Estimating the streamflow corresponding to a particular probability is of great importance in many hydrological studies, such as determining hydroelectric water potential, assessing water quality, and investigating sedimentation and drought. This paper aims to effectively estimate low-flow quantiles since hydrologic droughts motivate the study. The study illustrates a methodology, where droughts are characterized by the lower part of the flow–duration curve (FDC), and offers a perspective estimating low-flow quantiles related to the basin characteristics. Low-flow quantiles are derived both from traditional FDCs curves and median annual FDCs (AFDCs). As an innovation, the concept of areal scale factor, which represents a scaling ratio between the basin area and flow quantiles, was introduced. Unlike many other parametric approaches, this study models the streamflow quantiles depending on the basin characteristics instead of the parameters in the analytical equation of FDCs. The methodology was evaluated for the Western and Southwestern Anatolia regions in Turkey. The outcomes were compared for two types of FDCs in two regions. The approach gave similar results for both study regions. AFDCs provided a distinct advantage over traditional FDCs, especially for low-flow quantiles, due to the superiority of AFDCs in estimating streamflow quantiles of intermittent streams.

  • This research study offers a practical solution in terms of low-flow estimation.

  • It uses median annual flow–duration curves (AFDCs) instead of traditional FDCs, which were frequently used in the literature.

  • Instead of FDC used in many studies, it models the streamflow quantiles.

  • It proposes the areal scale factor, which describes the ratio between flow quantiles and basin area, as an innovation.

Hydrological studies can employ a range of methodologies to assess flow variability, taking into account elements such as droughts, low flows, high flows or floods, annual flows, and the seasonal flux of these flows. The variability of the flow in a basin and/or a stream is related to many climatic and hydrological factors (precipitation, evapotranspiration, infiltration, surface storage, land slope, vegetation, and soil type), and one of the most common tools used in hydrology and environmental sciences in examining this variability is the flow–duration curve (FDC). Flow period is also an important factor that significantly affects flow variability. The FDC graphs the relationship between a certain flow magnitude and a time period at which the flow magnitude is equalized or exceeded (Vogel & Fennessey 1995). Introduced by Clemens Herschel in 1880, FDCs have gained widespread acceptance and are used in environmental and hydrological studies around the globe.

There are two distinct types of FDCs mentioned in the literature with respect to their production. The first and the traditional one represents a long-term demonstration of the flow regime using all the long-term flow records and is reported in the literature as a period-of-record FDC (Vogel & Fennessey 1994). The second one is based on the annual interpretation of FDCs (LeBoutillier & Waylen 1993; Vogel & Fennessey 1994) and considers FDCs for separate years (annual FDCs, AFDCs); each one is constructed similar to the FDC, using only annual hydrometric information (Castellarin et al. 2013). A conjunctive evaluation of AFDCs provides a perspective of the interannual variability of the FDCs, which allows the estimation of the median of the AFDCs. The median AFDC is a hypothetical FDC representing the annual flow regime for a typical hydrological year and is not affected by the observations of abnormally wet or dry periods during the period of record (Vogel & Fennessey 1994; Castellarin et al. 2004a). The use of a traditional period-of-record FDC for regionalization limits the representation of the flow regime to the specific record period for which the FDC is constructed (Copestake & Young 2008; Taye & Willems 2011; Mendicino & Senatore 2013). AFDCs are presented to be less sensitive to the period-of-record than traditional FDCs, specifically in the region of low flows (Vogel & Fennessey 1994; Castellarin et al. 2013).

For similar reasons, Niadas (2005) suggested that for regional analysis involving multiple sites, AFDCs represent regional conditions better compared to traditional FDCs for which concurrent data are more important.

The estimation methods of regional FDCs are usually categorized into three groups (Castellarin et al. 2004b; Shu & Ouarda 2012):

Parametric methods have widely been used in various forms for basins in several parts of the world to derive FDCs in unmeasured or partially measured basins. Castellarin et al. (2004b) compared three different parametric approaches of the literature (Quimpo et al. 1983; Mimikou & Kaemaki 1985; Franchini & Suppo 1996). The approach of Franchini & Suppo (1996) outperformed all other parametric approaches due to the regionalization of flow quantiles (Q30, Q70, Q90, and Q95) instead of parameters and therefore inspired this study.

This study aimed to estimate the low-flow quantiles effectively. For this purpose, the FDC model of Franchini & Suppo (1996), based on an empirical regression equation between the streamflow quantiles and the basin characteristics, was improved here by also taking into account the median AFDCs. The relationship between the hydrological, geographical, and meteorological characteristics of the basin and the low-flow quantiles was examined. Low-flow quantiles are derived from the lower part of both traditional and median annual FDCs obtained from daily flow records. The findings revealed that the basin area is the most important basin characteristic to represent low-flow quantiles (Q30, Q40, Q50, Q60, Q70, Q80, Q90, Q95, and Q99). The relationship between low-flow quantiles and the basin area is defined by a ratio parameter called the areal scale factor (ASF). It is pointed out that low flows can be successfully estimated by using ASFs, which is presented as an innovation in this study. A case study for Western and Southwestern Anatolia regions in Turkey was implemented to show its applicability. Ungauged sub-basins are expected to benefit from the developed model.

Study area and dataset

This study was carried out for the basins located in the western and southwestern regions of the Anatolian peninsula, which forms a large part of Turkey (Figure 1). The climate of both regions is Mediterranean on the coasts where summers are hot and dry and winters are mild and rainy. Interiors are cold and usually see some snowfall during the winter. The climate is semi-arid continental. The average annual rainfall is about 596 mm (1938–2020) in Western Anatolia and 608 mm (1938–2020) in Southwestern Anatolia. Precipitations are concentrated in the winter period, and no precipitation usually occurs in July and August.
Figure 1

Study area and locations of flow gauging stations.

Figure 1

Study area and locations of flow gauging stations.

