stochastic theory of radiative transfer through …stochastic theory of radiative transfer through...

Stochastic theory of radiative transfer through generalized cloud fields

Dana E. Lane-Veron1 and Richard C. J. SomervilleClimate Research Division, Scripps Institution of Oceanography, University of California, San Diego, La Jolla,California, USA

Received 9 January 2004; revised 26 May 2004; accepted 12 July 2004; published 24 September 2004.

[1] We present a coherent treatment, based on linear kinetic theory, of stochasticradiative transfer in an atmosphere containing clouds. A brief summary of statisticalcloud radiation models is included. We explore the sensitivities inherent in thestochastic approach by using a well-known plane-parallel model developed byFouquart and Bonnel together with our own stochastic model which generalizes earlierwork of F. Malvagi, R. N. Byrne, G. C. Pomraning, and R. C. J. Somerville. Inovercast conditions, in comparison to the plane parallel model, the stochastic modelunderestimates transmittance at small optical depths (<7) and overestimatestransmittance at large optical depths. The stochastic model is strongly sensitive tocloud optical properties, including cloud water content and cloud droplet effectiveradius. The extension of the stochastic approach to an atmospheric general circulationmodel parameterization appears to be most appropriate for cloud fraction ranging from25 to 70%. We conclude that stochastic theory holds substantial promise as a modelingapproach for calculating shortwave radiative transfer through partially cloudy fields.Unlike cloud-resolving models and Monte Carlo cloud models, stochastic cloud modelsdo not depend on specific realizations of the cloud field. Instead, they calculate thetransfer of radiation through a cloudy atmosphere whose properties are knownstatistically in the form of probability density functions characterizing cloud geometryand cloud optical properties. The advantage of the stochastic approach is its theoreticalgenerality and its potential for representing a complex cloud field realistically atmodest computational cost. INDEX TERMS: 0360 Atmospheric Composition and Structure:

Transmission and scattering of radiation; 1610 Global Change: Atmosphere (0315, 0325); 0659

Electromagnetics: Random media and rough surfaces; KEYWORDS: clouds, climate, radiation, stochastic

Citation: Lane-Veron, D. E., and R. C. J. Somerville (2004), Stochastic theory of radiative transfer through generalized cloud fields,

J. Geophys. Res., 109, D18113, doi:10.1029/2004JD004524.

1. Introduction

[2] Inhomogeneities in actual clouds occur over manyscales [Lovejoy and Schertzer, 1990; Davis et al., 1999]ranging from the molecular scale to mesoscale systems.In general, modern atmospheric general circulation models(AGCMs) do not include the macroscopic geometry of cloudsize and spacing in the representation of the cloud field whenthe clouds are smaller than the size of a grid cell. The mostcommon approximation of partial cloudiness is that of cloudfraction, which is a convenient, but nonphysical, way ofdescribing the total horizontal cloud coverage. The lastseveral decades have seen an increased awareness of a needfor a robust theoretical description of radiative transferthrough media with statistically characterized parameters[i.e., Stephens, 1984; Ramanathan et al., 1989; Stephens

et al., 1991]. Stochastic theory is one approach towardrealizing this goal.[3] An overview of the literature on statistical radiative

transfer was given by Stephens et al. [1991]. The subjectwas initially of interest for interpretation of remotelysensed images of reflected solar radiation. The applicationof this technique in nuclear physics and engineering isfound in many places, including the work of Adams et al.[1989].[4] An earlier version of the present shortwave, sto-

chastic model was used to demonstrate the importance oftreating the fractional cloud problem with a statisticaltechnique [Malvagi and Pomraning, 1993]. The modelused in our study, DSTOC, is a more generalized form ofthat proposed by Titov [1990]. DSTOC can accuratelycalculate the heating rates through a complex cloud layerusing the macroscopic properties of the field, such as thecloud water content and cloud thickness, and a statisticaldescription of the horizontal distribution of the clouds inthe layer. Currently the model assumes the same statisticsto describe the vertical and horizontal distribution ofclouds. Evaluation of the stochastic model using observed

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, D18113, doi:10.1029/2004JD004524, 2004

1Now at Department of Environmental Sciences, Rutgers, StateUniversity of New Jersey, New Brunswick, New Jersey, USA.

Copyright 2004 by the American Geophysical Union.0148-0227/04/2004JD004524$09.00

D18113 1 of 14

cloud and radiation data is described by Lane et al.[2002].

2. Theoretical Basis

[5] There are large variations in all phases of water in theatmosphere on very small scales. For certain scales, it isappropriate to represent clouds and clear sky as a mixtureof two materials that do not interact with each other[Mullumaa and Nikolaev, 1972; Stephens et al., 1991]. Inparticular, when the changes in the optical properties ofliquid water and water vapor occur on scales much largerthan a photon mean free path in cloud, then cloud andclear sky may be treated as an immiscible binary mixture[Malvagi and Pomraning, 1993].[6] For the cloud radiation problem, photons can be

imagined as streaming through a clear sky filled with cloudswhere the cloudy and clear parcels of air do not mix.Stochastic theory, which is derived from linear kinetictheory, utilizes the statistics of a line. This does not indicatethat the three-dimensional effects of clouds are ignored. Theline under consideration is the path a photon travels. If thephoton scatters in a different direction, then s is the totalpath along which the statistics (the distribution of eachmaterial along the path of the photon) are calculated. Thevariables describing the scattering properties of each mate-rial are considered random variables.[7] There are two main approaches for applying the

stochastic theory to the representation of photons flowingthrough a mixture of clouds and clear sky. The first is todetermine the radiation field for each possible realization ofa cloud field that could occur with statistically equivalentcharacteristics. After summing over the infinite number ofpossible physical scenes, the resulting radiation field wouldbe robustly known. In an atmospheric general circulationmodel (AGCM) environment, this procedure is neitherrealistic nor economical. The second approach is to developa statistical system of equations that describes the ensemble-averaged intensity resulting from the photon flow. Thisreplaces the need to calculate an infinite number of specificphoton trajectories, and average over them.[8] The most basic form of this theory describes a ‘‘no-

memory’’ situation where at each point in time the photonis not influenced by previous interactions (Markovianstatistics). Recently, this theory has been modified toaddress the non-Markovian case [Levermore et al., 1987].Since the ultimate intended use of this approach is as anAGCM parameterization where the computation time willbe limited, the possible extension of the mixture to three ormore components will be postponed. For this paper, thenotation of Sanchez et al. [1993] will be used to describe thebasic probabilities required in developing a paradigm fordescribing the theory of stochastic radiative transfer.

