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Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Aerosol radiative forcing 71.1 The radiative transfer equation (RTE) . . . . . . . . . . . . . 7

1.1.1 Plane parallel approximation . . . . . . . . . . . . . . . 81.2 Aerosol direct radiative forcing . . . . . . . . . . . . . . . . . 10

1.2.1 Two stream approximation . . . . . . . . . . . . . . . . 111.2.2 TARFOX formulation of the direct aerosol forcings . . 13

2 Surface Reflectance 192.1 Bidirectional reflectance . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 Reflectance . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 Directional-Hemispherical reflectance ρ . . . . . . . . . 242.1.3 Bi-Hemispherical Reflectance (ρ) . . . . . . . . . . . . 27

2.2 Spectral albedo . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1 Spectral albedo in terms of hemispherical reflectances . 282.2.2 Integrated albedo . . . . . . . . . . . . . . . . . . . . . 29

2.3 Bidirectional reflectance models . . . . . . . . . . . . . . . . . 302.3.1 How to create a BRDF model . . . . . . . . . . . . . . 31

2.4 BRDF implementations . . . . . . . . . . . . . . . . . . . . . 332.4.1 Soil reflectance . . . . . . . . . . . . . . . . . . . . . . 332.4.2 Water reflectance . . . . . . . . . . . . . . . . . . . . . 422.4.3 Vegetation reflectance . . . . . . . . . . . . . . . . . . 452.4.4 Snow reflectance . . . . . . . . . . . . . . . . . . . . . 50

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Aerosol characterization 573.1 Aerosol properties . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 AERONET . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Aerosol models for the Flux Change LUT . . . . . . . . . . . . 68

3.3.1 Linear combination of 6S classes . . . . . . . . . . . . . 69

1

2 CONTENTS

3.3.2 Size distribution and complex refractive index . . . . . 743.4 Aerosol properties at the Lecce University’s AERONET station 74

3.4.1 Fifteen-day averages . . . . . . . . . . . . . . . . . . . 743.4.2 Case studies . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Methods and applications 834.1 The approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 The 6S radiative transfer code and the flux change evaluations 84

4.2.1 The choice of surface reflectance and aerosol properties 854.2.2 The program forcing . . . . . . . . . . . . . . . . . . 884.2.3 The scripts that support forcing . . . . . . . . . . . . 904.2.4 The program dforcing . . . . . . . . . . . . . . . . . . 94

4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3.1 Flux changes at Lecce for two vegetated surfaces (AK1

and AK4) . . . . . . . . . . . . . . . . . . . . . . . . . 974.3.2 Case studies at Lecce University . . . . . . . . . . . . . 984.3.3 Forcing evaluation at South East Italy . . . . . . . . . 1034.3.4 Differences with the lambertian case (ACE-2) . . . . . 1084.3.5 Look up tables . . . . . . . . . . . . . . . . . . . . . . 112

5 Conclusions 112

A List of symbol and acronyms 121

Bibliography 122

Acknowledgments 130

Introduction 3

Introduction

The Earth receives from the Sun the radiant energy that drives his cli-

mate. Radiative equilibrium is achieved when the spatial and temporal

average value of the shortwave flux incident at the top of the atmosphere

is equal to the sum of two terms, the first given by the shortwave flux

reflected by surface, atmosphere and clouds, and the second by the long-

wave flux emitted by the surface-atmosphere system, as expressed in eq. (1)

[Salby, 1995, Hansen at al., 2005],

πr2S0(1 − ap) = 4πr2σT 4 (1)

where ap is the average planetary albedo of the Earth (∼ 0.3), r theEarth radius (∼ 3640 km), S0 the solar constant (1368 Wm−2) and σ =5.67 · 10−8 Wm−2K−4 the Stefan-Boltzmann constant. The equilibrium tem-perature T , can be easily calculated finding the value of 255 K:

T =4

√

(1 − ap)S04σ

=4

√

(1 − 0.3) · 13684 · 5.67 · 10−8 = 255 K.

This value is considerably lower than the observed equilibrium temperature

of the atmosphere, which is estimated to be of around 288K. This differences

is due to the presence of the atmosphere and, in particular to the effects pro-

duced by the greenhouse gases such as CO2, CH4, H2O, O3, which absorb

the thermal radiation emitted toward the space trapping it within the atmo-

sphere. This effects lead to a warming of the system. On the other hand,

clouds and aerosols can be responsible of cooling effects. The wide variabil-

4 Introduction

ity of the physical and optical properties of these atmospheric constituents

is reflected in large uncertainties on the estimates of the effects that they

produce on the radiative balance of the surface-atmosphere system; for ex-

ample, absorbing aerosols and cirrus clouds can warm the Earth system as

well as scattering aerosols and highly reflective clouds can produce cooling

effects on it. Moreover, the poorly known feedback processes contribute to

the complexity of the problem.

The effects on the Earth radiative budget and, hence, on the climate

system induced by atmospheric aerosols are affected by larger uncertainties

than those regarding the climatic effects of the other atmospheric compo-

nents [IPCC, 2001]. Aerosols can modify the radiative balance of the Earth,

directly and indirectly. Direct effects are due to the scattering (elastic

process) of the solar radiation, reflecting a fraction of it back to space (back-

ward scattering), or to absorption of it (non-elastic process), causing both

the cooling of the surface and the warming of the atmosphere. These effects

change the temperature and humidity profiles of the atmosphere and influ-

ence the cloud formation, providing significant variations in the regional and

global distribution of the Earth energy budget [Chyleck and Coakley, 1974].

Figure 1: The radiation balance of a planet. The energy received in form ofshortwave radiation (SW ∼ 0.5 µm) has to be re-emitted backward to the spacein terms of longwave radiation (LW ∼ 10 µm), in order to achieve the energyequilibrium conditions.

Introduction 5

On the other hand, aerosols cause indirect effects acting as cloud conden-

sation nuclei controlling the cloud droplet formation and affecting the cloud

micro-physical properties, precipitation and radiative efficiency.

Understanding these phenomena cannot be achieved without observa-

tions: the role of sensors mounted on satellite platforms observing the Earth

and the complementary surface-based observations of the aerosol optical

properties are crucial. Only the view from satellite can provide a sufficient

coverage of the Earth for the computation of temporal and spatial based

averages, while only surface observation can provide the accuracy needed to

the knowledge of the aerosol radiative parameters involved in the problem,

reaching the correct validation of the satellite products.

Aerosol properties at the moment are routinely provided by different sen-

sors flying on-board various satellite platforms: the EOS TERRA and AQUA

polar orbiter satellites, carry on the MODIS 1 and the MISR2 spectroradiome-

ters; TOMS3, designed for O3 observation, offers the capability of a great

spectral resolution in despite of his low spatial resolution (100 km), proving

a widely used aerosol index product [Cacciari A., 2006]; also AVHRR 4, fly-

ing on the NOAA polar satellites, can give some information on the aerosol

optical depth due to their two band in the solar radiation wavelength range

[Hauser at al., 2005]. Other authors [Bush and Valero, 2003] approached the

computation of aerosol forcing with the measurement of the irradiance taken

from space (CERES5 project).

The most important set up of surface based observations of aerosol prop-

erties, is the NASA AERONET worldwide spread network of CIMEL sun

photometers, , which provides a complete and quality assured aerosol physical

and optical parameters dataset [Holben et al., 1998, Smirnov et al., 2000].

The direct aerosol radiative forcing is related to the surface reflectance

properties, being given as a result of the multiple reflections between the

1http://modis.gsfc.nasa.gov/2http://www-misr.jpl.nasa.gov/3http://jwocky.gsfc.nasa.gov/aerosols/aerosols v8.html4http://noaasis.noaa.gov/NOAASIS/ml/avhrr.html5http://shire.larc.nasa.gov/PRODOCS/ceres/SSF/Quality Summaries/ssf .html

6 Introduction

atmosphere and the Earth’s surface. Until now it was generally considered

to be a lambertian reflector. The aims of this thesis is to approach the issue

of the role of the anisotropy of the surface reflectance in this context.

Chapter 1 presents the radiative transfer equation along with the two stream

approximation analytical solution, that will help the reader to deal the prob-

lem, expressed in terms of the optical properties of the aerosol layer and sur-

face albedo. Chapter 2 defines some anisotropic reflectance models through

the description of different implementation of the so called bidirectional re-

flectance distribution function (BRDF). The aim of this work is to provide a

representation of the albedo of surfaces covered by water, bare soil, vegeta-

tion and snow-ice, with the most possible up-to-date available modeling. The

aerosol optical and physical parameters useful to develop this work and to de-

fine the corresponding method, are described in the Chapter 3, with a focus

on the measurements carried out at the AERONET Lecce University station.

In the Chapter 4, a description of the procedure developed for evaluating

the aerosol radiative effects is given. This is based on the code forcing,

which utilizes widely the 6S radiative transfer model [Vermote at al., 1997]

subroutines in order to compute the irradiances at the top and the bottom of

the atmosphere, for different geometrical configurations with respect to the

Sun position, and various aerosol and surface properties. Finally, different

applications are shown.

