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.     

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.     

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.     

Flood frequency analysis using historical data: accounting for random and systematic errors

Abstract

Flood frequency analysis based on a set of systematic data and a set of historical floods is applied to several Mediterranean catchments. After identification and collection of data on historical floods, several hydraulic models were constructed to account for geomorphological changes. Recent and historical rating curves were constructed and applied to reconstruct flood discharge series, together with their uncertainty. This uncertainty stems from two types of error: (a) random errors related to the water-level readings; and (b) systematic errors related to over- or under-estimation of the rating curve. A Bayesian frequency analysis is performed to take both sources of uncertainty into account. It is shown that the uncertainty affecting discharges should be carefully evaluated and taken into account in the flood frequency analysis, as it can increase the quantiles confidence interval. The quantiles are found to be consistent with those obtained with empirical methods, for two out of four of the catchments.

Citation Neppel, L., Renard, B., Lang, M., Ayral, P.-A., Coeur, D., Gaume, E., Jacob, N., Payrastre, O., Pobanz, K. & Vinet, F. (2010) Flood frequency analysis using historical data: accounting for random and systematic errors. Hydrol. Sci. J. 55(2), 192–208.     

Diagnostic evaluation of multiple hypotheses of hydrological behaviour in a limits‐of‐acceptability framework for 24 UK catchments

Testing competing conceptual model hypotheses in hydrology is complicated by uncertainties from a wide range of sources, which result in multiple simulations that explain catchment behaviour. In this study, the limits of acceptability uncertainty analysis approach used to discriminate between 78 competing hypotheses in the Framework for Understanding Structural Errors for 24 catchments in the UK. During model evaluation, we test the model's ability to represent observed catchment dynamics and processes by defining key hydrologic signatures and time step‐based metrics from the observed discharge time series. We explicitly account for uncertainty in the evaluation data by constructing uncertainty bounds from errors in the stage‐discharge rating curve relationship. Our study revealed large differences in model performance both between catchments and depending on the type of diagnostic used to constrain the simulations. Model performance varied with catchment characteristics and was best in wet catchments with a simple rainfall‐runoff relationship. The analysis showed that the value of different diagnostics in constraining catchment response and discriminating between competing conceptual hypotheses varies according to catchment characteristics. The information content held within water balance signatures was found to better capture catchment dynamics in chalk catchments, where catchment behaviour is predominantly controlled by seasonal and annual changes in rainfall, whereas the information content in the flow‐duration curve and time‐step performance metrics was able to better capture the dynamics of rainfall‐driven catchments. We also investigate the effect of model structure on model performance and demonstrate its (in)significance in reproducing catchment dynamics for different catchments. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.     

Bayesian methods for estimating multi-segment discharge rating curves

This study explores Bayesian methods for handling compound stage–discharge relationships, a problem which arises in many natural rivers. It is assumed: (1) the stage–discharge relationship in each rating curve segment is a power-law with a location parameter, or zero-plane displacement; (2) the segment transitions are abrupt and continuous; and (3) multiplicative measurement errors are of equal variance. The rating curve fitting procedure is then formulated as a piecewise regression problem where the number of segments and the associated changepoints are assumed unknown. Procedures are developed for describing both global and site-specific prior distributions for all rating curve parameters, including the changepoints. Estimation and uncertainty analysis is evaluated using Markov chain Monte Carlo simulation (MCMC) techniques. The first model explored accounts for parameter and model uncertainties in the interpolated area, i.e. within the range of available stage–discharge measurements. A second model is constructed in an attempt to include the uncertainty in extrapolation, which is necessary when the rating curve is used to estimate discharges beyond the highest or lowest measurement. This is done by assuming that the rate of changepoints both inside and outside the measured area follows a Poisson process. The theory is applied to actual data from Norwegian gauging stations. The MCMC solutions give results that appear sensible and useful for inferential purposes, though the latter model needs further efforts in order to obtain a more efficient simulation scheme.     

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.     

A method for uncertainty constraint of catchment discharge and phosphorus load estimates

River discharge and nutrient measurements are subject to aleatory and epistemic uncertainties. In this study, we present a novel method for estimating these uncertainties in colocated discharge and phosphorus (P) measurements. The “voting point”‐based method constrains the derived stage‐discharge rating curve both on the fit to available gaugings and to the catchment water balance. This helps reduce the uncertainty beyond the range of available gaugings and during out of bank situations. In the example presented here, for the top 5% of flows, uncertainties are shown to be 139% using a traditional power law fit, compared with 40% when using our updated “voting point” method. Furthermore, the method is extended to in situ and lab analysed nutrient concentration data pairings, with lower uncertainties (81%) shown for high concentrations (top 5%) than when a traditional regression is applied (102%). Overall, for both discharge and nutrient data, the method presented goes some way to accounting for epistemic uncertainties associated with nonstationary physical characteristics of the monitoring site.     

