Introduction to Geodesy



Introduction to Geodesy

D. Tortosa

Geodesy and the Global Positioning System

In order for the Global Positioning System to work, not only is it necessary to use satellites in order to triangulate and locate a position, but the calculated position has to be placed within a coordinate reference framework.

Geodesy is the mathematics of the shape and size of the earth from which a coordinate reference system is based. The greater the accuracy of the coordinate reference system, then the greater the accuracy of the GPS position relative to the real world.

The Ellipsoid

The earth's shape is best described by an ellipse (figure 1), or more correctly, by an Ellipsoid, which is formed by rotating an ellipse around its minor axis. The ellipsoid can best be defined by two parametres: 1. its semi-major axis, 2. its flattening (f).

The reference system for the ellipsoid is the Cartesian Coordinate System (X, Y, and Z). The Z axis is parallel to the mean rotation of the earth, the X-axis parallel to the defined zero meridian (usually Greenwich Meridian), and the Y-axis is perpendicular to both X and Z (figure 2).

Over the years there have been a variety of models for the ellipsoid of the earth. Different ellipsoids have been adopted for different parts of the globe as described in Table 1. The most recent model that is used by the NAVSTAR GPS is the World Geodetic System of 1984 (WGS 84).

The Geoid

The ellipsoid provides a mathematical model which is used to represent the earth; distortions of the earth's surface caused by gravitational variations produce a more detailed model which is referred to as the Mean Sea Surface also known as the Geoid. The mean sea surface equates very closely to Mean Sea Level, and so the geoid closely coincides with mean sea level (figure 3 and 4).

A important distinction between the ellipsoid and the geoid is that a true representation of the Geoid is created from real measurement data whereas the Ellipsoid is mathematically defined. The most recent derivations of the Geoid have been produced exclusively from direct satellite tracking observations (figure 5). GPS receivers such as the Magellan NAV 5000 PRO and the Garmin SRVY II contain a geoidal "model" which they use to determine Mean Sea Level. This model is a mathematical representation of the earth's gravity field or equipotential surface and is accurate to within a few metres.

Map Datums

In order to determine a position on the earth's surface, knowledge of the both the Ellipsoid and the Geoid are required for that part of the earth. The combination of the Ellipsoid/Geoid model is a datum (ie. North American Datum 27 or NAD-27). For first order geodetic surveys, knowledge of the both the Ellipsoid and the Geoid is required to determine an exact position. This may involve measuring the local gravity field and compensation for large masses of land (ie. mountains). For second, third and fourth order surveys, the ellipsoid provides sufficient accuracy.

Different parts of the world use different models of the Ellipsoid/Geoid to determine geodetic coordinates (figure 6). The most recent Ellipsoid model, GRS-80, is designed to be the world standard. For the purposes of this course we will use the NAD-27 datum which is based on the Clarke 1866 ellipsoid (Table 1).

The GPS Co-ordinate System

The Global Positioning System derives position using the World Geodetic System 1984 (WGS-84 Ellipsoid) coordinates. In order to relate these Cartesian Coordinates to a local geodetic coordinate system , such as the North American Datum (NAD-27), a GPS receiver has to perform a mathematical transformation to process the X, Y, Z coordinates to latitude, longitude, and elevation.

The GPS derived WGS-84 Cartesian coordinates can be recalculated to NAD-27 or any other datum as a Latitude and Longitude; the calculated elevation can be determined as either:

1. Height above the Ellipsoid

2. Height above the Geoid (external data required)

3. Height above Mean Sea Level (MSL)

The GPS receiver and post processing software should be set to the same datum as the map which will be used to represent the GPS position, otherwise large errors can result. For example NAD-27 coordinates can be offset by as much as 200 metres from NAD-83 coordinates. The NAD-83 geodetic coordinate system is currently the most accurate and best reflects the real world.

