Radar Meteorology

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The Complex Dielectric FactorK2

• The refractive index (and thus the value of K2)
  for liquid water depends on wavelength and
  temperature.
• For 3-10 cm wavelength and temps 0-20 C,
  K2 .93?0.004
• For Ice, K2 depends on density
• Pure ice has a density of .92 g/cm3
• Snowflakes may have a density of .05 g/cm3
• Using a density of 1 g/cm3, K2.197 (based on
  empirical data)
• Determined that using a value of 1 for density,
  resulted in ? ? Dm6 where Dm6 is the
  diameter of a sphere of water from the melted ice

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Scattering

• In deriving the radar equation, we've assumed
  that the targets scattering radiation back to the
  radar are Rayleigh scatterers.
• A target is a Rayleigh scatterer if Dltltl where D
  is the diameter of the target.
• If D gt l or D lt l or D is approximately equal to
  l, then the target is a Mie scatterer.
• If Mie scattering is occurring, you will not
  obtain a correct radar reflectivity factor value
  from the radar equation.

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Scattering

• Water Particles
• Ice Particles
• If l 10 cm (88-D) then D 5 mm or less for
  Rayleigh scattering
• If l 3 cm (X-band) then D 1.5 mm or less for
  Rayleigh scattering
• So, X-band radars (3 cm) are more susceptible to
  Mie scattering, and therefore violating the radar
  equation than S-band radars (88-D).

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Assumptions Made in Deriving the Radar EQN

• No intervening attenuation
• Target completely fills contributing region
• Antenna beam is adequately described as Gaussian
  (Probert- Jones Correction)
• Hydrometeors are spherical
• Hydrometeors are small relative to wavelength
  (Rayleigh Scattering)
• No multiple scattering
• Contributing region contains all water or ice,
  but not both
• Hydrometeors are evenly distributed in the
  sampling volume

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The Radar Equation

• If the targets are all Rayleigh scatterers, the
  above radar equation will give you an accurate
  value of the radar reflectivity factor (Z). 
• However, you need to know if the targets are
  water, ice or otherwise to use the proper value
  of the complex dielectric factor
• It is often the case that you have either
  non-water targets or Mie scatterers present
  within the pulse volume
• To account for these types of targets, we define
  the "equivalent radar reflectivity factor", Ze
• Thus the radar equation becomes

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The Radar Equation

• The final form
• Where Rc is a pre-determined radar constant
• Ze is

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Equivalent Radar Reflectivity

• Radar reflectivity factor of the hypothetical
  target that would produce the same backscatter as
  the target actually observed
• If the actual target meets Rayleigh and all other
  assumptions, ZZe (70-80 of real cases)
• Ultimately, we use Ze when we cannot assume
  Rayleigh scattering
• SLS

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Terms

• Average Power Returned (Pr)
• Computed over multiple pulses
• Necessary since Pr varies greatly from pulse to
  pulse
• 25 pulses are used by the 88D
• It is then averaged again in .13 nm resolution
  gates to produce the .54 nm resolution data
  displayed
• Maximum transmitted power (Pt)
• 750,000 watts for 88D
• Pr is DIRECTLY related to Pt
• Antenna Gain (G)
• Measure of the antennas ability to focus
  radiation
• 88D has an antenna gain of 35,481 (45 dB)
• Target is hit with 35,481 times more energy than
  if the antenna is isotropic
• Pr is directly related to the square of G

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Terms

• Angular beamwidth (?)
• Pr is directly related to the square of angular
  beamwidth
• The assumption that the radar beam is completely
  filled fails when ? gt 2 degrees and when the
  range gt 125 nm
• ? for 88D is .95 degrees
• Pulse length (h)
• Pr is directly related to pulse length
• Reflectivity factor (Z)
• Efficiency of returned power is related to (1)
  the number of drops and (2) the size of the drops
• Determined by the sum to the sixth power of all
  the drops in the sampled volume

