The geostatistical analysis of experiments at the landscape-scale

Abstract

In conventional field experiments inherent variability is managed by design. In all cases the inherent variation is treated as additive random variables (a residual and any block and/or covariate effects). This assumption is generally reasonable in most field experiments where our basic units are plots within a relatively small and uniform region of a field. It is less plausible when we wish to conduct experiments across heterogeneous landscapes, with inherent variation of the soil over a wide range of many variables. In this paper we present a geostatistical approach to the design and analysis of landscape-scale experiments. We no longer regard the treatment response as a fixed effect, but rather as a random variable. Therefore, at any target site we can estimate the response to different treatments and treatment contrasts. This could be done by ordinary kriging, or by cokriging after we model the treatment responses as coregionalized variables. The advantage of cokriging is that contrasts and their confidence limits may be estimated optimally, and are coherent with estimates of the responses (i.e. the difference between two optimal estimated responses is the optimal estimate of the contrast between them). Since only one treatment response can be observed at a particular location we use the pseudo cross-semivariogram [Papritz, A., Kunsch, H.R., Webster, R., 1993. On the pseudo cross-variogram. Mathematical Geology 25, 1015–1026] to model the cross-covariances of the responses. We compared ordinary kriging and three variants of cokriging; standardized ordinary cokriging, ordinary cokriging and the method of Papritz and Fluhler [Papritz, A., Fluhler, H., 1994. Temporal change of spatially autocorrelated soil properties: optimal estimation by cokriging. Geoderma 62, 29–43]. Standardized ordinary cokriging was found to be the best predictor of the treatment contrast and responses under different levels of spatial auto-correlation and structural correlation, and different experimental designs. The results showed that the plots of different treatments should be placed as close together as possible to give the best predictions of treatment contrasts. A field-scale nitrogen-response experiment was used to illustrate the proposed methods.