Evaluation of pollutant loads from stormwater BMPs to receiving water using load frequency curves with uncertainty analysis

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Evaluation of pollutant loads from stormwater BMPs to receiving water using load frequency curves with uncertainty analysis Daeryong Park a,*, Larry A. Roesner b a

Illinois State Water Survey, Prairie Research Institute, University of Illinois at Urbana-Champaign, 2204 Griffith Dr., Champaign, IL 61820-7463, USA b Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 80523-1372, USA

article info

abstract

Article history:

This study examined pollutant loads released to receiving water from a typical urban

Received 26 August 2011

watershed in the Los Angeles (LA) Basin of California by applying a best management

Received in revised form

practice (BMP) performance model that includes uncertainty. This BMP performance model

12 April 2012

uses the k-C* model and incorporates uncertainty analysis and the first-order second-

Accepted 15 April 2012

moment (FOSM) method to assess the effectiveness of BMPs for removing stormwater

Available online 22 April 2012

pollutants. Uncertainties were considered for the influent event mean concentration (EMC)

Keywords:

runoff model (STORM) was used to simulate the flow volume from watershed, the bypass

Stormwater

flow volume and the flow volume that passes through the BMP. Detention basins and total

Best management practices

suspended solids (TSS) were chosen as representatives of stormwater BMP and pollutant,

k-C* model

respectively. This paper applies load frequency curves (LFCs), which replace the exceed-

First-order second-moment

ance percentage with an exceedance frequency as an alternative to load duration curves

Storage, treatment, overflow and

(LDCs), to evaluate the effectiveness of BMPs. An evaluation method based on uncertainty

runoff model (STORM)

analysis is suggested because it applies a water quality standard exceedance based on

Load frequency curve

frequency and magnitude. As a result, the incorporation of uncertainty in the estimates of

Total daily maximum load

pollutant loads can assist stormwater managers in determining the degree of total daily

and the aerial removal rate constant of the k-C* model. The storage treatment overflow and

maximum load (TMDL) compliance that could be expected from a given BMP in a watershed. ª 2012 Elsevier Ltd. All rights reserved.

1.

Introduction

Urban stormwater runoff contains significant concentrations of a variety of pollutants and is a principal cause of the deterioration of receiving water quality in urban areas. Structural best management practices (BMPs) are widely applied to reduce nonpoint source pollution and attenuate peak runoff. However, the many uncertainties associated with BMP performance preclude models of BMP performance from

reliably simulating pollutant removal. Input flows and pollutant concentrations vary from storm to storm and within individual storms, and BMP pollutant removal mechanisms are not sufficiently accurate. Therefore, the computed (estimated) pollutant loads and concentrations emanating from a BMP model are uncertain. As regulatory agencies move toward BMP effluent criteria or total daily maximum load (TMDL) allocations for receiving waters, it becomes increasingly important to understand the certainty with which we

* Corresponding author. Tel.: þ1 970 988 0304. E-mail addresses: [email protected] (D. Park), [email protected] (L.A. Roesner). 0043-1354/$ e see front matter ª 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2012.04.023