2.1. Statistics on a Line

[9] In the development of this application of stochasticradiative transfer theory, the fundamentals of statistics on aline must first be explored. Not only is this the mathematicalbasis for the development of the stochastic approach, but itis also provides us with a conceptual framework. Thestochastic model described later in this article calculateswhat may happen to a photon as it travels from the top of

the atmosphere to the surface. In traditional radiativetransfer models only two situations are considered: thephoton is either in a cloud or it is not. The stochasticdescription allows for two additional states where thephoton is transitioning from cloud to clear sky or fromclear sky to cloud. These transitions may be described bylinear probability statistics.[10] One comprehensive description of linear probabil-

ity statistics is given by Sanchez et al. [1993]. This workis useful in understanding stochastic radiative transfer inthat it provides a link between simple statistics on a lineand the transition probabilities used in the stochasticshortwave model. In the notation of Sanchez and col-leagues, the line of interest (the photon path for thisapplication) is populated by a set {a} of M materials.They form a statistical set W = {w} where each w is onepossible realization. If one imagines traveling along thisline, the probability that the current segment beingtraversed is made of a particular material is describedusing probability densities. The probability densities p arenormalized to unity.[11] As this line is traversed several situations can

occur, which are described symbolically below and areillustrated in Figure 1. At a given position on the line, x,where there is a shift such that different materials lie tothe right and left of x, is called a point of transition(Figure 1a). Following the notation of Sanchez et al.[1993], a is one of the M possible materials in w, and (a)is any material in w except a.[12] Throughout this discussion, the notation (x, y) 2 a,

indicates that the segment (x, y) is in material a, but thepoints x and y are not (e.g., an open set). x is a point that isalways to the left of y although the line can be traversed ineither direction. Using these definitions, the incrementalpath that a photon travels can be described using intervalprobabilities and local probabilities: pa(x, y) is the proba-bility for the interval (x, y) to be in a, pa(a)[x, y)dx is theprobability of undergoing a transition a!(a) at x and have(x, y) 2 (a), pa(a)(x, y]dy is the probability of undergoing atransition a!(a) at y and have (x, y) 2 a, and pa(a)a[x, y]dyis the probability of undergoing a transition a!(a) at xand have (x, y) 2 (a) and undergoing a second transitionfrom (a)!a at y. These probabilities are illustrated inFigures 1b–1e. Although both linear kinetic theory andstochastic theory are derived using multiple materials, thestochastic model described in this manuscript is confined tohaving just two materials, cloud and clear sky. Thereforethe above set of probabilities can be written as pcld(x, y) isthe probability for the interval (x, y) to be in cloud,pcld!clr[x, y)dx is the probability of undergoing a transitionfrom cloudy to clear sky at position x where the openinterval (x, y) is entirely made of clear sky, pcld!clr(x, y]dy isthe probability of undergoing a transition from cloudy toclear sky at position y and have the entire interval (x, y) incloud, and pcld!clr!cld[x, y]dy is the probability of under-going a transition from cloud to clear sky at x and have theinterval (x, y) be clear sky, then undergoing a secondtransition from clear sky to cloudy at y. In Figure 1 thegray segments (chord lengths) would therefore representcloud and the black material, clear sky.[13] Several relationships can be drawn using these prob-

abilities. The likelihood that a given piece of the line (x, y) is

D18113 LANE-VERON AND SOMERVILLE: STOCHASTIC THEORY OF RADIATIVE TRANSFER

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in cloud, written pcld(x, y) is actually the sum of twoprobabilities: (1) the probability that the entire segment tothe left of y is already cloudy and (2) the probability thata transition occurs from clear sky to cloud at some pointto the left of y. If the transition occurs at position x thenthe probability density per unit length is expressed aspclr!cld[x, y).[14] In approximate models like the stochastic radiative

transfer model, local probabilities are of particular interest.In the limit that jy-xj goes to 0, the local material-to-materialtransition densities can be expressed as pcld!clr(x). Condi-tional probabilities can be used to relate these local prob-abilities to the stochastic treatment of radiative transferthrough a partially cloudy layer. For example, it is ofinterest to know that if at point x a transition is occurringfrom clear sky to cloud, the likelihood that as one continuestraversing the line the following segment is made of cloudcan be expressed as [Sanchez et al., 1993]

Qcld x; y½ Þ ¼ pclr!cld x; y½ Þ=pclr!cld xð Þ: ð1Þ

Another way to state this is that the probability that whentraversing the line from left to right there is a transition to

the left of x and that (x, y) is in cloud is equivalent to theprobability that position x is in cloud multiplied by theconditional probability that the segment (x, y) is in cloudgiven that there is a transition from clear sky to cloud at x,

pclr!cld x; y½ Þ ¼ pclr!cld xð ÞQcld x; y½ Þ:

There is a similar definition, Qcld(x, y], for when traversingthe line in the opposite direction with a transition occurringat position y.[15] To complete the relationship between line statistics

and stochastic theory, it is also necessary to derive therelated chord density distributions fcld[x, y]. Symbolically,these are conditional probability densities that express thelikelihood that a particular length (chord) will be made ofcloud if Qcld is known. When applied in the stochasticmodel, the physical geometry of the cloud field is describedby these chord density distributions, indicating that a photonhas a certain probability of traversing any of these horizon-tal cloudy segments when passing through the cloud field.The clear sky is described in a similar manner.[16] In this approach to cloud radiation modeling, the

statistics are homogeneous, meaning that it does not matterwhether the line is traversed from left to right or vice versa.This also indicates that all probability quantities definedover the interval (x, y) depend only on the length d = y � xand all local probabilities are constant.[17] Using this theoretical basis, the following situation

can be described. As the photon travels from the top of theatmosphere to the surface at a given position, it will be ineither cloud or clear sky. If the photon is in a cloud, themodel will calculate the likelihood that at the next step indistance, the photon will transition to clear sky versusremaining in cloud. Similar calculations are made forphotons in clear sky.[18] All quantities of interest can now be determined from

the chord length distribution fcld (d ), the local materialprobabilities pa and the local conditional transition proba-bilities. Because there are only two materials, cloud andclear sky, the transition probabilities will be equal to unitymeaning that at a transition point a change must be made tothe other material. With the initial condition that Qcld (0) = 1,

Qcld dð Þ ¼Z 1

d

fcld zð Þ@z; ð2Þ

Pcld dð Þ ¼ Pcld!clr

Z 1

d

Qcld zð Þdz: ð3Þ

The local transition probabilities become

Pcld!clr ¼Pcld

lcld

; ð4Þ

where

lcld ¼Z 1

0

fcld zð Þzdz ¼Z 1

0

Qcld zð Þdz ð5Þ

Figure 1. Illustration of transition probabilities for linestatistics. (a) Transition point from material (a) to a. (b)Example of an open interval in material a betweentransition points x and y. (c) Transition from material a to(a) at transition point x on the segment x-y which is closedat x and open at y. (d) Transition from material a to (a) atpoint y. (e) Transition from (a) to a at point x and then froma to (a) at point y. Open circles indicate an open interval,and solid circles indicate a closed interval.


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is the average chord in cloudy material. This is the quantitythat is input to the stochastic radiative transfer modeldescribed in section 3. In the literature, it is often written asgcld(x] = 1/lcld(x), and this notation will be used in thispaper.[19] For present purposes, only Markovian statistics, are

discussed. These statistics can be expressed in terms of theconditional transition probabilities and the chord lengthdistribution fcld,clr[x, y]. To do this, the line representingthe photon path will be filled progressively with alternatingsegments of the two materials, cloud and clear sky. Thissituation is visualized conceptually in Figure 2a, wherewhite indicates a cloudy segment of the path, and blackindicates a section of clear sky. The segments for both cloudand clear sky are randomly selected from the chord lengthdistribution shown in Figure 2b. For this situation, pclr(x) =1 � pcld(x). To determine how to sample at point x, the localprobabilities pcld(x) are used to determine which material is

placed at this point, and then using the density of probabilitylcld�1 to find the next transition point.[20] Markovian statistics are characterized as having

chord distributions with exponential behavior and beingdefined in terms of the Chapman-Kolmogorov equations.The Chapman-Kolmogorov relation for homogeneous sta-tistics can be written as

W rpcld!clr ¼�pclr

lclr

þ pcld

lcld

; ð6Þ

This term must be equal to zero, which leads to the resultthat

pcld ¼ pclr

lcld þ lclr

; ð7Þ

pclr ¼pcld

lclr þ lcld

:

For homogeneous Markovian statistics, the chord lengthdistribution can be represented as fi sð Þ ¼ 1

lie�s=li [Malvagi

and Pomraning, 1993], where s denotes the path followedby the photon including directional changes, and i = 0, 1indicates cloudy or clear sky, respectively.

2.2. Transport Equation

[21] The probabilities described above are now added tothe radiative transfer equation. The time-dependent, mono-energetic, conservative-scattering shortwave radiative trans-fer equation is written as follows:

c�1 @I

@tr;W; tð Þ þ W rI r;W; tð Þ þ s r; tð ÞI r;W; tð Þ

¼ 4pð Þ�1ss r; tð ÞZ4�

f r;W W0; tð ÞI r;W0; tð ÞdW0 ð8Þ

where W is the direction of travel, r is the position vector, sis the macroscopic total cross section, ss is the macroscopicscattering cross section, f(W W0) is the single-scatterangular redistribution function, I is the specific intensity ofradiation, and c is the speed of light. The single-scatterangular redistribution function indicates the likelihood ofscattering to direction W from W0. I(r, W) is defined such thatIdrdW is the number of particles in drdW at time t.[22] This relationship may be ensemble averaged to yield

two coupled formally exact transport equations that can bederived to describe particle flow in a binary statisticalmixture [Titov, 1990; Pomraning, 1997]. They are writtenbelow (with t and r dependencies dropped):

W r Iipið Þ þ siIipi ¼ ssi

Z4�

fi W W0ð Þ

piIi W0ð ÞdW0 þ Sipi þ IW rcih i ð9Þ

IW rcih i ¼ pj�Ijlj

� pi�Iili

; i; j ¼ 0; 1; i 6¼ j: ð10Þ

where ci is the characteristic function that indicates whetherthere is cloud or clear sky (material i or j) at position r. The

Figure 2. (a) Subset of 15 possible photon paths where thealternating cloudy (white) and clear sky (black) segmentswere taken from the same probability density function.(b) Distribution of cloud chord lengths from which thephoton paths in Figure 2a were selected.