Close modal

Although the two regions in the study show similar climatic features, they differ geologically. In Western Anatolia, 85% of topographic surfaces are composed of metamorphic and volcanic rocks, whereas in Southwestern Anatolia, 80% of the surface is composed of sedimentary rocks (limestone). The majority of groundwater resources in Southwestern Anatolia are due to the existence of complexes characterized by high infiltration. The geological formation of the region clearly highlights the significance of the outcropping of limestone complexes in relation to the infiltration and circulation of rainwater. In such basins, finding a direct relationship between the area and the amount of water flowing into the basin outlet is difficult (Franchini & Suppo 1996). Therefore, the effect of limestone complexes on the drainage mechanism and low-flow characteristics is significant and differs considerably to that within the non-karstic regions in Western Anatolia.

Daily streamflow data were provided for the study by State Water Works of Turkey. Only unregulated river basins with a minimum of 10 years of data were considered. In the study, 30 flow gauging stations in Western Anatolia and 21 in Southwestern Anatolia were used. The average daily flows of the basins range from 0.43 to 39.64 m3/s in Western Anatolia and from 0.08 to 27.47 m3/s in Southwestern Anatolia. A substantial number of the rivers in southwestern Anatolia are fed by groundwater. The minimum record length is 10 years, the maximum record length is 54 years, and the mean sample size is approximately 20 years in both regions. The basin areas range from 32 to 15,848 km2 in Western Anatolia and from 8 to 2,448 km2 in Southwestern Anatolia. The average annual rainfall varies between 556 and 658 mm in Western Anatolia and between 526 and 729 mm in Southwestern Anatolia. The main hydrological and geographical characteristics of the basins and flow gauging stations are given in Tables 1 and 2 for Western and Southwestern Anatolia, respectively.

Table 1

Basin characteristics in Western Anatolia

BasinDrainage area (km2)Mean flow (m3/s)Annual rainfall (mm)Altitude (m)Basin yield (mm/year)
Orhaneli Creek at Kucukilet 1,622 5.14 631 795 100 
Atnos Creek at Balikli 1,384 7.67 601 94 175 
3 Emet Creek at Dereli 1,126 4.61 579 557 129 
Kille Creek at Buyukbostanci 544 2.66 609 105 154 
5 Simav Creek at Osmanlar 1,254 7.74 571 271 195 
Dursunbey Creek at Sinderler 975 5.86 622 294 190 
7 Zeytinli Creek at Zeytinli 123 3.18 572 35 814 
Manastir Creek at Kavaklar 32 1.02 576 55 1,003 
Inonu Creek at Inonu 73 0.86 580 65 371 
10 Cumalidere Creek at Cumali 85 0.21 562 115 79 
11 Yagcili Creek at Yagcili 86 0.63 564 210 230 
12 Medar Creek at Kayalioglu 902 3.34 561 77 117 
13 Kum Creek at Killik 3,189 7.94 562 54 78 
14 Selendi Creek at Derekoy 690 2.38 598 345 109 
15 Deliinis Creek at Topuzdamlari 735 3.29 596 381 141 
16 Murat Creek at Sazkoy 176 1.66 585 790 297 
17 Yigitler Creek at Yigitler 64 0.72 579 158 357 
18 Tabak Creek at Caltili 81 1.09 600 137 425 
19 Gediz River at Muradiye Bridge 15,848 39.64 556 17 79 
20 Ahmetli Creek at Derekoy 95 0.74 590 125 245 
21 Uladi Creek at Yakapinar 69 0.43 572 120 198 
22 Cine Creek at Kayirli 948 4.79 636 262 159 
23 Büyük Menderes River at Burhaniye 12,799 21.40 630 120 53 
24 Büyük Menderes River at Citak Bridge 3,946 6.88 598 802 55 
25 Yenidere Creek at Calikoy 669 1.14 647 855 54 
26 Akcay Creek at Degirmenalani 855 7.66 653 397 283 
27 Sarhos Creek at Goktepe 236 1.16 648 390 156 
28 Büyük Menderes at Cogasli 4,664 7.28 610 617 49 
29 Mortuma Creek at Yemisendere 170 1.63 658 478 302 
30 Banaz Creek at Ulubey 2,286 4.85 605 531 67 
BasinDrainage area (km2)Mean flow (m3/s)Annual rainfall (mm)Altitude (m)Basin yield (mm/year)
Orhaneli Creek at Kucukilet 1,622 5.14 631 795 100 
Atnos Creek at Balikli 1,384 7.67 601 94 175 
3 Emet Creek at Dereli 1,126 4.61 579 557 129 
Kille Creek at Buyukbostanci 544 2.66 609 105 154 
5 Simav Creek at Osmanlar 1,254 7.74 571 271 195 
Dursunbey Creek at Sinderler 975 5.86 622 294 190 
7 Zeytinli Creek at Zeytinli 123 3.18 572 35 814 
Manastir Creek at Kavaklar 32 1.02 576 55 1,003 
Inonu Creek at Inonu 73 0.86 580 65 371 
10 Cumalidere Creek at Cumali 85 0.21 562 115 79 
11 Yagcili Creek at Yagcili 86 0.63 564 210 230 
12 Medar Creek at Kayalioglu 902 3.34 561 77 117 
13 Kum Creek at Killik 3,189 7.94 562 54 78 
14 Selendi Creek at Derekoy 690 2.38 598 345 109 
15 Deliinis Creek at Topuzdamlari 735 3.29 596 381 141 
16 Murat Creek at Sazkoy 176 1.66 585 790 297 
17 Yigitler Creek at Yigitler 64 0.72 579 158 357 
18 Tabak Creek at Caltili 81 1.09 600 137 425 
19 Gediz River at Muradiye Bridge 15,848 39.64 556 17 79 
20 Ahmetli Creek at Derekoy 95 0.74 590 125 245 
21 Uladi Creek at Yakapinar 69 0.43 572 120 198 
22 Cine Creek at Kayirli 948 4.79 636 262 159 
23 Büyük Menderes River at Burhaniye 12,799 21.40 630 120 53 
24 Büyük Menderes River at Citak Bridge 3,946 6.88 598 802 55 
25 Yenidere Creek at Calikoy 669 1.14 647 855 54 
26 Akcay Creek at Degirmenalani 855 7.66 653 397 283 
27 Sarhos Creek at Goktepe 236 1.16 648 390 156 
28 Büyük Menderes at Cogasli 4,664 7.28 610 617 49 
29 Mortuma Creek at Yemisendere 170 1.63 658 478 302 
30 Banaz Creek at Ulubey 2,286 4.85 605 531 67 

The basins used for the validation are in bold.

