1. Radar Equations
Radar Meteorology
M. D. Eastin

2. Radar Equations Outline Basic Approach to Radar Equation Development
Solitary Target
Power incident on target
Power scattered back toward radar
Amount of power collected by the antenna
Distributed (Multiple) Targets
Distributed (Multiple) Weather Targets
Radar Meteorology
M. D. Eastin

3. Radar Equation Development
Basic Approach:
Radar Observation: Measurement of echo power received from a target provides useful
information about the target (# raindrops ~ reflectivity)
Radar Equation: Provides a relationship between the received power, the characteristics of the target, and the unique characteristics of the antenna/radar design
Basic development is common to all radars!
Solitary Target: Develop radar equation for a single target (i.e., one raindrop)
1. Determine the transmitted power per unit area
(power flux density) incident on the target
2. Determine the power flux density scattered back
toward the radar
3. Determine the amount of back-scattered power
actually collected by the radar
Distributed Targets: Expand to allow for multiple targets with the volume
Radar Meteorology
M. D. Eastin

4. Transmitted Power Isotropic Antenna:
An isotropic radar transmits power equally in all directions
Power flux density at a given range from an isotropic antenna is:
(1)
where S = power flux density (W/m2)
Pt = transmitted power (W/m2)
r = range from antenna
Radar Meteorology
M. D. Eastin

5. Transmitted Power Directional Antenna:
Most radars attempt to focus all of the transmitted power into a narrow window (a beam)
This is NOT an easy task, but most radars come close
Gain Function:
Gain is the ratio of the power flux density at radius r, azimuth θ, and elevation φ for a
directional antenna to the power flux density for an isotropic antenna transmitting the
same total power:
(2)
Combining (2) with (1):
(3)
In practice, it also incorporates absorptive losses by the antenna and waveguide
The gain function is unique for each radar!
Radar Meteorology
M. D. Eastin

6. Transmitted Power What does the Gain Function look like? 3D Depiction
Main Lobe
Side Lobes
3D Depiction
of Gain (dB)
Radar Meteorology
M. D. Eastin

7. 2D Depiction of Gain (dB)
Transmitted Power
What does the Gain Function look like?
Main lobe (i.e. the beam) has a maximum
gain of 30 dB
Strongest side lobes are ~4 dB with the
majority less than 0 dB
All back lobes are less than 0 dB
Effective beam width (Θ) defined at
the location equivalent to 3 dB less
than the peak gain on main lobe
(in this case at 27 dB → Θ = 6°)
For the same total transmitted power,
a large peak Gain will correspond to
a narrow beam width → desired
2D Depiction of Gain (dB)
Radar Meteorology
M. D. Eastin

8. Transmitted Power
Relationship between Gain, Beam Width, Wavelength, and Antenna Size?
If the same antenna is used for both
transmitting and receiving, then the
antenna size is related to the Gain
(since the effective beam width is)
(4)
Where: Ae = effective antenna area
λ = transmitting wavelength
Thus: Large antennas → Large Gain
→ Small Beam Widths
→ Large Wavelengths
Desired: Small beam widths
Small antennas
Large gain
10 cm
0.8 cm
Radar Meteorology
M. D. Eastin

9. Transmitted Power Problems Associated with Side Lobes:
Echo from the side lobe is interpreted
as being from the main beam, but the
return power is weak because the
transmitted power was weak
Horizontal “spreading” of weaker echo
to sides of storm…
Radar Meteorology
M. D. Eastin

10. Transmitted Power Problems Associated with Side Lobes:
Echo from the side lobe is interpreted
as being from the main beam, but the
return power is weak because the
transmitted power was weak
Vertical “spreading” of weaker echo
to top of storm…
Radar Meteorology
M. D. Eastin

11. Transmitted Power Method to Minimize Side Lobes:
Use a parabolic antenna
Parabolic antennas allow for “tapered illumination”
which minimizes the transmitted power flux density
along the edges
Effects of Tapered Illumination:
Reduction of side lobe returns
Reduction of maximum Gain
Increased beam width
The last two are undesirable, but in practice
parabolic antennas reduce side lobes by ~80%,
reduce Gain by less than 5%, and increase beam
width by less than 25% → acceptable compromise
Beam Geometry
Outgoing Power
Flux Density
Radar Meteorology
M. D. Eastin

12. Backscatter Power Radar Cross Section:
Defined as the ratio of the power flux density
scattered by the target in the direction of the
antenna to the power flux density incident on
the target (both measured at the target radius)
(5)
PROBLEM: We don’t measure Sback at r,
we measure it at the radar
For practical purposes, redefined as the
power flux density received at the radar:
(6)
Radar Transmitted
Power Flux Density
Target Backscatter
Radar Meteorology
M. D. Eastin

13. Backscatter Power Radar Cross Section:
In general, the radar cross section
of a target depends on:
Target’s shape
2. Target’s size relative to
the radar’s wavelength
(more on this later …)
Complex dielectric constant
and conductivity of the target
(more on this later…)
Viewing aspect from the radar
Radar Meteorology
M. D. Eastin

