.NCGIA Core Curriculum in Geographic Information Science
URL: "http://www.ncgia.ucsb.edu/giscc/extra/e001/e001.html"
by Ahmad S. Massasati
United Arab Emirates University
This material is part of the
NCGIA Core Curriculum in Geographic Information Science. These materials may be used for study, research, and education, but please credit the author, Ahmad S. Massasati, and the project, NCGIA Core Curriculum in GIScience. All commercial rights reserved. Copyright 2000 by Ahmad S. Massasati.Your comments on these materials are welcome. A link to an
evaluation form is provided at the end of this document.This paper presents a practical way to teach about elevation models. These currently include solutions geographers refer to as digital elevation/terrain models such as point data, contour lines, triangular irregular networks, and mathematical models. The apparent complexity of data transfer in these methods, however, seems difficult to students and other first time users. In the authors classroom experience, the pyramids of Egypt have proved to be an excellent and efficient example for teaching digital elevation/terrain models.
Key words: geographic education, digital elevation models (DEM), digital terrain models (DTM), contour lines, triangular irregular network (TIN), geographic information systems (GIS).
Teaching elevation models in the classroom has always been a challenge. The difficulties come from the fact that data development is done with highly technical specialists and advanced computer technology while the use of the models is spread wide among different fields of sciences such as geology and geography.
Useful models for representing elevation data are point data, contour lines, triangular irregular network (TIN), digital elevation models (DEM) (Burrough, 1998) or digital terrain models (DTM) (Maguire, 1991), and mathematical models. A derivative of these models comes from traditional surveying practice and techniques in which a surveyor would collect the elevation of point data in the field, then transfer it to various methods of presentation on the map (Moffitt, 1982). Though these techniques are very precise, they are hard, require technical specialty, and are time consuming. The introduction of aerial photography made it possible to transfer the bulk of fieldwork to the office and derive elevation data using three-dimensional aerial photography. Recently the introduction of softcopy photogrammetry make it possible to derive elevation data with even more automation and less human involvement by using computers (Greve, 1996). For that, without hands on field experience, students of GIS might find it difficult to understand the variation between elevation models and how they compare to each other. The following exercise is designed to help understand these models, simplify the transformation from one to another, and clarify the differences among them.
The great pyramid of Khufu (Gizah, Egypt) (Fowler, 1999) is simple to imagine and calculate. The real dimensions of the pyramid are 755 feet. at the base with 481.4 feet height and a 52 degrees slope (Figure 1) (Edwards, 1993). Round these to 750 feet at the base and 450 feet height. The assumed orientation of the pyramid will be north-south and east-west. The coordinate of the southwest corner is zero east, zero north, and zero elevation.
.
Figure 1. SPIN-2 Panchromatic digital image of Gizah 1997
(from Fowler, 1999. Image provided courtesy of SPIN-2 at Aerial Images Inc.)
The task will be to represent the pyramid in each of the five elevation models. These models are:
The pyramid can be presented simply in five points, four at the base and one at the top (Figure 2). The x, y, and z coordinate of these points are;
A (0,0,0), B (0,750,0), C (750,750,0), D (750,0,0), and E (375,375,450).
Figure 2. The point model for the pyramid of Khufu.
Assuming a 50 meter interval, the four edges of the pyramid would be divided into 50 (450/50 = 9) resulting in ten contour line considering the top one as a point (Figure 3).
Figure 3. The contour lines for the pyramid of Khufu.
Standing on the top of the pyramid, the surface can be generalized and divided into four triangles (Figure 4). Each of the four triangles is identified with the coordinates of the three points forming it. It is important to note that any extra point on each of the four triangles is unnecessary to provide additional details of that surface.
Triangle one is A (0,0,0), B (0,750,0), and E (375,375,450). Triangle two is B (0,750,0), C (750,750,0), and E (375,375,450). Triangle three is C (750,750,0), D (750,0,0), and E (375,375,450). Triangle four is D (750,0,0), A (0,0,0), and E (375,375,450).
Figure 4. The TIN model for the pyramid of Khufu.
Assuming a 50x50 foot resolution, the elevation of the pixels at the bottom will have elevation ranges 0-50 feet and the single pixel at the top will have an elevation range from 400-450 feet (Figure 5). Stored in a computer file, the number of rows and columns are 17x17. If each pixel's elevation is set at the minimum elevation in each, the cell values of the file will vary between 0 and 400. Thus the minimum computer storage per file is 17x17x(9 bits per pixel) = 2601 bits of data. (Note: usually using an eight bit binary notion, you can express a number that varies between 0 and 255.) From a computer programming point of view, it is possible to reduce the file size using index value such as using 1 instead of 50, 2 instead of 100, and so on. This will allow the use of fewer bits per pixel therefore reducing the size of the computer file as well as processing time (Figure 6). For a small file, this saving might be minute, but DEM data is large and saving in file size is eventually significant (Burrough, 1998).
