Introduction

The suspended sediment transport in the rivers is one of the significant factors which has a fundamental influence on hydraulic structures and is a key challenge for surface water exploitation (Kisi and Yaseen 2019). As an essential issue in the sediment transport process, the precise estimation of suspended sediment concentration (SSC) is compulsory for the design of hydraulic structures. In addition, the impact of the river’s suspended sediment on the ecosystem, the pollution, and the quality of surface water resources gain a huge interest (e.g., Frings and Kleinhans 2008; Martinez et al. 2009; Shadkani et al. 2021; Shojaeezadeh et al. 2018; Samadianfard et al. 2021). The inaccurate estimation of SSC however leads to several unwanted impacts on the available resources, time, energy, and manpower (Meshram et al. 2021; Harun et al. 2021).

The management, measurement, and determining sediment transport process in the rivers are among the common subjects in sediment studies. Likewise, the deposition of sediment in the rivers is the most common phenomenal and costly challenge which could lead to considerable environmental, agricultural, and ecological problems. For instance, the sediment deposition in the dams would lower the active volume and increase its dead volume (Mohammadi et al. 2021). The SSC in rivers also has a significant impact on the drinking and agriculture water quality. For this, many scientists such as Jain (2001) believe that the association between the SSC, the water quality, and industrial water consumption needs to be determined prior to the advanced water resources planning and management. Also, Safari (2020) outlined that to discover the stochastic nature of sediment transport, the nonlinear attributes of its drivers and their contribution should be investigated through experimental and numerical studies.

By far, the application of time series analysis on sediment transport phenomena has been studied through different approaches (e.g., Andrews 1980; Walker and Hammack 2000; Wang et al. 2008; Yang et al. 2011). For instance, Tian et al. (2019) assessed the temporal and spatial variation of runoff and the SSC in the upper Yellow River, China. They depicted the abrupt changes within the applied variables of the post dam construction era, indicating an increase (decrease) in the runoff periodicities owing to the low (high) frequencies generated afterwards. Restrepo Lopez et al. (2017) examined the suspended sediment discharge variation and flux in Colombian Caribbean rivers using the wavelet spectrum. They revealed no significant trend on the SSC; however, the period 2000–2010 has the highest variations. Han et al. (2019) evaluated the precipitation-runoff-sediment transport relationship in the Wuding river basin in China using a long-term time series (1955–2012) and emphasized the drastic changes in the relationship of mentioned variables due to the several dam constructions along the river. Esteves et al. (2019) evaluated the probable links between rainfall discharge and suspended sediment transport during the flood events in a 20-km2 mountainous Mediterranean catchment. The seasonality of suspended sediment concentration is emphasized and it was suggested that a minimum of 5 years of consecutive dataset is needed to define a reliable relationship among the applied variables. Zakwan et al. (2021) assessed the applicability of the magnitude performance analysis to derive the suspended sediment transport in the Drava river, Croatia. They found that by considering the discharge with a 1-year return interval, the annual maximum discharge transports about 90% of the total sediment concentration of the lower Drava River. Sok et al. (2021) analyzed the temporal variation of suspended sediment concentration in Cambodia’s lower Mekong rivers and explained a decreasing trend over the period 1993–2018.

The sediment transport phenomena in the Rhone River (both in France and Switzerland) gained enormous attention in recent years. For instance, Stutenbecker et al. (2018) quantified the process of sediment generation and their spatial distribution in the Upper Rhone basin (URB) using various fingerprint techniques and pointed out that a considerable amount of glaciogenic sediments is currently linked to the retreating glaciers. From the geomorphological aspect, Schoch et al. (2018) tried to predict the location and spatial distribution of sediment and bedrock cover in the URB and revealed since the Little Ice Age, the amount of relative sediment cover has increased in de-glaciated areas. Costa et al. (2018b) assessed the changes in hydro-climatic drivers and suspended sediment at the Upper Rhone basin during 1975–2015. Their outcomes emphasized the dominance of erosive rainfall, ice melt, and snowmelt in the daily SSC prediction. They emphasized the need for analyzing the interactions between hydrological factors and sediment sources at URB in the future. Dabrin et al. (2021) assessed the suspended particulate matter transport at the Rhone basin (France) through different extraction methods including the fingerprinting method.

