Facets of uncertainty: epistemic uncertainty,non-stationarity,likelihood, hypothesis testing,and communication

ABSTRACT

This paper presents a discussion of some of the issues associated with the multiple sources of uncertainty and non-stationarity in the analysis and modelling of hydrological systems. Different forms of aleatory, epistemic, semantic, and ontological uncertainty are defined. The potential for epistemic uncertainties to induce disinformation in calibration data and arbitrary non-stationarities in model error characteristics, and surprises in predicting the future, are discussed in the context of other forms of non-stationarity. It is suggested that a condition tree is used to be explicit about the assumptions that underlie any assessment of uncertainty. This also provides an audit trail for providing evidence to decision makers.

Editor D. Koutsoyiannis; Associate editor S. Weijs     

Catchment‐scale sensitivity and uncertainty in water quality modelling

In this paper, we assess the performance of the catchment model SIMulated CATchment model (SIMCAT), to predict nitrate and soluble reactive phosphorus concentrations against four monitoring regimes with different spatial and temporal sampling frequencies. The Generalised Likelihood Uncertainty Estimation (GLUE) uncertainty framework is used, along with a general sensitivity analysis to understand relative parameter sensitivity. Improvements to model calibration are explored by introducing more detailed process representation using the Integrated Catchments model (INCA) water quality model, driven by the European hydrological predictions for the environment model. The results show how targeted sampling of headwater watercourses upstream of point discharges is essential for calibrating diffuse loads and can exert a strong influence on the whole‐catchment model performance. Further downstream, if the point discharges and loads are accurately represented, then the improvement in the catchment‐scale model performance is relatively small as more calibration points are added or frequency is increased. The higher‐order, dynamic model integrated catchments model of phosphorus dynamics, which incorporates sediment and biotic interaction, resulted in improved whole‐catchment performance over SIMCAT, although there are still large epistemic uncertainties from land‐phase export coefficients and runoff. However, the very large sampling errors in routine monitoring make it difficult to invest confidence in the modelling, especially because we know phosphorous transport to be very episodic and driven by high flow conditions for which there are few samples. The environmental modelling community seems to have been stuck in this position for some time, and whilst it is useful to use an uncertainty framework to highlight these issues, it has not widely been adopted, perhaps because there is no clear mechanism to allow uncertainties to influence investment decisions. This raises the question as to whether it might better place a cost on uncertainty and use this to drive more data collection or improved models, before making investment decisions concerning, for example, mitigation strategies. Copyright © 2016 John Wiley & Sons, Ltd.     

Input data resolution‐induced uncertainty in watershed modelling

The temporal‐spatial resolution of input data‐induced uncertainty in a watershed‐based water quality model, Hydrologic Simulation Program‐FORTRAN (HSPF), is investigated in this study. The temporal resolution‐induced uncertainty is described using the coefficient of variation (CV). The CV is found to decrease with decreasing temporal resolution and follow a log‐normal relation with time interval for temperature data while it exhibits a power‐law relation for rainfall data. The temporal‐scale uncertainties in the temperature and rainfall data follow a general extreme value distribution and a Weibull distribution, respectively. The Nash‐Sutcliffe coefficient (NSC) is employed to represent the spatial resolution induced uncertainty. The spatial resolution uncertainty in the dissolved oxygen and nitrate‐nitrogen concentrations simulated using HSPF is observed to follow a general extreme value distribution and a log‐normal distribution, respectively. The probability density functions (PDF) provide new insights into the effect of temporal‐scale and spatial resolution of input data on uncertainties involved in watershed modelling and total maximum daily load calculations. This study exhibits non‐symmetric distributions of uncertainty in water quality modelling, which simplify weather and water quality monitoring and reducing the cost involved in flow and water quality monitoring. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimation of catchment yield and associated uncertainties due to climate change in a mountainous catchment in Australia

