Environ. Sci. Technol. 1996, 30, 2312-2317
Kalman Filter Method for Estimating Organic Contaminant Concentrations in Major Chesapeake Bay Tributaries J. TERRY GODFREY* AND GREGORY D. FOSTER Department of Chemistry, George Mason University, 4400 University Drive, Fairfax, Virginia 22030
Concentrations of toxic organics (atrazine, metolachlor, fluoranthene, and total polychlorinated biphenyls) in fluvial transport were obtained for the Susquehanna, Potomac, and James Rivers during 1992-1993. These three tributaries constitute ca. 75% of the total riverine basin area supporting the Chesapeake Bay watershed. Across all basins a total of 26 samples was acquired. Owing to the recent focus on development of a predictive capability for assessing concentration levels in this watershed, the data were used as the basis for predicting concentrations using a Kalman filter. Prediction was obtained for months where the actual concentration was unknown and, using substitution, for selected months when the concentration was measured. A total of 92 comparisons were made across all tributaries and compounds using substitution. For known concentrations, a filtered or smoothed value was also obtained that reflects the correction due to the estimated noise in the measurements and the underlying environmental system. Examination of the contaminant concentration values in a predictive fashion, using substitution, suggests that they are mainly within a factor of 4-5 of the measured concentration in most cases. This is well within the accuracy goals for this watershed.
Introduction The Chesapeake Bay Fall Line Toxics Monitoring Program (CBFLP) has conducted on-going studies to determine the annual loads and transport fluxes of trace contaminants in various tributaries of Chesapeake Bay from 1990 through 1994, following specific requirements included in the Chesapeake Bay Toxics Reduction Strategy (1-6). A specific study initiated under the direction of the CBFLP was aimed at applying ultra-trace analytical methods for determining the concentrations and transport fluxes of selected organic contaminants above the river fall lines, regions where * Corresponding author present address: Environmental Sensing, Inc., 6878 McLean Province Cr., Falls Church, VA 22043; telephone: (703) 534-7249; e-mail address:
[email protected].
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upland streams meet the lowland river, of the Susquehanna, Potomac, and James Rivers (together accounting for ca. 75% of the total freshwater flow) for March 1992-February 1993 (3, 6). Data reported during the period stated above for the CBFLP, including organic contaminant data and ancillary environmental parameters, served as the basis set for the development of our Kalman filter. It is important to note that as part of the CBFLP sampling and analysis regime, it was necessary to concentrate selected organic contaminants up to 250 000-fold from surface water to determine low- or sub-nanogram per liter concentrations characteristic of the contaminants in fluvial transport (3). This has resulted in a monthly sampling frequency at the three river fall lines, providing ca. 15-20 samples collected each year with occasional omissions on an infrequent basis. Since the concentrations of organic contaminants in surface water are stochastic in nature (7), it is appropriate to consider estimation techniques that would allow inference of concentrations at times more frequent than those sampled. This prospect has been addressed by Johnson et. al (8) and is considered a major candidate for long-term estimations of contaminant fluxes in this ecosystem, given the realities of future reductions in sampling effort and costs in the CBFLP. The Kalman filter is particularly adaptable to this application because it allows the estimation of missing as well as projected concentration values (9). This is based on the underlying correlation and stochastic properties of the organic contaminant concentration when compared with other parameters. For purposes of comparison, stream discharge, water temperature, air temperature, and total suspended particulates (TSP) concentrations were selected as a set of environmental parameters (10, 11) against which concentration could be correlated. There exist well-known correlations between stream discharge and TSP (12, 13). Also, there exist known correlations between air and water temperatures. The correlation between concentration and stream discharge has been observed in this study (14). One of the important features of this data set was the availability of parameter data for the time frame in question. The goal of this study was to implement a Kalman filter for the purpose of providing a mathematical filter on existing river fall line concentrations versus time and to develop an approach for predicting concentrations of organic contaminants in fluvial transport in Chesapeake Bay tributaries. The model employed in this analysis assumes a known time series of concentration as well as environmental parameter data. The concentration data were gathered monthly, except for selected values when no data are reported. The model assumed stationary Gaussian statistics describing the performance of these quantities, which is reasonable within the limits of the central limit theorem. The Kalman filter is a linear predictive filter used to estimate the state of a linear stochastic system by employing measurements that are functions of the state of the system. Essentially, the model assumes linearity in its relationships among state transition matrix parameters. Surface water concentrations for atrazine, metolachlor, fluoranthene, and total polychlorinated biphenyls (t-PCBs) were assessed. The fluoranthene concentrations corresponded to values measured in the particulate phase while all the rest were
S0013-936X(95)00682-1 CCC: $12.00
1996 American Chemical Society
of the Susquehanna, Potomac, and James Rivers, as illustrated in Figure 1, according to sampling procedures previously described in detail for the 1992 CBFLP (3). The large-volume surface water samples were filtered to isolate suspended particulate matter. The filtered water was extracted using high-capacity C18 bonded-phase extraction cartridges. The particulate and filtered water phases were analyzed separately using gas chromatography or gas chromatography/mass spectrometry. Analytical method descriptions have been provided in a previous report (3).
