Quantification and relative comparison of different types of uncertainties in sewer water quality modeling

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Quantification and relative comparison of different types of uncertainties in sewer water quality modeling Patrick Willems Hydraulics Laboratory, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, BE-3001 Leuven, Belgium

art i cle info

ab st rac t

Article history:

Quantifiable sources of uncertainty have been identified for a case study of integrated

Received 15 November 2007

modeling of a sewer system with a more downstream wastewater treatment plant and

Received in revised form

storage sedimentation tank. The different sources were classified in model input and

25 March 2008

model-structure-related uncertainties. They were quantified and propagated towards the

Accepted 5 May 2008

uncertainty in the event-based prediction of sewer emissions (flow, and physico-chemical

Available online 20 May 2008

water quality concentrations and loads). Based on the concept of variance decomposition,

Keywords: Modeling Sewer emissions Uncertainty Water quality

the total prediction uncertainty was split into the contributions of the various uncertainty sources and the different submodels. Although the results strongly depend on the water quality variable considered, it is in most general terms concluded that the uncertainty contribution by the water quality submodels is an order of magnitude higher than that for the flow submodels. Future model improvement should therefore mainly focus on water quality data collection, which would reduce current problems of spurious model calibration and verification, but also of knowledge gaps in in-sewer processes. & 2008 Elsevier Ltd. All rights reserved.

1.

Introduction

Uncertainty analysis in environmental modeling allows modelers to derive knowledge on the sources of error in the modeling process, on their relative importance when propagated to the model outputs, and consequently on the priorities for model improvement. It also provides information on the accuracy and the value of the impact results for water engineering decision support (Reichert, 1997; Harremoe¨s, 2003a; Rauch, 2002; Refsgaard et al., 2005, 2007; Schaarup-Jensen et al., 2005; Krysanova et al., 2006). Estimation of the range of uncertainty in model predictions allows decision makers to assess the risk when model results are used on the basis of decisions (Reda and Beck, 1997; Novotny and Witte, 1997; Kreikenbaum et al., 2002; Korving et al., 2002b). Harremoe¨s (2003b) therefore recommends uncertainties to be recognized in post-normal urban water science and management. Tel.: +32 16 321658; fax: +32 16 321989.

E-mail address: [email protected] 0043-1354/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2008.05.006

Several studies analyzed and propagated uncertainties involved in sewer flow modeling, such as errors in the spatial rainfall input (Willems and Berlamont, 1999), errors in design rainfall or temporal limitations in rainfall series (Rauch and De Toffol, 2006), inputs from households (Butler et al., 1995) and leaking groundwater (Clemens, 2001), the model parameters including the runoff coefficient (Schaarup-Jensen et al., 2005), the system structure and related dimensions (Clemens, 2001; Korving et al., 2002a), numerical modeling errors (Clemens, 2001), etc. Korving et al. (2002b) applied Bayesian techniques for estimating the uncertainties in the overflow volumes of combined sewer systems into receiving surface water bodies. Few studies focus on sewer water quality modeling, which in comparison with sewer flow modeling involves several types of additional uncertainties in pollution load inputs and sewer quality processes. These additional uncertainties are important given the high complexity of sewer water quality

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processes or environmental processes in general, the consequent lack of knowledge and the limited amount of sewer quality sampling data (Beck, 1987; Ashley et al., 2004; Mannina et al., 2006). The urban drainage system is furthermore highly complex as it involves several subsystems including the sewer pipe network, the downstream wastewater treatment plant (WWTP), the combined sewer overflows and overflow ancillary structures, the various domestic, industrial and traffic-related pollution sources, etc. (Rauch et al., 2002). Despite this complexity and the large number of input and output variables involved, sampling data from monitoring campaigns are often limited in practice. In Mannina et al. (2006) and Freni et al. (2008), for instance, only five fully monitored events could be considered. In Willems and Berlamont (2002) and in this study only six monitored overflow events could be analyzed. Freni et al. (2008) applied the GLUE methodology of Beven and Binley (1992) to evaluate the uncertainty of the results from an integrated urban drainage model including a sewer network, a WWTP and a receiving water body. They found high uncertainties in the water quality model results, significantly higher than that in the results of the water quantity or flow modules. Given this high uncertainty, stochastic modeling was tested as an alternative approach to deterministic modeling by Rossi et al. (2006) and Willems (2006). Harremoe¨s (2003b) pointed out that not all uncertainty sources can be ‘quantified’, and that the fraction of uncertainty source terms being ‘ignored’ might be high in environmental investigations. This paper attempts to compare different ‘quantifiable’ uncertainty sources in sewer water quality modeling and their contribution to the total model output uncertainty. The combined sewer–WWTP system of Dessel (Belgium) is considered as case study. This system has a single combined sewer overflow and a storage sedimentation tank (SST) in parallel with the WWTP. Model output concentrations and

loads are considered at the influent and at the combined effluent of the WWTP and SST, for total suspended solids (TSS), settleable solids (SS), BOD and NH4-N.

2.

