Towards quantification of uncertainty in predicting water quality failures in integrated catchment model studies

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Towards quantification of uncertainty in predicting water quality failures in integrated catchment model studies A.N.A. Schellart a,*, S.J. Tait b, R.M. Ashley a a b

Pennine Water Group, Department of Civil & Structural Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Pennine Water Group, School of Engineering Design and Technology, University of Bradford, Bradford, BD7 1DP, UK

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abstract

Article history:

This paper describes the development and application of a method for estimating uncer-

Received 13 August 2009

tainty in the prediction of sewer flow quantity and quality and how this may impact on the

Received in revised form

prediction of water quality failures in integrated catchment modelling (ICM) studies. The

29 April 2010

method is generic and readily adaptable for use with different flow quality prediction

Accepted 4 May 2010

models that are used in ICM studies. Use is made of the elicitation concept, whereby expert

Available online 11 May 2010

knowledge combined with a limited amount of data are translated into probability distributions describing the level of uncertainty of various input and model variables. This type

Keywords:

of approach can be used even if little or no site specific data is available. Integrated

Uncertainty

catchment modelling studies often use complex deterministic models. To apply the results

Integrated modelling study

of elicitation in a case study, a computational reduction method has been developed in

Sewer emissions

order to determine levels of uncertainty in model outputs with a reasonably practical level

Receiving water impact

of computational effort. This approach was applied to determine the level of uncertainty in

Flow quality modelling

the number of water quality failures predicted by an ICM study, due to uncertainty asso-

Water quality failure

ciated with input and model parameters of the urban drainage model component of the ICM. For a small case study catchment in the UK, it was shown that the predicted number of water quality failures in the receiving water could vary by around 45% of the number predicted without consideration of model uncertainty for dissolved oxygen and around 32% for unionised ammonia. It was concluded that the potential overall levels of uncertainty in the ICM outputs could be significant. Any solutions designed using modelling approaches that do not consider uncertainty associated with model input and model parameters may be significantly over-dimensioned or under-dimensioned. With changing external inputs, such as rainfall and river flows due to climate change, better accounting for uncertainty is required. ª 2010 Elsevier Ltd. All rights reserved.

1.

Introduction

1.1.

Integrated urban catchment modelling

Discharges from urban drainage catchments can have a major impact on the quality of receiving surface waters. FWR (1998)

describes the advantages in using an integrated approach to managing urban wet weather discharges whereby the sewer system, the treatment plant and the receiving water are considered as a single interconnected system. An Integrated Catchment Model (‘ICM’) approach is therefore seen as an important technique for managing the impact of drainage and

* Corresponding author. E-mail addresses: [email protected] (A.N.A. Schellart), [email protected] (S.J. Tait), [email protected] (R.M. Ashley). 0043-1354/$ e see front matter ª 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2010.05.001

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waste water systems on the environment (FWR, 1998; Rauch et al., 1998; Muschalla et al., 2008; Willems, 2008). An ICM study typically uses a rainfall generation model, a rainfall runoff model, an urban drainage flow quantity and quality model, a waste water treatment model and a receiving water impact model. The sub-models used within an ICM study can range in complexity from conceptual models (Willems, 2010), to complex deterministic models that are composed of numerous interlinked empirically calibrated equations describing processes that affect water quality (Priestley and Barker, 2006; Benedetti et al., 2005). Commercial software packages with many linked deterministic equations are commonly used in engineering practice (Priestley and Barker, 2006; Osborne and Lau, 2003). These software packages are typically used without consideration of the uncertainty involved in the solution of their numerous deterministically based equations.

used can have a “weak” physical meaning. Various approaches have been developed to overcome the problem of long model run times when estimating uncertainty in model outputs. Benedetti et al. (2005) and Rousseau et al. (2001) describe a method whereby a ‘probabilistic’ shell is built around a deterministic model to quantify the uncertainty of the model predictions. Khuri and Cornell (1987) describe the principle of response surface methods, which can be used to describe the solution of more complicated models. The response surface methodology was formally developed by Box and Wilson in the 1950’s (Box, 1954). A response database was used by Dahal et al. (2005) to estimate the reliability of river dikes on a tidal river. It has also been used by Schellart et al. (2008, 2010) as a tool to calculate the uncertainty in sewer sediment deposit depth predictions.

