estimate of the incoherent-scattering contribution to lidar backscatter from clouds

i

Estimate of the incoherent-scattering contributionto lidar backscatter from clouds

David A. de Wolf, Herman W. J. Russchenberg, and Leo P. Ligthart

Lidar backscatter from clouds in the Delft University of Technology experiment is complicated by the factthat the transmitter has a narrow beam width, whereas the receiver has a much wider one. The issuehere is whether reception of light scattered incoherently by cloud particles can contribute appreciably tothe received power. The incoherent contribution can come from within as well as from outside thetransmitter beam but in any case is due to at least two scattering processes in the cloud that are notincluded in the coherent forward scatter that leads to the usual exponentially attenuated contributionfrom single-particle backscatter. It is conceivable that a sizable fraction of the total received powerwithin the receiver beam width is due to such incoherent-scattering processes. The ratio of this con-tribution to the direct ~but attenuated! reflection from a single particle is estimated here by means of adistorted-Born approximation to the wave equation ~with an incident cw monochromatic wave! and bycomparison of the magnitude of the doubly scattered to that of the singly scattered flux. The sameexpressions are also obtained from a radiative-transfer formalism. The ratio underestimates incoherentmultiple scattering when it is not small. Corrections that are due to changes in polarization are noted.© 1999 Optical Society of America

OCIS codes: 290.1350, 010.1310, 290.1310, 260.2110, 290.1090, 290.4210, 290.7050.

tdwitr

1. Introduction

In this research we treat the analysis of backscat-tered reflections from a small lidar range cell thatcontains liquid cloud particles well into the cloud. Ifit is accepted that electromagnetic power scattered atall but very small angles out of the beam, both to andfrom the range cell, cannot return to the receiver,then one can account for the effect of the particlesoutside the immediate environment of the line ofsight by means of an exponential attenuation factorin the backscatter cross section of the range cell.1,2

This contribution is considered coherent becausemultiple scattering from particle to particle at ~infin-tesimally! small angles does not change the phase of

When this research was performed, all the authors were with theInternational Research Centre for Telecommunications—Transmission and Radar, Faculty of Information Technology andSystems, Delft University of Technology, 2628 CD Delft, The Neth-erlands. D. A. de Wolf is now with the Bradley Department ofElectrical Engineering, Center for Stochastic Processes in Scienceand Engineering, Virginia Polytechnic Institute and State Univer-sity, 340 Whittemore Hall, Blacksburg, Virginia 24061-0111. Theemail address for D. A. de Wolf is [email protected].

Received 20 January 1998; revised manuscript received 15 Oc-tober 1998.

0003-6935y99y030585-09$15.00y0© 1999 Optical Society of America

he electric field from the free-space value. It is aominant contribution for particles large comparedith the wavelength because the scattering pattern

s highly peaked at zero scattering angle. This con-ribution resembles that which produces an effectiveefractive index for the medium3 but differs because

the interparticle distance is large compared with thewavelength.4 At infrared and optical frequencies,most cloud particles are also no longer large com-pared with the wavelength, and electromagnetic scat-tering then is closer to being isotropic. Hence thequestion arises to what extent electromagnetic powerscattered out of the line of sight can reenter it andcontribute to the backscatter cross section. Thispossibility is augmented in the clouds and radiationexperiments in The Netherlands5 by the fact that thereceiving beam width is much wider than that of thetransmitter so the receiver may process direct contri-butions from off-axis particles. These contributionsare incoherent because the large-angle scatteringsproduce sizable phase differences in the field contri-butions at the receiver, with a sum that tends to beincoherent. The optical thickness of the region tra-versed by the beam plays a significant role in theimportance of the incoherent contribution. Thesecontributions modify the coherent contribution byadding incoherent terms to the total received field,which may or may not yield significant contributions

20 January 1999 y Vol. 38, No. 3 y APPLIED OPTICS 585

a

aa

qqii

Ee

mcr

ptcbd

tmdmrmpeswgpboibno

tfi

bmoo

maet

Ts

pisf

5

to the irradiance, depending on the optical thicknessof the traversed region.

When the incoherent contributions dominate, it iscustomary to invoke radiative-transfer theory and tolabel them multiple-scattering contributions. Theconnection to Maxwell theory was discussed recentlyby Kravtsov and Apresyan.6 A recent comparisonwith the single-scattering contribution was made byMannoni et al.7 for Chebyshev particles, by Nicolas et

l.8 for ice crystals in cirrus clouds, by Xu et al.9 forvarious cloud-particle distributions, and by Flesiaand Starkov10 for clear-air versus ice-crystal contri-butions in a cloud. One of the earliest similar cal-culations was made by Bruscaglioni,11 who compareda double-scattering calculation with a single-scattering one. A difference from the present studyis that his does not include exponential attenuationfactors in the volume-scattering integral. These fac-tors are needed here because the present calculationsare based on a cw monochromatic incident sphericalwave. As explained below for assumption II, inclu-sion of the exponential factor allows us to estimatewhen multiple incoherent scattering must be consid-ered without incurring the difficulties of the usualpulsed-beam approach.

