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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. Outline: Clouds & Rain. Single sphere ( Mie vs. Rayleigh ) Sphere of rain, snow, & ice ( Hydrometeors )
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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
Outline: Clouds & Rain • Single sphere (Mie vs. Rayleigh) • Sphere of rain, snow, & ice (Hydrometeors) Find their ec, nc, sb • Many spheres together : Clouds, Rain, Snow a. Drop size distribution b. Volume Extinction= Scattering+ Absorption c. Volume Backscattering • Radar Equation for Meteorology • TB Brightness by Clouds & Rain
EM interaction with Single Spherical Particles Si • Absorption • 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
Mie Scattering: general solution to EM scattered, absorbed by dielectric sphere. • Uses 2 parameters (Mie parameters) • Size wrt. l : • Speed ratio on both media:
Mie Solution • Mie solution • Where am & bm are the Mie coefficients given by eqs 5.62 to 5.70 in the textbook.
Non-absorbing sphere or drop(n”=0 for a perfect dielectric, which is anon-absorbing sphere) c =.06 Rayleigh region |nc|<<1
Plots of Mie xe versus c 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 c=2.4 • Weakly conducting sphere is at c=.06
Rayleigh Approximation |nc|<<1 • Scattering efficiency • Extinction efficiency • where K is the dielectric factor
Absorption efficiency in Rayleigh region i.e. scattering can be neglected in Rayleigh region (small particles with respect to wavelength) |nc|<<1
Scattering from Hydrometeors >> particle size Rayleigh Scattering Mie Scattering • comparable to particle size --when rain or ice crystals are present.
Single Particle Cross-sections vs.c For small drops, almost no scattering, i.e. no bouncing from drop since it’s so small. • Scattering cross section • Absorption cross section In the Rayleigh region (nc<<1) =>Qa is larger, so much more of the signal is absorbed than scattered. Therefore
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/l<1/10), 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.
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.
Backscattering Cross-section • From Mie solution, the backscattered field by a spherical particle is Observe that • perfect dielectric (nonabsorbent) sphere exhibits large oscillations for c>1. • Hi absorbing and perfect conducting spheres show regularly damped oscillations.
Backscattering from metal sphere • Rayleigh Region defined as • For conducting sphere (|n|= ) where,
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 • For ice. • For snow, it’s a mixture of both above.
Sphere pol signature Co-pol Cross-pol
Snowflakes • Snow is mixture of ice crystals and air • The relative permittivity of dry snow • The Kds factor for dry snow
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. [Np/m]
Total number of drops per unit volume in units of mm-3
Volume Scattering Using... • It’s also expressed as • or in dB/km units, [Np/m] [s,e,b stand for scattering, extinction and backscattering.] [dB/km]
For Rayleigh approximation • Substitute eqs. 71, 74 and 79 into definitions of the cross sectional areas of a scatterer. D=2r =diameter
Volume extinction from clouds • Total attenuation is due to gases,cloud, and rain • cloud volume extinction is(eq.5.98) • Liquid Water Content LWC or mv ) • water density = 106 g/m3
Relation with Cloud water content • This means extinction increases with cloud water content. where and wavelength is in cm.
Volume backscattering from Clouds • Many applications require the modeling of the radar return. • For a single drop • For many drops (cloud)
Reflectivity Factor, Z • Is defined as so that • and sometimes expressed in dBZ to cover a wider dynamic range of weather conditions. • Z is also used for rain and ice measurements.
Reflectivity & Reflectivity Factor h Z (in dB) Reflectivity, h [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).
Precipitation (Rain) • Volume extinction • where Rr is rain rate in mm/hr • [dB/km] and b are given in Table 5.7 • can depend on polarization since large drops are not spherical but ~oblong. [dB/km] Mie coefficients
Rain Rate [mm/hr] • If know the rain drop size distribution, each drop has a liquid water mass of • total mass per unit area and time • rainfall rate is depth of water per unit time • a useful formula
Volume Backscattering for Rain • For many drops in a volume, if we use Rayleigh approximation • Marshall and Palmer developed • but need Mie for f>10GHz.
Rain retrieval Algorithms Several types of algorithms used to retrieve rainfall rate with polarimetric radars; mainly • R(Zh), • R(Zh, Zdr) • R(Kdp) • R(Kdp, Zdr) 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
Snow extinction coefficient • Both scattering and absorption ( for f < 20GHz --Rayleigh) • 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 is smaller that rain for the same R, but is higher for melting snow.
Snow Volume Backscattering • Similar to rain
Radar equation for Meteorology • For weather applications • for a volume
Radar Equation • For power distribution in the main lobe assumed to be Gaussian function.