predicting fecal coliform bacteria levels in the …introduction the charles river basin is one of...

ABSTRACT: In Massachusetts, the Charles River Watershed Asso-ciation conducts a regular water quality monitoring and public noti-fication program in the Charles River Basin during the recreationalseason to inform users of the river’s health. This program has reliedon laboratory analyses of river samples for fecal coliform bacterialevels, however, results are not available until at least 24 hoursafter sampling. To avoid the need for laboratory analyses, ordinaryleast squares (OLS) and logistic regression models were developedto predict fecal coliform bacteria concentrations and the probabili-ties of exceeding the Massachusetts secondary contact recreationstandard for bacteria based on meteorological conditions andstreamflow. The OLS models resulted in adjusted R2s ranging from50 to 60 percent. An uncertainty analysis reveals that of the totalvariability of fecal coliform bacteria concentrations, 45 percent isexplained by the OLS regression model, 15 percent is explained byboth measurement and space sampling error, and 40 percent isexplained by time sampling error. Higher accuracy in future bacte-ria forecasting models would likely result from reductions in labo-ratory measurement errors and improved sampling designs.(KEY TERMS: rivers and streams; nonpoint source pollution; sta-tistical analysis; water quality; recreational management; fecal col-iform; bacteria.)

Eleria, Anna and Richard M. Vogel, 2005. Predicting Fecal Coliform BacteriaLevels in the Charles River, Massachusetts, USA. Journal of the AmericanWater Resources Association (JAWRA) 41(5):1195-1209.

INTRODUCTION

The Charles River Basin is one of the most heavilyused recreational areas in the country. Upwards of20,000 people a day visit the river and the parklandalong both banks of the nine-mile (14.5 km) section ofriver (Figure 1). Historically known for its pollutedwaters, water quality in the river has improved

tremendously over the past 15 years as point sourcesof pollution from combined sewer overflows andindustrial plants have been reduced or treated priorto discharging to the river. Despite these efforts, thehealth of the river is impaired after a rainstormbecause stormwater discharges pollutants, such aspathogens from untreated combined sewage, water-fowl feces, wildlife feces, and domestic pet waste, thathave collected on parking lots, streets, driveways, andother impervious surfaces. Pathogens are the pollu-tant of greatest concern to human health.

Because of the enormous popularity of the river forrecreation, there is a need to inform the public of thepotential health risks involved with boating on theriver. In 1998, the Charles River Watershed Associa-tion (CRWA), one of the first watershed organizationsin the country, established the Flagging Program, awater quality monitoring and public notification pro-gram during the high use recreational season. On aroutine basis from June through October, CRWA staffhas collected river samples at four sites in the basin.A heuristic approach based primarily on the previousday’s fecal coliform bacteria levels, antecedent rainfallconditions and combined sewer overflow activation,enabled CRWA to qualitatively determine river waterquality and a color coded flag was hoisted at numer-ous boating centers located on the banks of the basin.A blue flag implies the river is safe for secondary con-tact recreation (i.e., boating, kayaking, canoeing) andmeets Massachusetts (MA) standard for bacteriawhile a red flag signifies elevated bacteria levels andthe associated potential health risks. Unfortunately,reporting of water quality conditions is often untimely

1Paper No. 03111 of the Journal of the American Water Resources Association (JAWRA) (Copyright © 2005). Discussions are open untilApril 1, 2006.

2Respectively, Engineer, Charles River Watershed Association, 48 Woerd Avenue, Suite 103, Waltham, Massachusetts 02453; and Profes-sor, Department of Civil and Environmental Engineering, Tufts University, Medford, Massachusetts 02155 (E-Mail/Vogel: [email protected]).

JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION 1195 JAWRA

JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATIONOCTOBER AMERICAN WATER RESOURCES ASSOCIATION 2005

PREDICTING FECAL COLIFORM BACTERIA LEVELSIN THE CHARLES RIVER, MASSACHUSETTS, USA1

Anna Eleria and Richard M. Vogel2

and inaccurate because the CRWA program cannotmonitor the river seven days a week owing to finan-cial and staffing constraints. In addition, time con-straints are imposed by the fact that laboratoryanalysis of fecal coliform bacteria requires a 24-hourincubation period.

With the advent of regular water quality monitor-ing and public notification programs at water relatedrecreational areas throughout the country, there hasbeen increased interest in developing models to pre-dict water quality conditions without relying on bacte-ria data and instead correlating precipitation or othereasily measured surrogate explanatory variables tobacteria concentrations. The goal was to create pre-diction models for bacteria at various locations in theCharles River Basin and to eliminate the dependenceon water quality sampling. The objectives of the pro-ject were to predict instantaneous bacteria levels frommeteorological and hydrological conditions using mul-tivariate regression, and to estimate the probability ofexceeding the secondary contact recreation standardfor bacteria using multivariate logistic regression.

LITERATURE REVIEW

Previous Statistical Studies of Bacteria

The following section reviews some studies thathave sought to develop multivariate statistical modelsto predict bacteria concentrations in rivers. Table 1summarizes the results of studies by Ferguson et al.(1996), Christensen et al. (2000), Clark and Norris,(2000), Francy et al. (2000, 2002), Crowther et al.(2001), and Rasmussen and Ziegler (2003), all ofwhom developed multiple linear regression models torelate bacteria concentrations to explanatory vari-ables. Also listed in Table 1 are the explanatory vari-ables as well as the overall goodness of fit associatedwith the regressions.