1.3. 1.2. Uncertainty in integrated urban catchment models and urban drainage models Even though the need to deal more explicitly with uncertainty of urban drainage systems is argued by Harremoe¨s and Madsen (1999), Bertrand-Krajewski (2006) and Ashley et al. (2005), relatively few studies deal with the quantification of uncertainty in urban drainage modelling (Willems and Berlamont, 1999; Clemens and Von der Heide, 1999; Thorndahl et al., 2008) and uncertainty in water quality processes in urban drainage (Bertrand-Krajewski and Bardin, 2002; Kanso et al., 2003; Mourad et al., 2005; McCarthy et al., 2008). Fewer studies deal with Integrated Catchment Models that include water quality processes, Freni et al. (2008), Mannina et al. (2006) and Willems (2008). Mannina et al. (2006), Freni et al. (2008) and Thorndahl et al. (2008) have all used the Generalised Likelihood Uncertainty Estimation (GLUE) method developed by Beven and Binley (1992) to evaluate overall uncertainty. Willems (2008) used variance decomposition to split total prediction uncertainty into contributions of various uncertainty sources and the different conceptual models within an integrated catchment model. The method described in this paper is different to the methods mentioned above, and is known as a ‘forward uncertainty propagation method’ in combination with a model reduction method, instead of a ‘conditioning of uncertainty on data’ method such as GLUE, as defined by the decision tree for selecting an uncertainty methodology as described by Pappenberger et al. (2006). For complex hydrodynamic models of typical urban drainage systems, computational resources may quickly become a limiting factor when estimating uncertainty in the model output, as described by Thorndahl et al. (2008). Thorndahl et al. (2008) needed to use 10 personal computers run for several weeks when applying the GLUE methodology on a small catchment in Denmark. Conceptual models are therefore often used, because of their relatively short computational run times. Haydon and Deletic (2009) describe that, even when using a simplistic modelling approach, running Monte Carlo simulations can take a week of run time per input variable or model parameter for a practical system. In conceptual models it is also difficult to gather expert judgement on parameter value ranges, as the parameters

Classification of uncertainty

Researchers have described different classification systems to identify uncertainty types (e.g. Harremoe¨s and Madsen, 1999; Slijkhuis et al., 1999; Korving, 2004). Many input parameters that are used to describe natural quantities are not fixed values. In this paper this type of uncertainty will be referred to as model input uncertainty. Most water quality relationships have been empirically calibrated using laboratory or field data. These equations are then implemented in many model studies, often without reference to the original circumstances in which the equations were developed. There is always a level of uncertainty as to how well these calibrated equations represented the original calibration data, and also how large the uncertainty is in the original measured data. In this paper this type of uncertainty will be referred to as model parameter uncertainty. Finally, there are also uncertainty types classified as ‘ignorance’ by Wynne (1992), which is nonreducible and cannot be quantified. This subdivision of uncertainty into these categories is pragmatic providing a logical structure to organise the uncertainty analysis presented here. This paper concentrates on the model input and model parameter related uncertainties, as these are quantifiable when sufficient data or expert knowledge is available.

1.4.

Elicitation of expert knowledge

There is often limited data and model development is not always well documented, however, relevant expert knowledge may be available. If this knowledge is used, uncertainty levels in model inputs and parameters can be estimated through elicitation. Garthwaite et al. (2005) describe the concept of eliciting probability distributions, or ‘the process of formulating a person’s knowledge and beliefs of uncertain quantities into a (joint) probability distribution for those quantities’. Garthwaite et al. (2005) and Kadane and Wolfson (1998) describe several methods for eliciting probability distributions. Kadane and Wolfson (1998) also describe elicitation examples from a wide range of areas such as economics, clinical trials, demography and macro-economics. O’Hagan (1998) describes two other elicitation examples, future capital maintenance of water treatment works and hydraulic conductivity of rocks at a potential nuclear waste disposal site.

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1.5.

Aim of this study

This paper describes the development of a method that estimates uncertainty in complex deterministic ICM predictions using prior elicited probability distributions combined with a model reduction method such as a response database. This method is applied to a small urban drainage catchment in the UK to estimate the uncertainty associated with the predicted number of water quality failures from combined sewer overflow discharges. Prior to the study described in this paper, an ICM study using deterministic models had been used by the sewer operator in order to assess compliance with receiving water quality standards. This paper describes a ‘didactical’ example, where uncertainty in the sewer flow quality and quantity component of the ICM is studied and described in statistical terms using a form of elicitation. The uncertainty in the sewer flow quality and quantity model inputs and model parameters is used to estimate the range and number of water quality failures. Monte Carlo simulations have been carried out using a response database. The uncertainty analysis methods used in this paper are not new in themselves, but their application to estimate uncertainty in the outputs of an ICM is novel. Although this paper only takes uncertainty in the sewer flow quality and quantity model inputs and parameters into account, the method is generic and can also be used to assess uncertainty in the ICM output based on uncertainty in other ICM components.

2. Description of case study and original deterministic ICM An ICM study had been carried out by a UK sewer operator in order to demonstrate compliance with the Fundamental Intermittent Standards (FIS), FWR (1998), that are used to set allowed discharges from Combines Sewer Overflows (CSOs) in England. The calibration of the original ICM was carried out to current industrial standards, e.g. WaPUG (2002). The ICM for this catchment comprised several sets of deterministic equations embedded in different commercial software packages (Fig. 1). Data was transferred manually between these software packages. A 10-year future rainfall series, from 2010 to 2019 with a 5min resolution, had been derived for this catchment using the stochastic rainfall generator STORMPAC (WRc, 2009). A future

