With the radar equation (or basic radar equation ), the power registered by the receiver can be determined as a function of the transmission power, the distance and the properties of the reflecting object. This makes it possible to estimate whether, under the given circumstances, the object in the receiver generates a sufficiently strong signal that it can be recognized. Due to the different reflection conditions for point targets (air surveillance) and volume targets (weather radar), different radar equations are formulated for these two cases. In general, the radar equation is derived from the distance law and, like this, only applies to unhindered emissions into free space.

## General overview

Illustration of the law of distance for energy quantities

Apart from a few special cases, the ratio of distance to wavelength in radar applications is so large that the transmitting antenna - regardless of its size - can be viewed as a mathematical point. Therefore, the energy is distributed over ever larger areas like a spherical wave . To determine the reflected energy, a distinction must therefore be made:

• In the case of a “point target” (e.g. an airplane) the absorption area is constant; therefore the received energy decreases with 1 / r². A part of it is (passively) reflected back to the radar device, whereby the reflected energy also decreases again with 1 / r²; the receiving antenna is also a point target. Overall, you have to multiply both factors and a 1 / r 4 relationship applies .
• In the case of very large, two-dimensional targets, such as clouds (not too far away), the illuminated volume increases with the factor r². That is, all of the energy received from the cloud is constant. A certain fraction of it is reflected back and the 1 / r² dependence of the energy only applies to this return path , so that overall only a 1 / r² relationship applies.
Interestingly, this magnification only applies to a semi-transparent volume - not to a reflective surface. With this surface, the reflected energy would be distributed over time, since the transit time in the middle of this surface is shorter than up to the edge of the surface. In the case of radar altimeters, this problem also leads to a significant temporal deformation of the echo pulse up to a multiplication of the duration of the transmission pulse.
• With increasing distance of the clouds, however, they fill an ever smaller portion of the radiation lobe, because a growing part of the transmitted energy no longer hits the cloud. Then neither of the two previous points applies anymore; In this case, there is a transition from a section with 1 / r² to a section with 1 / r 4 connection .

The radar equation only applies to primary radar. A secondary radar does not use the passive reflection of radio waves. Rather, it consists of a bidirectional radio link. The received power decreases in the distance according to a 1 / r² dependence by the simple distance law . When the transponder answers, its transmission power is not dependent on the power received. A 1 / r² dependency therefore applies separately to the return path.

## Radar equation for a point target

Point targets are called reflecting objects that the pulse volume do not completely fill a radar, which means that their geometrical extension much smaller than the product of the propagation speed and the transmission pulse length ( c 0 · τ ), and is much smaller than the width of the antenna pattern at the location of reflection are. These conditions exist , for example, in the case of air reconnaissance and target tracking radar .

P r = receiving power
P t = transmitting power
G t = antenna gain of the transmitting antenna
G r = antenna gain of the receiving antenna
λ = wavelength of the carrier frequency
σ = effective reflective surface (RCS), σ 0 = spherically scattering reference surface of 1 m 2
R t = distance from transmitting antenna to reflective object

R r = distance reflecting object - receiving antenna
${\ displaystyle P _ {\ text {r}} = P _ {\ text {t}} {{G _ {\ text {t}} G _ {\ text {r}} \ lambda ^ {2} \ sigma} \ over { {(4 \ pi)} ^ {3} R _ {\ text {t}} ^ {2} R _ {\ text {r}} ^ {2}}}}$Here means:

The equation assumes that the distance between the object and the transmitter is significantly greater than the wavelength of the radar. This means that the object must be in the far field of the transmitter. Furthermore, the power ratio assumes that the duration of the transmission pulse corresponds approximately to the duration of the echo signal in the signal processing, that is, that no pulse compression method is used.

By rearranging the above equation according to the range, one obtains a form of the radar equation that is often used in practice to assess the operational performance of radar systems:

${\ displaystyle R _ {\ text {max}} = {\ sqrt [{4}] {\ frac {P _ {\ text {t}} \ cdot G ^ {2} \ cdot \ lambda ^ {2} \ cdot \ sigma} {P _ {{\ text {r}} _ {\ text {min}}} \ cdot (4 \ pi) ^ {3} \ cdot L _ {\ text {ges}}}}}}$

Here, the antenna gain in the case of transmission and reception has been combined into G 2 : this is possible if a monostatic radar antenna ( R t  ≡  R r ) forms the same antenna diagram at the moment of transmission as it does during the reception time ( G t  =  G r ). The maximum range R max then depends on the receiver sensitivity P r, min . Various internal and external losses also flow into the practical application as L tot .

## Radar equation for a volume target

The radar equation for volume targets (read: for weather radar) uses the same parameters and relationships on the radar side. The main difference, however, is the characteristic properties of the reflection surface, which also change with increasing distance from the radar. When it rains, every single raindrop is much smaller than the wavelength of the radar device. Therefore, the effective reflecting area of ​​a raindrop is determined by Rayleigh scattering :

${\ displaystyle \ sigma _ {i} = {\ frac {\ pi ^ {5}} {\ lambda ^ {4}}} | K | ^ {2} D_ {i} ^ {6}}$With:${\ displaystyle | K | ^ {2} = \ left | {\ frac {\ varepsilon -1} {\ varepsilon +2}} \ right | ^ {2}}$

with D as the raindrop diameter and ε as the dielectric constant. For the frequency bands L to X commonly used in radar devices , water has the factor | K | 2 = 0.93 and for ice | K | 2 = 0.2.

With a volume target, the pulse volume is now completely filled by these reflective objects. The sum of this reflective area is denoted by the temporary variable η :

${\ displaystyle \ eta = {\ frac {\ pi ^ {5}} {\ lambda ^ {4}}} | K | ^ {2} Z}$With:${\ displaystyle Z = \ sum _ {i = 1} ^ {N} D_ {i} ^ {6}}$

The pulse volume V increases due to the divergence of the antenna beam with the distance from the radar:

φ = vertical opening angle of the antenna diagram
θ = horizontal opening angle
R = distance to the radar
τ = transmission pulse duration
c 0 = speed of light

${\ displaystyle V = {\ frac {\ pi \ phi \ theta R ^ {2} c_ {0} \ tau} {8}}}$Here mean:

The internal parameters of the radar as well as partially the free space attenuation are summarized for meteorological purposes in a factor C , which is:

${\ displaystyle C = {\ frac {P_ {t} \ tau G ^ {2} \ theta ^ {2} c_ {0} \ pi ^ {3}} {512 (2 \ ln 2) \ lambda ^ {2 }}}}$

is used further. Here it has already been taken into account that most weather radar devices use a symmetrical diagram form with φ = θ and therefore φ · θ = θ 2 . This leads to a greatly simplified form of the radar equation for volume targets, as used in meteorology:

${\ displaystyle P_ {r} = \ left [{\ frac {C} {R ^ {2}}} \ right] \ left [| K | ^ {2} \ sum D_ {i} ^ {6} \ right ]}$

With this equation, the reflectivity can now be deduced directly from the measured received power . This is a measure of the type and number of reflecting objects, although this conclusion is not yet clear: many small water droplets produce the same reflectivity as a few large ones. To partially resolve these ambiguities, polarimetric radar is used and a differential reflectivity is measured.

## literature

• RJ Doviak, DS Zrnic: Doppler Radar and Weather Observations , Academic Press. Second Edition, San Diego Cal. ISBN 978-0-12-221420-2 , page 562, 1993
• JR Probert-Jones: The radar equation in meteorology , Quarterly Journal of the Royal Meteorological Society, 1962, Volume 88, Issue 378, pages 485-495.