A literature survey indicates the lack of a study to examine a possible relationship between the total streamflow, direct runoff, and SSC in the URB. To this end, we have assessed the effect of rainfall washouts and sediments brought by direct precipitation over the Rhone River with sediments brought by the streamflow as a product of glacier or snow melts and channel erosion from the upstream. Then, by taking into consideration of the recent studies on SSC prediction at URB, this study aims to (i) evaluate the sediment concentration concerning direct runoff and streamflow in response to the monthly precipitation changes, and (ii) to conduct a multivariate stochastic approach in studying the suspended sediment concentration phenomenon at the URB.

Study area and data

The Upper Rhone basin (URB) is located between the Rhone Glacier and Lake Geneva in Switzerland with an approximate area of 5250 km2. Its altitude varies between 372 and 4521 m with a mean of 2124 m, which hosts several peaks higher than 4000 m (Fig. 1). The Rhone River’s watershed is a closed intra-montane which is one of the largest inner-orogeny watersheds in the European Alps that supplies around 75% of the total inflow and 85% of the supplied sediment to Lake Geneva (Loizeau et al. 2012; Klein 2016). The glacial erosion and mass wasting processes on steep hillslopes cause the supply and production of the sediment, which are largely controlled by precipitation, and snow and ice melt (Costa et al. 2018b). The URB was extensively influenced by anthropogenic impact. For instance, the Rhone River was channelized twice in the periods of 1863–1894 and 1930–1960 at the downstream area continued by the ongoing 3rd Rhone Correction Project.

Fig. 1
figure 1

Topographic map of the Upper Rhone basin with the applied gauging stations

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The annual mean precipitation in the URB is about 1400 mm with high spatial variability in response to the orographic effect (Costa et al. 2018a). The snowmelt and ice melt dominate the hydrological regime of the URB where the peak and low flows respectively take place in summer and winter. Mean discharge is about 320 m3/s during June–August and 120 m3/s during December–February, while the mean annual discharge is 181 m3/s.

The daily precipitation (P) data at Aigle station; the daily discharge (Q) data at the Reckingen, Brig, Sion, Branson, and Porte du Scex stations which are consecutively located at the most upstream to most downstream of the Rhone river; and twice a week recorded SSC data at the Porte du Scex station were acquired from the Swiss Federal Office of the Environment (FOEN). In this regard, the selected stations (Fig. 1) and the statistical properties of the data measured at each station are given in Table 1, which will be used through analysis in the subsequent sections. Although the period of acquired data varied among the streamflow stations, the common January 1981 to October 2020 time period between the allocated time series was chosen for further investigation. The statistical properties of the time series including average, standard deviation, frequency of records, drainage area, altitude, and glaciation extent are listed (Table 1). The time series plot of the raw data, initially provided for P and SSC (Fig. 2a, b) as well as the streamflow time series (Q) for the applied stations are presented (Fig. 2c–g).

Table 1 General statistical and geographical information of the applied time series
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Fig. 2
figure 2

Time series of (a) SSC at Porte du Scex station, (b) P at Aigle station together with the total streamflow in (c) Reckingen, (d) Brig, (e) Sion, (f) Branson, and (g) Porte du Scex stations

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Methods

The methodology used in this study consists of three main sections including; data regulation, data analysis and a mathematical simulation. Respectively, the subsequent sections detail the applied methods. First, the time series of suspended sediment concentration (SSC), streamflow (Q), and precipitation (P) are transformed into monthly time series (Fig. 3). The Q time series was converted to direct runoff (DF) and base flow (BF) time series to study the effect of precipitation. Then, the correlation analysis and sediment rating curves are obtained to study the spatial relationship between SSC and Q in the hydrometric stations and finally the allocated scenarios are used to generate the best simulation model using the Autoregressive Moving Average eXogenous (ARMAX) method. The procedure is shown in the flowchart given in Fig. 3.

