Comparison of the annual minimum flow and the deficit below ...

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Comparison of the annual minimum flow and the deficit below threshold approaches: case study for the province of New Brunswick, Canada Loubna Benyahya, Daniel Caissie, Fahim Ashkar, Nassir El-Jabi, and Mysore Satish

Abstract: A low-flow frequency analysis using the annual minimum flow (AMF) and the deficit below threshold (DBT) approaches was carried out for 30 hydrometric stations across the province of New Brunswick. The AMF method considers only the annual minimum events, and the DBT method considers all low flows below a certain threshold (or truncation level). In the present study, the DBT method characterizes low flow in terms of deficit intensity, which is becoming increasingly important in both water and aquatic resources management. The annual minimum series were fitted using the three-parameter Weibull distribution (AMF–WEI3), and the intensity data series were fitted using the two-parameter Weibull distribution (DBT–WEI2) and the generalized Pareto distribution (DBT–GP). All parameter estimates were obtained using the maximum likelihood method. The AMF–WEI3 and DBT–GP approaches provided a good fit to at-site data in terms of annual minimum flow and deficit intensity, respectively. However, the present study showed that the DBT–WEI2 approach underestimated the right tail of low-flow intensity. The Anderson–Darling statistic was also calculated for the data series and can provide insight into which distribution may adequately model the low-flow characteristics. A regionalization study was also performed using the AMF–WEI3 and DBT–GP methods. Key words: low flow, frequency analysis, annual minimum flow, deficit below threshold, regionalization. Re´sume´ : Une analyse fre´quentielle en pe´riode de faible ećoulement utilisant les me´thodes de l’ećoulement annuel minimal (« AMF ») et l’analyse de se´ries de de´passement incomple`te (« DBT – deficit below threshold ») a e´te´ reáliseé dans 30 stations hydrome´triques dans la province de Nouveau-Brunswick. La me´thode AMF ne conside`re que les e´veńements annuels minimums alors que la me´thode DBT tient compte de tous les ećoulements sous un certain seuil (ou niveau de coupure). Dans cette e´tude, la me´thode DBT caracte´rise le faible ećoulement en termes d’intensite´ de de´ficit, qui devient de plus en plus important en gestion de l’eau et des ressources aquatiques. Les se´ries minimales annuelles ont e´te´ ajusteés en utilisant une distribution de Weibull a` trois parame`tres (AMF-WEI3); les se´ries de donneés d’intensite´ ont e´te´ ajusteés en utilisant la distribution de Weibull a` deux parame`tres (AMF-WEI2) et la distribution de Pareto geńe´raliseé (DBT-GP). Toutes les estimations de parame`tres ont e´te´ obtenues en utilisant la me´thode de vraisemblance maximale. Les approches AMF-WEI3 et DBT–GP ont fourni un bon ajustement aux donneés sur le site en termes respectifs d’ećoulement minimum annuel et d’intensite´ de de´ficit. Toutefois, la pre´sente e´tude montre que l’approche DBT–WEI2 sous estime la queue droite de l’intensite´ de l’ećoulement faible. La statistique Anderson-Darling a e´galement e´te´ calculeé pour les se´ries de donneés et elle peut fournir un aperçu selon lequel la distribution peut ade´quatement mode´liser les caracte´ristiques de faible ećoulement. Une e´tude de re´gionalisation a e´galement e´te´ reáliseé en utilisant les me´thodes AMF–WEI3 et DBT–GP. Mots-cle´s : faible ećoulement, analyse fre´quentielle, ećoulement minimum annuel, analyse de se´ries de de´passement incomple`te, re´gionalisation. [Traduit par la Re´daction]

Received 14 February 2008. Revision accepted 10 May 2009. Published on the NRC Research Press Web site at cjce.nrc.ca on 24 September 2009. L. Benyahya.1 Department of Civil engineering, Dalhousie University, Halifax, NS B3J 2X4, Canada; Department of Fisheries and Oceans Canada, P.O. Box 5030, Moncton, NB E1C 9B6, Canada. D. Caissie. Department of Fisheries and Oceans Canada, P.O. Box 5030, Moncton, NB E1C 9B6, Canada. F. Ashkar. Department of Mathematics and Statistics Universite´ de Moncton, Moncton, NB E1A 3E9, Canada. N. El-Jabi. Department of Civil engineering, Universite´ de Moncton, Moncton, NB E1A 3E9, Canada. M. Satish. Department of Civil engineering, Dalhousie University, Halifax, NS B3J 2X4, Canada. Written discussion of this article is welcomed and will be received by the Editor until 31 January 2010. 1Corresponding

author (e-mail: [email protected]).