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rainfall series had been used, as the ICM study was used to determine capital investment requirements in order to restore or maintain compliance with the FIS. The MIKE NAM package (Rainfall runoff model; DHI, 2009) had been used to calculate the river flow quantity just upstream of the urban area and the river reach being examined for compliance. The river catchment rainfall runoff model, MIKE NAM (DHI, 2009) had been calibrated, using the following flow calibration parameters: Umax (maximum water content in surface storage); Lmax (Maximum water content in root zone); CQOF (overland flow runoff coefficient), CK12 (Time constant for routing interflow and overland flow); TOF (Root zone threshold for overland flow). These parameters were adjusted until the predictions of the NAM model visually matched the observed river flow rate data series and predicted and accumulated flows matched. A comparison of 6-years of 15-min gauged river flow data with modelled flow, showed that the modelled flow peaks fell within a þ or 20% range of the measured maximum flows. Infoworks CS (v 7.0) had been used to calculate runoff from the urban catchment surfaces, flows in the sewer system and hence the quantity and quality of combined sewer overflow emissions and quantity of the WwTW effluent entering the river. The combined sewer system serves an area of 294 ha and has around 11,000 contributing inhabitants. The system contained one Waste water Treatment Works (WwTW) and five combined sewer overflows (CSOs), all discharging into the same river. The Infoworks CS model was calibrated using flow quantity and water depth data obtained from a short term sewer flow survey. Three dry weather days and 6 storm events were used, following the selection criteria recommended in WaPUG (2002). The movement of the following pollutants had been modelled: sediments; Biochemical Oxygen Demand (BOD) in the dissolved phase as well as attached to sediment; and unionised Ammonia (NHþ 4 ). The case study Infoworks CS model consisted of 505 conduits, 530 nodes and 5 CSO structures. A river impact assessment tool developed by Priestley and Barker (2006) had been used for the assessment of the impact on the receiving water downstream of the urban area. The impacts of combined sewer overflow events and WwTW effluent on the receiving water was estimated over a 4.5 km river stretch downstream of the WwTW location. The river impact model calculates river water depths and velocities, assuming steady, uniform and fully turbulent flow. A StreeterPhelps type model (FWR, 1998) is used to estimate the decay of

Fig. 1 e Overview of the Integrated Catchment Modelling procedure used in the test catchment.

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Ammonia, BOD and re-aeration within the river reach. The input to the river impact model is (see also Fig. 1):

to the level of uncertainty in the outputs used for compliance testing.

 A time series of river flow quantity at the upstream end of the river reach for which compliance needs to be tested (derived from the NAM model output).  Mean and standard deviation of the concentration of flow quality parameters at the upstream boundary derived from 30 samples taken in the summer periods over 5 years (BOD mean 1.9 mg/L and standard deviation 1.75 mg/L, Ammonia mean 0.001 mg/L and standard deviation 0.002 mg/L, Ammonium 0.081 mg/L standard deviation 0.094 mg/L, Dissolved Oxygen mean 8.86 mg/L standard deviation 1.355 mg/L, pH mean 7.76 and standard deviation 0.249, Temperature 13.696  C and standard deviation 1.868  C).  A time series of WwTW effluent flow quantity and quality. WwTW effluent flow quantity is derived from the Infoworks CS model. WwTW effluent quality is derived from samples (BOD mean 9.20 mg/L and standard deviation 3.82 mg/L and NHþ 4 mean 1.26 mg/L and standard deviation 2.29 mg/L, based on 181 samples taken between January 2000 and August 2006).  Flow quantity and quality of all sewer system spill events (derived from Infoworks CS model).  Geometric data and re-aeration parameters of the river reach where compliance is to be tested (downstream of the urban area). (Channel slope was 4.4 m/km, the channel width 2.5 m, side slope 0.675 m/m and the Manning’s roughness 0.04).

3.3. Elicitation of probability distributions of model and input variables

Based on the inputs described above, the river impact model calculates the numbers of failures of the FIS for a defined time period.

3.

Uncertainty estimation method

The method used in this study for estimating levels of uncertainty in the outputs of an ICM such as described in Section 2 can be summarised as follows:

3.1. Mapping the processes within the integrated catchment model In order to create an overview of the deterministic modelling relationships and their integration, all equations, inputs and parameters are mapped. All possible sub-model outputs are listed, as well as the outputs necessary for testing compliance with water quality standards. As described by Rauch et al. (1998), existing models used as sub-models for ICM studies are often too complex, and not all their possible outputs need to be available as inputs to other sub-models. It is therefore important to clearly identify all the model input and model parameter variables that influence the values of the outputs required by an ICM study.

3.2.

Initial sensitivity check

An initial sensitivity check is carried out to eliminate, from further study, processes that will not contribute significantly

Knowledge on the listed model inputs and model parameters is collected from available experts, literature sources, and available data. Garthwaite et al. (2005), Kadane and Wolfson (1998) and O’Hagan (1998) describe a number of different elicitation methods to translate this kind of knowledge into probability distributions to describe uncertainty. Kadane and Wolfson (1998) defined two different types of elicitation e predictive and structural. In a predictive elicitation study experts are asked about their opinion on the uncertainty of a dependent variable, given an expected range of the predictor variables. In a structural elicitation study, experts are asked directly to define a priori distributions of the model and input parameters. The method followed in this paper is structurally based, experts were asked to define a confidence interval for identified model inputs and model parameters. The details of the elicitation method used to obtain the confidence intervals for the model and input parameters in the case study are described for each parameter in detail in Section 4.3.

3.4.

