ARCHIVED REPORTS_XR0002666 - Laserfiche WebLink

Home Browse Search

298 HYDROCARBON CONTAMINATED SOILS AND GROUNDWATER GROUNDWATER REMEDIATION LIMITSI 299 Each technique was applied to tine seven Lharacteristic data sets The two tech- iegression result% involved in selecting the asytriptotic region for this data set - niques were compared by a single criterion which technique wuld denio nstiatL i aie sumniaried in lable 17 1 The asymptotic- region is identified as theiperiod an existing asympttrtii_ condition will] tine least data hollowing day 157 1 he rLgicsuoii of the data subset following day 157 has tine following Lharacteristic.s (1) the apparent slope was not significantly different Method 1—Linear Asymptotic from zero, (2) both the best estimate of the slope and the standard error in the Slope were auniong the smallest, indurating trendlessness, and, as a LonsequenLc, Method I foLuses on the most recent portion of the time series data set, be- (3)regression had the smallest CV With 95% confidence, the slope is not greater cause that is the portion of tine record that is anticipated to be approaching an than 0 50 ppb/day asymptote This latter portion of the time series (e g , t > 150 days in Figure 17 I) is linear with a slope near zero(dCldt =s 0)and is referred to as the asymp- Method 2—Exponential Regression (otic region of tine time series reLord Method 1 seleLts a subset from the latter portion of the time series record and performs linear regression analysis of that In Method 2, tine data are assumed to be samples from an aquifer inl which subset to determine its asymptoticity In performing this linear regression of the tine conLentration is declining according to the following equation asymptotic region, it is assumed that the data fit an equation of the following form C = Cie-kt (2) C = a + bt (I) I where C„ = -on(critralion (ppb) when t equals 0, where a = conLentration intercept at t = 0 (ppb), e = 2 718, b = slope or trend of regression line[parts per billion per day(ppblday)E, k = first-order rate constant (day-'), and and t = time in days t = time (days) In such a system, the natural logarithm of the concentration, In C, would be ex- However, it is postulated that b is not significantly different from zero The iden- petted to be linear in time tification of the asymptotic region must be objective if the results of the statisti- cal analysis are to be reproducible An objective selection procedure was developed In C = In C„ -- kt (3) as follows Sequential linear regression analyses were performed on the most re- cent, consecutive data records (e g , subsets of the last five data points, then the It was hypothesized that the complete data record could be fit to this function last six data points, then the last seven data points, etc ) until the final data set With this hypothesis,the apparent asymptotic region was simply a region in which regressed included all the data Upper and lower 95% confidence intervals were concentrations were declining slowly, but the rate of decline could not be distin- calculated for the slope for each regression in the sequence For many regres- guished amid the noise associated with natural fluctuations,heterogeneity,or sam- sions, the upper bound was greater than zero and the lower bound was less than pling and analysis errors Benefits of this approach would be utilization of ail zero,signifying that the slope was not significantly different from zero For each the available data and the potential ability to anticipate a reduction in slope as regression with a slope not different from zero, the absolute values of the upper time increased and lower bound slopes were compared, and the larger was defined as a critical If the underlying aquifer contamination is declining according to Equation 2, value (CV) characterizing the trendlessness of the data subset for that regres- then the rate of decline decreases with time This function is asymptotic, but the sion The most trendless subset was the subset that had the smallest CV asymptote is zero If variability in the observed concentrations is large,the trend The uncertainty in a regression line slope decreases as more and more data may be obscured as the rate of decline decreases points are used in a regression, assuming the additional data points approximately By regressing the natural logarithm of concentration versus time and applying conform to a straight line When data from the nonasyniptotic, early region were classical rules of statistics and calculus,4 5 upper and lower confidence bounds included in regressions in the sequence,the absolute value and uncertainty in slopes on the slope were estimated based on Equation 2 Increased By finding the regression with the minimum upper bound on the slope The criterion used to compare the two methods was identification of trendless- (minimum CV), this procedure objectively selected the longest trendless subset ness at the earliest time(with the least data) Given a data set like that of Site I, of the data as defining the asymptotic, region which to most observers clearly exhibits an asymptotic tendency, which of these From the data set shown in Figure 17 1, the asymptotic region selected using methods rec.0gnl7es that the slope is insignificantly small at the earliest time(i e , the sequential regression procedure was the period from d-tv 157 to cI iv 969 The with the least data)? To answer this question each method was applied to each