Determination of the droplet effective size andoptical depth of cloudy media from polarimetricmeasurements: theory

Alexander Kokhanovsky and Reiner Weichert

We present the development of a semi-analytical algorithm for optical particle sizing in disperse media.The algorithm is applied to the specific case of water clouds. However, it can be extended with minormodifications to other types of light-scattering medium. It is assumed that the optical thickness � of themedium is large and the probability of photon absorption � is small. Thus the optical particle-sizingproblem is studied in the regime of highly developed multiple light scattering. It was found that thedegree of polarization in visible and near-infrared channels provides us with information both on theeffective size of droplets and on the optical thickness �. © 2002 Optical Society of America

OCIS codes: 010.0010, 030.5620.

1. Introduction

The influence of clouds on radiative fluxes in theterrestrial atmosphere has been studied by many au-thors both experimentally and theoretically.1–3 Theknowledge obtained was applied to the solution of theinverse problem, namely, to the determination of thethermodynamic phase, microstructure parameters�e.g., the effective radius of droplets�, single-scattering albedo, optical thickness, and liquid waterpath of cloudy media.4–10

However, cloudy media do not only absorb andscatter radiation. They can also be considered asmajor sources �along with aerosols, gases, and under-lying surfaces� of polarized light in the terrestrialatmosphere. Clouds introduce a new property of in-coming solar light, namely, the preferential directionof oscillation of the electric vector in a light beam.Generally speaking, both reflected and transmittedlight becomes elliptically polarized.

The polarization arises mainly because of the sin-gle scattering of a light beam by a water droplet, an

A. Kokhanovsky �alexk@iup.physik.uni-bremen.de� is with theInstitute of Environmental Physics, Bremen University, P.O. Box33 04 40, D-28334 Bremen, Germany, and the Stepanov Instituteof Physics, 70 Skarina Avenue, Minsk 220072, Belarus. R.Weichert is with the Institute of Particle Technology, Leib-nizstrasse 19, D-38678 Clausthal-Zellerfeld, Germany.

Received 5 June 2001; revised manuscript received 13 October2001.

0003-6935�02�183650-09$15.00�0© 2002 Optical Society of America

3650 APPLIED OPTICS � Vol. 41, No. 18 � 20 June 2002

aerosol particle, or an ice crystal. Molecular scatter-ing is another source of polarized light in the Earth’satmosphere. The phenomenon of multiple scatter-ing, which is of particular importance for clouds,leads to the randomization of both photon directionsand polarization states. This causes a decrease oflight polarization because of an increase in the opticalthickness of a cloud. Another effect of multiple scat-tering is the introduction of ellipticity in scatteredlight beams. It should be pointed out that the solarlight scattered by a spherical water droplet becomespartially linearly polarized. This is due to the factthat the ellipticity of singly-scattered light is equal tozero for spherical scatterers if the incident light isunpolarized.11

Studies of polarization characteristics of solar lighttransmitted and reflected by cloudy media have along history.12–17 However, the real burst of re-search in this area was given by a launch of thePOLDER �polarization and directionality of theEarth’s reflectances� instrument on board the Japa-nese Advanced Earth Observing Satellite I.18 ThePOLDER was able to transmit to the Earth a hugeamount of information about polarization character-istics of light reflected from cloudy media, aerosols,and underlying surfaces at several wavelengths.Specifically, the first three components of the Stokesvector Sr�I, Q, U, V� have been measured for wave-lengths � equal to 443, 670, and 865 nm. There is nodoubt that even more advanced polarimeters withwider spectral coverage will appear on board differ-ent satellites in the future, which makes further the-

oretical studies of polarization characteristics ofcloudy media extremely important. This is due topotential possibilities for global retrievals of the cloudmicrostructure,9 the shape of particles,18 and the op-tical thickness of clouds17 on the basis of polarizationmeasurements.

Of course, similar information can be obtained fromthe reflected intensity measurements. However, itcould well appear that the degree of polarization can beused as a source of additional information about thecloud particle size distributions close to the top of acloud.9 Indeed, the high proportion of photons scat-tered from a thin upper layer of a cloud in the creationof light polarization is quite understandable. Multi-ply scattered light fluxes from deep layers are hardlypolarized at all. Radiative characteristics, in con-trast, represent the cloud as a whole. Thus the effec-tive radius derived from radiative measurements is anaverage of large ensembles, of possibly quite differentparticle size distributions, presented in different partsof cloudy media.

