```Radar Equations
M. D. Eastin
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
M. D. Eastin
Basic Approach:
Radar Observation: Measurement of echo power received from a target provides useful
information about the target (# raindrops ~ reflectivity)
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
3. Determine the amount of back-scattered power
Distributed Targets:
Expand to allow for multiple targets with the volume
M. D. Eastin
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:
Siso 
Pt
4r 2
(1)
where S = power flux density (W/m2)
Pt = transmitted power (W/m2)
r = range from antenna
M. D. Eastin
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:
G( ,  ) 
Sinc ( ,  )
Siso
(2)
• Combining (2) with (1):
S inc 
GPt
4r 2
(3)
• In practice, it also incorporates absorptive losses by the antenna and waveguide
• The gain function is unique for each radar!
M. D. Eastin
Transmitted Power
What does the Gain Function look like?
Main Lobe
3D Depiction
of Gain (dB)
Side Lobes
M. D. Eastin
Transmitted Power
What does the Gain Function look like?
2D Depiction of Gain (dB)
• 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&deg;)
 For the same total transmitted power,
a large peak Gain will correspond to
a narrow beam width → desired
M. D. Eastin
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)
G
4Ae
2
10 cm
(4)
Where:
Ae = effective antenna area
λ = transmitting wavelength
Thus:
Large antennas → Large Gain
→ Small Beam Widths
→ Large Wavelengths
0.8 cm
Desired: Small beam widths
Small antennas
Large gain
M. D. Eastin
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…
M. D. Eastin
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…
M. D. Eastin
Transmitted Power
Method to Minimize Side Lobes:
Beam Geometry
• Use a parabolic antenna
Outgoing Power
Flux Density
• Parabolic antennas allow for “tapered illumination”
which minimizes the transmitted power flux density
along the edges
• Effects of Tapered Illumination:
1. Reduction of side lobe returns
2. Reduction of maximum Gain
3. 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
M. D. Eastin
Backscatter Power
• 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)
Target Backscatter
Power Flux Density
S (r )
  back
Sinc r 
(5)
PROBLEM: We don’t measure Sback at r,
we measure it at the radar
Power Flux Density
• For practical purposes, redefined as the
 Sr
 S inc
  4r 2 



(6)
M. D. Eastin
Backscatter Power
• In general, the radar cross section
of a target depends on:
1. Target’s shape
2. Target’s size relative to
(more on this later …)
3. Complex dielectric constant
and conductivity of the target
(more on this later…)
4. Viewing aspect from the radar
M. D. Eastin
Bringing it all together...
• Recall from before:
S inc 
GPt
4r 2
 Sr
 S inc
  4r 2 



(3)
Power flux density
incident on target
(6)
(7)
Power flux density
of target backscatter
at the antenna
• Substituting (3) into (6):
GPt
Sr 
16 2 r 4
• Accounting for the antenna area:
Pr  S r Ae 
AeGPt
16 2 r 4
(8)
Ae is the effective antenna area (m2)
Pr is the received power (W)
M. D. Eastin
Bringing it all together…
• Recall from before:
AeGPt
16 2 r 4
Pr  S r Ae 
G
4Ae

2
(8)
Power flux density
incident on target
(6)
Gain – Antenna size
relationship
• Substituting (6) into (8):
2G 2 Pt
Pr 
64 3 r 4
(9)
Radar equation for a single isolated target
(e.g. an airplane, ship, bird, one raindrop)
• Let’s rearrange and examine this equation in more detail…
M. D. Eastin
Radar Equation for a Solitary Target
1
Pr 
64 3
Constant
 
r4 
 
PG  
2
2
t
Characteristics
Target
Characteristics
• Written in terms of antenna effective area:
1
Pr 
4
Constant
 Pt Ae2 
 2 
  
Characteristics
 
r4 
 
Target
Characteristics
• What do these equations tell us about radar returns from a single target?
M. D. Eastin
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
• Spherical shell centered on the radar
• Radial extent determined by the pulse duration
• Angular extent determined by the antenna beam pattern
M. D. Eastin
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
the half power points of the beam
• Cross-sectional area of beam:
 r  r 


 2  2 

• Contributing volume length = half pulse length:
c h

2 2
• Volume of contributing region for a single pulse:
2
2
 h  r  r  hr  cr 
Vc     


2
2
2
8
8
  

(10)
M. D. Eastin
Distributed Targets
• Pulse duration → τ = 1.57 μs
• Angular circular beam width → 0.0162 radians
• Range from radar → r = 100 km
Vc 
cr 2
8




3.14 3.0 108 m s1 1.57106 s 105 m 0.0162

8
2
2
Vc  5.2 108 m3
• If the concentration of raindrops is the typical 1/m3, then the pulse volume contains:
520 million raindrops!!