A recent issue of Applied Physics B contains sev-eral related papers that deal with lidar backscatterfrom a laboratory aerosol. Bissonnette12 works out

radiative-transfer model in a paraxial-diffusionpproximation, whereas Bruscaglioni et al.13 per-

form Monte Carlo calculations on the same aerosol.Flesia and Schwendimann14 extend Mie theory ana-lytically, and Starkov et al.15 compare the transport-theoretical approach with a stochastic model by usingMonte Carlo techniques. Winker and Poole16 alsoperform a Monte Carlo backscatter calculation,whereas Zege et al.17 present a complete analyticaland a simplified semianalytical solution to theradiative-transfer equations. Finally, these effortsare compared in a summary paper by Bissonnette etal.18 The radiative-transfer methods generally re-uire some parameters to be adjusted to measureduantities. All these simulations are based on anncident pulsed beam. A different approach is takenn the present paper.

2. Analysis

This study is based on a renormalized version of themultiple-scattering equations19 for the propagation ofmonochromatic electromagnetic waves in particulatemedia:

E 5 E0 1 (a

t7

a z E0 1 (a

(bÞa

t7

a z t7

b z E0

1 (a

(bÞa

(gÞb

t7

a z t7

b z t7

g z E0 1 . . . . (1)

quation ~1! is a symbolic operator form of the actualquations. E0~r! is the field at r that is due to a

monochromatic plane wave emitted by the source inthe absence of a medium. The dyadic operator t

7a

represents the sum of all repeated scatterings that

86 APPLIED OPTICS y Vol. 38, No. 3 y 20 January 1999

occur inside particle a. Several assumptions areade for tractability of the ensuing analysis ~which is

onsiderably simpler than that for pulsed incidentadiation!:

Assumption I. The cloud is a uniform monodis-erse distribution of spherical particles between al-itudes z 5 R and z 5 R 1 L. The drop distributionertainly is not monodisperse, but the calculationsecome unnecessarily cumbersome if a polydisperseistribution is assumed.Assumption II. The reflection from a single par-

icle at z 5 R as the result of an incident monochro-atic spherical wave is a reasonable measure of the

esired effect. When pulses rather than monochro-atic waves are used, the reflection will come from a

egion of essentially a half-pulse width in height ~ifultiple scattering is ignored!. Multiply scattered

ower flux will take longer times to return and thusnter into other range cells. By the same token, theingly scattered power flux in the desired range cellill be contaminated by multiple scattering from re-ions that contribute to lower range cells. The sim-lified monochromatic cw calculation that we proposeelow is a reasonable measure of the degree of mixingf multiple scattering to ~attenuated! single scatter-ng. That is to say, if the multiple-scattering contri-ution is negligible here, then it certainly isegligible in the much more difficult to calculate casef reflections from pulsed signals.Assumption III. Details of antenna-beam pat-

erns and of Mie-scattering patterns will be simpli-ed at various places in the calculation.Assumption IV. The incoherent scattering contri-

ution will be estimated from the lowest-orderultiple-scattering contribution that excludes previ-

usly summed coherent contributions: the second-rder distorted-Born term.Assumption V. The particle to be considered isost distant from the lidar. The cloud is positioned

t 1–1.5-km altitude ~the crucial geometrical param-ter is the ratio of cloud thickness to altitude!, andhe lidar is either on the ground or on a satellite.

he actual form of the third term of Eq. ~1! can behown to be

Esc~2!~r! 5 k4 (

a(bÞa

g70~r 2 ra! z f7

a~kfa, kab!

z g70~ra 2 rb! z f7

b~kab, kb0! z E0~r!. (2)

Here k is the free-space wave number and g70~r 2 ra!' ~17

2 kfakfa!g0~r 2 ra!, where the scalar part is thefree-space Green’s function for propagation from ra tor ~the dyadic part ensures that the electric field is

erpendicular to the direction of propagation!. Thiss a high-frequency approximation that assumes, rea-onably for cloud particles, that each particle is in thear field of each other particle. The unit vector k21

represent a direction from r1 to r2; thus kfa representsthe direction from ra to r, etc. Likewise, higher-

athpaac

t

wplfl

cts

od

ci

order terms represent more intermediate scatteringsfrom particle to other particle. The scattering am-plitude f

7a~kfa, kab! is diagonal for a spherical particle,

nd it represents the change in magnitude and direc-ion ~by angle u! of an incident plane-wave time-armonic electric field immediately on exit from thearticle. The magnitudes of the three coefficientsre only functions of the scattering angle u, and theyre given by Mie theory. All the dyadics in Eq. ~2!an be considered 2 3 2 tensors1 with respect to two

orthogonal polarizations of the incident field E0, sothe two components of f

7a are fMie and fMie cos u,

where fMie~u, D! is the Mie-scattering coefficient forangle u from a spherical particle with diameter D.

