Comparing threshold level methods in development of stream flow drought Severity-DurationFrequency curves
Homa Razmkhah
Department of Water Engineering, College of Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran.
, [email protected], +989177038490
Abstract
Drought is a natural phenomenon occures in different climate regimes. This study
compares Severity-Duration-Frequency curves (SDF) derived from threshold level methods. For
this purpose hydrological drought of the Roudzard basin was investigated, based on run theory.
Daily runoff data of Mashin hydrometery station during 1970 to 2012 was assessed using 70%
(Q70), 90% (Q90) of mean daily and 70% of monthly average runoff (monthly) as threshold level
methods. Time series of the annual maxima values of duration and volume deficit showed
similar trend of increase and decreasing. Severity-Duration-Frequency (SDF) curves were
prepared, classifying drought durations to four intervals and fitting statistical distribution to each
one. Resulted SDF curves showed that, in each period, increasing of duration resulted to
increased value of the volume dificit with a non-linear trend. The duration and severities from
the threshold levels were different. Drought deficit volume increasing rate was also different in
each class of duration interval. For the additional analysis, the duration frequency and deficit
frequency curves were also derived to quantify the extent of the drought duration and deficit
more. SDF curves developed in this study can be used to quantify water deficit for natural stream
and reservoir. In addition, these curves will be extended to allow for regional frequency analysis,
which can estimate the stream flow drought severity at ungauged sites. They could be an
effective tool to identify hydrological drought using the severity, duration and frequency.
Keywords: Hydrological Drought, Severity-Duration-Frequency (SDF) curves, Threshold level
1. Introduction
Drought events are becoming increasingly world-wide. Drought impacts both surface and ground
water resources and can lead to reduced water supply and deteriorated water quality. Droughts
are of great importance in the planning and management of water resources. The American
Meteorological Society (1997) grouped the drought definition into four categories;
meteorological or climatological, agricultural, hydrological and socioeconomic droughts. Brief
explanation of drought categories is presented in Sung and Chung (2014).
Nalbantis and Tsakiris (2009) defined hydrological drought as a significant decrease in the
availability of water in all its form, as well as land phase of hydrologic cycle, and is reflected in
various hydrological variables such as stream flow, discharge from aquifers and base flow.
1
Tsakiris et al. (2013) described that stream flow is the key variable in describing hydrological
drought because it considers the outputs of surface runoff from the surface water subsystem,
subsurface runoff from the upper and lower unsaturated zones, and base flow from ground water
subsystem. It also crucially affects the socioeconomic drought for different activities like
hydropower, recreation and irrigation (Hein Jr., 2002). In another research, Tallaksen and
Vanlanen (2004) defined stream flow drought as an occurence of below average water
availability.
Considering drought hydrological aspects, many researches use runoff as the main indicator
(Dracup et al., 1980a). The main cause of drought is a precipitation deficit in an area compared
with the average precipitation in a time period (meteorological drought). Other characteristics are
influenced by meteorological drought and related spatiotemporal characteristics (Sung and
Chung, 2014).
There are a great number of methods, indices, used for defining drought. Detailed description of
drought indices is presented in Mishra and Sing (2010). Run analysis is a widely used method in
describing hydrological drought properties (Dracup et al., 1980b). A run is defined as succession
of the same kind of observation preseeded and succeeded by one or more observation of a
different kind (Chander et al., 1979). Yevjevich (1967) suggested that some elements of run
theory may be applied to the estimation of drought likelihood (Moye et al., 1988), though the
theory for identifying drought parameters as duration, intensity (average water deficiency) and
severity (cumulative water deficiency through a drought event) was proposed.
To define surplus and deficit amount of drought, the truncation level (threshold) should be
determined, called threshold approach. Threshold level method, defines the duration and severity
of a drought event explaining daily, monthly, seasonal and annual variation of drought time
series (Dracup et al., 1980). Bonacci (1993) compared methods of drought identification.
Kjeldson et al., (2000) applied three variable threshold level method using seasonal, monthly and
daily stream flow. Sung and Chung (2014) compared stream flow SDF curves derived from Q70,
daily, monthly and desired yield level for water use.
There has been a growing need for design of natural resources and environment based on the
afformentioned scientific trends. Many studies have integrated drought severity and duration
based on multivariate theory (Gonzalez and Valdes, 2004; Shiau, 2006; Shiau et al., 2007;
Nadarajah, 2009; Mishra et al., 2009; Kao and Govindaraju, 2010; Zhang et al., 2012; Lee and
Kim, 2013; Huang et al., 2014; Mallya et al., 2015). However, these studies cannot fully explain
droughts without considering frequency, which resulted in the development of drought isoseverity curves for certain return periods and durations for design purposes. In recent years there
have been a multitude of researches presenting methodologies for obtaining drought return
periods (Nadarajah, 2000; Fleig et al., 2006; Shiau et al., 2007; Panu and Sharma, 2009; Mishra
et al., 2009; Shiau and Modarres, 2009; Modarres and Sarhadi, 2010; Reddy and Ganguli, 2012;
Yoo et al., 2012;; Sharma and Panu, 2014).