Fig. 3
figure 3

Flowchart of the applied methodology

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Data regulation

The daily streamflow data were used to separate base flow and direct runoff to study the immediate effect of precipitation on the total streamflow on a daily time scale. For this, the base flow separation is conducted using the sliding interval method (Sloto and Crouse 1996) that determines the minimum streamflow discharge over half of the interval minus previous day streamflow [0.5(2 N*-1) days]. The sliding interval method can be illustrated with a moving bar 2 N* upward until it intersects with the daily streamflow hydrograph. The streamflow of any desired station is assigned to the median day within the assigned interval. Finally, the bar can be pushed over (being slid) to the next day and repeated over and over until the finalization of the procedure. Finally, each streamflow time series is evaluated as the total stream flow rate (Q), base flow (BF) separated with sliding interval method, and the direct runoff (DR) which is the residual between total streamflow and base flow time series at each interval (Fig. 4). According to the base flow separation analysis, more than 80% of the total streamflow in the river is the product of base flow. In order to clarify this issue, the base flow index (Vaheddoost and Aksoy 2018) as the percentage of ratio between BF to the Q is provided (Fig. 4).

Fig. 4
figure 4

Time series of DR and BF in the selected stations (BFI, the base flow index as the percentage of ratio of BF to the Q)

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Then, the daily precipitation data at the Aigle station and daily streamflow rate at Porte du Scex, Branson, Sion, Brig, and Reckingen stations, together with the daily suspended sediment concentration at the Porte du Scex station (refer to Table 1), are transferred respectively to the monthly average precipitation, average discharge rates, and the monthly total suspended sediment concentration for the common time January 1981 to October 2020 (478 months).

Data analysis

The correlation analysis is conducted based on Pearson’s correlation coefficient matrix applied to the ensemble of the time series. Therefore, the time series of precipitation, streamflow, and suspended sediment concentration are examined against each other to investigate the linear dependence between allocated time series. Similarly, the auto-correlation function (ACF) and the partial-auto-correlation function (PACF) of the individual time series are tested to conduct temporal analysis that is evaluated in the development of ARMAX models which is discussed broadly in the section “The mathematical simulation.”

Finally, the sediment rating curve (SRC) as a conventional method in SSC analysis is used (Asselman 2000; Mohammadi et al. 2021). It empirically represents the non-linear exponential functional relationship between the SSC and Q. Several alternative equations for the SRC have been reported in the literature; however, the most widely used SRC relationship has a form of

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

where a and b are constant values obtained from the data. It has to be noticed that SRCs are developed in three shapes in this study, i.e., as a product of the Q-SSC relationship as well as the products of DR-SSC and BF-SSC.

The mathematical simulation

Initially, the elasticity analysis is used to examine the flexibility of the models, their clustering capabilities, and the rate of influence for each predictor in space. It is commonly used in management practices to reveal the share of each predictor on prediction with consideration to its magnitude and significance. For this, usually a log-linear regression is used, while the outcome of such analysis details if the relationship between dependent and independent variables is inelastic (when the obtained coefficient |α|< 1, that the SSC has a small response to the variable), unit elastic (when |α|= 1, that the SSC has an exact response with the variable), or elastic (when |α|> 1, that the SSC is greatly under the influence of the variable). Interested readers may refer to Working (1943), Leser (1963), Intriligator et al. (1996), and Deaton and Muellbauer (1980a, b) for more details about the application of elasticity analysis in time series analysis. In this respect, the elasticity analysis is conducted between the time series of monthly SSC and monthly Q as the dependent and the independent variables, respectively. A simple log-linear elasticity regression can be determined as

$$log \left({SSC}_{t}\right)=\beta +\sum \left({\alpha }_{i}log\left({Q}_{t}^{i}\right) \right)$$
(2)

in which the intercept β and coefficients α are the parameters of the regression and can be evaluated with a t-test which defines if the obtained value is statically significant or not (t-critical is ± 1.96 at 0.05 significance level). Thereby, the effect of a non-significant coefficient can be ignored in the analysis regardless of the sign and/or value obtained in the analysis.

On the other hand, the inconsistency and non-homogeneity associated with the variables could induce the hydrologic features to change over time (Salas et al. 1980). Therefore, it is suggested to check the stationarity and homogeneity of the applied hydro-climatic variables in advance (Salas et al. 1980; Wang et al. 2009; Vazifehkhah and Kahya 2019). For this, the time series of the applied variables are checked for stationarity and homogeneity using the Augmented Dickey-Fuller (ADF: Dickey and Fuller 1979) and Pettitt’s (Pettitt 1979) tests respectively.