This paper examines the impacts of climate change on future water yield with associated uncertainties in a mountainous catchment in Australia using a multi‐model approach based on four global climate models (GCMs), 200 realisations (50 realisations from each GCM) of downscaled rainfalls, 2 hydrological models and 6 sets of model parameters. The ensemble projections by the GCMs showed that the mean annual rainfall is likely to reduce in the future decades by 2–5% in comparison with the current climate (1987–2012). The results of ensemble runoff projections indicated that the mean annual runoff would reduce in future decades by 35%. However, considerable uncertainty in the runoff estimates was found as the ensemble results project changes of the 5^th (dry scenario) and 95^th (wet scenario) percentiles by ?73% to +27%, ?73% to +12%, ?77% to +21% and ?80% to +24% in the decades of 2021–2030, 2031–2040, 2061–2070 and 2071–2080, respectively. Results of uncertainty estimation demonstrated that the choice of GCMs dominates overall uncertainty. Realisation uncertainty (arising from repetitive simulations for a given time step during downscaling of the GCM data to catchment scale) of the downscaled rainfall data was also found to be remarkably high. Uncertainty linked to the choice of hydrological models was found to be quite small in comparison with the GCM and realisation uncertainty. The hydrological model parameter uncertainty was found to be lowest among the sources of uncertainties considered in this study. Copyright © 2015 John Wiley & Sons, Ltd.     

Reliability analysis of hydraulic structures considering unit hydrograph uncertainty

Unit hydrographs (UHs), along with design rainfalls, are frequently used to determine the discharge hydrograph for design and evaluation of hydraulic structures. Due to the presence of various uncertainties in its derivation, the resulting UH is inevitably subject to uncertainty. Consequently, the performance of hydraulic structures under the design storm condition is uncertain. This paper integrates the linearly constrained Monte-Carlo simulation with the UH theory and routing techniques to evaluate the reliability of hydraulic structures. The linear constraint is considered because the water volume of each generated design direct runoff hydrograph should be equal to that of the design effective rainfall hyetograph or the water volume of each generated UH must be equal to one inch (or cm) over the watershed. For illustration, the proposed methodology is applied to evaluate the overtopping risk of a hypothetical flood detention reservoir downstream of Tong-Tou watershed in Taiwan.     

Reliability analysis of hydraulic structures considering unit hydrograph uncertainty

Unit hydrographs (UHs), along with design rainfalls, are frequently used to determine the discharge hydrograph for design and evaluation of hydraulic structures. Due to the presence of various uncertainties in its derivation, the resulting UH is inevitably subject to uncertainty. Consequently, the performance of hydraulic structures under the design storm condition is uncertain. This paper integrates the linearly constrained Monte-Carlo simulation with the UH theory and routing techniques to evaluate the reliability of hydraulic structures. The linear constraint is considered because the water volume of each generated design direct runoff hydrograph should be equal to that of the design effective rainfall hyetograph or the water volume of each generated UH must be equal to one inch (or cm) over the watershed. For illustration, the proposed methodology is applied to evaluate the overtopping risk of a hypothetical flood detention reservoir downstream of Tong-Tou watershed in Taiwan.     

Development of design flood hydrographs using probability density functions

Probability density functions (PDFs) are used to fit the shape of hydrographs and have been popularly used for the development of synthetic unit hydrographs by many hydrologists. Nevertheless, modelling the shapes of continuous stream flow hydrographs, which are probabilistic in nature, is rare. In the present study, a novel approach was followed to model the shape of stream flow hydrographs using PDF and subsequently to develop design flood hydrographs for various return periods. Four continuous PDFs, namely, two parameter Beta, Weibull, Gamma and Lognormal, were employed to fit the shape of the hydrographs of 22 years at a site of Brahmani River in eastern India. The shapes of the observed and PDF fitted hydrographs were compared and root mean square errors, error of peak discharge (EQ_P) and error of time to peak (ET_P) were computed. The best‐fitted shape and scale parameters of all PDFs were subjected to frequency analysis and the quartiles corresponding to 20‐, 50‐, 100‐ and 200‐year were estimated. The estimated parameters of each return period were used to develop the flood hydrographs for 20‐, 50‐, 100‐ and 200‐year return periods. The peak discharges of the developed design flood hydrographs were compared with the design discharges estimated from the frequency analysis of 22 years of annual peak discharges at that site. Lognormal‐produced peak discharge was very close to the estimated design discharge in case of 20‐year flood hydrograph. On the other hand, peak discharge obtained using the Weibull PDF had close agreement with the estimated design discharge obtained from frequency analysis in case of 50‐, 100‐ and 200‐year return periods. The ranking of the PDFs based on estimation of peak of design flood hydrograph for 50‐, 100‐ and 200‐year return periods was found to have the following order: Weibull > Beta > Lognormal > Gamma. Copyright © 2009 John Wiley & Sons, Ltd.     

