1.1 How weather radars work

A weather radar transmits a signal along a path called the radar beam, and the antenna rotates at a constant elevation angle to complete one sweep or elevation scan. The antenna makes a series of sweeps at increasing elevation angles, producing a set of nesting conical surfaces of three-dimensional data called a volume scan. When the radar beams encounter a backscattering target (e.g. rain drops, hail, snow, birds), some of the energy is scattered back to the radar receiver, and is then interpreted as the quantity reflectivity factor. This process is summarized by the radar equation (Hong and Gourley 2015):

\[\begin{equation} P_r = \frac{z}{r^2} \left ( \frac{P_t g^2 \theta \phi h}{\lambda^2}\right ) \left ( \frac{ \pi^3 } {1024 \ln(2)} \right ) |K|^2 l \end{equation}\]

where the non-numeric parameters can be classified into three categories:

• Derived quantities

  • \(P_r =\) power received by radar (watts)
  • \(r =\) range or distance to target (m)
  • \(z =\) radar reflectivity factor (\(mm^6/m^3\))

• Radar constants

  • \(P_t =\) power transmitted by radar (watts)
  • \(g =\) antenna gain
  • \(\theta =\) horizontal beam width (radians)
  • \(\phi =\) vertical beam width (radians)
  • \(h =\) pulse length (m)
  • \(\lambda =\) wavelength of radar pulse (m)

• Assumed values

  • \(|K|^2 =\) dielectric constant for radar targets (usually set at 0.93 for liquid water)
  • \(l =\) loss factor for beam attenuation (assumed to be 1 for if attenuation is unknown)

The equation can be simplified by combining the numeric values, the assumed values, and the radar-specific variables into a single constant \(c_1\), and solve for z, such that:

\[ \begin{equation} z = c_1 P_r r^2 \end{equation} \]

The constant \(c_1\) depends on a specific radar and its configuration, such that the reflectivity factor \(z\) is calculated based on the two parameters measured by the radar: the amount of power return (\(P_r\)) and the range (\(r\)). This reflectivity factor is a function of the distribution of the rainfall drop sizes within a unit volume of air measured. The reflectivity factor is derived as:

\[ \begin{equation} z = \sum_{vol} D^6 = D_1^6 + D_2^6 + D_3^6 + \ldots + D_N^6 \end{equation} \]

where D is the drop diameter in mm. The reflectivity factor can take on values across several orders of magnitudes (from 0.001 mm\(^6\)/m\(^3\) for fog to 36,000,000 mm\(^6\)/m\(^3\) for baseball-sized hail). To compress the range of magnitudes to a more comprehensible scale, the reflectivity factor is typically converted to decibels of reflectivity (dBZ) or simply \(Z\), given by:

\[ \begin{equation} Z = 10 \log_{10} \left (\frac{z}{mm^6/m^3} \right ) \end{equation} \]

Rain rate is also derived from drop-size distribution, such that we can relate reflectivity (\(Z\)) and rain-rate (\(R\)) into a so-called Z–R relation of the form:

\[ \begin{equation} Z = A \cdot R^b \end{equation} \]

where A and b are empirically derived constants. This bridge between the radar reflectivity measured aloft and the estimated rain-rate allows us to actively observe and monitor rainfall from distances far from the station (as far as 250 km) even before it hits the ground.

References

Hong, Yang, and Jonathan J. Gourley. 2015. Radar Hydrology: Principles, Models, and Applications. CRC Press, Taylor & Francis Group.