1 Introduction

Estimating sediments carried and transported by rivers is an important and essential issue in water-resources planning, design, and management since the information is required for reservoir sedimentation evaluation, hydraulic structures design, water-quality criterion and treatment costs assessment, ecological and recreation consideration, efficacy of watershed management, stable channels for navigation, and many other sediment-induced problems. Infrequent or periodically sampling of sediments, which is frequently met in most river basins, is insufficient to define a continuous record of sediment load and fail to determine the cumulative sediment load for a longer period. Sediment-load estimation thus heavily depends upon empirical approaches such as a sediment rating curve, which describes the relationship between sediment load and river discharge, to achieve these purposes. Although some researchers (e.g., Walling, 1977; McBean and Al-Nassri, 1988; Cohn et al., 1992; Kişi, 2005) had indicated that the regression and curve-fitting techniques are not adequate to describe the high degree of scatter existed between sediment load and discharge data, the sediment rating curve remains a popular and widely used approach to estimate sediment loads due to available streamflow data and calculation simplicity.

Many works had suggested various methods to improve predictive accuracy of the sediment rating curve because of its poor predictive performances. These methods include bias adjustment or bias correction factor (Ferguson, 1986; Cohn et al., 1989; Phillips et al., 1999; Asselman, 2000), multivariate rating curve (Cohn et al., 1992; Wang and Linker, 2008; Wang and Tian 2013), and different regression techniques (Hicks et al., 2000; Krishnaswamy et al., 2001; Tarras-Wahlberg and Lane, 2003). More recently, a considerable amount of studies have applied newly developed methods to construct the empirical relationship between sediment and discharge. These approaches include artificial neural network (Jain, 2001; Nagy et al., 2002; Alp and Cigizoglu, 2007; Rai and Mathur, 2008; Afan et al., 2015), genetic programming (Aytek and Kişi, 2008; Guven and Kişi, 2011; Kitsikoudis et al., 2014), fuzzy logic (Kişi, 2004; Lohani et al., 2007), wavelet − neural network (Partal and Cigizoglu, 2008), and support vector machines (Çimen, 2008; Lafdani et al., 2013).

Inherent complexity in the sediment transport process induces an inevitable scatter between sediment and discharge. Due to few samplings during high-flow periods, suspended sediment load estimation provided by a sediment rating curve for such periods has greater deviations generally. Besides, estimation errors may propagate from a short-period (daily) estimation to determine the cumulative load for a longer period (e.g., a year) using the sediment-rating-curve approach. Providing uncertainty information is as important as accurate sediment load estimation in water resources planning and management. Although a vast literature had proposed various approaches to improve estimation accuracy, uncertainty in sediment load estimation from the sediment rating curve is rarely considered. The uncertainty-related studies include McBean and Al-Nassri (1988) had attempted to establish a confidence interval of the mean suspended sediment concentration versus discharge. Clarke (1990a, 1990b) had discussed statistical characteristics such as bias and variance under the assumption of bivariate lognormal distribution of suspended sediment concentration and mean daily discharge. Rustomji and Wilkinson (2008) had used the bootstrap associated with Monte Carlo resampling techniques for estimating suspended sediment loads in order to quantify the uncertainty in the shape of the sediment rating curve. Wang et al. (2011) had adopted a generalized rating-curve approach with additional predictors to provide reliable load estimates with minimal bias and an associated measure of uncertainty. Vigiak and Bende-Michl (2013) had used bootstrap and Bayesian inference in estimating prediction intervals for daily and monthly constituent using improved eight-parameter rating curve.

This study aims to develop a probabilistic estimation scheme for daily and annual suspended sediment loads using quantile regression. Instead of constructing a single sediment rating curve at a specific quantile, a novel approach is proposed to derive a probabilistic distribution of daily suspended sediment loads through dense quantile-dependent sediment rating curves. Deriving probabilistic distribution of the cumulative suspended sediment load for a longer period such as a year is thus made possible by the derived daily suspended-sediment-loads distributions and utilization of convolution theorem. Advances of quantile regression for deriving probabilistic distributions of daily and annual suspended sediment loads are still novelties in literature. The proposed approach is applied to the Laonung station located in southern Taiwan for demonstration.

The remainder of this paper is organized as follows. Section 2 presents an introduction of sediment rating curve, quantile regression, and the approach for constructing probability distributions of daily and annual suspended sediment loads. Section 3 summaries the recorded daily data at the Laonung station located in southern Taiwan for the 1959–2008 period. Results and discussions are given in the Section 4, which is followed by the Section 5 of summaries and conclusions.

