- Understanding the radar horizon

For a sensor at height h, with the radius of the earth represented as r_earth, the distance to the visual horizon can be approximated as

The above formula is true for the visual band, for example, the frequency of sunlight. For somewhat lower frequencies, including almost all radars, there is another factor to take into account. The earth and its atmosphere acts to some extent like a giant waveguide. Radio waves are bent to some degree around the earth.

Extremely low frequency (ELF) signals are bent enough that they travel all the way around the world. At higher frequencies, they are bent less, but the bending is still significant. For most radar frequencies, we have a way to take into account the bending in the calculation of the horizon. When we calculate the horizon using these modified techniques, we are calculating the "radar" horizon.

During the 1940s, an effort was made to produce an equation that would enable easy calculation of the radar horizon. We reconstruct here the probable thinking that led to the radar horizon equation. The simplest technique was to alter the visual horizon equation by modifying the variable representing the earth's radius.

After an extensive set of measurements in the Eastern United States and Great Britain, it was decided that 1.33 was a reasonable multiplier for the earth radius to take into account radio wave bending for the frequencies of interest. The numbers were reported to vary from 1.2 to 1.4. 1.33 was still not easy to remember, so 4/3 was used in its place. In this way, the number could be remembered. The difficulty is that many people see the number 4/3 and think it is a derived number, rather than empirical.

The formula for the distance to the radar horizon is thus

By the way, the radius of the earth at the equator is 6378155 m from satellite measurements.

Here are the problems for the radar horizon

1. How much further is the radar horizon than the visual horizon?
2. A ground radar looking over the ocean is elevated to 50 meters. What is the radar horizon for this radar?
3. The same ground radar looks for ships that are 10 meters tall. At what range can it see the ships?
4. For a radar to see sea level targets out to a distance of 100 km, what is the minimum height of the radar?
5. An airborne early warning radar flies at a height of 20,000 feet, observing aircraft flying at 10,000 feet. What is the horizon limitation for this situation? Give the result in nautical miles.
   1 nautical mile (nmi) = 6076 feet
   1 foot = 0.3048 m

Once you understand the radar horizon equation, we encourage you to use the horizon calculator.