It was emphasized by Hansen12 30 years ago thatthe fact that the polarization data refer to cloud topsmay be not a drawback but an advantage. Indeed, itis easier to interpret observations that do not refer tosome average values over the entire cloud depth.

The polarization characteristics of cloudy mediacan be studied by application of numerical codes2,12

based on the vector radiative transfer equation solu-tion. However, one can also use the fact that cloudfields are optically thick in most cases.1 This allowsthe application of asymptotic analytical relations,19

derived for optically thick disperse media with arbi-trary phase functions and absorption. These solu-tions help us to clear physical mechanisms and mainfeatures behind the polarization change due to theincrease of the size of droplets or the thickness of acloud. Analytical solutions also provide an impor-tant tool for the simplification of the inverse prob-lem.20 They can be used, e.g., in studies of theinformation content of polarimetric measurements.

The drawback of the well-known vector asymptotictheory for optically thick layers19 is related to the factthat the main equations of this theory depend ondifferent auxiliary functions and parameters. Cal-culations of these functions and parameters arerather complex. This is due to their quite generalapplicability in terms of the single-scattering albedo.

The special case of weakly absorbing light-scattering media was considered by Kokhanovsky21

in the framework of the modified asymptotic theory.Formulas obtained are rather simple and allow forthe semi-analytical solution of the inverse problem.They can be applied to the case of water clouds. Thisis due to weak absorption of light by water droplets ina broad spectral range ��2.2 �m�.

The main task of this paper is to apply equations,obtained in the framework of the modified asymptotictheory in Ref. 21, to the inverse problem solution,namely, to the determination of the droplet size andoptical thickness of cloudy media. However, resultsobtained after minor changes are made can be ap-

plied to a much broader range of multiply scatteringmedia.

2. Reflection and Transmission Matrices

Let us introduce the reflection R̂��, �0, �� and trans-mission T̂��, �0, �� matrices of cloudy media. Re-flection and transmission matrices depend on theoptical thickness � of a cloud, microstructure of acloud, the solar angle 0, the viewing angle , and therelative azimuth � � � �0, where �0 and � areazimuths of the Sun and the observer, respectively.We also define �0 cos 0, � �cos 0�. The impor-tance of these matrices is due to the fact that they canbe used for the calculation of the Stokes vector ofreflected Sr�I, Q, U, V� and transmitted St�I , Q , U ,V � light under arbitrary illumination of a cloudylayer. Namely, it follows for plane-parallel cloudylayers15:

Sr��, �� �1� �

0

1

d� � �0

2�

d� R̂��, � , �

� � �S0�� , � �, (1)

St��, �� �1� �

0

1

d� � �0

2�

d� T̂��, � , � � � �

� S0�� , � � � S0��, ��exp�� �

�0� , (2)

where S0�� , � � represents the Stokes vector of theincident light beam. The second term in Eq. �2� issmall for most cloudy media �� �� 1� and can beneglected.

In a good approximation, the Sun is a source oflight incident from only one direction �0, �0. Then itfollows:

S0��, �� �1�0��� � �0���� � �0��F, (3)

where a vector F is normalized so that the first ele-ment of �F �namely, �F1� is the net flux of solar beamper unit area of a cloud layer. Equations �1� and �2�simplify if one accounts for Eq. �3�:

Sr��, �� � R̂��, �0, � � �0�F, (4)

St��, �� � T̂��, �0, � � �0�F, (5)

where only the first element of the column vector

F � �F1

F2

F3

F4

� (6)

differs from zero for unpolarized solar radiation.Thus only elements Rj1, Tj1� j 1, 2, 3, 4� are relevantto studies of solar light propagation, scattering, andpolarization in terrestrial atmosphere. Other ele-ments are of interest only in the case of polarized�e.g., laser� incident beams.

