6s manual v1.0

6S User Guide Version 1, November 3, 1995

1

Second Simulation of the Satellite Signalin the Solar Spectrum (6S)

AIR FRANCE

- KRAPS -

FANS CLUB

Vermote E.(1,* ), Tanré D.(2), Deuzé J. L.(2),Herman M.(2), and Morcrette J. J.(3)

(1) Department of Geography, University of Maryland

address for correspondence:

NASA-Goddard Space Flight Center-Code 923

Greenbelt, MD 20771, USA

(2) Laboratoire d'Optique Atmosphérique, URA CNRS 713

Université des Sciences et Technologies de Lille

59655 Villeneuve d'Ascq Cédex, France

(3) European Centre for Medium Range Weather Forecast

Shinfield Park, Reading

Berkshire RG2 9AX, United Kingdom

* Formerly affiliated to Laboratoire d'Optique Atmosphérique


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Contents

ABSTRACT 5

INTRODUCTION 7

DESCRIPTION OF ATMOSPHERIC EFFECTS IN SATELLITE OBSERVATIONS 111- Absorbing effects 112- Scattering effects 13

2.1 Case of a Lambertian homogenous target 132.2 Environment function 152.3 Intrinsic atmospheric reflectance 172.4 Airplane and elevated target simulations 192.5 Directional effect of the target 222.6 Atmospheric correction scheme 24

3- Interaction between absorption and scattering effects 24

CONCLUSION 27

FIGURES 28

ACKNOWLEDGMENTS 33

APPENDIX:I Description of the computer code 35II Examples of inputs and outputs 53III Description of the subroutines 57


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5

ABSTRACT

The remote sensing from satellite or airborne platform of land or sea surface in the visible

and near infrared is strongly affected by the presence of the atmosphere on the path Sun-target-

Sensor. The following manual presents 6S (Second Simulation of the Satellite Signal in the Solar

Spectrum), a code enabling simulation of the above problem. The 6S code is the improved version

of 5S developed by the Laboratoire d'Optique Atmospherique, 10 years ago. It enables to simulate

plane observations, to account for on elevated targets, to take into account for non lambertian

surface boundary conditions, and new gases (CH4,N2O,CO) are now integrated in the computation

of the gaseous transmission. The computation accuracy regarding Rayleigh and aerosols scattering

effects has been improved by the use of state of the art approximation and use of the successive

order of scattering method. The step used for spectral integration has been improved to 2.5

nanometers. Finally, all community approved results (e.g.: computation of Rayleigh optical depth )

have been integrated in the code.


6


7

INTRODUCTION

In the solar spectrum, sensors on meteorological or Earth remote sensing satellites measure

the radiance reflected by the atmosphere-Earth surface system illuminated by the sun. This signal

depends on the surface reflectance, but it is also perturbed by two atmospheric processes, the

gaseous absorption and the scattering by molecules and aerosols.

In the ideal case (without interfering atmosphere), the solar radiation illuminates the surface.

A fraction of the incoming photons is absorbed by the surface, whereas the remaining photons are

reflected back to space. Therefore, the measured radiance directly depends on the surface

properties: this radiance is the useful signal as much as it characterizes the actual surface

reflectance.

In the actual case, the signal is perturbed by the atmosphere. Only a fraction of the photons

coming from the target reaches the satellite sensor, typically 80% at 0.85 m and 50% at 0.45 m, so

that the target seems less reflecting. The missing photons have been lost through two processes:

absorption and scattering.

Some photons have been absorbed by aerosols or atmospheric gases (principally O3, H2O, O2,

CO2, CH4 and N2O). Generally, absorption by aerosols is small, and satellite sensor channels avoid

the molecular absorption bands. Thus, in this study, the absorption effect is a correction factor.

Nevertheless, we have corrected for the absorption as accurately as possible which allows this code


8

to be used to help to define new satellite sensor spectral channel. The absorption is computed using

statistical band models with a 10cm-1 resolution.

While some photons are absorbed others are scattered. The interaction with molecules or non-

absorbing aerosols is elastic, and the photons are immediately re-emitted in a direction other than

the incident one. After one or several such scattering processes, these photons leave the atmosphere

and must be counted in the budget of photons reaching the satellite sensor. However, their paths are

more complex than the previous direct paths as explained hereafter.

Firstly, let us consider the photons coming from the sun that do not reach the surface and are

backscattered toward the space. They take part in our radiative balance. This signal is an

interference term; it does not carry any information about the target.


9

On the sun-surface path, the remaining photons contribute to the illumination of the ground by

the way of scattered paths and compensate the attenuation of the direct solar paths.

This diffuse component has therefore to be considered in the useful signal.

In a second step, let us consider the photons scattered by the atmosphere on the surface-

satellite path. By the same way, a fraction will be scattered toward the sensor.

This component has to be carefully considered. If the surface is uniform, it is an useful

component but if the surface has a patchy structure, this term will introduce environment effects

that will be a perturbation.

Finally we have to consider the photons which are backscattered by the atmosphere to the

surface and that create a third component of its illumination; it is the trapping effect, the photons


10

successively interact with the surface and the atmosphere but, generally, the convergence is fast and

after one or two interactions, the phenomenon can be neglected.

From this qualitative description, we can then infer a modelization of the atmospheric effects

based upon:

• an accurate approach of the absorption by atmospheric gases

• a complete treatment of the scattering processes

• and an approximation for the interaction between the two processes.


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DESCRIPTION OF THE ATMOSPHERIC EFFECTS IN SATELLITE OBSERVATIONS

1 Absorbing effects

In the solar spectrum the atmospheric gaseous absorption is principally due to:

• oxygen (O2);

• ozone (O3);

• water vapor (H2O);

• carbon dioxide (CO2);

• methane (CH4);

• nitrous oxide (N2O).

O2, CO2, CH4, and N2O are assumed constant and uniformly mixed in the atmosphere, H2O and O3

concentrations depend on the time and the location. The latter two are the most important gases in

our study.

Gases absorb the radiation by changes of rotational, vibrational or electronic states. The

variations of rotational energy are weak and correspond to the emission or to the absorption of

photons of weak frequency, which are then located in microwave or far-infrared range. The

vibrational transitions correspond to greater energy which open to absorption spectrum in the near

infrared. They can also take place with rotational transitions and then give rise to vibrational-

rotational bands. Lastly, electronic transitions correspond to more important energy and give rise to

absorption or emission bands in the visible and the ultra-violet range. Since these transitions occur

at discrete values, the absorption coefficients vary very quickly with the frequency and present a

very complex structure.

Once the position, intensity and shape of each line in an absorption band is known, the

absorption can be then exactly computed using line-by-line integrations. Such a model requires a

very large computational time which makes it necessary to use an equivalent band model. We have

divided the solar spectrum into spectral intervals of 10cm-1 width using HITRAN database. This

width allows a good description of the spectral variations of the transmission and of the overlapping

between absorption bands of different gases. Moreover, this allows enough absorption lines to be

contained in order that the transmission models remain well adapted to the problem.


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We chose two random exponential band models, Goody (1964) for H2O and Malkmus (1967)

for the other gases. When several gases present absorption bands, we put the total transmission as

the product of each gas transmission.

A complete description of the statistical models is given in Appendix III (subroutine

ABSTRA), Here we present the results for each gas between 0.25 m and 4 m for a standard

atmosphere model corresponding to the US-1962 standard atmosphere and a sun at the nadir (air

mass is 1.0). These resuts are depicted in Figures I-1 to I-6

The H2O contribution (Fig. I-1) affects mainly wavelengths greater than 0.7 m. On the other

hand, O3 presents a significant absorption between 0.55 and 0.65 m and limits the earth

observations at wavelengths less than 0.35 m (Fig. I-2). The CO2 contribution occurs beyond 1 m,

but more weakly than H2O and only perturbing the water vapor windows (Fig. I-3). The O2

influence is limited to a very strong band about 0.7 m (Fig. I-4). The CH4 presents two absorption

bands at 2.3 and 3.35 m (Fig. I-5). Finally, the N2O contribution appears in two bands at 2.9 and

3.9 m (Fig. I-6).

In summary, we have, in the Solar Spectrum, good atmospheric windows for terrestrial

observations by satellites:

• in the visible: between 0.40 and 0.75 m

• in the near and middle infrared: at about 0.85, 1.06, 1.22, 1.60, 2.20 m.

Fig. I-7 to I-14 display these windows


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2 Scattering effects

2.1 Case of a Lambertian uniform target

In a first step, assuming that the surface is of uniform Lambertian reflectance and the

atmosphere horizontally uniform and various, the measured quantities will be expressed in terms of

equivalent reflectance, * defined as:

* L

s Es(1)

with L the measured radiance, Es the solar flux at the top of the atmosphere and s=cos( s)

where s is the sun zenith angle.

The optical thickness of the atmosphere will be noted , the viewing direction will be

referenced by the zenith angle V and the azimuth angle V, and the sun angles by s and s. t will

be the reflectance of the target. Absorption problems will not be considered in this part.

s v

n

s- v

It is convenient to express the signal received by the satellite as a function of successive

orders of radiation interactions in the coupled surface-atmosphere system.

For the surface illumination, we have by decreasing magnitudes:

• the direct solar flux attenuated by the atmosphere

/ se


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• the scattered flux on the first sun-surface path; it is independent from surfaceproperties and will be noted by a diffuse transmittance factor td( s) defined as

td( s )Esol

diff ( s)

s Es

where Esoldiff represents the downward diffuse solar irradiance

• a second scattered flux due to the trapping mechanism; it is depending on the

environment of the target and corresponds to the successive reflections and scattering

between the surface and the atmosphere. If the spherical albedo of the atmosphere is

noted S, we can write this terms as

e / s td ( s) t S t2 S2 ...

The total normalized illumination at the surface level is then written

T( s) / [1 t S] (2)

where T( s) is the total transmittance:

T( s) e / s td( s ) (3)

At the satellite level, the radiance results from:

• the contribution of the total (direct plus diffuse) solar radiation reflected by the surface

and directly transmitted from the surface to the sensor expressed by e- / v with v =

cos( v)

∆ω


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• the intrinsic atmospheric radiance expressed in terms of reflectances by function

a( s, v, s- V)

• the contribution of the environment which reflects the total (direct + diffuse) flux, the

photons reaching the sensor by scattering; we note this new atmospheric diffuse

transmittance td' ( v )

So, the apparent reflectance * at the satellite level can be expressed as

*( s , v , s v ) a( s , v, s v )T( s )

1 t S( t e / v

t td' ( v )) (4)

In actual fact, the functions td( s) and t'd( v) are identical in accordance with the reciprocity

principle and * can be rewritten

*( s , v , s v ) a( s , v, s v ) t

1 t ST( s)T( v ) (5)

with

T( v ) e / v td ( v ) (6)

2.2 Environment function

Let us assume that the surface reflectance is now not uniform. In a first approach, we shall

consider a small target M of reflectance c(M) with an uniform environment of reflectance e(M). In

this case, Eq.(4) remains interesting by its formalism, it gives the weight of each reflectance and *

can be now written as

*( s , v , s v ) a( s , v, s v )T( s)

1 e S( c (M)e / v

e (M)td ( v )) (7)


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If we now consider a patchy structure, we keep the same formalism as in Eq.(7), we only

have to define a new environment reflectance noted < (M)> which represents a spatial average of

each pixel reflectance over the whole surface.

*( s , v , s v ) a( s , v, s v )T( s)

1 (M) S( c (M)e / v (M) td ( v )) (8)

This spatial average must be weighted by an atmospheric function which takes into account

the efficiency of a point M' according to the distance from the point M, so < (M)> will be given by

(M)1

td ( v )'(x,y) e(x,y, v )dx dy (9)

M(0,0)

•M'(x,y)

x

y

where

• M is the origin of the x, y coordinate system;

• '(x,y) is the reflectance of the point M'(x,y);

• e(x,y, V) is the contribution to the diffuse transmission td( v) per unit area of an isotropic

unit source placed at M'(x,y).

The use of the same average reflectance < (M)> for the high orders of interaction (that is

the geometrical series) is incorrect but reasonable because in most cases the contribution from these

terms does not exceed 10-15% of the contribution of the first order term.

If we restrict the modeling to the case of vertical observations (or nearly, v<30°), Eq.(9)

can be written


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(M) '(r, )rp(r)drd00

2

(10)

with

r p(r)re(r, ,0)

td (0)(11)

where (r, ) are the polar coordinates of a given surface point with object pixel M at the origin. For

off-nadir observation, we will see in th subroutine ENVIRO how to handle the problem.

Actually, it is impossible to solve this problem exactly, using Eq. (10) because of the

computational time involved. Generally, we can define a characteristic size r of the target whose

reflectance can be assumed uniform. By the same way, we can compute the environment reflectance

e reflected by a simple arithmetical mean and we estimate < (M)> in the following manner.

Considers a circular target of radius r and reflectance c surrounded by homogeneous surface of

reflectance e. Eq.(10) becomes:

(M) c F(r) (1 F(r)) e (12)

with

F(r) 2 r'p(r')dr'0

r

(13)

which gives the relative contribution to < (M)> of surface points not farther than a distance r apart

from the origin.

2.3 Intrinsic atmospheric reflectanceThe intrinsic atmospheric reflectance observed over black target, r+ a is written here as the

sum of aerosols and Rayleigh contributions. This decomposition is not valid at short wavelengths

(less than 0.45 m) or at large sun and view zenith angles.


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2.3.1 Rayleigh

Atmospheric reflectance

For isotropic scattering, Chandrasekhar (1960) showed how solutions derived for small

optical thicknesses may be extended to larger values of . He expressed the atmospheric reflectancea( s v, v- s as

a( s , v, v s ) a1( s , v, v s ) (1 e / s )(1 e / v ) ( ) (14)

where a1( s , v, v s ) is the single-scattering contribution and the second term accounts

roughly for higher orders of scattering.

In 6S, we used this approach to compute the molecular scattering reflectance.

Transmission function

The transmission function refers to the normalized flux measured at the surface. There are

several approximate expressions based on the two-stream methods for computing the transmitted

flux. The accuracy of these expressions depends on the scattering properties of the atmospheric

layer (thick or thin clouds or aerosols) and on the geometrical conditions. The delta-Eddington

method, which has been proved to be well suited for our conditions, has been selected. Since

molecular scattering is conservative ( 0=1) and the anisotropy factor g is equal to zero, we may

write

T( )[2 / 3 ] [2/ 3 ]e /

4 / 3(15)

where is the cosine of the solar and/or observational zenith angle and is the optical thickness.

Spherical albedo

In conservative cases such as molecular scattering, the spherical albedo s is given by

s T( ) d0

1

(16)

where T( ) has been given in Eq.(15). Using Eqs.(15) and (16), the spherical albedo can be written as

s1

4 3[ E3( ) 6E4( )] (17)

where E3( ) and E4( ) are exponential integrals for the argument .


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2.3.2 aerosol

For the aerosols, the optical scattering parameters were computed using pre-defined models,

Continental, Maritime, Urban or user's model based on a mixture of 4 basic components, Dust-Like,

Oceanic, Water-Soluble, and Soot (provided by the World Climate Program). In 6S, these options

are still usable but we added 2 news models, the Background Desertic Model and the Stratospheric

model, and we also offer to the user the possibility of making up his own aerosol model (4

components maximum, see the description of the subroutine MIE for details).

In 5S, the scattering properties pertaining to the aerosol layer was performed using the

Sobolev (1975) approximation for the reflectance, Zdunkowsky et al. (1980) for the transmittance

and a semi-empirical formula for the spherical albedo. The reason was to provide the user having

limited computing ressources with a fast approximation. The drawbacks to using these

approximations were that the accuracy of the computations could be off by a few percent in

reflectance units especially at large view and sun angles or high optical thicknesses. In addition,

these approximations could be completely insufficient to handle the integration of the downward

radiance field with non-lambertian ground conditions, a problem in simulating BRDF. The new

scheme used to compute the aerosol+rayleigh coupled system relies on the Sucessive Order of

Scattering method. The accuracy of such a scheme is better than 1 10-4 in reflectance units. Other

improvements on the 5S model include an atmosphere divided into 13 layers which enables exact

simulations of airborne observations; the downward radiation field is computed for a quadrature of

13 gaussian emerging angles which will provide the necessary inputs for BRDF simulations (see

2.5).