Integrated urban drainage model

The combined sewer–STT–WWTP system has been modeled using a parsimonious modeling procedure, where simple conceptual models have been selected or developed for the different subsystems. An overview of these subsystems is given in Fig. 1. For the flow modeling of the sewer drainage network, a parsimonious model based on the piecewise linear reservoir model of Vaes et al. (1998) has been implemented. A serial connection of two reservoirs was considered: one reservoir for the sewer system and one reservoir for the WWTP (Fig. 2). The outflow from the first reservoir represents the throughflow to the WWTP (also input to the second reservoir). It is described by a piece-wise linear relationship between the static storage volume in the sewer system and the outflow discharge (see also Vaes et al., 1998). The maximum throughflow discharge equals the pumping capacity at the inlet of the WWTP and SST: 1197 l/s. Additional storages, related to inflow and overflow discharges, are added to the static storage and describe the dynamic storage in the sewer system and the additional storage by submerged overflow. In the second reservoir, the maximum outflow discharge is taken equal to the maximum discharge for biological treatment: 100 l/s. The throughflow discharge of this reservoir then represents the discharge in the biological treatment. The static storage equals the total volume of the SST and additional storage is added in relation to the overflow discharges. The overflow from the second reservoir represents the overflow from the SST. This overflow starts to spill when the WWTP+SST influent discharge is in excess of the 100 l/s maximum discharge for biological treatment.

Fig. 1 – Overview of subsystems and submodels involved in the integrated model of the sewer–SST–WWTP–receiving river system at Dessel (Belgium).

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Full hydrodynamic HydroWorks/InfoWorks-CS model:

WWTP & SST

Sewer system

Parsimonious reservoir model: Rainfall-runoff Emergency overflow Sewer system SST overflow Throughflow to WWTP

SST

Fig. 2 – Complementary use of a full hydrodynamic HydroWorks/InfoWorks model, and a parsimonious reservoir model for the Dessel urban drainage system.

The parsimonious model relationships for static, dynamic and submerged overflow have been calibrated based on simulation results with a full hydrodynamic model implemented in the HydroWorks/InfoWorks-CS software (Fig. 2). The parsimonious reservoir model thus has been used in a way complementary to the full hydrodynamic model. The detailed model is needed for assessing the runoff response of the sewer system, but has limitations in the length of the time series that can be simulated (due to computational time limitations). The conceptual model allows long-term simulations to be carried out. The input discharge to the first reservoir is calculated from point rainfall data and a simple rainfall-runoff model that is nearly the same in the detailed and the simplified model. This rainfall-runoff model is based on a constant runoff coefficient and constant depression storage (emptied by constant evapotranspiration). Their values are calibrated by comparing the peak discharges and the cumulative volumes of all storm runoff events continuously monitored with 10 min time step at the WWTP influent for the period Oct. 1996–Sept. 1998. The calibration procedure was detailed before in Vaes et al. (1998). Fig. 3 shows comparison of the simulation results with the parsimonious reservoir model and the HydroWorks/ InfoWorks model for a number of sewer overflow events in the simulated period.

For the water quality submodels of the sewer network and the SST, parsimonious models were implemented and calibrated based on the data collected during a water quality monitoring campaign. The water quality of the combined sewage (also the influent of the WWTP) was modeled after consideration of buildup and washoff processes for pollutants on the paved surfaces and the transport, sedimentation and resuspension of these pollutants in the sewer network. A parsimonious model has been developed for these processes, based on Bechmann et al. (1999), where the different processes take a strong macroscopic and lumped form. The combined pollution deposit z at the catchment surface and in the sewer system are modeled based on dz ¼ aðz  zÞ þ bðq  qÞ dt

(1)

During dry weather periods when the sewer flow q is smaller than q, the pollutant mass is built up exponentially to the maximum value zmax:    t zðtÞ ¼ zmax 1  exp  k    t ¼ ðz þ kbðqDWF  qÞÞ 1  exp  (2) k with t the duration of the dry weather period with dry weather flow qDWF and k the exponential growth rate. During rainy periods, mass is removed by washoff and resuspension

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0.8 Full hydrodynamic HydroWorks/InfoWorks-CS model

0.7

Parsimonious reservoir model Event cuts

Discharge [m3/s]

0.6 0.5 0.4 0.3 0.2 0.1 0

0

200

400

600 1000 1200 1400 800 Time [number of 10 min time steps]

1600

1800

2000

Fig. 3 – Verification of the parsimonious reservoir model for the Dessel urban drainage system: comparison of 10 min sewer throughflow discharges ( ¼ WWTP+SST influent) simulated by the detailed and the parsimonious model (for a number of events selected from the full simulation period).