Logistic regression models are useful when one’sinterest is in predicting the probability of the riverwater quality exceeding a threshold. Smith et al.(2001) employed logistic regression to show thatwatersheds with large proportions of urban land coveror agriculture on steep slopes had a very high proba-bility of being impaired by pathogens.

JAWRA 1196 JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION

ELERIA AND VOGEL

Figure 1. Charles River Basin, Massachusetts, USA.

There is a growing literature that has explored therelationship between land use and bacterial concen-trations in coastal estuaries (as opposed to riverswhich is the focus here). For example, Mallin et al.(2000) found that a simple regression using percentimpervious cover explained 95 percent of the variabil-ity in the geometric mean fecal coliform densityacross watersheds. Similarly, Kelsey et al. (2004)found that stormwater runoff from urban land useswere the primary source of fecal pollution. They foundthat proximity to areas with septic tanks and rainfallprior to the sampling date are good predictors of fecalpollution.

Factors That Influence Fecal Coliform BacteriaConcentrations

Bacteria levels in a river are a function of initialloading and the disappearance rate which, in turn, isa function of the time or the distance of travel fromthe source and of other factors including: tempera-ture, salinity, and light intensity. Auer and Niehaus(1993) found that the fecal coliform bacteria deathrate is impacted by both solar radiance and watertemperature.

Myers et al. (1998) found that the bacteria decayrate was a measure of the die-off of bacteria resultingfrom ultraviolet light and temperature stress, cell

starvation, predation by other bacteria and proto-zoans, and removal by filter feeders. They also deter-mined that transport, dilution, dispersion, andconcentrations of fecal coliform are strongly influ-enced by the timing, spatial distribution, and amountof rainfall, runoff, and streamflow and that light pen-etration, which is reduced by turbidity, is the mostimportant factor in determining decay rates. Youngand Thackston (1999) found that fecal bacteria countsin urban tributaries were much higher in seweredbasins than in nonsewered basins and in general were related to housing density, population, develop-ment, percent impervious area, and domestic animaldensity. Mallin et al. (2000) found that fecal coliformdensities were strongly correlated with turbidity (pos-itively) and salinity (negatively).

FECAL COLIFORM BACTERIA:INDICATOR OF HEALTH RISKS

Total coliform bacteria, present in the intestines ofwarm blooded animals, are excreted in the feces ofanimals and humans. Fecal coliform bacteria, a sub-set of the total coliform group, are a more specificindicator of warm blooded animal origin. Since thefederal Clean Water Act of 1972, fecal coliform bacte-ria have been established as an indicator of other


PREDICTING FECAL COLIFORM BACTERIA LEVELS IN THE CHARLES RIVER, MASSACHUSETTS, USA

TABLE 1. Previous Research on Linear Regression Modeling of Coliform Bacteria.

Goodness-of-FitCitation Independent Variable Explanatory Variables Statistic

Christensen et al. (2000) Fecal coliform bacteria Turbidity and time (month) 0.55 to 0.60(adjusted R2)

Clark and Norris (2000) Fecal coliform bacteria Discharge, specific conductance, pH, 0.012 to 0.775water temperature, dissolved oxygen (correlation coefficient)

Crowther et al. (2001) Fecal coliform bacteria Daily rainfall 0.10 to 0.50(R2)

Ferguson et al. (1996) Geometric mean fecal coliform bacteria Rainfall and sewage overflows 0.80(adjusted R2)

Francy et al. (2000) Total coliform bacteria Dissolved organic carbon, ammonia 0.20 to 0.40and organic nitrogen, total phosphorus, (Spearman’snitrate and nitrite, chloride, suspended correlationsediment and specific conductance coefficient)

Francy et al. (2002) E. coli bacteria Wave height, lake-current direction, 0.17 to 0.58turbidity, streamflow of nearby river, (adjusted R2)rainfall, number of birds on the beachat time of sampling

Rasmussen and Ziegler, (2003) Fecal coliform bacteria Turbidity 0.16 to 0.79(R2)

disease causing organisms that may pose a healthrisk to the public.

Massachusetts has established surface water quali-ty criteria for fecal coliform bacteria that sustain thedesignated uses of the waterbody (MADEP, 1997).The Charles River, classified as Class B Warm Water,is designated as a habitat for fish, other aquatic life,and wildlife, and is suitable for primary contact recre-ation, such as swimming and fishing, and secondarycontact recreation, such as boating. The MA primarycontact recreation standard for fecal coliform bacteriais a geometric mean threshold of 200 colony formingunits per 100 milliliters (cfu/100 mL) in any represen-tative sample set. No more than 10 percent of thesamples may exceed 400 cfu/100 mL. The MA sec-ondary contact recreation standard is equal to or lessthan a geometric mean of 1,000 cfu/100 mL in anyrepresentative sample set and 10 percent of the sam-ples shall not exceed 2,000 cfu/100 mL.

