IntroductionStormwater runoff from urban areas can have detrimental impacts to receiving waters due to the increased runoff volume and peak flow (Shuster et al. 2005; Bell et al. 2020; Hopkins et al. 2020) and pollutants (Nazari-Sharabian et al. 2019), compared to undeveloped conditions. Stormwater control measures (SCMs) are constructed facilities designed to mitigate the impacts of urban stormwater by reducing the rate, volume, and/or pollutant concentrations of runoff prior to being discharged downstream (Hopkins et al. 2020; Sadeghi et al. 2018; Jefferson et al. 2017).The literature is replete with studies that have quantified the effectiveness of SCMs through monitoring (e.g., Aguilar and Dymond 2019; Winston et al. 2012; Brown and Hunt 2011; Hathaway et al. 2009; Passeport et al. 2009; Hunt et al. 2006; Carleton et al. 2000; Comings et al. 2000; Greb and Bannerman 1997; Stanley 1996; Martin 1988; Grizzard et al. 1986; USEPA 1983). SCM effectiveness is typically reported as the difference in magnitudes of a particular metric [e.g., runoff volume, pollutant event mean concentration (EMC), pollutant load] entering (influent) and exiting (effluent) a SCM, and estimated benefits are often highly variable. Reasons for the large variability of SCM monitoring results include differences in influent pollutant concentrations (Barrett 2008), influent particle characteristics (Greb and Bannerman 1997), location and climate (Barrett 2008), SCM design (Brown and Hunt 2011), and presentation/aggregation of results (Strecker et al. 2001). The large variability of results presents challenges to extending SCM monitoring data directly to unmonitored SCMs.Mathematical models offer an alternative method for predicting the performance of SCMs under varying environmental and design conditions. One challenge that modelers face at the outset of a project is the selection of an appropriate model. Model selection criteria may include, among others: availability of calibration/testing data, model temporal and spatial resolution compared to modeling objectives, and the modeler’s familiarity with the model (ASCE 1998; Engel et al. 2007; USEPA 2002; Urbonas 2007; Heidari et al. 2020a, b; Ghanbari et al. 2020). The model selection criteria can conflict with one another when modeling SCM pollutant removal performance, especially for large-scale, planning-level modeling efforts such as those necessary for total maximum daily load (TMDL) allocation and implementation studies. Planning-level TMDL modeling, for example, may require the use of a watershed model to simulate pollutant loads discharged from the land surfaces, a SCM model to simulate pollutant load reductions from land surfaces under various SCM implementation scenarios, and a receiving water model to simulate fate and transport of pollutants discharged into the receiving water body. Watershed and receiving water modeling is often performed using dynamic models such as the United States Environmental Protection Agency (EPA) Stormwater Management Model (SWMM), United States Geological Survey (USGS) Hydrologic Simulation Program-Fortran (HSPF), and EPA Water Quality Analysis Simulation Program (WASP); which necessitates the use of dynamic SCM models in order to link dynamic watershed model outputs with dynamic receiving water model inputs (Stein et al. 2008).Dynamic models resolve hydrologic and water quality processes in response to time-varying climatic inputs. However, most data available to calibrate SCM pollutant removal models are event-averaged values (e.g., event mean concentrations). Thus, the most commonly used SCM pollutant removal models are typically calibrated assuming steady-state conditions (Choat and Bhaskar 2020). The practical application of steady-state SCM pollutant removal models within a dynamic modeling framework requires the modeler to (1) “pre-process” dynamic inputs to generate estimates of the influent pollutant EMC; (2) “post-process” the event-averaged SCM model outputs to generate dynamic inputs for the receiving water model; and (3) in some cases, perform separate dynamic hydrologic/hydraulic routing of influent flows to determine representative steady-state values of hydrologic/hydraulic flow metrics that are parameters in the steady-state SCM models. While not impossible tasks, these can be cumbersome and add to the cost and time resources needed to perform SCM modeling.Also, the variability of SCM performance from monitoring studies underlines the importance of explicit and statistically rigorous analysis of modeling uncertainty. Uncertainty analysis (UA) techniques quantify the uncertainty of model responses as a result of uncertainty in input forcing, model parameters, model structure, and/or field/laboratory measurements (Heidari et al. 2021). Compared to