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Effects of Size

• Pr is highly dependent on particle size
• A drop 3mm in diameter would return 729 times as
  much power as a drop 1 mm in diameter even though
  it contains only 27 times as much water
• Target reflectivity increases rapidly as drop
  size grows

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Terms

• Wavelength
• Pr is inversely proportional to the square of the
  wavelength
• Ratio of particle size to wavelength determines
  the amount of radiation attenuated
• Target Range (r)
• Pr is inversely proportional to the square of the
  range
• Same target returns 25 of the power at 100 km
  that it would at 50 km
• Physical Constant (K)
• Describes the ability of the target to transmit
  electrical currents (electrical conductivity)
• 88Ds set K squared to that of water
• Pr from water particles is 5 times greater than
  for ice
• Greatly underestimates water content of snow

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More on Radar Resolutions

• Range resolution
• Ability to differentiate between objects along a
  radial
• A point target becomes a bar with length h/2 on a
  radar scope
• The ½ factor occurs because the range is ½ the
  transmit distance traveled by the pulse
• Range resolutionh/2 tauc/2

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Range Resolution
Objects separated by more than h/2 appear as
separate objects
Objects separated by less than h/2 appear as a
single object
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Azimuth Elevation Resolution

• Two targets separated by less than a beamwidth
  will not be separately resolved
• See Figure in Handouts (Next slide)
• The size of the reflector is not the only factor
  in determining beamwidth
• Shorter waves are more easily focused
• For a circular reflector

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Meteorological Targets
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Minimum and Maximum Range

• Radar cant receive while it is transmitting
• Theoretical range limit is
• No first trip echo can be received after a second
  pulse has been transmitted

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Meteorological Targets

• The Big Unknown for a radar operator concerns
  the precise characteristics of the precipitation
  target.
• We discussed the many terms involved in the
  equation for the returned power in order to
  estimate the equivalent reflectivity factor.
• Recall the significance of the ?K2? term
  regarding ice and water
• The Diameter of drops in a sample volume in
  determining the true reflectivity factor Z was
  obtained using the expression Z S Ni Di6.
• But in the Probert Jones Radar equation we
  estimated this Z term calling it Ze (assuming
  Rayleigh scattering, liquid state and a
  completely filled beam etc.)
• The radar reflectivity factor (Z) is the measure
  in (mm6/m3) of the efficiency of a target in
  intercepting and backscattering radio wave energy
  of the C and S bands for precipitation sized
  particles.
  
  
• The efficiency of the target in backscattering
  depends on
  
• Size (diameter) State (frozen, liquid, mix)
  primarily...but also Concentration (number
  drops/m3) and Shape.

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Meteorological Targets

• Now we are interested in estimating precipitation
• Very valuable short-term (and climatological
  tool).
• Especially important in timely warnings of flash
  flooding.
• Each day the Missouri Basin River Forecast Center
  (MBRFC) sends out 1, and 3 hourly flash flood
  indices...that if equaled or exceeded (averaged
  over a basin) will result in flash flooding.
• Forecasts can be made from known basin averages
  and these indices for given streams or just
  areas.

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Meteorological Targets

• In the old days, some precipitation estimates
  could be made from DVIP values that were tracked
  and summed hourly by early computer programs.
• But the heart of the flash flood forecasting was
  the cooperative observer and his/her rain gage.
• The network needed to be as dense as possible,
  but could never be dense enough.
• Heavy amounts could be missed in convective
  rains, while the less threatening more stable
  stratiform rains were represented well by the
  gage network.

