Weather radars send directional pulses of microwave radiation, on the order of a microsecond long, using a cavity magnetron or klystron tube connected by a waveguide to a parabolic antenna. The wavelengths of 1 – 10 cm are approximately ten times the diameter of the droplets or ice particles of interest, because Rayleigh scattering occurs at these frequencies. This means that part of the energy of each pulse will bounce off these small particles, back in the direction of the radar station.
Shorter wavelengths are useful for smaller particles, but the signal is more quickly attenuated. Thus 10 cm (S-band) radar is preferred but is more expensive than a 5 cm C-band system. 3 cm X-band radar is used only for short-range units, and 1 cm Ka-band weather radar is used only for research on small-particle phenomena such as drizzle and fog.W-band weather radar systems have seen limited university use, but due to quicker attenuation, most data are not operational.
Radar pulses spread out as they move away from the radar station. Thus the volume of air that a radar pulse is traversing is larger for areas farther away from the station, and smaller for nearby areas, decreasing resolution at far distances. At the end of a 150 – 200 km sounding range, the volume of air scanned by a single pulse might be on the order of a cubic kilometer. This is called the pulse volume
The volume of air that a given pulse takes up at any point in time may be approximated by the formula {\displaystyle \,{v=hr^{2}\theta ^{2}}} \, {v = h r^2 \theta^2}, where v is the volume enclosed by the pulse, h is pulse width (in e.g. meters, calculated from the duration in seconds of the pulse times the speed of light), r is the distance from the radar that the pulse has already traveled (in e.g. meters), and {\displaystyle \,\theta } \,\theta is the beam width (in radians). This formula assumes the beam is symmetrically circular, "r" is much greater than "h" so "r" taken at the beginning or at the end of the pulse is almost the same, and the shape of the volume is a cone frustum of depth "h".
Listening for return signals
{\displaystyle {\text{Distance}}=c{\frac {\Delta t}{2n}},} \text{Distance} = c \frac{\Delta t}{2n},
If pulses are emitted too frequently, the returns from one pulse will be confused with the returns from previous pulses, resulting in incorrect distance calculations.
Determining height
The radar beam path with height
r = distance radar–target,
Scanned volume by using multiple elevation angles
Due to the Earth's curvature and change of index of refraction with height, the radar cannot "see" below the height above ground of the minimal angle (shown in green) or closer to the radar than the maximal one (shown as a red cone in the center).
{\displaystyle P_{r}=P_{t}{{G^{2}\lambda ^{2}\sigma _{0}} \over {{(4\pi )}^{3}R^{4}}}\propto {\frac {\sigma _{0}}{R^{4}}}} P_{r}=P_{t}{{G^{2}\lambda ^{2}\sigma _{0}} \over {{(4\pi )}^{3}R^{4}}}\propto {\frac {\sigma _{0}}{R^{4}}}
In this case, we have to add the cross sections of all the targets:
In combining the two equations:
{\displaystyle P_{r}=P_{t}{{G^{2}\lambda ^{2}} \over {{(4\pi )}^{3}R^{4}}}{\frac {c\tau }{2}}{\frac {\pi R^{2}\theta ^{2}}{4}}\eta =P_{t}\tau G^{2}\lambda ^{2}\theta ^{2}{\frac {c}{512(\pi ^{2})}}{\frac {\eta }{R^{2}}}} P_{r}=P_{t}{{G^{2}\lambda ^{2}} \over {{(4\pi )}^{3}R^{4}}}{\frac {c\tau }{2}}{\frac {\pi R^{2}\theta ^{2}}{4}}\eta =P_{t}\tau G^{2}\lambda ^{2}\theta ^{2}{\frac {c}{512(\pi ^{2})}}{\frac {\eta }{R^{2}}}
{\displaystyle P_{r}\propto {\frac {\eta }{R^{2}}}} P_r \propto \frac {\eta} {R^2}
Reflectivity (in decibel or dBZ)
