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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS
FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
JULIA KARLGÅRD
May 2008
Diploma work of 15 ESCTs
Department of Physics, Lund University
SWEDISH TITLE: Bestämning av molndroppsstorlek med hjälp av satellitmätningar
SUPERVISOR: Erik Swietlicki
CONTACT: [email protected]
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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ABSTRACT
A 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 of
aerosols and its role in global warming, it is important to be able to model the growth of cloud
drops.
In this study, cloud droplet effective radius is modelled from satellite reflectance
measurements in two MODIS wavebands, 0.65 µm (visible) and 2.1 µm (NIR). The effective
radius re is preferred to mean or mode radius since it accounts for the size distribution of the
droplets within the cloud. Reflectance data is taken from a satellite scene covering southern
Sweden, on May 9th, 2004. The approach is a relatively simple approximate forward model
called 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 pixels
with NIR reflectance R2 > 0.2 and optical thickness τ > 10. The underlying principle of the
model is the asymptotic theory. This theory is based on the fact that reflectance in the
nonabsorbing wavelengths (visible) is mainly a function of τ, while reflectance in the
absorbing wavelengths (near and mid infrared) is governed by re. For large optical thickness,
the reflection function is close to the known asymptotic equations. Also, when the optical
thickness is large enough, τ and re can be determined nearly independently of each other.
The results showed good agreement with near coastline clouds in the study by Rosenfeld and
Lensky (1998) for R2 > 0.3. Median and mode effective radius was 14.6 and 15 µm
respectively. A mode re of 15 µm indicates precipitating clouds or clouds close to
precipitation, since 14-15 µm is considered to be a rainout threshold. Smaller reflectance
values in the near infrared resulted in larger mode and median re (18 and 17-20 µm
respectively), and τ < 5. This implies that the model is not reliable for small reflectance values
in the near infrared. The error for pixels with τ > 10 is estimated to 10-20%, but varies with
viewing geometry, optical thickness and probably also wavebands used.
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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SAMMANFATTNING
Moln förknippas ofta med väder och kanske i synnerhet lågtryck. Men moln spelar också en
viktig roll i jordens uppvärmning. Moln både stänger ute värme, i form av inkommande
solstrålning, och isolerar, i form av utgående värmestrålning från jorden. Den totala effekten
beror bl. a. på molnets höjd ovanför marken, tjocklek och inre struktur, såsom droppstorlek
och droppkoncentration (antalet molndroppar per volymenhet). Droppstorleken tillsammans
med droppkoncentrationen avgör molnets s. k optiska täthet, en egenskap som beskriver hur
mycket av det inkommande solljuset som släpps igenom en viss tjocklek.
Molnens egenskaper, då framför allt droppstorleken, avgör också nederbörden. Ju större och
tyngre 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 sig
att aerosoler (små, luftburna partiklar t ex sot, damm, salter etc.) kan påverka droppstorleken.
Vissa aerosoler (hydrofila) fungerar som kondensationskärna för molndroppar genom att
utgöra en lämplig yta som atmosfärens vattenånga kan kondensera mot och bilda
vattendroppar. I ett luftpaket med hög aersolkoncentration finns således många
kondensationskärnor, vilket resulterar i många vattendroppar. Eftersom mängden vattenånga i
luftpaketet 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 inte
finns någon kondensationskärna). De mindre vattendropparna har lägre sannolikhet att falla ut
som regn och luftpaketet, eller molnet, får på sätt längre livslängd. Detta i sin tur kan leda till
förändrade nederbördsmönster och hydrologi, t ex genom att molnen hinner transportera bort
vattnet i atmosfären innan det regnar ut.
En effekt av att molndropparna i förorenade moln är mindre än normalt är att molnets
reflektiva egenskaper förändras. Då den inkommande solstrålningen passerar genom
molndropparna 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 inkommande
solstrålning än moln med lägre aerosolkoncentration och på så sätt har en kylande effekt på
klimatet. Detta kallas även aerosolers indirekta effekt. Enligt IPCC (2001) är den indirekta
effekten av aerosoler den enskilt största osäkerhetsfaktorn i dagens klimatmodeller. Man
menar att dagens aerosolhalter i luften troligtvis har en dämpande effekt på den globala
uppvärmningen, 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 hur
aerosoler påverkar molnets egenskaper, både i fråga om reflektans och nederbörd. Idealet är
att kunna jämföra molnegenskaper i luftmassor med låg respektive hög aerosolkoncentration
och 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 (eller
komplement) att använda sig av datamodeller, för att utifrån satellitmätningar av moln kunna
ta reda på molnets tjocklek och droppstorlek. Man utnyttjar här effekten av att moln med olika
droppstorlek och dropptäthet har olika reflektans (d v s reflekterar och absorberar olika
mycket). Tillsammans med kännedom om luftmassornas aersolhalt skulle dessa modeller
kunna ge en uppfattning om aersolers inverkan på moln och molnbildning.
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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I denna studie har en enklare modell använts på satellitdata från 9e maj 2004, för att beräkna
droppsstorleken i moln över ett område i södra Sverige. Modellen kallas Modified
Exponential Approximation (MEA) och utgår från den uppmätta reflektansen i två olika
våglängdsband från satelliten MODIS Aqua: 0.620-0.670 µm och 2.105-2.155 µm. Modellen
är en förenkling av den s k Asymptotiska teorin, en teori som bygger på att för tillräckligt täta
moln kan reflektansen beskrivas av kända funktioner, samt att den effektiva radien och den
optiska tätheten kan bestämmas näst intill oberoende av varandra. Istället för att beräkna
medelradien eller typradien beräknas den effektiva droppradien eftersom denna är ett viktat
mått 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 i
Bremen, 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 effektiva
droppstorleken och optiska tätheten beräknats för de pixlar som uppfyller vissa krav på
reflektansen. 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 14
μm, 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 optiska
täthet.
Den relativt enkla modellen har visat sig vara snabb i beräkningarna och samtidigt ge resultat
jämförbara med betydligt mer avancerade och beräkningstunga modeller. Målet med
modellen är att kunna använda denna tillsammans med data över molntoppstemperatur för att
kunna göra beräkningar av molnets tillväxt, och hur denna påverkas av aersolhalten i luften.
För en framtida användning skulle en noggrannare utvärdering av modellens tillförlitlighet
vara av stort värde, liksom en analys av modellens känslighet för vissa parametrar. En
utveckling av modellen som skulle kunna urskilja varma moln (bestående av enbart
vattendroppar) från kalla moln (enbart iskristaller) och mixed phased (blandning av
vattendroppar och iskristaller) skulle också vara intressant, eftersom modellen bygger på
antagandet om varma moln, d v s moln enbart uppbyggda av vattendroppar.
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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INDEX
ABSTRACT............................................................................................................................................................. i
SAMMANFATTNING.......................................................................................................................................... ii
1 INTRODUCTION .............................................................................................................................................. 3 1.1 BACKGROUND ............................................................................................................................................... 3 1.2 AIM ............................................................................................................................................................... 3 2 METHOD ........................................................................................................................................................... 4 2.1 THEORY......................................................................................................................................................... 4 2.1.1 Light scattering ..................................................................................................................................... 4 2.1.2 Radiative transfer ................................................................................................................................. 6 2.1.3 Asymptotic theory and Reflection function ........................................................................................... 9 2.1.4 Semi-analytical method and Modified Exponential Approximation ................................................... 12 2.2 DATA AND COMPUTATIONS......................................................................................................................... 16 3. RESULTS AND DISCUSSION ..................................................................................................................... 20 4. VALIDATION AND ERROR ANALYSIS ................................................................................................... 22 5. CONCLUSIONS ............................................................................................................................................. 24 APPENDIX .......................................................................................................................................................... 27 REFERENCES .................................................................................................................................................... 29 1
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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1 INTRODUCTION
1.1 Background
One of the major uncertainties in the models used for creating projections of future climate is
the 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 an
overall cooling effect on the global climate (IPCC, 2007). A high concentration of aerosols
increases the number of cloud condensation nuclei (CCN) and thus generating more, but
smaller cloud droplets. Hence aerosols affect cloud droplet size, optical thickness, growth rate
and 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 on
convective clouds such as cumulus clouds is believed to be significant, since these cloud
types are fed by air masses rising from below.