2.4 Airplane and elevated target simulations

2.4.1 Elevated target simulation

For a target not at sea level, Eq.(5) is modified as follows:*( s , v , s v, zt ) Tg( s , v ,zt ) [ r(zt ) a

t

1 S(zt ) tT( v ,zt )T( s ,zt )] (18)

The target altitude or pressure indicates the amount of scatterers above the target (moleculesand aerosols) and the amount of gaseous absorbants. The target altitude zt is handled in the

following manner: After the atmospheric profile and the target altitude or pressure are selected, a

new atmospheric profile is computed by stripping out the atmospheric level above target altitude

and interpolating if necessary. This way, an exact computation of the atmospheric parameters are

computed, without any kind of approximation which will account for coupled pressure-temperature

effect on absorption.


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Gaseous absorption

In most cases, only the integrated content may be modified. The user still has the option toenter the total amount of H2O and O3, but in this case the quantity entered must be representative

of the level measured or estimated at the target location. The influence of target altitude on Tg hasbeen evaluated. The absorption effect by O3 is not sensitive to target altitude, because the ozone

layer is located in the upper levels of the atmosphere, the variation of the altitude of the target doesnot modify this effect. The target's altitude has an important effect on the absorption by H2O

because most of the water vapor is located in the lower atmosphere. However the exact sensitivity

of the target's altitude on water vapor absorption cannot be generalized because water vapor profile

is variable.

Scattering function

The effect of target altitude on molecular optical thickness is accounted for in 6S. A goodapproximation, if needed, is to consider that r is proportionnal to the pressure at target level. As the

amount and types of aerosols are entered as parameters, the aerosol characteristics implicity depend

on target altitude because measured at target location.

2.4.2 Airborne sensor simulation

In case of a sensor inside the atmosphere (aircraft simulation), Eq (5) is modified as

following:*( s , v , s v, z ) Tg( s , v ,z) [ r (z) a (z) t

1 S(z) tT( v,z)T( s ,z)] (19)

Gaseous absorption

Gaseous absorption is computed with a technique similar to the one used in the case of a

target not at sea level. Only, the upward path is modified: the atmosphere level above the sensor

altitude are stripped, so the computation is done only to the altitude of the sensor (interpolation of

the atmospheric profile is conducted if necessary). The effect of altitude on gaseous transmissionhas been computed. For visible wavelength ranges, O3 absorption on the target-sensor path is no

longer accounted for due to the fact these molecules are located above most of aircraft sensors. H2O

absorption is very dependent on sensor altitude up to 4km. If the observed channel is sensitive to

water vapor absorption (as it is in the case of AVHRR channel 2) we recommend that additionnal

measurements of water vapor should be taken from the aircraft. An additional option has been set


21

up in 6S for this purpose and enables the user to enter ozone and water vapor contents (as well as

aerosol content) for the portion of the atmosphere located under the plane.

atmospheric reflectance and transmittance

In most cases the simple approximation "equivalent atmosphere", for atmospheric

reflectance and transmittance is sufficiently accurate, ie:

r(z) r(z , r (0 z)) (20)

T( r , v ,z) T( r(0 z), v, z ) (21)

which means that computations are performed for optical thicknesses corresponding to the

integration between the surface and the sensors.

In 6S, the computation is performed exactly by defining one of the multiple layers used in

the Sucessive Order of Scattering as the alitude of the sensor. This enables exact computation of

both reflectance and transmission term for a realistic mixing between aerosol and Rayleigh.

2.4.3 Non uniform target

In 5S, the case of a non-homogeneous target is solved using Eqs.(8) and (12) where the

function F(r) is defined as

F(r)tdr( v )Fr (r) td

a( v )F a(r)

td( v )(22)

The problem of a target not at sea level, as long as we consider that the target and

environment are at the same altitude, can be solved just by modifying the rayleigh optical thickness.

In the case of aircraft observations, first we have to take into account the reduction of the

amount of scatterers under the plane. This can be done just by adjusting tdr ( v ) and td

a ( v ) to

tdr (z, v ) and td

a (z, v ) . Once this has been done, the principal part of the effect accounted for is a

global reduction (factor 5-10 for an altitude of flight of 6 km) of the "environment effect". The

second effect is the dependence of Fr(r) and Fa(r) upon altitude of the sensor. Monte Carlo

simulation of Fr(r) and Fa(r) have been performed for different altitudes of the sensor (0.5,....12km)


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and included in 6S as a database. In case of a plane observation, the closest simulated altitudes are

used to interpolate the environment function at the aircraft altitude.

2.5 Directional effect of the target

2.5.1. BRDF

In 6S, the coupling of BRDF with the atmosphere is addressed, the contribution of the target

to the signal at the top of the atmosphere is decomposed as the sum of four terms:

• the photons directly transmitted from the sun to the target and directly reflected back to the

sensor

e / st ( s , v , )e / v (23)

s v

= s- v

• the photons scattered by the atmosphere then reflected by the target and directly transmitted

to the sensor,

td( s ) t ( s , v , ) / ve td ( s)

L ( s , , ' )0

1

0

2

t ( , v, ' ) d d '

L ( s , , ')0

1

0

2

d d '

/ ve (24)

s v

'

s

• the photons directly transmited to the target but scattered by the atmosphere on their way to

the sensor,td( v ) t

' ( s , v , ) e/ s td ( v ) t ( v, s , ) e

/ s (25)


23

s v

'

'

v

• the photons having at least two interactions with the atmosphere.td( v ) td ( s ) t td( v ) td ( s ) t

' ( s , v , ) (26)

s

'

'

v

In 6S, the first three contributions are computed exactly using the downward radiation field

as given by the Sucessive Order of Scattering method. The contribution which involves at least two

interactions between the atmosphere and the BRDF (Eq.(26)) is approximated by taking t equal to

the the hemispherical albedo of the target

t s0

1

t ( s ,0

2

v ,0

1

' )d v d d s (27)

This approximation is required because the exact computation needs a double integration

which would result in very long computation times. It is in addition justified by the limited impacton the total signal of this last contribution mitigated by td( s) td( v) and also because the radiation

field after one interaction trends to be isotropic.


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2.5.2. Sunglint

The solar radiation directly reflected by the water surface is computed exactly with the

Snell-Fresnel laws. For a rough sea surface, the reflection is conditioned by the wind and its

description is by necessity a statistical one. Thus, the model surface can be computed numerically

by many facets whose slopes are described by a Gaussian distribution (Cox and Munk, 1954). In

6S, this slope distribution is considered anisotropic (depending upon wind direction). Operationally,

the scheme to compute the reflection by the sunglint is identical to BRDF ones.

2.6 Atmospheric correction scheme.

An input parameter allows to activate atmospheric correction mode. In this case, the ground

is considered to be Lambertian , and as the atmospheric condition are known the code retrieves theatmospherically corrected reflectance value ac that will produce the reflectance (or the radiance)

equal to the apparent reflectance i* (Eq.(1)) (or apparent radiance) entered as input. Following Eq.

5, ac be determined as:

ac ( s, v, s v ) ac'

1 ac' s

(28)

with

ac'

i*( s , v, s v )

Tg a ( s , v , s v )

T( s ) T( v )(29)

where i* is the input parameter and where all the others parameters are computed in the

atmospheric conditions described by the user.

3 Interaction between absorption and scattering effects

The atmospheric radiation depends on atmospheric transmittances which can be modified by

the contribution of absorbing gases. The gaseous absorption must therefore be computed for each

scattering path. We can simplify the problem and separate the two processes.

Let us consider the O3 gas. It is localized at altitudes where molecules and aerosols are

sparse, then the incident and reflected radiation go through the O3 layer nearly without scattering.


25

So, we compute the atmospheric radiation * without accounting for O3. We modify the signal by

the only direct transmittance which corresponds to the twofold paths.

TgO3( s , v ,UO3

) (30)

where UO3 is the absorber amount.

In the case of H2O and CO2, the absorption bands occur in the near and middle infrared. In

this range, the molecular scattering becomes negligible and we only have to take into account the

aerosol scattering. Some simulations, made for realistic atmospheres, show that beyond 0.850µm,

the 1st and 2nd scattering constitute almost the whole diffuse radiation. Because the aerosol phase

function has a large forward scattering, the diffuse paths are not completely separate from direct

paths and we shall consider that the influence of CO2 or H2O upon scattered flux is the same as for

direct solar flux. So the diffuse and transmitted flux will be both reduced by the factor

TgH2O( s , v ,UH2O )TgCO2( s , v,UCO2

) (31)

For O2, the absorption is very localized and presents a very narrow band, so we take into

account this gas by the same way:

TgO2( s , v,UO2

) (32)

We have checked our approximation from a simulation in the visible Meteosat spectral

band. This band is large enough (0.35-1.10 m) to include both scattering effects and water vapor

absorption. To amplify the coupling effect, we have taken a tropical atmosphere model (H2O

content is about 4g/cm2) and a continental aerosol model with large optical depth (τ0.55 m =1). For

∆ν=20cm-1, we have:

s Texact ( S) Test ( s)

1.0 0.837 0.843

0.5 0.692 0.693

0.1 0.401 0.363

Estimation of the accuracy of our approximation

The comparison between exact computations and results obtained from our approach is

reported in the above table for three solar angles. Except for a grazing incidence ( s=0.1), we have a

good agreement and the error related to this approximation is smaller than one percent.


26

For atmospheric reflectance, the problem is slightly different. The diffuse paths are not close

to the direct paths. The photons are scattered at large angles away from the forward direction. For

O3, the same approximation is still justified because it is based upon the weak interaction between

the two processes. For H2O, the scale height is comparable to the aerosols scale height while CO2 is

uniformly mixed. Therefore, for these two gases our approximation is less valid but remains

sufficiently accurate as long as the gaseous absorption is small, i.e. in the atmospheric windows.

For getting the maximum uncertainties resulting from the approximation, we consider 3

extreme cases in 6S: the water vapor above the aerosol layer (maximum absorption, see (33-a) for

i=3), the water vapor under the aerosol layer (minimum absorption, see (33-a) for i=1), and an

average case where we consider that half of the water vapor present in the atmosphere absorbs the

aerosol path radiance (see (33-a) for i=2). In that respect, the satellite signal is written

TOAi 1,3( s , v, s v ) TgOG ( s , v ) [ R A ( R A R)Tg H2O ( s , v , i 1

2 UH2O )

T ( s) T ( v ) s

1 S sTgH 2O ( s , v,UH

2O ) ] (33-a)

where TgOG refers to the gaseous transmission for other gases than water vapor, TgH2O refers to

H2O absorption and R+A- R is an estimate of the aerosol intrinsic reflectance. For each cases, we

compute the top of the atmosphere reflectance, so the uncertainty due to the vertical distribtution of

aerosol versus water vapor can be considered.TgOG is computed on the sun-target-satellite direct

paths using

TgOG ( s , v ) Tgi ( s ,i 1

8

v,Ui ) (33-b)


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CONCLUSION

An accurate analytical expression of the reflectance measured by a satellite-sensor or by a sensor

aboard an aircraft has been established.

• In the case of Lambertian and uniform surface, we completely separate the surface-

atmosphere system and define three atmospheric functions.

- the intrinsic atmospheric signal component

- the total transmittance

- the spherical albedo.

• The coupled BRDF-atmosphere are accounted for according to the same scheme by

definition of mean angular reflectances which are depending on atmospheric properties.

• In the case of a non-uniform ground, we have defined a spatial average reflectance which

remains slightly dependent on atmospheric properties by the definition of the function F(r).

This function allows us to take into account the main blurring atmospheric effect for non-

uniform surfaces.

• In both cases, it is possible to consider elevated targets.

The absorption by atmospheric gases is computed separately as a simple multiplicative factor. The

solar spectrum is divided into spectral intervals of 10cm-1 width.

In addition, a scheme of atmospheric correction is available.

The computational aspect of our work is now detailed in the following appendix:

• Appendix I : General description of the computer code

• Appendix II : Example of inputs and outputs

• Appendix III : Complete description of the subroutines


28

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Fig. I-1: Spectral transmittance of H2O

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Fig. I-2: Spectral transmittance of O3


29

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Fig. I-3: Spectral transmittance of CO2

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Fig. I-4: Spectral transmittance of O2


30

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Fig. I-5: Spectral transmittance of N2O

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Fig. I-6: Spectral transmittance of CH4


31

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0.3

0.4

0.5

0.6

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0.6

0.7

0.8

Fig. I-7: Atmospheric window at 0.40 m Fig. I-8: Atmospheric window at 0.75 m

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0.7

0.8

0.9

1.0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0.9

1.0

1.1

1.2


0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1.1

1.2

1.3

1.4

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1.4

1.5

1.6

1.7

1.8

1.9


0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

2.8

3.0

3.2

3.4

3.6

3.8

4.0



32


33

ACKNOWLEDGEMENTS

The authors would like to thank Nazmi El Saleous from the GIMMS group for his help in putting

together this manual, Lorraine Remer from Code 923 for having kindly read the first draft of this

manual and help us to improve it. They are very grateful to Dr. Tucker and Justice from

GSFC/UMD for the logistic support essential to the realization of 6S. Finally, special thanks to all

the Beta tester of the code and in particular to Phil Teillet from Canadian Centre for Remote

Sensing for his interesting and useful remarks. The Laboratoire d'Optique Atmosphérique is

sponsored by CNRS (Centre National de Recherche Scientifique) and by CNES (Centre National

d'Etudes Spatiales).


34


35

APPENDIX I: DESCRIPTION OF THE COMPUTER CODE

This code predicts the satellite signal between 0.25 and 4.0 µm assuming cloudless

atmosphere. Let us recall that the apparent reflectance at the satellite level dor Lambertian surface

can be written,

'( s , v , v ) tg( s , v ) a ( s , v , v )T( s)

1 (M) S[ c (M)e / v (M) td ( v )]

This formalism takes into account the main atmospheric effects, gaseous absorption by H2O,

O2, CO2 and O3, scattering by molecules and aerosols and a non-homogeneous ground may be

considered. The general flow chart for 6S program is reported in Figure A-1

The following input parameters are necessary:

• geometrical conditions,

• atmospheric model for gaseous components,

• aerosol model (type and concentration),

• spectral condition,

• ground reflectance (type and spectral variation).

At each step, you can either select some proposed standard conditions or define your own

conditions. More details are given in Figures A-2 to A-9. Let us note that the satellite navigation is

done by a rough calculation from the nominal characteristics of the satellite orbit. It is obvious that,

without anchor points, the localization is not very accurate.

From these data, we compute the scattering atmospheric functions, at ten wavelengths which

are well distributed over the solar spectrum (Fig. A-11). At any wavelength, we interpolate from

these values by assuming the spectral dependance

f ( ) = f ( 0)0

−

and compute the apparent reflectance from the first equation (Fig.A-10)

We have reported on Fig. A-12 the general outputs and the optional ones.