Chapter 1

Aerosol radiative forcing

1.1 The radiative transfer equation (RTE)

The differential form of the radiative transfer equation represents the varia-

tion of the monochromatic radiance Lν passing through a small cylindrical

element of cross section dσ and length ds[m], due to the absorption occurring

for certain characteristics of the medium as the mass extinction coefficient

kν [kg−1m2] and the mass density ρ [kgm−3], in the form of the so called

Schwartzchild equation:

− dLνkνρds

= Lν − Jν , (1.1)

where, in the thermodynamic equilibrium, the source function Jν is equal to

the Planck function:

Bν(T ) =2hν3

c21

ehνkT − 1

. (1.2)

which depends upon the absolute temperature T of the medium; other quan-

tities are the Boltzmann constant k = 1.3805·10−23JK−1 and the Planck con-stant h = 6.6260693·10−34Jts, the speed of light in vacuum c = 299792458ms−1

(1) [Chandrasekhar, 1960]. In the cartesian system of coordinates the equa-

1http://www.physics.nist.gov/cuu/Constants/index.html

7

8 CHAPTER 1. AEROSOL RADIATIVE FORCING

tion of transfer can be written in the form

− 1kνρ

(

l∂

∂x+ m

∂

∂y+ n

∂

∂z

)

Lν(x, y, z; l, m, n) = Lν(x, y, z; l, m, n)−Jν(x, y, z; l, m, n)

(1.3)

where l, m, n expresse as usual the projections of the vector of length one,

associated with the direction of the beam of radiation, along the three orthog-

onal cartesian axes. The formal solution of the radiative transfer equation

is

Lν(s) = Lν(0)e−τ(s,0) +

∫ s

0Jν(s

′)e−τ(s,s′)kνρds

′ (1.4)

where τ(s, s′) is the optical thickness of the material between the points s

and s’:

τν(s, s′) =

∫ s′

skνρds

′′. (1.5)

The physical meaning of the solution is clear: its first term expresses the

attenuation of the incident radiance Lν(τν = 0) due to the whole mass of

the medium, and the second term yields the sum of the radiance emitted by

the medium itself and attenuated by the remaining path until reaching the

destination point s.

In the further discussion in the present and in the following sections, it is

convenient to remove the suffix ν from the various quantities τν , Lν , kν , and

so on. No ambiguity is likely to arise from this.

1.1.1 Plane parallel approximation

The atmosphere can be considered a medium having a great stability in the

horizontal direction with respect to the vertical, because the vertical gradi-

ents of physical quantities such as temperature, relative humidity and pres-

sure are considerably greater than the horizontal ones. This characteristic is

of basic importance for the approximation of the radiative transfer equation

for a plane parallel atmosphere which greatly reduces the complexity of the

1.1. THE RADIATIVE TRANSFER EQUATION (RTE) 9

solution of the differential equation:

− cos θdL(z, θ, φ)kρdz

= L(z, θ, φ) − J(z, θ, φ), (1.6)

z being the geometrical thickness along the direction perpendicular to the

parallelization. The angle θ represents the inclination of the optical path

with respect to the outward normal to the same planes and φ the azimuth

angle. Assuming the differential form of the optical thickness as given by

dτ = kρdz, (1.7)

eq.(1.6) becomes

µdL(τ, θ, φ)

dτ= L(τ, θ, φ) − J(τ, θ, φ) (1.8)

where µ = cosθ. For a plane parallel scattering atmosphere, the source

function can be written in the form

J(µ, φ; µ′, φ′) =1

4π

∫ 2π

0

∫ +1

−1p(µ, φ; µ′, φ′)L(τ, µ′, φ′)dµ′dφ′, (1.9)

and eq.(1.8) assumes the form

µdL(τ, µ, φ)

dτ= L(τ, µ, φ)− 1

4π

∫ 2π

0

∫ +1

−1p(µ, φ; µ′, φ′)L(τ, µ′, φ′)dµ′dφ′ (1.10)

where dµ′ = sin θ′dθ′. Assuming the radiance to be azimuth independent so

that L(µ, φ) ≡ L(µ, φ), eq.(1.10) can be further approximated to become

µdL(τ, µ)

dτ= L(µ, τ) − 1

2

∫ +1

−1p(µ, µ′)L(τ, µ′)dµ′. (1.11)

For a lambertian or isotropic scattering phase function p(µ, µ′) ≡ 1, eq.(1.11)assumes the form

µdL(τ, µ)

dτ= L(τ, µ) − 1

2

∫ +1

−1L(τ, µ′)dµ′ (1.12)


leading to the simplest form of the radiative transfer equation.

1.2 Aerosol direct radiative forcing

The radiative forcing is defined as the variation of the net radiant flux density

at a certain atmospheric level induced by the presence of matter having well

defined scattering and/or absorbing properties. The net radiant flux density

F is defined by the difference between the incoming irradiance E↓ and the

outgoing irradiance E↑

F = E↓ − E↑. (1.13)

The difference between the net radiant flux density2 F calculated (or mea-

sured) for an atmosphere containing the aerosol layer with respect to the net

radiant flux density calculated (or induced by measurements) in a pristine

atmosphere F0

∆F = F − F0 (1.14)

is defined as the instantaneous flux change. Finally, the average of this

quantity over a whole day

∆F =1

24

∫ sunset

sunrise∆Fdt (1.15)

defines the radiative forcing. Being widely used to estimate the radiative

effects of the aerosols, in the following paragraphs some classical approaches

are illustrated.

In Chapter 4, these concepts will be explained giving the explicit defini-

tion of the aerosol induced flux changes at the top-of-atmosphere (TOA) and

at the bottom-of-atmosphere (BOA) level, respectively.

2The terms Irradiance and Radiant flux density represents the same quantity.

1.2. AEROSOL DIRECT RADIATIVE FORCING 11

1.2.1 Two stream approximation

To determine the effects of atmospheric aerosol on the radiative heating of

the Earth-atmosphere system, [Chyleck and Coakley, 1974] asked themselves

which change in the top-of-atmosphere albedo is induced by the presence of

an aerosol layer above a surface-atmosphere system characterized by origi-

nal albedo a. The optical characteristics of the aerosol layer were defined in

terms of the backscattering coefficient b, single scattering albedo $ and opti-

cal depth τ1. The radiative transfer equation (1.11) is an integro-differential

equation, which cannot be solved analytically in general. Therefore, to deter-

mine the effects of a layer with phase function p(µ, µ′) and optical thickness τ ,

we have to solve numerically the problem or take some assumptions leading to

an approximated analytical solution. In the two-stream approximation, it is

commonly assumed that the radiance is isotropic over the upper hemisphere

(µ > 0) presenting the value L+, and over the lower hemisphere (µ < 0)

presenting the value L−. This means that the irradiances E+(τ) and E−(τ)

are equal to πL+(τ) and πL−(τ) regarding the incoming and outgoing direc-

tions respectively, and can be expressed in terms of the aerosol optical depth

from 0 to τ1 as shown schematically in figure 1.1. For such conditions, it is

possible to separate eq.(1.12) into two very similar differential equations for

an upwelling and a downwelling isotropic radiance, respectively. The system

of equations appears as:

−12

dL−(τ)dτ

= L−(τ) − (1 − b)$L−(τ) − b$L+(τ)12

dL+(τ)dτ

= L+(τ) − (1 − b)$L+(τ) − b$L−(τ)(1.16)

where

b$ =1

2

∫ 1

0dµ∫ 1

0dµ′p(µ,−µ′)

is assumed, in which the single scattering albedo $ is defined as the fraction

between the scattering coefficient and the extinction coefficient ks/ke of the

aerosol, and the backward scattering coefficient b, represents the backward

scattered fraction of total non-absorbed radiation.


Figure 1.1: Schematic representation of the two-stream approximation for theradiative transfer equation in a plane parallel atmosphere. An aerosol layer withoptical characteristics defined by parameters $ and b, superimposed to the surface-atmosphere system (with albedo a) is reached by an isotropic flux of irradiance$L. Boundary conditions allow to solve analytically the RTE problem.

Thus, the term (1 − $) yields the fraction of radiation absorbed by thelayer, $b represents the radiation scattered in the backward hemisphere and

$(1−b) gives the measure of the forward scattered radiation, as presented infig.1.1. Once the boundary conditions are defined, as given by L−(τ0) = L0,

L+(τ1) = L−(τ1) · a, where a is the original albedo, an analytical solution of

this system of differential equations can be written in terms of the variation

in the albedo of surface-atmosphere system from a to R′ = L+(0)/L−(0). It

is expressed by

a − R′ = 2a(1 − $) − (1 − a)2$b

(1 − $) + (1 − a)$b + a2 tanh(aτ)

(1.17)

The sign of eq.(1.17) indicates whether an aerosol layer heats or cools the


Earth-atmosphere system. If a − R′ < 0 , a < R′, the surface-atmospheresystem lost toward the space more energy than in the absence of the layer,

leading to a cooling of the system. On the other hand, when a > R′, the

effect is to warm of the system. Being the denominator always positive, the

raising or decreasing of the system albedo including the aerosol layer (R′)

with respect to that of the original system without aerosol (a), depends on

the the numerator sign, in sense that heating occurs when (1 − $)/b$ >(1 − a)2/2a.

A critical ratio can be defined as the fraction between the absorbing cross

section of the aerosol layer (1−$) and the backscattering cross section $b.A null effect ∆a = a − R′ = 0, is thus found when the critical ratio satisfiesthe following equation:

1 − $$b

=(1 − a)2

2a. (1.18)

Fixing the optical properties of the aerosol layer, the surface albedo becomes

the discriminant parameter defining the sign of the first term, so that it turns

out to be associated with cooling or warming of the coupled system, as shown

in fig.1.2.

Moreover, it can be observed that in the two stream approximation the

heating condition is independent from the aerosol optical depth, that acts

only to the strength of the cooling or warming effects as shown in fig.1.3. It

should be noticed that this form of the two stream approximation is applica-

ble only to globally averaged conditions. It does not include the dependence

of the heating (cooling) on solar zenith angle which is necessary for the study

of the regional heating (cooling) effects.