Can. J. Civ. Eng. 36: 1421–1434 (2009)

doi:10.1139/L09-077

Published by NRC Research Press

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Introduction Small streams and many segments of larger rivers are seasonally affected by major changes in their flow regime, particularly during low-flow events. Low flows are defined as flows present in streams and rivers during prolonged dry weather (WMO 1974). The natural flow regime and lowflow conditions are largely determined by the local climate, underlying geology, soils, topography, and vegetation, among others. Smakhtin (2001) provided a comprehensive review of low-flow conditions and their characteristics. The study indicated that low flow is governed by groundwater and surface water (e.g., lakes, marshes) depletion within a basin. The magnitude of low flow depends not only on initial moisture conditions prevailing at the onset of a period of dry weather but also on the duration of the event. The natural flow regime and low-flow conditions are of themselves naturally complex; however, they can also be affected by human activities such as irrigation and other water withdrawal activities. Low flows have also been observed to be impacted by climate change as reported in some studies (Lins and Slack 1999; McCabe and Wolock 2002). As suggested by Hamlet and Lettenmaier (1999), in regions where there is snow and rainfall, climate change could shift the balance between these two forms of precipitation, resulting in earlier snowmelt and reducing late summer streamflow, which could potentially intensify hydrologic droughts. Water resources managers have been increasingly concerned about the impacts of low flow on irrigation, water supplies, and other engineering works and on river ecosystems and instream flow requirements. For instance, several studies have examined responses of fish populations to lowflow events. Magoulick and Kobza (2003) observed an increase in the mortality rate, a decrease in the birth rate, and an increase in the migration rate of fish under low-streamflow conditions. Caruso (2001) reported that fish mortality increased during low flows because fish became stranded and isolated from the main flow and because of the stress associated with the decreased availability of deeper pool refuges. Notably, Covich et al. (2003) found that the extreme low flows affected shrimp species through a decrease in habitat and resource availability and subsequent increase in crowding. Moreover, low-flow events can adversely impact water quality in rivers by reducing their dilution capacity and increasing water temperatures (Nemerow 1985). To assess the availability of water for municipal and industrial supplies, irrigation, and wildlife conservation, analysis of streamflow characteristics and a thorough understanding of low flows are needed. Many techniques can be used for analyzing low flows, such as flow duration analysis (Vogel and Fennessey 1995), base flow recession curve analysis (Nathan and McMahon 1990), and low-flow frequency analysis (Vogel and Wilson 1996), among others. Low-flow frequency analysis using statistical methods is the most commonly used approach for estimating probabilities of low-flow events. Recently, Ouarda et al. (2008a) reviewed statistical models commonly used in the estimation of low flows and outlined potential improvements. In a related study, Whitfield (2008) suggested that in Canada the focus needs to be not only on the magnitude of low flows but also on their duration and persistence, as well as their seasonal and spatial variability.

