Time-series modeling of reservoir effects on river nitrate concentrations

Advances in Water Resources 32 (2009) 1197–1205

Contents lists available at ScienceDirect

Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres

Time-series modeling of reservoir effects on river nitrate concentrations Andrea L. Schoch a, Keith E. Schilling b,*, Kung-Sik Chan a a b

Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA, USA Iowa Geological and Water Survey, 109 Trowbridge Hall, Iowa City, IA 52242-1319, USA

a r t i c l e

i n f o

Article history: Received 30 December 2008 Received in revised form 2 April 2009 Accepted 11 April 2009 Available online 21 April 2009 Keywords: Reservoir Nitrate Time series Transfer function ARMA

a b s t r a c t Saylorville Reservoir is a 24.1 km2 impoundment of the Des Moines River located approximately 10 km north of the City of Des Moines, Iowa, USA. Surface water from the Des Moines River used for drinking water supply is impaired for nitrate–nitrogen. Monthly mean nitrate concentration data collected upstream and downstream of the reservoir for a 30-year period (1977–2006) were selected for time-series analysis. Our objectives were to (1) develop a model describing nitrate concentrations downstream of the reservoir as a function of the concentrations entering the reservoir and (2) use the model to provide a 1-month ahead forecast for downstream water quality. Results indicated that downstream nitrate can be effectively modeled using a transfer function approach that utilized inflow concentrations during the current and previous month as input variables. Inflow concentrations were modeled using an AR(20) model, with the higher order model consistent with temporal correlation noted by others. The transfer function model suggested that the reservoir is reducing nitrate concentrations by 22 ± 6%, a reduction that greatly exceeds previous estimates. Monthly nitrate forecasted with the model were nearly all within a 95% prediction interval of their actual measured values and did not appear greatly affected by flow variations. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Nonpoint source pollution from nitrate–nitrogen (nitrate) is a major cause of surface water impairment in the United States [19]. Impairments extend from excessive nutrient enrichment and eutrophication [6] to Gulf of Mexico hypoxia [14]. Public water supplies that utilize surface water intakes are threatened when nitrate concentrations exceed the US Environmental Protection Agency (USEPA) maximum contaminant level (MCL) of 10 mg/l. The Des Moines Water Works uses surface water from the Des Moines and Raccoon rivers as part of its drinking water supply for more than 400,000 people in central Iowa. Recent assessments have indicated that both rivers are impaired for their use as drinking water sources due to levels of nitrate that exceed the MCL [8]. Previous research has tended to focus more on the Raccoon River [16,17] than the Des Moines River because the Raccoon River is a free-flowing river. A large flood control reservoir (Saylorville Reservoir) is located on the Des Moines River upstream of the City of Des Moines (Fig. 1). Reservoirs have been shown to have significant effects on nitrate mass balances in river systems [4,7]. For a 4400-ha reservoir in Illinois, David et al. [4] showed that about 58% of the total nitrate input to the reservoir was removed (average of 4900 Mg N year1 removed). Annual flow-weighted nitrate con* Corresponding author. Tel.: +1 319 335 1575; fax: +1 319 335 2754. E-mail address: [email protected] (K.E. Schilling). 0309-1708/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2009.04.002

centrations decreased from about 10 mg/l in inlet water to less than 4–6 mg/l in outlet water. Garnier et al. [7] indicated that 40% of the nitrate entering three large reservoirs in France was removed. Denitrification was shown by both studies to be the dominant nitrate removal process, related, in part to the water residence time in the reservoir. In contrast, a modeling study by Seitzinger et al. [18] suggested that effects of reservoirs on nitrate removal may be minimal (less than 2%) if the ratio of reservoir water depth to water residence time is poor. Previous work conducted on Saylorville Reservoir suggested that the reservoir falls into the latter category. In a study conducted for 6-year period soon after the reservoir was completed (1977–1983), Okereke et al. [12] observed that there was only about a 5% difference in average nitrate concentrations between upstream and downstream locations, and differences in nitrate loading rates were considered negligible. However, since the early study of Okereke et al. [12], no further detailed study of the effects of Saylorville Reservoir on Des Moines River nitrate concentrations has been conducted, although water quality monitoring has been ongoing for more than 30 years [10]. Nitrate concentration patterns in surface water are amenable to empirical modeling using detailed time-series analysis [9,20,21]. Worrall and Burt [20] analyzed nitrate concentration patterns in three catchments in England using autoregressive (AR) and autoregressive moving average (ARMA) models and found that the time series showed a ‘‘memory effect”. Nitrate concentrations measured at particular time were positively correlated to concentrations

1198

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

Fig. 1. Location of Saylorville Reservoir and sampling sites.