when the flow is larger than the discharge q. The value of the parameter b is dependent on the average washoff or deposition concentration, and is considered dependent on the deposited pollutant mass in order to model the first flush effect (Bertrand-Krajewski et al., 1998; Skipworth et al., 2001):    z (3) b ¼ bmax 1  exp  kb The model thus describes the main sewer processes in a way qualitatively similar to what has been widely demonstrated in the literature: exponential increase in mass during build-up periods (e.g. Bertrand-Krajewski et al., 1993) and exponential decrease during wash-off, which becomes stronger with rain intensity or stormwater flow discharge (e.g. Alley and Smith, 1981; Deletic et al., 1997). The parameters z; q, a, bmax, k and kb are determined by calibration. bmax and kb are taken different during resuspension and deposition periods. The pollutant washoff can then be calculated as the pollution deposit reduction rate dz/dt, when it is positive and zero when it is negative. To calculate the total effluent loads and concentrations, the domestic loads need to be added to the pollutant washoff loads. These domestic loads are calculated as the DWF discharges multiplied by the average DWF concentrations. The parsimonious model structure is recommended for the case study, given the very limited amount of available water quality measurements. In comparison with highly detailed water quality models (i.e., with spatial detail at sewer pipe level), parsimonious models furthermore have advantages due to the lower level of overparameterization, the higher level of parameter identifiability and the higher robustness (Willems, 2006). Calibration of the model was based on water quality concentration measurements at the influent of the WWTP. These measurements were collected during a measurement campaign in the period 1996–1998 for six overflow events (see also next). Details on the parsimonious sewer quality model, as well as on the calibration results, were published before in Willems (2004).

The SST was modeled using an empirical–conceptual model. Advection and dispersion are described in this model on the basis of a conceptual reservoir model using a lag time for the input. The lag time represents the advection process, while the assumption of perfect mixing in the reservoir model mimics the dispersion process. During the residence time of the water in the system (the sum of the lag time and the reservoir constant), sedimentation occurs, which is modeled using the empirical model of De Cock et al. (1998) for the sedimentation efficiency Z of such a tank:  1:16 ! vs (4) Z ¼ 1  exp 1:22 v0 This efficiency represents the ratio of the solids after and before the settling process in the SST, where vs is the mean settling velocity of the solids and v0 the surface load. This surface load is for a rectangular SST calculated as the water depth divided by the residence time. Also for the WWTP, a parsimonious model was selected, with the treatment efficiency empirically derived from simultaneous influent–effluent observations during dry weather periods (see example Fig. 7).

3.

Quantification of uncertainties

The uncertainties were for each submodel decomposed in model input and model-structure-related uncertainties. Model input uncertainties are basically related to uncertainties in the data and estimations in input variables needed to feed the model. Model-structure uncertainties are the uncertainties due to the conceptualization and simplification of the physical processes by means of a limited number of model equations (the model-structure). They can be seen as the remaining uncertainties in the model output after use of error-free input in the model and after most optimal calibration of the model parameters to the available measurements (e.g. for a given model structure, by optimizing the

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selected goodness-of-fit statistics). In this study, model parameters were, for each of the submodels in Fig. 1, calibrated by minimizing the variance of the model output errors based on the available water quantity and quality measurements, as described next. The different sources of uncertainty involved in the sewer water quality modeling and their relative contribution to the total uncertainty in the model simulation results were quantified in the study by means of a step-wise procedure.

3.1.

Input uncertainties

Model input uncertainties were in the first step quantified based on detailed investigations on the input data. For rainfall, which is the main model input and the driving force of the temporal variability of the system processes, uncertainties in the calibration curves for the rain gauges, uncertainties by random rain losses due to wind effects and uncertainties in the areal rainfall estimation over the urban catchment were investigated before in Willems (2001). Description of the input uncertainties includes a probability distribution for the magnitude of the errors in the input variables, and the temporal correlation of these errors. Based on this description, series of error values were randomly

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simulated. They can be considered as the output (EX) of a stochastic submodel, and were added to the model input variables x to produce uncertain or erroneous input X (Fig. 4). In order to conduct the random error simulation, the time series (input series in this case) were separated in ‘‘independent’’ storm events (events leading to separate or independent sewer runoff events; thus, storm events separated by a dry weather flow period equal or larger than the concentration time of the sewer network). For each runoff event, peak values and cumulative event volumes can be obtained (Fig. 5). In the rainfall input uncertainty estimation procedure by Willems (2001), error distributions were obtained for both the peak rainfall intensities and the rain storm volumes. From the latter distribution, random volume errors were samples in this study. They were assumed independent from event to event, but perfectly dependent during the time span of one event. The errors in the event volume were in the next step distributed in time. Two limiting cases here assume constant absolute errors (total volume error divided by the event time span to obtain the absolute error per time step) or constant relative errors (volume error distributed in time proportional to the time series amplitude, i.e. rainfall intensities). The choice between both border cases or for an intermediate case was made after testing the distribution of peak event errors.

Fig. 4 – Stochastic error terms added to the deterministic sewer–SST–WWTP system.

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Original time series

Event peaks

Separation in events

amplitude

Time series after one example of random error perturbation

time Event volumes Original time series Confidence intervals on time series after MC random error perturbations

amplitude

time Fig. 5 – Schematic illustration of the random error simulation procedure (top: event separation, random sampling of event volumes and/or event peaks; bottom: confidence interval results after MC runs).

1 Normal distribution One set of 100 MC runs

0.9

95% confidence intervals sampling distribution

Cumulative probability

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Error on ln (rainfall volume [mm]) Fig. 6 – Error distribution on rain storm volumes, together with an example set of 100 MC runs, and the 95% confidence intervals on the error distribution when sampled based on 100 MC runs.