Then, the Autoregressive Moving Average eXogenous (ARMAX) method is used to evaluate a multi-variable stochastic model fit in the simulation of the desired relationship between SSC, P, Q, and DR. For this, the SSC in the river stream is examined for the source of SSC brought either by precipitation, direct runoff washouts, or the upstream glacier melt, snowmelt, or the channel erosion. For this, the ARMAX (p, q, b) consisting of order-p time lag (maximum 2 in this study), order-q moving average of residuals (maximum 2 in this study), and order-b variables depending on the amount of contributing independent variable (maximum 6 in this study based on the scenario) were proposed. In this respect, a related equation takes the form of

$${SSC}_{t}={\sum }_{k=1}^{b}\left({{\beta }_{k}x}_{t}^{k}\right)+{\sum }_{j=1}^{p}\left({{\phi }_{j}SSC}_{t-j}\right)-{\sum }_{j=1}^{q}\left({\theta }_{j}{\varepsilon }_{t-j}\right)-{\varepsilon }_{t}$$
(3)

in which t stands for the time interval, β is the coefficient or intercept obtained for the kth exogenous input, ϕ is the coefficient of jth order auto-regressive term, θ is the jth order moving average of residual series, ε. Therefore, by considering a different combination of p, q, and b orders, different scenarios can be generated. Then, the performance of models is evaluated using Akaike information criterion (AIC) and the determination coefficient (R2) defined as

$$AIC=Nln\left(\frac{N}{N-n-1}\right){\sigma }_{\varepsilon }^{2}+2\left(n+1\right)$$
(4)
$${R}^{2}={\left(\frac{Covariance \left(SSC,\widehat{ SSC}\right)}{\left({\sigma }_{SSC}\right)\left({\sigma }_{\widehat{SSC}}\right)}\right)}^{2}$$
(5)

in which n, N, and σε are respectively the number of parameters in the model, number of datasets, and standard deviation of the residual series. The AIC defines the parsimoniousness of the model, while R2 denotes the portion of the variance explained by the applied scenario.

Results

The following section details the results obtained in data analysis or model development as detailed in the flowchart (Fig. 3).

Correlation analysis

The correlation coefficients between all of the time series (i.e., SSC, Q, P, DR, BF) are obtained to evaluate the nature of the relationship between suspended sediment concentration (SSC) in Porte du Scex station (most downstream station) and total streamflow in the same station (Q5), Branson (Q4), Sion (Q3), Brig (Q2), and Reckingen (Q1) stations; direct runoff (DR5, DR4, DR3, DR2, DR1) and base-flow (BF5, BF4, BF3, BF2, BF1) in the aforesaid stations; and the precipitation (P) in the Aigle station (at downstream). The outcomes of the correlation analysis are in Fig. 5 which shows the SSC data in Porte du Scex is less correlated with the precipitation, while more correlated with total streamflow data over the applied stations (Fig. 1). The correlation coefficients between SSC and Q and BF (Fig. 5) are higher than DR on every station. On the other hand, it is seen that the correlation between Q and BF is higher than the correlation between Q and DR on every individual station. Likewise, it can be understood that the BF has a greater quota in total Q and the nature of the river is permanent being fed by the snowmelt and/or groundwater.

Fig. 5
figure 5

The correlation matrix of all variables including P, SSC, Q, BF, and DR related to selected stations