deterministic modeling, results of UA provide additional information that can enhance the decision making process; e.g., assessing the probability of an SCM discharging pollutants at a concentration/load greater than a regulatory threshold (e.g., Zhang et al. 2018; Park et al. 2015; Park and Roesner 2012).UA has been applied to a variety of stormwater quality modeling exercises in the literature (e.g., Ahmadisharaf et al. 2019; Hantush and Chaudhary 2014; Vezzaro et al. 2012; Park et al. 2011; Mannina and Viviani 2010; Avellaneda et al. 2009, 2010; Freni et al. 2008). However the application of UA outside of academia remains relatively limited (Camacho et al. 2018; Dilks and Freedman 2004; National Research Council 2001). One reason for this is that UA is impractical for the average practitioner to perform using current models (Pappenberger and Beven 2006; Reckhow 2003). To our knowledge, none of the most common models used by stormwater modelers have built-in UA capabilities; therefore, modelers must develop their own UA algorithms via computer programming efforts that they might not have the knowledge or budget to perform.Monte-Carlo (MC) and first-order variance estimation (FOVE) are two of the most common UA methods used (Mishra et al. 2018; Riasi et al. 2018; Shirmohammadi et al. 2006). MC methods involve realization of large numbers (typically thousands) of model responses with different combinations of model parameters. For a given set of model parameters, MC methods provide exact estimates of model output uncertainty, as long as a sufficient number of model simulations are conducted to properly explore the model parameter space (Shirmohammadi et al. 2006; Tung et al. 2006). Thus, MC methods can be used to computationally evaluate the uncertainty estimates provided by other UA methods, such as FOVE (Bates and Townley 1988; Burges and Lettenmaier 1975; Melching and Anmangandla 1992; Yu et al. 2001). Practically, MC-based UA can be burdensome because of the time and resources necessary not only to run thousands of model simulations (in some cases taking days/weeks of model run time), but also to develop code/algorithms to efficiently run these simulations and analyze thousands of model outputs.Alternatively, FOVE methods can produce model output uncertainty estimates with a single model simulation. The FOVE method uses properties of variance to propagate the variance of random input parameters to estimate the variance of the model outputs (McClarren 2018). While the FOVE method is computationally efficient, the accuracy of the uncertainty estimates can decrease with increasing non-linearity of the model and increasing variance of the random input parameters (Morgan and Henrion 1990). FOVE has been applied to several models with relatively high nonlinearity, including QUAL2E (Melching and Yoon 1996) and HSPF (Zhang and Yu 2004).The overall goal of this study is to evaluate the performance of new algorithms for dynamically simulating the pollutant removal of SCMs with uncertainty. Specifically, the objectives of the study are to (1) investigate the effects of applying steady-state SCM pollutant removal models within a dynamic SCM routing framework; and (2) compare uncertainty estimates of the dynamic model responses using MC, prediction interval, and FOVE UA methods. The results of the first portion of the study are intended to reveal how accurate (or inaccurate) model responses (i.e., SCM effluent EMC concentrations) might be when commonly used SCM pollutant removal models are applied within a dynamic routing framework. The results of the second portion of the study are intended to reveal how the uncertainty methods that require less computational resources compare to more computationally intensive MC methods.MethodsSteady-State SCM Pollutant Removal ModelsThree different SCM pollutant removal models were evaluated in this study. The models were selected primarily due to their prevalent use in other SCM modeling studies and recommendations as SCM pollutant removal algorithms by the Water Environment Research Foundation (Leisenring et al. 2013).Modified Fair and Geyer ModelThe Fair and Geyer model [Eq. (1), Fair and Geyer 1954] is suggested by USEPA (1986) as an appropriate model for simulating particle removal in stormwater detention basins under flow-through conditions (1) where R = the fraction of particles removed; v = particle settling velocity (m/s); A = basin surface area (m2); Q = steady-state flowrate through basin (m3/s); and n represents a hydraulic efficiency factor.The model was originally developed for application to water/wastewater treatment facilities where steady-state conditions apply, and the existence of hydraulic dead zones (i.e., non-ideal settling) within settling basins reduced particle settling efficiency