20 June 2002 � Vol. 41, No. 18 � APPLIED OPTICS 3651

One can see that the knowledge of matrices R̂��,�0, � � �0� and T̂��, �0, � � �0� is of primary im-portance for understanding radiative and polariza-tion characteristics of light fluxes that emerge from acloud body. As a matter of fact, they are Muellermatrices22,23 for turbid media. Note that Eq. �4� canbe written in the scalar form

I � R11F1, Q � R21F1, U � R31F1, V � R41F1

(7)

for the unpolarized incident light. Thus the inten-sity of the reflected light is determined by the elementR11 of the reflection matrix. This is the only elementthat has been a major concern of most cloud opticsstudies up to this date. The degree of polarization isgiven by

P � �Pl2 � Pc

2 , (8)

where

Pl � �R212 � R31

2�R11 (9)

is the degree of linear polarization and

Pc � R41�R11 (10)

is the degree of circular polarization. One canshow24 that for particular viewing geometries �e.g.,�0 1, � 1, ���0 0, �� the values of R31 and R41are equal to zero and light is partially linearly polar-ized. The degree of circular polarization Pc is equalto zero in this case. It is also common to use a dif-ferent definition of the degree of polarization, namely,

P � ��Q�I� (11)

for linearly polarized light beams. It follows fromEq. �11� at the nadir-viewing geometry �R31 R41 0� that

P � ��R21�R11�. (12)

We will use this definition of the degree of polariza-tion in this study. The second component of theStokes vector Q is proportional to the difference Il �Ir, where values of Il and Ir mean the intensities oflight polarized parallel and perpendicular to a merid-ional plane that contains the normal to a layer andviewing direction. Thus light polarized perpendicu-lar to the meridional plane is characterized by posi-tive polarization. Note that the sign of the degree ofpolarization strongly depends on the observation ge-ometry. For instance, it is positive in the main rain-bow region �see Fig. 1�. It could be both positive andnegative in the glory-scattering region �see Fig. 1�.Results presented in Fig. 1 were obtained by use ofthe Mie theory for polydispersions of water dropletsat the wavelength � equal to 443 nm. The particlesize distribution was given by the following equation:

f �a� � Aa� exp���� � 3�a�aef�, (13)

where aef is the effective radius of droplets, � is thehalf-width parameter, and

A ����1

a0��1��1 � ��

.

Note that the mode radius a0 is related to the effec-tive radius with the following equation:

a0 � aef��1 �3�� .

It should be pointed out that the Stokes vectorelements I, Q, U, V �see Eq. �7�� can be used for thedetermination of polarization ellipse parameterssuch as the ellipticity and the angle of preferential

Fig. 1. Degree of polarization of light singly scattered by a poly-dispersion of spherical water droplets characterized by the gammaparticle size distribution �Eq. �13�� with the half-width parameter� 6 and the effective radius aef 4, 6, 8, 10, and 16 �m as thefunction of the scattering angle at � 443 nm. �b� Same as in �a�but at aef 6 �m and values of the half-width parameter � 2, 4,6, 8, 10, and 12.

3652 APPLIED OPTICS � Vol. 41, No. 18 � 20 June 2002

oscillations of the electric vector in a light beam un-der study.22

Thus reflection and transmission matrices allowthe determination of both the intensity and the po-larization states of reflected and transmitted lightbeams under arbitrary illumination of a cloud layer.

3. Modified Asymptotic Theory

A. General Equations

The optical thickness of water clouds is large in mostcases.1 Some mixed and crystalline clouds are alsoextended in the vertical direction for many hundredsof meters and could be optically thick.25 Thus theasymptotic vector theory19,26 is quite applicable tocloudy media. The reflection matrix R̂��, �0, �, ��and transmission matrix T̂��, �0, �� of a homogeneousplane-parallel weakly absorbing turbid layer of alarge optical thickness � �extL��ext is the extinctioncoefficient and L is the geometrical thickness of alayer� can be written in the following form21:

R̂��, �0, �, �� � R̂���, �0, �� � T̂��, �0, ��

� exp��x � y�, (14)

T̂��, �0, �� � tK0���K0T��0�, (15)

where

x � k�, y � 4qk, q � �3�1 � g���1,

k � �3�1 � g��1 � �0��1�2,

�0 � 1 � �, � ��abs

�ext,

� � �extL, (16)

