Modeling probabilistic-based 1D riverbed elevation estimation model due to uncertainties in runoff and sediment-related factors

Recently, sediment transport in the river frequently causes the severe aggradation and erosion of the riverbed elevation to result in flood-derived disaster (e.g., Neuhold et al. 2009; Kim & Julien 2018; Paul & Biswas 2019; Pfeiffer et al. 2019; Cheor & Tachikawa 2020; Chen et al. 2022; Liu et al. 2022, 2023); however, over-developing the alluvial river possibly destroys the bed stability, lowering the safety of the hydraulic structures and storage of the reservoirs (e.g., Chang et al. 1993; Teraguchi et al. 2011; Helal et al. 2013; Liu et al. 2016, 2023; Song et al. 2020); whereas, over-sedimentation in the riverbed might markedly raise the water level to trigger the overtopping risk of the embankment during rainfall-induced flooding. Accordingly, the change in the riverbed due to the sediment transport significantly results in the failure risk of flood-mitigation performance of the hydraulic structures with severe damage to people's lives and property. Therefore, the riverbed's stability analysis should be an essential issue in alluvial river management and improvement plans.

In the past, the changes in the riverbed were mainly quantified and analyzed based on the historical measurements at the various cross-sections along a river (e.g., Schappi et al. 2010; Jagallah et al. 2019; Maddahi & Rahimpour 2023); however, the above measurements are frequently recorded through a time-consuming and expensive process; it is probably challenging to reflect the effect of a heavy rainfall event on the riverbed, thereby inducing an underestimation in flood hazard management due to the lack of details on sediment composition (e.g., Kampf et al. 2015; Riahi-Madvar & Seifi 2018; Liu et al. 2021). Also, they are generally taken at the specific cross-sections in the main river channel, meaning that it is difficult to understand the spatial change in the whole watershed. Thereby, with the enhancement of computation power, the numerical models could simulate the changes in the movable riverbed elevations via the well-known sediment transport approaches based on mass conservation under different hydrological and topographical conditions The physical-based mobile-bed elevation simulation models are generally developed by collaborating the hydrodynamic numerical model with the empirical sediment transport equations between the sediment discharge and river runoff (e.g., Mustafa et al. 2008; Chao et al. 2018). For example, HEC-6 (1993) is a well-known numerical mode for the long-term aggradation and erosion of the river bed by integrating the backwater routing module with the bed-load computation approach for steady flow. Nonetheless, as a result of responding to the effect of the temporal varying trend of the river runoff and downstream boundary (i.e., tide depth), the unsteady flow modules are used in the erosion and deposition simulation of the riverbed, e.g., MIKE11 (1992), SRH-1D (Huang & Greimann 2007), and CCHE1D (Wu et al. 2004a, 2004b). Additionally, artificial intelligence (AI) techniques are widely applied in the relevant hydrological and hydraulic analysis; hence, a number of well-known AI methods are used to develop the data-derived sediment transport models. For example, Riahi-Madvar & Seifi (2018) proposed an intelligent model for predicting the bed load transport (i.e., sediment discharge) by coupling the artificial neural network (ANN) with adaptation neuro-fuzzy inference system (ANFIS) to proceed with the model training and validation, a significant number of the corresponding sediment discharges to the combination of 6 model inputs were achieved via the Monte Carlo simulation approach.

Due to the occurrences of extreme hydrological events and climate change, the variations in the hydrological and topographical data probably cause uncertainties in the calibrated parameters of the sediment transport numerical models and corresponding simulations (e.g., Neuhold et al. 2009; Lin et al. 2014; Ann & Steihschneider 2017; Beckers et al. 2018). For instance, Ann & Steihschneider (2017) proposed a time-varying relationship between the runoff and sediment discharge to estimate the suspend-sediment concentrations model by means of the dynamic regression analysis used in response to climate change. Moreover, a group of hydrodynamic numerical models were presented for the sediment transport simulation; however, their accuracy and reliability are likely to be impacted attributed to the limitation of the model concepts and relevant measurements, including the assumptions and empirical equations of interest (e.g., Chang et al. 1993; Ruark et al. 2011). To cite an instance, Chang et al. (1993) employed the first-order method with Latin hypercubic sampling technique to quantify the sensitivity in the parameters of the sediment transport simulation model (HEC2-SR) to quantify their uncertainties, indicating that the river-channel roughness coefficients make a more significant contribution to the estimation of the water surface and sediment discharge with positive and negative impact, respectively.