Based on a number of random error simulations using the Monte-Carlo method, confidence intervals of random errors were obtained for each time step (Fig. 5). The number of MC runs was limited to 100 in this study. Fig. 6 shows for the error distribution on rain storm volumes by Willems (2001) one example set of 100 MC runs. Also the 95% confidence intervals on the sampling distribution are given. It is derived from 1000 random sets of 100 MC runs, and shows the sampling uncertainty on the rainfall quantiles due to the limited number of 100 MC runs. Given that the latter uncertainty is

of secondary importance in comparison with the width of the rainfall error distribution, it was considered inefficient to further increase the number of MC runs, in expense of a longer computational time.

3.2.

Model propagation of input uncertainties

The propagation of the random input errors (EX) or random inputs (X) to the model output variables was also done by Monte-Carlo simulation. Random simulations (100 runs) were

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carried out with the stochastic submodel for the model input error, and for each time step the propagated errors on the model output variables were calculated. With this procedure, distributions of random errors were obtained, reflecting the uncertainty in the model output variables caused by the total model input uncertainty. These distributions and corresponding error variances or confidence intervals could be obtained for each time step or averaged for all time steps in the simulation period (to obtain a ‘‘mean overall uncertainty’’ estimate). A similar procedure was followed for the random simulation and propagation of the other uncertainty sources considered.

3.3. Total model uncertainty, model-structure uncertainty and variance decomposition The variance of the total uncertainty in the model output variables was calculated, being the variance of the errors in the model results after comparison with observation data. The variance of the observation errors has to be subtracted from this variance. The standard deviation of the water quality measurement errors was assessed based on literature (Ahyerre et al., 1998; Bertrand-Krajewski et al., 2001; Kanso et al., 2003) to be as high as 30–40% for BOD and 15–20% for the other variables considered. The variance of the other model-structure-related uncertainties was then quantified as the rest term in the description of the total variance, making use of the concept of variance decomposition: s2Y;tot ¼

n X

s2Y;inpðXiÞ þ s2Y;str

(5)

i¼1

where s2Y;tot variance of total uncertainty in the model output variable Y, after subtracting the variance of the observation errors; s2Y;inpðXiÞ variance of the uncertainty contribution by model input variable Xi (i ¼ 1,y,n; n is the total number of input variables); s2Y;str variance of the contribution of the model-structure-related uncertainty. This variance decomposition equation makes use of the assumption that the input and model-structure-related uncertainties are independent as they have different and independent underlying causes. The variances s2Y;tot could be quantified at all locations where data were available from measurement campaigns (Fig. 1). Emission measurements are available at the influent and effluent of the WWTP from the monitoring network of the Flemish Environment Agency (VMM) and at the influent and effluent of the SST from a measuring campaign where 10 min concentrations of TSS, SS, biochemical oxygen demand (BOD) and ammonia (NH4-N) have been sampled during six overflow events (during a period of maximum 2 h per event). One example of the total variance results is given in Fig. 7 for the WWTP BOD effluent concentrations. As can be seen in Fig. 7(a), the error variances or the width of the confidence intervals typically increase with higher discharge or concentration values. For this reason, a Box–Cox (BC) transformation (Box and Cox, 1964) was applied to the model output Y: BCðYÞ ¼ ðYl  1Þ=l

(6)

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where the parameter l (0plp1) is calibrated to reach homoscedasticity in the errors (variance of errors nearly independent on the model output magnitude). The ln transformation can be seen as a limiting case of the BC transformation (for l ¼ 0). The l parameter was in this study calibrated by means of trial-and-error after visual judgment on the homoscedasticity of the model errors (see example in Fig. 7(b)). The BC transformation was applied to the model input and output variables before calculating the variances s2Y;tot , s2Y;inpðXiÞ and s2Y;str of Eq. (1). After this transformation, standard deviations or confidence interval widths could be obtained that are nearly uniform for all events or time steps (see Fig. 7(b)). The overall uncertainty could then be represented by the average of the variances for all time steps (after BC transformation). Using this methodology and above-mentioned measurements, total uncertainty could be quantified for the output variables of the following submodels (Fig. 1):

    

flow model sewer drainage network, flow model SST+WWTP system, sewer washoff and sewer transport model, WWTP biological treatment model, SST sedimentation model.

Consequently, for the same submodels model-structurerelated uncertainties could be quantified, decomposing the total uncertainty of the integrated model. This was done by propagating first the rainfall input uncertainties to the output of the sewer flow submodels, and calculating the modelstructure uncertainty of the sewer flow submodels as rest variance (considering the variance decomposition method described above; thus calculating the model error variance of the sewer flow submodels based on the available measurements, minus the variance contribution of the propagated rainfall input uncertainty). Next, these different uncertainties are propagated to the more downstream submodel, the model-structure uncertainty of this submodel quantified, etc. It is clear that this step-wise uncertainty quantification procedure has to be applied from up- to downstream, to enable propagation of the different uncertainties through the integrated model. The variance decomposition procedure thus was applied first for the most upstream subsystems and for the flow submodels. The model-structure-related uncertainties were simulated and propagated through the introduction of additional stochastic submodels, and the simulated errors added to the submodel output variables. Based on random simulations with both the input and model-structure uncertainty-related stochastic models, the probability distribution of the total uncertainty could be simulated for any model output variable and at any time step in the simulation. From this distribution, also the median value, the variance and confidence limits could be derived. A long-term simulation has been carried out with the model for the period 1996–1998. In this period, six overflow events occurred, labeled hereafter as C1–C6. In Fig. 9, the 95% confidence limits are shown for the WWTP influent discharges during the two overflow events C3 and C4.

