Over a long multi-year period, flood events can be classified according to their effectiveness in moving sediments. Efficiency depends both on the magnitude and frequency with which events occur. The effective (or dominant) discharge is the water discharge which corresponds to the maximum sediment supply. If its calculation is well documented in temperate or humid climates and large basins, it is much more difficult in small and semi-arid basins in which short floods with high sediment supplies occur. Using the example of 31 years of measurements in the Wadi Sebdou (north-west Algeria), this paper compares the two main statistical approaches to calculate the effective discharge (the empirical method based on histograms of sediment supply by discharge classes and an analytical calculation based on a hydrological probability distribution and on a sediment rating curve) to a very simple proxy: the half-load discharge, i.e. the flow rate corresponding to 50 % of the cumulative sediment yield. Three types of discharge subdivisions were tested. In the empirical approach, two subdivisions provided effective discharge close to the half-load discharge. Analytical solutions based on log-normal and log-Gumbel probability distributions were assessed but they highly underestimated the effective discharge, whatever the subdivision used to adjust the flow frequency distribution. Furthermore, annual series of maximum discharge and half-load discharge enabled the return period of hydrological years with discharge higher than the effective discharge (around 2 years) to be inferred and showed that more than half of the yearly sediment supply is carried by flows higher than the effective discharge only every 7 hydrological years. This study was the first to adapt the analytical approach in a semi-arid basin and to show the potentiality and limits of each method in a such climate.

Over a long multi-year period, flood events can be classified according to
their effectiveness in moving sediments. Efficiency depends both on the
magnitude and frequency with which events occur. According to Wolman and
Miller (1960), the efficiency can be examined through the “sediment
transport effectiveness curve”

Effective discharge curves.

To determine the effective discharge, the empirical approach proposed by
Benson and Thomas (1966) is based on the construction of a sediment supply
histogram as an alternative of the sediment-transport effectiveness
function,

Alternatively, Nash (1994) proposed an analytical approach to estimate the
effective discharge. He argued that for most rivers, the log-normal
distribution adequately represents the flow frequency, and that sediment
flow is commonly estimated from a power model,

Other methods are still proposed in the literature to estimate the effective
discharge. Ferro and Porto (2012), for example, associated it with the flow
rate corresponding to 50 % of the cumulative sediment yield, thus taking
up the concept of “half-load discharge” introduced by Vogel et al. (2003).
Since flows below this threshold carry 50 % of the total sediment
production and higher flow rates as much, this flow can also be called a
“median water discharge in the sense of sediment yield” (

These approaches to analysing sediment yield are less well adapted to
semi-arid environments that experience the alternation of very long periods
of drought or low flows and sporadic floods. Furthermore, Colombani et al. (1984)
and Castillo et al. (2003) emphasized practical difficulties in
controlling flows and associated matter in small catchments (10 to 10

In the context of current knowledge and methods, this article proposes to
adapt and compare these methods to the hydrology of a small semi-arid basin
on an example in northern Algeria. The application is carried out from
31 years of hydro-sedimentary measurements in the Wadi Sebdou (1973–2004), on
which floods last on average 7.78 % of the time. The questions dealt with
in this paper are the following:

How can we precondition data series in semi-arid environments?

What is the best subdivision of discharge classes adapted to the empirical method based on sediment yield histograms? Three types of subdivision are compared.

What are the analytical solutions following the Nash's (1994) method which fit statistical probability distributions to flow histograms to derive the dominant discharge? Theoretical solutions are established for two standard probability distributions (log-normal, log-Gumbel).

What are the different sources of errors in each approach?

Which lessons can we derive by comparing their results and the half-load discharge?

Which return periods regarding sediment supply over a long-term period can be derived from the annual series of hydrological parameters such as the annual maximum discharge and the half-load discharge?

