ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS ... - CAST

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUSFROM SATELLITE REFLECTANCE DATAModified Exponential ApproximationJULIA KARLGÅRDMay 2008Diploma work of 15 ESCTsDepartment of Physics, Lund UniversitySWEDISH TITLE: Bestämning av molndroppsstorlek med hjälp av satellitmätningarSUPERVISOR: Erik SwietlickiCONTACT: julia.karlgard.865@student.lu.se

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________ABSTRACTA major uncertainty in modelling future climate is the impact of aerosols on cloud formation,and its influence on the earth’s radiation fluxes. In order to study this indirect effect ofaerosols and its role in global warming, it is important to be able to model the growth of clouddrops.In this study, cloud droplet effective radius is modelled from satellite reflectancemeasurements in two MODIS wavebands, 0.65 µm (visible) and 2.1 µm (NIR). The effectiveradius r e is preferred to mean or mode radius since it accounts for the size distribution of thedroplets within the cloud. Reflectance data is taken from a satellite scene covering southernSweden, on May 9 th , 2004. The approach is a relatively simple approximate forward modelcalled the Modified Exponential Approximation (MEA), developed by Kokhanovsky et al.(2003). This model is valid for optically thick water clouds, and is here applied to cloud pixelswith NIR reflectance R 2 > 0.2 and optical thickness τ > 10. The underlying principle of themodel is the asymptotic theory. This theory is based on the fact that reflectance in thenonabsorbing wavelengths (visible) is mainly a function of τ, while reflectance in theabsorbing wavelengths (near and mid infrared) is governed by r e . For large optical thickness,the reflection function is close to the known asymptotic equations. Also, when the opticalthickness is large enough, τ and r e can be determined nearly independently of each other.The results showed good agreement with near coastline clouds in the study by Rosenfeld andLensky (1998) for R 2 > 0.3. Median and mode effective radius was 14.6 and 15 µmrespectively. A mode r e of 15 µm indicates precipitating clouds or clouds close toprecipitation, since 14-15 µm is considered to be a rainout threshold. Smaller reflectancevalues in the near infrared resulted in larger mode and median r e (18 and 17-20 µmrespectively), and τ < 5. This implies that the model is not reliable for small reflectance valuesin the near infrared. The error for pixels with τ > 10 is estimated to 10-20%, but varies withviewing geometry, optical thickness and probably also wavebands used.i

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________SAMMANFATTNINGMoln förknippas ofta med väder och kanske i synnerhet lågtryck. Men moln spelar också enviktig roll i jordens uppvärmning. Moln både stänger ute värme, i form av inkommandesolstrålning, och isolerar, i form av utgående värmestrålning från jorden. Den totala effektenberor bl. a. på molnets höjd ovanför marken, tjocklek och inre struktur, såsom droppstorlekoch droppkoncentration (antalet molndroppar per volymenhet). Droppstorleken tillsammansmed droppkoncentrationen avgör molnets s. k optiska täthet, en egenskap som beskriver hurmycket av det inkommande solljuset som släpps igenom en viss tjocklek.Molnens egenskaper, då framför allt droppstorleken, avgör också nederbörden. Ju större ochtyngre droppar, desto troligare att dessa faller ut som regn, snö eller hagel. Enligt studier(Rosenfeld och Lensky, 1998; Pinsky och Khain, 2002) ligger den kritiska (effektiva)droppradien innan molndropparna faller ut som nederbörd på ca 14-15 µm. Det har visat sigatt aerosoler (små, luftburna partiklar t ex sot, damm, salter etc.) kan påverka droppstorleken.Vissa aerosoler (hydrofila) fungerar som kondensationskärna för molndroppar genom attutgöra en lämplig yta som atmosfärens vattenånga kan kondensera mot och bildavattendroppar. I ett luftpaket med hög aersolkoncentration finns således mångakondensationskärnor, vilket resulterar i många vattendroppar. Eftersom mängden vattenånga iluftpaketet är begränsad, blir vattendropparna som bildas mindre än de normalt skulle bli i”ren” luft (i absolut ren luft bildas teoretiskt sett inga vattendroppar alls, eftersom det intefinns någon kondensationskärna). De mindre vattendropparna har lägre sannolikhet att falla utsom regn och luftpaketet, eller molnet, får på sätt längre livslängd. Detta i sin tur kan leda tillförändrade nederbördsmönster och hydrologi, t ex genom att molnen hinner transportera bortvattnet i atmosfären innan det regnar ut.En effekt av att molndropparna i förorenade moln är mindre än normalt är att molnetsreflektiva egenskaper förändras. Då den inkommande solstrålningen passerar genommolndropparna absorberas en del medan en del sprids genom reflektion och ljusets brytning.Förenklat kan man säga att ju mer vatten molnet innehåller, desto mer solstrålning absorberas,och att små partiklar (små i förhållande till ljusets våglängd) sprider ljus bättre stora partiklar.Det senare skulle innebära att moln med hög aerosolkoncentration bättre sprider inkommandesolstrålning än moln med lägre aerosolkoncentration och på så sätt har en kylande effekt påklimatet. Detta kallas även aerosolers indirekta effekt. Enligt IPCC (2001) är den indirektaeffekten av aerosoler den enskilt största osäkerhetsfaktorn i dagens klimatmodeller. Manmenar att dagens aerosolhalter i luften troligtvis har en dämpande effekt på den globalauppvärmningen, men att betydelsen av denna effekt i förhållande till andra faktorer är osäker.För att kunna förutspå framtida klimatscenarier är det alltså av stor vikt att känna till huraerosoler påverkar molnets egenskaper, både i fråga om reflektans och nederbörd. Idealet äratt kunna jämföra molnegenskaper i luftmassor med låg respektive hög aerosolkoncentrationoch utifrån detta kunna dra slutsatser om aerosolers molnpåverkan. Eftersom ”provtagningar”av moln m h a flygningar är dyra och tidskrävande att genomföra är ett alternativ (ellerkomplement) att använda sig av datamodeller, för att utifrån satellitmätningar av moln kunnata reda på molnets tjocklek och droppstorlek. Man utnyttjar här effekten av att moln med olikadroppstorlek och dropptäthet har olika reflektans (d v s reflekterar och absorberar olikamycket). Tillsammans med kännedom om luftmassornas aersolhalt skulle dessa modellerkunna ge en uppfattning om aersolers inverkan på moln och molnbildning.ii

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________I denna studie har en enklare modell använts på satellitdata från 9e maj 2004, för att beräknadroppsstorleken i moln över ett område i södra Sverige. Modellen kallas ModifiedExponential Approximation (MEA) och utgår från den uppmätta reflektansen i två olikavåglängdsband från satelliten MODIS Aqua: 0.620-0.670 µm och 2.105-2.155 µm. Modellenär en förenkling av den s k Asymptotiska teorin, en teori som bygger på att för tillräckligt tätamoln kan reflektansen beskrivas av kända funktioner, samt att den effektiva radien och denoptiska tätheten kan bestämmas näst intill oberoende av varandra. Istället för att beräknamedelradien eller typradien beräknas den effektiva droppradien eftersom denna är ett viktatmått på radien, som tar hänsyn till storleksfördelningen i molnet. MEA har tagits fram av bl.a. Alexander A. Kokhanovsky, docent i optik vid institutet för miljöfysik på universitetet iBremen, och har i flera studier använts i syfte att uppskatta molns optiska egenskaper.Studien resulterade i ett drygt 100-tal undersökta pixlar i satellitbilden, där den effektivadroppstorleken och optiska tätheten beräknats för de pixlar som uppfyller vissa krav påreflektansen. Droppstorleken för moln med optisk täthet större än 10 ligger i intervallet 11,4 –15,4 μm, med en medianstorlek på ca 14.6 μm. Detta ligger nära det förväntade värdet på 14μm, som enligt studier är typiskt för kustnära moln. Felet i modellen uppskattas till 10-20%,beroende på hur satelliten står i förhållande till det undersökta området, och molnets optiskatäthet.Den relativt enkla modellen har visat sig vara snabb i beräkningarna och samtidigt ge resultatjämförbara med betydligt mer avancerade och beräkningstunga modeller. Målet medmodellen är att kunna använda denna tillsammans med data över molntoppstemperatur för attkunna göra beräkningar av molnets tillväxt, och hur denna påverkas av aersolhalten i luften.För en framtida användning skulle en noggrannare utvärdering av modellens tillförlitlighetvara av stort värde, liksom en analys av modellens känslighet för vissa parametrar. Enutveckling av modellen som skulle kunna urskilja varma moln (bestående av enbartvattendroppar) från kalla moln (enbart iskristaller) och mixed phased (blandning avvattendroppar och iskristaller) skulle också vara intressant, eftersom modellen bygger påantagandet om varma moln, d v s moln enbart uppbyggda av vattendroppar.iii