It is well understood20 that a partial summation ofall the coherently acting terms in Eq. ~1! ~i.e., termsfor which scattering occurs at vanishingly small an-gles! leads to

E 5 E0 exp~2gur 2 r0u!,

g 5 ~2pyk! *r0

r

dz *0

`

dDn4~z, D!Im@ fMie~0, D!#. (3)

Here k 5 v=m0ε0 5 vyc, g is the attenuation coeffi-cient in nepers per meter, and n4~z, D! is the particle-diameter distribution density in meters to the minus-fourth power. The inverse, 1yg, is the equivalent ofa mean free path through a turbid medium. Theexponent gur 2 r0u is intended to be shorthand nota-ion for ~possibly complex! attenuation in the cloud

part of the path; see Fig. 1. We now make thedistorted-Born approximations21:

g0~r 2 ra!3 g~r 2 ra! ; exp~2gur 2 rau!g0~r 2 ra!,

g0~ra 2 r0!3 g~ra 2 r0! ; exp~2gura 2 r0u!g0~ra 2 r0!,

(4)

ith nonzero g only inside the medium. These ap-roximations seems to hold well in Eqs. ~1! and ~2! asong as particle densities are not so high that thear-field approximations in these equations are vio-ated. Use of g instead of g0 amounts to a reordering

Fig. 1. Sketch of geometry with one intermediate off-axis scat-tering.

of the terms in Eq. ~1! with partial summations of theoherent contributions. The renormalized seconderm of Eq. ~1!—which is the first term for the back-catter contribution at rp from a particle with diam-

eter D at r, as shown in Fig. 1—will be

Ebs~1! 5 g~r 2 rp! f ~2k0, k0!g~rp 2 r!E0, (5)

and the associated power will be

Pbs~1! 5

1~4pR!4 u f ~2k0, k0!u2 exp~24gDR!,

DR ; ur 2 r0u, k0 5 kz (6a)

r, in a slightly altered notation ~to indicate depen-ence on particle diameter!,

Pbs~1! 5

1~4pR!4 u fMie~p, D!u2 exp~24gDR!. (6b)

The first term of the incoherent contribution @thirdterm of Eq. ~1!# will be

Ebs~2! 5 (

a

g~r 2 ra! f ~ka0, kra!g~ra 2 rp!

3 f ~kra, k0!g~R!E0. (7)

Depolarization effects ~which will turn out to be arelatively small error in view of the fact that thereceiving beam width is less than a milliradian so theintermediate scattering occurs close to the polar axisz between the transmitter and the particle at z 5 R!will be ignored for the time being. This contributionis illustrated in Fig. 1 ~which shows an exaggeratedlywide receiving-beam pattern for clarity!. In Eq. ~7!we replace the summation by a volume integrationwith a particle-size distribution factor n4~D!, which isassumed to be independent of position. Under theassumption of particle incoherence, the associatedpower with this second-order process is

Pbs~2! 5

1~4p!6 * dva *

0

`

dDn4~D!

3exp@22g~l1 1 l2 1 R 2 R0!#

l32l2

2R2

3 u fMie~p 2 ura, D!u2u fMie~ua, D!u2u fbw~u0a!u2,

l1 ; ur1 2 rau, l2 ; ura 2 ru, l3 ; ura 2 rpu,

R0 ; ur 2 r0u, R 5 ur 2 rpu. (8)

The integrand has been multiplied by a factoru fbw~u0au2 to account for the beam-width pattern of thereceiving antenna. The following is a useful approx-imation for a lidar receiver with a half-power width of;0.0005 rad:

u fbw~uu2 5 exp~22.77 3 106u2! (9)

if u is given in radians. Obviously Eqs. ~8! are diffi-ult to evaluate because at least two integrations arenvolved in dva 5 2pla

2dla sin urad ura. Because atypical size distribution is sharply peaked and can be


c6

in

I

5

modeled by a gamma distribution, e.g., as n4~D! 56.863 3 10214 D6 exp~20.7D! mm24 ~if D is in mi-rometers; the numerical coefficient is chosen to yield00 particles per cubic centimeter!, it follows that we

might set D 5 ^D& 5 10 mm, the average diameter, inthe arguments of fMie in Eqs. ~8! to obtain

Pbs~2! 5

2pn3

~4p!6 *0

py2

dura sin ura

3 *0

lfdla

exp@22gR~l1 1 l2 1 R 2 R0!#

l32R2

3 u f#Mie~p 2 ura!u2u f#Mie~ua!u2u fbw~u0a!u2, (10)

where f#Mie~u! [ fMie~u, ^D&!, n3 is the particle densityin reciprocal cubic meters, and lf 5 R0ycos ura. Ondivision by Pbs

~1! and introduction of the new vari-ables z 5 zyR and h 5 ryR, we obtain the ratio

Pbs~2!