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Meteorological Targets

• The WSR-88D has filled a void in timely and
  fairly accurate rainfall estimations, and is best
  used in conjunction with the rain gage network as
  ground proof or adjustments.
• But the rainfall rate R (inches/hr) like Z is
  dependent on the drop diameters and distribution
  in the form of the equation R (p/6)S Ni Di3
  wt p/6 .5236
  
• wt is the terminal velocity of the hydrometeor in
  m/s,
• D - diameter in mm,
• N - number of hydrometeors of this diameter per
  cubic meter

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Meteorological Targets

• Snow falls around 0.2 to 0.5 m/s, while raindrops
  1-10 m/s
• Well examine some research by Marshall and
  Palmer that help us relate size distributions to
  reflectivity and then rainfall rates.
• And while there is a relationship between radar
  reflectivity and rainfall rates, there is no
  unique Z-R relationship that can be used in all
  cases because we just dont know enough about the
  drop size distribution and environment
• Thus various empirical Z-R relationships have
  been developed and tested and in the form the
  text shows (p142) as Z ? ARb with Z in mm6/m3,
  R in mm/hr with A and B as empirical constants
• The most commonly used Z-R equation for
  stratiform rain situations is used in the text to
  create the table 8.1 on page 145.
• This is z 200 R1.6 Lets try it!

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Reflectivity and Rainfall Measurements

• Relation of Z to other met quantities
• Wx radars measure Z or Ze (indirectly)
• They depend on number, shape, size and
  composition of the precip particlesDrop size
  distribution
• It is useful to relate Z to other met quantities
  dependant upon the same characteristics
• Rainfall (Precipitation) Rate (R)
• Precipitation Content (W)

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Precipitation Content (W)

• Mass of a substance (water or ice) present in the
  form of precipitation-size particles per unit
  volume of space
• Different from the liquid water content (LWC), M,
  which includes cloud water (M gt W)
• Smallest particles considered .2 mm
• Units for W are kg/m3 (or g/m3)
• A distribution of 1000 1-mm raindrops in a cubic
  meter (1 drop per liter) would have a
  precipitation content (W) of .52 g/m3
• Values range from lt .1 g/m3 in drizzle to gt 10
  g/m3 in heavy rain
• Average is 2-5 g/m3

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Precipitation Content (W)

• The contribution of any one particle to the total
  precipitation content is equal to its mass (?mj),
  which is proportional to its volume
• A spherical drop of diameter Dj has a
  volume?Dj3/6, while its contribution (?Zj) to
  the radar reflectivity factor is Dj6 (assuming
  R-scatt)
• Therefore, ?Zj is proportional to the square of
  the volume of the dropand ultimately to the
  square of its contribution to W
• W of a volume containing spherical particles is
  proportional to the sum of the cube of the
  particle diameter (Dj)

Where Vs is the sampling/pulse volume over which
the summation is performed
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Precipitation Content (W)

• Z on the other hand, is proportional to the sum
  of the sixth power of the diameters
• For a single particle
• Does that work for the summations?
• So, while the contribution ?Zj of a single drop
  is proportional to the square of its mass, this
  cannot be extrapolated to the case of many drops
  (I.e. z proportional to w squared)
• Empirically, it has been found that Z and W can
  be expressed in the form of a power law (Marshall
  and Palmer 1948), such that

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Precipitation Content (W)

• The exponent is usually not far from 2
• The exact relationship must be established
  empirically since W cannot be measured in the
  atmosphere
• For rain, Douglas (1964) found A2.4 x 104 and
  B1.82
• We can rewrite the above as

Where w is on g/m3 and Z is in mm6/m3
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Precipitation Rate (Rj)

• Volume of precipitation passing downward through
  a horizontal surface area, per unit area and per
  unit time
• m3m-2s-1 (or ms-1)
• The rate is usually given in mm/h
• Values range from lt .01 for drizzle to gt1000 mm/h
  for heavy rain
• Values gt 10 mm/hr (equivalent water content) are
  uncommon for snow
• Usually measured near the ground

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Precipitation Rate (R)

• The contribution, ?rj, of any one particle to the
  precipitation rate is proportional to its volume
  and fall velocity
• For a spherical drop
• The total R for an array of spherical raindrops
  is the sum of the individual contributions
  divided by the sampling volume considered

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Precipitation Rate (R)