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________iv

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________INDEXABSTRACT............................................................................................................................................................. iSAMMANFATTNING.......................................................................................................................................... ii1 INTRODUCTION .............................................................................................................................................. 31.1 BACKGROUND ............................................................................................................................................... 31.2 AIM ............................................................................................................................................................... 32 METHOD ........................................................................................................................................................... 42.1 THEORY......................................................................................................................................................... 42.1.1 Light scattering ..................................................................................................................................... 42.1.2 Radiative transfer ................................................................................................................................. 62.1.3 Asymptotic theory and Reflection function ........................................................................................... 92.1.4 Semi-analytical method and Modified Exponential Approximation ................................................... 122.2 DATA AND COMPUTATIONS ......................................................................................................................... 163. RESULTS AND DISCUSSION ..................................................................................................................... 204. VALIDATION AND ERROR ANALYSIS ................................................................................................... 225. CONCLUSIONS ............................................................................................................................................. 24APPENDIX .......................................................................................................................................................... 27REFERENCES .................................................................................................................................................... 291

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________2

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________1 INTRODUCTION1.1 BackgroundOne of the major uncertainties in the models used for creating projections of future climate isthe indirect effect of aerosols on global warming. In masses of air with high moisture content,aerosols may act as condensation nuclei and trigger the forming of cloud droplets, having anoverall cooling effect on the global climate (IPCC, 2007). A high concentration of aerosolsincreases the number of cloud condensation nuclei (CCN) and thus generating more, butsmaller cloud droplets. Hence aerosols affect cloud droplet size, optical thickness, growth rateand life time of clouds to name a few parameters. When it comes to anthropogenic aerosols,often produced close to the ground by industrial and engine combustion, the influence onconvective clouds such as cumulus clouds is believed to be significant, since these cloudtypes are fed by air masses rising from below.Studying the indirect effect of aerosols involves modelling cloud dynamics and cloudstructures. However, cloud modelling is a complex issue, since clouds are seldom, if everhomogeneous. A cloud cannot be considered as a single entity, but rather as a composition ofbillions of much smaller units, cloud droplets. The drop size distribution varies with heightand the phase may change from water to mixed phase to ice through the vertical profile of thecloud giving rise to different radiative characteristics. Since many satellites provide data ofphysical parameters such as reflectivity and emissivity at several wavelengths, covering mostparts of the world with frequent time intervals, satellite observations is a valuable source ofinformation. Through the development of technology and computer power, which has enabledmanaging large amounts of data and faster computations, satellite measurements have becomeextensively used in cloud and atmospheric research in the last decades.Numerous studies with focus on radiative characteristics of clouds have been performed, butbecause of the complexity of cloud modelling most algorithms are developed for planeparallel clouds, such as stratus clouds. The basic principle in the technique of using multispectralreflectance data for determining microphysical properties in clouds, such as opticalthickness and droplet sizes, is the variations in reflectance due to these two parameters atdifferent wavelengths. In the visible region the reflection function is primarily a function ofcloud optical thickness, while in the near or mid infrared the reflection function dependsprimarily on cloud droplet sizes (Nakajima and King, 1990; Liou, 1992; Kokhanovsky, 2006).1.2 AimIn this study the cloud effective radius of cumulus (water) clouds over southern Sweden ismodelled from satellite data. When modelling cloud droplet sizes the effective radius ispreferable to mode or mean radius since it better accounts for the size distribution within thecloud. The effective radius is defined as follows:∞∫∞∫32r e= r n( r)dr r n(r)dr(1.1)00where n(r) is the particle size distribution and r is the particle radius (Rosenfeld and Lensky,1998; Nakajima and King, 1990).3

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________The model used is a semi-analytical method called modified exponential approximation(MEA). The MEA is based on the asymptotic theory for radiative transfer problems, andrequires relatively few computations and input parameters. Reflectance data is taken fromband 1 and 7 (corresponding to wavelengths ~ 0.65 µm and 2.1 µm respectively) in MODISAqua observations on the 9 th of May 2004. The final product (cloud droplet effective radius)is meant to be used in combination with cloud top temperature data, with the purpose to studythe growth rate of cloud droplets and how this may be affected by the amount of aerosols inthe air.2 METHOD2.1 Theory2.1.1 Light scatteringScattering occurs when incident light interacts with matter such as an atmospheric molecule,aerosols or a water droplet. Depending on the wavelength of the incident light and the sizeand shape of the scattering particle, scattering patterns appear different in different cases. Forspherical particles a size parameter x may be defined; x = 2π⋅a/λ, where a is the particleradius and λ the wavelength of the incident light. When x ≥ 1 scattering is referred to asLorentz-Mie scattering or sometimes Mie theory (Liou 2002, p. 96). This is generally the casewhen light in the visible and near- and mid infrared (NIR, MIR) region interacts with aerosolsor cloud droplets (Liou, 2002 p. 97). The light is scattered in all directions, but the forwardscattering predominates. The intensity of light I after scattering in a direction θ (scatteringangle) is described byP(θ ) ⎛σs ⎞ P(θ )I( θ ) = I0Ωeff= I0 ⎜ ⎟2(2.1)4π⎝ r ⎠ 4πwhere I 0 denotes the intensity of incident light, Ω eff is the effective solid angle of scattering,P(θ) is the phase function, describing the probability of a photon being scattered in thedirection θ, σ s is the scattering cross section and r is the distance between the observer and thescattering particle (Liou, 2002, p. 96). The Mie theory is not a physical theory or law in itselfbut rather a complete analytical solution to Maxwell’s equations, describing the absorptionand scattering of light by spherical particles (Liou 1992, p. 262). The pattern of light intensitydue to scattering is complex and is governed by the size parameter and the refractive index(Liou 1992, p. 257). For a given particle size and wavelength the phase function P(θ) can bederived from the Mie solution, expressed in terms of Legendre polynomials, P l :N∑P(cosθ ) = c lP l(cosθ)(2.2)l=0where c l is the expansion coefficient. Because of the orthogonal property of P l we may writec l in the form12l+ 1cl= P(cosθ) Pl(cosθ) d cosθ2∫(2.3)−1For l = 0, c 0 = 1, representing the normalization of the phase function (the probability of thephoton being scattered in any direction being 1). When l = 1 Eq. (2.3) can be used to define4

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________the asymmetry parameter g, which denotes the relative strength of the forward scattering(Liou, 2002, p. 105; Kokhanovsky, 2006, p. 96):1c11g = = P(cosθ)cosθdcosθ3 2∫(2.4)−1Even with the help of computers, it is not a simple task to derive the Legendre polynomials P land the expansion coefficients c l from the Mie theory. Therefore an approximate analyticexpression of the phase function may be desirable as an alternative to the numerical methods.One such approximate equation that has been widely used is the Henyey-Greenstein phasefunction, where the phase function is expressed in terms of the asymmetry parameter g(Mishchenko et al., 1999; Liou, 1992, p. 127);pHG=(1 + g22(1 − g )− 2gcosθ)3 / 2=N∑l=0l(2l+ 1) g P (cosθ)l(2.5)The Henyey-Greenstein phase function is best suited for the case where the forward scatteringis less pronounced, i.e. for smaller size parameters, and one should bear in mind that one ofthe main features for light scattering by aerosols and cloud droplets is strong forwardscattering. Kokhanovsky (2004a) derived an approximate equation for the phase function,where the forward scattering peak is much stronger than in the Henyey-Greenstein-function,and where the smaller peak at 145° is distinguishable. This was used for retrieval of cloudmicrophysical properties and is given by:p(52−C⋅θ−βi( θ −θi)θ ) = Qe + ∑bie(2.6)i=1where Q = 17.7, C = 3.9. p is used instead of P to indicate that the method is anapproximation. b i , β i and θ i are derived by parameterization of Mie theory results, values aregiven in Table 1 below (Kokhanovsky, 2004a; Kokhanovsky et al., 2003).Table 1 Parameters b i , β i and θ i for approximate phase function(Kokhanovsky, 2004).i b i β i θ i1 1744.0 1200.0 0.02 0.17 75.0 2.53 0.30 4826.0 pi4 0.20 50.0 pi5 0.15 1.0 piFigure 1 illustrates the difference between the Henyey-Greenstein phase function (Eq. (2.5))and the approximate phase function by Kokhanovsky (Eq. (2.6)). The drawback of theKokhanovsky equation is that it does not account for the influence of the size parameter onthe phase function. It is merely an approximation for all clouds, with no dependence oneffective radius, optical thickness or even wavelength, but has proven to be useful when5