Figure 5. The DEM for the pyramid of Khufu with actual values. (click for a large image)
Figure 6. The DEM for the pyramid of Khufu with index values.
Each surface (triangle) can be presented with the following mathematical model
Z = a X + b Y + c
Where:
X, Y and Z are surface point coordinates and
a, b, and c are constants that can be determined for each surface by solving the equation using the three corner points (figure 7).
Calculation for surface 1;
0 = a * 0 + b * 0 + c (1)
0 = a * 750 + b * 0 + c (2)
450 = a * 375 + b * 375 + c (3)
By solving the three equations, b = 1.2 and the equation is
Z = 1.2 Y
Where:,
Likewise, the equations for surfaces 2, 3, and 4 are:
Z = 1.2 X Where:
, and
Z = 1.2 (750-Y) Where:, and
Z= 1.2 (750-X) Where:, and
The equations for the line that borders the triangles can be obtained by simply solving for any two surfaces.
Line AE:
Z = 1.2X=1.2Y or X=Y,
where:
Line BE:
Z= 1.2X=1.2(750-Y) or X+Y=750,
where:Line CE:
Z= 1.2 (750-X)=1.2(750-Y) or X=Y,
where:
Figure 7. The mathematical model for the pyramid of Khufu.
The mysteries behind elevation models seem to disappear when pyramids are used in examples at the classroom level. To derive elevation data is no longer limited to engineers or surveyors. Computers made it possible to use GIS and Softcopy Photogrammetry technology to bring elevation data and modeling development to interested students in all fields of science. Terms such as DEM or DTM were debated (Maguire, 1991). Advancement in computer technology will add more elements to such debate. Raster and vector data that are GIS terms need to be considered. Using the term Raster Elevation Data (RED) is an appropriate alternative to DTM or DEM in raster contents.
New terms and models for elevation data are emerging. Only recently, a term such as Voxel has been introduced where the principle unit is a volume cell rather than a grid cell (Burrough, 1998). The pyramid can be presented in a voxel model where each building block is accounted for. Errors and error analysis need to be addressed. When transformation is made between the various types of data models, generalization process takes place and some accuracy could be compromised (Joćo, 1998). Understanding the nature of errors is a key element in data transfer between the different models of elevation data. It is also important also to realize that these models are interchangeable but not without compromising accuracy (Maguire, 1991). This paper demonstrated the possibility of simplifying elevation models. It further suggest that using the pyramid as an education tool will make it possible to solve a more difficult problems such as voxel analysis and error analysis.
Burrough, P. A. and Rachael A. McDonnell 1998. Principles of Geographical Information Systems, New Edition. Oxford: Oxford University Press.
Clarke, Keith C. 1999. Getting started with geographic information systems, 2nd Edition. Upper Saddle River, NJ: Prentice Hall.
Edwards, I. E. S. 1993. The Pyramids of Egypt, Revised Edition. New York, NY: Penguin Books.
Fowler, Martin J. F. 1999. The Pyramids at Giza, Egypt. Satellite Remote Sensing and Archaeology. WWW: http://ourworld.compuserve.com/homepages/mjff/giza.htm. Winchester: UK.
Greve, Cliff 1996. Digital Photogrammetry: An Addendum to the Manual of Photogrammetry. Falls Church, Virginia: American Society of Photogrammetry and Remote Sensing.
Joćo, Elsa M. 1998. Causes and Consequences of Map Generalization, London; Bristol, PA: Taylor & Francis.
Maguire, D. J., Michael F. Goodchild, and David W. Rhind. 1991. Geographical Information Systems. New York, NY: Longman Scientific & Technical.
Moffitt, Francis H., Harry Bouchard 1982. Surveying, 7th Edition. New York: Harper & Row Publishers.
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Massasati, Ahmad S. (2000) An Exercise in Digital Elevation/Terrain Models: From Point to Mathematics, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/extra/e001/e001.html, accessed [today's date].
The correct URL for this page is: http://www.ncgia.ucsb.edu/giscc/extra/e001/e001.html.
Created: July, 2000. Last revised:
August 10, 2000.
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