Table 2

Basin characteristics in Southwestern Anatolia

BasinDrainage area (km2)Mean flow (m3/s)Annual rainfall (mm)Altitude (m)Basin yield (mm/year)
Yenice Creek at Zindan Bogazi 62 3.068 561 1,250 1,568 
Dim Creek Irrigation Channel 195 0.967 580 38 156 
3 Korkuteli Creek at Salamur Bogazi 131 0.974 595 1,190 235 
Kucukaksu Creek at Gebiz 239 4.36 587 62 576 
Aglasun Creek at Aglasun 49 0.566 582 1,100 367 
6 Duden Creek at Weir 1,782 16.587 685 96 294 
Sücüllü Creek at Dam Inlet 103 0.594 526 1,198 181 
Karpuz Creek at Uzunlar 303 4.192 576 100 436 
9 Kargi Creek at Turkler 336 6.801 567 16 638 
10 Oba Creek at Kadipinari 46 2.071 580 91 1,420 
11 Aksu River at Belence 349 4.88 584 1,000 441 
12 Degirmendere River at Sutculer 131 1.453 591 750 350 
13 Çandir Creek at Yemisli Pinar 164 1.788 700 160 344 
14 Basak River at Yanikkoy 223 3.315 584 1,085 469 
15 Doyran River at Doyran 106 0.773 700 145 230 
16 Esen Creek at Kinik 2,448 27.471 609 354 
17 Ballik River at Ballik 126.2 0.757 641 1,091 189 
18 Basgoz Creek at Gokbuk 222.2 3.47 554 208 492 
19 Akcay Creek at Gombe 114.5 1.243 602 80 342 
20 Cataloyuk River at Karaculha 0.075 637 1,340 296 
21 Geyik Creek at Nif 8.2 0.233 729 1,035 896 
BasinDrainage area (km2)Mean flow (m3/s)Annual rainfall (mm)Altitude (m)Basin yield (mm/year)
Yenice Creek at Zindan Bogazi 62 3.068 561 1,250 1,568 
Dim Creek Irrigation Channel 195 0.967 580 38 156 
3 Korkuteli Creek at Salamur Bogazi 131 0.974 595 1,190 235 
Kucukaksu Creek at Gebiz 239 4.36 587 62 576 
Aglasun Creek at Aglasun 49 0.566 582 1,100 367 
6 Duden Creek at Weir 1,782 16.587 685 96 294 
Sücüllü Creek at Dam Inlet 103 0.594 526 1,198 181 
Karpuz Creek at Uzunlar 303 4.192 576 100 436 
9 Kargi Creek at Turkler 336 6.801 567 16 638 
10 Oba Creek at Kadipinari 46 2.071 580 91 1,420 
11 Aksu River at Belence 349 4.88 584 1,000 441 
12 Degirmendere River at Sutculer 131 1.453 591 750 350 
13 Çandir Creek at Yemisli Pinar 164 1.788 700 160 344 
14 Basak River at Yanikkoy 223 3.315 584 1,085 469 
15 Doyran River at Doyran 106 0.773 700 145 230 
16 Esen Creek at Kinik 2,448 27.471 609 354 
17 Ballik River at Ballik 126.2 0.757 641 1,091 189 
18 Basgoz Creek at Gokbuk 222.2 3.47 554 208 492 
19 Akcay Creek at Gombe 114.5 1.243 602 80 342 
20 Cataloyuk River at Karaculha 0.075 637 1,340 296 
21 Geyik Creek at Nif 8.2 0.233 729 1,035 896 

The basins used for the validation are in bold.

Methods

Construction of FDCs and AFDCs

This study uses both traditional FDCs and AFDCs to estimate streamflow quantiles. The traditional method represents a long-term demonstration of the flow regime using all the long-term flow records obtained by sorting all records in descending order and plotting them against their exceedance probabilities. AFDCs, on the other hand, describe FDCs with a time span of one year, as advocated by Vogel & Fennessey (1994). Obtaining the median AFDCs consists of a few simple steps described below:

  • (a)

    sorting daily observed flows Qi for each year from largest to smallest to produce an ordered series (i= 1,2, … , 365),

  • (b)

    repeating the sorting procedure in the previous step for N observation years, and thus N number of AFDCs are obtained,

  • (c)
    calculating exceedance probabilities (percentiles) Pi for each sequential observation Qi with the Weibull plotting position,
    formula
    (1)
  • (d)

    generating the typical median AFDC of a stream (or a basin) by calculating the median of N AFDCs for each percentile.

Figure 2 shows the median AFDC obtained at a station. In the figure, N curves (AFDCs), each belonging to a separate year consisting of 365 daily flows sorted from largest to smallest, and their median (median AFDC) are illustrated.
Figure 2

Derivation of median AFDC.

Figure 2

Derivation of median AFDC.

Close modal

In particular, the methodology addresses the analysis of the lower part of the FDC that refers to low flows as hydrological droughts are the motivation for this paper. Therefore, the range of P= 0.30–0.99 of the probability of exceedance is considered. For similar reasons, Fennessey & Vogel (1990), Franchini & Suppo (1996), and Castellarin et al. (2004b) suggested describing the lower part of the FDCs (i.e., P ≥ 0.5 or P ≥ 0.3) for studying hydroelectric engineering, water quality assessment, sedimentation, low-flow analysis, or drought.