R̂���, �0, �� is the reflection matrix for a semi-infinitemedium with the same local optical properties as afinite one, L is the geometrical thickness of the me-dium, �ext is the extinction coefficient, �abs is theabsorption coefficient, g 1

2 �0� p���sin � cos �d� is the

asymmetry parameter of the phase function p���, and� is the scattering angle. Note that the vectorK0

T��0� in Eq. �15� is transposed to the vector K0��0�.The vector K0��� also occurs in the so-called Milneproblem.3 It describes the angular dependence andthe polarization characteristics of light emergingfrom semi-infinite turbid layers with sources of radi-ation located deep inside the medium. The value of

t �sinh y

sinh�x � �y�(17)

in Eq. �15� is the global transmittance,21 defined as3

t � 4 �0

1

�d� �0

1

�0d�0 T11��, �0, ��. (18)

Equation �18� also follows from Eq. �15�, taking intoaccount the integral relation3

2 �0

1

�d�K01��� � 1, (19)

where K01��� is the first component of the escapevector K0���. The value of � is approximately equalto 1.07.24

Equations �14� and �15�, derived in Ref. 21 for smallprobabilities of photon absorption, are much simplerthan initial asymptotic formulas, obtained byDomke.19 They allow us to calculate the reflectionand transmission matrices of thick weakly absorbingdisperse media by a simple means if the solution ofthe problem for a semi-infinite medium with thesame phase matrix as for a finite layer under consid-eration is available.24,26 This reduction of a generalproblem to the case of a semi-infinite medium is ofgeneral importance for the radiative transfer theory.

B. Nonabsorbing Media

It follows from Eqs. �12�, �14�, and �15� for the degreeof polarization of the reflected light at the nadir ob-servation in the case of nonabsorbing media21:

P��0� �P�

0��0�

1 � tU��0�, (20)

where

U��0� �K01�1�K01��0�

R�11��0, 1�, (21)

and we have instead of Eq. �17� at � 0:

t �1

� � 0.75��1 � g�. (22)

The value of

P�0��0� � �

R�210��0, 1�

R�110��0, 1�

(23)

is the degree of polarization for a semi-infinite non-absorbing medium at the nadir observation. HereR�11

0��0, 1� and R�210��0, 1� are correspondent ele-

ments of the reflection matrix �see Eq. �4�� of a non-absorbing semi-infinite medium. Note that wederived Eq. �20� by taking into account the equality26

K02�1� 0. Owing to this equality, the second com-ponent of the Stokes vector for optically thick mediacoincides with the value of Q for semi-infinite layersat the nadir observation geometry. This fact can beused for the retrieval of the size of particles fromspectral measurements of Q��� even if the preciseinformation on the optical thickness is not available.Thus an accurate estimation of the effective radius ofdroplets can even be obtained from measurements ofQ at a single wavelength in the visible.

Both functions K01��0� �see Fig. 2� and R�110��0, 1�

�see Zege et al.27, Fig. 3.3b� only slightly depend on

20 June 2002 � Vol. 41, No. 18 � APPLIED OPTICS 3653

the size of scatterers. They can be represented, e.g.,by the following approximate formulas27:

K01��0� �37�1 � 2�0�, R�11

0��0, 1� �1 � 4�0

2�1 � �0�.

(24)

Thus it follows from Eqs. �21� and �24�:

U��0� �54�1 � 2�0��1 � �0�

49�1 � 4�0�. (25)

The dependence U�0� is presented in Fig. 3. Onecan see that U � �1.03, 1.33�. Thus the functionU�0� does not change drastically with the angle andcan be substituted by a constant value �e.g., U 1.18�. Note that the function U��0� is multiplied bythe global transmittance t in Eq. �20�, the latter beinga small number for the thick clouds considered here.Thus one can see that small errors in the function

U��0� do not much influence the degree of polariza-tion calculated with Eq. �20�.

Thus approximate equations �20�, �22�, �25�, and�17� allow us to reduce the complexity of calculationsof the function P��0�. Note that exact calculationsrequire the solution of the four coupled integro-differential equations for the Stokes vector compo-nents of the vector radiative transfer theory,3 whichis a far more complex task than application of Eq.�20�.