Studying the indirect effect of aerosols involves modelling cloud dynamics and cloud
structures. However, cloud modelling is a complex issue, since clouds are seldom, if ever
homogeneous. A cloud cannot be considered as a single entity, but rather as a composition of
billions of much smaller units, cloud droplets. The drop size distribution varies with height
and the phase may change from water to mixed phase to ice through the vertical profile of the
cloud giving rise to different radiative characteristics. Since many satellites provide data of
physical parameters such as reflectivity and emissivity at several wavelengths, covering most
parts of the world with frequent time intervals, satellite observations is a valuable source of
information. Through the development of technology and computer power, which has enabled
managing large amounts of data and faster computations, satellite measurements have become
extensively used in cloud and atmospheric research in the last decades.
Numerous studies with focus on radiative characteristics of clouds have been performed, but
because of the complexity of cloud modelling most algorithms are developed for plane
parallel clouds, such as stratus clouds. The basic principle in the technique of using multispectral reflectance data for determining microphysical properties in clouds, such as optical
thickness and droplet sizes, is the variations in reflectance due to these two parameters at
different wavelengths. In the visible region the reflection function is primarily a function of
cloud optical thickness, while in the near or mid infrared the reflection function depends
primarily on cloud droplet sizes (Nakajima and King, 1990; Liou, 1992; Kokhanovsky, 2006).
1.2 Aim
In this study the cloud effective radius of cumulus (water) clouds over southern Sweden is
modelled from satellite data. When modelling cloud droplet sizes the effective radius is
preferable to mode or mean radius since it better accounts for the size distribution within the
cloud. The effective radius is defined as follows:
∞
re = ∫ r n(r )dr
3
0
∞
∫r
2
n(r )dr
(1.1)
0
where n(r) is the particle size distribution and r is the particle radius (Rosenfeld and Lensky,
1998; Nakajima and King, 1990).
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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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, and
requires relatively few computations and input parameters. Reflectance data is taken from
band 1 and 7 (corresponding to wavelengths ~ 0.65 µm and 2.1 µm respectively) in MODIS
Aqua observations on the 9th 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 study
the growth rate of cloud droplets and how this may be affected by the amount of aerosols in
the air.
2 METHOD
2.1 Theory
2.1.1 Light scattering
Scattering 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 size
and shape of the scattering particle, scattering patterns appear different in different cases. For
spherical particles a size parameter x may be defined; x = 2π⋅a/λ, where a is the particle
radius and λ the wavelength of the incident light. When x ≥ 1 scattering is referred to as
Lorentz-Mie scattering or sometimes Mie theory (Liou 2002, p. 96). This is generally the case
when light in the visible and near- and mid infrared (NIR, MIR) region interacts with aerosols
or cloud droplets (Liou, 2002 p. 97). The light is scattered in all directions, but the forward
scattering predominates. The intensity of light I after scattering in a direction θ (scattering
angle) is described by
P(θ )
⎛ σ ⎞ P(θ )
I (θ ) = I 0 Ω eff
= I 0 ⎜ 2s ⎟
(2.1)
4π
⎝ r ⎠ 4π
where I0 denotes the intensity of incident light, Ωeff is the effective solid angle of scattering,
P(θ) is the phase function, describing the probability of a photon being scattered in the
direction θ, σs is the scattering cross section and r is the distance between the observer and the
scattering particle (Liou, 2002, p. 96). The Mie theory is not a physical theory or law in itself
but rather a complete analytical solution to Maxwell’s equations, describing the absorption
and scattering of light by spherical particles (Liou 1992, p. 262). The pattern of light intensity
due 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 be
derived from the Mie solution, expressed in terms of Legendre polynomials, Pl:
N
P (cos θ ) = ∑ cl Pl (cos θ )
(2.2)
l =0
where cl is the expansion coefficient. Because of the orthogonal property of Pl we may write
cl in the form
1
2l + 1
cl =
P(cos θ ) Pl (cos θ )d cos θ
(2.3)
2 −∫1
For l = 0, c0 = 1, representing the normalization of the phase function (the probability of the
photon being scattered in any direction being 1). When l = 1 Eq. (2.3) can be used to define
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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the asymmetry parameter g, which denotes the relative strength of the forward scattering
(Liou, 2002, p. 105; Kokhanovsky, 2006, p. 96):
g=
c1 1 1
=
P(cos θ ) cos θ d cos θ
3 2 −∫1
(2.4)
Even with the help of computers, it is not a simple task to derive the Legendre polynomials Pl
and the expansion coefficients cl from the Mie theory. Therefore an approximate analytic
expression 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 phase
function, where the phase function is expressed in terms of the asymmetry parameter g
(Mishchenko et al., 1999; Liou, 1992, p. 127);
p HG
N
(1 − g 2 )
=
(2l + 1) g l Pl (cos θ )
=
∑
2
3/ 2
(1 + g − 2 g cos θ )
l =0
(2.5)
The Henyey-Greenstein phase function is best suited for the case where the forward scattering
is less pronounced, i.e. for smaller size parameters, and one should bear in mind that one of
the main features for light scattering by aerosols and cloud droplets is strong forward
scattering. 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 cloud
microphysical properties and is given by:
5
p (θ ) = Qe −C ⋅θ + ∑ bi e − βi (θ −θ i )
i =1
2
(2.6)
where Q = 17.7, C = 3.9. p is used instead of P to indicate that the method is an
approximation. bi, βi and θi are derived by parameterization of Mie theory results, values are
given in Table 1 below (Kokhanovsky, 2004a; Kokhanovsky et al., 2003).
Table 1 Parameters bi, βi and θi for approximate phase function
(Kokhanovsky, 2004).
i
bi
βi
θi
1
2
3
4
5
1744.0
0.17
0.30
0.20
0.15
1200.0
75.0
4826.0
50.0
1.0
0.0
2.5
pi
pi
pi
Figure 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 the
Kokhanovsky equation is that it does not account for the influence of the size parameter on
the phase function. It is merely an approximation for all clouds, with no dependence on
effective radius, optical thickness or even wavelength, but has proven to be useful when
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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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 of
droplet radius on the reflection function in two different wavelengths, so the HenyeyGreenstein function in this case is preferred. Since scattering is dominated by multiple
scattering (thus single scattering characteristics becoming less pronounced), the dependence
of the cloud reflection function on the phase function is however rather weak (Kokhanovsky
2006, p. 148). In the end the choice of phase function therefore mainly depends on the
purpose of use.
10
10
phase function
10
10
10
10
10
4
Kokhanovsky
Henyey-Greenstein
3
2
1
0
-1
-2
0
20
40
60
80
100
120
140
160
180
Θ
Figure 1 Comparison between the Henyey-Greenstein phase function (blue line) and the
parameterized phase function by Kokhanovsky (red line). g = 0.85.
