Sampling scale effects in random fields and implications for environmental monitoring

The concept of a sampling scale triplet of spacing, extent and support is used to define the spatial dimensions of a monitoring network or a field study. The spacing is the average distance between samples, the extent is the size of the domain sampled and the support is the averaging area of one sample. The aim of this paper is to examine what is the bias and the random error (uncertainty) introduced by the sampling scale triplet into estimates of the mean, the spatial variance and the integral scale of a variable in a landscape. The integral scale is a measure of the average distance over which a variable is correlated in space. A large number of two dimensional random fields are generated from which hypothetical samples, conforming to a certain sampling scale triplet, are drawn which in turn are used to estimate the sample mean, spatial variance and integral scale. The results indicate that the biases can be up to two orders of magnitude. The bias of the integral scale is positively related to the magnitude of any of the components of the scale triplet while the bias of the spatial variance is different for different components of the scale triplet. All sampling scale effects are relative to the underlying correlation length of the variable of interest which is closely related to the integral scale. The integral scale can hence be used for sampling design and data interpretation. Suggestions are given on how to adjust a monitoring network to the scales of the variables of interest and how to interpret sampling scale effects in environmental data.