For digital basemaps such as the National Topographic Series, Ontario Basemaps, and Canadian Hydrographic Charts the geodetic coordinate system is NAD-27. However, newer maps are being produced using NAD-83 (British Columbia's TRIM maps), and there are plans for the conversion of NAD-27 digital basemaps to a NAD- 83 system in the future.

Geodesy as applied to Desktop Mapping and GIS

Geometric Projections

As discussed previously the earth is the shape of an ellipsoid (also referred to as a spheroid) which can be easily represented as a globe using the system of latitude and longitude coordinates (figure 7). A globe, however, is not as convenient to use as a two dimensional map, therefore a mathematical transformation is required to represent a spherical surface on a flat map (Figure 8).

The transformation results in some distortion on the map. Map projections differ in the degree of distortion that is introduced in the representation of area, shape, distance, and direction. Some projections are more appropriate for one area of the earth than another. For example, in polar regions the polar stereographic projection would be the most appropriate since it offers the least distortion. The most commonly used projection for scales of 1:500,000 and larger is the Transverse Mercator or Universal Transverse Mercator (UTM). The transformation process to represent a spheroid (the Earth) on a 2 dimensional surface is described in figure 9; some commonly used projections are shown in figures 10 and 11.

The mathematical transformation for a projection is based on the choice of datum. The setup or configuration file for the desktop mapping/GIS software requires information on the datum being used in the form of an Equatorial Radius parameter and a Flattening parameter which define the ellipsoid/geoid model. In most cases, the datum is NAD-27 (digital NTS and OBM maps).

It is important to note that Computer Assisted Drafting (CAD) software such as AutoCAD and some Geographic Information System software such as ArcCAD are based on a Cartesian Coordinate system which allows for only the Universal Transverse Mercator projection. Other projections are not directly supported by CAD and CAD-based Geographic Information Systems software.

Datums and Surveying Accuracy

P. Beach

All surveying applications, including GPS, must be undertaken in relation to some form of coordinate reference framework. This reference framework is known as a datum. "Ideally a datum is a system of stable survey monuments at some convenient spacing with unique and invariable coordinates appreciably more accurate than any survey work that they might be used to control" [Jones, 1973]. A more complicated definition would state that a datum is a mathematically derived network of control stations which is based on an ellipsoid geoid model. We will be coming back to the idea of an ellipsoid geoid model a number of times.

In North America, all control survey work is related to what is known as the North American Datum of 1927 or NAD27. The starting point or origin point for NAD27 is located at a place called Meades Ranch, Kansas, which is the approximate center of the lower 48 states. All survey points and networks in North America are tied into this point.

All survey work must be referenced to an identifiable, consistent and continuous surface which defines the surface of the Earth. This is where the geoid comes into play. The geoid is an equipotential surface of gravity or in other words, the surface at which the direction of gravity is always perpendicular to. The level of the oceans closely approximates the geoidal surface. The geoid is a relatively smooth surface but it is also fairly irregular. This irregularity is due to anomalies or changes in the density of crustal materials which affects the equipotential gravity surface. For this reason, the level of the oceans itself varies by 100 to 200 metres from one region of the planet to another.

The geoid is an identifiable surface since it is based on gravity which is measurable. Since the geoid is identifiable and is continuous you would think that it would be the perfect surface to tie our survey work to. There is one major drawback though with the geoid, it is a surface which is fairly complicated and is difficult to model mathematically. Referencing our survey work to the geoid would involve horrendous mathematical calculations which makes it practically impossible. A datum cannot therefore be based on a geoid model entirely.

The solution to the above problem is to define an ellipsoid of revolution which closely approximates the geoid. An ellipsoid is a simple mathematical surface upon which mathematical computations can be performed. Using an ellipsoid to reference our survey work solves one problem but now creates a new one. An ellipsoid of revolution is not an identifiable or measurable object, or in other words it has no physical presence that can be directly detected with any instrumentation.

Now we have seen that a datum to reference our survey work to can not be based on a geoid model or an ellipsoid model. The solution to this problem is to use a combination geoid/ellipsoid model. If we can reference our survey work to the geoid (altitudes above or below sea level (geoid)) and we have gravity data to define the shape of the geoid, we can determine the geoid ellipsoid separation and thereby relate our survey work to the ellipsoid.

