The Radar Equation
The radar equation represents the physical dependences of the transmit power, the wave propagation up to the receiving of the echosignals. Furthermore can be assessed the performance of radar sets with the radar equation.
Argumentation/Derivation
At first we assume, that electromagnetic waves can propagate with ideal conditions without disturbing influences.
If highfrequency energy is emitted by an isotropic radiator, than the energy propagate evenly to all directions. Areas of same power density therefore form spheres ( A= 4 π R² ) around the radiator. At incremented spheric radius the same value of energy spreads out around on an incremented spherical surface. That means: the power density on an assumed area becomes lower with an increasing distance of the radiator.
Figure 1: nondirectional power density
So we get the formula to calculate the Nondirectional Power Density S_{u}

P_{S} = transmitted power [W] S_{U} = nondirectional power density R_{1} = Range Antenna – Target [m] 
(1) 
If the irradiation is limited on a spherical segment (at constant transmit power), then results an increase of the power density in direction of the radiation. This effect is called antenna gain. This gain is made by directional irradiation of the power. For the directional power density apply:
S_{g} = S_{u} • G

S_{g} = directional power density G = antenna gain 
(2) 
Of course in the reality radar antennas aren’t „partially radiating” isotropic radiators. Radar antennas have to have a small beam width and an antenna gain up to 30 or 40 dB. (e.g. parabolic dish antenna or phased array antenna).
The target detection isn’t only dependent on the power density at the target position. In addition of this it depends by the reduction how much is back reflected actually in direction of the radar equipment. To be able to determine the utilizable reflected power, the value of the radar cross section σ is needed. This difficultly comprehensible quantity is dependent on several factors. It is that way plausibly at first, a bigger area reflects more power than a little area. That means:
A Jumbo jet offers more radar cross section than a sporting aircraft at same flight situation. Beyond this the rereflecting area depends on design, surface composition and the using materials.
This is said summarized till now: At the final destination the reflected power P_{r} arises from the power density S_{u}, the antenna gain G and the very variable radar cross section σ:

Simplified a target can be regarded as a radiator in turn due to the reflected power. The reflected power Pr then becomes the emitted power.
Since there are the same conditions as on the way there on the way back of the echos yields himself for the power density at the receive place Se:
Figure 2: Connection between
formula 3 and 4

S_{e} = power density at receive place P_{r} = reflected power [W] R_{2} = range antenna – target [m] 
(4) 
At the radar antenna the received power is dependent on the power density at the receive place P_{E} and the effective antenna area A_{W} .
P_{E} = S_{e} • A_{W} 
P_{E} = power density at the receive place [W] A_{W} = effective antenna area [m²] 
(5) 
The effective antenna area arises from the fact that an antenna doesn’t work lossfreely i.e. the geometric measurements are available not quite as a receive area. As a rule, the effect of an antenna is smaller than these have geometric measurements suspected around the factor 0.6 to 0.7 (Factor K_{a}).
Applies to the effective antenna area:
A_{W} = A • K_{a} 
A_{W} = effective antenna area [m²] A = geometric antenna area [m²] K_{a} = Factor 
(6) 
For the power at the receive place P_{E} arises therefore:
(7) 

(8) 
The way there and way back was looked at separately at the argumentation till now. With the next step both ways being summarized: Since R_{2} (Target – Antenna) is the distance R_{1} (Antenna – Target) at once, this is taken into account now
(9) 
Another given equation (however, this one shall not be derived in this place) puts the antenna gain G in connection with the used wavelength λ.