36

Geometrical Conditions

Aerosol Model (type)P(θ,λ), g(λ), ωo(λ)

Aerosol Concentration

τA(550 nm)

Spectral Bandλinf, λsup, S(λ)

Atmospheric ConditionsP(z), T(z), UH2O(z), UO3(z)

Ground Reflectance

Computation of the Atmospheric functionsat discrete wavelength values

Integration over the spectral band

Computation of the gaseous absorption functions

Solar Spectral Irradiance

Interpolation of the scattering atmospheric functions

Environment reflectance

Computation of the apparent reflectance

Outputs

Computation of BRDF Atmospherecoupling at equivalent wavelength

Figure A-1 : General flow chart 6S computations

Atmospheric correction if requested


37

IGE

OM

θ s, φ

s, θ

v, φ

v, M

onth

, Day PO

SMT

OPO

SGE

POSG

WPO

SNO

APO

SSPO

POSL

AN

Non

e

12

30

67

45

θ s, φ

s, θ

v, φ

v

Sate

llite

Dat

a

Subr

outin

e

Figu

re A

-2

: D

etai

led

Flow

Cha

rt f

or G

eom

etri

cal C

ondi

tions

Non

eM

eteo

sat

Goe

s E

ast

Goe

s W

est

NO

AA

AM

NO

AA

PM

SPO

TL

ands

at

Mon

th, D

ay, H

our,

Nb.

col

s, N

b. r

ows

Mon

th, D

ay, H

our,

Lon

g, L

atM

onth

, Day

, Hou

r, N

b. c

ols,

Lon

g A

.N.,

Hou

r A

.N.


38

IDA

TM

01

23

45

67

8

Non

eSU

BSU

MSU

BW

INN

one

US6

2

P(z)

, T(z

), U

H2O

(z),

UO

3(z)

Subr

outin

e

Figu

re A

-3

: D

etai

led

Flow

Cha

rt f

or A

tmos

pher

ic C

ondi

tions

US6

2M

IDW

INM

IDSU

MT

RO

PIC

No

abso

rptio

nT

ropi

cal

Mid

latit

ude

Sum

mer

Mid

latit

ude

Win

ter

Suba

rctic

Sum

mer

Suba

rctic

Win

ter

US6

2U

SER

USE

RT

ype

Non

eN

one

Non

eN

one

Non

eN

one

Non

eU

H2O

,UO

3D

ata

P(z)

,T(z

),U

H2O

(z),

UO

3(z)


39

IAE

R

01

23

45

Non

e

Kex

t (λ)

, P(θ

,λ),

g(λ

), ω

0(λ)

Figu

re A

-4

: D

etai

led

Flow

Cha

rt f

or A

eros

ol M

odel

(T

ype)

Subr

outin

eNo

Aer

osol

Con

tinen

tal

Mar

itim

eU

rban

Des

ertic

Typ

eB

iom

ass

Stra

tosp

heri

cU

SER

usi

ngsi

ze d

istr

ibut

ion

USE

R u

sing

basi

c co

mpo

nent

s

Non

eN

one

Non

eN

one

Non

eC

(i)

i=1,

4D

ata

67

8 to

12

Non

eN

one

AE

RO

SOA

ER

OSO

AE

RO

SOA

ER

OSO

AE

RO

SOA

ER

OSO

AE

RO

SOA

ER

OSO

cf fi

gure

A-4

'


40

Subr

outin

e

Kex

t (λ)

, P(θ

,λ),

g(λ

), ω

0(λ)

Figu

re A

-4'

: D

etai

led

Flow

Cha

rt f

or A

eros

ol M

odel

(T

ype)

Typ

e

Dat

a

89

1011

12

Mul

ti M

odal

Log

-Nor

mal

edi

stri

butio

n(s)

(up

to 4

)

Mod

ifie

d G

amm

adi

stri

butio

nJu

nge

Pow

er-L

awdi

stri

butio

nFr

om S

un-p

hoto

met

erm

easu

rem

ents

From

an

exte

rnal

file

IAE

R-U

SER

with

siz

e di

stri

butio

n

r min

,rm

axα,

b, γ

r n(λ

), λ

=1,

10

r i(λ

), λ

=1,

10

r min

,rm

axα r n

(λ),

λ=

1,10

r i(λ

), λ

=1,

10

MIE

+A

ER

OSO

Res

ults

sav

ed in

to th

e fi

le F

ILE

.mie

IAE

RP

01

Typ

e

Dat

aFI

LE

(fi

le n

ame)

Non

e

irsu

nph

for

i=1

to ir

sunp

hr(

i), (

dV/d

log(

r))(

i)do

ner n

(λ),

λ=

1,10

r i(λ

), λ

=1,

10

r min

, rm

ax, i

cpfo

r i=

1 to

icp

r M(i

), σ

(i),

Cij(

i)r n

(λ,i)

, λ=

1,10

r i(λ

,i),

λ=1,

10

done

nam

e

of th

e ex

tern

al fi

le


41

V

Non

eτA

(550

)N

one

-10

Any

pos

itive

val

ueV

>5k

m

Non

eN

one

OD

A55

0

τA(5

50)

Dat

a

Subr

outin

e

Figu

re A

-5

: D

etai

led

Flow

Cha

rt f

or A

eros

ol M

odel

(C

once

ntra

tion,

τA

(550

))


42

XPS

Posi

tive

Val

ue

Tar

get a

t sea

leve

l

Neg

ativ

e V

alue

Tar

get a

t the

altit

ude

= -

xps

(in

km)

PRE

SSU

RE

UH

2O, U

O3,

τ aer

(550

),P(

z),T

(z)

Figu

re A

-6

: D

etai

led

Flow

Cha

rt f

or th

e ef

fect

of

the

altit

ude:

Tar

get

Subr

outin

eN

one

Tar

get

Alti

tude


43

-100

< x

pp <

0

Sens

or in

side

the

atm

osph

ere

xpp

= -

alti

tude

(in

kilo

met

ers)

XPP

-100

0

Sens

or a

boar

d a

Sate

llite

Dat

a

Subr

outin

e

Dat

a

Sens

or

Alti

tude

UH

2O, U

O3

belo

w s

enso

r

PRE

SPL

AN

E

if >

0

or

i

f <

0

τ aer

(550

) be

low

sen

sor

UH

2O, U

O3,

P(z)

,T(z

)U

H2O

, UO

3,P(

z),T

(z)

usin

g U

S 62

if >

0

orif

<0

PRE

SPL

AN

E +

US

62

τ aer

(z)

τ aer

(z)

assu

min

g sc

ale

heig

h of

2km

0

Sens

or a

tG

roun

d le

vel

defi

ned

by x

ps

UH

2O, U

O3,

τ aer

(550

),P(

z),T

(z)

Figu

re A

-6'

: D

etai

led

Flow

Cha

rt f

or th

e ef

fect

of

the

altit

ude:

Sen

sor

Non

e

Non

e

Non

e

Non

e

Non

e

Non

e


44

IWA

VE

12

3 -

425

- 3

0

λ inf

, λsu

p, S

(λ)

Subr

outin

e

Figu

re A

-7

: Det

aile

d Fl

ow C

hart

for

Spe

ctra

l Ban

d D

efin

ition

of

Ext

rem

e W

avel

engt

hs a

nd F

ilter

Fun

ctio

n

31 -

34

TM

HR

V

17 -

24

MSS

AV

HR

R

5 -

16

GO

ES

ME

TE

ON

one

Non

e

0

Non

e

-1-2

Non

eN

one

Non

eD

ata

Non

eN

one

Non

eλ

Non

e

outp

ut p

rint

edst

ep b

y st

ep (

2.5n

m)

λ inf

, λsu

pfo

r = inf,

sup,

2.5n

m

S(λ)

done

λ inf

, λsu

pw

ith

S()=

1.0

λ inf

, λsu

pw

ith

S()=

1.0


45

IHO

MO

01

non

hom

ogen

eous

IDIR

EC

T

IGR

OU

NIB

RD

F

-10

12

3

IGR

OU

1R

AD

IGR

OU

2

-11

43

20

R in

km

none

none

VE

GE

TA

CL

EA

RW

SAN

DL

AK

EW

VE

GE

TA

CL

EA

RW

SAN

DL

AK

EW

none

none

01

none

R (

in k

m)

Figu

re A

-8

: D

etai

led

Flow

Cha

rt f

or G

roun

d R

efle

ctan

ce, D

efin

ition

of

the

Tar

get a

nd E

nvir

onm

ent R

efle

ctan

ces

Subr

outin

e

Dat

a

cf fi

gure

A-8

'

ρ c(λ

),ρ e

(λ)

( ),

d( ),

p( ))

4

for

= inf,

sup,

d

ρ(λ)

wit

h d

=2.

5nm

ρno

neno

neno

neno

ne

hom

ogen

eous

none

none

none

none

ρ c,e

ρ c =

ρe

for

= inf,

sup,

d

ρ c,e

(λ)

wit

h d

=2.

5nm


46

IBR

DF

01

23

45

67

ρ(θ v

j ,ϕvi )

for

θ sρ(

θ vj ,ϕ

vi ) fo

r θ s

=θ v

sphe

rica

l alb

edo

(with

j:00

, 10

... 8

0, 8

5°

i

:00,

30,

... 3

60°)

(see

text

for

det

ails

)

ω,Θ

,S(0

),h

opt3

opt

4 op

t5st

r1 s

tr2

str3

str

4op

c.1

opc.

2 op

c.3

(see

text

for

det

ails

)

k 0,k

1,k 2

a,a'

,b,c

k,ρ l

ws,

ϕ w,C

sal,C

ild,ih

sL

t,2rΛ

r L,t L

,Rs

in-s

itum

easu

rem

ents

Hap

ke's

mod

elV

erst

raet

e's

mod

elR

ouje

an's

mod

elW

alth

all's

mod

elM

inna

ert's

mod

elO

cean

mod

elIa

quin

ta &

Pin

ty's

mod

el

ρ(θ v

j ,ϕvi )

for

θ sρ(

θ vj ,ϕ

vi ) fo

r θ s

=θ v

sphe

rica

l alb

edo

(with

j =

00°

to 9

0°

i =

00°

to 3

60°)

BR

DFG

RID

HA

PKB

RD

FH

APK

AL

BE

VE

RSB

RD

FV

ER

SAL

BE

RO

UJB

RD

FR

OU

JAL

BE

WA

LT

BR

DF

WA

LT

AL

BE

MIN

NB

RD

FM

INN

AL

BE

OC

EA

BR

DF

OC

EA

AL

BE

IAPI

BR

DF

IAPI

AL

BE

Dat

a

Subr

outin

e

Typ

e

Figu

re A

-8'

: Det

aile

d Fl

ow C

hart

for

BR

DF

8

Rah

man

'sm

odel

ρ 0,Θ

,k

RA

HM

BR

DF

RA

HM

AL

BE


47

RAPP

atmospherically corrected reflectance

Figure A-9 : Detailed Chart Flow for Atmospheric correction mode

RAPP <-1 -1 < RAPP < 0 RAPP > 0

No Atmospheric Correction

---- Atmospheric Correction mode ----

LTOA = RAPP (in W/m2/st/µm)

ρTOA = - RAPP

using atmospheric conditions previously described by the user


48

DIS

CO

M

OD

RA

YL

AT

MR

EF

SCA

TR

A

Figu

re A

-10

: D

etai

led

Flow

Cha

rt f

or D

iscr

ete

Com

puta

tions

of

the

Scat

teri

ng A

tmos

pher

ic F

unct

ions

, ρ, T

(θ)

and

S


49

Loop over wavelength

ABSTRA

SOLIRR

INTERP

ENVIRO

OUTPUTS

VARSOL

Figure A-11 : Detailed Chart Flow for Spectral Integration


50

INT

EG

RA

TE

D V

AL

UE

S O

F

CO

UPL

ING

AE

RO

SOL

-WA

TE

R V

APO

R

INT

EG

RA

TE

D N

OR

MA

LIZ

ED

VA

LU

ES

OF

% I

rrad

ianc

e at

gro

und

leve

l

Ref

lect

ance

at s

atel

lite

leve

l

Fig

ure

A-1

2 : D

escr

iptio

n of

the

outp

uts

(1/3

)

% D

irec

t Irr

.%

dif

fuse

Irr

.%

Env

iro.

Irr

.

S ()E

ss

d

S ()E

sd

S() E

st d

(s)[

1s

]

T(

s)d

S(

) Esd

S ()E

se

/s

[1s

]

T(

s)

d

S ()E

sd

Atm

. int

rin.

ref

.E

nvio

nmen

t ref

.T

arge

t ref

lect

ance

S() E

sTg

(s,

v)T

(s)e

/v

1s

d

S() E

sd

S() E

sTg

(s,

v)T

(s)t

d(

v)

1s

d

S() E

sd

S ()E

sTg

(s,

v)

ad

S ()E

sd

wv

abov

e A

er.:

wv

mix

ed w

ith A

er.:

wv

unde

r A

er.:

S ()E

s(1

) d

S ()E

sd

S() E

s( 2

)d

S() E

sd

S() E

s'

d

S() E

sd

App

aren

t Ref

lect

ance

:A

ppar

ent R

adia

nce:

Tot

al g

aseo

us T

rans

mitt

ance

:S(

) Es

'd

S() E

sd

1S(

) Es

'd

1S(

) EsT

gd

(1)

=T

g

Tg U

H2

O

R+

A+

(R

+A

−R

)Tg U

H2O

+s*

with

'=

Tg

Tg U

H2

O

R+

A+

(R

+A

−R

)Tg

1 2U

H2

O+

s*

( 2

)=

Tg

Tg U

H2O

R+

A+

s*(

)

whe

res*

=T

↓(

s)T

↑(

v)

s1

−s

sT

g UH

2O


51

AB

SOL

UT

E V

AL

UE

S O

F

Irra

dian

ce a

t gro

und

leve

l

(W

/m2 )

Rad

ianc

e at

sat

ellit

e le

vel

(W

/m2 /

st/µ

m)

Fig

ure

A-1

2: D

escr

iptio

n of

the

outp

uts

(2/3

)

Atm

. int

rin.

rad

.E

nvio

nmen

t rad

ianc

eT

arge

t rad

ianc

e

1S (

)EsT

g(

s,v)

a d1

S(

)EsT

g(

s,v

)T(

s)t

d(

v)

1s

d1

S(

)EsT

g(

s,v

)T(

s)e

/v

1s

d

Dir

ect S

olar

Irr

.A

tm. d

iffu

se I

rr.

Env

iro.

Irr

adia

nce

S(

) EsT

g(

s)T

(s

)s

1s

dS(

) EsT

g(

s)t

d(

s) d

S() E

sTg

(s

)e/

sd

Int.

Func

tion

filte

r (i

n µm

)In

t. So

l. Sp

ect.

(in

W/m

2 )S

() d

∫S

()

sE

d∫


52

D

OW

NW

AR

DU

PWA

RD

TO

TA

L

Gas

Tra

ns:

H2O

,CO

2,O

3,O

2

N2O

,CH

4,C

O

Ray

. Tra

ns.

Aer

. Tra

ns.

Tot

. Tra

ns.