Sensitivity analysis

After eliciting the probability distributions, a sensitivity analysis is carried out. The model inputs and model parameters are systematically varied, according to their estimated uncertainty ranges obtained from the elicitation study. Model simulations were carried out for a timescale relevant to the particular quality standards. Significant changes in predicted water quality failures based on the adjustments of single input and model parameters are thus identified. This approach was favoured because of the practical level of computational effort, which meant that the method could be applied to catchments using complex models. For example, for the case study sewer network the flow and quality simulation of a 5-month period using Infoworks CS takes approximately 5.5 h to run on a desktop computer (Pentium 4 3.0 GHz processor and 1 GB of RAM) and generates approximately 7 GB of model output. The case study is, however, relatively small as the Infoworks CS model consists only of 505 conduits and 530 nodes. For a larger urban area, the number of nodes and conduits modelled in a hydrodynamic sewer network model could easily exceed several thousand. Computation times for this type of urban drainage model are roughly proportionate to the number of nodes and conduits modelled.

3.5.

Estimation of uncertainty of the ICM outputs

The most influential model inputs and model parameters identified in the sensitivity analysis are used to estimate the uncertainty in the number of predicted water quality failures. A response database is used to describe the variation in frequency of water quality failures based on the estimated uncertainty ranges of input and model parameter values. The elicited probability distributions are then used in Monte Carlo simulations, thereby interpolating water quality failures from

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the response database instead of running a computationally expensive deterministic ICM. An estimate of the range and probability distribution of the frequency of water quality failure for selected water quality determinants is derived from the Monte Carlo simulation results.

4.

Results of the uncertainty analysis

This section describes how the uncertainty analysis method explained in Section 3 was applied to the case study.

4.1. Mapping of processes and equations in the original Integrated Catchment Model The uncertainty analysis described in this paper only concerns the Infoworks CS component of the ICM study. Fig. 2 shows diagrammatically how the water and pollutant flows are represented by a number of empirically calibrated relationships in the Infoworks CS model. The rainfall runoff volume and the rainfall runoff routing are calculated for each of the urban sub-catchments. Of the possible rainfall runoff and runoff routing model options in Infoworks CS, the New UK Percentage Runoff model (Packman, 1990) and the Double Linear Reservoir (Wallingford) model (Sarginson and Nussey, 1982) had been selected. In the flow quality module, a surface pollutant model calculates the mass inflow of sediment and pollutants from the sub-catchment surfaces and from the gully pots into the sewer system. The temporal build-up of pollutants in the gully pots during dry weather periods and the flushing of the gully pots during wet weather events is calculated via a number of empirically calibrated relationships (Gent et al., 1992). The amount of sediment building up on the sub-catchment surfaces and the amount of pollutants attached to the sediment washed off the subcatchment surfaces is estimated using empirically calibrated relationships (Bujon and Herremans, 1990). Dissolved pollutants are represented as entering the system through gully pots at each node. The amount of suspended sediment and dissolved and attached pollutants in the domestic waste water flow is based on historical data reported by Ainger et al. (1998). The erosion, transport and deposition of suspended sediment is calculated using the Ackers-White relationships (Ackers et al., 1996). Sediment can be deposited as well as reeroded in the network conduits throughout the model simulation. The sewer network model calculates hydraulic parameters in each conduit using the conservation of mass and momentum equations, combined with empirical energy loss equations. These are solved iteratively throughout the network. The conduit model estimates the transport of dissolved pollutants through the conduits based on simple advection and the conservation of mass of sediment, pollutants and water.

4.2.

Initial sensitivity check

The FIS (FWR, 1998) for a river reach comprise of threshold concentration values for DO and NH4þ, for 1 or 6 h durations, with return periods of 1, 3 and 12 months. The flows of BOD and NH4þ through the urban catchment and sewer system

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have therefore been included in this initial sensitivity check, Fig. 2 shows their potential flow paths. The original deterministic ICM, or ‘baseline model’ has its model and input parameters fixed using values obtained without consideration of uncertainty. Model runs whereby all pollutant sources except the one being examined are set to zero, are used to identify the relative mass of pollutants emitted into the receiving water from different pollutant sources. Several equations in the urban drainage model have an element of build-up, decay, erosion or depletion over time. All sensitivity analyses and uncertainty analyses therefore need to be carried out using ‘long term’ simulations and not single storm events. The use of time-series is also advocated by Rauch et al. (2002), whereas Thorndahl et al. (2008) advocates the use of a large enough number of events in order to reduce event specific uncertainties. The period used for analysis of FIS standards in the UK is commonly 10-years of summer periods. In order to reduce computation times to a reasonable level the sensitivity analysis for each pollutant was carried out for a single 5-month summer period obtained from a representative year from the 10-year rainfall time series. This representative year was selected by calculating the cumulative rainfall of each year, and taking the year where the cumulative rainfall was closest to the average yearly cumulative rainfall of the 10-year period. Table 1 includes all model inputs and parameters that have not been studied further because they did not significantly add to the total amount of pollutants entering the river. All the remaining processes were included in the elicitation process, Table 1 also includes the processes rejected during this elicitation process.

4.3.