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100

BOD concentration effluent [mg/l]

measurements 90

WWTP treatment efficiency model

80

+- st.dev. confidence limits

70 60 50 40 30 20 10 0 0

100

200

300

400

500

600

BOD concentration influent [mg/l]

10 measurements

BC (BOD concentration effluent [mg/l])

9

WWTP treatment efficiency model +- st.dev. confidence limits

8 7 6 5 4 3 2 1 0 0

5

10

15

20

BC (BOD concentration influent [mg/l]) Fig. 7 – Modeled versus measured BOD concentrations at the effluent of the WWTP, before and after BC transformation.

Discharge measurements were available from flow monitoring devices installed in the two downstream sewer pipes, just upstream of the WWTP. They are based on ultrasonic flow velocity and pressure depth measurements. The uncertainty

in this flow monitoring was assessed after testing the depth–velocity devices in the Hydraulics Laboratory of K.U. Leuven. The standard deviation of the flow per monitor ranges from 20% of the flow value for water levels as low as

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5 cm, and 6% for water levels up to 50 cm (Fig. 8). This standard deviation was propagated to confidence intervals on the monitored flow (see overflow events C3 and C4 in Fig. 9). The results in Fig. 9 show that for low flows, the measurement errors and model errors are of equal importance. For higher flow values, model results become more erroneous. The discharge values above the WWTP biological treatment capacity of 100 l/s could also be obtained from water level measurements available for the SST. These values are in Fig. 9 compared with the depth–velocity device measurements, and with the stochastic model results.

In Fig. 10, the 95% confidence limits are shown for the simulated TSS influent and effluent loads for all six overflow events cut from the long-term simulated series for 1996–1998. Also a comparison with the measured loads is given. The simulation results are hourly averaged loads, while for the measurements both hourly averaged and 10 min averaged values are given.

3.4.

Variance decomposition results

Table 1 shows the results of the variance decomposition. The mean relative contribution is calculated for the different

St.dev. discharge measurement errors [%]

60 Model 50

Average error of tested monitors

40

30

20

10

0 0

5

10

15

20 25 30 Water level [cm]

35

40

45

50

Fig. 8 – Relative error in the discharge measurements by the depth–velocity monitors, after calibration tests in the laboratory.

0.7 0.6

Discharge [m3/s]

0.5 0.4 0.3 0.2 0.1 0.0 1500

2000

2500

3000

3500 Time [h]

4000

4500

5000

5500

: 95% confidence interval on the model simulation results : depth-velocity devices based measurements (together with 95% confidence interval limits for events C3 and C4) : based on SST water level measurements Fig. 9 – Comparison of simulated and measured WWTP influent discharges (for overflow events C3, C4 and C5); urban drainage system Dessel.

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Influent WWTP / load TSS

200 180

Load TSS [g/s]

160 140 120 100 80 60 40 20 0 1

11

21

31 Time [h]

41

51

61

51

61

Effluent sewer - SST - WWTP system / load TSS

160 140

Load TSS [g/s]

120 100 80 60 40 20 0

1

11

21

31 Time [h]

41

Fig. 10 – Total uncertainty (in terms of 95% two-sided confidence limits) in the hourly TSS influent (top figure) and effluent (bottom figure) loads; for the six overflow events C1–C6 during the inter-university measurement campaign; urban drainage system Dessel.

uncertainty sources to the total variance in the modeled concentrations and loads of the integrated sewer–SST–WWTP system. The table thus reports variance ratios, expressed as % contribution to the total variance: s2Y;inpðXiÞ s2Y;tot

100% ði ¼ 1; . . . ; nÞ;

s2Y;str s2Y;tot

100%

(7)

The sum of each column equals 100%. In Table 1 the results are shown downstream in the sewer system (at the influent of the WWTP) and for the effluent concentrations and loads of the integrated sewer–SST–WWTP system (also the total emission towards the receiving river). The % variance contributions in the table are based on mean variances for all time steps in the simulation where measurements were available. The uncertainty sources are in the table lumped to the rainfall input uncertainty, the uncertainty related to the flow submodels (grouping the rainfall-runoff and sewer flow model uncertainties) and the model uncertainties related to the water quality submodels for the sewer system, the WWTP and the SST.

