Continuous surface representation and approximation: spatial analytical implications

Field-based continuous representation in a geographical information system (GIS) has long been important for describing complex spatially distributed phenomena, such as population, precipitation, air pollution, temperature elevation and land cover. Though theoretical knowledge and properties of continuous distributions can be employed, such surfaces are generally approximated or abstracted in practice due to a lack of complete information. That is, such surfaces are based on finite spatial samples, which is a practical necessity with regard to the infinite underlying attribute variability. These approximated surfaces are then used in various spatial analyses, yet impacts are not well understood. This article will examine theoretical properties and errors that result in practice when approximated continuous surfaces are relied on in spatial analysis.