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An error model for quantifying the magnitudes and variability of errors generated in the areas of polygons during spatial overlay of vector geographic information system layers is presented. Numerical simulation of polygon boundary displacements was used to propagate coordinate errors to spatial overlays. The model departs from most previous error models in that it incorporates spatial dependence of coordinate errors at the scale of the boundary segment. It can be readily adapted to match the scale of error-boundary interactions responsible for error generation on a given overlay. The area of error generated by overlay depends on the sinuosity of polygon boundaries, as well as the magnitude of the coordinate errors on the input layers. Asymmetry in boundary shape has relatively little effect on error generation. Overlay errors are affected by real differences in boundary positions on the input layers, as well as errors in the boundary positions. Real differences between input layers tend to compensate for much of the error generated by coordinate errors. Thus, the area of change measured on an overlay layer produced by the XOR overlay operation will be more accurate if the area of real change depicted on the overlay is large. The model presented here considers these interactions, making it especially useful for estimating errors studies of landscape change over time.
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Introduction
Geographic information systems (GIS) have become standard tools for investigators in virtually every segment of land resource planning and management. In spite of this widespread reliance on GIS products and analysis, relatively little attention has been given to assessing the accuracy of spatial databases (Chrisman 1987; Prisley, Gregorie, and Smith 1989; Keefer, Smith, and Gregorie 1991; Magnussen 1996; Zhang and Goodchild 2002). The overlay of multiple data layers is among the fundamental data transformation procedures available in GIS (Newcomer and Szajgin 1984). Because all source maps contain errors that accumulate in the final overlay product, the accuracy of overlay maps can potentially be degraded to the extent that they become useless for decision making (MacDougall 1975; Bailey 1988). Although reasonable map accuracy can be achieved through the careful use of modern techniques, the propagation of errors through the overlay process is poorly understood (Chrisman 1987; Veregin 1995). This article presents a new approach in quantifying error propagation during the spatial overlay of digital maps based on the arc-node vector data structure.
Most error models for vector data take a bottom-up approach, in which coordinate errors for individual points are evaluated and then propagated to more complex objects. Point-associated coordinate errors are introduced during database creation from a range of sources, including digitizing and line interpolation errors (Magnussen 1996), fuzzy boundary transitions (Edwards and Lowell 1996), distortion of the data medium (MacDougall 1975), and survey control errors (Chrisman 1987). All error sources contribute to the total uncertainty in point coordinates (Skidmore and Turner 1992; Zhang and Goodchild 2002). Contributing coordinate errors can be combined as the distance root mean square (dRMS), which is equal to the square root of the sum of the contributing error variances (Siouris 1993), and expressed in terms of error ellipses, or regions of uncertainty surrounding points. The error ellipse concept may be extended to lines by defining epsilon bands, which provide a basis for defining the widths of fuzzy boundaries (Blakemore 1984; Skidmore and Turner 1992). Various models have been proposed for propagating uncertainties in point positions to uncertainties in the lengths of lines or areas of polygons (Chrisman and Yandell 1988; Griffith 1989; Prisley, Gregorie, and Smith 1989; Skidmore and Turner 1992; Magnussen 1996; Zhang and Goodchild 2002).
A difficulty common to all error models of this type is that the characteristics of error propagation from points to more complex objects depends on the nature of spatial correlation between points. Some models, for example Chrisman and Yandell (1988) and Griffith (1989), are based on the assumption that all coordinate errors are independent, despite the fact that the assumption is widely regarded as being incorrect. Where spatial dependence is considered, it is invariably limited to correlations between coordinate errors at the scale of adjacent vertices and/or is vaguely defined. Keefer, Smith, and Gregorie (1991) modeled the spatial auto-correlation of digitizing errors as a first-order autoregressive process. Each digitized point along an arc was considered to be serially correlated with its closest neighbors only. In addition, the autoregressive model requires the specification of parameters for which no general guidance exists. The area variance model of Prisley, Gregorie, and Smith (1989) includes a fine-scale point correlation parameter, although the authors acknowledge that methods to determine its value have yet to be developed. An "arbitrary" autocorrelation model presented by Magnussen (1996) provides an interesting, but qualitative, description of how fine-scale error correlation effects might be incorporated in an area variance estimator.
None of the models noted above accommodates bias in the coordinate errors. The expected error in the areas of polygons on a single layer with unbiased coordinate errors is zero, and the models focus on estimating the variances of error in polygon area (Prisley, Gregorie, and Smith 1989; Magnussen 1996) or errors in boundary length (Keefer, Smith, and Gregorie 1991; Skidmore and Turner 1992). Zero error in polygon area, however, does not imply that polygon boundaries are positioned correctly. While this may be of little significance when extracting planimetric data from a single map layer, it is an important source of error when the layer is used in an overlay operation. Slivers, small polygons generated by the misalignment of polygon boundaries on different input layers, are perhaps the most common manifestation of error in vector overlay analysis (Bailey 1988; Chrisman 1989; Veregin 1989). Typical treatment of slivers includes merging polygons smaller than some threshold size with neighboring polygons, even though such procedures may not be appropriate (Edwards and Lowell 1996). Methods for assessing accuracy of...
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