Stevens and Urquhart 2000

Stevens, D. L., Jr., and N. S. Urquhart. 2000. Response designs and support regions in sampling continuous domains. Environmetrics 11(1):13-41. (Paper)

In many environmental samples, the target population is distributed over space in a more or less continuous manner, e.g., the waters of a lake or the trees in a forest. Attributes of such a population can be conceptualized as a continuous function defined on the spatial domain of the population. Some attributes (e.g., water temperature) can be observed at a point; others (e.g., species diversity) can only be determined over a finite extent or support region. A fixed-shape support with uniform weights leads to an unbiased estimator of the population total; however, it may be impossible to maintain a fixed shape near domain boundaries. From a purely formal standpoint, unbiasedness can be maintained by using differential weights or by changing the shape of the support region near the boundary. Both of these procedures raise some issues of interpretation that often are overlooked. We derive estimators that account for edge effects under several support strategies, and identify some interpretation issues, using examples from forestry and limnology.