measured 1-year later. One of the catchments was also found to have a 6-month memory effect. AR modeling of nitrate time series was used to evaluate long-term nitrate concentration patterns in paired forested catchments [21] and karst springs [9] and was used to decompose river water into urban and agricultural signals [23]. Time-series modeling of reservoir effects on river nitrate concentrations has not been documented. The goal of this research was to evaluate the effects of Saylorville Reservoir on nitrate concentrations in the Des Moines River using an empirical time-series model. Specifically, our objectives were to (1) develop a time-series model that described nitrate concentrations downstream of the reservoir as a function of the nitrate concentrations entering the reservoir; (2) assess model properties to identify temporal patterns of inflow and outflow nitrate concentrations; (3) validate model performance by forecasting nitrate concentrations for a time period extending beyond the model development range. Using a 30-year record of monthly nitrate measurements (1977–2006), we hypothesized that Saylorville Reservoir significantly alters the temporal dynamics of nitrate concentration in the Des Moines River, such that monthly water quality downstream of the reservoir can be modeled as a function of the current monthly upstream water quality and its past lags. Improved understanding of reservoir effects may then be used to better manage Saylorville Reservoir to mitigate nitrate concentrations in downstream receiving waters and reduce the impairment to the drinking water source for the City of Des Moines. 2. Study area The Des Moines River above Des Moines drains a watershed of 16,175 km2 flowing from the headwaters in Minnesota through north-central Iowa (Fig. 1). The watershed is located within the Des Moines Lobe landform region of Iowa, a region of recent glaciation (<12,000 years old) dominated by low relief and poor surface drainage [13]. Land use is predominantly agricultural, consisting of

78.5% row crops of corn and soybeans, 14.3% grass, 2.7% forest, 2.5% urban and 1.9% water and wetlands. Average annual precipitation in north-central Iowa is approximately 750–800 mm but this can vary substantially year-by-year. Greatest monthly rainfall typically occurs in May and June, although large convective storms can occur throughout the summer and deliver large amounts of rainfall over a short time. Temperatures vary widely across seasons, with maximum temperatures between June and September averaging over 20 °C, whereas maximum temperatures in December and January average less than 0 °C. Saylorville Reservoir is a 24.1 km2 impoundment of the Des Moines River located approximately 10 km north of the City of Des Moines (Fig. 1). The reservoir was completed in 1967 for the purpose of providing additional flood control capacity of Lake Red Rock downstream and reducing flood crests on the Des Moines and Mississippi Rivers. Streamflow into Saylorville Reservoir is derived primarily from the Des Moines River since the watershed narrows in the lower region between Boone and Des Moines and there are no major tributary inputs in this area above second order streams. Discharge into Saylorville Reservoir is measured at a US Geological Survey gage near Stratford, Iowa located approximately 25 km upstream of Site 1 (Fig. 1). From 1977 to 2006 (study period), Des Moines River discharge at Stratford averaged 206 ± 131 mm. The baseflow component of discharge, that is, the portion of streamflow derived from groundwater discharge, averaged 150 ± 97 mm. Hence, baseflow contributed approximately 73% of the discharge to the reservoir. Maximum monthly discharge in the Des Moines River has exceeded 70 mm during all months between March and July, with a peak monthly discharge of 146 mm occurring in July 1993. 3. Methods Nitrate concentrations are monitored upstream and downstream of the reservoir by the Army Corps of Engineers through a

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

18 16 14 12 10 8 6 4 2 0

and downstream locations so the time series data were considered stationary. However, to achieve normality, maximum likelihood estimation within the family of power transformations [1] indicated that a square root transformation of the monthly nitrate values was needed. Since it was assumed a priori that downstream nitrate concentration could be modeled as a function of the current upstream concentration and its past lags, a transfer function model [2,3] was utilized in this analysis. Let Yt and Xt represent the square root of the nitrate concentration downstream and the square root of the nitrate concentration upstream of the reservoir in the tth month. The transfer function model specifies that

Y t ¼ f ðX t ; X t1 ; . . .Þ þ et ;

e t ¼ ð1  p1 B  p2 B2      pk Bk ÞX t X ¼ pðBÞX t

e t is white noise: where X

ð2Þ

The order k of the preceding autoregressive filter was determined by AIC, with the coefficients estimated by the method of OLS. (B denotes the backshift operator in Eq. (2) such that BkXt = Xtk for k

Site 1

16 14

ð1Þ

where f is a linear function that may involve the current and all past X’s, and {et} is a zero-mean temporally correlated error process that may be modeled as an autoregressive moving average (ARMA) process. (The error process encapsulates the effects of other missing covariates which are generally correlated over time.) This analysis considered the simple case that f is a linear function dependent on Xtj, for j = 0, 1, 2, . . . , q where q is a fixed integer known as the transfer function order, and that {et} is an ARMA process, independent of the X’s. The order q was estimated from the data and indeed so was the lag structure, that is, which lags were significant. This approach facilitates a better understanding of the dynamic dependence of the downstream nitrate concentration on its upstream counterparts. In order to find the appropriate order of q, prewhitening was used to untangle the autocorrelation in the upstream and downstream nitrate concentrations from their linear regression relationship. Prewhitening is the process of transforming the upstream values to white noise using an autoregressive filter [3]:

Site 5

12 10 8 6 4 2 0 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Monthly mean nitrate-N concentration (mg/l)

cooperative agreement with Iowa State University (Des Moines River Water Quality Network or DMRWQN). Upstream samples are collected from the Des Moines River near Boone (DMRWQN Site 1) and the downstream samples are collected from the Des Moines River near Saylorville (DMRWQN Site 5). Water samples have been collected corresponding to weekly or bimonthly frequency from Sites 1 and 5 with more frequent samples collected during spring and summer months when streamflow and nitrate concentrations are typically more variable. Since the number of sampling events per month was variable, all nitrate concentrations measured in an individual month were combined by Lutz and Francois [10] into a monthly mean value to produce the same number of values per month. Sampling procedures and laboratory analytical methods were unchanged during the study period [10]. Monthly mean nitrate concentration data from Sites 1 and 5 for a 30-year period (1977–2006) were selected for time-series analysis (Fig. 2). Since the nearest USGS stream gage is located more than 50 km upstream of the reservoir, incoming flows to the reservoir cannot be estimated precisely. This presents difficulties when attempting to evaluate the effects of the reservoir on nutrient loads, considering that accurate discharge measurements are critical in load estimation schemes. Hence, this study has focused exclusively on nitrate concentration effects rather than loads. Inlet concentrations are measured much closer to the reservoir than discharge and they are not substantially altered in the river from their point of measurement to the limits of the reservoir downstream (through either in-stream processing or minor tributary inputs). Moreover, nitrate concentrations are subject to regulation (MCL) for drinking water supplies that use surface water. Monthly nitrate concentration data from 1977 to 2006 were analyzed with the GNU software R that is available free from the internet (http://www.r-project.org/). The R package TSA (time-series analysis, documented in [3]) was used for the analysis. Time series of monthly nitrate concentrations upstream and downstream of the reservoir from 1977 to 2006 were utilized in this study (n = 360). In this analysis, no increasing or decreasing trends were evident in nitrate concentrations over time at both upstream

1199

Calendar Year Fig. 2. Nitrate–N concentrations measured at Site 1 (upstream of Saylorville Reservoir) and Site 5 (downstream of Saylorville Reservoir).

1200

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

greater than 0.) The same filter was then applied to transform Yt to e t , and the transfer function order was determined by a cross-corY e t , where a sample cross-correlae t and Y relation analysis between X pffiffiffi tion is significant at 5% level if its magnitude is more than 1:96= n. A regression model of Yt on the suggested lags of Xt, based on the correlation analysis was then fitted, with the regression residuals used to assess the ARMA structure of the error process, {et}. The transfer function model was then specified accordingly and fit to the downstream nitrate concentration by maximum likelihood method. Model diagnostics were used to verify if the suggested model was a good fit to the downstream nitrate concentration time series. In practice, the identification of the final model is based on an iterative procedure of model fitting and examining model diagnostics popularized by Box et al. [2], as was the case for the current study. Additionally, out-of-sample month to month forecasts were made for the years of 2007 and 2008 to evaluate the empirical forecasting performance of the model. See Box et al. [2] and Cryer and Chan [3] for further discussions of transfer function modeling. 4. Results

linear relationships in the original data, the results indicate that the downstream nitrate concentration (Yt) is correlated with both the current upstream nitrate concentration (Xt) and the previous month’s upstream nitrate concentration (Xt1). Hence, the crosscorrelation results confirmed that a transfer function model was appropriate to model the nitrate concentrations patterns downstream of the reservoir. Regression was used to model downstream nitrate concentration as a function of the upstream nitrate concentration and the first lag of the upstream nitrate concentration. Residuals from the regression model were obtained and analyzed under the autocorrelation function (ACF), the partial autocorrelation function (PACF), and the extended autocorrelation function (EACF); see Cryer and Chan [3]. Based on these model identification tools, the residuals were initially fit with an ARMA(1, 0) process in the transfer function model. However, upon further examination, the ARMA(1, 0) failed to account for remaining significant seasonal lags. Thus, the final model of the residuals was fit using a stationary SARIMA(1, 0, 0)  (1, 0, 0)12 model to the error process (et). The final transfer function model is represented by the following equation:

4.1. Nitrate concentrations

Y t ¼ b0 þ

1 X

biþ1 X ti þ et ;

