Uncertainty in input can significantly impair parameter estimation in water quality modeling, necessitating the accurate quantification of input errors. However, decomposing the input error from the model residual error is still challenging. This study develops a new algorithm, referred to as the Bayesian Error Analysis with Reordering (BEAR), to address this problem. The basic approach requires sampling errors from a pre-estimated error distribution and then reordering them with their inferred ranks via the secant method. This approach is demonstrated in the case of total suspended solids (TSSs) simulation via a conceptual water quality model. Based on case studies using synthetic data, the BEAR method successfully improves the input error identification and parameter estimation by introducing the error rank estimation and the error position reordering. The results of a real case study demonstrate that, even with the presence of model structural error and output data error, the BEAR method can approximate the true input and bring a better model fit through an effective input modification. However, its effectiveness depends on the accuracy and selection of the input error model. The application of the BEAR method in TSS simulation can be extended to other water quality models.

For robust water management, uncertainty analysis is of growing importance in water quality modeling (Refsgaard et al., 2007). It can provide knowledge of error propagation and the magnitude of uncertainty impacts in model simulations to guide improved predictive performance (Radwan et al., 2004). However, the implementation of an uncertainty analysis in water quality models (WQMs) is still challenging due to complex interactions among sources of multiple errors, generally caused by a simplified model structure (structural uncertainty), imperfect observed data (input uncertainty and observation uncertainty in calibration data), and limited parameter identifiability (parametric uncertainty; Refsgaard et al., 2007).

Among them, input uncertainty is expected to be particularly significant in a WQM, which is interpreted here as the observation uncertainty of any input data. Observation uncertainty is different from other sources of uncertainty in modeling since these uncertainties arise independently of the WQM itself; thus, their properties (e.g., probability distribution family and distribution parameters) can, at least in principle, be estimated prior to the model calibration and simulation by analysis of the data acquisition instruments and procedures (Mcmillan et al., 2012). Rode and Suhr (2007) and Harmel et al. (2006) reviewed the uncertainty associated with selected water quality variables based on the empirical quality of observations. The general methodology developed in their studies can be extended to the analysis of other water quality variables. Besides the error coming from the measurement process, the error from surrogated data is another major source of input uncertainty (Mcmillan et al., 2012). Measurements of water quality variables often lack desirable temporal and spatial resolutions; thus, the use of surrogate or proxy data is necessary for improved inference of water quality parameters (Evans et al., 1997; Stubblefield et al., 2007). The probability distribution of the surrogate error is easy to estimate from the residuals between the measurements and proxy values. In this process, the measurement errors are ignored, given that the errors introduced from the surrogate process are commonly much larger than the measurement errors (Mcmillan et al., 2012). These estimated error distributions are prior knowledge of input uncertainty before any model calibration and can serve as the a priori uncertainty estimation in the modeling process.

Input uncertainty can lead to bias in parameter estimation in water quality modeling (Chaudhary and Hantush, 2017; Kleidorfer et al., 2009; Willems, 2008). Improved model calibration requires isolating the input uncertainty from the total uncertainty. However, the precise quantification of time-varying input errors is still challenging when other types of uncertainties are propagated through to the model results. In hydrological modeling, several approaches have been developed to characterize time-varying input errors, and these may hold promise for application in WQMs. The Bayesian total error analysis (BATEA) method provides a framework that has been widely used (Kavetski et al., 2006). Time-varying input errors are defined as multipliers on the input time series and inferred along with the model parameters in a Bayesian calibration scheme. This leads to a high-dimensionality problem which cannot be avoided (Renard et al., 2009) and restricts the application of this approach to the assumption of event-based multipliers (the same multiplier applied to all time steps of one storm event). In the Integrated Bayesian Uncertainty Estimator (IBUNE; Ajami et al., 2007) approach, multipliers are not jointly inferred with the model parameters but sampled from the assumed distribution and then filtered by the constraints of simulation fitting. This approach reduces the dimensionality significantly and can be applied in the assumption of data-based multiplier (one multiplier for one input data point; Ajami et al., 2007). However, this approach is less effective because the probability of co-occurrence of all optimal error values is very low, resulting in an underestimation of the multiplier variance and misidentification of the uncertainty sources (Renard et al., 2009).

