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8 Analysis of Rainfall Raindrops Investigations on the attenuation caused by rain and other hydrometers and their effects on terrestrial communications started as early as in the 1940s

8 Analysis of Rainfall

Raindrops

Investigations on the attenuation caused by rain and other hydrometers and their effects on terrestrial communications started as early as in the 1940s Subsequently, many theoretical and experimental results were obtained and used to predict the effects of interaction between hydrometers and microwave signals The theories for the prediction of rain attenuation on microwave signals are well established and widely used by many researchers Concep- tually, the specific attenuation due to raindrops depends on both the total (extinction) cross section and the raindrop size distribution

In 1908, Mie [161] f ormulated and put forth the exact formulation for the calculation of the total cross section (TCS) of an isotropic, homogeneous di- electric sphere of arbitrary size This is known as A&e theory Later, Stratton [86] expanded the scattered fields into a series of spherical vector wave func- tions to calculate the TCS from which the attenuation can readily be obtained upon knowing the raindrop size distribution (DSD)

In the work by Oslen et al [162], an empirical relationship between the specific attenuation A and a rain rate R was proposed as A = aRb, where

a = a(f) and b = b(f) are frequency-dependent parameters Based on this formulation, a and b can readily be obtained via regression analysis for dif- ferent frequencies, a known drop size distribution and a given atmospheric temperature The International Radio Consultative Committee (CCIR) (now known as CCITT) [163] recommended this relationship based on the Laws

227

Spheroidal Wave Functions in Electromagnetic Theory

Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic)

228 ANALYSIS OF RAlNFALL ATTENUATlON USING OBLATE RAINDROPS

and Parsons DSD [164] in the frequency range 1 to 1000 GHz Using interpo lation, the unknowns a and b can be determined using the formula presented

in the report However, the Laws and Parsons DSD is only a representa- tive one There are many other DSD models available in the literature, such

as the “Thunderstorm” distribution (J-T) by Joss et al., and the “Drizzle” distribution (J-D) [ 1651

However, these were generated from measurements taken in Europe, Canada, the United States and Japan that underestimated the rainfall attenuation and are unsuitable for a tropical climate such as that of Singapore In fact, stud- ies conducted [166] show that attenuation of microwave signals varies with geographical locations, even at the same rain rate and frequency Measure- ments over the years have shown that the specific attenuation in Singapore

is much higher than that predicted by CCIR at various frequencies Prom a two-year measurement of rainfall attenuation for a 21.225-GHz signal, Yeo et

al [167] generated a local DSD Generation of a new DSD based on multiple frequencies has been proposed, but has not yet been generated

The formulation for spherical raindrops is, however, valid for small sizes The earliest and best recognized contributions to theories in the prediction of rain attenuation of distorted raindrops were probably made by Oguchi in 1960 [168] and 1964 [169], using the boundary-perturbation and point-matching techniques He calculated the TCS values of raindrops with small eccentric- ity using vector wave functions and the first-order perturbation technique to account for drop-shape deformations at large sizes Using these calculations,

he predicted the attenuation for horizontally and vertically polarized waves

at 34.88 GHz However, this method is accurate only for small raindrops with small deformations As raindrops increase in size, deformations increase (eccentricity increases), leading to inaccuracies in the calculations He refined the method in 1963 using a second-order approximation to account for the shape deformations of large raindrops

Experimentally, photographic measurements of raindrops reviewed by Ogu- chi in 1981 [170] h s owed increased deformations in raindrops as they increase

in size A similar review of raindrop deformations was published in 1983 [171] Raindrops vary in size from very small to fairly large The smallest raindrops may be equivalent to those found in clouds The largest drops will not exceed

4 mm in radius, as otherwise they are hydrodynamically unstable and tend

to break up Based on previous investigations, the smallest raindrop is 0.25

mm in radius and the largest is 3.25 mm in radius

The shape of a water drop falling at terminal velocity may be determined theoretically by solving a nonlinear equation describing the balance of internal and external pressures at its surface In real life, it is impossible to solve such

an equation analytically, due to the unknown aerodynamic pressure around the surface, and such a solution is usually obtained numerically A popular