The Maghreb is a mountainous region with young relief, characterized by many
small watersheds. In these steep marl landscapes, rainfall erosivity is
particularly high (Heusch, 1982; Probst and Amiotte-Suchet, 1992). Located
in the north-west of Algeria, the Wadi Sebdou (or upper Tafna River) runs
along 29 km (Fig. 2). The upper reaches emerge through predominantly
carbonate Jurassic terrains at altitudes up to 1400 m. Then the wadi crosses
the plain of Sebdou composed of Plio-Quaternary alluviums, and a valley (the
gap of Tafna) made up of carbonate rocks (marl-limestone, limestone and
Jurassic dolomites) (Benest, 1972; Benest and Elmi, 1969). The Wadi Sebdou
flows into the Beni Bahdel reservoir, with a storage capacity of 63 million m

Previous studies on sediment dynamics in this basin proposed syntheses on the hydro-sedimentological dynamics and budgets, or on sediment processes at the origin of hysteresis phenomena during floods, based on the detailed analysis of short-term time variations of water and sediment discharge (Megnounif et al., 2013). Additional and detailed information on morphometric, geological and land use characteristics of the basin were reported in Bouanani (2004), Megnounif et al. (2013), and Megnounif and Ghenim (2016).

Discharge and concentration data were measured at the Beni Bahdel station by the National Agency of Hydraulic Resources (locally called ANRH; Agence Nationale des Ressources Hydrauliques, 2018), in charge of gauging stations and measurements in Algeria. These data cover a 31-year period from September 1973 to August 2004. When water level is low and stable, the operator takes water samples every other day. During flood periods, sampling is intensified, up to every half-hour. During low flow period, water samples are taken every 2 weeks. At each sampling, the operator reads the water level on a limnimetric scale or on a limnigraph which is then converted into a water discharge according to a stage–discharge relationship established for the station. The suspended sediment concentration is determined from a water sample taken from the streambank, after filtration (see Megnounif et al., 2013).

Location of the Wadi Sebdou in the Tafna watershed.

The product of discharge,

Over a duration

To analyse flow frequencies and associated sediment yields, the

Regarding the choice of discharge classes, the procedure is empirical and
varies according to the authors (Pickup and Warner, 1976; Andrews, 1980;
Lenzi et al., 2006). Biedenharn et al. (2001) recommended starting by the
use of 25 classes of equal lengths. If no measurement is assigned to a class
interval or the mode is isolated in the last histogram class corresponding
to the highest rates, the number of classes is changed. Crowder and Knapp (2005)
argued that each class must contain at least one flow of a flood
event. Thus, this procedure is subjective and remains dependent on the
measurement protocol and the watershed configuration (Sichingabula, 1999;
Goodwin, 2004). For example, Hey (1997) showed that it is necessary to
increase the number of classes to 250 for a suitable representation of the
distribution of the sediment yield brought by the Little Missouri River at
Marmarth and Medora. Yevjevich (1972) suggested that the number of classes
should be between 10 and 25, depending on the size of the sample. He
proposed that the length of the class interval does not exceed

In this study, we propose to compare three types of subdivision of discharge classes – classes of equal length, classes of equal water supply and classes in geometric progression:

In many rivers where flow variation is slow, water sampling required for
solid flow measurement is not carried out daily but at monthly or weekly
intervals (Horowitz, 2003). In this case, daily solid discharge is estimated
by interpolation between actual measurements. On the other hand, small
drainage basins (less than 1000 km

The measurement protocol of the ANRH services is based on a predefined
calendar. However, the high variability of the flows experienced by the Wadi
Sebdou is such that between two consecutive measurements the difference can
be significant, and one class or more may not be represented by any flow,
whatever the subdivision used to discretize the flow discharge into classes.
Moreover, such large differences cause an overestimate of the contributions
in the sampled classes and underestimate those that are not. A preliminary
data processing was thus performed in this study in order to improve the
distribution of elementary inputs amongst classes. To achieve this, liquid
and solid discharge is assumed to vary linearly as a function of time
between two measurements. When the discrepancy between two measured
discharge is large, an intermediate discharge is added at each increase of
0.2 m

The relevance of a subdivision was examined according to its ability to
represent the water and sediment supplies. Three aspects were considered:

A subdivision was considered suitable when histograms were informative on the three variables' (frequency, water supply and sediment supply) evolution over the whole flow range, from the weakest to the strongest.

