Weather radar equations ... The Weather radar equatio

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Weather radar equations

• To convert equations for distributed targets into
  weather radar equations, we must determine the
  radar reflectivity of arrays of precipitation
  particles.
• This problem can be divided into three parts
• Finding the radar cross of a single particle
• (b) Finding the total radar cross section for the
  entire contributing region
• (c) Dividing the total cross section by the
  effective volume of the contributing region to
  obtain the average radar reflectivity havg

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First Assumption Particles are all spheres
Small raindrops and cloud droplets Spherical Lar
ge raindrops Ellipsoids Ice
crystals Varied shapes Graupel and rimed
particles Can be spherical Hail May or may
not be spheres
The scattering properties and radar cross
sections of spherical particles have been
calculated and are well understood.
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Second assumption The particles are
sufficiently small compared to the wavelength of
the impinging microwaves that the scattering can
be described by Raleigh Scattering Theory
How small is small? From the figure above, the
radius of the particle, a, must be
( 1/6 of the wavelength)
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What is the fundamental difference between the
Rayleigh, Mie, and Optical regimes?
With Rayleigh scattering, the electric field is
assumed to be invariant in the vicinity of the
particle
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p
Einc
Dielectric Sphere (water drop)
incident plane wave
A plane wave with electric field Einc induces an
electric dipole p in a small sphere. The induced
dipole is parallel to the direction of Einc which
is also the direction of polarization of the
incident wave.
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The angular patterns of the scattered intensity
from particles of three sizes (a) small
particles, (b) large particles, and (c) larger
particles
Rayleigh scattering pattern
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From Rayleigh scattering theory, the dipole
moment p induced in a spherical particle is
proportional to the particles volume (D3), the
material the particle is made of (K ice or
water) and the magnitude of the incident electric
field (Einc).
(1)
And the intensity of the scattered electric field
at the location of the particle is
(2)
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Combining (1) and (2) we get
(3)
To determine the radar cross section
we

• divide (3) by Einc
• (b) Square both sides of the resulting equation
• (c) Multiply by 4pr2

(4)
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What is K? K is a complex number representing
the scattering (real part) and absorption
characteristics of the medium
Permittivity of medium
where
Permittivity of vacuum
Values of
Water
Ice
0.176 for ice particles (0.208 is used when
snowflake sizes are expressed as the diameters of
water drops obtained by melting the ice).
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(4)
The radar cross section
For an array of particles, we determine the
average radar cross section
(5)
Now we determine the radar reflectivity
(6)
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The quantity is of utmost
importance in radar meteorology
It is designated with the symbol Z, and is called
the radar reflectivity factor
In logarithmic units
It is the quantity that is displayed on a radar
screen.
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Relationship between the radar reflectivity and
the radar reflectivity factor
(7)
Recall the radar equation for a distributed
target
Combining
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THE RADAR EQUATION FOR WEATHER TARGETS
Radar characteristics
Target characteristics
constants
where Z in normally expressed in logarithmic units
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The Weather radar equation review of the
assumptions

• The precipitation particles are homogeneous
  dielectric spheres with diameters small compared
  to the radar wavelength
• 2. The particles are spread throughout the
  contributing region. If not then the equation
  gives an average reflectivity factor for the
  contributing region.
• 3. The reflectivity factor Z is uniform
  throughout the contributing region and constant
  over the period of time needed to obtain the
  average value of the received power.

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The Weather radar equation review of the
assumptions
4. All of the particles have the same dielectric
factor that is, they are all either water
droplets or ice particles. 5. The main lobe of
the antenna 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. Multiple scattering and attenuation
are related so if one is true the other is
too. 8. The incident and back-scattered waves
are linearly polarized.
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Validity of the Rayleigh Approximation for
weather targets
Valid
Raindrops 0.01 0.5 cm (all rain) Snowflakes
0.01 3 cm (most snowflakes) Hailstones 0.5
2.0 cm (small to moderate hail)
l 10 cm
Raindrops 0.01 0.5 cm (all rain) Snowflakes
0.01 1 cm (small snowflakes) Hailstones 0.5
0.75 cm (small hail)
l 5 cm
Raindrops 0.01 0.5 cm (all rain) Ice crystals
0.01 0.5 cm (single crystals) Graupel 0.1 --
0.5 cm (graupel)
l 3 cm
Raindrops 0.01 0.15 cm (cloud and drizzle
drops) Ice crystals 0.01 0.15 cm (single
crystals)
l 0.8 cm
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Validity of the Rayleigh Approximation for
weather targets
Invalid
l 10 cm
Hailstones gt 2.0 cm (large hail)
Snowflakes gt 1 cm (large snowflakes) Hailstones
gt 0.75 cm (moderate to large hail)
l 5 cm
Raindrops 0.01 0.5 cm (all rain) Snowflakes gt
0.5 cm Hail and large graupel
l 3 cm
Drops gt 100 microns All ice particles except
small crystals
l 0.8 cm
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When the assumptions built into the radar
equation are not satisfied, the reflectivity
factor is referred to as The Equivalent Radar
Reflectivity Factor, Ze
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Units of Z
One would think the standard units of Z would be
m6/m3 m3
But no
The standard units for Z are mm6/m3
If these units are not used, you will be off by
10-18
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Range of radar reflectivity factor in weather
echoes
WSR-88D Precipitation Mode
WSR-88D Clear Air Mode
75 dbZ giant hail
45-50 dbZ heavy rain
25 dbZ snow
-28 dbZ haze droplets
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Nebraska record hailstorm 2003 75 dBZ
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Heavy rain in Hurricane Andrews Eyewall 45 dBZ
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Snowstorm over Great Lakes 25-30 dBZ
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Clear air echoes (few small insects) -12 dBZ