Elicitation of probability distributions

The elicitation process was undertaken by the paper authors, the method was structurally based and distributions of selected input and model parameters were estimated based mainly on expert knowledge as field data was very limited. The paper authors are academic experts in urban drainage quality processes, they initially carried out the elicitation amongst themselves and the results were subsequently discussed with the urban drainage modelling experts at the sewer operator. After discussion with the paper authors and the modelling experts at the sewer operator, a consensus was reached so that for each elicited model input and model parameter a single probability distribution was selected. Published data and information available on the development of the equations incorporated in Infoworks CS was gathered, together with the limited amount of data available from the actual case study catchment. The model inputs and model parameters were assumed to be normally distributed unless the available data or literature sources indicated a lognormal distribution. Based on the limited amount of data as well as the information gathered, 95% confidence intervals (i.e. 2.5 and 97.5 percentiles) of model input and model parameter values had been estimated. These value ranges were then discussed with urban drainage modelling experts at the sewer operator, after which adjustment had to be made, if necessary. For example, the width of the distribution for soil moisture depth had to be adjusted because of the modelling experts’

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Fig. 2 e Schematic overview of the modelled water and pollutant flows in the sewer network model used in the Integrated Catchment Modelling study.

unease with the highest soil moisture depth found in literature in comparison with their experience of modelling catchments in the region. The distribution of domestic waste water flow particle size also had to be adjusted because of the boundaries of applicability of the sediment transport equation. Based on the 95% confidence intervals of input parameter and model parameter values, probability distributions were derived based on the assumption that the 95% confidence interval can be approximated by þ/ twice the standard deviation (s). Using this method, statistical distributions have been compiled for the relevant model inputs and model parameters identified by the initial sensitivity study. Table 2 summarises these inputs and parameters and their elicited probability distributions. After the initial sensitivity check, the following processes remained to be investigated: catchment surface runoff (water) and wash-off (sediments and soluble pollutants); surface pollutant build-up and decay; domestic waste water flow; erosion, deposition and transport of sediment through the sewer system; water flow through the sewer system. The process of estimating 95% confidence intervals of the input and model parameters associated with these processes is described in detail below. It must be appreciated that the elicitation process is not exact and requires the use of judgement. Previous work by O’Hagan (1998) has indicated that estimating 95% confidence intervals can lead to an underestimation of the extremes of the range. Consideration is also required to ensure the

distributions derived from an elicitation process do not contain physically impossible values, so a good knowledge of underlying physical processes is required. This is straight forward when parameter have a physical meaning, such as the input parameter particle size, but for several model parameters these are purely empirical and so do not have a direct physical meaning. A good understanding of the behaviour of the system is then required to ensure that the model parameters cannot take values that would predict behaviour that is physically impossible. In some cases there was still a very small probability of physically impossible negative values being drawn, when this occurred during the Monte Carlo simulations these values were changed to zero.

4.3.1.

Catchment surface runoff

The original data used to derive the New UK Runoff Model are described in Packman (1990). The soil moisture depth parameter had originally been used for calibrating the New UK Runoff Model, which led to a soil moisture depth of 200 mm being recommended as a default value in the UK. This is the default value currently incorporated in Infoworks CS. Seven catchments had been studied by Packman (1990), and the soil moisture depths ranged from 541 to 96 mm. It was assumed that the data would be normally distributed, the mean of the data is 236 mm and the standard deviation 144 mm. When discussing the Packman (1990) data with the engineering modellers, the highest moisture depth value of 541 mm was

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Table 1 e Sewer network model parameters & model inputs rejected during initial mapping of processes, initial sensitivity checks and the elicitation process. Model parameters/inputs

sub-catchment slope sub-catchment areas, runoff coefficients, sewer network geometry (Conduit lengths, pipe gradients, diameters, weir heights, orifices, manhole dimensions)

Soil type (‘New UK’ runoff model)

Rainfall, rainfall intensity, duration of dry periods

Rainfall erosion equation coefficients

Coefficients in the Ackers-White equations Trade flow quantity and quality Numerical parameters (number of computational nodes with each conduit and number of subdivision of each timestep)

Gully pot build-up equation coefficients

Reason for rejection after mapping & initial sensitivity checks Model output not sensitive to a 50% increase or decrease Clemens and Von der Heide (1999) indicated that reasonable levels of uncertainty in subcatchment areas and a sewer network geometry database would lead to up to 10% variations in predicted flow quantity. The volume of CSO spills was more sensitive to CSO weir heights. Data or expert knowledge on typical variation in CSO weir heights was not available. Packman (1990) indicated that only the soil moisture depth had been used to calibrate the ‘New UK’ runoff equation Future expected rainfall had been derived using the ‘STORMPAC’ software package, estimating the associated uncertainty fell outside the scope of this study. Artina et al. (2007) reported these to be very sensitive, but no expert knowledge on the development of the equation and its coefficients was found. Were hard-coded and hence could not be adjusted Contribution of traders was negligible in the case study According to Bouteligier et al. (2005), numerical dispersion can be a problem in Infoworks CS. The number of computational nodes had been increased from default 5 to the maximum possible number, 40. This gave a difference of 15% to þ20% spilled pollutant mass, but not in all CSOs As reported by Gent et al. (1992) outputs of these equations were not very sensitive to any of the coefficients

met with significant unease. After discussion, it had been decided that a more reasonable 95% confidence interval for describing soil moisture depth values in the UK would be between 50 mm and 350 mm, regardless of the soil type. This adjustment addressed the unease of the modellers.

4.3.2.