Ray

leig

hA

eros

ols

Tot

al

Sphe

rica

l Alb

edo

SA

ST

Opt

ical

Dep

thA

Ref

lect

ance

Phas

e Fu

nctio

nP

A

Sing

le S

catte

ring

Alb

edo

1.0

Fig

ure

A-1

0: D

escr

iptio

n of

the

outp

uts

(3/3

)

S ()

sE

Tg i

i=17

∏∫

(s)

d

S ()

sE

∫d

S ()

sE

Tg i

i=17

∏∫

(v)

d

S ()

sE

∫d

S ()

sE

Tg i

i=17

∏∫

(s,

v)d

S ()

sE

∫d

S ()

sE

T(

s)

∫d

S ()

sE

∫d

S ()

sE

T(

v)

∫d

S ()

sE

∫d

S ()

sE

T(

s)

∫T

(v)

d

S ()

sE

∫d

S ()

SE

SRd

∫S (

)S

Ed

∫ S ()

SE

Rd

∫S (

)S

Ed

∫ S ()

SE

Rd

∫S (

)SE

d∫ S (

)S

EPR

d∫

S ()

SE

d∫

RPR

+AP

A

R+

A

0A

A+

R R

R+

0AA

R+

A


53

APPENDIX II

EXAMPLE OF INPUTS AND OUTPUTS

INPUT FILE 0 (USER'S CONDITIONS) 40.0 100.0 45.0 50.0 7 23 8 (USER'S MODEL) 3.0 0.35 (UH2O(G/CM2) ,UO3(CM-ATM)) 4 (AEROSOLS MODEL) 0.25 0.25 0.25 0.25 (% OF:DUST-LIKE,WATER-SOL,OCEANIC,SOOT) 0 (NEXT VALUE IS THE AER. OPT. THICK. @550) 0.50 (AERO. OPT. THICK. @550) -0.2 (TARGET AT 0.2 km) -3.5 (AIRCRAFT AT 3.5 KM) -1.5 -0.35 (UH2o and UO3 under the plane not avail.) 0.25 (AERO. OPT. THICK. under the plane @550) 11 (AVHRR 1 (NOAA 9) BAND) 1 (GROUND TYPE,I.E. NON UNIFORME SURFACE) 2 1 0.50 (TARGET,ENV.,RADIUS(KM)) -0.10 (ATMOSPHERIC CORRECTION OF RAPP=0.10)


54

OUTPUT FILE (1/3)******************************* 6s version 4.0 ******************************** geometrical conditions identity ** ------------------------------- ** user defined conditions ** ** month: 7 day : 23 ** solar zenith angle: 40.00 deg solar azimuthal angle: 100.00 deg ** view zenith angle: 45.00 deg view azimuthal angle: 50.00 deg ** scattering angle: 146.49 deg azimuthal angle difference: 50.00 deg ** ** atmospheric model description ** ----------------------------- ** atmospheric model identity : ** user defined water content : uh2o= 3.000 g/cm2 ** user defined ozone content : uo3 = .350 cm-atm ** aerosols type identity : user defined aerosols model ** .250 % of dust-like ** .250 % of water-soluble ** .250 % of oceanic ** .250 % of soot ** optical condition identity : ** visibility : 8.49 km opt. thick. 550nm : .5000 ** ** spectral condition ** ------------------ ** avhrr 1 (noaa9) value of filter function ** wl inf= .530 mic wl sup= .810 mic ** ** target type ** ----------- ** inhomogeneous ground , radius of target .500 km ** target reflectance : ** spectral clear water reflectance .045 ** environmental reflectance : ** spectral vegetation ground reflectance .129 ** ** target elevation description ** ---------------------------- ** ground pressure [mb] 989.01 ** ground altitude [km] .200 ** gaseous content at target level: ** uh2o= 3.000 g/cm2 uo3= .350 cm-atm ** ** plane simulation description ** ---------------------------- ** plane pressure [mb] 657.54 ** plane altitude [km] 3.500 ** atmosphere under plane description: ** ozone content .008 ** h2o content 1.043 ** aerosol opt. thick. 550nm .292 ** ** atmospheric correction activated ** -------------------------------- ** input apparent reflectance : .100 ** ********************************************************************************


55

OUTPUT FILE (2/3)******************************************************************************** ** integrated values of : ** -------------------- ** apparent reflectance .0457 appar. rad.(w/m2/sr/mic) 17.623 ** total gaseous transmittance .937 ** ********************************************************************************* ** coupling aerosol -wv : ** -------------------- ** wv above aerosol : .046 wv mixed with aerosol : .046 ** wv under aerosol : .046 ** ** int. normalized values of : ** --------------------------- ** % of irradiance at ground level ** % of direct irr. % of diffuse irr. % of enviro. irr ** .791 .204 .006 ** reflectance at satellite level ** atm. intrin. ref. environment ref. target reflectance ** .021 .015 .014 ** ** int. absolute values of ** ----------------------- ** irr. at ground level (w/m2/mic) ** direct solar irr. atm. diffuse irr. environment irr ** 617.564 158.453 4.381 ** rad at satel. level (w/m2/sr/mic) ** atm. intrin. rad. environment rad. target radiance ** 7.935 5.846 5.565 ** ** ** int. funct filter (in mic) int. sol. spect (in w/m2) ** .1174545 185.589 ** ********************************************************************************


56

OUTPUT FILE (3/3)******************************************************************************** ** integrated values of : ** -------------------- ** ** downward upward total ** global gas. trans. : .94428 .98766 .93740 ** water " " : .98570 .99322 .98115 ** ozone " " : .96462 .99909 .96375 ** co2 " " : 1.00000 1.00000 1.00000 ** oxyg " " : .99343 .99533 .99178 ** no2 " " : 1.00000 1.00000 1.00000 ** ch4 " " : 1.00000 1.00000 1.00000 ** co " " : 1.00000 1.00000 1.00000 ** ** ** rayl. sca. trans. : .96434 .98767 .95244 ** aeros. sca. " : .70714 .80488 .56916 ** total sca. " : .67887 .79455 .53940 ** ** ** ** rayleigh aerosols total ** ** spherical albedo : .04966 .04532 .06717 ** optical depth total: .05550 .41782 .47332 ** optical depth plane: .01848 .24364 .26213 ** reflectance : .01081 .01323 .02171 ** phase function : 1.26026 .30427 .41637 ** sing. scat. albedo : 1.00000 .48167 .54245 ** ** ********************************************************************************

******************************************************************************** atmospheric correction result ** ----------------------------- ** input apparent reflectance : .100 ** measured radiance [w/m2/sr/mic] : 38.529 ** atmospherically corrected reflectance : .156 ** coefficients xa xb xc : .00513 .04025 .06717 ** y=xa*(measured radiance)-xb ; acr=y/(1.+xc*y) ********************************************************************************


57

APPENDIX III

DESCRIPTION OF THE SUBROUTINES

Contents

Description of the differents subroutines used to compute the geometrical conditions:

DAY_NUMBER 76

POS_FFT 76

POSGE 63

POSGW 64

POSLAN 65

POSMTO 66

POSNOA 72

POSSOL 76

POSSPO 80

Description of the subroutines used to compute the different atmospheric functions:

ABSTRA 83

AEROSO 91

ATMREF 97

CHAND 99

CSALBR 100

DISCOM 101

DISCRE 102

ENVIRO 103

EXSCPHASE 112

GAUSS 108

INTERP 109

ISO 110

KERNEL 111


58

MIE 112

ODA550 131

ODRAYL 133

OS 135

SCATRA 138

TRUNCA 140

Description of the subroutines used for BRDF ground

BRDFGRID 145

FRESNEL 155

GAULEG 148

GLITALBE 144

HAPKALBE 143

HAPKBRDF 146

IAPIALBE 144

IAPIBRDF 148

IAPITOOLS 148

INDWAT 155

LAD 148

MINNALBE 144

MINNBRDF 152

MORCASIWAT 156

OCEAALBE 144

OCEABRDF 153

OCEATOOLS 153

RAHMALBE 144

RAHMBRDF 163

ROUJALBE 144

ROUJBRDF 165

SOLVE 148

SUNGLINT 154

VERSBRDF 167

VERSALBE 144

VERSTOOLS 167

WALTALBE 144


59

WALTBRDF 169

Description of the subroutines used to update atmospheric profile (airplane, elevated target):

PRESPLANE 175

PRESSURE 176

Description of the different subroutines used to read the data:

AVHRR 181

BBM 201

BDM 201

CLEARW 190

DICA 1 to 3 194

DUST 201

GOES 184

HRV 185

LAKEW 191

METEO 187

METH 1 to 6 195

MIDSUM 202

MIDWIN 203

MOCA 1 to 6 196

MSS 188

NIOX 1 to 6 197

OCEA 201

OXYX 3 to 6 198

OZON 1 199

SAND 192

SOLIRR 179

SOOT 201

STM 201

SUBSUM 204

SUBWIN 205


60

TM 189

TROPIC 206

US 62 207

VARSOL 180

VEGETA 193

WATE 201

WAVA 1 to 6 200

Miscellaneous

EQUIVWL 211

PRINT_ERROR 212

SPECINTERP 213

SPLIE2, SPLIN2, SPLINE, SPLINT 214


61

DESCRIPTION OF THE SUBROUTINES USED

TO COMPUTE THE GEOMETRICAL CONDITIONS


62


63

SUBROUTINE POSGE

Function: Same as POSMTO but for GOES East satellite. We use exactly the same scheme,

only we add the longitude of the subsatellite point, namely 75W, at the retrieval longitude. Let us

also recall that the dimension of the frame is 17331 × 12997 and the altitude of the satellite is 35729

km.


64

SUBROUTINE POSGW

Function: Same as POSMTO but for GOES West satellite. We use exactly the same scheme

but we add the longitude of the subsatellite point, namely 135W, at the retrieval longitude. Let us

recall that the dimension of the frame is 17331 × 12997 and the altitude of the satellite is 35769 km.


65

SUBROUTINE POSLAN

Function: To compute the geometrical conditions for the LANDSAT satellite. As the

dimensions of the frame are 180 ×180 km, the maximum observation angle is 5.5°, so we put θV =

0. The incident conditions are taken from the latitude and the longitude of the centre of the scene.

Reference:

NASA, 1981, GSFC specification for the Thematic Mapper Subsystem and associated test

equipment. Revision C., GSFC 400-8-D.210C, NASA/GSFC, Greenbelt, Maryland, U.S.A..


66

SUBROUTINE POSMTO

Function: To compute the geometrical conditions from the knowledge of the line number and

the pixel in the line (in Meteosat Frame 2500*5000). Firstly, we compute the latitude and the

longitude of the pixel, so the solar position (with the time conditions) can be computed and

secondly, the observation angle.

y

NL

NC

M

W

N

S

P

5000

2500

x

S

x

y

Description: Let S be the satellite, M the subsatellite point, P the observed point, and the

orientation of the axes according to Fig. 1.

From Nc and N1, we obtain the two angles X, Y with respect to x and y axes.

If we refer now to the plane containing the points S, P, M and O the center of the earth (cf.

Fig. 2), we put H the altitude of the satellite and RE the earth radius in Equatorial plane.

The observed point P corresponds to the point P' on the earth surface and we have to determine

its coordinates x, y, z with respect to the axis-system centered at O.


67

Figure 2

θ

z

O

H

S

M

NPP'

RE

To obtain z, we put,

z = ON = (RE+H) - SN,

SN is obtained from the solutions of triangles OP'S and SNP'.

• By solving OP'S, we have

SP'2 - 2SP' cosθ OS = OP'2 - OS2 ,

so

SP' = cos(θ)(H+RE)- Error!

• By solving SNP' we have SN = SP' cos θ,

then SN=cos(θ)2 (H+RE)-cos(θ) (RE+H)2(cos(θ)2-1)-RE2

Therefore :

z = RE + H − cos θ( )2H + RE − RE

RE + H

RE

2

−

RE + H

RE

2

−1

cos(θ)2

To estimate cosθ, we use the deviations X and Y. A simple trigonometrical identity shows

that :

cos2θ=cos2X.cos2Y

where


68

cos2X=1

1+tan2X

and

cos2Y=1

1+(tanY(1+ε))2

with

ε=RE-RP

RE =

1297 ,

RE and RP being respectively the equatorial and polar radius, slightly different because of the

earth's oblateness.

O

y

x

z x1

P"yχ

λ

φ

x

z

N

y

To obtain x and y, we consider the Figure 3.

x = -SN tanY

and

y = SN/cosX.tanY

So we have the three coordinates x, y, z of the point P' and infer latitude φ and longitude λ .

To compute the longitude, we use, λ= arctan (x/z) (+ for East, - for West).

To compute the latitude, we have to consider the geoide with semiminor axis Rp and

semimajor axis RE (Figure 4) by solving triangle P'OP".As φ = atan(y/x1)where y is the ordinate of

the point located on the ellipsoïd, so we have to compute x1.


69

Figure 4

Rp

ψφ

x1 REx

y

The ellipse equation is written:

x21

R2E

+y2

R2P

=1

so

φ = atan (y

R2P-y2(RP/RE) ),

φ = atan ( tan ψ(RP/RE))

with ψ= asin(y/RP).

To obtain observations angles (azimuthal and zenithal), we use the following simple

geometrical considerations.

For the zenithal angle θv,


70

Figure 5 R

θvP'

S

Hθ

θv = asin ( (1+hR ) sinθ)

where θ is so as cos2 θ = cos2X cos2Y.

For the azimuthal angle φV, we solve the spherical triangle P'P''M

Figure 6

φv

y

x

z

S

λ

φ


71

where A = φV - π,

with tan A = tan λ (1/sin φ),

S

λ

φA

φv

so φV = arctan[ tan λ(1/sinφ)] + π.

From the line and column numbers in a MTOSAT Frame, we can compute the latitude (φ)

and the longitude (λ) of the point, and the viewing direction from the normal at the point (azimuthal

φV and zenithal θV angles). Moreover, if we know the date and the hour of the acquisition, we can

obtain the solar conditions (θs, φs) from the subroutine POSSOL.

Reference:

MORGAN, 1981, Introduction to the Meteosat System, ESOC, Darmstadt, Germany.


72

SUBROUTINE POSNOA

Function: To compute the geometrical conditions for the NOAA series satellites. Generally,

we know the pixel number on a line, the longitude and the time of the ascendant node at the

equator, and the time of the acquisition. We obtain latitude and longitude of the viewed point, the

viewing angles and with the knowledge of the date, the solar geometrical conditions.

Description: The altitude of NOAA satellite is about H = 860 km, the orbit inclination

98.96°and the time of one revolution is about 101.98 mn (6119 sec.). The 1/2 angle is of maximum

55.385°and you have 2048 pixels for each line.

Figure 1

Gre. Mer. P

Eq.

S

N

Na

98.96°

Let AN be the hour movement in rad/sec, HN the hour at the ascendant node, λN its longitude

and Nc the pixel number.


73

Figure 2

S

N

Na

NR

Na

i

P

UB

S

G.M.

ψ

Hδ

θv

Consider Figs.2, S is the subsatellite point, N the ascendant node and P the observed point.

The scan angle δ gives an angle noted ψ at the centre of the earth.

By solving the triangle PRN, we have the latitude φP so as:

sin φP = sin (i+B) sin (NP)

Now, in triangle PSN,

sin (NP) = sin ψ / sin B

and

tan (B) = tan ψ/ sin U

So,

sinφP= cos i sinψ +sin i cos ψ sinU

By solving PRN, we obtain the longitude λP with respect to λN,

sin λP = sin NPR sin NP


74

with

sin NPR = cos(i+B)

cosφP ,

so, we write

sinλP=cos(i+B)sinNP

cosφP

or

sinλP=cos(i+B)sinψ

cosφPsinB

sinλP=-sinisinψ+cosψcosisinU

cosφP .

To completly determine the longitude, we use the other relation which gives the cosine

cosλP=cosψcosU

cosφP

The absolute longitude (Greenwich, Meridian reference) is given by

λ = λP + λN - (T-HNa) 2π

86400 ,

where T is the time of the acquisition, the last term is for taking the rotation of the earth between T

and HNa into account. Let us recall that the mouvement angle U is calculated from U=AN.(T- HNa).

Consider again Figs. 2 to determine the azimuthal and zenithal observation angles.

δ is so as

δ = 55.385 Nc-1024

1024 in deg.

the zenithal viewing angle θV is defined by

θV=asin[(1+HR )sinδ]

The observation azimuthal angle φV is determinated by solving the triangle NSP,

sinφV=sin(λS-λP)cosφs

sinψ


75

and

cosφV=sinφS-sinφPcosψ

cosφPsinψ

where φS and λS are the latitude and the longitude of the subsatellite point P.

Reference:

The characteristics of the orbit have been taken from, NOAA POLAR ORBITER DATA USERS

GUIDE, 1985, U.S. Department of Commerce, NOAA: National Environment Satellite,

National Climatic Data Center, Satellite Data Service Division, World Weather Building,

Room 100, Washington DC 20233, U.S.A..


76

SUBROUTINE POSSOL

Function: To compute the solar azimuthal and zenithal angles (in degrees) for a point over

the globe defined by its longitude and its latitude (in dec. degrees) for a day of the year (fixed by

number of the month and number of the day in the month) at any Greenwich Meridian Time (GMT

dec. hour).

