Representing multiple spatial statistics in generalized elevation uncertainty models: moving beyond the variogram

Spatial data uncertainty models (SDUM) are necessary tools that quantify the reliability of results from geographical information system (GIS) applications. One technique used by SDUM is Monte Carlo simulation, a technique that quantifies spatial data and application uncertainty by determining the possible range of application results. A complete Monte Carlo SDUM for generalized continuous surfaces typically has three components: an error magnitude model, a spatial statistical model defining error shapes, and a heuristic that creates multiple realizations of error fields added to the generalized elevation map. This paper introduces a spatial statistical model that represents multiple statistics simultaneously and weighted against each other. This paper's case study builds a SDUM for a digital elevation model (DEM). The case study accounts for relevant shape patterns in elevation errors by reintroducing specific topological shapes, such as ridges and valleys, in appropriate localized positions. The spatial statistical model also minimizes topological artefacts, such as cells without outward drainage and inappropriate gradient distributions, which are frequent problems with random field-based SDUM. Multiple weighted spatial statistics enable two conflicting SDUM philosophies to co-exist. The two philosophies are 'errors are only measured from higher quality data' and 'SDUM need to model reality'. This article uses an automatic parameter fitting random field model to initialize Monte Carlo input realizations followed by an inter-map cell-swapping heuristic to adjust the realizations to fit multiple spatial statistics. The inter-map cell-swapping heuristic allows spatial data uncertainty modelers to choose the appropriate probability model and weighted multiple spatial statistics which best represent errors caused by map generalization. This article also presents a lag-based measure to better represent gradient within a SDUM. This article covers the inter-map cell-swapping heuristic as well as both probability and spatial statistical models in detail.