## C.7 Study Question 7: What are the contaminant attenuation rates in wells?

When contaminant concentrations are identified as having a trend over time (see Study Question 5), this question follows to estimate the rate of change over time (attenuation rate). You can then use the attenuation rate to evaluate whether the rate of decrease in concentration is adequate to achieve the site goals. In addition, the attenuation rate can be used to estimate future concentrations including when concentrations will reach a cleanup criterionGeneral term used in this document to identify a groundwater concentration that is relevant to a project; used instead of designations such as Groundwater Protection Standard, clean-up standard, or clean-up level. (see Study Question 4).

When using a data set from a single well or a set of wells in the source area, the attenuation rate determined from the concentration versus time data (sometimes called a "source attenuation rate") represents source depletion (Newell et al. 2002). In contrast, the decrease in concentration over distance downgradient of the source is called a "plume attenuation rate" (see Study Question 8).

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: Common Statistical Assumptions 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, or 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.

Determining whether two attenuation rates are different requires an evaluation of the uncertainty associated with each attenuation rate. The greater the difference in the attenuation rates relative to the uncertainty associated with each rate, the greater the confidence that the observed difference in the attenuation rates is real. See the case example in Appendix A.7.

Statistical Methods and Tools

Attenuation rates can be estimated 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). For groundwater monitoring data, exponential decay is a commonly observed long-term trend (Newell et al. 2002). Therefore, the attenuation rate is usually best represented by a first-order decay rate.

- Regression assumes a normal distributionSymmetric distribution of data (bell-shaped curve), the most common distribution assumption in statistical analysis (Unified Guidance). for the residuals (the variability not associated with the long-term trend is normally distributed). When this assumption is not satisfied, the accuracy of the results is reduced.
- Regression provides flexibility for the model fit to the data. Regression can be used with a linear model, exponential model, or a multivariate model that includes factors such as water table elevation in addition to time.
- Regression is sensitive to outliers.

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 (C_{t}=C_{0} e^{-kt}) to the concentration versus time data or by fitting a linear model for natural log concentration versus time data (ln(C_{t}) = ln(C_{0}) - kt). Many software packages also provide a 95% confidence intervalStatistical interval designed to bound the true value of a population parameter such as the mean or an upper percentile (Unified Guidance). for the slope of the model or the attenuation rate in the form described above. This confidence interval is useful for evaluating the uncertainty associated with the estimated attenuation rate. An example of regression applied to groundwater data is included in Appendix A.6.

- This test does not require a normal distribution for the residuals.
- This test is less sensitive than regression analysisA statistical tool for evaluating the relationship of one of more independent variables to a single continuous dependent variable (Kleinbaum et al. 2007). to outliers or extreme values.
- This test can only be used to evaluate linear trends (however, a first-order attenuation rate can be estimated by analyzing natural log concentration versus time data).

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 5.2.7 and Chapter 21.3, Unified Guidance. This confidence band is useful for evaluating the uncertainty associated with the estimated attenuation rate.

When comparing the attenuation rates for two different wells (or two data sets that each represent one or more wells) if the confidence bands for the attenuation rates do not overlap, then you can conclude with reasonable certainty that the attenuation rates are different. If the confidence bands do overlap, then you cannot conclude with confidence that the attenuation rates are different. Two examples of comparing attenuation rates are presented in Appendix A.7.

Interpretation of Results and Associated Uncertainty

When evaluating temporal trends in groundwater monitoring results, differences in results between wells are often present. Even at sites with overall decreasing contaminant concentrations, the trend analysis can identify some wells with statistically significant decreasing concentrations, some wells with apparently decreasing concentrations that are not statistically significant, and some wells with apparently increasing concentrations. In many cases, the apparent differences in concentration trends between wells 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 for multiple wells is determining whether these differences are due to random variability in monitoring results or due to true differences in attenuation between wells. 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 between wells, then it is likely that the observed differences are due to random variation. For a group of wells 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 medianThe 50th percentile of an ordered set of samples (Unified Guidance). 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).

When comparing attenuation rates, the evaluation of whether the attenuation rates are different is not an absolute yes or no. The results should influence the level of confidence in the conclusion. If the confidence bands for two attenuation rates almost overlap, then confidence in the conclusion that the attenuation rates are different should be lower than if the confidence bands are separated by a greater distance.

The 95% confidence band for the attenuation rate reflects the uncertainty associated with the estimate of the attenuation rate based on the variability in monitoring record that is not explained by the long-term trend. However, other sources of uncertainty may not be captured by the statistical analysis. For example, if a monitoring record was collected during an extended drought when the water table was dropping over time, then you may have less confidence that the observed rate of attenuation would continue during a subsequent period of normal precipitation. For an evaluation of whether two attenuation rates are different, consider the statistical analyses discussed, the complete 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.

See also Study Question 7, Section 4.5.1: Monitoring for Concentration Changes, and Section 4.6.2: Trends Toward Compliance Criteria.

Related Study Questions.

Study Question 8: How do contaminant concentrations change with distance from the source area?

Key Words: Attenuation, Contaminant attenuation rate, changing contaminant concentrations, Remediation, Monitoring, Closure

Publication Date: December 2013