from that predicted using Camp’s (1946) concept of ideal basins. Although such conditions rarely exist in stormwater basins due to the intermittent and highly variable nature of rainfall/runoff, basins that fill over a short period of time and empty over an extended period of time (i.e., extended detention basins, EDB) can be reasonably assumed to be operating at steady-state over the duration of an entire event.Here, a modified version of the Fair and Geyer model (MFG) is used that replaces the removal term (R) with influent (Ci) and effluent (Ce) TSS concentrations and simulates removal of particles with different settling velocities (Olson et al. 2020) (2) Ce=∑k=1KCi*psfk*(1+vkAnQ)−nwhere Ci = influent TSS concentration (mg/L); Ce = effluent TSS concentration (mg/L); and psfk = the fraction of particles in particle size bin k.Incorporating multiple particle bins into the model allows for a more representative analysis, considering the large distribution of particle sizes that are found in stormwater runoff (Selbig and Bannerman 2011; Roseen et al. 2011; Kim and Sansalone 2008; Greb and Bannerman 1997; USEPA 1986). We follow recent recommendations by Leisenring et al. (2013) to use five particle bins representing particle sizes in the following ranges; 2–10  μm, 11–30  μm, 31–60  μm, 61–100  μm, and >100  μm.k-C* ModelThe k-C* model was first proposed by Kadlec and Knight (1996) to model pollutant removal in wastewater treatment wetlands. The model uses a first-order decay coefficient (k in m/day) to reduce the influent concentration (Ci in mg/L) towards a background or irreducible concentration (C* in mg/L) as a slug of pollutants moves through a treatment device. The model assumes steady-state and plug-flow conditions exist with the treatment device. The effluent concentration (Ce) is estimated as: (3) Ce=C*+(Ci−C*)e−k/q′where q′=Q/A denotes the hydraulic loading rate (m/day), in which Q is the average inflow rate (and/or discharge rate, assuming steady-state conditions) in m3/s.Wong et al. (2006) recommended the k-C* model be adopted as a “unified approach” for simulating pollutant removal in SCMs, and demonstrated the model’s ability to be calibrated to measured data from several different types of SCMs for several different pollutants. It should be noted, however, that the calibration datasets in Wong et al. (2006) included spatially-variable measurements of the pollutants within the SCM. Park et al. (2011) calibrated the k-C* model to simulate TSS effluent discharged from EDBs using data obtained from the International Stormwater Best Management Practices (BMP) Database ( The authors assumed a C* value based on the smallest TSS effluent EMC in the dataset and calibrated k assuming steady-state conditions. Leisenring et al. (2013) included the k-C* model as a potential model for simulating pollutant removal in SCMs, but noted that the estimation of k and C* parameters can be difficult using published SCM data (e.g., the BMP Database).Linear Regression ModelFor a large number of SCMs studies, the only information available for assessing SCM performance are measured values of influent and effluent pollutant EMCs (Loáiciga et al. 2015). Recognizing the limitations of evaluating SCM performance using the percent removal metric (Strecker et al. 2001), Barrett (2005) suggested a new methodology for evaluating SCM pollutant removal performance by using linear regression of measured influent and effluent EMCs: (4) where m = the regression slope; b = the regression intercept; and ε denotes residuals between the simulated and measured values of Ce. Barrett (2005) discussed several useful attributes of this model for evaluating SCM pollutant removal. One is that the intercept value provides a reasonable approximation of the “irreducible minimum effluent concentration” that is frequently observed in SCM monitoring datasets. Another is that hypothesis testing can indicate whether the slope value is significantly different than zero. If it is not, then one may conclude that effluent concentrations are independent of influent concentrations for a particular BMP. Another benefit of this methodology can be readily applied to data contained in the BMP Database for any type of BMP and pollutant (Barrett 2008).SCM Pollutant Removal Model ParametersTable 1 presents the mean parameter values and their uncertainty used in the pollutant removal models and UA. All the parameter values and estimates of their uncertainty were obtained using data from the BMP Database, either as part of this study or others. Parameter values for the linear regression model were determined as part of this study using 137 pairs of influent/effluent TSS EMCs obtained from the BMP Database. Those data and results of the linear regression analysis are presented in