This article was downloaded by: 10.3.98.104 On: 05 Oct 2021 Access details: subscription number Publisher: CRC Press Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK

Weather Radar Polarimetry

Guifu Zhang

Wave Scattering by a Single Particle

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315374666-4 Guifu Zhang Published online on: 01 Aug 2016

How to cite :- Guifu Zhang. 01 Aug 2016, Wave Scattering by a Single Particle from: Weather Radar Polarimetry CRC Press Accessed on: 05 Oct 2021 https://www.routledgehandbooks.com/doi/10.1201/9781315374666-4

PLEASE SCROLL DOWN FOR DOCUMENT

Full terms and conditions of use: https://www.routledgehandbooks.com/legal-notices/terms

This Document PDF may be used for research, teaching and private study purposes. Any substantial or systematic reproductions, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 shown as apendulum, in and spring displacementof the back-and-forth movement by is caused aforce, that To awave, describe its with vibration. Avibration is mathematically wea must start 3.1.1 verse wave a called is also trans a vibrationdirection, the define to vector is needed Since atwo-dimensional of propagation. direction the to perpendicular (if present)electrons adirection in a called wave wave Alongitudinal describe its to reference is characteristics. needed is also displacement of variable from the propagation. such, As only aone-dimensional wave, wave of the a longitudinal molecules direction along the which air vibrates waves EM by which light, are atextremelyacoustic wave An frequencies. high is acoustic waves, which environment are sound, our see and through other each hear Waves lives. information. daily We and our energy in omnipresent are transporting A wave efficient propagation is an is of the and avibration or oscillation way of 3.1 described. also of hydrometeors are characteristics basic provided. scattering The are method, Rayleigh includingcalculations, the T-matrix theory, Mie and approximation, of waves representation EM cal wave approaches and for The scattering scattering. basic physicsthe mathemati and chapterdescribes behind modeled. This be can ors spheroid focusing the on which shapes sphere hydromete and particle, many on the wave with by chapterdeals asingle of hydrometeors. scattering This types various EM waveby understand to scattering conditions, specific important weather it is composition and shape, orientation, they relatehydrometeors of to the as for these signatures radar clouds in precipitation. To and are that ors polarimetric understand wave by hydromete the (EM) electromagnetic scattering measure Weather radars Based onBased Newton’s law, second as for equation displacement is written the

3 WAVE AND ELECTROMAGNETIC WAVE V scalar wave ib R ation a Single Particle Single a by Scattering Wave

and z . An EM wave, EM . An however, wave, is atransverse which vibrates ( κ is the restoring coefficient). restoring is the Examples vibrationincludeof a

w vector wave vector a V e fm == Figure 3.1 . dz dt 2 2 −κ . z ,

f = −κ z , in the opposite direction opposite the direction , in (3.1) 51 - - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 where

where 52 where FIGURE 3.2 displacement. the to FIGURE 3.1 obtained by changing the time time by the changing obtained which in is zero function, cosine of the argument is the The phase from peak to peak in peak to peak from from its equilibrium position, its equilibrium from where

quency, with Solving 3.2 Equation yields As awaveAs expression propagation is of the be avibration, its mathematical can A ω m v is the phase velocity phase is the wave, of the wavelength the and is is the amplitude, which is defined as the maximal displacement vibrationof maximal the as which is amplitude, defined is the 2 is the mass. We mass. 3.1 is the Equation rewrite can yield to =

κ / T Sketch of a vibration with instantaneous displacement displacement instantaneous with Sketch of avibration Conceptual illustration of a spring vibration. The restoring force is opposite force opposite is restoring The vibration. of a spring illustration Conceptual m . as the period), and period), and the as z Figure 3.3 Figure z m ( x,t ω t z ) = ). = 2 to to ( t ) = dz t dt A

ϕ − 2 π cos[ 2 A 0

f in Equation 3.3 is the initial phase at phase 3.3 initial Equation in is the x is the angular frequency ( frequency angular is the

cos ( cos +ω / v ω in Equation 3.3, Equation in yielding z ( 2 t ω z

− = t

+ x 0, / v ϕ ) +

0 ),

m ϕ 0 ], Weather Radar Polarimetry Weather Radar

z as a function of time. afunction as t

Figure 3.2 Figure f λ = 1/ = = vT T (as shown is the fre . t =0. (3.2) (3.3) (3.4) - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 tion of For example,manipulation. aderivative time to respect with complex the convenience in phasor is the representation it allows mathematical in the advantage of introducing The term. dependence time but the omits information, quantity complex Wave Scattering byaSingle Particle Wave Scattering tity tity form of an electric field electric ofform an wave EM mentionedAs earlier, an wave is a vector atransverse the in needs that 3.1.3 where example, as 3.4 rewritten Equation be can 3.4, Equation more convenientlylike be it can by acomplex represented quantity. For wave simply by be asinusoidal function represented can atime-harmonic Although 3.1.2 FIGURE 3.3 Using the analogy of the phasor for a scalar wave, for phasor ofUsing the ascalar of analogy representation the phasor the the field, and so on.yieldselectric a varying neticturn fieldin also changing, which is field mag This electric field.magnetic a achanging causes such that or amedium z ( p e x,t zx zx j () (, ω ) in representing the wave, the representing ) in except for omission of the [ z z z z M H : at( at( at( at( tA = a

)c ∂ S =ω w zx Ae o t t t t Sketch of wave propagation: asterisk (*) propagates in the (*)Sketch of wave the in asterisk propagation: propagates (, ∂ = = = = R t a t t t t R jx 4 3 2 1 V () t ) ) ) ) φ− z ) e os 0 ( ep ↔ω x ω ) is called a ) is called   R e / v S jz tx entation is acomplex quantity, quan which is equivalent real the to +φ Er () (, () xt 0 , −ω ) t . to represent it. An EM wave EM it. represent to An space in propagates phasor

of /R vA

t   i M = because it is acomplex because phase with quantity e -H eR   a RM ez jt oni (/ ω+ φ− C 0

w ωω xv a ) V   e t = S is simply amultiplica e e( j ω   t x ] in writing. The The writing. ] in -direction. xe ), jt x x x x  

(3.5) 53 - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 and where electric fieldelectric is 54 surfaces. in as itand is illustrated wave the with number polar waveplane solution is the of3.10, Equation specifically:

become fields 2.14)Maxwell (Equation equations magnetic and electric of for phasors the the ( ∇×∇× Figure 3.4 Taking the curl of 3.6curl Taking Equation the vector using the identity and of In the uniform medium where the permittivity and permeability are constant, a constant, are permeability and permittivity where the medium uniform the In In the case of a spherical wave, of case aspherical the fieldbe In electric expressed the by can ization, ization, η= EE =∇ ). µ ε kk () is the intrinsic impedance of the medium, medium, of the impedance intrinsic is the = ∇• Er () k ˆ is the wave is the vector, and ,R tE kk −∇ = =ω 2 e iue 3.5 Figure E   , and using Equation 3.7, using Equation , and we have µε () re =ω ∇+ ∇× ∇× Hk ∇= Ee Ee jt ∇= ω 22 , where the equal phase constitutes spherical spherical constitutes phase equal , where the Ek i = = =× µε   i . In the case of the time-harmonic wave, the time-harmonic of case the the . In Hj H Ej E 00 ˆ ˆ ˆ E Ee =− =ω 0 0 e 0 0 ˆ e E k −• ˆ ε=

. − E is the unit vector unit of is wave the propagation r

jk η r = ωµ jk ε 0 r E r 0 e

H

−•

jk 0

mm r , =

Weather Radar Polarimetry Weather Radar 2 λ π 0 e ˆ is the unit vector unit for is the .

(3.12) (3.10) (3.13) (3.14) (3.11) (3.6) (3.8) (3.7) (3.9) Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Wave Scattering byaSingle Particle Wave Scattering vibration direction, which is the locus which of tip of is the the the vibration direction, (vibration)time atagiven wave location. of isthe The adescription polarization polarization). If the locus is random, however, locus ispolarization). random, the If or Sunlight wave the is unpolarized. in as ellipse locus is an the if and loci in formacircle as Figure 3.6a equal on a plane that is perpendicular to the wave the to vector. perpendicular is that onaplane equal FIGURE 3.4 perpendicular to the wave the to vector. perpendicular 3.5 FIGURE coherent source or sources. If the locus of the the If or sources. coherent source wave EM An wave, is a transverse field electric the and 3.1.4 Figure 3.6 w a , the wave is said to be linearly polarized (or in linear polarization); if the polarization); the (or if linear wave in , the polarized linearly be to is said

V shows wave electric of an examples field polarization offrom the a e A sketch of a uniform plane wave propagation, where phases of the wave of wave the plane phases where are propagation, ofA sketch auniform A sketch of a spherical wave propagation, where equal phase surfaces are are surfaces phase wave equal where ofA sketch aspherical propagation,

p ola x R ization Figure 3.6b

and E R , it is circularly polarized (circular polarization); polarized , it is circularly Figure 3.6c O ep z k R e S entation E , it is elliptically polarized (elliptical , it polarized is elliptically vector is a straight line, as shown as in line, vector is astraight E E vector as time progresses. vector time as E varies sinusoidally with varies k y 55 Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 where ( 56 follows: as rewritten solution component 3.5), each followsand (Equation which is time-harmonic the wave field sources. incoherent and coherent it both if is from waves. polarized Awaveunpolarized partially be also can and (Born Wolf surface 1999)an is example of Lambertian from a light reflected polarization, and (c) and polarization. elliptical polarization, 3.6 FIGURE y

along the along the and and A mathematical representation of a polarized wave the by made describing be of representation can apolarized A mathematical z components, respectively and ω t

− x E -axis, the the -axis,

kx

vector in the polarization plane. Assuming that a plane wave a plane that plane. Assuming propagates polarization vector the in ) is the time–space variable phase term; term; phase variable time–space ) is the E A Polarization of an electric wave field. (a) Linear polarization, (b) circular (b) wave circular electric field. of an polarization, Polarization (a)Linear z z z (a) E Ex E vector will be in the the in be vector will (, y E tE )( E A E =+ z y y ( ( x,t x,t yz ) = ) = xt ,) A ϕ A z E y z 01 z y yE ˆ cos( cos( and and (0, (c) A ω ω z ) y ϕ (, t t

xt –

02 − − z

are their corresponding initial phases. initial phases. corresponding their are E

plane: kx kx ) y E zE ˆ + + = A ( A ϕ Re y ϕ and and y 02 (0, 01 , 0) () ), ) Weather Radar Polarimetry Weather Radar

A

E z A e ) z y z jt ω are the amplitudes for the amplitudes the are (b)

E y E ( A , 0) (3.15) (3.16) (3.17) y Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 3.15 Equations Combining 3.17 through polarized. conditions is elliptically yields waveThe of equations (3.11) by general the represented (3.12) and without further 3.1.4.3 δ pageof (the the in illustrated polarization circular the obtain we 3.17and 3.15 Equation into conditions, using we these and have straight line with aslope with of line straight locus is a the which and shows two components the have term, phase that same the Because the variable phase ( phase variable the Because ( term phase varying the is unity, ratio amplitude specifically, the and be difference phase the Let 3.1.4.2 by 3.19sented Equation being as is described Wave Scattering byaSingle Particle Wave Scattering be difference phase the Let 3.1.4.1 follows: as ences by 3.15sented Equations 3.17 through differ phase and ratios amplitude different with in illustrated such those as of polarization, types different = Equation 3.21Equation for is clearly equation a2D plane, hencerepresenting acircle an in Substituting Equations 3.16Substituting Equations 3.17 and 3.15 Equation into 3.18, using Equation and The tip of the vector forms an ellipse, which can also be represented by cancelling by cancelling represented be also ellipse, which vectorcan of tip an forms the The −π /2 represents right-hand circular polarization. right-hand circular represents /2

Linear Polarization Linear Circular Polarization Elliptical Polarization Elliptical Ex Ex (, (, x direction), direction), tA tA )c )c =ω =ω Ex ω () yz t os E A ,

os − ω tA y y 2 2 A ()

t kx () δ

=± +− z − / = δ tk A ) as ()

tk E A kx = −+ δ= y −+ , as shown, as in z z 2 2 π yz ) in Equations 3.16 Equations ) in 3.17 and the term, is acommon ϕ yA /2 represents left-hand circular polarization and and polarization left-hand represents /2 circular ˆ xy zy φ− 02 2c 02 Figure 3.6b

− ϕ φ± EE AA φ+ 01 yz yz 01 zt ˆ φ= 01 01 =0or co ˆ ˆ A os s, linearly polarized linearly () z ± At At Figure 3.6a ω− / δ= A si π 2 . Because the wave the . Because out is coming y cos( π. =1. 3.16 Substituting Equations n

si ()

kx ω− n, ω− 2 +δ δ Figure 3.6 kx

1 . Hence, the wave. Hence, the repre kz +φ

+φ . 01 02 ) z ˆ z , are then repre then , are ˆ . .

