## Probert-Jones Equation

where all constants related to the fixed variables are combined to create the radar constant, Cr and La are factors related to atmospheric attenuation. (Attenuation is the weakening of a radar beam as it moves downstream due to some of the energy being lost to scattering and absorption.). The unit for reflectivity is mm6m-3.

The Probert-Jones Equation states that the signal power returned by a scatterer is directly related to the percentage of energy scattered by the object, inversely related to how far the scatterer is to the radar receptor, and how much the signal is attenuated in transit.

A more conceptual version of the equation is obtained by simply solving for the returned power (that signal that reaches the radar receptor).

The power returned to the radar receptor is directly proportional to the reflectivity (of the scatterer) modulated by attenuation along the path back to the radar receptor, and inversely proportional to the distance of the scatterer from the radar.

Reflectivity vs. Decibels of Reflectivity (Z vs. dBZ)

Range-normalized values of reflectivity, Z, can range over many orders of magnitude. To compress this large range of values for operational use, Z is displayed in "echos strength of Z" or decibels of Z, that is, dBZ. Converting Z to dBZ is simply done by using

For example, if Z = 4000 mm6m-3, then dBZ = 10(log10 4000) » 10 x 3.6 = 36 dBZ.

Due to the WSR-88D's increased sensitivity, reflectivities as low as -32 dBZ can be detected in clear air mode. How can there be such a thing as a negative dBZ? If 0 < Z < 1, log10Z < 0 and thus dBZ < 0. Very low dBZ values indicate the presence of extremely small sized particles (e.g., dust, haze, smoke. Since dBZ contains a logarithm, negative values will be returned for very small reflectivities. Foro example, a reflectivity of 0.1 mm6m-3 will return dBZ = 10(log10 0.1) » 10 x -1 = -10 dBZ.

The WSR-88D can also detect reflectivity values as high as 95 dBZ. As an example, a one cubic meter volume containing just one 38.3 mm (~1.50 inch) diameter water-coated hailstone would yield a reflectivity value of approximately 95 dBZ. However, giant hail frequently occurs with reflectivities less than 70 dBZ. We'll discuss the radar's inability to correctly judge hail separatley because large hail, particularly when water coated, is not a Rayleigh scatterer.

Table 1 illustrates why dBZ is used instead of Z to portray reflectivity.

Table 1: dBZ and Z values

 dBZ Z(mm6m-3) dBZ Z(mm6m-3) -32 0.000631 30 1,000 -28 0.001585 41 12,589 -10 0.1 46 39,810 0 1 50 100,000 5 3.162 57 501.187 18 63.1 95 3,162,277,660

dBZ vs Video Integrator Processor (VIP) Levels

Categorized intervals of reflectivity which are computer processed by a Digital Video Integrator Processor (D/VIP). These intervals were very important before the installation of the 88-D Radar network. Some of the 88-D Radar products still have these intervals on them. The following table illustrates the various rainfall rates associated with VIPs:
VIP
Level
dBZ Precipitation Description Rainfall Rate (inches/hour)

inches millimeters
6 >57 Very heavy rain and hail; large hail possible     7.10 or more 180 or more
5 50-57 Very heavy rain; hail possible     4.50 - 7.09 114 - 179
4 44-50 Heavy rain     2.20 - 4.49 56 - 113
3 38-44 Moderate to heavy rain     1.10 - 2.20 28 - 55
2 30-38 Light to moderate rain     0.20 - 1.10 5 - 27
1 18-30 Light precipitation     0.05 - 0.19 1 - 4