• A distribution of 1000 1-mm raindrops /m3 falling
  at their terminal velocity of 4.03 m/s in the
  absence of vertical air motion would give a
  precipitation rate of 2.1 x 10-6 m/s or 7.5 mm/h
  using the previous equation
• A particles fall speed depends mainly on size
• Near the ground, particles fall at their terminal
  fall speeds since there is little air motion
• Above the ground, vertical air motion can alter
  the drop motion substantially

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Precipitation Rate (R)

• The relationship of Z to R is also in the form of
  a power law
• a and b are empirically determined, or estimated
• The most common solution (Marshall-Palmer 1948)
• For a stratiform region
• For a convective region

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Radar Estimated Precipitation

• The variability of the coefficients arises due to
  numerous factors which can be tentatively placed
  into 2 groups
• Place factors geographic and climatic local
  peculiarities of the atmosphere (depth of the
  trop, orographic effects, etc)
• Depending on the place and season, the dynamical,
  thermodynamical, and microphysical processes that
  are responsible for precip development change
• N(D) (Drop size distribution) changes
• Factors linked to cloud structure N(D) varies
  from one type of cloud to another, and for the
  same type, with the evolution of some processes
• Example coefficient a increases while b
  decreases when convective intensity increases

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Other Z-R Relationships

• Marshall and Palmer (1946) Stratiform
• Marshall and Palmer (1948) Convective
• Sekhon and Srivastava (1971) Convective
• Sekhon and Srivastava (1971) Snow

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Z-R Relationships

• The WSR-88D uses z 300 R1.4
• Works well for non-tropical convection if we
  remember the sources of Z error such as AP
  (which could cause overestimates), partial beam
  filling - especially at longer ranges and results
  in underestimating, attenuation and incorrect
  hardware calibration.

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Z-R Relationships

• Other sources of error in the Z-R relationship
  include
• Variations in the drop size distribution
• Mixed precipitation types of frozen, liquid, or
  both.
• Bright Banding due to precipitation near and
  below the melting level, and
• Hail (especially wet hailstones).
• Sleet, wet snow and the bright band can cause
  overestimations of precipitation amounts

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Z-R Relationships

• Below beam effects - Strong winds - for radar
  overestimate or underestimate depending on
  location.
• Evaporation is especially a problem as the range
  increases
• At farther ranges the radar observer is looking
  higher up and seeing the precipitation which will
  actually not all make it to the ground.
• This can be seen as a donut around the RDA, and
  the edge of the donut is the level of near total
  evaporation.
• This can be a tool in forecasting too - if the
  donut is getting smaller, the atmosphere is
  moistening and rain or snow will soon be falling.
  If our donut is getting larger, the dry air is
  deepening and no precipitation will fall.
• Coalescence - often because of overshooting storm
  core, or warm tropical rain pattern below beam

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Z-R Relationships

• Snow - No good Z-S relationship.
• All that can be done is relate reflectivity
  observed to reliable snow measurements from
  ground observers.
• A local study at Omaha suggests that in a near
  saturated environment below the beam accompanied
  by little surface fog, Z can be related to
  visibilities
• 20dBZ - visibilities down to at least a mile 24
  ? 1/2 mile, and ? 28 dBZ VIS ? 1/4 mile. Use
  this technique with VWP so that you can advect
  heaviest snow downstream using a fall rate of
  around 0.5 m sec. Fall time from radar level can
  then be computed. Then advect using the VWP
  velocity for the computed time. You will find
  the snow can fall up to 20 miles downstream -
  depending on the level of observation
• Note Snow crystal growth by deposition is
  greatest in the -12 to -17?C dendritic layer,
  where the difference between the SVP and
  supercooled water is at a maximum. Cloud physics
  is very important as SN falling below the beam
  (in a layer of supercooled cloud droplets) can
  grow and increase snowfall rates

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Z-R Relationships

• Precipitation Processing Algorithms or the
  precipitation processing subsystem (PPS) which
  provides estimates out to 124 nm.
• Preprocessing algorithm - corrects problems due
  to beam blockage (as in mountainous areas),
  spurious noise, Z outliers (like real high values
  of ?65 dBZ), ground returns (as in AP) - where
  the algorithm looks up from .5 to 1.5 degrees and
  if 75 or more of the echoes disappear - they are
  not processed. And Finally changes in beam
  height with range - chooses maximum Z if not
  eliminated as AP or clutter.
  