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Thus based on the typical drought characteristics, water deficit and duration, and threshold
levels, this study developed quantitative relations among drought parameters as well as severity,
duration and frequency using Q70, Q90 and monthly threshold level methods. The threshold
selection is analyzed because it is not clear that Q70, Q90 or monthly is a representative threshold
for rivers in all climates. Furthermore, this study proposed a stream flow SDF curves using the
traditional frequency analysis. In addition, this study also developed duration frequency and
deficit frequency curves of threshold level methods from the occurrence probabilities of various
duration events using a general frequency analysis because the deficit volume is not sufficient to
explain the extreme droughts. This framework was applied to the Roudzard river basin in
southwest of Iran.
2. Material and Methods
2.1. Procedure
This study consists of five steps. First: determining the threshold levels for the fixed level.
Second: calculating the severities, total water deficits, and durations for all drought events at
threshold level. Third: deriving the annual maxima of severity and duration, and determining the
best fitted probability distribution functions. Forth, calculating the steam flow drought severities
using the selected probability distributions, and finally preparing the SDF curves using an
appropriate probability distributions.
2.2. Threshold level selection
The threshold may be fixed or vary over the course of a year. A threshold is considered fixed, if
a constant value is used for the entire series, and variable if it varies over the year. (Hisdal and
Tallaksen, 2003). The threshold choice is influenced by the study objective, region and available
data. In general a percentile of the data can be used as the threshold (Sung and Chung, 2014).
Relatively low threshold in the range of Q70-Q95 are often used for perennial rivers (Kjeldsen et
al., 2000). The fixed threshold level in this study is the 70th and 90th percentile value, Q70 and Q90
of mean daily runoff, and 70th percentile value of monthly average, monthly, which are compiled
using all available daily stream flows. The threshold selection is analyzed because it is not clear
that Q70, Q90 or monthly is a representative threshold for rivers in all climates.
2.3. Stream flow drought severity
Use of run test has been proposed as a method to identify drought periods and evaluate the
statistical properties of drought. According to the run method a drought period coincides with a
negative run when a selected hydrological variable remains below a chosen truncation level or
3
threshold (Yevjevich, 1967). Such a threshold can be a fixed value or a seasonally or monthly
varying truncation level, so the truncation level in each time interval is somewhat arbitrary and
could be selected as a fixed or variable value. Usually it is assumed equal to the long period
mean, or median, of the variable of interest, but other possible choices include a fraction of the
mean, 70 to 95% or a level defined as one standard deviation below the mean (Tsakiris et al.)
The advantage of using run method for drought analysis consist in the possibility of deriving the
probabilistic features of drought characteristics as well as drought duration or deficit volume,
therefore the procedure to assess the return period of drought properties according to the run
method has been derived (Nadarajah, 2000).
The threshold level approach is one of the most widely used methods to estimate a hydrological
drought. One of the most important advantages of it is direct determination of drought
characteristics such as frequency, duration and severity without a prior knowledge of probability
distribution. A sequence of drought events can be obtained using the steam flow and threshold
levels. Each drought event is characterized by duration, deficit volume and time of occurence.
Figure 1 presents the basic concept of drought analysis based on run test.
Figure 1. Definition sketch of a drought event by run test (Sung and Chung, 2014)
2.4. Time unit of analysis
The time resolution of analysis, meaning weather to apply a series of annual, monthly, or daily
stream flow, depends on the hydrologic regime in the region of interest (Sung and Chung, 2014).
Different time resolutions may lead to different results. The most commonly used time unit in
drought analysis is the year (Sen, 1980), however this often means a loss of information. In a
temperate zone, a given year may include sever drought and months with abundant stream flow,
which indicates that the annual data do not often reveal sever droughts. This study used the daily
4
and mean monthly stream flow data to identify drought characteristics in order not to lose any
short time drought event.
2.5. Probability distribution function
kolmogrov Smirnov as goodness of fit technique was used to evaluate the best probability
distribution function for datasets in several studies (Massey Jr., 1951; Lilliefors, 1967;
Fermanian, 2005; Genest et al., 2009).