In the past though, before satellites and gravity surveys, it was impossible to develop an accurate and detailed shape for the geoid. Instead, geodesists defined an ellipsoid that closely approximated the geoid for there area on Earth as best they could so that the geoid ellipsoid separation would be as small as possible and then assume that the geoid ellipsoid separation would be zero. In different areas on Earth different ellipsoids were used to approximate the geoid because the geoid shape varies worldwide. The ellipsoid that best fit the geoid in North America was not the best fit for Europe and vice versa.

In North America the NAD27 datum was based on the Clarke 1866 ellipsoid which was the best approximation of the geoid for North America at that time. NAD27 assumes that the geoid ellipsoid separation is zero for all of North America.

As it turned out, the Clarke 1866 ellipsoid was a fairly good approximation of the geoid in much of the lower 48 states, but in Canada the geoid ellipsoid separations were significant, up to 50 metres. As the NAD27 survey control network spread across North America, random and systematic errors accumulated and resulted in distortions in the control network. Regional distortions in relative positioning developed that were in the order of 10 to 20 metres. The main reason for this was not taking the geoid ellipsoid separation into account in the NAD27 datum.

In an attempt to deal with the distortions in the NAD27 control network a number of regional readjustments were undertaken to lessen the effects of the distortions. These regional readjustments did not eliminate the distortions but simply distributed them evenly over an entire region. This improved the relative positioning accuracy within a region but caused significant problems between adjacent regions or regions not included in the readjustment. Continent wide continuity was sacrificed to improve local network accuracy.

In Ontario, two official readjustments are recognized. The first is known as the Ontario 1974 readjustment. In this readjustment some first order control points in Southern Ontario were held fixed while the rest of the province was readjusted. The second readjustment is known as the May76 adjustment. This was a Federal Government adjustment which used doppler satellites to determine the coordinates of some first order control points. Listed NAD27 coordinates in Ontario, such as in MNRs Cosine database will qualify whether a coordinate is 74 adjusted or May76 adjusted. The difference between an adjusted and a non adjusted coordinates may be in the order of 0 to 20 metres. Differences between 74 adjusted coordinates and May76 adjusted coordinates may be in the order of several metres as well.

As time passed the errors continued to accumulate within the NAD27 control network as well as within other datum reference networks around the world. In the last two decades, in an ongoing process, an attempt was, and is still being made to start fresh with a new datum, one which could be used anywhere on Earth. An ellipsoid was defined that was determined to be the best fit for the geoid for the whole planet as a whole, not just the best fit for one area. This ellipsoid is known as the GRS80 Ellipsoid. Unlike the continental or sub continental datums and their associated ellipsoids of the past, the center of this ellipsoid coincides with the center of mass of the Earth. This is important because the center of mass for the Earth can be located relative to the surface of the Earth through the use of satellites. The use of satellites and increasingly accurate and more abundant gravity surveys allows modern geodesists to develop a reasonably accurate geoidal surface model for the planet. Using determined geoidal ellipsoidal separations we can now relate our survey control networks to a reference ellipsoid that is common to the whole world. The geoid ellipsoid model based on the GRS80 ellipsoid is known as WGS84 (World Geodetic Reference System of 1984). WGS84 is an orthogonal X,Y,Z coordinate system which has its origin at the Earths center of mass, its X axis passing through the Greenwich meridian and the Z axis passes through the North pole and the Y axis is perpendicular to the X and Z axis.

In North America we are currently adopting a new datum to which all survey work in North America will be tied to. This datum is referred to as NAD83 and is based on WGS84.

Converting to from NAD27 to NAD83 is not a simple process. It is not just a matter of a simple uniform and constant conversion factor. It involves establishing transformation algorithms to deal with the differing reference ellipsoids, NAD27 distortions, adjustments and other problems. The Canadian Federal Government has developed a National Transformation Program to do this. The program has been completed to transform 1974 adjusted NAD27 coordinates to NAD83, but transforming May76 coordinates to NAD83 is currently in the works.