Surface sediment build-up and decay factors

Surface sediment build-up and decay factors express the build-up and decay rate of sediments on the catchment surface. According to Ellis (1986) and Ashley et al. (2004), the

Table 2 e Infoworks CS model parameters & model inputs, their expected uncertainty ranges and assumed statistical distributions (m [ mean and s [ standard deviation). Model parameters/inputs Infoworks CS taken into account for sensitivity analysis Soil moisture depth (‘New UK’) Hydraulic roughness of sewer conduits Potency factor (‘Kpn’) equation coefficients Particle size of suspended and deposited particles. (‘Particle size’)

Normal, m ¼ 200 mm, s ¼ 75 mm Normal, s ¼ 25% of m

For BOD C1: Lognormal, Logn_m ¼ 1.3, Logn_s ¼ 0.246 Domestic waste water flow particles: Normal, m ¼ 0.1 mm, s ¼ 0.02 mm Surface sediment particles: Lognormal, m ¼ 1.1 mm, s ¼ 0.45 mm Dry weather flow pattern, Dry weather flow pollutant quantity and quality concentration: Normal, s ¼ 5% of m

surface sediment build-up parameter (Ps) can vary between 2 kg/ha/day and 10 kg/ha/day in residential areas. Reliable literature sources or experts on the rate of decay were not found, hence the uncertainty in the decay rate could not be elucidated.

4.3.3. Surface sediment erosion and wash-off process, attachment of pollutants to surface sediment (potency factor equations) Wash-off is the process of erosion and transportation of sediments over the catchment surface with the rainfall runoff. Model parameter uncertainty is associated with the coefficients in the potency factor equations and the rainfall erosion equation. Potency factors are a measure of the ratio with which BOD is attached to surface sediments. Each surface sediment potency factor is described by the rainfall intensity and four calibration coefficients, in the form of: Potency factor ¼ C1(Intensity  C2)C3 þ C4. This equation was derived from field data by Bujon and Herremans (1990), this data only consisted of two rainfall events measured in a single catchment in France. Varying C1 between 0.56 and 0.14 creates an uncertainty range enveloping all potency factors derived from the limited field data described in Bujon and Herremans (1990). The limited amount of field data did suggest a lognormal distribution would be more suitable. Considering the uncertainty involved in using two events measured in France to describe the attachment of pollutants to sediment running off the surfaces in the UK, it was deemed sufficient to just vary C1 to get an estimate of the uncertainty in the model output caused by the spread in the Bujon and Herremans (1990) field data.

4.3.4.

Hydraulic flows through the sewer system

Rainfall is a considerable source of model input uncertainty associated with the amount of flow running off the urban catchment surface, and hence entering the urban drainage system. Future rainfall was, however, predicted by STORMPAC, the uncertainty in STORMPAC model output was not included in the scope of this study. Uncertainty associated with the catchment runoff and routing equations have been discussed in Section 4.3.1. Model input uncertainty can

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furthermore be associated with the hydraulic roughness of the sewer conduits as well as the sewer system geometry. Hydraulic roughness changes with pipe wall condition, sedimentation and water level (Wotherspoon, 1994). Uncertainty in the estimation of the hydraulic roughness of sewer pipes has been studied in Schellart (2007); the findings of this study suggested that the hydraulic roughness can vary between þ/50% of the estimated hydraulic roughness values in well calibrated hydraulic sewer network models. As the abovementioned studies suggested a symmetrical distribution of data, a normal distribution was selected with a 95% confidence interval of þ/50% of the hydraulic roughness values in the case study sewer network model.

4.3.5.

Dry weather flow pattern and pollutants

The baseline model, which did not incorporate any consideration of uncertainty, used a standard dry weather flow and pollutant pattern derived from a large field data set collected in the UK, see Ainger et al. (1998). A limited amount of field data available from the present field study site suggested that the dry weather flow pollution concentrations varied between þ/10% from the default average dry weather flow pollution concentrations that had been derived from Ainger et al. (1998). A normal distribution was selected with a 95% confidence interval of þ/10% of the dry weather flow pollution concentrations in the case study sewer network model.

4.3.6.

Sediments

Sewer sediments can originate from the catchment surface, domestic waste water flows, infiltrating water and trade flows. Model input uncertainty can be associated with the particle size and density of the sediment and model parameter uncertainty can be associated with the coefficients within the Ackers-White sediment transport equation. An overview of various different literature sources and shows that the particle size and density in domestic waste water flow can vary widely in both time and space e.g. Chebbo et al. (1990), Verbanck et al. (1990) and Wotherspoon (1994). Taking into account the boundaries for use of the Ackers-White equations, it was decided to select a normal distribution with a 95% confidence interval between 0.05 mm and 0.15 mm for the domestic waste water flow particle size, with a density of 2200 kg/m3. For the surface particles, the published data suggested a lognormal distribution would be suitable. A 95% confidence interval between 0.2 mm and 2 mm, and a density of 2600 kg/m3 was selected. It has to be noted that the limited amount of published data on domestic waste water flow particle size and density indicates that most of the particle size/density combinations lay outside the boundaries of the Ackers-White model. The model parameter uncertainty associated with Ackers-White has been studied in Schellart (2007) and this uncertainty can be significant. However, as the Ackers-White coefficients cannot be adjusted in Infoworks CS (v 7.0) a study into the network wide effect of this model parameter uncertainty could not be carried out.

4.4.