(3.20) (3.22) (3.23) (3.21) (3.19) (3.18) 57 - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 the by G. G. was Stokes introduced (1852a, parameterization This 1852b) is called and meteorology community. but not received radar has munity the much in attention sensing remote com widely the been which dimension, in has used same the with 1983). Kong polariza left-hand phase difference are random variables. Then, the definitions of the Stokes parameters the definitions Stokesof the parameters variables. Then, random are difference phase 58 where semiminor (b) axes of the ellipse and the orientation of orientation the and (b) ellipse of the axes semiminor 3.7 FIGURE is equivalent to the representation by the three parameters ( parameters by three is equivalent representation the the to wave.for polarized Stokes four-parameter Hence, afully the of notation [ as is defined angle ellipticity where the of (b) orientation the and ellipse of axes the semiminor (a) by semimajor the described be also ellipse, which and can atilted general in waveEquation field. polarized represented by The 3.23 elliptically ellipse is an Another way to represent an elliptically polarized wave wayisfour use polarized to parameters elliptically Another represent to an In the case of a partially polarized or unpolarized wave, the and amplitudes orthe unpolarized polarized of case a partially the In Note that there is a relationship among the four parameters, which is four is arelationship parameters, the among there Note that Stokes parameters Stokes δ = ϕ

02

Representation of an elliptically polarized wave by the semimajor (a) wave semimajor by the and polarized elliptically of an Representation − ϕ − tion and the negative the tion and (Shen sign for right-hand polarization and 01 is the phase difference. Three parameters ( parameters Three difference. phase is the UE QE VE IE : =− =+ == == 2I 2R yz yz m( e( 2 2 * yz * yz EA EA EA EA I

2 = 2 2 )2 )2 z =−= =+ ´ Q ba A 2 z + yz yz 22 22 z yz yz AI AI U AI A ta

2 si co ψ + n

ns χ χ= sc V δ= δ=

2 ψ

co A ± . y s2 a b in with the positive the with sign for os χψ Weather Radar Polarimetry Weather Radar y 2 ψ ´ co 2s χ , as shown, as in χψ A . s2

y y , in A z 2 ,

δ A ) in this case. this ) in y ,

A z , Figure 3.7 δ ) describe ) describe I,Q,U,V (3.26) (3.25) (3.27) (3.24) (3.28) - ] , Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 to theco-polarcrosscorrelationcoefficient ( biologicalobjects)—asshown in we usually deal with a high degree of polarization (except for those from clutter or tially polarizedwaves, and is thedegree ofpolarization.For ellipticallypolarized relative with constant dielectric where by permittivity represented are ties in described as particle, of the characteristics EM and physical wave the on the depend and properties properties absorption and scattering the Itthat is expected directions. out all to is scattered part other the and particle, the wave of wave the the When apart power particle, is incident on the by is absorbed 3.2.1 section. this by in which asingle is discussed particle, process quantitatively and scattering representthe understand how conceptually to Figure 3.8 hydrometeors), wave the directions. all into on incidence them redirect (e.g., Scattering 3.2 FIGURE 3.8 3.28 shown Equation be it by is replaced that can respective by averages. 3.27) replaced case, their 3.24 are through this In (Equations Wave Scattering byaSingle Particle Wave Scattering Let the particle have an arbitrary shape that will be specified later specified be will that shape have particle arbitrary the an Let The ratiobetweentheright-handandleft-handsides,

SCATTERING FUNDAMENTALS ε 0 S and and and C is a physical process in which an object or objects, called is a physical object which or objects, an called in process shows asketch of wave question by is asingleThe particle. scattering atte

S μ Conceptual sketch of wave scattering by a hydrometeor particle. by ahydrometeor of sketch wave scattering Conceptual C 0 R are free space permittivity and permeability, respectively; and permittivity space free are atte in G

R a in M G µ= plitude ε= C () () p rr r R = 0forunpolarizedwaves. Inweatherradarpolarimetry, o IQ SS 22 µ εε Chapter 4, ε′ , S r S 0 () >+ as its real part and and part its real as ,

e C ε C atte and permeability permeability and tion 00 = R () S in E ε− ρ the degree ofpolarization iscloselyrelated i UV ′

hv G () rj 22 ). M + k at i

= εε R kk ′′ ix 

() i waves,

ε″ r pQ , Chapter 2. ,

μ =+ as its imaginary part. its imaginary as as () p

22 = 1;0< E UV s

. Its EM proper . Its EM + p k < 1for s

= scatterers

22 kk ε  s r

is the is the (3.30) (3.29) (3.31) par I 59 - - ,

Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 3.14: Equation to similar equation by an represented be can wave plane sentation of the 3.12, is Equation which is free space with a permittivity and permeability of ( permeability and a permittivity with space free 60 FIGURE 3.9 where dent wave fieldand where medium and the unit vector unit the and medium shownas in onal components. Assuming that that components. Assuming onal information. phase and magnitude both complex general, is, in contains amplitude and scattering The particle. on the wave scattered of the field polarization phase, and unit for a plane waveincidence condition: the away is point far itobserved is called particle, the from wave excited vibrate,yielding the are areradiated and If or waves directions. all in kk ii = Assume that the incident wave is a linearly polarized plane wave plane incident wave the that in Assume propagating polarized is alinearly The scattering equation (Equation 3.33) can then be written in matrix formas 3.33) matrix in (Equation equation written be then scattering can The As discussed in in discussed As incident particle waveWith inside the charges the particle, on the impinging k ˆ e ˆ e sk ˆ i2 is the incident wave is the vector, with i is the unit vector unit for incident is the wave the polarization; () ˆ si r e ˆ ,

i1 >2 Figure 3.9 ks

ˆˆ Coordinate systems for the scattering matrix of a hydrometeor particle. of ahydrometeor matrix scattering for systems the Coordinate D = k    i 2

= / E () E λ Section 3.1.4 () kk kk s2 . In this case, the scattered fieldbehaves scattered the wave case, as a spherical and this . In ee s1 ˆ ˆ si s1 i

, , we have , Ee ˆ    ˆ ii s2 = Ee e == ˆ ss s e are the reference unit vectors for the scattered wave reference vectors the for unit scattered are the field, is the scattering amplitude, representing the amplitude, amplitude, the representing amplitude, scattering is the ˆˆ == − k ˆ r i jk Ee for propagation the direction. ˆˆ r Es 0i , wave polarization is fully described by two orthog described , wave is fully polarization E      ss −• 0s () = Ee e jk ˆ sk sk ee ˆ s1 11 ii 21 e i1 ii

rj − , = () r () () e ˆ jk ˆ kk s2 ˆ ˆ ˆ i2 r ˆ si si

() , , , Ee are the reference vectors the for unit inci the are ks Ee ks ˆ ˆˆ k ˆˆ 0 i0 () as the wave the as for number background the 1i k Ee E −• s

= s1 1i jk i kk + 12 e 22 s1 ˆ r − s Ee , r () () +

jk kk kk 2i r x si Ee si ˆ ε , , ,

s2 0 2 ˆ ˆ , μ ˆ e s2 Weather Radar Polarimetry Weather Radar 0 −• ). repre phasor Hence, the      . kr

   0 E φ far field far

E E z 0 is the amplitude; and amplitude; and is the θ i2 i1    v ˆ =– .

if it meets the the it if meets θ ˆ θ k ˆ ˆ h ˆ y =φ ˆ (3.36) (3.35) (3.34) (3.32) (3.33) - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 terms terms

is called the the is called plane wave the both to scattering vector the and vectorvectors. perpendicular The directions. vertical and zontal (ii) hori and plane the on (i) based scattering the are methods used two commonly where ways vectors unit orthogonal choose to the scattering plane are defined as defined are plane scattering and is generally elliptically polarized. The scattering matrix elements are discussed elements discussed are matrix scattering The polarized. elliptically is generally and passing through a unit area, which is the magnitude of the power of the magnitude density which flux is the vector. area, aunit passing through wave the by power the power density—the flux power matrix, is characterized tering following the in sections. further Wave Scattering byaSingle Particle Wave Scattering waves,scattered respectively. 3.36 Equation case, becomes this In and zontal off-diagonal terms off-diagonal terms

The scattering plane is the plane that contains the incident and scattered wave scattered incident and the contains that plane is the plane scattering The Hence, the scattered wave scattered Hence, the fieldincidentthe to related is wave field by Whereas awaveWhereas andscat field amplitude represented by is complex its scattering The matrix matrix The The other coordinate system to define the scattering matrix is based on the hori on is based matrix systemscattering the define to coordinate other The s 11 and and vertical perpendicular vector perpendicular s 22    []     S representing the wave scattering for co-polar components and the the wave components and for the representing co-polar scattering E E (more precisely described as longitudinal) directions with (more directions longitudinal) as precisely described E E s = s s 12 ⊥  sv sh    and and    vx ss ss ˆ     11 kx 21 = ˆ =− = h ˆ =θ s e =φ e 21 − si r − co 12 22 representing cross-polar scattering. There are alternate alternate are There scattering. cross-polar representing r jk ˆ nc jk r r sc =         θφ hx      ek ˆ ˆ is called the the is called zk zk i| ˆ ˆ os =− sk , defined as , defined sk |i sk sk os × × ⊥⊥  =× vh hh ⊥ φ+ ˆ ˆ () () si () () ˆ ˆ ˆ ˆ ˆ ˆ ˆ n an si si si si −θ , , , , φ+ si ks ks ks ks ˆˆ e d ˆˆ ˆˆ ˆˆ ˆ co ˆ ns i ⊥ () vk θφ ˆ and and ss ee ˆ 12 co =− ee , in ˆˆ 1 scattering matrix ˆ ⊥ vv hv s in   == θ= φ ˆˆ () () () () ek yz to define the scattering matrix— scattering the define to ˆ ˆ φ+ y kk ˆ kk kk kk s|

yz +θ si ˆ |s ⊥ si si si , , , , =× co ˆ ˆ ˆ ˆ × si ˆ s kk kk h n ˆˆ ˆˆ ˆ si si ,           θ

ˆ × ×     e   

ˆ ˆ s . ⊥

E E E E for incident and the iv ih i . The vectors on the vectors on the . The i ⊥  , with its diagonal its, with diagonal        .

