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Calculation of Height

Figure 1: simple triangular relation between elevation and height

Figure 1: simple triangular relation between elevation and height

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 + R2 Where: R = targets slant range
ε = measured elevation angle
re = earth's radius (about 6370 km)
(This equation is an approximation only!)
(3)

2 re

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:

R2 = re2 + (re + H)2 - 2re(re + H) · cos α (4)

(re 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° · Rtopogr. = α · 2π re (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 earth radius can be multiplied with a factor for an approximation of the influence of the refraction. An equivalent earth radius of 4/3 ·re ≈ 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

(6)

meaning at this:

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

Publisher: Christian Wolff (Revised by Karina Hoel)
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