Between each pulse, the radar station serves as a receiver as it listens for return signals from particles in the air. The duration of the "listen" cycle is on the order of a millisecond, which is a thousand times longer than the pulse duration. The length of this phase is determined by the need for the microwave radiation (which travels at the speed of light) to propagate from the detector to the weather target and back again, a distance which could be several hundred kilometers. The horizontal distance from station to target is calculated simply from the amount of time that lapses from the initiation of the pulse to the detection of the return signal. The time is converted into distance by multiplying by the speed of light in air:where c = 299,792.458 km/s is the speed of light, and n ≈ 1.0003 is the refractive index of air.[14]Assuming the Earth is round, the radar beam in vacuum would rise according to the reverse curvature of the Earth. However, the atmosphere has a refractive index that diminishes with height, due to its diminishing density. This bends the radar beam slightly toward the ground and with a standard atmosphere this is equivalent to considering that the curvature of the beam is 4/3 the actual curvature of the Earth. Depending on the elevation angle of the antenna and other considerations, the following formula may be used to calculate the target's height above ground:ke = 4/3,A weather radar network uses a series of typical angles that will be set according to the needs. After each scanning rotation, the antenna elevation is changed for the next sounding. This scenario will be repeated on many angles to scan all the volume of air around the radar within the maximum range. Usually, this scanning strategy is completed within 5 to 10 minutes to have data within 15 km above ground and 250 km distance of the radar. For instance in Canada, the 5 cm weather radars use angles ranging from 0.3 to 25 degrees. The image to the right shows the volume scanned when multiple angles are used.Calibrating intensity of return[edit]where {\displaystyle \scriptstyle P_{r}} \scriptstyle P_r is received power, {\displaystyle \scriptstyle P_{t}} \scriptstyle P_t is transmitted power, {\displaystyle \scriptstyle G} \scriptstyle G is the gain of the transmitting/receiving antenna, {\displaystyle \scriptstyle \lambda } \scriptstyle \lambda is radar wavelength, {\displaystyle \scriptstyle \sigma } \scriptstyle \sigma is the radar cross section of the target and {\displaystyle \scriptstyle R} \scriptstyle R is the distance from transmitter to target.{\displaystyle \sigma _{0}={\bar {\sigma }}_{0}=V\sum \sigma _{0j}=V\eta } \sigma_0 = \bar \sigma_0 = V \sum \sigma_{0j} = V \etaWhich leads to:Notice that the return now varies inversely to {\displaystyle \,R^{2}} \, R^2 instead of {\displaystyle \,R^{4}} \,R^4. In order to compare the data coming from different distances from the radar, one has to normalize them with this ratio.Return echoes from targets ("reflectivity") are analyzed for their intensities to establish the precipitation rate in the scanned volume. The wavelengths used (1–10 cm) ensure that this return is proportional to the rate because they are within the validity of Rayleigh scattering which states that the targets must be much smaller than the wavelength of the scanning wave (by a factor of 10).Precipitation rate (R), on the other hand, is equal to the number of particles, their volume and their fall speed (v[D]) as:So Ze and R have similar functions that can be resolved giving a relation between the two of the form:Where a and b depend on the type of precipitation (snow, rain, convective or stratiform), which has different {\displaystyle \Lambda } \Lambda , K, N0 and v.Since variation in diameter and dielectric constant of the targets can lead to large variability in power return to the radar, reflectivity is expressed in dBZ (10 times the logarithm of the ratio of the echo to a standard 1 mm diameter drop filling the same scanned volume).red: 52 dBZWhen describing weather radar returns, pilots, dispatchers, and air traffic controllers will typically refer to three return levels:level 2 corresponds to a yellow radar return, indicating moderate precipitation, leading to the possibility of very low visibility, moderate turbulence and an uncomfortable ride for aircraft passengers.Some displays provided by commercial weather sites, like The Weather Channel, show precipitation types during the winter month : rain, snow, mixed precipitations (sleet and freezing rain). This is not an analysis of the radar data itself but a post-treatment done with other data sources, the primary being surface reports (METAR).