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.     

Total least square method applied to rating curves

The ordinary least square method (OLS) has been the most frequently used least square method in hydrological data analysis. Its computational algorithm is simple, and the error analysis is also simple and clear. However, the primary assumption of the OLS method, which states that the dependent variable is the only error‐contaminated variable and all other variables are error free, is often violated in hydrological data analyses. Recently, a matrix algorithm using the singular value decomposition for the total least square (TLS) method has been developed and used in data analyses as errors‐in‐variables model where several variables could be contaminated with observational errors. In our study, the algorithm of the TLS is introduced in the evaluation of rating curves between the flow discharge and the water level. Then, the TLS algorithm is applied to real data set for rating curves. The evaluated TLS rating curves are compared with the OLS rating curves, and the result indicates that the TLS rating curve and the OLS rating curve are in good agreement. The TLS and OLS rating curves are discussed about their algorithms and error terms in the study. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A linear approach to the influence of discharge measurement error on flood estimates

ABSTRACT

A linear approach is presented for analysing flood discharge series affected by measurement errors which are random in nature. A general model based upon the conditional probability concept is introduced to represent random errors and to analyse their effect on flood estimates. Flood predictions provided by quantiles are shown to be positively biased when performed from a sample of measured discharge. Though for design purposes such an effect is conservative, this bias cannot be neglected if the peak discharges are determined from stage measurements by means of the extrapolated tail of the rating curve for the gauging station concerned. Monte Carlo experiments, which have been carried out to analyse small sample effects, have finally shown that the use of the method of maximum likelihood is able to reduce the bias due to measurement errors in discharge data.     

Discharge and nutrient uncertainty: implications for nutrient flux estimation in small streams

The measurement of discharge is fundamental in nutrient load estimation. Because of our ability to monitor discharge routinely, it is generally assumed that the associated uncertainty is low. This paper challenges this preconception, arguing that discharge uncertainty should be explicitly taken into account to produce robust statistical analyses. In many studies, paired discharge and chemical datasets are used to calculate ‘true’ loads and used as the benchmark to compare with other load estimates. This paper uses two years of high frequency (daily and sub‐hourly) discharge and nutrient concentration data (nitrate‐N and total phosphorus (TP)) collected at four field sites as part of the Hampshire Avon Demonstration Test Catchment (DTC) programme. A framework for estimating observational nutrient load uncertainty was used which combined a flexible non‐parametric approach to characterising discharge uncertainty, with error modelling that allowed the incorporation of errors which were heteroscedastic and temporally correlated. The results showed that the stage–discharge relationships were non‐stationary, and observational uncertainties from ±2 to 25% were recorded when the velocity–area method was used. The variability in nutrient load estimates ranged from 1.1 to 9.9% for nitrate‐N and from 3.3 to 10% for TP when daily laboratory data were used, rising to a maximum of 9% for nitrate‐N and 83% for TP when the sensor data were used. However, the sensor data provided a better representation of the ‘true’ load as storm events are better represented temporally, posing the question: is it more beneficial to have high frequency, lower precision data or lower frequency but higher precision data streams to estimate nutrient flux responses in headwater catchments? Copyright © 2015 John Wiley & Sons, Ltd.     

Experimental catchments in the Pampa biome: Database on hydrology in grasslands and eucalyptus plantations in subtropical Brazil

The introduction of exotic, fast-growing forest species in the Pampa biome (Southern Grasslands) is a controversial topic, considering the potential effect on water and soil resources. This repository contains hydrologic data (rainfall, discharge and turbidity) collected since 2011 in three small (≤1.1 km²), paired experimental catchments of the “Ponta da Canas” site, in the Pampa biome in subtropical Brazil. Two catchments are predominantly covered with eucalyptus plantations, and one with livestock-grazing degraded grassland. For each catchment, the collected data include 10-min resolution rainfall, streamflow, and turbidity (except for one of the eucalyptus catchments), automatically recorded in 10-min intervals. In each catchment, rainfall is measured with an automatic tipping-bucket rain gauge; stream depth is determined with a pressure transducer at the spillway, and a rating curve is used to estimate discharge; and turbidity is measured with a turbidimeter. The collected data are being used to understand water balance and sediment production under the distinct land uses, to improve forest management, and comply with State legislation.     

Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?

In recent years, a strong debate has emerged in the hydrologic literature regarding what constitutes an appropriate framework for uncertainty estimation. Particularly, there is strong disagreement whether an uncertainty framework should have its roots within a proper statistical (Bayesian) context, or whether such a framework should be based on a different philosophy and implement informal measures and weaker inference to summarize parameter and predictive distributions. In this paper, we compare a formal Bayesian approach using Markov Chain Monte Carlo (MCMC) with generalized likelihood uncertainty estimation (GLUE) for assessing uncertainty in conceptual watershed modeling. Our formal Bayesian approach is implemented using the recently developed differential evolution adaptive metropolis (DREAM) MCMC scheme with a likelihood function that explicitly considers model structural, input and parameter uncertainty. Our results demonstrate that DREAM and GLUE can generate very similar estimates of total streamflow uncertainty. This suggests that formal and informal Bayesian approaches have more common ground than the hydrologic literature and ongoing debate might suggest. The main advantage of formal approaches is, however, that they attempt to disentangle the effect of forcing, parameter and model structural error on total predictive uncertainty. This is key to improving hydrologic theory and to better understand and predict the flow of water through catchments.     

Response to comment by Keith Beven on “Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?”

In recent years, a strong debate has emerged in the hydrologic literature regarding what constitutes an appropriate framework for uncertainty estimation. Particularly, there is strong disagreement whether an uncertainty framework should have its roots within a proper statistical (Bayesian) context, or whether such a framework should be based on a different philosophy and implement informal measures and weaker inference to summarize parameter and predictive distributions. In this paper, we compare a formal Bayesian approach using Markov Chain Monte Carlo (MCMC) with generalized likelihood uncertainty estimation (GLUE) for assessing uncertainty in conceptual watershed modeling. Our formal Bayesian approach is implemented using the recently developed differential evolution adaptive metropolis (DREAM) MCMC scheme with a likelihood function that explicitly considers model structural, input and parameter uncertainty. Our results demonstrate that DREAM and GLUE can generate very similar estimates of total streamflow uncertainty. This suggests that formal and informal Bayesian approaches have more common ground than the hydrologic literature and ongoing debate might suggest. The main advantage of formal approaches is, however, that they attempt to disentangle the effect of forcing, parameter and model structural error on total predictive uncertainty. This is key to improving hydrologic theory and to better understand and predict the flow of water through catchments.     

Simulating temporal variability in catchment response using a monthly rainfall–runoff model

Abstract

Temporal variability can result from shifts in climate, or from changes in the runoff response due to land- or water-use changes, and represents a potential source of uncertainty in calibrating hydrological models. Parameter values were determined using Monte Carlo parameter sampling methods for a monthly rainfall–runoff model (Pitman model) for different sub-periods on four catchments, with different types and degrees of temporal variability, in Australia and Africa. For some catchments, parameters were not dependent upon the sub-period used and fell within expected ranges given the relatively high degree of model equifinality. In other catchments, dependencies can be identified that are associated with signals contained within the sub-periods. While the Pitman model is relatively robust in the face of temporal variability, it is concluded that better simulations will always be obtained from calibration data that include signals representing the total variability in climate, land-use change and catchment responses.     

Uncertainty in stage–discharge rating curves: application to Australian Hydrologic Reference Stations data

The purpose of this paper is to determine uncertainty in the gauged range of the stage–gauged discharge relationship for 622 rating curves from 171 Australian Bureau of Meteorology Hydrologic Reference streamgauging Stations (HRS). Water agencies use many methods to establish rating curves. Here we adopt a consistent method across all stations and develop rating curves based on Chebyshev polynomials, and estimate uncertainties from standard regression errors in which residuals from the polynomials are adjusted to ensure they are homoscedastic and normally distributed. Uncertainty in input water level is also taken into account. The median uncertainties in mean response of the available gauged discharge relationship at median daily discharges for the HRS dataset range from +4.5 to ?4.2% (95% confidence band) and for individual gaugings from +29 to ?22% incorporating a water level uncertainty of ±4 mm. The uncertainties estimated are consistent with values estimated in Australia and elsewhere.