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Scattering from Hydrometeors: Clouds, Snow, Rain Microwave Remote Sensing INEL 6069 Sandra Cruz Pol Professor, Dept. of Electrical & Computer Engineering, UPRM, Mayagüez, PR 1 Outline: Clouds & Rain 1. Single sphere (Mie vs. Rayleigh) 2. Sphere of rain, snow, & ice (Hydrometeors) Find their ec, nc, sb 3. Many spheres together : Clouds, Rain, Snow a. Drop size distribution b. Volume Extinction= Scattering+ Absorption c. Volume Backscattering 4. Radar Equation for Meteorology 5. TB Brightness by Clouds & Rain 2 Clouds Types on our Atmosphere 3 Cirrus Clouds Composition 70 60 hexagonal 50 plates 40 bullet rosettes % 30 dendrites 20 others 10 0 Ice Crystals 4 EM interaction with Single Spherical Particles Absorption Si – Cross-Section, Qa =Pa /Si – Efficiency, xa= Qa /pr2 Scattered – Power, Ps – Cross-section , Qs =Ps /Si – Efficiency, xs= Qs /pr2 Total power removed by sphere from the incident EM wave, xe = xs+ xa Backscatter, Ss(p) = Sisb/4pR2 5 Mie Scattering: general solution to EM scattered, absorbed by dielectric sphere. Uses 2 parameters (Mie parameters) – Size wrt. l : 2pr lb – Speed ratio on both media: np n nb 6 Mie Solution Mie solution 2 x s (n, ) 2 (2m 1)(| am |2 | bm |2 ) m 1 2 x a (n, ) 2 (2m 1) Re{a m 1 m bm } Where am & bm are the Mie coefficients given by eqs 5.62 to 5.70 in the textbook. 7 Mie coefficients Am m n Re{Wm } Re{Wm 1} am Am m 2pr 2πr n Wm Wm 1 ec lp λo m nAm Re{Wm } Re{Wm 1} bm np e pc ( j ) m n ec nAm Wm Wm 1 nb e bc ko where Wo sin j cos n n ' jn" 8 Non-absorbing sphere or drop (n”=0 for a perfect dielectric, which is a non-absorbing sphere) Re call ( j ) n n' jn " ko =.06 k o oe o Rayleigh region |n|<<1 9 Conducting (absorbing) sphere =2.4 10 Plots of Mie xe versus Four Cases of sphere in air : n=1.29 (lossless non-absorbing sphere) n=1.29-j0.47 (low loss sphere) n=1.28-j1.37 (lossy dielectric sphere) n= perfectly conducting metal sphere As n’’ increases, so does the absorption (xa), and less is the oscillatory behavior. Optical limit (r >>l) is xe =2. Crossover for – Hi conducting sphere at =2.4 – Weakly conducting sphere is at =.06 11 Rayleigh Approximation |n|<<1 Scattering efficiency 8 4 x s | K |2 ... 3 Extinction efficiency 8 4 x e 4 Im{ K } | K |2 ... 3 where K is the dielectric factor n2 1 e c 1 K 2 n 2 ec 2 12 Absorption efficiency in Rayleigh region x a x e x s 4 Im{ K } x e i.e. scattering can be neglected in Rayleigh region (small particles with respect to wavelength) |n|<<1 13 Scattering from Hydrometeors Rayleigh Scattering Mie Scattering l >> particle size l comparable to particle size --when rain or ice crystals are present. 14 Single Particle Cross-sections vs. For small drops, almost Scattering cross section no scattering, i.e. no 2l2 6 Qs | K |2 [m 2 ] bouncing from drop since 3p it’s so small. Absorption cross section l2 3 Qa Im{ K } [m 2 ] p In the Rayleigh region (n<<1) =>Qa is larger, so much more of the signal is x s x a absorbed than scattered. Therefore 15 16 Rayleigh-Mie-GeometricOptics 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 diameter to the wavelength (d/l) of the radiation. When this ratio is less than about one-tenth (d/l1/10), Rayleigh scattering occurs in which the scattering coefficient varies inversely as the fourth power of the wavelength. At larger values of the ratio of particle diameter to wavelength, the scattering varies in a complex fashion described by the Mie theory; at a ratio of the order of 10 (d/l>10), the laws of geometric optics begin to apply. 