Despite numerous investigations proceeding with the uncertainty analysis regarding the movable riverbed elevation simulation in a river, they focused on either the variation in the hydrological and morphological factors (e.g., precipitation and runoff) (Lin et al. 2014; Ann & Steihschneider 2017; Riahi-Madvar & Seifi 2018; Pfeiffer et al. 2019) or the model parameters adopted in the process of simulating the change in the riverbed elevations triggered by the rainfall-induced runoff, including the rainfall–runoff modeling, river routing and sediment transport (Chang et al. 1993; Ruark et al. 2011; Sabatine et al. 2015; Beckers et al. 2018). Furthermore, it is noted that the time series of the river runoff and sediment discharge are used in the parameter calibration and application of the sediment transport model; the resulting riverbed elevation estimates at the various cross-sections during a rainstorm should be regarded as the spatial and temporal variables. Therefore, this study aims to develop a one-dimension (1D) probabilistic-based model for estimating the movable riverbed elevations at various cross-sections along a river under consideration of the uncertainties in the hydrological data and model parameters in time and space, named the PM_MBEE_1D model. The proposed PM_MBEE_1D model is anticipated to quantify the resulting reliability of the estimated riverbed's elevations at the specific cross-sections from the different conditions of the hydrological data, sediment features, and topographical factors. Additionally, the estimated riverbed elevations with a desired reliability by the proposed PM_MBEE_1D model could be considered as a reference to the flood-induced hazard assessment and the watershed treatment and management plan.

To quantify and evaluate the effect of the uncertainties in the hydrological data, sediment features, and topographical factors on the estimation of the riverbed elevation in a river, the proposed PM_MBEE_1D model is developed by coupling the uncertainty analysis with a considerable number of simulated riverbed elevations at various cross-sections; the resulting mobile-bed elevation simulations are from the 1D hydrodynamic numerical model with the sediment transport approach under the different conditions of the runoff and sediment caused by the rainfall events; this implies that hydrological and topographical variables (e.g., precipitation and roughness coefficient) and model parameters (e.g., rainfall–runoff model and sediment transport approach) related to the estimation of the river runoff and sediment could be defined as the runoff-related and sediment-related uncertainty factors, respectively.

Accordingly, while developing the proposed PM_MBEE_1D model, a significant number of the uncertainty factors are simulated using the Multivariate Monte Carlo simulation (MMCS) based on the statistical properties of their uncertainty. After that, a 1D sediment transport numerical model is configured under the study area's hydrological, hydraulic, topographic, and riverbed load conditions. Thus, the resulting simulations of the riverbed elevations at the cross-sections along the river could be achieved by the 1D sediment transpiration simulation numerical mode with a noticeable number of generated uncertainty factors; their corresponding reliabilities could be quantified via the uncertainty and risk analysis. Eventually, to efficiently assess the reliabilities of the estimated riverbed elevations under various uncertainty factors, the logistic regression analysis could derive a functional relationship of the reliabilities of the specific riverbed elevations with the uncertainty factors given.

In summary, the development of the proposed PM_MBEE_1D model could be grouped into six components: (1) identification of the uncertainty factors related to the estimation of the riverbed elevation; (2) configuration of the sediment transport numerical model; (2) generation of uncertainty factors; (4) simulation of the riverbed elevations at the various cross-sections (5) reliability assessment of the mobile-bed elevations at the various cross-sections; and (6) establishment of the exceedance-probability calculation equation. The detailed methods and concepts adopted in the development of the proposed PM_MBEE_1D model are addressed below:

When deriving the proposed PM_MBEE_1D model, the change in the riverbed elevations due to the uncertainty factors considered could be quantified by the 1D hydrodynamic numerical model under the various conditions of the river runoff and corresponding sediment discharge; also, the riverbed elevation could be regarded as the spatial and temporal variates; thus, the uncertainty factors considered in the proposed PM_BMEE_1D model could be identified within the hydrological and hydraulic analysis for estimating the river runoff and sediment in reaction to the change of the riverbed elevation in time and space, respectively.

Regarding the runoff estimation via the 1D river routing model, the variates adopted in the rainfall–runoff and hydraulic models should be treated as the uncertainty factors (Wu et al. 2011a, 2011b); that is, the event-based rainfall characteristics (i.e., event-based duration, rainfall depth, and storm pattern), baseflow, tide depth, external inflow, and roughness coefficients in the river channel (i.e., Manning coefficient) are supposed to be uncertainty factors. Specifically, the lateral and upstream runoff are frequently estimated via the rainfall–runoff modeling with the subbasin-based areal average rainfall; thus, the uncertainty in the lateral and upstream runoff could be attributed to the variations in the parameters of the rainfall–runoff routing and the areal average rainfall (Wu et al. 2011a, 2011b; Douinot et al. 2015).

