Reference:  http://earth.esa.int/applications/data_util/SARDOCS/spaceborne/Radar_Courses/Radar_Course_III/radar_equation.htm

The radar equation

The fundamental relation between the characteristics of the radar, the target, and the received signal is called the radar equation. The geometry of scattering from an isolated radar target (scatterer) is shown in the figure, along with the parameters that are involved in the radar equation.

When a power Pt is transmitted by an antenna with gain Gt , the power per unit solid angle in the direction of the scatterer is Pt Gt, where the value of Gt in that direction is used. At the scatterer,
(1)

where Ss is the power density at the scatterer. The spreading loss is the reduction in power density associated with spreading of the power over a sphere of radius R surrounding the antenna.
To obtain the total power intercepted by the scatterer, the power density must be multiplied by the effective receiving area of the scatterer:
(2)

Note that the effective area Ars is not the actual area of the incident beam intercepted by the scatterer, but rather is the effective area; i.e., it is that area of the incident beam from which all power would be removed if one assumed that the power going through all the rest of the beam continued uninterrupted. The actual value of Ars depends on the effectiveness of the scatterer as a receiving antenna.

Some of the power received by the scatterer is absorbed in losses in the scatterer unless it is a perfect conductor or a perfect isolater; the rest is reradiated in various directions. The fraction absorbed is fa, so the fraction reradiated is 1- fa, , and the total reradiated power is
(3)

The conduction and displacement currents that flow in the scatterer result in reradiation that has a pattern (like an antenna pattern). Note that the effective receiving area of the scatterer is a function of its orientation relative to the incoming beam, so that Ars in the equation above is understood to apply only for the direction of the incoming beam.

The reradiation pattern may not be the same as the pattern of Ars, and the gain in the direction of the receiver is the relevant value in the reradiation pattern. Thus,
(4)

where Pts is the total reradiated power,Gts is the gain of the scatterer in the direction of the receiver, and is the spreading factor for the reradiation.

Note that a major difference between a communication link and radar scattering is that the communication link has only one spreading factor, whereas the radar has two. Thus, if Rr = Rt, the total distance is 2Rt; for a communication link with this distance, the spreading factor is only:

whereas for the radar it is:

Hence, the spreading loss for a radar is much greater than for a communication link with the same total path length.

The power entering the receiver is
(5)

where the area Ar is the effective aperture of the receiving antenna, not its actual area. Not only is this a function of direction, but it is also a function of the load impedance the receiver provides to the antenna; for example,Pr would have to be zero if the load were a short circuit or an open circuit.
The factors in the eq. 1 through the eq. 5 may be combined to obtain

The factors associated with the scatterer are combined in the square brackets.
These factors are difficult to measure individually, and their relative contributions are uninteresting to one wishing to know the size of the received radar signal. Hence they are normally combined into one factor, the radar scattering cross section:
(7)

The cross-section s is a function of the directions of the incident wave and the wave toward the receiver, as well as that of the scatterer shape and dielectric properties.

The final form of the radar equation is obtained by rewriting th eq. 6 using the definition of the eq. 7:
(8)

The most common situation is that for which receiving and transmitting locations are the same, so that the transmitter and receiver distances are the same. Almost as common is the use of the same antenna for transmitting and receiving, so the gains and effective apertures are the same, that is:


Rt= Rr =R

Gt= Gr =G

At= Ar =A

Since the effective area of an antenna is related to its gain by:
(9)

we may rewrite the radar equation (eq. 8) as
(10)

where two forms are given, one in terms of the antenna gain and the other in terms of the antenna area.

The radar equations (eq. 8 and eq. 10) are general equations for both point and area targets. That is, the scattering cross-section s is not defined in terms of any characteristic of a target type, but rather is the scattering cross-section of a particular target.

The form given in the equation 10 is for the so-called monostatic radar, and that in eq. 8 is for bistatic radar, although it also applies for monostatic radar when the conditions for R, G, A given above are satisfied.

 

Reference: http://earth.esa.int/applications/data_util/SARDOCS/spaceborne/Radar_Courses/Radar_Course_III/radar_equation.htm