STUDY DESCRIPTION

Description of Fecal Coliform Bacteria Data

Charles River Watershed Association staff collectedriver samples at four sites in the Charles River Basinduring the high use recreational season; however, theLarz Anderson Bridge site (Figure 1) is the only siteconsidered here. For further information on othermonitoring locations, see Eleria (2002). At the LarzAnderson Bridge site, 141 samples were collected overthe two-year sampling period from 2000 to 2001 witheach sampling period occurring from mid-Junethrough mid-October. Between the hours of 7:00 a.m.and 8:30 a.m., instantaneous grab samples were col-lected mid-stream between the riverbanks and sixinches below the water surface via decontaminatedand sterilized buckets. Samples were then transferredinto sterile, opaque 125 mL plastic containers. TheCRWA followed strict water quality control and assur-ance measures outlined in the Flagging ProgramQuality Assurance Project Plan (QAPP) approved bythe U.S. Environmental Protection Agency-New Eng-land (CRWA, 1999). Duplicate samples were collectedfor at least 10 percent of the total samples and equip-ment blanks collected for at least 5 percent of thesamples. Immediately after collection, samples wereplaced on ice and cooled to a temperature of at least4˚C. Finally, samples were delivered within the six-hour holding time for bacteria to a State-certified laboratory for bacteria analyses. Samples were ana-lyzed for fecal coliform bacteria using the membrane filtration method (Method No. 9222-D), described in

Standard Methods for the Examination of Water andWastewater (APHA, 1998).

Potential Explanatory Variables

The selection of explanatory variables to predictbacteria concentrations is based on several factors:prior knowledge of the explanatory variables’ rela-tionships with fecal coliform bacteria in the CharlesRiver Basin, previous findings in the literature con-cerning factors that influence microbiological organ-isms, and the ease of obtaining explanatory variableson a daily basis. Table 2 lists the explanatory vari-ables considered. The following subsections outlinethe explanatory variables and their rationale forinclusion.

Meteorologic Variables.

Rainfall – Stormwater runoff is a significant sourceof pollutants to the river, which can include bacteria,viruses, and sediment, to which the substrate pollu-tants attach. Storm rainfall characteristics and condi-tions prior to the storm are significant factors in thetransport and concentration of pollutants in the river.The U. S. Geological Survey (USGS) collected hourlyrainfall from June 2000 through October 2001 atWatertown Dam, located several miles upstream ofthe site. Total volume (inches), duration (hours), andintensity (inches/hour) of rainfall in a storm eventwere considered. In addition, antecedent storm char-acteristics, such as time (hours) since storms greaterthan 0.01 inches (0.25 mm), 0.10 inches (2.5 mm),0.25 inches (6.4 mm), 0.50 inches (13 mm), and 1.0inches (25 mm) of rainfall and amount of rainfall(inches) that fell in the previous 24 hours, 48 hours,72 hours, and 168 hours, were extracted from thehourly precipitation data sets using an unofficialUSGS computer program called METCOMP (A.M.Lumb and J.L. Kittle, Jr., 1995, unpublished report).

Seasonality – Due to the flushing effect mechanismassociated with bacteria transport (see McDonald andKay, 1981; and Kelsey et al., 2004), one expects bacte-ria to vary seasonally. To accommodate the influenceof seasonality, the following term was introduced

Seasonality term = β1 sin(ωt) + β2 cos(ωt)

where ω is 2π/365, t is the Julian day, and β1 and β2are model coefficients to be estimated using multi-variate regression.


ELERIA AND VOGEL

(1)

Net Radiation – Average daily net solar radiation,expressed in langleys, from the National ClimateData Center (NCDC) was considered as an explanato-ry variable because light intensity is known to affectthe die-off rate of fecal coliform bacteria.

Sky Cover – As a measure of light intensity, aver-age daily sky cover was considered. The NationalWeather Service (NWS) at Logan Airport in Bostonrecords average daily sky cover.

Wind Speed – Average daily wind speed (miles perhour), also measured by NWS, reflects a transportmechanism for bacteria at the water surface.

Hydrologic Variables.

Streamflow – River flow is the primary transportmedium of fecal coliform bacteria. Daily streamflow(discharge) measurements at 7:00 a.m. from theUSGS Waltham gauge were employed. Bacteria con-centrations in the river tend to increase during the

hydrograph rise and decrease during the hydrographrecession due to watershed washoff processes. Toaccount for this phenomenon, known as hysteresis, adummy variable set to either 1 or 0 was employed tosignify either the hydrograph rise or recession,respectively. Bacteria concentrations exhibit persis-tence from one day to the next, hence lagged bacteriaand streamflow data were considered as predictorvariables. Because samples were not collected onweekends and holidays, the data set reduced to 78observations with inclusion of this explanatory vari-able. In addition, because it was postulated that thereis a strong relationship between bacteria data andprevious rainfall, interaction terms between laggedbacteria data and rainfall in the previous 24, 48, and72 hours were considered by multiplying the laggedbacteria data with each antecedent rainfall character-istic.

Combined Sewer Overflow Activation – Boston andCambridge are served by combined sewer systems,where both wastewater and stormwater flow in the



TABLE 2. Explanatory Variables.