The integrated effects of climate and hydrologic uncertainty on future flood risk assessments

This work examines future flood risk within the context of integrated climate and hydrologic modelling uncertainty. The research questions investigated are (1) whether hydrologic uncertainties are a significant source of uncertainty relative to other sources such as climate variability and change and (2) whether a statistical characterization of uncertainty from a lumped, conceptual hydrologic model is sufficient to account for hydrologic uncertainties in the modelling process. To investigate these questions, an ensemble of climate simulations are propagated through hydrologic models and then through a reservoir simulation model to delimit the range of flood protection under a wide array of climate conditions. Uncertainty in mean climate changes and internal climate variability are framed using a risk‐based methodology and are explored using a stochastic weather generator. To account for hydrologic uncertainty, two hydrologic models are considered, a conceptual, lumped parameter model and a distributed, physically based model. In the conceptual model, parameter and residual error uncertainties are quantified and propagated through the analysis using a Bayesian modelling framework. The approach is demonstrated in a case study for the Coralville Dam on the Iowa River, where recent, intense flooding has raised questions about potential impacts of climate change on flood protection adequacy. Results indicate that the uncertainty surrounding future flood risk from hydrologic modelling and internal climate variability can be of the same order of magnitude as climate change. Furthermore, statistical uncertainty in the conceptual hydrological model can capture the primary structural differences that emerge in flood damage estimates between the two hydrologic models. Copyright © 2014 John Wiley & Sons, Ltd.     

Model uncertainty analysis by variance decomposition

Errors and uncertainties in hydrological, hydraulic and environmental models are often substantial. In good modelling practice, they are quantified in order to supply decision-makers with important additional information on model limitations and sources of uncertainty. Several uncertainty analysis methods exist, often with various underlying assumptions. One of these methods is based on variance decomposition. The method allows splitting the variance of the total error in the model results (as estimated after comparing model results with observations) in its major contributing uncertainty sources. This paper discusses an advanced version of that method where error distributions for rainfall, other inputs and parameters are propagated in the model and the “rest” uncertainties considered as model structural errors for different parts of the model. By expert knowledge, the iid assumption that is often made in model error analysis is addressed upfront. The method also addresses the problems of heteroscedasticity and serial dependence of the errors involved. The method has been applied by the author to modelling applications of sewer water quantity and quality, river water quality and river flooding.     

Predictive ground motion scaling in Switzerland: Best estimates and uncertainties

Spectral ground motion (1 to 15 Hz) as a function of distance is modeled for events spanning 3.0 <M_w ≤ 7.0 in Switzerland. The parameters required to simulate ground motion with a stochastic approach are inverted from 2958 horizontal and vertical component waveforms of small to moderate size events (2.0 ≤ M_{L} ≤ 5.2) in the distance range 10 to 300 km recorded on hard rock sites. Using a Monte Carlo simulation, we establish a significantly different amplification of about a factor of 1.9 between the Alpine Foreland and the Alps. To assess the trade-off between the free parameters of our stochastic model and their influence on the predictive ground motion relationship, we perform a grid search over the five-dimensional solution space. The uncertainties are separated into epistemic and aleatory parts; the main epistemic uncertainty is attributed to the lack of data forM > 5. To constrain the viable models at large magnitudes, results from worldwide scaling studies are evaluated in light of the Swiss data. The model that explains best the low observed stress drops at small magnitudes (Δσ ≅ 3 bar) yet matches observed intensities of historical earthquakes assumes a stress drop increasing with moment asM₀^0.25. For three sites in Switzerland we evaluate the sensitivity of the epistemic uncertainty by computing probabilistic hazard curves. Our model offers the most comprehensive and detailed study of spectral ground motion for Switzerland to date.     