Fig. S1 and Table S1. Parameter values for the MFG model are adopted from the results of Olson et al. (2020), which used the same BMP Database dataset as the linear regression analysis described previously. The parameter values for the k-C* model are adopted from the results of Park et al. (2011), which used a smaller subset of the BMP Database dataset (approximately 46 pairs of influent/effluent TSS EMCs), but which were also included in the larger dataset used for linear regression and MFG model parameterization.Table 1. SCM pollutant removal parameter values and probability distributionsTable 1. SCM pollutant removal parameter values and probability distributionsModelParameterUnitsMeanVarianceProbability distribution functionMFGapsf1—0.424.90×10−2Normalpsf2—0.357.70×10−3Normalpsf3—0.146.30×10−3Normalpsf4—0.052.30×10−3Normalpsf5—0.043.50×10−3Normalv1m/s6.29×10−61.50×10−11Normalv2m/s1.01×10−44.00×10−9Normalv3m/s5.10×10−41.00×10−7Normalv4m/s1.61×10−31.00×10−6Normalv5m/s2.52×10−32.50×10−6Normalk-C*bkm/day0.8280.191NormalC*mg/L1.00×1014.00×10NormalLinear regressioncm—0.158——bmg/L14.95——εmg/L0276NormalDynamic Pollutant RoutingFor dynamic modeling, we assume pollutants are routed through the SCM using a variable-volume, continuously stirred tank reactor (CSTR) model. This reactor model assumes that the pollutant concentration within the reactor is equal throughout (vertically and horizontally) and that the pollutant concentration discharged from the reactor is equal to the pollutant concentration within the reactor. The mass balance equation for this type of reactor can be written as (Chapra 1997): (5) ∂V∂C∂t=Qi,t·Ci,t−Qe,t·Ce,twhere V = volume of runoff within the SCM (m3); C = concentration of pollutant within the SCM (mg/L) (and also discharging from the SCM due to the assumption of a CSTR); Qi = rate of runoff entering the SCM (m3/s); Qe = rate of runoff discharging from the SCM (m3/s); ∂V/∂t = rate of change of runoff stored within the SCM (m3/s); and ∂C/∂t = rate of change of pollutant concentration within the SCM in mg/L per second. In typical reactor applications, Eq. (5) will also include additional terms describing the time-rate of pollutant removal within the reactor; however, such rates cannot be quantified for SCMs without intra-event pollutant concentration data. Instead, we apply the steady-state pollutant removal equations [Eqs. (2)–(4)] to runoff as it enters the reactor, such that the values of the Ce terms in those equations become the Ci term in Eq. (5). In order to apply Eqs. (2)–(4) to this dynamic model, the values for A, Q, and q′ in those equations are computed at the timestep (t) that runoff enters the SCM, i.e., A=A(t), Q=Q(t), and q′=q′(t).Eq. (5) can be rewritten as a finite-difference approximation over a timestep interval Δt : (6) Vt+1Ct+1−VtCt=Qi,t+Qi,t+12Ci,t+Ci,t+12Δt−Qe,t+Qe,t+12Ct+Ct+12ΔtEq. (6) can further be simplified if the dynamic model is operated over relatively small increments of Δt (i.e., on the order of minutes), where inflows and outflows are assumed to not change significantly over the timestep: (7) Vt+1Ct+1−VtCt=Qi,tCi,tΔt−Qe,tCtΔtThis assumption is justified under the conditions of this study because (1) the precipitation data used to generate the inflow hydrographs are aggregated on 1-h increments, so runoff rates generated from the watershed model assume that precipitation is equal within each 1-h increment; and (2) the inflow pollutographs are assumed constant throughout the entire storm event, as is typically performed for planning-level studies. More discussion on these assumptions is provided in the following sections.Finally, Eq. (7) can be rearranged to solve for Ct+1, which is the pollutant concentration within the SCM at the end of the simulation timestep and the concentration of pollutant discharging from the SCM during the next timestep: (8) Ct+1=VtCt+Qi,tCi,tΔt−Qe,tCtΔtVt+1Dynamic runoff routing through the SCM is based on the continuity equation [Eq. (9)], where the value for Qe,t is estimated using the storage-indication method (Viessman and Lewis 1996) and a priori knowledge of the stage-volume-discharge relationship of the SCM, characterized as: (9) Dynamic Watershed Modeling for Inputs to Dynamic SCM AlgorithmsInputs for the SCM models were generated using SWMM. SWMM is a widely used dynamic stormwater model capable of simulating runoff and pollutant concentrations from urban areas (Rossman 2010). Runoff quality and quantity were simulated from a hypothetical 12.1 ha (30 acre) watershed in Fort Collins, CO (Table 2) for three different precipitation events. All events had a total precipitation depth of 12.7 mm (0.5 in.), but different durations were used (Table 3). These represent actual events obtained from the hourly National Climatic Data Center ( record of Gage 