#### Calculation of Height

Figure 1: simple triangular relation between elevation and height

Figure 1: simple triangular relation between elevation and height

Figure 1: simple triangular relation between elevation and height

The height of a target over the earth's surface is called height or altitude. This is denominated by the letter H (like: Height) in the following formulae and figures. The altitude can be calculated with the values of distance R and elevation angle ε.

sin α = | opposite side | (1) |

Hypothenuse |

The values of distance R and elevation angle ε we insert in the equation:

H = R · sin ε | (2) |

The altitude cannot be so simply calculated on a flying airplane, while refraction is caused when electromagnetic waves cross airlayers at different density and the earth's surface has a bend. The calculation of the targets altitude is not only a trigonometrically calculation. The current location grounding bend must also be taken into consideration. Both factors are compensated for with an integrated altitude calculation by extensive equations.

H = R · sin ε + | R^{2} |
Where: | R = targets slant range ε = measured elevation angle r _{e} = earth's radius (about 6370 km)(This equation is an approximation only!) |
(3) |

2 r_{e} |

Figure 2: relation between elevation and height under consideration of the earth bend

Figure 2: relation between elevation and height under consideration of the earth bend

Figure 2: relation between elevation and height under consideration of the earth bend

One can take the mathematical rule for the calculation from Figure 2. A triangle arises between the points: center of the earth, the radar site and the position of the aircraft. The sides of this triangle are described by the cosine theorem and therefore by the equation:

R^{2} = r_{e}^{2} + (r_{e} + H)^{2} - 2r_{e}(r_{e} + H) · cos α |
(4) |

(r_{e} is the earth radius here).

##### Calculation of the Down Range

Under the assumption that the earth is a sphere, the section of the circumference of the earth can be calculated with help of a simple ratio from the complete circumference of the earth from the angle α:

360° · R_{topogr.} = α · 2π r_{e} |
(5) |

This segment of the circumference of the earth can be considered an approximation of the actual topographical range (here though still without considering refraction).

##### Height Calculation with the Influence of Refraction

In practice, however, the propagation of electromagnetic waves is also subject to a refraction, this means, the transmitted beam of the radar unit isn't a straight side of this triangle but this side is also bent and it depends on:

- the transmitted wavelength,
- the barometric pressure,
- the air temperature and
- the atmospheric humidity.

The earth radius can be multiplied with a factor for an approximation of the influence
of the refraction. An equivalent earth radius of
^{4}/_{3} ·*r _{e}* ≈ 8500 km
is often used and can additionally modified by a manual input of a correction factor to consider temporarily
changed weather conditions or changed barometric pressure.

E.G. the following equation is used to calculate the height into the height finder PRW–16

meaning at this:

- height without including the earth's radius
- term including the earth's equivalent radius (about 8500 km)
- term including the refraction into the atmosphere
- term including the refraction’s dependence on temperature.