Charles R. Schwarz, Editor
National Geodetic Survey

Charting and Geodetic Services

National Ocean Service
Rockville, MD 20852
December 1989

Charles R. Schwarz

This chapter addresses the differences between the North American Datum of 1983 and the World Geodetic System of 1984 (WGS 84) of the U.S. Defense Mapping Agency (DMA). Both NAD 83 and WGS 84 were defined (in words) to be geocentric, and oriented as the BIH Terrestrial System. In principle, the three-dimensional coordinates of a single physical point should therefore be the same in both systems; in practice, small differences are sometimes found. The original intent was that both systems would also use the Geodetic Reference System of 1980 (GRS 80) as a reference ellipsoid. As it happened, the WGS 84 ellipsoid differs very slightly from GRS 80.

22.1 THE CONCEPT OF A GEODETIC DATUM

To understand the sources and importance of these differences, it is necessary to take a close look at the concept of a datum and at how the coordinates in a datum are actually computed. The concept of a horizontal geodetic datum actually involves several ideas. A definition almost always begins with some form of specification of a reference surface. This involves the specification of the dimension of a reference ellipsoid, as well as quantities which determine the origin and orientation of the ellipsoid with respect to the Earth. (See, for instance, National Geodetic Survey, 1986.)

22.1.2 A Datum as Ellipsoid

The WGS 84 ellipsoid differs very slightly from the GRS 80 ellipsoid which was used for NAD 83. The differences can be seen in tables 22.1 and 22.2. These differences arise because DMA used the normalized form of the coefficient of the second zonal harmonic of the gravity field as a fundamental constant, while GRS 80 had used the unnormalized form. Furthermore, the normalized value used by DMA was obtained by using the mathematical relationship

and rounding the result to eight significant figures (Defense Mapping Agency' 1987). Thus quantities depending directly on the form factor, such as the flattening, generally differ after the eighth significant digit, while linear quantities, such as the semiminor axis, generally differ after the tenth significant digit. These differences, while small, can cause confusion among users who attempt to compare computations in the two systems. Most analysts agree that these differences will be of no significance for practical applications.

22.1.3 A Datum as Coordinates

The specification of a reference surface defines a datum only in an idealized sense. This specification is usually supplemented by a second definition which states that a horizontal geodetic datum is composed of the adopted horizontal coordinates of a set of physical points in that datum. This is the operational definition. It is from this second definition-the adopted coordinates-that we actually determine the origin and orientation of a datum. In this sense, the first definition is more a statement of intention than a statement of reality.

There are other qualities connoted by the concept of a datum. The idea that there are adopted coordinates implies that a datum is stable-the coordinates seldom change. Furthermore, a datum must be extensible-there must be some way of computing the coordinates of new points. Often there are preferred or expected ways to determine these new coordinates. For instance, it is expected that new NAD 83 points will be established by running new horizontal surveys using theodolites and distance measuring equipment. It is also expected that if one uses Global Positioning System (GPS) observations in the single point positioning mode, together with a satellite ephemeris given in the WGS 84 coordinate system, then the resulting coordinates will also be in WGS 84.

The idea of extending a datum by adding new points implies that there are some fundamental points from which the process is begun. By definition, these are the points that participate in the initial network adjustment, irrespective of accuracy or order. All of the points that participated in the NAD 83 adjustment are thus fundamental points of that datum. New points that will be added are not. In most geodetic datums, the distinction between fundamental and non-fundamental points has been lost. Typically a new point surveyed to first-order accuracy and adjusted into the network has been treated as equal in usefulness to a fundamental first-order point, and superior to a fundamental second-order point. This common, but incorrect, practice has often misled users as to the accuracy of a point's coordinates.

Some physical points are fundamental to both NAD 83 and WGS 84. The coordinates of these points in the two systems may differ because the two adjustments which produced the coordinates of the two sets of fundamental points were based on two different sets

of observations. For instance, a Doppler survey may have been performed at a point by either DMA or NGS, and the data may have been exchanged, so that both agencies had exactly the same data set. Furthermore, the two agencies agreed on all the details of data processing, so that both agencies determined the same set of Doppler-derived three-dimensional coordinates. Even further, the agencies agreed extly on how to transform the Doppler-derived NWSC 9Z-2 coordinates into the BIH Terrestrial System. However, in the NAD 83 adjustment these coordinates received corrections due to interactions with other observations (mostly classical triangulation and traverses), while no such corrections were made in the determination of the WGS 84 coordinates. These corrections can amount to a meter or more. However, both adjustments are still thought to be valid. The differences of coordinates are thought to be simply the effect of small random measurement errors in the two sets of observations. Even though differences as large as several meters are found occasionally, the expected value of these differences is zero.

Other physical points are derived, rather than fundamental. For these points, coordinates in the two datums may differ for two reasons:

1. The two coordinate determinations are based on different fundamental points.

2. The observations used to extend the datum may differ.

The method of labeling the datum for derived points is mainly a matter of convention. The actual physical observations (such as angles or distances) are themselves independent of any datum. When a new point is surveyed for the purpose of determining its coordinates, the survey must be tied to one or more old points. If the coordinates of the old point in the NAD 83 system are used in the computations, the coordinates of the new point are also said to be in NAD 83. Similarly, if the coordinates of the old point in WGS 84 are used, the coordinates of the new point are said to be in WGS 84.