Figure 1

S

N

φO

δ

t

SP

Description: Let P be the point determined by the latitude φ and δ the declination of the sun

at this period of the year, the hour angle is noted t. So the incident angle θS can be determined by

spherical trigonometry expression


77

cos θS = cos (π2 - δ) cos (

π2 - φ) + sin (

π2 - δ) sin (

π2 - φ) cos t

or

cos θv = sin δ sinφ + cosδ cosφ cos t

The solar declination depends upon the day of the year. We used the decomposition in Fourier

series of the declination based on astronomical data with the expression:

δ = β1 - β2 cos(A)+β3 sin(A) -β4 cos(2A)+β5 sin(2A)-β6 cos(3A)+β7 sin(3A)

where A=2πJ365 and J is the julian day

β1=.006918, β2=.399912, β3=.070257,

β4=.006758, β5=.000907, β6=.002697, β7=.001480

The hour angle is computed from the following considerations. From the GMT time, we

compute the mean solar time (or local time) for the longitude λ

MST = GMT +λ15 (dec.hour) .

The length of the day changes within the year (differences between +30 s and - 20 s), so we

have to correct the local time to obtain the true solar time (TST).

TST = MST + ET

where the equation of time ET is given by:

ET=(α1+α2cos(B)-α3sin(B)-α4cos(2B)-α5sin(2B))12

π (dec.hour)

with

B = 2πJ365 , α1=.000075, α2=.001868, α3=.032077, α4=.014615, α5=.040849

We obtain the hour angle t

t = 15 π

180 (TST-12) (radians)

and can compute θS.


78

Figure 2

N

t

S P

χθs

π/2 - δ π/2 - φ

To determine the azimuthal angle φS, we solve the spherical triangle NSP:

sinχ

sin(π2-δ)

=sint

sinθS

where χ is the solar azimuthal angle measured from the south through the west.

Or

sinχ=cosδ sint

sinθS .

To determine the sign of χ we use the cosine

cosχ=cosφsinδ+cosδcost

sinθS

so χ is completly defined.

To define the solar azimuthal angle φS with respect to North, we write,

φS = π + asin χ.

References:

Ch. PERRIN DE BRICHAMBAUT, Rayonnement Solaire et Echanges Radiatifs Naturels,

Monographies de Météorologie, Gauthier-Villars, Paris, France, 1963.

N. ROBINSON, Solar Radiation, Elsevier Publishing Company, New-York, N.Y., 10017, 1966.

Figure 3: Simulation of solar angles for the 1rst May, at different Latitudes, versus Universal Time.


79

0

20

40

60

80

100

120

140

160

180

0 4 8 12 16 20 24

Lat: 45° southLong: 0°

Sola

r ze

nith

ang

le

Universal time

0

40

80

120

160

200

240

280

320

360

0 4 8 12 16 20 24

Long: 0°

Sola

r az

imut

h an

gle

Universal time

Lat: 45° south

0

20

40

60

80

100

120

140

160

180

0 4 8 12 16 20 24

Long: 0°

Sola

r ze

nith

ang

le

Universal time

Lat: 10° south

0

40

80

120

160

200

240

280

320

360

0 4 8 12 16 20 24

Long: 0°

Sola

r az

imut

h an

gle

Universal time

Lat: 10° south

0

20

40

60

80

100

120

140

160

180

0 4 8 12 16 20 24

Long: 0°

Sola

r ze

nith

ang

le

Universal time

Lat: 45° north

0

40

80

120

160

200

240

280

320

360

0 4 8 12 16 20 24

Long: 0°

Sola

r az

imut

h an

gle

Universal time

Lat: 45° north


80

SUBROUTINE POSSPO

Function: To compute the geometrical conditions for the SPOT satellite. As the dimensions

of the frame are 60 × 60 km with an observation angle of maximum 2.06°, we have considered that:

• the zenithal observation angle is nul, so the azimuthal angle is not defined,

• the incident conditions are the same that those computed for the center of the frame.

Note: We have not considered the off nadir viewing.

Reference:

M. CHEVREL , M. COURTOIS, G. WEILL (1981). The SPOT Satellite Remote Sensing Mission,

Photogrammetric Engineering and Remote Sensing, 47, 1163-1171.


81

DESCRIPTION OF THE SUBROUTINES USED

TO COMPUTE THE ATMOSPHERIC CONDITIONS


82


83

SUBROUTINE ABSTRA

Function: To compute the gaseous transmittance between 0.25 and 4 µm for downward,

upward and total paths. We consider the six gases (O2, CO 2, H 2O, O3, N2O and CH4) separately.

The total transmission is put equal to the simple product of each ones. The spectral resolution is

equal to 10 cm-1.

Description: We have used two random exponential band models (Goody for H2O and

Malkmus for O 2, CO2, O 3, N2O and CH4) to compute the gaseous transmissions. If we consider an

homogeneous path, the transmission function is written,

for H2O

t ∆νG = exp −

N0 k m

∆ν1 +

k m

Π α0

− 12

(01)

for the other gases

t ∆νM = exp −

2Π α0 N0

∆ν1 +

k m

Π α0

12

−1

, (02)

where m is the absorber amount, N0 the total line number in the frequency interval ∆ν, k the

average intensity and α0 the average Lorentz half width, obtained from intensity S j and half width

αj of the jth spectral line by

k =S j

j=1

N0

∑

N0(03)

k

Πα=

1

4

S jj=1

N0

∑

Sj α j( )1/2

j=1

N 0

∑

2

(04)


84

The spectral resolution of 10 cm -1 is sufficient and contains enough spectral lines to use a

random band model transmission function.

From a general point of view, the width of a spectral line corresponds to the convolution

product of the two shapes, Lorentz and Doppler and is therefore called a Voigt line. For anatmospheric gas (O2, CO2, H2O, O3, N2O and CH4 ) the altitude where the Lorentz width and

Doppler width are equivalent, is about 30 km. So, according to the vertical distribution, only O3

requires a more complex treatment to take into account a Voigt profile. The O3 visible transmission

is computed by an other method detailed in the next part and the absorption in the solar infrared (3.3

µm) is very small (cf. Fig. I-2 of the chapter 1, §1). Therefore, we have used the same formalism for

all gases. The approximation contributes no consequential error.

Equations (1) and (2) are valid for a homogeneous path, where pressure and temperature are

assumed to be constant. To take into account the variations of temperature and pressure along the

atmospheric path, we use the Curtis-Godson approximation which associates an amount m

weighted by temperature (thereby related to the line intensity), and a amount mφ weighted by

pressure and temperature (thereby related to the intensity and half width line)

m (z,z') = Φ(T) duz

z'

∫ , (05)

mφ(z,z') = Ψ(T) φ duz

z'

∫ , (06)

with• φ =p/p0 (p0 is the standard pressure at which the measurements of spectroscopic parameters have

been made)

• du=ρg(dz/µ) (g is the gaseous density and µ the cosine of the viewing angle).

The functions Φ(T) and Ψ(T) are given by

Φ(T) =Sj (T)

j=1

N 0

∑

Sj (Tr )j=1

N 0

∑, (07)


85

Ψ(T) =S j(T) α j0 (T)[ ]1/2

j=1

N 0

∑

S j(T r ) α j0 (T r )[ ]1/2

j=1

N 0

∑

2

,(08)

with Tr the reference temperature and αj0 the half-width at temperature Tr and pressure p0.

To simplify, we fit these functions with,

Φ(T) = exp a(T − Tr ) + b(T − Tr )2[ ] , (09a)

Ψ(T) = exp a'(T − Tr ) + b'(T − Tr )2[ ] . (09b)

The spectroscopic data are taken from the AFGL atmospheric absorption line parameters

compilation (1991 edition). We have selected the following parameters,

• the position (in cm-1),• the integrated line strength Sj (Tr) at 296 K (in cm-1/(molecules-cm2)),

• the half width αj0 at 296 K and 1013 mb (in cm-1),

• the energy of the lower transition state.

The half width at any temperature and pressure is obtained by

α j (p,T) =α j0p

p0

Tr

T

1/2

(10)

and the intensity at any temperature can be computed from the vibrational and rotational partition

and the energy of the lower transition state.

Subsequently, we have taken Tr = 250 K and computed Φ(T) and Ψ(T) for 3 temperatures

(200, 250 and 300 K) to determine the coefficients a, a', b and b'.

Now we have a series of eight coefficients by steps of 10 cm-1:


86

• k

δ=

S jj=1

N0

∑ (T r )

∆ν(11)

• Πα0

δ=

4

∆ν

S j(T r ) α j0 (T)[ ]1/2

j=1

N 0

∑

2

Sj (Tr )j=1

N 0

∑(12)

• a, a'

• b, b'• νlow the lower frequency of the interval, and νsup = νlow + 10 cm-1

These coefficients are read in the subroutines, WAVA1 to 6 for H2O, OZON1 for O3,

OXYG3 to 6 for O2 and DICA1 to 3 for CO2

The weighted absorber amounts m and mφ are computed according to Eq. (5) and (6) and the

transmission functions (which correspond to Eq. (1) and (2) for a homogeneous path) are written

t ∆νG = exp −

k m

δ1 +

k

Π α0

m 2

mφ

− 12

, (13)

t ∆νM = exp −

Π α0

δmφ2m

1 +4 k

Π α0

m 2

mφ

12

−1

. (14)

Due to the deficiency of spectroscopy data, the visible ozone transmission function is written,

tO3(∆ν) = exp(−AO3

(∆ν) uO3) (15)

where uO3 is the absorber amount, AO3

the absorption coefficient given by Kneizys and al. (1980).

These coefficients are given in steps of 200 cm-1 between 13000 and 24200 cm-1 and by step of

500 cm-1 between 27500 - 50000 cm-1.


87

To take into account the water vapor continuum, we use the same expression with the

coefficients A H2 Oc are given in step of 5 cm-1 between 2350 and 2420 cm-1.

A comparison between MODTRAN2 and our results (6S) is shown in the following figures

(1-3). The difference observed at roughly 3.1µm is due to the fact that we have not taken into

account the N2O continuum. This spectral range is already contaminated by water vapor and is not

an atmospheric window. Therefore, the 3.1µm region is not used in remote sensing and its emission

in 6S generally unimportant.

References:

A.R. CURTIS, The computation of radiative heating rates in the atmosphere, Proc. Roy. Soc.

London, A236, p. 156-159, 1956.

R.G. ELLINGSON, J.C. GILLE, An infrared radiative transfer model. part 1: Model description

and comparison of observations with calculations, J. Atmos. Sci. 35, p. 523-545, 1978.

R.M. GOODY, Atmospheric Radiation 1, Theoretical Basis, Oxford University Press, 436 pp, 1964.

F.X. KNEIZYS, E.P. SHETTLE, W.O. GALLERY, J.H. CHETWYND Jr.L.W. ABREU, J.E.A.

SELBY, R.W. FENN, R.A. Mc CLATCHEY, Atmospheric Transmittance/Radiance:

Computer Code LOWTRAN 5, AFGL-TR-80-0067, Air Force Geophysics Laboratory,

Bedford, Mass. 1980.

W. MALKMUS, Random Lorentz Band Model with Exponential- Tailed S-1 Line-intensity

Distribution Function, J. Opt. Soc. Am. 57, 3, p. 323-329, 1967.

J.J. MORCRETTE, Sur la Paramétrisation du Rayonnement dans les Modèles de la Circulation

Génèrale Atmosphérique, Thèse d'Etat no 630, Université de Lille.

D.C. ROBERTSON, L.S. BERNSTEIN, R. HAIMES, J. WUNDERLICH, L. VEGA, 5 cm-1. Band

Model Option to Lowtran 5, Applied Optics, 20, p. 3218-3226, 1981.


88

C.D. RODGERS, C.O. WALSHAW, The Computation of Infrared Cooling Rates in Planetary

Atmospheres, Quart. J. Roy. Meteor. Soc., 92, p. 67-92, 1966.

L.S. ROTHMAN, R.R. GAMACHE, A. BARBE, A. GOLDMAN, J.R. GILLIS, L.R. BROWN,

R.A. TOTH, J.M. FLAUD, C. CAMY-PEYRET, AFGL Atmospheric Absorption Line

Parameters Compilation: 1982 Edition, Applied. Optics, 22, p. 2247-2256, 1983.


89

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0 1.2

6S

MODTRAN

Tg

Wavelength (µm)

Figure 1: Gaseous transmission between 0.25 and 1.2µm (mid. lat. summer atmosphere)

0.0

0.2

0.4

0.6

0.8

1.0

1.2 1.4 1.6 1.8 2.0 2.2 2.4

6S

MODTRAN

Tg

Wavelength (µm)

Figure 2: Gaseous transmission between 1.20µm and 2.40µm (mid. lat. summer atmosphere).


90

0.0

0.2

0.4

0.6

0.8

1.0

2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

6S

MODTRAN

Tg

Wavelength (µm)

Figure 3: Gaseous transmission between 2.40µm and 4.00µm (mid. lat. summer atmosphere).


91

SUBROUTINE AEROSO

Function: To compute the optical scattering parameters (extinction and scattering

coefficients, single scattering albedo, phase function, assymetry factor) at the ten discrete

wavelengths for the selected model (or created model) from:

(1) the characteristics of the basic components of the International Radiation Commission.

(1983).

• dust-like component (D.L., SUBROUTINE DUST)

• oceanic component (O.C., SUBROUTINE OCEA)

• water-soluble component (W.S., SUBROUTINE WATE)

• soot component (S.O., SUBROUTINE SOOT)

(2) pre-computed caracteristics,

now available are the desertic aerosol model corresponding to background conditions, as

described in Shettle(1984), a stratospheric aerosol model as measured Mona Loa (Hawaii)

during El Chichon eruption and as described by King et al. (1984), and a biomass burning

aerosol model as deduced from measurements taken by sunphotometers in Amazonia.

(SUBROUTINES BDM, STM and BBM)

(3) computed using the MIE theory with inputs (size distribution, refractive indexes...) given

by the user (see SUBROUTINES MIE and EXSCPHASE).

These models don't correspond to a mixture of the four basic components.

Description: From the MIE theory (see SUBROUTINE MIE), we have computed the phase

function P(Θ), the extinction and scattering coefficients, the assymetry factor g for the basic

components defined by their size distributions and their refractive index. The computations were

performed at 10 wavelengths and 83 phase angles (80 Gauss angles, 0°, 90°, 180°)

Note: We compute the resultant phase function for the scattering angle by linear interpolation

in the table of 83 values.

From the four basic components, three tropospheric aerosols types models have been selected

by mixing with the following volume percentages. By mixing, we suppose an idea of "external


92

mixing" in the model construction, so the resultant values are obtained by a weighted average using

the volume percentages Cj given by:

D.L. W.S. O.C. S.O.

Continental 0.70 0.29 0.01

Maritime 0.05 0.95

Urban 0.17 0.61 0.22

For each component, we know the volume concentration Vj and the particle number

concentration Nj (particle/cm3):

D.L. W.S. O.C. S.O.

Vj µ3/cm3113.98352 113.98352 10−06 5.14441 59.777553 10−06

Nj part/cm3 54.73400 1.86850 10+06 276.0500010 1.805820 10+06

where

V j =4Π3

r3 dN j(r)

dr0

+∞

∫ dr

and Nj is computed so as to normalize the extinction coefficient at 550 nm .

If Cj is the aerosol fraction by volume of the component j, we have Cj =v j / v with vj = n j Vj

where nj is the number of particles in the mixing so

n = n j = vC j

V jj

∑j

∑

then we can obtain the percentage density of particles

n j

n=

C j

V jC j

V jj∑

so for example nj/n for the 3 selected models:


93

D.L. W.S. O.C. S.O.