2.1.2 Radiative transfer
For optically thin media a large part of the scattering events is due to single scattering, i.e. the
incident 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 cumulus
clouds, scattering is dominated by multiple scattering, i.e. photons are scattered several times
between the cloud drops before escaping the media. (Liou, 2002, p. 105). The optical depth τ
is defined from the light extinction coefficient ke (Liou, 2002, p. 103):
∞
τ = ∫ k e dz
(2.7)
z
The radiative transfer theory is used to predict how the intensity of light changes as incoming
direct solar light (solar flux density) F0 is scattered and transmitted through the media, in this
case the cloud. The basic theory in radiative transfer is an assumption of a plane parallel
atmosphere or cloud, sliced into homogeneous layers with a differential thickness Δz. As
incoming solar flux F0 from the direction (-v0, ϕ0) (minus sign denotes downward direction) is
traversed through the layer Δz it undergoes several processes affecting the intensity of the
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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outgoing light in the direction (v, ϕ) (Figure 2). The main processes are single and multiple
scattering, emission and absorption (Liou, 2002, p. 152). A portion of the light is scattered
and emitted downwards, entering the underlying layer where the same processes occur.
Outgoing light is here re-entering the layer above or scattered further downwards. This
procedure is run for all the layers within the cloud and eventually the light is either
transmitted through the whole cloud or the intensity becomes negligible (Figure 3).
v0
v
φ
ϕ
N
ϕ0
Δz
Figure 2 Viewing geometry of incoming and reflected light.
I(0,µ,φ)
τ=0
Top
I(τn,µ,φ)
I(0,-µ,φ)
n
τ = τn
I(τn,-µ,φ)
I(τc,µ,φ)
τ = τc
Bottom
I(τc,-µ,φ)
Figure 3 Principle of radiative transfer. The intensity is expressed in terms of optical thickness τ , (τc is the
cloud optical thickness) incident angle µ ≡ cos(v), and relative azimuth angle φ = ϕ0 - ϕ
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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How deep into the medium one must look before the intensity is reduced by a certain degree
is given by the optical depth τ, which depends on the wavelength of the incident light and the
structure of the media (effective radius, liquid water path). In the simplest case with no
scattering, the reduction in intensity I at a given wavelength λ is due only to absorption. The
attenuation in intensity can then be expressed in terms of the intensity and the extinction
coefficient ke (assumed to be constant within a homogeneous layer Δz) (Kokhanovsky 2006,
p. 114):
dI λ = − k e ,λ I λ dz
(2.8)
For a more realistic, non homogeneous media (with finite volume and thickness Δz) and the
particle (or droplet) number density n, the extinction coefficients ke is defined by
ke =
∫
σ e ( z ) n( z )
Δz
Δz
(2.9)
dz
with σe denoting the cross section for extinction.
In most cases the attenuation is however due to both absorption and scattering so that
ke = ks + ka. In resemblance with Eq. (2.8) the scattering and absorption coefficient, ks and ka,
can be defined
σ s , a ( z ) n( z )
k s ,a = ∫
dz
(2.10)
Δz
Δz
where σs,a denotes the cross section for scattering and absorption respectively. From ke, ks and
ka the single scattering albedo is defined.
ω0 =
ks
ke
or
1 − ω0 =
ka
ke
(2.11)
The single scattering albedo is thus the ratio between the scattering coefficient and the total
extinction coefficient. In the visible absorption is negligible and ω0 = 1. This is sometimes
referred to as conservative scattering, i. e. where no energy is lost due to absorption.
For ω0 < 1 part of the extinction is due to absorption (non conservative scattering), seen in the
NIR and MIR wavebands. The problem becomes more complex when considering the
increase in intensity caused by emission and multiple scattering (Liou, 2002, p. 27). It is
necessary to define the change in intensity, dI in such a way that it accounts for both the loss
in intensity by scattering and absorption and the increase in intensity due to multiple
scattering and emission:
dI = −σ e,λ nIdz + j λ ndz
(2.12)
Here the extinction coefficient is exchanged by the extinction cross section σe,λ (units of area
per number) and the number density n (units per volume) of the material. jλ is the source
function coefficient, which has the corresponding physical meaning as extinction cross
section. From jλ and σe,λ the source function is defined as the ratio
J λ ≡ j λ σ ext ,λ
(2.13)
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
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Dividing Eq. (2.12) by σe,λ⋅n⋅dz generates the general radiative transfer equation (Liou 2002,
p. 28; Liou 1992, p. 109).
dI
(2.14)
= −I λ + J λ
σ e,λ n ⋅ dz
This equation is the fundamental relationship in radiative transfer problems, where the
solution may give information on several radiative parameters such as optical thickness,
single scattering albedo, effective radius etc.
2.1.3 Asymptotic theory and Reflection function
The discrete ordinate method, developed by Chandrasekhar in 1950, is an exact method for
solving radiative transfer problems numerically (Rybicki, 1996). The method is often used for
accurate calculations of radiative properties of cloudy atmospheres (Liou, 1992, p. 108). One
of the main products from the discrete ordinate method is the reflectance function (in remote
sensing often referred to as Bidirectional Reflection Distribution Function or simply BRDF)
which gives the reflected intensity in relation to incident solar flux F0 density. The reflectance
is defined as follows:
R (τ ; µ, µ0 , φ ) =
πI r (τ = 0; µ, φ )
µ0 F0
(2.15)
Here Ir denotes the intensity of the reflected light at top of the atmosphere (TOA), in the
direction (µ,φ), where µ = |cos(v)|, µ0 = cos(v0) and φ = |ϕ-ϕ0| (Kokhanovsky et al., 2003). In
remote sensing studies of clouds the reflectance Ir, and hence R, is however measured in nonor weakly absorbing wavelengths, therefore the loss in intensity due to water vapour
absorption is often neglected and Ir is approximated to the intensity at cloud top.
Knowing parameters such as cloud optical depth τ, single scattering albedo ω0 and effective
radius re 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 a
certain viewing geometry is given, while information on optical depth and effective radius is
desired. The fact that reflectance in the visible (nonabsorbing) wavelengths is mainly a
function of optical thickness while reflectance in the NIR and MIR (absorbing) wavelengths
is governed by effective radius makes it possible to determine the optical thickness and
effective radius using reflectance data from two or more wavebands. For large optical
thickness (≥12) the sensitivity of the nonabsorbing and absorbing wavelengths to these two
parameters are almost orthogonal, so τ and re can be determined nearly independently of each
other (Nakajima and King, 1990). Figure 4a and 4b illustrates how the reflection function in
the 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 make
pre-calculated look-up tables of the reflection function at given viewing geometry and cloud
parameters. This method is slow, and if conditions (ground surface albedo, phase function
representation) are changed, new look-up tables must be created. Thus, when dealing with
satellite images, with thousands of pixels being analysed with varying conditions, a simpler,
analytical method may be desirable. Therefore another approach called the asymptotic theory
is often used for optically thick clouds. The underlying principle is that when optical
thickness is large enough the numerical solution of radiative transfer is close to the known
9
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
asymptotic equations presented below. King (1987) found that the accuracy of the asymptotic
theory for both conservative and non conservative scattering is within 1% if the scaled optical
thickness τ*= τc(1-ω0g) is equal or larger than 1.45 (τc denoting cloud optical thickness).
Typically for water clouds g ≈ 0.85 which corresponds to a cloud optical thickness equal to or
larger 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 nadir
observation angle for different values of optical thickness τ and effective radius ref.