To complete this goal, this study attempts to add a reordering strategy into the IBUNE framework and names this developed algorithm as the Bayesian Error Analysis with Reordering (BEAR). The derivation and details of the BEAR algorithm in quantifying input errors are described in Sect. 2. Section 3 introduces the build-up/wash-off model (BwMod) to illustrate this approach. Its model input, streamflow, often suffers from observational errors from a rating curve. By comparing the results with other calibration frameworks, the ability of the BEAR method is explored in two synthetic case studies and one real case study. In this way, the new algorithm is tested in a controlled situation (with the knowledge of the true error and data value) and in a realistic situation (with the interference of multiple error sources), respectively. Section 3 describes the setup of these case studies, and Sect. 4 demonstrates their results. Section 5 evaluates the BEAR method and its implementation. Finally, Sect. 6 outlines the main conclusions and recommendations for this work.

A WQM in the ideal situation without any error can be described as follows:

In reality, the model input

It should be noted that this study focuses on identifying the input errors in the process of parameter estimation, and the derivation of the BEAR method is based on the assumption that the model only suffers from input errors and parameter errors, but other sources of error (i.e., model structural error and output observational error) are inevitable in the WQM and can impair the effectiveness of input error identification and parameter estimation. Considering this realistic situation, the ability of the BEAR method will be tested later in one synthetic case study and one real case study, where the interference of other sources of error has been considered.

To counter the influence of input errors in a traditional calibration, a
common approach is to subtract estimated errors

The above approach does not improve the input error model itself but improves the WQM specification to have parameters closer to what would be achieved under no-error conditions. Then the model can be more effectively used for scenario analysis (where we may know the hydrologic regime of a catchment in a hypothetical future), for forecasting under the assumption of perfect inputs (where the driving hydrologic forecast is independently obtained via a numerical weather prediction and a hydrologic model), or for the regionalization of the WQM (where the model is transferred to a catchment without data). In all of these cases, an ideal model should have unbiased parameter estimates. This is our goal in identifying the optimal input errors and not to use the model for predictions with input data suffering from the same errors.

Unlike directly estimating the input error value via existing methods, this
study attempts to transform the input error quantification into the rank
domain. Here, the rank is defined as the order of any individual value
relative to the other sampled values and determines the relative magnitude
of each error in all data errors. For example, in the first iteration in
Table A1, the error at the 15th time step,

In the rank domain, the challenge becomes finding a way to effectively adjust
the input error rank to push the residual error equal to zero. The secant
method can be applied to address this problem. It is an iterative process to
produce better approximations to the roots of a real-valued equation (Ralston and Jennrich, 1978). Here, the root is the optimal rank
of each input error, and the equation is the corresponding model residual
equal to zero. The secant method (Ralston and Jennrich, 1978) can
be repeated as follows:

After calculating Eq. (4), it is possible that the rank

From the above, estimating the rank of input errors via the secant method can be described as the following two equations.

We update the rank of each input error

Thus, the procedure of input error quantification has been developed via the following key steps:

Sample the errors (the number is equal to the number of input data) from the assumed error distribution to maintain the overall statistical characteristics of the input errors and allocate them randomly to all the time steps.

Update the input error ranks to force the model residual close to zero via the secant method (Eqs. 5 and 6).

Reorder these sampled errors according to the updated error ranks.

Repeat steps (2) and (3) for a few iterations until a defined target is achieved. This new algorithm is referred to as the Bayesian Error Analysis with Reordering (BEAR). An example to illustrate how the BEAR method works is presented in Appendix A.