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probability of finding material i at position r and time t ispi(r, t) = hci(r, t)i. �Ii and Ii are conditional ensemble–averaged intensities which are governed by whetherposition r is in material i, and whether r is an interfacebetween clear and cloudy sky. Each material must have aspecific set of properties, written as [si, ssi, fi, Si] where i =0 for clouds and i = 1 for clear sky, which will be consideredas discrete random variables. As the photon moves along apath it encounters alternating segments of cloud and clearsky.[23] The angular dependence (W) of the transition lengths

(l) can be used to introduce directionally dependent cloudsize and spacing. For an ensemble of scenes (clear sky withclouds), l is the probability distribution function of amaterialalong a given path. The stochastic model characterizes cloudsas ellipses using g ¼ D

HwhereH is the vertical dimension and

D is the horizontal dimension [Malvagi et al., 1993]. Thechord density distribution is then represented as

1

lcld

¼ m2

H2þ 1� m2

D2

� �1�2

ð11Þ

where m is the cosine of the polar angle between the z axisand direction W. The volume fraction of cloud is thendefined as pclr = 1 � pcld. It is important to note that cloudyand clear sky can have different statistics, although thisoption is not currently employed in the model.[24] For spherical clouds the transition probabilities

become independent of the direction W. Also, the Markovianmodel can be completely described by the independentparameters pi and li. These two quantities must be specifiedto calculate radiative transfer in the layer.

Ih i ¼ pclrIclr þ pcldIcld ð12Þ

[25] In principle, all quantities are dependent on the threespatial coordinates of x, y and z. However, for thesecalculations the restricted problem of three-dimensionalclouds embedded in a planar layer is considered. This limitsthe clouds to a single characteristic horizontal dimensionalong both x and y directions. Furthermore, if the physicalproperties of the two materials are assumed to be constant inthe horizontal dimension for each cell the solutions �Ii and Iiare assumed to be independent of x and y. This means thatthe direction W is only defined by m and the azimuthal anglej. This conveniently allows integration over j in order toobtain a planar radiative transfer equation:

m@

@zpi zð Þyi z; mð Þ½ � þ si zð Þpi zð Þyi z; mð Þ

¼ ssi zð ÞZ 1

�1

gi z; m; m0ð Þpi zð Þyi z; m0ð Þdm0

þpj zð Þyj z; mð Þ

lj z; mð Þ � pi zð Þyi z; mð Þli z; mð Þ ; i ¼ 0; 1; j 6¼ i ð13Þ

where

yi mð Þ ¼Z 2p

0

Ii m;jð Þdj; ð14Þ

yi mð Þ ¼Z 2p

0

I i m;jð Þdj ð15Þ

gi m; m0ð Þ ¼Z 2p

0

fi W W0ð Þdj: ð16Þ

The last two coupling terms on the right hand side ofequation 13 represent the conditional probabilities describ-ing photons transitioning from one material to the other.This one-dimensional equation accounts for the three-dimensional cloud geometry under assumed horizontalinvariance of the chord material.[26] There is no general exact mathematical solution for

the stochastic transport problem. However, there are numer-ous approximations [Pomraning, 1997]. An entire subset ofsolutions and closures has been developed for climateresearch that addresses the highly idealized situation of abinary mixture of clouds and clear sky [Malvagi et al.,1993]. In the present simulations the simplest form of theclosure is employed, which is reasonable for most scales.

3. Model Formulation

[27] The stochastic model [Byrne et al., 1996] is com-prised of a spectral radiative transfer model and a modelatmosphere. The spectral model is a band model based onthe exponential sum fitting scheme of Wiscombe and Evans[1977]. A discrete ordinate method is used with an approx-imate iterative technique to solve the radiative transferequation. Only isotropic scattering is considered. Detailedinformation about the characteristic geometry and physicalproperties of the cloud field and the clear sky is ingested.The output of the stochastic model is the ensemble-averagedradiative components at each atmospheric layer for eachband, in addition to path length information for clear andcloudy sky. The path length information is of interest toexperimentalists who observationally determine photon pathlength and the relationship to cloud optical depth [Min andHarrison, 1999], as well as the relationship of path length toabsorption [Daniel et al., 1999].

3.1. Atmospheric Characteristics

[28] In order to calculate the radiative transfer in arealistic atmosphere, the model is initialized with profilesof pressure, temperature, moisture, carbon dioxide andozone. For the following sensitivity studies, the profilesare taken from McClatchey et al. [1972] climatologicalvalues for midlatitude summer (Figure 3). The modelatmosphere is divided into 32 layers, with a reflectivesurface. The model is applied to an area of approximately250 km by 250 km, similar to the area of an AGCM gridcell.[29] For a cloudless layer, the gaseous volume absorption

coefficient is determined as a function of the absorptioncoefficient and the optical path. The optical paths arecalculated based on absorber concentrations, taken fromthe atmospheric profiles, and then scaled by pressure andtemperature, independent of wavelength [Kneizys et al.,1988]. The volume scattering coefficient is given bybscaclear = Nssca where ssca is the scattering cross sectionand N is the total number of air molecules per unit volume.


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[30] The effect of cloudiness is parameterized in terms ofcloud water content. The characteristic scale of the clouds,both horizontal and vertical, is also specified as input. Thecloud thickness, the cloud water path and the cloud fractionare used to calculate cloud water content (CWC). If thecloud fills at least half of the model layer, the layer isconsidered cloudy, and the cloud fraction is used along withthe cloud dimensions to give a volume fraction occupied bycondensate.[31] For water clouds, the distribution of cloud droplets

follows a modified gamma distribution function of the form,

n rð Þ ¼ Ncr1�3bð Þ=b½ �e�r=reb; ð17Þ

where b = 0.1 represents the variance of the distributionabout re [Ackerman and Stephens, 1987]. From thisinformation, the volume extinction (bext) and absorption(babs) coefficients can be approximated by

bcloudext ¼ 3

2rw

CWC

reð18Þ

bcloudabs ¼ 3

4rw

CWC

reþ pNc

kimL 1�1

bð Þ 1

b� 2

� �rebþ

L2kim

� �� 1

2kim

� �

ð19Þ

where

kim ¼ 4pmim

l; ð20Þ

L ¼ 2kimrebþ 1; ð21Þ

and mim is the imaginary part of the refractive index. Iceclouds are not considered in these simulations.

3.2. Radiative Transfer Module

[32] The shortwave band model is a medium-resolutionmodel with 38 bands (Figure 4). The extinction processesinclude gaseous absorption, cloud scattering and absorptionand Rayleigh scattering [Shi, 1992]. The input parameters

Figure 3. Standard atmospheric profiles of (a) ozone, (b) water vapor, (c) temperature, and (d) densityas described by McClatchey et al. [1972]. In these simulations the climatological values for midlatitudesummer were employed.