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 DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
Equations (2.16) and (2.17a) are valid for plane parallel homogeneous layer overlying a
Lambertian surface with albedo Ag (King, 1987; Kokhanovsky and Nauss, 2006; Nakajima
and King, 1990). For conservative scattering with ω0 = 1 (i.e. in the visible) the reflectance
function of R is written
R (τ c ; µ, µ0 , φ ) = R∞ ( µ, µ0 , φ ) −
4(1 − Ag ) K ( µ) K ( µ0 )
3(1 − Ag )(1 − g )(τ c + 2q 0 ) + 4 Ag
(2.16)
where, R∞ denotes the reflected intensity of a semi-infinite layer (infinite in the downward
direction, but with an upper surface) with the same optical properties (single scattering albedo
and phase function) as the finite layer (Nakajima and King, 1990, 1992; Kokhanovsky and
Nauss, 2006), τc is the cloud optical thickness, q0 ≈ 0.714/(1-g), and K(µ) and K(µ0) the
escape functions, describing the angular distribution of light escaping a semi-infinite cloud
from sources located deep inside the medium.
For non-conservative scattering in weakly absorbing layers, i.e in NIR and MIR outside the
water vapour absorption bands, R is written
R (τ c ; µ, µ0 , φ ) = R∞ ( µ, µ0 , φ )
−
[
]
m (1 − Ag A*)l − Ag mn 2 e −2 kτ c K ( µ) K ( µ0 )
(1 − Ag A*)(1 − l 2 e − 2 kτ c ) + Ag mn 2 le − 2 kτ c
(2.17a)
It should be noted that the influence on Ag is rather small and sometimes can be left out which
gives
ml exp[− 2k eτ c ]
K ( µ) K ( µ 0 )
R (τ c ; µ, µ0 , φ ) = R∞ ( µ, µ0 , φ ) −
(2.17b)
1 − l 2 exp[− 2k eτ c ]
with the extinction (or diffusion) coefficient ke (defined in Eq 3.8), A* is the spherical albedo
of a semi-infinite atmosphere (Nakajima and King, 1990; Kokhanovsky et al., 2003). The
escape functions are related to the phase function and can be derived from solutions of
numerical radiative transfer solutions, but it can also be expressed as
∞
K ( µ) = ∑ k e K n ( µ)
n
(2.18)
0
where ke is the extinction coefficient (Kokhanovsky, 2006, p. 141). For the non-absorbing
case K(µ) can be approximated by the first term K0(µ). Because multiple scattering dominates,
typical features of single scattering becomes less pronounced, and K0 can be well represented
by the function
3
K 0 ( µ) =
(1 + 2µ)
(2.19)
4
This approximation is valid for isotropic scattering, but the error is less than 2% if µ ≥ 0.2 in
the visible (nonabsorbing) (Kokhanovsky et al., 2003; Kokhanovsky, 2006, p. 154). However,
since ke0 = 1, Eq. (3.19) does not take into account differences in ω0 in the absorbing case.
Figure 4 illustrates the dependence on µ of the escape function for different values of ω0.
11
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
A*, l, m, n and ke in Eq. (2.17a,b) are all constants and can be parameterized for the general
case (Kokhanovsky and Nauss, 2006). Because they are all strongly dependant on the so
called similarity parameter s, which in turn depends on the single scattering albedo and
asymmetry parameter, the asymptotic theory equations thus depend primarily on two
parameters; the optical thickness and the effective radius (Nakajima and King, 1990):
s=
1 − ω0
1 − ω0 g
(2.20)
This is also in agreement with numerical simulations (King, 1987).
Figure 5. The dependence of escape function on µ and single scattering albedo ω0.
The K-values are derived from asymptotic fitting model, with the phase function
represented by the Henyey-Greenstein function, assuming g = 0.85 (King, 1987)
2.1.4 Semi-analytical method and Modified Exponential Approximation
Ideally the constants of the asymptotic theory should be derived using the exact numerical
solution. Again, it is a slow process that is not always applicable to large datasets, and where
the high accuracy may not be very useful due to relatively large uncertainties in remote
sensing images. Instead one may use approximate equations, such as the semi-analytical
retrieval model called the Exponential Approximation developed by Zege, and modified for
12
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
larger absorption by Kokhanovsky et al. (2003, 2005a). The exponential approximation is
based on the asymptotic theory for diffusion, and made valid for the radiative transfer
equation for plane parallel water clouds with optical thickness τ > 10 (for quick first order
estimations and lower accuracy requirements it is applicable to clouds with optical thickness
as low as 5). It is basically a further simplification of the asymptotic equations (2.16 and
2.17a,b) that allows faster computations, with an accuracy of 10-15% (depending on optical
thickness and viewing geometry). The use of the modified exponential approximation shows
that the errors in the computation of the reflection function is generally smaller than errors
related to uncertainties in the forward model and errors due to calibration uncertainties of the
optical instruments (Kokhanovsky and Nauss, 2006). Together with parameterizations of the
radiative transfer solutions, the MEA was used by Kokhanovsky et al. (2005a) to derive
analytical 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 and
effective radius and is here described in detail.
In the exponential approximation parameters in Eq. (2.16) and (2.17b) are approximated by
analytical functions (ignoring the underlying surface albedo) assuming no or weak absorption:
R∞ ( µ, µ0 , φ ) = R∞0 ( µ, µ0 , φ ) − 4
β
3(1 − g )
K ( µ) K ( µ0 )
⎛
β ⎞⎟
K ( µ) = K 0 ( µ)⎜⎜1 − 2α
3(1 − g ) ⎟⎠
⎝
(2.21)
(2.22)
where (β = 1 - ω0). For conservative scattering β = 0 and it follows that R∞ = R∞0 and K(µ) =
K0(µ). In the visible with zero absorption the reflection function of Eq. (2.16) can thus be
written
t1 (re , w) 1 − Ag ,1 K 0 ( µ) K 0 ( µ0 )
∗
(2.23)
R1 (τ c ; µ, µ0 , φ ) = R∞0 ( µ, µ0 , φ ) −
1 − Ag ,1 [1 − t1 (re , w)]
[
]
where R1* is the (measured) reflectance, R0∞ is the reflection function of an idealized semiinfinite nonabsorbing water cloud, Ag,1 is the ground surface albedo in the nonabsorbing
waveband and t1 is the diffused transmittance of a cloud. t1 is governed by the effective radius
re and the liquid water path w, but can be expressed in terms of asymmetry parameter g and
optical thickness τ1:
1
(2.24)
t1 =
3
α + τ 1 (1 − g1 (re ) )
4
where α is nearly constant at 1.072 for water clouds (Kokhanovsky et al., 2003; Kokhanovsky
et al. 2005a). The liquid water path w is correlated to the optical thickness through the
extinction coefficient (Kokhanovsky et al., 2003; Kokhanovsky et al. 2005a):
τ = wk e (λ , re )
where
ke =
1.5
ρ ⋅ re
(2.25)
⎛
⎞
⎜1 + 1.1 ⎟
2
/
3
⎜ (k ⋅ r ) ⎟
e
⎝
⎠
13
(2.26)
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
with the liquid water density ρ and k = 2π/λ (not to be confused with the extinction coefficient
ke). 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 follows
τ1 =
4(t1−1 − 1.072)
3(1 − g1 (re ) )
(2.27)
⎛ K ( µ) K ( µ )
A ⎞
t1 = ⎜⎜ 0 0 0 ∗ 0 − 1 ⎟⎟
1 − A1 ⎠
⎝ R∞ − R1
where
−1
(2.28)
For weak absorption ke, l and m in the exponential approximation are given by
l = 1 − 4α
k e = 3(1 − g ) β ,
β
3(1 − g )
,
m=8
β
3(1 − g )
(2.29)
These formulae together with Eq. (2.21)-(2.22) are valid for the case of small probability of
absorption (β ≤ 0.0001). But when absorption is stronger (for water clouds β can be close to
or even larger than 0.1), the next terms in the expansion of these constants must be
considered. However, the corresponding expressions for stronger absorption become
extremely complicated. Instead the following exponential expressions can be used for the
reflection function in absorbing media (Kokhanovsky et al., 2003; Kokhanovsky et al. 2005a,
Kokhanovsky, 2006, p.183):
where
R∞ ( µ, µ0 , φ ) = R∞0 ( µ, µ0 , φ ) ⋅ exp[− y (re )u ( µ, µ0 , φ )]
(2.30)
mK ( µ) K ( µ0 ) = (1 − exp[2 y (re )])K 0 ( µ) K 0 ( µ0 )
(2.31)
l = exp[-αy]
(2.32)
yi = 4
β
3(1 − g i )
,
u ( µ, µ0 , φ ) =
K 0 ( µ0 ) K 0 ( µ)
R∞0 ( µ, µ0 , φ )
(2.33)
With Eq. (2.30)-(2.33) Eq. (2.17b) can be written
R2 (τ 2 ; µ, µ0 , φ ) = R∞0 exp[− y 2 u ] − t c exp[− x 2 − y 2 ]K 0 ( µ) K 0 ( µ0 )
(2.34)
with the global transmittance tc:
tc =
sinh y 2
sinh(αy + x)
(2.35)
and xi = k e,iτ i , where ke is the extinction coefficient: k e ,i = 3(1 − g i ) β
The optical thickness τ in the absorbing wavelength can be determined from the optical
thickness in the visible. It follows from Eq. (2.25) and (2.26) that
14
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
τ 2 σ e (λ 2 , re ) ⎛ λ2 ⎞
=
=⎜ ⎟
τ 1 σ e (λ1 , re ) ⎜⎝ λ1 ⎟⎠
2/3
1.1 + ς 22 / 3
1.1 + ς 12 / 3
(2.36)
with ζj = 2π⋅re /λj .