2 Methodology

2.1 Sediment Rating Curve

The sediment rating curve denotes the empirical relationship between measured sediment load and river discharge. The power law function is commonly adopted to model this relationship, which is expressed as

$$ {Q}_s=a{Q}^b $$
(1)

where Q s denotes the suspended sediment load; Q denotes the river discharge; a and b are constants which are estimated from measured data. The nonlinear least squares regression (Asselman, 2000) is employed in this study to obtain sediment rating curve constants in order to avoid the need of applying a bias correction factor to the estimated loads (Rustomji and Wilkinson, 2008).

The obtained suspended sediment rating curve transforms the recorded daily streamflow series into a sequence of daily suspended sediment load estimations. The cumulative suspended sediment load over a longer period such as a month or a year is thus calculated as the sum of daily suspended sediment load over such period. That is,

$$ {L}_n={\displaystyle \sum_{i=1}^n{\widehat{Q}}_{s,\ i}}={\displaystyle \sum_{i=1}^na{Q}_i^b} $$
(2)

where L n denotes the cumulative suspended sediment load over an n-day period; \( {\widehat{Q}}_{s,\ i} \) denotes the estimated suspended sediment load of the ith day through sediment rating curve; Q i is the ith-day discharge; n denotes the number of days.

2.2 Quantile Regression

Quantile regression, initiated by Koenker and Basset (1978), is a statistical technique for regression on various quantiles rather than regression only on the mean. Since quantile regression can capture the variability of all parts of data distribution, it recently has been used in a broad range of applications such as economics and finances (Meligkotsidou et al., 2009; Gaglianone et al., 2011; Alagidede and Panagiotidis, 2012), biology and ecology (Cade and Noon, 2003; Austin, 2007; Cozzoli et al., 2013), and environmental science (Jagger and Elsner, 2009; Hirschi et al., 2011; Barbosa et al., 2011; Shiau and Huang, 2015). Detailed description of the theory of quantile regression can be found in Koenker (2005).

The quantile regression-based power-law sediment rating curves for various quantiles are expressed as

$$ {Q}_{s,\ q}={a}_q{Q}^{b_q} $$
(3)

where q is quantile which is constrained between 0 and 1; Q s, q is a conditional suspended sediment load at quantile q given a discharge of Q; a q and b q are quantile-dependent regression coefficients, which are determined by the selected quantile q and minimizing the sum of asymmetrically weighted absolute deviations, i.e.,

$$ \min {\displaystyle \sum_{i:\ {Q}_{s,\ i}\ge {a}_q{Q}_i^{b_q}}\kern1em q\left|{Q}_{s,\ i}-{a}_q{Q}_i^{b_q}\right|}+{\displaystyle \sum_{i:\ {Q}_{s,\ i}<{a}_q{Q}_i^{b_q}}\kern1em \left(1-q\right)\left|{Q}_{s,\ i}-{a}_q{Q}_i^{b_q}\right|} $$
(4)

There is no analytic solution for the quantile-dependent regression coefficients of a q and b q . The minimization problem can be formulated as a linear program and solved by an algorithm developed by Koenker and D’Orey (1987). Calculations in this study are carried out using a free-access software quantreg (Koenker, 2014). Instead of obtaining a single estimation of suspended sediment load for a given discharge, quantile regression-based model estimates suspended sediment loads at various quantiles for a given discharge. Examining entire distribution of suspended sediment load for a given discharge is thus made possible using quantile regression.

2.3 Probability Distribution of Daily Suspended Sediment Loads

All the paired sediment-discharge recorded data are used to construct suspended sediment rating curves at various quantiles (quantiles of 0.95, 0.9, 0.7, 0.5, 0.3, and 0.1 demonstrated in Fig. 1a). These dense quantile-dependent sediment rating curves are employed in order to have a finer resolution in probability distribution. Given a daily discharge of Q, the conditional suspended sediment \( {\widehat{Q}}_{s,\ q} \) for each quantile-dependent sediment rating curve is calculated (various discharges of Q 1 and Q 2 demonstrated in Fig. 1a). These estimated conditional \( {\widehat{Q}}_{s,\ q} \) associated with the corresponding quantile q constitute a rough estimation of the empirical cumulative distribution function (CDF) of the daily suspended sediment load (CDFs of Q 1 and Q 2 demonstrated in Fig. 1b). In this study, suspended sediment load is considered as a discrete random variable, the empirical probability mass function (PMF) is then constructed using the obtained CDF (PMFs of Q 1 and Q 2 demonstrated in Fig. 1c). Figure 1 illustrates the entire procedure for deriving empirical CDFs and PMFs of daily suspended sediment load given various discharges from estimated quantile regression-based sediment rating curves. The obtained PMF describe the probabilities of different suspended-sediment-load intervals for a given discharge, which offers rich-information than the single estimation obtained from the traditional sediment rating curve described by equation (1). The derived quantile regression-based suspended sediment rating curves thus transform the recorded daily streamflow series into a sequence of PMFs of daily suspended sediment load.