In more general cases �see Eqs. �14� and �15�� thefunctions K0��� and R̂�

0��, �0, �� for a nonabsorbingsemi-infinite light-scattering medium should befound numerically. They depend on the phase ma-trix of a random medium in question. However,they do not depend on the single-scattering albedo �0and the optical thickness � by definition. Anotherinteresting feature of these functions is their weakdependence on the microstructure parameters �size,shape, and chemical composition of particles� of amedium under study in the broad range of angularparameters �, �0, �. This is due to the randomiza-tion of both polarization states and propagation di-rections of photons in highly scattering nonabsorbingsemi-infinite layers irrespective of the local opticalproperties of the medium in which they propagate.

C. Weakly Absorbing Media

Equation �20� is easily generalized to account for theabsorption of light in a cloudy medium. Namely, itfollows from Eqs. �12� and �14�21:

P��0� �P�*��0�

1 � U*��0�t exp��x � y�, (26)

where the global transmittance t is given by Eq. �17�and

U*��0� �K01�1�K01��0�

R�11*��0�. (27)

Values of P�*��0� and R�11*��0, 1� represent the de-gree of polarization and the reflection function of asemi-infinite weakly absorbing cloud at the nadir-viewing geometry. They can be expressed via thecorrespondent functions for nonabsorbing media.Indeed, it follows for the reflection function R� �R�11*27:

R���, �0, �� � R�0��, �0, ��exp��yU��, �0, ���,

(28)

where

U��, �0, �� � K0���K0��0��R�0–1��0, �, ��, (29)

K0 � K01, and R�0 is the reflection function of the

semi-infinite nonabsorbing light-scattering medium.Our calculations show that the accuracy of Eq. �28�

can be improved at y� 1 if we substitute the value of

Fig. 2. Dependence of the escape function K01 on the cosine of theincidence angle for the Henyey–Greenstein phase function withthe asymmetry parameter g 0, 0.5, 0.75, 0.8.28 Results of cal-culations with Eqs. �24� are presented by a dashed line.

Fig. 3. Dependence U�0� according to Eq. �25�. The dashed linerepresents the average value of U 1.18.

3654 APPLIED OPTICS � Vol. 41, No. 18 � 20 June 2002

U by s �1 � 0.05y�U. Thus we have for the valueof R�11*��0, 1� in Eq. �27�:

R�11*��, �0, �� � R�0��, �0, ��exp��sy�. (30)

Substituting Eqs. �27� and �30� into Eq. �26�, weobtain

P��0� �P�*��0�

1 � U��0�t exp��x � y�1 � s��, (31)

where we also used Eq. �21�.Let us find now the relationship between functions

P�*��0� and P�0��0�. For this we will use Eq. �12�

and the fact that the value of R�21*��0� for semi-infinite weakly absorbing media does not differ muchfrom the same value for the nonabsorbing case.24

Then we have by using Eqs. �12� and �30�

P�*��0� � P�0��, �0, ��exp�sy�. (32)

4. Inverse Problem

A. Algorithm Description

Equations �20� and �31� can be used for the semi-analytical retrieval of droplet sizes from the two-wavelength polarimetric inversion algorithm. Letus rewrite these equations here, explicitly showingthe wavelength dependence:

P��1� �P�

0��1�

1 � Ut��1�, (33)

P��2� �P�

0��2�exp�s��2�y��2��

1 � Ut��2�exp��x��2� � y��2��1 � s��2���,

(34)

where �1 is the wavelength in the visible and �2 is thewavelength in the infrared. We also accounted forEq. �32�. A closer look at these equations shows thatvalues P��1� and P��2� depend only on the solar ele-vation, the optical thickness at each wavelength, thedroplet size distribution, and the refractive index. Itis known, however, that the influence of the type ofthe particle size distribution on the integral light-scattering characteristics, such as x and y in Eqs. �33�and �34�, is weak.29 So we will neglect this depen-dence and characterize the size of droplets by just onenumber—the effective radius of droplets. We alsoignore the possible variability of the half-width of thedroplet size distribution and use only the distribution�Eq. �13�� at � 6 in the discussion, which follows.The dependence of the degree of polarization of thesingly scattered light on the value of � is presented inFig. 1�b�. We see that this dependence can be ne-glected for most scattering angles. It is significant,however, at � � 150–155 degrees.