Sensitivity of hydrological models to uncertainty in rainfall input

Abstract

Different approaches used in hydrological modelling are compared in terms of the way each one takes the rainfall data into account. We examine the errors associated with accounting for rainfall variability, whether in hydrological modelling (distributed vs lumped models) or in computing catchment rainfall, as well as the impact of each approach on the representativeness of the parameters it uses. The database consists of 1859 rainfall events, distributed on 500 basins, located in the southeast of France with areas ranging from 6.2 to 2851 km². The study uses as reference the hydrographs computed by a distributed hydrological model from radar rainfall. This allows us to compare and to test the effects of various simplifications to the process when taking rainfall information (complete rain field vs sampled rainfall) and rainfall–runoff modelling (lumped vs distributed) into account. The results appear to show that, in general, the sampling effect can lead to errors in discharge at the outlet that are as great as, or even greater than, those one would get with a fully lumped approach. We found that small catchments are more sensitive to the uncertainties in catchment rainfall input generated by sampling rainfall data as seen through a raingauge network. Conversely, the larger catchments are more sensitive to uncertainties generated when the spatial variability of rainfall events is not taken into account. These uncertainties can be compensated for relatively easily by recalibrating the parameters of the hydrological model, although such recalibrations cause the parameter in question to completely lose physical meaning.

Citation Arnaud, P., Lavabre, J., Fouchier, C., Diss, S. & Javelle, P. (2011) Sensitivity of hydrological models to uncertainty of rainfall input. Hydrol. Sci. J. 56(3), 397–410.     

A critical examination of seismic response uncertainty analysis in earthquake engineering

The last decade of performance‐based earthquake engineering (PBEE) research has seen a rapidly increasing emphasis placed on the explicit quantification of uncertainties. This paper examines uncertainty consideration in input ground‐motion and numerical seismic response analyses as part of PBEE, with particular attention given to the physical consistency and completeness of uncertainty consideration. It is argued that the use of the commonly adopted incremental dynamic analysis leads to a biased representation of the seismic intensity and that when considering the number of ground motions to be used in seismic response analyses, attention should be given to both reducing parameter estimation uncertainty and also limiting ground‐motion selection bias. Research into uncertainties in system‐specific numerical seismic response analysis models to date has been largely restricted to the consideration of ‘low‐level’ constitutive model parameter uncertainties. However, ‘high‐level’ constitutive model and model methodology uncertainties are likely significant and therefore represent a key research area in the coming years. It is also argued that the common omission of high‐level seismic response analysis modelling uncertainties leads to a fallacy that ground‐motion uncertainty is more significant than numerical modelling uncertainty. The author's opinion of the role of uncertainty analysis in PBEE is also presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Effect of uncertainty in Digital Surface Models on the extent of inundated areas

The impact of uncertainty in ground elevation on the extent of areas that are inundated due to flooding is investigated. Land surface is represented through a Digital Surface Model (DSM). The effect of uncertainty in DSM is compared to that of the uncertainty due to rainfall. The Monte Carlo method is used to quantify the uncertainty. A typical photogrammetric procedure and conventional maps are used to obtain a reference DSM, later altered to provide DSMs of lower accuracy. Also, data from the Shuttle Radar Topography Mission are used. Floods are simulated in two stages. In the first stage, flood hydrographs for typical return periods are synthesized using generated storm hyetographs, the Soil Conservation Service–Curve Number method for effective rainfall, and the Soil Conservation Service synthetic unit hydrograph. In the second stage, hydrographs are routed via a one‐dimensional hydraulic model. Uncertainty in DSM is considered only in the second stage. Data from two real‐world basins in Greece are used. To characterize the inundated area, we employ the 90% quantile of the inundation extent and inundation topwidth for peak water level at specific river cross‐sections. For topwidths, apart from point estimates, also interval estimates are acquired using the bootstrap method. The effect of DSM uncertainty is compared to that for rainfall. Low uncertainty in DSM is found to widen the inundated area; whereas, the opposite occurred with high uncertainty. SRTM data proved unsuitable for our test basins and modelling context.     

