C.8 Study Question 8: How do contaminant concentrations change with distance from the source area?
This question continues the progression of Study Question 7. Study Question 7 characterizes the temporal trends at different wells. Study Question 8 expands the comparison to the overall mass of a plume as you move away from the source area. As discussed in Study Question 7, comparing concentration trends in different wells is useful for evaluating the existence of spatial differences in attenuation rates and if concentrations are changing as you move further from the source area at a specific time. However, the question often involves the distance the plumes extends, and the overall behavior of the plume. If the slopes in concentration at each well are not statistically different, then the magnitude of those trend lines represents a consistent rangeThe difference between the largest value and smallest value in a dataset (NIST/SEMATECH 2012). of the plume attenuation rate. The rate of change of concentrations over distances is calculated by concentration difference over the distance. If there is not homogeneity of trends of wells along the centerline of the plume, then quantifying the decrease of contaminant concentrations as the plume moves downgradient of the source area becomes a complex spatial statistical problem. Comprehensive spatial analysis of change in concentration requires geostatisticsA branch of statistics that focuses on the analysis of spatial or spatiotemporal data, such as groundwater data (Gilbert 1987)., which is beyond the scope of this document.
The use of mass flux across an aquifer, however, is a better method to estimate the characteristics of the plume (Feenstra, Cherry, and Parker 1996). The uncertainty of the spatial variations of concentrations are handled by quantifying the mass discharge at transects of a plume as described in the ITRC document Use and Measurement of Mass Flux and Mass Discharge (ITRC 2010). This method appropriately gives manageable values of the mass discharge for an area of the plume. Then by evaluating the mass discharge between transects of a plume, you can understand the trend of the contaminant concentration over distance. The attenuation rate of the plume is the slope of the differences in the mass discharge between transects.
Comparing the trends in mass discharge across transects of a plume as you move further from the source area is similar to comparing the attenuation rates between wells (see Study Question 7). Determining whether two attenuation rates are different requires an evaluation of the uncertainty associated with each attenuation rate. If the slopes to the mass discharge trends of a plume are not statistically different as contamination moves away from the source area, then the slope of the trend lines will represent the range of the attenuation rate for the overall plume.
This question is usually relevant in the remediation, monitoring, and closure stages of the project life cycle.
Selecting and Characterizing the Data Set
Verify that the data set can support trend analyses and modeling. Refer to Section 3.4 for further discussion of how the following requirements may impact statistical analysis results.
- Check for outliersValues unusually discrepant from the rest of a series of observations (Unified Guidance). using box plots, probability plots, Dixon’s Test, and Rosner’s Test.
- Check for autocorrelation between successive sampling events.
- Ability to detect trends can be impacted by pooling data across wells.
- In general, you can obtain better detection of trends using longer records of data, but in many cases attenuation rates will differ based on remedial methods.
- See also Section 4.1: Considerations for Statistical Analysis.
When you evaluate multiple time intervals from a single monitoring record in order to identify changes in attenuation rates, be sure to evaluate the uncertainty in the attenuation rate estimates (the confidence bands) in order to determine whether an apparent difference in attenuation rates is most likely to be associated with true change in the source attenuation rate or an artifact of shorter-term random variability.
ITRC 2010 discusses the selection of points to define the transect and the calculation of the mass discharge. Each transect becomes a unique data point along the longitudinal axis of the plume, instead of looking at one location over time. This is a data intensive process, where at least four transects are needed to calculate the slope over the distance between transects.
Statistical Methods and Tools
The statistical methods of estimating the attenuation rates from monitoring wells along the centerline can be done by using parametricA statistical test that depends upon or assumes observations from a particular probability distribution or distributions (Unified Guidance). or nonparametricStatistical test that does not depend on knowledge of the distribution of the sampled population (Unified Guidance). methods. Both methods require assumptions about the concentration trend for example zero order (linear trend over time) or first order (exponential decay). See Study Question 7 for details.
- Regression is an easy procedure that shows the relationship between two variables (distance and mass discharge).
- The slope of the trend line is an estimate of the change in the meanThe arithmetic average of a sample set that estimates the middle of a statistical distribution (Unified Guidance). of the mass discharges over distance between transects.
- To apply a linear regression appropriately, the relationship must be linear (monotonic and noncyclical trends to the mass discharge).
Regression is an easy procedure to apply and shows the relationship of pairs of data (time and concentration) to obtain a fit to a model (such as for linear regression, the slope and intercept of a line). A best estimate of the first-order attenuation rate (k) can be obtained by fitting a first-order decay model (Ct=C0 e-kt) to the concentration versus time data or by fitting a linear model for natural log concentration versus time data (ln(Ct) = ln(C0)-kt). With the use of a bootstrapping method, many software packages also provide a 95% confidence band for the attenuation rate as described in Section 5.5.3 and Chapter 21.3.1, Unified Guidance. This confidence intervalStatistical interval designed to bound the true value of a population parameter such as the mean or an upper percentile (Unified Guidance). is useful for evaluating the uncertainty associated with the estimated attenuation rate.
