RADAR EQUATION

Dr. Brad Muller

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 quantified in the Radar Equation.

The Radar Equation is given as follows:

where

p_t= power transmitted by radar (watts)

p_r= 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. (Note that for an equivalent mass of frozen precipitation, much less power is returned from the ice than from liquid precipitation; thus snow with the same water content is less reflective than rain). For this reason, NEXRAD’s clear air mode rather than precipitation mode is sometimes used to monitor snow situations because of its greater sensitivity).

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.

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).

z = radar reflectivity factor (mm⁶/m³) and can be expressed as

where D is the drop diameter and the summation is over the total number of  drops (of varying sizes) within a unit volume within the beam; in the equation it gets multiplied by the radar volume [defined by the beam width, height, pulse length and distance from the radar]

Therefore z is a function of the diameters and number of drops in unit volume, i.e., the drop size distribution.

Note that z is an inherent property of the drop size distribution sampled and is not radar-dependent.

In other words, it is a property of the precipitation characteristics within a given storm, and whatever drop sizes happen to be in that storm, and has nothing to do with the radar.

[Note: The radar equation can also be written in this slightly different form

where h, the pulse length in distance units, is replaced by c, the speed of electromagnetic radiation, and t, the pulse duration.]

However, the drop size distribution in the measured volume is unknown. Therefore, we calculate the radar reflectivity factor, z, from the return power, p_r, by solving the above equation for z:

We can combine the known (radar-specific) variables like beamwidth, gain etc., the numerical values, and the assumed values (|K|² is assumed as the liquid water value, 0.93, since we don’t know a priori what the precipitation type is, even though it may actually be snow or hail, and l is usually assumed to be 1) in the above equation into a single known constant, c₁, to arrive at the simplified expression:

z = c₁ p_r r²

Thus, given a specific radar and configuration, the only real variables in this problem are the amount of power returned, p_r, and the range, r, to the echoes, which are both measured by the radar based on the elapsed time since the pulse was sent out.

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)

[Note: be sure to distinguish between capital Z and lower case z here!]

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)	10^x = 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	10^0	0	0
10	10^1	1	10
100	10^2	2	20
1,000	10^3	3	30
10,000	10^4	4	40
100,000	10^5	5	50
1,000,000	10^6	6	60
10,000,000	10^7	7	70

Color-filled contours of  22 levels of reflectivity, Z, are what is plotted on NEXRAD radar data displays. Prior to NEXRAD, reflectivity from the WSR-57’s or WSR74’s was plotted as 6 levels of reflectivity called VIP (Video Integrator Processor) levels. NEXRAD data plotted as VIP levels can be found on an Aviation Digital Data Service radar page showing radar coded messages .

Rain rates from Reflectivity

Rain rates can be calculated from radar reflectivity data if we assume that we know the drop size distribution within a volume of air measured by the radar. So-called z-R relationships relate the radar reflectivity factor, z (mm⁶/m³), to the rain rate, R (mm/hr) based on different assumed drop sizes for different kinds of storms, and are of the form

where A and b are empirically-derived constants. Over the years, many z-R relations have been obtained by researchers. A few z-R relations recommended by the Radar Operations Center are listed in this table borrowed from

https://www.ou.edu/radar/z_r_relationships.pdf

Relationship	Optimum for:	Also recommended for:
Marshall-Palmer	General stratiform precipitation
(z = 200R^1^.^6)
East-Cool Stratiform	Winter stratiform precipitation -	Orographic rain -East
(z = 130R^2.0)	east of continental divide
West-Cool Stratiform	Winter stratiform precipitation -	Orographic rain -West
(z = 75R^2^.^0)	west of continental divide
WSR-88D Convective	Summer deep convection	Other non-tropical
(z = 300R^1^.^4)		convection
Rosenfeld Tropical	Tropical convective systems
(z = 250R^1^.^2)

Note that this form of the z-R relation is not useful for calculating the rain rate—we must solve the equation for R and convert from radar reflectivity factor, z, to reflectivity, Z, then we can plug in the reflectivity measured by the radar.

The result for a Marshall-Palmer drop size distribution is:

R = C 10^{(0.0625 Z)}

where R is rain rate in mm/hr, C = 0.036 mm/hr, and Z is the reflectivity in dBZ.

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