Generalized Likelihood Uncertainty Estimation Methodology Research Articles

Coupled modelling of hydrological processes and grassland production in two contrasting climates

Abstract. Projections of global climate models suggest that ongoing human-induced climate change will lead to an increase in the frequency of severe droughts in many important agricultural regions of the world. Eco-hydrological models that integrate current understanding of the interacting processes governing soil water balance and plant growth may be useful tools to predict the impacts of climate change on crop production. However, the validation status of these models for making predictions under climate change is still unclear, since few suitable datasets are available for model testing. One promising approach is to test models using data obtained in “space-for-time” substitution experiments, in which samples are transferred among locations with contrasting current climates in order to mimic future climatic conditions. An important advantage of this approach is that the soil type is the same, so that differences in soil properties are not confounded with the influence of climate on water balance and crop growth. In this study, we evaluate the capability of a relatively simple eco-hydrological model to reproduce 6 years (2013–2018) of measurements of soil water contents, water balance components and grass production made in weighing lysimeters located at two sites within the TERENO-SoilCan network in Germany. Three lysimeters are located at an upland site at Rollesbroich with a cool, wet climate, while three others had been moved from Rollesbroich to a warmer and drier climate on the lower Rhine valley floodplain at Selhausen. Four of the most sensitive parameters in the model were treated as uncertain within the framework of the GLUE (generalized likelihood uncertainty estimation) methodology, while the remaining parameters in the model were set according to site measurements or data in the literature. The model satisfactorily reproduced the measurements at both sites, and some significant differences in the posterior ranges of the four uncertain parameters were found. In particular, the results indicated greater stomatal conductance as well an increase in dry-matter allocation below ground and a significantly larger maximum root depth for the three lysimeters that had been moved to Selhausen. As a consequence, the apparent water use efficiency (above-ground harvest divided by evapotranspiration) was significantly smaller at Selhausen than Rollesbroich. Data on species abundance on the lysimeters provide one possible explanation for the differences in the plant traits at the two sites derived from model calibration. These observations showed that the plant community at Selhausen had changed significantly in response to the drier climate, with a significant decrease in the abundance of herbs and an increase in the proportion of grass species. The differences in root depth and leaf conductance may also be a consequence of plasticity or acclimation at the species level. Regardless of the reason, we may conclude that such adaptations introduce significant additional uncertainties into model predictions of water balance and plant growth in response to climate change.

Scale-dependence of observational and modelling uncertainties in forensic flash flood analysis

The mismatch between the space–time scales of flash flood occurrence and those of the typical hydro-meteorological monitoring networks has stimulated the development of forensic flash flood analysis, which involves post-flood indirect peak discharge estimation in ungauged channels and flood response modelling driven by weather-radar rainfall estimates. However, both approaches are potentially affected by significant uncertainties. Assessment of scale dependence of such uncertainties is important to identify how uncertainty affecting forensic flash flood analysis increases with decreasing basin size. In this work, we apply the forensic methodology to the flash flood of November 18, 2013 in Sardinia (Italy). We introduce the ‘flash flood forensic consistency index’ as a tool to compare the probability distribution of peak discharge uncertainties from observational and model estimates, and to determine the scale effects of the forensic analysis concept. Uncertainties in field-based peak flow estimates are evaluated through a first-order error analysis of the Taylor-series approximation of the slope-conveyance method, whereas uncertainties in flash flood modelling are based on the Generalized Likelihood Uncertainty Estimation methodology using a distributed hydrologic model. Results show no significant relationship between observational and modelling uncertainties, considered independently, and basin area and channel bed slope. Conversely, when considering the interaction between the two uncertainty distributions, a relationship arises between their degree of overlap and basin size. In particular, with decreasing basin area or increasing channel bed slope, the absolute relative bias between the estimated peak flow values increases more than their relative uncertainties, decreasing the consistency index. This calls for more robust approaches for the analysis of flash flood response in small-sized rugged-relief mountain basins, which are of high interest for flood risk management.