1.2.2 TARFOX formulation of the direct aerosol forc-

ings

Following [Charlson et al., 1991, Nemesure at al., 1995, Haywood and Shine, 1995,

Russell et al., 1999] the global average change in the upwelling flux, at the

top-of-atmosphere (TOA), caused by absorbing ($ < 1) aerosols, can be


0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1

Crit

ical

Rat

io

a

Ocean RuralVegetation

SoilDesertClouds Ice

WARMING

COOLING

a)b)c)

Figure 1.2: Value of the critical ratio for various solutions of the plane parallelradiative transfer equation. Critical ratio is defined in eq.(1.18). Typical valuesare lower than 0.1 for marine particles, between 0.5 and 2 for continental particles,and greater than 4 for urban aerosols. The curves correspond to three differentsolutions from: (a) (1− a)2/2a is for the two stream approximation, (b) (1− a)/aand (c) (1 − 2a)/2a are for optically thin layers. The abscissa represents thealbedo of the Earth-atmosphere system containing a superimposed aerosol layerfor different surface covers; cfr. [Chyleck and Coakley, 1974].

expressed as

∆F = −12S0(1 − Ac)T 2a [$b(1 − as)2 − 2(1 − $)as]τ (1.19)

where S0 = 1368W/m2 is the solar constant at the mean Earth-Sun distance,

$ the single scattering albedo,

b the fraction of radiation scattered upward by the aerosol, averaged over the

Sun-ward hemisphere of the planet,

Ta the transmittance of the atmosphere above the aerosol layer,

Ac the fractional cloud cover,


Figure 1.3: Two stream approximation results for the dependence of the heatinga-R’ on the optical depth τ of the aerosol layer, for different albedo a (labels) of theunderlying system. Solid curves are for single scattering albedo $ of 0.9 (moderateabsorbing particles), while dashed curves are for $=0.99 (non absorbing particles).Both sets of curves represent the heating due to an aerosol, which backscatter thefraction of radiation b$=0.1 at each scattering.

as the surface reflectance, and

τ the optical depth of the aerosol layer (usually taken at the 550 nm wave-

length).

The variation of the surface albedo as can cause a shift from cooling


(negative) to warming (positive) aerosol effects, when the single scattering

albedo $ of the layer exceeds a critical value determined by the condition

(1 − $)/$b = (1 − as)2/2as [Russel et al., 2002].It can be noticed the great similarity of eq.(1.19) with the formulation

given by [Chyleck and Coakley, 1974] and expressed in eq.(1.17).

Nevertheless, this equation is valid for annual mean conditions only. For

example, it does not take into account the effects due to the increase of the

upscatter fraction b and the decrease of atmospheric transmission Ta with

respect to the air mass. A variation of the single scattering albedo $ of 0.07

causes a variation of 21% in the aerosol induced flux change at the top-of-

atmosphere; the sensitivity over common land surfaces can be much larger

[Bergstrom and Russell, 1999]. This is illustrated in figure 1.4, which shows

that, over dark vegetation (surface albedo 0-0.2), the change of $ from 1.0 to

0.9 can lower the flux change at TOA by 50% or more. Over desert and snow

fields (albedo > 0.4), the same change in $ can reduce ∆F by more than

100%, thus changing the sign of the aerosol effects from cooling to heating.

Also, flux changes within and below the aerosol layer, which can affect

the atmospheric stability, heating rates, surface temperatures, and cloud for-

mation and persistence, can be even more sensitive to the aerosol single

scattering albedo $. This increased sensitivity can cause the critical single

scatter albedo, where cooling shifts to warming, to exceed the values implied

by fig.1.4.

Local aerosol direct radiative forcing

A formulation expressed in terms of the variation in the system albedo ∆a

was proposed by [Russell et al., 1997], see eq.(5’):

∆a =τ(

$b(µ)(1 − as(µ)) − 2µbas(µ)(1 − as) − (1 − $)as(µ)(1 + 2µ))

µ(1 + b(µ)τ /µ)(1.20)

where µ = cos θ; here, the values of the instantaneous up-scatter fraction b

can be expressed in terms of the asymmetry parameter g using the approxi-


Figure 1.4: Aerosol induced change in the top-of-atmosphere upwelling flux. Re-sults are from eq.(1.19), using τ = 0.1, aerosol upscatter fraction b = 0.17, noclouds Ac = 0, and atmospheric transmission T = 0.75. Critical single scatter-ing albedo is the dotted curve labeled as 0: it divides the graph into two partswhere the effects of the aerosols are of cooling (in the upper part) or warming(lower part). This is a 3D reproduction of the original illustration presented by[Russel et al., 2002] for the same parameters.


mated simple expressions for b

b ≈ 12

(

1 − 74gµ)

and for its daily mean value b

b ≈ 12

(

1 − 78g)

according to [Wiscombe and Grams, 1976].

Chapter 2

Surface Reflectance

The strong surface-atmosphere coupling in the radiative transfer equation

implies the use of the BRDF physical models in place of the simplicistic

albedo concept, which can be used to define a boundary condition in approx-

imated solutions of the radiative transfer problem, such as the two-stream

approximation described in the Chapter 1. In order to have an idea of how

anisotropy can affect the forcing, it could be thought that a variation in

the albedo in the simple Russel equation of the radiative forcing eq.(1.19),

can produce a significant change, also in the sign of this important quantity,

leading to the occurrence of opposite radiative effects induced by the aerosol

population. It can’t be neglected that the albedo varies strongly with the

geometry of illumination and observation; this is mandatory in the satellite

observation for the retrieval of surface properties, but still often neglected

in the radiative forcing evaluations [Ricchiazzi el al., 2005]. In the present

chapter the utility of the so called ’bidirectional reflectance distribution func-

tion’ (BRDF) for an accurate characterization of the surface albedo will be

described.

19

20 CHAPTER 2. SURFACE REFLECTANCE

2.1 Bidirectional reflectance

The reflectance ρλ is a spectral property of the surface, which is physically de-

fined as the fraction between the radiant flux Φr[W ] reflected to the incident

flux Φi reaching the surface. Nevertheless, the geometry of the incident and

reflected fields of radiance determines the value of the reflectance. A specific

nomenclature was developed in order to describe this geometric configura-

tion, a pair of terms in the set directional, conical and hemispherical being

arranged in order to represent the geometry of the incident and reflected flux

respectively. For example the directional-hemispherical reflectance is used for

a direct beam incident on a surface and a reflected diffuse radiation. For uni-

form irradiance over a large enough area of a uniform and isotropic surface,

the basic quantity that characterize (geometrically) the reflecting properties

of that surface is the function

frλ(θi, φi; θr, φr) =dLλ(θr, φr)

dEλ(θi, φi)[sr−1] (2.1)

called bidirectional reflectance distribution function (BRDF).

In eq.(2.1) Lλ(θr, φr) is the reflected radiance [Wm−2sr−1nm−1] in the di-

rection (θr, φr), while dEλ(θi, φi) = L(θi, φi) cos θi sin θidθidφi is the incident

irradiance from the direction (θi, φi), which is measured in [Wm−2nm−1].

Therefore bidirectional reflectance dimensions are sr−1. Since for a pair of

directions, the BRDF fr is a concentration of reflectance (per steradian) it

may take on any value from zero to infinity [Nicodemus et al., 1977].

In order to economize on writing, we will represent the product of an

element of solid angle dω with the cosine of the angle θ between the normal

Figure 2.1: Bidirectional effects of a water surface.

2.1. BIDIRECTIONAL REFLECTANCE 21

Figure 2.2: Geometrical definitions of the angles representing the bidirectionalreflectance concept. The principal plane contains the source-target beam, and isorthogonal to the surface. The polar zenith θ and azimuth φ angles are shown forthe incident beam i and the reflected beam r.

to the surface and the direction associated with dω by dΩ, the element of

projected solid angle, so that

dΩ = cosθ · dω = cosθ · sinθdθdφ. (2.2)

Moreover, the index λ taking into account that all quantities reported in this

chapter present spectral variations will be here in after omitted.

2.1.1 Reflectance

To define the BRDF (eq. 2.1), the concepts of both radiance and irradiance

are already used. Therefore it is useful to provide the definitions of these

quantities for a better clarity of the text.

As defined in [Raschke E.,1978], the radiant energy Q [Joule] is the whole

energy emitted or reflected by a body, and the radiant flux Φ = dQ/dt [W]

is the energy emitted per unit time [s]. The radiance is the flux per unit


surface and unit solid angle, as defined by the equation

dL(θ, φ) =d2Φ

dA cos θdω. (2.3)

The irradiance or radiant flux density is the radiant flux of any origin, inci-

dent onto an area element (dA),

E =dΦ

dA(2.4)

and can be expressed in terms of the hemispherical integral of the radiance,

as expressed by

E =∫ 2π

0

∫ π/2

0L(θ, φ)dΩ (2.5)

or, representing dΩ in explicit form, by

E =∫ 2π

0

∫ π/2

0L(θ, φ) cos θ sin θdθdφ. (2.6)

Being the reflectance (ρ), the ratio of reflected to incident flux, it follows,

from conservation of energy, that it may vary only within the interval 0 to 1

[Nicodemus et al., 1977]. Mathematically, the reflectance is expressed by

ρ = dΦr/dΦi. (2.7)

An arbitrary incident radiant flux (dΦ = dQ/dt = ’radiant energy / unity

of time’ [W ]) can be expressed in terms of the radiance field by the equation

dΦi = dA∫

ωiLi(θi, φi) sin θi cos θidθidφi (2.8)

and the reflected one by the similar equation

dΦr = dA∫

ωrLr(θr, φr) sin θr cos θrdθrdφr. (2.9)

Using the definition of BRDF, as expressed by eq.(2.1), together with the

notation for the element of projected solid angle, eq.(2.9) can be written in


the form

dΦr = dA∫

ωr

∫

ωifr(θi, φi, θr, φr)Li(θi, φi)dΩidΩr. (2.10)

Thus, using eq. (2.8) and (2.9), the general expression of reflectance in terms

of radiance fields becomes

ρ(θr, φr; θi, φi) =

∫

ωi

∫

ωr fr(θi, φi, θr, φr)Li(θi, φi)dΩidΩr∫

ωiLi(θi, φi)dΩi

(2.11)

The reflectance factor (R) is defined as the ratio of the radiant flux re-

flected by a sample surface to that which would be reflected into the same

reflected-beam geometry by an ideal (lossless) perfectly diffuse (lambertian)

standard surface irradiated in the same way as the sample.