Computational Procedures This section describes the implementation of a Kalman filter used as the basis for filtering and estimation of concentration values. The starting point for the Kalman filter is the parameters discussed earlier plus concentration:
xk ) [xk(conc)xk(disc)xk(WT)xk(AT)xk(TSP)]T
(1)
Here, the superscript T indicates transpose. This vector (eq 1), at discrete time k, describes the state of the system where the components represent organic contaminant concentration and other environmental parameters. The equations of motion describing the state of the system are
xk ) Φk-1xk-1 + wk-1
(2)
z k ) H kx k + v k
(3)
and FIGURE 1. Map of the middle Atlantic region of the United States illustrating Chesapeake Bay, the James, Susquehanna, and Potomac Rivers, and the location of Cartersville, Conowingo Dam, and Chain Bridge, where the river fall line samples were collected, respectively. (Courtesy USGS, Towson, MD.)
measured in the filtered or dissolved phase.
Experimental Section Study Sites. The Susquehanna, Potomac, and James Rivers are among three of the largest tributaries feeding Chesapeake Bay. Figure 1 illustrates the fall line regions for each tributary within the Chesapeake Bay watershed, the point at which the lowland streams meet the upland river. The James River fall line is located in a rural region approximately 100 mi downstream from a large population center, Lynchburg, VA, and the James River basin has a drainage of 16 206 km2 above the fall line. The Potomac River fall line is adjacent to the Washington, DC area at Chain Bridge and has a basin drainage of 29 966 km2 above the fall line. The Susquehanna River fall line is located at Conowingo Dam in a rural part of Maryland just prior to entry to the Chesapeake Bay and has a drainage of 70 189 km2 above the fall line. Analytical Methods. Organic contaminant concentrations were obtained from the Chesapeake Bay Fall Line Toxics Monitoring Program database for the periods March 1992-February 1993 (3). The compounds examined in the CBFLP for 1992-1993 included an extensive list of organonitrogen and organophosphorus pesticides, organochlorine pesticides, PCBs, and PAHs. The subset atrazine, metolachlor, fluoranthene, and total PCBs (t-PCBs) was used in this study because of the high detection frequency found in the tributaries investigated. It was felt that these organic contaminants would provide a good opportunity to assess the predictive capabilities of the Kalman filter. Large volume (i.e., 40-140 L) surface water samples were collected from March 1992 to February 1993 at the fall lines
where Φk-1 is the state transition matrix (STM), Hk is the measurement sensitivity matrix, wk-1 and vk are noise vectors associated with the internal system dynamics and the measurement, respectively, and zk is the measurement vector. It is useful to define the system states at time k before the measurement (the a priori values), xk(-), and after the measurement (the a posteriori values), xk(+). The expectation value of these states is given by
xˆ k(-) ) E[xk(-)]
(4)
xˆ k(+) ) E[xk(+)]
(5)
It has been assumed that the noise processes are zeromean Gaussian processes with variances at time k given by Q and V, respectively for wk and vk. Note the processes are assumed stationary and independent in this study. Following Grewal and Andrews (11), the state estimate is extrapolated according to the following:
xˆ k(-) ) Φk-1xˆ k-1(+)
(6)
which provides a mechanism for prediction. Using
x˜k(-) ) xk - xˆ k(-)
(7)
it follows that the a priori error covariance, the error covariance before the update, satisfies
Pk(-) ) E[x˜k(-)x˜k(-)T]
(8)
with similar expressions for the a posteriori error covariance. Then the error covariance extrapolation is defined by
Pk(-) ) Φk-1Pk-1(+)ΦTk-1 + Q
(9)