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for the radiative transfer calculations are the optical depth,single scattering albedo and the asymmetry parameter foreach subinterval and each atmospheric layer. The opticaldepth of a layer is defined as the extinction coefficientmultiplied by the thickness of the layer. The single-scattering albedo is expressed as the ratio of the scatter-ing to the extinction. The asymmetry parameter forspherical cloud droplets is fit with a regression, 4ceff mim,which is a continuous function of wavelength andeffective radius ceff ¼ 2pre

l

where ceff is the effective

size parameter [Smith and Shi, 1995]. Using this regres-sion the asymmetry parameter can be expressed as anonlinear polynomial

gDu ¼X5k¼1

ak log10 4c2eff mim

� h ik�1

ð22Þ

with the coefficients of a1 = 0.858092431749, a2 =�0.131213669100 � 10�3, a3 = 0.777674540789 �10�2, a4 = 0.319546605618 � 10�2, and a5 =0.304210434786 � 10�3 for water clouds. This fit isappropriate for a range of 4ceff mim between 10�4 and 102.[33] DSTOC’s bands range in wave number from

2500 cm�1 to 50,000 cm�1 in unequally spaced intervals.Gaseous absorption is calculated by band; each one has twopossible absorbers. The two most important absorbers arewater vapor and ozone. Carbon dioxide and molecularoxygen are included as well. In bands where CO2 or O2 isactive, the band intervals are chosen between the boundariesof the active absorber region. The band passes for allspectral intervals and the active absorbers are given bySmith and Shi [1995].[34] The effective absorption cross section ki and the

absorption coefficient ai for each absorbing gas [Goodyand Yung, 1989] are input each time that the model isinitialized. The spectrally integrated flux is approximatedby the sum of fluxes from M monochromatic calculationswith appropriate weights and absorption coefficients. Forthis physical interpretation, ai and ki must meet the

requirements that ai > 0, ki � 0 andPMi¼1

ai = 1. (Warren

Wiscombe provided the absorption coefficients andweights for bands with H2O and CO2 absorption. Similarinformation for bands having O3 and O2 absorption wasprovided by Ming-Dah Chou [Shi, 1992].)[35] Each band is divided into subbands for which the

values of the opacity, k, are similar. For each subband, aweight is determined for the average k, taken as themiddle of the subband. The bands are assumed to besmall enough that the solar input, the droplet interaction,and the surface albedo are nearly constant across theband, so that the only change in any subband is due tothe variation of the opacity. Each absorber has a uniquevariation in opacity that is nearly uncorrelated with theother absorbers. The subbands are fixed such that theweights and opacities must be determined for nonidealsubbands.[36] For these simulations, DSTOC has been updated

with a modern CO2 concentration, an accurate solarconstant for calculations in the year 1998, and moreinternally consistent atmospheric characteristics. Themodel was also modified to specify multiple cloud layersthat have different characteristics, both in terms of cloudfield geometry and cloud microphysical properties. In thisway, the situation of cumulus or stratocumulus cloudsunder cirrus clouds, which occurs often in nature, can besimulated.[37] The current formulation of the stochastic model,

while more computationally efficient than a detailedapproach like Monte Carlo modeling, is not efficientenough to be considered directly as an AGCM parameter-ization. Instead off-line calculations with the stochasticmodel would serve as the basis of a parameterization. Forexample, when model results are evaluated using observa-tions and the plane-parallel technique [Lane et al., 2002] alook-up table or curve fit may be constructed to providecorrections to a conventional shortwave radiative transferparameterization. Currently, the amount of time required torun one simulation depends directly on the amount of liquidwater present in the cloud field.

3.3. Stochastic Nature of the Model

[38] ‘‘The goal of any statistical model of cloud radiationinteraction is to obtain a relatively simple and accurate set ofequations for the ensemble-averaged intensity’’ [Malvagi etal., 1993]. The fundamental equations for the present modelare expressed as equations 13–16 in section 2. Nonstochas-tic boundary conditions are assigned that correspond to anisotropic intensity that is incident on a planar system at z =0. The radiation incident at the top of the atmosphere is notconsidered to be stochastic. The stochastic nature of theproblem enters through the description of the partiallycloudy atmosphere as described above. Alternating cloudyand clear segments along the photon path are randomlyselected from predetermined Markovian probability distri-butions that describe the horizontal scales of the populationsof cloud and clear sky. The clouds do not possess internalvariability.[39] The simple closure, �Ii = Ii, is employed, which

assumes that the radiation field and the underlying geometrycan be described as a Markovian process. The closure isexact for the case of no photon scattering, but has beenshown to be a quite good and robust approximation for

Figure 4. Representation of the solar irradiance inDSTOC for the top of the atmosphere (dashed line) andthe surface (solid line) for a solar zenith angle of 60� in aclear atmosphere.


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scattering with Markovian statistics [Adams et al., 1989] atmost scales, differing from the exact solution by no morethan 10%. In most cases, the solution with this simpleclosure slightly overestimates the transmission [Malvagi etal., 1993].[40] A standard discrete ordinate technique is used to

solve the resulting differential equations [Byrne et al.,1996]. The solver uses an approximate iterative tech-nique, which calculates on a mesh that is small relativeto a photon path length. The model solution is theintensity averaged over an ensemble of cloudy sceneswith equivalent statistics. The intensity is also averagedover horizontal variations. The result is not specific to asingle small area or to only the cloudy portion of an area,but represents the domain-averaged radiation field in thegrid cell.

