Calculation and Use of First-Order Rate Constants for Monitored ...

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Appendix I. Uncertainty in Rate CalculationsUsing Statistics to Estimate the Time Frame toAchieve Remediation ObjectivesAs with any remediation method, one of the fundamental questionsthat arises is “How much time will be required before remediationobjectives are achieved?” At the current state of practice, theonly practical approach available uses a statistical analysis oflong-term monitoring data from wells in the source area of thecontaminant plume. Many practitioners will calculate the Pearsonproduct moment correlation coefficient (R 2 ) for the regression usedto extract the Point Decay Rate constant (k point). If the coefficientis near one (e.g., greater than 0.9 or 0.95), the regression isaccepted as being useful in a qualitative way. There are twoproblems with this approach; it does not allow the user to select alevel of confidence for the comparison, and it does not give morevalidity to regressions with many points compared to regressionswith only a few points.The slope of the regression is the rate constant. A better approachis to calculate a confidence interval on the slope of the regression.The following data from Kolhatkar et al., 2000 will be used toillustrate this approach. They collected long-term ground-watermonitoring data from three wells at a gasoline release site in NewJersey. Their original data displayed extreme oscillations withconcentrations bouncing from a high value down to the analyticaldetection limit of 1µg/L, and then back to a high value oversequential sampling intervals. Although the scatter in the dataset is typical of the variation seen at many other sites, the influenceof these outliers on the statistical estimate of the rate of attenuationwas removed by editing the data set to remove those points wherethe concentration of MTBE was less than the detection limit.Table I-1. Sources of Uncertainty in Calculated Rate ConstantsType of Uncertainty Applies to Type of Effect Ways to ManageMonitoring WellLocation(horizontal andvertical location)Seasonal EffectsSeepage VelocityEstimatePoint Decay Rate (k point)Bulk Attenuation RateConstant (k )Biodegradation RateConstant (λ)Point Decay Rate (k point)Bulk Attenuation RateConstant (k )Biodegradation RateConstant (λ)Bulk Attenuation RateConstant (k )Wells not in strongest source area maynot give repres entative indication of howlong entire plume will persist.Wells not on centerline of plume can givemis leading indications aboutconcentration profile in plume.A poorly des igned monitoring wellnetwork may give misleading informationabout source s trength, source size, andcenterline plume concentrations used forcalibration.Can introduce additional s catter in dataus ed to develop k pointrate c ons tant.Ty pically not a problem as all data arecollected at the same time.Can be a problem if seas onal effects aresignificant and the data used forcalibration are not c ollec ted(concentration vs. distance) at the sametime.Increases ov erall uncertainty incalculation.Characterize source with several wells.Estimate and report uncertainty in final result(estimated time to reach clean-up s tandards ).Us e a well-designed monitoring well networkwith transects of wells in rows across theplume rather than one set of wells down theinferred centerline. Estimate and reportunc ertainty in final res ult (es timated plumelength).The source and plume need to be wellc haracterized to ensure representativemodeling res ults . Perform sensitivity analysison model.Addres s as part of an uncertainty calculation(s ee below). For s trong seasonal effects, useof data from the same season c an bec onsidered.Not applicable.For strong seasonal effects, use data froms ame s eas on to help ensure representativ emodeling res ults . Perform sensitivity analysison model.Average results from multiple seepageestimates along plume centerline. Improv eseepage velocity estimate. Estimate andreport uncertainty in final result (estimatedplume length).PlumeHeterogeneityAll rate constantc alc ulationsIncreases apparent uncertainty.Use worst-case data. Use transects toc apture plume heterogeneity. For regressionbased rate cons tants (k and k point), estimateand report uncertainty in final result. Formodeling studies designed to determine λ,perform sensitivity analy sis on model byc hanging k ey variables to their upper andlower expected range and evaluate howmodeling res ults c hange.12
Because there is natural scatter in the long-term monitoring data,there is uncertainty in the estimate of the Point Decay Rate (k point),and in the projected time frame to achieve cleanup in thatmonitoring well. To account for this uncertainty, a confidenceinterval was calculated for each estimate of the Point Decay Rate(k point) at a pre-determined level of confidence of 90% and 95%.The level of confidence is simply the probability that the true rateis contained within the calculated confidence interval. Aconfidence level of 90% is reasonable for many sites. At othersites, a more stringent confidence level (e.g., 95%) may be moreappropriate, depending upon the level of risk that is acceptable.In most applications of regression, the user wishes to calculateboth an upper boundary and lower boundary on the confidenceinterval that will contain the true rate at the pre-determined levelof confidence. This is termed a “two tailed” confidence intervalbecause the possibility of error (the tail of the probability frequencydistribution) is distributed between rates above the upper boundaryand below the lower boundary of the confidence interval. As aconsequence, tables of critical values in statistical reference booksand computer applications provide a “two-tailed” confidenceinterval. At a level of confidence of 80%, the estimate will be inerror 20% of the time. The true rate will be contained within thecalculated confidence interval 80% of the time, 10% of the timethe true rate will be faster than the upper boundary of theconfidence interval, and 10% of the time the true rate will be slowerthan the lower boundary of the confidence interval. Using thedata provided above from MW-5, the slope of a regression of thenatural logarithm of concentration of MTBE on time is -0.188 peryear. The Point Decay Rate (k point) is +0.188 per year. Theboundaries of the “two tailed” confidence interval on the rate at80% confidence are 0.248 per year and 0.127 per year. Thismeans that 80% of the time the true rate will be between 0.248and 0.127 per year, that 10% of the time the true rate is greaterthan 0.248 per year, and 10% of the time the true rate is less than0.127 per year. The true rate will be greater than 0.127 per year90% of the time.There is little value in estimating the shortest possible time thatwould be required to reach the goals for cleanup; remedial optionsare compared and evaluated based on the greatest time requiredto reach goals. At the selected level of confidence, all thepossibility of error should be assigned to rates that are slowerthan the lower boundary of the confidence interval. This is a “onetailed”confidence level; it includes all true rates that are fasterthan the lower boundary of the confidence interval. A “one tailed”Table I-2. MTBE Concentrations in the Three Most Contaminated Monitoring Wells at a Gasoline Spill SiteMW-5 MW-6 MW-11DateConcentration(µg/liter)Concentration(µg/liter)Concentration(µg/liter)9/17/93 1,900 2709/23/94 1,800 200 22005/17/96 1,300 120 8808/10/96 980 12011/7/96 620 66 66012/8/97 500 3393/27/98 635 71.2 4267/23/98 470 4199/18/98 1,210 4412/16/98 379 1443/1/99 700 42.2 1236/21/99 574 4649/7/99 792 43.2 1959/7/99 1,050 15512/30/99 525 2203/20/00 501 36 1736/22/00 420 51.2 14613
Page 3 and 4: plots serve to characterize the dis
Page 5 and 6: to the different attenuation proces
Page 7 and 8: Table 2.Continued...HOW TO USE:Poin
Page 10 and 11: EXAMPLE 2. Use of Concentration vs.
Page 14 and 15: confidence interval can be calculat
Page 16 and 17: Bulk Attenuation Rate (k) as a Func
Page 18 and 19: Bulk Attenuation Rate (k) as a Func
Page 20 and 21: Appendix III. Effect of Dispersion,
Page 22 and 23: Effect of Sorption on Concentration
Page 24 and 25: Effect of Source Decay (k source) o
Page 26: ReferencesAmerican Society for Test

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