The state estimate observational update satisfies, where zk is the input measurement vector,
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xˆ k(+) ) xˆ k(-) + Kk[zk - Hkxˆ k(-)]
(10)
Kk ) Pk(-)HkT[HkPk(-)HkT + V]-1
(11)
Finally, the a posteriori error covariance update becomes
Pk(+) ) [I - KkHk]Pk(-)
(12)
where I is the identity matrix. These equations complete the definition of the filter used in this analysis. A more symmetrical form of eq 12 is available to simplify analysis (10). From eq 6, it is clear that the state transition matrix relates the a priori states at time k to the a posteriori states at time k - 1. It was useful to determine the relationships
Φk ) R(k)R-1(k - 1)
(13)
among the concentration and environmental parameters where R is the correlation matrix for xk. Additionally, Hk ) I, Po(+) ) 0, Po(-) ) 0, and V was estimated to be roughly 60% of the system noise. Q was calculated based on the scatter in the data. The measurement vector, zk, is simply the input data values as discussed above. For predicted concentration at time k, with components of zk missing, the filter is used to carry forward the component value based on
zk z Hkxk-1(+)
(14)
For predicted concentration with substitution, a known value of concentration is set to zero and eq 14 is used to extrapolate the value. This is then compared with the known value. It is this approximation, eq 14, that is used to achieve the predictive capability of the filter.
Results The data employed in this analysis included censored values that were adjusted to the detection limit of the analytical method when the measured values were either below the detection limit or not detected in a collected surface water sample (15). Across all basins, 26 samples were acquired. To estimate the significance of this number of samples, Richards and Holloway (16) determined that loadings (concentration times discharge) for such a sampling regime result in a study bias error on the order of 30-35%. Predicted and Measured Concentrations. Measurements were made during most months as indicated in Tables 1-3. These served as input to the Kalman filter analysis process. For the months in which no measurement existed, eq 14 was used to obtain an estimate of the concentration. This constituted a set of estimated concentrations for which no measurement existed; hence, these values could not be used to calibrate the model. A second set of estimated concentrations was obtained by taking known concentration values in the sequence and arbitrarily assuming that they were unknown. The concentration was then calculated using eq 14, and the result was compared with the actual known value. This latter approach was referred to as substitution. The experimental values used for the environmental parameters were also obtained monthly. Figures 2-5 represent results for the measured (solid black), filtered or smoothed (white), and predicted or
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TABLE 1
zk Values for the Susquehanna Rivera
with the Kalman gain matrix defined by
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atra meto fluo t-PCB disc WT AT TSP k (ng/L) (ng/L) (ng/L) (ng/L) (m3/s) (°C) (°C) (mg/L) 1 1.3 11.5 1.0 2 15.7 21.2 1.0 3 71.2 39.5 11.8 4 255.0 121.0 1.0 5 66.3 27.5 1.0 6 7 43.4 12.1 2.5 8 9 33.2 17.0 2.5 10 11 1.3 1.4 3.9 12
0.1 0.5 0.6 5.7 9.6 2.6 8.9 0.3
966 2503 1359 459 496 383 866 681 1651 1380 3171 1506
12.0 7.0 16.0 26.0 28.0 28.0 26.0
date
12.0 8.0 20.0 23.0 33.0 27.0 21.0
10.0 22.0 13.0 5.0 2.0 8.0 8.0
3/6/92 4/3/92 5/12/92 6/19/92 7/15/92
6.0 8.0 5.0 15.0 6.0 9.0
10.0 12.0 25.0
11/18/92
9/2/92
1/8/93
a atra, atrazine, meto, metolachlor; fluo, fluoranthene; t-PCB, total polychlorinated biphenyls; disc, discharge; WT, water temperature; AT, air temperature; TSP, total suspended particulates.
