A comprehensive environmental monitoring program based on a sound statistical design is necessary to provide estimates of the status of, and changes or trends in, the condition of ecological resources. A sampling design based upon a systematic grid can adequately assess the condition of many types of resources and retain flexibility for addressing new issues as they arise. The randomization of this grid requires that it be regular and retain equal-area cells when projected on the surface of the earth. After review of existing approaches to constructing regular subdivisions of the earth's surface, we propose the development of the sampling grid on the Lambert azimuthal equal~area map projection of the earth's surface to the face of a truncated icosahedron fit to the globe. This geometric model has less deviation in area when subdivided as a spherical tessellation than any of the spherical Platonic solids, and less distortion in shape over the extent of a face when used for a projection surface by the Lambert azimuthal projection. A hexagon face of the truncated icosahedron covers the entire conterminous United States, and can be decomposed into a triangular grid at an appropriate density for sampling. The geometry of the triangular grid provides for varying the density, and points on the grid can be addressed in several ways.

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We describe a statistical sampling approach to characterizing and comparing map projection distortion within irregular areas. Statistical measures of distortion, coupled with traditional distortion isoline maps, give a clear picture of map projection distortion for irregularly shaped areas, like the United States or portions of it. We calculate cumulative distribution functions and several descriptive statistics from the distortion measures. In our example, we compare two common projections, the Lambert azimuthal equal area and the Albers conic equal area. over the conterminous United States and over two subregions. In addition to scale and angle distortion, we develop a new measure of shape distortion. Our analyses show that the Lambert projection has lower mean and median shape distortion when compared over the conterminous U.S., whereas the Albers projection has a lower maximum distortion and distortion variance for all three distortion measures. The cumulative distribution functions are substantially different and show that the Lambert projection has lower distortion values for approximately 80% of the sample points. We also compare a large unrestricted random sample with a systematic random sample. The sample size is large enough that the unrestricted and systematic samples give virtually identical results.