Evaluating Subsurface Parameterization to Simulate Hyporheic Exchange: The Steinlach River Test Site

Hyporheic exchange is the interaction of river water and groundwater, and is difficult to predict. One of the largest contributions to predictive uncertainty for hyporheic exchange has been attributed to the representation of heterogeneous subsurface properties. Our study evaluates the trade-offs between intrinsic (irreducible) and epistemic (reducible) model errors when choosing between homogeneous and highly complex subsurface parameter structures. We modeled the Steinlach River Test Site in Southwest Germany using a fully coupled surface water-groundwater model to simulate hyporheic exchange and to assess the predictive errors and uncertainties of transit time distributions. A highly parameterized model was built, treated as a “virtual reality” and used as a reference. We found that if the parameter structure is too simple, it will be limited by intrinsic model errors. By increasing subsurface complexity through the addition of zones or heterogeneity, we can begin to exchange intrinsic for epistemic errors. Thus, the appropriate level of detail to represent the subsurface depends on the acceptable range of intrinsic structural errors for the given modeling objectives and the available site data. We found that a zonated model is capable of reproducing the transit time distributions of a more detailed model, but only if the geological structures are known. An interpolated heterogeneous parameter field (cf. pilot points) showed the best trade-offs between the two errors, indicating fitness for practical applications. Parameter fields generated by multiple-point geostatistics (MPS) produce transit time distributions with the largest uncertainties, however, these are reducible by additional hydrogeological data, particularly flux measurements.     

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.     

Potentials and limitations of modelling vertical distributions of root water uptake of an Austrian pine forest on a sandy soil

Root water uptake patterns are often studied with simulation models of the unsaturated soil water flow, as they are difficult to measure directly. Calibration of these models is not straightforward and causes uncertainties in simulated uptake distributions. In this paper we study how uncertainties in the calibration of the SWIF model affect uncertainty intervals in simulated uptake patterns of an Austrian pine stand (Pinus nigra var. nigra) on a sandy soil. After calibrating and validating SWIF with a large data set of more than 125 000 measured soil water contents over a three year period, uncertainty ranges in simulated soil water dynamics and root water uptake distributions were estimated with a Monte Carlo analysis. In general, uncertainties in root uptake patterns were small (typically <2 10⁻⁴ m³ m⁻³ day⁻¹) and were higher for trees with a shallow rooting system (0·8 m) than for trees with a deep rooting system (2·5 m). Uncertainties arose mainly from uncertainties in simulated soil water fluxes and from variations in the reduction of uptake during periods of drought. Uncertainties in soil water contents were far higher (typically 0·01 m³ m⁻³) than uncertainties in uptake, illustrating that uncertainties in uptake parameters and those in the distribution of water uptake hardly affect the modelling of soil water dynamics. Root water uptake models should therefore be validated against measured uptake distributions, which can be determined on sandy soils during dry periods with a high water use when soil fluxes are negligible to uptake. Copyright © 2000 John Wiley & Sons, Ltd.     

Effectiveness of BIOECODS for peak flow attenuation: an appraisal using InfoWorks SD

The Bio-Ecological Drainage System (BIOECODS) is a sustainable drainage system, which adopts a “control at source” approach for urban storm water management in Malaysia. This study attempts to model a small-scale BIOECODS using InfoWorks SD. New modelling techniques are used to fully integrate the surface and on-line subsurface conveyance system, in which overland flow routing is described by a storm water management model that uses a nonlinear reservoir method and the kinematic wave approximation of the St Venant equation, and subsurface flow is described by the Horton method in conjunction with the Soil Conservation Service (SCS) curve number (CN) method. The observed water levels at primary outlets are compared with those obtained from model simulation. The modelling approach has been proven successful as the hydrographs (predicted and observed) match each other closely, with a mean error in the range of 4.58–7.32%. Results from the model showed that the BIOECODS is able to attenuate peak flow by 60–75%, and increase the lag time by 20 min within an area of <28?300 m² when compared with a traditional drainage system.     

