Design-based Generalized Least Squares estimation of status and trend of soil properties from monitoring data

Abstract

This paper introduces and demonstrates design-based Generalized Least Squares (GLS) estimation of spatial means at selected time points from data collected in repeated soil surveys with partial overlap, such as a rotating and a supplemented panel. The linear time trend of the spatial means can then be obtained as a linear combination of the estimated spatial means. The GLS estimator is the minimum variance linear unbiased estimator. Five space–time designs were compared under a first order autoregressive time series model for the spatial means, through the average and generalized sampling variances of the estimated spatial means, and through the sampling variance of the estimated linear trend. None of the designs scored best on all three quality measures. If the aim of soil monitoring is estimation of both status and trend, then these two conflicting aims must be prioritized in order to choose an efficient space–time design. The methodology is demonstrated by a case study on eutrophication and acidification of forest soils. The linear trends in the spatial means of pH and the ammonium and nitrate concentrations at three depths in the soil profile, as estimated from a rotational design with four sampling times at an interval of one year, were small and not significant. Exceptions were pH in the subsoil (− 0.06 pH units yr − 1 ) and ammonium in the middle soil horizon (− 0.086 mg N kg − 1 yr − 1 ). The linear trend is here defined as a linear combination of the true and unknown, but fixed spatial means. In quantifying the uncertainty of the estimated trend, only the sampling error in the estimated spatial means is accounted for. If there is a need to include uncertainty due to fluctuations of the true spatial means around a linear trend, then a super-population or time series model for the spatial means must be postulated which comprises a model error term. The linear trend is then defined as a model parameter, that can be estimated by Generalized Least Squares as in generalized linear modeling.