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Sediment rating curves

The sediment rating curve between SSC at Porte du Scex station and streamflow in five stations (Q5, Q4, Q3, Q2, Q1), and the SRCs for SSC and direct runoff (DR5, DR4, DR3, DR2, DR1) and base flow (BF5, BF4, BF3, BF2, BF1) are developed (Fig. 6). The SRC for SSC and total streamflow data at Porte du Scex station (Q5) yields an R2 of 0.73 demonstrating the high applicability of the SRC for the applied cases. Developing the SRCs for SSC and streamflow at Branson, Sion, Brig, and Reckingen stations (Q4, Q3, Q2, Q1) shows that SSC has also a reasonable relationship between upstream streamflow which is more tangible for Sion, Brig, and Reckingen (Q3, Q2, Q1). In this regard, the relationships between the SSC and Q3, Q2, and Q1 are higher in contrast to the Q5 and Q4. Therefore, it may be interpreted as the effect of erosion occurring in the hillslope, human activities, glaciers, and/or snowmelt, etc., at the upstream of the river. It is seen that the SSC at Porte du Scex station has higher correlations with the BF and the Q compared to the DR. Similar to the results obtained for total streamflow, the SRCs established on BF for upstream stations give higher R2 in comparison to the downstream stations. It is noteworthy that the incline of the rating curves increases from upstream (Porte du Scex, Fig. 6e) to downstream (Porte du Scex, Fig. 6a) indicating that the Q and the corresponding BF and DR have a higher effect on the SSC in the downstream.

Fig. 6
figure 6

Sediment rating curves related to suspended sediment concentration (SSC) at Porte du Scex station and Q, BF, and DR at (a) Reckingen, (b) Brig, (c) Sion, (d) Branson, and (e) Porte du Scex stations

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Elasticity of the data

The elasticity analysis results are also given (Table 2) with a log transformation revealing that the Q5 (same as for the SSC) has the highest effect on the SSC. Based on the regression performance of the performed elasticity analysis, the model explains 98.6% of the total variance (R2) in the SSC time series. For instance, a 1% change in Q5 causes in a 0.64% change in SSC. A similar association between Q4 and SSC can be observed while the Q1, Q3, and Q2 respectively have 0.59%, 0.58%, and 0.25% effects on the SSC. However, when the t-value is considered (Table 2), the t-value associated with Q1 exceeds 1.96 which indicates that the coefficients of elasticity obtained for Q1 are significant, while the remaining coefficients are not statistically significant and could be ignored in the elasticity analysis. For instance, the Q4 (Branson) owns a negative coefficient unlike the other stations, but due to the non-significant t-value it can be ignored or assumed with zero effect in the analysis.

Table 2 Elasticity analysis of the log-linear regression with R2 0.986
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As detailed above, the relationship between SSC and the predictors (Q1, Q2, Q3, Q4, and Q5) is quite inelastic, but it still interpreted about 99% of the variance which urges the need for detailed investigation about the micro-hydrology and morphology of the basin in future. In other words, it can be concluded that the changes in the SSC are extremely related to the discharge rates, but the portion and the direction the relationship may differ due to the micro- and macro-hydrological events at the basin. In this respect, the relationship between SSC and the streamflow in the river is found to be quite complicated and nonlinear that needs to be simulated with a deterministic-stochastic approach that interprets the effect of time and space, simultaneously.

The mathematical simulation

To develop Autoregressive Moving Average eXogenous (ARMAX) best-fit equations, different scenarios are developed. The urge for using these scenarios relies on the examination of the role of DR, P, and Q on the SSC (i.e., BF is not expected to be the source of SSC when it cannot be recognized as the streamflow when DR exists). Therefore, the most relevant variable sets in the form of suggested scenarios would reveal the most reliable scenario that is probably taking place. To this end, considering the obtained results (Table 2), the scenarios are suggested as follows:

$${SSC}_{t}=f\left({SSC}_{t-j},{\varepsilon }_{t}, {Q}_{t}^{5},{P}_{t}\right)$$
(6)
$${SSC}_{t}=f\left({SSC}_{t-j},{\varepsilon }_{t}, {Q}_{t}^{1},{P}_{t}\right)$$
(7)
$${SSC}_{t}=f\left({SSC}_{t-j},{\varepsilon }_{t},{Q}_{t}^{1}, {Q}_{t}^{2},{Q}_{t}^{3},{Q}_{t}^{4}, {Q}_{t}^{5},{P}_{t}\right)$$
(8)
$${SSC}_{t}=f\left({SSC}_{t-j},{\varepsilon }_{t}, {DR}_{t}^{5}, {P}_{t}\right)$$
(9)
$${SSC}_{t}=f\left({SSC}_{t-j},{\varepsilon }_{t}, {DR}_{t}^{1}, {P}_{t}\right)$$
(10)
$${SSC}_{t}=f\left({SSC}_{t-j},{\varepsilon }_{t},{DR}_{t}^{1}, {DR}_{t}^{2},{DR}_{t}^{3},{DR}_{t}^{4},{DR}_{t}^{5}, {P}_{t}\right)$$
(11)