14. Power Received at Antenna
Bringing it all together...
Recall from before:
(3)
(6)
Substituting (3) into (6):
(7)
Accounting for the antenna area:
(8)
Power flux density
incident on target
Radar cross section
Power flux density
of target backscatter
at the antenna
Ae is the effective antenna area (m2)
Pr is the received power (W)
Radar Meteorology
M. D. Eastin

15. Power Received at Antenna
Bringing it all together…
Recall from before:
(8)
(6)
Substituting (6) into (8):
(9)
Let’s rearrange and examine this equation in more detail…
Power flux density
incident on target
Gain – Antenna size
relationship
Radar equation for a single isolated target
(e.g. an airplane, ship, bird, one raindrop)
Radar Meteorology
M. D. Eastin

16. Radar Equation for a Solitary Target
Written in terms of antenna effective area:
What do these equations tell us about radar returns from a single target?
Constant
Radar
Characteristics
Target
Characteristics
Constant
Radar
Characteristics
Target
Characteristics
Radar Meteorology
M. D. Eastin

17. Distributed Targets Distributed Target:
A target consisting of many scattering elements, for example, the billions of raindrops
that might be illuminated by a single radar pulse
Contributing Region:
Volume containing all objects from which the scattered microwaves arrive back
at the radar simultaneously
Spherical shell centered on the radar
Radial extent determined by the pulse duration
Angular extent determined by the antenna beam pattern
Radar Meteorology
M. D. Eastin

18. Distributed Targets Single Pulse Volume: Azimuthal coordinate (Θ)
Beam width in the azimuthal direction
is rΘ, where Θ is the arc length between
the half power points of the beam
Elevation coordinate (Φ)
Beam width in the elevation direction
is rΦ, where Φ is the arc length between
Cross-sectional area of beam:
Contributing volume length = half pulse length:
Volume of contributing region for a single pulse:
(10)
Radar Meteorology
M. D. Eastin

19. Distributed Targets Single Pulse Volume: NEXRAD Radar
Pulse duration → τ = 1.57 μs
Angular circular beam width → radians
Range from radar → r = 100 km
If the concentration of raindrops is the typical 1/m3, then the pulse volume contains:
520 million raindrops!!
Radar Meteorology
M. D. Eastin

20. Distributed Targets Caveats to Consider:
The pulse volume is not a perfect cone
Recall the antenna beam (gain) pattern
About half the transmitted power
fall outside the 3 dB cone
The gain function is not uniform
with the cone – targets along the
beam axis received more power
than those off axis
Radar Meteorology
M. D. Eastin

21. Distributed Targets Radar Cross-Section: Assumptions
The radial extent (h/2) of the contributing region is small compared to the
range (r) so that the variation of Sinc across h/2 can be neglected
(good assumption)
Sinc is considered uniform across the conical beam and zero outside – the spatial variation of the gain function can be ignored.
(not good, but we are stuck with this one)
Scattering by other objects toward the contributing region must be small so that interference effects with the incident wave do not modify its amplitude
(good for wavelengths > 3 cm)
Scattering or absorption of microwaves by objects between the radar and contributing region do not modify the amplitude of Sinc appreciably
Radar Meteorology
M. D. Eastin

22. Distributed Targets Radar Cross-Section:
Equal to the average radar cross for the random array of individual particles that
comprise the target (e.g. average radar cross section of 520 million drops)
(11)
Recall radar equation for a single target:
Radar equation for a distributed target:
(12)
Radar Meteorology
M. D. Eastin

23. Distributed Targets Radar Reflectivity (ηavg):
Since the radar cross-section is valid for all targets within the contributing volume,
and that volume is non-uniform (i.e. its a function of range and beam shape), we
need to account for this variability in each possible contributing volume:
(13)
Using (13) and the definition of the contributing volume (10), our radar equation
for a distributed target becomes:
(14)
Radar Meteorology
M. D. Eastin

24. Radar Equation for Distributed Targets
Accounting for Gain Function Shape:
Since the gain function maximizes along the beam axis, and decreases with angular
distance from the axis, we can approximate this shape as a Gaussian function
The equation above assumes uniform beam
A correction factor of 1/[2ln(2)] is needed for
Gaussian-shaped beams
Radar equation for a distributed target
(multiple birds) (multiple aircraft) (multiple raindrops)
Radar
Characteristics
Target
Characteristics
Constant
Radar Meteorology
M. D. Eastin

25. Distributed Weather Targets
Modifying our Radar Equation for Weather Targets:
Since meteorologists are interested in weather targets, we can develop special forms
of the distributed radar equations for typical collections of precipitation particles.
Three tasks must be completed:
Find the radar cross section of a single precipitation particle
Find the total radar cross section for the entire contributing region
Obtain the average radar reflectivity from all particles in that region
First Assumption: All particles are spheres!
Small Raindrops Spheres
Large raindrops Ellipsoids
Ice Crystals Variety of shapes
Gaupel / Hail Variety of shapes
Radar Meteorology
M. D. Eastin