(3.37) (3.40) (3.38) (3.39) (3.42) (3.41) 61 - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 where density 3.45, by incident power the power flux we differential normalized have scattered the the particle and dissipated. This part can be calculated by volume of the the calculated be integral can part This dissipated. and particle the 3.46, of Equation specifically integral is the angles the going all to angle. solid power 4 the in scattered total the with compared be itso that can

we have Poynting vectors mean the the power The also called density flux is 62 (Shen Kong and 1983). waves averaged power the between powerdifference peak for the time-harmonic and wave.for scattered the factor 3.43 of Equations The 1/2 in for 3.44 and accounts the area area section cross radar backscattering and abistatic multiplied obtain to (section).power of aunit area has density applications, and afactor of 4 radar In wavesents scattered power solid for angle aunit incident wave in an having aunit where for incident wave the and Besides the scattered power loss, another part of the wave of the power part Besides loss, scattered power the another by is absorbed The scattering cross section that represents power represents power that section cross loss for scattered the scattering The Hence, the differential power differential Hence, the da σ= d = Ω ds () r = sin = kk ˆ 2 d , Ω ˆˆ is is θ SE d σ= ss θ sk bi =× d () ϕ () 1 2 kk is the differential solid angle. Using differential is the ˆ , si Re dP dP k , ˆ S i ˆ i s ss () SE 2 == =Ω is called the the is called ii Sd =× sk 4 σ= () πσ 1 2 H sd ˆ a dP si Re s , * ds kd ˆˆ 4 () ∫ s () for the scattered wave through a differential wave adifferential for scattered the through = π kk sk ˆ σΩ () 2 , 2 E Poynting vector Poynting ˆ ˆ η si r () s ib differential cross section differential , 2 kk 0 2 ˆ H k ˆ ,4 si =σ k , i ˆ * s ˆ 2 σ= = = ds 2 E d () sk η 2 0 E kk ( 0 η 2 0 . ˆ ,

0 si r rd 2 πσ ˆ , 2 i 2 k k ˆ ˆ i Weather Radar Polarimetry Weather Radar . For a time-harmonic wave,. For atime-harmonic d di Ω )

Ω () 2 , − ,

2 E

kk ˆ η 0 S , 0 i 2 ˆ i = k ˆ , s

because it rep because 2 E π η steradians of a steradians 0 0 2 in Equation Equation in (3.46) (3.44) (3.47) (3.45) (3.48) (3.43) π re is is of of - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 section: cross absorption and section cross scattering of sum the is the section cross total the process; then scattering the Next, we discuss the scattering calculations in the aforementioned regimes. the in calculations Next, wescattering the discuss the wave the is called usand tells theory intuition what between our difference section. This cross its to responds geometric exactly that cor area extinct an object which blocks makes an in intuition and our to is contrary This for section cross particles. large very geometric of the that twice (or total the extinction) regime, optical geometric is section the cross in that, note to optics of applies. geometric It is interesting theory where the approaches a constant, wavelength the with ( large compared very which is called wavelength of size the about the ( the is called This increases. size where where as crosssection incidentabsorption the powerthe density flux to obtain powers, power absorption the peak is normalized and mean ference the between voltage: and current different powerpower differential is aproductof Consider the loss particle. that inside the follows: as ized 1983)and Huffman (Bohren crosssections normal the as defined are absorption and is called conductivitywhere the is Wave Scattering byaSingle Particle Wave Scattering ticle is very small ( ticle small is very ( size electric of the functions as sections cross normalized the scattering cross section (Equation 3.48) (Equation section cross extinction section, cross the and scattering tude; it the as is known ampli scattering forward of the part imaginary the to is proportional section cross crosstion section The sum of the scattering and absorption represents the total power total loss the represents due to absorption and scattering of sum the The For convenience efficiency the offactors comparison, for scattering, extinction, Figure 3.10 σ sk () g is the geometric cross section, and and section, cross geometric is the albedo ˆ ii , k ˆ is the forward scattering amplitude; scattering forward is the shows aplot efficiency of the factors for spheres, water specifically, . resonance . The second equity of 3.51 Equation equity second . The shows extinction the that ka dP dP σ= << 1),the as efficiency the increase monotonically factors aa optical theorem = Q σ= σ dI dI or the or the t ts PS = = × ωε″ε σ σ+ dV t i0 / σ =ω =( Mie scattering regime Mie scattering g ka 0 Rayleigh scattering regime Rayleigh scattering , ∫ σ= . Taking the factor. Taking of the Q ai

1 2 Jda ∼ s = 1), the normalized cross sections oscillate, oscillate, sections cross 1), normalized the (see εε 4 ) σ k

′′ extinction paradox π × s /

ka σ Im σ ( Ed g g Appendix 3AAppendix Er , >> 1), efficiency of each the factors =   Q in ℓ sk t π a ) = () () = a ˆ σ ′′ 2 , σ for asphere of radius σ t k

2 ˆ a ≡ σ ≡ E i / dr σ 2   . When the particle size is size particle the . When dad g , , 1 2 e

is also called the the called is also into account for account into the S ). The ratio between the ). between the ratio The i ℓ . (Van Hulst De 1957).

,

. When a particle is aparticle . When ka ). apar When w a = . extinc (3.50) (3.52) (3.49) (3.51) σ s dif / σ 63 by by t ------, Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 scattering cross section Q section cross scattering 64 follows: as physical the to related be it to parameters was because ratio suspected tude power.” themselves inverse lights ofintensity the the fourth as wavelength of inversely the incident light varies and square the the as and scattered wavelengths, any ofwith the vibrations of of the the amplitudes of ratio the the compared small very which are by particles light is scattered statement:“When ing (1983) which Huffman was matching, dimension by and cited Bohren follow the in law developedThe of Rayleigh was initially by Rayleigh scattering (1871) through 3.3.1 follow. that representation mathematical and statement conceptual on the based discussed awavelength than smaller as is known whose is much mentioned size As earlier, wave particle by asmall scattering 3.3 FIGURE 3.10 The lawThe of Rayleigh was for derived ampli by the matching dimension scattering Normalized cross sections o 10 10 RAYLEIGH SCATTERING RAYLEIGH 10 10 –2 –1 1 0 R i

G inal Rayleigh Normalized extinction cross section section cross extinction Normalized 10 –1 S tate s of spheres. water M ent A A s i :, Vr Rayleigh scattering ,, 10 λρ Mie 0 ka c ,, Q e t , absorption cross section Q section cross , absorption

Weather Radar Polarimetry Weather Radar . Rayleigh is scattering 10 Optics 1 Q Q Q t a s = σ = σ = σ a , and total total , and t s a /σ /σ /σ g g g (3.53) - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 tains the dimension of time [T]. Hence, because the amplitude ratio is dimensionless is dimensionless ratio amplitude the Hence, [T]. dimensionofbecause time the tains ponent of dimensionin time out cancel to the term is no other there and In Equation 3.53, velocityIn Equation the length, where * wave EM it an ate is excited 1991). after by incident wave the (Ishimaru radi and adipole as act to antenna particle continues, causing the process tions. This field electric of the changes),direction the movecharges and the direc opposite in negative move to charges wave end. The other the to (i.e., changes phases then the positiveand causes fieldthe particle to moveto charges is applied and to one end circle), (represented by dashed particle the spherical electric incident an on asmall in illustrated adipole As as radiation. understood be can wavelength, the with Rayleigh compared the is small particle scattering the When 3.3.2 Maxwell equations. on the based amore rigorous formulation of wave through issues wereThese addressed scattering was. were, absorption dependences what the and polarization and angular what the were phenomena, unresolved there issues such as natural explaining in successful eyes. our to orange appears long the of wavelength sight, line what remains therefore sun the direct lightand the wavelength short the because sun, lightis removed atthe from directly when looking is stronger for wavelength short scattering (blue) is perceived sky blue; as the light, sky, wave the to isthe why orange: is looking blue sun sky because the When the and power wavelength. of the explanation law the for provides, things, This other among law of fourth Rayleighis inversely ratio intensity the to the proportional scattering: Wave Scattering byaSingle Particle Wave Scattering dependence on dependence However,

range the to tional volumewave the to of particle is proportional amplitude wave incidentof the Equation 3.53. scattered and the between ratio amplitude the If

thought to propagate within a medium called “ether,” like a sound wave propagates in air. in wave asound propagates “ether,” like called amedium within propagate to thought It rather was space. free in wave propagate EM can the that yet it not was understood time, At that Although the law the was ofAlthough dimension-matching Rayleigh derived from scattering expression 3.54Hence, 3.55 Equations and mathematical for the constitute the Taking the square of both sides of of3.54 both Equation gives square ofTaking ratio intensity the an

V c is the volume is the particle, of the is the velocity is the wave, EM of the and S there is no other term that contains the dimensionof mass; the therefore, the contains that term is no other there C A A atte s i . Similarly, density ether the ρ R in e should removed. leaves be This r G , the wavelength squared is needed in the denominator, yielding the in wavelength, the is needed squared

a S

d ipole c R has a dimension ofT has [L adiation I I A A s i r s i is the range from the particle, particle, the from range is the ∝ ∝ λ V λ ρ 42 V 2 r 2 e r contains the dimension of [M]. the mass contains ρ . e .

is the density of the ether medium. density ether of is the the V , r , and , and Figure 3.11

−1 V ], but no other term con ], but term no other and is inversely and propor λ on the right-hand side on the c , , when awave is c is not acom λ is the wave is the (3.54) (3.55) 65 ------*

Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 and 66 205): 1941, particle. excitation only due the to which particle inside exists charges the inside of the the 1978, 1997) as (Ishimaru radiation. and dipole vibration (b) for and process sketched (a) scattering configuration 3.11 FIGURE Green’s space free is function where is source equivalent the current and The internal field can be represented by an incidentan fieldrepresented by be wave can internal fieldThe as follows (Stratton wave scattered Therefore, the the potential field represented by be vector can r is the observation and point is the

Illustration of wave scattering from a small sphere as a dipole radiation; radiation; adipole as sphere asmall from of wave scattering Illustration Gr t () =0

E Jr

i ,e

Er eq rj AG ′ s () () =µ =− ′ = =− 0e xp

k

ˆ EE ∫

i

in j r V () t ωµ t jr = ′ ωε

= 1 T is the source location inside the particle. The The particle. location inside the source is the () 00 kr /4 rr ε+ 0r ε (a) ,, r p   3 (b) ε − ∇×∇× ′′ 2 Jr rr () ′ q χ ′ () i t .

= − () 4 T 1, π− Ar /2 Θ dr   () E E ′

s in Weather Radar Polarimetry Weather Radar

t

r

′ t , =3

T

k

ˆ

/4 s

(3.56) (3.57) (3.60) (3.58) (3.59) A

Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 sin as section cross ing wavefield,scatter scattered the powerdifferential the represented by distribution is of waves. EM Wave Scattering byaSingle Particle Wave Scattering on the particle’son the physical property, volume depends amplitude scattering the sphere. expected, of As on incidence asmall wave scattered of the amplitude fieldunit which the the wave represents with projected onto the scattered wave of scattered onto the projected polarization unit waveunit of using far-field incidence, and ( approximation as amplitude scattering the with direction direction over 4 integral the is the section cross 3.63) scattering (Equation section total cross the known, are 1993). et al. (Wurman polarization receiver (bistatic network) BINET like vertical uses but radar abistatic polarization, horizontal use to tends (H-plane). monostatic is why radar asingle This polarization fieldmagnetic the in field is isotropic (E-plane),plane electric plane scattering the Figure 3.12 the incident wavethe field scattered wave propagation direction of wave propagation direction scattered tering direction and incident direction incident direction and direction tering Whereas the scattering amplitude represents the property of the scattered wave scattered of the property the represents amplitude scattering the Whereas Substituting Equations 3.58Substituting Equations 3.59 and 3.56 Equations 3.57, into and a assuming Once the scattering amplitude (Equation 3.62) and the differential scattering 3.62) (Equation scattering amplitude differential the and scattering the Once in are plotted power patterns andscattering field patterns scattering The χ term can be understood as the unit vector unit of incident the as wave understood be can polarization term k ˆ σ= s . Whereas the scattering pattern appears in the sin the in appears pattern scattering the . Whereas and the incident wave the and polarization sd = ∫ ka 4 46 π π steradians of asolid angle, yielding steradians σΩ sk ε+ ε− ( () r r kk ˆ ˆ si σ= e si ˆ , , i ds 1 2 . It is noted that . It that is noted k ˆ ˆ () 2 ) kk ˆ d ∫ = = Er , 0 π s0 ˆ ka 4 k () i ∫ 23 π 2 0 2 π 3( = si () ε+ ε+ ε− ε− ka e r ns r r r 23 − 2 r jk χχ () 2 1 2 r 1) kk ˆ k ˆ Es si 2 si in s Vk , , demonstrating the transverse nature nature transverse the , demonstrating χ n ε+ ε− ˆ V is the angle between the scattering scattering the between angle is the   r χ r , electric property property , electric () dd −× e , and the polarization direction of direction polarization the , and kk ˆ ˆ s χφ si ˆˆ 1 2 , ss , e ˆ e ˆ i ˆ 2 , as shown, as in s , which is perpendicular to the the to , which is perpendicular si () =

ke n 2 8 × χ πε r . ka ˆ 3

>> >> i 46  

r χ ′ ε+ distribution in the the in distribution ), we have we ), r r ε Figure 3.11 − r , both the scat the , both 1 2 2 .