053005 recorded in Fort Collins (Table 3). A total precipitation depth of 12.7 mm was selected because the stormwater drainage criteria for Fort Collins requires that SCMs be designed to capture and treat the runoff that occurs from approximately 12.7 mm of precipitation (City of Fort Collins Colorado 2018). The three different storm durations were selected to evaluate if storm duration affects the performance of the dynamic algorithms.Table 2. SWMM subcatchment parameter values used to generate runoff from a synthetic watershedTable 2. SWMM subcatchment parameter values used to generate runoff from a synthetic watershedParameterUnitsValueAreaha12.1Widthm2,655Slope%2Imperviousness%75N-Imperv—0.012N-Perv—0.25Dstore-Impervmm2.54Dstore-Pervmm5.08% Zero-Imperv%25Horton (Max Infiltration Rate)mm/h76.2Horton (Min Infiltration Rate)mm/h12.7Horton (decay constant)1/h4Table 3. Information about three different storm events evaluated during this studyTable 3. Information about three different storm events evaluated during this studyEventStart date (time)End date (time)Total precipitation depth [mm (in.)]Event duration (h)1July 2, 2013 (14:00)July 5, 2013 (15:00)12.7 mm (0.5)12August 5, 1993 (11:00)August 5, 1993 (17:00)12.7 mm (0.5)63May 20, 1951 (18:00)May 21, 1951 (18:00)12.7 mm (0.5)24Pollutant concentrations were simulated using the EMC method in SWMM. The SWMM EMC method applies a constant pollutant concentration to runoff. While it is possible to calibrate the SWMM buildup/washoff algorithms to generate intra-event variability of pollutant concentrations in runoff, it is more common to apply the EMC method for planning-level studies, due to lack of available intra-event calibration data. In this study, we applied TSS EMCs of 50  mg/L, 125  mg/L, and 200  mg/L.Fig. 1 shows the hydrographs and pollutographs generated from SWMM for the three different precipitation events. These hydrographs and pollutographs were used as inputs to the dynamic SCM models.Extended Detention Basin DesignThe objectives of this study were evaluated using an EDB SCM. The simulated EDB was designed according to design criteria published by the Colorado Department of Transportation Urban Drainage and Flood Control District (UDFCD 2015), which describe methods for calculating the water quality capture volume (WQCV) and drawdown time (40 h) of EDBs implemented along the Front Range of Colorado. We assumed the EDB has vertical side slopes and a WQCV depth of 0.91 m. Table 4 presents the EDB design parameters used in the SCM simulations.Table 4. Design parameters of the EDB evaluated in this studyTable 4. Design parameters of the EDB evaluated in this studyParameterDescriptionUnitsValueNotesWQCVWater quality capture volumem3923.7Computed according to design criteria in UDFCD (2010).ASurface aream21,010Surface area is constant (assumed vertical side slopes).QAverage discharge ratem3/s6.41×10−3Used in steady-state version of the MFG model.q′Average hydraulic loading ratem/day0.549Used in steady-state version of the k-C* model.SWMM was used to generate a stage-surface area-discharge table for the EDB, assuming a single 102 mm (4 in.) circular orifice is used to control the drawdown of the WQCV. The stage-surface area-discharge curve (Fig. 2) was used as input to the dynamic SCM model.Aggregation of Dynamic OutputsThe dynamic model generates estimates of effluent flows, pollutant concentrations, and uncertainty of the pollutant concentrations at every timestep of the simulation. To compare the dynamic results to steady-state model results, the TSS effluent EMC and variance of the EMC were computed using Eqs. (10) and (11): (10) EMC(Ce)=∑Qe(t)Ce(t)∑Qe(t)(11) VAR[EMC(Ce)]=∑Qe(t)VAR(Ce(t))∑Qe(t)Uncertainty AnalysisEstimates of TSS effluent EMC uncertainty were generated using three different UA methods: MC, prediction interval for linear regression, and FOVE. The parameters for which uncertainty was propagated are listed in Table 1.Monte Carlo Method with Latin Hypercube SamplingParameter uncertainty for the MFG and k-C* models was propagated using 1,000 simulations of the steady-state and dynamic models. (Initial testing of higher numbers of simulations showed that 1,000 simulations of both models were sufficient to generate stable results.) Random parameter combinations were generated using the Latin Hypercube Sampling (LHS) method (McKay et al. 1979). The LHS method is a stratified sampling method that divides the multi-variate space into a specified number of equally probable intervals, from which one sample is generated randomly. The primary benefit of using the LHS method is that it provides similar estimates of uncertainty to traditional random sampling methods, but uses fewer model simulations. Fifth and 95th percentile values were obtained for each scenario