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can estimate the performance of a BMP or, conversely, to understand uncertainty associated with an estimated effluent concentration from a BMP. To evaluate the TMDL, load duration curves (LDCs) have been developed by Stiles (2001). Several researchers (Stiles, 2001; Cleland, 2002, 2003; Bonta and Cleland, 2003; and O’Donnell et al., 2005) have utilized LDCs to estimate the TMDL because this method is capable of identifying daily loads that account for the variable nature of water quality with time. A maximum concentration standard and a hydrologic flow duration curve (FDC) can identify a TMDL that is appropriate for the full range of streamflow conditions, and the maximum daily load can be verified for any given day (US EPA, 2007). Although LDCs have become more widely used and accepted for TMDL estimation, it is necessary to take the pollutant-reducing physical model into consideration. The current LDC method only accounts for flow variables and does not consider other variables because it does not incorporate certain physical models (Shen and Zhao, 2010). The purpose of this paper is to present an approach for estimating the pollutant load exceedance frequencies resulting from BMPs with estimates of the certainties (or uncertainties) of those estimates. If a TMDL is specified as an average load and an upper limit on that load which is not to be exceeded more than n times per year, the algorithm presented will assist in the design of a BMP that will meet the criteria with 95% certainty. The method is simple, but it is a step forward in linking BMP performance to receiving water quality. The conceptual model used is the storage, treatment, overflow and runoff model (STORM) as illustrated in Fig. 1 (US Army Corps of Engineers, 1977). Urban runoff (VR) is introduced to the BMP at a rate equal to the lowest inflow value or the average drawdown rate specified for the BMP. When the BMP is full, the flow is bypassed around the BMP and discharged directly to the receiving water. The total volume of storm flow that passes through the BMP is designated as VBMP. The difference (VR  VBMP) is VO, the volume of runoff that bypasses the BMP, An advantage of STORM is that it can simulate above processes very quickly, estimating VR, VBMP and VO based on storm events for long time periods faster than other computer models (Lee et al., 2005; Park et al. in press). The total pollutant load from any storm is calculated as the sum of pollutant load discharged from the BMP plus the pollutant load bypassed directly to the receiving water. In this model a “storm” is defined as a period of rainfall that is preceded and succeeded by a period of 6 or more hours of no rainfall. The interevent time is set by the user, but most analysts choose a 6-h interevent time because the body of experience indicates that a 6-h interevent time produces the most reasonable results.

Uncertainty is introduced into this system in the following ways: 1. The uncertainty of the pollutant concentration of the runoff, Cin is calculated using a log-normal distribution of event mean concentrations from the International BMP Database (www.bmpdatabase.org) or local data. 2. The uncertainty in BMP treatment effectiveness is accounted for by associating the uncertainty with the key performance parameters of the k-C* model (Kadlec and Knight, 1996). The uncertainty in the pollutant load discharged to the receiving water reflects the combined uncertainties of each of the two sources indicated above. The inflow to the BMP is the average runoff rate during the event, and treatment efficiency is based on the average inflow during the event. Pollutant loads discharged to the receiving water are the total masses discharged over the course of the event. Details are provided in the following sections.

2.

Methods

2.1.

The flow model

Storage-release urban stormwater management systems, or volumetric BMPs, are used extensively for controlling urban stormwater runoff and nonpoint source pollutants. In the 1970s, the US Army Corps of Engineers developed the STORM, which is capable of computing the stormwater runoff for a storage-treatment control structure (US Army Corps of Engineers, 1977). STORM is practical and can be easily understood. It has been used extensively to estimate the quantity and quality of watershed runoff on the basis of watershed land use (Roesner et al., 1974). The underlying method is capable of long-term continuous simulations, whereas other more complicated models produce only singleevent simulations or take large amounts of computer memory and time to run in continuous simulation mode. The STORM algorithm has been applied to both explicit processes using spreadsheets (Lee et al., 2005) and to analytical methods (Adams and Papa, 2000). The STORM computes the percent of runoff captured and produces overflow volume exceedance frequency data for various BMP capture volumes. STORM has been used extensively in the US to determine water quality capture volume (WQCVs) for BMPs. For example, the STORM algorithm adopted in NetSTORM was used to determine the stormwater water quality capture volume for the California Stormwater Best Management Practice Handbook (California Stormwater Quality Association, 2003; Heineman, 2004; Park et al. in press). This paper uses QuickSTORM, which contains the same algorithm as the STORM in the DOS version of NetSTORM.

2.2.

Fig. 1 e Schematic of an urban stormwater system.

