The Photogrammetric Record 21(116): 342–354 (December 2006) LOCAL ACCURACY MEASURES FOR DIGITAL TERRAIN MODELS Karl Kraus Institute of Photogrammetry and Remote Sensing Vienna University of Technology, Austria Wilfried Karel (wk@ipf.tuwien.ac.at) Christian Briese (cb@ipf.tuwien.ac.at) Gottfried Mandlburger (gm@ipf.tuwien.ac.at) Christian Doppler Laboratory for ‘‘Spatial Data from Laser Scanning and Remote Sensing’’ at the Institute of Photogrammetry and Remote Sensing Vienna University of Technology, Austria Abstract Airborne laser scanning and image matching can today form the basis for the generation of digital terrain models (DTMs). In addition to the DTM, quality parameters are needed that describe the accuracy at a high level of detail, at best for every interpolated DTM point. Furthermore, other parameters are of interest, for example, the distance from each DTM point to the data point next to it. This paper presents a method to derive accuracy measures from the original data and the DTM itself. Its application is demonstrated with an example. The quality measures are suitable for informing users in detail about DTM quality and warning them of weakly determined areas. Keywords: accuracy, digital terrain model, laser scanning, precision, quality Introduction Digital terrain models (DTMs) form a fundamental data-set in many geographical information systems (GIS). Unfortunately, the quality of these DTMs is rarely communicated to GIS users (Wood and Fisher, 1993). During the past decade, interest in DTM generation was stimulated through the development of airborne laser scanning (ALS). In the process of DTM creation from ALS data, the elimination of off-terrain points on buildings, trees and bushes is of crucial importance. This removal is also described as filtering (Sithole and Vosselman, 2004). This may result in large data voids of which users must be made aware. In a similar way, matching procedures in digital stereo photogrammetry encounter difficulties with off-terrain objects. Furthermore, it has to be considered that the point density decreases significantly in poorly textured areas. The best way of informing users about DTM quality is by means of accuracy measures. Examples for global DTM accuracy measures can be found in Li (1993). However, global Professor Karl Kraus died in Berlin on 5th April 2006 while this paper was still under review after revision. An obituary is published elsewhere in this issue. Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. Blackwell Publishing Ltd. 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street Malden, MA 02148, USA. The Photogrammetric Record accuracy measures are not satisfactory as they provide only one parameter for a rather large area. This is because, for derived quantities such as volumes, lines of equal slope or boundaries of flood risk areas, the spatial variation of quality is needed. For example, a quality value may be given for every point of the regular grid which frequently forms the output of a DTM computation. At best, this grid is enhanced with structural information, leading to a hybrid DTM. The advantages of a hybrid DTM over a DTM consisting only of a triangular irregular network (TIN) have been published recently (Ackermann and Kraus, 2004). The calculus for DTM generation is the interpolation. McCullagh (1988) compiled and analysed all well-established methods of that period. The accuracy of every interpolated point may be deduced by error propagation during the DTM generation process. In the theory of least squares adjustment, this accuracy is called precision (Mikhail, 1976). However, the precision derived this way does not help to increase the confidence of DTM users, because they perceive these algorithms as ‘‘black boxes’’. This paper presents a simple method for the estimation of the accuracy at every grid point, which considers all influencing factors. On the demand of the DTM user these factors can also be visualised. Thus this approach provides confidence to DTM users. It may also be used to analyse existing DTMs. The following factors influence the accuracy of the interpolated grid points: (a) (b) (c) (d) the the the the number and alignment of the neighbouring original points, whereby distance to the respective grid point is of special importance, terrain curvature in the neighbourhood of the grid point,* and accuracy in height of the original points. These factors form the input of a simple interpolation method for the estimation of the accuracy of a grid point. As stated above, in least squares theory this accuracy is called precision. From the point of view of DTM users this term sounds unfamiliar as they are accustomed to the term relative accuracy. In order to obtain absolute accuracies, supplementary terms must be added to the relative accuracies. These terms mainly stem from the remaining errors of the georeferencing which precedes the DTM generation. In aerial photogrammetry, this is achieved through