ð3Þ

i¼0

Monthly nitrate concentrations varied widely during the 360month monitoring period from 1977 to 2006 (Fig. 3). Site 1 concentrations (upstream of reservoir) ranged from 0.01 to 15.4 mg/l and averaged 6.23 mg/l. Maximum nitrate concentrations at Site 5 (downstream) were lower than Site 1 (13.7 mg/l) but the minimum and average were very similar (0.02 and 6.13 mg/l, respectively). Monthly concentrations exceeded the MCL (10 mg/l) approximately 16.4% of the time at Site 1 and 13.1% at Site 5. Overall, for the entire period, nitrate concentrations downstream of Saylorville Reservoir were about 1.6% lower than incoming nitrate concentrations from the Des Moines River. However, it should be noted that inflow concentrations at Site 1 do not include any additional nitrate contributions added between Site 1 and Saylorville Reservoir. 4.2. Transfer function model Monthly nitrate concentrations downstream of Saylorville Reservoir were clearly influenced by inflow nitrate concentrations (Fig. 3). The prewhitened cross-correlation function of the transformed upstream nitrate concentration and its downstream counterpart revealed that the upstream value influences the downstream value by the current month and the previous month. Since prewhitening serves as a linear operator and preserves the

where Yt is the monthly nitrate concentration (square root transformed) downstream of Saylorville Reservoir, b0 is the intercept, bi+1 are model coefficients on the previous month’s upstream nitrate concentration (Xt1, square root transformed) and et is the model error that is modeled as a SARIMA(1, 0, 0)  (1, 0, 0)12 process:

et ¼ /et1 þ Uet12  /  Uet13 þ at ;

ð4Þ

where at are white noise of zero-mean and finite variance. Model coefficients and standard errors are summarized in Table 1. (The estimates of / and U are both less than 1 in magnitude, so that the regression errors {et} are stationary.) One innovative outlier (IO) existed within the model for the month of December 2001, i.e. at at that month was shifted by a constant estimated from the data (denoted as IO in Table 1). Model diagnostics suggested that the model fit was adequate, with no remaining outliers. The ACF plot indicated that there were no remaining autocorrelations present in the residuals, which was also confirmed by the Ljung–Box test where all the p-values of the Ljung–Box statistics (K = 13– 100) were above the 0.05 level, indicating no evidence against the independence assumption of the innovations {at} in Eq. (4) (Fig. 4). Because the transfer function model assumes a linear relationship between Yt and the Xt’s, the linear relationship was verified (Fig. 5). Results indicate that the linear relationship holds true even when Xt is low or high. Further Fig. 5 indicates that the best-fit line through the relation does not pass through the origin but has a yintercept of about 0.7 on the square root transformed scale (Yt versus Xt). Overall, the transfer function model provided a good fit to the downstream nitrate concentration time series across a range of concentration values over time (Fig. 6). Model performance examined by plotting measured downstream nitrate concentrations versus fitted values from the transfer function model revealed that the plotted points clustered around a 1:1 line passing through the ori-

Table 1 Transfer function model parameters.

Fig. 3. Sample cross-correlation function of prewhitened upstream and downstream nitrate concentrations.

Estimates S.E. t-Statistic p-Values

/

U

b0

b1

b2

IO

0.5063 0.0466 10.8648 <0.001

0.1863 0.0567 3.2857 0.0011

0.5421 0.0807 6.7175 <0.001

0.4285 0.0236 18.1568 <0.001

0.3554 0.0249 14.2731 <0.001

1.2527 0.2804 4.4675 <0.001

1201

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

1

2

3

4

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0

Monthly Mean Site1 Nitrate-N Concentration (mg/l)

Monthly Mean Site 5 Nitrate-N Concentration (mg/l)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Monthly Mean Site 5 Nitrate-N Concentration (mg/l)

Fig. 4. Model diagnostics from transfer function model: (top) standardized residuals; (middle) ACF plot of residuals; (bottom) Ljung–Box statistics.

0

1

2

3

4

Lag 1 Monthly Mean Site1 Nitrate-N Concentration (mg/l)

Fig. 5. (left) Nitrate concentrations at Site 5 (downstream) versus Site 1 (upstream) of Saylorville Reservoir (square root transformed). (right) Nitrate concentrations at Site 5 versus lag 1 nitrate concentrations at Site 1 (square root transformed).

gin (Fig. 7). In general, the model appeared to fit slightly better for high concentrations than low concentrations, with more variability observed at low concentrations. Improved performance at high concentrations is important when considering that an MCL for nitrate concentrations exists for nitrate at the downstream drinking water intake. Results from the transfer function model were compared to a simple model where downstream nitrate concentrations were predicted by simply reducing the upstream (Site 1) concentrations by 1.6% (the average nitrate concentration difference between Site 1 and Site 5 for the measurement period). Results indicated that the transfer function model substantially reduced the prediction

error. Over the 30-year monthly record, the prediction standard error was 1.0 mg/l for the simple model and 0.44 mg/l for the transfer function model. Reducing the sample standard error by 0.56 mg/l can be significant. Considering that the mean annual concentration at Site 5 is 6.13 mg/l, using the transfer function model would reduce the prediction error by 9.1% compared to the simple model estimation. The final transfer function model can be used to quantify the nitrate concentration reduction that occurs between Site 1 and Site 5. Specifically, the summation of the b1 and b2 coefficients in the transfer function model equals the fraction of nitrate concentration upstream contributing to nitrate concentration downstream, on