The treatment model

Most storage-release models for pollutant removal in the literature are first-order kinetic models that directly compute pollutant loading but not concentration (Roesner, 1982; Nix

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and Heaney, 1988; Patry and Kennedy, 1989; Segarra-Garcia and Loganathan, 1992; Segarra-Garcia and Basha-Rivera, 1996; Lee et al., 2005). These studies applied a first-order kinetic equation to express BMP performances and represented the results with percentages of pollutant removal as functions of BMP storage volumes and release rates. However, the hydraulic loading rate (HLR) is strongly correlated to pollutant removal and is a function of both the inflow rate and BMP surface area (Kadlec, 2000). The background concentration (C*) of outflow from the BMP can be considered constant from storm to storm, as has been verified by many researchers (e.g. Schueler, 1996; Wong and Geiger, 1997; Minton, 2005). This paper also uses the k-C* model to characterize BMP performance. The k-C* model is defined by the following equation (Kadlec and Knight, 1996): Cout ¼ C þ ðCin  C Þek=q

(1)

where Cout ¼ effluent EMC (mg/L), Cin ¼ influent EMC (mg/L), C* ¼ background EMC or “irreducible minimum concentration” (mg/L), k ¼ aerial removal rate constant (m/day), and q ¼ BMP hydraulic loading rate, defined as the ratio of the average inflow rate Q to the surface area A of the BMP, i.e., (Q/ A) (m/day). The k-C* model has been used to model wetland performance, and many references have verified that this model characterizes the removal of pollutants by wetlands very well (Kadlec and Knight, 1996; Kadlec, 2000, 2003; Braskerud, 2002; Rousseau et al., 2004; Lin et al., 2005). Recently, the k-C* model was used by Wong et al. (2002, 2006) and Huber (2006) to simulate stormwater BMPs because the characteristics of wetlands, detention basins and retention ponds are similar. However, it is difficult to obtain a reliable prediction of pollutant removal with the k-C* model because the determination of the parameters C* and k includes large intersystem variabilities (Kadlec, 2000) and additional parameters, such as temperature (Kadlec and Reddy, 2001), Damko¨hler number (Carleton, 2002), flow velocity and residence time (Carleton and Montas, 2007). Therefore, it is necessary to develop a more simplified and generalized model for predicting pollutant removal. Park et al. (2011) applied uncertainty analysis to the k-C* model and successfully presented Cout as a probability density function in detention basins depending on Cin and q. k was represented as a function of q (Schierup et al., 1990; Lin et al., 2005). This paper adopts this method to simulate BMP performance.

2.3.

Incorporation of uncertainty into the model

Uncertainty analysis has been used to quantify reliabilities or probabilistic risks for a variety of engineering problems. The first-order second-moment (FOSM) method, also known as the first-order error (FOE) or first-order variance estimation (FOVE) method, is one of the most general and simple methods of uncertainty analysis. It is a first-order approximation method