Uncertainty Analysis of Stage-Discharge Curves by Generalized Likelihood Uncertainty Estimation (GLUE) Method

Calibration of the rating curve is a challenge due to uncertainty in the parameters. This problem increases in an area with considerable seasonal vegetation diversity. In this study, the generalized likelihood uncertainty estimation (GLUE) method was combined with the proposed stage-discharge model, introduced by Maghrebi et al., to compute the parameter uncertainty of the rating curve in the Main River in the UK and the Colorado River in Argentina. In GLUE methodology, the sensitivity and uncertainty analysis of each parameter were investigated by comparing the prior and posterior distribution of the effective parameters in the proposed method. It should be noted that all the parameters studied in the proposed model, such as power function parameters (a1, a2, and a3) and Manning’s roughness coefficient can be acquired with an optimal parameter domain using the calibration method. The results demonstrated low sensitivity of roughness parameters and high sensitivity of power function parameter a1 in comparison with other parameters. In order to evaluate the uncertainty results, two average relative interval length (ARILCI) and the percent of observations bracketed by the 95CI (PCI95%) factors at 95% confidence level were used. The results showed that if the observational data were selected from the middle level, a more favorable result would be obtained and demonstrated less uncertainty in the output range of the model.

Hydrological performance of lined permeable pavements in Norway

Abstract Lined permeable pavements (LPPs) are types of sustainable urban stormwater systems (SUDs) that are suitable for locations with low infiltration capacity or shallow groundwater levels. This study evaluated the hydrological performance of an LPP system in Norway using common detention indicators and flow duration curves (FDCs). Two hydrological models, the Storm Water Management Model (SWMM)-LID module and a reservoir model, were applied to simulate continuous outflows from the LPP system to plot the FDCs. The sensitivity of the parameters of the SWMM-LID module was assessed using the generalized likelihood uncertainty estimation methodology. The LPP system was found to detain the flow effectively based on the median values of the detention indicators (peak reduction = 89%, peak delay = 40 min, centroid delay = 45 min, T50-delay = 86 min). However, these indicators are found to be sensitive to the amount of precipitation and initial conditions. The reservoir model developed in this study was found to yield more accurate simulations (higher NSE) than the SWMM-LID module, and it can be considered a suitable design tool for LPP systems. The FDC offers an informative method to demonstrate the hydrological performance of LPP systems for stormwater engineers and decision-makers.

Hybrid Coupled Multifracture and Multicontinuum Models for Shale Gas Simulation by Use of Semi-Analytical Approach

Combining the advantages of multicontinuum and multifracture representations provides an easy-to-use tool to adequately capture the characteristic of the multiscaled fracture system in shale gas reservoir. A hybrid model is established on the basis of simplified conceptual productivity assumption, where the matrix volume is divided into two sub-domains (triple-porosity model and dual-depletion flowing model) and the fracture volume is represented by discrete finite conductivity fracture. In addition, the mechanisms of instant desorption, viscous flow and dual-depletion in matrix are taken into account. The rate transient responses are then obtained by use of semi-analytical approach. Based on the model, type curves are plotted and verified by comparing with alternative reliable methods. Different flow regimes in shale gas reservoirs can be identified and detected. The Generalized Likelihood Uncertainty Estimation methodology, based on probabilistic aggregation theory, is employed to integrating those two productivity models together such that the production can be predicted more accurately. A field example is applied to validate the applicability of this new model. Finally, it is concluded that the proposed model can predict the rate and cumulative rate more easily and practically.

