Inferences from fluctuations in the local variogram about the assumption of stationarity in the variance

Geostatistics is commonly used to describe and predict the variation of soil properties over the landscape. However, many geostatistical methods require the assumption that our observed data are a realization of a random function which is intrinsically stationarity. Under stationarity, observations of a single realization of the random function at different positions can be treated as a form of replication. There are various ways in which a random function may breach the assumption of intrinsic stationarity and numerous geostatistical techniques have been developed that are able to cope with some forms of non-stationarity. What is currently needed is a set of diagnostic tools capable of detecting and identifying when data cannot plausibly be treated as a realization of a process which is stationary in the variance.In this paper, we propose an inferential method that can identify when stationarity in the variance cannot plausibly be assumed. The basis of our approach is to obtain a model for the random function under the assumption of intrinsic stationarity. If the global dataset can be regarded as a realization of a Gaussian process (perhaps after transformation), then the global variogram is sufficient for this purpose. By using a window-based method to locally estimate variograms, we can define some statistic of homogeneity of the sample variation of the data. This allows us to obtain a sample distribution for this statistic, under the null hypothesis of intrinsic stationarity, by generating multiple realizations of the postulation random function at the original sample points using Monte Carlo methods and recomputing the statistic for each realization. We selected as statistics the interquartile ranges of: i) the spatial dependence ratio (s), the proportion c₁ / (c₀ + c₁), ii) a distance parameter (a), which is the maximum lag over which the random function is autocorrelated for variograms like the spherical, and iii) the local variances (v; c₀ + c₁), where (c₀) is the nugget component and (c₁) the spatially structured component. We demonstrated this method using data from the large scale sampling (n = 1341 over 8248 km²) of the Florida Everglades, United States.