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________modelling single and multiple scattering in cloudy atmospheres in a general sense(Kokhanovsky, 2004a). However the indirect purpose of this paper is to study the influence ofdroplet radius on the reflection function in two different wavelengths, so the Henyey-Greenstein function in this case is preferred. Since scattering is dominated by multiplescattering (thus single scattering characteristics becoming less pronounced), the dependenceof the cloud reflection function on the phase function is however rather weak (Kokhanovsky2006, p. 148). In the end the choice of phase function therefore mainly depends on thepurpose of use.10 4 Θ10 3KokhanovskyHenyey-Greenstein10 2phase function10 110 010 -110 -20 20 40 60 80 100 120 140 160 180Figure 1 Comparison between the Henyey-Greenstein phase function (blue line) and theparameterized phase function by Kokhanovsky (red line). g = 0.85.2.1.2 Radiative transferFor optically thin media a large part of the scattering events is due to single scattering, i.e. theincident photon is only scattered once before escaping the media (scattered back to space).However, for media with larger optical depth, such as thick stratus clouds or large cumulusclouds, scattering is dominated by multiple scattering, i.e. photons are scattered several timesbetween the cloud drops before escaping the media. (Liou, 2002, p. 105). The optical depth τis defined from the light extinction coefficient k e (Liou, 2002, p. 103):∞∫zτ = k edz(2.7)The radiative transfer theory is used to predict how the intensity of light changes as incomingdirect solar light (solar flux density) F 0 is scattered and transmitted through the media, in thiscase the cloud. The basic theory in radiative transfer is an assumption of a plane parallelatmosphere or cloud, sliced into homogeneous layers with a differential thickness Δz. Asincoming solar flux F 0 from the direction (-v 0 , ϕ 0 ) (minus sign denotes downward direction) istraversed through the layer Δz it undergoes several processes affecting the intensity of the6

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________outgoing light in the direction (v, ϕ) (Figure 2). The main processes are single and multiplescattering, emission and absorption (Liou, 2002, p. 152). A portion of the light is scatteredand emitted downwards, entering the underlying layer where the same processes occur.Outgoing light is here re-entering the layer above or scattered further downwards. Thisprocedure is run for all the layers within the cloud and eventually the light is eithertransmitted through the whole cloud or the intensity becomes negligible (Figure 3).vv 0φNϕϕ 0ΔzFigure 2 Viewing geometry of incoming and reflected light.I(0,µ,φ)Top τ = 0I(τ n ,µ,φ)I(0,-µ,φ)nτ = τ nI(τ n ,-µ,φ)I(τ c ,µ,φ)Bottomτ = τ cI(τ c ,-µ,φ)Figure 3 Principle of radiative transfer. The intensity is expressed in terms of optical thickness τ , (τ c is thecloud optical thickness) incident angle µ ≡ cos(v), and relative azimuth angle φ = ϕ 0 - ϕ7

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________How deep into the medium one must look before the intensity is reduced by a certain degreeis given by the optical depth τ, which depends on the wavelength of the incident light and thestructure of the media (effective radius, liquid water path). In the simplest case with noscattering, the reduction in intensity I at a given wavelength λ is due only to absorption. Theattenuation in intensity can then be expressed in terms of the intensity and the extinctioncoefficient k e (assumed to be constant within a homogeneous layer Δz) (Kokhanovsky 2006,p. 114):dIλ= −ke,λIλdz(2.8)For a more realistic, non homogeneous media (with finite volume and thickness Δz) and theparticle (or droplet) number density n, the extinction coefficients k e is defined byke=∫Δzσe( z)n(z)dzΔz(2.9)with σ e denoting the cross section for extinction.In most cases the attenuation is however due to both absorption and scattering so thatk e = k s + k a . In resemblance with Eq. (2.8) the scattering and absorption coefficient, k s and k a ,can be definedσs,a( z)n(z)ks,a= ∫ dz(2.10)ΔzΔzwhere σ s,a denotes the cross section for scattering and absorption respectively. From k e , k s andk a the single scattering albedo is defined.ksω0=koreka1− ω 0=(2.11)keThe single scattering albedo is thus the ratio between the scattering coefficient and the totalextinction coefficient. In the visible absorption is negligible and ω 0 = 1. This is sometimesreferred to as conservative scattering, i. e. where no energy is lost due to absorption.For ω 0 < 1 part of the extinction is due to absorption (non conservative scattering), seen in theNIR and MIR wavebands. The problem becomes more complex when considering theincrease in intensity caused by emission and multiple scattering (Liou, 2002, p. 27). It isnecessary to define the change in intensity, dI in such a way that it accounts for both the lossin intensity by scattering and absorption and the increase in intensity due to multiplescattering and emission:dI = −σe, λnIdz+ jλndz(2.12)Here the extinction coefficient is exchanged by the extinction cross section σ e,λ (units of areaper number) and the number density n (units per volume) of the material. j λ is the sourcefunction coefficient, which has the corresponding physical meaning as extinction crosssection. From j λ and σ e,λ the source function is defined as the ratioJ ≡ j(2.13)λ λσ ext,λ8

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________Dividing Eq. (2.12) by σ e,λ ⋅n⋅dz generates the general radiative transfer equation (Liou 2002,p. 28; Liou 1992, p. 109).dI= −Iλ + J λ(2.14)σ n ⋅ dze,λThis equation is the fundamental relationship in radiative transfer problems, where thesolution may give information on several radiative parameters such as optical thickness,single scattering albedo, effective radius etc.2.1.3 Asymptotic theory and Reflection functionThe discrete ordinate method, developed by Chandrasekhar in 1950, is an exact method forsolving radiative transfer problems numerically (Rybicki, 1996). The method is often used foraccurate calculations of radiative properties of cloudy atmospheres (Liou, 1992, p. 108). Oneof the main products from the discrete ordinate method is the reflectance function (in remotesensing often referred to as Bidirectional Reflection Distribution Function or simply BRDF)which gives the reflected intensity in relation to incident solar flux F 0 density. The reflectanceis defined as follows:R(τ;µ , µ0πIr( τ = 0; µ , φ), φ)= (2.15)µ F00Here I r denotes the intensity of the reflected light at top of the atmosphere (TOA), in thedirection (µ,φ), where µ = |cos(v)|, µ 0 = cos(v 0 ) and φ = |ϕ-ϕ 0 | (Kokhanovsky et al., 2003). Inremote sensing studies of clouds the reflectance I r , and hence R, is however measured in nonorweakly absorbing wavelengths, therefore the loss in intensity due to water vapourabsorption is often neglected and I r is approximated to the intensity at cloud top.Knowing parameters such as cloud optical depth τ, single scattering albedo ω 0 and effectiveradius r e one should be able to predict the reflection function from the viewing geometry.However, often the problem is the reverse. From satellite observations the reflectance of acertain viewing geometry is given, while information on optical depth and effective radius isdesired. The fact that reflectance in the visible (nonabsorbing) wavelengths is mainly afunction of optical thickness while reflectance in the NIR and MIR (absorbing) wavelengthsis governed by effective radius makes it possible to determine the optical thickness andeffective radius using reflectance data from two or more wavebands. For large opticalthickness (≥12) the sensitivity of the nonabsorbing and absorbing wavelengths to these twoparameters are almost orthogonal, so τ and r e can be determined nearly independently of eachother (Nakajima and King, 1990). Figure 4a and 4b illustrates how the reflection function inthe visible and NIR is governed mainly by optical thickness and effective radius, respectively.Running the numerical methods backwards is often not possible, so the approach is to makepre-calculated look-up tables of the reflection function at given viewing geometry and cloudparameters. This method is slow, and if conditions (ground surface albedo, phase functionrepresentation) are changed, new look-up tables must be created. Thus, when dealing withsatellite images, with thousands of pixels being analysed with varying conditions, a simpler,analytical method may be desirable. Therefore another approach called the asymptotic theoryis often used for optically thick clouds. The underlying principle is that when opticalthickness is large enough the numerical solution of radiative transfer is close to the known9