INTRODUCTION 229

model for simulating shapes of raindrops was developed earlier by Pruppacher and Pitter (the P-P model) [172] B ase on this, the shape of raindrops of d various sizes was determined theoretically by solving a nonlinear equation The calculations show that small raindrops are spherical in shape As they grow in size, they become spheroids and gradually become “Hamburger” shaped (i.e., bottom flattened in the side view but caved-in in the cross- sectional view) In reality, the shape of a raindrop is not determined only by its size but is a complex function of other variables, such as wind direction and air pressure

In 1974, Morrison and Cross [58] computed the TCS of an oblate rain- drop using a least squares fitting technique They made modifications to the method introduced by Oguchi, applying the perturbation method to a sphere equal in volume to the raindrop and of suitable eccentricity

Later, originating from the ellipsoidal scattering problem, further develop ments were made by Asano and Yamamoto [24], who described the fields in terms of spheroidal wave functions and solved the problem using the variable- separation method and point-matching techniques Alternatively, Holt et al [59] used an integral equation technique in his approach Other methods were considered by various researchers Details as to the theories and applicability

of various methods of prediction of raindrop scattering and attenuation are also available in the review papers by Oguchi [ 170,171]

As indicated by Oguchi and mentioned here earlier, it is almost impossible

to solve the nonlinear equation for the P-P model analytically To simplify calculations, a cosine series was utilized [ 1721 To further simplify the formu- lation in applications, Li et al [126] implemented a new model using different expressions to describe the upper and lower portions of a realistically distorted nonaxisymmetric raindrop Based on this new model, a formula for calculation

of the TCS is desired The formula contains terms representing zeroth-order approximation (Mie scattering) and first-order approximation (sphere distor- tion or perturbation theory), plus two additional analytical terms to account for spheroid-based distortion of raindrops This model provides a simple ana- lytical expression It is equivalent in simplicity to that of Oguchi’s first-order perturbation methods in the calculation of total cross section, but produced far more accurate results

Raindrop size is a major factor that determines the shape of raindrops For small drop sizes, solutions of P-P nonlinear model shows that Mie theory can

be used to accurately determine the TCS due to its spherical shape However,

as the raindrop size gets larger, distortions of the drop occur from a balance

of pressure inside and outside the drop A major portion of this chapter is

to model plane-wave scattering by raindrops of various sizes in the spher- oidal coordinate system by expanding fields inside and outside the raindrop

in spheroidal wave functions Much work on EM scattering by spheroids has

230 ANALYSIS OF RAINFALL ATTENUATION USlNG OBLATE RAlNDROPS

been done in this area, with many different configurations explored The for- mulations presented in this chapter closely follow the work presented in [24], implying our simplification to that of scattering by a single oblate raindrop

Consider the geometry where an incident plane EM wave is scattered by an oblate spheroidal raindrop in a homogeneous, isotropic medium, as shown

in Fig 8.1 The E field is in the plane of incidence in the case of vertical polarization (TM) mode The Ii field is in the plane of incidence in the case

of horizontal polarization (TE mode) It is no doubt that both E and H fields inside and outside the raindrop satisfy Maxwell’s equations

It is obvious that the EM fields here can be expressed in the same way -

as the spheroidal wave function expansions - as in previous chapters In this chapter we expand the incident, scattered, and transmitted EM fields

in terms of another type of oblate spheroidal wave vector, M(r) and N(r) Their complete definitions in terms of spheroidal coordinates T are given in Appendix A Two cases of polarizations are considered, the TE mode and the

TM mode

8.2.2.1 Case 1: TE Mode

Incident field

Ei = 2 2 in [9,,(-ic,O)M~~~n(-ic,iF)

n=m m=O

(&la)

