In this month's column, I will discuss the geoid models developed by the United States National Geodetic Survey (NGS). It is possible to determine orthometric heights to a fair degree of accuracy using real-time kinematic (RTK) technology with the geoid model installed in a GPS receiver. I don't intend to get into the accuracy of these heights-there are too many variables.



The Horizontal Coordinate System

In order to understand geoid models, it's necessary to understand the horizontal and vertical coordinate systems we use in the United States. The horizontal coordinate system used in North America is based on a theoretical ellipsoid of revolution that closely approximates the shape of the Earth and is centered near the center of mass of the Earth. The ellipsoid shape is used rather than that of a sphere because although the Earth is round, the polar axis (the rotational axis) is shorter than the equatorial axis; the equatorial radius of the Earth is approximately 6,378 km-longer than the polar radius by approximately 21 km. Projecting surveying observations made on the Earth's actual surface either up or down to the surface of the ellipsoid makes it possible to survey large areas without significant distortions. Figure 1 below shows an ellipsoid of revolution that I used in my dissertation at Ohio State University in 1973. This is a map projection so the parallels of latitude are curved.

Referring to Figure 1 below, the lines north and south of the equatorial plane are called parallels of latitude. Measured in angular coordinates, latitude ranges in value from 0 degrees at the equator to +90 degrees at the North Pole and -90 degrees at the South Pole. We will use the notion ±(0 to 90)°.

Again referring to Figure 1, the north-south lines on the ellipsoid are called meridians of longitude; all meridians pass through the North and South Poles. The meridian that passes through the Greenwich Observatory in England is the origin of these longitudinal meridians, which range in value from 0° to 180° east and west.

Coordinates of points on the ellipsoid are noted by latitude and longitude. For example, I live in Las Cruces, New Mexico. There is a GPS control point at the Las Cruces airport named CRUCESAIR that has the following coordinates:

Latitude = 32° 16' 54.63123" North

Longitude = 106° 55' 22.24784" West.

In geodesy, when an ellipsoid of revolution has an established origin and an orientation, it's referred to as a horizontal datum. For North America, the ellipsoid is the Geodetic Reference System of 1980 (GRS 80) with the origin at the center of mass of the Earth and oriented by many fixed stations selected by NGS. The datum is named the North American Datum of 1983 (NAD 83).

An ellipsoid of revolution showing latitude and longitude lines.

The Vertical Coordinate System

The vertical coordinate goes by many names; not all of them are correct. A point on the ground can have an elevation, altitude, orthometric height, ellipsoid height, etc. The obvious question is, elevation above what?

In geodesy, the geoid serves as the height reference surface for describing surface topography. C.F. Gauss first described the geoid in 1828 as a model for the figure of the Earth. Gauss defined the "˜mathematical figure of the Earth' as the equipotential surface of the Earth's gravity coinciding with the mean sea level of the oceans. In 1873, J.F. Listing coined the term "geoid" to describe this mathematical surface.

Sea surfaces vary with time, and this variation can only be partially reduced by averaging over time or modeling. Perhaps a better definition is "˜the geoid can be defined as the equipotential surface that best fits mean sea level at a certain epoch.' The height above the geoid, properly called orthometric height, is the vertical coordinate of the vertical coordinate system.

Vertical coordinates, like horizontal coordinates, are referenced to a datum. The datum in use in North America today is the North American Vertical Datum of 1988 (NAVD 88). The point of origin is a bench mark called Father Point on the St. Lawrence River in Quebec, Canada.



Geoid-ellipsoid relationships.

The Geoid-Ellipsoid Relationship

The horizontal and vertical coordinate systems are independent of each other. Up until the late 1970s, transits, theodolites and total stations were the instruments used for establishing horizontal control, and levels were used for establishing vertical control. Determining the geoid height and deflection of the vertical would give the relationship between the systems. Figure 2 above shows the geoid-ellipsoid relationships.

Why not use the geoid as the reference surface for both horizontal and vertical control systems? It can't be seen in Figure 2, but the geoid is not a mathematical surface. Irregular mass distributions in the Earth's crust cause the geoid to undulate. Mountains cause the geoid to rise and areas with low density like oceans, large lakes and salt deposits cause the geoid to recess. The opposite question can be asked: why not use the ellipsoid as the reference surface for both horizontal and vertical control systems? There are scientists who would like to see that happen, but the problem is that water flows downhill, and downhill is defined with respect to the geoid and other equipotential above and below it.

Now let's look at Figure 2 in detail. Notice the notation "GEOID UNDULATION," over the water surface on the left side and then again under the land surface on the right side. Also notice the deflection of the vertical; it is the angle between the perpendicular to the geoid and the perpendicular to the ellipsoid. We don't need to know the deflection of the vertical when using GPS, but we do need to know about the two perpendiculars.