When the variances are based on model residuals after comparison with measurements, the measurement errors explain part of the variances. Measurement errors do, however, not contribute to the total uncertainty in the model results, and, therefore, their variance was subtracted from the total variance of the model residuals to derive the model prediction uncertainties in the table. It is concluded from Table 1 that for the flow model approximately 30–40% of the total uncertainty in the modeled flow volumes is explained by the rainfall input uncertainty. For the water quality concentrations in the effluent of the combined system, this contribution reduces down to 10–25%. The water quantity submodels have the same order of magnitude contribution. Previous analysis reported in Willems and Berlamont (1999) has shown that the runoff model of the sewer catchment explains approximately 40–50% of the total uncertainty in the flow model results, and that this mainly originates from the uncertainty in the estimation of the runoff contributing areas. These results are in agreement with Schilling and Fuchs (1986), Lei and Schilling (1994) and Korving et al. (2002a, b),

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WAT E R R E S E A R C H

Table 1 – Mean % contribution of different uncertainty sources considered to the total variance in the modelled WWTP influent concentrations and effluent concentrations and loads of the integrated sewer–SST–WWTP system Uncertainty sources

Mean relative contribution (%) to the total variance in the modelled WWTP influent concentrations Model residuals

Rainfall input uncertainty Model uncertainty flow submodels Model uncertainty water quality submodels: Washoff and sewer transp. model WWTP model SST model Water quality measurement errors

Model prediction uncertainty

BOD

NH4

TSS

SS

BOD

NH4

TSS

SS

15 19

17 20

21 25

16 20

21 26

19 24

26 31

19 23

38 – – 28

49 – – 14

35 – – 19

50 – – 14

53 – – –

57 – – –

43 – – –

58 – – –

Mean relative contribution (%) to the total variance in the effluent concentrations of the sewer–SST–WWTP system Model residuals

Rainfall input uncertainty Model uncertainty flow submodels Model uncertainty water quality submodels: Washoff and sewer transp. Model WWTP model SST model Water quality measurement errors

Model prediction uncertainty

BOD

NH4

TSS

SS

BOD

NH4

TSS

SS

12 15

10 12

7 9

9 12

17 21

13 16

9 11

12 15

9 19 15 30

10 34 12 22

7 E0 55 21

6 27 23 23

13 27 22 –

13 43 16 –

9 E0 77 –

8 35 30 –

Mean relative contribution (%) to the total variance in the effluent loads of the sewer–SST–WWTP system Model residuals

Rainfall input uncertainty Model uncertainty flow submodels Model uncertainty water quality submodels: Washoff and sewer transp. model WWTP model SST model Water quality measurement errors

Model prediction uncertainty

BOD

NH4

BOD

NH4

15 18

14 18

17 21

13 16

14 20 17 16

16 32 10 10

20 29 24 –

21 41 13 –

who also concluded that the input of rainfall, the estimation of the runoff contributing area, as well as the rainfall-runoff model (including estimation of the antecedent conditions) are far the most important sources of uncertainty in urban stormwater modeling. The use of the simplified reservoir model instead of the full hydrodynamic model accounted only for 5% of the total uncertainty in the flow model results. This uncertainty together with the total sewer runoff model uncertainty and the uncertainty in the full hydrodynamic model is in the table denoted as ‘Model uncertainty flow submodels’. For the water quality concentrations at the effluent of the combined system, the complete quantitative modeling (including the rainfall input uncertainty) takes 20–40% of the total uncertainty, whilst the remaining fraction of 60–80% is to be explained by the water quality submodels.

The uncertainty-source contributions of these water quality submodels strongly differ from one water quality variable to another. For TSS, for instance, the SST model takes a much higher uncertainty contribution in comparison with the WWTP and sewer water quality model. This is due to the influence of the sedimentation process. For the other water quality variables, the SST model explains 15–30% of the total uncertainty, the WWTP model 30–40% and the sewer water quality model (washoff and sewer transport model) 10–20%. Based on these values, the statement made by others, such as by Harremoe¨s (1988) and Ashley et al. (1999), can objectively be affirmed: the uncertainty in the results of sewer water quality models is an order of magnitude higher than that for the flow models. As also discussed in Ashley et al. (2004), this is mainly caused by lack of water quality data, leading to spurious model calibration and verification, but also to knowledge gaps in in-sewer processes.

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Conclusions

Based on a specific urban drainage system case and a parsimonious modeling technique, different types of uncertainties related to the rainfall input, the flow submodels, and the sewer, WWTP and SST water quality submodels were quantified, and their relative importance assessed. These were for the rainfall input and the water quantity submodels based on long series of rainfall and flow measurements. The sewer and SST overflow water quality measurements were, however, more limited (six overflow events monitored). The error distributions for the sewer and SST water quality submodels consequently might be subject to important uncertainties. The relative uncertainty contributions, quantified in this study, were therefore interpreted in terms of orders of magnitudes. Although the results strongly depend on the water quality variable considered, it is in most general terms concluded that the uncertainty contribution by the water quality submodels is an order of magnitude higher than that for the flow submodels. Future model improvement should therefore mainly focus on water quality data collection, which would reduce current problems related to model calibration and verification and increase knowledge of in-sewer processes. Finally, it is important to recap from the introduction that the uncertainty analysis only focussed on the ‘‘quantifiable uncertainty’’. The modeler has to consider, however, that the real uncertainty might be larger than this quantifiable uncertainty. It indeed might occur that the system structure is depending on specific influences, which were not observed in the past, but which may occur in an unpredictable manner in the future. This is a specific and special type of ‘lack of knowledge’, which could not be included in the uncertainty predictions. Such additional non-quantifiable uncertainty is usually referred to as ‘ignorance’, and occurs when we are missing relevant knowledge. These non-quantifiable uncertainties also contribute to the sewer model uncertainty, and may in some case be more serious than the quantifiable uncertainties (Harremoe¨s, 2003b).

