Figure 1. Height relationships.<

The previous article in this series, “Physical Geodesy 101,” (POB August 2007) described the methods of collecting gravity measurements.

In this article I will describe how the geoid can be determined from gravity measurements only. It’s a complicated process, but easy to describe. The final product is the determination of the geoid height, N.

Let’s begin this discussion by showing how surveyors today can determine geoid heights from GPS observations on established bench marks. Figure 1 shows a section of the Earth, the ellipsoid and the geoid. It also shows the relationship between three different heights. The orthometric height, H, denoted as height above sea level, is the distance along the plumb line from a point on the Earth’s surface to the geoid. The ellipsoid height, h, is the distance from a point on the Earth’s surface down to the ellipsoid. H is perpendicular to the geoid, h is perpendicular to the ellipsoid; the two lines are not in the same plane. The geoid height, N, is the distance from the geoid to the ellipsoid. It can’t be seen in the figure, but N is not in the same plane as either H or h.

Looking again at Figure 1, if the point on the Earth’s surface is a bench mark with a known orthometric height (established by differential leveling), and if the point was occupied by a GPS receiver and the ellipsoid height determined, the geoid height can be calculated by the equation

N = h – H.

With enough bench marks occupied by GPS receivers, it’s possible to generate a geoid model from this information only. The problem with this technique is the orthometric heights must be accurate, not hit by a farmer with his plow or disturbed by a construction crew. And if orthometric heights are needed in mountains, heavily forested areas or massive swamps, the method of differential leveling can be difficult. GPS-derived orthometric heights will eventually be used in these difficult locations.

The National Geodetic Survey (NGS) 10-year plan states: “The era of using geodetic leveling for continental-scale vertical datum definition comes to an end. The gravimetric geoid, long used as the foundation for hybrid geoid models, becomes the most critical model produced by NGS.”1 What is a hybrid geoid model? It’s a combination of a gravity model and GPS observations on established bench marks. That’s what we have today, GEOID03, and perhaps earlier geoid models produced by NGS.

Gravimetric Geoid

Most readers of my column are unfamiliar with gravity observations, so the explanation that follows is classical (used before the computer age). This is a basic description of how gravity observations only are used to determine geoid heights.

Let’s start by looking at the following equation on the right.

This is Stokes’ Formula published by George Gabriel Stokes in 1849. It is the most important formula of physical geodesy because it makes it possible to determine geoid heights from gravity data. Don’t be intimated by a formula that contains a double integral; this can be solved numerically.

I’ll come back to the formula, but notice it contains the term g. The term g, as defined in the first article of this series, is observed gravity. g is called a gravity anomaly. This is the only observable in the formula, and I’ll try to define it in simple terms. A gravity anomaly is a combination of two different gravity values: the observed gravity reduced to sea level and normal gravity. There are several methods of reducing observed gravity, but I’ll discuss only one: normal gravity.


Figure 2. Δg template.

Normal Gravity

If you want to get false information on normal gravity, go to the Internet. There are pseudo-geodesists who, for whatever reason, write equations that are completely in error or obsolete. Normal gravity, denoted by the symbol , is a fictitious number based on an earth of equal density. Fictitious is my word; normal gravity is an important value. It’s a geodetic reference system based on the theory of a level ellipsoid. It was first introduced in the 1920s based on the International ellipsoid. Since it assumes an earth of equal density, the only variable can be latitude, . The equation for normal gravity, at that time, was
γ = 9.78049 ( 1 + 0.0052884 sin2 φ – 0.0000059 sin2 2φ)ms-2.

In 1979, at a General Assembly meeting of the International Union of Geodesy and Geophysics (IUGG) held in Canberra, Australia, the IUGG selected the Geodetic Reference System 1980 (GRS 80) as the ellipsoid of reference. This, as you know, is the ellipsoid used by NGS for our National Spatial Reference System (NSRS). The equation, now more complicated, is

γ = 9.7803267715 ( 1 + 0.0052790414 sin2 φ + 0.0000232718 sin4 φ +                                                    0.0000001262 sin6 φ + 0.0000000007 sin8 φ) ms-2.

When a gravity observation is made in the field, the latitude must be known so that normal gravity can be calculated at a later date.


Gravity Reduction

Gravity, measured on the surface of the Earth, is not comparable with normal gravity referred to the surface of an ellipsoid. A reduction of the observed gravity to sea level is necessary. Since there are masses above sea level, the reduction methods differ, depending on the way in which topographic masses are dealt with. As stated earlier, I will discuss only one reduction method, and that is the Free Air reduction.

The Free Air reduction, simply stated, assumes that the distance from the observed point on the Earth’s surface to the geoid is air, not rock, soil or water. To calculate this distance, the orthometric height, H, of the observed point must be known. The equation for the reduction is

gF = g + 0.3086 H milligal,

where H is in meters. For positive elevations on land, this correction is always added to the observed gravity because the closer the attracting mass, the greater the gravity.

Gravity Anomaly

The gravity anomaly, g, is needed to solve the Stokes formula. The gravity anomaly is defined as

     ΔgF   = γ - g F.

Solving for N

Looking again at Stokes’ Formula, this function is solved for a single geoid height. In order for this function to work, there must be a dense network of gravity anomalies. These gravity anomalies are plotted on a sheet of paper as “point values” and a contour map is generated at an appropriate scale.
 

A special template (shown in Figure 2) made of transparent material is placed on the contour map, centered over the point where N is to be determined. Referring to the formula and the template, γm is the mean value of normal gravity, ψ is the angular distance between the point where N is to be determined (m1) and the area where the effect of Δg is being considered, α is the azimuth from the affected point (m1) to the causing effect (dm), and

 

                S(ψ) = csc ψ/2 + 1 – 6 sin ψ/2 – 5 cos ψ – 3 cos ψ ln(sin ψ/2 + sin2 ψ/2).

 

Figure 2 is often referred to as Rice’s Rings. There are, in this example, 24 equal sectors radiating from the computation point. Each sector is subdivided into compartments by rings concentric to the computation point. The value of Δg is estimated for a compartment, recorded on a form, S(ψ) calculated, then the formula solved for one compartment. The process is continued for every compartment; this is numerical integration. The major contribution to N is due to values of Δg within a few hundred miles of the point. All of this work and the value of N is determined just for one point. But don’t be intimated by the mathematics. As a graduate student, I worked with two others to do this computation using only a hand calculator.
Today, with modern gravity meters and massive computers, NGS will be establishing an extensive network of gravity observations. Instead of manually solving the problem using Rice’s Rings, computer software will perform the numerical integration.
The problem will be gathering the gravity data. Many years ago, a friend who worked for the U.S. Department of Defense asked me, “Do you know how many gravity observations NGS has?” I didn’t know; he said 1,000,000. Then he asked, “Do you know how many gravity observations the Department of Defense has?” Again, I didn’t know; he said 10,000,000-9,000,000 are classified. These numbers may or may not be correct, but gravity is an important component of the Department of Defense, and soon to be for the NGS.

It’s important that you know more about gravity anomalies. In my next column in December I will describe other gravity anomalies.