aerotriangulation, nowadays assisted by the global positioning system (GPS) and perhaps an inertial measurement unit (IMU) (Jacobsen, 2004). In ALS, a related method is used (Kager, 2004). The absolute accuracy is computed using check points of a much better accuracy than the DTM under investigation. The estimation of absolute accuracies, which frequently involves costly field measurements, is not discussed in this paper. Instead, it is presumed that the georeferencing is carried out with a high accuracy and a reliability such that the difference between relative and absolute accuracy can be ignored. In the following, it is assumed that both the DTM and the data employed for DTM interpolation are available. Sophisticated databases provide both the original data and the DTM (Warriner and Mandlburger, 2005). In the subsequent development of theory a DTM with hybrid structure is assumed. However, in the example presented no structural information is available. Approach for the Derivation of the Height Accuracy of Each Grid Point For accuracy estimations, simple functional models can be used, even if more complex functional models have been applied for DTM generation. As a simple and easily *Curvature is related to the well-known spectral analysis, which is sometimes used for the global accuracy estimation of DTMs (Frederiksen, 1980; Tempfli, 1980). Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 343 Kraus et al. Local accuracy measures for digital terrain models understandable interpolation method ‘‘moving least squares’’ (MLS) with a first order polynomial (plane) was chosen. By means of a tilted plane, the height ZG at grid point G can be computed as Z G ¼ a0 þ a1 X þ a2 Y : ð1Þ Having chosen the origin of the X, Y coordinate system at grid point G (Fig. 1), equation (1) becomes simpler: ZG ¼ a0 : ð2Þ The coefficients a0, a1 and a2 can be computed in a least squares adjustment, using points in the neighbourhood of grid point G (Fig. 1): 3 2 1 X1 v1 6 .. 7 6 .. .. 6 . 7 6. . 6 . 7¼6. . 4 .. 5 4 .. .. 1 Xn vn 2 2 3 3 Y1 Z1 2 3 .. 7 a0 6 .. 7 . 7 4 5 6 . 7  a1  6 . 7 .. 7 4 .. 5 . 5 a2 Zn Yn v ¼ A  x  1: ð3aÞ ð3bÞ The a priori weights pi applied to the observation equations of the adjustment are pi ¼ 1 1 ¼ 2 2 1 þ si =rai 1 þ s2i ð4Þ where si is the distance from the grid point to the original point and rai is the radius of curvature of the DTM at the original point towards the grid point. The weight pi of an original point will be greatest when it is near to the location of an interpolated grid point and diminishes to zero with increasing distance (McCullagh, 1988). This threshold controls the selection of points that participate in the interpolation. Additionally breaklines should be considered as barriers in the selection of original points. This way, the density and alignment of original points in the neighbourhood of the respective grid point are considered. Furthermore, equation (4) indicates that the distance si is normalised with the Z G Y X Fig. 1. Interpolation of a tilted plane using six neighbouring original points. 344 Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. The Photogrammetric Record radius of curvature rai, which is computed using the DTM at the original point in the direction ai towards the grid point G. Thus si si ¼ : ð5Þ rai Through this normalisation, original points with a large radius of curvature tend to affect the result more than those with a small radius. The weights pi in equation (4) are arranged in a diagonal matrix Pll. The authors are not interested in the interpolation result ZG (equation (1) or (2), respectively), but only in an estimation of the height accuracy of the grid point G, which is named r^DTM . This r^DTM can be calculated as follows: 2 P 3 P P p P pX P pY P AT Pll A ¼ 4 P pX P pX 2 P pXY 5 ¼ N: ð6Þ pY pXY pY 2 The a posteriori weight coefficient qa0 for the unknown a0 is P   pX 2 P pXY  P  P  pXY pY 2  qa0 ¼ : jNj ð7Þ The spatial variation of the weight coefficient qa0 depends on the following factors: (a) the DTM curvature through the weights pi, (b) the alignment of the original points and the distances between them and the respective grid point through the weights pi and through their X, Y coordinates in equations (6) and (7) (if the neighbouring original points almost form a straight line and the grid point is located aside, then the weight coefficient qa0 becomes very large), and (c) the number of original points that are selected using the threshold stated above, through the summations in equations (6) and (7). With an estimation for the reference standard deviation r^0 , which will be described in the following section, the accuracy r^DTM of the grid points is computed as pffiffiffiffiffiffi ð8Þ r^DTM ¼ r^0 qa0 : Representative Accuracy of the Original Points The theory of least squares adjustment applied to the moving tilted plane estimates the variance of the observation with unit weight, r^20 , as r^20 ¼ vT Pll v nu ð9Þ where v is the vector of residuals (equation (3)), Pll is the diagonal matrix of the weights (equation (4)), n is the number of original points participating in the adjustment of the tilted plane and u is the number of unknowns, in this case: u ¼ 3. In place of the residuals vi of the least squares adjustment of the tilted plane, the discrepancies di between the original points and the DTM to be evaluated (Fig. 2) are used. Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 345 Kraus et al. Local accuracy measures for digital terrain models di M DT G ce rfa su Si ZG (X,Y ) Fig. 2. Calculation of the height accuracy of original points in the neighbourhood of a grid point G (di are the discrepancies, si are the distances between the original points and the grid point G). This is because the discrepancies di better represent the errors of the original points with respect to the DTM than do the residuals vi. The reference standard deviation derived from the discrepancies di is denoted as r^0;d . It is computed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi dT Pll d ð10Þ r^0;d ¼ nu where d is the vector of discrepancies (Fig. 2), Pll is the diagonal matrix of the weights (equation (4)), n is the number of original points participating in the adjustment and u is the number of unknowns, in this case: u ¼ 0. The discrepancies are calculated using the points employed for DTM interpolation, thus without the eliminated points on buildings, trees and bushes. They are influenced by the calculus used for DTM interpolation, and by the performance of the filtering method used to eliminate the outliers (such as points on bushes in ALS data). The reference standard deviation r^0;d (equation (10)) varies from grid point to grid point. It should stay greater than a certain threshold. This threshold can be estimated through the specifications of the measurement device that was employed for data capture. Data points from an airborne laser scanner, for instance, have an accuracy of about 5 cm, if high quality devices are used and if advanced georeferencing is applied (Kager, 2004). In general, the accuracy of measurement of stereo photogrammetry is worse and depends mainly on the flying height and the camera type (Kraus, 2000). Before the presentation of an example for the spatial variation of the accuracy of original points r^0;d , the calculation of curvature and the alignment analysis of the original points will be illustrated. Curvature DTM accuracy is influenced remarkably by terrain curvature. Curvature is regarded as 1/rai in equation (4). The terrain curvature is computed using the DTM. Although wellestablished GIS routines are applied, some details need to be stated. The curvatures along the grid lines are calculated at each grid point. In doing so, the breaklines integrated into a hybrid DTM (Ackermann and Kraus, 2004) must be considered, as denoted in Fig. 3. Consequently, the minimum and maximum curvatures can be derived at each grid point. Using Dupin’s Indicatrix, curvatures or radii of curvature rai can be computed for arbitrary directions ai. The radius of curvature rai at a certain data point is computed for the direction to a usually rather 346 Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. The Photogrammetric Record Grid Point G Original Breakline Point Grid Intersection Point Fig. 3. Computation of curvature at grid point G. Grid points and grid intersection points of breaklines that are used for the computation are coloured black. distant grid point. As the original points rarely coincide with the grid points, the curvature at a data point has to be interpolated from the four surrounding grid points of the respective cell. There is plenty of literature about differential geometry, for example, Kreyszig (1964) or Novikov and Fomenko (1990). The application of this theory to the DTM can be found in Briese (2004) and Gajski (2005). Arrangement and Density of Original Data Points The accuracy of a DTM is strongly influenced by point density. For the visualisation of point density, the following method is preferred: an analysing grid with cell area A is laid over the area of interest. Usually, A is chosen to be larger than the cell area of the DTM grid. Using the number ni of original points in each cell, the point density ni ¼ ni =A can be computed. More complex than the computation of point density is the modelling of the arrangement of original points. The distance from the data points to each grid point, or their influence on grid point heights, respectively, is of special interest. For this purpose, the Chamfer function (Borgefors, 1986) may be applied. It facilitates the measurement of distances smi between each grid point and its nearest original point with high efficiency (Fig. 4). Fig. 4. Distances smi between each grid point (grey) of the DTM and its nearest data point (black) in the original data-set. Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 347 Kraus et al. Local accuracy measures for digital terrain models Fig. 5. Shaded DSM, derived from airborne laser scanner data. Example Fig. 5 shows the shaded digital surface model (DSM) computed with a set of airborne laser scanner data that originates from EuroSDR (IPF, 2005). Using the method described by Kraus and Pfeifer (1998), a DTM with a grid width of 1 m was deduced from the data. The mathematical background of this method is the least squares interpolation, which is equivalent to simple kriging (Kraus, 1998). For the processing of ALS data, the least squares interpolation was adapted, resulting in a robust estimator with an eccentric and asymmetric weight function. Using the robust estimator, points on buildings and high vegetation are eliminated (filtered out). Details can be found in the literature mentioned above (Kraus and Pfeifer, 1998; Briese et al., 2002). Fig. 6 illustrates the DTM in a shaded view together with contour lines. This already existing DTM is evaluated using the approach of this paper. Before the presentation of the final, spatially varying DTM accuracy r^DTM , some intermediate results and diverse influencing factors are visualised. Fig. 6. Shaded DTM, together with contour lines (2 m interval), derived from airborne laser scanner data. 348 Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. The Photogrammetric Record Fig. 7. Original points that form the basis of DTM interpolation (Fig. 6). The first of these figures (Fig. 7) shows the original points that were employed for the interpolation of the DTM (Fig. 6). Areas with large data voids are distinguishable. These areas originate on the one hand from the missing overlap of the airborne laser scanner strips. On the other hand, they stem from zones where many points on high vegetation and on buildings have been eliminated in the filtering process (compare Fig. 5 and Fig. 7). The next visualisation is the point density  ni (Fig. 8). It also reflects the holes in the original data that originate from the filtering process and the missing overlap of the airborne laser scanner strips. These holes must be quantified. In order to do so, the distances between the grid points and their nearest data point smi are computed (see Fig. 9). These distances can be confronted with an adequately chosen threshold (in Fig. 9, its value amounts to seven times the DTM grid width, or 7 m). Grid points that hold too large minimum distances are classified as unusable. This information is very important for the DTM user. Unaffected by large data voids, powerful software packages provide DTMs without gaps (see Fig. 6). For many applications, for example, the modelling of the boundary of a flood risk area, a closed DTM is required. However, the result is unusable if the boundary proves to lie on a data void. In this case, that part of the boundary must not be passed on to users such as local governments or property owners. Fig. 8. Point density ni of the original points (Fig. 7), computed with a cell size A of 100 m2. Cells that do not contain any point are coloured black. Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 349 Kraus et al. Local accuracy measures for digital terrain models Fig. 9. Minimum distances smi between each grid point and its nearest data point (Fig. 7). Grid points where the threshold of seven times the DTM grid width (7 m) is exceeded are marked in red. A further visualisation, namely, terrain curvature, appears to be less interesting for DTM users, but still of importance for analysts. In the approach of this paper, the terrain curvature 1/rai is used for weighting (equation (4)). In Fig. 10, the maximum main curvature at each grid point is presented on a scale from )1Æ0 to 1Æ0 m)1. Obviously, DTM curvature is a good indication of terrain relief. However, DTM curvature is also very sensitive to DTM artefacts. Wood and Fisher (1993) have investigated the spatial variation of DTM accuracy, using GIS operators that employ neighbouring grid points (similar to the computation of DTM curvature in this paper). However, the focus of that article is set on the visualisation of DTMs derived from contour lines using four different interpolation methods. Unlike in the present paper, quantitative accuracy measures are not a central issue. The next important visualisation of the presented approach displays the accuracy of the original points, derived from their discrepancies to the DTM. The reference standard deviation defined in equation (10) is outlined in Fig. 11. The value of r^0;d varies between ±5 and ±40 cm. The lower value should not be smaller than the a priori known accuracy addressed above. In the present example, this lower boundary amounts to 5 cm. Red areas denote zones where no Fig. 10. Maximum main curvature at the grid points of the DTM from Fig. 6. 