Comparing sampling patterns for kriging the spatial mean temporal trend

Abstract

In monitoring the environment one often wishes to detect the temporal trend in a variable that varies across a region. A useful executive summary is then the temporal trend in the spatial mean. In this article, the best linear unbiased predictor of the spatial mean temporal trend and its variance are derived under a universal kriging model. Five different, spatially explicit sampling patterns are compared in terms of this variance. For small spatial ranges, the time-separation pattern is optimal, regardless of the temporal range. For larger spatial ranges, the best pattern depends on the temporal range. If the temporal range is small (less than ca. 1/20 of the spatial range), then the always-revisit pattern is best. For larger temporal ranges (between ca. 1/20 and ca. 1/4 of the spatial range) the serially alternating pattern is best, whereas the time-separation pattern is best for larger temporal ranges. The gain in using the serially alternating pattern or time-separation pattern instead of the always-revisit pattern can be substantial. The potential loss with respect to the always-revisit pattern is only minor for the serially alternating pattern, but can be substantial for the time-separation pattern. Unless one has good knowledge of the spatial and temporal range, the serially alternating pattern is thus a good choice. This work extends that by Urquhart and Kincaid and gives further support to their plea for the serially alternating pattern.