ASF-based regionalization

Considering previous studies on the scale invariance and self-similarity in hydrologic process (see, for example, Gupta & Waymire 1990; Haltas & Kavvas 2011), one can observe that river flows in either form of mean, high-, or low-flow forms are significantly correlated on the drainage area. Most possibly, this property of flows appears due to the fact that statistical moments of river flows are proportional with the drainage area as:
formula
(2)
where k= 1 replaces for the population mean and β is a common characteristic for the hydrologic region (Gupta & Waymire 1990).

Furthermore, it has been shown that the p-quantiles of the flows are proportional to the drainage area, that is, . If A is not too large, the characteristic power β is often close to 1. It tends theoretically to 1 as the drainage area is less than a few km2.

Flood Estimation Handbook (FEH 1999), for example, has derived the following empirical relation for the median flood for river basins in the UK:
formula
(3)
where A is in km2.
Considering the above facts, we introduced in this paper a very simple and useful concept, the ASF, based on the assumption that . Under this assumption, we expect that p-quantiles of low flows scale with the drainage area as . Thus, p-quantiles at a station in the hydrologic region can be regionalized as in the following:
formula
(4)
where is a function of probability of exceedance and its analytical form should be established for the regions under consideration.

Parametric approach for modeling streamflow quantiles using regression relationship

In the parametric studies, regression models defined by the geographical, hydrological, and meteorological characteristics of the relevant basin were used to determine the analytical form of FDCs. Parametric models have often used multiple linear, exponential, or logarithmic regressions to regionalize FDCs. Within the scope of this study, multiple linear regression relations were considered in defining the streamflow quantiles obtained from FDCs. The relationship between the low-flow quantiles (Q30, Q40, … ,Q99) and the basin characteristics is investigated using a multiple linear regression relationship in the following form:
formula
(5)
where QP is the quantile (or percentile) of a probability P, Xi are the explanatory variables for i= 1,2, …, n, and is the residual of the model.

Validation

Within the scope of the study, the streamflow quantiles of the basins reserved for validation will be compared with the estimates obtained from the regression relationship. NSE, Nash–Sutcliffe efficiency (Nash & Sutcliffe 1970), RSR (the ratio of RMSE and standard deviation of measured data) (Moriasi et al. 2007) criteria, and mean percentage difference (MPD) are calculated to evaluate the model performance. NSEP, RSRP, and are the criteria of the Pth quantile and are calculated as follows:
formula
(6)
formula
(7)
formula
(8)
where s is the basin number (s= 1,2, …, S), QP is the empirical flow quantile, is the estimated flow quantile, and is the mean value of the empirical streamflow quantiles. The classification of NSE and RSR values according to model performance is given in Table 3 (Moriasi et al. 2007).
Table 3

Performance criteria categorizations

PerformanceNSERSR
Very good 0.75 < NSE ≤ 1.00 0.00 ≤ RSR ≤ 0.50 
Good 0.65 < NSE ≤ 0.75 0.50 < RSR ≤ 0.60 
Adequate 0.50 < NSE ≤ 0.65 0.60 < RSR ≤ 0.70 
Inadequate NSE ≤ 0.50 RSR > 0.70 
PerformanceNSERSR
Very good 0.75 < NSE ≤ 1.00 0.00 ≤ RSR ≤ 0.50 
Good 0.65 < NSE ≤ 0.75 0.50 < RSR ≤ 0.60 
Adequate 0.50 < NSE ≤ 0.65 0.60 < RSR ≤ 0.70 
Inadequate NSE ≤ 0.50 RSR > 0.70 

Traditional FDCs and median AFDCs

In this section, traditional FDCs and median AFDCs were produced in both study regions, as described in Section 2.2.1. As an example, traditional FDCs and median AFDCs calculated for the Western Anatolia region are shown in Figure 3. Quantiles corresponding to 80% probability (Q80) are illustrated in the figure as an example.
Figure 3

Traditional FDCs and median AFDCs in Western Anatolia.

Figure 3

Traditional FDCs and median AFDCs in Western Anatolia.

Close modal

There is greater variability between the traditional FDCs than the median AFDCs, because median AFDCs are less affected by long dry/wet periods.

Modeling low-flow quantiles

Daily streamflow observations of 30 Western Anatolia basins and 21 Southwestern Anatolia basins, with observation lengths varying between 10 and 54 years, are compiled to prepare FDCs. The low-flow quantiles (Q30, Q40, … ,Q99) of each basin are calculated using the lower part of both traditional and median annual FDCs.

The variables of multiple linear regression relationship in Equation (2) are replaced by the basin characteristics and Equation (2) turns into the following:
formula
(9)
where A is the basin area (km2), Q0 is the average flow (m3/s), MAP is the mean annual precipitation (mm), h is the altitude (m), Lat is the latitude (°), and Long is the longitude (°) of the flow gauging station. The significance of the basin characteristics in the regression relationship is examined, and the results are given for two regions and two types of FDCs in Tables 47.
Table 4

P-values of the variables in multiple linear regression in Western Anatolia for streamflow quantiles calculated from traditional FDCs

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.995R2= 0.993R2= 0.986R2= 0.983R2= 0.976R2= 0.959R2= 0.899R2= 0.863R2= 0.598
p-values
A 1.4 × 10−9 6.8 × 10−9 6.8 × 10−8 1.75 × 10−7 4.55 × 10−8 1.59 × 10−8 1.21 × 10−7 6.37 × 10−7 3.29 × 10−3 
Q0 2.2 × 10−10 1.5 × 10−8 2.0 × 10−5 4.04 × 10−3 0.38 0.05 3.77 × 10−4 4.58 × 10−4 0.03 
MAP 0.36 0.21 0.24 0.13 0.03 0.01 0.01 0.01 0.13 
h 0.82 0.16 0.15 0.48 0.91 0.14 0.03 0.05 0.44 
Lat 0.34 0.35 0.34 0.21 0.07 0.03 0.03 0.02 0.02 
Long 0.94 0.07 0.08 0.22 0.52 0.45 0.08 0.10 0.11 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.995R2= 0.993R2= 0.986R2= 0.983R2= 0.976R2= 0.959R2= 0.899R2= 0.863R2= 0.598
p-values
A 1.4 × 10−9 6.8 × 10−9 6.8 × 10−8 1.75 × 10−7 4.55 × 10−8 1.59 × 10−8 1.21 × 10−7 6.37 × 10−7 3.29 × 10−3 
Q0 2.2 × 10−10 1.5 × 10−8 2.0 × 10−5 4.04 × 10−3 0.38 0.05 3.77 × 10−4 4.58 × 10−4 0.03 
MAP 0.36 0.21 0.24 0.13 0.03 0.01 0.01 0.01 0.13 
h 0.82 0.16 0.15 0.48 0.91 0.14 0.03 0.05 0.44 
Lat 0.34 0.35 0.34 0.21 0.07 0.03 0.03 0.02 0.02 
Long 0.94 0.07 0.08 0.22 0.52 0.45 0.08 0.10 0.11 