The Kolmogorov-Smirnov statistic (D) is based on the largest vertical difference between the
theoretical and empirical cumulative distribution function (CDF):
𝐷 = 𝑚𝑎𝑥 (𝐹(𝑥𝑖 ) −
𝑖−1 𝑖
𝑛
, − 𝐹(𝑥𝑖 )) 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛 (1)
𝑛
where 𝑥1 , … . . 𝑥𝑛 are random samples from some distribution with CDF of F(X). The empirical
CDF is denoted by:
𝐹𝑛(𝑥) =
1
. [𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 ≤ 𝑥]
𝑛
(2)
The test was performed using Easyfit software. Some widely used probability distribution
functions (PDFs) are presented in Table 1.
Table 1. Some probability distribution functions (PDFs)
Name
Burr
PDF
𝑓(𝑥) =
Pearson
𝑥 𝛼−1
𝛼𝑘 ( )
𝛽
𝑘+1
𝑥 𝛼
𝛽 (1 + ( ) )
𝛽
((𝑥 − 𝑦)⁄𝛽 )𝛼1 −1
𝑓(𝑥) =
𝛽𝐵(𝛼1⁄𝛼2)(1 + (𝑥 − 𝑦)⁄𝛽 )𝛼1 +𝛼2
Fatigue Life
𝑓(𝑥) =
√(𝑥 − 𝑦)⁄𝛽 + √𝛽 ⁄(𝑥 − 𝑦)
1
𝑥−𝑦
. ∅ ( (√
2𝛼(𝑥 − 𝑦)
𝛼
𝛽
𝛽
−√
))
𝑥−𝑦
Gen. Extreme Value
1
1
1
𝑒𝑥𝑝 (−(1 + 𝑘𝑧)− ⁄𝑘 ) (1 + 𝑘𝑧)−1− ⁄𝑘
𝑓(𝑥) = {𝜎
1
𝑒𝑥𝑝(−𝑧 − 𝑒𝑥𝑝(−𝑧))
𝜎
Dagum
𝛼𝑘 (
𝑓(𝑥) =
5
𝑘=0
𝑥 − 𝛾 𝛼𝑘−1
)
𝛽
𝑘+1
𝑥−𝛾 𝛼
) )
𝛽
Quantile function:
𝛼
𝛾
𝑥(𝐹) = 𝜀 + (1 − (1 − 𝐹)𝛿 ) − (1 − (1 − 𝐹)−𝛿 )
𝛽
𝛿
𝛽 (1 + (
Wakeby
𝑘≠0
Parameters
k: continuous shape parameter
𝛼: continuous shape parameter
𝛽: continuous scale parameter
Conditions
(𝑘>0)
(𝛼 >0)
(𝛽 >0)
𝛼1: continuous shape parameter
𝛼2 : continuous shape parameter
𝛽: continuous scale parameter
𝛾: continuous location parameters
(𝛼1>0), (𝛼2 >0), (𝛽 >0)
(𝛾 ≡ 0)𝑦𝑒𝑖𝑙𝑑𝑠 𝑡ℎ𝑒 𝑡ℎ𝑟𝑒𝑒 −
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑃𝑒𝑎𝑟𝑠𝑜𝑛 6 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
𝛾 ≤ 𝑥 ≤ +∞
𝛼: continuous shape parameter
𝛽: continuous scale parameter
𝛾: continuous location parameters
∅: PDF of standard Normal
Distribution
(𝛼>0)
(𝛽 >0)
(𝛾 ≡ 0)𝑦𝑒𝑖𝑙𝑑𝑠 𝑡ℎ𝑒 𝑡𝑤𝑜 −
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝐹𝑎𝑡𝑖𝑔𝑢𝑒 𝐿𝑖𝑓𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
𝛾 ≤ 𝑥 ≤ +∞
𝑥−𝜇
𝜎
𝑘: continuous shape parameter
𝜎: continuous scale parameter
𝜇: continuous location parameter
𝑘: continuous shape parameter
𝛼: continuous shape parameter
𝛽: continuous scale parameter
𝛾: continuous location parameters
(𝜎 >0)
(𝑥 − 𝜇)
1+𝑘
> 0 𝑓𝑜𝑟 𝑘 ≠ 0
𝜎
−∞ < 𝑥 < +∞ 𝑓𝑜𝑟 𝑘 = 0
𝛼, 𝛽, 𝛾, 𝛿, 𝜀 (𝑎𝑙𝑙 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠)
𝛼 ≠ 0 𝑜𝑟 𝛾 ≠ 0
𝛽 + 𝛿 > 0 𝑜𝑟 𝛽 = 𝛾 = 𝛿 = 0
𝐼𝑓 𝛼 = 0, 𝑡ℎ𝑒𝑛 𝛽 = 0
𝐼𝑓 𝛾 = 0, 𝑡ℎ𝑒𝑛 𝛿 = 0
𝑧≡
(𝑘 >0), (𝛼 >0), (𝛽 >0)
(𝛾 ≡ 0) 𝑦𝑒𝑖𝑙𝑑𝑠 𝑡ℎ𝑒 𝑡ℎ𝑟𝑒𝑒 −
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝐷𝑎𝑔𝑢𝑚 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
𝛾 ≤ 𝑥 ≤ +∞
𝛾 ≥ 0 𝑎𝑛𝑑 𝛼 + 𝛾 ≥ 0
𝜀 ≤ 𝑥 ≤ ∞ 𝑖𝑓 𝛿 ≥ 0 𝑎𝑛𝑑 𝛾 > 0
𝜀 ≤ 𝑥 ≤ 𝜀 + 𝛼 ⁄𝛽 − 𝛾⁄𝛿 𝑖𝑓 𝛿 ≤ 0 𝑜𝑟 𝛾 = 0
Beta
𝑓(𝑥) =
Weibull
(𝑥 − 𝑎)𝛼1 −1(𝑏 − 𝑥)𝛼2 −1
1
(𝑏 − 𝑎)𝛼1 +𝛼2−1
𝐵(𝛼1 , 𝛼2 )
𝑓(𝑥) =
𝛼 𝑥 − 𝛾 𝛼−1
𝑥−𝛾 𝛼
(
)
𝑒𝑥𝑝 (− (
) )
𝛽
𝛽
𝛽
1⁄
𝑘
(1 + 𝑘𝑧)−1−
Generalized
Logistic
𝑓(𝑥) =
−1⁄𝑘
𝜎 (1 + (1 + 𝑘𝑧)
2
𝑘=0
2
{ 𝜎(1 + 𝑒𝑥𝑝(−𝑧))
Phased Bi-Weibull
Log-Logistic