TABLE 2
zk Values for the Potomac Rivera atra meto fluo t-PCB disc WT k (ng/L) (ng/L) (ng/L) (ng/L) (m3/s) (°C)
AT TSP (°C) (mg/L)
1 33.5 12.9 2 36.4 23.5 3 206.7 84.9 4 542.4 344.7 5 6 58.7 51.5 7 8 48.5 25.3 9 10 19.8 9.1 11 9.6 0.7 12
13.0
30.0
22.0 24.0 26.0 27.0 25.0
48.0 7.0 11.0 5.0 7.0
12.0
4.0
1.0 1.0 1.4 8.3
1.1 1.3 2.0 3.6
2.7
0.5
1.0
0.5
5.7 1.0
2.2 0.5
340 886 199 105 156 142 113 66 225 600 479 242
8.0 16.5 17.0 25.0 16.0 29.5 24.0 17.0 10.0 5.0 3.5 2.5
date 3/20/92 4/22/92 5/29/92 6/30/92 8/4/92 10/6/92 12/19/92 1/26/93
10.0
a atra, atrazine; meto, metolachlor; fluo, fluoranthene; t-PCB, total polychlorinated biphenyls; disc, discharge; WT, water temperature; AT, air temperature, TSP, total suspended particulates.
TABLE 3
zk Values for the James Rivera atra meto fluo t-PCB disc WT AT TSP k (ng/L) (ng/L) (ng/L) (ng/L) (m3/s) (°C) (°C) (mg/L) 1 1.3 2 1.3 3 58.2 4 31.5 5 7.4 6 7 4.6 8 23.6 9 3.9 10 1.3 11 1.3 12
0.7 2.5 29.6 9.4 1.4
73.5 1.0 22.2 1.0 1.0
11.0 0.1 4.3 1.4 0.1
10.7 0.7 0.7 0.7 0.7
1.0 1.0 5.9 33.2 1.0
0.1 4.5 0.3 1.1 0.1
555 133 323 155 59 56 46 50 279 881 260 262
10.0 15.0 19.0 22.0 29.0
9.0 18.0 18.0 24.0 25.5
28.0 5.0 42.0 6.0 3.0
date 3/13/92 4/10/92 5/20/92 6/24/92 7/22/92
25.0 24.5 3.0 9/3/92 12.0 15.0 3.0 10/28/92 12.0 15.5 138 11/25/92 3.0 8.0 433 12/11/92 3.5 2.0 14.0 1/28/93 5.0 0.5
a atra, atrazine; meto; metolachlor; fluo, fluoranthene; t-PCB, total polychlorinated biphenyls; disc, discharge; WT, water temperature; AT, air temperature; TSP, total suspended particulates.
estimated (striped) organic contaminant concentrations for the Susquehanna River for atrazine, metolachlor, fluoranthene, and t-PCBs. The filtered results represent removal of the noise inherent to the actual measurement and system. The predicted values in the figures correspond to estimates of concentration for which no measurements exist; hence, substitution is impossible. Figures 6 and 7 present similar information for the Potomac and James
FIGURE 2. Actual (solid black), filtered (white), and predicted (striped) concentration for atrazine in the Susquehanna River versus month: March 1992 (month 1)-February 1993 (month 12).
FIGURE 3. Actual (solid black), filtered (white), and predicted (striped) concentration for metolachlor in the Susquehanna River versus month: March 1992 (month 1)-February 1993 (month 12).
FIGURE 4. Actual (solid black), filtered (white), and predicted (striped) concentration for fluoranthene in the Susquehanna River versus month: March 1992 (month 1)-February 1993 (month 12). River fall lines, respectively, for atrazine. For the Susquehanna River fall line, no samples were collected for months 6 (August 1992), 8 (October 1992), 10 (December 1992), and 12 (February 1993). Hence, the filter was used to predict values for these months without substitution. For the
FIGURE 5. Actual (solid black), filtered (white), and predicted (striped) concentration for t-PCBs in the Susquehanna River versus month: March 1992 (month 1)-February 1993 (month 12).
FIGURE 6. Actual (solid black), filtered (white), and predicted (striped) concentration for atrazine in the Potomac River versus month: March 1992 (month 1)-February 1993 (month 12).
FIGURE 7. Actual (solid black), filtered (white), and predicted (striped) concentration for atrazine in the James River versus month: March 1992 (month 1)-February 1993 (month 12).
