Outline: Clouds & Rain

1. Single sphere ( Mie vs. Rayleigh ) 2. Sphere of rain, snow, & ice ( Hydrometeors ) εεε σσσ Find their c, n c, b Scattering from Hydrometeors: 3. Many spheres together : Clouds, Rain, Snow Clouds, Snow, Rain a. Drop size distribution b. Volume Extinction= Scattering+ Absorption Microwave Remote Sensing INEL 6069 c. Volume Backscattering Sandra Cruz Pol 4. Radar Equation for Meteorology Professor, Dept. of Electrical & Computer Engineering, UPRM, Mayagüez, PR 5. TB Brightness by Clouds & Rain

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Clouds Types on our Atmosphere Cirrus Clouds Composition 70

60 hexagonal 50 plates

bullet rosettes % 40

30 dendrites

20 others 10

0 Ice Crystals

3 4 EM interaction with Mie Scattering: general solution to EM Single Spherical Particles scattered, absorbed by dielectric sphere ..

S
Absorption i
Uses 2 parameters (Mie parameters) – Cross-Section, Q =P /S a a iii – Size wrt. λλλ : – Efficiency, ξξξ = Q /πππr2 π a a χ = 2 r
Scattered λ b – Power, Ps

– Cross-section , Qs =P s /S iii – Speed ratio on both media: ξξξ πππ2 – Efficiency, s= Qs / r

Total power removed by sphere from the np ξξξ ξξξ ξξξ n = incident EM wave, e = s+ a nb

πππ σσσ πππ 2 5 6
Backscatter , Ss( ) = S iii b/4 R

Mie Solution Mie coefficients

 A m   m +  −   Re{ Wm} Re{ Wm−1}  n χ  a =
Mie solution m  A m  2πr 2πr m + − χ = = ε ∞  Wm Wm−1 c 2 n χ λ λ ξ (n, χ) = 2( m +1)(| a |2 + | b 2 )|   p o s χ 2 ∑ m m m=1 ∞  m  2 nA +  Re{ W }− Re{ W } ξ χ = + +  m χ  m m−1 a (n, ) 2 ∑ 2( m )1 Re{ am bm}   ε χ b = np p (β − jα) m=1 m   n = = c = ε = + m − ε c nA m Wm Wm−1 nb b ko  χ  c
Where am & b m are the Mie coefficients given by eqs 5.62 to 5.70 in the textbook. = χ + χ where Wo sin j cos 7 n = n' − jn " 8 NonNon--absorbingabsorbing Conducting (absorbing) sphere sphere or drop ((n”= 0 for a a perfect dielectric , which is a nonnon--absorbingabsorbing sphere)

Re call (β − jα) n = n'− jn "= ko χ =.06 = ω µ ε ko o o χ =2.4 Rayleigh region | nχ|<<1

9 10

ξξ χχ Plots of Mie ee versus Rayleigh Approximation | nnχχ|<<1

Four Cases of sphere in air :
Scattering efficiency n=1.29 (lossless non-absorbing sphere) 8 ξ = χ 4 | K |2 +... n=1.29-j0.47 (low loss sphere) s 3 n=1.28-j1.37 (lossy dielectric sphere) n= ∞ perfectly conducting metal sphere
Extinction efficiency 8 ξ = 4χ Im{ −K}+ χ 4 | K |2 +... ξ e
As n’’ increases, so does the absorption ( a), and less is the 3 oscillatory behavior .
Optical limit (r >> λ) is ξ =2. e
where K is the dielectric factor
Crossover for – Hi conducting sphere at χ =2.4 n2 −1 ε −1 K = = c – Weakly conducting sphere is at χ =.06 2 + ε + 11 n 2 c 2 12 Absorption efficiency in Rayleigh Scattering from Hydrometeors region

Rayleigh Scattering Mie Scattering

ξ = ξ −ξ = χ − ≅ ξ a e s 4 Im{ K} e

i.e. scattering can be neglected in Rayleigh region (small particles with respect to wavelength) λ |nχ|<<1 >> particle size λ comparable to particle size --when rain or ice crystals 13 are present. 14

Single Particle CrossCross--sectionssections vs. χχ

For small drops, almost
Scattering cross section no scattering, i.e. no 2λ2 Q = χ 6 | K |2 [m 2 ] bouncing from drop since s 3π it’s so small .
Absorption cross section λ2 Q = χ 3 Im{ K} [m 2 ] a π χ In the Rayleigh region ( n <<1) => Qa is larger, so much more of the signal is ξ << ξ absorbed than scattered . Therefores a

15 16 RayleighRayleigh--MieMie--GeometricOpticsGeometricOptics Mie Scattering (d/(d/ λλ≈≈11),