Pbs~1! 5

n3 R2

8p *0

py2

d ura sin ura

3 *0

l#fdla

exp@22gR~l#1 1 l#2 2 1 1 z0!#

l#32

3u f#Mie~p 2 ura!u2u f#Mie~ua!u2u fbw~u0a!u2

u fMie~p, D!u2, (11)

with l# 5 lyR for all values of l and z0 5 R0yR. It isuseful to replace the particle density by the attenu-ation coefficient g. We do this by using

g 5 n3

2p

kÎm@ fMie~0, D!#&,

Îm@ fMie~0, D!#& 5

*0

`

dDn4~D!Im@ fMie~0, D!#

*0

`

dDn4~D!

. (12)

The denominator in Eq. ~12! is the density n3 as anntegral over size distribution. In that way we fi-ally obtain

Pbs~2!

Pbs~1! 5

k~2gR!

32p2 *0

py2

dura sin ura

3 *0

lf

dla

exp@2~2gR!~l#1 1 l#2 2 1 1 z0!#

l#32

3u f#Mie~p 2 ura!u2u f#Mie~ua!u2u fbw~u0a!u2

Îm@ fMie~0!#&u fMie~p, D!u2,

sin u0a 5l#2 sin ura

l#3, tan ua 5 tan~ura 1 u0a!,

l#3 5 @~l#2 sin ura!2 1 ~1 2 l#2 cos ura!

2#1y2,

l#1 5 S1 2z0

1 2 l#2 cos uraDl#3. (13)


To obtain Eqs. ~13! we used the monodisperseparticle-distribution relationship for the attenuationcoefficient, which depends only on Im@ f ~k0, k0!# [m@ fMie~0, ^D&!#:

g 52p

kn3 Im@ fMie~0!# (14)

to replace n3 implied in Eq. ~8! by g. Equations ~13!are the basis for numerical calculations discussedbelow.

3. Numerical Calculations

A basic input to Eqs. ~13! is evaluation of the Miecoefficients Re@ fMie~u, ^D&!# and Im@ fMie~u, ^D&!# at thewavelengths of interest, in this case 1.064 and 0.532mm. At 10 °C, the refractive index for water22 at thiswavelength is n ' 1.328 1 1.3E 2 06i. A FORTRAN

program derived by Bohren and Huffman1 was usedto produce an angular pattern of the magnitudeu fMie~u, ^D&!u of the Mie coefficients in Figs. 2 and 3.Here, ^D& has been chosen to be 10 mm. Carefulstudy of the integrand in Eqs. ~13! reveals that it issizable only for not-large angles when the lidar is on

Fig. 2. Absolute value of the Mie-scattering coefficient as a func-tion of scattering angle at l 5 1.064 mm: ~a! small angles, ~b! allangles.

T2~b1

oaacs

a

o

o

the ground and for angles close to py2 for a satellite-based receiver. This is due largely to the narrow-ness of the receiving beam width @see Eq. ~9!#. Areasonable approximation at l 5 1.064 mm ~see Fig.2!, for 0 , ~u in degrees! , 30° ~and for angles close to180°!, is

u f#Mie~u!u < 8.5 3 1025 usin~0.4u 1 0.6!uu0.4u 1 0.6u

. (15)

he approximation is shown as dotted curves in Fig.. The receiving beam-width factor is given in Eq.9!. The value of ^Im@ fMie~0!#& can be approximatedy 9.23 3 1025 m, and that of u fMie~p, D!u2 by 1.79 30212 m, at this wavelength and refractive index.

Figure 4 then depicts the ratio Pbs~2!yPbs

~1! as a functionof attenuation 8.686gR at the above-mentioned wave-length. The dashed curves are the result of using asmoothed polynomial approximation instead of ap-proximation ~15!:

lnu f#Mie~u!u < 210.2 2 0.0878u 1 0.000618u2

2 1.282 3 1026u3. (16)

Fig. 3. Absolute value of the Mie-scattering coefficient as a func-tion of scattering angle at l 5 0.532 mm: ~a! small angles, ~b! allangles.