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________asymptotic equations presented below. King (1987) found that the accuracy of the asymptotictheory for both conservative and non conservative scattering is within 1% if the scaled opticalthickness τ*= τ c (1-ω 0 g) is equal or larger than 1.45 (τ c denoting cloud optical thickness).Typically for water clouds g ≈ 0.85 which corresponds to a cloud optical thickness equal to orlarger than 9, which makes the theory applicable to most types of large clouds.Figure 4a The reflection function of a plane parallel homogeneous cloud at nadirobservation angle for different values of optical thickness τ and effective radius r ef .λ = 0.65 µm. (Kokhanovsky and Rozanov, 2003)Figure 4b Same as in figure 4a, but for λ = 1.55 µm. (Kokhanovsky and Rozanov, 2003)10

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________Equations (2.16) and (2.17a) are valid for plane parallel homogeneous layer overlying aLambertian surface with albedo A g (King, 1987; Kokhanovsky and Nauss, 2006; Nakajimaand King, 1990). For conservative scattering with ω 0 = 1 (i.e. in the visible) the reflectancefunction of R is writtenR(4(1 − A ) K(µ ) K(µ )g0τc; µ , µ0, φ)= R ∞( µ , µ0, φ)−3(1 − Ag)(1 − g)(τc+ 2q0) + 4A(2.16)gwhere, R ∞ denotes the reflected intensity of a semi-infinite layer (infinite in the downwarddirection, but with an upper surface) with the same optical properties (single scattering albedoand phase function) as the finite layer (Nakajima and King, 1990, 1992; Kokhanovsky andNauss, 2006), τ c is the cloud optical thickness, q 0 ≈ 0.714/(1-g), and K(µ) and K(µ 0 ) theescape functions, describing the angular distribution of light escaping a semi-infinite cloudfrom sources located deep inside the medium.For non-conservative scattering in weakly absorbing layers, i.e in NIR and MIR outside thewater vapour absorption bands, R is writtenR( c ∞ 0τ ; µ , µ0, φ)= R ( µ , µ , φ)2[(1− A A*)l − A mn ]−2kτcmgge K(µ ) K(µ0)−2 −2kτc2 −2kτ(2.17a)c(1 − A A*)(1− l e ) + A mn legIt should be noted that the influence on A g is rather small and sometimes can be left out whichgivesml exp[ − 2keτc]R( τ c; µ , µ0, φ)= R ∞( µ , µ0, φ)− K(µ ) K(µ )2 0 (2.17b)1−l exp − 2kτg[ ]with the extinction (or diffusion) coefficient k e (defined in Eq 3.8), A* is the spherical albedoof a semi-infinite atmosphere (Nakajima and King, 1990; Kokhanovsky et al., 2003). Theescape functions are related to the phase function and can be derived from solutions ofnumerical radiative transfer solutions, but it can also be expressed as= ∑ ∞nK( µ ) ke Kn( µ )(2.18)0where k e is the extinction coefficient (Kokhanovsky, 2006, p. 141). For the non-absorbingcase K(µ) can be approximated by the first term K 0 (µ). Because multiple scattering dominates,typical features of single scattering becomes less pronounced, and K 0 can be well representedby the function3K0( µ ) = (1 + 2µ)(2.19)4This approximation is valid for isotropic scattering, but the error is less than 2% if µ ≥ 0.2 inthe visible (nonabsorbing) (Kokhanovsky et al., 2003; Kokhanovsky, 2006, p. 154). However,since k e 0 = 1, Eq. (3.19) does not take into account differences in ω 0 in the absorbing case.Figure 4 illustrates the dependence on µ of the escape function for different values of ω 0 .ec11

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________A*, l, m, n and k e in Eq. (2.17a,b) are all constants and can be parameterized for the generalcase (Kokhanovsky and Nauss, 2006). Because they are all strongly dependant on the socalled similarity parameter s, which in turn depends on the single scattering albedo andasymmetry parameter, the asymptotic theory equations thus depend primarily on twoparameters; the optical thickness and the effective radius (Nakajima and King, 1990):1−ω0s = (2.20)1−ω gThis is also in agreement with numerical simulations (King, 1987).0Figure 5. The dependence of escape function on µ and single scattering albedo ω 0 .The K-values are derived from asymptotic fitting model, with the phase functionrepresented by the Henyey-Greenstein function, assuming g = 0.85 (King, 1987)2.1.4 Semi-analytical method and Modified Exponential ApproximationIdeally the constants of the asymptotic theory should be derived using the exact numericalsolution. Again, it is a slow process that is not always applicable to large datasets, and wherethe high accuracy may not be very useful due to relatively large uncertainties in remotesensing images. Instead one may use approximate equations, such as the semi-analyticalretrieval model called the Exponential Approximation developed by Zege, and modified for12

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________larger absorption by Kokhanovsky et al. (2003, 2005a). The exponential approximation isbased on the asymptotic theory for diffusion, and made valid for the radiative transferequation for plane parallel water clouds with optical thickness τ > 10 (for quick first orderestimations and lower accuracy requirements it is applicable to clouds with optical thicknessas low as 5). It is basically a further simplification of the asymptotic equations (2.16 and2.17a,b) that allows faster computations, with an accuracy of 10-15% (depending on opticalthickness and viewing geometry). The use of the modified exponential approximation showsthat the errors in the computation of the reflection function is generally smaller than errorsrelated to uncertainties in the forward model and errors due to calibration uncertainties of theoptical instruments (Kokhanovsky and Nauss, 2006). Together with parameterizations of theradiative transfer solutions, the MEA was used by Kokhanovsky et al. (2005a) to deriveanalytical relationships of the reflection function that depend only on the effective radius.Their approach will be used in this study for retrieval of the cloud optical thickness andeffective radius and is here described in detail.In the exponential approximation parameters in Eq. (2.16) and (2.17b) are approximated byanalytical functions (ignoring the underlying surface albedo) assuming no or weak absorption:R∞0β( µ , µ0, φ ) = R ∞( µ , µ0, φ)− 4 K(µ ) K(µ0)(2.21)3(1 − g)⎛ β ⎞K( µ ) = K ⎜⎟0( µ )1−2α (2.22)⎝ 3(1 − g)⎠where (β = 1 - ω 0 ). For conservative scattering β = 0 and it follows that R ∞ = R 0 ∞ and K(µ) =K 0 (µ). In the visible with zero absorption the reflection function of Eq. (2.16) can thus bewritten∗t1(r , )[ 1 ] ( ) ( )0ew − Ag,1K0µ K0µ0R1( τc; µ , µ0,φ)= R ∞( µ , µ0, φ)−(2.23)1−A 1−t ( r , w)g,1[ ]where R * 1 is the (measured) reflectance, R 0 ∞ is the reflection function of an idealized semiinfinitenonabsorbing water cloud, A g,1 is the ground surface albedo in the nonabsorbingwaveband and t 1 is the diffused transmittance of a cloud. t 1 is governed by the effective radiusr e and the liquid water path w, but can be expressed in terms of asymmetry parameter g andoptical thickness τ 1 :1t1=(2.24)3α + τ1( 1−g1(r e))4where α is nearly constant at 1.072 for water clouds (Kokhanovsky et al., 2003; Kokhanovskyet al. 2005a). The liquid water path w is correlated to the optical thickness through theextinction coefficient (Kokhanovsky et al., 2003; Kokhanovsky et al. 2005a):τ = wk eλ,r )(2.25)(e1ewhereke1.5ρ ⋅ r⎛⎜1+⎝1.1=3e⎞( ) ⎟ 2 /k ⋅ re⎠(2.26)13

