Academic journal article Cartography & Geographic Information Systems

Statistical Modeling Uncertainties of Three-Dimensional GIS Features

Academic journal article Cartography & Geographic Information Systems

Statistical Modeling Uncertainties of Three-Dimensional GIS Features

Article excerpt

ABSTRACT. This article presents a newly developed statistical approach for modeling positional uncertainties of geometric features in three-dimensional geographical information systems (GIS). These features include points, line segments and linear features. The developed approach provides a confidence volume model for each feature type. The true locations of these measured features are included within these volumes with probability larger than a pre-defined confidence level. With the assumptions that positional errors of the two endpoints are independent and follow two three-dimensional normal distributions, the error of any point on the line segment is derived and described by the distribution of the point. The confidence volume model of a line segment is hence derived. Based on this model, the confidence volume model of linear features is further derived.


Three-dimensional geographical information systems (GIS) are developed for applications such as urban planning and management, geological subsurface modeling, mining, and oil exploration. In a three-dimensional GIS, an object can be described in horizontal and vertical spaces. This spatial coordinate system allows more than one object located at the same horizontal location to have different vertical positions.

Most of the current research about positional uncertainties of GIS features focuses on two-dimensional GIS; little research has been conducted for modeling uncertainties in three-dimensional GIS. Techniques in modeling uncertainties of points in one dimension and two dimensions in surveying can be directly adopted for GIS. This is because the nature of the problems for these situations is similar for GIS and conventional surveying. Uncertainties of two-dimensional lines have not been investigated very much in surveying. Investigation is, however, essential for GIS and cartography. Perkal (1966) proposed the epsilon-band model for describing uncertainty of two-dimensional lines. This model was further discussed and applied to describe error of thematic maps by Chrisman (1982). Dutton (1992) investigated error of two-dimensional lines using the Monte Carlo simulation approach. Caspary and Scheuring (1992) discussed random error of two-dimensional line segments using the error propagation law. Shi (1994) derived the confidence region model for two-dimensional line segments and probability distribution of line segments.

The nature of uncertainties for three-dimensional GIS features are of increasing interest, given the growth in three-dimensional GIS applications. This article presents results that are a further development of the confidence model (Shi 1994) in three-dimensional space.

Geometric Features of Three-dimensional GIS

Three-dimensional computer models can be classified into two major groups: surface-based and volume-based representations. Volume-based geometric representations describe the interior of objects by using solid information. The typical volume-based model may include three-dimensional array, needle model, octree, and constructive solid geometry. On the other hand, surface-based geometric representations describe geometric characteristics of objects by using micro surface cells or surface primitives. The typical surface-based models include grid, shape model, facet model, and boundary representation. This article deals with uncertainties in surface-based three-dimensional GIS.

In a surface-based three-dimensional GIS, the geometric features can be classified as points, line segments, and linear features. A point is the most primitive element in three-dimensional GIS. A line segment is composed of two endpoints. A geometric feature is composed of more than one line segment. When these line segments form a closed shape, it is called a polygon--a special linear feature. Accordingly, the scope of this article is identified as modeling positional uncertainties of points, line segments, and linear features. …

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