𝑓(𝑥)
𝛼1 𝑥 − 𝛾1 𝛼1 −1
𝑥 − 𝛾1 𝛼1
(
)
𝑒𝑥𝑝 (− (
) ) 𝛾1 ≤ 𝑥 ≤ 𝛾2
𝛽1
𝛽1
𝛽1
=
𝛼2 𝑥 − 𝛾1 𝛼2 −1
𝑥 − 𝛾1 𝛼2
(
)
𝑒𝑥𝑝 (− (
) ) 𝛾2 ≤ 𝑥 ≤ +∞
𝛽2
𝛽2
{ 𝛽2
𝑓(𝑥) =
Johnson SB
𝑓(𝑥) =
𝛼 𝑥 − 𝛾 𝛼−1
𝑥 − 𝛾 𝛼 −2
(
)
(1 + (
) )
𝛽
𝛽
𝛽
2
𝛿
𝜗√2𝜋𝑧(1 − 𝑧)
1
𝑧
𝑒𝑥𝑝(− (𝛾 + 𝛿𝑙𝑛 (
)) )
2
1−𝑧
−1
Cauchy
𝑓(𝑥) = (𝜋𝜎 (1 + (
(𝛼1 > 0)
(𝛼2 > 0)
𝑎< 𝑏
𝑎≤𝑥≤𝑏
𝛼: continuous shape parameter
𝛽: continuous shape parameter
𝛾: continuous location parameters
(𝛼 > 0), (𝛽 > 0)
(𝛾 ≡ 0 ) 𝑦𝑒𝑖𝑙𝑑𝑠 𝑡ℎ𝑒 𝑡𝑤𝑜 −
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑊𝑒𝑖𝑏𝑢𝑙𝑙 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
𝛾 ≤ 𝑥 ≤ +∞
𝑥−𝜇
𝜎
𝑘: continuous shape parameter
𝜎: continuous scale parameter
𝜇: continuous location parameter
(𝜎 >0)
(𝑥 − 𝜇)
1+𝑘
>0
𝜎
−∞ < 𝑥 < +∞
𝛼1: continuous shape parameter
𝛽1 : continuous scale parameter
𝛾1 : continuous location parameters
𝛼2 : continuous shape parameter
𝛽2 : continuous scale parameter
𝛾2 : continuous location parameters
(𝛼1>0), (𝛽1 >0)
(𝛼2 >0), (𝛽2 >0)
(𝛾2 > 𝛾1)
𝛾2 − 𝛾1 𝛼1
𝛾2 − 𝛾1 𝛼2
(
) =(
)
𝛽1
𝛽2
𝛾1 ≤ 𝑥 ≤ +∞
𝛼: continuous shape parameter
𝛽: continuous scale parameter
𝛾: continuous location parameters
(𝛼>0), (𝛽 >0)
(𝛾 ≡ 0 ) 𝑦𝑒𝑖𝑙𝑑𝑠 𝑡ℎ𝑒 𝑡𝑤𝑜 −
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝐿𝑜𝑔 − 𝐿𝑜𝑔𝑖𝑠𝑡𝑖𝑐 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
𝛾 ≤ 𝑥 ≤ +∞
(𝛿 >0)
(𝜗 >0)
𝜀 ≤𝑥 ≤𝜀+𝜗
𝑧≡
𝑘≠0
)
𝑒𝑥𝑝(−𝑧)
𝛼1: continuous shape parameter
𝛼2 : continuous shape parameter
𝑎, 𝑏:
continuous
boundary
parameters
𝑥−𝜇 2
) ))
𝜎
𝛾: continuous shape parameter
𝛿: continuous shape parameter
𝜗: continuous scale parameters
𝜀: continuous location parameter
𝑥−𝜀
𝑧≡
𝜗
𝜎: continuous scale parameter
𝜇: continuous location parameter
𝑘≠0
𝑘=0
(𝜎>0)
−∞ ≤ 𝑥 ≤ +∞
2.6. Development of the SDF relationship
Statistical frequency analyzes are frequently used for drought analysis. However, this method
cannot fully explain droughts without considering the severity and duration, which resulted in the
development of the SDF curves. The SDF curves can be defined to allow calculation of the
average drought intensity, or depth, for a given exceedance probability over Range of durations.
Thus, extreme drought events can be specified using the frequency, duration, and either depth or
mean intensity, severity. The frequency is usually described by the return period of the drought.
To estimate the return periods of drought events of a particular depth and duration, the frequency
distributions can be used (Dalezios et al., 2000).
2.7. Study region
The study region is a part of Hendijan -Jarrahi watershed located in south-west of Iran between
49°, 39' and 50°, 10 E longitude and '31°, 21' and 31°, 41'N latitude with approximately 909.7
square kilometers area. The elevation varies from 340 m at the Mashin hydrometric station
(outlet of the Roud Zard basin) to 3300 m on the eastern mountains. The area includes five main