(10) 
This is convert to the antenna area A and put into the upper equation. After the simplification it yields:

(11) 
After the converting to the range R the classic form results for the radar equation:
(12) 
All quantities which have influence on the wave propagation of the radar signals were taken into account at this radar equation. Beyond this the dependences of the sizes were illustrated and summarized in the classic radar equation at least.
Until after this theoretical attempt can be used the radar equation very well also in the practice e.g. to determine the efficiency of radar sets. The form of the classic radar equation isn’t, however, suitable for these extended considerations yet. Some further considerations are necessary.
Obtained on a given radar equipment most sizes (P_{s}, G , λ) can be regarded as constant since they are only in very little ranges variable parameters. Against this the radar cross section represents a quantity to be described heavily and therefore 1 m² is assumed as a practical oriented value mostly.
(13) 
Under this condition that value of the received power P_{E} is interesting, which in the radar receiver causes an even still visible signal. This received power is called P_{E min}. Smaller received powers aren’t usable since they would lost in the noise of the receiver. The occurred into the radar equation received power P_{E min} causes that with the equation the theoretically maximum range R_{max} can be calculated now.
An application of this radar equation is the determination of the performance of radar units to compare each other.
Influences on the maximum range of a radar unit
All considerations in connection with the radar equation were made under the prerequisite till now that the electromagnetic waves can propagate under ideal conditions without disturbing influences. In the practice a number of losses appears, though. These cannot remain unconsidered since they partly reduce the effectiveness of a radar unit considerably.
To this, at first the radar equation is extended by the loss factor L_{ges}.
(14) 
This factor summarizes the following listed kinds of losses:
 L_{ D} = internal attenuation factors of the radar unit on the transmitting path and the receive path
 L_{ f} = fluctuation losses during the reflection
 L_{ Atm} = atmospheric losses during the propagation of the electromagnetic waves in direction of the target (and the way back)
Piece of internal losses arise in the main thing at high frequency components, like waveguides, filters but also by a radome. Obtained on a given radar unit this kind of loss is relatively constant and also well measurable in it’s value.
As permanent influence, still has to be called the atmospheric attenuation and reflections at the earth’s surface.
Influence of the earth’s surface
An extended, lesserused form of the radar equation considers additional factors, like the influence of the Earth’s surface and does not continue to classify receiver sensitivity and the atmospheric absorption.
In this formula, in addition to the already wellknown quantities:
K_{α}=  Dissipation factor in place of L_{ges.}  A_{z}=  effective reflection surface in place of σ 
T_{i}=  Pulse length  K=  Boltzmann’s constant 
T_{0}=  absolute Temperature in °K  n_{R}=  Noise figure of the receiver 
d=  Clarity factor of the display terminal  γ=  Reflected beam angle 
δ_{R}=  Breakeven factor  R_{e}=  Distance of the absorbing medium 
The factor with the trigonometric functions represents the influence of the Earth’s surface. The earth plane immediately surrounding a radar antenna has a significant impact on the vertical polar diagram. Radar Reflections from Flat Ground
By the combination of the direct with the reflected echo, the transmitting and receiving patterns of the antenna change. This influence is substantial in the VHF range and decreases with increasing frequency. For the detection of targets at low heights, a reflection at the Earth’s surface is necessary. This is possible only if the ripples of the area within the first Fresnel zone do not exceed the value 0.01 R (i.e.: Within a radius of 1000 m no obstacle may be larger than 1 m!). 
Specialised Radars at lower ( VHF) frequency band make use of the reflections at the Earth’s surface and lobing to maximise cover at low levels. At higher frequencies these reflections are more disturbing. The following picture shows the lobe structure caused by ground reflections. Normally this is highly undesirable as it introduces intermittent cover as aircraft fly through the lobes. The technique has been used in ATC ground mounted radars to extend the range but is only successful at low frequencies where the broad lobe structure permits adequate cover at higher elevations.
Free space vertical pattern diagram  
Effect of ground reflections  
Gray, my dear friend, is every theory: here it is the idealized cosecant squared diagram! 
Raising the height of the antenna has the effect of making the lobbing pattern finer. A fine grained lobing structure is often filled in by irregularities in the ground plane. Specifically, if the ground plane deviates from a flat surface then the reinforcement and destruction pattern resulting from the ground reflections breaks down. Avoidance of lobe effects is one of the prime considerations when selecting a radar location and the height of the antenna.