Fig. 1
figure 1

Schematic illustration of constructing probabilistic distributions of daily suspended sediment loads, (a) quantile-dependent suspended sediment rating curves, (b) empirical cumulative distribution function (CDF) of various discharge Q1 and Q2, and (c) empirical probability mass function (PDF) of discharge Q1 and Q2

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2.4 Probability Distribution of Annual Suspended Sediment Loads

The probabilistic description of cumulative suspended sediment loads for a year period can be constructed by the obtained PMFs of daily suspended sediment load and convolution theorem, which is described below.

Let S 1 and S 2 be the discrete random variables which represent the daily suspended sediment loads for day 1 and day 2, respectively. The derived PMFs of S 1 and S 2 are denoted by \( {f}_{S_1}\left({s}_1\right) \) and \( {f}_{S_2}\left({s}_2\right) \), respectively. The cumulative suspended sediment load of these two successive days L 2 is the sum of S 1 and S 2, i.e., L 2 = S 1 + S 2. Under the assumption of independence between S 1 and S 2, the PMF of L 2 can be derived by the convolution theorem (Ross, 2007). That is,

$$ {f}_{L_2}\left({l}_2\right)={\displaystyle \sum_{s_1=0}^{\infty }{f}_{S_1}\left({s}_1\right){f}_{S_2}\left({l}_2-{s}_1\right)} $$
(5)

where \( {f}_{L_2}\left({l}_2\right) \) denotes the PMF of L 2.

Repeating the procedure again, the PMF of cumulative suspended sediment load for three successive days can be derived. Let L 3 = S 1 + S 2 + S 3 = L 2 + S 3. After deriving the PMF of L 2, convolution is implemented between L 2 and S 3 to obtain the PMF of L 3. Considering the recursive formula of L n  = L n − 1 + S n , where n = 2, 3, …, N, the above procedure can be used to construct the PMF of the annual suspended sediment load when N = 365.

3 Data Used

The recorded daily suspended sediment load and river discharge data of Laonung station located in southern Taiwan are used in this study to demonstrate the proposed approach. River discharge and suspended sediment load data have been measured by Water Resource Agency in Taiwan. The daily streamflow sequence analyzed in this study has been collected between years 1959 and 2008. With a drainage area of 812 km2 and mean annual rainfall of approximate 3,600 mm, the mean daily streamflow of Laonung station of the 1959 − 2008 period is 85.2 m3/s, which is ranged between the maximum daily streamflow of 4,090 m3/s and the minimum one of 0.35 m3/s. However, infrequent sampling of daily suspended sediment load leading to the number of recorded suspended sediment load is approximate 6 % of recorded discharge data. A total of 1,110 paired sediment-discharge measured data are used to construct the empirical relationship between sediment load and river discharge, which are shown in Fig. 2.

Fig. 2
figure 2

Scatter plot of recorded daily suspended sediment load and discharge data of Laonung station, Taiwan for the 1959 − 2008 period and the corresponding suspended sediment rating curve

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Approximate 75 % of these 1,110 paired data are collected below mean daily discharge of 85.2 m3/s, while only 17 daily suspended sediment loads are collected during the greater-than-1,000 m3/s period. The mean value of these recorded daily suspended sediment load is 7.32 × 104 ton, which is ranged between 9.02 million ton (for measured discharge of 1,820 m3/s) and 0.65 ton (for measured discharge of 10.8 m3/s). Table 1 summaries the basic statistics of these recorded data of Laonung station. Although the recorded data shown in Fig. 2 illustrate a positive correlation, considerable scatter at the high-flow stage between recorded data cannot be ignored.

