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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: (a)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 avg

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First Assumption: Particles are all spheres Small raindrops and cloud droplets:Spherical Large raindrops:Ellipsoids Ice crystalsVaried shapes Graupel and rimed particlesCan be spherical HailMay 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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E inc incident plane wave Dielectric Sphere (water drop) A plane wave with electric field E inc induces an electric dipole p in a small sphere. The induced dipole is parallel to the direction of E inc which is also the direction of polarization of the incident wave. p

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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 (D 3 ), the material the particle is made of (K: ice or water) and the magnitude of the incident electric field (E inc ). (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: To determine the radar cross section we (a) divide (3) by E inc (b) Square both sides of the resulting equation (c) Multiply by 4 r 2 (3) (4)

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What is K? K is a complex number representing the scattering (real part) and absorption characteristics of the medium where Permittivity of medium Permittivity of vacuum Values of Temperature = 10 cm = 3.21 cm = 1.24 cm = 0.62 cm 20C0.92800.92750.91930.8926 10C0.93400.92820.91520.8726 0C0.93400.93000.90550.8312 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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Recall the radar equation for a distributed target: Relationship between the radar reflectivity and the radar reflectivity factor: (7) Combining:

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THE RADAR EQUATION FOR WEATHER TARGETS constants Radar characteristics Target characteristics where Z in normally expressed in logarithmic units

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The Weather radar equation: review of the assumptions 1.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 = 10 cm = 5 cm = 3 cm = 0.8 cm 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) Raindrops: 0.01 – 0.5 cm (all rain) Snowflakes: 0.01– 1 cm (small snowflakes) Hailstones: 0.5 – 0.75 cm (small hail) Raindrops: 0.01 – 0.5 cm (all rain) Ice crystals: 0.01– 0.5 cm (single crystals) Graupel: 0.1 -- 0.5 cm (graupel) Raindrops: 0.01 – 0.15 cm (cloud and drizzle drops) Ice crystals: 0.01– 0.15 cm (single crystals)

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Validity of the Rayleigh Approximation for weather targets Invalid = 10 cm = 5 cm = 3 cm = 0.8 cm Hailstones: > 2.0 cm (large hail) Snowflakes > 1 cm (large snowflakes) Hailstones: > 0.75 cm (moderate to large hail) Raindrops: 0.01 – 0.5 cm (all rain) Snowflakes > 0.5 cm Hail and large graupel Drops > 100 microns All ice particles except small crystals

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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, Z e

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Units of Z One would think the standard units of Z would be m 6 /m 3 = m 3 But no… The standard units for Z are mm 6 /m 3 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 -28 dbZ = haze droplets 45-50 dbZ = heavy rain 25 dbZ = snow

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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

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