Can. J. Civ. Eng. Vol. 36, 2009

The more traditional low-flow frequency analysis consists of analyzing annual minimum average flows over a period of k days, where the k values most frequently used are 1, 7, or 10 days. This method was the subject of several low-flow studies (Kroll and Vogel 2002; He´bert et al. 2003). However, one potential limitation of the annual minimum flow (AMF) approach results from the fact that it considers only the most severe low-flow events within a given time period (year, season) and eliminates secondary low-flow events that may be important. An alternative to the AMF approach is to analyze all flows below a given threshold (or truncation level), an approach known as the partial duration series approach. In fact, most partial duration series studies have focused on high-flow events and carried out the analysis through the use of the peak over threshold (POT) approach (Ashkar and El-Jabi 2002), whereas only a few studies have considered the partial duration analysis for low-flow analysis (Tallaksen et al. 1997; El-Jabi et al. 1997; Ashkar et al. 1998, 2004). In the present study, the partial duration series approach applied to low flows is referred to as the deficit below threshold (DBT) method. To our knowledge, no studies in the literature have attempted to compare both AMF and DBT approaches on a regional basis. Therefore, the present study has the following objectives for the province of New Brunswick: (i) compare results from AMF and DBT approaches to determine which method best represents low-flow events in the province, and (ii) provide regional low-flow frequency characteristics to predict low flow for ungauged catchments within the province. The present study was carried out using 30 hydrometric stations across the province of New Brunswick.

Methods Annual minimum flow (AMF) approach The first approach that was used to describe the frequency of low-flow events was the annual minimum flow (AMF) approach, in which the minimum annual discharge (Qmin) is analyzed for the period from January to December. The minimum annual discharges were extracted from the daily mean streamflow time series for each year of record. Various statistical frequency models have been used to represent the distribution of low flows. In the present study, the minimum annual discharges were fitted using the threeparameter Weibull (WEI3) distribution (or extreme value type III). This distribution was chosen because it is theoretically the parent model of extreme low flows (Gumbel 1954), and hence it was one of the main models considered by Matalas (1963) and Loganathan et al. (1985), among others, as suitable for low-streamflow analysis. The WEI3 cumulative distribution function (cdf) is of the form " # xt b ½1 FðxÞ ¼ 1 exp ;x t h where x represents discharge, t is a threshold parameter, h > 0 is a scale parameter, and b is a shape parameter. Several methods have been used to estimate the parameters of a distribution, but the methods of moments and of maximum likelihood are the most common ones. The maximum likelihood method (MLM) is used in this study. The Published by NRC Research Press

Benyahya et al. Fig. 1. Hydrograph representing low-flow event characteristics by the deficit below threshold (DBT) approach.

1423 Fig. 2. Location of hydrometric stations in the province of New Brunswick. Station names are abbreviated (e.g., 01AD002 is given as 1AD2).

advantage of using the MLM over other methods is that under quite general conditions it gives asymptotically optimal estimators of the parameters (i.e., unbiased with minimum variance). Once the parameters of the Weibull distribution are estimated, the cdf is then used to calculate estimates of low flows QT associated with different return periods T (e.g., T = 2, 5, 10, 20, or 50 years). These WEI3 quantiles QTT are calculated by the following equation: ½2

QT ¼ hfln ½1 FðxÞ1=b g þ t

where the relationship between the cdf F(x) and the return period T is given by ½3

FðxÞ ¼

1 T

which means that the lower the annual flow, the higher the return period. Deficit below threshold (DBT) approach The deficit below threshold (DBT) approach consists of analyzing all flows below a given reference discharge (Qr), or threshold, taking into account their times of occurrence. It should be noted that the threshold (Qr) is different from that described in eq. [1], which is a WEI3 location parameter. The DBT approach characterizes the nth low-flow event (En) in terms of its intensity In (maximum flow deficit during the event in m3/s), duration Tn (length of time during which the flow is below the reference value in days), and volume Dn (cumulative streamflow deficit for the duration of flow below threshold in m3). The characteristics of a typical En event are illustrated in Fig. 1, in which the difference between the time of the beginning of an event and the time of the end of the previous event is denoted by DTn,n+1 (in days). In the present study, the strategy for choosing the optimal threshold level (reference discharge) was to try several successive values of Qr and select the value that gave the best fit between observed and estimated intensity (In) distributions. Following the selection of Qr, some low-flow events may be close to each other and therefore statistically dependent. To minimize this dependency, events were com-

bined when the time separating two consecutive events DTn,n+1 was less than or equal to 6 days, as per Savoie et al. (2004). In the case where events were combined, their low-flow characteristics (intensity, duration, and volume) were calculated based on the combined events. Several distributions have been suggested to model the exceedances in the partial duration series approach. These include the gamma distribution (Zelenhasic 1970), the Weibull distribution (Ekanayake and Cruise 1993), the logarithm-normal distribution (Rosbjerg et al. 1991), and the generalized Pareto distribution (Wang 1991; Madsen et al. 1997). In the present study, two two-parameter distributions were selected for the analysis, namely the Weibull (WEI2) and generalized Pareto (GP) distributions. The WEI2 distribution was chosen because its three-parameter counterpart, Published by NRC Research Press