0. 0

0.5

1.0

1.5

2.0

2.5

3. 0

3.5

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

Monthly Mean Nitrate-N Concentration (mg/l)

1202

1980

1985

1990

1995

2000

2005

Time

2

3

4

the upper limits of Saylorville Reservoir, the intercept term can be interpreted as the net reservoir-moderated effect of additional nitrate concentration that is added to the Des Moines River between Site 1 and the entrance to the reservoir. The value of b0 and the fact that Saylorville Reservoir modulates nitrate concentration and passes on average about 78.39% of inflow nitrate concentration to the outflow implies that 0.48 mg/l (0.6915 = 0.5421/0.7839 on the square root scale) of additional nitrate concentration is effectively added to the Des Moines River between Site 1 and Site 5.

1

4.3. Forecasting

0

Monthly Mean Site 5 Nitrate-N Concentration (mg/l)

Fig. 6. Modeled nitrate concentrations at Site 5 downstream of Saylorville Reservoir (line) compared to measured values (points) (square root transformed).

0

1

2

3

4

Fitted Values Fig. 7. Measured nitrate concentrations at Site 5 versus fitted nitrate concentrations from the transfer function model (square root transformed).

the average. This is because, assuming stationarity, Eq. (3) implies that

EðY t Þ ¼ b0 þ ðb1 þ b2 ÞEðX t Þ

ð5Þ

or, in other words, the average nitrate concentration downstream equals b0 plus the product of (b1 + b2) times its counterpart upstream, on square root scale. The variance of b1 and b2 equaled to 5.586  104 and 6.2188  104, respectively, whereas the covariance of b1 and b2 equaled 1.075  104. Eq. (5) decomposes the mean nitrate concentration downstream into the sum of two sources: the mean effect of additional nitrate input between Site 1 and Saylorville Reservoir (see below) and the mean effect of nitrate concentration upstream as measured at Site 1. The fraction, b1 + b2, is estimated to equal 0.7839 with a 95% confidence interval of (0.7230, 0.8448). Since the value of 1 is not in the confidence interval, this implies that the reservoir is reducing nitrate concentrations from Site 1 to Site 5. Specifically, the portion of nitrate concentrations downstream due to nitrate concentration upstream at Site 1 is between 72.30% and 84.48% as high as upstream concentrations measured at Site 1. Alternatively, it can be restated that outflow nitrate concentrations are approximately 22 ± 6% lower compared to inflow upstream nitrate concentrations. The intercept term b0 in Eq. (3) provides additional insights into nitrate concentration inputs into the reservoir. Since the upstream monitoring site (Site 1, Fig. 1) is located approximately 25 km from

A month by month out-of-sample forecast was completed for nitrate monitoring data that extended from January 2007 to November 2008, a period of 23 months beyond the sampling record used to develop the time-series model. The objective of the forecast analysis was to evaluate the predictive capability of the model against measured data. In order to generate a month by month forecast from the transfer function model, the inflow nitrate concentrations to Saylorville Reservoir from the Des Moines River must be considered. The square root of the upstream nitrate concentration time series was modeled with an AR(20) model (Table 2), with the AR order chosen by AIC, and the model estimated by maximum likelihood estimation. The AR model showed highest loading on the first coefficient (0.7053) with significantly positive coefficients identified at lags of 8, 12 and 17 months and significantly negative coefficients at lags of 7 and 20 months. It should be noted that the concentration mean is given in the output. Thus, the intercept was calculated by taking the mean times one minus the sum of the AR coefficients, 2.31 * (1  0.6681) = 0.7658. Based on the fitted transfer function model and the fitted AR(20) model for the upstream series, routine algebra yields the following 1-step ahead forecasting formula:

Y tþ1 ¼ 0:5421 þ 0:4285  0:765812 þ ð0:4285  0:7053 þ 0:3554Þ  X t þ ð0:0247Þ  X t1 þ ð0:0091Þ  X t2 þ ð0:0311Þ  X t3 þ ð0:0814Þ  X t4 þ ð0:0038Þ  X t5 þ ð0:1631Þ  X t6 þ ð0:1285Þ  X t7 þ ð0:0936Þ  X t8 þ ð0:0938Þ  X t9 þ ð0:1157Þ  X t10 þ ð0:1655Þ  X t11 þ ð0:0803Þ  X t12 þ ð0:0735Þ  X t13 þ ð0:0740Þ  X t14 þ ð0:1190Þ  X t15 þ ð0:1406Þ  X t16 þ ð0:0037Þ  X t17 þ ð0:0394Þ  X t18 þ ð0:1413Þ  X t19 þ 0:5063  et þ 0:1863  et11 þ ð0:5063  0:1863Þ  et12 ;