Explanatory Variable Notation Range Units

Volume of rainfall vol (in) 0 to 1.8 inches

Duration of storm event dur (hr) 0 to 40 hours

Intensity of storm event int (in/hr) 0 to 0.17 inches per hour

Time since storm greater than 0.01 inches >0.01 in 0 to 231 hours





Amount of rainfall in previous 24 hours 24 hr 0 to 3.29 inches


Amount of rainfall in previous 72 hours 72 hr 0 to3.29 inches


Seasonality Sine+Cosine -1 to 0.32 radians

Average daily net radiation net rad 3.18 to 23.11 langleys

Average daily sky cover sky cov 0 to 1.0 percent

Average daily wind speed win spd 5.3 to 17.6 miles per hour

Discharge at time t (day) Q (t) 24.0 to 744.0 cfs

Discharge at time (day) t-1, t-2, t-3, t-4, t-5 Q(t-1, t-2, t-3, t-4, t-5) cfs

Natural log discharge (cfs) at time t LN Q(t) 3.18 to 6.61 NA

Natural log discharge (cfs) at time t-1, t-2, t-3, t-4, t-5 LN Q(t-1, t-2, t-3, t-4, t-5) NS

Hydrograph dummy variable HYDRO 0 or 1 NA

Interaction term – bacteria concentration and rainfall C(t-1)*24 hr 0 to 9.91 NAover the previous 24 hours

Combined sewage overflow dummy variable COM 0 or 1 NA

Notes: cfs = cubic feet per second; NA = Not applicable.

same conveyance pipes to the nearby wastewatertreatment plant. When it rains heavily, the hydrauliccapacities of the combined sewer pipes are exceededand a portion of the untreated combined sewage dis-charges to the Charles River Basin, raising bacterialevels in the river. Combined sewer overflow activa-tion was included using a dummy variable.

EXPLORATORY DATA ANALYSES

In this section the stochastic and probabilisticstructure of the daily streamflow and bacteria dataare examined. The measures of central tendency ofboth bacteria and streamflow vary dramatically as isshown in Table 3. Figures 2(a) and 2(b) illustrate log-normal probability plots for the fecal coliform bacteriaand discharge (Helsel and Hirsch, 1992). In bothcases, a lognomal distribution provides a first approx-imation to the probability distribution of both bacteriaand streamflow and as a result, logarithmic transfor-mations are employed in all future analyses.

The decay rates of bacteria, combined with the nat-ural persistence associated with streamflow, result inbacteria concentrations that exhibit memory or auto-correlation. Because of gaps in the bacteria data set,only one-day lags were considered. Figure 3 illus-trates a plot of bacteria concentrations at time t ver-sus concentrations the previous day (t-1). With a fewexceptions, high concentrations of bacteria on one daytend to be followed by high concentrations the nextday. The lag-1 serial correlation for bacteria concen-trations is 0.51.


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TABLE 3. Statistics of Fecal Coliform Bacteria at Larz AndersonBridge and Discharge at USGS Waltham Gauge.

Fecal ColiformBacteria Discharge

(cfu/100 mL) (cfs)

Sample Size 141 254

Mean 910 147

Median 100 100

Geometric Mean 110 101

Harmonic Mean 38 77

Standard Deviation 6,640 146

Minimum 2.5 24

First Quartile 38.8 55

Third Quartile 265 171

Maximum 79,000 747

Figure 2. Lognormal Probability Plot of (a) Fecal ColiformBacteria and (b) Discharge at USGS Waltham Gauge.

Figure 3. Persistence of Fecal Coliform Bacteria.

Figure 4 compares the autocorrelation function ofthe observed daily discharges with a Markov process.The autocorrelation coefficient measures the strengthof association between the daily streamflow at time tand t-1. The lag-1 and lag-2 correlation coefficients ofobserved discharge are r1 = 0.96 and r2 = 0.90. For aMarkov or first-order autoregressive process, oneexpects r2 = r1

2 = 0.962 = 0.922, which is close to theobserved lag-2 discharge value of r2 = 0.90. Thisdemonstrates that the discharges are roughly Markov,hence one only needs to consider yesterday’s stream-flow to approximate the complete memory of dailystreamflow.

REGRESSION METHODS

Multivariate Linear Regression

Multivariate linear regression models of the follow-ing form were fit to the daily bacteria concentrations

yj = β0 + β11x11 + β12x12 + …. + βi,jxi,j + εi

where yj is the natural logarithm of daily fecal col-iform bacteria concentration (cfu/100 ml) on day j; βi,jis the slope coefficient of explanatory variable xi; xi,j isthe ith explanatory variable on day j; and εi is themodel error or residual on day j.