The parametrizations for the integral light-scattering characteristics �Eqs. �16�� as well as func-tions P�

0��1�, P�0��2� at wavelengths �1 0.65 �m

and �2 1.55 �m are presented in Appendices A andB. We will use these two wavelengths in the devel-opment of the inversion algorithm.

Thus we have the system of the two Eqs. �33� and�34� to find the optical thickness in the visible and thesize of droplets. We proceed as follows. First of all,we express the value ���1� in Eq. �33� via the mea-sured degree of polarization P��1�:

���1� �4�t�1��1� � ��

3�1 � g��1��, (35)

where

t��1� �1U 1 �

P�0��1�

P��1� , (36)

and we also used Eq. �22�. Second, we use the factthat the ratio �aef� ���2�����1� depends only on theeffective radius of droplets �see Appendix A�. So wecan substitute the value of ���2� by �aef����1� in Eq.�34�. It gives us a single transcendent equation �Eq.�34�� for the effective radius determination. Thisequation can be easily solved with standard tech-niques. The value of ���1� can be obtained from Eq.�35� afterward. This completes the proposed algo-rithm description.

B. Accuracy of the Retrieval Algorithm

Let us study the errors associated with the retrievalalgorithm proposed. We will use the fixed viewinggeometry �0 37°, 0°�, in which the sensitivityof the measured degree of polarization in the visibleto the value of the effective radius of droplets is high�see Fig. 1�. However, one can also choose anothergeometry. This is mostly due to the fact that theinformation on sizes of droplets is contained in theparameter y, which does not depend on the observa-tion geometry. This is an important point.

First, let us study the accuracy of Eqs. �33� and�34�. The results of comparisons with the exact ra-

Fig. 4. Dependence of the degree of polarization on the opticalthickness at � 0.65 �m, 1.55 �m, 0 37°, 0°, aef 6 �m,and � 6 �see Eq. �13�� according to the exact radiative transfercalculations �symbols� and Eqs. �33�, �34�, and �B1�.

20 June 2002 � Vol. 41, No. 18 � APPLIED OPTICS 3655

diative transfer code are presented in Fig. 4. Onecan see that errors of approximate formulas at opticalthicknesses larger than 5 are smaller than possiblemeasurement and forward-model assumption errors.So approximate equations can be applied to the in-verse problem solution.

Let us consider now the error in the retrieval of aefand � that is due to possible measurement errors.For this, we have calculated the values of P��1� andP��2� at aef 6 �m and � 10 with the exact radi-ative transfer code for wavelengths 0.65 and 1.55 �m.Then we assumed that the actual values of P��1� andP��2� differ from the calculated values by the relativepositive error !, which was changed from 0% to 10%.The results are presented in Fig. 5. We see that theerrors in the retrieved values of aef and � are smallerthan 12% in this case. This shows that our algo-rithm is not particularly sensitive to the measure-ment errors at aef 6 �m and � 10. Clearly, theerrors of retrieval of the optical thickness increasewith the optical thickness. This is due to the factthat the degree of polarization for extremely thickclouds is not dependent on the optical thickness at all�see Eqs. �33� and �34� at t 0�. So it cannot beretrieved in principle. This is not the case as far asthe effective radius of droplets is concerned. It canbe derived for clouds of arbitrarily large thicknesses.However, for large clouds the transmission t is ap-proximately equal to zero and the transcendent equa-tion �Eq. �34�� transforms into more simple equation�Eq. �32��.

Solving this single transcendent equation numeri-cally, one obtains the value of aef, which subsequentlycan be used to find the optical thickness and liquidwater path of clouds, if they are not thick. Cloudswith optical thicknesses around 15 and larger couldbe considered as semi-infinite for wavelengths largerthan 1.55 �m. Of course, this is not the case in thevisible. Knowing the values of aef and �, one can findit easy to obtain many other parameters of cloudy

media, including their transmittance, reflection func-tion, and spherical albedo.