(3.64) (3.63) (3.62) (3.61) . The . The 67 e ˆ - - i

Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 is proportional linearly to the volume the to ( linearly is proportional and the right column is the plane perpendicular to the incident wave polarization (H-plane). wave incident the to polarization perpendicular plane the is column right the and (E-plane), wave incident the polarization containing plane row). the in is column left The 3.12 FIGURE 68 to the volume squared ( volume the to squared (Equations 3.63 3.66), (Equations apply derived through formulas electrically the only the to 3.62) (Equation amplitude sections cross derivations and scattering of the the in stant waveconbe to Rayleigh internal assumed field the Because been has scattering. equator. atthe occur direction backscattering the and maximum as 4 is defined section cross 3.64). radar the (Equation section cross is because This scattering total Whereas the scattering cross sections (Equations 3.64 and 3.65) are proportional 3.64 3.65) and proportional (Equations sections cross are scattering the Whereas is then section cross radar backscattering The Following 3.49 Equations is section cross 3.50, absorption and the 3.65) the (Equation section cross than is larger radar backscattering the Note that 180 180 150 210 150 210

π Normalized scattering field patterns (top row) and power pattern (bottom (bottom row) (top fieldpowerpatterns and pattern scattering Normalized 120 240 120 times the differential scattering cross section, for which both the for section, cross the which both scattering differential the times 240 σ= a E- σ= 270 plane 270 90 90 ∫ ∼ bd 0.5 a 0.5 1 2 1 6 1 =( ωε 4 πσ 300 300 0 60 60 D ε ′′ /2) S () i Er −= 6 330 330 30 30 () kk ), the absorption cross section (Equation 3.66) (Equation section cross ), absorption the ˆ ii 0 0 ′′ , ˆ 2 a dr 3 =( 4 18 =π 18 π D 210 150 00 210 150 ka 4 3 00 /2) 46 ak 3 ) of the particle, in the case of case the in particle, ) of the 240 120 120 3 240 ε+ ε− r ε r ′′ Weather Radar Polarimetry Weather Radar 1 2 ε+ H- r 2 270 270 3 plane 90 90 .

0.5 0.5 2 1 1 2 .

300 300 60 60 330 330 30 30 (3.66) (3.65) - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Section 3.4 in discussed theory Mie the with more clearly seen comparison be in can tering inside the sphere is inside the

Wave Scattering byaSingle Particle Wave Scattering netic fields must be continuous. Let us use a spherical coordinate system ( netic fieldscoordinate Let a ususe continuous.be mustspherical mag and components electric of the tangential the that Maxwell require equations small scatterers/particles ( scatterers/particles small boundary problem. That is, under the wave problem. the is, under That incidence of boundary is to solve for representation EM scattering Mie the mathematical of the purpose The 3.4.2 of Rayleigh that than complex scattering. pattern field henceits radiation scatterer, and a more has the in resonating are that modes well in (a ahexapole illustrated and dipole, are aquadrupole, hexapole, dipolenot quadrupole, onlymode, buthigher-order the pole the and as modes allows for wave the theory Mie and the Byinsidethe sphere contrast, fieldfor to vary it adipole if were from antenna. as radiation wave is treated the and constant scattering Rayleigh of approximation. the regime valid of the scattering determination 1983; Kerker 1969). allow for also calculation a scattering Mie of results the The have Huffman calculations multipleand well in been (Bohren documented textbooks these constant; dielectric of sphere of wave and any size by auniform scattering the is called and Mie exact solutionThe for wave by asphere was developed scattering 1908 in by Gustav 3.4.1 3.4 dent wavedent dipole field as (b), (a),hexapoleand quadrupole (c)radiation. FIGURE 3.13 wave field We origin. the sphereas have center the In general, the wave order-pole the by general, superposition of a sphere is In all the scattering wave internal fieldbe the Rayleighto approximation, assumed the is In scattering

MIE SCATTERING THEORY SCATTERING MIE C M on E . at i and the scattered wave scattered the field and

H C eptual Conceptual sketch of the electric field inside a scatterer excited by the inci the excited by fieldscatterer inside a electric of sketch the Conceptual e M ati E C Mie theory Mie

in al (a) d – + t , and the wave the , and incident fieldthe of outsidethe summation is

e e SCR xp ka rH ˆˆ rE ˆˆ R ×= iption << 1, i.e., a << ×= e SS . The Mie theory allows for the accurate calculation calculation allows for accurate the theory Mie . The in ion in tr tr =

= ai and ai rH + rE – E S ×+ ×+ s , specifically , specifically a     (b M λ )( ). The valid regime for). regime valid Rayleigh The scat ple + R – H E s s e   S   ult ra ra = = EE

. S

is E + Figure 3.13Figure i , the internal wave internal field, the – + . The first and second andsecond first . The + – c) ). + – r , θ , ϕ ) with ) with (3.67) (3.68) 69 - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 respectively. respectively.

with with field, and ( and field, dent, scattered, and internal wave internal fieldsas are and expressed dent, scattered, component the formin 70 and where wave scattered the fields Besselare Then, function. spherical the coefficients: Equationsfields from 3.67 and yieldsandequations 3.68 solvingscattering the the respectively. sions in where where Because the wave fields can be expanded in vector spherical harmonics, the inci the wave the Because harmonics, in fieldsbe spherical vector expanded can 3.67 of Equations 3.68 part Each and it two if is expressed represents equations in Using 3.69 3.71 Equations through derived expressions their and for magnetic P nn 1 C ψ= () Appendix 3B Appendix n co nn =( () xx a sc θ= n − , M j b ) n s mn n s E πθ jx ) and ( ) and  ⊥⊥ d and and d Er 0 n () () θ (2 θ= Er Er () () in     θ= co P s t i n b a () () () ), and the subscripts subscripts ), the and =π E +1)/[ n E n s () θ s N c s − = = ⊥  and and n =− = =θ 12 os mn jk 12 , jk mk ψ = mk ζ ∑ ∑ ∑ xJ     d are the vector spherical harmonics (see expres their harmonics vector spherical the are n n n θ ∑ ψ ∞ ∞ ∞ ∑ nn nn n = = = n n ζ = /2 n P ) are those for the scattered and internal wave internal fields, and for scattered those the ) are ∞ 1 1 1 () () = ∞ ( ϕ = and and nn nn n ka ka 1 n 1 1 Cb Cc CM e () directions, totaling aset of four equations. totaling directions, () si () +1)] expansion the coefficientsincident are the for − nn nn nn nn n co r n n ak ak jk (1 ψ− ψ− + (1       n r θ 1/ n P s ψ− ψ− ′ ′ 2 n     + + θ + + () () () Mk () ′ o1 ′ (1 km km 1 x 1 co () () s ) Mk nn ) ) an o1 (1 ⊥⊥ ( is the Riccati–Bessel function, and and function, Riccati–Bessel is the [] ma ma [] am am s ) n kr o1 0( ab (4 ab dc θ () (, nn n ) e nn θ ,,) τθ πθ ( τθ and and as the the as ′ nn rd rj () φ+ () () θφ ψ ψ ψ ψ ,,) co co s θφ os ,)  nn nn nn nn o 0 sc sc () () () () θ km km km km indicate the even the modes, odd indicate and jN n + ) −θ th order Legendre polynomial. Legendre order th = +π e1 +τ ak (1 ak ak ak     n ) aN d     ζ Nk ζ ψ ψ d ( Weather Radar Polarimetry Weather Radar n nn θ nn kr ′ ′ e1 (1 ′ ′ E () () E P () () ) n ,,) e1 i (4 i θφ () 1 ⊥ ( a a  () a n a ) () ( co os ′ os r kr     ,

,,)

,   s θ θφ θ ,,)

θ ,

φ ,

 

,  

j n (3.69) (3.77) (3.76) (3.75) (3.70) (3.72) (3.73) (3.74) (3.71) ( x ) is - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Wave Scattering byaSingle Particle Wave Scattering stronger in the forward direction than the pattern in the backward direction (the pat direction backward the in pattern the than direction forward the stronger in Figure 3.14 (top of 2mm water row) diameters spheres with row) (bottom 2cm and shown in are for a sphere with a 2-mm diameter and (b) bottom row for a sphere with a 2-cm diameter. row (b) a2-cm for with and bottom asphere diameter a2-mm for with asphere FIGURE 3.14 follows: 3.47,Equations 3.48, 3.51, and we as have for sections cross scattering Mie the 2cm. to 2mm from size in sphereby whenincreases orders the many increases section cross scattering total the is because This direction. forward the in is energy scattered main sphere,the although by 2-mm the backscattering the than two larger orders sphere is still by 2-cm the backscattering the that Note also tering. of backscat that than larger order is of and an is dominant pattern scattering forward invalid.) become to of isthe evident, As 2cm, forstarts adiameter sphere with the Rayleigh approximation the 2mm, scattering than larger hydrometeors of diameters of Rayleigh shown in patterns those to scattering somewhat similar is ~10% magnitude but is scattering), backward tern and forward between different The sample calculation results for the amplitude patterns at the X-band at the of two patterns forsample results calculation amplitude The the The radar backscattering cross section is section cross backscattering radar The 3.75 (Equations amplitude Using scattering 3.76) the and expressions the in in D D =2 cm =2 mm . It is apparent that the pattern for the sphere with a diameter of 2 mm is of 2 mm for sphereadiameter with the pattern the that . It is apparent

Sample amplitude patterns for water sphere scattering at X-band: at (a) scattering for sphere Top water patterns row amplitude Sample 180 180 150 210 150 210 120 240 240 120 σ= b E- k 270 270 π plane 90 90 2 5 0.005 0.01 10 ∑ 15 n ∞ = 1 300 (2 60 300 60 na +− 330 30 330 30 1) 0 0 (1 (b) (a) ) n 180 180 () 210 150 210 150 nn − 240 240 120 120 b 2 H- ,

270 270 plane 90 90 5 0.01 0.005 10 15 300 300 Figure 3.12 60 60 330 30 30 330 0 0 (3.78) . (For . (For 71 - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 in free space is small. space free in awave and awave between particle difference the phase the passing through that cally cally for specifi valid, Rayleigh be requirement to is amore stringent there scattering sizes, atsmaller differ to extinction start and of absorption However,constant. as a wavetreated the be can results fieldthe insidethe particle extinction is section cross the and 72

FIGURE 3.15 sphere whose diameter sphere whose diameter for is Rayleigh valid approximation a means That scattering of 2mm. a diameter well toup agree efficiencytheory thoseMie and of Rayleighfactors of scattering λ in compared approaches and two using aforementioned the calculated are section, cross by geometric the ized 3.78 (Equations 3.80).theory through efficiency The cross sections normal factors, rigorous given 3.66 calculation given the 3.64 and by Mie Equations through in the = 3 cm) with a dielectric constant of (44 constant adielectric with − = 3 cm) The total scattering cross section is section cross scattering total The The calculations are performed for water spheres atX-band performed (wavelength: are calculations The Now, we Rayleigh approximation have the both from sections scattering cross D < λ /50. value is close This to

Q factors 10 10 10 Dependence of efficiency factors on water sphere diameter at X-band. diameter sphere on of efficiencywater factors Dependence 10 10 –3 –2 –1 0 1 σ= t 4 k D π σ= < Im s 10 λ Figure 3.15    0 /16, which is equivalent 2 to 2 k s π 2 (0 jk ∑ )2 n ∞ =    1 = k (2 ( Diameter, mm m k na . π 2 ′ ++ − 1) − ∑ 1) n ∞ = 1 () D (2 j 43). It is scattering evident the that nn <π/4 (with na 10 22 ++ 1 1) b Re Weather Radar Polarimetry Weather Radar []

Q Q Q Q Q Q Q Q nn kD m a t b s a t b s , Rayleigh , Mie , Mie , Rayleigh , Rayleigh , Mie , Mie , Rayleigh D ′ b < π/4, meaning that that <π/4, meaning = 7), which requires requires which 7), = < 1 mm. Typically, <1mm.