When the BEAR method is applied in a realistic situation, the model
structural error and output observational error will be lumped into residual
error, which is often assumed to follow a Gaussian distribution with mean 0
and variance

Unlike the formal Bayesian inference, the BEAR method does not update the
posterior distribution of the input errors. Considering that

According to Eq. (3), the residual error is related to the deterministic
variables

The core strategy of the BEAR method is to identify the input errors by estimating their ranks, which can be easily integrated into formal Bayesian inference schemes (for example, Markov chain Monte Carlo – MCMC; Marshall et al., 2004; sequential Monte Carlo – SMC; Jeremiah et al., 2011; Del Moral et al., 2006) and other calibration schemes (for example, the generalized likelihood uncertainty estimation – GLUE; Beven and Binley, 1992). Based on the traditional calibration approach, the BEAR method works by replacing the observed input with a modified input that is obtained through the estimated input error rank via the secant method. This study applies the SMC sampler for updating the model parameter. In the SMC approach, the model parameter is first sampled from a prior distribution and then propagated through a sequence of intermediate populations by repeatedly implementing the reweighting, mutation, and resampling processes until the desired posterior distribution is achieved (Del Moral et al., 2006). The details of the SMC algorithm can be found in the study of Jeremiah et al. (2011).

Figure 1 demonstrates the integration of the BEAR method into the SMC sampler. In the SMC scheme,

Flowchart of the algorithm to quantify the input errors via the Bayesian Error Analysis with Reordering (BEAR) method in the SMC calibration scheme. The gray charts demonstrate the BEAR method, while the white charts demonstrate the SMC algorithm. For details on the BEAR method, refer to Appendix A. For details on the SMC algorithm, refer to the study of Jeremiah et al. (2011), including the mutation step, the reweighting step, and calculating the acceptance probability. The term “rand(0, 1)” means a number randomly sampled from 0 to 1.

In this study, three methods, including the traditional method, IBUNE method, and BEAR method, are compared to evaluate the ability of the BEAR method to estimate the model parameters and quantify input errors. The traditional method regards the observed input as being error free, without identifying input errors (i.e., Eq. 2), while the other two methods employ a latent variable to counteract the impact of input error and derive a modified input (i.e., Eq. 3). In the IBUNE method, potential input errors are randomly sampled from the assumed error distribution and filtered by the maximization of the likelihood function (Ajami et al., 2007). Although the comprehensive IBUNE framework additionally deals with model structural uncertainty via Bayesian model averaging (BMA), this study only compares the capacity of its input error identification. The BEAR method adds a reordering process into the IBUNE method to improve the accuracy of the input error quantification.

This study tests the BEAR algorithm in the context of the build-up/wash-off model (BwMod), which is a group of models used to simulate two processes in sediment dynamics, including the build-up of sediments during dry periods and the wash-off process during wet periods. The two formulations were developed in a small-scale experiment (Sartor et al., 1974), while, in applications at the catchment scale, the conceptualized parameters largely abandon their physical meanings, and the formulations can be considered a black box (Bonhomme and Petrucci, 2017). This study chooses Eq. (9) to describe the build-up process and Eq. (10) to express the wash-off of sediments, representing the nonlinear relationship between the wash-off load (output) and the runoff rate (input). These two equations were applied in the research of Sikorska et al. (2015) and, in this study, are integrated with the BEAR method. This study will test the BEAR algorithm in a case of simulating the daily sediment dynamics of one catchment; thus, we use daily time steps and consider the catchment to be a single, homogeneous spatial unit. This version of BwMod has four parameters (Table 1). The model input is streamflow, which typically comes from the observation of a rating curve. As discussed in the introduction, the error distribution can be estimated prior to the model calibration via a rating curve analysis. The output of the BwMod is the concentration of total suspended solids (TSSs), whose transport can be efficiently simulated by the conceptualization of the build-up/wash-off process (Bonhomme and Petrucci, 2017; Sikorska et al., 2015). Although BwMod is relatively simple compared with process-based WQMs, its nonlinearity and the use of surrogates for the input data can make it a typical WQM scenario to test the BEAR algorithm.