Table 1 Summaries of basic statistics of recorded daily suspended sediment load and discharge data of Laonung station, Taiwan for the 1959 − 2008 period
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4 Results and Discussions

4.1 Estimated Daily and Annual Suspended Sediment Loads by Sediment Rating Curve

The suspended sediment rating curve of Laonung station is derived for the measured 1,110 paired sediment-discharge daily data of the 1959 − 2008 period. The obtained power-law rating curve is described below and also shown in Fig. 2.

$$ {\widehat{Q}}_s=2.22\times {10}^{-4}\times {Q}^{1.8819} $$
(6)

where \( {\widehat{Q}}_s \) denotes the estimated daily suspended sediment load in 104 ton per day; Q denotes the daily river discharge in m3/s.

Inherent scatter between paired sediment-discharge measured data shown in Fig. 2 induces estimation errors when the derived sediment rating curve is employed to estimate daily suspended sediment load for a given discharge. The mean absolute error between this set of 1,110 measured and estimated daily suspended sediment load is approximate 5.1 × 104 ton. However, the discharge of 1,820 m3/s occurred in Aug. 9, 1994 leads to the maximum estimation error of 599.2 × 104 ton, which is caused by the difference between measured value of 902.2 × 104 ton and estimation of 303.0 × 104 ton. Estimation error is generally declined with decreasing discharge. For example, a mean absolute error of 1.6 × 104 ton is obtained when river discharge is below 500 m3/s, while this value is increased to 56.8 × 104 ton for the condition of discharge exceeding 500 m3/s. It is worth to note that the discharge of 1,840 m3/s occurred in Jul. 31, 1982 carries a suspended sediment load of 192.9 × 104 ton and has an estimation of 309.3 × 104 ton using the derived sediment rating curve. This overestimation of 116.4 × 104 ton for 1,840 m3/s associated with the underestimation of 599.2 × 104 ton caused by 1,820 m3/s reveal that an estimation error is inevitable and great uncertainty is existed in estimations even though the very close discharges.

The derived sediment rating curve is used to estimate the sequence of 18,263 daily suspended sediment load for the 1959–2008 period through recorded daily discharge. The yearly cumulative suspended sediment load is thus obtained by summing the estimated daily suspended sediment loads over the yearly period. Figure 3 illustrates the estimated yearly suspended sediment load and the mean daily discharge of Laonung station for the 1959–2008 period. Generally greater annual streamflow results in greater annual suspended sediment load. However, the maximum annual suspended sediment load and the maximum mean daily discharge do not occur in the same year. This phenomenon is attributed to the nonlinear power-law feature of the empirical sediment rating curve. The maximum annual suspended sediment load of 26.3 million ton occurred in 1960 is caused by three daily discharges exceeding 1,000 m3/s. The mean daily streamflow of 143.1 m3/s occurred in 2005 is ranked first during the 1959–2008 period. However, daily streamflow all below 500 m3/s in 2005 leads to an annual suspended sediment load of 13.4 million ton, which is ranked 14th within the 50-year period. It is hard to quantify the uncertainty of the estimated yearly suspended sediment loads without measured data, although the uncertainty actually exists.

Fig. 3
figure 3

Annual suspended sediment load estimated by sediment rating curve and the time series plot of mean daily discharge of Laonung station, Taiwan for the 1959 − 2008 period

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4.2 Probabilistic Estimation of Daily Suspended Sediment Loads

To quantify the estimation uncertainty of daily suspended sediment load, a probabilistic estimation scheme using quantile regression is proposed in this study. The regression constants of power-law quantile regression model are estimated first at dense quantiles which range from 0.01 to 0.99 with an increment of 0.01. Figure 4 shows the measured daily sediment-discharge data associated with the derived quantile-dependent sediment rating curves at quantiles of 0.9, 0.7, 0.5, 0.3, and 0.1 from top to bottom for illustration. For a given discharge, the conditional suspended sediment loads are calculated for each quantile and then integrated to obtain the empirical CDF and PMF for that discharge. The PMF is derived in a fine interval of 1 × 104 ton in order to have a finer resolution in probability distribution.

