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Radar Meteorology M. D. Eastin
Radar Equations
Radar Meteorology M. D. Eastin
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
Radar Equation DevelopmentBasic 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
Transmitted PowerIsotropic 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
24 r
PS tiso
Radar Meteorology M. D. Eastin
Transmitted PowerDirectional 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!
iso
inc
S
SG
),(),(
24 r
GPS tinc
Radar Meteorology M. D. Eastin
Transmitted PowerWhat does the Gain Function look like?
Main Lobe
Side Lobes
3D Depictionof Gain (dB)
Radar Meteorology M. D. Eastin
Transmitted PowerWhat 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
Transmitted PowerRelationship 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 areaλ = transmitting wavelength
Thus: Large antennas → Large Gain → Small Beam Widths → Large Wavelengths
Desired: Small beam widthsSmall antennasLarge gain
2
4
eAG
10 cm
0.8 cm
Radar Meteorology M. D. Eastin
Transmitted PowerProblems 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
Transmitted PowerProblems 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
Transmitted PowerMethod 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:
1. Reduction of side lobe returns2. Reduction of maximum Gain3. 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 GeometryOutgoing Power
Flux Density
Radar Meteorology M. D. Eastin
Backscatter PowerRadar 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)
rS
rS
inc
back )(
inc
r
S
Sr 24
Radar TransmittedPower Flux Density
Target BackscatterPower Flux Density
Radar Meteorology M. D. Eastin
Backscatter PowerRadar Cross Section:
• In general, the radar cross section of a target depends on:
1. Target’s shape
2. Target’s size relative to the radar’s wavelength (more on this later …)
3. Complex dielectric constant and conductivity of the target
4. Viewing aspect from the radar
Radar Meteorology M. D. Eastin
Power Received at Antenna
inc
r
S
Sr 24
24 r
GPS tinc
Power flux density incident on target
Power flux densityof target backscatter
at the antenna4216 r
GPS tr
Radar cross section
4216 r
GPAASP teerr
Ae is the effective antenna area (m2)
Pr is the received power (W)
Radar Meteorology M. D. Eastin
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 sizerelationship
4216 r
GPAASP teerr
Radar equation for a single isolated target (e.g. an airplane, ship, bird, one raindrop)
2
4
eAG
43
22
64 r
PGP tr
Power Received at Antenna
Radar Meteorology M. D. Eastin
• Written in terms of antenna effective area:
• What do these equations tell us about radar returns from a single target?
422
364
1
rGPP tr
Radar Equation for a Solitary Target
Constant RadarCharacteristics
TargetCharacteristics
42
2
4
1
r
APP etr
Constant RadarCharacteristics
TargetCharacteristics
Radar Meteorology M. D. Eastin
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
Distributed Targets
Radar Meteorology M. D. Eastin
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:
• Contributing volume length = half pulse length:
• Volume of contributing region for a single pulse:
(10)
Distributed Targets
22
rr
22
hc
88222
22
rchrrrhVc
Radar Meteorology M. D. Eastin
Single Pulse Volume: NEXRAD Radar
• Pulse duration → τ = 1.57 μs• Angular circular beam width → 0.0162 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!!
Distributed Targets
8
0162.0101057.1100.314.3
8
2256182 msmsrcVc
38102.5 mVc
Radar Meteorology M. D. Eastin
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
Distributed Targets
Radar Meteorology M. D. Eastin
Radar Cross-Section: Assumptions
1)The radial extent (h/2) of the contributing region is small compared to therange (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 > 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 > 3 cm)
Distributed Targets
Radar Meteorology M. D. Eastin
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)
Distributed Targets
j
j
422
364
1
rGPP tr
4
22364
1
rGPP j
j
tr
Radar Meteorology M. D. Eastin
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)
Distributed Targets
avgcc
jj
c VV
V
222
2512 rGP
cP avg
tr
Radar Meteorology 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
• The equation above assumes uniform beam• A correction factor of 1/[2ln(2)] is needed for Gaussian-shaped beams
Radar Equation for Distributed Targets
222
2512 rGP
cP avg
tr
222
2)2ln(1024 rGP
cP avg
tr
Radar equation for a distributed target
(multiple birds) (multiple aircraft)
(multiple raindrops)Constant Radar
CharacteristicsTarget
Characteristics
Radar Meteorology M. D. Eastin
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:
1) Find the radar cross section of a single precipitation particle2) Find the total radar cross section for the entire contributing region3) Obtain the average radar reflectivity from all particles in that region
First Assumption: All particles are spheres!
Small Raindrops SpheresLarge raindrops EllipsoidsIce Crystals Variety of shapesGaupel / Hail Variety of shapes
Distributed Weather Targets
Radar Meteorology M. D. Eastin
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:
RayleighMieOptical
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
Distributed Weather Targets
Radar Meteorology M. D. Eastin
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
Distributed Weather Targets
2
30 incEKD
p
r
pEr
02
r
EKDE incr 2
32
2
Radar Meteorology M. D. Eastin
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:
Distributed Weather Targets
2
2
inc
r
inc
r
E
E
S
S
4
625
DK
inc
r
S
Sr 24
Proportional to the sixth power of the diameter
Proportional to the inverse fourth power of the radar wavelength
Radar Meteorology M. D. Eastin
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 0.930 (spheres)
ICE 0.176 (spheres) 0.202 (snow flakes)
Distributed Weather Targets
2
1
r
rK
0
1
r
Radar Meteorology M. D. Eastin
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:
Distributed Weather Targets
4
625
DK
j
jj
j DK 64
25
c
jj
c
jj
avg V
DK
V
6
4
25
Radar Meteorology M. D. Eastin
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…
Distributed Weather Targets
4
25
ZK
avg c
jj
V
D
Z
6
36 /1log10
mmm
ZdBZ Note the required units for Z
to have a unitless dBZ
Radar Meteorology M. D. Eastin
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:
Distributed Weather Targets
4
25
ZK
avg
222
2)2ln(1024 rGP
cP avg
tr
2
2
2
23
)2ln(1024 r
ZKGPcP tr
2
2
2
2
3
)2ln(1024
K
rP
GPcZ r
t
Constant Radar
CharacteristicsTarget
Characteristics
Radar Meteorology M. D. Eastin
Review of Assumptions:
1)The precipitation particles are homogeneous dielectric spheres with diameterssmall 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,
Weather Radar Equation
Radar Meteorology M. D. Eastin
Validity of the Rayleigh Approximation:
Valid:
Invalid:
Weather Radar Equation
λ = 10 cmRaindrops: 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 cmRaindrops: 0.01 – 0.5 cm (all rain)Ice crystals: 0.01– 0.5 cm (single crystals)Ice stones: 0.1 - 0.5 cm (graupel)
λ = 10 cmIce stones: > 2 cm (large hail)
λ = 3 cmIce crystals: > 0.5 cm (snowflakes)Ice stones: > 0.5 cm (hail and large graupel)
Radar Meteorology M. D. Eastin
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.
Weather Radar Equation
2
2
2
2
3
)2ln(1024
K
rP
GPcZ r
te
Radar Meteorology M. D. Eastin
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