Descriptions of BwMod parameters.

The overall BwMod equations are as follows:

The output TSS concentration

To test the capability of the secant method in identifying the input error
ranks in the process of the model parameter estimation, the BEAR method is
first implemented in a controlled situation with synthetic data. The true
input

Summary of the calibration scenarios in case studies.

Case study 1 is an ideal situation in that the model calibration only suffers from input and parameter errors. However, in real-life cases, other sources of errors (i.e., model structural error and output data error) will impact this effectiveness. To explore the ability of the BEAR method with the interference of other sources of errors, the output observational errors with the increasing standard deviations are considered to build the synthetic data based on scenarios 3 and 4 in case study 1 (details in Table 2).

To sum up, two synthetic case studies have been analyzed. Case study 1 generates synthetic data only suffering from input errors to evaluate the effectiveness of the BEAR method in isolating the input error and the model parameter error. Case study 2 additionally considers output observation errors via synthetic data generation to evaluate the impacts of other sources of error on the BEAR method. Each scenario in the synthetic case studies is calibrated via the traditional method, the IBUNE method, and the BEAR method, respectively. Their algorithms are described in Sect. 2.5. Considering the unknown initial sediment loads in real applications, the calibration sets 90 d as a warm-up period to remove the influence of antecedent conditions.

To explore the ability of the BEAR method in real-life applications, a real
case of one catchment located in southeastern Wisconsin, USA, is demonstrated. Table 3 is a description of the test catchment and data (Baldwin et al., 2013). The daily TSS concentration and streamflow data are collected from the United States Geological Survey (USGS) database on National Real-Time Water Quality (

Characteristics of the study catchments and calibration data.

To evaluate the ability of different calibration methods (i.e., traditional method, the IBUNE method, and the BEAR method) in identifying the input error and model parameter, the following statistical characteristics are selected to compare the results of case study 1, which only suffers from input and parameter errors. The SD of the estimated input errors represents the accuracy of the input error distribution (0.5 is the reference value). The correlation between the estimated input error and the true input error evaluates the capability of the method to catch the temporal dynamics of input error. The Nash–Sutcliffe efficiency (NSE) of the modified input vs. true input measures the precision of the input data after removing the estimated input errors. In the calibration part, the simulated output corresponds to the modified input and estimated model parameters, and its NSE, compared to the true output, measures the goodness of fit. In the validation part, the simulated output corresponds to the true input and estimated model parameters, and its NSE, compared to the true output, can assess the accuracy of the model parameter estimation. These statistical characteristics are calculated as the weighted-average values, considering the weights of each estimation in the posterior distribution and compared in Fig. 2. Figures B1–B4 in Appendix B demonstrates the temporal dynamics of the input estimations and model simulations of synthetic case 1. In Fig. B1, the reliability is the ratio of observations caught by the confidence interval of 2.5 %–97.5 %, and the average width of this interval band is referred to as sharpness (Yadav et al., 2007; Smith et al., 2010).

Comparison of the statistical characteristics of four calibration scenarios in synthetic case 1 (including add-fixed, add-inferred, mul-fixed, and mul-inferred; notations are given in Table 2) via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; their algorithms are explained in Sect. 2.4).

Comparison of the Nash–Sutcliffe efficiency (NSE) of the modified input vs. the true input under the interference of the output observational errors, with the increasing standard deviations in two calibration scenarios in the synthetic case 2 (including mul-fixed and mul-inferred; notations are given in Table 2) via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; their algorithms are explained in Sect. 2.4).