The OLS method in Minitab® (Minitab, Inc., 2000),was used to estimate the model coefficients and sta-tistical tests were performed to ensure that the modelresiduals were approximately normally distributedand that each explanatory variable increased thegoodness-of-fit in a statistically significant fashion.In addition, variance inflation factors (VIF) were esti-mated to ensure against excess multicollinearityamong the explanatory variables. A total of six modeltypes were considered and ranged from simple to complex. The first model considered only antecedent

rainfall characteristics. For Model 2, the hydrologicalvariables were added to Model 1, while for Model 3,the other meteorological explanatory variables,including seasonality, net radiation, cloud cover, andwind speed, and the combined sewer overflow (CSO)activation variable were added to Model 2. Models 4through 6, similar to Models 1 through 3, respectively,also included lagged bacteria concentrations. As aresult, all explanatory variables were considered inModel 6. Unusual combinations of explanatory vari-ables can strongly influence the regression model andreduce its predictive power, hence they were identifiedusing standard influence statistics, such as Cooks’ D(see Helsel and Hirsch, 1992), and subsequentlyremoved. Stepwise multivariate linear regression wasapplied to select combinations of the independentvariables. The resulting models were evaluated usingprediction type goodness-of-fit metrics, such as predic-tion R2, and the predicted residual sums of squares(PRESS) statistic as well as various graphical diag-nostic evaluations of the behavior of the model residu-als outlined by Helsel and Hirsch, (1992).

Multivariate Logistic Regression

Multivariate logistic regression is useful for deter-mining the relationship between categorical or dis-crete model responses, in this case, the probability ofbacteria levels exceeding the State secondary contactrecreation (boating) standard, and a variety of predic-tor variables (see Chapter 12 in Helsel, 2005). Thevalues of the bacteria response above or below thethreshold are designated using binary variables.Logistic regression transforms estimated probabilitiesinto a continuous response variable. The transformedresponse is predicted from one or more explanatoryvariables, and subsequently retransformed back to avalue between 1 and 0. For this project, a value of 1signifies that there is a greater than 50 percent prob-ability of the river exceeding the secondary contactrecreation standard for bacteria, while a 0 impliesthere is a less than 50 percent chance of exceeding thesecondary contact recreation standard.

The odds ratio is defined as the ratio of the proba-bility of obtaining a 1 divided by the probability ofobtaining a 0.

where p is the probability of a response of 1.The natural log of odds ratio (termed the logit)

transforms a variable constrained between 0 and 1 into a continuous and unbounded variable. To



Figure 4. Correlogram of Daily Dischargeat USGS Waltham Gauge.

=−p

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odds ratio

estimate logistic regression, the logit is modeled as alinear function of one or more explanatory variablesso that

where β0 is the intercept and β is the slope coeffi-cients for each explanatory variable.

Exponentiation of Equation (4) leads to

which can be rewritten as

where p is the probability of the river exceeding thesecondary contact recreation standard for bacteria.Maximum likelihood estimates of the model parame-ters in Equation (6) were obtained using Minitab®.Akaike’s Information Criteria was employed to evalu-ate the goodness-of-fit of alternative logistic models.

Unlike multivariate linear regression methods, astepwise approach for selecting explanatory variablesbased on goodness-of-fit is currently not available forlogistic regression. Instead, the stepwise approach ofmultivariate linear regression models served as ascreening tool for suitable explanatory variables forthe logistic regression models.

MODELING RESULTS

Multivariate Linear Regression Results

Table 4 summarizes the results of the multivariatelinear regression models and lists the explanatoryvariables, their corresponding coefficient values, andgoodness-of-fit statistics. In terms of overall goodness-of-fit, the ‘best’ model at Larz Anderson Bridge wasModel 6, which had an adjusted R2 of 60.4 percentand four significant predictor variables: lag-1 bacteriaconcentration, the interaction term between the lag-1bacteria concentration and the amount of rainfall inthe previous 24 hours, time since rainfall greater than0.10 inches (2.5 mm), and average daily wind speed.The standard error and PRESS statistic for Model 6were the lowest among all the models evaluated. Figure 5 illustrates the modest linear relationship

between observations and predictions of bacteria con-centrations at Larz Anderson Bridge.

The best model that did not include lag-1 bacteriaconcentrations was Model 3, the model category thatconsidered all meteorologic and hydrologic variables.This model had the second highest adjusted R2, 56percent; yet, it also required eight explanatory vari-ables. The significant variables included average rain-fall intensity, amount of rainfall in the previous 168hours, time since rainfall greater than 0.10 inches(2.5 mm), natural log of discharge, natural log of lag-4discharge, hydrograph, average wind speed, and CSOactivation in the previous 24 hours.

Split Sample Validation Experiment. A splitsample validation experiment was performed to testthe predictive power of Model 4, which included lag-1bacteria data as an explanatory variable. Althoughthis model did not have the highest adjusted R2, thismodel was selected for validation because of the easeof applying this model by CRWA. The procedures forthe split sample validation experiment were: (1) themultivariate regression model was fit to the first halfof the data; (2) the fitted multivariate regressionmodel from Step 1 was used with the second half ofthe data to compute predicted values; and (3) the performances of the model over the calibration andvalidation portions of the dataset were compared inTable 5 and Figure 6.

The multivariate regression equation for Model 4based on 78 observations was

ln(Ct) = 2.16 + 0.39Ct-1 + 2.89*24 hr + 0.61*168 hr

where ln(Ct) is the natural log of fecal coliform bacte-ria concentration; Ct-1 is the previous day’s bacteriaconcentration (cfu/100 mL); 24 hr is the amount ofrainfall in the previous 24 hours (in); and 168 hr isthe amount of rainfall in the previous 168 hours (in).