22.2 USING NAD 83 AND WGS 84 POINTS

NAD 83 and WGS 84 should be thought of" a! geographically overlapping datums (in the sense of datum as adopted coordinates). There will be points with coordinates in both datums. The action to take when confronted with two sets of coordinates for a single point is up to the user. If neither position determination contains a blunder, then the differences of coordinates should be small. In fact, the expected size of these differences can be computed from the uncertainties of the two determinations. If the differences are smaller than the accuracy required, then the user may select either determination (or some combination of the two).

"Small" differences must be properly understood here. The actual difference between coordinates may quite possibly be a meter or more. Although this might be disturbing to some, this is actually the magnitude of the uncertainty of the differences that would be computed from the uncertainties of the two coordinate determinations. It reflects the fact that the two coordinate determinations are independent and uncorrelated.

22.2.1 Mixing Coordinates

Surveyors are familiar with the limitations imposed when mixing the results of two independent surveys (or two datums) in a single positioning problem. Within a single survey, the relative coordinates of nearby points are much more accurate than the coordinates of either. This is not the case if the two sets of coordinates come from different surveys.

Suppose that within a local area there is both an NAD 83 point and a WGS 84 point. Suppose also that a survey is run to determine the distance between the points. The measured distance could differ from the value computed from the coordinates by a meter or more. Some might find this difference to be disturbing, but it is only a reflection of the fact that the variance of relative coordinates from two different surveys is much larger than the variance of the relative coordinates of two points from the same survey.

We thus say that the most common reason that we find differences between the NAD 83 and the WGS 84 coordinates of a point is that we are dealing with two independent determinations of the same thing. Both determinations are affected by the small statistical variations which are inherent in any measurement process. Each has its own associated standard deviation, but each is valid in its own way. The user may chose either, but must be careful about mixing coordinates.

22.2.2 Area of Validity

Some investigators have suggested that a difference between NAD 83 and WGS 84 is that NAD 83 is valid only within North America, while WGS 84 is valid worldwide. This is incorrect. If one has an accurate method of extending NAD 83 outside of North America, then there is no reason not to do so, nor is there any reason to think that the resulting coordinates would differ from WGS 84 coordinates. In fact, as

part of the NAD 83 adjustment, Doppler observations were used to extend the datum outside of the contiguous survey networks to isolated areas such as Greenland, Puerto Rico, and Hawaii.

22.2.3 Extending the Datum Offshore

The case of a ship navigating offshore is of particular interest to the hydrographic and bathymetric surveying activities of the National Ocean Service. If the ship navigates with a radio navigation system using shore-based transmitters, and if the coordinates of the transmitters are known in NAD 83 coordinates, then the navigated position will also be in NAD 83. The ship may also navigate with a satellite-based system which yields coordinates in the WGS 84 system. We expect both navigation systems to provide the same coordinates at each instant of time; but due to unavoidable measurement errors we may find small differences. The existence of such differences should not be interpreted to mean that there is a difference in the two datums. Unless there is some reason to suspect that one or the other navigation system is producing serious errors, the differences between the coordinates produced by the two systems should be attributed simply to measurement error. The navigator may choose to use either set of coordinates. Only the navigator with extraordinarily demanding accuracy requirements will need to worry about computing some combination of the two sets of coordinates.

22.2.4 Computational Differences

There are some differences between NAD 83 and WGS 84 which may arise because of approximations made in a particular method of computing coordinates. For most applications, the effect of these approximations is considerably smaller than the effect of observational errors. These differences are important only if one is testing the accuracy of a set of equations or a method of computing coordinates.

One such set of approximations concerns the different ellipsoids used for NAD 83 and WGS 84. This difference has no effect on the three-dimensional coordinates of a point computed by satellite surveying. If such a set of three-dimensional Cartesian coordinates is converted to latitude and longitude using the two coordinate systems, there would be no difference in the longitudes, and the latitude difference would be

which reaches a maximum value of 0.000003 second of arc (or 0.0001 meter) at a latitude of 45 degrees. It is assumed that most users will ignore this very small difference.

Another approximation concerns the datum shifts computed for map sheets. The National Geodetic Survey has computed a latitude and longitude shift for every map sheet published by the U.S. Geological Survey. These pairs of numbers were computed by meaning the actual shifts from NAD 27 to NAD 83 at all points falling on the map sheet. These mean shifts are then assumed to be correct for the entire map sheet. Thus a very small error, amounting to the difference between the actual datum shift and the mean datum shift for the map sheet, is committed at each point. This error is everywhere much smaller than the observational errors committed when coordinates are scaled from maps.

22.3 REFERENCES

Moritz, Helmut, 1984: "Geodetic Reference System 1980." Bulletin Geodesique, vol. 54, No. 3 (also republished in vol. 58, No. 3). International Association of Geodesy, Paris.

National Geodetic Survey, 1986: Geodetic Glossary. National Geodetic Information Branch, NOAA, Rockville, MD 20852, 274 pp.

Defense Mapping Agency, 1987: "Department of Defense World Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems." DMA Technical Report 8350.2. Defense Mapping Agency, Hydrographic/Topographic Center, Washington, DC 20315.