Spectral measurements at several wavelengthscan improve the accuracy of the algorithm. How-ever, remarkably, as we have discussed before, thereis even a possibility17 of finding the size of particlesand the liquid water path of clouds from measure-ments of the reflection function and the degree ofpolarization at the single wavelength in the visible.For this, one should make measurements at the rain-bow geometry.

5. Conclusions

We presented here a system of simple equations thatcan be used for parametrizations of radiative andpolarization characteristics of cloudy media. The al-gorithm of the inverse problem solution was pro-posed. It is based on spectral measurements of thedegree of polarization of reflected light in the visibleand the infrared at the rainbow geometry. Notethat the method proposed is not significantly influ-enced by the choice of the observation geometry.This is not the case, e.g., when only measurements inthe visible are used.9,17

The reflection of light from the surface and scatter-ing by molecules and aerosols was neglected in thisstudy. However, they can easily be taken into ac-count. The method described can also be used formultiple light-scattering disperse media other thanwater clouds. It is also applicable with slight mod-ifications for the retrieval of the spectral dependenceof the imaginary part of the refractive index of drop-lets, which can differ from the refractive index of purewater because of various sources of contamination.

Appendix A: Local Optical Characteristics ofCloudy Media

We use the parametrizations of the extinction coeffi-cient �ext, the asymmetry parameter g, and the ab-sorption coefficient �abs of cloudy media:

�ext �3C2aef

1 �1.1�xef�

2�3 , (A1)

�abs �4�"C� #

n0

4

pn�xef�n, (A2)

1 � g �#n0

4

qn�xef��2n�3. (A3)

Here C is the volumetric concentration of droplets, xef 2�aef��, " is the imaginary part of the refractiveindex of particles. The parameters pn at the1.55-�m wavelength, where there is an appreciableamount of light absorption by water droplets, arep0 1.671, p1 0.0025, p2 �2.365 $ 10�4, p3 2.861 $ 10�6, and p4 �1.05 $ 10�8. The param-eters qn depend on the refractive index of particles.They are given in Table 1 for water droplets at the0.65- and 1.55-�m wavelengths. The accuracy ofEqs. �A1�–�A3� is better than 3% for the effective

Fig. 5. Error of the retrieval as the function of the errors ofmeasurements.

3656 APPLIED OPTICS � Vol. 41, No. 18 � 20 June 2002

radius of droplets in the range of 4–16 �m, which ismost common for water clouds.

Clearly, the ratio of optical thicknesses for se-lected wavelengths is given by ratios of values �1 ��1.1��xef�

2�3�� for both wavelengths.

Appendix B: Degree of Polarization at theRainbow Angle

We present here the results of the parametrization ofthe degree of polarization at the rainbow geometry.It is assumed that the incidence angle is equal to 37deg and the observation angle is zero, which meansthe nadir observation. Also, we consider the case ofa semi-infinite nonabsorbing medium. Then the de-gree of polarization depends only on the effective ra-dius of droplets at this geometry �see Fig. 1�. It canbe presented by the following equation:

P�0 � � �

%

1 � exp�& � �aef�, (B1)

where the parameters �, %, &, � are presented in Table2 at the 0.65- and 1.55-�m wavelengths. The accu-racy of these equations is better than 6% at the ef-fective radii in the range of 4–16 �m �see Fig. 6�.

This research was supported by Volkswagen Fel-lowship, Germany. The author is grateful to T.Takashima and K. Masuda for providing their vectorradiative transfer code.

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Fig. 6. Degree of polarization, calculated with Eq. �B1� �line� andexact radiative transfer code �symbols� at �a� � 0.65 �m, �b� 1.55�m. Other parameters are � 0, 0 37°, 0°, aef 6 �m,and � 6.

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�, �m q0 q1 q2 q3 q4

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Table 2. Parameters in Eq. �A4� at the Wavelengths 0.65 and 1.55 �m

�, �m � % & �

0.65 9.1 22.0 0.37 0.171.55 5.5 8.4 �1.0 0.29

20 June 2002 � Vol. 41, No. 18 � APPLIED OPTICS 3657

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