WHAT IS A GEOID? - PHYSICAL - 2025

The geoid or figure of the Earth is the theoretical surface of our planet, determined by the average level of the oceans and with a fairly irregular shape. Mathematically it is defined as the equipotential surface of the effective gravitational potential of the Earth, at sea level.

As it is an imaginary (non-material) surface, it crosses continents and mountains, as if all the oceans were connected by water channels that pass through the land masses.

Figure 1. The geoid. Source: ESA.

The Earth is not a perfect sphere, since the rotation around its axis turns it into a kind of ball flattened by the poles, with valleys and mountains. This is why the spheroid shape is still inaccurate.

This same rotation adds a centrifugal force to the Earth's gravity force, whose resultant or effective force does not point to the center of the Earth, but has a certain gravitational potential associated with it.

Added to this, geographical accidents create irregularities in density, and therefore the gravitational force of attraction in some areas definitely ceases to be central.

So scientists, beginning with CF Gauss, who devised the original geoid in 1828, created a geometric and mathematical model to represent the Earth's surface more accurately.

For this, an ocean is assumed at rest, without tides or ocean currents and with constant density, whose height serves as a reference. The Earth's surface is then considered to ripple gently, rising where local gravity is greatest and sinking when it decreases.

Under these conditions let the effective gravity acceleration always be perpendicular to the surface whose points are at the same potential and the result is the geoid, which is irregular since the equipotential is not symmetric.

Physical foundation of the geoid

To determine the shape of the geoid, which has been refined over time, scientists have carried out many measurements, taking into account two factors:

- The first is that the value of g, the Earth's gravitational field equivalent to the acceleration of gravity , depends on latitude: it is maximum at the poles and minimum at the equator.

- The second is that, as we said before, the density of the Earth is not homogeneous. There are places where it increases because the rocks are denser, there is an accumulation of magma or there is a lot of ground on the surface, such as a mountain for example.

Where the density is higher, so is g. Note that g is a vector and that is why it is denoted in bold.

The Earth's gravitational potential

To define the geoid, the potential due to gravity is needed, for which the gravitational field must be defined as the gravitational force per unit mass.

If a test mass m is placed in said field, the force exerted by the Earth on it is its weight P = mg, therefore the magnitude of the field is:

Force / mass = P / m = g

We already know its mean value: 9.8 m / s ² and if the Earth were spherical, it would be directed towards its center. Similarly, according to Newton's law of universal gravitation:

P = Gm M / r ²

Where M is the mass of the Earth and G is the universal constant of gravitation. Then the magnitude of the gravitational field g is:

g = GM / r ²

It looks a lot like an electrostatic field, so a gravitational potential can be defined that is analogous to electrostatic:

V = -GM / r

The constant G is the universal constant of gravitation. Well, the surfaces on which the gravitational potential always has the same value are called equipotential surfaces and g is always perpendicular to them, as said before.

For this particular class of potential, the equipotential surfaces are concentric spheres. The work required to move a mass on them is zero, because the force is always perpendicular to any path on the equipotential.

Lateral component of the acceleration of gravity

Since the Earth is not spherical, the acceleration of gravity must have a lateral component g _l due to the centrifugal acceleration, caused by the rotational motion of the planet around its axis.

The following figure shows this component in green, whose magnitude is:

g _l = ω ² a

Figure 2. Effective gravity acceleration. Source: Wikimedia Commons. HighTemplar / Public domain.

In this equation ω is the angular speed of rotation of the Earth and is the distance between the point on the Earth, at a certain latitude and the axis.

And in red is the component that is due to planetary gravitational attraction:

g _o = GM / r ²

As a result, by vectorly adding g _o + g _l, a resulting acceleration g (in blue) is originated, which is the true acceleration of gravity of the Earth (or effective acceleration) and which, as we see, does not point exactly to the center.

Furthermore, the lateral component depends on the latitude: it is zero at the poles and therefore the gravitational field is maximum there. At the equator it opposes gravitational attraction, reducing effective gravity, whose magnitude remains:

g = GM / r ² - ω ² R

With R = equatorial radius of the Earth.

It is now understood that the equipotential surfaces of the Earth are not spherical, but take a shape such that g is always perpendicular to them at all points.

Differences between geoid and ellipsoid

Here is the second factor that affects the variation of the Earth's gravitational field: the local variations of gravity. There are places where gravity increases because there is more mass, for example on the hill in figure a).

Figure 3. Comparison between the geoid and the ellipsoid. Source: Lowrie, W.

Or there is an accumulation or excess of mass below the surface, as in b). In both cases there is an elevation in the geoid because the greater the mass, the greater the intensity of the gravitational field.

On the other hand, over the ocean, the density is lower and as a consequence the geoid sinks, as we see on the left of figure a), above the ocean.

From figure b) it is also noted that local gravity, indicated with arrows, is always perpendicular to the surface of the geoid, as we have said. This does not always happen with the reference ellipsoid.

The undulations of the geoid

The figure also indicates, with a bidirectional arrow, the difference in height between the geoid and the ellipsoid, which is called undulation and is denoted as N. Positive undulations are related to excess mass and negative ones to defects.

The undulations hardly ever exceed 200 m. Actually, the values ​​depend on how the sea level that serves as a reference is chosen, since some countries choose differently according to their regional characteristics.

Advantages of representing the Earth as a geoid

-On the geoid the effective potential, the result of the potential due to gravity and the centrifugal potential, is constant.

-The force of gravity always acts perpendicular to the geoid and the horizon is always tangential to it.

-The geoid offers a reference for high precision cartographic applications.

-Through the geoid, seismologists can detect the depth at which earthquakes occur.

-The positioning of the GPS depends on the geoid to be used as a reference.

-The surface of the ocean is also parallel to the geoid.

-The elevations and descents of the geoid indicate the excesses or defects of mass, which are gravimetric anomalies. When an anomaly is detected and depending on its value, it is possible to infer the geological structure of the subsoil, at least to certain depths.

This is the foundation of gravimetric methods in geophysics. A gravimetric anomaly can indicate accumulations of certain minerals, structures buried underground, or even empty spaces. The salt domes in the subsoil, detectable by gravimetric methods, are indicative in some cases of the presence of oil.

References

1. THAT. Euronews. Gravity's grip on Earth. Recovered from: youtube.com.
2. JOY. Geoid. Recovered from: youtube.com.
3. Griem-Klee, S. Mining explorations: gravimetry. Recovered from: geovirtual2.cl.
4. Lowrie, W. 2007. Fundamentals of Geophysics. 2nd. Edition. Cambridge University Press.
5. NOAA. What is the geoid ?. Recovered from: geodesy.noaa.gov.
6. Sheriff, R. 1990. Applied Geophysics. 2nd. Edition. Cambridge University Press.