Evaluating the model simulation, the BEAR method always produces the best
output fit in all scenarios, as supported by the highest green bars in Fig. 2d. Although its correlations with the true error series are much higher than the IBUNE method (red bars) in all scenarios (in Fig. 2b), the BEAR method cannot ensure a better input estimation (in Fig. 2c), and its ability depends on the prior information of the input error parameter. When the error parameters are fixed at the reference values (in the scenarios add-fixed and mul-fixed), the BEAR method always outperforms the other two methods in the input modification and model parameter estimation, as its NSE is the highest (green bars in Fig. 2c and e). Without the reordering strategy, the IBUNE method even gives worse input modification, model simulation, and parameter estimation than the traditional method, as demonstrated by the lower red bars than blue bars in Fig. 2c, d and e. When the error parameters are inferred (in the scenarios of add-inferred and mul-inferred), then the IBUNE method can improve the
input data and the model parameter estimation compared with the traditional
method (in Fig. 2c and e), although the estimations of

The Nash–Sutcliffe efficiency (NSE) is selected to measure the difference between the modified input in case study 2 and the true input. Figure 3 demonstrates that, in the mul-fixed scenario, where the prior information of the standard deviation of input errors is accurate, the BEAR method always brings a better input modification than other methods, although its ability is impaired by the impact of the output observational errors as the NSE reduces with the increasing SD of the output observational error. The IBUNE method leads to an even worse modified input than the input data without modification in the traditional method. In the mul-inferred scenario, where the standard deviation of input errors cannot be pre-estimated accurately and given in a wide range, the BEAR method brings worse input data, while the IBUNE method can modify the input data.

Figure 4 compares the SDs of estimated input errors, the variances of model residual errors, and reliability and sharpness of model simulations among the four calibration scenarios and three calibration methods in the real case study. Figure 4b demonstrates that the BEAR method always produces a better fit to the output data than the IBUNE method, which is consistent with the synthetic case shown in Fig. 2d. In Fig. 4c, except for the

Comparison of the statistical characteristics of four calibration
scenarios in the real case (including

The novelty of the BEAR method lies in transforming a direct error value
estimation to an error rank estimation. In a continuous sequence of data,
the potential error values have an infinite number of combinations, while
the error rank has limited combinations that are dependent on the data length. For example, in Table A1, the estimated error at the first time step could be any value. Even under a constraint of the range from the minimized to the maximized sampled errors (i.e., [

However, the rank estimation will suffer from the sampling bias problem. For
the same error distribution and the same cumulative probability
(corresponding to the same error rank), the errors in different samplings
could be largely different, especially for a small sample size (depending on
the data length) or a large

In addition, the rank estimation can make better use of the knowledge of the input error distribution. In a direct value estimation, it is difficult to keep the overall error distribution constant when the errors are updated in the calibration. The estimated errors are more likely to compensate for other sources of errors to maximize the likelihood function and, subsequently, be overfitted. By contrast, in a rank estimation, the errors at all of the time steps are sampled from the pre-estimated error distribution first and are then reordered. Whatever the error rank estimates are, they always follow the pre-estimated error distribution, and the compensation effect will be limited. In the IBUNE framework (Ajami et al., 2007), the errors are also sampled from the error distribution but not reordered. In the BEAR method, adjusting the sampled errors according to the inferred error rank reduces the randomness of the error allocation in the IBUNE framework (Ajami et al., 2007), which significantly improves the accuracy of the error estimation (as demonstrated by much higher correlations than the IBUNE method in Fig. 2b).

Unlike the formal Bayesian inference, the rank estimation does not update the posterior distribution of the input errors but optimizes their time-varying values through the relationship between the input error rank and the corresponding model residual error. The rank estimation is implemented after the model parameters have been updated, and the model residual error depends on the input error estimation. Thus, the reordering strategy identifies the optimal input error rank conditional to the model parameters, effectively considering the interaction between the input error and the parameter error. This is akin to calibrating the input errors along with the model parameters in the BATEA framework (Kavetski et al., 2006).