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________with the liquid water density ρ and k = 2π/λ (not to be confused with the extinction coefficientk e ). The accuracy of Eq. (2.25) and (2.26) is better than 8% for λ < 2.2. From Eq. (2.23) and(2.24) the optical thickness in the visible can thus be expressed as followswhere−1( t −1.072)41τ1=(2.27)3 1⎛ K⎜⎝( − g ( ))1( µ ) Kr e( µ)A⎞⎟⎠−10 0 0 1t1=⎜−0 ∗1−⎟(2.28)R∞− R A11For weak absorption k e , l and m in the exponential approximation are given byk e= 3(1− g)β,βl = 1−4α ,3(1 − g)m = 8β3(1 − g)(2.29)These formulae together with Eq. (2.21)-(2.22) are valid for the case of small probability ofabsorption (β ≤ 0.0001). But when absorption is stronger (for water clouds β can be close toor even larger than 0.1), the next terms in the expansion of these constants must beconsidered. However, the corresponding expressions for stronger absorption becomeextremely complicated. Instead the following exponential expressions can be used for thereflection function in absorbing media (Kokhanovsky et al., 2003; Kokhanovsky et al. 2005a,Kokhanovsky, 2006, p.183):[ − y(r ) u(µ , , )]0R ∞( µ , µ0, φ ) = R ∞( µ , µ0, φ)⋅ expeµ0φ(2.30)( 1−exp[ 2y(r )]) K ( µ ) K ( )mK( µ ) K(µ0) =e 0 0µ0(2.31)l = exp[-αy] (2.32)whereyi= 4β3(1 − gi),u(µ , µ0K0( µ0) K0( µ ), φ)= (2.33)0R ( µ , µ , φ)∞0With Eq. (2.30)-(2.33) Eq. (2.17b) can be written[ − y u] − t exp[ − x y ] K ( µ ) K ( )0R2 ( τ2;µ , µ0, φ)= R ∞exp2 c 2−2 0 0µ0 (2.34)with the global transmittance t c :t sinh y= c sinh( αy+2x)(2.35)and x= τ , where k e is the extinction coefficient: k = 3(1 − g ) βik e,ii14e, iiThe optical thickness τ in the absorbing wavelength can be determined from the opticalthickness in the visible. It follows from Eq. (2.25) and (2.26) that

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________where F(θ) ≈ p(θ), and p(θ) is the phase function. The phase function of a cloudy medium ishere approximated using the Henyey-Greenstein function for g 1 , given by Eq. (2.5) where(Kokhanovsky et al., 2003; Kokhanovsky, 2004a).θ = arccos( −cosµ cos µ0+ sin µ sin µ0cosϕ)(2.41)Eq (2.30-2.41) generates the reflection function in the NIR and MIR wavelengths. For anaccuracy of R ∞ 0 (Eq. (2.40)) better than 7%, the observation angles must be smaller than 30°and incident angles lower than 70° (Kokhanovsky, 2004b). This is generally not a problemwith remote sensing images, but must be checked for when running the model, especially ifthe pixels are located far from the centre of the image. In the selected area of study the sensorzenith angles were all smaller than 17°, and the solar zenith angles were within the range 38°-41°.2.2 Data and ComputationsBy comparing the calculated reflection function with the measured value in the correspondingwavebands, the effective radius can be determined. For the retrieval of the effective radius incumulus cloud tops, reflectance data from at least two wavebands (visible and NIR/MIR) isrequired. The light intensity measured from space is however both reflected and emitted light.Satellite sensors cannot distinguish between reflected and emitted light, the photons areidentical if in the same wavelength, so how does one come around this problem?Emitted light is due to absorption by in this case water vapour present in the cloud andsurrounding atmosphere. The easiest way to discriminate reflected light from emitted light isprobably to use wavebands with minimal water vapour absorption. For the purpose ofdetermining the cloud droplet radius we are interested in comparing the reflected light in thevisible region with no absorption, with the reflected light in the near/mid infrared region,where cloud drops absorb some of the incident light, depending on the liquid water content. Inthe visible with almost no absorption there is no problem. But in the near and mid infrared(~ 0.7 – 1.3 and 1.3 – 3.0 µm respectively (Lillesand et al., 2004, Remote Sensing and ImageInterpretation, p. 6)) water vapour absorption occur at 0.72, 0.82, 0.94, 1.1, 1.38, 1.87, 2.7 and3.2 µm with the strongest peaks at 1.38, 1.87 and 2.7 µm (Liou 1992, p.157, Liou 2002, p. 83,371). Therefore one must choose wavebands in between these wavelength bands where theliquid water absorption is still sufficiently strong to show differences in reflection comparedto the visible (for the same reason the cloud optical thickness must not be too small). Suitablewavelengths for these requirements, often used in remote sensing analysis of clouds are 0.5 –0.7 µm in the visible and 1.6, 2.2 and 3.7 µm in the NIR and MIR (Platnick, 2000). Becausethese three wavelengths show different sensitivity to underlying surface reflection andatmospheric water vapour absorption one may wish to use more than one of thesewavelengths in order to reduce the uncertainty.Rosenfeld et al. (2004) studied NOAA and METOP satellite measurements of cloudmicrophysical properties and found that 1.6 and 3.7 are equally (un)suitable for deriving clouddroplet effective radius. The 3.7 µm waveband is more sensitive to cloud top reflectance.However, in this waveband solar reflection and thermal emission occur simultaneously, andso it is more sensitive to atmospheric water vapour. On the other hand, the 1.6 µm is moreaffected by underlying surface reflection, which is a relevant problem in smaller clouds.Another disadvantage with 1.6 µm waveband is that the radiation may originate from deepinside the cloud, and must therefore be corrected if cloud top radiation is wanted. This also16

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________indicates to that the 1.6 µm waveband is less suitable for studying smaller clouds(Kokhanovsky et al. 2005b).In this study the MODIS aqua satellite wavebands 0.620-0.670 µm and 2.105-2.155 µm areused for reflectance measurements in the visible and NIR region respectiely. The scene usedis from the 9 th of May, 2004, and a frame of 200×200 pixels over southern Sweden (figure 6)is chosen for the study. Because the whole scene is not covered by clouds, a simple “cloudmask” was applied in order to discriminate fully cloud covered pixels. It is assumed that theclouds are water clouds, i. e. clouds consisted of liquid water droplets, as opposed to iceclouds or mix phased clouds. In each model run, the effective radius of cloud droplets will becomputed only for pixels meeting the following conditions:1. The reflectance in the visible must be larger than in the NIR2. Reflectance in 2.1 µm waveband > 0.2; 0.25; 0.3The first condition is obvious, since the principal theory of the model is the fact that waterabsorbs energy in the NIR/MIR spectrum, the visible is non-absorbing. Pixels where thereverse is true, i.e. where R 2 > R 1 indicates ground surface reflectance, where bare soil,vegetation or urban areas have different reflectance signatures. A third condition, to accountfor the limitation of optical thickness in the model, is thereafter applied:3. τ 1 > 10Four model runs are conducted, the first three with R 2 values of 0.2, 0.25 and 0.3 respectively.The last run, with R 2 > 0.3, was performed with τ 1 > 5.320340360380400420440460480300500700 720 740 760 780 800 820 840 860 880 900Figure 6 MODIS Aqua Satellite image over the area selected for studying, coveringthe most southern parts of Sweden and eastern parts of Zealand (Denmark). Cloudsare seen in pink colour, land areas in green and open water in dark brown.17

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________Besides the measured reflectance in the visible and NIR, data on solar and satellite zenithangles and azimuth angles are provided for every scene pixel by MODIS. These data are themain inputs in the model described in the previous section. With the reflectance and viewinggeometry given by MODIS, a few more input parameters are required. The ground surfacealbedo for every pixel can be received as a MODIS product. Since the clouds are blocking thesurface, the ground surface albedo must be taken from another date with clear sky conditions.In early May when vegetation period is at its beginning, ground surface albedo may varysignificantly from one week to another, depending on the vegetation cover and the soilmoisture content. Therefore it is at this time of the year difficult to produce accurate values ofA g . However, the ground surface albedo has a relatively small influence on the final result,and for simplicity, standard values of A g in the two wavelengths is set for all pixels coveringland. Typically, surface albedo for bare soil is about twice as large for wavelengths > 0.7 µmthan for wavelengths < 0.7 µm (Post et al., 2000). For the two wavebands λ = 1.6 µm and λ =2.1 µm the ground surface albedos are set to 0.2 and 0.4 respectively. These values areaverage values of surfaces with forest cover or bare soil (Ahrens, 2003, p. 46). The surface isalso assumed to be Lambertian, i.e. the reflectance is equal in all directions.The asymmetry parameter g and the single scattering albedo ω 0 must be known, but they areboth depending on the effective radius. The asymmetry parameter is defined in Eq. (2.4) butsince the Legendre polynomials and expansion coefficients are not known, g can instead befound by following equation:1−g = 0.12 + 0.5 3−2( kr e) − 0.15κre(2.42)where k = 2π /λ and κ = 4πχ /λ. χ is the imaginary part of the refractive index of water(Kokhanovsky et al., 2003; Kokhanovsky et al. 2005a) and set to 1.64⋅10 -8 and 4.00⋅10 -4 forχ 1 and χ 2 respectively (Kokhanovsky, 2006, p. 259; Kokhanovsky et al., 2006, unpublished).Kokhanovsky and Zege (1995) also present a parameterisation of the absorption andextinction coefficients k a and k e which are used to derive the single scattering albedo in theNIR (in the visible ω 0 is assumed to be 1):ka**= kaCv, kekeCv= , (2.43)where⎛ ⎞∗ 1.5⎜1.1ke =1 +(2.44)re⎝( ) ⎟ 3kr2 /e ⎠k∗a( 1−κr)5πχ ⎛ ⎛⎞⎞e ⎜⎡⎜8λ⎤=1 + 0.34⎟⎟1 − exp⎢− ⎥λ⎝ ⎝ ⎣ re⎦⎠⎠(2.45)and C v is the volumetric concentration of droplets. However, since 1 – ω 0 = k a / k e (Eq. 2.11)ω 0 is given by*kaω0= 1−(2.46)*ke18

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________Equation (2.42)-(2.45) was used by Kokhanovsky et al. (2006, unpublished) in their semianalyticalcloud retrieval algoritm for SCIAMACHY/ENVISAT, and has proved to beaccurate within 5-8% error for λ < 2.2 µm when compared to exact Mie calculations(Kokhanovsky et al., 2003). The imaginary part of the refractive index, χ, for λ = 0.65 and 2.2µm is 1.64⋅10 -8 and 2.89⋅10 -4 respectively (Kokhanovsky, 2006).Following the procedure in the modified exponential approximation and using the equationsfor g and ω 0 presented above, the expected reflectance value in the second waveband (NIR) iscalculated for r e values ranging from 3 to 30 µm. This range of values was set because dropletsizes outside this range are very unlikely, especially droplets larger than 30, since these wouldfall out as precipitation due to gravity. By comparing the model value of R 2 with the measuredR 2 the effective radius is given by the value producing the R 2 -value closest to the onemeasured. This is performed for every pixel (in the chosen 200×200 pixel area) that meets theconditions of the cloud mask. For a detailed description of the procedure, see Appendix.An example of the model output is illustrated in figure 7. The effective radius in one pixel iscalculated with the MODIS inputs as listed:R 1 * : Measured reflectance in the visible (λ 1 ) = 0.5624R 2 * : Measured reflectance in the NIR/MIR waveband (λ 2 ) = 0.3610v: Sensor zenith angle = 2.73°v 0 : Solar zenith angle = 43.09°ϕ: Sensor azimuth angle = -104.63°ϕ 0 : Solar azimuth angle = -165.49°0.650.60.55R 2modelledR 2satellite0.5NIR Reflectance0.450.40.350.30.250.20 5 10 15 20 25 30Effective radius (µm)Figure 7 Exampel of calculatedeffective radius from one pixel.r e = 10.60 µm, τ 1 = 13.119

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________3. RESULTS and DISCUSSIONFour runs were performed, with different initial conditions for the reflectance in the 2.1 µmband(R 2 ) and optical thickness in the visible (τ 1 ). The results are summarized in table 2 and 3.Figure 6 illustrates the droplet size distribution in the four runs.It is obvious that lower set up conditions result in more pixels being calculated. Lowreflectance in the second waveband results in values τ 1 and τ 2 as low as 3.6 (R 2 > 0.25) and2.6 (R 2 > 0.2), and mode optical thickness at 8 and 5 respectively. Since the model is suitedfor clouds with optical thickness > 10, this implies that the model is not applicable to pixleswith R 2 -values < 0.3, where the optical thickness appears to be too small. For R 2 > 0.3 theoptical thickness lies in the range 4.6 – 50.9, with mode optical thickness at 13, which isacceptable.Many pixels in the first two runs, with R 2 > 0.2, result in much larger r e –values than in thelast two runs. Maximum effective radius is at 27(20) µm for R 2 > 0.2(0.25) and 15.5 for R 2 >0.3 (Table 3, Figure 8). Also the median and mode effective radius is higher in the first tworuns. The last two runs produced median and mode effective radius at 14.6 and 15 µmrespectively, which is in line with the expected result.The results can be compared with results from Rosenfeld and Lensky (1998). According toRosenfeld and Lensky the cloud top median of effective radius is 14 µm for clouds near thecoastline and 9 µm over inland, therefore both 20(17) µm (R 2 > 0.2(0.25)) and 14.6 µm (R 2 >0.3) appear relatively high for median value for continental clouds. However, one should bearin mind that the continental area studied by Rosenfeld and Lensky covers southern Malaysiaand central Sumatra. Southern Sweden is thus relatively small in comparison and most partscan be considered to be located near the coastline. The areas over Malaysia and Sumatra alsohave higher population density. Likely the air is more polluted (due to traffic and forestburning) here than over southern Sweden. If so, the retrieved median value of 14.6 µm,although higher than 9 µm, is realistic. Still, the median effective radius of 17-19 µm, fromthe first two runs, can be considered overestimated, confirming that the model is not suitablewhere R 2 < 0.3. Thus, the results from this study are in agreement with the same fromRosenfeld and Lensky (1998). It is interesting that the use of the 2.1 µm waveband instead ofthe 3.7 µm (as in the study by Rosenfeld and Lensky) cause no major differences in theresults.Table 3 Summary over the results of optical thickness τ from the four runs. Since the only difference betweenthe two last runs is the restriction of τ 1 , the statistics of τ are identical, why both cases of τ 1 for R 2 > 0.3 arerepresented in the last row.conditionsTotal no. ofpixels computedmin τ max τ median τ 1 median τ 2 mode τR 2 > 0.2 3903 2.65 65.42 9.52 9.89 5R 2 > 0.25 1721 3.61 68.25 10.95 11.10 8R 2 > 0.3 202 4.62 48.64 10.91 11.43 1320

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________Table 3 Summary over the results of the effective radius r e from the four runs. The number of pixels fulfillingthe conditions is listed in the first three columns. The last four columns show the statistics of the retrievedeffective radius, including minimum, maximum, mean, and mode effective radius.conditionsTotal no. ofpixels computedpixlesτ 1 >5pixlesτ 1 >10min(µm)r emax(µm)r emedian(µm)r emode(µm)r eR 2 > 0.2τ 1 >10R 2 > 0.25τ 1 >10R 2 > 0.3τ 1 >10R 2 > 0.3τ 1 >53903 3293 1876 11.39 27.27 19.91 181721 1589 910 11.39 20.22 17.35 18202 199 116 11.39 15.42 14.64 15202 199 116 8.010 15.54 14.60 15freq200180160140120100806040200R2 >0,3; t>5R2 > 0,3; t>10R2 > 0,25; t>10R2 > 0,2; t>103 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30effective radius (µm)Figure 8. The drop size distribution in the four runs. Each colour represents one runwith the set up conditions given in the box to the upper left.The sharp drop in frequency after the peak at 15 µm (Figure 8) for the two last runs (R 2 > 0.3),implies that this is an upper limit of the drop size before the cloud drops fall out asprecipitation, indicating that the clouds are precipitating or close to precipitating. Rosenfeldand Lensky (1998) found that 14 µm is a typical precipitation threshold for continentalclouds. In accordance to this, Pinsky and Khain (2002) found 15 µm to be the threshold fordrizzle. At this threshold, droplets continue to grow through coalescence, but this is balancedout by the larger droplets falling out as precipitation due to gravity. This pattern is not seen inthe first two runs, where the size distribution is more symmetric, and the effective radiusranges up to well above 20 µm.21

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________It is probable that the differences in results from the first two and the last two runs is at leastpartly due to cloud optical thickness being too small when R 2 < 0.3. But there are otherfactors that may have impact on the results too. The model assumes water clouds, hence thepresence of mixed phase clouds or ice clouds can give rise to incorrect or misleading results.Water clouds with effective radius > 15 µm are rare, while ice crystals have much largereffective radius according to Kokhanovsky (2006, p. 5). Information on cloud top temperaturewould be relevant to include in the cloud mask in order to discriminate water clouds from iceor mixed phase clouds.Another source of error may be the so called 3D effect. As most cloud models, the MEA isdeveloped for plane parallel, semi-infinite homogeneous clouds. That is, it is assumed that thecloud has no borders horizontally, which is a very coarse approximation for cumulus clouds,characterized by its finite and irregular forms. Light being reflected inside a cumulus clouddoes not necessarily escape the cloud body either upward or downward, but may just as wellescape in a horizontal direction. Compared to a semi-infinite plane parallel cloud with thesame microphysical properties, less light would be reflected into the satellite sensor. In themodel this “missing” light is assumed to be absorbed by the cloud, which would lead tomiscalculation (overestimation) of the effective radius. Also, in addition to the direct solarradiation, some incoming radiation may be light reflected from nearby cloud tops within thesame cumulus cloud.4. VALIDATION and ERROR ANALYSISThe error analysis here is rather brief and based on the results from extensive error analysisperformed by Kokhanovsky et al. (2005a,b), Kokhanovsky (2004b), Kokhanovsky andRozanov (2003), and Kokhanovsky et al. (2003). Many of the equations used in the MEA areapproximations of asymptotic theory, where the accuracy has been studied closer for eachequation individually. Where computed, the error is found by comparing the results of theapproximate equation with the same from the exact (numerical) methods, and given togetherwith the equation in the text.Validation of the model is difficult since it requires real measurements of the effective dropradius in the studied clouds. These measurements must be done by aircraft and is not availablefor the date of interest. Another method for validating the model is to compare the retrievedcloud drop radius with the MODIS cloud product. This does however not give anyinformation on how accurate the method is compared to reality, but shows rather the degree ofagreement or discrepancy between the MEA model and the MODIS cloud product algorithm.Such a validation was done in the study by Nauss et al. (2005), where the MEA model (in thestudy called SACURA) was compared with two other models, one of them the MODIS cloudproduct. Nauss et al. concluded that the relatively simple MEA model produced resultscomparable to the more complex models.On average the MEA effective radius is within 10% from the MODIS cloud product for mostpixels, with somewhat larger biases for r e < 13 µm and r e > 20 µm. The agreement betweenthe MODIS cloud product and the MEA model was higher over ocean than over land, with r 2as high as 0.94 for pixels with τ > 10. The corresponding figure over land was 0.79. Over ocean thetwo models used the same waveband (0.86 µm). However, for pixels over land the MODIScloud product used the 0.65 µm waveband, while the MEA model used the same waveband asover ocean (0.86 µm). Nauss et al. explain the larger discrepancy over land to be caused bythe fact that the 0.86 µm waveband is much more sensitive to vegetation covering the22

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________underlying surface than the 0.65 µm waveband. The r 2 –values of the MEA given here arecalculated using the 0.86 µm and 1.6 µm waveband, both more sensitive to the underlying surface thanthe two wavebands used in this study (0.65 µm and 2.1 µm respectively). It is probable that if the0.65 µm and 2.1 µm wavebands were used in the MEA model comparison, the agreementcould have been increased over land. Over all Nauss et al. (2005) showed that the error of theapproximate solution of the reflection function in the visible and near-infrared is less than 5%for most pixels at nadir observation.Other studies of the MEA method (Eq. (2.38a)) have shown similar results. Kokhanovsky andRozanov (2003) found the error to be less than 5% for nadir observations at λ 2 (λ 2 ) = 0.65(1.65) µm and incident angle 15° < v 0 < 65°. For non-nadir observation angles, the accuracydepend on the accuracy of the reflection function R 0 ∞ (Eq. (2.40)). Kokhanovsky (2004b)showed that the accuracy of the R 0∞ -function was better than 95% for incident andobservation angles smaller than 55° and 30° respectively when compared to the numericalsolution.The application of the MEA method to SCIAMACHY/ENVISAT data has also been validatedagainst the conventional LUT approach by Kokhanovsky et al. (2005a, 2005b). Here thevalidation showed relatively large differences in retrieved effective radius between the LUTapproach and the MEA, but it did not exceed 20%. The large differences are partly explainedby the fact that the LUT approach used the 3.7 µm waveband, while the MEA used the 1.6µm waveband. Another factor that may result in poor agreement between the two approachesis the low resolution of the SCIAMACHY data, which could not guarantee 100% cloudcovered pixels. They therefore conclude that the MEA (SACURA) method gives resultscomparable to the LUT approach. When using the 0.865 µm and 2.13 µm wavebands theaccuracy for the nadir observation conditions and the solar zenith angle 60° was better than94% for τ ≥ 4, see Figure 9 (Kokhanovsky, 2006).Figure 9. The error of the MEA approach for nadir observation and solar zenithangle = 60°. The dotted line represents the 2.13 µm waveband, while the solid line23

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________corresponds to the 0.865 µm wavenband (Kokhanovsky, 2006)According to Kokhanovsky et al. (2006, unpublished) the accuracy of r e is strongly dependenton the correct information with respect to cloud fraction. One can expect r e to beoverestimated if retrieved over broken cloud field, due to unknown contribution from theground. However, the larger the optically thickness, the smaller the contribution from theground, and even more so if the 0.65 µm waveband is used instead of the 0.865 µmwaveband. From the different error analysis conducted in the studies by Kokhanovsky et al.the overall error is estimated to be in the range 5-20%, depending on viewing geometry andcloud optical thickness.5. CONCLUSIONSThe results from the four runs show that the modified exponential approximation can be usedwith relatively high accuracy on satellite data under certain conditions. The model is restrictedto optically thick clouds where τ > 10 (5 if only a first order estimation is required). However,since the optical thickness is not previously known but computed in the model, the conditionsmust be given in terms of reflectance. The results from the two first runs where R 2 > 0.2(0.25)showed relatively high values of effective radius and low values of optical thickness (τ < 5),which confirms that low reflectance values indicate optically thin clouds, thus making theresults less reliable.The model performed well (in relation to what was expected) where R 2 > 0.3, for both τ > 10and τ > 5. The median effective radius of 14.6 µm was high in comparison to the results fromcontinental clouds (9 µm) in the study by Rosenfeld and Lensky (1998). However, the resultis rather similar to the near coastline clouds, which had a median effective radius of 14 µm.A reason for this difference may lie in the geographical differences. The highly continentalclouds studied by Rosenfeld and Lensky (1998) was located over southern Malaysia andcentral Sumatra, a much larger area with dense population and likely more air pollution thanover the forests of southern Sweden. From this aspect, the “continentality” of southernSweden area is not comparable to the same of Malaysia and Sumatra, and it would perhaps bemore relevant to compare the results from southern Sweden with the near coastline clouds.The model has been validated against numerical methods in a few studies. The validation byNauss et al. (2005) showed that the approximate model, despite its simplicity, tended toproduce results comparable to results from conventional methods (numerical methods, LUT).The error varies from 5 to 20%, depending on which wavebands that are used, viewinggeometry and pixel resolution. The studies stress the importance of fully cloud covered pixelsfor good results, and good knowledge about ground surface albedo where the cloud cover isbroken, or where clouds are optically thin. This is even more important when usingwavebands sensitive to ground surface reflectance. However, in this study the wavebandsused were chosen to minimize the influence by the underlying surface.The model is fast, simple and straight forward, which is an advantage if the purpose is tounderstand the processes involved in cloud formation. The aim of the study was to produce amodel that can be used together with cloud top temperatures to study changes in the verticalprofile of the cloud, induced by aerosols. In this case, exact values of the cloud droplet radiusare perhaps not essential, but rather the ability to study differences and changes, and to do so24

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________for many sites and dates, which requires a fast approach, where input variables are easilychanged. If the purpose is to produce accurate and reliable projections of future weather andclimate scenarios, a more exact approach may be desirable. One should bear in mind thoughthe geographical uncertainty in the remote sensing images, where the resolution may be lowwith pixels of 1000×1000 m. The accuracy of the model can never be better than theuncertainty in the input data. Thus the high accuracy of numerical methods cannot always befully benefited from. Another uncertainty is the influence of the 3D-effect.For future work, a proper sensitivity test of the model would be of great value. Only throughrunning the model it has turned out to be very sensitive to the value of imaginary refractiveindex, while the influence of ground surface albedo is small, if A g < 0.5. Also, a moreadvanced cloud mask with the ability to discriminate pure water clouds from ice or mixphased cloud would be appropriate, to support the assumption of pure water clouds beingmodelled.For deeper knowledge and understanding of cloud remote sensing and modelling I refer to thetwo excellent books by Liou; “Radiation and Cloud Processes in the Atmosphere” (1992) and“Introduction to Atmospheric Radiation” (2002). The MEA model is carefully described andevaluated in the many articles written by Kokhanovsky et. al., see References.Acknowledgements:Thanks to Erik Swietlicki for support and helpful ideas and to Anders Wigren for help with MODIS data.25