Furthermore, the estimated sediment discharge in a river is generally obtained via a relationship with the river runoff Q, called the sediment rating curve (i.e., Q–Q_s relationship) (Isik 2013; Tfwala & Wang 2016; Sitorus & Susanto 2019); namely, the sediment discharge could be calculated in the case of the river discharge given. Nonetheless, the river sediment discharge possibly exhibits a considerable variation in time caused by the uncertainties in the river runoff and hydraulic characteristics (e.g., velocity, stress force, and energy slope) (Riahi-Madvar & Seifi 2018). By so doing, the coefficients of the Q_s–Q relationship and parameters adopted in the sediment transport approaches are regarded as the uncertainty factors. Moreover, the initial condition of the riverbed elevation is commonly needed and obtained from historical digital elevation map in advance for the movable riverbed elevation simulation (Guo & Jin 2002); however, the initial riverbed elevation undergoes an apparent change with the variation in the hydrological (e.g., soil moisture) and topographical conditions (e.g., spatial resolution of DEM) in the watershed (Kim & Ivanov 2014; Hancock et al. 2016). Accordingly, the initial riverbed elevation should be considered as an uncertainty factor for the development of the proposed PM_MBEE_1D model.

Overall, the uncertainty factors corresponding to the riverbed elevation simulation in a river could be classified into runoff-related and sediment-related factors. The runoff-related uncertainty factors include the rainfall characteristics, baseflow, tide depth, river-channel roughness coefficient, and parameters of the rainfall–runoff modeling, whereas, the initial riverbed elevation, coefficients of the Q–Q_s relationship, and parameters of the sediment transport simulation model are listed in Table 2.

The computational hydroscience and engineering one-dimensional model (CCHE1D) aims to simulate the change in the riverbed elevations at the compound cross-sections under the condition of non-uniform and unsteady flow via the diffusion wave approximation equation within the ungauged watershed systems (Wu et al. 2004a, 2004b; Shen et al. 2016). Also, the CCHE1D model can quantify the effects of the hydraulic structures along a river-channel system, such as culverts, measuring flumes, drop structures, and bridge crossings, on the river runoff and sediment.

As shown in Table 2, a group of the runoff-related and sediment-related factors could be defined as the uncertainty factors; in this study, to quantify their effect in the estimation of the riverbed elevations, a considerable number of the runoff-rated and sediment-related uncertainty factors of interest are generated and used in the development of the proposed PM_MBEE_1D model. According to the results from the sensitivity analysis with the standard normal regression equation, the rainfall depth and initial riverbed elevation play a more critical role in estimating the riverbed elevations; thus, changing the rainfall depth and initial elevation account for the uncertainties in the rainfall and geographical factors.

The Monte Carlo simulation algorithm with the variate's statistical properties is widely applied in the uncertainty and induced risk quantification in the relevant hydrological analysis (Wu et al. 2017, 2022). Nevertheless, the hydrological, hydraulic, and geographical features generally exist in somewhat inherent correlation in time and space (Simpson & Schlunegger 2003; Wu et al. 2011a, 2011b, 2021a, 2021b). Therefore, this study employs the Monte Carlo simulation with the non-normal correlated multi-variates (Chang et al. 1997) to reproduce the runoff-related and sediment-related uncertainty factors; the relevant model concept and framework are briefly addressed below.

The runoff-related and sediment-related uncertainty factors (see Table 2) could be generated using the MMCS method with the associated statistical properties given in advance. In particular, the simulated rainfall characteristics should be combined as the hyetographs while developing the proposed PM_MBEE_1D model. In short, the simulation of the uncertainty factors subject to the riverbed elevation estimates would be executed through the rainfall characteristics and the remaining uncertainty factors.

To quantify the uncertainty of rainfall in time and space, a significant number of event-based rainstorms at all grids reproduced by the stochastic modeling of the gridded short-term rainstorms (called SM_GSTR model) (Wu et al. 2021a, 2021b) are used in the development of the proposed PM_MBEE_1D model. Within the SM_GSTR model, the event-based rainstorm is characterized in terms of three rainfall characteristics: the rainfall duration, gridded rainfall depths (regarded as the spatial variates), and gridded storm depths comprised of the dimensionless rainfalls at the various dimensionless times (treated as the Spatio-temporal correlated variates); specifically, the gridded storm pattern is supposed to be separated into two components, the areal average of the dimensionless rainfalls (i.e., the storm pattern) and the associated deviations at the various dimensionless times.

In addition to the generation of the rainfall characteristics, the MMCS approach would be implemented in the simulation of the remaining runoff-related and sediment-related uncertainty factors based on the corresponding statistical properties of the observation. Among these uncertainty factors subject to the riverbed elevation estimation, the model parameters, including the rainfall–runoff model, river routing, and sediment transport approach, should first be calibrated with the relevant observed rainfall–runoff-sediment discharge data. Also, concerning the remaining uncertainty factors (i.e., the upstream baseflow and tide depth), the statistical analysis is supposed to be carried out with the observation recorded in the historical events to quantify their uncertainties.