Fig. 4
figure 4

Recorded daily suspended sediment load and discharge data of Laonung station, Taiwan for the 1959 − 2008 period associated with quantile-dependent suspended sediment curves at quantiles of 0.9, 0.7, 0.5, 0.3, and 0.1 (from top to bottom)

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Left panel of Fig. 5 shows the obtained CDFs of daily suspended sediment loads given that discharge respectively equaling to 250, 496, and 998 m3/s for demonstration. The corresponding PMFs for these three discharges are shown in the right panel of Fig. 5. Probabilistic description of various sediment intervals are evidently provided by the obtained PMFs. For instance, daily suspended sediment load for the discharge of 250 m3/s ranges between 0 and 60 × 104 ton, which is shown in upper right panel of Fig. 5. However, a highly uneven and right-skewed PMF is observed in this figure. Daily suspended sediment load with a cumulative probability of 0.94 is distributed between 0 and 13 × 104 ton and the highest probability of 0.27 occurs in 0.5 × 104 ton (interval of 0 − 1 × 104 ton). This obtained PMF of daily suspended sediment load is useful to explain why the recorded daily suspended sediment loads for 250 m3/s are different in three different days. That is, 32.4, 1.8, and 4.9 × 104 ton respectively occur in Jul. 14, 1961, Aug. 10, 1961, and Aug. 12, 1969. The PMFs of daily suspended sediment load for 496 and 998 m3/s, respectively occurred in Jun. 12, 1995 and Aug. 6, 1962, also behave a right skewness.

Fig. 5
figure 5

Derived cumulative distribution function (CDF) and probability mass function (PMF) for the given discharge of 250, 496, and 998 m3/s

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The box-plot showing the minimum (first percentile), first quartile, medium (second quartile), third quartile, and the maximum (99th percentile) estimations of daily suspended sediment load for a given discharge is a useful approach to illustrate the probability distribution and compare the range of distribution for various discharges. Figure 6 demonstrates the box-plot of daily suspended sediment load for various discharges. It is worth to note that these distributions are highly right-skewed, which are in line with the quantile-dependent sediment rating curves shown in Fig. 4. That is, lower-quantiles sediment rating curves are denser than the higher-quantiles sediment rating curves. The increasing range and interquartile range with increasing discharge exhibit greater uncertainty for higher streamflow. For example, the interquartile range increases from 23.4 to 88.0 × 104 ton when discharge increases from 500 to 1000 m3/s. This value further increases to 338.7 × 104 ton when discharge reaches 2000 m3/s. The greater range of daily suspended sediment load for higher discharge also indicates a possibility of extreme events. For instance, the maximum daily suspended sediment load for 2,000 m3/s is 1,286 × 104 ton. Although the extreme value is associated with a small probability, it is a potential threat to damage hydraulic-facility operation. Incorporating the risk of such extremes into water-resources planning and management is essential, but this information is not provided by the traditional sediment-rating-curve approach.

Fig. 6
figure 6

Box-plot of daily suspended sediment load given various discharges for Laonung station

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Since the PMF of daily suspended sediment load is derived, the single estimation such as the popular central-tendency measures of mean, median, and mode can also be derived. For instance, the mean, median, and mode of daily suspended sediment load for the discharge of 250 m3/s are 5.8, 2.9, and 0.5 × 104 ton, respectively, which are less than the single estimation of 7.2 × 104 ton obtained by the traditional sediment rating curve. Repeating the same procedure to construct the CDF and PMF from the quantile-dependent rating curves for each recorded daily discharge, the single estimations are calculated and compared with the recorded daily suspended sediment loads. The mean absolute errors of mean, median, and mode for these 1,110 recorded data are 4.8, 4.3, and 6.9 × 104 ton, respectively. The proposed mean and median estimation schemes derived from the quantile regression-based model evidently outperform the traditional sediment rating curve, which induces the mean absolute error of 5.1 × 104 ton. Since the median estimation scheme has the minimum mean absolute error for the Laonung station, this method is suggested if the single estimation is needed for planning or management purpose.

The quantile regression-based approach provides not only the probabilistic description of daily suspended sediment load, but also a single estimation. Besides, the single estimations of mean and median have less mean absolute errors than the traditional sediment-rating-curve approach for Laonung station. Furthermore, probabilities of extreme events threatening hydraulic-facility operation are also provided by the proposed approach for planners and managers to assess risk of such extreme events.

4.3 Probabilistic Estimation of Annual Suspended Sediment Loads

The continuous records of daily discharge for the 1959–2008 period are employed to construct the sequence of PMF of suspended sediment load for that period. The PMF of cumulative suspended sediment load for a yearly period can be constructed by the derived daily PMFs and convolution theorem.

