A Simple Model of the Mmwave Scattering Parameters of Randomly Oriented Aggregates of Finite Cylindr

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A Simple Model of the Mm-wave Scattering
Parameters of Randomly Oriented Aggregates of
Finite Cylindrical Ice Hydrometeors An
End-Run Around the Snow Problem?

• Jim Weinman
• University of Washington
• Min-Jeong Kim
• NASA GSFC/UMBC GEST

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• Terrestrial snowfall may be dry or wet.
• The dielectric constant will be affected.
• Land surface and snow accumulation
• affect emission from the surface.
  

Solution Utilize absorption by water vapor in
the boundary layer to screen mm-wave emission
from snow-covered surfaces.
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March 5-6, 01 New England Blizzard
NOAA NWS NEXRAD Data
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Model Assumptions

• We assume that snow is dry and that the
  refractive index, m 1.78.
• We let the mass of the particles M ? L? where
  
• ? (.026 .001) N 0.930.02 and ? 1.88
  0.03, for 1 lt N lt 4 and L is the length of the
  constituent cylinders.
• That corresponds to an aspect ratio, a 0.20
  L-0.59 , which comes from Auer Veal S 0.20 L
  0.41 for L gt 1 mm, and where S is the diameter
  of the constituent cylinders.

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Aspect ratios,? , depend on the habit, they are
not always constant
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DDA Calculated Single Scattering Parameters
95 GHz
183 GHz
340 GHz
Extinction cross section
Asymmetry factor
Accurate bench mark, but very time consuming.
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Theoretical Considerations

• Equivalent spheres have been used to represent
  randomly oriented aggregates of prisms.
• Spheres with large real refractive indices are
  probably the worst models to represent
  irregularly shaped particles. (Infinitely long
  cylinders are not much better)
• Because spheres are high Q spherical resonators,
  surface waves produce artificial ripples that
  distort the results.
• Surface waves can be attenuated by complex
  refractive indices, but how to define the
  imaginary part?

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• Use Equivalent Finite Cylinders to Represent
  Numerous Aggregates of Cylinders.
• The trick is to define the equivalent dimension,
  ?

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• Anomalous Diffraction Theory smoothes the surface
  wave resonances by neglecting reflection at the
  surfaces. The scattering efficiency is determined
  by the phase delay parameter between the incident
  and scattered waves, ? 2 ? (m - 1) ? ? /c. (van
  de Hulst)
• The definition of the effective diameter, ?, is
  crucial
• Assume that definition is ? 4 V / p A- ,
  where V is the volume and A- is the area
  perpendicular to the incident radiation.
• For a single randomly oriented cylinder,
  ? 4 a L / p (1 a /
  2) . Life gets more complicated for aggregates
  with N gt 1.

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Scattering Efficiency , Q, as a Function of
Phase Delay Parameter, ?
T-Matrix
DDA
(o) C-1, (x) C-2, (?) C-3, ( ) C-4, (--) TMM,
(- -) TMM ? 0.24
? 0.4 ? 0.6
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Asymmetry Factor as a Function of Phase Delay
Parameter, ?
(o) C-1, (x) C-2, (?) C-3, ( ) C-4, (--) TMM,
(- -) TMM ? 0.24
? 0.4 ? 0.6
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• Once we have Q(?) / ?, we can compute, Cext / M ,
  the extinction cross section (mm2) per mass (mg)
• Cext / M 0.008 (m - 1) ? / c d Q(?)/ ?
• where ? is frequency (GHz), d is density
  (gm/cm3), c (300 mm/s) and m 1.78 for ? lt 3.
  We can fit
• Q(?)/ ? 0.34 ?2 / (1 0.02 ?4.12 )
• Q(?)/ ? c d / 8 (m - 1) Cext / (M . ?)
• Similarly, the asymmetry factor can be fitted by
• g 0.25 ?2 / (1 0.14 ?2.5 )
• Computing the scattering parameters for the
  idealized aggregates that were displayed is thus
  greatly simplified.

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Asymmetry Factor as a Function of Phase Delay
Parameter, ?
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Scattering Efficiency , Q, as a Function of
Phase Delay Parameter, ?
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Table 1 Extinction cross section (mm2) per mass
(mg), Cext / M, at ? 183 GHz for a 0.20
L-0.59 L (mm) \ N 1 2 3
4 1 0.98 1.12 1.36
1.66 2 1.33 1.49 1.69
1.91 3 1.56 1.72 1.89
2.04 4 1.73 1.88 2.01
2.10 Table 2 Asymmetry factor, g,
L (mm) \ N 1 2 3
4 1 0.11 0.15 0.21
0.30 2 0.20 0.25 0.31
0.40 3 0.27 0.32
0.39 0.46 4 0.33 0.39
0.45 0.51 Table 3 Irradiance
attenuation factor / mass, (1- g) Cext / M
L (mm) \ N
1 2 3 4 1
0.87 0.95 1.07 1.16 2
1.06 1.12 1.17 1.15
3 1.14 1.17 1.15
1.10 4 1.16 1.15
1.11 1.03 Mean value 1.10 .05
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Conclusions

• Mm-wave scattering properties of randomly
  oriented ice cylinders and aggregates can be
  computed from the phase delay parameter using the
  T-Matrix method or a simple analytic
  approximation.
• Mm-wave properties of snow need to be measured at
  the same time as particle volumes and 2-D
  projected areas.
• Scattering parameters in optically thick snow
  clouds may not be sensitive to particle models,
  but absorption may prevent establishment of the
  diffusion regime where (1-g) Cext could be
  effective. This requires radiative transfer model
  runs.
• Other shapes may produce different scattering
  parameters.