Significant variables for = 0.05 are in bold.

Table 5

P-values of the variables in multiple linear regression in Western Anatolia for streamflow quantiles calculated from median AFDCs

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.989R2= 0.995R2= 0.988R2= 0.982R2= 0.980R2= 0.975R2= 0.972R2= 0.968R2= 0.961
p-values
A 3.4 × 10−6 7.3 × 10−11 1.8 × 10−7 1.10 × 10−6 9.71 × 10−9 1.05 × 10−8 5.81 × 10−8 1.96 × 10−7 6.71 × 10−6 
Q0 8.1 × 10−7 3.1 × 10−9 2.3 × 10−5 7.48 × 10−4 0.26 0.97 0.71 0.68 0.22 
MAP 0.88 0.73 0.36 0.32 0.09 0.05 0.10 0.12 0.29 
h 0.24 0.38 0.15 0.14 0.42 0.77 0.35 0.27 0.16 
Lat 0.63 0.11 0.38 0.39 0.10 0.05 0.06 0.03 0.08 
Long 0.05 0.88 0.08 0.07 0.27 0.43 0.28 0.40 0.27 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.989R2= 0.995R2= 0.988R2= 0.982R2= 0.980R2= 0.975R2= 0.972R2= 0.968R2= 0.961
p-values
A 3.4 × 10−6 7.3 × 10−11 1.8 × 10−7 1.10 × 10−6 9.71 × 10−9 1.05 × 10−8 5.81 × 10−8 1.96 × 10−7 6.71 × 10−6 
Q0 8.1 × 10−7 3.1 × 10−9 2.3 × 10−5 7.48 × 10−4 0.26 0.97 0.71 0.68 0.22 
MAP 0.88 0.73 0.36 0.32 0.09 0.05 0.10 0.12 0.29 
h 0.24 0.38 0.15 0.14 0.42 0.77 0.35 0.27 0.16 
Lat 0.63 0.11 0.38 0.39 0.10 0.05 0.06 0.03 0.08 
Long 0.05 0.88 0.08 0.07 0.27 0.43 0.28 0.40 0.27 

Significant variables for = 0.05 are in bold.

Table 6

P-values of the variables in multiple linear regression in Southwestern Anatolia for streamflow quantiles calculated from traditional FDCs

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.994R2= 0.988R2= 0.983R2= 0.979R2= 0.976R2= 0.966R2= 0.634R2= 0.618R2= 0.640
p-values
A 0.05 3.8 × 10−3 1.8 × 10−3 1.06 × 10−3 3.97 × 10−4 4.87 × 10−4 2.45 × 10−2 2.61 × 10−2 2.47 × 10−2 
Q0 3.7 × 10−5 0.04 0.35 0.82 0.60 0.29 0.09 0.09 0.09 
MAP 0.55 0.68 0.74 0.81 0.93 0.58 0.26 0.24 0.16 
h 0.74 0.57 0.49 0.43 0.46 0.57 0.89 0.89 0.82 
Lat 0.67 0.65 0.66 0.67 0.76 0.95 0.76 0.74 0.68 
Long 0.33 0.49 0.47 0.50 0.73 0.88 0.18 0.17 0.12 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.994R2= 0.988R2= 0.983R2= 0.979R2= 0.976R2= 0.966R2= 0.634R2= 0.618R2= 0.640
p-values
A 0.05 3.8 × 10−3 1.8 × 10−3 1.06 × 10−3 3.97 × 10−4 4.87 × 10−4 2.45 × 10−2 2.61 × 10−2 2.47 × 10−2 
Q0 3.7 × 10−5 0.04 0.35 0.82 0.60 0.29 0.09 0.09 0.09 
MAP 0.55 0.68 0.74 0.81 0.93 0.58 0.26 0.24 0.16 
h 0.74 0.57 0.49 0.43 0.46 0.57 0.89 0.89 0.82 
Lat 0.67 0.65 0.66 0.67 0.76 0.95 0.76 0.74 0.68 
Long 0.33 0.49 0.47 0.50 0.73 0.88 0.18 0.17 0.12 

Significant variables for = 0.05 are in bold.

Table 7

P-values of the variables in multiple linear regression in Southwestern Anatolia for streamflow quantiles calculated from median AFDCs

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.992R2= 0.988R2= 0.983R2= 0.979R2= 0.976R2= 0.969R2= 0.972R2= 0.974R2= 0.977
p-values
A 0.03 0.01 3.7 × 10−3 1.37 × 10−3 4.14 × 10−4 5.01 × 10−4 3.06 × 10−4 3.33 × 10−4 6.74 × 10−4 
Q0 3.0 × 10−5 0.01 0.18 0.77 0.57 0.35 0.31 0.38 0.79 
MAP 0.59 0.65 0.52 0.67 0.82 0.91 0.95 0.92 0.57 
h 0.65 0.42 0.43 0.43 0.48 0.53 0.54 0.56 0.51 
Lat 0.48 0.48 0.49 0.63 0.66 0.76 0.72 0.68 0.54 
Long 0.44 0.47 0.33 0.41 0.53 0.82 0.69 0.52 0.20 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.992R2= 0.988R2= 0.983R2= 0.979R2= 0.976R2= 0.969R2= 0.972R2= 0.974R2= 0.977
p-values
A 0.03 0.01 3.7 × 10−3 1.37 × 10−3 4.14 × 10−4 5.01 × 10−4 3.06 × 10−4 3.33 × 10−4 6.74 × 10−4 
Q0 3.0 × 10−5 0.01 0.18 0.77 0.57 0.35 0.31 0.38 0.79 
MAP 0.59 0.65 0.52 0.67 0.82 0.91 0.95 0.92 0.57 
h 0.65 0.42 0.43 0.43 0.48 0.53 0.54 0.56 0.51 
Lat 0.48 0.48 0.49 0.63 0.66 0.76 0.72 0.68 0.54 
Long 0.44 0.47 0.33 0.41 0.53 0.82 0.69 0.52 0.20 