Continental 2.26490 10−06 0.938299 0.0616987

Maritime 0.999579 4.20823 10−04

Urban 1.65125 10−07 0.592507 0.407492

To obtain the extinction coefficient of the resultant model, we compute

K ext(λ) =n j

nK j

ext (λ)j

∑

and we normalize also this coefficient at 550 nm. So we have to compute the equivalent

number N of particles by :

N =1

n j

nK j

ext(550)j

∑

Since Kextj (550) = 1/Nj, we obtain

1

N=

n j

nj

∑1

N j

The other optical parameters are computed by the same way:

• scattering coefficient :

Ksca (λ) = Nn j

nj

∑ K jsca (λ)

• assymetry factor :

g(λ) =N

Ksca (λ)

n j

nj

∑ g j(λ) K jsca (λ)

• phase function :

Pλ (Θ) =N

Ksca (λ)

n j

nj

∑ Pλj (Θ)K j

sca (λ)

• the single scattering albedo is directly obtained by the ratio

ω0(λ) =Ksca (λ)

K ext(λ)


94

Notes:

- The data for extinction or scattering coefficients are in km-1

- The following figures give us an order of magnitude of these terms for the 3 selected aerosol

models plus the desert aerosol model.

Reference:

World Meteorological Organization (CAS)/Radiation Commission of IAMAP Meeting of experts

on aerosols and their climatic effects, WCP 55, Williamsburg, Virginia, U.S.A., 28-30 March

1983.

E. P. SHETTLE, Optical and radiative properties of a desert aerosol model. Symposium of

Radiation in the atmosphere, 1 Deepak publishing), pp. 74-77, 1984.

M. KING, HARSHVARDHAN, and ARKING, A, A model of the Radiative Properties of the El

Chichon Stratospheric Aerosol Layer, J. Appl. Meteor., 23, (7), pp. 1121-1137, 1984.


95

Figure 1: Spectral dependence of the extinction coefficient for various aerosol models.

0.01

0.1

1

0.4 0.6 0.8 1.0 2.0 3.0

ContinentalMaritimeUrbanDeserticBiomassStratospheric

Ext

inct

ion

Coe

ffic

ient

Wavelength (mm)

Figure 2: Spectral dependence of the single scattering albedo for various aerosol models.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.4 0.6 0.8 1.0 2.0 3.0


Sing

le S

catte

ring

Alb

edo

Wavelength (mm)


96


97

Figure 3: Spectral dependence of the assymetry parameter for various aerosol models.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.4 0.6 0.8 1.0 2.0 3.0


Ass

ymet

ry P

aram

eter

Wavelength (µm)

Figure 4: Phase function at 0.550µm versus scattering angle for various aerosol models.

0.01

0.1

1

101

102

103

0.0 30.0 60.0 90.0 120.0 150.0 180.0


Phas

e Fu

nctio

n

Scattering Angle (°)


98


99

SUBROUTINE ATMREF

Function: To compute the atmospheric reflectance for the molecular and aerosol atmospheres

and the mixed atmosphere. In 6S instead of an approximation as in 5S, we use the scalar Successive

Order of Scattering method (subroutine OS.f). The polarization terms of aerosol or rayleigh phase

are not accounted for in the computation of the aerosol reflectance and the mixed Rayleigh-aerosol

reflectance. The polarization is addressed in computing the Rayleigh reflectance (Subroutine

CHAND.f) by semi-empirical fitting of the vectorized Sucessive Order of Scattering method (Deuzé

et al, 1989).

Description: Three reflectance terms have to be computed by ATMREF.f, the aerosol

reflectance (ρA), the rayleigh reflectance (ρR) and the reflectance of the mixed Rayleigh-aerosol

(ρR+A). In addition three different configurations of sensor position are possible, ground based

observation, satellite sensor or airborne sensor.

In the case of ground based observations, we consider that there is no contribution of the

atmosphere below the sensor and the three reflectances are simply set to zero.

For the case of satellite based observations, we can consider that all the molecules and aerosol

are below the sensor. In that case, we use the subroutine OS.f to compute ρA and ρR+A and the

subroutine CHAND.f to compute ρR. The subroutine OS.f is able to deal with a mixture of

molecules and aerosols or with aerosol only or molecules only, by computing the signal in a set of

layers for which the proportion of molecules and aerosol can be adjusted. The computation of the

proportion of aerosol and molecules in each layer is optimized by the subroutine DISCRE.f to

divide the entire atmosphere in equal optical depth layers, the proportion depends on the aerosol

profile which is assumed to be exponential with a scale heigh of 2km.


100

For the case of airborne observation, the three components are computed by the OS.f

subroutine. In OS.f, a special layer is set so that the top of the layer corresponds to the aircraft's

altitude. When aerosol optical depth below the plane is provided by the user as encouraged, the

scale height of aerosol is computed again to match the total aerosol optical depth, the aerosol optical

below the plane and the plane altitude. If in that case, the scale heigh is found to be greater than

7km a warning message is issued and computation are aborted.

References:

Radiation Commission of IAMAP, Standart Procedures to compute Atmospheric Radiative

Transfer in a scattering Atmosphere. Edited by J. LENOBLE, Available from Dr. S. Ruttenberg,

NCAR, Boulder Colorado 80307, U.S.A., 1977.

D. TANRE, M. HERMAN, P.Y. DESCHAMPS, A. DE LEFFE, Atmospheric Modeling for Space

Measurements of Ground Reflectances including bidirectional properties, Appl. Opt . 18, no 21,

p. 3587-3594, 1979.

J. L. DEUZÉ, M. HERMAN AND R. SANTER, Fourier series expansion of the transfer equation in

the atmosphere-ocean system. Journal of Quantitative Spectroscopy and Radiative Transfer, 41,

6, 483-494, 1989.


101

SUBROUTINE CHAND

Function: To compute the atmospheric reflectance for the molecular atmosphere in case of satellite

observation.

Description: In 6S, to save computer ressources but maintain a good accuracy, we used the same

approach to compute the molecular scattering reflectance as it is detailled in a recent paper

(Vermote and Tanré 1992). The molecular reflectance, as computed from 6S is plotted vs the

reflectance computed from the SOS method for τ=0.35 in Fig. 1. Four values of the solar zenith

angle (0°, 53°, 66° and 70°), 17 values of the viewing zenith angle (from 0° to 60° with a step of

3.3°) and 19 values of the difference of the azimuth angles (from 0° to 180° with a step of 10°),

covering a large range of possible geometrical conditions, have been selected. Multiple points fall

on the 45-degree line; the right-hand scale, which gives absolute differences between the two

results, clearly shows that the accuracy of 0.001 is achieved for the full range of geometric

conditions.

References:

E. F. VERMOTE and D. TANRE , Analytical Expressions for Radiative Properties of Planar

Rayleigh Scattering Media Including Polarization Contribution. Journal Of Quantitative

Spectroscopy and Radiative Transfer, 47, 4, 305-314, 1992.

Figure 1: Accuracy of CHAND.f

0.10

0.20

0.30

0.40

0.50

0.60

0.000

0.001

0.002

0.003

0.004

0.005

0.10 0.20 0.30 0.40 0.50 0.60

Reflectance

Absolute Difference

APP

RO

XIM

ATE

REF

LEC

TA

NC

E

ACTUAL REFLECTANCE

ABSO

LUTE D

IFFERENC

E


102

SUBROUTINE CSALBR

Function: To compute the spherical albedo of the molecular layer.

Description: We integrate the transmission function of the different incident directions to

calculate the spherical albedo, s, that is:

s = 1 − µ T(µ) dµ0

1

∫ (01)

Using the expression of T(µ) derived in SCATRA (Eq. 01.), it can be shown that s reduces to:

s =1

4 + 3τ[3τ − 4E3(τ) + 6E 4(τ)] (02)

where E3(τ) and E4(τ) are exponential integrals for the argument τ. These functions are easily

computable from expressions given in the reference below.

Figure 1 shows that the differences between the exact results and Eq. (02) are approximately

0.003 for τ=0.35 which results in an error of 0.0003 for a surface albedo of 0.10. In the red part of

the solar spectrum for which the surface albedo may be larger, the error is still below 0.001.

References

M. ABRAMOWITZ AND I STEGUN, Handbook of Mathematical Functions (New-York: Dover

Publications,Inc), 1970.

0.00

0.05

0.10

0.15

0.20

0.25

0.000

0.002

0.004

0.006

0.008

0.010

0.05 0.10 0.15 0.20 0.25 0.30 0.35

exact results

analytical expression

Error

SPH

ERIC

AL

ALB

EDO

ABSO

LUTE D

IFFERENC

E

OPTICAL THICKNESS

Figure 1: Accuracy of Eq. 02.


103

SUBROUTINE DISCOM

Function : To compute the optical properties of the atmosphere at the 10 discrete

wavelengths.

Description :

The 10 wavelengths, 0.400, 0.488, 0.515, 0.550, 0.633, 0.694, 0.860, 1.536, 2.250, 3.750,

have been selected because they correspond to the atmospheric windows used in remote sensing.

The computed quantities are

• molecular optical depth (subroutine ODRAYL)

• aerosol optical depth(subroutine ODA550)

• atmospheric reflectances (subroutine ATMREF)

• scattering transmittances (subroutine SCATRA)

• spherical albedos (subroutine SCATRA).

The computations have been made respectively for the 3 types of atmosphere:

• molecular

• aerosols only

• complete atmosphere with the two components.


104

SUBROUTINE DISCRE

Function: Decompose the atmosphere in a finite number of layers. For each layer, DISCRE

provides the optical thickness, the proportion of molecules and aerosols assuming an exponential

distribution for each constituants. Figure 1 illustrate the way molecules and aerosols are mixed in a

realistic atmosphere. For molecules, the scale height is 8km. For aerosols it is assumed to be 2km

unless otherwise specified by the user (using aircraft measurements).

Figure 1: Molecules and aerosol mixing in atmosphere.molecule - aerosol content

layer2

layer3

∞

groundsurface

molecule

aerosol

τ=0

τ

τ=τ1

τ(z)=τ(0)exp(-z/h)


105

SUBROUTINE ENVIRO

Function: To compute the environment functions F(r) which allows us to account for an in-

homogeneous ground.

Description: For an accurate evaluation of F(r), Monte Carlo computations are necessary to

take into account

• the altitude dependence of the phase function

• the dependence of the phase function upon the aerosols type

• the scaling factors which are different for the aerosols and the molecules.

Simulations for some different vertical distributions and phase functions show that the

variability of the environment function F(r) can be rather tractable.

The molecular scattering which is a major factor for the enlarged contribution of the

background can be linearized and accounted for by:

F(r) =tdR (θv )F R (r) + td

P (θv) FP (r)

tdR (θv) + td

P (θv)(01)

where tdR (θv ) and td

P(θv ) are the diffuse fractions in the transmission functions respectively

for Rayleigh and aerosols.

FR(r) and FP(r) correspond to the environment functions estimated for Rayleigh and aerosols

taken into account separately, these functions are slightly dependent upon the wavelength.

We have computed these 2 functions for a mean atmosphere (Mc Clatchey et al, 1971) and we

propose the following approximations:

FP (r) =1 − 0.375 E−0.20r + 0.625E −1.83r[ ] (02)

FR (r) =1 − 0.930 E −0.08r + 0.070 E−1.10r[ ] (03)

where r is in km.


106

If the actual aerosol model (type and vertical distribution) does not differ much from the mean

model, these approximations are reasonnable and we account for major part of the environment

effect. Figure 1 shows the two functions FR(r) and FPr. We note that the horizontal scales of the

environment effect are typically 1km for aerosol scattering and 10km for molecular scattering.

For the case of an airborne observation, we computed the altitude dependence of the

Rayleigh and aerosol environment function. For several typical altitude we have computed FR(r,z)

and FP(r,z) by the Monte Carlo method and we have derived an approximate expression (Eq. (2)

and (3)). Figures 2 and 3 show, for the selected altitudes, the environment functions For a plane

flying at an arbitrary altitude, we perform a linear interpolation between the closest simulated

altitudes in 6S to get the environment function at the altitude of the plane.

Effect of the view zenith angle.

For 6S, we look at the dependence of these two environment functions as a function of the

view zenith angle. Fig. 4a and 4b show for several values of the view zenith angle the environment

function of Rayleigh and aerosol. As it can be observed on Fig. 4a-b, there is a dependence of the

function F(r) on the view direction for view zenith angle larger than 30°. In order to account for

this effect, we chose to fit the environment function at the desired view angle solely as a function of

the environment function computed for a nadir view as it is suggested by Fig. 4a-b. The results

presented on Fig. 4a-b (symbols) show that a simple polynomial function of nadir view

environment function whose coefficients depend on the logarithm of the cosine of view angle is

adequate. For molecules, the F function is fitted by the simple expression:

F R(θv ) = F R(θv = 0° ). ln(cos(θv ). 1 − F R(θv = 0° )( ) +1[ ] (04)

for aerosol, a polynomial of a higher degree is needed, that is:

FA (θv) = FA (θv = 0°).

1 + a0 ln(cos(θ)) + b0 ln(cos(θ))2[ ] +

FA (θ v = 0°). a1ln(cos(θ)) + b1 ln(cos(θ))2[ ] +

FA (θ v = 0°)2 . (−a1 − a0)ln(cos(θ)) + (−b1 − b 0) ln(cos(θ))2[ ]

(05)

with a0=1.3347, b0=0.57757, a1=-1.479, b1=-1.5275

However, it has to be pointed out that if the approximations (04 and 05) enable to take into account

adjacency effect for an arbitrary view angle, they implied uniformity of the background as a

function of azimuth. As contributions of the adjacent pixels for a large view angle don't comply to

the symmetry in azimuth, the 6S results, in case of large view angles, have be interpreted more like


107

a sensitivity test to the problem of adjacency effect rather than an actual way to perform adjacency

effect correction .

Reference:

D. TANRE, M. HERMAN and P.Y. DESCHAMPS, Influence of the background contribution upon

space measurements of ground reflectance, Appl. Opt., 20, p. 3676-3684, 1981.

0.8

Radius (km)

F (r)P

F (r)R

0 2 4 6 8 10

0.6

0.4

0.2

0.0

1.0

Figure 1: Environment fucntion at satellite level for Rayleigh and Particules.


108

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

z=0.5km

z=2km

z=5km

z=7kmEn

viro

nm

ent

Fu

nct

ion

FR

(r)

Radius (km)

Figure 2: Variation of Rayleigh environment fucntion wih sensor altitude.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

z = 0.5km

z=2km

z=5km

z=7km

Environm

ent

Functi

on F

p(r

)

Radius (km)

Figure 3: Same as figure 2 but for particles.


109

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10

v=0°

v=30°

v=60°

fit Fr(r,

v=30°) (10a)

fit Fr(r,

v=60°) (10a)

F r(r)

r [km]

Figure

4a: Environment function for a pure molecular atmosphere (lines) for different view zenith angle(θv) compared to approximation used in 6S (symbols)

as a function of the distance to the imaged pixel (r).

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

v=0°

v=30°

v=60°

fit Fa(r,

v=30°) (10b)

fit Fa(r,

v=60°) (10b)

F a(r

)

r [km]

Fi

gure 4b: same as Figure 4a but for aerosol.


110

SUBROUTINE GAUSS

Function: Compute for a given n, the gaussian quadrature (the n gaussian angles and the

their respective weights). The gaussian quadrature is used in numerical integration involving the

cosine of emergent or incident direction zenith angle.


111

SUBROUTINE INTERP

Function: To estimate the different atmospheric functions ρ(µS,µv,φS,φv), T(θ) and S at any

wavelength from the 10 discret computations.(subroutine DISCOM)

Description: The different atmospheric functions (noted f) have been assumed linear as a

function of optical depth τ, so the interpolation scheme is written,

f(τ) = A λ-α

The constants A and α are interpolated between 0.4 and 3.7 µm and extrapolated for the two

extreme intervals 0.25-0.4 and 3.7-4 µm.

The spectral dependances for Rayleigh (α = 4) and aerosols (α = 1) are quite different and we

considered the two types of atmosphere separately .