ð6Þ

1203

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205 Table 2 Coefficients and standard error for AR(20) model for the square root of the upstream time series of monthly nitrate concentrations. /4

/5

0.7053 0.0522 13.5115 <0.001

0.0247 0.0641 0.3853 0.7002

0.0091 0.0644 0.1413 0.8877

0.0311 0.0641 0.4852 0.6279

0.0814 0.0639 1.2739 0.2036

/8

/9

/10

/11

/12

/13

/14

0.1655 0.0646 2.5619 0.0108

0.0803 0.0647 1.2411 0.2154

0.0735 0.0642 1.1449 0.2531

/19

/20

Mean

0.0394 0.0661 0.5961 0.5515

0.1413 0.0543 2.6022 0.0097

2.3075 0.0914 25.2462 <0.001

0.1285 0.0649 1.9800 0.04852

0.0936 0.0645 1.4512 0.1477

/15

/16

0.0740 0.0646 1.1455 0.2528

0.1190 0.0649 1.8336 0.0676

0.0938 0.0644 1.4565 0.1462

0.1157 0.0645 1.7938 0.0737

/17

/18 0.0037 0.0658 0.0562 0.9552

0.0038 0.0644 0.0590 0.9530

/7 0.1631 0.0645 2.5287 0.0119

15

0.1406 0.0652 2.1564 0.0318

/6

10

Monthly Downstream Nitrate-N (mg/l) Concentration

Coefficient S.E. t-Statistic p-Value

/3

5

Coefficient S.E. t-Statistic p-Value

/2

0

Coefficient S.E. t-Statistic p-Value

/1

2006.0

2006.5

2007.0

2007.5

2008.0

2008.5

Time

Fig. 8. One-month ahead forecasts for 2007 and 2008 (through November 2008); measured data (open circles), predicted nitrate concentration (green line) and 95% confidence interval (red lines). (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

where et are the residuals. Results indicated that the transfer function model predicted downstream nitrate concentrations quite well (Fig. 8). Only one observation was outside the 95% prediction bands (September 2007), which is well within the expected 5% error rate (one error out of twenty-three 1-month ahead predictions). Further, the forecast values were not overly affected by extreme river flows. Major flooding occurred in the Des Moines River in June 2008 when outflow from Saylorville Reservoir topped the emergency spillway. June 2008 nitrate concentrations predicted by the model (7.98) were very similar to measured values (7.21 mg/l), however during July, measured nitrate concentrations were higher (7.74 mg/l) than predicted values (4.80 mg/l). By August, measured and modeled concentrations were similar (4.12 and 4.38 mg/l, respectively). 5. Discussion 5.1. Reservoir effects Study results indicate that nitrate concentrations downstream of Saylorville Reservoir can be effectively modeled using a transfer function that utilized inflow concentrations during the current and previous month as input variables. The transfer function model provided a very good fit to the downstream nitrate concentration

time series and improved quantification of reservoir effects on river nitrate concentrations. Previous research [12], and simple comparisons of mean inflow and outflow concentrations, have led to the common belief that nitrate concentrations are not significantly different upstream or downstream of the reservoir. For example, a comparison of annual means over a 30-year period suggested that Saylorville Reservoir reduced nitrate concentrations approximately 1.6%. However, using the transfer function model in this study, a 22 ± 6% reduction in nitrate concentration was indicated, primarily because (a) the model considered the temporal averaging inherent within the reservoir system, and (b) the model contained nitrate added to the reservoir system between Site 1 and Site 5 (intercept term b0). Temporal averaging of nitrate concentrations is important to consider because of the seasonal nature of nitrate concentration patterns in Iowa rivers. For example, nitrate concentrations in rivers typically peak in late spring and early summer but decrease substantially in late summer and early fall [16]. In contrast, nitrate concentrations discharged from Saylorville Reservoir are out of phase with high inflow concentrations, typically peaking in July, August and September. By using the current and previous month’s inflow nitrate concentrations to predict the outflow nitrate, the transfer function model better incorporates the temporal structure associated with nitrate moving through