(&lb)

where M$zn ( -ic, is) and N$&( -ic, it) are presented in Appendix A (in which the barameters c and c are replaced by -ic and it, respectively, for the

PROBLEM FORMULATlON 231

232 ANALYSIS OF RAINFALL ATTENUAT/ON WNG OBtATE RAINDROPS

oblate case), and

49-n =)I

mn r=O,l

with Pr m+r (cos t9) denoting the associated Legendre functions and

00

A mn =

x

2(r + 2m)!

o : (2r + 2m + l)r!

pmn(-~c)j2e

’

r

= 7

When 0 = 0, only the terms with m = 1 remain, and hence

fin(O) = gin(O) = $ gfd:n(-iC)

mn r=O,l

Scattered field

E, = 2 5 in [&,mn(-ic,e)M:~~n(-~c,i~)

n=m m=O

+ ial,mn(-iC,8)N$zn(- l ) l

9

w 41 )

(8 3)

(8.4a)

H, = & 2 2 in [OLl,mn(-iC,B)M~~~n(-ic,iE)

n=m m=O

- @l ,mn(-ic7e)N~‘$-jn(-’ 9

where al,mn and Pl,mn are unknown coefficients to be determined from the boundary conditions

Transmitted field

Et = 5 2 in [61,mn(-i~,0)M~~~n(-i~,i~)

n=m m=O + iyl,mn(-ic, e)NTg(zn(- 7 zc 4] ’ ) ’ ) (8.5a)

Ht = JG 5 2 in [n,,,(-ic,t?)M$&(-ic,iJ)

n=m m=O

- iSl,mn(-iC, e)NL(zn(- 7 2c zs,] ’ 9 ’ 7 (8.5b)

where Yl,mn and Sl,mn are unknown coefficients to be determined from the boundary conditions

PROBLEM FORMULATION 233

8.2.2.2 Case 2: TM Mode

Incident field

(8.6a)

H i= - fi 5 2 in [gmn(-ic,B)M~~~n(-ic,i~)

n=m m=O

+ ifmn(-iC,tY)N~‘~,(-’ , l

where fmn and gmn are expressed in Eqs (8.2a) and (8.2b), respectively

Scattered field

n=m m=O

H s= - & 2 2 in [Rz,,,

n=m m=O

(-ic, e)M$Z$( 1 -ic, i<)

(8.7a)

(8.7b)

WhtXX3 a2,mn and P2,mn are unknowns to be determined from the boundary conditions

Transmitted field

Et = E 2 in [~2,mn(-ic,B)M~~~n(-ic,iE)

n=m m=O

- i62,mn(-iC,0)N~(~n(-* , l

1

Ht = - ,/z F 5 in [62,mn(-iC,e)M~~~n(-iC,i5)

n=m m=O

+ iYP,mn(-iC,e)NP,(~n(- 9 l

Where Y2,mn and 62,mn are also unknown coefficients to be determined from the boundary conditions

234

At the surface of a spheroidal raindrop (i.e., < = <e),

i x (Ei + Es) = t x Et, (8.9a)

The unknown coefficients can be solved for for TE and TM modes respec- tively by substituting the EM field expressions in Eqs (8.la) to (8.8b) into the boundary conditions above The eigenfunctional expansion technique de- scribed in Chapters 3 to 7 is employed to handle the nonorthogonality There- fore, the subsequent matrix equations are finally obtained:

TE mode

n=m

+ fmnV$!~‘t (-ice) + g 00 mn uQJvt (AC()) mn 1

- ,mnV~Jyt ( -iCl) + 61,mnU,~t(-iCl) 1 7 (8.lOa)

n=m

00

l n

x [ 2 crl,mnY~$‘t(-iC()) + Pl,mnX.)$f~t(-iC())

n=m + fmnY$dyt 00 (AC()) + g mn X(y mn -ice) 1

-

- x [ in Yl,mnY$A’t(-iC1) + 61,mnX~~t(-iCl) 1 , (8.10b)