Global sampling to assess the value of diverse observations in conditioning a real‐world groundwater flow and transport model

AbstractThe use of additional types of observational data has often been suggested to alleviate the ill‐posedness inherent to parameter estimation of groundwater models and constrain model uncertainty. Disinformation in observational data caused by errors in either the observations or the chosen model structure may, however, confound the value of adding observational data in model conditioning. This paper uses the global generalized likelihood uncertainty estimation methodology to investigate the value of different observational data types (heads, fluxes, salinity, and temperature) in conditioning a groundwater flow and transport model of an extensively monitored field site in the Netherlands. We compared model conditioning using the real observations to a synthetic model experiment, to demonstrate the possible influence of disinformation in observational data in model conditioning. Results showed that the value of different conditioning targets was less evident when conditioning to real measurements than in a measurement error‐only synthetic model experiment. While in the synthetic experiment, all conditioning targets clearly improved model outcomes, minor improvements or even worsening of model outcomes was observed for the real measurements. This result was caused by errors in both the model structure and the observations, resulting in disinformation in the observational data. The observed impact of disinformation in the observational data reiterates the necessity of thorough data validation and the need for accounting for both model structural and observational errors in model conditioning. It further suggests caution when translating results of synthetic modeling examples to real‐world applications. Still, applying diverse conditioning data types was found to be essential to constrain uncertainty, especially in the transport of solutes in the model.

Global Sensitivity and GLUE-Based Uncertainty Analysis of a 2D-1D Dual Urban Drainage Model

AbstractThis paper presents a global sensitivity and uncertainty analysis of a dual drainage model applied to a 0.049 km2 (4.9-ha) urban catchment based on generalized likelihood uncertainty estimation (GLUE). The model solves the nonlinear bidimensional (2D) and one-dimensional (1D) Saint-Venant equations to compute respectively the surface runoff and the flow in the sewer network. The sensitivity of the outlet hydrograph to the input parameters of the model was determined by using variance-based Sobol sensitivity indices. These indices highlight the great effect of the overland flow parameters on the model output, in contrast with the limited effect of the sewer-network parameters. The Manning coefficient of the impervious terrains was identified as the most influential model parameter. The effect on model calibration of two subjective factors of the GLUE methodology (the acceptability threshold and the shaping factor of the likelihood function) was analyzed. From this analysis, a NSE acceptability thre...

Uncertainty in a monthly water balance model using the generalized likelihood uncertainty estimation methodology

Hydrological models are simplified representations of natural processes and subject to errors. Uncertainty bounds are a commonly used way to assess the impact of an input or model architecture uncertainty in model outputs. Different sets of parameters could have equally robust goodness-of-fit indicators, which is known as Equifinality. We assessed the outputs from a lumped conceptual hydrological model to an agricultural watershed in central Chile under strong interannual variability (coefficient of variability of 25%) by using the Equifinality concept and uncertainty bounds. The simulation period ran from January 1999 to December 2006. Equifinality and uncertainty bounds from GLUE methodology (Generalized Likelihood Uncertainty Estimation) were used to identify parameter sets as potential representations of the system. The aim of this paper is to exploit the use of uncertainty bounds to differentiate behavioural parameter sets in a simple hydrological model. Then, we analyze the presence of equifinality in order to improve the identification of relevant hydrological processes. The water balance model for Chillan River exhibits, at a first stage, equifinality. However, it was possible to narrow the range for the parameters and eventually identify a set of parameters representing the behaviour of the watershed (a behavioural model) in agreement with observational and soft data (calculation of areal precipitation over the watershed using an isohyetal map). The mean width of the uncertainty bound around the predicted runoff for the simulation period decreased from 50 to 20 m3s−1 after fixing the parameter controlling the areal precipitation over the watershed. This decrement is equivalent to decreasing the ratio between simulated and observed discharge from 5.2 to 2.5. Despite the criticisms against the GLUE methodology, such as the lack of statistical formality, it is identified as a useful tool assisting the modeller with the identification of critical parameters.