350 Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. The Photogrammetric Record Fig. 11. Height accuracy r^0;d of original points (Fig. 7) in the neighbourhood of grid points of the DTM. Areas where smi exceeds the user-defined threshold of 7 m (Fig. 9) are coloured red. Fig. 12. Cofactor in height of the adjustment of the tilted plane. original points surround the grid points. These areas were excluded with a threshold of 7 m for the minimum distances smi. In addition to the reference standard deviation r^0;d , the weight coefficient qa0 (equation (7)) is needed for the computation of the DTM accuracy r^DTM (equation (8)). For the present example, the weight coefficients are shown in Fig. 12. Red areas once more denote zones where there are no original points near the grid points. These areas were excluded using a threshold of 7 m for the minimum distances smi. The weight coefficients vary between 0Æ3 and 30. If the weight coefficient qa0 equals 1, then the height of the DTM grid point has the accuracy of the neighbouring original points that is expressed by the reference standard deviation r^0;d (Fig. 11). If qa0 is greater than 1, then r^DTM is worse than the reference standard deviation r^0;d . If qa0 is smaller than 1, which should be aimed at, then r^DTM is better than the reference standard deviation r^0;d . The last but most important visualisation is the spatial variation of the height accuracy of the DTM, computed with equation (8), employing the reference standard deviation r^0;d (Fig. 11) and the weight coefficients qa0 (Fig. 12). Fig. 13 shows the height accuracy r^DTM of Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 351 Kraus et al. Local accuracy measures for digital terrain models Fig. 13. Standard deviation in height of the DTM. the DTM. Areas that have been excluded using the minimum distances smi are marked as unusable in red. The height accuracy r^DTM varies between ±0Æ01 and ±2Æ00 m. Recommendations and Outlook Where the original points used for the interpolation of the DTM are available, the spatial variation of DTM quality in height can be derived. Fig. 13 is such a quality layer. Unusable areas should be marked in the visualisation—for example, in red. In the future, this quality layer should be provided in addition to the DTM. Such information can be generated for both new and existing DTMs. GIS users will welcome its application. Moreover, the software should allow interested GIS users to view the following information: (a) (b) (c) (d) (e) point density (Fig. 8), minimum distances smi between each grid point and its nearest data point (Fig. 9), maximum main curvature (Fig. 10), reference standard deviation in height r^0;d computed using equation (10) (Fig. 11), weight coefficients qa0 computed using equation (7) (Fig. 12). These additional visualisations are of special interest to DTM experts who analyse DTMs. The DTM theory presented in this paper will be tested with diverse data-sets in the future. A test series with both ALS and photogrammetric data has already been evaluated. The results will soon be released in a EuroSDR publication (Karel and Kraus, 2007). These test series will also provide feedback for the choice of adequate thresholds, and for the enhancement of software performance (for example, not computing the standard deviation r^DTM at every grid point, but only at every fourth). As stated above, this paper is focused on relative accuracy. It is presumed that the data is free from systematic errors. Whether this assumption is admissible may be tested using external check points with an accuracy superior to that of the DTM. However, the analysis of zones of overlapping ALS strips also shows up systematic errors, see Kager (2004) or Maas (2002). Presumably, additional processes will be investigated in the near future that allow for the estimation of absolute accuracies on the basis of relative ones. Concluding, it has to be mentioned that the presented quality measures focus on height accuracies. In future research, planimetry will also be considered. 352 Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. The Photogrammetric Record references Ackermann, F. and Kraus, K., 2004. Grid based digital terrain models. Geo Informatics, 7(6), 28–31. Borgefors, G., 1986. Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34(3): 344–371. Briese, C., 2004. Three-dimensional modelling of breaklines from airborne laser scanner data. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 35(B3): 1097–1102. Briese, C., Pfeifer, N. and Dorninger, P., 2002. Applications of the robust interpolation for DTM determination. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 34(3A): 55–61. Frederiksen, P., 1980. Terrain analysis and accuracy prediction by means of the Fourier transformation. International Archives of Photogrammetry and Remote Sensing, 23(4): 284–293. Gajski, D., 2005. Rasterbasierte Geländeoberflächenanalysen. Dissertation, Vienna University of Technology, Vienna, Austria. 167 pages. IPF, 2005. http://www.ipf.tuwien.ac.at/eurosdr/download.htm [Accessed: 19th September 2005]. Jacobsen, K., 2004. Direct/integrated sensor orientation—pros and cons. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 35(B3): 829–835. Kager, H., 2004. Discrepancies between overlapping laser scanner strips—simultaneous fitting of aerial laser scanner strips. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 35(B1): 555–560. Karel, W. and Kraus, K., 2007. Quality parameters of digital terrain models. Seminar on Automatic Quality Control of Digital Terrain Models, Aalborg, Denmark (Ed. J. Höhle). EuroSDR Publication (in press). Kraus, K., 1998. Interpolation nach kleinsten Quadraten versus Krige-Schätzer. Österreichische Zeitschrift für Vermessung und Geoinformation, 86(1): 45–48. Kraus, K., 2000. Photogrammetry. Volume 1: Fundamentals and Standard Processes. Fourth edition. Dümmler, Bonn, Germany. 397 pages. Kraus, K. and Pfeifer, N., 1998. Determination of terrain models in wooded areas with airborne laser scanner data. ISPRS Journal of Photogrammetry and Remote Sensing, 53(4): 193–203. Kreyszig, E., 1964. Differential Geometry. Revised edition. University of Toronto Press, Toronto. 377 pages. Li, Z., 1993. Theoretical models of the accuracy of digital terrain models: an evaluation and some observations. Photogrammetric Record, 14(82): 651–660. Maas, H.-G., 2002. Methods for measuring height and planimetry discrepancies in airborne laserscanner data. Photogrammetric Engineering & Remote Sensing, 68(9): 933–940. McCullagh, M., 1988. Terrain and surface modelling systems: theory and practice. Photogrammetric Record, 12(72): 747–779. Mikhail, E. M., 1976. Observations and Least Squares. With contributions by F. Ackermann. IEP, New York. 497 pages. Novikov, S. P. and Fomenko, A. T., 1990. Basic Elements of Differential Geometry and Topology. Kluwer, Dordrecht. 490 pages. Sithole, G. and Vosselman, G., 2004. Experimental comparison of filter algorithms for bare-earth extraction from airborne laser scanning point clouds. ISPRS Journal of Photogrammetry and Remote Sensing, 59(1/2): 85–101. Tempfli, K., 1980. Spectral analysis of terrain relief for the accuracy estimation of digital terrain models. I.T.C. Journal, 1980–3:478–510. Warriner, T. and Mandlburger, G., 2005. Generating a new high resolution DTM product from various data sources. Photogrammetric Week 05, Stuttgart, Germany. 339 pages: 197–206. Wood, J. and Fisher, P., 1993. Assessing interpolation accuracy in elevation models. IEEE Computer Graphics and Applications, 13(2): 48–56. Résumé Pour établir des modèles numériques du terrain (MNT) on peut considérer qu’on dispose actuellement, comme techniques de base, du scannage par laser aéroporté et de l’appariement d’images. Accompagnant le MNT, il est souhaitable d’avoir en supplément des paramètres de qualité qui en fournissent la précision avec un grand niveau de détail, au moins en chacun des points du MNT obtenus par interpolation. D’autres paramètres sont également intéressants, comme par exemple la distance de Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 353 Kraus et al. Local accuracy measures for digital terrain models chaque point du MNT au point de base voisin. On décrit dans cet article une méthode permettant de déterminer la précision à partir des données-source et du MNT lui-même. Un exemple vient en appui de cette méthode. La connaissance de la qualité d’un MNT est une information très utile pour les utilisateurs de ce MNT et qui les met en garde dans les zones où celui-ci présente des faiblesses dans sa formation. Zusammenfassung Flugzeuggetragenes Laserscanning und automatische Bildzuordnung dienen heutzutage als Grundlage für die Erstellung von digitalen Geländemodellen (DGM). Zusätzlich zum DGM werden Qualitätsparameter benötigt, die seine Genauigkeit sehr detailliert, am besten für jeden interpolierten DGM-Punkt, beschreiben. Außerdem sind weitere Parameter hilfreich, zum Beispiel die Distanz von jedem Gitterpunkt zum nächstgelegenen Datenpunkt. Dieser Artikel beschreibt eine Methode, um aus den Originaldaten und dem zu beurteilenden DGM seine Genauigkeit abzuleiten. Die Anwendung der Methode wird an einem Beispiel veranschaulicht. Mit Hilfe der präsentierten Qualitätsparameter können Benutzer sehr genau über die DGM-Qualität informiert und vor schlecht bestimmten Bereichen gewarnt werden. Resumen En la actualidad la altimetrı́a de barrido láser y la correlación automática de imágenes son la base para la obtención de modelos digitales de elevación (MDE). Además del MDE, se necesitan parámetros de calidad que describan detalladamente su exactitud, a ser posible en cada punto interpolado del MDE. Además, hay otros parámetros de interés, por ejemplo la distancia entre cada punto del modelo y el punto original más próximo. Este artı́culo describe un método para obtener medidas de calidad de los datos originales y del propio MDE. Su aplicación se demuestra con un ejemplo. Mediante dichos parámetros de calidad se puede informar a los usuarios acerca de la calidad del MDE y advertirles acerca de las áreas peor estimadas. 354 Ó 2006 The Authors. Journal Compilation Ó 2006 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd.