The Radar Equation

Conventional weather radars measure only two basic quantities to derive information about atmospheric targets:

  1. Elapsed time (to provide range information).
  2. Bearing of the radar beam.
  3. Return power (to estimate reflectivity).

Since reflectivity information provided by the radar depends upon the power returned, it is important to understand how the characteristics of the target, the radar, and the distance between then determine the amount of power received.

The power received by the radar is given by the radar equation:

where

Pr = average echo (return) power (average received power)
Pt = peak transmitted power
G = antenna gain
θ = beamwidth
c = speed of light 3x108 m/s
τ = Pulse length (sec)
λ = wavelength
r = target range
l2(r) = loss factor due to attenuation by precipitation, cloud or atmospheric gases.
KW = complex refractive index of water
Z = the reflectivity factor of the target.

Particle interpretation of the variables in the radar equation relates different characteristics of radar and target to the amount of power return and, therefore, the indicated intensity.

Average echo (return) power

(Pr) The weather radar averages the power returned from a number of pulses (typically 16 or 32). Averaging is necessary since power returned from a meteorological target varies greatly from pulse to pulse.

Transmitted power

(Pt) This is the power transmitted in the outgoing radar pulse. The average return power varies directly with the transmitted power. Thus, doubling the transmitted power and changing nothing else would double the average target return.

Antenna gain

(G) The gain of the antenna is a measure of antenna capability to focus outgoing energy into the beam and capture energy returning from a target. Power received from a given target varies directly with the square of antenna gain.

Beamwidth

(θ) The return power increases directly with the square of the beamwidth, just as for antenna gain. However derivation of the radar equation assumes the target fills the radar beam. This will only be valid if the angular extent of the precipitation target is greater than the radar beamwidth in both azimuth and elevation.

Pulse length

(cτ) Power received from a target varies directly with pulse length. The total amount of power received from a given target using a 2 μs pulse would be four times greater than for a 0.5 μs pulse. Recall however that the range resolution decreases as the pulse length increases.

Reflectivity factor

(Z) If the assumptions of the radar equation are met, then there are two important characteristics of a liquid precipitation target which determine how efficiently it returns power to the radar:

a. The number of drops within the target volume.
b. The size distribution of drops.

The reflectivity factor Z is given by:

where

Z = reflectivity factor (mm6/m3)
ni = number of drops with a given diameter per cubic metre.
Di = drops diameter (typically, in millimetres)
Σ = summation over all size classifications encountered per unit volume.

The reflectivity factor of a precipitation target is determined by the sum of the sixth power of all drop diameters in the sampled volume, and is usually expressed in millimetres to the sixth power per cubic meter (mm6/m3).

The power received from a precipitation target is highly dependent upon particle size. A drop 3 mm in diameter returns 729 times the power of a drop 1 mm in diameter – even though it contains only 27 times as much liquid!

Attenuation loss factor

l2(r) The loss in received power due to attenuation of the radar beam by precipitation, cloud or atmospheric gases. This was discussed in the module on radar beam propagation. For weather radar display systems it is assumed that no attenuation loss occurs. In heavy rain situations however there can be significant attenuation of the radar beam particularly for 3 cm and 5 cm radars. The ratio of particle size to wavelength affects the degree of attenuation experienced. The larger this ratio becomes, the worse the attenuation.

Wavelength of transmitted energy

(λ) The amount of power returned from a precipitation target varies inversely with the square of the wavelength. This means that a C-band radar (5 cm. wavelength) will receive four times as much power from a target as an S-band (10 cm. wavelength) radar with all other characteristics the same. However, short wavelengths (less than 10 cm) are subject to significant attenuation by precipitation in the atmosphere, and this introduces a further uncertainty into the measurement of reflectivity.

Range of target

(r) As with wavelength, the average power return from a target varies inversely with the square of the range from the radar site. Thus, the same target at 200 km range would return only one fourth as much power as it does at 100 km. This occurs because power density is proportional to 1/r2. This drop-off in power density with rang is referred to as range attenuation and it is compensated by Swept Gain or digital processing.

Complex refractive index KW

¦KW¦ The magnitude of this term ¦KW¦ is proportional to the induced electric dipole moment of the rain drops. The power received by the radar is proportional to ¦KW¦2. For liquid water, ¦KW¦2 ranges from 0.91 to 0.93 as the radar wavelength varies from 1 to 10 cm. For ice, it is equal to 0.18 and is relatively insensitive to any variation in wavelength. Weather radars interpret received power using a ¦KW¦2 value which assumes liquid water. For this reason, power received from particles composed entirely of ice (except for those which are extremely large) is about one fifth that which would be received from the water equivalent in liquid form. This produces a large underestimate of the water content of snow and ice crystals, and may result in the failure to detect ice and snow at long distances.

Assumptions of the radar equation

For the user to properly interpret radar presentations it is necessary to know the limitations of radar. Some of these limitations result from the assumptions which are made when applying the radar equation.

These include:

  1. The target is composed of a large number of relatively small spherical water drops, such that Rayleigh scattering is occurring. Studies have shown that Rayleigh scattering can be assumed if D<0.07λ where D is the drop diameter and λ is the radar wavelength.
  2. No attenuation occurs between the radar and the target.
  3. The drops which comprise the target are evenly distributed and fill the entire radar sampling volume.
Often these assumptions are not met, and consequently radar displays can be difficult to interpret correctly. Little can be done to eliminate the effects of these assumptions, although the radar design can somewhat compensate for them. It is evident the radar operator must be aware of the assumptions which are made and the limitations of radar.