The Influence of Objective Function and Acceptability Threshold on Uncertainty Assessment of an Urban Drainage Hydraulic Model with Generalized Likelihood Uncertainty Estimation Methodology

Urban drainage model is an important computer-aided tool in stormwater management and drainage planning and designing. A popular urban drainage hydraulic model, stormwater management model (SWMM), was applied in a pump lifting combined sewer system for a high-intensity urban catchment located in Shanghai, China. Uncertainty of SWMM water quantity parameters was assessed with generalized likelihood uncertainty estimation (GLUE) methodology. The sensitivity of parameters was discussed and compared based on the results of uncertainty analysis. To discuss the influence of the acceptability threshold on model parameter sensitivity and the margin of uncertainty band, the GLUE approach was applied several times varying acceptability threshold. The results indicated that a higher acceptability threshold value is contributed to achieve a stricter verification with a high confidence level, and the uncertainty analysis significant level can be featured by the value of acceptability threshold. The selection of acceptability threshold value can be regarded as a tradeoff process. Both reducing the low efficient simulation and reducing computation cost should be considered for the selection of acceptability threshold. Moreover, the GLUE approach was applied several times varying different objective functions with corresponding acceptability thresholds. The results indicated that some parameters may be sensitive to a specific objective function, and other parameters may be sensitive to another objective function. Some parameters cannot easily identified when a single objective function was used within the GLUE approach, and a multiple-objective function combined different objective functions requirements, may be a alternative approach to reduce the model prediction uncertainty.

Uncertainty assessment of water quality modeling for a small-scale urban catchment using the GLUE methodology: a case study in Shanghai, China.

There is often great uncertainty in water quality modeling for urban drainage systems because water quality variation in systems is complex and affected by many factors. The stormwater management model (SWMM) was applied to a small-scale urban catchment with a simple and well-maintained stormwater drainage system without illicit connections. This was done to assess uncertainty in build-up and wash-off modeling of pollutants within the generalized likelihood uncertainty estimation (GLUE) methodology, based on a well-calibrated water quantity model. The results indicated great uncertainty of water quality modeling within the GLUE methodology. Comparison of uncertainties in various pollutant build-up and wash-off models that were available in SWMM indicated that those uncertainties varied slightly. This may be a consequence of the specific characteristics of rainfall events and experimental sites used in the study. The uncertainty analysis of water quality parameters in SWMM is conducive to effectively evaluating model reliability, and provides an experience base for similar research and applications.

Sensitivity of Subjective Decisions in the GLUE Methodology for Quantifying the Uncertainty in the Flood Inundation Map for Seymour Reach in Indiana, USA

Generalized likelihood uncertainty estimation (GLUE) is one of the widely-used methods for quantifying uncertainty in flood inundation mapping. However, the subjective nature of its application involving the definition of the likelihood measure and the criteria for defining acceptable versus unacceptable models can lead to different results in quantifying uncertainty bounds. The objective of this paper is to perform a sensitivity analysis of the effect of the choice of likelihood measures and cut-off thresholds used in selecting behavioral and non-behavioral models in the GLUE methodology. By using a dataset for a reach along the White River in Seymour, Indiana, multiple prior distributions, likelihood measures and cut-off thresholds are used to investigate the role of subjective decisions in applying the GLUE methodology for uncertainty quantification related to topography, streamflow and Manning’s n. Results from this study show that a normal pdf produces a narrower uncertainty bound compared to a uniform pdf for an uncertain variable. Similarly, a likelihood measure based on water surface elevations is found to be less affected compared to other likelihood measures that are based on flood inundation area and width. Although the findings from this study are limited due to the use of a single test case, this paper provides a framework that can be utilized to gain a better understanding of the uncertainty while applying the GLUE methodology in flood inundation mapping.

Parameter sensitivity and uncertainty analysis of a stormwater runoff model

In this study, Generalized Likelihood Uncertainty Estimation (GLUE) methodology was used to perform uncertainty analysis of a stormwater model, which randomly generates parameter sets and identifies behavioral ones with higher likelihood. Latin Hypercube Sampling (LHS) is used to generate parameter sets. The Model for Urban Stormwater Improvement Conceptualization was chosen as an appropriate model for stormwater runoff. Prediction limits are determined by selecting the cutoff threshold for likelihood function. Study area is Goonja Drainage Basin located in the city of Seoul, Korea. From the results, maximum likelihood value for calibration is 0.78 and 0.73 for validation. The p-factors for the calibration and validation are 87 and 83%, respectively. The p-factor for all storm events is 85%. These are all acceptable values as the results are considered good when 60% or more of the observed data are bracketed by prediction limits. Overall, it was shown that, using GLUE methodology with LHS, the model calibrated well for the basin considered in this study.