to define if the DR or the Q (BF is the same as Q due to the permanent nature of the river and not likely to be an alternative source in the production of the suspended sediment concentration) reveals the most relevant results associated with the SSC while jth time lag and randomness ε either in moving average or random walk is considered at time t. In other words, the proposed scenarios investigate the effect of upstream (glacier-melt, snow-melt, channel erosion etc.), rainstorm washouts (at the downstream), dust brought by the immediate precipitation over the channel in the most downstream station, and the temporary deposition and/or motion of the sediment (lag times) on the total suspended sediment concentration at the Port du Scex station.

Since ARMAX uses autoregressive (AR) and moving average (MA) components for the SSC, the autocorrelation function (ACF; Fig. 7a) and partial autocorrelation function (PACF; Fig. 7b) of the SSC time series are applied to define the order of the AR and MA process in ARMAX models. Based on the analysis (Fig. 7), both AR and MA components exist while the nature of the phenomenon is based on the dragged sediment components before time t and also randomness like the phenomenon itself.

Fig. 7
figure 7

The ACF and PACF related to SSC time series

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The combinations of models and scenarios are generated (Eqs. 6–11) to study the effect of DR or Q on the SSC (Fig. 8 and Table 3). The developed equations for the proposed scenarios are given as follows:

Fig. 8
figure 8

Time series of observation versus ARMAX model

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Table 3 Results of the conducted models based on the scenarios given in Eqs. (12–17)
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$${SSC}_{t}=\left(-317.598+2.771{Q}_{t}^{5}+0.869{P}_{t}\right)+\left(-0.352{SSC}_{t-1}+0.263{SSC}_{t-2}+0.622{\varepsilon }_{t-1}+{\varepsilon }_{t}\right)$$
(12)
$${SSC}_{t}=\left(-86.790+27.776{Q}_{t}^{1}+1.209{P}_{t}\right)+\left(-0.293{SSC}_{t-1}+0.549{\varepsilon }_{t-1}+0.188{\varepsilon }_{t-2}+{\varepsilon }_{t}\right)$$
(13)
$${SSC}_{t}=\left(-180.409+1.004{Q}_{t}^{5}+1.976{Q}_{t}^{4}-4.837{Q}_{t}^{3}+4.549{Q}_{t}^{2}+21.738{Q}_{t}^{1}+1.201{P}_{t}\right)+\left(-0.415{SSC}_{t-1}+0.616{\varepsilon }_{t-1}+0.192{\varepsilon }_{t-2}+{\varepsilon }_{t}\right)$$
(14)
$${SSC}_{t}=\left(-106.305+13.795{DR}_{t}^{5}+0.946{P}_{t}\right)+\left(1.730{SSC}_{t-1}-0.998{SSC}_{t-2}-1.682{\varepsilon }_{t-1}+0.951{\varepsilon }_{t-2}+{\varepsilon }_{t}\right)$$
(15)
$${SSC}_{t}=\left(73.795+133.247{Q}_{t}^{1}+0.819{P}_{t}\right)+\left(-0.352{SSC}_{t-1}+0.263{SSC}_{t-2}+0.622{\varepsilon }_{t-1}+{\varepsilon }_{t}\right)$$
(16)
$${SSC}_{t}=\left(-10.365+5.649{DR}_{t}^{5}-14.236{DR}_{t}^{4}+32.644{DR}_{t}^{3}\pm 13.249{DR}_{t}^{2}+73.629{DR}_{t}^{1}+0.362{P}_{t}\right)+\left(0.565{SSC}_{t-1}-0.299{SSC}_{t-2}-0.1842{\varepsilon }_{t-1}+0.229{\varepsilon }_{t-2}+{\varepsilon }_{t}\right)$$
(17)