00

l n

x [ 2 Cl!l,mnU,~t(-iC()) + Pl,mnVi$‘t(-iC())

+ fmnU it (40) + g WJyt (-ice) mn mn J 7

(8.10~)

PROBLEM FORMULATION 235

+ fmnx:;t (-ice) + g mn Y(1)9t (-ic()) mn 1

n=m

(8.10d)

TM mode

00

l n

x 2 [ 12Z,mnU,,it(-iC~) + &,mnVi$9t(-iC())

n=m + fmnUzkt 00 (-ice) + g 77-m mn Wq-ic()) 1

-

n=m

+ fmnXzkt (-ic()) + g mn Y(Qt (-ic()) mn

J

00

-

n=m

00

l n

x a [ *~,mnV~~‘t (-iC()) + pz,mnUI.t (-iC())

n=m

+ fmnV$$‘t (-ice) + g mn wq-ic()) mn 1

= 2 in [ &~~,mnVY$!iyt(-iCl) + J;;a,,mnU~~t(-icl,] 3

n=m

(8.11~)

n=m

n=m

(8.11d)

236 ANALYSIS OF RAINFALL ATTENUATION USING OBLATE RAINDROPS

In Eqs (8.lOa) to (8.lld), Ugkt, Viit, X$& and Yii” are defined as: for m > 1, -

ug;t (-ic) = m&)R$Q-ic,&) l [(g + 1)21$yic)

- 2(<,2 + l)I$n( ic) + I;n(-ic)] , (8.12a)

c R;;(-ic,i&,) t2 + 1

x [($ + 1)21tyy(-ic) - 2(g + l)Ity~n(-ic)

+ IpT( ic)] 7 + R,(-ic,iJo) Ix,n(-ic) (4 > 2

-

0 -

x [-2I3-ic) + ($f + l)Iygy-ic) - I$yic)]

+ R&(‘ic,i<,) [(<; + l)21~;(-ic)

x,$.+ic) = coR$$(-ic, i<o)I~;(-ic) 7

YiI(-ic)

and for m = 0,

On -

(8.12~)

im

1

R$& (-4, i&))

- - -

-

C 502 + 1

x [Ir;(-ic) + Ir;(-ic)] + R~~(-i~,i~~)I~~~(-ic)

(8.12d)

UW 0

f { RtA [X,,

C - (tie,“] [CEO2 + WE 9 - I$]

+ Rg [(g + 1>“It”; - (35; + l)I$vj] } ) 7 7 (8.13b)

PROBLEM FORMULATION 237

(813c)

where Itmtn is defined in Appendix B, where parameter c for a prolate spheroid should be replaced by -ic for an oblate spheroid

The extinction total cross section (TCS) is defined as the ratio of the sum of absorbed and scattered energy flow of the incident waves, or alternatively, as the sum of absorption and scattering cross sections For spherical (or small) raindrops, calculation of the TCS is straightforward and is given by

Q = 27r O”

k2 ~(2772 + 1) Re[S& + Sh],

0 m=l

where

S a- &&P) [~j&)l - j&d Kh-dCdl

me-

(8.14)

(8.15a)

Sb jn (p> KPjn (Cdl - C2jn (Cd [Pjn WI

while 5 is the complex refractive index of the raindrop that is calculated from Etay’s FORTRAN program [4]; p = Icea, where IQ is the complex propagation constant of the raindrop and a is its radius, jn is the Bessel function of the first kind, and hp) is the Hankel function of the second kind

For an oblate-shaped raindrop, the TCS is defined as follows:

TE mode

C1,ext = -TRe g 2 [al,mn l arnn + Pl,mn l Xmn] y (8.16)

n=m m=O

TM mode

c 2,ext = 2 2 [QIZ,mn ’ ornn + P2,mn ’ xmn] 7 (8.17)