Despite the model inputs and parameters commonly treated as uncertainty factors based on the model concept and configuration, sensitivity analysis should be implemented to find the uncertainty factors contributing more sensitivity to the model output in order to efficiently conduct hydrological reliability analysis (Chang et al. 1993; Wu et al. 2021a, 2021b). Thereby, while developing the proposed PM_MBEE_1D by coupling the Monte Carlo simulation with the uncertainty analysis algorithm, the sensitivities of the runoff-related and sediment-related uncertainty factors to the estimated riverbed elevation (see Table 2) are supposed to be quantified to identify the sensitive uncertainty factors exhibiting significant impact to the estimation of the riverbed elevation.

Two types of sensitivity analysis methods, standardized regression equation and analysis of the variance methods, could be widely applied in identifying the sensitive model inputs to the model outputs (Wu et al. 2011a, 2011b). As mentioned earlier, a group of the runoff-related and sediment-related factors are summarized for the estimation of the riverbed elevations in this study; thus, the standardized regression equation for the multivariate model inputs is adopted to specify the uncertainty factors with more sensitivity to the riverbed elevation estimates.

Therefore, within the proposed PM_MBEE_1D model, a sizeable number of the estimated riverbed elevations (i.e., the model outputs) under various combinations of the simulated runoff-related and sediment-related uncertainty factors as the model inputs are utilized in the derivation of the standard normal regression Equation (5). In the end, the identification of the sensitive uncertainty factors regarding the river-based elevation estimates could be achieved based on the results from the parameter calibration of the standard normal equation.

In this study, P in Equation (9) is regarded as the exceedance probability of the estimated riverbed elevation, i.e., ⁠, directly assigned or estimated by the numerical model; and X_i could be selected factors among the runoff-related and sediment-related factors (see Table 2). Although a group of the runoff-related and sediment-related uncertainty factors are of concern in the model development, the goal of the proposed PM_MBEE_1D is to effectively quantify the estimated riverbed elevation based on the observation data related to the hydrological and topographical features at the location of interest. Therefore, while deriving the exceedance-probability equation, the runoff-related and sediment-related uncertainty factors could be used in Equation (11) subject to the hydrological and topographical features. Therefore, using the resulting exceedance-probability equation, the reliability analysis for the estimated riverbed elevation under various river runoff and sediment conditions could be efficiently carried out without proceeding with the AFOSM approach with a complicated analysis process.

According to the concepts and methods addressed in the previous sections, the proposed 1-D probabilistic-based mobile-bed elevation estimation model (PM_MBEE_1D) comprises the five main components: the identification of uncertainty factors, generation of uncertainty factors, estimation of the riverbed elevation, calculation of the exceedance probability for the riverbed elevation estimate, and derivation of the exceedance-probability calculation equation; the aforementioned exceedance probability accounts for the risk of underestimating the riverbed elevation due to the uncertainties in the runoff-related and sediment-related factors (see Table 2) required in the well-known rainfall–runoff routing (SAC_SMA) and sediment-transportation simulation numerical modeling (CCHE1D). In short, the proposed PB_MBEE_1D model mainly provides the stochastic estimation of the riverbed elevations along the river under consideration of the uncertainties related to the runoff and sediment characteristics; thus, while developing the proposed PM_MBEE_1D model, two equations could be established and utilized for estimating the riverbed elevations and quantifying the corresponding exceedance probability (i.e., underestimated risk). The thorough procedure of model development and application could be expressed as follows:

• Step [1]: Collect the hydrological data (rainfall, river runoff, and stage, baseflow, and tide depth), hydraulic features (river-channel cross-section and roughness coefficient), and local soil properties information (riverbed grain size and sediment discharge) in the study area to configure the rainfall–runoff model (SAC-SMA) and the numerical sediment transport simulation for the movable riverbed (i.e., CCHE1D).

• Step [2]: Calibrate the parameters of the runoff-runoff model (SAC-SMA) and numerical mobile-bed sediment transport model (CCHE1D).

• Step [3]: Extract the corresponding gridded rainfall characteristics from the historical radar-rainfall data and calculate their statistical properties in time and space; reproduce a significant number of gridded rainstorms.

• Step [4]: Generate the runoff-related and sediment-related factors, excluding the gridded rainfall characteristics.

• Step [5]: Simulate the runoff hydrographs at the branches with the generated rainfall hyetographs and SAC-SMA parameters as the boundary conditions for the CCHE1D model.

• Step [6]: Obtain the simulations of the temporally-based riverbed elevations at the specific locations (i.e., cross-section) along the river under consideration of generated lateral and upstream runoff hydrographs as well as the sediment-related factors used in the CCHE1D model.

