputs and it is thus important to look for evidence of temporal changes in streamflow regimes in general, and in lowflows in particular. A temporal change could be in the form of a monotonic trend or a sudden jump or a combination of both. Low-flow regimes corresponding to various durations should be considered because the conclusions drawn from the analysis of 1-day low flows may not hold true for low flows of 30-day duration. In a study investigating the temporal evolution of 1-, 7-, 15- and 30-day annual and seasonal (summer and winter) low-flow regimes of pristine river basins of the Canadian reference hydrometric basin network (RHBN), Khaliq et al. (2008) considered three timeframes: 1974–2003, 1964–2003 and 1954– 2003. Nonparametric trend detection and bootstrap resampling approaches were used for the assessment of at-site temporal trends under the assumption of shortterm persistence. The results of the study demonstrated that both statistically significant increasing and decreasing trends were noticed in low-flow regimes in different parts of Canada. Furthermore, the results indicate that detected changes seem to be more significant for the more recent time windows. However, the exact nature of causes of these trends and the interaction between climatic factors and low-flow regimes were not investigated.

These results indicate that it is no longer realistic to ignore change signals in low-flow regimes when carrying out local and regional estimation activities. If a change signal is identified in low-flow records, non-stationary estimations should be produced at least as a possible scenario (rather than a unique scheme) and considered in the modelling effort. The use of statistical distributions for which the parameters are functions of covariates (time or other variables such as low frequency climate oscillation indices) should permit non-stationary estimation both on the local and regional scale. This type of non-stationary model is already used in the local and regional estimation of the right tail of the distribution of hydrological variables (El-Adlouni et al., 2007; Leclerc and Ouarda, 2007). Several recent developments in statistics will have application in the field of low-flow estimation. One particularly important approach is to use combination techniques such as the parametric Bayesian combination method to combine local and regional information. This type of approach would combine deterministic and stochastic models for regional estimation of low-flow characteristics; such approaches use many sources of information: [historic data, remotely sensed information, expert knowledge, other subjective information (see for instance Burn et al., 2004; Bonin and Burn, 2005)]. There are also approaches that might improve predictions of low flows using the inter-relationships between variables. These include modelling the spatial correlation between data of different sites in a region; developing non-stationary regional estimation models for low-flow characteristics by using probability distributions with covariates; and adapting powerful non-parametric approaches such as artificial neural networks (ANNs), adaptive neuro-fuzzy inference systems (ANFIS), ensemble-ANNs, CCA-ANNs for regional modelling of low flows. There are also several techniques that may be appropriate for improving how we include uncertainty. Improvements in uncertainty modelling may be gained from such tools as fuzzy logic and Bayesian modelling; this should translate into a reduction of the effective uncertainty. Finally, copulas, a general way of formulating a multivariate distribution with various general types of dependence, and in particular trivariate copulas, could be used to estimate all three low-flow characteristics (duration, volume and magnitude) at ungauged sites. © 2008 Canadian Water Resources Association

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Conclusions and Future Perspectives To meet the increasing demands for accurate estimations of low-flow processes for the various environmental and economic activities, the improvement of existing estimation methods and the development of more efficient and more appropriate modelling approaches are imperative. It is clear that more energy should be devoted to the development of improved tools for the estimation and analysis of low flows and resulting long-term droughts. Major advances in the estimation of extreme low flows will only result from the combination of the optimal adoption of recently developed advanced statistical tools and the improved understanding and physical modelling of the processes that contribute to the generation of these events at the catchment scale, along with the use of all available sources of information.

Acknowledgements The financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. The authors wish also to thank all the participants and organizers of the Low-Flow PUB Workshop (Quebec City, April 12-13, 2007). Thanks are also due to the Co-Editor, Paul Whitfield and the three anonymous reviewers of the paper.

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