According to the obtained results (Table 3) in terms of R2 and AIC, the third scenario could be recognized as the best-fit scenario model. Hence, the time series of the best models obtained from the six scenarios are presented (Fig. 9), which shows their acceptable performances in capturing the sediment concentration in the river streams. However, some recent extremes in the SSC values exceeded the predictions. As also detailed previously (Fig. 8 and Table 3), such a model explains about 63% of the volatilities in the time series, while most of the deviations occurred due to outliers of the extreme events that could be probably in response to the high flows caused by the temperature increase and the sudden ice and/or snowmelts. What is more, and based on the obtained coefficients for the best model (Eq. 14), both of the exogenous variables (first parentheses of Eq. 14) and the stochastic variables (second parentheses of Eq. 14) alter the simulations. By considering the discharge rates in Eq. (14), the Q1 has the highest coefficient which also confirms the results of the elasticity analysis that were already explained (Table 2). Thereby, it could be interpreted that the most upstream parts of the URB are the main source (i.e., ice and snowmelt upstream) of the SSC at Rhone River, but the streamflow has a dominant effect against direct runoff in the production of SSC at the Port du Scex station. It is also concluded from the lower P coefficient that the best-fit model conceptualizes a non-significant effect of precipitation over the SSC at the Port du Scex station, which is in line with the correlation analysis results that were explained earlier. Alternatively, the weak relationship between the SSC and DR also emphasizes the less impact of the P (at Aigle) on the SSC (at Porte du Scex). For testing this hypothesis, temperature and precipitation time series at Grächen and Visp stations (located upstream of the URB) are compared with the SSC. In this respect, the correlation coefficients between SSC and temperature in Grächen and Visp stations respectively depict 0.66 and 0.65, while the correlations associated with precipitation in the same stations were respectively 0.26 and 0.20. Hence, the main source of the SSC in the upstream brought by Q is probably the glacier or snowmelt rather than rainfall washouts or dust brought by the precipitation. A similar result is obtained by the sediment rating curve in the Reckingen station which depicts a higher correlation with the SSC at the Porte du Scex station.

Fig. 9
figure 9

The time series of the observed and the best-fit models

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As given in the scatter plot below (Fig. 10), the observed values contain more outliers that could not be simulated by the models. However, other properties of the distribution including median, average, quartile ranges, minima, and maxima are preserved adequately. In this respect, the results of the model fit can be considered acceptable enough for further application.

Fig. 10
figure 10

Scatter plots and box-whiskers of the proposed best-fit models

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Discussion

The suspended sediment concentration studies in rivers could provide valuable information about the source of sediment transport for theoretical and practical uses. The amount of sediment being transported through the Rhone River to Lake Geneva is about 500,000 tons and reaches up to almost five million tons per year (Loizeau and Dominik 2000), and therefore needs to be studied carefully to avoid probable hazards and damages to the infrastructure and the environment of the surrounding regions. The process of SSC transport into the Rhone River is a challenging issue in particular with the increase of extreme precipitation events and heatwaves. Although the most recent glacial cover in the URB is estimated to be around 11% (FOEN, 2021), the investigation of the SSC process in the mountainous URB which hosts some of the largest Alpine glaciers gains more importance. The hydrological regime of URB results in the rainfall-runoff interactions having a non-linear and complex mechanism. For instance, during the winter as the freezing level falls below 900 m above sea level for a long period where the mean altitude of the basin reaches up to 2100 m above sea level, snowfall is a dominant precipitation type. This fact is confirmed through the sediment rating curve and the performed data analysis where SSC has a higher correlation with the base flow rather than the direct runoff. The base flow could be recognized as the dominant contributor to the streamflow. Therefore, the source of SSC seems to be related to glacier or snowmelt, or channel erosion rather than washouts from the overland flow or the dust brought by the immediate precipitation. This hypothesis is then examined by comparing the SSC with temperature and precipitation time series at the Grächen and Visp stations in the upstream areas. Results denote that correlation between the SSC and temperature is more significant than the SSC and precipitation; and therefore, the source of SSC is more likely the ice, glacier, or snowmelt at the upstream of the river. In this respect, as the temperature rises in March–May, the number of rainfall events increases which is followed by the decrease in the snow cover area over the basin. So, the simultaneous increase in discharge driven by the snowmelt in the rivers and channels, and the ice melt from the glaciers as well as the increase in the erodible area under rainfall, results in the peak SSCs in June–July. Therefore, the major glaciers of the URB that are located upstream of the Reckingen and Brig stations are probably the main source of SSC to the river throughout the year. This was also emphasized by Costa et al. (2018b) who showed the role of glaciers on the upstream regions of the URB as the main suspended sediment contributors of the Rhone River. However, the considerable glaciers’ existence in the URB and the amount of SSC which is contributed and being transferred to the main river through the current erosion in the small valleys and channels make the phenomenon very complex which needs to be studied through various time series analysis techniques and methods. Although the amount of precipitation decreases in August–September, the ice and snowmelt from glaciers continue to feed the Rhone River which keeps the SSC values significant until the autumn and enhance the significant relationship of SSC with the base flow (i.e., low flow) throughout the year. This phenomenon generates several extreme values resulting in the outliers in the SSC time series causing a negative impact on the performance of the proposed models.

