Inspection of Topographic Distances Measurements within ArcGIS
Penn State GEOG 596B Capstone Project
Steven Rick Bradshaw
November, 2013
Background
In a non-projected global coordinate system environment, GIS software measures geodetic distances using a spheroid model of the earth called a reference ellipsoid. The OpenGIS Consortium (2007) defines geodetic distance as “…the length of the shortest curve between those two points along the surface of the earth model being used by the spatial reference system. Geodetic distance behaves well for wide areas of coverage, and takes the earth's curvature into account.” However, this geodetic distance measurement using any reference ellipsoid, does not take into account the changes in elevation between two points. Consider two cities (A and B) separated by a mountain range as shown in Figure 1. When traveling from City A to City B, a traveler changes elevation crossing over the summit of the range. When GIS software calculates the distance between City A and B, the resultant distance is calculated using geodetic (or planar) distance but does not include the changes in topography.
Figure 1- The comparison between the topographic distance and reference ellipsoid distance between City A and City B.
There is a growing need to incorporate topographical distances in the GIS environment. Currently, GIS software functions offered by major vendors, do not provide an accurate analysis of topographic distances. Topographic distance measurements are necessary when traveling over changes in elevation. Changes in elevation will necessarily increase actual distance traveled. Extreme changes in elevation will impact measurements of even short distances.
Goals and Objectives
Thegoal of thisproject was to determine the need for topographic distances within a GIS environment. A successful project willserve as a call for further research by academics and GIS software vendors in developing a topographic distance utility.
To help complete the project goal, objectives were established to guide the project. The key objectives were:
1. Establish a study area to simulate the scenario discussed above.
2. Survey the routes of study area with GPS. A section of the route had the topographic distance measured with a device (measuring wheel) that measured the distance resulting from changes in elevation.
3. Calculate the slope distances of the routes in the coordinates of two reference ellipsoids to determine the topographic distances.
4. Determine the differences in distance of the topographic routes to the geodetic distance method.
5. Determine the causes of the differences and the impact of the differences on GIS spatial analysis.
Project Methodology
Research approach-
The key objective of this study was to explore the need for developing topographic distance capabilities in GIS software. The hypothesis of this study:Given the same route, topographic distances would be greater than geodetic distance. To test the hypothesis, several comparisons of two stretches of roads near White Sands Missile Range were made.
First, a route with moderate elevation changes was surveyed. This route is a portion of Highway 70 that traverses over the Organ Mountain Range in New Mexico.
Second, a route with little elevation change was mapped. This second route, called the Flats Site, isa section of White Sands Range Road 7 that runs across an extinct lakebed.
Figure 2 shows the surveyed routes. Each route was mapped using relative GPS surveying methods to simulate the routes of the theoretical traveler shown in Figure 1. The survey data of the routes was collected in World Geodetic System of 1984 and then transformed into North American Datum of 1927.
Figure 2- Map showing the study area of Highway 70 and Range Road 7 that was surveyed with GPS.
Both sites were surveyed to collect horizontal and vertical coordinates, using relative GPS surveying methods. According to Wolf and Ghilani (2006), relative surveying involves two GPS units. One unit measures GPS signals over a known positionand another unit measures unknown positions. The study site contains survey control monuments with known coordinates. Survey control monuments are necessary for precise measurement since one GPS unit is only accurate within 10 meters. Known coordinates are used to correct errors introduced by atmospheric conditions. The GPS unit sits on top of the survey control monument (known position), and corrects the data gathered by the second GPS unit. The errors are corrected by using the difference in distance and azimuth between the known coordinatesand the position the GPS unit over the known position is calculating. Even though the GPS unit is over the known point, atmospheric conditions and other factors will impact the calculations. The same distance and azimuth will be used to correct the coordinates of the unknown positions.