Along with absorption, scattering is a major cause of the attenuation of radiation by the atmosphere for visible.
Scattering varies as a function of the ratio of the particle
Mie theory : A complete mathematical-physical theory of the diameter to the wavelength ( d/ λ) of the radiation. scattering of electromagnetic radiation by spherical particles, developed by G. Mie in 1908.
When this ratio is less than about one-tenth ( d/ λ<1/10 ),
In contrast to Rayleigh scattering, the Mie theory embraces all Rayleigh scattering occurs in which the scattering possible ratios of diameter to wavelength. The Mie theory is very coefficient varies inversely as the fourth power of the important in meteorological optics, where diameter-to- wavelength. wavelength ratios of the order of unity and larger are characteristic of many problems regarding haze and cloud
At larger values of the ratio of particle diameter to scattering. wavelength, the scattering varies in a complex fashion
When d/ λ ≈ 1 neither Rayleigh or Geometric Optics Theory described by the Mie theory ; applies . Need to use Mie.
at a ratio of the order of 10 ( d/ λ>10 ), the laws of geometric
Scattering of radar energy by raindrops constitutes another optics begin to apply. significant application of the Mie theory. 17 18

Backscattering CrossCross--sectionsection Backscattering from metal sphere
From Mie solution, the backscattered
Rayleigh Region defined as ξ = χ 4 2 b 4 | K | field by a spherical particle is χ < 2 for n 5.0 1 ∞ σ ξ (n, χ) = ()−1 m 2( m + )(1 a −b ) = b b χ 2 ∑ m m π 2 m=1 r

Observe that where,  perfect dielectric K = (nonabsorbent) sphere exhibits large oscillations for χ>1.

 Hi absorbing and perfect conducting spheres show regularly damped oscillations. ξ = χ 4 19
For conducting sphere (| n|= ∞ ) 20b 9 Scattering by Hydrometeors Liquid water refractivity, n’

Hydrometeors (water particles)
In the case of water , the index of refraction is a function of T & f . (fig 5.16)
@T=20C  9 − j.25 @ 1 GHz = − =  − nw n' jn ''  2.4 j 5.2 @ 30 GHz   4.2 − j 47. @ 300 GHz =
For ice. n'i 78.1
For snow, it’s a mixture of both above. 21 22

Sphere pol signature Sizes for cloud and rain drops

Co-pol

Cross-pol

23 24 Snowflakes Volume Scattering

Snow is mixture of ice crystals and air
Two assumptions: ρ = 3 ρ = ≤ ρ ≤ 3 – particles randomly distributed in volume-- i .0 916 g/cm a 0 05.0 s 3.0 g/cm incoherent scattering theory .
The relative permittivity of dry snow – Concentration is small-- ignore shadowing. ε ' −1 ρ  ε ' −1  ds = s  ds 
Volume Scattering coefficient is the total ε ' ρ  ε ' − ε '  3 ds i  i 2 ds  ε −1 scattering cross section per unit volume . K = i
The K factor for dry snow i ε + 2 ds i κκ == σ η = κb = ppN((r(rD))Q)bσ(r()Drd )dD [Np/m] K 1.1 K π 5 D 6 π 5 D 6 sb ∫∫∫ s b ds ≅ i ≈ 5.0 σ = ξ πr 2 = | K |2 ≈ | K |2 ρ ρ bs b s λ4 ds λ 4 i ξ = π 2 ξ = π 2 ξ = σ π 2 ds i o 25 4 o s Q s / r a Q a / r b 26 b / r

Total number of drops per unit volume Volume Scattering

λ Using... χ = 2πr / λ , ξ = Q /πr 2 and dr = o dχ o s s 2π = =
It’s also expressed as Nv ∫ p(r) rd ∫ N(D)dD in units of mm -3 λ3 ∞ κ = o χ 2 χ ξ χ χ [Np/m] s,e,b 2 ∫ p( ) s,e,b ( )d α −α γ 8π p(r) = ar e / 0 c [s,e,b stand for scattering, extinction and backscattering.] − N(D) = N e D / Do
or in dB/km units, o ∞ η dB = Κ = × 3 σ [dB/km] b 34.4 10 ∫ N(D) b (D)dD 0 27 28 Noise in Stratus cloud image For Rayleigh approximation --scanning K aa--bandband radar

Substitute eqs. 71, 74 and 79 into definitions of the cross sectional areas of a scatterer. 2π 5 D 6 Q = ξ πr 2 = | K |2 s s 3λ 4 w π 2 D 3 Q = ξ πr 2 = Im( − K ) a a λ w π 5 D 6 σ = ξ πr 2 = | K |2 b b λ 4 w

D=2r =diameter 29 30

Volume extinction from clouds Relation with Cloud water content

Total attenuation is due to gases,cloud, and rain κ = κ +κ +κ a g ec ep
This means extinction increases with
cloud volume extinction is (eq.5.98) cloud water content. π 2 κ = = − 3 e QadD Im{ K w} D dD κ = κ m c ∫ λ ∫ ec 1 v o where
Liquid Water Content LWC or m ) v 6π − − π π κ = .434 Im( −K) [dBkm 1g 1m3 ] 4 3 6 3 1 λ m = ρ r dr =10 D dD o v w ∫ 3 6 ∫
ρ = water density = 10 6 g/m 3 w 31 and wavelength is in cm. 32 Raindrops symmetry Volume backscattering from Clouds