It confirms that approximation ~15! is reasonablever all angles. Figure 4 indicates, for this geometrynd wavelength, that multiple scattering becomesppreciable at a one-way attenuation of 50 dB. Nowonsider the following results for l 5 0.532 mm. Thecattering pattern for u fMie~u, ^D&!u is shown in Fig. 3.

A numerical fit similar to approximation ~15!, showns dashed curves, is

u f#Mie~u!u < 1.8 3 1024 usin~0.78u 1 0.6!uu0.78u 1 0.6u

, (17)

and a smoothed polynomial approximation ~yieldinglittle difference in Fig. 5! is

lnu f#Mie~u!u < 29.92 2 0.1076u 1 0.0008756u2

2 2.257 3 1026u3. (18)

The calculation of Eq. ~13! then yields the curves ofFig. 5, where the solid curve represents the fit withapproximation ~17! and the dashed curve the fit with

Fig. 4. Ratio of double-scattering to single-scattering power as afunction of attenuation 8.686g~R 2 R0! at l 5 1.064 mm: groundbservation.

Fig. 5. Ratio of double-scattering to single-scattering power as afunction of attenuation 8.686g~R 2 R0! at l 5 0.532 mm: groundbservation.


una

i

o

o

5

approximation ~18!. The dominant process for bothfrequencies and small optical thickness ~large g! ap-pears to be incoherent diffusion of electromagneticpower, which may be the reason that radiative-transfer calculations yield good results for this type ofproblem at optical frequencies for optically dense par-ticulate media.

As a check, we have calculated a curve for amillimeter-wave frequency ~94 GHz! for which we can

se the Rayleigh approximation, Re@ fMie~u, ^D&!# 54.14 3 10210 m, Im@ fMie~u, ^D&!# 5 8.26 3 10211 m.The result of Eqs. ~13! then is shown in Fig. 6.Clearly, the diffusion ~incoherent multiple scattering!effect is negligible at this frequency.

In what follows, the calculations are repeated forsatellite viewing of a 10-mm particle that is embedded0.5 km into a cloud that is 300 km below the satellite,so R 5 300 km and R0 5 299.5 km ~but point rp nowlies high above the layer, whereas r0 is at the lowerboundary!. The results at two optical wavelengthsare shown in Figs. 7 and 8. The ratios are larger ~at

Fig. 6. Ratio of double-scattering to single-scattering power as afunction of attenuation 8.686g~R 2 R0! at f 5 94 GHz: groundbservation.

Fig. 7. Ratio of double-scattering to single-scattering power as afunction of attenuation 8.686g~R 2 R0! at l 5 1.064 mm: satellitebservation.


given 2gR! than they are in the ground-viewing case.The main reason for this seems to be the wide inter-section of the reception cone with a 0.5-km layer at300-km distance, which gives a much larger regionfor secondary scatterings in a horizontal direction.

4. Depolarization ~Vectorial! Effects

To account for the nonscalar quantities in Eq. ~7! weeed to know that a single spherical particle scattersfield E0 5 E0xx 1 E0yy in the far field into two new

orthogonal components1:

SE1u

E1wD 5 Ff ~u!cos u 0

0 f ~u!GF cos w sin w2sin w cos wGSE0x

E0yD ,

or

SE1u

E1wD 5 Ff1 cos u 0

0 f1GSE0r

E0wD , (19)

given that the spherical-coordinate scattering anglesare ~r, u, w!, r 5 r sin u, and f1 5 f ~u!. The geometrys depicted in Fig. 9, where two scatterings occur:

Fig. 8. Ratio of double-scattering to single-scattering power as afunction of attenuation 8.686g~R 2 R0! at l 5 0.532 mm: satelliteobservation.

Fig. 9. Wave number and field vectors with relevant geometricscattering angles.

a

yus

i

f

e

i

Hm

from direction k0 to k1 and from direction k1 to k2.The field E0r is in the r 5 x cos w 1 y sin w direction.The scattered field amplitudes are E1u in direction und E1w in direction w. The connection between E0r

and E0w and the Cartesian coordinates E0x and E0y isobvious from Eq. ~19!, the notation of which is spe-cifically for the first scattering.

For the second scattering, the following relation-ships can be inferred from Fig. 9: The new sphericalangles after that scattering are u2 5 2r 5 2x cos w 2ˆ sin w and w2 5 w 5 2x sin w 1 y cos w. Moreover,2 5 p 2 u and w2 5 py2. Thus f2 5 f ~p 2 u!. Theecond scattering is

SE2u

E2wD 5 Ff2 cos~p 2 u! 0

0 f2GF cos~py2! sin~py2!