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Can. J. Civ. Eng. Vol. 36, 2009 Table 1. Location, identification (ID) number, drainage area, and record length of selected hydrometric stations analyzed in New Brunswick. Location Saint John River at Fort Kent St. Francis River at outlet of Glacier Lake Fish River near Fort Kent Grande River at Violette Bridge Meduxnekeag River near Belleville Big Presque Isle Stream at Tracey Mills Becaguimec Stream at Coldstream Shogomoc Stream near Trans-Canada Highway Nackawic Stream Nashwaak River at Durham Bridge Narrows Mountain Brook near Narrows Mountain North Branch Oromocto River at Tracy Salmon River at Castaway Canaan River at East Canaan Kennebecasis River at Apohaqui Lepreau River at Lepreau Restigouche River below Kedgwick River Upsalquitch River at Upsalquitch Jacquet River near Durham Centre Restigouche River, Rafting Ground Brook Middle River near Bathurst River Caraquet at Burnsville Big Tracadie River at Murchy Bridge Crossing Southwest Miramichi River at Blackville Little Southwest Miramichi River at Lyttleton Northwest Miramichi River at Trout Brook Coal Branch River at Beersville Petitcodiac River near Petitcodiac Turtle Creek at Turtle Creek Point Wolfe River at Fundy National Park

Station ID 01AD002 01AD003 01AE001 01AF007 01AJ003 01AJ004 01AJ010 01AK001 01AK007 01AL002 01AL004 01AM001 01AN002 01AP002 01AP004 01AQ001 01BC001 01BE001 01BJ003 01BJ007 01BJ010 01BL002 01BL003 01BO001 01BP001 01BQ001 01BS001 01BU002 01BU003 01BV006

Drainage area (km2) 14 700 1 350 2 260 339 1 210 484 350 234 240 1 450 3.89 557 1 050 668 1 100 239 3 160 2 270 510 7 740 217 173 383 5 050 1 340 948 166 391 129 130

Period of record 1927–1999a 1952–1999 1981–1999a 1977–1999 1968–1999 1968–1999 1974–1999 1919–1999a 1968–1999 1962–1999 1972–1999 1963–1999 1974–1999 1926–1999a 1962–1999 1919–1999 1963–1999 1919–1999a 1965–1999 1969–1999 1982–1999 1970–1999 1971–1999 1919–1999a 1952–1999 1962–1999 1965–1999 1962–1999 1963–1999 1965–1999

Record length (years) 72 48 18 23 32 32 26 78 32 38 28 37 26 52 38 81 37 70 35 31 18 30 29 51 48 38 35 38 37 35

a

Data are missing for some years during the period of record.

WEI3, has been widely applied for studying low flows (Matalas 1963). It was also noted that the WEI2 distribution has been used in peak over threshold (POT) flood frequency modeling in the past (Miquel 1984; Groupe de recherche en hydrologie statistique (GREHYS 1996) and in the frequency analysis of low flows (Loganathan et al. 1985). Moreover, the generalized Pareto distribution was chosen because it has been widely suggested for exceedance distribution (Davison and Smith 1990; Wang 1991; Madsen et al. 1997). The WEI2 distribution has the following cdf: " 0# x b ;x 0 ½4 HðxÞ ¼ 1 exp 0 h where h’ > 0 is the scale parameter, and b’ is the shape parameter. The GP distribution has the following cdf: h x i1=k ½5 HðxÞ ¼ 1 1 k ; a 0 x < 1for k < 0; 0 x a=k for k > 0 where a > 0 and k are the scale and shape parameters, respectively. For hydrologic engineering applications, Hosk-

ing and Wallis (1987) restricted attention to –0.5 < k < 0.5, values most commonly encountered in practice. The distribution of the annual maximum deficit intensity (from which the annual minimum low flow is calculated as described later in the paper) is given by the following equation when using the DBT approach: ½6