17 Mie Scattering (d/l1), Mie theory : A complete mathematical-physical theory of the scattering of electromagnetic radiation by spherical particles, developed by G. Mie in 1908. In contrast to Rayleigh scattering, the Mie theory embraces all possible ratios of diameter to wavelength. The Mie theory is very important in meteorological optics, where diameter-to- wavelength ratios of the order of unity and larger are characteristic of many problems regarding haze and cloud scattering. When d/l 1 neither Rayleigh or Geometric Optics Theory applies. Need to use Mie. Scattering of radar energy by raindrops constitutes another significant application of the Mie theory. 18 Backscattering Cross-section From Mie solution, the backscattered field by a spherical particle is 2 sb x b (n, ) 2 1 (2m 1)(am bm ) 2 1 m m1 pr Observe that perfect dielectric (nonabsorbent) sphere exhibits large oscillations for >1. Hi absorbing and perfect conducting spheres show regularly damped oscillations. 19 Backscattering from metal sphere Rayleigh Region defined as x 4 4 | K |2 b for n 0.5 where, K For conducting sphere (|n|= ) x20 9 4 b Scattering by Hydrometeors 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' ' 4.2 j 2.5 @ 30 GHz 2.4 j.47 @ 300 GHz For ice. n'i 1.78 For snow, it’s a mixture of both above. 21 Liquid water refractivity, n’ 22 Sphere pol signature Co-pol Cross-pol 23 Sizes for cloud and rain drops 24 Snowflakes Snow is mixture of ice crystals and air i 0.916 g/cm a 0 0.05 s 0.3g/cm 3 3 The relative permittivity of dry snow e ds 1 s e ds 1 ' ' ' 3e ds ' ' i e i 2e ds ei 1 Ki The Kds factor for dry snow ei 2 K ds 1.1K i 0.5 p 5 D6 p 5 D6 s bs x bprs 2 | K ds |2 | K i |2 ds i l4 o 4l4 o 25 Volume Scattering Two assumptions: – particles randomly distributed in volume-- incoherent scattering theory. – Concentration is small-- ignore shadowing. Volume Scattering coefficient is the total scattering cross section per unit volume. bsb p(rD)bs (b ( Dr d D N ( )s s r )d ) Q [Np/m] x s Qs / pr 2 x a Qa / pr 2 x b s b / pr 2 26 Total number of drops per unit volume Nv p(r )dr N ( D)d D in units of mm-3 / p(r ) arc e D / Do N ( D) N o e 27 Volume Scattering lo Using... 2pr / lo , x s Qs / pr and dr d 2 2p It’s also expressed as 3 l s , e ,b p( )x s ,e,b ( )d o 2 [Np/m] 8p 2 0 [s,e,b stand for scattering, extinction and backscattering.] or in dB/km units, dB b 4.34 103 N ( D)s b ( D)d D [dB/km] 0 28 For Rayleigh approximation Substitute eqs. 71, 74 and 79 into definitions of the cross sectional areas of a scatterer. 