Variance Estimation for Spatially Balanced Samples of Environmental Resources

The spatial distribution of a natural resource is an important consideration in designing an efficient survey or monitoring program for the resource. We review a unified strategy for designing probability samples of discrete, finite resource populations, such as lakes within some geographical region; linear populations, such as a stream network in a drainage basin; and continuous, two-dimensional populations, such as forests. The strategy can be viewed as a generalization of spatial stratification. In this paper, we develop a local neighborhood variance estimator based on that perspective, and examine its behavior via simulation. The simulations indicate that the local neighborhood estimator is unbiased and stable. The Horvitz-Thompson variance estimator based on assuming independent random sampling (IRS) may be two times the magnitude of the local neighborhood estimate. An example using data from a generalized random-tessellation stratified design on the Oahe Reservoir resulted in local variance estimates being 22 to 58 percent smaller than Horvitz-Thompson IRS variance estimates. Variables with stronger spatial patterns had greater reductions in variance, as expected 1. Introduction Environmental studies invariably involve populations distributed over space. Traditionally, such studies tended to focus on relatively small and well-delimited systems. However, some of the environmental issues that we face today, such as global warming, long-range transport of atmospheric pollutants, or habitat alteration, are not localized. Understanding and quantifying the extent of symptoms of widespread concerns requires large-scale study efforts, which in turn needs environmental sampling techniques and methodology that are formulated to address regional, continental, and global environmental issues. Stehman and Overton (1994) give an overview of some statistical issues associated with environmental sampling and monitoring, and Gilbert (1987) has an extensive discussion of sampling methods for monitoring environmental pollution.