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because it considers only the first order of the Taylor series. The FOSM method has been applied to various hydro-systems, including storm drainage (Yen and Tang, 1976) and levee systems (Tung and Mays, 1981). In environmental engineering, several researchers have applied the FOSM method to the StreeterePhelps equation, which is used to estimate dissolved oxygen in streamflows (Burges and Lettenmaier, 1975; Tung and Hathhorn, 1988; Song and Brown, 1990; Melching and Anmangandla, 1992). The FOSM method has been used to estimate the margin of safety (MOS) with respect to the TMDL (Zhang and Yu, 2004; Franceschini and Tsai, 2008). Shirmohammadi et al. (2006) integrated several uncertainty analysis methods, including the Monte Carlo simulation (MCS), FOE analysis, Latin hypercube sampling (LHS), and generalized likelihood uncertainty estimation (GLUE), into the soil and water assessment tool (SWAT) to represent the cumulative density function (CDF) of monthly sediment reduction as a measure of BMP effectiveness. They suggested using uncertainty analysis to improve estimates of the MOS and TMDL. Arabi et al. (2007) characterized BMP effectiveness in terms of estimated monthly sediment reduction, total phosphorus (TP), and total nitrogen (TN) with two types of uncertainty analysis methods: onefactor-at-a-time (OAT) sensitivity analysis and GLUE using SWAT. In addition, they suggested using a probabilistic estimation of the MOS for TMDL development. In urban stormwater modeling, Kleidorfer et al. (2009) and Dotto et al. (2011) applied uncertainty analysis to the rainfall/ runoff process by the model for urban stormwater improvement conceptualization (MUSIC) and the KAREN model and build-up/wash-off processes using simple regression and build-up/wash-off equations, respectively. The Bayesian Monte Carlo Markov Chain (BMCMC) method was applied for uncertainty analysis. Kleidorfer et al. (2009) focused on measurement errors for uncertainty analysis but Dottos et al. (2011) considered the predictive uncertainty resulting from parameter uncertainty in the MUSIC, KAREN and build-up/ wash-off models. These studies focused on uncertainties in runoff and pollutants from a watershed. Therefore, it was necessary to study the uncertainty of water quality BMP performance for the next step. As mentioned above, Park et al. (2011) studied the BMP performance incorporating uncertainty analysis and compared the results with observed data. This study focused on the predictive uncertainty due to parameter uncertainties (Cin and k) in the BMP performance model (the k-C* model) and did not consider other sources of uncertainty. The results showed that the FOSM method is exceptionally easy to apply to uncertainty analysis of BMP performance compared with the derived distribution method (DDM) and LHS because it requires only the mean and variance of the data if the probability density function is similar to a two-parameter distribution and that its accuracy is only slightly different (approximately 5 mg/L for confidence limits [CLs]) from that of the LHS method. The applicability of the FOSM method increases if the variable distribution is a known two-parameter distribution.

2.3.1.

Uncertainty analysis in Cin and k

Fig. 2 and Table 2 provide the two-parameter log-normal probability plots and the goodness-of-fit statistics for the

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Table 2 e Results of goodness-of-fit tests; observed Cin and Cout from Table 1. Test

Chi-Square KolmogoroveSmirnov AndersoneDarling

Fig. 2 e Log-normal probability plots of observed Cin and Cout in detention basins.

observed Cin and Cout of TSS in detention basins for the BMP sites listed in Table 1. Fig. 2 shows that the TSS distributions for both Cin and Cout are well represented as log-normal probability plots. Table 2 shows the results of three goodness-of-fit tests using the well-known chi-squared, KolmogorveSmirnov, and AndersoneDarling tests. To apply these tests for normality, all Cin and Cout values were transformed using the natural logarithm (base e) (D’Agostino and Stephens 1986; Kottegoda and Rosso, 1997). All tests at a significance level of 0.1 showed that a log-normal distribution can be accepted for both observed Cin and observed Cout. For TSS in BMPs, Park et al. (2011) showed that k and q in Eq. (1), exhibit a power regression relation following the equation k ¼ 1.4841q0.9721, as shown in Fig. 3. A regression of k versus q for each storm event (as defined in the Introduction) was performed with a 95% prediction interval of 0.4370. This is regarded as the uncertainty in k. Thus, for Eq. (1), uncertainty in k can be generated based on the given q. The distribution generating k is considered a two-parameter log-normal distribution depending on q because k exhibits a power