by sorting the 1,000 model outputs lowest to highest and retrieving the value of the 50- and 950-ranked values.First Order Variance Estimation MethodThe FOVE method was also used to generate uncertainty estimates for all three pollutant removal models applied to the dynamic algorithms. The FOVE method propagates the variance of random parameters using a Taylor-series expansion of the model function to generate an estimate of the variance of the model output. Consider a function Y=g(X), where X is a series of random variables. X={X1,X2,…,Xk} and Xo={x¯1,x¯2,…,x¯k} denotes the set of mean parameter values. The value of the function at Xi={x1,x2,…,xk} can be approximated using a first-order Taylor series approximation (Morgan and Henrion 1990; Tung et al. 2006): (12) Y(Xi)=Y(Xo)+∑j=1k[∂Y∂xj]Xo(xj−x¯j)According to properties of variance, the variance of function Y=g(X) can be expressed as: (13) VAR(Y)=∑j=1k[∂Y∂xj]Xo2VAR(xj)+2∑i=1n∑j>1n[∂Y∂xi]Xo[∂Y∂xj]XoCOV(xi,xj)The partial derivative terms for each pollutant removal model response with respect to each model parameter were determined analytically and are provided in Table 5. The probability distribution function, expected value, and variance of each random parameter assumed in this study are provided in Table 1. Additionally, it was assumed that all random parameters were independent and uncorrelated; hence, the second term in the right-hand side of Eq. (13) was dropped from the final analysis.Table 5. Partial derivatives used in the FOVE application of the SCM pollutant removal modelsTable 5. Partial derivatives used in the FOVE application of the SCM pollutant removal modelsModelParameterAnalytical partial derivativeMFGpsf∂Ce∂pf=Ci(1+AvnQ)nv∂Ce∂v=−Ci×psf×A×(1+AvnQ)−n−1Qk-C*C*∂Ce∂C*=1−e−k/q′k∂Ce∂k=(C*−Ci)e−k/q′q′Linear regressionε∂Ce∂ε=1Recalling that Eq. (8) is the finite-difference approximation of effluent pollutant concentrations for the dynamic algorithm, the values of both Ct and Ci,t are considered random parameters that affect the variance of Ct+1. The FOVE derivation of the dynamic pollutant routing model is (14) VAR(Ct+1)=a2VAR(Ct)+b2VAR(Ci,t)+2abCOV(Ct,Ci,t)where a=(Vt−Qe,tΔt/Vt+1); b=(Qi,tΔt/Vt+1); COV(Ct,Ci,t)=rσCtσCi,t; and r is the correlation coefficient between Ct and Ci,t). It is reasonable to assume that Ct and Ci,t are perfectly and positively correlated (i.e., r=1) because if Ci,t is greater than the expected value (at all timesteps of the simulation) then Ct will also be greater for all timesteps of the simulation. The 5th and 95th percentiles of Ce were estimated assuming that Ce was lognormally distributed (Williams et al. 2014).Prediction Interval for Linear Regression ModelThe prediction interval method (Walpole et al. 1998) is used to estimate uncertainty intervals for the linear regression model. The prediction interval provides the uncertainty of the dependent variable (i.e., effluent EMC) of a single, unknown event as a function of the independent variable (i.e., influent EMC), and represents how well the best-fit linear regression parameters fit the data. For an individual storm event with influent concentration Ci, the uncertainty of the predicted effluent concentration Ce can be estimated using Eq. (15), which is the (1−α)100% prediction interval (Walpole et al. 1998): (15) Ce±tα/2,n−2s1+1n+(Ci−Ci¯)2∑j=1n(Ci,j−Ci¯)2where tα/2,n−2 denotes the t-distribution value at the α significance level, with (n−2) degrees of freedom; n = the number of data point pairs; s represents the standard error of the linear regression model; Ci¯ denotes the mean of the influent concentration data points; and Ci,j denotes the individual influent concentration data point. The 5th and 95th percentiles of Ce were obtained using α=0.1. A flowchart depicting the steps involved in the methodology is provided in Fig. S2.Results and DiscussionDeterministic Comparison of the Three EDB Pollutant Removal ModelsThe results of deterministically modeling SCM effluent EMCs using the mean value of pollutant removal model parameters are shown in Table 6. The linear regression model produced approximately the same (<1% difference) effluent EMCs for both the steady-state and dynamic models. This was expected, because the linear regression equation is only a function of the influent concentration, and the influent concentration was constant throughout the duration of the storm events. These results demonstrate that linear regression models calibrated to EMC data can be applied to SCM models with dynamic routing without introducing additional error to the model results. It is likely that the results may be different if the influent concentration varies throughout the duration of the event; however, calibrating pollutant buildup/washoff models at the planning level is generally not possible due