Great expectations and even greater exceedances from spatially referenced data

2020 Elsevier B.V. We clear land for agricultural purposes, we draw water from streams and aquifers, and we build houses in coastal regions for their ready access to the sea. Our need for food, water, and shelter is basic, but so is variability in our natural environment. Understanding this is key to the long-term sustainability of the anthropocene. At any location in the environment, long-term temporal averages may indicate regions that exceed sustainability thresholds. For example, environmental thresholds might be used by insurance companies to set the prices of insurance premiums. In this article, we present the problem of spatial-statistical inference on exceedance regions, defined as a set of locations whose long-term environmental condition is above a given threshold. An example is given of rainfall in Paraná, Brazil, averaged over a period of three decades.