FðxÞ ¼ 1 expfl½1 HðxÞg

where l represents the mean number of events per year, and H(x) is the cdf of the deficit intensity In. By substituting cdfs eq. [4] (WEI2) and eq. [5] (GP) for H(x), the distribution of the annual maximum deficit intensity by the DBT approach is given by eqs. [7] and [8], respectively: ( " 0 #) x b ½7 FðxÞ ¼ 1 exp lexp 0 h

½8

h x i1=k FðxÞ ¼ 1 exp l 1 k a

The deficit intensity for different recurrence intervals T (in years) is then given for WEI2 and GP by eqs. [9] and [10], respectively: Published by NRC Research Press

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Table 2. Maximum likelihood estimates of AMF characteristics for the studied catchments. Shape parameter, b 2.06 1.96 1.59 2.61 1.00 2.10 2.54 1.25 1.00 1.35 1.36 1.03 1.23 1.66 2.34 1.29 2.05 2.04 1.81 2.10 1.42 3.08 1.62 2.60 2.48 2.21 2.06 1.77 2.46 1.28

Station ID 01AD002 01AD003 01AE001 01AF007 01AJ003 01AJ004 01AJ010 01AK001 01AK007 01AL002 01AL004 01AM001 01AN002 01AP002 01AP004 01AQ001 01BC001 01BE001 01BJ003 01BJ007 01BJ010 01BL002 01BL003 01BO001 01BP001 01BQ001 01BS001 01BU002 01BU003 01BV006

½9

½10

Scale parameter, h 22.720 2.240 2.410 0.550 1.103 0.860 0.523 0.300 0.063 2.060 0.006 0.550 1.060 0.495 2.240 0.506 5.560 4.030 0.657 15.400 0.209 0.510 0.950 13.600 4.500 1.700 0.200 0.382 0.236 0.281

Threshold parameter, t 12.820 1.530 4.320 0.106 0.424 0.060 0.047 0.045 0.014 2.300 0.002 –0.012 1.110 0.044 0.741 0.025 4.850 2.050 0.439 9.590 0.091 0.220 0.730 7.150 1.200 1.230 0.045 0.084 0.116 0.104

Fig. 3. Observed and fitted probability distribution (WEI3) of minimum annual discharge data for four selected stations in New Brunswick: (a), (b) best fit; (c), (d) less adequate fit.

0 ln½1 FðxÞ 1=b XT ¼ h ln l 0

! a ln½1 FðxÞ k 1 XT ¼ k l

eqs. [9] and [10] follow directly from eqs. [3], [7], and [8]. It should be noted that the annual maximum deficit intensity XT in eqs. [9] and [10] represents the difference between the threshold value (Qr) and the annual low flow QT. Therefore, the low flows for different recurrence intervals are given by ½11

QT ¼ Qr X T

As with the WEI3 distribution, the MLM was used for parameter estimation of the WEI2 and GP models. The use of the MLM for the WEI2 model was recently recommended by Mahdi and Ashkar (2004) after a comparison with several competing estimation methods. All analyses and parameter estimations for the WEI3, WEI2, and GP distributions were performed using Minitab and Matlab software versions 15.1 (2006) and 7.2 (2006). Published by NRC Research Press

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Can. J. Civ. Eng. Vol. 36, 2009 Table 3. AMF quantile estimates associated with return periods of 2, 5, 10, 20, and 50 years for the studied catchments. Return period Station ID 01AD002 01AD003 01AE001 01AF007 01AJ003 01AJ004 01AJ010 01AK001 01AK007 01AL002 01AL004 01AM001 01AN002 01AP002 01AP004 01AQ001 01BC001 01BE001 01BJ003 01BJ007 01BJ010 01BL002 01BL003 01BO001 01BP001 01BQ001 01BS001 01BU002 01BU003 01BV006

2 years 31.83 3.39 6.24 0.58 1.19 0.78 0.50 0.27 0.06 3.87 0.500 0.456 >0.500 0.433 >0.500 >0.500

WEI2 0.130 >0.250 >0.250 >0.250 0.097 0.250 0.250 >0.250 0.071 >0.250 >0.250 >0.250 >0.250 >0.250 0.185

Fig. 7. Regression analysis (AMF and DBT approaches) of daily low flow for return periods of 2, 10, 20, and 50 years. The solid lines represent the AMF (WEI3) approach, and the broken lines the DBT (GP) approach. Equations are applicable to basins with drainage area from 129 to 14 700 km2.

