Radar Equation for Distributed Targets

• Before we derive the radar equation for the distributed targets situation, we need to make some assumptions:

  1. The beam is filled with targets.  Q:  Where/when would this assumption break down?
  2. Multiple scattering is ignored.
  3. Total average power is equal to the sum of powers scattered by individual particles.

Recall the radar equation for a single target:

   (1)

• P_r is the average received power
• P_t is the transmitted power
• G is the gain for the radar
• l is the radar's wavelength
• s is the targets scattering cross section
• r is the range from the radar to the target

For multiple targets, (1) can be written as:

    (2)

where the sum is over all targets within the pulse volume. 

If we assume that h/2 << r_i,

then (2) can be written as:

    (3)

It is advantageous to sum the backscattering cross sections over a unit volume of the total pulse volume.

Hence the sum in (3) can be written as:

  (4)

where the total volume is the volume of the pulse.

Thus, (4) can be written as:

  (5)

Substituting (5) into (3) gives:

  (6)

Note that:

 P_r is proportional to R^-2 for distributed targets

P_r is proportional to R^-4 for point targets.

WHY?????  Answer

RADAR REFLECTIVITY 

The sum of all backscattering cross sections (per unit volume) is referred to as the radar reflectivity (h).  In other words,

    (7)

Q:  What are the units of h?  Answer

In terms of the radar reflectivity, the radar equation for distributed targets (6) can be written as:

    (8)

all variables in (8), except h are either known or measured.

Now, we need to add a fudge factor due to the fact that the beam shape is gaussian (we could derive this, but it's not worth the trouble.  Check out the reference books if you are interested).

Hence, (8) becomes;

  (9)

COMPLEX DIELECTRIC FACTOR

The backscattering cross section (s_i) can be written as:

  (10)

where:

• D is the diameter of the target
• l is the wavelength of the radar
• K is the complex dielectric factor
  • is some indication of how good a material is at backscattering radiation

For water, = 0.93

For ice, = 0.197.

Notice that the value for water is much larger than for ice.  All other factors the same, this creates a 5 dB difference in returned power.

So, let's incorporate this information into the radar equation.

Recall from (7) that .  Using (10), h can be written as:

    (11)

Taking the constants out of the sum;

    (12)

Remember that the sum is for a unit volume.  Substituting (12) into (9) gives:

    (13)

Simplifying terms gives:

(14)

Note the D_i⁶ dependence on the average received power.

RADAR REFLECTIVITY FACTOR

In Equation (14), all variables except the summation term, are either known or measured.

We will now defined the radar reflectivity factor, Z, as:

  (15)

Q:  What are the units of Z?  Answer

 Substituting (15) into (14) gives the radar equation for distributed targets:

  (16)

• That's it!!!!  
• Note the relationship between the received power, range, and radar wavelength.
• Q:  To maximize the received power, how would one adjust the wavelength of radiation transmitted by the radar??
• Everything in Equation (16) is measured or known except Z, the radar reflectivity factor.
• Since the strength of the received power can span many orders of magnitude, then so does Z.
• Hence, we take the log on Z according to:

  (17)

• The dBZ value calculated above is what you see displayed on the radar screen or on imagery accessed from the web.
• Q:  What dBZ value will be observed if one raindrop 1mm in diameter is found within a volume of 1 m3?  Answer
• Q:  What dBZ value will be observed if one raindrop 2 mm in diameter is found within a volume of 1 m3?  Answer
• Q:  Can the dBZ value be negative?  Answer