RADAR EQUATION

RADAR is an acronym that stands for "RAdio Detection And Ranging." This is an active remote sensing technique because it involves a transmitter sending out pulses of electromagnetic radiation, then measuring the amount of power reflected (scattered) back to the radar antenna.  This process can be quatified in the Radar Equation.

The Radar Equation is given as follows:

pr=(p3ptg2qfh|K|2 l z) / (1024 ln(2) l2r2)

where

pt= power transmitted by radar (watts)
pr= power received back by radar (watts)
g = gain of the antenna (ratio of power on the beam axis to power from an isotropic [i.e., radiating equally in all directions] antenna at the same point);  it is a measure of how focused the radar beam is.
q = horizontal beamwidth (radians)
f = vertical beamwidth (radians)
h = pulselength (m)
|K|2 = dielectric constant for hydrometeors; usually taken as 0.93 for liquid water, 0.197 for ice
l = loss factor for attenuation of radar beam, varies between 0 and 1, usually near 1. Since the attenuation of the beam is often unknown, it is often ignored.
z = radar reflectivity factor (mm6/m3); it  is a function of the diameters and number of drops in a unit volume
l = wavelength of radar pulse (m)
r = range or distance to the target (i.e., the distance to an area of precipitation that reflects the originally transmitted pulse back to the radar).
The radar reflectivity factor, z, may be calculated from the return power, pr, by solving the above equation for z:

z = pr (1024 ln(2) l2r2)/(p3ptg2qfh|K|2 l)


We can combine the known variables and numerical values in the above equation to arrive at the simplified expression:
z = c pr r2

Because the radar reflectivity factor spans a huge range of magnitudes (from 0.001 mm6/m3 for fog, to 36,000,000  mm6/m3 for softball-sized hail, it is usually expressed in decibels (dB) of reflectivity or dBZ as follows:

Z = 10log10(z / 1 mm6/m3)

The logarithmic transformation here is used to compress the large range of magnitudes into a more comprehensible scale of values. Logarithms are actually just exponents, so the "log10 of z" is just the exponent that 10 would be raised to to obtain a value of z.

The following table shows interrelationships between z, Z, exponents, logs, and the decibel scale:

z
Radar reflectivity factor from the radar equation.
(linear scale of reflectivity)
 10x = z
 x = log10 z
 Z
dBZ = 10 log10 z
(decibel scale of reflectivity)
0.001
 10-3 -3
 -30
0.01
 10-2 -2
 -20
0.1
 10-1 -1
 -10
1
 100 0
 0
10
 101 1
 10
100
 102 2
 20
1,000
 103 3
 30
10,000
 104 4
 40
100,000
 105 5
 50
1,000,000
 106 6
 60
10,000,000
 107 7
 70


Contours of  Z plotted as VIP (Video Integrator Processor) levels is what is plotted on radar data displays. For more information on converting between the pre-NEXRAD six-VIP level format, and the current NEXRAD 16-level reflectivity display, click here.

Rainrates from Reflectivity

The rain rate (RR) in mm/hr can be calculated from the reflectivity (Z in dbZ) according to the following formula (based on a Marshall-Palmer drop size distribution):

RR = C 10(0.0625 Z)

where C = 0.036 mm/hr

Example: For a reflectivity of 39 dBZ, the rain rate is

RR = 0.036 x 10(0.0625 x 39)

= 9.86 mm/hr x (1 cm/10 mm) x (1 in/2.54 cm)

= 0.39 inches/hour