sub-watersheds upstream of the outlet where Mashin hydrometric station is installed. The
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boundary of main sub-watersheds and basin location in Hendijan -Jarrahi watershed and Iran are
shown in Figure 2 and the rainfall gauges and Mashin hydrometery station location of Roud Zard
basin are shown in Figure 3.
Figure 2. Roud Zard basin in Hendijan-Jarrahi basin and IRAN
Figure 3. Roud Zard record and nonrecording rain gauges
3. Results
3.1. Determination of the threshold level and drought characteristics
This study used two threshold level methods. The fixed threshold level in two scenarios, Q70 and
Q90 of mean daily runoff, which resulted from 42 years daily stream flow data, and monthly
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thresholds which are twelve Q70 values of monthly average of all daily stream flow from January
to December for 42 years (1970-2013) of Mashin hydrometery station at outlet of the Roudzard
river basin. Table 2 presents the summary of daily runoff statistical properties, and Figure 4
shows a comparison between threshold levels in this study.
Table 2. Statistical properties of Mashin station daily runoff
Statistic
N
Daily runoff (m3/s) 15892
Range (m3/s) Minimum (m3/s) Maximum (m3/s) Mean (m3/s) Std. Deviation
673.97
.03
674.00
9.1850
20.42586
Q70
Q90
14
Monthly
Threshol level (m3/s)
12
10
8
6
4
2
0
0
50
100
150
200
250
300
350
Day
Figure 4. Comparison of the threshold levels in this study
The monthly threshold level, which is flactuated because of the natural stream flow variations
was the largest among three threshold levels in March, April, December and January, because of
considering long average monthly runoff in winter and spring period, rainy seasons in the region,
and smallest in May, June, July, August, September and December, drought months with rare
precipitation events. The Q70 and Q90 levels were fixed all the year.
3.2. Calculation of the stream flow drought severity and duration
The durations and severities of all stream flow drought events were calculated based on the
stream flow drought concept and threshold level. The time series of annual maximum values of
duration and severity for three threshold levels are shown in Figure 5 and 6. It could be seen that
maximum drought durations and deficits were derived from Q90 threshold, when minimums
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derived from monthly level. This could be resulted from different drought threshold determining
of the threshold methods. Q70 level durations were also much higher than those from monthly,
similar to Q90. The graphs also revealed that drought deficit and duration followed similar
increase and decreasing trend.