Critical value (a ¼ 0.10) Cin

Cout

0.663 0.789 0.567

0.860 0.852 0.685

Decision

Accept Accept Accept

regression relationship with q (Schierup et al., 1990; Lin et al., 2005). Detailed descriptions of the prediction interval application for the k and q regression line are included in Park et al. (2011). To summarize, the distribution of Cout was estimated using the k-C* model (Eq. (1)) with two log-normally distributed input parameters: Cin and k. The geometric (BMP surface area, A (m2)) and the hydrological parameter (inflow, Q (m3/day)), has been considered known and unaffected by errors to focus on the performance uncertainty from the k-C* model. In other words, this study neglected systematic uncertainties such as measurement uncertainties and only considered only statistical uncertainties because statistical uncertainties such as Cin and k in the k-C* model affected to Cout were greater than the measurement uncertainties such as A and Q (Rousseau et al., 2004). C* was fixed at a value of 10 mg/L based on consistent recommendations for this value by Kadlec and Knight (1996), Barrett (2005), and Crites et al. (2006) because the unification of C* throughout all stormwater events is necessary for convenient computation (Park et al., 2011). Table 3 shows the required information given for the input variables for the uncertainty analyses of both Cin and k. The FOSM method was applied to the k-C* model with parameters defined according to the protocol above, assuming that the two variables Cin and k are independent because the correlation coefficient between k and Cin computed from the observed data is 0.027. This value is small enough to allow for Cin and k to be regarded as independent. The FOSM method is described as follows (Salas et al., 2004):

Table 1 e Selected best management practices (Park et al., 2011). BMP type

BMP Name, location

Detention 15/78, Basin Escondido, CA 5/605 EDB, Downey, CA 605/91 edb, Cerritos, CA Manchester, Encinitas, CA

BMP size Number Volume Surface Length of (m3) area (m) datasets (ha) 17

1122.54

0.0977

60.96

2

364.66

0.0598

47.24

5

69.57

0.0114

22.86

12

252.79

0.0304

22.86

Fig. 3 e Estimated k versus q based on individual storm events for detention basins (Park et al., 2011).

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Table 3 e Required parameter information for uncertainty analysis of both Cin and k (Park et al., 2011). Input parameters

k

Cin

Log-transformed statistical properties Value used in paper Cases of uncertainty analysis

Mean

Standard deviation

Mean

Standard deviation

5.038

0.6083

Log (1.4841q0.9721)

0.437

*

*

Uncertainty in Cin and k

*

*

*Required information.

EðYÞ ¼ E½gðX1 ; /; Xn Þzgðm1 ; /; mn Þ VarðYÞ ¼ Var½gðX1 ; /; Xn Þz

(2)

2 n  X vg j¼1

vXi

VarðXi Þ

(3)

m

where X indicates a random variable and Y specifies the general function y ¼ g (x).

2.4.

Process for estimating TSS load with uncertainty

The following steps demonstrate the method for computing a pollutant-load frequency curve on an event basis from Fig. 1: 1. STORM simulates VR for an event. 2. The pollutant mass in the runoff (MR) is computed by multiplying VR and the TSS EMCs in the runoff from the watershed Cin ¼ (Cin  εin ) where Cin is the average inflow concentration and εin is a random variable taken from the normalized log-normal distribution of Cin from Table 3. 3. VR is divided into VBMP and Vo by STORM. 4. The pollutant mass that bypasses the BMP (Mo) is computed by multiplying Vo and Cin (see Step 2 above). 5. The pollutant mass that leaves the BMP (Mout) is computed by multiplying VBMP and the Cout estimated by the k-C* model. 6. The total pollutant mass discharged to the receiving waters for the event (MTOT), is computed by adding Mo and Mout. The median and the 95% CLs of MTOT are computed using the 500th, 25th, and 975th sorted samples for 1000 generations. 7. A long-term hourly rainfall record is input to STORM to generate a time series of the mass loads identified in steps 2e6. 8. The exceedances per year are computed by ranking the TSS event loads. Load frequency curves (LFCs) are plotted.

where MR and VR are substituted for M and V for runoff, and as MO and VO are substituted for bypass calculations. To compute 95% CLs for Cout, which is computed from the k-C* model, we must estimate mlnCout and slnCout were estimated as shown in Appendix A. The 95% CLs of Cout were estimated as follows:   Cout;95% UCL ¼ exp mlnCout þ 1:96slnCout

(7)

  Cout;95% LCL ¼ exp mlnCout  1:96slnCout

(8)