GLUE based marine X-band weather radar data calibration and uncertainty estimation

The Generalized Likelihood Uncertainty Estimation methodology (GLUE) is investigated for radar rainfall calibration and uncertainty assessment. The method is used to calibrate radar data collected by a Local Area Weather Radar (LAWR).In contrast to other LAWR data calibrations, the method combines calibration with uncertainty estimation. Instead of searching for a single set of calibration parameters, the method uses the observations to construct distributions of the calibration parameters. These parameter sets provide valuable knowledge of parameter sensitivity and the uncertainty.Two approaches are analyzed; the static calibration approach, where the LAWR is calibrated once for a long period and the dynamic approach, where the estimate is continuously adjusted based on ground observations.The analysis illustrates that the static calibration performs insufficiently, whereas the dynamic adjustment improves the performance significantly.It is found that even if the dynamic adjustment method is used the uncertainty of rainfall estimates can still be significant.

Informal uncertainty analysis (GLUE) of continuous flow simulation in a hybrid sewer system with infiltration inflow – consistency of containment ratios in calibration and validation?

Abstract. Monitoring of flows in sewer systems is increasingly applied to calibrate urban drainage models used for long-term simulation. However, most often models are calibrated without considering the uncertainties. The generalized likelihood uncertainty estimation (GLUE) methodology is here applied to assess parameter and flow simulation uncertainty using a simplified lumped sewer model that accounts for three separate flow contributions: wastewater, fast runoff from paved areas, and slow infiltrating water from permeable areas. Recently GLUE methodology has been critisised for generating prediction limits without statistical coherence and consistency and for the subjectivity in the choice of a threshold value to distinguish "behavioural" from "non-behavioural" parameter sets. In this paper we examine how well the GLUE methodology performs when the behavioural parameter sets deduced from a calibration period are applied to generate prediction bounds in validation periods. By retaining an increasing number of parameter sets we aim at obtaining consistency between the GLUE generated 90% prediction limits and the actual containment ratio (CR) in calibration. Due to the large uncertainties related to spatio-temporal rain variability during heavy convective rain events, flow measurement errors, possible model deficiencies as well as epistemic uncertainties, it was not possible to obtain an overall CR of more than 80%. However, the GLUE generated prediction limits still proved rather consistent, since the overall CRs obtained in calibration corresponded well with the overall CRs obtained in validation periods for all proportions of retained parameter sets evaluated. When focusing on wet and dry weather periods separately, some inconsistencies were however found between calibration and validation and we address here some of the reasons why we should not expect the coverage of the prediction limits to be identical in calibration and validation periods in real-world applications. The large uncertainties result in wide posterior parameter limits, that cannot be used for interpretation of, for example, the relative size of paved area vs. the size of infiltrating area. We should therefore try to learn from the significant discrepancies between model and observations from this study, possibly by using some form of non-stationary error correction procedure, but it seems crucial to obtain more representative rain inputs and more accurate flow observations to reduce parameter and model simulation uncertainty.