10 2 (3.80) (3.79) - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 D result shown in by a single nonspherical particle is fundamental to understanding and interpreting the the interpreting and understanding to is fundamental by particle asingle nonspherical scattering Wave measurements. radar polarimetric from for information additional allow and returns give they that radar different meaning not spherical, are meteors waves. polarized However, vertically and for same horizontally the are most hydro sections cross radar the and amplitudes backscattering the that meaning polarization, by ahomogenous earlier, backscattering in discussed sphere exhibitsAs no difference 3.5 condition ofthe creeping andwave reflection in model of awavelength. of number ahalf odd is an difference Using the if the minimum integer wavelength, of is an the difference path the but a structively meaning add, direction. opposite comes the waveoutsecond then in half-sphere and the propagatesaround back edge.the In from reflected is then sphere and the into penetrates other edge, front the the from reflectionmodel. model The is shownin models: (a)conceptual (b) reflectionand creepingmodel the and reflection the wave waves of paths. scattered different summation going through intuitive the explanation section, cross destructive isand constructive scattering wave full the theory. and intuition our between is adifference Hence, there account. into optics not does take geometric that wave causediffraction propagation can and alter anyhow field,and far scatterer, matter no can that edgebig the an in it is,has (ii) extinction is defined section and cross the of scattering is considered part direction resolved be removed can tion we forward paradox (i) if the that note from anything the result is called unexpected which This section, is counterintuitive. cross geometric the as large as is twice section ( tion and creeping wave creeping model. and tion FIGURE 3.16 Wave Scattering byaSingle Particle Wave Scattering D = 1.17 which cm, is close of ~1.0 that to figure. shown cm, the in > 100 > In either case, we can expect a maximum if the two waves con the and if phase in eitherare we case, expect amaximum In can back normalized down effect up forof the and As changes resonance the in the wavelength end extreme, of the other when the At asphere the is much than larger

SCATTERING CALCULATIONS FOR ANONSPHERICAL PARTICLE λ ), optics isHowever, a good approximation. geometric extinction cross the Path 2 Path 1

Conceptual sketches for resonance effect: (a) effect: (b) forand sketches resonance reflec model reflection Conceptual Figure 3.15 k ( ℓ Figure 3.16b 2

−

ℓ 1 ) = (a) for X-band an with k ( D extinction paradox , Path 1 is the same as that in in that as same 1is the , Path + π D /2) =2 Figure 3.16b Figure 3.16a π Path 1 Path 2 , results in in , results λ , as noted in in noted , as = 3 cm, we have the maximum at we have =3 cm, maximum the , the first maximum that that satisfies maximum first , the : one wave back is reflected D (b) = 0.39 = Section 3.2 Figure 3.16 Figure 3.16a λ . In the case of the of case the the . In . The extinc . The shows two , but the , but the 73 - - - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 the ellipse rotates around its major axis, we its major have axis, around rotates ellipse aprolate spheroidthe ( approximation, T-matrix method, or other numerical approach, depending on its elec depending approach, numerical or other T-matrixapproximation, method, Rayleigh using the calculated be scattering can amplitudes scattering the determined, wave-mine two key the deter factors that are orientation Shape and polarimetric measurements. 74 late spheroid. late FIGURE 3.17 costs) for ( large solved numerically solved and (with scatterers for relatively small low computation problem analytically be can is one basic of for shapes the scattering EM which the spheroid model the to it hydrometeors is used that is because reason Another used. spheroid the is one model reason is often This data. radar polarimetric interpreting model, longshape sufficient sphere, as it is usually as on aperfect is notfor based hydrometeors. about the Hence, asimple information spheroidal structural detailed hydrometeors of without the requiring properties statistical main the capture can measurements These measurements. only afewof radar all, types after are, there because polarimetry radar in Such is rigorousunnecessary modeling parameters. with a of shape fewand snowflakeshailstones rigorouslyto irregular model the plate, cylinder, disk, or acombination shapes. Itneedle, of is difficult dendrite, these the shape of a in be asnowflake in can ice crystal an Furthermore, hailstones. and shape for snowflakes irregular an to base raindrops, for large aflattened with shape oblate spheroidal an to drops, for shape rain clouda spherical small and droplets in described As 3.5.1 complexity the and of size its shape. tric dles and columns of columns ice crystals. and dles representnee to used prolate spheroids be whereas can dendrites, plates and crystal in illustrated are ellipse rotates around its minor axis, we obtain an oblate spheroid an we ( obtain axis, its minor around rotates ellipse its principal around ellipse an by rotating obtained be which can Oblate spheroids are used to model raindrops, snowflakes, hailstones, and ice snowflakes,hailstones, model to raindrops, Oblate used spheroids are A spheroid is a special case of an ellipsoid; its two semidiameters are equal, equal, are ellipsoid; of case A spheroidan its two semidiameters is aspecial b a scattering characteristics. Once the shape and orientation of a scatterer are are of orientation ascatterer and shape the Once characteristics. scattering S x i

C

Basic shape models for hydrometeor scattering: (a) scattering: for spheroid; (b) models oblate Basic hydrometeor shape pro n D Figure 3.17 on Chapter 2, the 2, Chapter

∼

S λ a p ) scatterers. 1 H z e b R (a) i C al a . 2 S re are a variety of for shapes hydrometeors: avariety are re from H ape : S y p H e R x oid a 1 z Weather Radar Polarimetry Weather Radar b (b) a 2 a z a -axis. When the the When -axis. 1 1 = = y a a 2 2 > < b b ); when ). Both - - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Hulst 1957): following the we (Van obtain axis, De component on its symmetry polarization by following the (Stratton 1941): However,orientation. wave field polarized the be eachcomponent for expressed can particle’s incident the the becauseficult and on find depends it to wave polarization wave field (Equation field3.60), anellipsoidinternal the for (spheroid)dif is more Section 3.3.2 of representation Rayleigh by asphere was provided mathematical in scattering The 3.5.2 factors of shape follow the that equality an shown be scatterer. of It shape (Stratton the can on the depend 1941) factors that are where field is internal where the as rewritten be can Wave Scattering byaSingle Particle Wave Scattering

Combining Equations 3.82 Equations 3.83Combining field and solving and foreach internal for the Although the internal fieldincident the easilyrepresented beby internal for a can sphere the Although E

in tx R aylei = 1( . The expression of scattering amplitude provided there (Equation 3.62) (Equation provided expression there amplitude . The of scattering +ε GH L L L E xr E L sk S x y p z () ix C = = ˆ = = si atte − , ∫ ∫ ∫ ∫ k ˆ 1) 0 0 S 0 ∞ ∞ ∞ 4 R ; 2 2 2 = ≈ πε E in () () () dq sa sa in 4 sb kV G 0 t + + k πε 2 + r y

a 2 2 = 0 (1 4 1 () 2 2 2 pp 2 ε− L −= 1( π ∫ r EE x R rp     V +ε ˆˆ   + () () oxi in () {} i sa sa L ti sa −× L ) +++ +++ =+ yr E +++ kk M y ˆˆ   + iy ss −× ation 1 1 aa aa 2 2 1 aa 2 kk 12 12 − ˆˆ L 12 − ss   () () () z ε 1) sa sa =1. sa L E b b 0 b () p

;a × PL . fo

Pr nd 2 2

22 22 =− 2 22 × R () S () () () E E sb sb sb ′ p in in H (1 t   tz ε− e   ri ed R = , oid jk s 1(       i 1/ 1/ 1/ r ) +ε ′ 2 2 E S 2 L ds ds r ds nt zr E ′ . , ,

iz

− 1) ,

(3.87) (3.86) (3.85) (3.84) (3.88) (3.82) (3.83) (3.81) 75 - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 with polarization on the major axis ( major on the axis polarization with amplitude scattering the As figure. shownexpected, the also in of are afew raindrops in plotted are axes major minor on the and izations Equations 3.81Equations 3.83 letting through and by 3.84 substituting Equation into obtained are axes major minor atthe and aligned the Consider awave calculated. be can amplitude incident on aspheroid in scattering the for factors shape ( aprolate spheroid. the Once the minor axis ( axis minor the for oblate spheroid an ( 3.88 and performing the integral of3.87 Equation ( with integral the 3.88 performing and

differ from the unity too much too (0.5 unity the < from differ rationot does axis the is sufficientlythat is, spheroidal, scatterer the dimension. If 76

and with size increases. The Rayleigh scattering approximation for Rayleigh approximation The spheroids for is valid increases. size scattering with more oblate become raindrops as increases two polarizations the between amplitudes scattering the in dipole difference alarger moment The because is formed. scattering more accurate calculation than is provided by Rayleigh than approximations. calculation more accurate scattering require precipitation particles X-band wave all frequencies, from or higher scattering At and C-band amplitudes. for calculations scattering more accurate which require but snow is not applicable S-band, atthe and melting dry for and clouds, hail snow, rain, forward forward equal. are amplitudes scattering forward and backward tion. The different with with different sin term the with same the are patterns amplitude of3.62 Equation amplitude for scattering normalized asphere. the to The In the case of spheroids, the shape factors can be calculated by using Equation calculated of be case spheroids, factors shape can the the In In general, the shape factors are inversely proportional to the corresponding corresponding inversely the to factors shape are the general, proportional In Using the axis ratio of2.16 ratio Equation Using 3.90 axis the 3.91), (Equations and for raindrops the It is evident that the scattering amplitude of3.93 Equation amplitude forIt isscattering evident aspheroid the that is similar

x -direction: the scattering amplitudes in the scattering plane for plane wave scattering the in polarization amplitudes scattering the -direction: () kk ˆˆ si =Θ L 3[ ;0 s z 1( b : dashed line); the larger dimension of the scatterer causes stronger causes line); : dashed scatterer dimensionof larger the the = +ε L = 1 z L sk ε− + = a, gg a, r br bs 2 a or backward or backward g 1 () > 2 1 − ee ˆ    b 2 e , − 1 kk ) and ) 2 ˆ − 1) i    L ] 1 −+ = , instead of , instead x LL 1 : ar L xy ctan 22 y == : ab 2 () L 1 s kk ˆˆ z b a si

L : solid line) is larger than for polarization on on for: solid line) polarization than is larger ln / ≈ gg

χ a 3[ =− y    1/ = as shown as in <2), relation exists approximate an 1 1 1( an 1 2 − + a +ε L 1 () ε+ ε− e e d1 L a :1/ 1 L ; ε− r and and r ) and the dielectric constant are known, known, are constant dielectric the ) and    Θ= a, r − a br an 2 L 2 1 2 :1/ 1 = z , which depends on the polariza on, which the depends L d1 π Figure 3.18    z b

− e = scattering amplitudes for amplitudes polar scattering . b a 2

1) Figure 3.11    L =− ] Weather Radar Polarimetry Weather Radar 2 b a si as follows: as 1 −= n = χ    a e ˆ a b 2 s . The effective. The shapes . = γ   

1 2 2 . The magnitude is magnitude . The a

), giving ), − 1

(3.90) (3.89) (3.92) (3.93) (3.91) - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 particle and the internal wave internal field the and particle also known. That leaves That known. also four of sets expansion coefficients,includingscattering the Under wave the incidence the in following. which is brieflydescribed transitionmatrix, expansion a the through coefficients conditions determine to ary and (ii) to use bound extended harmonics their spherical vector of fieldsterms in wave internal and is (i) T-matrixof method the incident, scattered, expand to the 1975; 1976; Seliga Bringi and 1991; et al. Vivekanandan Waterman 1965). idea The Yeh and (Barber meteorology community developed radar widely the and in used successfully been has that method is anumerical T-matrix method The scattering. wave calculate for numerically to aneed arigorous is method thus not exist. There solution does analytical above an and invalid, introduced becomes approximation ahydrometeor’sWhen its to is comparable size wavelength, Rayleigh the scattering 3.5.3 polarization at major at ( polarization FIGURE 3.18 Wave Scattering byaSingle Particle Wave Scattering harmonics vector in expanded spherical

Wave scattering by an irregularly shaped particle is illustrated in in is illustrated particle shaped Wave irregularly by an scattering Because the incident wave the Because expansion the is known, coefficients ( t -M

at

Scattering amplitudes of raindrops as a function of equivolume diameter for of equivolume diameter afunction as of raindrops amplitudes Scattering Scattering amplitudes |s(0/π)|, mm 10 10 10 10 10 10 R 10 ix Er –6 –5 –4 –3 –2 –1 Er Er 0 0 i M in s () s () t a () ) and minor ( minor ) and et = =θ H =θ ∑ 1234567 mn od ∑ mn ∑ , Er mn , , i   ()   eM   aM Rayleigh scatteringapproximation mn cM mn , there is the scattered wave scattered is the field, there mn s b ) axes, respectively.) axes, (1 mn Er mn (4 )( mn (1 in ( )( )( kr t (, (, () kr ′ kr ,,) inside the particle. These wave These particle. inside the fieldsare θφ D , mm ,) ,) φ+ φ+ + bN fN dN mn mn mn mn mn 1) mn 4) 1) (, (, (, kr kr kr ′ θφ θφ θφ ,) ,) ,)      

.