Water quality at Larz Anderson Bridge during thisstudy period was fairly good; only eight out of 78observations exceeded the secondary contact recre-ation standard for bacteria. Table 5 summarizes thenumber of correct and incorrect predictions of meetingor exceeding the standard at Larz Anderson Bridgeover the validation period. The number in the paren-theses equals the percent of the time the model cor-rectly or incorrectly predicted when the river was safeor unsafe for boating. The predictive power of themodel was very good when the river was safe for boat-ing, leading to correct predictions 97 percent of thetime. On the other hand, when observed bacteria con-centrations were greater than 1,000 cfu/100 mL, themodel predicted those violations with less accuracy(64 percent of the time). Figure 6 compares the time


ELERIA AND VOGEL

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series of observed bacteria concentrations (diamonds)with the time series of bacteria concentrations pre-dicted by Model 4 (the dotted black line) for 78 obser-vations. In general, the model performed well inpredicting concentrations between 100 cfu/100 mLand 1,000 cfu/100 mL but tended to overestimate thelow concentrations and underestimate the high con-centrations.

Additional Validation Experiments. Models 4,5, and 6 required observations of lagged bacteria con-centrations, yet such information is not always avail-able in practice. These models are not nearly asaccurate as they appear because if the program isunable to collect bacteria samples, modeled estimatesof lagged concentrations are needed for their applica-tion and such modeled estimates contain significantadditional model error. Therefore, an additional

experiment was conducted to verify the accuracy ofthese models when lagged bacteria concentrationsmust also be estimated from a regression model.First, a single bacteria observation was used in Model4 to obtain a regression estimate of the next day’s bac-teria concentration. From that day on, the values oflagged bacteria became the regression estimatesobtained from Model 4. The results of this experimentin Table 6 showed that the use of estimates of bacte-ria concentrations in Model 4 led to lower predictionaccuracy. The model was 98 percent accurate at pre-dicting when the river met the secondary contactrecreation standard, while the model was only accu-rate 44 percent of the time, for predicting violations tothe standard (Table 6), which is worse than a simpleguess.

In addition, recognizing that the original hope wasto eliminate dependence on indicator bacteriological


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Figure 5. Observed Versus Predicted Fecal ColiformBacteria Concentrations at Larz Anderson Bridge.

TABLE 5. Split Sample Experiment:Model 4 of Larz Anderson Bridge.

ObservationsExceeded

Met Boating BoatingStandard Standard

Met Boating 68 (97%) 3 (37%)Standard

PredictionsExceeded Boating 2 (3%) 5 (63%)Standard

Figure 6. Larz Anderson Bridge Split Sample Model Experiment: ObservedBacteria Concentrations Versus Predicted Bacteria Concentrations.

monitoring and that practitioners may need a modelwithout a lagged bacteria explanatory variable, theaccuracy of Model 3 was tested. This model had thesecond highest adjusted R2 and did not rely on thepersistence structure of bacteria to estimate bacterialevels. Model 3’s predictions were similar to the previ-ous experiment. Ninety-eight percent of the time, themodel predicted when the river met the secondarycontact recreation standard; however, its accuracydecreased to 44 percent for predicting when the riverdid not meet the standard.

Multivariate Logistic Modeling Results

Similar to the ‘best’ linear regression model, the‘best’ logistic model for Larz Anderson Bridge alsoincluded the lag-1 bacteria concentration variable.The probability of the river exceeding the secondarycontact recreation standard at Larz Anderson Bridgewas estimated from the following equation

where P is the probability of the river exceeding thesecondary contact recreation standard at Larz Ander-son Bridge; θ = -4.451 + 0.828* ln(Ct-1) * 24 hr +(1.592 * 168 hr); ln(Ct-1) * 24 hr is the interactionterm between the natural log of lag-1 bacteria concen-tration and rainfall in the previous 24 hours; and 168hr is the rainfall (inches) in the previous 168 hours.

The accuracy of the logistic model in predictingwhen the river meets or exceeds the secondary con-tact recreation standard is presented in Table 7. Themodel had very accurate predictions (97 percent)when the river met the secondary contact recreationstandard, but the model was less successful (64 per-cent) when violations to the standard occurred.

Discussion and Comparison of Linear Regression andLogistic Regression

Both regression approaches had excellent successin predicting when the river met the secondary con-tact recreation standard at Larz Anderson Bridge, yetonly fair to poor success in predicting violations of thestandard. The ‘best’ models of both the linear andlogistic regression approaches predicted correctly atleast 95 percent of the time when the river met thesecondary contact recreation standard. The linearregression models correctly predicted violationsbetween 44 and 63 percent of the time and logisticregression models correctly predicted violations 64percent of the time. The explanatory powers of bothmodels are much higher during dry weather events(e.g., rainfall less than 0.10 inches (2.5 mm) in theprevious 72 hours) than wet weather events.

Lower accuracies in the models’ predictive capabili-ties of the higher bacteria concentrations (>1,000cfu/100 mL) were anticipated because of fewer obser-vations of elevated bacteria concentrations than thelower bacteria concentrations. In addition, theobserved bacteria concentrations may not be repre-sentative of bacteria concentrations in the river at thetime of sampling because of the significant uncertain-ties discussed in the next section.