An Approach Using a 1D Hydraulic Model, Landsat Imaging and Generalized Likelihood Uncertainty Estimation for an Approximation of Flood Discharge

Collection and investigation of flood information are essential to understand the nature of floods, but this has proved difficult in data-poor environments, or in developing or under-developed countries due to economic and technological limitations. The development of remote sensing data, GIS, and modeling techniques have, therefore, proved to be useful tools in the analysis of the nature of floods. Accordingly, this study attempts to estimate a flood discharge using the generalized likelihood uncertainty estimation (GLUE) methodology and a 1D hydraulic model, with remote sensing data and topographic data, under the assumed condition that there is no gauge station in the Missouri river, Nebraska, and Wabash River, Indiana, in the United States. The results show that the use of Landsat leads to a better discharge approximation on a large-scale reach than on a small-scale. Discharge approximation using the GLUE depended on the selection of likelihood measures. Consideration of physical conditions in study reaches could, therefore, contribute to an appropriate selection of informal likely measurements. The river discharge assessed by using Landsat image and the GLUE Methodology could be useful in supplementing flood information for flood risk management at a planning level in ungauged basins. However, it should be noted that this approach to the real-time application might be difficult due to the GLUE procedure.

Water balance modeling of Upper Blue Nile catchments using a top-down approach

Abstract. The water balances of twenty catchments in the Upper Blue Nile basin have been analyzed using a top-down modeling approach based on Budyko's hypotheses. The objective of this study is to obtain better understanding of water balance dynamics of upper Blue Nile catchments on annual and monthly time scales and on a spatial scale of meso scale to large scale. The water balance analysis using a Budyko-type curve at annual scale reveals that the aridity index does not exert a first order control in most of the catchments. This implies the need to increase model complexity to monthly time scale to include the effects of seasonal soil moisture dynamics. The dynamic water balance model used in this study predicts the direct runoff and other processes based on the limit concept; i.e. for dry environments since rainfall amount is small, the aridity index approaches to infinity or equivalently evaporation approaches rainfall and for wet environments where the rainfall amount is large, the aridity index approaches to zero and actual evaporation approaches the potential evaporation. The uncertainty of model parameters has been assessed using the GLUE (Generalized Likelihood Uncertainty Estimation) methodology. The results show that the majority of the parameters are reasonably well identifiable. However, the baseflow recession constant was poorly identifiable. Parameter uncertainty and model structural errors could be the reason for the poorly identifiable parameter. Moreover, a multi-objective model calibration strategy has been employed to emphasize the different aspects of the hydrographs on low and high flows. The model has been calibrated and validated against observed streamflow time series and it shows good performance for the twenty study catchments in the upper Blue Nile. During the calibration period (1995–2000) the Nash and Sutcliffe efficiency (E NS) for monthly flow prediction varied between 0.52 to 0.93 (dominated by high flows), while it varied between 0.32 to 0.90 using logarithms of flow series (indicating the goodness of low flow simulations). The model is parsimonious and it is suggested that the calibrated parameters could be used after some more regionalization efforts to predict monthly stream flows in ungauged catchments of the Upper Blue Nile basin, which is the vast majority of catchments in that region.

Processes influencing model-data mismatch in drought-stressed, fire-disturbed eddy flux sites

[1] Semiarid forests are very sensitive to climatic change and among the most difficult ecosystems to accurately model. We tested the performance of the Biome-BGC model against eddy flux data taken from young (years 2004–2008), mature (years 2002–2008), and old-growth (year 2000) ponderosa pine stands at Metolius, Oregon, and subsequently examined several potential causes for model-data mismatch. We used the Generalized Likelihood Uncertainty Estimation methodology, which involved 500,000 model runs for each stand (1,500,000 total). Each simulation was run with randomly generated parameter values from a uniform distribution based on published parameter ranges, resulting in modeled estimates of net ecosystem CO2 exchange (NEE) that were compared to measured eddy flux data. Simulations for the young stand exhibited the highest level of performance, though they overestimated ecosystem C accumulation (−NEE) 99% of the time. Among the simulations for the mature and old-growth stands, 100% and 99% of the simulations underestimated ecosystem C accumulation. One obvious area of model-data mismatch is soil moisture, which was overestimated by the model in the young and old-growth stands yet underestimated in the mature stand. However, modeled estimates of soil water content and associated water deficits did not appear to be the primary cause of model-data mismatch; our analysis indicated that gross primary production can be accurately modeled even if soil moisture content is not. Instead, difficulties in adequately modeling ecosystem respiration, mainly autotrophic respiration, appeared to be the fundamental cause of model-data mismatch.

