# 7.2.6 Decision Mechanism 6: Extrapolating from Sampled to Unsampled Areas

Extrapolation from a sampled area to an unsampled area requires an assumption that the distributions of contamination in the unsampled areas are sufficiently similar to the sampled areas.

This decision mechanism entails using estimates of the mean obtained from areas where ISM samples are taken to make decisions regarding other DUs that are unsampled. The fundamental assumption made with this mechanism is that the distributions of contaminant concentrations in the unsampled areas are essentially the same as in the sampled areas. The most common rationale for this assumption is that the source of contamination, mechanism(s) of transport, etc. are similar for each of the areas and that these conditions should lead to similar levels of contamination and similar variances. This decision mechanism is typically considered when large tracts of land or large volumes of soil must be assessed with a limited budget.

The key to this decision mechanism is confidence that the fundamental assumptions are valid and that there are no significant differences in contaminant distribution among the sampled and unsampled areas. In the absence of data to verify the assumption, that confidence is subjective. There is nothing unique about ISM that enables this extrapolation with reduced uncertainty—the same issue of whether or not to extrapolate exists whether the sampled areas are evaluated with ISM or discrete samples. Based on feedback obtained in development of this report, this decision mechanism is not acceptable for many states.

A distinction should be made between extrapolation between DUs and extrapolation within a DU. It is sometimes suggested that because there is precedence for using results from discrete samples to make inferences about unsampled areas within a DU, the same uncertainty applies to ISM. In this context, there is a difference between how information from discrete and ISM data may be used. With discrete data, spatial interpolation methods (e.g., geostatistics, inverse distance weighting) or discretization methods (e.g., Thiessen polygons) can be used to provide more reliable estimates of the mean and standard deviation throughout the DU. These methods also have the advantage of using information across DUs (i.e., when a site is split into multiple DUs) to derive estimates of the mean and standard deviation within each individual DU. With ISM, this degree of spatial resolution is lost because the increments are composited, so there is no basis for estimating concentrations in subareas of the DU or for developing a mathematical model that uses data from across the DUs. One exception would be for a site that is divided into many DUs—if a sufficient sample size is available, each estimate of the mean may be considered representative of a portion of the site such that spatial patterns and interpolation method may be explored.

A variation on this approach is to collect replicates in subset of the DUs and extrapolate the estimate of the variance (or the CV) to DUs with a single ISM sample. Although this approach appears to be a less uncertain way to extrapolate findings among DUs, the extent to which the distributions may be comparable across DUs must be considered. The chance that the distributions differ among DUs increases as the number of sources and the complexity of the contaminant transport mechanisms increase. In addition, sites with multiple subareas of elevated concentrations can be expected to introduce inherent variability within and between DUs, making a successful extrapolation of the variance more difficult. In general, the greater the number of DUs where replicate ISM samples are collected, the more likely that the average measure of variance will be representative of DUs with single ISM results (see Section 4.2).

As noted in Section 4, it is unclear whether the appropriate statistic for extrapolation is the SD or CV. The CV is preferred if it can be reasonably assumed or demonstrated that there is a positive correlation between the mean and SD. Based on the proportionality effect, the mean and SD are expected to be positively correlated for positively skewed distributions (Goovaerts 1997). If replicate data are available for multiple DUs, plots of the SD vs. the mean should be developed to explore patterns in the relationship between the sample statistics.

A related situation exists when a DU is subdivided into SUs and only a fraction of the SUs are actually sampled. In this approach, the results from each of the sampled SUs are compared with the action level(s). If all are lower than the action level(s), the entire DU passes. The same assumptions and considerations discussed above apply in this situation as well. If one or more SUs are above the action level, the DU does not pass, and the systematic planning team should be reconvened to plan the next steps, which may include additional sampling.