These equations are general and can be improved by adding correction terms derived from
numerical radiative transfer solutions. Kokhanovsky et al. (2003) and Kokhanovsky and
Rozanov (2003) found that for cloudy media Eq. (2.30) – (2.35) can be modified by following
correction terms in order to improve the accuracy (hence the Modified Exponential
Approximation):
u → u(1 – 0.05y)
(2.37a)
tc → tc – Δ = t2
(2.37b)
2 2
(4.86 13.08μ 0 μ + 12.76μ 0 μ ) exp[x 2 ]
Δ=
(2.37c)
3
τ2
The final reflection function for the absorbing case can thus be written (corresponding to Eq.
(2.17b)) on the MEA form
R2 (τ 2 ; µ, µ0 , φ ) = R∞0 exp[− y 2 (1 − 0.05 y 2 )u ]
− (t − Δ) exp[− x 2 − y 2 ]K 0 ( µ) K 0 ( µ0 )
(2.38a)
Considering ground surface albedo and spherical albedo a2, Eq. (2.17a) is written on the MEA
form
R2 (τ c ; µ, µ0 , φ ) = R∞0 ( µ, µ0 , φ ) ⋅ exp[− y 2 (re ) ⋅ (1 − 0.05 y 2 (re ) ) ⋅ u ( µ, µ0 , φ )]
⎛
t 2 (re , w) ⋅ Ag , 2 ⎞
⎟t 2 (re , w) K 0 ( µ) K 0 ( µ0 )
− ⎜ exp[− x 2 (re , w) − y 2 (re )] −
⎜
⎟
A
a
r
w
−
⋅
1
(
,
)
g ,2
2 e
⎝
⎠
(2.38b)
where x2 and y2 are functions of optical thickness and effective radius. The spherical albedo is
given by a2 = exp(-y2) – tcexp(x2 –y2). Thus, knowing τ2 and R∞0 the reflection function in the
absorbing wavelength can be calculated. For isotropic scattering R∞0 is approximated by the
form (Kokhanovsky, 2004b):
A + B (µ + µ0 ) + Cµµ0
R∞0 =
(2.39)
4( µ + µ 0 )
where A ≈ 1, B ≈ 2 and C ≈ 4. For other cases the constants are assumed to depend on the
asymmetry parameter. The parameterization of A, B and C was done by fitting Eq. (2.39) to
the exact radiative transfer equation assuming µ = 1 and ω0 = 1 and using Henyey-Greenstein
phase function with g = 0.85 (Kokhanovsky, 2004b). This gives for water clouds A = 3.944,
B = -2.5 and C = 10.664. These constants can be used when µ-values are close to 1, but in
order to improve the accuracy for smaller µ, a small term F can be added to the numerator in
Eq. (2.39):
A + B (µ + µ0 ) + Cµµ0 + F (θ )
R∞0 =
(2.40)
4(µ + µ0 )
15
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
where F(θ) ≈ p(θ), and p(θ) is the phase function. The phase function of a cloudy medium is
here approximated using the Henyey-Greenstein function for g1, given by Eq. (2.5) where
θ = arccos(− cos µ cos µ0 + sin µ sin µ0 cos ϕ )
(2.41)
(Kokhanovsky et al., 2003; Kokhanovsky, 2004a).
Eq (2.30-2.41) generates the reflection function in the NIR and MIR wavelengths. For an
accuracy 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 problem
with remote sensing images, but must be checked for when running the model, especially if
the pixels are located far from the centre of the image. In the selected area of study the sensor
zenith angles were all smaller than 17°, and the solar zenith angles were within the range 38°41°.
2.2 Data and Computations
By comparing the calculated reflection function with the measured value in the corresponding
wavebands, the effective radius can be determined. For the retrieval of the effective radius in
cumulus cloud tops, reflectance data from at least two wavebands (visible and NIR/MIR) is
required. The light intensity measured from space is however both reflected and emitted light.
Satellite sensors cannot distinguish between reflected and emitted light, the photons are
identical 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 and
surrounding atmosphere. The easiest way to discriminate reflected light from emitted light is
probably to use wavebands with minimal water vapour absorption. For the purpose of
determining the cloud droplet radius we are interested in comparing the reflected light in the
visible 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. In
the 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 Image
Interpretation, p. 6)) water vapour absorption occur at 0.72, 0.82, 0.94, 1.1, 1.38, 1.87, 2.7 and
3.2 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 the
liquid water absorption is still sufficiently strong to show differences in reflection compared
to the visible (for the same reason the cloud optical thickness must not be too small). Suitable
wavelengths 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). Because
these three wavelengths show different sensitivity to underlying surface reflection and
atmospheric water vapour absorption one may wish to use more than one of these
wavelengths in order to reduce the uncertainty.
Rosenfeld et al. (2004) studied NOAA and METOP satellite measurements of cloud
microphysical properties and found that 1.6 and 3.7 are equally (un)suitable for deriving cloud
droplet 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, and
so it is more sensitive to atmospheric water vapour. On the other hand, the 1.6 µm is more
affected 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 deep
inside the cloud, and must therefore be corrected if cloud top radiation is wanted. This also
16
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified 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 are
used for reflectance measurements in the visible and NIR region respectiely. The scene used
is from the 9th 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 “cloud
mask” was applied in order to discriminate fully cloud covered pixels. It is assumed that the
clouds are water clouds, i. e. clouds consisted of liquid water droplets, as opposed to ice
clouds or mix phased clouds. In each model run, the effective radius of cloud droplets will be
computed only for pixels meeting the following conditions:
1. The reflectance in the visible must be larger than in the NIR
2. Reflectance in 2.1 µm waveband > 0.2; 0.25; 0.3
The first condition is obvious, since the principal theory of the model is the fact that water
absorbs energy in the NIR/MIR spectrum, the visible is non-absorbing. Pixels where the
reverse is true, i.e. where R2 > R1 indicates ground surface reflectance, where bare soil,
vegetation or urban areas have different reflectance signatures. A third condition, to account
for the limitation of optical thickness in the model, is thereafter applied:
3. τ1 > 10
Four model runs are conducted, the first three with R2 values of 0.2, 0.25 and 0.3 respectively.