In eq.(2.10), we already have a general expression for the reflected flux

dΦr of any sample surface element, characterized by a BRDF fr. This same

equation gives also the reflected flux dΦr,id of an element of the ideal standard

surface for fr,id = 1/π, see eq.(2.14) at 23. Therefore, the ratio is given by

R(θr, φr; θi, φi) =dΦr

dΦr,id=

dA∫

ωi

∫

ωr fr(θi, φi, θr, φr)Li(θi, φi)dΩidΩr

(dA/π)∫

ωr

∫

ωiLi(θi, φi)dΩidΩr

.

(2.12)

Perfectly diffuse reflectance

For a perfectly diffuse or lambertian surface element dA, the reflected ra-

diance is isotropic. Thus, Lr assumes the same value for all the viewing

directions (θr, φr), regardless of how it is irradiated. This is possible only

when fr = fr,d is a constant, so that

Lr,d = fr,d

∫

ΩiLi(θi, φi)dΩi = fr,dEi. (2.13)

An ideal diffuse standard reflector returns the whole incident flux, so that

ρi = ρid = 1 (2.14)


Quantity Value∫ 2π0

∫ π/2−π/2 dω 4π

∫ 2π0

∫ π/2−π/2 cos θdω 2π

∫ 2π0

∫ π/20 cos θdω π

Table 2.1: Values of hemispherical integrals of the element of solid angle dω =sin θdθdφ and of its projection along the θ direction. They are widely used in thedefinition of the -/- reflectances as normalization factors.

and

fr,id = 1/π. (2.15)

In the following paragraphs, the definitions for some of the reflectance

configurations illustrated in fig.2.3 will be adopted. They can be derived

from the integration of eq.(2.11) for different values of the integration limits

and for a isotropic radiance illumination field Li. These will be useful in

order to compute the spectral albedo aλ, defined as the ratio of reflected to

incident flux for natural illumination conditions.

In this context, it is useful to remember the constants derived from geo-

metric integrals of various combinations of the trigonometric functions of the

polar angles θ and φ. They are reported in table 2.1.

2.1.2 Directional-Hemispherical reflectance ρ

This quantity describes an hemispherical reflectance of the surface irradiated

by a localized point source in the space, with the irradiance field composed

by just the direct component. This quantity can be often referred to as

Black Sky Albedo. The planetary albedo, computed at the TOA, belongs to

this category, the Sun being the only source of radiation that cannot be

neglected for practical purposes.

Mathematically, the directional-hemispherical reflectance can be obtained

from eq.(2.11) for finite integration limits, i.e. by

ρ(dωi, 2π) =

∫

dωidΩi

∫

2π fr(θi, φi, θ′r, φ

′r)Li(θi, φi)dΩ

′r

∫

dωiLi(θi, φi)dΩi

(2.16)


Figure 2.3: Different geometrical configurations adopted to define the reflectance.Here, the symbol ~ωx indicate the solid angle element, ∆ ~ωx the conical configurationand 2π the whole hemisphere. Lewis albedo defined in eq.(2.29) is represented interms of the directional-hemispherical and the bi-hemispherical reflectances. Fora whole description of these quantities see [Nicodemus et al., 1977].


which becomes

ρ(dωi, 2π) =

∫

dωidΩi

∫

2π fr(θi, φi, θ′r, φ

′r)dΩ

′r

∫

dωidΩi

. (2.17)

if one considers that Li is assumed constant. In this way,

ρ(dωi, 2π) =∫ 2π

0

∫ π/2

0fr(θi, φi, θ

′r, φ

′r)dΩ

′r (2.18)

can be written with dΩr = cos θr sin θrdθrdφr, the index i referred to the

incident direction, and the index r to the reflected one.

Hemispherical-Directional reflectance ρ′

The physical meaning of the hemispherical-directional reflectance is oppo-

site with respect to that described by the quantity discussed in the previous

paragraph: it is the reflected radiation in the direction ωr of a surface ir-

radiated with a perfectly diffuse field of radiance (lambertian). As in the

previous paragraph, the use of eq.(2.11) allows to express the hemispherical-

directional reflectance in mathematical form

ρ′(2π, dωr) =

∫

dωr dΩr∫

2π dΩ′ifr(θ

′i, φ

′i, θr, φr)Li(θ

′i, φ

′i)

∫

2π Li(θ′i, φ

′i)dΩ

′i

(2.19)

that becomes

ρ′(2π, dωr) =

∫

dωrdΩr

∫

2π dΩ′ifr(θ

′i, φ

′i, θr, φr)

∫

2π dΩ′i

. (2.20)

considering that Li is assumed constant.

Thus, bearing in mind that the value of the denominator is π (see also

table 2.1), and that∫

dωr dΩr = dΩr, eq.(2.20) becomes as follows:

ρ′(2π, dωr) =dΩrπ

∫ 2π

0

∫ π/2

0fr(θ

′i, φ

′i, θr, φr)dΩi. (2.21)

2.2. SPECTRAL ALBEDO 27

2.1.3 Bi-Hemispherical Reflectance (ρ)

The so called Bi-Hemispherical quantity, describes the reflectance of a surface

irradiated by a diffuse field of radiation. It can be obtained by integrating the

directional-hemispherical reflectance over the whole intervals of illumination

angles θi and φi. For an isotropic diffuse radiance field eq.(2.22) is commonly

used to represent this quantity, called White Sky Albedo:

ρ =1

π

∫

2π

∫

2πfr(θ

′i, φ

′i, θ

′r, φ

′r)dΩ

′idΩ

′r. (2.22)

2.2 Spectral albedo

This quantity is used to characterize the balance of the shortwave (0.2−4µm)energy of the surface and can be expressed in terms of the BRDF. The albedo

a, which is the fraction between the incoming and outgoing shortwave fluxes,

can be expressed in terms of irradiances, by splitting the incoming radiance

field into two terms relative to the diffuse and direct components. Thus,

the albedo can be expressed in terms of the white sky albedo (eq.2.22) and

the black sky albedo (eq.2.18) [Lewis, 1995]. The albedo depends on the field

of illumination at least described in terms of the diffuse to global fraction

of radiation as shown in fig.2.4. So, the spectral albedo is not an intrinsic

property of the surface as in the case of the bidirectional reflectance fr.

Mathematically, the spectral albedo of a natural surface can be repre-

sented as the the fraction of the upwelling to downwelling spectral irradiances

aλ(θs) =E↑λ(θs, surface properties...)

E↓λ(θs)(2.23)

where θs is the solar zenith angle. In order to express the spectral albedo

in terms of the BRDF, the concept of radiance can be used, as defined by


eq.(2.3). Thus, this quantity is given by

aλ =

∫ 2π0

∫ π/20 L

↑λ(θ

′r, φ

′r)dΩ

′r

∫ 2π0

∫ π/20 L

↓λ(θ

′i, φ

′i)dΩ

′i

(2.24)

which is similar to eq.(2.11). Now, considering that the incident irradiance

E↓λ(θs) is composed by the direct E↓λ(θs)dir and diffuse E

↓λ(θs)dif components

and that the reflected irradiance can be expressed in terms of the BRDF fr,

E↑(θi, ...) =∫

2πL↑λdΩ

′r =

∫

2πfrλL

↓λ(θ

′i, φ

′i)dΩ

′r (2.25)

or, splitting the reflected radiance into two components L′, due to the

direct field, and L′′ due to the diffuse irradiance field, it can be written

L↑r(ωr) = frEs cos θs︸︷︷︸

L′

+1

π

∫

2πfrL

↓(θ′i, φ′i)dΩ

′i

︸︷︷︸

L′′

(2.26)

where Es is the direct solar irradiance, θs the solar zenith angle and fr the

bidirectional reflectance function. Integrating over the viewing angles, it is

possible to write the reflected irradiance in terms of the BRDF fr:

E↑(θi) =∫

2π

[

frEs(θs) cos θs +1

π

∫

2πfrL


′i

]

dΩ′r (2.27)

or, expanding the square parenthesis

E↑(θi) = Es(θs) cos θs

∫

2πfrdΩ

′r +

∫

2π

∫

2πfrL


′idΩ

′r. (2.28)

2.2.1 Spectral albedo in terms of hemispherical re-

flectances

Following [Lewis, 1995], it is possible to represent approximately the spectral

albedo as the sum of two terms, which have been previously referred to as the

black and white sky albedos, defined by eq.(2.18) and eq.(2.22), respectively.

2.2. SPECTRAL ALBEDO 29

Figure 2.4: Lewis albedo expressed in terms of hemispherical reflectances. D(λ)represents the diffuse fraction of the global downwelling irradiance.

aL(λ) ≈ (1 − Dλ)ρλ + Dλρλ (2.29)

where the weight parameter Dλ represents the fraction of diffuse over the

global irradiance, and depends upon the atmospheric conditions, in particular

the aerosol content (τ), and from the solar zenith angle (θi). This approxi-

mation can be obtained from eq.(2.28), considering constant the diffuse field

of radiance L↓λ(θ′i, φ

′i).