Acknowledgement The research was funded through a postdoctoral scholarship for the Flemish Fund for Scientific Research F.W.O.-Vlaanderen, currently being extended by the Research Fund of K.U. Leuven. The measuring campaign at the combined sewer system of Dessel was part of an inter-university project on ‘Ancillaries to combined sewer overflows’ supported by the Flemish Authorities VMM and AMINAL. R E F E R E N C E S

Ahyerre, M., Chebbo, G., Tassin, B., Gaume, E., 1998. Storm water quality modelling, an ambitious objective? Water Sci. Technol. 37 (1), 205–213. Alley, W.M., Smith, P.E., 1981. Estimation of accumulation parameters for urban runoff quality modeling. Water Resour. Res. 17 (6).

Ashley, R.M., Hvitved-Jacobsen, T., Bertrand-Krajewski, J.L., 1999. Quo Vadis sewer process modelling? Water Sci. Technol. 39 (9), 9–22. Ashley, R.M, Tait, S., Clemens, F., Veldkamp, R., 2004. Sewer processes—problems and new knowledge needs. In: Krebs, P., Fuchs, L. (Eds.) Proceedings of the Sixth International Conference on Urban Drainage Modelling-UDM’04, Dresden, Germany, 15–17 September 2004, Conference Proceedings. Bechmann, H., Nielsen, M.K., Madsen, H., Poulsen, N.K., 1999. Grey-box modelling of pollutant loads from a sewer system. Urban Water 1 (1). Beck, M.B., 1987. Water quality modeling: a review of the analysis of uncertainty. Water Resourc. Res. 23 (8), 1393–1442. Bertrand-Krajewski, J.L., Scrivener, O., Briat, P., 1993. Sewer sediment production and transport modelling: a literature review. J. Hydraul. Res. 31 (4), 435–460. Bertrand-Krajewski, J.L., Chebbo, G., Saget, A., 1998. Distribution of pollution mass vs. volume in stormwater discharges and the first flush phenomenon. Water Res. 32 (8), 2341–2356. Bertrand-Krajewski, J.L., Bardin, J.P., Mourad, M., Boulanger, Y., 2001. Accounting for sensor calibration, concentration heterogeneity, measurement and sampling uncertainties in monitoring urban drainage systems. In: Proceedings of International Conference on Automation in Water Quality Monitoring (AutMoNet), Vienna, Austria, pp. 171–178. Beven, K.J., Binley, A.M., 1992. The future of distributed models—model calibration and uncertainty prediction. Hydrol. Processes 6 (3), 279–298. Box, G.E.P., Cox, D.R., 1964. An analysis of transformations. J. R. Stat. Soc. 26, 211–243 discussion 244–252. Butler, D., Friedler, E., Gatt, K., 1995. Characterizing the quantity and quality of domestic wastewater inflows. Water Sci. Technol. 33 (2), 1–15. Clemens, F.H.L.R., 2001. Hydrodynamic models in urban drainage: Application and calibration. Ph.D. Dissertation, TUDelft, The Netherlands. De Cock, W., Vaes, G., Blom, P., Berlamont, J., 1998. The efficiency of a storage sedimentation tank: numerical simulation and physical modeling. In: Proceedings of International Conference on Developments in Urban Drainage Modelling-UDM’98, Imperial College of Science, Technology and Medicine, London. Deletic, A., Maksimovic, Cˇ., Ivetic, M., 1997. Modelling of storm wash-off of suspended solids from impervious areas. J. Hydraul. Res. 35, 99–118. Freni, G., Mannina, G., Viviani, G., 2008. Uncertainty in urban stormwater quality modelling: the effect of acceptability threshold in the GLUE methodology. Water Res 42 (8–9), 2061–2072. Harremoe¨s, P., 1988. Stochastic models for estimation of extreme pollution from urban runoff. Water Res. 22 (8), 1017–1026. Harremoe¨s, P., 2003a. The need to account for uncertainty in public decision making related to technological change. Integrated Assessment 4 (1), 18–25. Harremoe¨s, P., 2003b. The role of uncertainty in application of integrated urban water modeling. In: Proceedings of the International IMUG Conference, Tilburg, 23–25 April 2003. Kanso, A., Gromaire, M.C., Gaume, E., Tassin, B., Chebbo, G., 2003. Bayesian approach for the calibration of models: application to an urban stormwater pollution model. Water Sci. Technol. 47 (4), 77–84. Korving, H., van Gelder, P., van Noortwijk, J., Clemens, F., 2002a. Influence of model parameter uncertainties on decisionmaking for sewer system management. In: Hydroinformatics 2002: Proceedings of the Fifth International Conference on Hydroinformatics, Cardiff, UK. Korving, H., Clemens, F., Van Noortwijk, J., Van Gelder, P., 2002b. Bayesian estimation of return periods of CSO volumes for