IMPACT OF UNCERTAINTY INMODEL, MEASUREMENTS, AND

REPRESENTATIVENESS OF DATA

To better understand the challenges of predictinginstantaneous bacteria concentrations, this sectionexplores the uncertainty associated with bacteriameasurements and the ability of those measurementsto reflect the true bacteria concentrations in the riverat the time of sampling. Ideally, samples would be col-lected at several locations across a river cross sectionand in six-hour intervals over a 24-hour period



P =+exp( )

exp( )θ

θ1(8)

TABLE 6. Model 4 Experiment With ModeledEstimates of Lagged Bacteria Concentrations.





TABLE 7. Number of Observations Versus Number of Predictions:‘Best’ Logistic Regression Model at Larz Anderson Bridge.





(absent of rain) to capture the spatial and temporalvariability associated with bacteria measurements.Unfortunately, the available bacteria data only cap-tured an instantaneous snapshot of bacteria concen-trations at a specific location in the river, hence thedata may not be representative of true water qualityconditions of the entire river cross section during theday of sampling. In addition, bacteria measurementsare known to contain significant laboratory measure-ment error. The overall bacteria modeling problemcan be written as

yi = f(xi) + εi

where yi is the natural logarithm of the ith bacteriaconcentration observation; f(xi) is the multivariate lin-ear regression model where xi denotes logarithm ofthe ith independent variable; and εi is the ith errorrealization in log space.

The goal of the regression model is to describe mostof the variability in yi, with the errors describing onlya small portion of that variability. However, in thisapplication, the error terms play a central role. Theoverall error may be disaggregated into the followingsources

εi = model error + measurement error+ spatial and temporal representation error

Model error characterizes one’s inability to selectthe correct form of the multivariate regression modeland the correct explanatory variables to include inthat model. Measurement error represents the errorassociated with laboratory measurements of bacterialevels. Spatial and temporal representation errorreflects the inability of a single instantaneous bacte-ria measurement to represent the behavior of the bac-teria concentrations over the full spatial (river crosssection) and temporal range (daily) considered. In thefollowing section, an attempt is made to quantifythese three sources of error.

Error Sources and Their Variability

Since the regression model is fit in log space, it isinstructive to compare the variability of the variousterms in Equation (10) by simply comparing theirvariances. This is analogous, but not equivalent, tocomparing the coefficient of variations of the variousterms, because for a lognormal variable

where σy2 = 2.74 is the variance of the natural loga-

rithms of the bacteria concentrations and Cv = 3.8 isthe coefficient of variation of bacteria in log space.

Assuming that all sources of error are independent,Equation (9) yields

where Var(y) is the variance of logs of bacteria;Var(f(x)) is the variance of logs of model error; andVar(ε) is the variance of log of three additional sourcesof variability resulted from laboratory measurement,spatial sampling, and temporal sampling errors. Theoverall variability, Var(y) = 2.74, results from at leastthree sources of error in addition to the variabilityexplained by the model: laboratory measurementerror, spatial sampling error, and temporal samplingerror. Note that in most situations (when developingprediction models for other parameters), the primarysource of error is model error, and even that error canbe quite small. When modeling instantaneous bacte-ria concentrations, these additional error sourcesserved to further confound the ability to reproduceobserved bacteria observations.

The variance of the logarithm of the model errorterm is equal to the variance of the log space residu-als for the regression model, which was equal toVar(f(x)) = 1.23. Clearly, Model 4 was able to explain agood portion of the original variability of the bacteriaconcentrations or 100(1.23/2.74) equals 44.9 percentof that variability. The question addressed here iswhether the remaining 55.1 percent variability can beexplained by the additional sources of error describedabove in Equation (10).

Assuming independence among the individualerrors in Equation (10), the standard deviation of thetotal errors, εi, is

where σε is the standard error associated with bacte-ria observations, σm is the standard error of laborato-ry duplicate errors, σs is the standard error ofbacteria data collected over space; and σt is the stan-dard error of bacteria data over time.

Laboratory Measurement Error. Because of theknown variability associated with laboratory mea-surement of water quality constituents in river sam-ples, laboratories routinely conduct duplicatemeasurements to quantify this variability and ensurethat an estimated precision criterion is met. The CRWA laboratory derives yearly the precision


ELERIA AND VOGEL

(9)

(10)

Cv y= ( ) −exp σ2 1 (11)

Var y Var f x Vary( ) ( ) ( )= = ( ) +σ ε2 (12)

σ σ σ σε = + +m s t2 2 2 (13)

criterion, the index for comparison of duplicate bacte-riological data, according to the Standard Methods forthe Examination of Water and Wastewater (APHA,1998). The CRWA laboratory conducted duplicateanalyses of 130 observations from March 2000 toDecember 2001. The duplicate error, or laboratorymeasurement error, is the difference between the firstsample and the duplicate measurement. The mean,mm, and standard deviation, sm, of the laboratorymeasurement error, in real space, equaled 11.9cfu/100 mL and 380, respectively. The standard errorof the mean was 33.3, hence the estimated mean mea-surement error is not significantly different from zeroas expected.