2.2.2 Integrated albedo

The concept of spectral albedo implies that the corresponding evaluations

can be applied only to narrow spectral regions. The spectrum of the solar

radiation covers the wavelength range from approximately 0.25 µm to 4 µm.

In order to evaluate the balance for the whole radiant energy, it is neces-

sary to integrate the spectral albedo over this range. The weight function

E↓λ representing the irradiance reaching the Earth surface depends on the

thermodynamic and composition characteristics of the atmosphere and the

albedo as well, because of the multiple reflection processes occurring between

the surface and the atmosphere. Thus, the integrated albedo can be written,

as simpler es possible, in the form

a =

∫∞0 E

↑λdλ

∫∞0 E

↓λdλ

(2.30)


or, alternatively, using the Lewis approximation for the upwelling irradiance,

in the form

a =

∫∞0 aλE

↓λdλ

∫∞0 E

↓λdλ

. (2.31)

2.3 Bidirectional reflectance models

During the last two decades, various BRDF models were developed in order to

describe the spectral and geometrical features of the field of radiance reflected

by different natural surfaces covered by vegetation, soil and rock, water and

snow-ice. It is important to take into account the anisotropy of the natural

surfaces reflectance in all cases where the Earth images from swapping satel-

lite sensors are analyzed. These models are often developed for other appli-

cations, such as the monitoring of the canopy cover [Rahman et al., 1993a],

the extension of a desert and so on, but in the present work they are used for

evaluate the effects of surface reflectance anisotropy on the direct radiative

aerosol effects.

A certain number of parameterizations of different physical and hyper-

spectral models will be defined in order to represents:

• vegetation [Kuusk, 1994]

• arid soil [Rahman et al., 1993a, Rahman et al., 1993b]

• marine environment [Morel, 1988, Cox and Munk, 1954]

• snow and ice [Wiscombe and Warren, 1980, Warren and Wiscombe, 1980]

An hyperspectral BRDF function allows to describe the variable features

of the anisotropic reflectance field along the solar spectrum, using a lim-

ited set of structural, chemical and optical parameters. Models for vegeta-

tion and water reflectance are already implemented in the Second Simula-

tion of Satellite Signal in Remote Sensing radiative transfer code (6S RTC)

[Vermote at al., 1997], which was developed for the simulation of the radi-

ance reflected by the surface atmosphere system toward aircrafts or satellites.

2.3. BIDIRECTIONAL REFLECTANCE MODELS 31

Figure 2.5: Examples of the reflectance anisotropy of uniform natural targets thatproduces a variation of the brightness of the image that can be related to differentviewing angles. In particular, the hot spot (left) and the phase function (right)effects can be noticed in this photos. These effects are often described by differentkernels functions in the same mathematical formulation of the BRDF fr.

The BRDF model for snow and ice covered surfaces was implemented in the

6S code, in the frame of the present work.

2.3.1 How to create a BRDF model

A good BRDF implementation should satisfy a number of characteristics such

as the simplicity, the inversion properties, the physical content, as deeply

discussed by [Pinty and Verstraete, 1992]. We can distinguish the models

into three categories, with respect to the approach followed to create them,

which can be of the following kinds:

1. empirical,

2. numerical,

3. physical.

The empirical approach consists generally of a fitting procedure using

a simple parametric formula, capable of reproducing some typical features


of the anisotropy, such as the hot spot or the asimmetry : the fit of the for-

mula with laboratory or field measurements of the reflected radiance field

allow to define the desired parameters. This approach is followed when we

don’t need to obtain surface physical parameters from the reflectance mea-

surements, only attempting to achieve informations on the non lambertian

features, concerning the normalization of the reflectance to a common illu-

mination and viewing geometry.

The numerical approach is generally very complex, since it is based on

an accurate mathematical implementation describing carefully the geometry

and the physical properties of the medium. Inn these cases, the path of

photons with different energy is computed following methods such as the

Monte-Carlo, Ray-Tracing, or radiative transfer ones. These models waste

a lot of computational time, but are very realistic and can be useful for the

validation of simpler BRDF implementation, also because measurements of

this kind are poorly available.

The physical approach is widely used since it offers versatility and sim-

plicity. These models are physically derived, being often implemented through

the well known radiative transfer theory. This fact favours the retrieval of

some geometric and physical properties of the medium from a limited set

of radiance/reflectance measurements. Thus, once the parameters such a

BRDF implementation are defined, the whole field of reflected radiance can

be computed correctly.

For example, in the case of a canopy model, approximated assumptions

can be made with respect to various structural properties, such as the dis-

tribution of the orientation, dimensions and reflectance of the leaves, and

the background radiance reflected by the soil shadowed area. The physical

approach allows to convert these characteristics into quite simple equations,

similar to those used in the empirical models but providing the additional

capability of describing the structure of the medium through radiative mea-

surements.

A detailed list of these models can be found in [Verstraete et al, 1990].

2.4. BRDF IMPLEMENTATIONS 33

In the following paragraph, the model used in the frame of this work will

be described with more details.

2.4 BRDF implementations

Other works approaches the calculation of the radiative forcing by aerosol

particles for anisotropic surface reflectance conditions [Ricchiazzi el al., 2005].

The differences from one method to another are mainly due to the choice of

the BRDF models. These choices, in the present work, are closely related to

the use of the well known 6S radiative transfer code, used to calculate the so-

lar radiance field reflected by the surface-atmosphere system at the top of the

atmosphere (TOA) and at its bottom (BOA). The choice of the BRDF models

was addressed to use the hyperspectral physical models proposed by Kuusk

[Kuusk, 1994] for the vegetation, [Cox and Munk, 1954] and [Morel, 1988]

for the marine surfaces, [Wiscombe and Warren, 1980] for the snow and ice

covered surfaces. The spectral model proposed by [Rahman et al., 1993a] is

used to represent arid soil reflectance. A hyperspectral model represents the

anisotropic features of the reflected radiance field with only a limited set of

structural and optical parameters, because the spectral curves of the complex

refractive index of the absorbing chemical species are included in the code

itself, as shown in fig.(2.6).

2.4.1 Soil reflectance

We adopted the parameterization developed by [Rahman et al., 1993a], in-

cluding a procedure adopted in order to extend the knowledge of the spec-

tral dependent parameters to the whole spectra between 0.25µm and 4.0µm.

In fact, these parameters are obtained experimentally and, thus, provided

within only the spectral regions in which the radiometer operates. In the

case of [Rahman et al., 1993a] these regions are 0.58−0.68µm 0.73−1.1µm,corresponding to the CH1 (vis) and CH2 (near-IR) channels of the NOAA

AVHRR radiometers.


Figure 2.6: Differences between multispectral and spectral BRDF models: multi-spectral models permit to describe the bidirectional reflectance distribution func-tion frλ(θi, φi, θr, φr) with a limited set of geometrical and/or optical parame-ters pi, i = 1, ..,M , because the spectral dependencies are included in the modelthrough, for example, the spectral refractive index. On the other hand the spectralmodels requires a set of parameter pi for each wavelength j, j = 1, .., N where Nis the number of spectral region to be considered.


A similar method was used also by [Ricchiazzi el al., 2005].

RPV and Modified RPV model

This is a semi-empirical, non linear model [Rahman et al., 1993a], which was

subsequently linearized by [Engelsen et al., 1996] in the version used in pro-

cedures adopted by the EOS-MISR (NASA) project. An important applica-

tion of this model was the normalization of the normalized difference vege-

tation index (NDVI) derived from AVHRR sensors [Rahman et al., 1993b].

As usual, independent variables are the solar and viewing zenith angles (θi,

θr), and the difference φ between the azimuthal angles, when assuming a

cylindric symmetry of the medium. The asymmetry is driven through the

variation of three spectral parameters ρ0λ, kλ and Θλ (fig.2.6), having the

physical meaning described below. In fact, they are not structural proper-

ties of the medium, but their use can represent with appropriateness some

typical features of the anisotropy of the reflectance, modeling the shapes of

the BRDF.

The RPV BRDF is defined by the equation is

fr(θi, θr, φ; ρ0, k, Θ) = S(θi, θr; ρ0, k)F (ξ; Θ)[1 + R(G; ρ0)]. (2.32)

The first kernel is given by

S(θi, θr, k) = ρ0cosk−1θicos

k−1θr(cos θi + cos θr)1−k

, (2.33)

as adopted by [Minnaert, 1941] who used it to describe the moon reflectance.

By varying parameter k throughout [0, 1], the function models the bell shape

of the reflected radiance field, being equal to unity for lambertian reflection

conditions. The second kernel

F (ξ; Θ) =1 − Θ2

[1 + Θ2 − 2Θ cos (π − ξ)] 32(2.34)

adopted by [Henyey and Greenstein, 1941], regulates the fraction of forward


scattered radiation, using Θ as key parameter varying from -1 for total back-

ward scattering, to 1 in the opposite case. The scattering angle ξ is defined

by eq.(2.35) and is often used in the mathematical formulation of the bidirec-

tional reflectance to represent the angle between the two ωi and ωr directions

of the incoming and outgoing beams of radiation:

cos ξ = cos θi cos θr + sin θi sin θr cos (φi − φr) (2.35)

The HG function F does not allow to linearize the RPV model, through

the use of the logarithms 1. Therefore, in order to add this property to

the model, [Martonchik et al., 1998] substitutes F with the following simple

implementation

F ′(ξ; bM) = exp(−bM cos ξ) (2.36)

where the parameter bM is used in place of Θ. Making use of the logarithms

of the multiplied kernels, a sum of kernels was obtained, each one depending

by one parameter only. This allows to linearize the model and to invert it

with a simple and fast linear algebraic system, instead of following a slower

iterative inversion method.