The TSS load (Mout) that leaves the BMP and its 95% CLs are determined from Eqs. (4)e(6). Mout and Cout for M and C. Finally, the pollutant load discharged to the receiving water (MTOT) can be estimated as the sum of the BMP load (Mout) and the bypass load (Mo). To determine the uncertainty in MTOT, Monte Carlo simulations were applied. Because this study generated 1000 samples for each Cin and k value to estimate Mtot by summing Mo and Mout, the 95% CLs are represented by the 25th and 975th sorted samples, and the median value is represented by the 500th sorted sample, as shown in Fig. 7. This study used LFC instead of LDC. As water quality regulation moves toward TMDLs that contain exceedance frequency criteria of storm events for instream concentrations and/or BMP loads the certainty (or uncertainty) of meeting these criteria becomes important and the proper design point on the load exceedance frequency curve becomes an issue. The LFC generation in steps 7 and 8 the frequency and the event TSS loads can be estimated by the Cunnane (1978) formula as follows: T¼

N þ 1  2A MA

(9)

where

To develop the statistical characteristics of the mass loads for an event, the median, 95% upper confidence limit (UCL), and lower confidence limit (LCL) for TSS loads in the runoff (MR) and in the bypass (MO) were computed as:

T ¼ return period (years), N ¼ number of years of record, M ¼ rank of the event (in descending order of magnitude), and A ¼ plotting position parameter (0.4).

Mmedian ¼ Cin;median $V

(4)

The number of exceedances per year (E ) can be calculated from the return period as follows:

M95% UCL ¼ Cin;95% UCL $V

(5)

M95% LCL ¼ Cin;95% LCL $V

(6)

E¼

1 T

(10)

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Fig. 4 e Hourly rainfall, Los Angeles International Airport (LA), 1950e2009.

3.

Application and results

The following example illustrates how this HLR uncertainty analysis might be applied. To create the example, sixty years of continuous hourly rainfall data for Los Angeles International Airport (LA) were obtained from the National Climate Data Center (NCDC) and input into STORM to simulate VR, Vo, and Vout. This location was chosen because all of the detention basins shown in Table 1 are located around LA, and this sampling duration is the minimum time unit used by STORM. The rain gauge NCDC COOP ID number is 045114, and the records span the period from January 1, 1950 to December 31, 2009. There is no record history from Jan 1, 1968 to May 28, 1968 and hourly and daily rainfall data are provided in this station. Figs. 4 and 5 represent the hourly and annual rainfall amounts in LA. Values of over 30 and 20 mm/h were recorded twice and 15 times, respectively, over 60 years as shown in Fig. 3. The average annual rainfall was 310 mm, the highest was 748 mm and the lowest was 106 mm as shown in Fig. 5. A box and whisker plot describing monthly rainfall across the 60 year record is presented in Fig. 6. These figures indicate that LA is a dry region with a clearly distinguished wet season from

Fig. 5 e Annual total rainfall, LA, 1950e2009.

Fig. 6 e Box and whisker plot for monthly rainfall, LA, 1950e2009.

November to April and a clearly distinguished dry season from May to October. Event based urban runoff quality data used to develop the statistical characteristics Cin and k were taken from Park et al. (2011) because the available pollutant data in the International Stormwater BMP Database (www. bmpdatabase.org) are limited. Table 1 lists the locations, number of datasets, and sizes of the four detention BMPs used in this paper. A 6-h interevent time and a minimum threshold runoff depth of 0.01 inches were specified to separate the flow data into individual events. For this example, a the watershed was defined as having an area of 1 acre (4045 m2, imperviousness of 40%, a BMP volume of 0.2 inches (z5.0 mm), and a BMP surface area to watershed area ratio of 0.01, as indicated in Table 4. The runoff coefficient for STORM was calculated using the watershed imperviousness ratio (i) between 0 and 1, as described by WEF and ASCE (1998) as follows: C ¼ 0:85i3  0:78i2 þ 0:774i þ 0:04

(11)

Fig. 7 e Load frequency curves, including confidence limits, for the requirements of load and exceedance for the example.