The last run, with R2 > 0.3, was performed with τ1 > 5.
500
700
720
740
760
780
800
820
840
860
880
900
480
460
440
420
400
380
360
340
320
300
Figure 6 MODIS Aqua Satellite image over the area selected for studying, covering
the most southern parts of Sweden and eastern parts of Zealand (Denmark). Clouds
are seen in pink colour, land areas in green and open water in dark brown.
17
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
Besides the measured reflectance in the visible and NIR, data on solar and satellite zenith
angles and azimuth angles are provided for every scene pixel by MODIS. These data are the
main inputs in the model described in the previous section. With the reflectance and viewing
geometry given by MODIS, a few more input parameters are required. The ground surface
albedo for every pixel can be received as a MODIS product. Since the clouds are blocking the
surface, 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 vary
significantly from one week to another, depending on the vegetation cover and the soil
moisture content. Therefore it is at this time of the year difficult to produce accurate values of
Ag. However, the ground surface albedo has a relatively small influence on the final result,
and for simplicity, standard values of Ag in the two wavelengths is set for all pixels covering
land. Typically, surface albedo for bare soil is about twice as large for wavelengths > 0.7 µm
than 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 are
average values of surfaces with forest cover or bare soil (Ahrens, 2003, p. 46). The surface is
also assumed to be Lambertian, i.e. the reflectance is equal in all directions.
The asymmetry parameter g and the single scattering albedo ω0 must be known, but they are
both depending on the effective radius. The asymmetry parameter is defined in Eq. (2.4) but
since the Legendre polynomials and expansion coefficients are not known, g can instead be
found by following equation:
1 − g = 0.12 + 0.5(kre )
−2
3
− 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 for
χ1 and χ2 respectively (Kokhanovsky, 2006, p. 259; Kokhanovsky et al., 2006, unpublished).
Kokhanovsky and Zege (1995) also present a parameterisation of the absorption and
extinction coefficients ka and ke which are used to derive the single scattering albedo in the
NIR (in the visible ω0 is assumed to be 1):
k a = k a*C v ,
where
k e = k e*C v ,
(2.43)
k e∗ =
1.5 ⎛⎜
1.1 ⎞⎟
+
1
re ⎜⎝ (kre )2 / 3 ⎟⎠
(2.44)
k a∗ =
⎛
⎡ 8λ ⎤ ⎞ ⎞
5πχ (1 − κre ) ⎛⎜
1 + 0.34⎜⎜1 − exp ⎢− ⎥ ⎟⎟ ⎟
⎜
λ
⎣ re ⎦ ⎠ ⎟⎠
⎝
⎝
(2.45)
and Cv is the volumetric concentration of droplets. However, since 1 – ω0 = ka / ke (Eq. 2.11)
ω0 is given by
k a*
ω0 = 1 − *
(2.46)
ke
18
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
Equation (2.42)-(2.45) was used by Kokhanovsky et al. (2006, unpublished) in their semianalytical cloud retrieval algoritm for SCIAMACHY/ENVISAT, and has proved to be
accurate 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 2.2
µm is 1.64⋅10-8 and 2.89⋅10-4 respectively (Kokhanovsky, 2006).
Following the procedure in the modified exponential approximation and using the equations
for g and ω0 presented above, the expected reflectance value in the second waveband (NIR) is
calculated for re values ranging from 3 to 30 µm. This range of values was set because droplet
sizes outside this range are very unlikely, especially droplets larger than 30, since these would
fall out as precipitation due to gravity. By comparing the model value of R2 with the measured
R2 the effective radius is given by the value producing the R2-value closest to the one
measured. This is performed for every pixel (in the chosen 200×200 pixel area) that meets the
conditions 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 is
calculated with the MODIS inputs as listed:
R1*:
R2*:
v:
v0:
ϕ:
ϕ 0:
Measured reflectance in the visible (λ1) = 0.5624
Measured reflectance in the NIR/MIR waveband (λ2) = 0.3610
Sensor zenith angle = 2.73°
Solar zenith angle = 43.09°
Sensor azimuth angle = -104.63°
Solar azimuth angle = -165.49°
0.65
R2 modelled
0.6
R2 satellite
0.55
NIR Reflectance
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0
5
10
15
20
Effective radius (µm)
25
19
30
Figure 7 Exampel of calculated
effective radius from one pixel.
re = 10.60 µm, τ1 = 13.1
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
3. RESULTS and DISCUSSION
Four runs were performed, with different initial conditions for the reflectance in the 2.1 µmband (R2) and optical thickness in the visible (τ1). 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. Low
reflectance in the second waveband results in values τ1 and τ2 as low as 3.6 (R2 > 0.25) and
2.6 (R2 > 0.2), and mode optical thickness at 8 and 5 respectively. Since the model is suited
for clouds with optical thickness > 10, this implies that the model is not applicable to pixles
with R2-values < 0.3, where the optical thickness appears to be too small. For R2 > 0.3 the
optical thickness lies in the range 4.6 – 50.9, with mode optical thickness at 13, which is
acceptable.
Many pixels in the first two runs, with R2 > 0.2, result in much larger re –values than in the
last two runs. Maximum effective radius is at 27(20) µm for R2 > 0.2(0.25) and 15.5 for R2 >
0.3 (Table 3, Figure 8). Also the median and mode effective radius is higher in the first two
runs. The last two runs produced median and mode effective radius at 14.6 and 15 µm
respectively, which is in line with the expected result.
The results can be compared with results from Rosenfeld and Lensky (1998). According to
Rosenfeld and Lensky the cloud top median of effective radius is 14 µm for clouds near the
coastline and 9 µm over inland, therefore both 20(17) µm (R2 > 0.2(0.25)) and 14.6 µm (R2 >
0.3) appear relatively high for median value for continental clouds. However, one should bear
in mind that the continental area studied by Rosenfeld and Lensky covers southern Malaysia
and central Sumatra. Southern Sweden is thus relatively small in comparison and most parts
can be considered to be located near the coastline. The areas over Malaysia and Sumatra also
have higher population density. Likely the air is more polluted (due to traffic and forest
burning) 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, from
the first two runs, can be considered overestimated, confirming that the model is not suitable
where R2 < 0.3. Thus, the results from this study are in agreement with the same from
Rosenfeld and Lensky (1998). It is interesting that the use of the 2.1 µm waveband instead of
the 3.7 µm (as in the study by Rosenfeld and Lensky) cause no major differences in the
results.
Table 3 Summary over the results of optical thickness τ from the four runs. Since the only difference between
the two last runs is the restriction of τ1, the statistics of τ are identical, why both cases of τ1 for R2 > 0.3 are
represented in the last row.
conditions
Total no. of
pixels computed
min τ
max τ
median τ 1
median τ 2
mode τ
R2 > 0.2
3903
2.65
65.42
9.52
9.89
5
R2 > 0.25
1721
3.61
68.25
10.95
11.10
8
R2 > 0.3
202
4.62
48.64
10.91
11.43
13
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
Table 3 Summary over the results of the effective radius re from the four runs. The number of pixels fulfilling
the conditions is listed in the first three columns. The last four columns show the statistics of the retrieved
effective radius, including minimum, maximum, mean, and mode effective radius.
