Radar MeteorologyM. D. Eastin Radar Equations. Radar MeteorologyM. D. Eastin Radar Equations Outline...

Radar Meteorology M. D. Eastin

Radar Equations


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 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


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


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


Transmitted PowerWhat does the Gain Function look like?

Main Lobe

Side Lobes

3D Depictionof Gain (dB)


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)


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


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…


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…


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


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


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

(more on this later…)

4. Viewing aspect from the radar


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)


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


• 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


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


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


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


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 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 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 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


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


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


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



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


2

30 incEKD

p

r

pEr

02

r

EKDE incr 2

32

2


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:


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


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)


2

1

r

rK

0

1

r


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:


4

625

DK

j

jj

j DK 64

25

c

jj

c

jj

avg V

DK

V

6

4

25


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…


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 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:


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


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


Validity of the Rayleigh Approximation:

Valid:

Invalid:


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)


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.


2

2

2

2

3

)2ln(1024

K

rP

GPcZ r

te


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 MeteorologyM. D. Eastin Radar Equations. Radar MeteorologyM. D. Eastin Radar Equations Outline...

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