The two sets of WGS-84 survey data collected was transformed to the North American Datum of 1927 for comparison to the original reference ellipsoid, creating a total of 4 sets of data. The vertical data of both ellipsoids for the Organ Mountain Pass Site was manipulated to create additional datasets thatrepresent extreme elevation changes. The vertical datawas manipulated by adding approximately 10 percent of the elevation value to itself. This created two more data sets with large changes in elevation, for a total of 6 datasets. This was necessary since there is not an area with extreme changes in elevation close to White Sands Missile Range.The Flats Site vertical data value was not manipulated. The spatialdistances of each route were determined and represent the topographic distance along the surface of the earth. These distances were then compared to the geodetic distances of each reference ellipsoid to show the differences between the geodetic and topographic distances.
Research Steps-
Step 1:Both survey sites were surveyed with a Trimble Global Positioning System (GPS) R-8 receiver.
The GPS receivers occupied a known position within the White Sands Missile Range (WSMR) surveying control network. The GPS receiver collecting the unknown positions along the route was mounted onto a truck, driving both survey sites while collecting coordinates every 150 meters. These coordinates were used to generate topographic distances. The actualsurface distance of the Flats Site was also surveyed with a manual measuring wheel. This distance was used as a baseline of the surface distance to be compared to the measured geodetic distance and the GPS collected Topographic distance.
The initial surveys of the two routes was collected in the World Geodetic System of 1984 (WGS-84) reference ellipsoid. Ellipsoidal heights were used to determine the elevation of the routes mapped in the WGS-84 reference ellipsoid.
In addition to the GPS survey, the Flats Site actual surface distance was measured with a measuring wheel. This distance will be used as a baseline of the surface distance to be compared to the measured geodetic distance and the GPS collected Topographic distance.
Step 2:To illustrate how different reference ellipsoids affect topographic distances, the horizontal and vertical coordinates collected during the survey, was transformed into the North American Datum of 1927 (NAD27). Transforming the points to the NAD27 reference ellipsoid is more precise and easier than resurveying the route into NAD27. The same exact points on the surface were represented in the study. If the route was resurveyed into NAD27, additional error in differences in distance was introduced, since the very same exact path of the driver cannot be replicated. The GPS surveying methodology called “Continuous Topo” by Trimble,was used to collect the coordinates. Collecting coordinates of the same exact point would nearly be impossible. Continuous Topo collects coordinates of points along a path of travel, which are separated by either amount of distance traveled or by a set amount of time from the previous point.
Step 3:The vertical data for the Organ Mountain Pass Site was manipulated to create another dataset that represented extreme elevation changes. The Flats Site was not manipulated. This step yielded a total of six datasets, separate from the surface distance measured with the measuring wheel. The Organ Mountain Pass site has two original datasets representing moderately changing topographic features cast on different reference ellipsoids, WGS-84 and NAD27. The Organ Mountain Pass Site also has two additional modified datasets representing extremes in changes in elevations cast on different reference ellipsoids, WGS-84 and NAD27. The Flats Site has two datasets representing a flat surface with little change in elevation cast on different reference ellipsoids, WGS-84 and NAD27.
Step 4:The slope distances of all datasets was determined using an in-house Department of Defense surveying software package. This software package calculates the slope distance between separate points at different elevations. The software calculates the slope distance by determining the 3 dimensional coordinates (x,y,z) of two points on a reference ellipsoid, then calculating the distance between those points. Using this software the topographic distance of each route was compiled and totaled.
Figure 3- Demonstration of how the in-house DoD software generates topographic distances. The survey points are illustrated by the green circles. Linear Spatial Distances were calculated by finding the distance between the 3 dimensional coordinates. The spatial distances are shown by the red lines, which represent the topographic distance generated by the software. The actual topography surface distance is illustrated by the blue lines. The graphic shows the red lines or spatial distance is much closer to the surface distance than the geodetic distance or black line.
