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.     

Estimating river discharge from very high‐resolution satellite data: a case study in the Yangtze River,China

The measurement of river discharge is necessary for understanding many water‐related issues. Traditionally, river discharge is estimated by measuring water stage and converting the measurement to discharge by using a stage–discharge rating curve. Our proposed method for the first time couples the measurement of water‐surface width with river width–stage and stage–discharge rating curves by using very high‐resolution satellite data. We used it to estimate the discharge in the Yangtze (Changjiang) River as a case study. The discharges estimated at four stations from five QuickBird‐2 images matched the ground observation data very well, demonstrating that the proposed approach can be regarded as ancillary to traditional field measurement methods or other remote methods to estimate river discharge. Copyright © 2004 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.     

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.     

A multi‐temporal analysis of AMSR‐E data for flood and discharge monitoring during the 2008 flood in Iowa

The objective of this work is to demonstrate the potential of using passive microwave data to monitor flood and discharge conditions and to infer watershed hydraulic and hydrologic parameters. The case study is the major flood in Iowa in summer 2008. A new Polarisation Ratio Variation Index (PRVI) was developed based on a multi‐temporal analysis of 37 GHz satellite imagery from the Advanced Microwave Scanning Radiometer (AMSR‐E) to calculate and detect anomalies in soil moisture and/or inundated areas. The Robust Satellite Technique (RST) which is a change detection approach based on the analysis of historical satellite records was adopted. A rating curve has been developed to assess the relationship between PRVI values and discharge observations downstream. A time‐lag term has been introduced and adjusted to account for the changing delay between PRVI and streamflow. Moreover, the Kalman filter has been used to update the rating curve parameters in near real time. The temporal variability of the b exponent in the rating curve formula shows that it converges toward a constant value. A consistent 21‐day time lag, very close to an estimate of the time of concentration, was obtained. The agreement between observed discharge downstream and estimated discharge with and without parameters adjustment was 65 and 95%, respectively. This demonstrates the interesting role that passive microwave can play in monitoring flooding and wetness conditions and estimating key hydrologic parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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 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.     

An analytical method of stage–fall–discharge rating

This paper presents an analytical method for establishing a stage–fall–discharge rating using hydraulic performance graphs (HPG). The rating curves derived from the HPG are used as the basis to establish the functional relation of stage, fall and discharge through regression analysis following the USGS procedure. In doing so, the conventional trial‐and‐error process can be avoided and the associated uncertainties involved may be reduced. For illustration, the proposed analytical method is applied to establish stage–fall–discharge relations for the Keelung River in northern Taiwan to examine its accuracy and applicability in an actual river. Based on the data extracted from the HPG for the Keelung River, one can establish a stage–fall–discharge relation that is more accurate than the one obtained by the conventionally used relation. Furthermore, the discharges obtained from the proposed rating method are verified through backwater analysis for measured high water level events. The results indicate that the analytical stage–fall–discharge rating method is capable of circumventing the shortcomings of those based on single‐station data and, consequently, enhancing the reliability of flood estimation and forecasting. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimating suspended sediment concentrations in areas with limited hydrological data using a mixed‐effects model

Sediment rating curves are commonly used to estimate the suspended sediment load in rivers and streams under the assumption of a constant relation between discharge (Q) and suspended sediment concentrations (SSC) over time. However, temporal variation in the sediment supply of a watershed results in shifts in this relation by increasing variability and by introducing nonlinearities in the form of hysteresis or a path‐dependent relation. In this study, we used a mixed‐effects linear model to estimate an average SSC–Q relation for different periods of time within the hydrologic cycle while accounting for seasonality and hysteresis. We tested the performance of the mixed‐effects model against the standard rating curve, represented by a generalized least squares regression, by comparing observed and predicted sediment loads for a test case on the Chilliwack River, British Columbia, Canada. In our analyses, the mixed‐effects model reflected more accurate patterns of interpolated SSC from Q data than the rating curve, especially for the low‐flow summer months when the SSC–Q relation is less clear. Akaike information criterion scores were lower for the mixed‐effects model than for the standard model, and the mixed‐effects model explained nearly twice as much variance as the standard model (52% vs 27%). The improved performance was achieved by accounting for variability in the SSC–Q relation within each month and across years for the same month using fixed and random effects, respectively, a characteristic disregarded in the sediment rating curve. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of freshwater and material fluxes from the Yangtze River into the East China Sea by using TOPEX/Poseidon altimeter data

In order to determine material fluxes in rivers by non‐contact methods, it is essential to estimate river discharge first. Although developed and optimized for open oceans, satellite radar altimetry has the potential to monitor variations in the levels of inland waters such as lakes and rivers. Making use of the concept of an ‘assumed reference point’, we converted TOPEX/Poseidon satellite altimetry data on water level variations in the Yangtze River (Changjiang) to ‘water level’ data. We also used ‘water level’ time‐series data and in situ river discharge to establish a rating curve. By use of the rating curve, we converted data on ‘water level’ derived from 7 years (1993–99) of TOPEX/Poseidon data to actual river discharge. On the basis of statistical correlation between discharge and nutrient concentration data collected in 1987–88 and in 1998–99, we estimated the total amounts of freshwater and material fluxes transferred by the Yangtze River during the 1990s. The result reveals that an overall, but very slight, increase in freshwater and material fluxes occurred during the 1990s. Copyright © 2005 John Wiley & Sons, Ltd.     