• Step [7]: Identify the sensitive uncertainty factors via the standard regression analysis with the simulations of riverbed elevations and corresponding generated uncertainty factors.

• Step [8]: Derive the nonlinear functional relationship of the corresponding simulated riverbed elevations to the sensitive uncertainty factors determined at the previous step.

• Step [9]: Calculate the exceedance probabilities of specific magnitudes of the riverbed elevations via the advanced first order and second-moment (AFOSM) approach with the mean and variance of the sensitive runoff-related and sediment-related factors given in advance.

• Step [10]: Establish the exceedance-probability calculation equations for the specific riverbed elevations under consideration of the sensitive uncertainty factor concerned via the logistic regression analysis.

When the proposed PM_MBEE_1D is developed, the reliability analysis could be carried out via the exceedance-probability calculation equations derived within the proposed model under the runoff-related and sediment-related factors given; also, the riverbed elevation could be estimated with a desired likelihood in case of the uncertainty factors assigned. Accordingly, the procedure mentioned above of estimating the riverbed elevation with an exceedance probability can be addressed:

• Step [1]: Ensure the sensitive uncertainty factors related to the hydrological and topographical characteristics subject to the change in the riverbed elevation.

• Step [2]: Calculate the exceedance probability of the specific riverbed elevations at the different cross-sections under various conditions of the hydrological and topographical factors given. Accordingly, the reliabilities of the estimated riverbed elevations could be achieved with varying the uncertainty factors concerned.

• Step [3]: Estimate the riverbed elevations at various locations corresponding to the desired reliability to determine the riverbed scour and siltation.

As mentioned in the model development framework, the sediment-transport simulation numerical model (CCHE1D) is used in the estimation of the riverbed elevations with the generated runoff-related and sediment-related uncertainty factors; thus, the CCHE1D model should be configured in advance based on the hydrological and topological characteristics as well as hydraulic facilities within the study area. According to the introduction on the study area with two main branches, the schematic model structure of the CCHE1D could then be set up; note that the riverbed elevation simulation carried out via the CCHE1D model mainly focuses on the Zhuoshui River and resulting lateral runoff hydrographs from the two branches, Chenyoulan and Chingshui Creek (i.e., boundary conditions). Altogether, while proceeding with the sediment transport simulation via the CCHE1D model to simulate the change in the riverbed elevations with time, the lateral and upstream runoff hydrographs are reproduced via the SAC-SMA model with the subbasin-based average rainfall.

According to the aforementioned model concept and framework, the parameters of the rainfall–runoff model (SAC_SMA) and sediment transport simulation numerical model (CCHE1D) should be calibrated with the relevant observations as addressed in Section 3. The relevant results are expressed as:

According to the introduction on the 19-parameter SAC-SMA model in Section 2.3.2, 11 calibrated parameters could be obtained with the observed rainfall-induced runoff of 10 typhoon events (Figure 2); their statistical properties could then be calculated as listed in Table 4; it can be seen that there is a significant difference in the statistical properties, especially for the high skewness coefficients (from −0.5 to 2.1) and correlation coefficients (from −0.6 to 0.65). Specifically, the above negative coefficient indicates that the two SAC-SMA parameters make a completely different contribution to the runoff estimation; for instance, the parameter UZTWN and LZTWN are related to the surface runoff and infiltration, respectively; accordingly, they should have a negative correlation coefficient. As a result, the SAC-SMA parameters could be treated as non-normal correlated multivariate.

As introduced in Section 2.3.1, the sediment transport equations should be determined in advance when configuring the CCHE1D model to simulate the change in the riverbed elevation at the various cross-sections along a river. In general, the four commonly used sediment transport equations, SEDTRA formula (Garbrecht et al. 1995), Wu-Wang-Jia formula (Wu et al. 2000), modified Ackers-White and modified Engelund-Hansen formula (Proffitt & Sutherland 1983). In this study, subject to the investigations (e.g., Wu et al. 2004a, 2004b; Ding & Langendoen 2016; Norouzi et al. 2022), the SEDTRA model is selected to describe the runoff-induced change in the riverbed elevation, i.e., scoring and silting, within the study area. In addition, the adaptation length (Ls) markedly contributes to estimating the riverbed elevation. Although adopting a shorter adaptation length causes a significant variation in riverbed scour and siltation with a large simulation error in comparison to observations, it could result in a higher correlation between the simulated and measured riverbed elevations. Bed-load adaptation lengths could be given from 4,000 to 20,000 m. Thus, the mean and standard deviation could be assigned as 5,920 and 3,870 m, respectively, in the simulations of the uncertainty factors for the model development. Note that the remaining CCHE1D parameters (e.g., wash-load adaptation coefficient and mix-layer height) mainly contribute to the runoff-induced riverbed elevation estimation (see Table 5).