• Precipitation Rate Algorithm, where rainfall rate
  is capped at 4.09 in/hr, which corresponds to 53
  dBZ and uses the WSR-88D Z-R equation z 300
  R1.4, and is converted from a .54 nm X 1 degree
  product to 1.1 nm X 1 degree by averaging.

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Z-R Relationships

• Precipitation Accumulation Algorithm - makes up
  for data missing by 30 minutes or less by
  averaging or extrapolation.
• The products generated are One hour Product
  (OHP), updated each volume scan, the Three hour
  product (THP) updated at the top of each hour and
  useful as it corresponds to commonly used flash
  flood guidance, and the Storm Total Product
  (STP), which is updated every volume scan.
• This product will clear itself after 1200Z if no
  significant precipitation is occurring.
• It is useful to time lapse the OHP and STP in
  tracking storm movement, precipitation
  accumulation, and short term forecasting.
• There is also the operator generated User
  selectable product (USP) updated at the top of
  each hour.
• All products have a resolution of 1.1 nm X 1
  degree.

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OHP
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THP
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STP
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Radar Reflectivity Loop (Frances)
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Z-R Relationships (Bright Banding)

• The bright band is a region of relatively high
  equivalent reflectivity that usually appears as
  an elevated layer at the height where falling ice
  particles begin to melt and thus become water
  coated. This bright band region depicts the
  melting layer and is often about 3000 feet in
  depth (Green 1993).
• As frozen precipitate fall into an area with
  temperatures in the -5C to 0C range there is a
  rapid increase in the coalescence of individual
  snow crystals.

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Bright Banding

• Then, as snowflakes exit sub-freezing they
  gradually melt to raindrops and increase their
  fall speed. Ice particles exhibit about one-fifth
  of the reflectivity as the equivalent amount of
  liquid water (Green 1993).
• So, when the snowflakes fall into near-freezing
  air, the bigger flakes begin to exhibit greater
  radar reflectivity.
• As the particles fall below the melting level
  they become coated with liquid water.
• This causes a fivefold increase in the efficiency
  of the particles to return energy to the radar
  (Green 1993). This is the primary cause of the
  bright band. Change in k
• After the particles completely melt into rain,
  their fall speed increases rapidly resulting in a
  decrease of precipitation particle concentration.
  Radar reflectivity again decreases (Martner, et
  al. 1993).

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Bright Banding

• Bright Banding - is a function primarily of
  differential fall velocity, the coalescence
  factor, and the changed reflectivity from snow to
  wet snow to rain (the most significant factor).
• The Diagram was developed theoretically based on
  work by Austin and Bemis (1950).
• In looking at the diagram...notice the increased
  reflectivity as 0?C is approached due to greater
  aggregation at warmer temperatures, the maximum
  is 15 to 30 times as great as the snow above it
  (5 to 15 db according to Rinehart), and 4 to 9
  times as great as the rain below it (5 to 10 dB).

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Bright Banding

• Is the bright band always this theoretical depth?
  No!
• Can range from around 500 feet to gt3000 feet, or
  may not even show up around zero degrees C is
  there is a lot of supercooled water as in
  convective clouds, though Rinehart mentions the
  bright band showing up in dissipating convective
  systems.
• Process actually takes energy out of air below
  (latent heat associated with the change of state)
  and creates a downward directed vertical pressure
  gradient.

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Bright Banding
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Bright Banding

• RHI
• PPI