- The slope of the trend line is a measurement of the change in the medianThe 50th percentile of an ordered set of samples (Unified Guidance). of the mass discharges over distance between transects.
- The trend line is constructed by combining the median pair-wise slope with the median mass discharge and distance between transects.
When the Theil-Sen trend line is used for a data set of natural log concentration versus time, the estimated slope is the negative of estimated the first-order attenuation rate with units of time-1. In other words, if the slope is -0.25 and the time units for the data set is years, then the estimated attenuation rate is 0.25 yr-1. With the use of a bootstrapping method, many software packages also provide a 95% confidence band for the attenuation rate as described in Section 5.5.3 and Chapter 21.3.1, Unified Guidance.
Interpretation of Results and Associated Uncertainty
You can estimate the change in concentration over time (attenuation rate). Regression analysis (parametric) or Theil-Sen trend line (nonparametric) can be used to estimate attenuation rates. Then, by calculating a confidence interval around the median at a point in time, you can generate a nonparametric confidence band around the attenuation rate with the use of bootstrapping (see Section 5.5.3 and Chapter 21.3.1, Unified Guidance). This approach provides an estimate of the uncertainty you have with the magnitude of the slope.
If the confidence bands do not overlap, then the attenuation rates are statistically different (see Study Question 7 for details). By dividing the concentration difference by the distance between the wells you can estimate the concentration change over distance.
In order to compare the attenuation rates for two different transects, estimate the attenuation rates and provide confidence bands for the attenuation rates. If the confidence bands do not overlap, then you conclude that the attenuation rates are statistically different and you will need addition geostatistics methods to evaluate the overall plume.
The confidence band around trend lines reflects the uncertainty associated with the estimate of the attenuation rate. The uncertainty associated with the attenuation rate depends on a number of factors including the length of the monitoring record and the magnitude of variability not associated with the long-term trend (relative to the magnitude of the long-term trend). In data sets with higher variability and shorter monitoring, records will have more uncertainty (larger confidence bands) compared to data sets with lower variability and longer monitoring records.
When evaluating trends in groundwater monitoring results, differences in results between wells and different plume areas are often present. Even at sites with overall decreasing contaminant concentrations, the trend analysis can identify some wells or plume areas with statistically significant decreasing concentrations, some wells or plume areas with apparently decreasing concentrations that are not statistically significant, and some wells or plume areas with apparently increasing concentrations. In many cases, the apparent differences in concentration trends between wells or plume areas can be attributed to random variability in the monitoring data rather than real differences in attenuation rates between wells.
A key challenge in the evaluation of concentration trends is determining whether these differences are due to random variability in monitoring results or due to true differences in attenuation. This determination should be based on lines of evidence such as:
- Are the differences in attenuation rate statistically significant? If the 95% confidence bands for the attenuation rates overlap, then the difference is not significant.
- Does the variation in attenuation rates exhibit a spatial pattern? In other words, are wells with increasing concentrations or slower attenuation clustered together? Are wells with faster attenuation clustered together?
- Is there a potential mechanistic explanation for the observed differences? For example, are the wells with faster attenuation rates located closer to an active remediation system? Or are the wells with slower attenuation rates screened in lower permeability soils?
Unless a majority of the lines of evidence suggest true differences in attenuation rates, then it is likely that the observed differences are due to random variation. For wells or plume areas without clear differences in attenuation rates, the best estimate of the overall plume attenuation rate can be obtained by either using a midpoint attenuation factor such as the average or median attenuation factor or evaluating the attenuation rate for groundwater concentrations that are representative of the group of wells (such as the average, median, or maximum concentration for each monitoring period).
An evaluation of whether two attenuation rates are different should include consideration of the statistical analyses discussed, the site conceptual site model (CSM)A living collection of information about a site which considers factors such as environmental and land use plans, site-specific chemical and geologic conditions, and the regulatory environment (ITRC 2007b). and any other available information that may be relevant to the determination.
Related Study Questions
Study Question 7: What are the contaminant attenuation rates in wells?
Key Words: Attenuation, Remediation, Monitoring, Closure
ITRC. 2010. "Use and Measurement of Mass Flux and Mass Discharge." In MASSFLUX-1. Washington, D.C.: Interstate Technology & Regulatory Council. https://itrcweb.org/Guidance/ListDocuments?topicID=14&subTopicID=11.
Feenstra, S., J.A. Cherry, and B.L. Parker. 1996. "Conceptual Models for the Behavior of DNAPLs in the Subsurface." In Dense Chlorinated Solvents and Other DNAPLs in Groundwater. Pankow, J.F. and J.A. Cherry, Eds. Portland OR: Waterloo Press.
Publication Date: December 2013