Given the development procedure of the proposed PM_MBEE_1D model, while carrying out the sediment transport analysis to quantify the change in the riverbed elevation caused by the river runoff, the rainfall-derived runoff should be estimated in advance; hence, the simulations of the runoff from the upstream basin (upstream runoff) and two branches (i.e., lateral runoff) could be obtained by configuring the SAC-SMA model with the generated gridded rainstorms, baseflow and tide depths; thus, the resulting lateral and upstream runoff should be used as the boundary conditions for carrying out the sediment-transportation simulation via the CCHE1D model. Moreover, to take into account the effect of the uncertainty in the initial riverbed elevation, the simulated elevations at the last time step for the specific cross-sections regarding the ith simulation case are treated as the simulated initial elevations for the (i + 1)th case, except for the first simulation case with the measured cross-section elevation in 2012. As mentioned earlier, the estimated riverbed elevation should be defined as the spatiotemporally-based variates; however, this study focuses on generally evaluating the effect of temporally- and spatially-varied uncertainty factors on estimating the riverbed elevations. Therefore, the riverbed elevations at the last time step (named final riverbed elevation) are mainly adopted in the model development and demonstration.

In total, the spatial and temporal uncertainties in the runoff-related and sediment-related factors could significantly impact the changes in the riverbed elevations at the specific cross-sections within the study area. Hence, it is necessary to quantify and evaluate their effect on estimating the riverbed elevations at the particular cross-sections. As a result, this study aims to develop a probabilistic-based model that can provide the section-based elevations of the movable riverbed with a corresponding likelihood under consideration of the spatial and temporal uncertainties.

Despite a group of runoff-related and sediment-related uncertainty factors triggering the dramatic variation in the riverbed elevation, it probably causes a complicated process of quantifying the reliability of the riverbed elevation estimate using the AFOSM approach for developing the proposed PM_MBEE_1D model. This is because a functional relationship of the riverbed elevation with the uncertainty factors (i.e., Equation (10)) should be derived in advance while conducting the reliability analysis via the AFOSM approach with the runoff-related and sediment-related uncertainty factors. Therefore, in this study, using the standard normal equation addressed in Section 2.5, the sensitive uncertainty factors could be identified among the runoff-related and sediment-related uncertainty factors based on their corresponding regression coefficients of the relationship calibrated regarding the riverbed elevation (see Equation (9)) at the specific cross-sections selected from the downstream to the upstream. Also, the sensitivity assessment would be carried out based on four groups classified under their types, i.e., the rainfall characteristics (5 factors), SAC-SMA parameters (11 factors), CCHE1D parameters (4 factors), and remaining factors (3 factors); of which the uncertainty factors considered, the rainfall characteristics are mainly generated to produce the hyetograph in which the rainfall duration (Dur_rain), depth (Depth_rain) and the maximum rainfall intensity (RI_max) which could account for the change in the rainfall in time (Wu et al. 2011a, 2011b, 2015). In the end, the sensitive uncertainty factors would be found out individually from four groups at the six cross-sections selected from the downstream to the upstream, including the first (CS_1st), 72nd (CS_72th), 100th(CS_100th), 124th (CS_120th), 136th (CS_136th) and 146th (CS_146th). Therefore, the rainfall duration, depth, and maximum rainfall intensity are the rainfall factors used in the reliability/uncertainty quantification via the AFOSM method. Similarly, the tide depth hydrograph's maximum value is also treated as the uncertainty factor for the model development.

In the case of the lateral and upstream runoff hydrograph, the average regression coefficients of the SAC-SMA parameters approximate 0.012, but they primarily contribute to the estimation of the river runoff triggered by the rainfall. Accordingly, the SAC-SMA parameters are necessarily used in developing the proposed PM_MBEE_1D model. Moreover, the baseflow is frequently required to estimate the river runoff via the rainfall–runoff routing, especially in drought periods. Since the baseflow in the study area mainly comes from the upstream catchment and two branches (Chenyoulan and Chingshui Creeks) by observing Figure 12, the regression coefficients of the baseflows at the upstream catchment (i.e., upstream baseflow) (BF1) (about 0.021) noticeably exceed those at the two branches (BF2 and BF3) (around 0.008); thus, the baseflow from the upstream catchment is adopted as the uncertainty factor.