Uncertainty assessment of a model for biological nitrogen and phosphorus removal: Application to a large wastewater treatment plant

In the last few years, the use of mathematical models in WasteWater Treatment Plant (WWTP) processes has become a common way to predict WWTP behaviour. However, mathematical models generally demand advanced input for their implementation that must be evaluated by an extensive data-gathering campaign, which cannot always be carried out. This fact, together with the intrinsic complexity of the model structure, leads to model results that may be very uncertain. Quantification of the uncertainty is imperative. However, despite the importance of uncertainty quantification, only few studies have been carried out in the wastewater treatment field, and those studies only included a few of the sources of model uncertainty. Seeking the development of the area, the paper presents the uncertainty assessment of a mathematical model simulating biological nitrogen and phosphorus removal. The uncertainty assessment was conducted according to the Generalised Likelihood Uncertainty Estimation (GLUE) methodology that has been scarcely applied in wastewater field. The model was based on activated-sludge models 1 (ASM) and 2 (ASM2). Different approaches can be used for uncertainty analysis. The GLUE methodology requires a large number of Monte Carlo simulations in which a random sampling of individual parameters drawn from probability distributions is used to determine a set of parameter values. Using this approach, model reliability was evaluated based on its capacity to globally limit the uncertainty. The method was applied to a large full-scale WWTP for which quantity and quality data was gathered. The analysis enabled to gain useful insights for WWTP modelling identifying the crucial aspects where higher uncertainty rely and where therefore, more efforts should be provided in terms of both data gathering and modelling practises.

Modified Generalized Likelihood Uncertainty Estimation (GLUE) Methodology for Considering the Subjectivity of Likelihood Measure Selection

The generalized likelihood uncertainty estimation (GLUE) methodology has been widely used in many areas as an effective and general strategy for model calibration and uncertainty estimation associated with complex models. The application of GLUE requires a formal definition of a likelihood measure. However, it has been recognized that the choice of a likelihood measure is inherently subjective. This, in turn, introduces a new kind of uncertainty—the uncertainty owing to the lack of knowledge in choosing the true likelihood measure in the GLUE methodology. This study proposes a practical framework to address this uncertainty by using multiple likelihood measures, analogous to considering multiple expert opinions. The final uncertainty probability estimates are then obtained by combining the estimates from individual likelihood measures based on probability theory.

Uncertainty assessment of a membrane bioreactor model using the GLUE methodology

A mathematical model for the simulation of physical-biological organic removal by means of a membrane bioreactor (MBR) has been previously developed and tested. This paper presents an analysis of the uncertainty of the MBR model. Particularly, the research explores the applicability of the Generalised Likelihood Uncertainty Estimation (GLUE) methodology that is one of the most widely used methods for investigating the uncertainties in the hydrology and that now on is spreading in other research field. For the application of the GLUE methodology, several Monte Carlo simulations have been run varying the all model influential parameters simultaneously. The model was applied to an MBR pilot plant located at the Acqua dei Corsari WWTP (Palermo, IT) where water quality data were gathered. In particular, the MBR pilot plant consists of a lab-scale hollow fibre membrane module in submerged configuration. The GLUE methodology enabled us to gain useful insight about the robustness of the model approach. Particularly, the results showed that the biological process is influenced mainly by the parameters characterising the formation and degradation of Soluble Microbial Products, whereas the fouling phenomenon is mainly influenced by the backwashing efficiency. The application of the GLUE methodology shows that the model considered for the MBR simulation is somehow too simple in order to predict plants performances. Indeed, GLUE enabled us to identify the main model components that needs to be improved and where much attention has to be paid both in terms of model algorithms and quality data to be gathered. This studies confirmed the suitability of the GLUE methodology as a powerful tool for simplified screening methodology to assess the uncertainty also in the field of wastewater treatment.