1204

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

the reservoir than a simple comparison of means. Further, although the intercept term was suggested by a simple plot of Yt versus Xt (Fig. 5), the intercept is not considered in simple upstream versus downstream comparisons of annual means. For example, reducing the mean annual nitrate concentration at Site 5 by the intercept (0.48 mg/l) yields a value of 5.65 mg/l. Comparing this value to the upstream average (6.23 mg/l) suggests that 10% reduction in nitrate concentration may be occurring between Site 1 and Site 5 if there were no added nitrate. Thus, nitrate concentrations cannot simply be compared upstream and downstream of the reservoir (i.e., Fig. 2) without considering the lag time associated with water flow through the impoundment and any nitrate added to the system between measurement points. Perhaps more importantly, the transfer function model provides a means to predict nitrate concentrations in the Des Moines River downstream of Saylorville Reservoir 1-month ahead (and even for longer forecast horizon). Monthly nitrate concentrations forecasted with the model were nearly all within a 95% prediction interval of their actual measured values and did not appear greatly affected by flow variations. Since inflow concentrations were used as model input, factors that would normally contribute to nitrate concentration variations, such as climate, flow variability and seasonal nitrate patterns, were already taken into account for the prediction. Simply modeling outflow concentrations as a function of time, would not be capable of predicting nitrate concentrations during unusual events because the events would not have been captured in the recent record. Since the Des Moines Water Works utilizes different surface and subsurface water sources as part of their regional public water supply, improved prediction of nitrate concentrations in the Des Moines River downstream of Saylorville Reservoir will allow the utility to better plan for use of the Des Moines River water source in their system-wide water supply. Should nitrate concentrations in the Des Moines River downstream of Saylorville Reservoir be forecasted to approach or exceed the MCL in the next month, the DMWW can make provisions for shifting their water supply intake away from the river to other water sources. Although the transfer function model effectively captured the influence of inflow nitrate concentrations on outflow values, the model did not identify specific reservoir characteristics that may be causing the 22 ± 6% reduction in concentration. The degree of nitrate reduction estimated by the model was consistent with concentration and load reductions measured in other reservoirs, where reservoir processing occurs through biological assimilation, abiotic sedimentation, dissimilatory nitrate reductions and denitrification [4,15]. Although this study focused on nitrate concentration reductions, the amount of nitrate removal provided by Saylorville Reservoir was estimated using monthly mean values of reservoir outflow and nitrate concentration data and assuming the same 22% reduction in nitrate load as estimated for nitrate concentrations. However, it should be noted that our model did not include discharge so the estimated load reduction is approximate and presented only to provide additional context to evaluate reservoir effects. If incoming nitrate loads are assumed to be 22% greater than outflow loads, the reservoir is serving to reduce average annual nitrate loading rates from 19.2 to 15.7 kg ha1 year1, or 5590 Mg nitrate year1. Assuming an average Saylorville pool elevation of 840 ft that corresponds to a surface area of 2691 ha, the average annual nitrate removal was estimated to be 208 g N m2 year1. The amount of nitrate removal by Saylorville Reservoir is similar to nitrate removal rates reported by David et al. [4] for a 4400-ha reservoir in Illinois (average of 4900 Mg N year1 or 108 g N m2 year1). Future research directions include time-series modeling that incorporates covariates such as reservoir volume, surface area and residence time in the transfer function model to shed light on reservoir processes that might be contributing to the nitrate concentration reduction. This consideration is expected to reduce

the remaining error process from its current status as a SARIMA(1, 0, 0)  (1, 0, 0)12 model to white noise. 5.2. Inflow nitrate concentrations In order to conduct the forecasting, it was first necessary to model incoming nitrate concentrations at Site 1. While not the central focus of this study, it was interesting to note that monthly nitrate concentrations in the Des Moines River were best modeled using an AR(20) model. The AR(20) model suggests that nitrate concentrations in the river are related in time to concentrations measured the previous 20 months. The relation of current values to past values has been termed a ‘‘memory effect” [20,11] which implies that nitrate concentrations from 1-month ago and 1-year ago have a large effect on the present value. The negative coefficient at lag 7 and 20 months implies that concentrations today are opposite to what they were 7 and 20 months ago. The alternating pattern of positive and negative correlation is consistent with the typical 6-month pattern of nitrate concentrations in rivers that show elevated concentrations during late spring and late fall, and lower concentrations in the summer when biological uptake in the river is particularly evident [16,22]. The 20-month temporal correlation is similar to the 2-year temporal correlation identified in the Raccoon River using variograms and spectral analysis [22]. The similar pattern of temporal correlation observed in these two systems suggests that there may be regional consistency in nitrate concentration patterns in Iowa rivers. The memory effect in the Iowa rivers is longer than the memory effect of nitrate concentrations measured in English rivers (maximum of 15 months; [20]) or in North Carolina streams (13 months; [21]). Jones and Smart [9] observed positive long-term lags in nitrate concentrations extending to 20 months at some karst springs. Further work is needed on nitrate time series from other Iowa rivers to determine if the 2-year memory of nitrate concentrations is perhaps a diagnostic feature of highly agricultural watersheds in the glaciated Midwest. 6. Conclusions Time-series analysis of 30 years of monthly nitrate concentrations showed that reservoir effects on river nitrate concentrations can be effectively modeled using a transfer function approach that considers the effects of inflow concentrations on outflow concentrations. In the case of Saylorville Reservoir, the transfer function model suggested that the reservoir is reducing nitrate concentrations by 22 ± 6%, a reduction that greatly exceeds previous estimates. Moreover, the transfer function model provided a method to predict nitrate concentrations downstream of the reservoir 1month ahead given the current and previous month’s inflow concentrations. For water supplies that use surface water downstream of reservoirs, the improved prediction capability may enable better management of water supplies. Above the reservoir, nitrate concentrations in the Des Moines River were modeled using an AR(20) model, with the higher order model consistent with a typical long memory noted by others. Future work on the effects of reservoirs on river nitrate concentrations will focus on refining the transfer function model by including additional covariates, such as stage, volume, and residence time, to better identify reservoir processes that might be contributing to the nitrate concentration reduction. Acknowledgements We gratefully acknowledge partial support for KS and KSC from the Iowa Water Center.