Er

s s a b s () 8 e iue 3.19 Figure outside the mn , f mn (3.96) (3.95) (3.94) ) are ) are 77 - . Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 where Substitution 3.95 of harmonics. Equations spherical 3.96 and 3.97 Equation into yields waveas field,written which is wave scattered sphere, the field inner incidentscatterer.the the Inside out cancels the outer sphere circumscribing the sphere and inner the lines: dashed in drawn are in Huygens illustrated and As used. theorem are principle by conditions Waterman (1965, was introduced boundary 1969), extinction the and wave of dependence the angular fields. at conditions theory,Mie to butsolve is more difficult apply continuity the because we cannot problem the to set up by problem the conditions. is similar This boundary using the 78 coefficients ( conditions. boundary extended FIGURE 3.19 have we wave field.the FromHuygens principle, havewe field. nal expansionincident the the coefficientsbetween wavewhich links inter the field and To address the difficulty in solving the boundary problem, conceptthe of extended in solvingboundary the To difficulty the address Solving ( Solving Using 3.94 Equations 3.95 and 3.99, Equation in we obtain Outside the scatter, by contrast, the scattered wave scattered scatter, the byOutside fieldthe contrast, the surface is caused by Er ii () Gr Er si () () +ω , ∫ r c

′ S =ω mn a r da = constant as we could in the case of case asphere, we as which the complicates couldthe in =constant is the dyadic is Green’s the by vector the represented be can that function mn Illustration of wave scattering by a nonspherical scatterer and the concept of concept the and scatterer by anonspherical of wave scattering Illustration , ∫ d S ,   mn da in b ) from Equation 3.98 Equation 3.100,) from Equation into substituting them and mn µ×       ) and the internal coefficients ( internal the ) and in ˆ a b µ× mn mn Hr ˆ     nt E Hr i =

()         [] nt ′′ BA e a b f () i mn mn mn mn [] Gr ′′ () i         − Gr = 1 = , rn     () [] [] E A B int e , S f +× rn mn mn         ˆ +× d c d c     mn mn mn mn Er ˆ = in [] t T         Er () , .

in     ′ c t Weather Radar Polarimetry Weather Radar () mn i e f E ∇× mn mn , ′ Figure 3.19 s

d i mn ∇×     Gr , ), to be determined ), determined be to

() Gr ,0 r () ′ ,   r , two spheres = ′   . ,

(3.100) (3.101) (3.97) (3.99) (3.98) - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 magnitude (left column) (left (right phase and column)magnitude for wave by spheroid scattering be found. can amplitude/matrix wave scattered the known, fieldscattering the and and this method is known as the the as is known method this and wavetered field coefficientsthe incident the to of wave the called field. It is size at S-band (a), S-band at size (b), C-band X-band (c) and frequencies. FIGURE 3.20 where [ Wave Scattering byaSingle Particle Wave Scattering iue 3.20 Figure s(π) magnitude, mm s(π) magnitude, mm s(π) magnitude, mm 10 10 10 10 10 10 T 10 10 10 –4 –2 –4 –2 ] =[ –4 –2 0 0 0 02468 02468 02468 S-band C-band X-band

B shows the T-matrix calculations for backscattering amplitudes in in shows amplitudes T-matrix for calculations the backscattering ][ Magnitudes and phases of scattering amplitudes as a function of raindrop of raindrop a function as amplitudes of scattering phases and Magnitudes A ] −1 is the transition matrix that relates the coefficientsscat the relates the of that matrix transition is the D D D , mm , mm , mm s s s s s s s s s s s s a b a a b b a a b a b b : T-matrix : T-matrix : T-matrix : Rayleigh : Rayleigh : Rayleigh : T-matrix : T-matrix : T-matrix : Rayleigh : Rayleigh : Rayleigh T-matrix method (b) (a) (c) –40 –30 –20 –10 –30 –20 s(π)–10 phase, degree s(π) phase, degree –30 –20 s(π)–10 phase, degree 10 10 20 30 10 20 30 40 40 0 0 0 . Once the coefficients ( the . Once 02468 02468 02468 D D D , mm , mm , mm a mn T-matrix , b mn ) are ) are 79 - , Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 amplitudes (d) as a function of raindrop size at S-band, C-band, and X-band frequencies. and C-band, S-band, at size (d) of raindrop afunction amplitudes as scattering forward of the (c), part differences real phase the and (b),ratios backscattering 3.21 FIGURE well with their corresponding Rayleigh approximation results until the drop diam Rayleigh drop the corresponding until results well approximation their with agree T-matrix-calculated magnitudes the for At calculation. used S-band, the of 10°C was Rayleigh Atemperature approximation. the with scattering obtained amplitudes backscattering the with compared as frequencies, atdifferent raindrops 80 normalized backscattering cross section (upper section cross ratios left), magnitude backscattering backscattering normalized own plots. their pare com T-matrix download and results the can readers interested properties; similar have amplitudes scattering forward The invalid even raindrops. for median-sized even at an ( sizes smaller appear wavelength phase the as even scattering effect becomes and resonance shorter, the At amplitudes. X-band, for phase scattering the substantial effect and resonance the ( size tric elec- the because of approximately 3mm, atdiameters differ to start approximation however,C-band, Rayleigh the T-matrix and calculations the between results the At small. very are amplitudes scattering of phases the The 6mm. to increases eter Figure 3.21

2 4 4 2 6 δ = δa–δb, degree |sa| ×(4λ )/(π |K| ), mm 10 –10 10 10 10 10 10 10 20 30 –2 0 0 2 4 6 8 ka 02 02 ) at C-band is double that at S-band. The T-matrix show calculations is The double) atC-band atS-band. that

shows a summary of the results given results of the in shows asummary Normalized backscattering cross section (a), backscattering magnitude (a), magnitude section cross backscattering backscattering Normalized D D , mm , mm (c) (a) 46 46 D X-band C-band S-band ~ 2 mm); Rayleigh becomes then approximation 8 80

2 Re[sa(0)–sb(0)], mm (|sa|/|sb|) ,dB 0.1 0.2 0.3 0.4 0.5 0.6 0 0 2 4 6 8 0246 Weather Radar Polarimetry Weather Radar Figure 3.20 24 D D , mm , mm (d) (b) by plotting the by the plotting 6 8 8 - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Wave Scattering byaSingle Particle Wave Scattering ferential reflectivity,respectively, There distributions. drop for size monodispersion dif and reflectivity the factor are ratio magnitude and section cross backscattering right). (bottom amplitudes scattering forward of the left), part (upper (bottom real differences the right), phase and backscattering The cases for both dry and wet snow/hail are calculated. For wet calculated. wet snow/hail, 20% and snow/hail are for dry cases both The ( decorrelation signal causes factor that is amain difference phase The increases. frequency and/or size drop as increases difference phase scattering the general, In effect.ratio, which by resonance is caused the magnitude C-band the in is a peak as a function of particle size at S-band for snow row). S-band at (top size row) (bottom of particle hail afunction and as FIGURE 3.22 flakes is assumed to be to 0.1assumed is g/cmflakes density: density is the of modeling the snow snow the between hail and difference of ratio oblate be to axis spheroids an assumed with are hailstones shown in are results the and T-matrix method, the Chapter 4. tudes is associated with the specific differential phase ( phase differential specific the with is associated tudes ρ hv The scattering amplitudes of snowflakes and hailstones were also calculated using also calculated ofand snowflakeshailstones were amplitudes scattering The ) between the dual-polarizations. The real part of the forward scattering ampli scattering forward of the part real The dual-polarizations. the ) between 10 10 10 10 s(π) magnitude, mm 10 10 s(π) magnitude,10 10 mm 10 10 10 10 –3 –2 –1 –0 –4 –3 –2 –1 1 2 0 1 01 02

Magnitudes (left column) and ratios (right column) of scattering amplitudes amplitudes (right column) ratios of scattering column) and (left Magnitudes 04 02 D D , mm , mm (a) (c) 06 03 3 s s s s s s s s , whereas hailstones have hailstones , whereas adensity of 0.92 g/cm a a b b a a b b : Wet : Wet : Dry : Dry : Wet : Wet : Dry : Dry 00 00

–10 2 2 –5 10 (|sa |/|sb |) , dB –4 –2 (|sa |/|sb |) , dB Chapter 4 s Chapter 4 0 5 0 2 4 Figure 3.22 Wet Wet Dry Dry K 20 DP 10 hows that the normalized hows normalized the that ). These are described in in described ). are These D D . Both snowflakes and snowflakes . Both , mm , mm (d) (b) 40 20 γ = 0.75. = The 60 30 81 3 - - - . Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 water as background (seewater background as Problem 3.4). in As formulawith Maxwell-Garnett the from effective of an to water–ice mixture particle At lowparticle. water-coated shown itbe the ice isfrequencies, equivalent that can a water-coated as ice (two-layer) treated be melting can A melting is used. hailstone 82 can be represented by either local (body) polarization reference by (body) either represented local be polarization can of angle an with plane polarization the in cants ascatterer When 3.6.1 amplitudes scattering The 3.6 biological 1998; objects (Zhang 1996). et al. Zhang vegetation, and objects such terrain, shaped as waveculate by irregularly scattering 1991; Pennypacker and 1973). cal to used be Purcell can methods numerical The 1968), et al. dipole (Goodman approximation of discrete moment and (Harrington include calculations.These physicalWolf and optics (Born scattering 1999), method have developed methods been also Besides numerical for other T-matrix methods, 3.5.4 X-bands), even are greater. differences these (C- frequencies At higher axis. and minor on the that than is smaller major axis that on the for polarization which 2cm, couldthe yield backscattering than greater adiameter with for effect pronounced is very hail resonance The cases. dry and wet the between difference but asmaller two polarizations, the between difference forlarger snow. wet for snow that dry than is much two polarizations the between ratio decibels. the It in that is clear squared magnitudes amplitude scattering of ratio the the contains column ference, right the snow. for dry small is very axes major minor and To dif show polarization better the the in snow. wet polarizations between dry the nitudes between and difference The mag the in of order difference an is almost cases: there wet dry the between and amplitude is evident scattering the difference in axes. Asubstantial major minor and the with aligned polarizations with amplitudes backscattering of the magnitudes the below. simple derived the approaches discussed with be can particle by acanted scattering cases, waveIn special condition configurations. for such scattering boundary EM the solved problem be to the general, needs by In needed. are orientation for arbitrary amplitudes scattering Hence, yielding orientations. the random fall, they as tumble mostly horizontally, hailstones aligned major axis their with fall raindrops Whereas vertical). and (typically horizontal directions base polarization radar the with align don’t axes major minor and their ever, and orientations, random necessarily in occur of axes a spheroid major minor ( the and The scattering characteristics of hail are more complicated: there is a greater is a greater more complicated:there are of hail characteristics scattering The

o SCATTERING FOR ARBITRARY ORIENTATIONS ARBITRARY SCATTERING FOR S C t atte H e R R

n in u G M

f e o R i RM C al ulation s M a and and et H od s

b t HR represent wave scattering for polarizations in in represent wave for polarizations scattering S

fo ou Figure 3.23 R GH S C C atte Figure 3.20 oo R R in dinate ). Natural hydrometeors, how). Natural G C Weather Radar Polarimetry Weather Radar al

