Kriging with External Drift in Model Localization

Abstract

When modelling a large area, models that can take into account the variation from the general mean in small sub-areas could perform better in prediction than a general model fitted to entire dataset. One method for adjusting the large-area models for such variation is kriging, in which the predictions are corrected with the aid of neighbouring observations. A variogram represents the spatial correlation between neighbouring observations as a function of distance. The predictions are obtained using a drift model that describes the general mean, and the selected variogram. The aim of this study was (1) to test for a spatial correlation in the residuals of a global form height model fitted over a large study area and (2) to use this correlation in prediction of the same variable. The dataset consisted of measurements from 19 175 Scots pines (Pinus sylvestris L.) from the 9th National Forest Inventory of Finland. Nested spherical and Bessel variograms were selected for the kriging calculations. In nested models the short-range (intra-stand) correlation and long-range (inter-stand) correlation are modelled separately. We used cross-validation to evaluate the variogram models selected. At the global level, 30 neighbours were needed for stable estimates, and with 60 neighbours the root mean squared prediction errors (RMSPE) of kriging were lower than those of the global model. At the regional level, we obtained better estimates than with regionally re-fitted models when the number of neighbours was 60 for both variogram models. The mean biases (i.e., average difference between actual and predicted values) at the regional level in the kriging predictions were small (0.8% of the regional RMSPE). In conclusion, there was an app. 6-km spatial correlation in the residuals, but due to relay effect the size of the kriging neighbourhood required for improving prediction was larger than the range.