Significant variables for = 0.05 are in bold.

When Tables 47 are examined, it is noteworthy that most of the stream flow quantiles were successfully described by multiple linear regression relationships for both regions. In Western Anatolia, the determination coefficients (R2) of the multiple linear regression for the flow quantities Q90, Q95, and Q99 calculated from the traditional FDC were 0.899, 0.863, and 0.598, while the coefficients of determination calculated for the quantiles obtained from the median AFDC were 0.972, 0.968, and 0.961, respectively. Similarly, the determination coefficients of the multiple linear regression for the streamflow quantiles Q90, Q95, and Q99 calculated from the traditional FDC in Southwestern Anatolia were 0.634, 0.618, and 0.640, while the coefficients of determination calculated from quantiles obtained from the median AFDC were 0.972, 0.974, and 0.977, respectively. It is noteworthy that the streamflow quantiles obtained from median AFDCs have stronger regression relationships than the quantiles obtained from FDCs, especially for the quantiles of higher probabilities (P > 80%). Only one (A, basin area) of the six basin characteristics has a significant effect on the regression relationship for each of the nine quantiles. This result raises the question of whether streamflow quantiles can be determined solely in terms of basin area.

Areal scale factor

To answer the question ‘Can streamflow quantiles be determined solely in terms of basin area?’ the relationship between streamflow quantiles QP and basin area A is examined.

As explained in Subsection 2.2.2, s defining the relationship between flow quantiles and basin areas were calculated for corresponding probabilities P (0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, and 0.99) using Equation (4). The s of each quantile and the results of the regression are given in Tables 811 in Western Anatolia and Southwestern Anatolia for streamflow quantiles calculated from traditional FDCs and median AFDCs.

Table 8

and P-values of regression relationship for streamflow quantiles calculated from traditional FDCs in Western Anatolia

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.971R2= 0.974R2= 0.975R2= 0.975R2= 0.973R2= 0.945R2= 0.801R2= 0.734R2= 0.358
P-values 7.35 × 10–24 1.84 × 10–24 7.07 × 10–25 8.48 × 10–25 2.62 × 10–24 7.90 × 10–20 1.08 × 10–11 7.65 × 10–10 3.78 × 10–4 
ASFP 0.00225 0.00178 0.00148 0.00122 0.00100 0.00074 0.00043 0.00028 0.00007 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.971R2= 0.974R2= 0.975R2= 0.975R2= 0.973R2= 0.945R2= 0.801R2= 0.734R2= 0.358
P-values 7.35 × 10–24 1.84 × 10–24 7.07 × 10–25 8.48 × 10–25 2.62 × 10–24 7.90 × 10–20 1.08 × 10–11 7.65 × 10–10 3.78 × 10–4 
ASFP 0.00225 0.00178 0.00148 0.00122 0.00100 0.00074 0.00043 0.00028 0.00007 
Table 9

and P-values of regression relationship for streamflow quantiles calculated from median AFDCs in Western Anatolia

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.960R2= 0.977R2= 0.974R2= 0.970R2= 0.979R2= 0.972R2= 0.969R2= 0.965R2= 0.956
P-values 7.12 × 10–22 2.97 × 10–25 1.81 × 10–24 1.44 × 10–23 8.86 × 10–26 3.76 × 10–24 2.06 × 10–23 9.65 × 10–23 2.89 × 10–21 
 0.00219 0.00172 0.00152 0.00134 0.00114 0.00095 0.00078 0.00060 0.00054 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.960R2= 0.977R2= 0.974R2= 0.970R2= 0.979R2= 0.972R2= 0.969R2= 0.965R2= 0.956
P-values 7.12 × 10–22 2.97 × 10–25 1.81 × 10–24 1.44 × 10–23 8.86 × 10–26 3.76 × 10–24 2.06 × 10–23 9.65 × 10–23 2.89 × 10–21 
 0.00219 0.00172 0.00152 0.00134 0.00114 0.00095 0.00078 0.00060 0.00054 
Table 10

and P-values of regression relationship for streamflow quantiles calculated from traditional FDCs in Southwestern Anatolia

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.979R2= 0.984R2= 0.989R2= 0.985R2= 0.983R2= 0.976R2= 0.688R2= 0.658R2= 0.645
P-values 2.40 × 10–18 1.64 × 10–19 3.67 × 10–18 7.15 × 10–17 3.73 × 10–16 1.10 × 10–14 4.07 × 10–4 8.76 × 10–4 1.18 × 10–3 
 0.01182 0.01004 0.00881 0.00756 0.00639 0.00518 0.00214 0.00154 0.00098 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.979R2= 0.984R2= 0.989R2= 0.985R2= 0.983R2= 0.976R2= 0.688R2= 0.658R2= 0.645
P-values 2.40 × 10–18 1.64 × 10–19 3.67 × 10–18 7.15 × 10–17 3.73 × 10–16 1.10 × 10–14 4.07 × 10–4 8.76 × 10–4 1.18 × 10–3 
 0.01182 0.01004 0.00881 0.00756 0.00639 0.00518 0.00214 0.00154 0.00098 
Table 11

and P-values of regression relationship for streamflow quantiles calculated from median AFDCs in Southwestern Anatolia

Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.972R2= 0.980R2= 0.976R2= 0.970R2= 0.965R2= 0.957R2= 0.955R2= 0.954R2= 0.953
P-values 4.71 × 10–17 1.46 × 10–18 1.02 × 10–17 9.51 × 10–17 5.03 × 10–16 3.65 × 10–15 6.38 × 10–15 7.17 × 10–15 8.98 × 10–15 
 0.01156 0.01010 0.00894 0.00780 0.00667 0.00568 0.00510 0.00482 0.00425 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
R2= 0.972R2= 0.980R2= 0.976R2= 0.970R2= 0.965R2= 0.957R2= 0.955R2= 0.954R2= 0.953
P-values 4.71 × 10–17 1.46 × 10–18 1.02 × 10–17 9.51 × 10–17 5.03 × 10–16 3.65 × 10–15 6.38 × 10–15 7.17 × 10–15 8.98 × 10–15 
 0.01156 0.01010 0.00894 0.00780 0.00667 0.00568 0.00510 0.00482 0.00425 

For the probabilities between 0.30 and 0.80 in both regions, the streamflow quantiles obtained from traditional FDCs and median AFDCs showed successful regression results. On the other hand, for higher probabilities (P > 0.8), streamflow quantiles obtained from median AFDCs indicate significantly stronger regression relationships than those obtained from FDCs. The determination coefficients (R2) of the multiple linear regression for the streamflow quantiles Q90, Q95, and Q99 obtained from the traditional FDC in Western Anatolia were 0.801, 0.734, and 0.358, whereas the determination coefficients of the regression for the quantiles obtained from the median AFDC were 0.969, 0.965, and 0.956, respectively. Similarly, the determination coefficients (R2) of the multiple linear regression for the streamflow quantiles Q90, Q95, and Q99 obtained from the traditional FDC in Southwestern Anatolia were 0.688, 0.658, and 0.645, whereas the determination coefficients of the regression for the quantiles obtained from the median AFDC were 0.955, 0.954, and 0.953, respectively. It can be argued that the shape of traditional FDCs is negatively affected by drought and/or long dry periods or zero flow periods causing negative jumps on the FDC. As an example, the traditional FDC and the median AFDC of Esen Creek at Kinik station are shown in Figure 4(a) and 4(b), respectively. For the sake of dramatization, the FDC and AFDC of the station with the largest jump are given.
Figure 4

(a) The traditional FDC and (b) the median AFDC of Esen Creek at Kinik station.

Figure 4

(a) The traditional FDC and (b) the median AFDC of Esen Creek at Kinik station.

Close modal

In Figure 4(a), a sudden jump is observed in the low-flow region corresponding to high probabilities of traditional FDC. This is caused by the long dry periods and/or zero flows in the observation series. On the other hand, the shape of AFDC in Figure 4(b) indicates a relatively stable slope. This confirms the regression results in Tables 811 and makes the median AFDC more reliable against the traditional FDC.

It is also observed that ASFs are at higher levels in Southwestern Anatolia than in Western Anatolia. This is caused by the fact that basin yield in Southwestern Anatolia is on average twice as large as in Western Anatolia (Tables 1 and 2), which is due to the contribution of groundwater to the streams in the karstic region. The reason why the significance of ASFs in Western Anatolia is higher than the significance of ASFs in Southwestern Anatolia is that ‘the flow quantile-basin area’ relationship can be constructed more successfully in the region where there is no groundwater effect.

Validation

To test the significance of the proposed model, 10 of the 30 basins in Western Anatolia and seven of the 21 basins in Southwestern Anatolia are initially retained for validation. In Tables 1 and 2, these stations are in bold. The success of the developed model is tested for 10 validation basins in Western Anatolia and seven validation basins in Southwestern Anatolia for streamflow quantiles estimated using traditional FDCs and median AFDCs. Tables 12 and 13 show the validation results of the proposed model. NSE and RSR values indicating the success of the model performance as ‘very good’ are given in bold.

Table 12

Model performance criteria of streamflow quantiles calculated from traditional FDCs and median AFDCs in Western Anatolia

Q30Q40Q50Q60Q70Q80Q90Q95Q99
Traditional FDC NSE 0.974 0.978 0.975 0.968 0.966 0.938 0.503 0.187 0.119 
RSR 0.161 0.148 0.159 0.180 0.184 0.249 0.705 0.902 0.960 
MPD (%) 15 18 22 
Median AFDC NSE 0.952 0.977 0.975 0.964 0.967 0.963 0.959 0.961 0.951 
RSR 0.220 0.151 0.157 0.189 0.183 0.193 0.204 0.197 0.221 
MPD (%) 10 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
Traditional FDC NSE 0.974 0.978 0.975 0.968 0.966 0.938 0.503 0.187 0.119 
RSR 0.161 0.148 0.159 0.180 0.184 0.249 0.705 0.902 0.960 
MPD (%) 15 18 22 
Median AFDC NSE 0.952 0.977 0.975 0.964 0.967 0.963 0.959 0.961 0.951 
RSR 0.220 0.151 0.157 0.189 0.183 0.193 0.204 0.197 0.221 
MPD (%) 10 
Table 13

Model performance criteria of streamflow quantiles calculated from traditional FDCs and median AFDCs in Southwestern Anatolia

Q30Q40Q50Q60Q70Q80Q90Q95Q99
Traditional FDC NSE 0.951 0.987 0.985 0.975 0.963 0.939 0.555 0.529 0.527 
RSR 0.220 0.115 0.123 0.158 0.193 0.247 0.667 0.686 0.688 
MPD (%) 11 12 15 16 20 
Median AFDC NSE 0.917 0.974 0.977 0.974 0.961 0.947 0.946 0.949 0.951 
RSR 0.289 0.163 0.152 0.162 0.197 0.230 0.232 0.226 0.222 
MPD (%) 10 11 11 11 10 
Q30Q40Q50Q60Q70Q80Q90Q95Q99
Traditional FDC NSE 0.951 0.987 0.985 0.975 0.963 0.939 0.555 0.529 0.527 
RSR 0.220 0.115 0.123 0.158 0.193 0.247 0.667 0.686 0.688 
MPD (%) 11 12 15 16 20 
Median AFDC NSE 0.917 0.974 0.977 0.974 0.961 0.947 0.946 0.949 0.951 
RSR 0.289 0.163 0.152 0.162 0.197 0.230 0.232 0.226 0.222 
MPD (%) 10 11 11 11 10 