In Ontario, the MNR has now updated the Cosine database of control points in Ontario to NAD83. There are plans in the works to transform all of the OBMs in Ontario to NAD83 after the May76 to NAD83 portion of the National Transformation program is completed and tested. Currently, OBMs are May76 adjusted NAD27. Many GIS software packages will do rough conversions from one datum to another and may be suitable for most applications.

The latitude and longitude coordinates and UTM coordinates will be significantly different for any given control point when going from NAD27 to NAD83. Latitude and longitude shifts may be as much as 120 metres on the West Coast to 70 metres on the East Coast. UTM coordinates will differ by as much as 245 metres mostly in a north south direction. These shifts result from using the Earth centered (geocentric) ellipsoid associated with the WGS84 datum (GRS80 Ellipsoid) as opposed to the non geocentric ellipsoid associated with NAD27 (1866 Clarke Ellipsoid) and to the removal of other NAD27 errors, distortions and readjustments.

The introduction of the WGS84 coordinate reference system is very important because for the first time we have a world wide reference system that is continuous and consistent everywhere. The reference framework will still have small inaccuracies within it, but it will be a definite improvement over existing coordinate reference systems. Any future improvements or adjustments to WGS84 or NAD83 can be done simultaneously and uniformly throughout the framework rather than in a piecewise fashion as in the past.

GPS determines its initial positions as WGS84 coordinates and then must transform them to latitudes and longitudes or UTM coordinates in the datum of your choice (such as NAD27). The GPS receivers transform coordinates between datums by using offsets between the coordinate axes (delta X, delta Y, delta Z) and the change in size (delta A) and shape (delta F) of the reference ellipsoid. The transformation formula and the delta values for different datums are commonly listed in an appendix in the GPS receiver manual. Most GPS receivers have a list of possible datums to choose from, or have an option to type in the delta values for a datum.

For the Garmin GPS unit, many datums are listed to choose from, depending where you are on Earth. For North America there are three possible datums to choose from: NAD27, NAD27 Canada, and NAD83. NAD27 is also referred to as NAD27 CONUS (Continental USA). NAD27 Canada is basically the same thing as NAD27 except that the Canadian government has determined slightly different delta values from WGS84 that agree better with Canadian survey fabric. The difference between NAD27 and NAD27 Canada is between approximately 1 and 8 metres. The Magellan GPS receiver does not include a NAD27 Canada option, though delta values for it are included in an appendix in the Users Manual.

NAD83 (or WGS84) is an option on all GPS receivers and you would think that it would be the datum of choice for most users. In reality, this is not the case. The reason for this is because most of the paper and digital maps that are currently available are based on NAD27. The datum for your GPS data must be setup to match the datum of the base map that you are going to plot that data on. Any GIS map or hardcopy map will generally represent coordinate information as latitudes and longitudes or UTMs in reference to a map datum. Throughout North America the process has begun to convert existing maps to NAD83, but this will take a number of years.

As a GPS receiver or its post processing software transform its initial WGS84 coordinates to the datum of your choice such as NAD27 Canada, only the shifts between the two reference ellipsoids ( in this case GRS80 and Clarke 1866) is taken into account. NAD27 distortions and readjustments are not taken into account. A GPS receiver does not know anything about regional readjustments. Consequently, errors up to 20 metres may result during the transformation between NAD27 and NAD83 (Junkins 1990). Only a very complicated program like the National Transformation Program will take all of these other things into account to convert from one datum to another accurately. The information in this paragraph also applies to conversions between datums in GIS software packages, though ArcInfo now does include an initial release of the National Transformation Program to be used internally for datum conversions (74 adjustment to NAD83 only).