, the left column shows left column , the t C R ulation an φ S () fo hv , wave scattering ˆ ′′ , RM ˆ S ation (symmetry (symmetry - - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 minor axis, and and axis, minor reference system as is references of angle an axis. scatterer’sthe symmetry expressed reference, by be wave can the polarization local the with scattering Wave Scattering byaSingle Particle Wave Scattering where scatterer) reference of or global the (radar) axis polarization angle orientation an with plane FIGURE 3.23 is the scattering matrix with the canting angle taken into account. into taken angle canting the with matrix scattering is the Using 3.103 Equations 3.102, and we wave have scattered the fieldglobalthe in reference local global reference of is arotation the the Because system by reference is on polarization the because case this in is no cross-polarization There SR ER ss =ϕ =ϕ − () 1( φ

() , the relation wave, the of the twothe polarization fieldin represented () Coordinate systems for wave scattering by a scatterer canted in polarization polarization in canted by ascatterer for systems wave scattering Coordinate E hv ˆ SR , ′     ˆ =ϕ b)     is the radar polarization reference. polarization radar the is E E e E E sv sh ′ ′ −− () r ϕ= v h jk rj     v     ˆ RS = ´ =

()     e     −− φ r ss () jk ; a ss co si rj O () ab (b co hv nc ˆ     −ϕ ss ) ′′ v , E ˆ ϕϕ ss ϕ− ˆ s 2 i 0 ′ a ϕ+ =ϕ is the local polarization reference on its major and onits major and reference polarization local the is si e s nc 0 os b r in b kr     ϕ in RS os     () 2     ϕϕ E E φ ϕ−     iv ih ′ ′ E E (b ss ()     v h ′ ′ a )1 ss RE = ab si     − nc e =ϕ 2 () h ˆ r h ϕ≡ ˆ ´ RE kr

() +ϕ si SE () i hv nc ˆ (b , b ϕϕ ) ˆ ′ e . os . As shown. As earlier, i

′ os − r .

jk 2 r SE     i

,

(3.105) (3.104) (3.102) (3.103) 83 Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 as system body coordinate system. coordinate body system ( global coordinate scattering body coordinate system ( coordinate body scattering entation angle ( angle entation FIGURE 3.24 tion can be described through the projection of follows. the as adipole radiation through described be tion can orienta Rayleigh of case the the In wave approximation, for arbitrary an scattering 3.6.2 84 and the scatterer polarization polarization scatterer the and as written be can 3.81 amplitude 3.84, scattering the through As shownAs in According to the Rayleigh scattering approximation, as expressed in Equations Equations expressed as in Rayleigh approximation, the to According scattering

G ene     

θ R x z y ˆ ˆ ˆ b Coordinate systems for wave scattering by a spheroidal scatterer with ori with scatterer by for aspheroidal systems wave scattering Coordinate al b b b Figure 3.24 , ϕ

b      e x ); ( ); = xp x      R , x e y      b SS co si , sk z nc ion x p p ) is the global coordinate system, and ( and system, coordinate global the ) is p sc −φ () , a scatterer has its principal axis in the the in axis its principal has , ascatterer , θφ x z y θφ si ˆ y bb bb si ,

nc ,      fo p z k os ˆ os = ). The two coordinate systems are related by related systems are ). two coordinate The can be expressed in the scatterer body coordinate coordinate body scatterer the expressed be in can bb R      R = x α O 4 b aylei 00 00 , φ z x k πε y b co si 2 b θ 0 , ns α ss b GH z 00   θφ θφ os b y pk bb ), which has an orientation ( ), orientation which an has bb S − φ in in z α C b ˆˆ z atte ss () kp           R i −θ in co E E E y si b G iy ix iz Weather Radar Polarimetry Weather Radar s 0 n   θ ,

b      b ,

     x b      , y y x z y b ˆ ˆ ˆ z , b z direction of the of the direction      b ) is the scatterer scatterer the ) is .

θ b , ϕ b ) in the the ) in (3.106) (3.107) (3.108) - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

let let 3.113sions Equations in 3.116 through incidence, horizontal with for backscattering wavefor scattered the projectionthe field,matrix and directions polarization

where In the case of case aspheroid, the weIn have 3.6.3 projectionwhere matrix the matrix scattering the we form, obtain matrix Wave Scattering byaSingle Particle Wave Scattering

for incident wave the field.be each shown that element It can is s s s s Projecting the scattering amplitude to the reference (horizontal and vertical) vertical) and reference the to (horizontal amplitude scattering the Projecting vv hv hh vh θ i = = = = = b π 4 4 4 4 /2, /2, k k k k πε πε πε πε 2 2 2 2 a 0 0 0 0 C ϕ k () () () () i = α+ α+ α+ α+ SC xs xs xs xs atte π () () () () , hx hx vx vx ˆˆ ˆˆ ˆ ˆ θ s ii ii ii ii R = in ˆˆ ˆ bi b bi bi π G () /2, and and /2, () () () hv () M ˆ P hx hx vx vx , ˆˆ ˆ ˆ ib s ˆ at = defined in Equation 3.38 and then writing them in them Equationwriting then in and3.38 defined C ˆ ˆ ˆˆ α=     R by by by i ix S j0 ϕ = hx vx ˆˆ ˆ

s sb sb = hy =0; we have fo α ˆ       hy α+ α+ α+ α+ ˆ V ss ii ii ii x 4 = ys == ε R ˆ hx hy hz =− ˆ ˆ ˆ k πε () () () ()

ib ib ib a hy hy vy vy 2 ˆ ˆˆ ˆ ˆ ii ii α 1( ii ; sb sb sb S 0 y ˆ ˆˆ ˆˆ ˆ +ε hy vz vy ii ii ii ii ˆ ˆˆ = ˆ ; PP p sb sb ˆˆ ˆ L ˆ vz ε− H ˆ s bi α r α jr e vx ˆ vy vz = ˆ R () () () a () () ˆ ib ib ib b and and oid hy hy vy vy . ˆˆ ˆ ˆ 1

ˆ ib iz ib −

ˆ ˆ ˆ hz vz ˆ ˆ i 1) sb sb ˆ ˆ ˆˆ , α

b bz i i .      

z ˆ ˆ

= α α α α α     zs zs

b () () () () . To expres the simplify hz hz vz vz ˆ ˆ ˆ sb sb ii ii ii ii ˆ ˆ ˆ ˆ bi bi () () () () hz hz vz vz ˆ ˆ ˆ ˆ ib ib ˆ ˆ ˆ ˆ b b

.

(3.109) (3.115) (3.116) (3.117) (3.112) (3.110) (3.113) (3.118) (3.114) (3.111) 85 - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

elements amplitude follows: as scattering the obtain

86 back scattering alignment (BSA) conventions. alignment scattering back FIGURE 3.25 reference with and vectors of system coordinate of the far,Thus origin we location the haveas scatterer the used 3.6.4 This is called the the is called This 2001; Elach 1990), Ulabyand Chandrasekar and shown as in (Bringi wave and reference the convenient achange polarization in vector it requires because applications, however, monostatic radar calculations.In is not scattering retical this vectors,reference aset of vectors radar-based wave To back direction. the reference in of the avoid directions the changing the Applying 3.117 Equations 3.121 through 3.113 Equations to 3.116, through we From Equation 3.106,From Equation we have vectors orientation body the In the case of case FSA, the both In f o R

wa x ss ss ss Diagram of coordinate systems for forward scattering alignment (FSA) and alignment scattering for systems forward of coordinate Diagram vv hh R hv d forward scattering alignment scattering forward =θ =− =− S () yx zx C xx ˆ ˆ ˆ () a bb bb bb atte (s vh =− =θ h ˆˆ a =θ i in (c

=− / si co h R os 2 nc si in r sc hk ˆ n () k 2 ( b ss ˆ ss G v ˆ () θφ s φ+ ab i +φ

an hv

/ ˆ b a v −θ ii os ˆ co si , ˆ os d r li

/ ˆ nc k φ+ s) G ˆ φ+ ˆ , 2 2 s bb co ) k n bb haveincident of opposite the that to directions ˆ si k ˆˆ ˆ i ˆ M i b s

nc bb h / and and z ˆ φ k ent +φ s si +θ b r co φ bb θ y ns ˆ s os s s

ss θφ os V () θφ 2 hv co e () ˆ RS hv ss θφ ˆ h ˆ , (FSA: (FSA: in rr s bb s , u ˆ BSA :( in 2 )s si ˆ S +θ , n

bb k , b ˆ v b ˆ yz s bb kh s ˆ s ˆˆ . yz a rs ˆ b +θ

, which is convenient theo in h ˆ

C () −θ r in co hv =− (–

k Weather Radar Polarimetry Weather Radar , v ˆ FSA :( si ss ˆ SC 2 h () , r ˆˆ s k ˆ

n , k s ˆ s

atte , ˆ b v r ˆ , ) r si k

ˆ h ˆ , – ˆ , ˆ . s s n vk

ˆ

R , v k ss 2 ) convention.) ˆ in r , s ) φ −

, k G b ˆ ˆ s

)

y a are used, and and used, are li Figure 3.25 G n M (3.124) (3.120) ent (3.122) (3.123) (3.121) (3.119) - . Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 and cross-to-co-polar ratio are plotted in plotted are ratio cross-to-co-polar and ratio co-polar components. The henceyielding cross-polarization axis, symmetry wave electric the and axes, is fieldno longer minor alonginsidescatterer the and the major of those the change from directions vertical and horizontal the in dimensions projected the because is expected, due This canting. to is cross-polarization there and axes, major minor on the and polarizations on the based amplitudes those from FSA in bythat ment Wave Scattering byaSingle Particle Wave Scattering it follows (b) cross-to-co-polar ratio. (b) cross-to-co-polar FIGURE 3.26 ( polarization the in is canted scatterer the If direction. the canting angle in the polarization plane be be plane polarization the in angle canting the S = B     This shows that both the horizontal and vertical scattering amplitudes change amplitudes scattering vertical and horizontal shows the both This that (|s |/|s |)2, dB aspecific in is oriented discussing two scatterer It when extreme is worth cases the Using 3.122 Equations 3.124 through 3.125, Equation in we obtain hh vv ss (BSA) convention. Hence, the scattering amplitude/matrix in BSA in (BSA) to is related convention. amplitude/matrix scattering Hence, the ab –6 –4 –2 2 (c 0 4 6 02     os () ss ss 2 hv ˆ vh hh θφ rr ,

b () ˆ si ss Dependences of polarization ratio on canting angle: (a) co-polar ratio and angle: (a) and ratio oncanting ratio co-polar of polarization Dependences s s Canting angle,degree 04 S , ab a a nc vv hv kh / / B ˆˆ −θ s s 2 ri b b =

= 1.5 = 2     b =     B +φ si () 06 = ss (a) () nc ab ss os    , ab co vk ˆ bb 2 −ϕ ii − 01 , ss 10 os 22 ˆ b 08 ϕ+ )s θφ +θ in backscattering, called the the called backscattering, in si ba nc si    n( B in 00 in     os 2 ss ss ϕθ Figure 3.26 ϕ− vh hh b si n( 2 θ ss 2 b () (|shv|/|shh|) , dB ab vv hv

ss φ− ≡ –40 –30 –20 –10 si ab bb

φ nc 0     22 ; Equation 3.126; Equation to reduces ss = y .     – si si ss +ϕ z nc ab ) plane, meaning ) plane, meaning −− nc ss 2 20 ss ϕϕ b vh θ+ Canting angle,degree hh b s s os os a a )s / / s s back scattering align scattering back b b in