2sin~py2! cos~py2!G3 SE1x

E1yD . (20a)

Now use E1w 5 2E1x and E1u 5 2E1y to convert Eq.~20a! into

SE2u

E2wD 5 Ff2 cos u 0

0 f2GSE1u

E1wD . (20b)

Use the transformation

SE2u

E2wD 5 F2cos w 2sin w

2sin w cos w GSE2x

E2yD (21)

to obtain for the cascaded two scatterings

SE2x

E2yD 5 f1 f2F2cos w 2sin w

2sin w cos w GFcos u 00 1G

3 Fcos u 00 1GF2cos w 2sin w

2sin w cos w GSE0x

E0yD

5 f1 f2Fsin2 u cos2 w 2 cos 2w21⁄2~1 1 cos2 u!sin 2w

21⁄2~1 1 cos2 u!sin 2wsin2 u cos2 w 1 cos 2wGSE0x

E0yD . (22)

So, symbolically, we write Eq. ~22! as

E2x 5 f1 f2@~a 2 b!E0x 2 cE0y#,

E2y 5 f1 f2@2cE0x 1 ~a 1 b!E0y#, (23)

such that a [ sin2 u cos2 w, b [ cos 2w, and c [ 1⁄2~1 1cos2 u!sin 2w, as a result of which we obtain for therradiance uE2u2 5 uE2xu2 1 uE2yu2:

uE2u2 5 u f1 f2u2$~a2 1 b2 1 c2!@uE0xu2 1 uE0yu2#

2 2ab@uE0xu2 2 uE0yu2# 2 4ac Re@E0xE0y*#%. (24)

The 2ab and 4ac terns vanish after integration of dwrom 0 to 2p. The remaining integrations, including

that over sin udu from 0 to py2 ~note that we integratessentially over a half-space!, finally yield

*0

2p

dw *0

py2

du sin uuE2u2 5 1.6167pu f1 f2u2

3 ~uE0xu2 1 uE0yu2!. (25)

Thus the net result of incorporating the polarizationproperties is that the ratio uE2u2yuE0u2 changes afterangular integration in Eqs. ~13! by a factor of 1.6167pinstead of by 2p, so Eqs. ~13! requires a correctionfactor 1.6167py2p 5 0.8083.

5. Radiative-Transfer Formalism

Equation ~8! also can be derived by means ofradiative-transport theory,11 and this is done brieflyhere. The basic integral equation for the specificintensity I~l, V! is

dI~t, V!

dt5 2I~t, V! 1

1st * d2V9u f ~V, V9!u2I~t, V9!.

(26)

Here, t 5 * dln3~l !st~l ! is the optical distance in thedirection indicated by unit vector V, which in turndenotes the ~straight! line along which the integralequation is being considered; f ~V, V9! is the scatter-ng amplitude for direction V9 into direction V; and st

is the extinction cross section, which is the sum ofabsorption cross section sa and scattering cross sec-tion ss 5 * d2Vu f ~V, V9u2. As before, n3 is the par-ticle density. More generally, t 5 * dl * d2V9 * dDn4~l, D!u f ~V, V9, D!u2 if we distinguish between dif-ferent particle sizes. For a uniform monodispersemedium, one may set t 5 ln3st, as a consequence ofwhich Eq. ~25! can be written more simply as

dI~l, V!

dl5 2n3st I~l, V! 1 n3 * d2V9u f ~V, V9!u2I~l, V9!.

(27)

Application of the optical theorem to a unit volume ofparticles yields

n3st 54p

kn3 Im@ f ~V, V!#

54p

k * dDn4~D!Im@ f ~V, V, D!# ; 2g. (28)

ere g is the extinction coefficient ~in nepers pereter! for the medium, and consequently

dI~l, V!

dl5 22gI~l, V! 1 n3 * d2V9u f ~V, V9!u2I~l, V9!.

(29)


S

T

r

ˆ ˆ 2

T

a

5

This can serve as an adequate starting point. Thisintegral equation, with the aid of an integrating fac-tor and a slight change of notation, becomes

I~rP, V! 5 exp~22glQP!I~rQ, V!

1 n3 *Q

P

dl exp@22g~lP 2 l !#

3 * d2V9u f ~V, V9!u2I~l, V9!. (30)

Now we apply Eq. ~30! several times to the sketch ofFig. 10. Let the received electric flux at point rR be

~R! 5 * d2VI~rR, V!u fbw~V!u2, where we use the samereceiving beam width factor as in Eq. ~9!. Apply Eq.~30! to this flux integral along the line QR to obtaintwo contributions, one from each part of Eq. ~30!.

he first of these is

S1~R! 5 * d2Vexp~22glQR!I~rQ, V!u fbw~V!u2. (31)

It represents the contribution of the reduced incidentintensity ~Ref. 10, p. 158!. However, I~rQ, V! is non-zero, for the purpose of calculating two successivescatterings, only when point Q is in the particle atpoint P ~as there is no zeroth-order source other thanthe particle at P to produce radiation in direction V!,at which point u fbw~V!u2 ' u fbw~0!u2 5 1 andexp~22glQR! ' exp~22glPR! 5 exp~22gR!. Further-more, in the immediate vicinity of the particle at P,the solid-angle increment can be written as d2V 5d6yR2, where surface element d6 is 'k0. The di-ection V ' 2k0, and therefore

S1~R! 5exp~22gR!