In addition to the SAC-SMA and CCHE1D parameters, the tide depth is needed as the downstream conditions in the river routing. According to Figure 9, although the average regression coefficient is nearly 0.007, the associated river runoff is commonly impacted by the tide depth at the cross-sections close to the estuary (e.g., the first and 70th sections). Likewise, the average regression coefficients of the river-channel roughness coefficients are merely about 0.0014; nonetheless, the roughness coefficients should be given to estimate the runoff and induced water level for calculating the corresponding sediment discharge. As a result, the tide depth and river-channel roughness coefficient should both be considered in the model development.

To sum up the results from the sensitivity analysis, the uncertainty factors, including the rainfall depth, maximum rainfall intensity, tide depth, baseflow from the upstream, river-channel roughness coefficient, the SAC-SAM and CCHE1D parameters (i.e., initial riverbed elevation and adaptation length), defined as the sensitive uncertainty factors. By so doing, the above-derived functional relationships for estimating the riverbed elevations at various sections are adopted in the AFOSM method. The resulting reliability of the simulated riverbed elevation (i.e., exceedance probability) could be quantified and utilized in developing the proposed PM_MBEE_1D model.

Based on the model framework (see Section 2.8), while developing the proposed PM_MBEE_1D model via the AFOSM algorithm, 1,000 simulations of the riverbed elevations at various cross-sections are used in advance to establish the corresponding relationship with the sensitive runoff-related and sediment-related uncertainty factors (see Equation (10)); the derived relationship of the riverbed elevation would then be utilized for calculating the probability of the riverbed elevation exceeding a given magnitude named the exceedance probability. Eventually, the resulting exceedance probabilities of the riverbed elevations and corresponding uncertainty factors selected would be applied in the establishment of the exceedance-probability calculation equation; thus, the reliability analysis for the rivebed elevation given at the particular cross-section could be efficiently carried out under the known magnitudes of the uncertainty factors. Overall, the detailed model development process could be addressed in three parts: establishment of the riverbed elevation estimation equation, the reliability quantification of the estimated riverbed elevation, and derivation of the exceedance-probability calculation.

In conclusion, the exceedance probability of the riverbed elevation exhibits a noticeable variation in space; namely, the results from reliability analysis for the estimated riverbed elevations should change with space. Therefore, it is proven that the estimation of the riverbed elevation is significantly impacted due to the uncertainties in the runoff-related and sediment-related factors; that is, the inherent spatial variations of the riverbed elevations should be considered in the sediment-transport simulation for the mobile riverbed.

In this study, the AFSOM approach is employed to reproduce a significant number of the exceedance probabilities of the riverbed elevations at the various cross-sections under different conditions of various rainfall and topographical factors given with the statistical properties of the remaining runoff-related and sediment-related uncertainty factors; then, their coefficients in Equation (13) could be calibrated via the multivariate regression analysis. Table 7 lists the regression coefficients of uncertainty factors of interest and deterministic coefficient (R²) of the exceedance-probability calculation equations at the 1st, 72nd, 100th, 124th, 136th, and 146th cross-sections located from the downstream and upstream within the study area. In referring to Table 7, the corresponding R² value to the various cross-sections, on average, reaches 0.95, meaning that the resulting exceedance-probability calculation equations could reasonably reflect the change in the reliability of the estimated riverbed elevation with the rainfall depth, baseflow, initial riverbed elevation, and river-channel roughness coefficient. In particular, according to the absolute values of the regression coefficients of hydrological and topographical factors, the absolute coefficient of the initial elevation is significantly superior to those of the remaining factors, especially for the first cross-section (about 193), implying that the reliabilities of the estimated riverbed elevations at the downstream cross-sections are mainly affected due to the corresponding initial elevations.

In total, despite the identical uncertainty factors being utilized in the reliability quantification of the estimated riverbed elevation, their variation due to the varying hydrological factor (rainfall depth and baseflow) and topographical features (roughness coefficient and initial elevation) could be accordingly achieved; this is advantageous to assess the corresponding reliability of the estimated riverbed elevation to the change in the hydrological and topographical factors via the derived exceedance-probability equations without using the AFOSM approach to save expensive computation time.

In this study, to validate the applicability of the proposed PM_MBEE_1D model in the reliability quantification of the estimated section-based riverbed elevation via the model application procedure (see Section 3.8.2), the estimated riverbed elevations of three validation events at the specific sections via the CCHE1D model with the corresponding runoff-related and sediment-based factors given (see Table 8). The model validation could then be made by comparing the initial riverbed elevation with the final estimated elevation based on the associated exceedance probability, which is calculated via Equation (12) under consideration of the hydrological and geographical factors known. In addition, by varying rainfall and geographical factors used in Equation (12), the logistic regression equation regarding the estimated riverbed elevation could be applied in the evaluation of the effect of their spatial variations on the reliability of the specific riverbed elevations at various cross-sections. As well as the reliability assessment for the estimated riverbed elevation of interest via Equation (12), the estimation of the riverbed elevation at a specific cross-section with a likelihood could be achieved via Equation (13). Furthermore, through the proposed PM_MBEE_1D model with the runoff-related factor given, the riverbed scour and siltation could be identified by comparing the resulting final estimated riverbed elevation with the initial values; in detail, in case of the final estimated riverbed elevation at a cross-section being more significant than the initial elevation, the riverbed would be stilted; on the contrary, the scouring riverbed is possibly detected based on the lower final estimated elevation than initial one. The detailed results and discussion could be referred to as follows:

To demonstrate the applicability of the exceedance probability calculation equations within the proposed PM_MBEE_1D model on estimating the stochastically-based final riverbed elevations, three validation events with various simulations of the final estimated elevations at different cross-sections are simulated in advance via the CCHE1D model under consideration of the uncertainty factors of interest. In detail, the exceedance probability of the initial riverbed elevation could be achieved via Equation (12) based on the rainfall and geographical factors given (see Table 8).

However, in contrast with the cross-sections CS_1st and CS_100th, the exceedance probabilities of the initial riverbed elevations at the remaining cross-sections (i.e., CS_72nd, CS_124th, and CS_136th) reach from 0.75 to 0.999, indicating that the estimated riverbed elevations are less than the initial ones; thus, it can be verified that the scouring riverbeds could be found at the particular cross-sections. However, with respect to the upstream cross-section CS_100th, the associated exceedance probabilities of the initial riverbed elevations vary from 0.46 to 0.96 for different validation events; this reveals that the riverbed at this cross-section might be both scoured and silted based on the known rainfall and geographical conditions.

In total, the quantified reliabilities of the estimated riverbed elevations by the proposed PM_MBEE_1D model could successfully account for the result from the comparison between the initial riverbed elevations and final estimated ones; namely, when a low exceedance probability is obtained, the final estimated riverbed elevation should be less than the initial one with high likelihood. That is to say, the results from the proposed PM_MBEE_1D model have a good agreement with the variation of the riverbed attributed to the spatial and temporal variation of the rainfall-induced runoff quantified via the CCHE1D model. As a result, the resulting exceedance probability of the final estimated riverbed from the proposed PM_MBEE_1D model could reasonably reflect the change in the riverbed due to uncertainties in the runoff-related and sediment-related factors on the elevation of the movable riverbed.

In addition to the rainfall depth, the geographical factor (i.e., initial riverbed elevation) is verified to quantify the effect of its uncertainty on the estimation of the riverbed elevation, as shown in Figure 15(2), in which the final riverbed elevation is estimated by adding an increment of 0.5 m into the elevation (273 m) under the condition of the rainfall depth of 500 mm. Given Figure 15, at the cross-section CS_136th, the exceedance probability of the final riverbed elevation of 274 m has a significant decrease with the initial riverbed elevation. In detail, the corresponding exceedance probability to the initial riverbed elevation over 274 m (i.e., silting probability) significantly reaches 0.999 under an initial elevation of 274 m given; in contrast, in the case of the initial elevation being less than 274 m, the exceedance probability (i.e., scouring probability) sharply decline to 0.001. This implies that at the cross-section CS_136th, siltation takes place with a high likelihood attributed to its mild slope at the downstream.

In summary, the riverbed elevations at the downstream cross-sections appear to have a stable varying trend in space, scouring (CS_1st and CS_100th), and silting (CS_72nd), whereas about the cross-sections at the upstream (CS_124th, CS_136th and CS_146th), scouring riverbed is frequently detected, but the riverbed might be scoured with a low chance. As a result, the probabilistic-based riverbed elevations given within the study area are significantly consistent with the results from the uncertainty analysis via the AFOSM approach as a reference to riverbed improvement.

This study aims to develop a probabilistic movable riverbed elevation estimation (PM_MBEE_1D) model to quantify the effect of uncertainties in the runoff-related and sediment-related factors on the estimation of the riverbed elevations at various cross-sections in terms of the corresponding exceedance probability. The 1,000 simulations of the riverbed elevations and corresponding uncertainty factors are achieved via the MMCS with the historical topographical, hydrological, and sediment data in the Jhuosdhuei River watershed for the model development and validation. As a result, the exceedance probability calculated via the proposed PM_MBEE_1D model could reasonably respond to the change in the riverbed elevation (scouring and siltation) in comparison to the given initial elevation under consideration of the hydrological and geographical factors given with high accuracy. Also, under consideration of the uncertainties in the riverbed elevations at the cross-sections with a desired reliability (named probabilistically-based riverbed elevation) could be effectively achieved by means of the proposed PM_MBEE_1D model with given runoff-related and sediment-related factors.

This published paper was funded by the National United University Projects (grant 112-NUUPRJ-04).

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.