Spatial Variability of Fecal Coliform Bacteria.The spatial variability of bacteria concentrations inthe vertical and horizontal cross sections of the riverwas not captured in this monitoring project. Sampleswere collected at the same location, at the same timeof day, representing only an instantaneous picture ofthe river’s health. As part of another study, the USGSquantified the spatial variability of bacteria in rivercross sections during two separate storm events inJuly 2002, which was the only spatial data availablefor this area and time period. The USGS collectedthree bacteria samples each at several points: themiddle, near the right bank and near the left bank,and across the horizontal cross section. One USGSsite corresponded to the Larz Anderson Bridge moni-toring location. The spatial error was estimated bycalculating the difference between the middle of theriver sample and the right or left bank sample. Themean and standard deviation of the spatial errorswere equal to µs = 29.2 cfu/100 mL and σs = 64.3,respectively. In addition, the standard error of themean spatial error was equal to 18.6, hence the meanspatial error is not significantly different from zero.

Temporal Variability of Fecal Coliform Bacte-ria. Temporal conditions may vary substantiallybetween daily bacteria samples due to the naturaldie-off of bacteria, additional inputs of bacteria to theriver, and/or the transport of pollutants within a 24-hour period. High frequency data over time has onlybeen collected in the basin during wet weather eventsto determine the response of the river to variousstorm volumes, durations, and intensities. Therefore,quantification of the temporal variability of bacteriameasurements is not possible in the following analy-sis.

Summary Comparison of Sources of Uncertainty

Recall that the mean bacteria concentration is

cient of variation of the measurement and space sam-pling errors in real space are Cv(ε) = 385/910 = 0.423,which corresponds to a log space variance of Var(ε) =0.406. Recall from Equation (10) that Var(y) =Var(f(x)) + Var(ε), which becomes 2.74 = 1.23 + 0.406 +x, where Var(y) = 2.74, Var(f(x)) = 1.23 and Var(ε) =0.406, and x = 1.1 is the remaining variance explainedby time sampling error, which was ignored in theabove analysis. Thus, of the overall variability of theobserved bacteria concentrations, 45 percent isexplained by the model, 15 percent is explained byboth measurement and space sampling error, and,apparently, 40 percent is explained by time samplingerror.

CONCLUSIONS AND FUTURE WORK

This project was an effort to predict bacteria con-centrations in the Charles River Basin using easilymeasured and readily available explanatory vari-ables. Multivariate regression models were developedbetween fecal coliform concentrations and a variety ofhydrologic, environmental, and meteorologic vari-ables. The linear regression models employing meteo-rologic and hydrologic explanatory variables (Models1 through 3) could only moderately explain theobserved variance in bacteria (adjusted R2 valuesranged from 46 percent to 56 percent). Models thatincluded the observed persistence structure of bacte-ria (Models 4 through 6) led to slight improvementswith the highest adjusted R2 equal to 60 percent forModel 6.

Although Model 6 had the highest adjusted R2, itwas not preferred for application in the CRWA Flag-ging Program because it requires a large number ofexplanatory variables that originate from differentdata sources. Model 4 was the preferred linear regres-sion model because it requires explanatory variablesfrom only two different data sources. During a splitsample validation experiment, the predictive capabili-ty of Model 4 was excellent for concentrations belowthe secondary contact recreation standard of 1,000cfu/100 mL but only fair for concentrations greaterthan 1,000 cfu/100 mL. The regression model accu-rately predicted when the river met the bacteria



σ σ σε = +m s2 2910 cfu/100 mL. From Equation (13),

is obtained. Hence the coeffi-= + =2 2380 64 3 385. ,

secondary contact recreation standard over 90 percentof the time. However, the percentage decreased to 63percent when predicting violations to the secondarycontact recreation standard.

Logistic regression models were also developed topredict the probability of the river being safe or notsafe for secondary contact recreation, which is ofgreater concern to recreational users than the actualbacteria levels of the river. The best logistic regres-sion model showed no improvements to predictions.It accurately predicted when the river met the sec-ondary contact recreation standard over 90 percent ofthe time and when the river exceeded the standardabout 60 percent of the time.

Finally, the impact of additional sources of variabil-ity on the accuracy of model results was explored. Ofthe total variability of fecal coliform bacteria concen-trations, 45 percent is explained by the ordinary leastsquares regression model, 15 percent is explained byboth measurement and space sampling error, andapparently 40 percent is explained by time samplingerror. Clearly, future improvements to such modelsare likely to come from reductions in both the timeand space sampling errors, which are under the mod-eler’s control.

The relationships developed here, although onlymodestly successful, are an improvement over modelsdeveloped previously. For CRWA’s Flagging Program,either model with or without lagged bacteria data asan explanatory variable is used on a daily basis, pro-viding a useful quantitative tool for predicting thesuitability of the river for secondary contact recre-ation. The application of this statistical approach withone or more of the same explanatory variables usedhere has already been tested in other freshwaterrecreational rivers in the country.

ACKNOWLEDGMENTS

This project was also conducted in cooperation with the U.S.Geological Survey and was partially funded by the U.S. Environ-mental Protection Agency.

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