Geodesy is not a dead science as some people think. GPS satellites are geodetic satellites. Their orbits have origins at the center of mass of the earth, and are tracked from stations located on a geodetic datum or reference frame.

The classic geodesy textbook, Physical Geodesy,1 states in the first paragraph of the preface: “Almost every geodetic measurement depends in a fundamental way on the earth’s gravity field. Therefore, the study of the physical properties of the gravity field and their geodetic applications, which are the subject of physical geodesy, form an essential part of the geodesist’s education.” This article is the first of two that will explain in layman’s terms how a gravimetric geoid is determined. I’ll be using my geodetic library as the main reference, but will also cite the National Geodetic Survey (NGS).

The geoid is an equipotential surface of the earth’s gravity field that best fits global mean sea level. The famous mathematician and geodesist Carl Friedrich Gauss (1777-1855) stated, “What we call the surface of the earth in the geometric sense is nothing more than the surface which intersects everywhere the direction of gravity at right angles, and part of which coincides with the surface of the oceans.”2 Figure 1 shows the geoid and other equipotential surfaces with plumb lines perpendicular to each equipotential surface.

Physical geodesy, simply stated, is that branch of geodesy concerned with the determination of coordinates that have a direct dependence on the gravity field of the earth. For this article, the coordinates of concern are heights, specifically orthometric heights and geoid heights. I will also discuss ellipsoid heights, relative to geometric ellipsoids, since they play a part in gravimetric height determination.

Figure 1. The geoid and other equipotential surfaces with plumb lines perpendicular to each equipotential surface.

Vertical Control

Since the 1800s, vertical control networks were established by geometric leveling within the framework of the national geodetic surveys but independent from the horizontal control system. Heights were referenced to a level surface close to the geoid and defined by the mean sea level as observed at a tide gauge. The geoid was not needed in this separate treatment of horizontal position and height, but it played a major role as a geometric representation of the earth’s gravity field.

The geoid is used in geodesy as a reference surface for heights and depths. The orthometric height, H, is defined as the linear distance between the surface point and the geoid, measured along the curved plumb line. This definition corresponds to the common understanding of “height above sea level.”

The NGS is currently concerned with this topic and has listed “Improve gravity field modeling” as one of five technical improvements it will focus on in the next 10 years.3 Quoting from the NGS 10-year plan, “The acceleration of gravity at the surface of the Earth is usually requested at accuracies of 0.1 – 1.0 mGal and occasionally higher. The creation of an accurate geoid model requires highly accurate and continentally consistent gravity measurements. This means that the acceleration of gravity at points used in defining the NSRS [National Spatial Reference System] should have an absolute accuracy of 10 microGals at any time.”4

Most POB readers know that the acceleration of gravity is approximately 32 feet/sec2. For scientific work, metric units are preferred, and the unit of measure is the Gal (named after Galileo Galilei). A Gal is 1 cm/sec2. Gravity varies from 978 Gals at the equator to 983 Gals at the poles. 1 mGal = 0.001 Gal and 10 microGals = 0.000010 Gal. There are instruments that can measure gravity to those accuracies.

Quoting again from the NGS 10-year plan, “NGS recognizes the strong relative accuracies of geodetic leveling, but also recognizes its impracticality as the primary tool for defining a country-wide consistent orthometric height network that is maintainable to an accuracy that meets all user needs. For years the goal has been to achieve such a network via GPS and an accurate gravimetric geoid, but at what accuracy? NGS is not aware of either any achievable method or practical application of absolute orthometric height accuracies better than 1 centimeter. Certainly local relative accuracies of 1 millimeter or better exist and are needed. These can be achieved via geodetic leveling if NGS provides access to an accurate height at a starting point. As such the gravimetric geoid used in defining the NSRS should have an absolute accuracy less than 1 centimeter anyplace at any time.”

Gravity Measurements

The classical method of determining gravity was with a pendulum. Geodesists and other scientists designed and built sophisticated pendulum devices that worked in a vacuum, and then traveled to Potsdam, Germany, to calibrate their instruments. Upon returning home, a gravity station was established as a national standard, and other pendulum devices would be taken to that station for calibration. This was a time-consuming process because stations with a known value of gravity were few and far between.

In the early 1900s, a graduate student at the University of Texas at Austin developed a relative gravity meter that could determine the difference in gravity. Julian LaCoste and his advisor, Professor Romberg, opened a factory in Austin that manufactured this instrument, which is still in business today. I have used these instruments on gravity surveys in the States and in Egypt. The instrument is positioned over a point of known gravity and then moved to a station where gravity is to be determined. Observation time is less than 30 minutes. Because the instrument would “drift,” observers had to return to the starting gravity station at the end of the survey to eliminate the drift. Just like running a traverse, you have to start and end at known positions.

In the 1950s, an absolute gravity meter was developed in England; it was large and not portable, but significantly more accurate than the pendulum devices. A transportable instrument was designed in 1958 by Dr. Faller, then a student at Princeton University. I had the pleasure of attending lectures by Dr. Faller when he visited Ohio State University when I was in graduate school.

Measuring gravity with an absolute gravity meter is time-consuming, but not as time-consuming as using a pendulum device. A firm in Arvada, Colo., Micro-g Solutions Inc., manufactures three free fall gravity meters. Its FG5 instrument weighs 320 kilograms (704 lbs) and is in six containers. It can determine gravity to an accuracy of 15 mGal in 3.75 minutes and 0.1 microGal in 6.25 hours. Interestingly, the firms of Micro-g solutions and LaCoste and Romberg have combined, operating under the name Micro-g LaCoste.

The common practice is to use an absolute gravity meter to establish accurate gravity stations, and then use a relative gravity meter to densify the area around the known point. Measuring gravity on the oceans and from an aircraft has been a challenge due to the unstable mounts for the instruments. Progress has been made, and I’ve been told that 1 mGal can be achieved. NGS will be experimenting with airborne gravity meters in places where it would be difficult to perform conventional gravity surveys.

Measuring Gravity in New Mexico

About 15 years ago, I approached the Department of Defense surveying office at Holliman Air Force Base in New Mexico about the possibility of determining a value of absolute gravity at two stations on the New Mexico State University campus. The office agreed to the idea and sent two men to make the measurements.

Holliman Air Force Base is about 70 miles from the campus. The two men who made the measurements used a LaCoste and Romberg instrument, which had to be operated at a certain temperature and kept under power for the entire survey. The night before the survey, the instrument was plugged into an electrical outlet to charge the battery and keep the instrument at operating temperature. On the morning of the survey, the observing procedure was as follows:

1. A reading was taken at a known gravity station on the air force base.

2. With the instrument under power, they drove toward the university. About 15 miles away from the campus, they stopped and made an observation called a “swing point.” This established a known gravity station at the top of the mountain pass leading into the city.

3. They drove into the city, went directly to city hall and made an observation on a known gravity station.

4. They drove to the university and made observations on the two stations we requested.

5. They reversed the process by returning to city hall, the swing point and the gravity station at the Air Force Base, making observations on each station.

It took the two men about four hours to complete all observations. Two weeks later, they sent a letter giving the absolute gravity for the two stations on campus.

In my next column in October, I will continue on this topic, explaining how gravity measurements are used to determine geoid height, from which a geoid model can then be created.