ARTICLE IN PRESS WAT E R R E S E A R C H

42 (2008) 3539 – 3551

decision-making in sewer system management. In: Proceedings of the Ninth International Conference on Urban Drainage, September 2002, Portland, Oregon, USA. Kreikenbaum, S., Krejci, V., Rauch, W., Rossi, L., 2002. Probabilistic modeling as a new planning approach to stormwater management. In: Proceedings of the Ninth International Conference on Urban Drainage, September 2002, Portland, Oregon, USA. Krysanova, V., Hattermann, F., Wechsung, F., 2006. Implications of complexity and uncertainty for integrated modelling and impact assessment in river basins. Environ. Modelling Software, in press. Lei, J.H., Schilling, W., 1994. Parameter uncertainty propagation analysis for urban rainfall runoff modelling. Water Sci. Technol. 29 (1–2), 145–154. Mannina, G., Freni, G., Viviani, G., Saegrov, S., Hafskjold, L.S., 2006. Integrated urban water modelling with uncertainty analysis. Water Sci. Technol. 54 (6–7), 379–386. Novotny, V., Witte, J.W., 1997. Ascertaining aquatic ecological risks of urban stormwater discharges. Water Res. 31 (10), 2573–2585. Rauch, W., 2002. On risk, uncertainty and incertitude in integrated wastewater management. In: Proceedings of the Third International Conference on Water Resources and Environment Research, Dresden, 22–25 July 2002, pp. 121–125. Rauch, W., De Toffol, S., 2006. On the issue of trend and noise in the estimation of extreme rainfall properties. Water Sci. Technol. 54 (6–7), 17–24. Rauch, W., Bertrand-Krajewski, J.-L., Krebs, P., Mark, O., Schilling, W., Schutze, M., Vanrolleghem, P.A., 2002. Deterministic modeling of integrated urban drainage systems. Water Sci. Technol. 45 (3), 81–94. Reda, A.L.L., Beck, M.B., 1997. Ranking strategies for stormwater management under uncertainty: sensitivity analysis. Water Sci. Technol. 36 (5), 357–371. Refsgaard, J.C., Nilsson, B., Brown, J., Klauer, B., Moore, R., Bech, T., Vurro, M., Blind, M., Castilla, G., Tsanis, I., Biza, P., 2005. Harmonised techniques and representative river basin data for assessment and use of uncertainty information in integrated water management (HarmoniRiB). Environ. Sci. Policy 8 (3), 267–277.

3551

Refsgaard, J.C., Van der Sluijs, J.P., Højberg, A.L., Vanrolleghem, P.A., 2007. Uncertainty in the environmental modelling process—a framework and guidance. Environ. Modelling Software 22, 1543–1556. Reichert, P., 1997. On the necessity of using imprecise probabilities for modelling environmental systems. Water Sci. Technol. 36 (5), 149–156. Rossi, L., Krejci, V., Rauch, W., Kreikenbaum, S., Frankhauser, R., Gujer, W., 2006. Stochastic modeling of total suspended solids (TSS) in urban areas during rain events. Water Res. in press. Schaarup-Jensen, K., Johansen, C., Thorndahl, S., 2005. Uncertainties related to extreme events statistics of sewer system surcharge and overflow. In: Proceedings of the 10th International Conference on Urban Drainage, Copenhagen, Denmark, 21–26 August 2005. Schilling, W., Fuchs, L., 1986. Errors in stormwater modelling— a quantitative assessment. J. Hydraul. Eng. 112 (2), 111–123. Skipworth, P.J., Tait, S.J., Saul, A.J., 2001. The first foul flush in combined sewers: an investigation of the physical causes. Urban Water 2 (4), 317–326. Vaes, G., Willems, P., Berlamont, J., 1998. Assessment of combined sewer overflow emissions. In: Proceedings of the International Conference on Developments in Urban Drainage ModellingUDM’98, Imperial College of Science, Technology and Medicine, London. Willems, P., 2001. Stochastic description of the rainfall input errors in lumped hydrological models. Stochastic Environ. Res. Risk Assessment 15, 132–152. Willems, P., 2004. Parsimonious pollutant washoff and sewer transport modelling. In: Krebs, P., Fuchs, L., (Eds.), Proceedings of the Sixth International Conference on Urban Drainage Modelling-UDM’04, Dresden, Germany, 15–17 September 2004, Conference Proceedings, pp. 461–468. Willems, P., 2006. Random number generator or sewer water quality model? Water Sci. Technol. 54 (6–7), 387–394. Willems, P., Berlamont, J., 1999. Probabilistic modelling of sewer system overflow emissions. Water Sci. Technol. 39 (9), 47–54. Willems, P., Berlamont, J., 2002. Probabilistic emission and immission modelling: case-study of the combined sewerWWTP-receiving water system at Dessel (Belgium). Water Sci. Technol. 45 (3), 117–124.