The third kernel R(G) represents one of the peculiarity of the reflectance

anisotropy, consisting in the existence of a local maximum of reflectance for

ωi = ωr, which is called hot spot effect. It is due to the fact that the observer

cannot see directly the shadowed, and therefore darker, regions of the scene

in this case, which turns out in a local brightness of the reflectance function.

The RPV model describes analytically such an effect as

1 + R(G) = 1 +1 − ρ01 + G

, (2.37)

where

G =√

tan2θi + tan2θr − 2 tan θi tan θr cos (φi − φr)

1If a function is the product of kernels F =∏

Ai, its logarithm is the sum: log F =∑Ai.


Parameter visible near-IRρ0 0.026 0.238Θ -0.007 0.015k 0.536 0.668fr0 0.038 0.319Iλ 0.08 0.52R = Iλ/fro 2.10 1.63

Table 2.2: Values of the parameters ρ0,Θ and k defined within the two spectralregions, as obtained by [Rahman et al., 1993b], and corresponding value of thebidirectional reflectance function fr0 definded in eq.(2.38). Iλ are the referencereflectance of a lawn surface determined using a GER SFOV spectroradiometerduring a field campaign carried out at Nonantola (Italy, 2000) [Lanconelli, 2002].Visible and near-IR pertain to bands 1 (0.58-0.68 µm) and 2 (0.73-1.1 µm) of theAVHRR sensor.

is a geometric factor called angular distance. It can be noticed that for the

ωi = ωr case, G is null and R(G) assumes correspondingly its maximum,

equal to 1 + R(0) = 2 − ρ0. The linearized version (MRPV) is given by thefollowing equation,

ln fr,MRPV (λ) =

ln ρ0 + (k − 1) ln[cosθicosθr(cosθi + cosθr)]−bMcosξ + ln(1 + 1−ρHotSpot1+G )

(2.38)

in which the additional parameter ρHotSpot can be used in place of ρ0 to

describe the hot-spot effect.

In order to extend the wavelength range of the spectral parameters given

only within a few bands by [Rahman et al., 1993b] for certain surfaces, the

following approach was developed: the RPV function for nadir observation

(θr = 0) and zenith illumination (θi = 0, φ = 0) assumes the very simple

expression

fr0 = fr0(θs = 0, θv = 0, φ = 0) = ρ0(2 − ρ0)2k−11 − Θ

(1 + Θ)2. (2.39)

The knowledge of the parameters within certain spectral channels, like


Figure 2.7: Spectral curves of surface reflectance parameters for lawn, as adaptedfrom [Rahman et al., 1993b] k (white circles), Θ (black circles), ρ0 (white trian-gles) for the spectral visible and near-infrared intervals of the AVHRR radiometer.Continue lines yield the spectral values of the parameters obtained by means of thepresent procedure for kλ (green), Θλ (black), ρ0λ (red); input curve I0(λ) (cyan)and spectral reflectance at nadir fr0(λ) (green) are also shown .


the AVHRR visible (0.58 − 0.68 µm) and the near-infrared (0.73 − 1.1 µm)ones [Rahman et al., 1993b], enabled us to define the values of fr0 for these

channels only, which are given in table 2.2.

As shown also by other authors in recent studies [Ricchiazzi el al., 2005]

a characteristic spectral reflectance curve Iλ suitable for use is available in

the USGS Digital Spectral Library splib05a [Clark et al, 2003].

Some of them are presented in fig.2.9 in order to extend to the whole

spectrum the set of parameters found in some spectral regions only, by means

of some a priori assumptions.

With regard to this, it was supposed that kλ and Θλ are subject to vary

according to the following equation

y(λ) = y(λi) +y(λi+1) − y(λi)

λi+i − λi(λ − λi), (2.40)

throughout the range between the two spectral regions, and assume constant

values outside them as shown in fig.2.7. We took into account the wavelength-

dependence of the reflectance provided by the USGS curves Iλ, during the

definition of the spectral variability of ρ0λ obtained from the eq.(2.39). Nev-

ertheless, we need an expression for fr0,λ in order to invert this equation with

respect to ρ0. For this purpose, the fraction

Rλ = Iλ/fr0,λ

given by the ratio of Iλ to the nadir reflectance fr0,λ, was assumed to vary

with the same features of Θλ and kλ described in eq.(2.40), being the values

of Rλi within the two spectral regions of the visible and the near-IR, those

given in table 2.2. Therefore fro,λ = RλIλ was determined over the whole

spectrum and eq.(2.39) can be written in terms of ρ0λ as

ρ20λ − 2ρ0λ + c = 0, (2.41)


Figure 2.8: Example of the dependence of the directional hemispherical (coloredsurface) and bi-hemispherical (gray surface) reflectance for the lawn on wavelengthand solar zenith angle.

where

c = RλIλ(1 + Θλ)

2

1 − Θλ2kλ−1.

The solution of this second degree equation with respect to ρ0(λ) provides

a spectral dependence for the parameter ρ0 which agrees with a typical re-

flectance curve as that shown in fig.2.7. A representation of the anisotropy

characteristics is correspondingly given in the fig.2.8, in terms the directional-

hemispherical and bi-hemisperical reflectances.


Parameter visible near-IRρ0 0.076 0.095Θ -0.29 -0.268k 0.648 0.668

Table 2.3: Values of the RPV BRDF’s parameters retrieved for the plowed fieldby [Rahman et al., 1993b] whithin the two spectral region 0.58 − 0.68 µm and0.73 − 1.1 µm.

Models from USGS spectral library used for the creation of the

LUT

Making use of the previously described method, a set of spectral curves was

selected from the USGS archive, shown in fig.(2.9), in order to represent

diverse spectral features for the Earth surface, covered by soil or rocks, by

means of the procedure described above. Parameters for the soil at two

fixed wavelengths in the visible and in the near-infrared were taken from

[Rahman et al., 1993b], who provided the retrieved values for a plowed field

given in table (2.3). Moreover, fig. 2.9 gives the spectral curves used for

extending these parameters to the whole wavelength range. Four different

models named BS (bare soil) were defined, numbered from 1 to 4 for the re-

flectance properties of surfaces composed by sand (BS1), illite (BS2), alunite

(BS3) and montmorillonite (BS4), respectively.

The values of the Lewis albedo aL expressed by eq.(2.29), together with

those of the directional-hemispherical and of the bi-hemispherical reflectances

are given in table 2.9 (pag.42) at the end of this chapter and compared

with the evaluations of the same quantities derived for all the other models

examined and defined in the rest of the chapter.

Considering the relatively weak variability of the ρ values it is evident that

the spectral features of the visible range exert the predominant influence

on the albedo. The strong differences in the reflectance observed in the

spectral range beyond 1 µm (fig.2.9), do not turn out to affect the resulting

integrated albedo, because the solar irradiance reaching the Earth surface at


those wavelengths is considerably weaker than at the visible wavelength.

2.4.2 Water reflectance

Ocean albedo is usually considered very low, not exceeding a few percents at

all wavelength. In many retrievals of atmospheric properties from satellite

observation, the sea is assumed to be a dark target, leading to the assump-

tion that the signal measured by the sensor can be considered as produced by

the atmospheric constituents only, and in particular by the aerosol particles.

Nevertheless, the reflectance of the water exhibits some important geometric

dependence aspects (such as the sun glitter effect). Moreover, its spectral be-

havior is related to the chemical composition of sea water, depending mainly

on the salinity and the chlorophyll content.

Figure 2.9: USGS spectral curves selected for the definition of the soil RPVBRDF models used in the creation of the flux change look up tables presented inthe chapter 4. Four different surfaces are composed by sand (BS1), illite (BS2),alunite (BS3), montmorillonite (BS4) respectively. Their spectral characteristicreflectances Iλ are taken from the USGS database, except the curve for sand(BS1), that is adopted from the 6S code.


We adopted the parameterization developed by [Cox and Munk, 1954,

Morel, 1988, Koepke, 1984] that is already implemented in the 6S RTM.

[Cox and Munk, 1954] made measurements of the sun glitter from aerial pho-

tographs defining a many-faceted model surface whose wave-slopes vary ac-

cording to isotropic and anisotropic Gaussian distribution with respect of

the surface wind speed v. In the solar spectral range, the reflectance of the

ocean surface ρos(λ) can be assumed for a set of geometrical conditions θi,

θr and φ, given as the sum of three components, each wavelength dependent

according to

ρos(θi, θr, φ, λ) = ρwc(λ)+(1−W )ρgl(θi, θr, φ, λ)+(1−ρwc(λ)ρsw(θi, θr, φ, λ),(2.42)

where ρwc(λ) is the reflectance due to the whitecaps, ρgl(λ) is the specu-

lar reflectance at the ocean surface and ρsw(λ) is the scattered reflectance

emerging from the sea water. Parameter W is the relative area covered with

whitecaps for a water temperature higher than 14◦C. It can be expressed

in terms of the wind speed v by W = 2.9510−6v3.52 [Koepke, 1984]. The 6S

user guide contains a detailed description of this model. Particular attention

must be paid to the set of parameters used to drive the variation of the re-

flectance properties: they are the wind speed at the surface (v), the salinity

(S) and chlorinity (C) of the water; default values are S=34.3 ppt for the

salinity and C=19 mgm−3 for the chlorinity. The spectral information are

introduced with the complex index of refraction of salted water.

The major effect is that produced by the variation in the wind speed

(v) that affects the formation of the highly reflecting white caps on the sea

surface.