= 1.5 = 2 os vv 40 θθ hv 2 (b)     φ+ co .     b

. )c ss

60 bb ϕ b b = = in os φ π (3.126) (3.125) (3.127) 2 /2. Let Let /2. 80 θ b 87     - . Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

“ points are addressed in in addressed are points should characterized—these statistics be their and variables, random as treated be however, ( angles canting the orientated; randomly hydrometeors are 3.129using Equations 3.133 through reality, In any orientation. with for aparticle

88 dependent angle factors are canting where the To 3.127 combine Equations 3.128, and we replace “ tion expressed by 3.126 either Equation 3.127 Equations or by combining 3.128. and orienta for arbitrary an matrix scattering the last subsection, we the In obtained 3.6.5 subscript we the omit book, can this in alignment ing

canting in the scattering plane be be plane scattering the in canting tering for the horizontal polarization, and there is no cross-polarization component. is no cross-polarization there and polarization, for horizontal the tering plane; is no scat change hence, in there scattering the change when in it is canted s a S

sin Once the canting angles ( angles canting the Once By contrast, if the scatterer is canted in the scattering ( scattering the in is canted scatterer the if By contrast, This makes physical scatterer’s not dimensiondoes the makes sense, because This horizontal ≡ = =             2

ϑ () ss ss ss a a + + S ab (c As co −+ C os s s( ab atte b 2 +− () 22

cos ϕ+ ss ϕ+ Bs () ab ss −ϑ BC R ab 2 si in −ϑ

ϑ a ns si G ” based on Equation 3.128. on Equation ” based backscatter using we the Because are co nc M ϑϕ () co 2 ss ss 2 Cs ϑ+ ab in ss Chapter 4. at S 2 2 ba B ss R in b = ix )c in Ds ϕϕ ϑ +ϑ os

    co ϕϕ BC , by D C B A b φ 22 co ss ϑϕ = cos =

s = cos = = sin = = cos = ) are known, the scattering matrix can be calculated calculated be can matrix scattering the known, ) are 0s os a a s( )s θ     S 22 in ≡ b

p ≡     H si ss 2 2

2 2 φ a ϑ ϑ ϑ e n( φ ss ss R vh hh + sin ; Equation 3.126; Equation becomes sin + sin cos in oid ϕ− ss a 2 2 2 φ in ϑ+ φ vv hv ss

wit a

2

2 2 ϑ () ϑ 0 ϕ+ si     cos ab sin , ns H b −ϑ 22 (s

co ϕ+ s a 2 2 a φ φ ss s ny ab . in co

2

b in 2 and have and ϑ ss

s Weather Radar Polarimetry Weather Radar θ+ o 2 b ϑϕ )c ” in Equation 3.127 Equation ” in with z R co     s – os ientation in b . x

co s) ) plane, 2 ϕϕ 2 ϑϕ s) co 22 si ϑϕ +ϑ s nc co b co os s ϕ b sc ϑ ϕ = 0. Let the the 0. = Let 22 ,     φ os ) should (3.128) (3.130) (3.129) (3.132) (3.133) (3.131) ϕ     - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

Wave Scattering byaSingle Particle Wave Scattering is is FIGURE 3A.1 Consider wave aplane (FORWARD SCATTERING THEOREM) APPENDIX 3A: DERIVATION OFOPTICAL THEOREM

Noting Noting

E Then, the total absorbed power absorbed is total the Then, Substitution of 3A.1 Equation 3A.2 Equation into yields s , and the total wave total the , and field is SE ii =× PE 1 2

ai Re =− =− A sketch of wave scattering by a nonspherical scatterer. by anonspherical ofA sketch wave scattering ()   ∫ ∫ PS PS A A ss ii 1 1 2 2 ======E H Re Re E  i ∫∫  2 2 incident on a particle ( incident on aparticle ∫∫ i * i EE 1 1 η η AA AA ()    and and 0 0 EH () =+ PE a ii Er Ek i i ×+ =− i i da da is +× SE 2 2 ss EH  ∫ ∫  * ∫ =× EH si 0 S π A 1 2 1 1 2 2 ∫ ˆ 1 2 EH i 0 Re 2 () i ss an Re π da ×+ E co () Re d int Re A () ss + θ () * () Hd × EH EH i s ss Figure 3A.1 ii H Hd EH =+ 2 × × * s * * is    HH in ×+ , we have we , i i is * * θθϕ a a dd i * E i . i da da

+ E . E EH

. s ); the scattered wave); scattered the field

s si = × 0

* i da .

(3A.5) (3A.2) (3A.3) (3A.4) (3A.1) 89 Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

90

have we

Substitution 3A.10 of Equation 3A.6 Equation into yields Using vector formula the Combining Equations 3A.8 and 3A.9, taking the real part, we have 3A.8 Equations part, 3A.9,Combining and real the taking Using 3A.4 Equations 3A.5 3A.3, and Equation in we have In the forward direction, Equation 3A.12 Equation direction, forward the In becomes From Equation 2.20 (Ishimaru 1997), (Ishimaru 2.20 From Equation Re es ˆ ss =−   () ∇× kk ˆ PP ∇× ∇× , i Im as () ˆ i i i += EH PP () () as es   i * ˆ EH EH = ωε ii += ∇× i * 4 () =− =− =− i 0r kk π ˆ k () −× () E 2 , AB ε− +× i * 1 1 1 1 2 2 2 2 0 ˆ i Im EH    ∫ ∫ ∫ ∫ ∫ =− =ω =∇ =ω = V V 1. A A A ∫ HE Hj jH Re   Re Re Re 4 V EE −× =∇ jH i i ωµ π µ− k i 1 2 * kkE ˆˆ i ∇× () BA () () () E 2 0i ss ωε EH EH EH   0 i i 0i ×− i * is i i * 00 =− () ∫ µ−   ×+ i EH ×+ V Re 0i Hj ×− i i i * * (1 (1 HE × Hj ε− ε− * * *   ri r * 0( EH +× * () +ω EH re i * EH EH AB ωε +× ) i )( × si   jE EE Er i ∇× × EH ωε (1 0i ∇× i * εε ε− i EE r0 i r 0r i * ) () i * Weather Radar Polarimetry Weather Radar i * i * −ω ε− EE i dv i i * i jE , da ) * i dv

da i . da

i − ε 1) . dv jk

i * ˆ s i . r

i i E dv i *   .

(3A.12) (3A.10) (3A.11) (3A.6) (3A.8) (3A.7) (3A.9) Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

Wave Scattering byaSingle Particle Wave Scattering

Problems on based wave. scattered defined wave the vectorrepresents are spherical The harmonics of phasor wave on the whether or outgoing.depending is incoming we Because the use where eigen the be Let solution wave coordinates of ascalar spherical in equation WAVE SPHERICAL APPENDIX VECTOR 3B: HARMONICS

Hence, Hence, Substitution 3A.13 of Equation 3A.11 into to leads 3.1

z represented by represented chapter, be wave aplane this can in discussed As any polarization with n ( ψ kr e mn ) is one of the four spherical Bessel) is one four of functions: spherical the j ω as t , Ex zj (, nn (1 ) = tA ∫ )c V =ω (1

ε− represents the incident and internal wave, internal incident and and the represents r PP y1 as += os ψ= ) PS EE () ti mn σ= =σ i = ti tk Mr N −+ 4 4 * i k mn mn zk π πω ii 4 nn k xy dv ti π () =∇ =∇ 2 2 E ε = k η Im rP 0 00 1 k = δ+ S 2 0 2 E ×ψ 4   Im mj 4 sk π ˆ k () × 2 2 π (c () k E   M Im ˆ os Im mn 2 sk 0 At , () mn k 2 θ z ˆ   ˆ   i ) . cos( sk ii sk sk

e

, () ()   () k ˆ . m ˆ ˆ

ˆ ii ii φ ii ω− , , , ,   k

ke ˆ k ˆ ˆ     ˆˆ kx

ii .

e +δ . j

n , 2 ) y z ˆ n . ,

h n (1 ) , and , and zh nn (4 (3A.15) (3A.16) (3A.13) (3A.14) (3.134) )( (3B.2) (3B.3) (3B.1) = h n (2 91 2) ) ,

Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4

92

3.4 3.3 3.2

d. b. b. a. a. is equal to that of a mixture sphere calculated using the Maxwell-Garnett Maxwell-Garnett using the sphere calculated of a mixture that to is equal sphere (with of acoated expression the Show polarizability below) the that give the name for each polarization for following the forgive name polarization each differences: phase the Let Let following: (80, relative be to the constant 17), dielectric assuming and do the form):(represented polynomial in oblate be to spheroids following the ratio with raindrops axis Assuming extinction paradox. explain the and Rayleigh for regime valid Rayleigh the Discuss approximation. scattering, the and theory Mie for the albedo both show and scattering calculate the addition, theory. Mie In the from results the with them compare tion and Rayleigh approxima Qfactors using the the Calculate is 3cm. scattering relative is (41, the constant that is known dielectric 41) wavelength the and respectively. section, cross backscattering and section, cross It scattering extinction section, cross millimeters, in diameter particle the are umns four col diameter. The ory, particle of plot the functions Qfactors as the Using given the (hw2p2.dat) the sections Mie cross the from calculated likely transmit? to 88D radar (of polarization wave the above) WSR-What described polarimetric is the c. γ

= δ− δ− δ− 10%. change why occurred. Explain this volume snow of afractional by of with is replaced dry that constant δ ratio of the scattering amplitudes | amplitudes scattering ofratio the how discuss your Write and in calculation the down formula used the differences. and similarities explain the and phase and well as magnitude as parts nary imagi and of real terms provided in results the your and results both of magnitudes yourmethod), Show calculations. verify scattering the T-matrix rigorously the with provided (hm4.dat, results the calculated 0.1 from of Using equivolumefunction ranging 8mm. to diameter (2.8 atS-band GHz) plot a raindrops of and as the them amplitudes scattering the Using calculate Rayleigh approximation, the scattering A b 2 21 21 21 y /

= a − =0.995 +0.251

δ δ= δ= δ= A 1 z = 0 and =0and , plot the trace of the tip of the electric fieldthe electric of tip in of the the , plot trace the ± ± ± 3 π π 4 2 4 π . δ 2

−

δ D 1

= − 0.0364 π D 2 + 0.005303 + s a / s b | changes when the dielectric | changes dielectric when the Weather Radar Polarimetry Weather Radar D 3 − 0.0002492 − y – z plane and and plane D 4 - - - - Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Wave Scattering byaSingle Particle Wave Scattering

3.5

outer shell, respectively. volume fractional is The sphere, polarizability is in formula is discussed (the its 2 as background medium formula Maxwell-Garnett with mixing expression expression a terer with a canting angle of angle acanting with terer for ascat polarizations vertical and reference the of in horizontal matrix and minor axes are are axes minor and major on the for amplitudes polarizations backscattering the that Assume where ′ are the radius for the inner and outer spheres, respectively. and for inner radius the the are Use the     ε where 1 and and ss ss α= vh hh α= V ε ε× ε 2 vv hv are the relative dielectric constant of the inner sphere and sphere and relative inner of the the constant are dielectric V 0 e is the effective dielectric constant for the mixture sphere. effective for mixture is the constant the dielectric ε 0     3 B 3( (2 (2 s (1 ε+ = ε+ a ε− ε− e and and 21 e 21     Chapter 2). Th Chapter 2). ss () 1) a ss ) )( )( ab s co for the effective polarizability of the coated coated of the for effective the polarizability b φ ε+ ε+ −ϕ , respectively. Show backscattering the that ss in the polarization plane follows plane polarization the in 2 ϕ+ 2) 2) si ε+ ε+ nc 2v 2v b e expression sphere coated of the in os f f 2 (2 () ϕϕ ϕ− ε− 12 ε− 21 ss () ε+ a ss ab 2) si fa (1 v nc () ε− 2 = 2) +ϕ si 33 ε ε nc 2 2 a b ϕϕ ′ , , where os os 2     a . and and - 93 Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4