Figure 3 below is similar to Figure 2, but now we see that the geoid undulation has a value labeled N. N is Geoid Height, and that will be the subject of the remainder of this article.

Figure 4 on page 52 shows the measurements between the Earth's crust (or the Earth's surface), the geoid and the ellipsoid. The symbol H, which is labeled "˜Height Above Sea Level,' is orthometric height. This is the height of bench marks. The definition of orthometric height is the perpendicular distance from the Earth's surface, along the curved vertical, to the geoid. It is a distance, and the units are in meters; the engineering community usually converts meters to feet. Orthometric heights on bench marks have been, traditionally, established by leveling. Geoid height is labeled N, and is the distance (separation) between the ellipsoid and the geoid. The sign convention (+ and -) denotes that if the geoid is below the ellipsoid, N is negative (-) and if the geoid is above the ellipsoid, N is positive (+).

Now you will see why it was necessary to define both horizontal and vertical coordinate systems. The label h denotes ellipsoid height. Ellipsoid height, at a point, is the sum of the orthometric height and the geoid height, i.e.

h = H + N (1)

With GPS, we must rearrange this equation. With two GPS receivers observing in the static mode, the results of the survey are the "˜differences' in coordinates; difference in latitude (df), difference in longitude (dl) and the difference in ellipsoid height (dh). GPS has no information on the location of the geoid. Rearranging the above equation and using coordinate differences, we get the following equation.

dN=dh-dH (2)

This is how the difference in N can be determined with GPS. Occupy two bench marks with GPS receivers. The results will give dh, the difference in ellipsoid height. The difference in orthometric height is obtained by subtracting one orthometric height from another. With equation 2, you can determine dN. In the 1980s and early 1990s, using equation 2 was the only way surveyors could use GPS to determine orthometric heights. They would occupy at least three bench marks surrounding the area to be surveyed and interpolate dN.

Geoid undulation.

Geoid Models

Around 1991 NGS introduced a geoid model for the conterminous United States (CONUS) called GEOID90. I received a copy on six 3 1/2" floppy disks of the DOS program with a database. It worked as follows: the input was NAD 83 latitude and longitude of a point. When this information was entered, the program would display on the screen the geoid height (N) to 1/100 of a meter. For CONUS, all geoid heights were negative. Surveyors could use a rearranged version of equation 2,

dH= dh - dN (3)

Working in the relative positioning mode, they could calculate the difference in orthometric height between two observing GPS receivers. I might add that the accuracy of N determined by GEOID90 may have been accurate only to ±1 meter, but the accuracy of dN was ± a couple of centimeters. The "Read Me" document that came with GEOID90 gave accuracy estimates based on the distance between points. Important: the geoid heights determined were valid for NAD 83 horizontal coordinates and NAVD 88 vertical coordinates only.

GEOID90 was revolutionary. It made it possible to take instruments designed for establishing horizontal control and use them to get a fair estimate of the orthometric height of observed points.

Three years later, NGS introduced GEOID93. This was an updated version of GEOID90. Both GEOID90 and GEOID93 were gravimetric geoid models, meaning that they were derived from gravimetric data. GEOID93 also covered Hawaii, Puerto Rico and the American Virgin Islands.

In 1997, GEOID96 was introduced. This covered Alaska, as well as Hawaii, Puerto Rico and the American Virgin Islands. This model was different in two ways: it was no longer necessary to subtract geoid heights to get the 2-4 cm accuracy; each geoid height calculated already had that accuracy. In addition, NGS added GPS observations on bench marks to the gravitationally derived results. NGS named this a Hybrid Geoid Model. With GEOID96, the GPS equipment manufacturers began incorporating the geoid model into their RTK receivers; it was now possible to use RTK GPS for topographic surveys, etc., and get three-dimensional coordinates that were suitable for mapping.

In 2000, GEOID99 was introduced. This is the current geoid model available on the NGS website at www.ngs.noaa.gov. Like GEOID99, this is a hybrid geoid model; there are 6,200 GPS observations on bench marks in this model.

Relationship between height above sea level, ellipsoid height and geoid height

The New Geoid Model

NGS has upgraded the geoid models approximately every three years. I spoke to Daniel R. Roman, PhD, research geodesist with the NGS, who said NGS is trying to introduce a new model by the end of this calendar year. This will also be a Hybrid Geoid Model with 11,000 GPS observations on bench marks. It will use a revised approach to collocation.

This new geoid model will have a different name, USHG2003, which is United States Hybrid Geoid 2003. A new gravimetric geoid, USGG2003, will also be introduced.

Surveyors can't expect the same accuracy from geoid models as they would get from differential leveling. But with each new geoid model the expected accuracy will increase. For projects that do not require centimeter or less vertical accuracy, the geoid model will do the job.