When setup on a control point for which you know the 1974 adjusted coordinates from the Cosine database, and you take very long averages, you cannot expect the GPS unit to give you the same value as the control point because the unit knows nothing of the 1974 adjustment. When doing differential GPS, and you setup the base on a NAD83 control point or use the US Coast Guard differential signal (NAD83 control point) and then compare the averaged, corrected reading at a remote control point, with the known (adjusted coordinate), you will not get the same result, just as in the nondifferential example above. If your base and remote control points are both NAD83 you will get very good results since there are no regional readjustments or transformations to deal with when dealing with NAD83, which is the same WGS84.

When doing differential GPS between two NAD27 readjusted coordinates you will get reasonably good results if both control points are within the same regional readjustment and there is not too much distortion between the points. The reason for this is that you are entering a 1974 readjusted NAD27 coordinate (for example) as the known base coordinate into the GPS receiver post processing software, though the software does not know that it is a readjusted coordinate and treats it as if it is a NAD27 Canada coordinate, and the differential processing is determining corrections given that this coordinate is correct, then this correction is being applied to the remote, and since the difference between the readjusted coordinate and a straight NAD27 Canada coordinate should be the same for both the control and remote locations, a good result should be obtained at the remote, unless there are significant distortion errors in the readjusted coordinates. The determined remote coordinate should be reasonably accurate relative to the base coordinate that is within the same regional readjustment area, but it may not be accurate relative to other areas not in the readjustment or to base maps that do not take the readjustments or NAD27 distortions into account.

As well, as the post processing software does its differential correction it must internally transform your entered NAD27 Canada base coordinate to a WGS84 coordinate since it works on a WGS84 basis. The software then differentially corrects your remote position producing a WGS84 coordinate that it must transform back to your selected datum (NAD27 Canada). Since the software applies the same transformation equation going in and out the inaccuracies of the transformation process described earlier will be the same at the base and remote and should drop out.

Differential GPS can sometimes eliminate coordinate errors common to both the remote and control locations and give good relative results, but non differential GPS cannot thereby introducing the 20 metre possible inaccuracy mentioned earlier on top of all of the other errors which occur in non differential GPS work.

As a final point, every GPS receiver contains an internal geoid model. The geoid model is used to display your GPS positions relative to the geoid or in other words to give your altitudes above sea level. The geoid model within the GPS unit is a worldwide model that does not contain some of the much more detailed gravity work that is available in certain parts of the world. For very accurate sub metre work it is important to use the more detailed gravity work and apply your own geoid ellipsoid separations rather than use the internal geoid information contained within the GPS unit.

Sources

Brown, A. 1992. The GPS Coordinate System Explained. GPS World, March 1992, pp.70-71.

Erickson, C and P. Heroux. 1994. GPS Locations for GIS: Getting them Right the First Time. Geodetic Survey Division, Natural Resources Canada.

Goadsby, J.M. 1990. NAD83 in Ontario: Pre-adoption Concerns and Policies. Moving to NAD83 the new address for georeferenced data in Canada.. The Canadian Institute of Surveying and Mapping, Ottawa, pp. 141-158.

Jiwani, Z.A. and K.M. Kelly. 1987. History of NAD27 and Subsequent Readjustments in Ontario. Northpoint, Spring 1987, pp. 21-25.

Jones, H.E. 1973. Geodetic Datums in Canada. The Canadian Surveyor (27), 2.

Junkins, D. 1990. The National Transformation for Converting Between NAD27 and NAD83 in Canada. Moving to NAD83 the new address for georeferenced data in Canada.. The Canadian Institute of Surveying and Mapping, Ottawa, pp. 16-40.

Pinch, M.C. 1990. Differences Between NAD27 and NAD83. Moving to NAD83 the new address for georeferenced data in Canada.. The Canadian Institute of Surveying and Mapping, Ottawa, pp. 1-15.

Schwarz, K.P. and M.G. Sideris. 1993. Heights and GPS. GPS World, February 1993, pp. 50-51.

Smith, J.R. 1988. Basic Geodesy. Landmark Enterprises, Rancho Cordova, CA.

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