Propagation of uncertainties in interpolated rainfields to runoff errors

Conditional daily rainfields were generated using collocated raingauge radar data by a kriging interpolation method, and disaggregated into hourly rainfields using variants of the method of fragments. A geographic information system (GIS)-based distributed rainfall–runoff model was used to convert the hourly rainfields into hydrographs. Using the complete radar rainfall as input, the rainfall–runoff model was calibrated based on storm events taken from nested catchments. Performance statistics were estimated by comparing the observed and the complete radar rainfall simulated hydrographs. Degradation in the hydrograph performance statistics by the simulated hourly rainfields was used to identify runoff error propagation. Uncertainty in daily rainfall amounts alone caused higher errors in runoff (depth, peak, and time to peak) than those caused by uncertainties in the hourly proportions alone. However, the degradation, which reduced with runoff depth, caused by the combined uncertainties was not significantly different from that caused by the uncertainty of amounts alone.     

Validation of a full hydrodynamic model for large‐scale hydrologic modelling in the Amazon

A key aspect of large river basins partially neglected in large‐scale hydrological models is river hydrodynamics. Large‐scale hydrologic models normally simulate river hydrodynamics using simplified models that do not represent aspects such as backwater effects and flood inundation, key factors for some of the largest rivers of the world, such as the Amazon. In a previous paper, we have described a large‐scale hydrodynamic approach resultant from an improvement of the MGB‐IPH hydrological model. It uses full Saint Venant equations, a simple storage model for flood inundation and GIS‐based algorithms to extract model parameters from digital elevation models. In the present paper, we evaluate this model in the Solimões River basin. Discharge results were validated using 18 stream gauges showing that the model is accurate. It represents the large delay and attenuation of flood waves in the Solimões basin, while simplified models, represented here by Muskingum Cunge, provide hydrographs are wrongly noisy and in advance. Validation against 35 stream gauges shows that the model is able to simulate observed water levels with accuracy, representing their amplitude of variation and timing. The model performs better in large rivers, and errors concentrate in small rivers possibly due to uncertainty in river geometry. The validation of flood extent results using remote sensing estimates also shows that the model accuracy is comparable to other flood inundation modelling studies. Results show that (i) river‐floodplain water exchange and storage, and (ii) backwater effects play an important role for the Amazon River basin hydrodynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

Sensitivity analysis of EUROSEM using Monte Carlo simulation I: hydrological,soil and vegetation parameters

Knowledge about model uncertainty is essential for erosion modelling and provides important information when it comes to parameterizing models. In this paper a sensitivity analysis of the European soil erosion model (EUROSEM) is carried out using Monte Carlo simulation, suitable for complex non‐linear models, using time‐dependent driving variables. The analysis revealed some important characteristics of the model. The variability of the static output parameters was generally high, with the hydrologic parameters being the most important ones, especially saturated hydraulic conductivity and net capillary drive followed by the percentage basal area for the hydrological and vegetation parameters and detachability and cohesion for the soil erosion parameters. Overall, sensitivity to vegetation parameters was insignificant. The coefficient of variation for the sedigraph was higher than for the hydrograph, especially from the beginning of the rainstorm and up to the peak, and may explain difficulties encountered when trying to match simulated hydrographs and sedigraphs with observed ones. The findings from this Monte Carlo simulation calls for improved within‐storm modelling of erosion processes in EUROSEM. Information about model uncertainty will be incorporated in a new EUROSEM user interface. Copyright © 2000 John Wiley & Sons, Ltd.