A.L. Schoch et al. / Advances in Water Resources 32 (2009) 1197–1205

References [1] Box GEP, Cox DR. An analysis of transformations. J R Stat Soc Ser B 1964;26:211–46. [2] Box GEP, Jenkins GM, Reinsel GC. Time series analysis: forecasting and control. Englewood Cliffs, NJ: Prentice-Hall; 1994. [3] Cryer JD, Chan KS. Time series analysis with applications in R. 2nd ed. Springer Science + Business Media; 2008. [4] David MB, Wall LG, Royer TV, Tank JL. Denitrification and the nitrogen budget of a reservoir in an agricultural watershed. Ecol Appl 2006;16:2177–90. [6] Dodds WK, Welch EB. Establishing nutrient criteria in streams. J N Am Benth Soc 2000;19:186–96. [7] Garnier J, Leporcq B, Sanchez N, Philippon X. Biogeochemical mass-balances (C, N, P, Si) in three large reservoirs of the Seine Basin (France). Biogeochemistry 1999;47:119–46. [8] Iowa Department of Natural Resources (IDNR). 2002 Impaired waters list. Des Moines, IA: Iowa Department of Natural Resources; 2004. [9] Jones AL, Smart PL. Spatial and temporal changes in the structure of groundwater nitrate concentration time series (1935–1999) as demonstrated by autoregressive modeling. J Hydrol 2005;310:201–15. [10] Lutz DS, Francois B. Water quality studies – Red Rock and Saylorville Reservoirs, Des Moines, IA. Annual report, Engineering Research Institute, ISU-ERI-Ames-02321, Iowa State University, Ames, IA; 2007. [11] Kirchner JW, Feng X, Neal C. Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 2000;403:524–7. [12] Okereke VI, Bauman ER, Austin TA, Schulze Lutz D. Midwest (USA) reservoir water quality modification. III. Soluble nutrients. Water Air Soil Pollut 1988;37:343–54.

1205

[13] Prior JC. Landforms of Iowa. Iowa City, IA: University of Iowa Press; 1991. [14] Rabalais NN, Turner RE, Scavia D. Beyond science into policy: Gulf of Mexico hypoxia and the Mississippi River. BioScience 2002;52:129–42. [15] Saunders DL, Kalff J. Nitrogen retention in wetland, lakes and rivers. Hydrobiologia 2001;443:205–12. [16] Schilling KE, Lutz DS. Relation of nitrate concentrations to baseflow in the Raccoon River, Iowa. J Am Water Res Assoc 2004;40:889–900. [17] Schilling KE, Zhang Y-K. Baseflow contribution to nitrate–nitrogen export from a large, agricultural watershed, USA. J Hydrol 2004;295:305–16. [18] Seitzinger SP, Styles RV, Boyer EW, Alexander RB, Billen G, Howarth RW, et al. Nitrogen retention in rivers: model development and application to watershed in the northeastern USA. Biogeochemistry 2002(57/57):199–237. [19] US Environmental Protection Agency (USEPA). National section 303(d) list fact sheet; 2003. . [20] Worrall F, Burt TP. A univariate model of river water nitrate time series. J Hydrol 1999;214:74–90. [21] Worrall F, Swank WT, Burt TP. Changes in stream nitrate concentrations due to land management practices, ecological succession, and climate: developing a systems approach to integrated catchment response. Water Resour Res 2003;39:1177. doi:10.1029/2000WR000130. [22] Zhang Y-K, Schilling KE. Temporal variations and scaling of streamflow and baseflow and their nitrate–nitrogen concentrations and loads. Adv Water Resour 2005;28:701–10. [23] Worrall F, Burt TP. Decomposition of river water nitrate time-seriescomparing agricultural and urban signals. Sci. Tot. Env. 1998(210/ 211):153–62.