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Table 4 e Employed values for STORM parameters. Parameters Area (acre) Depression storage (in) Evaporation (in/day) Interevent time (hours) First flush depth (in) Time of concentration (hours)

Employed value 1 (¼4046.86 m2) 0.1 (¼2.54 mm) 0.18 (¼4.57 mm) 6 0 0.1

This equation was first suggested by Urbonas et al. (1990) to investigate the condition of more than 60 urban watersheds described by the US EPA (1983) and has been widely used in many US municipalities. As mentioned above, this study focused on the parameter uncertainty in the k-C* model. Thus, uncertainty in imperviousness was not considered. Fig. 7 shows the LFCs of TSS event loads resulting from the 60-year simulation. Each point is a storm event load computed using the procedures described in this paper. The blue points and curves result from simulations of the watershed loads without BMPs. Because there is no treatment, the scatter in the event loads is entirely due to the uncertainty in the TSS concentration in the runoff. Note that the log-transformed median line is located in the middle between the 95% UCL and 95% LCL over the entire range of exceedances, which is to be expected because MR and MTOT are similar to the log-normal distribution. The effectiveness of the BMP in reducing runoff loads can be observed by comparing the blue median line, which is the runoff load and the pink line, which is the sum of the effluent load from the BMP plus the flow bypassed around the BMP once it has filled. The median curves are parallel for exceedance frequencies greater than 3 times per year because the watershed runoff only exceeds the BMP capacity 3 times per year on the average. As the exceedance frequency decreases (larger storms), the median concentration for the watershed with BMP tends to converge on the untreated discharge as expected because the BMP treats less and less of the total runoff. For the 10-year storm, the median loads are different by a factor of approximately 1.7, while for the 100-year storm, they differ by a factor of less than 1.3. Fig. 7 reveals interesting information regarding the design of BMPs. It was assumed that a TMDL had been set for the example watershed used in this paper requiring the TSS load from the watershed not to exceed 0.23 g/m2 more than 4 times per year. Fig. 7 shows that the median untreated watershed runoff load that is exceeded 4 times per year is approximately 0.85 g/m2, so a BMP is required to reduce this median load by 73%. Fig. 7 also shows that if a BMP is designed with the parameter values assigned in this paper, the median discharge from the watershed-BMP system will be equal to the allowable TMDL; however, 50% of the time, the TMDL will be exceeded by as much as a factor of approximately 3.8 (the value of the 95% UCL). Thus, to be 95% certain that we will not exceed the TMDL, it is necessary to design the BMP so that the 95% UCL meets the target, which means that the design point for the median value must be a factor of approximately 3.8 smaller than the TMDL or 0.06 g/m2. Conceptually, by sizing

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the BMP such that its 95% CLs meet the target criteria, it is fairly certain that the load-frequency target value for the watershed TSS load will be met. These results agree with the concept of the reliability-based MOS suggested by Zhang and Yu (2004) and Franceschini and Tsai (2008).

4.