Monthly
Annual maximum Duration
400
Q90
Annual maximum Duration (Day)
350
Q70
300
250
200
150
100
50
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
Year
Figure 5. The time series of annual maxima values of duration
Monthly
Annual maximum Deficit
Q90
Annual maximum Deficit (million m3)
200
Q70
150
100
50
0
1965
1970
1975
1980
1985
1990
1995
Year
2000
2005
2010
2015
2020
Figure 6. The time series of annual maxima values of severity
Figure 7 (a-e) was prepared to compare difference among drought characteristics as well as event
number, deficit and average deficit, minimum and average runoff, derived from threshold levels.
Drought durations were classified to 4 time intervals, 1-10 days, 11-30, 31-120 and over 120
9
days, in order to better understand of the drought characteristics and prepare SDF curves.
Maximum drought event number was derived from monthly method, when comparatively the
number of short duration intervals, 1 to 10, and 11 to 30 days were much higher than those from
other methods, Fig. 7 (a). Maximum deficit and average deficit volume were also resulted from
Q90 method, Fig. 7 (b,c), as well as annual maxima, Fig. 6, but the trend was a little different in
short and long duration intervals, Fig. 7 (c). Maximum of the average and minimum runoff in
drought events were derived from monthly method, Fig. 7 (d,e) with a decreasing trend from
short to long duration intervals. It could be resulted from the fluctuated threshold level
determining of this method which considers summer and winter periods in.
Q70
Q90
Monthly
Chart Title
Event Number
400
300
200
100
Mont…
0
Q90
1 to 10
11 to 30
Q70
31 to 120
Over than
120
Duration (Day)
Total
Duration
a
Deficit
Q70
Q90
Monthly
500.000
Average.
Max
Min
Max
31 to 120
Average.
11 to 30
Min
Max
Min
Average.
1 to 10
Average.
Max
0.000
Min
Deficit (million m3)
1000.000
Over than 120
Duration (Day)
b
10
Q70
Average Deficit
Monthly
6.000
4.000
2.000
31 to 120
Duration (Day)
Max
Min
Average.
11 to 30
Average.
Max
Min
Average.
1 to 10
Max
Min
Average.
Max
0.000
Min
Average Deficit (million m3)
Q90
Over than 120
c
Q70
Q90
Monthly
15.00
10.00
5.00
Max
Min
Average.
Average.
31 to 120
Duration (Day)
d
11
Max
Min
Min
Max
11 to 30
Average.
1 to 10
Average.
Max
0.00
Min
Minimum Runoff (m3/s)
Minimum Runoff
Over than 120
Q70
Average Runoff
Q90
Monthly
10.00
5.00
31 to 120
Duration (Day)
Average.
Max
Min
Average.
Min
11 to 30
Max
Max
Min
Average.
1 to 10
Average.
Max
0.00
Min
Average Runoff (m3/s)
15.00
Over than 120
e
Figure 7 (a-e). Drougth characteristics in different duration intervals
For better comparison of the relation between duration and deficit in threshold level methods,
Pearson correlation coefficients were calculated (Chung and Stenson, 1990), Table 3. Maximum
value of 0.949 was resulted from Q70 and the minimum from Q90, so Q70 levels’ deficit and
duration had more correlation in comparison with other methods.
Table 3. Pearson correlation coefficient of drought deficit and duration
Threshold level method
Q70
Q90
Monthly
Drought Characteristic
Duration
Deficit
0.949**
0.545**
0.898**
** Correlation is significant at the 0.01 level (2-tailed)
3.3. Determination of the probability distribution function
The Kolomogrov Smirnov goodness of fit criteria was used to evaluate the best probability
distribution function fitted to the data sets. To forecast duration and deficit in different return
periods (frequency), and develop SDF curves, the proper probability distribution function was
determined, based on the statistical result of the goodness of fit criteria.
3.4. Development of drought duration-frequency curve
To forecast drought duration in different return periods, probability distribution functions were
fitted to the duration data, and the best one was determined, using Kolomogrov Smirnov
12
goodness of fit criteria. Table 4 shows forecasted drought duration, in different return period,
with Burr distribution for Q70, Pearson 6 for Q90 and Fatigue Life distribution function for
monthly threshold as the best fitted ones. It could be seen that Q70 forecasted durations were the
largest specially in large return periods and Q90 is smallest.