Sensitivity analysis e case study

All sensitivity analysis model runs have been compared with the predictions obtained from the ‘baseline’ deterministic

ICM, using the same 5-months rainfall as the ‘baseline’ model run. For the sensitivity analysis, all model inputs/parameters listed in Table 2 have been changed individually according to their estimated 2.5 and 97.5 percentile values, whilst keeping all other model parameters set at their default values and model inputs at their baseline values. The results are shown in Fig. 3, for clarity, the scatter graphs only show the more sensitive model inputs/parameters (soil moisture depth, dry weather flow for NHþ 4 and domestic waste water particle size and soil moisture depth for BOD). The emitted masses of pollutants predicted by the model runs have been cumulated over 6-h periods, in order to ensure that the sensitivity of Infoworks CS output is compared over the same time-frame as the FIS standards. The highest relative as well as actual over- and under prediction of BOD emitted from all outfalls, when compared with the baseline run, was caused by changing the soil moisture depth to 50 mm and the domestic waste water particle size to 0.1 mm. The amount of NHþ 4 emitted from the sewer system is most sensitive to the soil moisture depth parameter, and, as expected, a 10% increase in the amount of NHþ 4 in the domestic waste water flow leads to a 10% increase in NHþ 4 spilled.

4.5.

Estimation of the ICM output uncertainty

Following the statistical distributions for the most sensitive parameters, simulations were conducted using the full ICM (Fig. 1), in order to create the response database. The full ICM procedure has been run for a range of possible combinations of soil moisture depth (50e350 mm in steps of 50 mm) and particle size (0.05e0.15 mm in steps of 0.025 mm), i.e. 25 model runs in total. For all these simulations, the same 5-month rainfall series was used. For each quality standard a separate response database can then be populated with the number of predicted failures; Table 3 shows one example response database. During a Monte Carlo analysis, 10,000 random draws were made from the elicited probability distributions of particle size and soil moisture depth. The accompanying incidents of water quality failures are linearly interpolated from the response database (taking into account that the number of failures is an integer value), removing the need to run the original ICM an additional 10,000 times. The actual probability of failure of a standard can then be derived by comparing the 10,000 Monte Carlo simulation results with the surface water quality standards, i.e. the allowed number of failures in a given period of time. If nf is the number of model runs where the system fails, and n is the total number of runs then Pf ¼ nf/n, where Pf is the probability of system failure. Fig. 4 shows the probability of failure of the 6-h, 1, 3 and 12months return period cyprinid FIS standards for DO and NHþ 4, due to uncertainty in the particle size and the soil moisture depth of the Infoworks CS model. In the baseline model run, 26 failures of the DO 6-h standard with a 12-month return period were predicted. When uncertainty in particle size and soil moisture is taken into account, the number of failures can vary between 22 and 31 (Fig. 4). Thus, the uncertainty in particle size and soil moisture depth has caused a possible range of 35% variation in the number of predicted failures.

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Fig. 3 e Relative over- or under prediction in the mass of Biochemical Oxygen Demand (BOD) and unionised ammonia (NHD 4 ) released from all Combined Sewer Overflows compared with model predictions obtained without considering uncertainty, Section 4.4.

Fig. 4 also indicates that estimated probability of the number of failures is generally not normally distributed. The range of variation in the predicted failures of the other FIS standards due to uncertainty in soil moisture depth and particle size is summarised in Table 4. Application of the uncertainty estimation method indicated that due to uncertainty in the soil moisture depth and the domestic waste water flow particle size in the sewer model, there was a potential uncertainty range in the number of predicted water quality failures of between 45% and 32% respectively, for DO and NHþ 4 standard failures in the CSO spills.

5.

Discussion

A reduction in computer simulation time of approximately two orders of magnitude was made by using response databases. Creation of the response databases took 6 days of run time on one desktop computer (Pentium 4 3.0 GHz processor and 1 GB of RAM), compared with an estimated run time of over 6 years if the full ICM were to be run 10,000 times on a similar desktop computer. It has to be noted that the Infoworks CS model runs relatively slowly when water quality prediction is included; 5.5 h for a single representative 5-month simulation period for the case study catchment. Computational constraints have been noted even by researchers using simpler and faster running conceptual

Table 3 e Example Response Database (for the number of failures of the 6 h DO cyprinid FIS standard with a return period of 12 months, for the case study). Soil moisture depth (mm)

Particle size (mm)

