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

1. 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

2. 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.

3. AMSU-B 89 GHz AMSU-B 183.3 ± 7 GHz • Surface effect (Ocean/Land) • Hard to discriminate snow storm from ocean surface Surface effect is screened out. March 5-6, ‘01 New England Blizzard NOAA NWS NEXRAD Data

4. Model Assumptions • We assume that snow is dry and that the refractive index, m = 1.78. • We let the mass of the particles M = ζ Lηwhere • ζ = (.026 + .001) N 0.93+0.02and η = 1.88 + 0.03, for 1 < N < 4 and L is the length of the constituent cylinders. • That corresponds to an aspect ratio, α = 0.20 L-0.59 , which comes from Auer & Veal S = 0.20 L 0.41 for L > 1 mm, and where S is the diameter of the constituent cylinders.

5. Aspect ratios, , depend on the habit, they are not always constant

6. Sample imagery from the PMS 2D-C probe aboard the UW Convair-580 aircraft on 13–14 Dec 2001. Solid line with arrow heads shows flight track. The sample particle images were observed at the points indicated by the blue arrows. The region of ice-phase precipitation is indicated by gray fallstreaks, and the top of the cloud liquid water region is indicated by the gray-scalloped cloud outline. Height is indicated on the left axis and temperature is indicated by the labeled horizontal line segments.

7. Schematic views of model aggregates C-1 C-2 C-3 C-4

8. DDA Calculated Single Scattering Parameters 95 GHz 183 GHz 340 GHz Extinction cross section Asymmetry factor Accurate bench mark, but very time consuming.

9. 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?

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

11. 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,ρ = 2  (m - 1) Δ ν /c. (van de Hulst) • The definition of the effective diameter, Δ, is crucial • Assume that definition is: Δ = 4 V / π A┴ , where V is the volume andA┴is the area perpendicular to the incident radiation. • For a single randomly oriented cylinder, Δ = 4αL /π(1 +α / 2) .Life gets more complicated for aggregates with N > 1.

12. 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

13. 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

14. Once we have Q(ρ) / ρ, we can compute, Cext / M , the extinction cross section (mm2) per mass (mg): • Cext / M = {0.008 (m - 1) ν / c δ } Q(ρ)/ ρ • where ν is frequency (GHz), δ is density (gm/cm3), c (300 mm/s) and m = 1.78 for ρ < 3. We can fit • Q(ρ)/ ρ = 0.34 ρ2 / (1 + 0.02 ρ4.12 ) • Q(ρ)/ ρ ={ c δ / 8 (m - 1) } Cext / (M . ν) • Similarly, the asymmetry factor can be fitted by: • g = 0.25 ρ2 / (1 + 0.14 ρ2.5 ) • Computing the scattering parameters for the idealized aggregates that were displayed is thus greatly simplified.

15. Empirical Fit Asymmetry Factor as a Function of Phase Delay Parameter, 

16. Empirical Fit Scattering Efficiency , Q, as a Function of Phase Delay Parameter, 

17. Table 1: Extinction cross section (mm2) per mass (mg), Cext / M, at ν =183 GHz for α = 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 0.870.95 1.07 1.16 • 2 1.06 1.12 1.17 1.15 • 3 1.14 1.17 1.15 1.10 • 41.16 1.15 1.11 1.03 • Mean value: 1.10 + .05

18. 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.