Figure 7a shows the PMF of annual suspended sediment load of 1970, which is roughly ranged between 200 and 800 × 104 ton. The fine interval of 1 × 104 ton replaced by a course interval of 100 × 104 ton leads to the histogram shown in Fig. 7b. This figure evidently shows that annual suspended sediment load of 350 × 104 ton (interval of 300 − 400 × 104 ton) has the highest probability of 0.51. In order to easily compare PMF of annual suspended sediment load for each year, Fig. 7a is transferred into a box-plot and shows the minimum (first percentile), first quartile, medium, third quartile, and the maximum (99th percentile) values, which is shown in Fig. 7c. This figure shows that the interquartile range ranges between 312 and 418 × 104 ton, which illustrates the most likely sediment load with a probability of 0.5 to occur within this range. The single estimations of annual suspended sediment load for the mean, median, and mode in 1970 are 378, 355, and 333 × 104 ton, respectively, which are greater than the value of 287 × 104 ton estimated by the traditional sediment rating curve.

Fig. 7
figure 7

Derived probability mass function (PMF) of annual suspended sediment load of 1970 for interval width of (a) 1 × 104 ton, (b) 100 × 104 ton, and (c) the corresponding box plot (red dot denoting the value estimated from the traditional sediment rating curve)

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Figure 8 illustrates the box-plot of annual suspended sediment load of Laonung station over the 1959–2008 period. Comparing with the single annual suspended sediment load estimated from the traditional sediment rating curve shown in Fig. 3 (also shown in Fig. 8 as red dots), this proposed scheme provide a uncertainty information. For instance, the annual sediment load in 1990 roughly ranges between 623 and 2,352 × 104 ton, while the most likely 50 % of annual sediment load ranges between 990 and 1,501 × 104 ton, and the estimated mean, median, and mode of annual sediment load are 1,270, 1,203, and 1,069 × 104 ton, respectively. Although no recorded annual sediment loads can be used to check which single estimation has less error, the median estimation of annual sediment load is suggested because of the minimum mean absolute error obtained by this scheme for daily suspended sediment load.

Fig. 8
figure 8

Box-plot of annual suspended sediment load of Laonung station over the 1959 − 2008 period (red dot denoting the value estimated from the traditional sediment rating curve)

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The proposed approach provides probabilistic description of daily and annual suspended sediment loads. These derived empirical PMFs offers water-resources planners and managers useful information to assess risk or uncertainty of sediment-related problems. For example, useful life of reservoirs determined by sedimentation can be expressed in terms of probability distribution using the derived sequence of PMFs of annual suspended sediment load.

5 Summaries and Conclusions

Providing accurate suspended sediment load estimation together with uncertainty information is an important and essential issue in water-resources planning, design, and management. Quantile-dependent sediment rating curves associated with a probabilistic description scheme for daily and annual suspended sediment loads using quantile regression are proposed in this study to achieve this purpose.

Paired sediment-discharge recorded data are firstly used to construct dense quantile-dependent sediment rating curves. The estimated conditional suspended sediment loads at various quantiles associated with the corresponding quantiles for a given discharge are then integrated to build the empirical probability mass function (PMF) of daily suspended sediment loads. Derivation of the PMF of annual suspended sediment loads is made possible by the derived PMFs of daily suspended sediment loads and the convolution theorem.

A total of 1,110 recorded daily suspended sediment load and discharge data during the 1959–2008 period of the Laonung station located in southern Taiwan are used for illustrating the proposed novel scheme. The derived empirical PMFs of daily suspended sediment load for discharge equaling to 250, 486, and 998 m3/s are shown in Fig. 5 for demonstration. The proposed probabilistic description for daily suspended sediment load evidently describes the inherent scatter between sediment and discharge data. The results of single estimations indicate that mean and median estimation schemes have mean absolute errors of 4.8 and 4.3 × 104 ton, respectively, which outperform the traditional sediment-rating-curve approach with a mean absolute error of 5.1 × 104 ton. The empirical PMFs in terms of box-plot of annual suspended sediment loads over the 1958–2008 period are shown in Fig. 8, which offer rich-information than the single estimation obtained from the traditional sediment-rating-curve approach.

Quantile regression-based approach provides not only the probabilistic description of daily and annual suspended sediment loads, but also a single estimation as well as the extremes. The obtained results offer water-resources planners and managers useful information to assess risk or uncertainty of sediment-induced problems such as the risk of water quality failing to meet the sediment-concentration criterion or the probabilistic description of useful life of reservoirs determined by sedimentation. These sediment-related problems are not considered in this work and remain as future topics for extending this research.