0.05 0.075 0.1 0.125 0.15

50

125

200

275

350

22 22 22 22 22

27 26 26 26 25

27 26 27 27 27

31 31 31 31 31

30 30 30 30 30

models (Haydon and Deletic, 2009) as well as authors using hydrodynamic sewer models without water quality processes (Thorndahl et al., 2008). The time saving that can be made by using a response database is dependent on the range of uncertain input and model parameters included, as the number of simulations necessary to populate the response databases increases with the number of parameters included in the response database. A second issue is that only model outputs that can be expressed as a discrete result can be included in the response database; the database cannot be populated with time series. Water quality standards are, however, usually expressed in terms of concentrationedurationefrequency and thus ‘standard’ failures can be captured in a single number. Finally, there is also a level of uncertainty involved in using a response database, instead of the full ICM in the Monte Carlo simulations, Schellart et al. (2010) demonstrated that the level of uncertainty inherent in using a well constructed response database was significantly less than the predicted modelling uncertainty. The comprehensive uncertainty mapping procedure used from the outset of the analysis also included all other potential sources of uncertainty, and explicitly mentions those that have been ignored/could not be taken into account (Table 1). It therefore provides key information for application to similar future uncertainty studies on other catchments, as well as future deterministic ICM studies. Conceptualisation of the catchment as a series of reservoir models is common in a number of studies (e.g. Freni et al., 2008; Mannina et al., 2006; Willems, 2008), however, this approach was not used here. A network model approach, where networks of sewer conduits were modelled was selected instead, in order that the spatial and temporal differences in sediment and pollutant transport capacity, as well as deposition and erosion between different sewer sections could be taken into account. The case study system was relatively steep, with an average gradient of 0.0218 for all the modelled conduits. In sewer systems that are less steep, uncertainty due to sediment erosion, transport and deposition can be far more influential, Schellart et al. (2008). The expert elicitation process used requires careful application, if it is to provide a reasonable estimate of output uncertainty (Garthwaite et al., 2005; Kadane and Wolfson,

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Fig. 4 e aef. Histograms of the prediction of the number of Fundamental Intermittent Standard failures due to uncertainty in soil moisture depth and particle size.

1998; O’Hagan, 1998). During the case study described in this paper several difficulties in the elicitation process were encountered. The development of some empirically calibrated quality equations incorporated in the sewer network model used were not documented. Hence little expert knowledge could be found on the development and original calibration of the model parameters and their probability distributions could not be elicited or included in the sensitivity analysis. There were also limitations in the sediment transport equations incorporated in the sewer network model. Literature on sediment particle size in domestic waste water flow (e.g. Chebbo et al., 1990; Verbanck et al., 1990; Wotherspoon, 1994) indicates that the elicited probability distribution for domestic waste water flow particle size is almost certainly too narrow. However, a wider distribution could not be selected, as the sediment transport equations are only valid within a restricted

Table 4 e Range of variation in the number of 6-h cyprinid FIS standard failures predicted when uncertainty in soil moisture depth and particle size is taken into account, compared with the number of failures predicted with the baseline model. FIS standard

Range of variation

DO 1-month 10% to þ4% rp DO 3-month 8% to þ5% rp DO 1215% to þ19% month rp

FIS standard for NH4þ (return period)

Range of variation

1-month

11% to þ3%

3-month

12% to þ6%

12-month

37% to þ12%

particle size/particle density combination range, an issue first reported by Bouteligier et al. (2002). Another issue arose when examining the soil moisture depths reported in Packman (1990) with the modelling experts at the sewer operator. The highest moisture depth value of 541 mm was met with significant unease, and after discussion with the modelling experts a narrower distribution was selected. It is difficult to test the judgement of these experts as there is little reliable UK urban soil moisture depth data. Based on the experience of elicitation described in the literature, as well as the specific issues encountered in this case study it was believed that the widths of the elicited probability distributions derived for this case study are under-estimates. The uncertainty ranges found in the ICM water quality outputs are therefore expected to be under-estimates, not only due to limitations in the elicitation process, but also because they only account for uncertainty traced back to the Infoworks CS model. Uncertainty related to the rainfall generator, the hydrological rainfall runoff model and the river pollutant transport and transformation model have not been taken into account.

6.

Conclusions

This study has identified and reviewed all the processes in the sewer flow quantity and quality model component of an integrated catchment modelling study. For these processes, an elicitation process was used to estimate appropriate probability distributions of model inputs and parameters to quantify their uncertainty. This information was then used to estimate the impact of these uncertainties on the levels of uncertainty associated with the number of water quality

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failures predicted when the method was applied to a small UK catchment for which a purely deterministic ICM was already available. The following conclusions were drawn:  Expert elicitation can be used to estimate the level of uncertainty in model inputs and parameters for urban drainage systems even when field data is limited.  By combining descriptions of uncertainty from selected model and input parameters an estimate of the uncertainty in the predicted number of water quality failures can be obtained in an ICM study.  A Monte Carlo approach whereby ICM predictions are interpolated from a response databases can be used to reduce significantly computation times and so make the estimation of the uncertainty in ICM predictions possible when using complex deterministic models of real catchments.  The sensitivity and uncertainty analysis method described in this paper can be readily adapted for use with other complex deterministic commercial software packages used to model aspects of real catchments in ICM studies.  Analysis of the uncertainty levels associated with predicted values can be used to optimise resources spent on data collection and the implementation of any solution. It is expected that the levels of uncertainty in ICM outputs found in this study are under-estimates. This is because only the uncertainty in the sewer network model has been taken into account, several model and input parameters could not be elicited and some of the elicited probability distributions were expected to be too narrow. With the levels of uncertainty reported any solutions designed based on this deterministic ICM are likely to be significantly over- or under-dimensioned. This level of uncertainty therefore has considerable significance in a time of increasing uncertainty in external inputs such as rainfall and runoff and the ability of receiving watercourses to assimilate polluting inputs due to a changing climate.

Acknowledgement This study has been funded by Yorkshire Water Services (YWS) Ltd. The views expressed in this paper are, however, not necessarily the views of YWS Ltd. The authors would like to thank YWS Ltd. and MWH UK Ltd. and their staff for sharing their experience and knowledge thereby providing valuable input to this project.

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