Table 4. Forecasted drought duration in different return periods
Duration (day)
Q70
Q90
Monyhly
Burr
Pearson 6
Fatigue Life
T
P=1/T
1-P
𝑘=0.43736
𝛼=1.5308
𝛽=3.8644
𝛼1 =2.5336
𝛼2 =0.73174
𝛽=2.0314
𝛼=1.5757
𝛽=9.2581
2
0.5
0.5
9.3676
10.214
9.2581
5
0.2
0.8
42.057
45.926
32.13
10
0.1
0.9
120.02
124.58
54.701
20
0.05
0.95
338.87
327.32
79.63
25
0.04
0.96
473.05
445.4
87.993
50
0.02
0.98
1332.5
1154.6
114.72
80
0.0125
0.9875
2688.9
2198.1
133.35
100
0.01
0.99
3752.6
2983.3
142.31
Figure 8 shows duration-frequency curves of threshold levels, which shows the relation between
forecasted duration and frequency (return period). For this plot 2, 5, 10, 20, 25, 50, 80 and 100
year frequency durations were calculated. The duration for the Q70 and Q90 were much higher
than monthly threshold, where the maximum values were resulted from Q70 level.
Q70
Q90
Monthly
Chart Title
4000
3500
Duration (Day)
3000
2500
2000
1500
1000
500
0
0
20
40
60
Return Period (Year)
Figure 8. Duration - frequency curve
13
80
100
3.5. Development of drought deficit volume-frequency curve
The probability distribution functions were fitted to the drought deficit data in order to forecast
drought deficit volume in different return periods. Table 5 presents forecasted drought deficit
volume in different return period by Generalized (Gen.) Extreme Value distribution for Q70,
Dagum for Q90 and Wakeby distribution function for monthly threshold method, as the best one
selected by Kolomogrov Smirnov criteria. It could be declared that maximum deficits were
resulted from Q90 level and minimums from monthly.
Table 5. Forecasted drought deficit in different return periods
Deficit (million m3)
Q70
Q90
Monthly
Gen. Extrem Value
Dagum (4p)
Wakeby
T
P=1/T
1-P
𝑘=0.58783
𝜎=4.4816
𝜇=1.8085
𝑘=0.7907
𝛼=0.77366
𝛽=4.1036
𝛾=0.00346
𝛼=-7.1837
𝛽=0.10548
𝛾=7.585
𝛿=0.37456
𝜀=-0.0124
2
0.5
0.5
3.6414
2.653
1.189
5
0.2
0.8
12.597
17.472
6.1069
10
0.1
0.9
22.805
50.909
13.024
20
0.05
0.95
37.881
135.01
23.479
25
0.04
0.96
44.158
182.93
27.746
50
0.02
0.98
69.748
461.86
44.371
80
0.0125
0.9875
94.017
857.43
59.064
100
0.01
0.99
108.09
1148.3
67.179
Figure 9 shows the deficit-frequency curves of threshold level methods. The deficits for the Q90
were much higher than those from the other methods, when the minimum values derived from
monthly level.
14
Q70
Chart Title
Q90
1400
Monthly
Deficit (million m3)
1200
1000
800
600
400
200
0
0
20
40
60
80
100
Return Period (Year)
Figure 9. Deficit – frequency curves
3.6. Development of SDF curves
Stream flow drought SDF curves of threshold level methods, were developed using derived
probability functions, fitted to each drought interval, Table 6. For a specific duration, this table
compares all severities with specific frequencies and duration, in the different duration intervals.
When the duration increased the severity differences among the return periods significantly
increased. It could be seen that the best probability distribution fitted to the drought deficit
(severity) were different in each interval, as the Bata distribution was the best for 1 to 10 days
interval, Weibull for 11 to 30 days, Generalized Logistic distribution for 31 to 120 days and
Dagum for over 120 days interval, for the Q70 level. This trend could be seen for other threshold
levels too.
