Existence of the frequentistic estimate for power-law regression with a location parameter, with applications for making discharge rating curves

Practical application of the power-law regression model with an unknown location parameter can be plagued by non-finite least squares parameter estimates. This presents a serious problem in hydrology, since stream flow data is mainly obtained using an estimated stage–discharge power-law rating curve. This study provides a set of sufficient requirements for the data to ensure the existence of finite least squares parameter estimates for a power-law regression with an unknown location parameter. It is shown that in practice, these requirements act as necessary for having a finite least squares solution, in most cases. Furthermore, it is proved that there is a finite probability for the model to produce data having non-finite least squares parameter estimates. The implications of this result are discussed in the context of asymptotic predictions, inference and experimental design. A Bayesian approach to the actual regression problem is recommended.     

Parameter and rating curve uncertainty propagation analysis of the SWAT model for two small Mediterranean catchments

Abstract

The SWAT model was tested to simulate the streamflow of two small Mediterranean catchments (the Vène and the Pallas) in southern France. Model calibration and prediction uncertainty were assessed simultaneously by using three different techniques (SUFI-2, GLUE and ParaSol). Initially, a sensitivity analysis was conducted using the LH-OAT method. Subsequent sensitive parameter calibration and SWAT prediction uncertainty were analysed by considering, firstly, deterministic discharge data (assuming no uncertainty in discharge data) and secondly, uncertainty in discharge data through the development of a methodology that accounts explicitly for error in the rating curve (the stage?discharge relationship). To efficiently compare the different uncertainty methods and the effect of the uncertainty of the rating curve on model prediction uncertainty, common criteria were set for the likelihood function, the threshold value and the number of simulations. The results show that model prediction uncertainty is not only case-study specific, but also depends on the selected uncertainty analysis technique. It was also found that the 95% model prediction uncertainty interval is wider and more successful at encompassing the observations when uncertainty in the discharge data is considered explicitly. The latter source of uncertainty adds additional uncertainty to the total model prediction uncertainty.

Editor D. Koutsoyiannis; Associate editor D. Gerten

Citation Sellami, H., La Jeunesse, I., Benabdallah, S., and Vanclooster, M., 2013. Parameter and rating curve uncertainty propagation analysis of the SWAT model for two small Mediterranean watersheds. Hydrological Sciences Journal, 58 (8), 1635?1657.     

Rating curve estimation under epistemic uncertainty

River discharge values, estimated using a rating curve, are subject to both random and epistemic errors. We present a new likelihood function, the ‘Voting Point’ likelihood that accounts for both error types and enables generation of multiple possible multisegment power‐law rating curve samples that aim to represent the total uncertainty. The rating curve samples can be used for subsequent discharge analysis that needs total uncertainty estimation, e.g. regionalisation studies or calculation of hydrological signatures. We demonstrate the method using four catchments with diverse rating curve error characteristics, where epistemic uncertainty sources include weed growth, scour and redeposition of the bed gravels in a braided river, and unconfined high flows. The results show that typically, the posterior rating curve distributions include all of the gauging points and succeed in representing the spread of discharge values caused by epistemic rating errors. We aim to provide a useful method for hydrology practitioners to assess rating curve, and hence discharge, uncertainty that is easily applicable to a wide range of catchments and does not require prior specification of the particular types and causes of epistemic error at the gauged location. Copyright © 2014 John Wiley & Sons, Ltd.     