26. Distributed Weather Targets
Second Assumption: All particles are sufficiently small compared to the wavelength
of the transmitted radar pulse such that the back scatter can
be described by Rayleigh Scattering Theory
Types of scattering:
Rayleigh
Mie
Optical
How small? Why Raleigh scattering?
Radius less than λ/20
Since the particle is much smaller than
the variability associated with the radar
pulse E-field (a sine wave), then we can
assume the E-field across the particle
will be uniform
Radar Meteorology
M. D. Eastin

27. Distributed Weather Targets
Impact of Radar Pulse on a Water Particle:
The radar pulse, and its associated E field, will induce an electric dipole within
any homogeneous dielectric sphere (i.e. a water drop or ice sphere)
The induced dipole vector → Points in the same direction as the pulse’s E field
→ Magnitude is the product of the incident field and
the polarization of the sphere:
where: ε0 = permittivity of free space
K = dielectric constant for water/ice
D = diameter of sphere
Einc = amplitude of incident E field
The sphere then scatters that portion of the E field equivalent to the dipole magnitude
The backscatter received at the radar is: OR
Radar Meteorology
M. D. Eastin

28. Distributed Weather Targets
Radar Cross Section of a Small Dielectric Sphere:
Recall the definition of radar backscatter:
The power flux densities and the E-field are related via:
Using the previous three equations, the radar cross section for a single sphere:
Proportional to the sixth power of the diameter
Proportional to the inverse fourth power of the radar wavelength
Radar Meteorology
M. D. Eastin

29. Distributed Weather Targets
What is the Dielectric Constant (K2)?
A measure of the scattering and absorption properties of a medium (water or ice)
where: Permittivity of medium
Permittivity of vacuum
Values of K2:
WATER (spheres)
ICE (spheres)
(snow flakes)
Radar Meteorology
M. D. Eastin

30. Distributed Weather Targets
Radar Cross Section of Multiple Dielectric Spheres:
Following the same methods as before:
For an array of particles, we compute the average radar cross section:
We then determine the average radar reflectivity:
Radar Meteorology
M. D. Eastin

31. Distributed Weather Targets
Radar Reflectivity Factor (Z) for Multiple Dielectric Spheres:
Most practitioners of radar use this quantity to characterize precipitation
Thus…
It is regularly expressed in logarithmic units
and displayed on radar screens…
Note the required units for Z
to have a unitless dBZ
Radar Meteorology
M. D. Eastin

32. Distributed Weather Targets
Radar Equation:
We can then insert our radar reflectivity into our radar equation for a distributed target:
to get our desired radar equation for weather targets:
Solving for Z:
Constant
Radar
Characteristics
Target
Characteristics
Radar Meteorology
M. D. Eastin

33. Weather Radar Equation
Review of Assumptions:
The precipitation particles are homogeneous dielectric spheres with diameters
small compared to the radar wavelength
Particles are spread through the contributing region. If not, then the equation gives an average radar reflectivity factor for the contributing region
The reflectivity factor (Z) is uniform throughout the contributing region and constant over the period of time required to obtain the average value of the received power
All of the particles have the same dielectric constant. We assume they are all either water or ice spheres
5) The main lobe of the radar pulse is adequately described by a Gaussian function
Microwave attenuation over the distance between the radar and the target is negligible.
Multiple scattering is negligible. Since attenuation and multiple scattering are related, if one is true, both are true.
8) The incident and backscattered waves are linearly polarized,
Radar Meteorology
M. D. Eastin

34. Weather Radar Equation
Validity of the Rayleigh Approximation:
Valid:
Invalid:
λ = 10 cm
Raindrops: 0.01 – 0.5 cm (all rain)
Ice crystals: 0.01– 3 cm (all snow)
Ice stones: 0.5 – 2.0 cm (small to moderate hail)
λ = 3 cm
Raindrops: 0.01 – 0.5 cm (all rain)
Ice crystals: 0.01– 0.5 cm (single crystals)
Ice stones: cm (graupel)
λ = 10 cm
Ice stones: > 2 cm (large hail)
λ = 3 cm
Ice crystals: > 0.5 cm (snowflakes)
Ice stones: > 0.5 cm (hail and large graupel)
Radar Meteorology
M. D. Eastin

35. Weather Radar Equation
Equivalent Radar Reflectivity Factor (Ze):
If one or more of the assumptions built into the radar equation are not satisfied, then
the reflectivity factor is re-named
In practice, one or more of the assumptions is almost always violated, and we
regularly use Z and Ze interchangeably.
Radar Meteorology
M. D. Eastin

36. Radar Equations Summary: Basic Approach to Radar Equation Development
Solitary Target
Power incident on target
Power scattered back toward radar
Amount of power collected by the antenna
Distributed (Multiple) Targets
Distributed (Multiple) Weather Targets
Radar Meteorology
M. D. Eastin