The free-space radar range equation relating radar maximum detection range to the key physical parameters describing a radar can be written in the form [1, E. 1.28]

,

where

is the maximum range for which the radar can detect the target,

is the transmit power of the radar,

is the transmit antenna gain,

is the receive antenna gain,

is the target radar cross section,

is the wavelength of the radar carrier frequency,

is the receiver pattern propagation factor,

is the transmitter pattern propagation factor,

is Boltzmann's constant

),

is the system noise temperature in K,

is the noise bandwidth of the receiver, and

is the system loss factor. The quantity

in the radar equation is the effective noise spectral density measured in units of power/Hz and referenced to a point just after the receiver antenna output and before the receiver transmission line.

An extremely common mistake in applying the radar equation is to assume that the system noise temperature

is the reference temperature

, corresponding to the physical temperature of ordinary objects on the surface of the Earth. Actual system noise temperatures can range from 10 K to several thousand degrees K depending on the radar design and operating frequency.

For many radar applications, the radar system noise temperature can be computed using an approach developed in [1] and refined in [2]:

,

where

is the antenna noise temperature,

is the thermal temperature of the radar transmission line,

is the power loss in the radar transmission line,

is the reference temperature 290 °K and

is the receiver noise figure. The geometry of this situation is shown in snapshot 1. The antenna noise temperature

contains components from the atmosphere, the Sun, and the cosmos as well as the ground. It can be written in the form

,

where

is the sky noise temperature,

is the fraction of antenna power that is radiated onto the surface of the ground,

are the ohmic power losses in the antenna,

is the amplitude reflection coefficient from the ground,

is the physical temperature of the ground, and

is the physical temperature of the antenna. The first term in the preceding equation is the contribution to the antenna noise temperature that comes from the sky via the main lobe of the radar receive beam pattern. The second term represents sky noise that is reflected from the ground. The third term represents radiation from the ground that couples into the system via the receive beam pattern side lobe structure. The fourth term represents noise that is generated within the antenna by virtue of ohmic losses. If there are no ohmic losses in the antenna, then

and the fourth term is zero. If, on the other hand, there are large ohmic losses, then the fourth term in the limit is

, the physical temperature of the antenna, and the first three terms are zero.

The received power

from a radar antenna in a lossless medium at wavelength

due to radiation from a black body at temperature

located at a distance

from the radar is

,

is the Planck spectral radiance (

),

is frequency,

is Planck's constant,

is the speed of light,

is the effective area of the radar antenna, and

is the bandwidth of the radar receiver. If

denotes the area subtended by the radar receiver beam pattern on the black body and

is the gain of the radar receive beam, then

, where

is the radar receive beam solid angle. Also,

. Thus it follows that

at radar frequencies. If the radar antenna is polarized, then the received power is simply

.

The sky noise temperature

arises from contributions from the atmosphere, the cosmos, and the Sun. Sky noise temperature

can be written in the form

,

where

is the atmospheric (also called tropospheric) noise temperature,

is the cosmic noise temperature,

is the brightness temperature of the quiet Sun, and

are the absorptive propagation losses in the atmosphere. The factor

in the preceding equation reflects the fact that the Sun is much smaller in angular extent than the sky or atmosphere.

An approximation to the brightness temperature from a quiet Sun based on data presented in [1] and [3] is

.

The cosmic noise temperature can be written in the form

,

where

is the radar frequency in

and

is the cosmic noise temperature at 100 MHz, a quantity that varies between 500 K and 18650 K. The term 2.7 °K in this equation represents isotropic background radiation.

The atmospheric noise temperature

is related to loss-weighted average physical noise temperature

in the atmosphere and the propagation power loss in the atmosphere via

.

Thus if there are no power losses due to propagation through the atmosphere, then

, and the atmospheric noise temperature is 0. On the other hand, if there are large absorption losses in the atmosphere, the antenna will see the average physical noise temperature

in the atmosphere. The atmospheric noise temperature is also referred to as the tropospheric noise temperature since only the troposphere is absorptive at frequencies above about 100 MHz. If rain is present, then absorption increases and there is a corresponding increase in the atmospheric noise

. The key parameters that determine the effects of rain rate on system noise are the rain rate

in mm/hr and the height of the rain-producing column

in km.

Snapshot 1: the radar is modeled as a combination of a receiver, a receiver (Rx) transmission line, and an antenna, with each of these three components contributing to system noise in addition to noise contributions from the sky and the ground

[1] L. V. Blake,

*Radar Range-Performance Analysis*, Silver Spring, MD: Munro Publishing Co., 1991.

[2] D. K. Barton,

*Radar Equations for Modern Radar*, Boston, MA: Artech House, 2013.

[3] F. I. Shimabukuro and J. M. Stacey, "Brightness Temperature of the Quiet Sun at Centimeter and Millimeter Wavelengths,"

*The Astrophysical Journal*,

**152**, 1968 p. 777.