M. D. Eastin
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
than those off axis
M. D. Eastin
Distributed Targets
1) 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)
2) 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)
3) 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 &gt; 3 cm)
4) Scattering or absorption of microwaves by objects between the radar and
contributing region do not modify the amplitude of Sinc appreciably
(good for wavelengths &gt; 3 cm)
M. D. Eastin
Distributed Targets
• 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)
   j
(11)
j
• Recall radar equation for a single target:
1
Pr 
64 3
PG  
2
2
t
 
r4 
 
• Radar equation for a distributed target:
Pr 
1
64 3
P G  
2
t
2
  j 
 j

 r4 


(12)
M. D. Eastin
Distributed Targets
• 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:
  j

  Vc  j
 Vc



  Vc avg


(13)
• Using (13) and the definition of the contributing volume (10), our radar equation
for a distributed target becomes:
c
Pr 
512 2
PG  
2
t
2
 avg 
 2 
 r 
(14)
M. D. Eastin
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
c
Pr 
512 2
 avg 
 2 
 r 
PG  
2
2
t
• The equation above assumes uniform beam
• A correction factor of 1/[2ln(2)] is needed for
Gaussian-shaped beams
c
Pr 
1024ln(2) 2
Constant
PG  
2
2
t
Characteristics
 avg 
 2 
 r 
Target
Characteristics
a distributed target
(multiple birds)
(multiple aircraft)
(multiple raindrops)
M. D. Eastin
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.
1) Find the radar cross section of a single precipitation particle
2) Find the total radar cross section for the entire contributing region
3) Obtain the average radar reflectivity from all particles in that region
First Assumption: All particles are spheres!
Small Raindrops
Large raindrops
Ice Crystals
Gaupel / Hail
Spheres
Ellipsoids
Variety of shapes
Variety of shapes
M. D. Eastin
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?
• 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
M. D. Eastin
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
p
 0 KD 3 Einc
→ 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:
2
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
p
Er  2
  0r
OR
 2 KD 3 Einc
Er 
22 r
M. D. Eastin
Distributed Weather Targets
Radar Cross Section of a Small Dielectric Sphere:
• Recall the definition of radar backscatter:
 Sr
 S inc
  4r 2 



• The power flux densities and the E-field are related via:
Sr
Er2
 2
Sinc Einc
• Using the previous three equations, the radar cross section for a single sphere:
 5 K 2 D6

4
Proportional to the sixth power of the diameter
Proportional to the inverse fourth power of the
M. D. Eastin
Distributed Weather Targets
What is the Dielectric Constant (K2)?
• A measure of the scattering and absorption properties of a medium (water or ice)
 r 1
K
r  2
where:
r 
1
0
Permittivity of medium
Permittivity of vacuum
Values of K2:
WATER
0.930 (spheres)
ICE
0.176 (spheres)
0.202 (snow flakes)
M. D. Eastin
Distributed Weather Targets
Radar Cross Section of Multiple Dielectric Spheres:
• Following the same methods as before:
 5 K 2 D6

4
• For an array of particles, we compute the average radar cross section:
 5K 2
   j  4  D6j

j
j
• We then determine the average radar reflectivity:
avg 
 j
j
Vc
6
D
 j
 5K 2 j
 4

Vc
M. D. Eastin
Distributed Weather Targets
Radar Reflectivity Factor (Z) for Multiple Dielectric Spheres:
• Most practitioners of radar use this quantity to characterize precipitation
Z
6
D
 j
j
Thus…
 avg
Vc
 5K 2Z

4
• It is regularly expressed in logarithmic units


Z

dBZ  10 log
6
3 
 1m m / m 
Note the required units for Z
to have a unitless dBZ
M. D. Eastin
Distributed Weather Targets
• We can then insert our radar reflectivity into our radar equation for a distributed target:
 avg
 5K 2Z

4
c
Pr 
1024ln(2) 2
PG  
2
2
t
 avg 
 2 
 r 
to get our desired radar equation for weather targets:
c 3  PtG 2  K 2 Z 

 2 
Pr 
2
1024ln(2) 

 r 
• Solving for Z:
 Pr r 2 
1024ln(2) 
2

 2 
Z
3
2
c
 PtG   K 
Constant
Characteristics
Target
Characteristics
M. D. Eastin
Review of Assumptions:
1) The precipitation particles are homogeneous dielectric spheres with diameters
small compared to the radar wavelength
2) Particles are spread through the contributing region. If not, then the equation gives
an average radar reflectivity factor for the contributing region
3) 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
4) 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
6) Microwave attenuation over the distance between the radar and the target is
negligible.
7) 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,
M. D. Eastin
Validity of the Rayleigh Approximation:
Valid:
λ = 10 cm
λ = 3 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)
Raindrops: 0.01 – 0.5 cm (all rain)
Ice crystals: 0.01– 0.5 cm (single crystals)
Ice stones: 0.1 - 0.5 cm (graupel)
Invalid:
Ice stones: &gt; 2 cm (large hail)
λ = 10 cm
λ = 3 cm
Ice crystals: &gt; 0.5 cm (snowflakes)
Ice stones: &gt; 0.5 cm (hail and large graupel)
M. D. Eastin
• If one or more of the assumptions built into the radar equation are not satisfied, then
the reflectivity factor is re-named
 Pr r 2 
1024ln(2) 
2

 2 
Ze 
3
2
c
 PtG   K 
• In practice, one or more of the assumptions is almost always violated, and we
regularly use Z and Ze interchangeably.
M. D. Eastin
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