Modelling rating curves using remotely sensed LiDAR data

Accurate stream discharge measurements are important for many hydrological studies. In remote locations, however, it is often difficult to obtain stream flow information because of the difficulty in making the discharge measurements necessary to define stage‐discharge relationships (rating curves). This study investigates the feasibility of defining rating curves by using a fluid mechanics‐based model constrained with topographic data from an airborne LiDAR scanning. The study was carried out for an 8m‐wide channel in the boreal landscape of northern Sweden. LiDAR data were used to define channel geometry above a low flow water surface along the 90‐m surveyed reach. The channel topography below the water surface was estimated using the simple assumption of a flat streambed. The roughness for the modelled reach was back calculated from a single measurment of discharge. The topographic and roughness information was then used to model a rating curve. To isolate the potential influence of the flat bed assumption, a ‘hybrid model’ rating curve was developed on the basis of data combined from the LiDAR scan and a detailed ground survey. Whereas this hybrid model rating curve was in agreement with the direct measurements of discharge, the LiDAR model rating curve was equally in agreement with the medium and high flow measurements based on confidence intervals calculated from the direct measurements. The discrepancy between the LiDAR model rating curve and the low flow measurements was likely due to reduced roughness associated with unresolved submerged bed topography. Scanning during periods of low flow can help minimize this deficiency. These results suggest that combined ground surveys and LiDAR scans or multifrequency LiDAR scans that see ‘below’ the water surface (bathymetric LiDAR) could be useful in generating data needed to run such a fluid mechanics‐based model. This opens a realm of possibility to remotely sense and monitor stream flows in channels in remote locations. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling River Stage‐Discharge Relationships Using Different Neural Network Computing Techniques

One of the most important problems in hydrology is the establishment of rating curves. The statistical tools that are commonly used for river stage‐discharge relationships are regression and curve fitting. However, these techniques are not adequate in view of the complexity of the problems involved. Three different neural network techniques, i. e., multi‐layer perceptron neural network with Levenberg‐Marquardt and quasi‐Newton algorithms and radial basis neural networks, are used for the development of river stage‐discharge relationships by constructing nonlinear relationships between stage and discharge. Daily stage and flow data from three stations, Yamula, Tuzkoy and Sogutluhan, on the Kizilirmak River in Turkey were used. Regression techniques are also applied to the same data. Different input combinations including the previous stages and discharges are used. The models' results are compared using three criteria, i. e., root mean square errors, mean absolute error and the determination coefficient. The results of the comparison reveal that the neural network techniques are much more suitable for setting up stage‐discharge relationships than the regression techniques. Among the neural network methods, the radial basis neural network is found to be slightly better than the others.     

Predicting suspended sediment concentrations in the Meuse river using a supply‐based rating curve

A rating curve provides a reasonable estimate of the suspended sediment concentration at a given discharge. However, analysis of a detailed 9‐year time‐series of suspended sediment concentration (SSC) and discharge Q of the Meuse River in The Netherlands indicates that SSC is (besides discharge) controlled by exhaustion and replenishment of different sediment sources. Clockwise hysteresis and other effects of sediment exhaustion can be observed during and after flood events, and the effects of stockpiling of sediment in the river bed during low‐discharge periods are obvious in the SSC of the next flood. In a single regression equation we have implemented a parameter that represents the presence or absence of stock for sediment uptake. In comparison with a rating curve of SSC and Q, adding this parameter is shown to be a more reliable and comprehensive method to predict SSCs at all discharge regimes with all preceding discharge conditions, for single‐peaked and multi‐peaked runoff events as well as for low flow conditions. The method is probably applicable to other small‐ to medium‐scaled river basins. Copyright © 2007 John Wiley & Sons, Ltd.     

Using width‐based rating curves from spatially discontinuous satellite imagery to monitor river discharge

Remote estimation of river discharge from river width variations is an intriguing method for gauging rivers without conventional measurements. Entirely cloud‐free imagery of an entire river reach is often rare, but partial coverage is more frequent. Discharge is estimated from spatially discontinuous imagery via construction of multiple width–discharge rating curves within a 62‐km reach of the Tanana River, Alaska. The resulting discharge error is as low as 6.7% root mean squared error. Imagery covering <20% of the study reach can be used. Copyright © 2014 John Wiley & Sons, Ltd.     

Remote sensing of river stage using the cross‐sectional inundation area‐river stage relationship (IARSR) constructed from digital elevation model data

Remote sensing of discharge and river stage from space provides us with a promising alternative approach to monitor watersheds, no matter if they are ungauged, poorly gauged, or fully gauged. One approach is to estimate river stage from satellite measured inundation area based on the inundation area – river stage relationship (IARSR). However, this approach is not easy to implement because of a lack of data for constructing the IARSR. In this study, an innovative and robust approach to construct the IARSR from digital elevation model (DEM) data was developed and tested. It was shown that the constructed IARSR from DEM data could be used to retrieve water level or river stage from satellite‐measured inundation area. To reduce the uncertainty in the estimated inundation area, a dual‐thresholding method was proposed. The first threshold is the lower limit of pixel value for classifying water body pixels with a relatively high‐level certainty. The second threshold is the upper limit of pixel value for classifying potentially flooded pixels. All pixels with values between the first threshold and the second threshold and adjacent to the classified water body pixels may be partially flooded. A linear interpolation method was used to estimate the wetted area of each partially flooded pixel. In applying the constructed IARSR to the estimated inundation areas from 11 Landsat TM images, 11 water levels were obtained. The root mean square error (RMSE) of the estimated water levels compared with the observed water levels at the US Geological Survey (USGS) gauging station on the Trinity River at Liberty in Liberty County, Texas, is about 0.38 m. Copyright © 2012 John Wiley & Sons, Ltd.