Reflectivity perceived by the radar (Ze) varies by the sixth power of the rain droplets' diameter (D), the square of the dielectric constant (K) of the targets and the drop size distribution (e.g. N[D] of Marshall-Palmer) of the drops. This gives a truncated Gamma function, [19] of the form:
{\displaystyle Z_{e}=\int _{0}^{Dmax}|K|^{2}N_{0}e^{-\Lambda D}D^{6}dD} Z_e = \int_{0}^{Dmax} |K|^2 N_0 e^{-\Lambda D} D^6dD
{\displaystyle R=\int _{0}^{Dmax}N_{0}e^{-\Lambda D}{\pi D^{3} \over 6}v(D)dD} R = \int_{0}^{Dmax} N_0 e^{-\Lambda D} {\pi D^3 \over 6} v(D)dD
Z = aRb
As the antenna scans the atmosphere, on every angle of azimuth it obtains a certain strength of return from each type of target encountered. Reflectivity is then averaged for that target to have a better data set.
magenta: 65 dBZ (extremely heavy precipitation, possible hail)
Aviation conventions
level 1 corresponds to a green radar return, indicating usually light precipitation and little to no turbulence, leading to a possibility of reduced visibility.
Precipitation types
Until dual-polarization (section Polarization below) data are widely available, any precipitation types on radar images are only indirect information and must be taken with care.
{\displaystyle H={\sqrt {r^{2}+(k_{e}a_{e})^{2}+2rk_{e}a_{e}\sin(\theta _{e})}}-k_{e}a_{e}+h_{a},} H = \sqrt{r^2+(k_ea_e)^2+2rk_{e}a_{e}\sin(\theta_e)} - k_{e}a_{e} + h_{a},
where:
ae = Earth radius,
θe = elevation angle above the radar horizon,
ha = height of the feedhorn above ground.
Because the targets are not unique in each volume, the radar equation has to be developed beyond the basic one. Assuming a monostatic radar where {\displaystyle \scriptstyle G_{t}=A_{r}(or\ G_{r})=G} \scriptstyle G_t=A_r (or\ G_r) =G:
{\displaystyle {\begin{cases}V\quad =scanned\ volume\\\qquad =pulse\ length\ \ X\ beam\ width\\\qquad ={\frac {c\tau }{2}}{\frac {\pi R^{2}\theta ^{2}}{4}}\end{cases}}} {\begin{cases}V\quad =scanned\ volume\\\qquad =pulse\ length\ \ X\ beam\ width\\\qquad ={\frac {c\tau }{2}}{\frac {\pi R^{2}\theta ^{2}}{4}}\end{cases}}
where {\displaystyle \,c} \,c is the light speed, {\displaystyle \,\tau } \,\tau is temporal duration of a pulse and {\displaystyle \,\theta } \,\theta is the beam width in radians.
How to read reflectivity on a radar display[edit]
Radar returns are usually described by colour or level. The colours in a radar image normally range from blue or green for weak returns, to red or magenta for very strong returns. The numbers in a verbal report increase with the severity of the returns. For example, the U.S. National Doppler Radar sites use the following scale for different levels of reflectivity:
yellow: 36 dBZ
green: 20 dBZ (light precipitation)
Strong returns (red or magenta) may indicate not only heavy rain but also thunderstorms, hail, strong winds, or tornadoes, but they need to be interpreted carefully, for reasons described below.
level 3 corresponds to a red radar return, indicating heavy precipitation, leading to the possibility of thunderstorms and severe turbulence and structural damage to the aircraft.
Aircraft will try to avoid level 2 returns when possible, and will always avoid level 3 unless they are specially-designed research aircraft.
Over the area covered by radar echoes, a program assigns a precipitation type according to the surface temperature and dew point reported at the underlying weather stations. Precipitation types reported by human operated stations and certain automatic ones (AWOS) will have higher weight. Then the program does interpolations to produce an image with defined zones. These will include interpolation errors due to the calculation. Mesoscale variations of the precipitation zones will also be lost. More sophisticated programs use the numerical weather prediction output from models, such as NAM and WRF, for the precipitation types and apply it as a first guess to the radar echoes, then use the surface data for final output.
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