2p 5 D 6 Qs x spr 2 | K w |2 3l4 p D 2 3 Qa x apr 2 Im( K w ) l p 5 D6 s b x bpr 2 | K w |2 l4 D=2r =diameter 29 Noise in Stratus cloud image -scanning Ka-band radar 30 Volume extinction from clouds Total attenuation is due to gases,cloud, and rain a g e c ep cloud volume extinction is (eq.5.98) p 2 e c Qa dD Im{ K w} D3dD lo Liquid Water Content LWC or mv ) 4p 3 6 p mv w r dr 10 D 3dD 3 6 w water density = 106 g/m3 31 Relation with Cloud water content This means extinction increases with cloud water content. e c 1mv where 6p 1 .434 Im( K ) 1 1 [dBkm g m ] 3 lo and wavelength is in cm. 32 Raindrops symmetry 33 Volume backscattering from Clouds Many applications require the modeling of the radar return. For a single drop p 5 D6 s b x bpr 2 | K w |2 l4 For many drops (cloud) 5 p s vc s b N ( D )dD 4 | K w |2 N(D)D 6 dD l p5 4 | K w |2 Z l 34 Reflectivity Factor, Z Is defined as p5 Z D6 N ( D)dD so that s vc 4 | K w |2 Z lo and sometimes expressed in dBZ to cover a wider dynamic range of weather conditions. dBZ 10 log Z Z is also used for rain and ice measurements. 35 Reflectivity in other references… p5 10 12 2 | Kw | Z lo4 where is in cm and -1 6 3 Z is expressedin mm /m 36 Reflectivity & Reflectivity Factor Z (in dB) Reflectivity, [cm-1] dBZ for 1g/m3 Reflectivity and reflectivity factor produced by 1g/m3 liquid water Divided into drops of same diameter. (from Lhermitte, 2002). 37 Cloud detection vs. frequency 38 Rain drops 39 Precipitation (Rain) Volume extinction 3 l er p( )x e ( )d 1 R o 2 b 8p2 r [dB/km] 0 Mie coefficients where Rr is rain rate in mm/hr 1 [dB/km] and b are given in Table 5.7 can depend on polarization since large drops are not spherical but ~oblong. 40 W-band UMass CPRS radar 41 Rain Rate [mm/hr] If know the rain drop size distribution, each drop has a liquid water mass of m p D 3 w 6 total mass per unit area and time 0 N ( D)m( D)dD dAdt ( w p / 6) D 3 N ( D)vt dD rainfall rate is depth of water per unit time Rr p / 6 vt ( D) N ( D) D dD 3 a useful formula (-6.8D2 4.88D) vt ( D) 9.251 - e 42 Volume Backscattering for Rain For many drops in a volume, if we use Rayleigh approximation p5 p5 s vr s br dD 4 | K w |2 D 6 dD 4 | K w |2 Z l l Marshall and Palmer developed Z 200 Rr .6 1 but need Mie for f>10GHz. p 5 s vr 4 | K w |2 Z e l 43 Rain retrieval Algorithms Several types of algorithms used to retrieve rainfall rate with polarimetric radars; mainly R(Zh), R(Zh, Zdr) ˆ ( K ) 11.62K 0.937 for S band R dp R(Kdp) dp R(Kdp, Zdr) ˆ R( K dp ) 40.5K dp85 for X band 0. where R is rain rate, Zh is the horizontal co-polar radar reflectivity factor, Zdr is the differential reflectivity Kdp is the differential specific phase shift a.k.a. differential propagation phase, defined as dp (r2 ) dp (r1 ) K dp 2(r2 r1 ) 44 Snow extinction coefficient Both scattering and absorption ( for f < 20GHz --Rayleigh) e s 4.34 10 3 Q dD Q dD a s for snowfall rates in the range of a few mm/hr, the scattering is negligible. At higher frequencies,the Mie formulation should be used. The e s is smaller that rain for the same R, but is higher for melting snow. 45 Snow Volume Backscattering Similar to rain p5 p5 s vs 4 | K ds |2 D6 dD 4 | K ds |2 Z s l lo 1 1 Z s D s N ( D )dD D dD 6 6 Zi 2 s i 2 s 46 Radar equation for Meteorology For weather applications Pt G l 2 2 g e c ep dr R 2 Pr se o o 4p R 3 4 o R c p 2 for a volume s s vV V p 2 2 2 Pt G l c p e 2 2 2 Pr o o sv 324pR 2 47 Radar Equation For power distribution in the Pt Go2 l2 oo c p Lr 2 s v Pr o main lobe L 2 1024 p ln 2 2 R assumed to be Gaussian function. where, s v radar reflectivity and Lr receiver losses And the two - way atmospheric losses are defined here as L2 e 2 48

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