to lack of sufficient data.Table 6. TSS effluent EMCs generated using steady-state and dynamic applications of the SCM pollutant removal modelsTable 6. TSS effluent EMCs generated using steady-state and dynamic applications of the SCM pollutant removal modelsModelInfluent EMCSteady-stateDynamic% differenceMFG501014.94912524.937.349.820038.959.753.5k-C*5018.828.953.712535.464.381.62005299.791.7Linear regression5022.922.8−0.412534.734.6−0.320046.646.4−0.4MFG501013.63612524.934.13720038.954.640.4k-C*5018.82743.612535.45966.7200529175Linear regression5022.922.8−0.412534.734.6−0.320046.646.4−0.4MFG5010122012524.930.120.920038.948.223.9k-C*5018.823.826.612535.449.840.72005275.845.8Linear regression5022.922.7−0.912534.734.4−0.920046.646.1−1.1The results of applying the MFG model showed that effluent EMCs are higher when applied to a dynamic model compared to a steady-state model. The dynamic model produced effluent EMCs that were approximately 50%, 38%, and 22% higher than the steady-state model for storm durations of 1 h, 6 h, and 24 h, respectively. The effect of influent concentration on the difference between steady-state and dynamic effluent EMCs was considerably smaller (1%–4%) than the effect of storm duration.The dynamic k-C* model also produced much larger effluent EMCs compared to the steady-state model, with differences ranging from 27% to 92% higher. The largest difference (92%) was produced by applying an influent concentration of 200  mg/L during a 1-h storm duration, while the smallest difference (27%) was produced by applying an influent concentration of 50  mg/L during a 24-h storm duration. Overall, the differences between dynamic and steady-state model results increased with larger influent concentrations and decreased with longer storm durations.Because the EDB was assumed to have constant surface area and influent concentration, the only difference between the steady-state and dynamic models is the value of the discharge rate (Q). All other parameters being equal, a higher value of Q will result in a higher effluent concentrations for both the MFG and k-C* models. For the steady-state model, the value of Q was assumed to be constant and equal to the WQCV divided by the design drawdown time (40 h), an assumption also used by Park et al. (2011). The dynamic models assumed that the discharge rate in an EDB is not constant, as the discharge rate increases with increasing depth of storage due to hydrostatic head on the outlet orifice. Fig. 3 shows the discharge rate generated from the dynamic model compared to the assumed discharge rate for the steady-state model for all three precipitation events tested. Clearly, the discharge rates simulated using the dynamic model are higher than the average discharge rate during a considerable portion of all storm events. Most importantly, the discharge rates are generally higher than the average discharge rate when the discharge hydrograph is rising, which represents conditions when runoff is entering the EDB. The dynamic models apply the pollutant removal equations instantaneously at the time that runoff enters the EDB, using the discharge rate computed for that timestep. When the instantaneous discharge rate is larger than the average discharge rate when runoff enters the EDB, the MFG and k-C* models generate higher effluent concentrations of pollutants. Fig. 3 also shows that the difference between the instantaneous discharge rates and the average discharge rate is much lower for the 24-h storm event than for the 1-h and 6-h storm events. This explains why the difference between the dynamic and steady-state model results are lower for the 24-h storm duration compared to the results of the 1-h and 6-h storm durations.Modeling Uncertainty in the Assessment of the Effects of EDBsThe 95% prediction interval (PI) TSS effluent concentrations are presented in Fig. 4 for the MFG model, Fig. 5 for the k-C* model, and Fig. 6 for the linear regression model. The marker on each figure represents the mean effluent TSS EMC, while lines show the PI. Each panel represents the results for a specific influent TSS concentration (e.g., 50  mg/L, 125  mg/L, or 200  mg/L). For the MFG and k-C* models, three different UA results are presented for each storm duration (e.g., 1 h, 6 h, or 24  h). The first interval for each storm duration category (circle marker) represents the results of 1,000 Monte Carlo simulations of the respective steady-state models using the random parameter distributions shown in Table 1. The second interval (square marker) represents the results of 1,000 Monte Carlo simulations of the respective dynamic model. The third interval (diamond marker) represents the results of applying the FOVE method to the respective dynamic models.For the Monte Carlo scenarios, the percentiles