Total no. of
pixels computed
pixles
τ1>5
pixles
τ1>10
min
(µm)
re max
(µm)
R2 > 0.2
τ1 >10
3903
3293
1876
11.39
R2 > 0.25
τ1 >10
1721
1589
910
R2 > 0.3
τ1 >10
202
199
R2 > 0.3
τ1 >5
202
199
conditions
re median
(µm)
re mode
(µm)
27.27
19.91
18
11.39
20.22
17.35
18
116
11.39
15.42
14.64
15
116
8.010
15.54
14.60
15
re
200
180
160
140
freq
120
100
R2 >0,3; t>5
80
R2 > 0,3; t>10
60
R2 > 0,25; t>10
40
R2 > 0,2; t>10
20
0
3
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 30
effective radius (µm)
Figure 8. The drop size distribution in the four runs. Each colour represents one run
with 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 (R2 > 0.3),
implies that this is an upper limit of the drop size before the cloud drops fall out as
precipitation, indicating that the clouds are precipitating or close to precipitating. Rosenfeld
and Lensky (1998) found that 14 µm is a typical precipitation threshold for continental
clouds. In accordance to this, Pinsky and Khain (2002) found 15 µm to be the threshold for
drizzle. At this threshold, droplets continue to grow through coalescence, but this is balanced
out by the larger droplets falling out as precipitation due to gravity. This pattern is not seen in
the first two runs, where the size distribution is more symmetric, and the effective radius
ranges up to well above 20 µm.
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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It is probable that the differences in results from the first two and the last two runs is at least
partly due to cloud optical thickness being too small when R2 < 0.3. But there are other
factors that may have impact on the results too. The model assumes water clouds, hence the
presence 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 larger
effective radius according to Kokhanovsky (2006, p. 5). Information on cloud top temperature
would be relevant to include in the cloud mask in order to discriminate water clouds from ice
or mixed phase clouds.
Another source of error may be the so called 3D effect. As most cloud models, the MEA is
developed for plane parallel, semi-infinite homogeneous clouds. That is, it is assumed that the
cloud 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 cloud
does not necessarily escape the cloud body either upward or downward, but may just as well
escape in a horizontal direction. Compared to a semi-infinite plane parallel cloud with the
same microphysical properties, less light would be reflected into the satellite sensor. In the
model this “missing” light is assumed to be absorbed by the cloud, which would lead to
miscalculation (overestimation) of the effective radius. Also, in addition to the direct solar
radiation, some incoming radiation may be light reflected from nearby cloud tops within the
same cumulus cloud.
4. VALIDATION and ERROR ANALYSIS
The error analysis here is rather brief and based on the results from extensive error analysis
performed by Kokhanovsky et al. (2005a,b), Kokhanovsky (2004b), Kokhanovsky and
Rozanov (2003), and Kokhanovsky et al. (2003). Many of the equations used in the MEA are
approximations of asymptotic theory, where the accuracy has been studied closer for each
equation individually. Where computed, the error is found by comparing the results of the
approximate equation with the same from the exact (numerical) methods, and given together
with the equation in the text.
Validation of the model is difficult since it requires real measurements of the effective drop
radius in the studied clouds. These measurements must be done by aircraft and is not available
for the date of interest. Another method for validating the model is to compare the retrieved
cloud drop radius with the MODIS cloud product. This does however not give any
information on how accurate the method is compared to reality, but shows rather the degree of
agreement 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 the
study called SACURA) was compared with two other models, one of them the MODIS cloud
product. Nauss et al. concluded that the relatively simple MEA model produced results
comparable to the more complex models.
On average the MEA effective radius is within 10% from the MODIS cloud product for most
pixels, with somewhat larger biases for re < 13 µm and re > 20 µm. The agreement between
the MODIS cloud product and the MEA model was higher over ocean than over land, with r2
as high as 0.94 for pixels with τ > 10. The corresponding figure over land was 0.79. Over ocean the
two models used the same waveband (0.86 µm). However, for pixels over land the MODIS
cloud product used the 0.65 µm waveband, while the MEA model used the same waveband as
over ocean (0.86 µm). Nauss et al. explain the larger discrepancy over land to be caused by
the fact that the 0.86 µm waveband is much more sensitive to vegetation covering the
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
underlying surface than the 0.65 µm waveband. The r2 –values of the MEA given here are
calculated using the 0.86 µm and 1.6 µm waveband, both more sensitive to the underlying surface than
the two wavebands used in this study (0.65 µm and 2.1 µm respectively). It is probable that if the
0.65 µm and 2.1 µm wavebands were used in the MEA model comparison, the agreement
could have been increased over land. Over all Nauss et al. (2005) showed that the error of the
approximate 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 and
Rozanov (2003) found the error to be less than 5% for nadir observations at λ2 (λ2) = 0.65
(1.65) µm and incident angle 15° < v0 < 65°. For non-nadir observation angles, the accuracy
depend 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 and
observation angles smaller than 55° and 30° respectively when compared to the numerical
solution.
The application of the MEA method to SCIAMACHY/ENVISAT data has also been validated
against the conventional LUT approach by Kokhanovsky et al. (2005a, 2005b). Here the
validation showed relatively large differences in retrieved effective radius between the LUT
approach and the MEA, but it did not exceed 20%. The large differences are partly explained
by the fact that the LUT approach used the 3.7 µm waveband, while the MEA used the 1.6
µm waveband. Another factor that may result in poor agreement between the two approaches
is the low resolution of the SCIAMACHY data, which could not guarantee 100% cloud
covered pixels. They therefore conclude that the MEA (SACURA) method gives results
comparable to the LUT approach. When using the 0.865 µm and 2.13 µm wavebands the
accuracy for the nadir observation conditions and the solar zenith angle 60° was better than
94% for τ ≥ 4, see Figure 9 (Kokhanovsky, 2006).
Figure 9. The error of the MEA approach for nadir observation and solar zenith
angle = 60°. The dotted line represents the 2.13 µm waveband, while the solid line
23
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
corresponds to the 0.865 µm wavenband (Kokhanovsky, 2006)
According to Kokhanovsky et al. (2006, unpublished) the accuracy of re is strongly dependent
on the correct information with respect to cloud fraction. One can expect re to be
overestimated if retrieved over broken cloud field, due to unknown contribution from the
ground. However, the larger the optically thickness, the smaller the contribution from the
ground, and even more so if the 0.65 µm waveband is used instead of the 0.865 µm
waveband. 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 and
cloud optical thickness.
5. CONCLUSIONS
The results from the four runs show that the modified exponential approximation can be used
with relatively high accuracy on satellite data under certain conditions. The model is restricted
to 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 conditions
must be given in terms of reflectance. The results from the two first runs where R2 > 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 the
results less reliable.
The model performed well (in relation to what was expected) where R2 > 0.3, for both τ > 10
and τ > 5. The median effective radius of 14.6 µm was high in comparison to the results from
continental clouds (9 µm) in the study by Rosenfeld and Lensky (1998). However, the result
is 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 continental
clouds studied by Rosenfeld and Lensky (1998) was located over southern Malaysia and
central Sumatra, a much larger area with dense population and likely more air pollution than
over the forests of southern Sweden. From this aspect, the “continentality” of southern
Sweden area is not comparable to the same of Malaysia and Sumatra, and it would perhaps be
more 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 by
Nauss et al. (2005) showed that the approximate model, despite its simplicity, tended to
produce results comparable to results from conventional methods (numerical methods, LUT).