112

SUBROUTINE ISO

Function: Compute the atmospheric transmission for either a satellite or aircraft observation

as well as the spherical albedo of the atmosphere.

Description: The subroutine performs the computation on the basis of the Sucessive Orders

of Scattering method (see subroutine OS). The transmission is obtained directly by initially setting

the bottom of the atmosphere to a isotropic source of radiation. The spherical albedo is computed by

numerical integration (gaussian quadrature) of the transmission function (see Eq. (01) of

CSALBR.f).


113

SUBROUTINE KERNEL

Function: Compute the values of Legendre polynomials used in the successive order of

scattering method.


114

SUBROUTINE MIE (and EXSCPHASE)

Function: To compute, using the scattering of electromagnetic waves by a homogeneous

isotropic sphere, the physical properties of particles whose sizes are comparable to or larger than the

wavelength, and to generate mixture of dry particles.

Description: The interaction of an electromagnetic wave with a absorbing sphere is

described and expressed by the Mie theory (Mie, 1908). This theory has been particularly discussed

by Van de Hulst (Van de Hulst, 1981) and also in part by many other authors (as example Aden,

1951; Deirmendjian et al., 1961; Wyatt, 1962; Kattawar and Plass, 1967; Dave, 1969; Hansen and

Travis, 1974; Liou, 1980). Here, we outline the basic equations of the Mie scattering behind the

computation procedures.

1. Mie Scattering

Let represents the wavelength, r the radius of the sphere, x the Mie's parameter ( x 2 r ), m

the complex index of refraction (m nr in i), and the direction of scattered radiation measured

from the forward direction. From the Maxwell's equations, we can defined two complex functions

S1(x,m, ) and S2(x,m, ) related to the amplitude of the scattered radiation, respectively,

perpendicular and parallel to the plane of scattering

S1(x,m, )(2n 1)

n(n 1)n 1

an(x,m) n (cos ) bn (x,m) n (cos )

and

S2 (x,m, )(2n 1)

n(n 1)n 1

an (x,m) n (cos ) bn (x,m) n(cos )

1.1 Computation of a n (x,m) and b n (x,m)

The complex functions an(x,m) and bn(x,m) are given by

an (x,m) n' (mx) n(x) m n (mx) n

' (x)

n' (mx) n(x) m n (mx) n

' (x)

and

bn (x,m)m n

' (mx) n (x) n (mx) n' (x)

m n' (mx) n (x) n (mx) n

' (x)


115

where the prime denotes derivative of the function with respect of the argument (x or mx), andwhere n (z x or mx) and n (z x) are the Ricatti-Bessel functions defined by

n (z) 12 z

12 Jn 1

2(z) z jn(z)

n(z) 12 z

12 Nn 1

2(z) -z nn (z)

n (z) 12 z

12 H

n 12

(2) (z) z hn(2) (z) n (z)+i n (z)

where Jn 12, Nn 1

2 and Hn 1

2

(2) are respectively the Bessel functions of first, second, and third kind,

and where jn , nn and hn(2) are the corresponding spherical Bessel functions. Nn 1

2 is also called the

Neumann functions and Hn 12

(2) the half integral order Hankel function of the second kind.

In order to make the computational work more convenient, it is useful to introduce the logarithmic

derivative of the Ricatti-Bessel functions (Infeld, 1947; Aden, 1951; Kattawar and Plass, 1967)

Dn(z)d

dzln n(z)

Gn (z)d

dzln n (z)

making use of these equations, an(x,m) and bn(x,m) may be rewritten

an (x,m) n (x)

n (x)

Dn (mx) mDn (x)

Dn (mx) mGn (x)

and

bn (x,m) n (x)

n (x)

mDn(mx) Dn (x)

mDn (mx) Gn (x)

Expressions of an(x,m) and bn(x,m) are now reduced to a ratio of Ricatti-Bessel functions involving

real arguments and a ratio of "Dn(mx or x) and Gn (x) " functions which are easily computable. We

reported Figures 1 and 2, examples of an(x,m) and bn(x,m) for m = 1.33 - i 0.001 and for x=10 and

x=50 (which means respectively r ≈ 0.8 µm and r ≈ 4.0µm at 0.50 µm).

Also in order to save time, we use in 6S the criterion defined by Deirmendjian et al., 1961:"the quantities an and bn are terminated either when (an an

* + bn bn* )/ n < 10-14 "

1.1.1 Computation of the Ricatti-Bessel function.

The ratio of Ricatti-Bessel functions can be reduced to a ratio of spherical Bessel functions with a

real argument x as follown (x)

n (x)

jn (x)

hn(2) (x)

jn (x)

jn (x) inn (x)

The spherical Bessel functions jn (x) , nn (x) or hn(2) (x) , have different behaviors following they are

below or above the transition line defined by x2 = n(n + 1). Below the transition line


116

(n(n + 1) < x2 ), they behave as oscillating functions of both order and argument, whereas the

behavior becomes monotonic above the transition line (n(n 1) x 2 ).

It has been shown by many authors that nn (x) or hn(2) (x) can be processed using an upward

recurrence (what ever values of n and x). Functions nn (x) are computed using

nn 1 (x) = 2n+1

xnn (x) + nn 1(x)

with n0 (x)cos(x)

xn1(x)

cos(x)

x2

sin(x)

xFigures 03 show examples of the nn (x) function for x=10 and x=50.

For jn (x) , we use a similar recurrence

jn+ 1(x) = 2n+1

xjn (x) + jn− 1(x)

but, has it is explained in the paper of Corbató and Uretsky, 1959, the function jn (x) cannot be

computed by an upward recurrence "since upward recursion (except in the region of the x-n planewhere jn oscillate) would bring about a rapid loss of accuracy". Then, a downward recurrence is

called for, but we have to define the starting value of n, and for that purpose we use the work of

Corbato and Uretsky which is summurized hereafter. Let N be the starting order of the recursionwith N(N 1) x 2 , in their paper, they show "that rather than accurately evaluate jN (x) and

jN 1(x) to start the process, a very approximately starting the recursion at a higher order will give

a set of numbers which are accurately proportional to the jn over the desired range of n from 0 to

N". Let j n be one of these numbers.

They propose to define the higher order by

= N' −ln N

ln2A +

B u ' (2-u '2 )

2(1-u '2 )

where A=0.10 and B=0.35,

N = 2−30 (this value comes from the fact that generally computers can store floating

points numbers with a 30 binary digit mantissa),

and u' = 2x (2N ' + 1) with N' =N or N'= x − 12 + − ln N

ln2 Bx such that be

the lower, with however N ' ≥ N .

To avoid computational difficulties above the transition line, Corbato and Uretsky worked with theratio r n = j n+1 j n using the recurrence relation

r n-1 =x

2n + 1 − xr n


117

with the starting condition r = 0 . The recursion is continued downward until a ratio r n which

exceeds unity is reached. Then, they set j n+ 1 = r n and j n = 1 , and continue downward using the

recurrence relation

j n− 1(x) = 2n+1

xj n (x) + j n+ 1(x)

The positive number j n is defined by j n (x) = jn(x) with a constant of proportionality obtained

from the relation=( j 0 (x)- x j 1(x))cos(x) + x j 0 (x)sin(x)

Figures 03 also show examples of the jn (x) function for x=10 and x=50.

1.1.2 Computation of the Dn(mx or x) and Gn (x) function.

As Kattawar and Plass, 1967, have pointed out, the procedure of computing Dn(z) by an upward

recurrence is unstable, then a downward process is needed, and Dn(z) is defined using

Dn 1(z) n z1

Dn (z) n z

Calculations have to be started at an order n = ' >> z with a starting value which is not reallyimportant because the serie converges rapidly to the exact value (then D ' (z) = 0 is a convenient

value). When n z , Dn(z) becomes oscillatory, and then there is no problem for the calculation in

using the recurrence relation. For practical reasons, we selected in 6S ' = as defined for jn .

Kattawar and plass have also shown that Gn (x) may always be calculated using an upward process

with a starting value G0 (x) = −i

Gn (x) = −n x −1

Gn−1 (x) − n x

Figure 04 reportes examples of the Dn(x) function, Figures 05 the Dn(mx) function, and Figures 06

theGn (x) function.

1.2 Computation of n (cos ) and n (cos )

Functions n and n depend of only the scattering angle . They are related to the associated

Legendre polynomials Pn1 (cos )

n (cos ) = 1

sinPn

1(cos )

n (cos ) =d

dPn

1(cos )

and are computed from upward recurrence relations defined as follow

n n+1(cos ) = (2n+1) cos n(cos ) - (n+1) n-1 (cos )

n+1(cos ) = (n+1) cos n+1(cos ) - (n+2) n (cos )


118

with the starting values

0 (cos ) = 0 1(cos ) = 1

Examples of functions n and n are shown Figures 07 for n=1 to 6 and for 0°<θ<90°.

2. Computation of the physical properties of a particle (see for example Liou, 1980).

2.1 Extinction The extinction cross section e , which denotes the amount of energy removed (scattered and

absorbed) from the original beam by the particle, is obtained considering a point in the forward

direction ( =0) in the "far field". If we consider an isotropic homogeneous sphere, the extinction

cross section is given by

e ( ,r,m) =4

2( )2 ℜe S(x,m, = 0)[ ]

with

S(x,m, 0) S1(x,m, 0) S2(x,m, 0)1

2(2n 1)[an (x,m) bn (x,m)

n 1

]

Thus the extinction efficiency Qe is defined by

Qe( ,r,m) e ( ,r,m)

r 2

2

x 2 (2n 1) e[an (x,m) bn(x,m)n 1

]

2.2 Scattering The scattering cross section s is derived by a similar way, but considering a scattered light in an

arbitrary direction, by

s ( ,r,m)x r

2 [S1(x,m, )S1*(x,m, ) S2(x,m, )S2

*(x,m, )]sin d0

Owing of the functions n and n , we have to integrate products of the associated Legendre

polynomials. Using the orthogonal and recurrence properties of these polynomials, the scattering

cross section can be written

s ( ,r,m)2

x r2 (2n 1)[an (x,m)an

*(x,m) bn (x,m)bn*(x,m)]

n 1

where the asterisk denotes the complex conjugate value, and the scattering efficiency Qs can be

evaluated by the relation

Qs ( ,r,m) s ( ,r,m)

r 2

2

x2 (2n 1)[an (x,m)an* (x,m) bn (x,m)bn

* (x,m)]n 1

2.3 Absorption The absorption cross section a and the absorption efficiency Qa can be deduced from


119

a ( ,r,m)= e ( ,r,m) s ( ,r,m) Qa ( ,r,m) Qe( ,r,m)-Qs ( ,r,m)

2.4 Phase function

On the basis of the Stokes parameters, the intensity I of the electromagnetic waves at each point and

in any given direction can be related to the incident intensity I0 by

I =M11 I0

with

M11( ,r,m, )=1

2x 2 [S1(x,m, )S1*(x,m, )+S2(x,m, )S2

* (x,m, )]

The angular distribution of the scattered energy for a single sphere (also called Phase function)P11( , r,m, ) can be defined by

M11( ,r,m, ) = s ( ,r,m)

4 r 2 P11( ,r,m, )

then

P11( ,r,m, )=2

(x r)2s ( ,r,m)

[S1(x,m, )S1* (x,m, )+S2(x,m, )S2

* (x,m, )]

It can be check that

P11( ,r,m, )0∫

0

2

∫ sin d d = 4

3. Physical properties of a sample of identical particles

We now consider a sample of identical particles whose size is described by the size distributionn(r) (in cm-3 µm-1) such that

n(r) dr0

∞

∫ =dN(r)

drdr

0

∞

∫ =1

where d N(r) represents the number of particle per unit volume having a radius between r and

r+dr.

In 6S, we selected several possibilities to represent the size distribution, thus the user will be

allowed to choice between 4 options:

1- a Junge power-law function . Junge, 1952, showed that the size distribution of aerosols whose

radii are larger than 0.1µm may be described by

dN(r)

dlogr= ln(10) c r0

α 1

r

α−1or

dN(r)

dr= c r0

α 1

r

α

with α varying between 3 and 5, c the number density of particles with radius r0 and r0 an

arbitrary radius.

Figure 08-a shows an example of Junge Power-Law function which is the "Model C"

defined by Deirmendjian, 1969, for c.r0α=1 and α=4


120

2- a Modified Gamma distribution function. Used by Deirmendjian, 1964, to compute scattering

properties of water clouds and haze and to fit aerosol measurements. Also employed by Mie

in the Mie and diffraction calculations.dN(r)

dr= A r r0( ) exp -b r r0( )( ) with r0=1µm

An example of Modified Gamma distribution function is given Figure 08-b (Volcanic Ash

defined in WCP 112, A=5461.33, α=1.0, γ=0.5, b=16).

3- a Log-Normal distribution function . Based on the Junge power-law function, Davies, 1974,

introduced this function to take into account large particles.

dN(r)

dlogr=

N

2 log exp -

1

2

log r -log rM

log

2

where rM is the mean radius of the particle, and the standard deviation of r.

We reported Figure 08-c examples of Log-Normal distribution functions which are the 3

three components of the "Continental Model" defined in WCP 112 (see AEROSO to find rM

and ).

4- sun photometer measurements . You enter directly d V(r)/ dlogr ≈ r 4dN(r) / dr .

The Figure 09 shows the same function than Figures 08 but for dV(r)/dlog(r).

Under the assumption of "independent scattering" which means that particles are sufficiently far

from each other compared to the incident wavelength to consider just one scattering, it is possible to

add scattered intensities independently of the phase of the wave. Then we can defined the radiative

characteristics upon the particle size distribution by

•The extinction (e), scattering (s) and absorption (a) coefficient

ke,s,a ( ,m) Qe,s,a ( ,r,m) r 2

rmin

rmax dN(r)

drdr

•The normalized phase function

P( ,m, ) =1

ks ( ,m)M11( ,r,m, ) 4

rmin

rmax

∫ r 2 dN(r)

drdr

•We now introduce the single scattering albedo 0 which represents the percentage of

energy removed from the incident beam which will reappear as a single scattered

radiation.

0 ( , m)ks ( , m)

ke( ,m)


121

Computationally, ke,s,a ( ,m) and P( ,m, )are integrated step by step following:

ke,s,a ( ,m) = Qe,s,a ( ,r,m) r 2

r min

rmax

∑ dN(r)

drr

and

P( ,m, ) =1

ks ( ,m)M11( ,r,m, ) 4

rmin

rmax

∑ r 2 dN(r)

drr ,

where r is defined by

logr + r

r

= 0.03

The value 0.03 has been selected in order to preserve a good accuracy with a reasonable

computational time. For example D'Almeida used a very small step width, 0.011, for the

computations given in his book (D'Almeida et al., 1991). The logarithmic expression of ∆r comes

from the fact that size distributions can be frequently described by a logarithmic shape (Junge,

1952; Davies, 1974).

Finally, and, in order to save computational time, we defined a criterion on the summation such that

the computations are not performed either whenni

n r 2 dN(r)

drr

1< 10−8

where ni n is the percentage density of particles (cf. subroutine AEROSO for some examples). The

latter criterion has been tested between 0.4 and 4.0 µm.

4. Physical properties of a mixture of aerosol type

We now consider a mixture of particles originating from different sources (4 max.). The mixing is

treated in the same way that the one used to generated the data base in the AEROSO subroutine.

Let us recall that the mixture of individual components (or type) of an aerosol is characterized bythe percentage density of particles ni n , and if we assume that the particles are spherical, each type

i is described by its size distribution (then by its microphysical identity: rM i and i see Table 1 for

some examples (Shettle and Fenn, 1976; World Climate Programme, 1986), and by its complexrefractive index mi (see Table 2, from (Shettle and Fenn, 1976; Shettle and Fenn, 1979; World

Climate Programme, 1986; D'Almeida et al., 1991). For the size distribution, the Log-Normal

distribution is well adapted to emphasize the individual components of a mixture (Davies, 1974,

D'Almeida et al., 1991).