Step 5: A profile of each reference ellipsoid was then identified and compared to the other reference ellipsoids. The differences between the WGS-84 and NAD27 reference ellipsoid was determined by comparing the semi-major axis, semi-minor axis and the flattening ratio as shown in Table 1.
The Defense Mapping Agency (1984) reports that WGS-84 is a geocentric reference system ellipsoid with an origin at the center of Earth’s mass. The alignment of the system is oriented and fixed to astronomic observations. The size and shape of the ellipsoid have the semi-major axis is 6,378,137.0 meters, semi-minor axis is 6,356,752.314 meters and has a flattening ratio of 1/298.257223563. The semi-major axis of an ellipsoid is defined as the radius of the ellipsoid along the equatorial plane. The flattening ratio is defined as how closely the ellipsoid approaches a spherical shape. The semi-minor axis length is the radius between the geodetic poles of the reference ellipsoid also helps to visualize the shape of the ellipsoid. The National Imaging and Mapping Agency (2000) explains that since the WGS84 reference ellipsoid is fixed to astronomic observations, the coordinate values are not affected by tectonic movements and places on Earth have different coordinates as tectonic plates move through time.
NAD27 is a horizontal control datum that utilizes the Clarke Spheroid of 1866. The origin of the datum is located at a single survey marker at Meades Ranch, Kansas. NAD27 is oriented to an azimuth from Meades Ranch to Waldo. NAD27 also defines the geoid value to be 000, which means NAD 27 defines Meades Ranch as sea level. (NOAA, 2009) The Clarke Spheroid of 1866 is a reference ellipsoid, which semi-major axis is 6,378,206.4 meters, semi-minor axis is 6,356,583.8 meters and has a flattening ratio of 1/294.9786982 (DMA, 1984).
NAD27s semi-major axis along the equatorial plane is 69.4 meters wider. WGS-84 semi-minor axis has 168.514 meters in height over NAD27. This means that NAD27 is shorter in height, but wider in width than WGS-84.
Ellipsoid / Semi-Major Axis / Semi-minor Axis / FlatteningWGS-84 / 6,378,137.0 m / 6,356,752.314 m / 1/298.257223563
NAD27 (Clarke 1866) / 6,378,206.4 m / 6,356,583.8 m / 1/294.9786982
Comparison / NAD27 is 69 m wider at equatorial plane / WGS-84 is 168.514m taller between poles
Table 1- Table showing the differences of reference ellipsoid definitions of WGS-84 and NAD27.
Step 6:
The topographic distance results of all six routes were compared to the geodetic distances. Additionally, the topographic distance data gathered using the WGS-84 ellipsoid was compared to the topographic distance data gathered using the NAD27 ellipsoid as shown in Table 2.
Dataset / Surface Dist. (m) / Spatial Dist. (m) / Geodetic Dist. (m) / Additional Topographic Dist. (m)Flats Site / 3228.083
Flats Site WGS84 / 3218.8352 / 3218.1719 / 0.663326
Flats Site NAD27 / 3218.8368 / 3218.2148 / 0.622006
Difference / 0.0016 / 0.0429
Organ Site WGS84 / 18391.4434 / 18369.5695 / 21.873945
Organ Site NAD27 / 18391.4425 / 18369.4668 / 21.975684
Difference / 0.0009 / 0.1027
Organ 10% WGS84 / 18395.5594 / 18369.5695 / 25.989945
Organ 10% NAD27 / 18395.5676 / 18369.4668 / 26.100784
Difference / 0.0082 / 0.1027
Table 2- Table showing the measurement values of the surface distance of the Flats Site, and the Spatial and Geodetic Distances of the Flats Site, Organ Site and the Organ Site with the modified elevation data.
The Additional Topographic Distance Column illustrates the difference in topographical distances and geodetic distances. The difference in the Flats Site was around 6 decimeters, while the difference in the Organ Mountain site was close to 22 meters. The topographic distance is closer to the actual surface distance measured by the measuring wheel, than the geodetic distance.
