Probert-Jones Radar Equation

Nearly all weather radars transmit a series of discrete pulses of energy that have been concentrated into a relatively narrow beam. Each pulse or packet of energy contains a certain amount of energy that propagates through Earth's atmosphere and illuminates targets of various sizes. The amount of energy returned to the radar is very small (~10-8 to 10-14 watts) and can be as much as 20 orders of magnitude smaller than the peak transmitted power.

The Probert-Jones (P-J) radar reflectivity equation will help to quantify the physical aspects of pulsed E-M energy and the associated limitations of target (e.g., precipitation) detection. The P-J equation is described below as

  Equation 1.  Probert-Jones Radar Equation.  Click for Long Description.  
(Equation 1)

where:

  Pr = power returned to the radar from a target (watts)   Pt = peak transmitted power (watts)
  G = antenna gain   greek symbol theta = angular beamwidth
  H = pulse length   greek symbol pi = pi (3.141592654)
  K = physical constant (target character)   L = signal loss factors associated with attenuation and receiver detection
  Z = target reflectivity   greek symbol lambda = transmitted energy wavelength
  R = target range    

Simplified Radar Equation

For the WSR-88D, the only variables that are not fixed are returned power (Pr), reflectivity (Z), attenuation factor (La), and range (R). The fixed variables are combined to create a new term which we will refer to as the radar constant, Cr. By combining the fixed variables into a radar constant, Equation (1) simplifies into

 
Equation 3. Simplified Radar Equation. Click for long description.
 
(Equation 3)

where Cr is the radar constant. Solving for Z, the above equation, Equation (3), becomes

 
Equation 4. Simplified Radar Equation solved for Z. Click for long description.
 
(Equation 4)

By knowing the returned power and range (based on timing), the above equation, Equation (4), estimates target reflectivity.

Equivalent Reflectivity vs. Reflectivity (Ze vs. Z)

Since we technically don't know the drop-size distribution or physical makeup of all targets within a sample volume, radar meteorologists oftentimes refer to radar reflectivity as equivalent reflectivity, Ze. The assumption is that all backscattered energy is coming from liquid targets whose diameters meet the Rayleigh approximation. Obviously, this assumption is invalid in those cases when large, water-coated hailstones are present in a sample volume. Hence, the term equivalent reflectivity instead of actual reflectivity is more valid.

However, to be consistent with WSR-88D product mnemonics and the assumption that all backscattered energy is from Rayleigh targets, the term reflectivity (Z) will be used.