R2 * d6I~rP, 2k0!. (32)

The surface integral * d6I~rP, 2k0!, which extendsonly over the surface area of the particle subtendedperpendicular to the direction k0, must be the powerscattered per unit solid angle in direction 2k0 and

Fig. 10. Geometry relevant to the radiative-transfer calculation.


hence must equal u f ~2k0, k0!u S0~P!, where S0~P! isthe incident power flux: S0~P! 5 S0~P!k0. Thus

S1~R! 5exp~22gR!

R2 u f ~2k0, k0!u2S0~P!. (33)

Now return to the received flux at rR: S~R! 5 * d2VI~rR, V!u fbw~V!u2, and insert the second term of Eq.~30! to obtain

S2~R! 5 n3 * d2Vu fbw~V!u2 *a

P

dla

3 exp@22g~lR 2 la!#

3 * d2V9u f ~V, V9!u2I~ra, V9!. (34)

he last factor in Eq. ~34! is a specific intensity at ra

in the direction V9 for which a separate expressionthat resembles Eq. ~30! can be written down:

I~ra, V9! 5 exp~22glba!I~rb, V9! 1 n3

3 *g

a

dl exp@22g~la 2 l !#. . . (35)

and for which specific intensity we have not writtendown the full expression, as only the first term of Eq.~35! will be inserted into Eq. ~34! to yield an expres-sion for two scatterings. Referring again to Fig. 9,we note that I~rb, V9! is nonzero only when rb ' rP, so

* d2V9u f ~V, V9!u2exp~22glba!I~rb, V9!

< u f ~kf, ka!u2exp~22glPa! * d2V9I~rP, V9!

< u f ~kf, ka!u2exp~22glPa!

lPa2 * d6I~rP, V9!, (36)

in which expressions d6 ' ka and V9 hardly deviatesfrom direction ka. For reasons similar to those thattransform Eq. ~32! into Eq. ~33!, it follows that

* d6I~rP, V9! 5 u f ~ka, k0!u2S0~P! (37)

nd therefore that approximation ~36! becomes

* d2V9u f ~V, V9!u2exp~22glba!I~rb, V9!

< n3

exp~22glPa!

lPa2 u f ~kf, ka!u2u f ~ka, k0!u2S0~P!. (38)

w

tn

m

t

b

2. H. C. van de Hulst, Light Scattering by Small Particles ~Wiley,
The net result of all this is that Eq. ~34! becomes
S2~R! 5 n3 * d2V * dla

exp@22g~lR 2 la! 2 2g(lPa!#

lPa2

3 u fbw~V!u2u f ~kf, ka!u2u f ~ka, k0!u2S0~P!. (39)

The final step is a replacement of d2V by d6aylaR2,

here d6a ' V, so we obtain d6adla, which is thevolume element dva at point ra. The result is

S2~R! 5 n3 * dva

exp@22g~laR 1 lPa!#

laR2lPa

2

3 u fbw~V!u2u f ~kf, ka!u2u f ~ka, k0!u2S0~P!. (40)

This expression is substantially the same as Eq. ~8!;there are some differences in notation @e.g., the nota-tion does not show explicitly that laR is nonzero onlyin the medium, and n3 can be replaced by an integra-ion over particle diameter with the distribution4~D! such that the Mie factors can be those for a

polydisperse rather than a monodisperse set of par-ticles#. Also, a factor ~4p!26 is missing in approxi-

ation ~39! because Ishimaru’s13 @ fMie# differs fromthe present fMie by a factor of 4p. Therefore, radia-tive transfer theory leads to the same expressions@Eqs. ~6! and ~8!# for direct backscatter and backscat-er with one indirect intermediate scattering.