The observed low-flow quantiles and the low-flow quantile estimates of the parametric model are compared for the validation basins. Validation results show that the estimates of median AFDCs performed ‘very good’ for all streamflow quantiles, but quantiles obtained from traditional FDCs led to ‘inadequate’ model performance for high probabilities (P > 0.8). The validation results confirm the conclusion that ‘median AFDCs are more stable than traditional FDCs especially for intermittent streams’ as explained in the previous section.

In addition, MPDs between the observed low-flow quantiles and the low-flow quantile estimates of the parametric model are calculated. While the percentage differences of median AFDC model predictions generally are less than 10%, model predictions of traditional FDCs showed percentage differences of 15% and above, especially for higher probabilities (P > 0.8).

Regionalization of ASFs

Based on the successful results of validations in estimating the low-flow quantiles using ASFs obtained from median AFDCs, the relationship between ASFs and exceedance probabilities, P, for two study regions is investigated. Detailed analysis shows that ASFs can be successfully regionalized based on their exceedance probability by a logarithmic regression given in Equations (10) and (11) for Western Anatolia and Southwestern Anatolia, respectively:
formula
(10)
formula
(11)
P is the exceedance probability in percent. Figures 5 and 6 show the regional curves that indicate the relationship between regional ASFs and exceedance probabilities.
Figure 5

Regional curve of ASFs related to the exceedance probabilities in Western Anatolia.

Figure 5

Regional curve of ASFs related to the exceedance probabilities in Western Anatolia.

Close modal
Figure 6

Regional curve of ASFs related to the exceedance probabilities in Southwestern Anatolia.

Figure 6

Regional curve of ASFs related to the exceedance probabilities in Southwestern Anatolia.

Close modal
Considering that the flow quantiles can be successfully represented by the basin area A, and that the low-flow quantiles obtained from median AFDCs show more successful relationships than those obtained from traditional FDCs, the application steps of the model proposed in the study can be summarized with the flow chart in Figure 7.
Figure 7

Steps of the model.

Figure 7

Steps of the model.

Close modal

The main purpose of this paper is to reveal the regional relationships that enable effective estimations of low-flow quantiles. While most parametric approaches in the literature involve the regionalization of the parameters that define the FDCs, this study deals with the regional modeling of the streamflow quantiles. Most previous studies on the estimation of streamflow quantiles used traditional FDCs, which are undesirably affected by long dry periods or zero flows causing negative jumps on the FDC (Figure 4(a) and 4(b)).

This study proposes to use median AFDCs in addition to the classical FDC curves encountered in the literature for modeling streamflow quantiles. The performance of the proposed model is compared for two different types of FDCs in two different regions in Turkey. As the study takes low flows into account, streamflow estimations are made for nine fixed percentage points (30, 40, 50, 60, 70, 80, 90, 95, and 99%) of FDCs and median AFDCs. The relationship between low-flow quantiles and the basin characteristics is examined. Among the basin characteristics, the most significant contribution to the statistical definition of flow quantiles was determined to be the basin area, A. This result, as expected, confirms the relationship described in Section 2.2.2. A strong linear relationship was determined between the low-flow quantiles estimated from median AFDCs and the basin area for all fixed percentage points.

Based on the strong relationship between low-flow quantiles and the basin area, ASFs explaining this relationship were obtained. In addition, regional curves have been generated that effectively represent the relationship between ASFs and the exceedance probabilities for both study regions.

Although the two regions in the study have similar meteorological characteristics, they are of completely different types geologically. Two different regions with dissimilar geological structure, drainage mechanism, and low-flow characteristics are selected. Thus, the proposed parametric model is tested in terms of adapting to different geohydrological conditions.

The main results of our paper can be summarized as follows:

  • 1.

    The relationship between the low-flow quantiles and the hydrological and geographical characteristics of the basins is examined, and it was demonstrated that the low-flow quantiles are successfully represented depending on the basin area.

  • 2.

    As a novelty of the study, ASF ratios that describe low-flow quantiles related to the basin area are proposed.

  • 3.

    It has been expressed for both study regions that median AFDCs successfully represent streamflow quantiles for all percentage points compared to traditional FDCs.

  • 4.

    Regional curves describing the relationship between ASFs and exceedance probabilities allow efficient estimations of ASFs.

  • 5.

    The low-flow quantile, which is a crucial design criterion for a water resources project, can be estimated simply and practically depending on project basin area A, when the ASF curve is determined for a region.

Thus, the successful representation of the model for these two geologically different regions is promising in terms of the robustness of the model and its use in future studies. However, further examination in different geographical areas of dissimilar meteorological characteristics is needed to generalize the results of the study. Another limitation of the study is that basin characteristics such as evapotranspiration, vegetation cover, and soil type are not included in the model.

The ability to estimate low-flow quantities in a practical way, depending only on the basin area, emerges as important information in many topics of hydrology, such as the design of hydroelectric power plants, calculations of irrigation projects, studies on river pollution, and management of water resources.

The authors are grateful to the Editor in chief and four anonymous reviewers for their valuable comments and suggestions, and to Prof. Dr. E. Benzeden for support in the revision process.

Both authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by TN and OLA. The first draft of the manuscript was written by OLA, and both authors commented on the manuscript. Both authors reviewed the results and approved the final version of the manuscript.

The authors declare that no fund, grants, or other support were received during the preparation of this manuscript.

All relevant data are available from an online repository or repositories: https://www.dsi.gov.tr/Sayfa/Detay/744.

The authors declare there is no conflict.

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