Potomac River fall line, sampling months 5 (July 1992), 7 (September 1992), 9 (November 1992), and 12 (February 1993) constituted months in which no sampling was acquired, and prediction was applied for these months, again with no substitution. No measurements for the James
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FIGURE 8. Plot of the logarithm of predicted versus measured concentration for the James (9), Susquehanna (O), and Potomac (.) Rivers. Substitution was used to qualify the data.
River fall line were available for month 6 (August 1992); hence, a predicted value was calculated with no substitution. The values recorded generally corresponded to base-flow events as opposed to storm-flow events. Figure 8 provides an indication of the estimation accuracy that the Kalman filter achieved. Plotted in this figure is the predicted value for a concentration versus the actual measured value. To accomplish this, a known monthly concentration value is treated as unknown (substitution) and a predicted value is estimated. Data for all three tributaries and all four contaminants in each tributary are plotted along with a best fit straight line. The agreement between the predicted and measured values is reasonable and is usually within a factor of 4-5.
Discussion Concentration and Environmental Values. Tables 1-3 illustrate the input data values used as the measurement state for the system. The vector zk is 1 × 5 and consists of one concentration value followed by discharge, water temperature, air temperature, and total suspended particulate concentration (TSP). These, of course, occur at the month denoted by k. The concentration values were calculated from the 26 samples per compound. The remaining environmental parameters are not necessarily complete or optimum but represent known data values for the basins in question. Each table corresponds to a single tributary and yielded four sets of vectors, one for each contaminant. The contaminants were selected from the overall CBFLP database based on their frequency of detection and presence in the data sets. The environmental parameters were selected because of their availability. Some correlation was assumed among these parameters, but this is taken into consideration with the state transition matrix estimates. These parameters are not necessarily comprehensive but represent an initial attempt at categorizing these basins. It was our intent that part of the study would be to consider parameters that were available and to look for agreement with the resulting concentration estimates. It appeared that these parameters
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yielded reasonably good performance, allowing estimation to within a factor of 4-5 for the known concentrations when substitution is employed. This is adequate for estimating loadings, for example, as discussed by Richards and Holloway (16). System and Measurement Noise. The system noise Q was estimated by calculating the variance of the input data based on experiment and assigning these values to each state, respectively, constant across all time periods. The measurement noise, V, was estimated as the Q value times an estimated percentage error value. Model Considerations. The Kalman filter used in this analysis assumes that the relationships among the components of each basin are linear and that the noise influencing the system and measurements is Gaussian within the confines of the central limit theorem. The assumption of linearity tended to weigh nearest neighbor effects the most heavily in the predictive estimates for concentration because the model implementation is firstorder Markoff. Censored values were replaced with the detection limit value. Missing samples for a given monthly period k were predicted by setting the measured input value at time k to the transformed a posteriori state at time k - 1 (eq 14). The Kalman filter assumes system noise, underlying the overall environmental system, and measurement noise intrinsic to the experimental process. Examples of system noise would be rain or point dumping of contaminant. An example of measurement noise would be the loss of contaminant during evaporation. The filtered result is intended to correct for these problems. The predicted result is intended to extrapolate across missing values. Because of the extrapolation constraint expressed in eq 14, the predicted values tend to be weighted toward the previous epoch. Quantitative Implications of the Kalman Filter Approach. In this paper, the Kalman filter applies STM estimates in its analysis of time series data for fall line concentrations. This approach models the data as stochastic and provides a mechanism for the smoothing or filtering of the data as well as the estimation or prediction of “missing” data values. The filtered concentration values simply provide estimates of the concentration minus the presence of basin and measurement noise effects, hence are noise-free estimates of the concentration. The predicted concentration values are estimates of concentration where no values are known (or substitution has been used as in Figure 8) based on the underlying time series for concentration and other environmental parameters. One measure of the significance of these concentration measurements lies in the estimation of river loadings (17). Such loadings represent the product of discharge and concentration and are measures of the rate at which contaminants flow across the fall line point. For filtered data, the loadings more correctly reflect the true concentration component when using Kalman filter output. For predicted or estimated concentrations in the absence of measurement data, the Kalman filter provides a mechanism for estimating loadings. Figures 2-7 illustrate that the filtered concentration data are not significantly different from measured estimates to have major impact on the loadings estimates, except for small concentration situations. This filtered value is generally within a factor of 2 of the measured concentration. Hence, some improvement in loadings can be expected for filtered data.