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________26

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________APPENDIXSummary of the Modified Exponential Approximation approach:Input parameters: λ 1 : Peak wavelength in the first waveband (visible): 0.65 µmλ 2 : Peak wavelength in the second waveband (NIR/MIR): 2.1 µmR 1 * : Measured reflectance in the visible (λ 1 )R 2 * : Measured reflectance in the NIR/MIR waveband (λ 2 )v: Sensor zenith angle (radians)v 0 : Solar zenith angle (radians)ϕ: Sensor azimuth angle (radians)ϕ 0 : Solar azimuth angle (radians)χ 1 : Imaginary part of the refractive index for liquid water forχ 2 : Imaginary part of the refractive index for liquid water λ 2A 1 : Ground surface albedo in λ 1A 2 : Ground surface albedo in λ 2Output: R 2 as a function of effective radius r e .Calculated effective radius r e .Approach: 1. Calculate µ = |cos(v)| and µ 0 = cos(v 0 )2. Calculate relative azimuth angle: φ = ϕ - ϕ 03. Calculate k λ = 2π/λ for λ 1 and λ 24. Calculate κ λ = 4πχ /λ for λ 1 and λ 25. Calculate escapefunctions K 0 (µ) and K 0 (µ 0 ) using Eq. 3.196. Create a vector of the effective radius r e from 3 µm to 30 µmFor each value r e (i), calculate following microphysical andradiative properties:7. Calculate asymmetry parameter g 1 and g 2 (1 and 2 representingλ 1 and λ 2 ), using Eq. (3.42)8. Calculate single scattering albedo ω 0 for λ 2 using (3.43)-(3.46).ω 0 in the visible is set to 19. Calculate scattering angle θ : Eq. (3.41)Calculate Henyey-Greenstein phase function from g 1 : Eq. (3.5)10. Calculate R 0 ∞ (in the visible only) using Eq. (3.39)-(3.40)11. Calculate t 1 Eq (3.28)12. Calculate optical thickness τ 1 : Eq. (3.27)13. Calculate optical thickness τ 2 using Eq (3.36).14. Calculate x 2 = k e τ 2 where = 3(1− g)βk e27

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________15. Calculate y 2 using Eq. (3.33)16. Calculate global transmittance t c using Eq. (3.35)17. Calculate t 2 using Eq. (3.37b-c)18. Calculate spherical albedo a 2 = exp(-y 2 ) – t c exp(x 2 –y 2 )19. Calculate u using Eq. (3.33) and (3.37a)20. Calculate R 2 using Eq. (3.38b)21. Compare the theoretically R 2 -value with the measured valueR 2 * , and find the index i where R 2 (i) = R 2 * to retrieve the clouddroplet effective radiusThe code is written in MATLAB 7.5.0 and available on request.Please contact Julia Karlgård on julia.karlgard.865@student.lu.se or Erik Swietlicki onErik.Swietlicki@nuclear.lu.se28

ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATAModified Exponential Approximation___________________________________________________________________________________________________REFERENCESAhrens, C. D. 2003. Meteorology Today: An Introduction to Weather, Climate, and the Environment. 7 th Ed.Thompson, Brooks/Cole. Pacific Grove, United States. 537 pp.IPCC. 2007. Climate Change 2007: Synthesis Report. Contribution of Working Groups I, II and III to the FourthAssessment Report of the Intergovernmental Panel on Climate Change [Core Writing Team, Pachauri, R.Kand Reisinger, A.(eds.)]. IPCC, Geneva, Switzerland, 104 pp.King, M. D.. 1987. Determination of the Scaled Optical Thickness of Clouds from Reflected Solar RadiationMeasurements. Journal of the Atmospheric Scinece, 44(13). 1734-1751.Kokhanovsky, A. A.. 2002. Simple approximate formulae of the reflection function of a homogeneous semiinfiniteturbid medium. Journal of Optical Society of America, 19(5). 957-960.Kokhanovsky, A.A.. 2004a. Optical properties of terrestrial clouds. Earth-Science Reviews, 64. 189-241.Kokhanovsky, A. A.. 2004b. Reflection of light from nonasbsorbing semi-infinite cloudy media: a simpleapproximation. Journal of Quantitative Spectroscopy & Radiative Transfer, 85(1). 25-33.Kokhanovsky, A. A.. 2006. Cloud Optics. Springer. Dordrecht, Netherlands. 276 pp.Kokhanovsky, A.A. and Nauss, T.. 2006. Reflection and transmission of solar light by clouds: asymptotictheory. Atmospheric Chemistry and Physics, 6. 5537-5545.Kokhanovsky, A.A. and Rozanov, V.V.. 2003. The reflection function of optically thick weakly absorbingturbid layers: a simple approximation. J. Quantitative Spectroscopy & Radiative Transfer, 77, 165-175.Kokhanovsky, A. A., Rozanov V. V., Burrows J. P., Eichmann K.-U., Lotz W., and Vountas M.. 2005a. TheSCIAMACHY cloud products: Algorithms and examples from ENVISAT. Advances in Space research, 36.789-799.Kokhanovsky, A. A., Rozanov V. V., Nauss T., Reudenbach C., Daniel J. S., Miller H. L., and Burrows J. P..2005b. The semianalytical cloud retrieval algorithm for SCHIAMACHY – I. The validation. AtmosphericChemistry and Physics Discussions, 5. 1995-2015.Kokhanovsky, A.A., Rozanov, V.V., Vountas, M., Lotz W., Bovensmann H., Burrows, J.P., 2006(unpublished). Semi-analytical cloudretrieval algorithm for SCIAMACHY/ENVISAT. Algorithm TheoreticalBasis Document. Pp 40. http://www.knmi.nl/samenw/sciamachy/products/clouds/clouds_IFE_AD.pdf. pagelast visited 24 th April 2008.Kokhanovsky, A. A., Rozanov, V. V., Zege, E. P., Bovensmann, H. and Burrows, J. P.. 2003. A semianalyticalcloud retrieval algorithm using backscattered radiation in 0.4-2.4 µm spectral region. Journal of GeophysicalResearch, 108(D1), 4008. doi: 10.1029/2001JD001543.Kokhanovsky, A. A. and Zege, E.. 1995. Local parameters of spherical polydispersions: simple approximations.Applied optics, 34(24). 5513-5519.Lillesand, T. M., Keifer, R. W., and Chipman, J. W.. 2004, Remote Sensing and Image Interpretation. 5 thedition. Wiley & Sons Inc. Crawfordsville. United States of America. 704 pp.Liou, K.N. 1992. Radiation and Cloud Processes in the Atmosphere. Oxford University Press. New York. 504pp.Liou, K.N. 2002. An Introduction to Atmospheric Radiation. 2 nd edition. Academic Press. London. 248 pp.Nakajima, T. and King, M.D.. 1990. Determination of the optical thickness and effective radius of clouds fromreflected solar radiation measurements. Part I: Theory. Journal of the Atmospheric Science, 6(15). 1878-1893.Nakajima, T. and King, M.D.. 1992. Asymtotic theory for optically thick layers: application to the discreteordinates method. Applied Optics, 31(36). 7669-7683.Nauss T., Kokhanovsky A. A., Nakajima T. Y., Reudenbach C., and Bendix J.. 2005. The intercomparison ofselected cloud retrieval algorithms. Atmospheric Research, 78. 46-78.Pinsky N., B. and Khain A. P.. 2002. Effects of in-cloud nucleation and turbulence on droplet spectrumformation in cumulus clouds. Quarterly Journal of the Royal Meteorological Society, 128(580), 501-533Platnick S. 2000. Vertical transport in cloud remote sensing problems. Journal of Geophysical Research,105(D18), 22,919-22,935.Post, D. F., Fimbres A., Matthias A. D., Sano E. E., Accioly L., Batchily A. K., and L. G. Ferreira. PredictingSoil Albedo from Soil Color and Spectral Reflectance Data. Soil Science Society of America journal,64(3). 1027-1034Rosenfeld, D., Cattani, E., Melani, S., and Levizzani, V.. 2004. Considerations on Daylight Operation of 1.6-VERSUS 3.7-µm Channel on NOAA and Metop Satellites. Bulletin of the American Meteorological Society,85(6). 873-881.Rosenfeld, D., and Lensky, I. M.. 1998. Satellite–Based Insights into Precipitation Formation Processes inContinental and Maritime Convective Clouds. Bulletin of the American Meteorological Society, 79. 2457-2476.Rybicki, G. B.. 1996. Radiative Transfer. Journal of Astrophysics and Astronomy, 17. 95-112.29