#### Slant Range

Figure 1: A different height causes a different range

#### Slant Range

Because the radar equipment measures a slant range, two aircraft flying on top of each other (i.e., having the same topographical distance to the radar equipment, the so-called Down Range) will have different ranges.

In modern radars such as the AN/FPS-117, this error is corrected by a software module. However, these software modules must then also be specially adapted to the geographic setup location of the radar antenna. The calculation is very complicated and also requires some weather data for correction.

Unfortunately, typical 2D radars used in air traffic control cannot do this.

Here, the operator must know and automatically take into account in his work that the echo signal of an aircraft flying farther away is displayed at a greater geographical distance than the aircraft is!

In practice, however, this measurement error of the order of hardly one percent has a minor effect. However, it becomes problematic in airborne or space-borne SAR, where complex computational operations are necessary to compensate for distortions in the image due to the measured oblique distance.

##### Calculation of the Down Range

Figure 2: trigonometrical connections without consideration of the earth’s bend

Figure 2: trigonometrical connections without consideration of the earth’s bend

Figure 2: trigonometrical connections without consideration of the earth’s bend

It is much important to know at which topographic point on the earth the located airplane is. For this reason, an electronic map is always projected into the radar image, which is called a “video map” and is expected to be as accurate as possible. However, contrary to expectations, calculating the actual topographic distance of a located target in a radar image is very complicated.

Using the trigonometric relationships indicated in figure 2, the measured topographic distance is

R_{topogr.} = R · cos ε

However, this would only be valid if the earth were a flat disk. In addition, however, the earth’s radius also has an effect, as shown in figure 3. Thus, the actual topographic distance concerning the slant distance measured by the radar depends on:

- the measured slant range,
- the actual height of the aim, and
- the earth radius, which is valid for the location of the radar unit.

Figure 3: Trigonometric correlations considering the curvature of the earth.

Figure 3: Trigonometric correlations considering the curvature of the earth.

Figure 3: Trigonometric correlations considering the curvature of the earth.

From figure 3 one can see the solution approach.
A triangle between the points: Center of the earth, the location of the radar unit, and the location of the flight target,
whose sides defines the cosine theorem and thus by the equation:

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

(r_{e} is the equivalent radius of the earth).

Under the assumption that the earth is a sphere, from the angle α,
the part of the earth’s circumference can be calculated with a simple ratio calculation from the total earth circumference:

360° · R_{topogr.} = α · 2π r_{e}

This partial section of the earth circumference can be regarded as an approximation
(here still without consideration of the refraction) to the actual topographic distance.

In practice, however, the propagation of electromagnetic waves is also subject to refraction, i.e. the transmitted beam of the radar is not a rectilinear side of this triangle, but this side is additionally also curved depending on

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

Since all these parameters cannot be included in the radar video map, the map is inevitably inaccurate if the radar software does not take into account the relationship between slant range and topographic range. And this is unfortunately always the case with 2D radar devices since these lack the height information compellingly necessary for these computations!