Figure 5 demonstrates the mechanics of input error reordering in the BEAR method and input error filtering in the IBUNE method to understand their effects on the input error realizations and model parameter estimation. The first sequence represents the situation where the raw input errors are randomly sampled from the pre-estimated error distribution; therefore, their marginal means and standard deviations are the same as the parameters of overall error distribution (cyan lines in column (c) of Fig. 5). In the later sequence, these errors are optimized via different methods. In the IBUNE method, these sampled input error series are selected by the maximized likelihood function; the interval of input errors becomes a little converged (Fig. 5b1), and their marginal standard deviations reduce slightly (Fig. 5c1). However, in the BEAR method, these input errors are reordered according to the inferred ranks via the secant method, and the reordered errors gradually converge to the true values (represented by the blue interval are near the red line in Fig. 5b2). Therefore, their marginal means are similar to the true values, and their
marginal standard deviations reduce to zero (Fig. 5c2). In the BEAR method,
the promotion of the input error identification in the sequential updating
will improve the model parameter estimation, represented by the posterior
distribution of model parameter

Comparison of the results for scenario 1 in synthetic case 1 and
the ensemble particle size of

The data length can affect the efficacy of the BEAR method but impose little effect on the IBUNE method. The IBUNE method takes advantage of the stochastic errors and keeps the marginal error distribution almost constant. The input error realization at each time step seems independent, only being filtered by the overall likelihood function. Therefore, the number of sampled errors does not matter in the IBUNE method. However, in the BEAR method, the input errors at all the time steps are not sampled independently, and they are from one sample set. Therefore, before or after reordering, all errors will keep the same statistical features of the input error distribution, and only their marginal distribution changes due to the convergence to the unknown true values. Figure 5b2 demonstrates that, when the data length (the same as the error number) is small, the input error estimation might be biased from the true values. This likely arises from the abovementioned sampling bias or the impacts of the model parameter error because the sampling bias reduces with the larger number of error samples, and the impacts of parameter error are more likely to be offset when the data length is longer.

The IBUNE method takes advantage of stochastic error samples to modify the
input observations (Ajami et al., 2007). In the real case study,

However, the findings in the BEAR method are quite different. Accurate prior
information about the input error model is important in the BEAR method.
Figure 3 demonstrates that the fixed scenarios calibrated via the BEAR method always produce a higher NSE of the modified input than the inferred scenarios. This is likely because the prior information can constrain the input error distribution and reduce the impacts of other sources of errors. The availability of prior information of the input error relies on studies about benchmarking the observational errors of water quality and hydrologic data, and the selection of a proper input error model is important. Comparing the results in Fig. 2, when the input error model is an additive formulation, the BEAR method consistently brings the best performance regardless of the prior information of the error

The observation uncertainty in input data is independent of the model process, and the input error model can be estimated prior to the model calibration and simulation by analyzing the data itself. Taking advantage of the prior information of an input error model, a new method, the Bayesian Error Analysis with Reordering (BEAR), is proposed to approach the time-varying input errors in WQM inference. It contains the following two main processes: sampling the errors from the assumed input error distribution and reordering them with the inferred ranks via the secant method. This approach is demonstrated in the case of TSS simulation via a conceptual water quality model, BwMod. Through the investigation of synthetic data and real data, the main findings are as follows:

The estimation of the BEAR method focuses on the error rank rather than the error value in the existing methods, which can take advantage of the constraints of the known overall error distribution and then improve the precision of the input error estimation by optimizing the error allocation in a time series.

The introduction of the secant method addresses the nonlinearity in the WQM transformation and can effectively update the error rank of each input data, minimizing its corresponding model residual.

The ability of the BEAR method in decomposing the input error from model residual error is limited by the accuracy and selection of the input error model and is impacted by model structural uncertainty and output observation uncertainty.

