The first two articles in this series (POB August 2007 and October 2007) explained how to solve for geoid heights, N, using Stokes’ equation. The gravity anomaly in that equation, Δg, has many different forms.

In this article, I will provide a brief introduction to gravity anomalies.

Gravity Reduction

Gravity reduction serves as a tool for three main purposes:
  1. Determination of the geoid
  2. Interpolation and extrapolation of gravity
  3. Investigation of the Earth’s crust.

The main purpose for surveyors is No. 1--the determination of the geoid. I’m sure that the National Geodetic Survey (NGS) wants enough gravity data so that it doesn’t have to do much extrapolation per No. 2. Geologists and geophysicists are primarily interested in No. 3.

The use of Stokes’ formula for the determination of the geoid requires that gravity anomalies, Δg, represent boundary values on the geoid. This means that gravity, g, must refer to the geoid, and there must be no masses outside the geoid. There are masses outside the geoid everywhere except on the oceans. For this reason, topographic masses outside the geoid are removed and shifted below sea level, and the gravity station is lowered from the Earth’s surface to the geoid.

In my last column, “Physical Geodesy 201,” I gave the formula for the most basic reduction, the free air reduction. Let’s go one step further and define the Bouguer Reduction, which is the complete removal of the topographic masses outside the geoid.

The Bouguer Plate

Assume the area around the gravity station to be flat, and let the masses between the geoid and the Earth’s surface have a constant density of 2.67, a good value for rock. Without deriving the formula, we have the equation

for h in meters. This is called the Incomplete Bouguer Reduction. To complete the gravity reduction we must lower the gravity station to the geoid. This is the free air reduction, which, as shown in the previous article, is +0.3086h mgal, with h in meters. This combined process of removing topographic masses and applying the free air reduction is called the Complete Bouguer Reduction. Simply stated,

The Bouguer anomaly is

Before the computer age, the major gravity anomalies were the two described earlier: free air anomalies (used by the geodetic community) and Bouguer anomalies (used by the geophysical community). But there are others. Gravity reductions differ by the manner in which the topographic masses are displaced.

The Bouguer plate assumes the area around the gravity station to be completely flat; in Iowa, yes, but not in Colorado. For high precision gravity field modeling, the topography has to be smoothed by a terrain correction. Before the computer age, a template similar to Rice’s Rings was used, adding the effects of the individual compartments. The terrain correction is much smaller than the Bouguer reduction. An example given in Heiskanen and Vening Meinenz’s bookThe Earth and its Gravity Fieldreads: “Even for mountains 3000 meters in height the terrain correction is only of the order of 50 mgals.”1In modern-day geodesy this correction cannot be ignored.

There is a reduction mentioned in modern geodesy texts called Helmert’s Condensation Method that results in terrain-corrected Bouguer anomalies. From what I understand this is a method used when the gravity anomalies are determined in a grid. Another reduction mentioned in modern geodesy texts is the Faye anomaly, a terrain-corrected free air anomaly.

Another gravity reduction is the isostatic reduction. According to Heiskanen and Moritz in their bookPhysical Geodesy, “One might be inclined to assume that the topographic masses are simply superposed on an essentially homogeneous crust. If this were the case, the Bouguer reduction would remove the main irregularities of the gravity fields, so that the Bouguer anomalies would be very small and would fluctuate randomly around zero. However, just the opposite is true. Bouguer anomalies in mountainous areas are systematically negative and may attain large values, increasing in magnitude on the average by 100 mgals per 1,000 meters of elevation. The only explanation possible is that there is some kind of mass deficiency under the mountains. This means the topographic masses arecompensatedin some way.”2I will cover more on the topic of isostatic reduction in a future column.

To repeat, the determination of the geoid requires that the gravity anomalies are given everywhere on the geoid. The gravity reductions differ by the manner in which topographic masses are displaced. The gravity anomalies (free air, Bouguer, etc.) are the “Δg”s used in Stokes’ equation.

We now have to wait and see what NGS will do in its quest to improve on the determination of the geoid, since this goal is included in its 10-year plan.

Who Was Bouguer?

To conclude this article, I want to answer the question, “Who was Bouguer?” Born in 1698, Pierre Bouguer (pronounced bu:gei) was a French hydrographer and mathematician. He was a member of the French Academy of Sciences expedition to Perú (now Ecuador) in 1735-1744 that measured an arc of the meridian in an attempt to determine whether the Earth was flattened at the pole or flattened at the equator. The Perú data, combined with information of an arc measured in Lapland, led to a flattening at the poles of 1/210.