Conclusion

This paper investigated TSS load reduction in stormwater extended detention basins using a mathematical BMP performance model that incorporates uncertainty. The STORM was used to simulate watershed runoff from a typical urban watershed in the LA Basin of California and route it through an extended detention BMP/overflow system. STORM was used because it is simple and capable of continuous simulation. BMP performance was modeled by incorporating uncertainties in the runoff concentration (Cin) and in the rate coefficient (k) in the k-C* model. The LFC method was used to describe receiving TSS loads for a stormwater system to assess watershed performance in terms of meeting a TMDL, expressed as an allowable load frequency in terms of exceedances per year, of the watershed TSS load. The LFC CLs produced the MOS in TMDL estimation. LFCs were integrated with uncertainty analysis to estimate the reduction of storm event loads from the BMP/overflow system with and without treatment. The appropriate BMP size (volume and surface area) for the target water quality standard (load) can then be determined from the k-C* model and BMP performance can be quantified by this method. The results showed that the uncertainty in TSS loads in the untreated runoff is one order of magnitude and independent of the frequency of load. The same magnitude of uncertainty was found for a BMP/overflow system, which means that the uncertainty in the load to the receiving water is principally due to the uncertainty in specifying the pollutant concentration in stormwater runoff. For the k-C* model algorithm tested, the uncertainty in k added only about a 24% increase in uncertainty over the range of storms that are captured entirely by the BMP. The implications for design are that if the median EMC for watershed runoff is used with the k-C* model, the median discharge from the resulting system might meet the TMDL target, but 50% of the time the load will be exceeded with a 2.5% exceedance of the TMDL by a factor of approximately 3.8. If the BMP is sized such that its 95% CLs, derived from the load uncertainty estimation, meet the target criteria, it is fairly certain that the load-frequency target value for the watershed TSS load will be met. These results agree with the concept of the reliability-based MOS suggested by Zhang and Yu (2004) and Franceschini and Tsai (2008). However, for the case illustrated here, it seems unlikely that an extended detention BMP could be designed that could meet the specified TMDL because the median TSS load reduction of the BMP would need to be about 94% which far exceeds the capability of state of the practice BMPs today. These results are based on an application of the k-C* model to a typical urban watershed located in the LA Basin area. It is recommended that the uncertainty approach described and applied here be tested in other hydrologic areas, and with

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different BMP algorithms, which may yield better results. It seems certain that sizing BMPs based on median EMCs of runoff will not provide reliable pollutant removal from stormwater discharges to receiving waters.

 2 scout ¼ expðkmedian =qÞ $s2Cin  

2 1=2 Cin;median  C expðkmedian =qÞ $s2k þ q

Appendix A. Derivation of mlnCout and slnCout

Using Eq. (16), the mean value of Cout (mCout ) can be estimated as follows:

First, it is necessary to log-transform the original EMC data. x represents an element of the original EMC data and y represents its respective log-transformed result as described below, assuming that the data are normally distributed. y ¼ lnðxÞ

(12)

The mean (mx) and standard deviation (sx) of the log-normal EMC distribution are related to the log-transformed mean (my) and standard deviation (sy) by the method of moments as follows (Salas et al., 2004): mx;median

  ¼ exp my

mx ¼ exp my þ

s2y

2 (21)

The log-transformed standard deviation of Cout can be determined by combining Eq. (17) with Eqs. (20) and (21) as follows:   2 1=2 sCout slnCout ¼ ln 1 þ mCout

(22)

! (14)

2

(15)

1

1 B m2x C my ¼ ln@  2 A 2 sx 1þ mx   2  sx s2y ¼ ln 1 þ mx

(16)

(17)

where m ^ x ¼ the mean of the EMC data, sx ¼ the standard deviation of the EMC data, m ^ x;median ¼ the median of the EMC data, my ¼ the mean of the log-transformed EMC data, and sy ¼ the standard deviation of the log-transformed EMC data. Only the bypass-overflow volume is needed because the pollutant concentration is assumed to be equal to Cin. The effluent pollutant concentration in the BMP (Cout) is calculated as the estimated pollutant concentration from the k-C* model as follows:   mlnCout ¼ ln Cout;median    ¼ ln C þ Cin;median  C $expðkmedian =qÞ

(18)

If Cout,median is log-transformed, it becomes the mean of the log-transformed values and is called mlnCout . The standard deviation of k can be calculated from the log-transformed mean and standard deviation from Eq. (15), as follows: sk ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     uexp2m 2 exp 4mlnCout þ 4sCout exp 2mlnCout t lnCout þ

(13)

n   o   s2x ¼ exp s2y  1 exp 2my þ s2y 0

mCout ¼

(20)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ffi exp s2lnk  1 $exp 2mlnk þ s2lnk

(19)

It is assumed that Cin and k are independent. Therefore, the standard deviation of Cout can be evaluated using Eqs. (1) and (3), respectively, as follows:

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