For predicted concentrations, however, the estimated concentration values provide new insight into the loadings estimates, and based on Figure 8 for example, it is clear that the predicted loadings can be expected to be within a factor of 4-5 of the actual loadings. Storm-flow corresponds to random-like conditions when higher than normal flow exists in response to storm events. Base-flow corresponds to the normal seasonal river flow in the absence of storm conditions. It is this latter component that is amenable to Kalman filter analysis and the component that was employed in this analysis using time series data. Some storm-flow data were used in the analysis because of the nature of the sampling; however, these data constituted less than 8% of the overall samples used.
(6)
(7)
(8)
(9) (10)
(11)
Acknowledgments The authors would like to extend their appreciation to Katrice A. Lippa for providing the Chesapeake Bay tributary fall line data used in this analysis.
Literature Cited (1) Chesapeake Bay Program Office. Chesapeake Bay Basinwide Toxics Reduction Strategy. Agreement Commitment Report, Chesapeake Executive Council: Annapolis, MD, 1988. (2) Chesapeake Bay Program Office. Chesapeake Bay Fall Line Toxics Monitoring Program: 1990-1991 Loadings; U.S. EPA-Chesapeake Bay Program: Annapolis, MD, 1993; CBP/TRS 98/93. (3) Chesapeake Bay Program Office. Chesapeake Bay Fall Line Toxics Monitoring Program: 1992 Final Report; U.S. EPA-Chesapeake Bay Program: Annapolis, MD, 1994; CBP/TRS 121/94. (4) Chesapeake Bay Program Office. Chesapeake Bay Basinwide Toxics Reduction Strategy Reevaluation Report; U.S. EPAChesapeake Bay Program: Annapolis, MD, 1994; CBP/TRS 117/ 94. (5) Chesapeake Bay Program Office. Chesapeake Bay Basin Toxics Loading and Release Inventory; Basinwide Toxics Reduction
(12) (13) (14) (15) (16) (17)
Strategy Commitment Report; U.S. EPA-Chesapeake Bay Program: Annapolis, MD, 1994; CBP/TRS 102/94. Chesapeake Bay Program Office. Fall Line Toxics Program 1994 Final Report. U.S. EPA-Chesapeake Bay Program: Annapolis, MD, 1994. Thomann, R. V.; Mueller, J. A. Principals of Surface Water Quality Modeling and Control; Harper Collins Publishers, Inc.: New York, 1987. Johnson, W. E.; Kroll, R. B.; Plimmer, J. R.; Pait, A. S. Perspectives on Chesapeake Bay, 1994: Advances in Estuarine Sciences; Chesapeake Research Consortium, Inc.: Edgewater, MD, 1994; pp 105-146. Grewal, M. S.; Andrews, A. P. Kalman Filtering; Prentice Hall: Englewood Cliffs, NJ, 1993. James, R. W.; Simmons, R. H.; Strain, B. F. Water Resources Data Maryland and Delaware Water Year; Water-Date Reports MDDE-92-1 and MD-DE-93-1; Towson, MD, 1992-1993. Prugh, B. J., Jr.; Herman, P. E.; Belval, D. L. Water Resources Data Virginia Water Year; Water-Date Reports VA-92-1 and VA-93-1; Richmond, VA, 1992-1993. Hemond, H. F.; Fechner, E. J. Chemical Fate and Transport in the Environment; Academic Press: New York, 1994. Schottler, S. P.; Eisenreich, S. J.; Capel, P. D. Environ. Sci. Technol. 1994, 28, 1079-1089. Squillace, P. J.; Thurman, E. M. Environ. Sci. Technol. 1992, 26, 538-545. Slymen, D. J.; de Peyster, A. Environ. Sci. Technol. 1994, 28, 898-902. Richards, R. P.; Holloway, J. Water Resour. Res. 1987, 23, 19391948. Godfrey, J. T.; Foster, G. D.; Lippa, K. A. Environ, Sci. Technol. 1995, 29, 2059-2064.
Received for review October 5, 1995. Revised manuscript received January 18, 1996. Accepted March 14, 1996.X ES9506824 X
Abstract published in Advance ACS Abstracts, May 1, 1996.
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