The error varies from 5 to 20%, depending on which wavebands that are used, viewing
geometry and pixel resolution. The studies stress the importance of fully cloud covered pixels
for good results, and good knowledge about ground surface albedo where the cloud cover is
broken, or where clouds are optically thin. This is even more important when using
wavebands sensitive to ground surface reflectance. However, in this study the wavebands
used 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 to
understand the processes involved in cloud formation. The aim of the study was to produce a
model that can be used together with cloud top temperatures to study changes in the vertical
profile of the cloud, induced by aerosols. In this case, exact values of the cloud droplet radius
are perhaps not essential, but rather the ability to study differences and changes, and to do so
24
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
for many sites and dates, which requires a fast approach, where input variables are easily
changed. If the purpose is to produce accurate and reliable projections of future weather and
climate scenarios, a more exact approach may be desirable. One should bear in mind though
the geographical uncertainty in the remote sensing images, where the resolution may be low
with pixels of 1000×1000 m. The accuracy of the model can never be better than the
uncertainty in the input data. Thus the high accuracy of numerical methods cannot always be
fully 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 through
running the model it has turned out to be very sensitive to the value of imaginary refractive
index, while the influence of ground surface albedo is small, if Ag < 0.5. Also, a more
advanced cloud mask with the ability to discriminate pure water clouds from ice or mix
phased cloud would be appropriate, to support the assumption of pure water clouds being
modelled.
For deeper knowledge and understanding of cloud remote sensing and modelling I refer to the
two excellent books by Liou; “Radiation and Cloud Processes in the Atmosphere” (1992) and
“Introduction to Atmospheric Radiation” (2002). The MEA model is carefully described and
evaluated 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.
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
26
ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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APPENDIX
Summary of the Modified Exponential Approximation approach:
Input parameters:
λ1: Peak wavelength in the first waveband (visible): 0.65 µm
λ2: Peak wavelength in the second waveband (NIR/MIR): 2.1 µm
R1*: Measured reflectance in the visible (λ1)
R2*: Measured reflectance in the NIR/MIR waveband (λ2)
v:
v0 :
ϕ:
ϕ0 :
Sensor zenith angle (radians)
Solar zenith angle (radians)
Sensor azimuth angle (radians)
Solar azimuth angle (radians)
χ1: Imaginary part of the refractive index for liquid water for
χ2: Imaginary part of the refractive index for liquid water λ2
A1: Ground surface albedo in λ1
A2: Ground surface albedo in λ2
Output:
R2 as a function of effective radius re.
Calculated effective radius re.
Approach:
1.
2.
3.
4.
5.
6.
Calculate µ = |cos(v)| and µ0 = cos(v0)
Calculate relative azimuth angle: φ = ϕ - ϕ0
Calculate kλ = 2π/λ for λ1 and λ2
Calculate κλ = 4πχ /λ for λ1 and λ2
Calculate escapefunctions K0(µ) and K0(µ0) using Eq. 3.19
Create a vector of the effective radius re from 3 µm to 30 µm
For each value re(i), calculate following microphysical and
radiative properties:
7. Calculate asymmetry parameter g1 and g2 (1 and 2 representing
λ1 and λ2), using Eq. (3.42)
8. Calculate single scattering albedo ω0 for λ2 using (3.43)-(3.46).
ω0 in the visible is set to 1
9. Calculate scattering angle θ : Eq. (3.41)
Calculate Henyey-Greenstein phase function from g1: Eq. (3.5)
10. Calculate R∞0 (in the visible only) using Eq. (3.39)-(3.40)
11. Calculate t1 Eq (3.28)
12. Calculate optical thickness τ1: Eq. (3.27)
13. Calculate optical thickness τ2 using Eq (3.36).
14. Calculate x2 = keτ2 where k e = 3(1 − g ) β
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
___________________________________________________________________________________________________
15. Calculate y2 using Eq. (3.33)
16. Calculate global transmittance tc using Eq. (3.35)
17. Calculate t2 using Eq. (3.37b-c)
18. Calculate spherical albedo a2 = exp(-y2) – tcexp(x2 –y2)
19. Calculate u using Eq. (3.33) and (3.37a)
20. Calculate R2 using Eq. (3.38b)
21. Compare the theoretically R2-value with the measured value
R2*, and find the index i where R2(i) = R2* to retrieve the cloud
droplet effective radius
The code is written in MATLAB 7.5.0 and available on request.
Please contact Julia Karlgård on [email protected] or Erik Swietlicki on
[email protected]
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ESTIMATING CLOUD DROPLET EFFECTIVE RADIUS FROM SATELLITE REFLECTANCE DATA
Modified Exponential Approximation
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REFERENCES
Ahrens, C. D. 2003. Meteorology Today: An Introduction to Weather, Climate, and the Environment. 7th 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 Fourth
Assessment Report of the Intergovernmental Panel on Climate Change [Core Writing Team, Pachauri, R.K
and Reisinger, A.(eds.)]. IPCC, Geneva, Switzerland, 104 pp.
King, M. D.. 1987. Determination of the Scaled Optical Thickness of Clouds from Reflected Solar Radiation
Measurements. Journal of the Atmospheric Scinece, 44(13). 1734-1751.
Kokhanovsky, A. A.. 2002. Simple approximate formulae of the reflection function of a homogeneous semiinfinite turbid 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 simple
approximation. 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: asymptotic
theory. Atmospheric Chemistry and Physics, 6. 5537-5545.
Kokhanovsky, A.A. and Rozanov, V.V.. 2003. The reflection function of optically thick weakly absorbing
turbid 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. The
SCIAMACHY 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. Atmospheric
Chemistry 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 Theoretical
Basis Document. Pp 40. http://www.knmi.nl/samenw/sciamachy/products/clouds/clouds_IFE_AD.pdf. page
last visited 24th April 2008.
Kokhanovsky, A. A., Rozanov, V. V., Zege, E. P., Bovensmann, H. and Burrows, J. P.. 2003. A semianalytical
cloud retrieval algorithm using backscattered radiation in 0.4-2.4 µm spectral region. Journal of Geophysical
Research, 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. 5th
edition. 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. 504
pp.
Liou, K.N. 2002. An Introduction to Atmospheric Radiation. 2nd edition. Academic Press. London. 248 pp.
Nakajima, T. and King, M.D.. 1990. Determination of the optical thickness and effective radius of clouds from
reflected solar radiation measurements. Part I: Theory. Journal of the Atmospheric Science, 6(15). 18781893.
Nakajima, T. and King, M.D.. 1992. Asymtotic theory for optically thick layers: application to the discrete
ordinates method. Applied Optics, 31(36). 7669-7683.
Nauss T., Kokhanovsky A. A., Nakajima T. Y., Reudenbach C., and Bendix J.. 2005. The intercomparison of
selected 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 spectrum
formation in cumulus clouds. Quarterly Journal of the Royal Meteorological Society, 128(580), 501-533
Platnick 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. Predicting
Soil Albedo from Soil Color and Spectral Reflectance Data. Soil Science Society of America journal,
64(3). 1027-1034
Rosenfeld, D., Cattani, E., Melani, S., and Levizzani, V.. 2004. Considerations on Daylight Operation of 1.6VERSUS 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 in
Continental and Maritime Convective Clouds. Bulletin of the American Meteorological Society, 79. 24572476.
Rybicki, G. B.. 1996. Radiative Transfer. Journal of Astrophysics and Astronomy, 17. 95-112.
29