When the height of the Organ Mountain Site was increased by 10%, the difference between the topographic distance and geodetic distance increased. The topographic distance of the mountain range is a full 26 meters more than the geodetic distance.
The geodetic distance column shows the difference between the WGS-84 and NAD27 ellipsoids. The shapes of the ellipsoids and the location of the site on the ellipsoids create the small differences of approx. 4 centimeters and 1 decimeter.
It was anticipated there would be differences in topographical distances between the ellipsoids WGS-84 and NAD27. The differences are small and insignificant, most likely caused by truncating errors in the software. The reason for the small difference is because although each ellipsoid uses different coordinates, they represent points in space accurately.
Causes of Geodetic Distance Differences
The differences in the geodetic distance between WGS-84 and NAD27 are greater than the differences in topographical distance. The larger differences in geodetic distance measurements highlight the importance of selecting the proper reference ellipsoid and its application for a particular geographic region. The physical characteristic differences of each reference ellipsoid will cause differences in geodetic distance measurements. The flattening ratio of each reference ellipsoid is determined by its defined lengths of the semi-major axis, semi-minor axis, and the axis of symmetry. Other differences between WGS-84 and NAD27 are the origin and orientation of the ellipsoids. The origin or the center of NAD27’s reference ellipsoid, Clarke 1866 is approximately 236 meters away from WGS-84 Earth-centered Earth-fixed reference ellipsoid. The orientation of these reference ellipsoids differ by nearly 11 minutes of angle, with NAD27 tilted counter-clockwise to WGS 84. (Roman, 2007) Additionally, WGS-84 was developed to fit the overall shape of the whole Earth, and is not a customized to represent the United States as a localized reference ellipsoid. NAD27, which utilizes the Clarke 1866 ellipsoid, is more of a custom fit to the United States. (Wolf & Ghilani, 2006)
Study Results
The results of this study are as follows.
1. The greater the elevation changes between two locations, the greater the difference between the topographic and geodetic distance.
2. The topographical distance of datasets with extreme elevation changes will be much greater than the geodetic distance.
3. Datasets with smaller elevation changes will have similar topographic and geodetic distances.
These resultsillustrate the need for a tool within GIS software to measure typographical distance. The topographic distanceof datasets with elevation changes are greater than the geodetic distances; this resultwarrants further study into the necessity of using topographic distance analysis within GIS software.
Suggested Distance Utilities within ArcGIS
After conducting this research, it is suggested for ESRI to develop the following utilities:
1- Topographic Distance for Polylines- A utility that generates topographic distances between polyline nodes from elevation values derived from Digital Elevation Models.
2- Topographic Distance Buffers- A utility to calculate radial topographic distances from a given point or object.
The following studies may be useful for utility development.
Meyer (1994) developed a Distance Watershed Algorithm that utilizes watershed lines to calculate geodetic distances. It also uses slope to calculate the topographic distances of water traveling toward the minimum watershed line values from a chosen starting point. The discussion by Meyer of topographic distances within a digital environment may be useful in creating topographic distances within a GIS environment.
Yang et al (2010) proposed a cell based algorithm for evaluating Directional Distance. This algorithm is used to identify nearest object borderlines to a given point within the map. The algorithm pre-processes a map into a grid, identifies the nearest object borderline and creates radiating “fetch lines” in multiple directions. Yang identifies a fetch line as a”… distance from a point to the nearest (object) in a given direction.” The fetch line segment of this algorithm may be useful in creating radial topographic distances such as topographic buffers.
References
Defense Mapping Agency. 1984. Geodesy for the Layman. Washington, D.C. Available at
ESRI. (n.d.). GIS Dictionary. Retrieved from ellipsoid of 1866
Open Geospatial Consortium, T. C. (2007, December 28). Schema for Coverage Geometry and Functions. Retrieved from
Meyer, F. (1994). Topographic Distance and Watershed Lines. Signal Processing, 113 - 125. Retrieved from