Uncertainty of spatial averages and totals of natural resource maps

Global, continental and regional maps of concentrations, stocks and fluxes of natural resources provide baseline data to assess how ecosystems respond to human disturbance and global warming. They are also used as input to numerous modelling efforts. But these maps suffer from multiple error sources and, hence, it is good practice to report estimates of the associated map uncertainty so that users can evaluate their fitness for use. We explain why quantification of uncertainty of spatial aggregates is more complex than uncertainty quantification at point support because it must account for spatial autocorrelation of the map errors. Unfortunately, this is not done in a number of recent high-profile studies. We describe how spatial autocorrelation of map errors can be accounted for with block kriging, a method that requires geostatistical expertise. Next, we propose a new, model-based approach that avoids the numerical complexity of block kriging and is feasible for large-scale studies where maps are typically made using machine learning. Our approach relies on Monte Carlo integration to derive the uncertainty of the spatial average or total from point support prediction errors. We account for spatial autocorrelation of the map error by geostatistical modelling of the standardized map error. We show that the uncertainty strongly depends on the spatial autocorrelation of the map errors. In a first case study, we used block kriging to show that the uncertainty of the predicted topsoil organic carbon in France decreases when the support increases. In a second case study, we estimated the uncertainty of spatial aggregates of a machine learning map of the above-ground biomass in Western Africa using Monte Carlo integration. We found that this uncertainty was small because of the weak spatial autocorrelation of the standardized map errors. We present a tool to get realistic estimates of the uncertainty of spatial averages and totals of natural resource maps. The method presented in this paper is essential for parties that need to evaluate whether differences in aggregated environmental variables or natural resources between regions or over time are statistically significant.