The water and sediment inputs assigned to each discharge class can be
quantified by the “standard” elementary contributions (Eq. 5) or
alternatively estimated using the midpoint discharge and the mean sediment
concentration of each class (Eq. 6). Discrepancies are expressed as a
percentage by the ratios

An additional criterion was considered to determine the effective discharge
from analysis. The suspended sediment concentration assigned to each class

Probability density functions representing flow frequencies from
instantaneous values are left-skewed distributions. The most commonly used
is the log-normal distribution (Wolman and Miller, 1960; Nash, 1994).
However, for irregular flows such as those encountered in semi-arid environments with
long periods of very little discharge, more pronounced asymmetric
distributions are recommended. Hence, in addition to the log-normal
distribution, the log-Gumbel distribution was examined. The theoretical
density functions were fitted to the discharge frequency histogram. The
dominant discharge was deduced from the analytical solution of

The two-parameter log-normal distribution has a probability density function:

The two-parameter log-Gumbel distribution is defined through its probability
density function:

Cumulative frequency, water and sediment inputs assigned to ordinal discharge in the Wadi Sebdou (1973–2004).

The function

In their study, based on 27 stream gauge stations located in three regions
of southern Italy, Ferro and Porto (2012) liken the dominant discharge to
the median discharge in terms of sediment yield (

The series of hydrologic data,

An effective discharge recurrence interval is traditionally derived from the probability distribution fitted to the annual maximum discharge series (Biedenharn et al., 2001; Simon et al., 2004; Crowder and Knapp, 2005; Ferro and Porto, 2012; Gao and Josefson, 2012; Bunte et al., 2014). To complete this parameter, which relies only on hydrological measurements and does not consider the associated sediment supplies, we also calculate in this study the recurrence interval of the effective discharge estimated from a probability distribution fitted to the series of annual half-load discharge and investigate its additional information.

In the Wadi Sebdou, discharge is greater than

Duration, water and sediment supplies, as well as the sediment rating curve with
a subdivision into classes of equal length (1 m

Interquartile discharge for water supplies, [0.66; 9.68 m

The first and third quartiles for sediment production are delimited by

Discretization of the Wadi Sebdou discharge into classes of length equal to
1 m

Characteristics and performance of various subdivisions: class of
dominant discharge range CDD; effective discharge; flow frequency

Sediment yields for subdivisions of equal lengths: 2, 4, 6 and 8 m

For such a subdivision, a change in class length necessarily affects the
representativeness of the flow characteristics, in particular the magnitude
and position of the effective discharge

Sediment supply per class, and sediment rating curves, for a subdivision into classes of equal water supplies of 1 % (left panels) and 4 % (right panels).

The comparison between the water and sediment inputs estimated from class
representatives (

Recurrence intervals, R.I.

Duration, water and sediment input per class, as well as sediment rating curves, for the subdivision into classes of geometric progression of common ratio 1.2.

Subdivision into classes of equal water input of 4 % results in 25 classes
(Fig. 6). The choice of 4 % allows 25 classes to be obtained, as recommended by
Biedenharn et al. (2001) and Crowder and Knapp (2005). The upper class
concerns discharge higher than 66.8 m

Although this subdivision describes a physical reality, allowing a rather
detailed reading of the frequency variations and water and sediment inputs
at low flows, it remains basic and provides little detail on the efficiency
of moderate to high flows. The difference (Eq. 13) between direct calculation
of water inputs (Eqs. 2 and 4) and the one based on discharge of each class
(Eq. 5), despite being low globally (3.3 %), was shown to be high for some
classes (Table 1). The corresponding rating curve

A calculation performed with a subdivision into 100 classes of equal water
contributions of 1 % (Fig. 6) reduced the errors made on

The subdivision into classes of geometric progression was chosen so that
from one class to another, the amplitude of the class increases by 20 %.
Thus, discharge in the same class is within 10 % of the class centre. In
this case, on a logarithmic scale, classes have a length equal to

The determination coefficient and Nash–Sutcliffe (1970) coefficient of the
rating curve,

Adjustment to the log-normal distribution of maximum annual discharge,

The annual series of maximum flow rate series,

Adjustment of the frequency distribution of flows to the log-Gumbel
probability distribution:

The analytical approach requires a probability density function

Half-hour sampling carried out by the ANRH is unsuitable during the Wadi
Sebdou flash floods, which produce more than 80 % of the total sediment
load in 1 % of the time, with an estimated average concentration of
10.3 g L

The quality of graphs and the error on water and sediment supplies made it possible to compare subdivisions and select those that are able to represent the flows and to identify the effective discharge. Several studies dedicated to dominant discharge class focused exclusively on the graphical aspect by readjusting the interval amplitude with equal classes until a dominant class appears outside the first and last classes (Benson and Thomas, 1966; Pickup and Warner, 1976; Andrews, 1980; Hey, 1997; Lenzi et al., 2006; Roy and Sinha, 2014). However, this approach remains subjective (Sichingabula, 1999; Biedenharn et al., 2001; Goodwin, 2004) and poses a dilemma. Reducing the class amplitude can make the dominant class emerge outside the two extreme classes, but this can bring up empty classes which, conversely, require the amplitude for each class that is to be covered to be increased. Where appropriate, the series is considered non-compliant with the selection criteria and does not allow the dominant class to be identified (Crowder and Knapp, 2005). To avoid such situations, Bienderhan et al. (2000) recommended the use of adequately provided datasets covering at least 10 years of measurements.

Yevjevich's (1972) proposal, based on statistical concepts, to use between
10 and 25 classes of amplitude less than

In this study, two types of subdivisions other than the classical
subdivision with classes of equal amplitude were examined: discharge classes
corresponding to equal water supply, and a geometric progression of flows.
The subdivisions into classes of equal amplitude 1 m

Comparison between the analytical sediment supply by class given from
the rating curve (

The sediment supply calculated from data (

Sediment load histogram established using the sediment rating curve:
for a subdivision into classes of equal amplitudes 1 m

In the Wadi Sebdou, despite a correct estimate of the total sediment supply
for the two subdivisions of equal amplitude 1 m

This result may be site-specific. Indeed, sediment–discharge rating curves fail to properly reproduce the dynamics of suspended sediment flows in the Wadi Sebdou due to the hysteresis phenomena, studied in Megnounif et al. (2013). Such errors “of the first type”, high in the wadi Sebdou, may be reduced in other semiarid basins.

Analysis of errors (difference and ratio) between observed and
theoretical frequencies of water discharge: for the log-normal distribution

When the flow frequency is represented by a probability distribution, the
sediment load histogram can be built from this distribution and the sediment
rating curve. However, it should be remembered that for a continuous random
variable such as water discharge, the theoretical probability at a point
does not exist in the probabilistic sense, but necessarily refers to an
interval. Thus, the contribution of a class,

Since the function

However, the analysis of errors associated with the subdivision in geometric
progression and the log-normal distribution (Fig. 12) shows that above
18.3 m

Three sediment load histograms obtained from the dataset, from the product of the rating curve times to the log-normal distribution of discharge, and from the product of the rating curve times to the log-Gumbel distribution.

The analytical expressions giving

In summary, the pronounced asymmetric probability distributions which seemed to be adapted to the Wadi Sebdou failed to reproduce good frequencies of high discharge associated with flash floods. Consequently, the empirical method by decomposition of histogram classes is the most suitable in a semi-arid environment. This had never been tested in the literature. It is an original result of this paper.

Another point deserves a remark in the calculation of the effective
discharge from

Finally, it should be noted that introducing a density function necessarily gives a monomodal sediment transport efficiency curve, whereas this is not necessarily the case. Pickup and Warner (1976), Carling (1988), Phillips (2002), Lenzi et al. (2006) and Ma et al. (2010) reported the existence on some sites of a bimodal dominant flow. Hudson and Mossa (1997) pointed out that sediment load histograms present a variety of forms, including bimodal and complex forms, that differ from the unimodal form identified by Wolman and Miller (1960). In addition to the monomodal sediment load histograms, Ashmore and Day (1988) distinguished three other kinds of histograms: bimodal, multimodal and complex. Of the 55 basins studied by Nash (1994), 29 are bimodal and 9 are multimodal.