4. Application of the Stochastic Approach

[41] Several different studies have investigated thestochastic approach to representing clouds in a clearsky. Before exploring the sensitivities inherent in DSTOCit is useful to understand the evolution of the stochasticcloud model. Mullumaa and Nikolaev [1972] made acareful study with spectral methods to relate the cloudvariability to the statistical properties of the radiationfield. The clouds were assumed to be internally uniform.The authors used plane-parallel clouds with a Gaussian-modulated upper boundary to investigate cumulus andstratiform clouds. Ronholm et al. [1980] demonstratedthat small vertical fluctuations in average optical proper-ties enhance the transmitted radiation through plane-parallel clouds, and reduce the reflection and absorption.The authors used a modified Gaussian probability densityfunction to distribute the single-scattering albedo, opticaldepth and asymmetry parameter. Stephens [1988a, 1988b]began to explore radiative transfer through a layer usingvarious perturbations of the medium’s optical properties.[42] Malvagi et al. [1993] demonstrated that stochastic

theory is readily applied to the atmospheric radiativetransfer problem. The approach is appropriate ‘‘wherethe size of the spatial numerical cell does not allow forthe resolution of individual clouds, even if the description(size, space, and location) of such clouds were known’’[Malvagi and Pomraning, 1993]. It is inappropriate toapply this method to stratiform cloud fields with largecloud fractions. In the study by Malvagi et al. [1993] amodification is made for ‘‘arbitrary statistics’’ wherecloud spacing and distribution are random (in the verticaland horizontal directions) and both directionally andspatially dependent. This allows for the introduction ofnon-Markovian statistics where the greatest variability isin the z direction.[43] An earlier version of DSTOC was used to inves-

tigate the influence of increased mean free paths incomplex cloud fields with respect to the absorption ofsolar radiation [Byrne et al., 1996]. The clouds aredistributed at random, scene by scene, as part of thebinary mixture. Cloud differs from clear sky primarily inthe liquid water content. The probability distributionfunction is the same throughout the layer. The cloudsoccupy on average a fraction pcld of the volume of the

cloudy layer. Cloud shape is not as important as theaspect ratio. The authors took a liquid water path of200 g m�2, a 10 mm effective droplet radius, a 40� solarzenith angle and a constant surface reflectivity of 20%.The clouds are constrained to a layer between one andtwo km in altitude. In the case where the cloud size andspacing is small relative to the photon mean free path, theproblem is reduced to a plane-parallel calculation. Asthe scale increases, inhomogeneities start to occur in thescene, such as holes developing in clouds. This causesthe albedo of the cloud to drop and the transmission toincrease. However, the authors found that for cloud sizessimilar to the scale of a photon mean free path, absorp-tion may be enhanced by laterally traveling photonsrelative to the predictions of nonstochastic models. Verylarge-scale clouds are localized, and a fractional cloudcover model describes them accurately. The fractionalcloud cover model is a weighted summation of twoplane-parallel calculations, one clear and one overcast.Simplistic homogeneous models were found to overesti-mate the decrease in absorption with an increase in cloudscale. The output reflectance at the top of the atmosphere,column absorption and surface absorption were comparedto the same quantities calculated by DISORT [Stamnes etal., 1988] for three scenarios: no scattering, Rayleighscattering and thick clouds. The models were found tobe in complete agreement.[44] Lane et al. [2002] explored the realism of the

stochastic approach to radiative transfer in partiallycloudy fields by inputting observed cloud field character-istics from days when fair weather cumulus or cumulushumilis were present. They compared the output from anupdated version of DSTOC with that from SUNRAY andwith independent observations from the Oklahoma Mes-onet. The authors determined that for circumstances

Figure 5. Diagram showing the difference in how cloudfraction is treated in (left) SUNRAY and (right) DSTOC. InSUNRAY the typical cloud fraction model is employed.DSTOC calculates the domain-averaged radiative flux overall possible cloudy scenes with the same statistics (threepossible scenes).


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where the cloud fraction was intermediate and the cloudswere in thin layers, the stochastic model provided a morephysical representation of the actual cloud field presentand supplied an estimate of DWSR that compared morefavorably with observations than the simple plane-parallelmodel.

5. Sensitivity Tests

[45] Until now, the ability of the stochastic model torepresent shortwave radiative transfer through complexcloud field geometries has been tested using highly ideal-ized cloud fields [Malvagi and Pomraning, 1993; Byrne etal., 1996] or a limited number case studies [Lane, 2000;Lane et al., 2002]. If the stochastic approach is to beexplored further as a possible modification of an AGCMparameterization, understanding the sensitivities inherent inthe model, and sensitivities to input cloud field parametersis critical. To this end several numerical tests were per-formed to assess these model characteristics. An analysis ofthe model’s effectiveness can be made by providing iden-tical input to DSTOC and a typical two-stream shortwaveradiative transfer code, SUNRAY, and comparing theresults. SUNRAY was developed by Fouquart and Bonnel[1980], and has two spectral bands, one in the shortwavepart of the spectrum (0.25–0.68 mm), and one in the nearinfrared part of the spectrum (0.68–4.00 mm). The modelatmosphere in SUNRAY was divided into 30 atmosphericlayers. This algorithm is typical of the shortwave radiationroutines utilized in modern AGCMs.

[46] In order to make a direct comparison between thetwo models, the vertical structure of SUNRAY’s atmo-sphere was modified to match that of DSTOC. This assuresthat the profiles of pressure, temperature, ozone and watervapor as a function of height are equivalent in both models.The stochastic model has 32 standard pressure layers, and ahorizontal domain equivalent to that of an AGCM grid area.Sensitivity tests are performed using the McClatchey et al.[1972] profiles of the atmosphere for midlatitude summer(Figure 3).[47] All clouds in these simulations are confined to a

single model layer, the second pressure layer from thesurface for both models, except when testing modelsensitivity to the height of the clouds. The inputs toSUNRAY are cloud optical depth and cloud fraction. Theoptical depth can be determined from the liquid waterpath and the effective radius using 3/2(LWP/re) [Liou,1992]. SUNRAY outputs the two-stream shortwave radi-ative flux at each level. The stochastic model ingestsliquid water path and effective particle radius and con-verts this to a volume extinction coefficient using thecloud fraction. An illustration describing the differingtreatments of cloud fraction is shown in Figure 5. InFigure 5 (left), utilized in SUNRAY, a fractional cloudmodel is employed. The flux at the surface is a summa-tion of a clear sky calculation and an overcast calculation,weighted by the cloud fraction. Figure 5 (right) indicateswhat three possible cloud distributions with the samestatistical features might be in the stochastic model fora cloud fraction equivalent to that in SUNRAY. DSTOC

Figure 6. (a) Relationship used to convert liquid water path to optical depth for input to SUNRAY.(b) Downwelling shortwave radiation at the surface (DWSR) and (c) upwelling shortwave radiation(UWSR) at the top of the atmosphere from each model are compared as a function of optical depth forovercast conditions.