Evaluations of the Lewis albedo were obtained using eq.(2.29) for different

solar zenith angles from 0◦ to 60◦ and presented in fig.2.10. It can be noticed

that water surfaces albedo is a growing function of the solar zenith angle θs

and a decreasing function of the chlorophyll content C. Its variation with

the salinity cannot be appreciable for the scale adopted in the figure. The


Figure 2.10: Lewis albedo for various ocean models and different solar zenithangles θs. The model number is given in abscissa. For models in the range 0-3,the wind speed is 2 m/s, (4-7): 5 m/s, (8-11): 10 m/s, (12-15): 20 m/s. The effectof the chlorophyll content growth, that varies from C = 0.01− 0.1− 1− 10mg/m3within each group of four models, leads to a decrease of the albedo with increasingconcentration. For the higher wind speeds, the difference in the reflectance due tothe solar zenith angle is weaker because the sun glitter effect is hidden by brokenwaves.


Model C S v[mg/m3] ppt m/s

OC1 0.1 34.3 2OC2 0.1 34.3 5OC3 0.1 34.3 10OC4 0.1 34.3 20

Table 2.4: Chlorophyll content C, salinity S and wind speed v used for the OCBRDF models.

wind speed has the effect of smoothing the anisotropy, being the reflectances

at various zenith angles more similar for the higher wind speeds (20 m/s)

with respect to the case pertaining lower wind speed conditions (2 m/s).

Table 2.4.2 presents the values of the parameters of the four models named

OC, which were used to create the look up tables of the flux change described

in carefully in Chapter 4. Table 2.7 shows the values of the Lewis albedo at

three different solar zenith angles.

2.4.3 Vegetation reflectance

The anisotropic reflectance of vegetated surface was parameterized adopting

the model developed by [Kuusk, 1994]. The description of the mathematical

implementation of this model is outside the purposes of the present work;

nevertheless, some qualitative descriptions are provided herein, in terms of

the parameters used in this hyperspectral implementation of the BRDF de-

voted to represent the canopy reflectance by means of the radiative transfer

theory.

The Kuusk model parameters can be subdivided into three classes: i)

structural canopy parameters, ii) radiometric leaf parameters and iii) soil

reflectance parameters 2. Structural parameters are (1) the Leaf area index

LAI m2/m3) that represents the foliage area per unit volume of the canopy,

(2) the fraction of the leaf linear dimensions to the height of the canopy sL,

2The underlying soil is described in terms of the modified Walthall bidirectional re-flectance model and the Price spectral model [Walthall et al., 1985]


(3) the average of the leaf inclination with respect to the horizontal θm, and

(4) the eccentricity e of the elliptic distribution of the leaf inclination.

The optical parameters are (1) the leaf pigment concentration CAB, (2)

the leaf water equivalent thickness Cw, (3) the number N of effective ele-

mental layer of a leaf and (4) the fraction cn between the refractive index

of the wax of the leaf surface and its internal. In addition the underlying

soil reflectivity is expressed through a linear combination of two spectral

functions F1 and F2, which are derived from a principal component analysis

(PCA) of approximately 500 spectral curves. Thus, the weights si in the lin-

ear combination ρsoil(λ) = s1F1(λ)+ s2F2(λ) [Price, 1990] are two additional

parameters.

In this work, some crucial parameters were modified in order to obtain

a variability of the albedo within the range 0.1 − 0.3. In particular thereflectance was assumed to vary appreciably with respect to the parameters

CAB and Cw. Starting form the set of parameters provided by [Kuusk, 1994]

for a corn covered surface, some simulations were performed for increasing

values of CAB from 50 mg/m3 to 150 mg/m3, and Cw from 0.13 cm to 0.52 cm,

considering an intermediate value of 0.26 cm. The original value for the corn

surface was CAB = 100 mg/m3 and Cw = 0.26 cm. The effects produced are

shown and discussed shortly in figure 2.11 for a solar zenith angle θs = 40◦,

a standard atmosphere US62 and a continental aerosol loading yielding a

visual range of 23 km.


Figure 2.11: Part a). Effects of the variation of the water content Cw of thecanopy. It can be noticed that the effects produced by the water content areappreciable within the water absorption bands. Part b). Effects of the variationof CAB on the Lewis albedo. The effects of the chlorophyll content results to berelevant only in the visible spectral region. The solar zenith angle is θs = 40

◦ forboth figures. The calculations have been made for the US62 standard atmosphere,and a continental aerosol content yielding a visibility of 23 km.


Parameter Corn Soybean RangeLAI(m2m−3) 1.2 2.9 0.6 - 4

W .39 .99 0 - 1H (m) .56 1.02 -

sL = dL/H .1 .1 -CAB(mg m

−3) 100. 82.2 10 - 100Cw(cm) .026 .005 0.005 - 0.05

N 1.09 1.24 1.0 - 2.5Cn .9 .9 -θm 10.7

◦ 45.8◦ 0◦ - 90◦

e 0.972 .965 0 - 0.989s1 .213 .225 0.05-0.4s2 .100 .100s3 .013 .037s4 .015 -0.002

Table 2.5: Variability range of the structural and optical parameters obtained by[Ranson et al., 1984]. The physical meaning of the parameters is described in thetext.

Parameter AK1 AK2 AK3 AK4LAI(m2m−3) 0.1 3.6 2.5 5.0W - - - -H (m) 0.56 0.56 1.02 1.02sL = dL/H 0.1 0.1 0.1 0.1CAB(mg m

−3) 100. 100. 82.2 82.2Cw(cm) 0.04 0.026 0.005 0.005N 1.09 1.09 1.24 1.24Cn 0.9 0.9 0.9 0.9θm 10.7

◦ 10.7◦ 45.8◦ 45.8◦

e 0.972 0.972 0.965 0.965s1 0.213 0.213 0.225 0.225

Table 2.6: Parameters used in the AK vegetation models defined in terms of theKuusk’s BRDF.


Figure 2.12: Spectral variation of the Lewis albedo for the AK models. The calcu-lations have been made for a solar zenith angle θs = 0

◦, the standard atmosphereUS62, and a continental aerosol content giving total optical depth 0.1 at the 550nm wavelength.


2.4.4 Snow reflectance

To define this physical quantity the parameterization originally developed by

[Wiscombe and Warren, 1980, Warren and Wiscombe, 1980], was adopted us-

ing the Mie theory to describe the propagation of the a solar radiance field in

a medium consisting of a mixture of spherical particles of snow polluted by

dust or particulate matter (soot) of various sizes and refractive index. The

spectral dependence of the complex refractive index for the ice is shown in

fig.2.13.

Following this approach, a series of three surface models covering the

albedo range from 0.45 to 0.85 was defined. The key parameters for this

modeling approach are (1) the mean radius and the variances of the multi-

modal distribution expressed in terms of log-normals of the snow particles

(r and σ), and (2) the optical properties of the layer in terms of the sin-

gle scattering albedo SSA and the asymmetry factor g, which can be com-

puted on the basis of the Mie theory. In order to obtain these values, the

mie subroutine of the 6S code was modified extending the wavelength set

from the default number of 10 to 217 values, due to the need of more exten-

sive information on the complex refractive index of ice and snow provided

by [Warren, 1984]. Varying the radius of the distribution, it was possible to

determine the curves of reflectance, some of which are shown in fig.(2.14)

to give evidence of the strong variability occurring in the near-IR region.

In fact, it can be noticed that appreciably differences in reflectance appear

only in a spectral region where the solar radiation is strongly attenuated by

the atmosphere. The albedo in the visible does not significantly vary as a

function of the radius of the snow or ice spherical particles. As pointed out

by [Warren and Wiscombe, 1980] a more marked effect on the snow surface

reflectance values can be produced by a small contamination of the snow

layer by particles of different origins like dust or soot particles. Different

works provided evidence of the presence of both these particle types in mid-

latitude glaciers, as well in the polar ice-sheet, the Arctic being considerably

more polluted than Antarctica, with the concentration varying from 1 to 105


Figure 2.13: Spectral dependence curves of the (a) imaginary part and (b) realpart of refractive index for ice.

ppmw (1 ppmw=103 mg/g).

The absorption effect of 1 ppmw of soot particles is comparable with

that produced by 100 ppmw of dust, due to the strong absorption properties

of carbonaceous substances. Both the species cause a reduction in the re-

flectance of pure snow below 1µm, being the curve for the soot more stable,

and that for the desert dust presenting a local maximum at a 0.6 − 0.7µm.Soot particles, presenting a fixed radius of 0.1 µm, a density of 2.05g/cm3 and

a complex refractive index n=1.8-i0.5, have been used in different concentra-

tions in order to create three snow reflectance models labeled with PS (polar

surface). The concentration are expressed in terms of the relative number C

of the giants snow particles with respect to the smaller soot particles. They

are 0.5, 0.05 and 0.005 for PS1, PS2 and PS3 reflectance models, respectively.

Their Lewis albedo aL are shown in the fig.2.18 along with those obtained

for the other models.


Figure 2.14: Spectral reflectance of pure snow for different radii of the sphericalparticle monodispersions.

Figure 2.15: Spectral curves of (a) the asymmetry factor g and (b) of the singlescattering albedo SSA , resulting from the Mie theory calculation for a mixture ofsnow and soot spherical particles for two different values of the relative number ofsnow particles C. The radius of the snow is 100 µm, while that of soot particles isassumed to be 0.1 µm.


Figure 2.16: Spectral reflectance curves for soot contaminated snow; the sootrefractive index is adopted from the 6S subroutines instead of the fixed assumption(see text). Curves represent the black sky albedo for various concentration of snowand for θs = 0, The red curve is the white sky albedo for a snow relative numberconcentration C=0.005. Decreasing in the relative number of snow particles turnsout to a lower albedo.

Figure 2.17: Anisotropy of the snow albedo. The spectral curves represent thedirectional-hemispherical reflectance for different values of the cosines of the solarzenith angle of 0.1 (θs = 85

◦), 0.5 (θs = 60◦), and 0.9 (θs = 26

◦). Red curve is thebi-hemispherical reflectance that is independent from θs.


