## C.4 Study Question 4: When will contaminant concentrations reach a criterion?

This question, associated with projecting future contaminant concentrations, is closely related to Study Question 5 and Study Question 7 regarding trends and attenuation rates. The attenuation rate determined for a chemical in a monitoring well (or for a data set representative of a group of monitoring wells) is useful for understanding how quickly concentrations are changing over time. The attenuation rate, estimated from existing monitoring data, can be used to predict concentrations in the future. The methods used to estimate how long it would take to reach a 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. could also be used to project concentrations at some future time.

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 autocorrelationCorrelation of values of a single variable data set over successive time intervals (Unified Guidance). The degree of statistical correlation either (1) between observations when considered as a series collected over time from a fixed sampling point (temporal autocorrelation) or (2) within a collection of sampling points when considered as a function of distance between distinct locations (spatial autocorrelation). between successive sampling events.
- Verify that significant temporal trends using time series plots.
- Ability to detect trends can be impacted by aggregating 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 in remedial methods.
- See also Section 4.1: Considerations for Statistical Analysis.

Statistical Methods and Tools for this Question

Estimating concentrations at a future time involves constructing a statistical model of chemical concentrations over time. Such models can reflect linear or nonlinear trends. These statistical models are closely related to attenuation rates and can be estimated by linear regression analysis (parametricA statistical test that depends upon or assumes observations from a particular probability distribution or distributions (Unified Guidance).) or a Theil-Sen trend line (nonparametricStatistical test that does not depend on knowledge of the distribution of the sampled population (Unified Guidance).).

- Linear regression assumes a normal distributionSymmetric distribution of data (bell-shaped curve), the most common distribution assumption in statistical analysis (Unified Guidance). for the residuals (that is, 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 is sensitive to outliers.
- Regression as a general tool provides flexible ways to develop models for your data. You may transform the data to be normally distributed using a log or other type of data transformation. In addition, regression can be used with a linear model, exponential model, or a multivariate model that includes multiple factors such as water table elevation in addition to time.

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 method does not require a normal distribution for the residuals.
- Theil-Sen line analysis 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 method 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 the estimated 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 in years, then the estimated attenuation rate is 0.25 yr^{-1}. Many software packages also provide a 95% confidence interval for the attenuation rate. This confidence interval is useful for evaluating the uncertainty associated with the estimated attenuation rate.

Interpretation of Results and Associated Uncertainty

Any prediction of future concentrations that is made using an attenuation rate estimated from past data implicitly relies on several assumptions. The key assumptions include:

- Future site conditions will be the same as past conditions (same remedy, same groundwater flow conditions).
- The attenuation rate is determined using an appropriate model. For example, if a linear model was used to determine the attenuation rates, then the attenuation is assumed to be monotonically decreasing along a straight line.

For most sites, it is unlikely that these assumptions will be completely satisfied. For example, matrix diffusion effects may cause the attenuation to deviate from first order. In this case, the rangeThe difference between the largest value and smallest value in a dataset (NIST/SEMATECH 2012). of future concentrations or cleanup times calculated from the 95% confidence interval of the attenuation rate should not be considered a true 95% confidence interval for the prediction. Instead, the calculated future concentrations and cleanup times are reasonable estimates based on the available data. The predictions should be interpreted in the context of 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)..

For any case where the 95% confidence interval for the attenuation rate includes zero (meaning the difference between the attenuation rate and zero is not statistically significant), any predictions made using the attenuation rate are highly uncertain.

See also Study Question 7,Section 4.5.1: Monitoring for Concentration Changes, Section 4.6.2: Trends Toward Compliance Criteriaand Section 5.9: Time Series Forecasting.

Related Study Questions

Study Question 3: Are concentrations above or below a criterion?

Study Question 5: Is there a trend in contaminant concentrations?

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

Key Words: Cleanup Time, Concentration trends, Attenuation Rate, Remediation, Monitoring, Closure

Publication Date: December 2013