6. Comment

The scattering amplitude fMie~u, l, ε, D! is a functionnot only of angle and wavelength but also of permit-tivity ~or refractive index! and diameter of the spher-ical particle. As a result, Eq. ~8! needs to becalculated without replacing D by an average value inthe Mie coefficients. The expression then is more dif-ficult to evaluate because the three Mie factors in theintegral over diameter D are functions of h and z, asindicated in Eqs. ~13!, so the numerator is essentiallya triple integral. However, this expression still wouldlead directly to Eqs. ~13! for a monodisperse collectionbecause the particle-size distribution n4~z, D! would bea Dirac delta function in the diameter of the particles.Also, the widely different small-angle and smooth-polynomial approximations in Figs. 2 and 3—whichlead to small differences in Figs. 4, 5, 7, and 8—seemto affirm that the lobed details of the Mie coefficientsare not crucial here. Finally, the assumption thatn4~D! is independent of z is probably reasonable.The calculations of this study therefore lead to areasonable estimate of conditions under which dif-fusionlike processes ~radiative-transfer calcula-tions! are more suited to obtaining estimates of

ackscatter from cloud particles.

References and Notes1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of

Light by Small Particles ~Wiley-Interscience, New York, 1983!,Chap. 3.

New York, 1957!, Chap. 4.3. M. Born and E. Wolf, Principles of Optics, 3rd ed. ~Pergamon,

Oxford, 1965!, Sec. 2.4.2.4. The denominator of the Lorentz–Lorenz formula for the effec-

tive refractive index reduces to a factor of 3.5. A. Van Lammeren, H. Russchenberg, A. Apituley, H. Ten

Brink, and A. Feijt, “CLARA: a data set to study sensorsynergy,” presented at the Workshop on Synergy of Radarand Lidar in Space, Geesthacht, Germany, 12–14 November1997.

6. Yu. A. Kravtsov and L. A. Apresyan, “Radiative transfer: newaspects of the old theory,” in Progress in Optics, E. Wolf, ed.~Elsevier, Amsterdam, 1996!, Vol. 36, pp. 200–212.

7. A. Mannoni, C. Flesia, P. Bruscaglioni, and A. Ismaeli, “Mul-tiple scattering from Chebyshev particles: Monte Carlo sim-ulations for backscattering in lidar geometry,” Appl. Opt. 36,7151–7164 ~1996!.

8. F. Nicolas, L. R. Bissonnette, and P. H. Flamant, “Lidar effec-tive multiple scattering coefficients in cirrus clouds,” Appl.Opt. 36, 3458–3468 ~1997!.

9. L. S. Xu, G. T. Zhang, and L. B. Cheng, “Parameterization ofthe shortwave radiative properties of water clouds for use inGCMS,” Theor. Appl. Climatol. 55, 211–219 ~1996!.

10. C. Flesia and A. V. Starkov, “Multiple scattering from clearatmosphere obscured by transparent crystal clouds in satelliteborne lidar sensing,” Appl. Opt. 35, 2637–2641 ~1996!.

11. P. Bruscaglioni, “On the contribution of double scattering tothe lidar returns from clouds,” Opt. Commun. 27, 9–12~1978!.

12. L. R. Bissonnette, “Multiple scattering of narrow lightbeams inaerosols,” Appl. Phys. B. 60, 315–323 ~1995!.

13. P. Bruscaglioni, A. Ismaeli, and G. Zaccanti, “Monte-Carlocalculations of lidar returns: procedure and results,” Appl.Phys. B. 60, 325–329 ~1995!.

14. C. Flesia and P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the lidar equation,”Appl. Phys. B. 60, 331–334 ~1995!.

15. A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “Astochastic model and a variance-reduction Monte-Carlomethod for the calculation of light transport,” Appl. Phys. B.60, 335–340 ~1995!.

16. D. M. Winker and L. R. Poole, “Monte-Carlo calculations ofcloud returns for ground-based and space-based radars,” Appl.Phys. B. 60, 341–344 ~1995!.

17. E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solutionto lidar return signals from clouds with regard to multiplescattering,” Appl. Phys. B. 60, 345–353 ~1995!.

18. L. R. Bissonnette, P. Bruscaglioni, A. Ismaeli, G. Zaccanti, A.Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P.Schendimann, A. V. Starkov, M. Noormohammadian, U. G.Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, and I. N. Polon-sky, “Lidar multiple scattering from clouds,” Appl. Phys. B. 60,355–362 ~1995!.

19. A. Ishimaru, Wave Propagation and Scattering in RandomMedia ~Academic, New York, 1978!, Vol. 2, Chap. 14.

20. L. Tsang, J. A. Kong, and R. T. Shin, Theory of MicrowaveRemote Sensing ~Wiley, New York, 1985!, Chap. 6.

21. Ref. 19, Vol. I, Chaps. 7 and 8; see also Ref. 11, pp. 381–382.22. P. S. Ray, “Broadband complex refractive index of ice and

water,” Appl. Opt. 2, 1836–1844 ~1972!.


estimate of the incoherent-scattering contribution to lidar backscatter from clouds

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