Table 6. Severity - duration - frequency value of Roudzard river basin
Deficite
(million
m3)
T
2
5
10
20
25
50
80
100
15
Q70
Duration
Q90
Duration
1 to 10
Days
11 to 30
Days
31 to 120
Day
Over 120
Days
Beta
Weibull
Gen.Logistic
Dagum
𝛼1=0.595
𝛼2 =3.942
a=7.6E16
b=4.7561
α=1.827
β=3.772
𝑘=0.2109
𝜎=5.7422
𝜇=11.104
𝑘=2.1511
=4.2482
=32.929
0.3797
1.0954
1.6105
2.0721
2.2084
2.5942
2.8250
2.9264
3.0867
4.8940
5.9535
6.8756
7.1512
7.9566
8.4661
8.6994
11.104
20.351
27.154
34.542
37.101
45.748
52.307
55.643
41.347
55.447
66.593
79.124
83.541
98.696
110.39
116.39
1 to 10
Days
Phased BiWeibull
𝛼1=1.0388
𝛼2 =0.6608
𝛽1 =0.2962
𝛽2 =0.8807
𝛾1 =0
𝛾2 =0.0440
0.5058
1.8097
3.1115
4.6335
5.1656
6.9387
8.2384
8.8814
11 to 30
Days
31 to 120
Day
Beta
Wakeby
𝛼1=1.2294
𝛼2 =2.1255
a=0.02646
b=11.938

𝛽



4.0459
6.9661
8.3926
9.3988
9.6561
10.299
10.627
10.758
14.172
22.915
30.954
40.516
43.964
55.990
65.416
70.295
Monthly
Duration
Over 120
Days
LogLogistic
1 to 10
Days
Wakeby
11 to 30
Days
Johnson
SB
31 to 120
Day
Over 120
Days
Wakeby
Cauchy



𝛽








𝛽





70.165
98.476
120.07
144.14
152.61
181.72
204.23
215.81
0.3324
1.2308
2.0538
2.9366
3.2299
4.1647
4.8173
5.1323
2.9786
6.5276
8.4680
9.7776
10.094
10.840
11.193
11.328
13.262
22.694
29.086
35.244
37.182
43.070
46.951
48.763
80.259
107.53
141.24
205.35
237.10
395.18
584.53
710.72
The SDF curves described the stream flow drought severities with respect to durations and
frequencies (return period), Figure 10 (a-c). It could be seen that the severity increased with
increasing frequency and duration in three threshold levels but the increasing trend was different
for threshold levels. For these plots, 2, 5, 10, 20, 25, 50, 80 and 100 year frequency severities
were calculated for 1-10, 11-30, 3-120 and over 120 days duration intervals. Because the amount
of variable data only corresponds to 42 years, the calculation was up to a 100-year return period.
Q70
T=2
Deficit volum (million m3)
T=5
120
T=10
100
T=20
80
T=25
60
T=50
40
T=80
T=100
20
0
1 to 10
Days
11 to 30
Days
31 to
120 Day
Duration (Day)
Over
120
Days
a
Q90
T=2
T=5
T=10
Deficit volum (million m3)
200
T=20
T=25
150
T=50
100
T=80
T=100
50
0
1 to 10
Days
11 to 30
Days
31 to
120 Day
Duration (Day)
b
16
Over
120
Days
Monthly
T=2
Deficit volum (million m3)
T=5
700
T=10
600
T=20
500
T=25
400
T=50
300
T=80
200
T=100
100
0
1 to 10
Days
11 to 30
Days
31 to 120
Day
Over 120
Days
Duration (Day)
c
Figure 10. SDF curves in Roudzard river basin
The SDF from the monthly level showed largest water deficits for much longer durations, as well
as over than 120 days. In addition, the water deficits from the monthly level were much higher
than those from other levels, for the long duration.
Figure 11 (a-c) shows the 3D surface, fitted to the SDF values of threshold levels methods, to
better understanding of severity–duration–frequency properties. It has been plotted by Surfer
software, using kriging method for interpolation. Iso-deficit contour lines of drought were also
drawn on the surfaces.
17
a. Q70
b. Q90
18
c. Monthly
Figure (a-c). 3D surface fitted to the SDF values
All of drought characteristics, including long and short durations, deficits and the trends have
been summarized and reported in this Figure.
4. Conclusion
This study developed a useful concept to describe the characteristics of stream flow droughts
using fixed threshold, Q70 and Q90 and monthly level approach, which derives the deficiencies or
anomalies from the average of historical stream flow. Derived drought characteristics from
threshold levels revealed that maximum drought durations and deficits were resulted from Q90,
when minimums were from monthly level. This could be resulted from different drought
threshold determining of methods. The Q70 level durations were also much higher than those
from monthly, similar to Q90. Drought deficit and duration followed similar increase and
decreasing trend.
The SDF curves for stream flow droughts were developed to quantify a specific volume based on
desired duration and frequency, using threshold level data. This study compared the SDF curves
of threshold level methods. The severities increased with increasing duration and frequency,
however these values were different. The SDF from the monthly level showed the most water
deficits for much longer durations as well as over than 120 days. In addition, the water deficits
from the monthly level were much higher than those from other levels for the long duration.
19
The duration–frequency and deficit-frequency curves of threshold levels were also developed to
quantify the stream flow drought duration and deficit. The Q70 forecasted durations were the
largest especially in large return periods and Q90 were smallest. Maximum deficits were resulted
from Q90 level and minimums from monthly. It could be concluded that monthly threshold level
results are more reliable, considering seasonal increase and decreasing of stream flow.
Drought SDF curves developed in this study can be used to quantify water deficit for natural
stream and reservoir. In addition, these curves will be extended to allow for regional frequency
analysis, which can estimate the stream flow drought severity at ungauged sites. They could be
an effective tool to identify hydrological drought using the severity, duration and frequency.
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