In order to find the possible relationship between DR and SSC, the sliding interval approach is applied and then the BF and DR time series are analyzed together with Q in five stations. Furthermore, P in one station (most downstream station) is incorporated into the analysis to study its possible impact on SSC. Correlation analysis among the variables illustrates the higher correlation between the SSC at Porte du Scex station and not only streamflow in the same station but also streamflow in other four upstream stations where correlation coefficients between SSC and, Q5 and Q4 are found to be equal. But, the effect of Q1 (the most upstream station) on the SSC at Port du Scex station is higher than that of the remaining streamflows when Q5 (in the same station where SSC is measured) is neglected. This fact can be justified by the effect of erosion in the area in which the considered sediment supply may be probably located upstream of the basin. It also confirms the findings of Schoch et al. (2018) who revealed the major contribution of glaciers on the upstream of the URB. Considering the relationship between the BF and DR with SSC, findings obtained in the SRCs analysis tend to support the correlation analysis results where, interestingly, the SSC is found more correlated to the BF (standing for the low flow through the year instead of Q) in contrast to the DR. It can also be linked to the nature of the Rhone River that is a permanent river and may be fed by alternative sources such as natural springs, groundwater, ice, and snowmelt. Alternatively, in permanent rivers such as the Rhone River, the base flow provides more water than direct runoff. Therefore, the continuous channel erosion, temporary deposition and motion of the sediments through the river stream, and geometry of the river result in a higher correlation coefficient between base flow, as stands for the largest portion of the streamflow, and the suspended sediment yield.

Conducted elasticity analysis shows that streamflow at Porte du Scex station (Q5) is the most important contributor among the selected variable that induces the SSC where changing 1% in Q5 quantity causes a 0.64% change in SSC. But interestingly, the Q1 was found to be the most significant source of SSC downstream. Relying on the correlation and elasticity analysis and ACF and PACF, six different scenarios are developed to construct the best-fit relationships for SSC computation by applying the ARMAX approach. Examination of different scenarios indicates that the scenarios which incorporate all upstream streamflow and precipitation variables generate better results. This agrees with foregone results that SSC in Porte du Scex station is greatly dependent on the upstream events rather than the most downstream station. The results of SRC and ARMAX models also revealed a stronger relationship between the discharge at Reckingen station (located upstream of the URB) and SSC at the Porte du Scex station (downstream of the URB). Eventually, it can be concluded that in total agreement with the previous studies that the source of suspended sedimentation are the upstream stations that gain water from glaciers and ice melt.

Conclusions

The following conclusions are made from the conducted analysis detailed above.

The effect of upstream events surpasses the effect of DR and direct precipitation over the river channel in the evaluation of the SSC at the most downstream station.

The temperature disturbances and the corresponding snow and ice melts upstream are likely to be the main source of SSC downstream, while the effect of channel erosion or bedded sediment is not clear. The morphology of the study area may support the finding where the largest Alpine glaciers exist in the URB.

The applied ARMAX models are capable of interpreting the relationship between SSC, DR, P, and Q while a similar application is suggested for further investigation. The models with more flexible and consistent behavior against extreme values, seasonality, and outliers would present a robust performance.

Overall, our results showed that future studies must incorporate the effect of glaciers and their retreat in suspended sediment concentration due to global warming which challenges the analysis process in the URB.