Biedenharn et al. (2001) suggest to carefully study long (over 30 years) data series (liquid flow, sediment concentration and flow frequency) and to ensure that the hydrological regime of the watershed did not undergo a significant change in flow rates or sediment production in the long term. Change can be attributed to climate change (Zhang and Nearing, 2005; Ziadat and Taimeh, 2013; Liu et al., 2014; Achite and Ouillon, 2016) or anthropogenic actions (Cerdà, 1998a, b; Liu et al., 2014), such as intensification of agriculture (Montgomery, 2007; Lieskovský and Kenderessy, 2014), deforestation (Walling, 2006), forest fires (González-Pelayo et al., 2006; Cerdà et al., 2010) or urbanization (Graham et al., 2007; Whitney et al., 2015). In the study area, and like northern Africa and the Maghreb, there has been a continuous drought since the mid-1970s (Giorgi and Lionello, 2008; Achite and Ouillon, 2016; Zeroual et al., 2016). Overall, decreasing rainfall is more concentrated over time (Ghenim and Megnounif, 2016), which increases the susceptibility of soils to erosion (Shakesby et al., 2002; Bates et al., 2008; Vachtman et al., 2012). Megnounif and Ghenim (2016) showed that sediment production, which is increasing with increasing rainfall variability (Achite and Ouillon, 2007), increased significantly in the late 1980s, with a pivot in 1988. After 1988, the annual sediment yield was on average 7 times higher compared to the previous period (Megnounif and Ghenim, 2013).

Sediment supply by class for the subdivision in geometric progression:
for 1973–1988

The application of a subdivision of discharge classes into geometric
progression at the Wadi Sebdou for the two periods 1973–1988 and 1988–2004
confirmed the change in the watershed functioning, with a bimodal sediment
supply distribution for the first period (Fig. 14). For 1973–1988, the class
[6.1; 7.4 m

The half-load discharge in 1973–1988,

From a time series of flow and concentration data, a direct calculation
provides estimates of water and sediment supplies by summing the elementary
contributions. This gives access to seasonal or annual values, and to the
analysis of their variability. Sediment dynamics can also be analysed from
discharge and sediment yield histograms by water discharge classes. In the
Wadi Sebdou, we have shown that an appropriate choice of subdivisions makes
it possible to minimize the difference between the flows estimated and
measured at less than 10 % (

The introduction of a rating curve between the

In the Wadi Sebdou, the coupled use of a sediment rating curve and the log-normal and log-Gumbel probability distributions were most likely to reproduce the observed regime, characterized by a very weak mode. However, they failed to properly estimate the flow frequency of flash floods which are typical in semi-arid environments, and the corresponding sediment yield.

Two return periods of the effective discharge were identified: one (from the
annual maximum flow rate series,

Flows of the dominant class carry the most sediment in the watercourse. It should be possible to link them to major processes of erosion, transport and deposition that occur in the watershed. Lenzi et al. (2006), who have observed a bimodal sediment contribution for a mountain river in the Alps in Italy, attributed the first modal class, of low but more frequent flow, to the shaping of channel and suggested that the second-class flows, larger in magnitude but of low occurrence, would be responsible for the macroscale shape of the watercourse. In the Wadi Sebdou, we also observed a bimodal distribution. However, it is difficult to conclude because the secondary mode of distribution obtained for the first period (1973–1988) became the dominant mode of distribution later (1988–2004). The modes moved with the hydrological regime towards higher and higher sediment yields, in line with what has been observed in most watersheds studied over recent decades in the semi-arid environments of Algeria (e.g. Achite and Ouillon, 2016). Applying this method to other watersheds will undoubtedly allow us to go further in the analysis of dominant discharge and in their dynamics, in a context of global change.

Discharge and suspended sediment concentrations are available
on request for scientific purposes from the National Agency of Hydraulic
Resources of Algeria (locally called ANRH,

AM and SO designed the study. AM computed and processed the results. Both authors performed all analyses, prepared the manuscript, revised it and approved the final version.

The authors declare that they have no conflict of interest.

Two anonymous reviewers are warmly thanked for their reviews and comments on previous versions of this paper. The editor, Thomas Kjeldsen, is gratefully acknowledged. Edited by: Thomas Kjeldsen Reviewed by: two anonymous referees