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calculates the average radiative flux over all possiblecloudy scenes with the same statistics (of which thereare three shown).

5.1. Sensitivity to Cloud Optical Depth

[48] In the calculation of radiative transfer throughclouds, the most influential property is the cloud opticaldepth. In the stochastic model, the optical depth is calcu-lated by combining the information about the cloud fieldgeometry with liquid water path and droplet effectiveradius. To understand the sensitivity of DSTOC to changesin liquid water path (optical depth) both models were runfor overcast conditions (100% cloud fraction), an overheadsun and a cloud layer at the same height and of the samevertical extent (0.7 km thick with the base located at0.5 km). It should be noted that the effective radius washeld constant in both models at 6 mm. The liquid waterpath (LWP), and thereby the optical depth, was varied overa range of values from 10–1000 g m�2 in LWP (0–140 inoptical depth; see Figure 6a). The downwelling shortwaveradiation at the surface (DWSR) and upwelling shortwaveradiation (UWSR) at the top of the atmosphere from eachmodel are compared as a function of optical depth(Figure 6). The DWSR predicted by DSTOC is in closeagreement with that for SUNRAY for small optical depths(<7). There is a slightly larger discrepancy in the UWSRin this optical depth range. For optical depths greater than15 (LWP > 60 g m�2) the difference in transmissionbetween the two models asymptotes to 35 W m�2

(Figure 7). The difference between the two models in

UWSR is never greater than 15 W m�2 at any opticaldepth, and tends to �15 W m�2 at large optical depths.There is complete agreement between the models for clearsky conditions.

5.2. Sensitivity to Number of Radiative Streams

[49] Most modern AGCMs radiation parameterizationsperform their calculations with two or four streams.Therefore it is of interest to document the sensitivityintroduced by reducing the number of streams used in thestochastic predictions. Futhermore, the studies describedin this manuscript are typically performed using 8 streams(4 upward and 4 downward scattering angles) in DSTOC,while SUNRAY only has two streams. As seen inFigure 8, the difference in the downwelling shortwaveflux at the surface simulated by the stochastic model dueto varying the number of streams is much smaller thanthe difference between the downwelling fluxes at thesurface predicted by DSTOC versus SUNRAY. In general,the difference between calculations of DWSR performedby DSTOC with 8 streams and those performed with 2 isapproximately 5 W m�2 in the downwelling shortwaveradiation at the surface and 10 W m�2 in the upwellingshortwave radiation at the top of the atmosphere. Ofcourse, some of the differences between the stochasticand plane-parallel calculations are likely to come fromdifferences in how absorption and scattering are handledin the two models, as well as from differences in thespectral resolution. However, quantifying the impact thishas on the calculation of DWSR is beyond the scope of

Figure 7. (a) Difference in model prediction of the downwelling shortwave radiation at the surfaceand upwelling shortwave radiation at the top of the atmosphere as a function of optical depth.(b) Same as Figure 7a but shown as a ratio to the original flux. Note the large sensitivity at verysmall optical depths.


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this work. The greatest difference between the twomodels occurs in the liquid water path range between100 and 200 g m�2.

5.3. Droplet Effective Radius

[50] Droplet effective radius is an important microphys-ical characteristic of clouds. However, this quantity isdifficult to retrieve and AGCM predicted values oftenvary greatly from observed ones. In the stochastic model,the droplet effective radius is used to calculate thevolume scattering coefficient and in SUNRAY it is usedto convert liquid water path to optical depth. Therefore itis important to investigate the sensitivity of the simulatedfluxes to changes in the droplet effective radius. Tounderstand this sensitivity more completely, DSTOCwas run with three different droplet effective radii (5, 6and 7 mm) for the range of overcast conditions describedabove. This range of droplet effective radii is representa-tive of the error involved in retrieving this property fromground-based remote sensors. Figure 9 shows the result-ing downwelling shortwave radiation at the surface andthe outgoing shortwave radiation at the top of theatmosphere. A change of 1 mm in cloud droplet effectiveradius has a relatively small radiative impact of 10 W m�2

at low optical depths. However, as optical depthincreases, this impact can be as large as 70 W m�2

in transmitted shortwave radiation and as much as50 W m�2 in the reflected shortwave radiation. This isless than a 10% error in the outgoing radiation at thetop of the atmosphere, but can cause as much as a 20%

error in the downwelling radiation at the surface for largeoptical depths.

5.4. Sensitivity to Cloud Height

[51] The altitude of a cloudy layer in any atmosphericmodel can greatly change the calculation of the domain-averaged radiative flux. Errors in the specification ofcloud base height and cloud top height, whether throughinstrument error or poor model or observational resolu-tion, will cause errors in the prediction of the radiativefluxes. Therefore the sensitivity of the stochastic model tocloud base height was tested in the stochastic model forovercast conditions. The cloud base was initially set to0.2 km with a cloud thickness of 1 km. Then the cloudbase height was changed by 100-m increments whilemaintaining the cloud thickness. Initial simulationsrevealed several numerical problems with DSTOC relatedto the vertical placement of the cloudy layer (not shown).This issue has been rectified by introducing an adaptivegrid that allows for the addition of model layers if thecloudy layer placement (specified in height) does notcoincide with preexisting atmospheric layers. Furthersensitivity tests indicated that DSTOC was relativelyinsensitive to small changes in cloud base height. A100-m change in height altered the DWSR by no morethan 3 W m�2.[52] It is important to note that for cloud fractions

smaller than 1.0 (completely overcast skies), dividingthe cloud layer between multiple model pressure layerseffectively results in a random overlapping of clouds in

Figure 8. (top) Difference in the downwelling shortwave flux at the surface simulated by the stochasticmodel due to varying the number of streams and that calculated by the plane-parallel model as a functionof liquid water path. (bottom) Same as Figure 8 (top) for an expanded vertical axis showing that thegreatest discrepancies occur at liquid water paths between 200 and 600 g m�2.