GP >0.500 >0.500 0.500 0.500 >0.500 0.250 >0.500 >0.500 >0.500 >0.500 >0.500 >0.500 >0.500 >0.500 >0.250 >0.500 >0.500 >0.500 >0.500

tail of the DBT–WEI2 model is not as flexible as that of the DBT–GP model, so the DBT–GP model is generally better at fitting the extreme values of the data series (lowest annual minimum flows). To confirm the findings obtained from the visual graphical evaluations of the AMF and DBT methods, we calculated the Anderson–Darling (AD) goodness-of-fit statistic. This statistic (D’Agostino and Stephens 1986) is known to be more sensitive and therefore more powerful than other commonly used goodness-of-fit statistics to variations in distribution tails. The AD statistic values, which are presented as box plots in Fig. 6, are for the WEI3 fit (three parameters) that was used in the AMF approach and for the GP and WEI2 fits (each with two parameters) that were used in the DBT approach. The AD statistics for the WEI3 and WEI2 models are conveniently obtained in Minitab. For the GP model they were calculated using Matlab. The AD statistics for the WEI3 model were, on average, lower than those for the GP and WEI2 models (Fig. 6). This is to be expected because the WEI3 model has one more parameter than the other two models. The GP and WEI2 models, on the other hand, are more comparable between themselves because they are applied to the same data and have the same number of parameters. It is clear from Fig. 6 that between these two Published by NRC Research Press

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Can. J. Civ. Eng. Vol. 36, 2009 Table 7. Regression analysis of daily low flow as a function of drainage area for recurrence intervals of 2, 10, 20, and 50 years (log(QT) = log(a) + b log(DA)). AMF method Return period (years) 2 10 20 50

b 1.12 1.21 1.24 1.26

Log(a) –3.04 –3.54 –3.69 –3.84

GP method

R2 0.91 0.87 0.85 0.83

models, the GP showed lower AD statistic values on average. Generally, the lower AD statistic values represent a better fit. Results for the AD statistic in the form of median, first quartile, and third quartile for the three models are as follows: AMF–WEI3, 0.26, 0.23, and 0.39; DBT–GP, 0.39, 0.27, and 0.59; DBT–WEI2, 0.45, 0.26, and 0.95. It is important to add that Minitab also provides probability p values for the AD goodness-of-fit test, which aids in deciding on whether or not to reject the AMF–WEI3 or DBT–WEI2 fit (cdfs eqs. [1] and [4]). For testing the GP distribution cdf (eq. [5], similar p values are given by Choulakian and Stephens (2001) for samples of size greater than 25. The p values are presented in Table 6 for the catchments under study. A p value less than 0.05 means that the fit should be rejected at the 5% significance level. From Table 6, it can be concluded that the DBT–WEI2 hypothesis is rejected in 30% (9 of 30) of the stations, the DBT–GP hypothesis is rejected in 23% (7 of 30) of the stations, and the AMF–WEI3 hypothesis is rejected in 7% (2 of 30) of the stations. It is important to note, however, that in a few cases (e.g., stations 01AK001 and 01AK007), even though the AD test rejected the null hypothesis of a GP distribution, both the DBT–GP and AMF–WEI3 fitted equally well the tail of the distribution (Figs. 5c and 5d). This was not the case for the DBT–WEI2 distribution function. In summary, the results of the present study showed that the cdfs F(x) obtained under the AMF–WEI3 and DBT–GP models are more flexible than the cdf under the DBT–WEI2 model and are thus better capable of representing the right tail of the distribution of low flows. The third part of the study pertains to the regionalization of low-flow frequency estimates. The hydrometric station 01AL004 was excluded from regionalization calculations because its drainage area was small (