5 EM interaction with Single Spherical Particles AbsorptionCross-Section, Qa =Pa /SiEfficiency, xa= Qa /pr2ScatteredPower, PsCross-section , Qs =Ps /SiEfficiency, xs= Qs /pr2Total power removed by sphere from the incident EM wave, xe = xs+ xaBackscatter, Ss(p) = Sisb/4pR2
6 Mie Scattering: general solution to EM scattered, absorbed by dielectric sphere. Uses 2 parameters (Mie parameters)Size wrt. l :Speed ratio on both media:
7 Mie Solution Mie solution Where am & bm are the Mie coefficients given by eqs 5.62 to 5.70 in the textbook.
11 Plots of Mie xe versus cFour 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 sphereAs n’’ increases, so does the absorption (xa), and less is the oscillatory behavior.Optical limit (r >>l) is xe =2.Crossover forHi conducting sphere at c =2.4Weakly conducting sphere is at c =.06
12 Rayleigh Approximation |nc|<<1 Scattering efficiencyExtinction efficiencywhere K is the dielectric factor
13 Absorption efficiency in Rayleigh region i.e. scattering can be neglected in Rayleigh region(small particles with respect to wavelength)|nc|<<1
14 Scattering from Hydrometeors Rayleigh ScatteringMie Scattering >> particle sizecomparable to particle size--when rain or ice crystals are present.
15 Single Particle Cross-sections vs.c Scattering cross sectionAbsorption cross sectionIn the Rayleigh region (nc<<1) =>Qa is larger, so much more of the signal is absorbed than scattered. ThereforeFor small drops, almost no scattering, i.e. no bouncing from drop since it’s so small.
17 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.
18 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 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.
19 Backscattering Cross-section From Mie solution, the backscattered field by a spherical particle isObserve thatperfect dielectric(nonabsorbent) sphereexhibits largeoscillations for c>1.Hi absorbing and perfectconducting spheres showregularly damped oscillations.
20 Backscattering from metal sphere Rayleigh Region defined asFor conducting sphere (|n|= )where,
21 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=20CFor ice.For snow, it’s a mixture of both above.
25 Snowflakes Snow is mixture of ice crystals and air The relative permittivity of dry snowThe Kds factor for dry snow
26 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]
27 Total number of drops per unit volume in units of mm-3
28 Volume Scattering It’s also expressed as or in dB/km units, Using... [Np/m][s,e,b stand for scattering, extinction and backscattering.][dB/km]
29 For Rayleigh approximation Substitute eqs. 71, 74 and 79 into definitions of the cross sectional areas of a scatterer.D=2r =diameter
30 Noise in Stratus cloud image -scanning Ka-band radar
31 Volume extinction from clouds Total attenuation is due to gases,cloud, and raincloud volume extinction is (eq.5.98)Liquid Water Content LWC or mv )water density = 106 g/m3
32 Relation with Cloud water content This means extinction increases with cloud water content.whereand wavelength is in cm.
37 Reflectivity & Reflectivity Factor hZ (in dB)Reflectivity, h [cm-1]dBZ for 1g/m3Reflectivity and reflectivity factor produced by 1g/m3 liquid waterDivided into drops of same diameter. (from Lhermitte, 2002).
40 Precipitation (Rain) Volume extinction where Rr is rain rate in mm/hr [dB/km] and b are given in Table 5.7can depend on polarization since large drops are not spherical but ~oblong.[dB/km]Mie coefficients
42 Rain Rate [mm/hr]If know the rain drop size distribution, each drop has a liquid water mass oftotal mass per unit area and timerainfall rate is depth of water per unit timea useful formula
43 Volume Backscattering for Rain For many drops in a volume, if we use Rayleigh approximationMarshall and Palmer developedbut need Mie for f>10GHz.
44 Rain retrieval Algorithms Several types of algorithms used to retrieve rainfall rate with polarimetric radars; mainlyR(Zh),R(Zh, Zdr)R(Kdp)R(Kdp, Zdr)whereR is rain rate,Zh is the horizontal co-polar radar reflectivity factor,Zdr is the differential reflectivityKdp is the differential specific phase shift a.k.a. differential propagation phase, defined as
45 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.