Therefore, the study identifies several areas which need further analysis. First, the availability of prior knowledge of the input error model is important. When this information is not reliable or cannot be estimated, this causes a significant issue with the selection of a suitable error distribution. Thus, a general measure should be found to judge whether an error model is appropriate, especially in real cases where the true information is limited. Second, this study focuses on identifying the input errors in model calibration, and the derivation of the BEAR method is based on the assumption that the input error is dominant in the residual error. If the reordering strategy is developed within a more comprehensive framework to quantify multiple sources of error, then this assumption will be relaxed, the interactions amongst these error sources might be well identified, and the quantification of individual errors might be improved. This study provides a starting point for developing the rank estimation via the secant method to identify input error. Further study is necessary to modify the algorithm and improve confidence in extended case studies or model scenarios.

The BEAR method for identifying the input errors is implemented after generating the model parameters and contains two main parts, i.e., sampling the errors from an assumed error distribution and reordering them with the inferred ranks via the secant method. An example is illustrated in Table A1, and the explanation about the specific steps is presented in the following.

In the first iteration (

Repeat step (1) in the second iteration (

At the first time step in the third iteration (

Sort all the pre-ranks to obtain the integral error rank (row 8).

According to the updated error ranks (row 8), the sampled errors in the second iteration (row 4) are reordered. The example for the first time step is demonstrated in dotted boxes in Table A.1. The error rank at the first time step is updated as 6, and the rank 6 corresponds to the error value

The reordered input error will lead to new input data, a new model simulation, and a new model residual. The residual result and its MSE statistic are shown in rows 10 and 11, respectively.

Last, check the convergence. If the objective function or likelihood function meets the convergence criterion, then stop, and the input error estimation is accepted. Otherwise,

An example illustrating the BEAR method.

Demonstration of the input error estimated in Table A1.

Comparison of the time series of synthetic data and uncertainty bands estimated via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; algorithms are explained in Sect. 2.4) for a select period of the add-fixed scenario in synthetic case 1 (notations are given in Table 2).

Comparison of the time series of synthetic data and uncertainty bands estimated via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; algorithms are explained in Sect. 2.4) for a select period of the add-inferred scenario in synthetic case 1 (notations are given in Table 2).

Comparison of the time series of synthetic data and uncertainty bands estimated via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; algorithms are explained in Sect. 2.4) for a select period of the mul-fixed scenario in synthetic case 1 (notations are given in Table 2).

Comparison of the time series of synthetic data and uncertainty bands estimated via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; algorithms are explained in Sect. 2.4) for a select period of the mul-inferred scenario in synthetic case 1 (notations are given in Table 2).

Comparison of the time series of synthetic data and uncertainty bands estimated via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; algorithms are explained in Sect. 2.4) for a select period of the

Comparison of the time series of synthetic data and uncertainty bands estimated via three calibration methods (including the traditional method, the IBUNE method, and the BEAR method; algorithms are explained in Sect. 2.4) for a select period of the

The daily streamflow and TSS concentration data for real case catchment (USGS ID 04087030) can be accessed from the National Real-Time Water Quality website of the USGS at

LM and AS designed the research. XW developed the research code, analyzed the results, and prepared the paper, with contributions from all co-authors.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Frontiers in the application of Bayesian approaches in water quality modelling”. It is a result of the EGU General Assembly 2020, 3–8 May 2020.

The authors are thankful to the editor and reviewers, for their constructive comments which helped us substantially improve the quality of the paper. This work has been supported by the Australian Research Council (grant no. FT120100269) and the Australian Research Council (ARC) Discovery Award (grant no. DP170103959) to Lucy Marshall, the National Natural Science Foundation of China (grant no. 52109015) and the Jiangsu Postdoctoral Research Funding (grant no. 2021K046A) to Xia Wu, and the National Natural Science Foundation of China (grant nos. 51979004 and 41830752).

This research has been supported by the Jiangsu Postdoctoral Research Funding (grant no. 2021K046A), the National Natural Science Foundation of China (grant nos. 52109015, 51979004 and 41830752) and the Australian Research Council (grant nos. DP170103959 and FT120100269).

This paper was edited by Lorenz Ammann and reviewed by three anonymous referees.