were obtained directly from the 1,000 simulation outputs while the percentiles for the FOVE scenario were obtained assuming a lognormal distribution of the outputs. For the linear regression model, the left interval (circle marker) was generated using the steady-state model, with uncertainty estimated using the prediction interval method and the right interval (square marker) was the result of applying the FOVE method to the dynamic model.MFG ModelThe widths of the uncertainty intervals generated using Monte Carlo simulations were approximately the same for both the steady-state and dynamic MFG models. The primary difference between the results of those two scenarios is that the dynamic mean EMC values and uncertainty intervals were consistently higher. As discussed in the previous section, this shift in outputs from the dynamic model is likely due to the use of the instantaneous discharge rate to compute pollutant removal in the dynamic model. The difference between the steady-state and dynamic model results was smaller for larger duration storms, but the overall PI remained mostly unchanged.Application of the FOVE method to the dynamic MFG model produced a slightly narrower PI compared to the MC method under all combinations of influent concentration and storm duration. The 5th percentile values generated using the FOVE were all larger than those generated using the MC simulations, and the 95th percentile values generated using the FOVE were all smaller than those generated using the MC simulations. The maximum absolute difference between FOVE- and MC-generated 95th percentile values was approximately 24  mg/L, which occurred with influent concentration at 200  mg/L and storm duration of 24 h. The maximum relative difference between those values was approximately 21%, which occurred with influent concentration at 125  mg/L, and storm duration of 24 h. The width of PI generated by the FOVE method tended to decrease with increasing storm duration.The PI values for the FOVE method were narrower than the steady-state MC results for all storm durations and it appears that the narrower intervals generated using the FOVE method somewhat compensated for the dynamic model’s tendency to predict higher effluent concentrations compared to the steady-state model. However, the difference between those values increased with longer storm durations, with the FOVE method generating upper PI values approximately 20% smaller for a 24-h storm duration.k-C* ModelMC simulations of the dynamic k-C* model produced noticeably narrower uncertainty intervals compared to MC simulations of the steady-state k-C* model. However, most of the difference in uncertainty results between those applications is in the estimation of the lower percentiles, as the 95th percentile values are similar under all conditions. The maximum absolute difference between 95th percentile values is approximately 13  mg/L, which occurs with influent concentration at 200  mg/L and storm duration of 1 h. The maximum relative difference between those values is approximately 7%, which occurs with influent concentration at 125  mg/L, and storm duration of 1 h. This is an important finding when considered within the context of UA, as decision makers may be interested in planning for worst-case scenarios to reduce the probability that discharged pollutants exceed a regulatory threshold. The 95th percentile value, for example, has a 5% chance of being exceeded under the simulated conditions. Thus, one may conclude that applying the dynamic k-C* model with MC methods produces estimates of higher percentile values that are reasonably close to the steady-state model with MC methods.The FOVE method generally produces narrower uncertainty intervals compared to the MC method, when applied to the dynamic model. One exception is for the case of influent concentration of 50  mg/L and storm duration of 24 h, where the FOVE uncertainty estimates are slightly wider. Similar to the results generated using the MFG model, the FOVE method produced lower values of 95th percentiles and higher values of 5th percentiles. The maximum absolute difference between 95th percentile values is approximately 33  mg/L and the maximum relative difference between those values is approximately 20%, which both occur with influent concentration of 200  mg/L and storm duration of 24 h. In general, the FOVE uncertainty intervals tend to increase in width with increasing storm duration.Linear Regression ModelOverall, the FOVE method produces narrower uncertainty intervals than the prediction interval for all combinations of influent concentration and storm durations; however, the primary effect of the smaller uncertainty intervals is the estimate of the 5th percentile value. 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