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the stochastic model that artificially increases the hori-zontal cloud coverage. The random overlapping in thestochastic model creates a greater impact on the down-welling shortwave radiation in the stochastic model thanin the plane-parallel model. This is due to the fact thatthe assumed Markovian distribution creates a large num-ber of very small, yet opaque clouds that increases thenumber of opportunities that a photon has to be absorbed.This can be envisioned using the idea in Figure 5 withmultiple layers of clouds.

5.5. Cloud Fraction

[53] As mentioned above, cloud fraction is utilized inmost AGCMs to indicate the horizontal cloud coverage of aspecified area, usually a model grid cell. Although thestochastic model does employ more realistic cloud fieldgeometry, it still requires information about the total volumecloud amount. The broadband upwelling and downwellingshortwave radiation were compared for DSTOC andSUNRAY over a range of cloud fractions from 0 to 1(Figure 10). The liquid water path was set to 100 g m�2

Figure 9. (a) Downwelling shortwave radiation at the surface shown as a function of optical depth andinput droplet effective radius. (b) Outgoing shortwave radiation at the top of the atmosphere for the sameconditions.

Figure 10. (a) The broadband upwelling shortwave radiation at the top of the atmosphere and(b) downwelling shortwave radiation at the surface predicted by DSTOC and SUNRAY over a range ofcloud fractions from 0 to 1.


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for all simulations and the effective droplet radius was inputas 6 mm, a reasonable value for low-level, continentalclouds. DSTOC and SUNRAY are in closest agreement atvery low and very high cloud fractions. The two modelsdisagree by nearly 50 W m�2 for much of the cloud fractionrange. This suggests that a stochastic approach to modelingcloud radiation interactions will have the largest impact insituations where the cloud fraction is between 0.2 and 0.8.This effect was also noted by Byrne et al. [1996].

6. Summary and Conclusions

[54] This paper describes the theoretical basis behind thecurrent version of our stochastic radiative transfer model,DSTOC, and indicates some of the sensitivities inherent inthis technique. The described simulations are an initial stagein the evaluation of this technique with the ultimate goalbeing the development of an AGCM parameterization. Laneet al. [2002] describes additional evaluation of this tech-nique using statistics derived from observed cloud fields.One benefit of simulating photons passing through cloudsusing a stochastic approach is that it is possible to calculateimproved heating rates and radiation fields without requir-ing specific information about individual clouds. However,the stochastic model is not appropriate for all cloud sit-uations, such as when cloud fraction is less than 0.2 andgreater than 0.8, and should be explored only when the scaleof the clouds is much smaller than the area over which theradiative transfer is being calculated. The stochastic modelrequires more specific information about the cloud fieldthan most current atmospheric general circulation modelsprovide and therefore is evaluated as a stand-alone modelfor this preliminary study. The model outputs the domain-averaged, broad-band, shortwave radiation fields.[55] This diagnostic study using the stochastic model

indicates that simulated values of the radiation fields displaya marked sensitivity to several input parameters. The sto-chastic predictions of the radiation fluxes at the surface andthe top of the atmosphere were compared to the equivalentinformation from a plane-parallel model, SUNRAY. Thismodel is representative of the type of shortwave radiationparameterizations found in modern AGCMs. At low opticaldepths, the stochastic model predicted less radiationreaching the surface than in the plane-parallel model, wherefor optical depths greater than 7 the opposite is apparent.This is consistent with the results of Adams et al. [1989]. Inaddition, the stochastic model indicated sensitivity to thedroplet effective radius and cloud fraction.[56] The stochastic approach has the advantage that, in

principle, it is capable of being made more general, morecomprehensive, and more physically realistic than conven-tional techniques for handling radiative transfer throughcloud fields with complex geometry. The goal in this workhas been to take the first steps toward developing acorrection to existing cloud parameterizations in climatemodels. The purpose of such a correction is to account forthe influence of cloud population and cloud geometrycharacteristics on the radiation field. The next steps willinvolve devising means of deriving the necessary stochasticmodel inputs from those attributes of cloud fields that canbe inferred from climate model simulations. Among thecandidate routes to further progress is the development of

context-sensitive parameterizations in which the cloud sta-tistics will be appropriate to given circumstances, such aslocation, season and synoptic regime.

[57] Acknowledgments. This work was inspired in part by discus-sions with the late G. R. Pomraning. This work was supported in part bythe Office of Naval Research under grant N00014-97-1-0554, by theDepartment of Energy under grant DE-FG03-97ER62338 and grantDE-FG02ER63314, by the National Oceanic and Atmospheric Administra-tion under grant NA77RJ0453, by the National Aeronautics and SpaceAdministration under grant NAG5-8292, and by the National ScienceFoundation under grant ATM-9612764 and grant ATM-9814151. This workis based in part on the Ph.D. dissertation of Dana E. Lane (University ofCalifornia, San Diego, 2000).

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��D. E. Lane-Veron, Department of Environmental Sciences, Rutgers, State

University of New Jersey, 14 College Farm Road, New Brunswick, NJ08901-8551, USA. ([email protected])R. C. J. Somerville, Climate Research Division, Scripps Institution of

Oceanography, University of California, San Diego, La Jolla, CA 92093,USA.


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stochastic theory of radiative transfer through …stochastic theory of radiative transfer through...

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