2.5 Summary

A complete and periodically updated table 2.7 of the albedo obtained for

different BRDF models can be found here:

http://www.isac.cnr.it/∼radiclim/forcing lut/index.htm.Password is required, please contact the author at [email protected]

in order to obtain the access.

Figure 2.18: Graphycal representation of the Lewis albedo obtained for the 15BRDF models, continental aerosol with optical depth at 550 nm of 0.1, and for theUS62 standard atmosphere. OC are ocean models implemented by means of theMorel model (see section 2.4.2). BS are bare soil surfaces modeled by means of theRPV’s BRDF (section 2.4.1). AK are vegetated surfaces modeled with the Kuusk’sBRDF (section 2.4.3). The reflectance model of Wiscombe and Warren describedin the section 2.4.4 was used to create the three polar surface (PS) models.

2.5.SU

MM

ARY

55

Model aL ρ ρθ0 0 30 60 0 30 60 0 30 60AK1 0.1349 0.1385 0.1552 0.1337 0.1375 0.1553 0.1531 0.153 0.1542AK2 0.2216 0.2279 0.2619 0.2195 0.2261 0.2626 0.2534 0.2532 0.2559AK3 0.2031 0.2133 0.2575 0.2008 0.2114 0.2587 0.243 0.2428 0.2453AK4 0.2527 0.264 0.3063 0.2502 0.2621 0.3075 0.2922 0.2919 0.2949BS1 0.2259 0.2276 0.2399 0.2251 0.2268 0.24 0.2372 0.2374 0.2387BS2 0.4339 0.4377 0.4606 0.4309 0.4348 0.4612 0.4571 0.4572 0.4579BS3 0.4591 0.4632 0.4888 0.4561 0.4604 0.4894 0.4841 0.4844 0.4858BS4 0.4263 0.4302 0.455 0.424 0.428 0.4555 0.4496 0.45 0.4522OC1 0.0297 0.0348 0.1929 0.0258 0.031 0.2153 0.0698 0.0698 0.0696OC2 0.0297 0.0332 0.158 0.026 0.0293 0.1741 0.0691 0.0691 0.0689OC3 0.0311 0.0343 0.1265 0.0275 0.0305 0.1369 0.0693 0.0693 0.0691OC4 0.0475 0.0509 0.1046 0.0443 0.0476 0.109 0.0806 0.0806 0.0804PS1 0.8265 0.834 0.8542 0.824 0.8324 0.8559 0.8466 0.8465 0.8448PS2 0.7257 0.739 0.7752 0.7196 0.7351 0.7793 0.7605 0.7604 0.759PS3 0.472 0.4958 0.5636 0.4609 0.4883 0.5715 0.5357 0.5356 0.5345

Table 2.7: Lewis albedo aL, black sky albedo ρ and white sky albedo ρ, obtained for the BRDF models and different valuesof the solar zenith angle θs. The white sky albedo presents some slights variations due to the dependence of the weightingfunction E↓(λ) on θs eq.(2.31).


Acknoledgements Results shown in this chapter were obtained in the

frame of a grant with the title “Studio delle caratteristiche di riflettività di

superfici vegetate di diverso tipo sia mediante misure in campo sia attraverso

opportune parametrizzazioni di modelli di riflettanza bidirezionale (BDRF)

- Bando n126.226.AR.41”, supported by the Italian Space Agency.

Chapter 3

Aerosol characterization

3.1 Aerosol properties

Aerosol particle extinction prevails on the other atmospheric attenuation

processes (absorption) only within the narrow spectral intervals which sepa-

rate the various bands, commonly called atmospheric windows. Within these

windows, all presenting particularly high transparency conditions, airborne

aerosol particles extinguish the solar radiation through both scattering and

absorption processes, whose intensity is more marked in the lower part of

the troposphere, where most of the atmospheric particulate matter is con-

centrated.

Considering the approximate solution of the radiative transfer equation ex-

pressed in the first chapter by eq.(1.17), it can be pointed out that the atmo-

spheric parameters influencing the radiation values are basically the single

scattering albedo $ and the backscattering coefficient b. These two crucial

parameters depend on the aerosol physical properties which can be defined

following the inversion methods based on the Mie theory for spherical parti-

cles, to determine the number density size distribution N(r) and the complex

refractive index m=n+ik.

Measurements of direct solar irradiance taken within the main atmo-

57

58 CHAPTER 3. AEROSOL CHARACTERIZATION

Figure 3.1: Multispectral solar photometry scheme adopted form the web sitehttp://www.isac.cnr.it/∼radiclim.

spheric windows using multispectral sun photometers (fig. 3.6) can be con-

veniently examined in order to obtain reliable estimates of the direct solar

radiation attenuation caused by aerosol particles at the visible and near-

infrared wavelength.

The analysis of these measurements is usually made by following the

so-called multispectral sun-photometric method described in fig.3.1 in order

to calculate the values of aerosol particle optical thickness at the various

sun-photometric wavelengths, from which the atmospheric turbidity param-

eters (α and β) can be easily determined according to the Ångström formula

[Tomasi et al., 1983].

3.1. AEROSOL PROPERTIES 59

The the multi-spectral solar photometry technique, is based on the well

known Bouguer-Lambert-Beer law

Vλ = RVλ0exp[−mτt(λ)]

where V is the signal measured at the ground, proportional to the direct

spectral solar irradiance, V0 is the signal measured outside the terrestrial

atmosphere, air mass m is the relative optical air mass varying as a function

of the solar zenith angle θs and, τtλ is the total atmosphere optical depth at

wavelength λ.

The total optical depth τt(λ) can be expressed as a sum of different terms,

τt(λ) = τ(λ) + τO3(λ) + τNO2(λ) + τH2O(λ) + τRay(λ)

due to the gaseous absorption, the Rayleigh scattering and the aerosols

(τ(λ)).

The electromagnetic Mie theory describes the scattering and absorption of

an electromagnetic wave by a spherical particle, assuming that the radiation

field of the particle is the sum of various fields created by quadrupoles and

multipoles of higher order. This theory shows that aerosol particles scatter

and absorb the incoming solar radiation with features closely related to:

• the particle size distribution;

• the ratio between particle radius r and incident wavelength λ;

• the complex refractive index m(λ) of the particulate matter, whichdepends on the physico-chemical properties of the particles and, hence,

varies as function of wavelength.

The angular distribution function of the scattered intensity usually presents

a pronounced forward lobe, a less marked backward lobe and numerous peaks

and troughs on sides. The shape of such a scattering angular diagram widely

varies as a function of the particle size and wavelength ratio. It is subject


Figure 3.2: Example of scattering angular diagram for a spherical particle. θ isthe scattering angle. In a) the ratio r/λ is equal to 0.25, in b) r/λ > 1.

to vary appreciably as a function of the absorption coefficient of particulate

matter. Two examples of the scattering angular diagram are shown in fig.3.2

for different values of the ratio between particle radius and wavelength . As a

result of scattering and absorption by aerosol particles, an atmospheric par-

ticle layer exhibits spectral extinction, scattering and absorption coefficients

per unit path length, which depends on the shape of particle size distribution

curve, the particle number concentration and the particle refractive index,

as well as on the vertical profile of these parameters.

The refractive index of aerosol particles is a complex number function of

wavelength expressed as follows:

m(λ) = n(λ) − ik(λ) (3.1)

where n(λ) is the real part of the refractive index and k(λ) is the imaginary

part, which characterizes substantially the absorption features.

The airborne aerosol particles have sizes varying by several order of mag-

nitude. Statistically averaged optical properties in a unit volume are com-

puted using analytical aerosol size distributions, N(r).

3.1. AEROSOL PROPERTIES 61

The must commonly used size distributions for atmospheric aerosol parti-

cles are the log-normal distributions and the modified Gamma distributions.

For a log normal size distribution, [Levoni et al., 1997]

N(r) =C

log10σ√

2πexp

{

− [log(r/rm)]2

2(σ)2

}

(3.2)

where C is the total number of aerosol particles in a unit volume, σ is the

geometric standard deviation, and rm is the geometric mean radius. The

size-distributions of thus kind are asymmetric. The radius of 95% of these

particles lies between rm − 2σ ≤ r ≤ rm + 2σ).

For the modified Gamma size distribution [Deirmendjian, 1969], the par-

ticle number density is given by

N(r) = a0rαe

(−αγ

( rr0

)γ)(3.3)

where a0, α and γ are the shape parameters of the distribution.

The computation of the optical properties of an individual aerosol par-

ticle is based on Mie theory [McCartney, 1976], in which the particles are

assumed to be all spherical and the radiative properties are represented in

terms of the complex refractive index. Parameters typically used to describe

the aerosol optical properties are the extinction, scattering and absorbing ef-

ficiency factors, which are given by the ratio between extinction (scattering

and absorption) cross section and geometrical cross section, respectively. In

other words, by multiplying the efficiency factor by the particle geometrical

cross section πr2, the volume extinction scattering and absorbing coefficients

of radiation lost are, obtained respectively, i.e.:

Qext(λ) =σext(λ)

πr2(3.4)

Qsca(λ) =σsca(λ)

πr2(3.5)


Figure 3.3: Variation of extinction efficiency, Qext as a function of the particleradius for individual particles, [Twomey, 1977].

Qabs(λ) =σabs(λ)

πr2(3.6)

Considering that extinction is produced by both scattering and absorption,

the extinction coefficient is expressed as:

Qext(λ) = Qsca(λ) + Qabs(λ) (3.7)

These efficiency factors are all functions of wavelength, particle radius and

refractive index. It is important to highlight that scattering and extinction

efficien

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