DIGITAL TERRAIN MODELS 1
Digital Terrain Models Dr. Mohsen Mostafa Hassan Badawy Remote Sensing Center GENERAL: A Digital Terrain Models (DTM) is defined as the digital representation of the spatial distribution of terrain characteristics in addition to an algorithm for the reconstruction of the terrain surface from these digital values. If the terrain characteristics represented digitally are only elevations, the DTM is called a digital elevation model (DEM). In this case the DEM may be defined as the digital representation of terrain relief in terms of a set of spatial coordinates obtained by sampling the surface, and an algorithm to reconstruct other elevations using these sampled values. The study of digital elevation models includes three main areas, namely; data acquisition, conversion processes and performance evaluation. Data acquisition or sampling includes all the processes which are applied to continuous surfaces in order to obtain a set of discrete values that describe these surfaces. The area of data acquisition includes: Sampling methods, Sampling patterns and Sampling density determination. The data conversion phase includes all the operations performed on the acquired data to convert them into the required output. If these outputs are in the form of continuous surfaces, the conversion processes may be considered the inverse process of sampling. The operations performed in the conversion process area are usually defined by a mathematical model and the corresponding rules of implementation. In the performance evaluation phase the fidelity and the accuracy of the given DEM are studied. There are two approaches to perform this phase, the experimental approach and the analytical approach. Throughout this presentation a brief illustration of the three areas of DEM is given. Fig. (1) The three areas of DEM 2
Fig. (2) The main governing factors of DEM SAMPLING METHODS The most common sampling method of DEM is the Photogrammetric method. However, this method should be completed by the digitization of the existing conventional maps. This is performed by digitizing the contour layer of the topographic map, then converting contours into a DEM. Less important methods of DEM data acquisition are used for small areas or for special purposes. These methods may include Land surveying, Satellite surveying or Airborne profile recording. SAMPLING CONTROL Data acquisition requires measurements of single points in a stereoscopic model. The adjustment of the floating mark of these points may be done in three different ways: Manually, Semi-automatic or Fully automated. In the manual control systems the three coordinates of each point are selected manually by the operator, while in the semi-automatic systems two coordinates are determined automatically and the machine moves to the point that corresponds to these coordinates. The operator, in this case, has only to adjust the third dimension. In the fully automated systems the correlation techniques are utilized to correlate corresponding points in the stereo model and measure their three coordinates automatically. The operator is only needed to guide the instrument in difficult areas. 3
Fig. (3) Sampling methods for DTM POINT PATTERNS DEM data can be sampled in different patterns, the most common of which is the regular grid pattern which can be a square grid or regular triangles. Other forms are profiles, irregular point patterns or contour lines. The irregular pattern is the most interesting pattern at present as it represents the morphology of the terrain surface better than any other point pattern. Irregularly distributed points are then connected in an irregular network of triangles called TIN. There are three factors that govern the choice of the point pattern, these factors are: i) The purpose of constructing a DEM. ii) Terrain type characteristics. iii) The operational aspects. Fig 4 Point configuration or sampling pattern of DTM 4
SAMPLING DENSITY The determination of the sufficient density of points for a DEM in the planning stage is one of the main problems of elevation models. There are some sophisticated methods for density determination based on the sampling theory and the Time Series Analysis techniques. Sampling density can either be determined before measurement begins and considered as a constant for a given area, or it can be continuously adjusted during measurements. A fixed sampling density may result in redundancy of elevations in undesired areas or lost of information in other areas. The continuous adjustment techniques can be subjective or objective. Fig 5 Sampling Density determination in DTM DATA STORAGE Storage methods depend on three factors: the type of data, the type and limitations of the data base and the user requirements. Data may either be stored as discrete values or as mathematical models that represent the surface. Discrete data may be in the form of points, vectors or polygons as in the case of dead areas. The data structure depends on the logical as well as the physical format of the stored data. 5
Fig 6 Data storage in DEM Fig 7 The USGS method of data storage 6
CONVERSION PROCESSES The computer processing that is performed on DEM data can be divided into two main phases: pre-processing and data processing. The pre-processing phase includes: Editing, Data Compression, Coordinate Transformation, Merging, Partitioning, Smoothing, Trend Analysis and Filtering. The editing step may include Gross Error Detection and Elimination, Completing Missing Data, Deleting Redundant Data and Edge Matching. The processing phase includes: Interpolation, Densification Surface Fitting, Intersections with planes, Intersection with lines, Volume Computations etc. One of the main conversion processes in DEM is surface fitting. Usually Patch wise polynomials are used as the mathematical model of the fitting process. In some cases least squares or the more complicated linear least squares interpolation and filtering techniques are used. Collocation fitting in which a distance function is used instead of the gaussian covariance function is also implemented in DEM surface fitting. The technique of patch wise polynomial fitting is given here as a simple example. Fig 8 Conversion Processes SURFACE FITTING When a polynomial is used to fit a set of data points, the deviations between the measured and the fitted elevations must be kept reasonably 7
P ip YP jp (1) small. The simplicity of polynomials permits this goal to be performed in various ways such as Direct polynomials, oscillation polynomials or Least Squares Surface Fitting. Polynomials can be fitted with or without boundary conditions. Boundary conditions provide smooth transitions between patches. The general form of the bi-variate polynomial is: Z = Ʃ Cij XP i j Polynomials as simple as plane surfaces or as complicated as the 16- term bicubic polynomial can be used in DEM surface fitting. It is clear that the variety of polynomials in addition to the various choices of the point distribution model and the size of the patch give way to many mathematical models that can be utilized in this technique. One of the main advantages also is the use of the socealled local coordinate system which results in the same matrix of coefficients for each patch. This simplifies the solution considerably and at the same time reduces the computer time and the required storage memory. The output of the fitting process is a surface that can be graphically presented in different forms. Fig 9 Research Plan: Processing and Testing PERFORMANCE EVALUATION The performance of DEM can be evaluated either experimentally or analytically. The experimental approach depends on real data that can be obtained from a test area. These data are statistically analyzed and the 8
statistical parameters that indicates the accuracy of the data and the conversion process are determined and judged. On the other hand the analytical approach depends on the evaluation of the transfer functions. These transfer functions represent the fidelity of the DEM as a representation of the real terrain. The analytical approach uses sophisticated techniques from the Fourier analysis theory and its applications in the three dimensional domain. Fig 10 Performance studies in DEM Fig 11 Some of the practical applications of DTM 9