PUBLICATIONS OF THE IAHS

Abstract

Gauging stations where the stage—discharge relationship is affected by hysteresis due to unsteady flow represent a challenge in hydrometry. In such situations, the standard hydrometric practice of fitting a single-valued rating curve to the available stage—discharge measurements is inappropriate. As a solution to this problem, this study provides a method based on the Jones formula and nonlinear regression, which requires no further data beyond the available stage—discharge measurements, given that either the stages before and after each measurement are known along with the duration of each measurement, or a stage hydrograph is available. The regression model based on the Jones formula rating curve is developed by applying the monoclinal rising wave approximation and the generalized friction law for uniform flow, along with simplifying assumptions about the hydraulic and geometric properties of the river channel in conjunction with the gauging station. Methods for obtaining the nonlinear least-squares rating-curve estimates, while factoring in approximated uncertainty, are discussed. The broad practical applicability and appropriateness of the method are demonstrated by applying the model to: (a) an accurate, comprehensive and detailed database from a hydropower-generated highly dynamic flow in the Chattahoochee River, Georgia, USA; and (b) data from gauging stations in two large rivers in the USA affected by hysteresis. It is also shown that the model is especially suitable for post-modelling hydraulic and statistical validation and assessment.     

Uncertainty in streamflow rating curves: methods,controls and consequences

Discharge time series' are one of the core data sets used in hydrological investigations. Errors in the data mainly occur through uncertainty in gauging (measurement uncertainty) and uncertainty in determination of the stage–discharge relationship (rating curve uncertainty). Thirty‐six flow gauges from the Namoi River catchment, Australia, were examined to explore how rating curve uncertainty affects gauge reliability and uncertainty of observed flow records. The analysis focused on the deviations in gaugings from the rating curves because standard (statistical) uncertainty methods could not be applied. Deviations of greater/lesser than 10% were considered significant to allow for a measurement uncertainty threshold of 10%, determined from quality coding of gaugings and operational procedures. The deviations in gaugings were compared against various factors to examine trends and identify major controls, including stage height, date, month, rating table, gauging frequency and quality, catchment area and type of control. The analysis gave important insights into data quality and the reliability of each gauge, which had previously not been recognized. These included identification of more/less reliable periods of record, which varied widely between gauges, and identification of more/less reliable parts of the hydrograph. Most gauges showed significant deviations at low stages, affecting the determination of low flows. This was independent of the type of gauge control, with many gauges experiencing problems in the stability of the rating curve, likely as a result of sediment flux. The deviations in gaugings also have widespread application in modelling, for example, informing suitable calibration periods and defining error distributions. This paper demonstrates the value and importance of undertaking qualitative analyses of observed records. Copyright © 2012 John Wiley & Sons, Ltd.     

Uncertainty issues of a conceptual water balance model for a semi‐arid watershed in north‐west of China

Hydrological models are useful tools for better understanding the hydrological processes and performing the hydrological prediction. However, the reliability of the prediction depends largely on its uncertainty range. This study mainly focuses on estimating model parameter uncertainty and quantifying the simulation uncertainties caused by sole model parameters and the co‐effects of model parameters and model structure in a lumped conceptual water balance model called WASMOD (Water And Snow balance MODeling system). The validity of statistical hypotheses on residuals made in the model formation is tested as well, as it is the base of parameter estimation and simulation uncertainty evaluation. The bootstrap method is employed to examine the parameter uncertainty in the selected model. The Yingluoxia watershed at the upper reaches of the Heihe River basin in north‐west of China is selected as the study area. Results show that all parameters in the model can be regarded as normally distributed based on their marginal distributions and the Kolmogorov–Smirnov test, although they appear slightly skewed for two parameters. Their uncertainty ranges are different from each other. The model residuals are tested to be independent, homoscedastic and normally distributed. Based on such valid hypotheses of model residuals, simulation uncertainties caused by co‐effects of model parameters and model structure can be evaluated effectively. It is found that the 95% and 99% confidence intervals (CIs) of simulated discharge cover 42.7% and 52.4% of the observations when only parameter uncertainty is considered, indicating that parameter uncertainty has a great effect on simulation uncertainty but still cannot be used to explain all the simulation uncertainty in this study. The 95% and 99% CIs become wider, and the percentages of observations falling inside such CIs become larger when co‐effects of parameters and model structure are considered, indicating that simultaneous consideration of both parameters and model structure uncertainties accounts sufficient contribution for model simulation uncertainty. Copyright © 2012 John Wiley & Sons, Ltd.     

Impacts of uncertain river flow data on rainfall‐runoff model calibration and discharge predictions

In order to quantify total error affecting hydrological models and predictions, we must explicitly recognize errors in input data, model structure, model parameters and validation data. This paper tackles the last of these: errors in discharge measurements used to calibrate a rainfall‐runoff model, caused by stage–discharge rating‐curve uncertainty. This uncertainty may be due to several combined sources, including errors in stage and velocity measurements during individual gaugings, assumptions regarding a particular form of stage–discharge relationship, extrapolation of the stage–discharge relationship beyond the maximum gauging, and cross‐section change due to vegetation growth and/or bed movement. A methodology is presented to systematically assess and quantify the uncertainty in discharge measurements due to all of these sources. For a given stage measurement, a complete PDF of true discharge is estimated. Consequently, new model calibration techniques can be introduced to explicitly account for the discharge error distribution. The method is demonstrated for a gravel‐bed river in New Zealand, where all the above uncertainty sources can be identified, including significant uncertainty in cross‐section form due to scour and re‐deposition of sediment. Results show that rigorous consideration of uncertainty in flow data results in significant improvement of the model's ability to predict the observed flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Simple vs complex rating curves: accounting for measurement uncertainty,slope ratio and sample size

Multiple segmented rating curves have been proposed to better capture the variability of the physical and hydraulic characteristics of river–floodplain systems. We evaluate the accuracy of one- and two-segmented rating curves by exploiting a large and unique database of direct measurements of stage and discharge data in more than 200 Swedish catchments. Such a comparison is made by explicitly accounting for the potential impact of measurement uncertainty. This study shows that two-segmented rating curves did not fit the data significantly better, nor did they generate fewer errors than one-segmented rating curves. Two-segmented rating curves were found to be slightly beneficial for low flow when there were strong indications of segmentation, but predicted the rating relationship worse in cases of weak indication of segmentation. Other factors were found to have a larger impact on rating curve errors, such as the uncertainty of the discharge measurements and the type of regression method.     

Uncertainty assessment of a process-based integrated catchment model of phosphorus

Despite the many models developed for phosphorus concentration prediction at differing spatial and temporal scales, there has been little effort to quantify uncertainty in their predictions. Model prediction uncertainty quantification is desirable, for informed decision-making in river-systems management. An uncertainty analysis of the process-based model, integrated catchment model of phosphorus (INCA-P), within the generalised likelihood uncertainty estimation (GLUE) framework is presented. The framework is applied to the Lugg catchment (1,077 km²), a River Wye tributary, on the England–Wales border. Daily discharge and monthly phosphorus (total reactive and total), for a limited number of reaches, are used to initially assess uncertainty and sensitivity of 44 model parameters, identified as being most important for discharge and phosphorus predictions. This study demonstrates that parameter homogeneity assumptions (spatial heterogeneity is treated as land use type fractional areas) can achieve higher model fits, than a previous expertly calibrated parameter set. The model is capable of reproducing the hydrology, but a threshold Nash-Sutcliffe co-efficient of determination (E or R ²) of 0.3 is not achieved when simulating observed total phosphorus (TP) data in the upland reaches or total reactive phosphorus (TRP) in any reach. Despite this, the model reproduces the general dynamics of TP and TRP, in point source dominated lower reaches. This paper discusses why this application of INCA-P fails to find any parameter sets, which simultaneously describe all observed data acceptably. The discussion focuses on uncertainty of readily available input data, and whether such process-based models should be used when there isn’t sufficient data to support the many parameters.     

Accuracy of discharge determined from a rating curve

ABSTRACT

An error analysis shows that three types of errors influence the random error of a single discharge measurement determined from a rating curve. They are rating curve error, water level measurement error and an error caused by ignoring all physical parameters, other than water level, that affect discharge. Methods in the literature for evaluating the first two types of errors are reviewed and a method for evaluating the third type is given. The error of average discharge for an arbitrary period is also considered.     

Time‐varying,nonlinear suspended sediment rating curves to characterize trends in water quality: An application to the Upper Hudson and Mohawk Rivers,New York

This work proposes two modelling frameworks for diagnosing temporal variations in nonlinear rating curves that describe suspended sediment–discharge relationships. A variant of the weighted regression on time, discharge, and season model is proposed and is compared against dynamic nonlinear modelling, a newly developed nonlinear time series filter based on sequential Monte Carlo sampling. Both approaches estimate a time series of rating curve parameters, with uncertainty, that can be used to diagnose variability in the sediment–discharge relationship over time. We evaluate the models with a variety of synthetic scenarios to highlight their ability to estimate signals of known rating curve change. Results reveal important bias‐variance trade‐offs unique to each approach, and in general, suggest that dynamic nonlinear modelling is better suited for rapid rating curve changes, whereas the weighted regression on time, discharge, and season variant more precisely estimates slow change. The techniques are then applied in two case studies in the Upper Hudson and Mohawk Rivers in New York. We conclude with a discussion of the implications of dynamic rating curves for the management of water quality in riverine and estuary systems.     

Stage‐discharge uncertainty derived with a non‐stationary rating curve in the Choluteca River,Honduras

Uncertainty in discharge data must be critically assessed before data can be used in, e.g. water resources estimation or hydrological modelling. In the alluvial Choluteca River in Honduras, the river‐bed characteristics change over time as fill, scour and other processes occur in the channel, leading to a non‐stationary stage‐discharge relationship and difficulties in deriving consistent rating curves. Few studies have investigated the uncertainties related to non‐stationarity in the stage‐discharge relationship. We calculated discharge and the associated uncertainty with a weighted fuzzy regression of rating curves applied within a moving time window, based on estimated uncertainties in the observed rating data. An 18‐year‐long dataset with unusually frequent ratings (1268 in total) was the basis of this study. A large temporal variability in the stage‐discharge relationship was found especially for low flows. The time‐variable rating curve resulted in discharge estimate differences of ? 60 to + 90% for low flows and ± 20% for medium to high flows when compared to a constant rating curve. The final estimated uncertainty in discharge was substantial and the uncertainty limits varied between ? 43 to + 73% of the best discharge estimate. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian analysis of stage–fall–discharge models for gauging stations affected by variable backwater

Parsimonious stage–fall–discharge rating curve models for gauging stations subject to backwater complications are developed from simple hydraulic theory. The rating curve models are compounded in order to allow for possible shifts in the hydraulics when variable backwater becomes effective. The models provide a prior scientific understanding through the relationship between the rating curve parameters and the hydraulic properties of the channel section under study. This characteristic enables prior distributions for the rating curve parameters to be easily elicited according to site‐specific information and the magnitude of well‐known hydraulic quantities. Posterior results from three Norwegian and one American twin‐gauge stations affected by variable backwater are obtained using Markov chain Monte Carlo simulation techniques. The case studies demonstrate that the proposed Bayesian rating curve assessment is appropriate for developing rating procedures for gauging stations that are subject to variable backwater. Copyright © 2009 John Wiley & Sons, Ltd.     

Exposure estimation using repeated blood concentration measurements

Physiologically based toxicokinetic (PBTK) modeling has been well established to study the distributions of chemicals in target tissues. In addition, the hierarchical Bayesian statistical approach using Markov Chain Monte Carlo (MCMC) simulations has been applied successfully for parameter estimation. The aim was to estimate the constant inhalation exposure concentration (assumed) using a PBTK model based on repeated measurements in venous blood, so that exposures could be estimated. By treating the constant exterior exposure as an unknown parameter of a four-compartment PBTK model, we applied MCMC simulations to estimate the exposure based on a hierarchical Bayesian approach. The dataset on 16 volunteers exposed to 100 ppm (≅0.538 mg/L) trichloroethylene vapors for 4 h was reanalyzed as an illustration. Cases of time-dependent exposures with a constant mean were also studied via 100 simulated datasets. The posterior geometric mean of 0.571, with narrow 95% posterior confidence interval (CI) (0.506, 0.645), estimated the true trichloroethylene inhalation concentration (0.538 mg/L) with very high precision. Also, the proposed method estimated the overall constant mean of the simulated time-dependent exposure scenarios well with slightly wider 95% CIs. The proposed method justifies the accuracy of exposure estimation from biomonitoring data using PBTK model and MCMC simulations from a real dataset and simulation studies numerically, which provides a starting point for future applications in occupational exposure assessment.     

Exposure reconstruction using a physiologically based toxicokinetic model with cumulative amount of metabolite in urine: a case study of trichloroethylene inhalation

The use of a physiologically based toxicokinetic (PBTK) model to reconstruct chemical exposure using human biomonitoring data, urinary metabolites in particular, has not been fully explored. In this paper, the trichloroethylene (TCE) exposure dataset by Fisher et al. (Toxicol Appl Pharm 152:339–359, 1998) was reanalyzed to investigate this new approach. By treating exterior chemical exposure as an unknown model parameter, a PBTK model was used to estimate exposure and model parameters by measuring the cumulative amount of trichloroethanol glucuronide (TCOG), a metabolite of TCE, in voided urine and a single blood sample of the study subjects by Markov chain Monte Carlo (MCMC) simulations. An estimated exterior exposure of 0.532 mg/l successfully reconstructed the true inhalation concentration of 0.538 mg/l with a 95% CI (0.441–0.645) mg/l. Based on the simulation results, a feasible urine sample collection period would be 12–16 h after TCE exposure, with blood sampling at the end of the exposure period. Given the known metabolic pathway and exposure duration, the proposed computational procedure provides a simple and reliable method for environmental (occupational) exposure and PBTK model parameter estimation, which is more feasible than repeated blood sampling.     

Uncertainty in high and low flows due to model structure and parameter errors

This paper aims to investigate the uncertainty in simulated extreme low and high flows originating from hydrological model structure and parameters. To this end, three different rainfall-runoff models, namely GR4J, HBV and Xinanjiang, are applied to two subbasins of Qiantang River basin, eastern China. The Generalised Likelihood Uncertainty Estimation approach is used for estimating the uncertainty of the three models due to parameter values, henceforth referred as parameter uncertainty. Uncertainty in simulated extreme flows is evaluated by means of the annual maximum discharge and mean annual 7-day minimum discharge. The results show that although the models have good performance for the daily flows, the uncertainty in the extreme flows could not be neglected. The uncertainty originating from parameters is larger than uncertainty due to model structure. The parameter uncertainty of the extreme flows increases with the observed discharge. The parameter uncertainty in both the extreme high flows and the extreme low flows is the largest for the HBV model and the smallest for the Xinanjiang model. It is noted that the extreme low flows are mostly underestimated by all models with optimum parameter sets for both subbasins. The largest underestimation is from Xinanjiang model. Therefore it is not reliable enough to use only one set of the parameters to make the prediction and carrying out the uncertainty study in the extreme discharge simulation could give an overall picture for the planners.     

Rating curve uncertainty and change detection in discharge time series: case study with 44‐year historic data from the Nyangores River,Kenya

The intersection of the developing topic of rating curve and discharge series uncertainty with the topic of hydrological change detection (e.g., in response to land cover or climatic change) has not yet been well studied. The work herein explores this intersection, with consideration of a long‐term discharge response (1964–2007) for a ~650‐km² headwater basin of the Mara River in west Kenya, starting with stream rating and daily gauge height data. A rating model was calibrated using Bayesian methods to quantify uncertainty intervals in model parameters and predictions. There was an unknown balance of random and systemic error in rating data scatter (a scenario not likely unique to this basin), which led to an unknown balance of noise and information in the calibrated statistical error model. This had implications on testing for hydrological change. Overall, indications were that shifts in basin's discharge response were rather subtle over the 44‐year period. A null hypothesis for change using flow duration curves (FDCs) from four different 8‐year data intervals could be either accepted or rejected over much of the net flow domain depending on different applications of the statistical error model (each with precedence in the literature). The only unambiguous indication of change in FDC comparisons appeared to be a reduction in lowest baseflow in recent years (flows with >98% exceedance probability). We defined a subjective uncertainty interval based on an intermediate balance of random and systematic error in the rating model that suggested a possibility of more prevalent impacts. These results have relevance to management in the Mara basin and to future studies that might establish linkages to historic land use and climatic factors. The concern about uncertain uncertainty intervals (uncertainty²) extends beyond the Mara and is relevant to testing change where non‐random rating errors may be important and subtle responses are investigated. Copyright © 2013 John Wiley & Sons, Ltd.     

Can climate variability information constrain a hydrological model for an ungauged Costa Rican catchment?

Long‐term hydrological data are key to understanding catchment behaviour and for decision making within water management and planning. Given the lack of observed data in many regions worldwide, such as Central America, hydrological models are an alternative for reproducing historical streamflow series. Additional types of information—to locally observed discharge—can be used to constrain model parameter uncertainty for ungauged catchments. Given the strong influence that climatic large‐scale processes exert on streamflow variability in the Central American region, we explored the use of climate variability knowledge as process constraints to constrain the simulated discharge uncertainty for a Costa Rican catchment, assumed to be ungauged. To reduce model uncertainty, we first rejected parameter relationships that disagreed with our understanding of the system. Then, based on this reduced parameter space, we applied the climate‐based process constraints at long‐term, inter‐annual, and intra‐annual timescales. In the first step, we reduced the initial number of parameters by 52%, and then, we further reduced the number of parameters by 3% with the climate constraints. Finally, we compared the climate‐based constraints with a constraint based on global maps of low‐flow statistics. This latter constraint proved to be more restrictive than those based on climate variability (further reducing the number of parameters by 66% compared with 3%). Even so, the climate‐based constraints rejected inconsistent model simulations that were not rejected by the low‐flow statistics constraint. When taken all together, the constraints produced constrained simulation uncertainty bands, and the median simulated discharge followed the observed time series to a similar level as an optimized model. All the constraints were found useful in constraining model uncertainty for an—assumed to be—ungauged basin. This shows that our method is promising for modelling long‐term flow data for ungauged catchments on the Pacific side of Central America and that similar methods can be developed for ungauged basins in other regions where climate variability exerts a strong control on streamflow variability.     

Observational and predictive uncertainties for multiple variables in a spatially distributed hydrological model

In this study, uncertainty in model input data (precipitation) and parameters is propagated through a physically based, spatially distributed hydrological model based on the MIKE SHE code. Precipitation uncertainty is accounted for using an ensemble of daily rainfall fields that incorporate four different sources of uncertainty, whereas parameter uncertainty is considered using Latin hypercube sampling. Model predictive uncertainty is assessed for multiple simulated hydrological variables (discharge, groundwater head, evapotranspiration, and soil moisture). Utilizing an extensive set of observational data, effective observational uncertainties for each hydrological variable are assessed. Considering not only model predictive uncertainty but also effective observational uncertainty leads to a notable increase in the number of instances, for which model simulation and observations are in good agreement (e.g., 47% vs. 91% for discharge and 0% vs. 98% for soil moisture). Effective observational uncertainty is in several cases larger than model predictive uncertainty. We conclude that the use of precipitation uncertainty with a realistic spatio‐temporal correlation structure, analyses of multiple variables with different spatial support, and the consideration of observational uncertainty are crucial for adequately evaluating the performance of physically based, spatially distributed hydrological models.     

Effect of observation errors on the uncertainty of design floods

This study investigates the uncertainty in the estimation of the design flood induced by errors in flood data. We initially describe and critically discuss the main sources of uncertainty affecting river discharge data, when they are derived using stage-discharge rating curves. Then, different error structures are used to investigate the effects of flood data errors on design flood estimation. Annual maxima values of river discharge observed on the Po River (Italy) at Pontelagoscuro are used as an example. The study demonstrates that observation errors may have a significant impact on the uncertainty of design floods, especially when the rating curve is affected by systematic errors.