Radar Equations

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Radar Equations
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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

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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
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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

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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 ?, and elevation f for a
• directional antenna to the power flux density
  for an isotropic antenna transmitting the
• same total power
• (2)
• Combining (2) with (1)
• (3)

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Transmitted Power
What does the Gain Function look like?
3D Depiction of Gain (dB)
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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 (T) defined at
• the location equivalent to 3 dB less
• than the peak gain on main lobe
• (in this case at 27 dB ? T 6)
• For the same total transmitted power,
• a large peak Gain will correspond to
• a narrow beam width ? desired

2D Depiction of Gain (dB)
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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
• ? transmitting wavelength
• Thus Large antennas ? Large Gain
• ? Small Beam Widths
• ? Large Wavelengths
• Desired Small beam widths
• Small antennas

10 cm
0.8 cm
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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

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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

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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

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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 dont measure Sback at r,
• we measure it at the radar
• For practical purposes, redefined as the
• power flux density received at the radar
• (6)

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Backscatter Power

• Radar Cross Section
• In general, the radar cross section
• of a target depends on
• Targets shape
• 2. Targets size relative to
• the radars wavelength
• (more on this later )
• Complex dielectric constant
• and conductivity of the target
• (more on this later)
• Viewing aspect from the radar

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Power Received at Antenna

• Bringing it all together...
• Recall from before
• (3)
• (6)
• Substituting (3) into (6)
• (7)
• Accounting for the antenna area

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)
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Power Received at Antenna

• Bringing it all together
• Recall from before
• (8)
• (6)
• Substituting (6) into (8)
• (9)

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)
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Radar Equation for a Solitary Target

• Written in terms of antenna effective area

Constant
Radar Characteristics
Target Characteristics
Constant
Radar Characteristics
Target Characteristics
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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

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Distributed Targets

• Single Pulse Volume
• Azimuthal coordinate (T)
• Beam width in the azimuthal direction
• is rT, where T is the arc length between
• the half power points of the beam
• Elevation coordinate (F)
• Beam width in the elevation direction
• is rF, where F is the arc length between
• the half power points of the beam
• Cross-sectional area of beam
• Contributing volume length half pulse length

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Distributed Targets

• Single Pulse Volume NEXRAD Radar
• Pulse duration ? t 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!!

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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

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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 gt 3 cm)
• 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)

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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)

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Distributed Targets

• Radar Reflectivity (?avg)
• 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)

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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
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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

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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 ?/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

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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 pulses E field
• ? Magnitude is the product of the incident
  field and
• the polarization of the sphere
• where e0 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

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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
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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 0.930 (spheres)
• ICE 0.176 (spheres)
• 0.202 (snow flakes)

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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

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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

Note the required units for Z to have a unitless
dBZ
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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
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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,

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Weather Radar Equation
Validity of the Rayleigh Approximation Valid
Invalid
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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.

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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