# Vertical Datums, Elevations, and Heights

A National Imagery and Mapping Agency publication.

Have you ever considered what it means to say that Mt. Elbert, the highest peak in Colorado, has a height of 14,433 feet? Where is this measured from -- the center of the Earth, the surface of the Earth, or somewhere else? How do we define zero height or zero elevation?

The zero surface to which elevations or heights are referred is called a vertical datum. Traditionally, surveyors and mapmakers have tried to simplify the task by using the average (or mean) sea level as the definition of zero elevation, because the sea surface is available worldwide. The mean sea level (MSL) is determined by continuously measuring the rise and fall of the ocean at "tide gauge stations" on seacoasts for a period of about 19 years. This averages out the highs and lows of the tides caused by the changing effects of the gravitational forces from the sun and moon which produce the tides. The mean sea level (MSL) then is defined as the zero elevation for a local or regional area. But what do you do for Mt. Elbert? Where is the MSL in Colorado? There is no tangible surface of the ocean from which to measure height.

It turns out that MSL is a close approximation to another surface, defined by gravity, called the geoid, which is the true zero surface for measuring elevations. Because we cannot directly see the geoid surface, we cannot actually measure the heights above or below the geoid surface. We must infer where this surface is by making gravity measurements and by modeling it mathematically. For practical purposes, we assume that at the coastline the geoid and the MSL surfaces are essentially the same. Nevertheless, as we move inland we measure heights relative to the zero height at the coast, which in effect means relative to MSL.

## Height Measurement

Aviators, sailors, soldiers, and astronauts use a variety of techniques and equipment to measure heights. Because of assumptions made, there is no guarantee that each technique will produce the same height measurement. Although the map says heights are with respect to MSL, height measurement systems may not give exactly equivalent results as the following example illustrates.

A sailor aboard an anchored destroyer turns on his Navstar Global Positioning System (GPS) receiver and reads that his present height is minus 25 meters. Knowing the ship hasn't sunk, he questions the accuracy of his equipment. The manual says his receiver has a vertical accuracy of plus or minus 9 meters, so he decides to record the measurements for several hours to see if the elevation will ever be positive. Later, when he compares the readings, he finds that all are negative with a mean value of minus 20 meters.
What's the problem here -- is the receiver broken? Probably not. Standard GPS receivers and software have built-in flexibility to meet a variety of applications. Different applications need heights or elevations with respect to different zero surfaces. Most GPS receivers have built-in software to do these transformations. The user needs to specify the correct vertical reference system. If the user makes wrong assumptions, the results can be very confusing at best. For the pilot trying to make an instrument landing (precision approach), having the wrong height information can have dire consequences!

## Zero Surfaces of Height Systems

To understand the differences in height measurements and their representation on maps and charts or on the display of a piece of equipment, it is necessary to understand the differences between topographic surface, ellipsoidal surface, and the geoid as illustrated below.

The topographic surface is the actual visible surface of the Earth. To represent horizontal positions on maps and charts, we need a mathematical model of the Earth which includes a set of numbers for the size and shape of the Earth. Because the Earth is slightly flattened at the poles, a sphere won't work. A flattened sphere, which is called an ellipsoid, is used to represent the geometric model of the Earth. As mentioned earlier the geoid, which is approximated by MSL, is the zero surface as defined by the Earth's gravity. The answer to the question posed in the diagram is, "Depending on the application, the ellipsoid, the geoid, and the center of the Earth are all used." In fact, for the standard GPS receiver to output MSL elevation all three of these zero references must be used!

## Calculating Height or Elevation

We can calculate height or elevation, using any zero reference as a starting point, including the center of the Earth. The diagram below shows the relationships among height systems.

• MSL elevation is roughly equivalent to orthometric height (H), the technical name for height above the geoid.
• Geoid height (N) is the separation between the geoid and the ellipsoid. It can be plus or minus.
• Ellipsoid height (h) is the distance above or below the ellipsoid (plus or minus). Ellipsoid height is also called geodetic height.

For example, to get MSL elevation from ellipsoid height, you simply have to subtract the geoid height
(H equals h minus N).

## How Good is MSL?

All of our maps say "elevations are based on mean sea level," so we're all using the same standard, right? Unfortunately for mapmakers and surveyors, sea level is not a simple surface. Because the Earth is ellipsoidal, you might expect the ocean to form a smooth, even "ellipsoidal" surface around the Earth, but it doesn't. Variations in the topography and the different densities within the Earth's crust produce slight variations in the gravity field, described by the dips and peaks of the geoid. Since the sea surface conforms to that gravity field, MSL also has slight hills and valleys in it, similar to the topographic surface although much smoother. Depending on where you are, sea level may be closer to or farther from the center of the Earth than at some other location. That's why locally-defined vertical datums differ from each other. Zero elevation (MSL) as defined by China differs from zero elevation (MSL) as defined by Chile.

By definition, the geoid describes this irregular gravity shape. Because there is no way to accurately measure the geoid, it has been roughly approximated in the past by MSL. Technically, the geoid differs from MSL by several meters due to other ocean effects beyond the scope of this pamphlet.

## Modern Elevation Standards

Remember our sailor's problem? He was expecting to find his height above mean sea level, but GPS receivers automatically calculate the height above or below the ellipsoid instead. The sailor is simply in one of those valleys in the geoid. Many receivers can be programmed to give you an approximate MSL elevation based on a model of the geoid stored internally, but you have to choose this output option. In fact if our sailor had chosen MSL output, the average value would have been approximately zero.

The MSL elevation from the geoid model is not very accurate in some areas, because until recently, we just didn't know exactly how the geoid was shaped. Now we have the technology to improve the geoid model to about one-meter level of accuracy. NIMA has an ongoing project to develop an improved geoid by mid-1996, which in turn will be used as a zero surface to establish consistent and accurate elevations worldwide. In the meantime, realize that if you ignore vertical datum issues errors of up to 100 meters could result!

The Defense Mapping School provides educational services to warfighters throughout the Services. Mobile Training Teams can travel to your location at NIMA expense to assist you. Call DSN 655-3206, or (703) 805-3206 commercial for more information about the Mapping, Charting and Geodesy for the Warrior course.

Each Unified Command has an MC&G officer and a NIMA Liaison. Call the headquarters of your MAJCOM to get in touch with them.

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