Many applications require the modeling of the radar return.
For a single drop π 5 D 6 σ = ξ πr 2 = | K |2 b b λ 4 w

For many drops (cloud) π 5 η = σ = σ N (D )dD = | K |2 N(D)D 6 dD = vc ∫ b λ4 w ∫ π 5 η = 2 | K w | Z 33 λ4 34

Reflectivity Factor, Z Reflectivity in other references…

Is defined as π 5 π 5 = 6 σ = | K |2 Z −12 2 Z ∫ D N (D )dD so that vc 4 w η = λ o 10 4 | Kw | Z λ o
and sometimes expressed in dBZ to cover a wider dynamic range of weather conditions. where η is in cm 1- and dBZ = 10 log Z Z is expressed in mm 6/m 3
Z is also used for rain and ice measurements.

35 36 Reflectivity & Reflectivity Factor Cloud detection vs. frequency η Z (in dB) ] -1 [cm 3 η dBZ for 1g/m 1g/m forfordBZdBZ Reflectivity, Reflectivity, Reflectivity,

Reflectivity and reflectivity factor produced by 1g/m3 liquid water Divided into drops of same diameter. (from Lhermitte , 2002). 37 38

Rain drops Precipitation (Rain)

Volume extinction λ3 ∞ o 2 b κ = χ p(χ)ξ (χ)dχ = κ R [dB/km] er π 2 ∫ e 1 r 8 0 Mie coefficients
where Rr is rain rate in mm/hr κ
1 [dB/km] and b are given in Table 5.7
can depend on polarization since large

39 drops are not spherical but ~oblong40 . WW--bandband UMass CPRS radar Rain Rate [mm/hr]

If know the rain drop size distribution, each π drop has a liquid water mass of m = D3 ρ 6 w

total mass per unit area and time ∞ = ρ π 3 ∫ N(D)m(D)dD dAdt ( w )6/ ∫ D N(D)vt dD 0
rainfall rate is depth of water per unit time = π 3 Rr 6/ ∫ vt (D)N(D)D dD

a useful formula 2 v (D) = 9.25 [ e-1 (-6.8D +4.88D) ] 41 t 42

Volume Backscattering for Rain Rain retrieval Algorithms Several types of algorithms used to retrieve rainfall rate with polarimetric radars; mainly
R(Zh),
For many drops in a volume, if we use
R(Zh, Zdr) Rˆ(K ) =11 62. K .0 937 for S band Rayleigh approximation
R(Kdp) dp dp π 5 π 5
R(Kdp, Zdr) ˆ = 85.0 σ = σ dD = | K |2 D 6 dD = | K |2 Z R(Kdp ) 40 .5Kdp for X band vr ∫ br λ4 w ∫ λ4 w where R is rain rate,
Marshall and Palmer developed Zh is the horizontal co-polar radar reflectivity factor, Z = 200 R 6.1 r Zdr is the differential reflectivity
but need Mie for f>10GHz. Kdp is the differential specific phase shift a.k.a. π 5 differential propagation phase, defined as 2 φ −φ σ = | K | Z dp (r2 ) dp (r1 ) vr 4 w e K = λ dp − 43 (2 r2 r1 ) 44 Snow extinction coefficient Snow Volume Backscattering

Both scattering and absorption ( for f < 20GHz --Rayleigh)
Similar to rain 3 π 5 π 5 κ = × [ + ] 2 6 2 e s 34.4 10 ∫ Qa dD ∫ Qs dD σ = | K | D dD = | K | Z vs λ4 ds ∫ λ 4 ds s o
for snowfall rates in the range of a few mm/hr, the scattering is negligible.
At higher frequencies,the Mie formulation 1 1 Z = D 6 N (D )dD = D 6 dD = Z should be used. s ∫ s ρ 2 ∫ i ρ 2 i κ s s
The e s is smaller that rain for the same R, but is higher for melting snow. 45 46

Radar equation for Meteorology Radar Equation

For weather applications 2λ2 R
For power PG −2τ = t o o σ τ = ()κ +κ +κ dr distribution in the 2λ2θ φ τ σ Pr e ∫ g ec ep PtGo o o oc p Lr 2 ()π 3 4 P = L v 4 R o main lobe r 1024 π 2 ln 2 R2 assumed to be  Rβ 2  cτ  where,
for a volume σ = σ V V = π    p  Gaussian function. v   σ =η = radar reflectivi ty  2   2  v = and Lr receiver losses PG 2λ2 β 2cτ e−2τ = t o o p σ And the two - way atmospheri c losses are defined here as Pr 2 v 32 ()4πR 2 −2τ 47 L = e 48