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The Shape of Earth and Reference Ellipsoids

Elevation and the Ellipsoid

GPS receivers use a smoothed model of sea level to calculate elevation. The model is called a reference ellipsoid. This means that the elevation reading from the GPS receiver is not the vertical distance from true sea level, but rather the distance from the reference ellipsoid. In this tutorial, we introduce the concept of ellipsoids and reference ellipsoids in a way suitable for both teachers and students. Please use this and subsequent tutorials to help you learn more about accuracy, scientific models, and how GPS receivers work.

The Shape of the Earth: An Introduction to the Ellipsoid

The Earth is not a perfect sphere, but is instead more like a slightly squashed sphere called an ellipsoid. An ellipsoid looks like a basketball when someone is sitting on it. Instead of being perfectly round, it is squashed down from top to bottom, and bulged out from side to side. Because of mountains and valleys on Earth’s surface, the Earth isn’t a perfect ellipsoid either, but an ellipsoid is a good enough approximation of Earth’s overall shape.

Not squished

Illustration of a perfectly round basketball.

Squished

Illustration of a basketball that has been compressed vertically, bulging outward horizontally.

The GPS Receiver and the Reference Ellipsoid Model

A GPS receiver gives latitude, longitude, and elevation. Let’s focus specifically on how a GPS receiver calculates elevation. The information required is:

  1. The location of at least four satellites in space
  2. The current time
  3. The reference ellipsoidal shape of sea level for the entire Earth

There are many different reference ellipsoids, but all GPS receivers use the same one, and it is called the WGS-84 Reference Ellipsoid. Below is a picture of this ellipsoid with some lengths in meters for:

  1. The distance from the North Pole to the Center
  2. The distance from any point on the Equator to the Center
Figure of the Earth labeling the North Pole and Equator. The text says Semi Minor Axis (b) Pole to Center: 6356752.3142 meters. Semi Major Axis (a) equator to center: 6378137.0 meters. Flattening (f), (a-b)/a: 1/298.257223563.

Notice that the Semi Major Axis (a) is LONGER than the Semi Minor Axis (b), Physically, this means the distance from side to side on the Earth is LONGER than from top to bottom, just like the squashed basketball. The lengths shown in the image define the exact shape of this ellipsoid, and are used by the GPS receiver to help it calculate elevation.

So, if a GPS receiver reports an elevation of 1600 meters, what that means is that you are 1600 meters above the reference ellipsoid.

Illustration of a mountain next to the ocean, with an X labeled "you are here" at the peak of the mountain. An arrow labels the vertical distance between the peak of the mountain and sea level as "elevation".

Models and Their Usefulness in Science

The reference ellipsoid is a simplified model of the world around us. It is a smoothed mathematical representation of Earth’s sea level surface and ignores the effects of tides, seasonal currents, and waves. On land, the reference ellipsoid forms the mathematical surface that would be taken by sea level if the land areas were crisscrossed by canals. Why, then, would a model be useful in science? In the case presented here, a model (the reference ellipsoid) is helpful in measuring elevation because describing the true shape of sea level is complicated and would involved storing a lot of information within a GPS receiver.

Accuracy and Using Averages to Increase Accuracy

You may now be wondering, “If the GPS receiver is using a model that only approximates sea level to determine elevation, what effect does the shape of the reference ellipsoid have on an elevation measurement?” Usually this effect is expressed in terms of accuracy.

Two images of dartboard, one with an arrow pointing to the center labeled "most accurate", the other with an arrow off to the side labeled "less accurate". Text reads: The Bullseye Example. To be good at darts, you have to have accuracy. If you throw the dart, the most accurate throw would be right on the bullseye for 20 points. A less accurate throw might get you on the board, but you will get less points. An accurate throw is closest to the desired target. An accurate elevation reading is closest to the actual real height above sea level.

Since a GPS receiver uses the reference ellipsoid, and not true sea level, to calculate elevation, then the elevation measurement is sometimes less accurate. We say “sometimes,” because there are times when the reference ellipsoid matches true sea level.

How can you increase accuracy? Let’s say you took GPS measurements for a study site and you have taken 15 readings, 1 each minute for 15 minutes. As you did this exercise, the GPS readings probably seemed to wander, and so you used an average of all 15 measurements to get the position. This averaging increased the accuracy of your GPS coordinates from 100 meters to 30 meters.

Drawing of three dartboards, one with an arrow near the outer ring labeled "first throw", one with arrows in five locations around the board labeled "many throws", and one with a single arrow near the center labeled "average throw". Text reads: The bullseye example 2. Imagine that you are not so good at playing darts. Your first throw is far from the bullseye in the center of the target. After many throws, you see that none of your darts is very close to the center. What happens when you take the average of all your throws? Do you notice that the average throw is closer to the bullseye than any of the individual throws? This is what you did with your GPS receiver measurements when you reported the average latitude, longitude, and elevation.

Now we know the GPS receiver uses a model to figure out your elevation. To learn more about what using this model means about your elevation measurements, continue to the Geoid and Receiver Measurements tutorial.

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Reference Frames

Geoid and Receiver Measurements Tutorial

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