Both this article, referred to as Article I, and another one, Article II, entitled “Geometric Quality in Geographic Information IFSAR DEM Control”, published in this encyclopedia propose a method to evaluate the geometric integrity of Digital Elevation Models obtained by different techniques. Therefore, the theoretical aspect of the method to evaluate the geometric integrity and different stochastic hypotheses will be presented in both of them. Herein, we consider the classical topographic or aerial photogrammetry stereo images method (included ASTER or SPOT images) and we assume consistent stochastic hypotheses. In the Article II we consider Interferometry SAR (IFSAR) techniques and the stochastic hypotheses are specific according to the particular geometry involved in this technique.
In the literature, several unsatisfactory solutions were proposed for the DEM control with respect to a reference. A critical problem in the error estimation (evaluated using the difference referred to in the previous paragraph) is to establish for each selected point of the DEM the corresponding homologous point in the R-DEM. Other kinds of problems and errors are related to the existence of aberrant points, systematization, etc. These problems were studied for horizontal errors in maps in (Abbas, 1994; Grussenmeyer, 1994; Hottier, 1996a). These authors found that the dissymmetry model-reference was the most important factor to determine homologous pairs.
Several solutions have been proposed for the ‘punctual control method’ (recognition algorithms, filtering methods, adjustment of histograms to theoretical laws) without obtaining completely satisfactory results (Dunn et al, 1990; Lester & Chrisman, 1991). Later, (Abbas, 1994; Grussenmeyer, 1994; Hottier, 1996a) present an alternative to the punctual control method: the ‘linear control method’ based on the dissymmetry of the Hausdorff distance.
Habib (1997) analyzes precision and accuracy in altimetry and mentions some of the proposals of the last decade for the elevation control of quality.
In the case of the DEM’s, to assess the difference that gives rise to the error we wish to compute, we need to identify without ambiguity each point M in the DEM with its homologous point P in the R-DEM.
Two reasons make this task difficult:
Many points M are not identifiable, those situated on regular sides, which are indistinguishable from their neighbors. Potentially identifiable points are those located on sharp slope variations, and possibly those with zero slope (tops, bottoms and passes).
A point identifiable on the DEM is not necessarily identifiable in the R-DEM, because of a difference in scale (“generalization”) or aberrant errors.
Key Terms in this Chapter
Digital Elevation Model (DEM): Set of points in a 3 dimensional coordinate system modeling a real object surface.
Spatial Data: information related to a real system involving geometric (dimension, shape) and topological (relative position) aspects of it.
Covariance Matrix: the square n x n of which the entries are the pair wise correlations of the variables of a random vector of length n; the (i, j) th entry is the correlation between the ith and the jth variables.
Stereo Images: two or more images of a real object’s surface obtained from different points of view.
Revolution Ellipsoid: The 3 dimensional solid obtained from revolving an ellipse about one of its axes.
Accuracy: The degree of conformity of a measured or calculated quantity to its actual (true) value. Accuracy is the degree of veracity.
Variance (Estimation): The average of the sum of the squares of the differences of the values attained by a random variable from it’s mean.
Geographic Information System (GIS): Tool for processing localized information. A GIS will model and locate the spatial data of a real phenomenon.
DEM Geometric Quality: The geometric precision measured in terms of the difference between a digital elevation model (DEM) and a reference DEM (R-DEM).
Photogrammetry: Technique permitting the set up of the coordinates of points (DEM) of an object surface by processing stereo images of the object surface.
Homologous Points: The point in the DEM and the point in the R- DEM modeling the same point in the real surface. Their distance is the error of the DEM point, assuming the points in the R-DEM are without error.
Precision: The degree of reproducibility or repeatability. The measure to which, further measurements or calculations show the same or similar results. In many cases precision can be characterized in terms of the standard deviation of the measurements, sometimes incorrectly called the measurement process’s standard error. The results of calculations or a measurement can be accurate but not precise; precise but not accurate; neither; or both. A result is called valid if it is both accurate and precise.