122

5. Examples and comparisonsThe comparison of the computed normalized at 550 nm Ke , Ks and 0 values that obtained by 6S

with those given by World Climate Programme, 1986 are reported for a Continental and an Urban

dry aerosol model respectively Table 3 and Table 4. Also reported in theses tables, are the

asymmetry parameters g. In 6S, this parameter is already computed in an another subroutine, but we

can compute it here using

g =cos P( ,m, )

−1

+1

∫ dcos

P( ,m, )−1

+1

∫ dcos

The comparison of the phase function of a Continental model (WMO/WCP-112) computed by the

MIE subroutine with those by a precise code (AEROSO subroutine) is reported Figure 10 for

several wavelengths. Also, we show, Figure 11, the phase function computed using the volumic

distribution dV/dlogr provided by a CIMEL sunphotometer during the SCAR-A field experiment

(Sulfate Clouds Aerosol and Reflectances - America) that took place in July 1993 in the Eastern

USA.


123

The parameters are:

For IAER=8 Multimodal Log Normal (up to 4 modes)

rmin,rmax,icp

then for k=1 to icp, enter:

σ, rM, Cij

rn(λj), j=1,10

ri(λj), j=1,10

where rmin and rmax are the radii min and max of the aerosol

icp the number of mode (component)

σ and rM are parameters of the Log-Normale size distributions

Cij is the percentage density of particles (see SUBROUTINE AEROSO)

rn and ri are the real and imaginary index of refraction of each component

with r=rn-i ri. You have to enter these parameters for the 10 wavelengths

used to compute the atmospheric signalwhich are:

0.400, 0.488, 0.515, 0.550, 0.633, 0.694, 0.860, 1.536, 2.250, 3.750

For IAER=9 Modified gamma distribution

rmin,rmax

α, b, γrn(λj), j=1,10

ri(λj), j=1,10

where α, b, and γ are the parameters of the Modified Gamma size distribution

For IAER=10 Junge Power-Law distribution

rmin,rmax

αrn(λj), j=1,10

ri(λj), j=1,10

where α is the parameter of the Junge Power-Law size distribution

For IAER=11 Sun Photometer distribution (50 values max)

irsunph

for k=1 to irsunph enter: r and dV/dlogr

rn(λj), j=1,10

ri(λj), j=1,10

where irsunph is the number of value and dV/dlogr is usually profided by sunphotometers.


124

Table 1: Microphysical characteristics of the aerosol type (dry particles)

used for the comparisons shown Tables 3 and 4 (from WMO-WCP112).

Dust-Like Water Soluble Oceanic Soot

rMi (µm) 0.500 0.0050 0.30 0.0118

σi 2.990 2.990 2.51 2.00

Table 2: Complexe refractive indexes of the aerosol types (dry particles)

used for the comparisons shown Tables 3 and 4 (from WMO-WCP112).

λ Dust-Like Water Soluble Oceanic Soot

(µm) nr ni nr ni nr ni nr ni

0.400 1.530 8.00E-3 1.530 5.00E-3 1.385 9.90E-9 1.750 0.460

0.488 1.530 8.00E-3 1.530 5.00E-3 1.382 6.41E-9 1.750 0.450

0.515 1.530 8.00E-3 1.530 5.00E-3 1.381 3.70E-9 1.750 0.450

0.550 1.530 8.00E-3 1.530 6.00E-3 1.381 4.26E-9 1.750 0.440

0.633 1.530 8.00E-3 1.530 6.00E-3 1.377 1.62E-8 1.750 0.430

0.694 1.530 8.00E-3 1.530 7.00E-3 1.376 5.04E-8 1.750 0.430

0.860 1.520 8.00E-3 1.520 1.20E-2 1.372 1.09E-6 1.750 0.430

1.536 1.400 8.00E-3 1.510 2.30E-2 1.359 2.43E-4 1.770 0.460

2.250 1.220 9.00E-3 1.420 1.00E-2 1.334 8.50E-4 1.810 0.500

3.750 1.270 1.10E-2 1.452 4.00E-3 1.398 2.90E-3 1.900 0.570


125

Table 3: Comparison between 6S and WMO-WCP112

for a Continental model (normalized value-dry particles)

Kext Ksca ω0 g

λ 6S WMO 6S WMO 6S WMO 6S WMO

0.400 1.40 1.40 1.27 1.27 0.902 0.901 0.643 0.646

0.488 1.14 1.14 1.03 1.03 0.900 0.898 0.637 0.640

0.515 1.08 1.08 0.967 0.967 0.899 0.897 0.635 0.638

0.550 1.00 1.00 0.893 0.891 0.893 0.891 0.634 0.637

0.633 0.849 0.849 0.755 0.754 0.890 0.888 0.629 0.633

0.694 0.760 0.760 0.671 0.669 0.881 0.879 0.628 0.631

0.860 0.577 0.577 0.487 0.486 0.844 0.841 0.629 0.633

1.536 0.282 0.283 0.212 0.212 0.753 0.750 0.641 0.645

2.250 0.150 0.151 0.115 0.115 0.765 0.761 0.738 0.741

3.750 0.101 0.103 0.0796 0.0805 0.790 0.785 0.777 0.779

Table 4: Comparison between 6S and WMO-WCP112

for an urban model (normalized value-dry particles)

Kext Ksca ω0 g

λ 6S WMO 6S WMO 6S WMO 6S WMO

0.400 1.48 1.48 0.980 0.976 0.664 0.660 0.600 0.600

0.488 1.16 1.17 0.766 0.762 0.658 0.654 0.594 0.593

0.515 1.09 1.09 0.715 0.711 0.655 0.651 0.592 0.592

0.550 1.00 1.00 0.651 0.647 0.651 0.647 0.591 0.591

0.633 0.828 0.829 0.535 0.532 0.646 0.641 0.587 0.587

0.694 0.733 0.733 0.466 0.462 0.635 0.631 0.585 0.585

0.860 0.542 0.542 0.322 0.319 0.593 0.588 0.584 0.583

1.536 0.242 0.243 0.111 0.111 0.460 0.455 0.564 0.565

2.250 0.123 0.124 0.0428 0.0426 0.347 0.342 0.583 0.585

3.750 0.0647 0.0659 0.0177 0.0181 0.274 0.274 0.579 0.587


126

Figures 01: Examples of a an(m,x) function.

0.0

0 .20

0 .40

0 .60

0 .80

1.0

1.2

0 10 20 30 40 50 60 70

Re[an(m,x=10)]

Re[an(m,x=50)]

Re[

a n(m,x

)]

Order n

m = 1.33 - i 0.001

-0.60

-0.40

-0.20

0.0

0 .20

0 .40

0 .60

0 .80

0 10 20 30 40 50 60 70

Im[an(m,x=10)]

Im[an(m,x=50)]

Im[a

n(m,x

)]

Order n

m = 1.33 - i 0.001

Figures 02: Examples of a bn(m,x) function.

0.0

0 .20

0 .40

0 .60

0 .80

1.0

1.2

0 10 20 30 40 50 60 70

Re[bn(m,x=10)]

Re[bn(m,x=50)]

Re[

b n(m,x

)]

Order n

m = 1.33 - i 0.001

-0.60

-0.40

-0.20

0.0

0 .20

0 .40

0 .60

0 .80

0 10 20 30 40 50 60 70

Im[bn(m,x=10)]

Im[bn(m,x=50)]

Im[b

n(m,x

)]

Order n

m = 1.33 - i 0.001

Figures 03: Examples of spherical Bessel functions jn(x) and nn(x)

-0.15

-0.10

-0.05

0 .00

0 .05

0 .10

0 .15

0 10 20 30 40 50 60 70

jn(x=10)

jn(x=50)

j n(x)

Order n

-0.20

-0.15

-0.10

-0.05

0 .00

0 .05

0 .10

0 .15

0 .20

0 10 20 30 40 50 60 70

nn(x=10)

nn(x=50)

nn(x

)

Order n


127

Figure 04: Examples of a Dn(x) function.

-20.00

-15.00

-10.00

-5.00

0 .00

5 .00

10.00

15.00

20.00

0 20 40 60 80 100

Re [Dn(x=10)]

Re [Dn(x=50)]

Re

[Dn(x

)]

Order n

Figures 05: Examples of a Dn(mx) function.

-10.00

-5.00

0 .00

5 .00

10.00

0 20 40 60 80 100

Re [Dn(m,x=10)]

Re [Dn(m,x=50)]

Re

[Dn(m

x)]

Order n

m=1.35 - i 0.001

10- 3

10- 2

10- 1

100

101

102

0 20 40 60 80 100

Im [Dn(m,x=10)]

Im [Dn(m,x=50)]

Im [

Dn(m

x)]

Order n

m=1.35 - i 0.001

Figures 06: Examples of a Gn(x) function.

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0 .00

0 20 40 60 80 100

Re[Gn(x=10)]

Re[Gn(x=50)]

Re[

Gn(x

)]

Order n

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0 .00

0 20 40 60 80 100

Im[Gn(x=10)]

Im[Gn(x=50)]

Im[G

n(x)]

Order n


128

Figures 07: Examples of functions Πn(θ) and τn(θ) for n=1 to 6.

-5

0

5

10

15

20

25

0° 30° 60° 90°

Πn

Scattering Angle

1

2

3

4

5

6

-20

-10

0

10

20

0° 30° 60° 90°

τ n

Scattering Angle

1

234

5

6

Figure 08-a: Junge Power-law function: Model C (Deirmendjian, 1954)

10-10

10-7

10-4

10-1

102

105

10-4 10-3 10-2 10-1 100 101 102

dN(r

)/dr

r (µm )

Model C (Deirmendjian 1969)

Figure 08-b: Modified Gamma distribution function: Volcanic Ash (WCP 112)

10-10

10-7

10-4

10-1

102

105

10-4 10-3 10-2 10-1 100 101 102

dN(r

)/dr

r (µm)

Volcanic Ash(WMO/WCP 112)


129

Figure 08-c: Log-Normal distribution function: Continental Model (WCP 112)

10-10

10-7

10-4

10-1

102

105

10-4 10-3 10-2 10-1 100 101 102

dN(r

)/dl

ogr

r (µm)

Dust Like

Soot

Water Soluble

Continental Model(WMO/WCP 112)

Figure 09: Same as Figures 8 but represented for dV/dlogr

10- 6

10- 5

10- 4

10- 3

10- 2

10- 1

100

101

10- 3 10- 2 10- 1 100 101 102

Model C (Deirmendjian 1954)

Volcanic Ash (WMO/WCP 112)

Continental Model (WMO/WCP 112)

dV /

dlog

r

r (µm)


130

Figure 10: Phase function (dry particle) as computed by the MIE subroutine and

by the one generated by AEROSO subroutine (exact case).

0.1

1

10

100

0° 30° 60° 90° 120° 150° 180°

Continental Model

From AEROSO (Exact Case)

Phas

e fu

nctio

n

Scattering angle

3.750µm

0.860µm

0.440µm

From MIE

Figure 11: Phase function as computed by MIE subroutine using the dV/dlogr provided

by a sunphotometer CIMEL during the SCAR-A experiment (Hog Island, July 11, 1993).

0.1

1

10

100

0° 30° 60° 90° 120° 150° 180°

441 nm873 nm

Phas

e fu

nctio

n

Scattering Angle


131

REFERENCES

A.L. ADEN, Electromagnetic Scattering from Spheres with Sizes Comparable to the Wavelength. J.

Appl. Phys., 22, 5, 601-605, 1951

F.J. CORBATO and J.L. URETSKY, Generation of Spherial Bessel Functions in Digital

Computers. J. Assoc. Computing Machinery, 6, 366-375, 1959

G.A. D'ALMEIDA, P. KOEPKE, and E.P. SHETTLE, Atmospheric Aerosols Global Climatology

and Radiative Characteristics, A. Deepak Publishing, Hampton, 1991.

J.V. DAVE, Scattering of Visible Light by Large Water Spheres. Appl. Opt., 8, 1, 155-164, 1969

C.N. DAVIES, Size distribution of atmospheric particles. J. Aerosol Sci., 5, 293-300, 1974

D. DEIRMENDJIAN, Scattering and Polarization properties of water clouds and hazes in the

visible and infrared. Appl. Opt., 3, 187-196, 1964

D. DEIRMENDJIAN, R. CLASEN, and W. VIEZEE, Mie Scattering with Complex Index of

Refraction. J. Opt. Soc. Am., 51, 6, 620-633, 1961

D. DEIRMENDJIAN, Electromagnetic scattering on spherical polydispersion. Elsevier Ed. New-

York, 1969

J.E. HANSEN and L. TRAVIS, Light scattering in planetary atmospheres. Space Sci. Rev., 16, 527-

610, 1974

L. INFELD, The Influence of the Width of the Gap upon the Theory of Antennas. Quart. Appl.

Math., 5, 2, 113-132, 1947

C.E. JUNGE, Gesetzmäßigkeiten in der Droßenverteilung atmosphärischer Aerosole über dem

Kontinent. Ber d. Deirsch Wetterdienst U.S.-Zone, 35, 261-277, 1952

G.W. KATTAWAR and G.N. PLASS, Electromagnetic Scattering from Absorbing Spheres. Appl.

Opt., 6, 8, 1377-1382, 1967

K.N. LIOU, An Introduction to Atmospheric Radiation, Academic Press Inc., San Diego, 1980.

G. MIE, Beigrade zur Optik trüber Medien, speziell kolloidaler Metallösungen. Ann. Physik., 25,

377-445, 1908

E.P. SHETTLE and R.W. FENN, Models of atmospheric aerosols and their optical properties, in:

Optical Properties in the Atmosphere, AGARD-Cp-183, NTIS, ADA 028615, 1976.

E.P. SHETTLE and R.W. FENN, Models for the aerosol of the lower atmosphere and the effect of

humidity variations on their optical properties, AFGL-TR-79-0214, Environmental Research

Paper No 675, NTIS, ADA 085951, 1979.

H.C. VAN DE HULST, Light Scattering by Small Particles, Dover Publications, New York, Dover,

1981.


132

WORLD CLIMATE PROGRAMME, WCP-55, Report of the expert meeting on aerosols and their

climatic effects (Eds. A. Deepak and H.E. Gerber), World Meteorological Organization,

Geneva, 1983.

WORLD CLIMATE PROGRAMME, WCP-112, A preliminary cloudless standard atmosphere for

radiation computation, World Meteorological Organization, WMO/TD-No 24, Geneva, 1986.

P.J. WYATT, Scattering of Electromagnetic Plane Waves from Inhomogeneous Spherically

Symmetric Objects. Phys. Rev., 127, 5, 1837-1843, 1962


133

SUBROUTINE ODA550

Function: To compute the extinction cross section and the aerosol optical depth at λ = 550

nm from the vertical distributions of the particle density (in particules/cm3).

Description: We have considered the 2 profiles, suggested by Mc Clatchey et al (1971),

corresponding to a visibility of 23 (clear) and 5 km (hazy) at ground level. The total numbers of

aerosols for the clear atmosphere have been adjusted so that the total extinction coefficient at λ =

550 nm becomes identical to the values used by Elterman (1964).

This total extinction coefficient K (in km-1) is obtained from

K550 (z) = σ550 10−3 N(z)

where σ is the extinction cross section in µm2 and N(z) the particules density (in part/cm3)

(the factor 10 -3 is to obtain an extinction coefficient in km-1). σ was computed with the same

aerosol model as the one defined by Mc Clatchey, index of refraction equal to 1.50 and size

distribution similar to Deirmendjian's model "C" (1969) (cut off has been extended from 5 to 10

µm). The computed value of σ550 is 0.056032.

The optical thickness τ is defined by

τ550 = K550(z) dz0

+∞

∫

We obtain the optical thicknesses at 550 nm, 0.235 and 0.780 respectively for the two

standard visibilities 23 and 5km. For another visibility, we compute a new profile particle density

from those defined for 23 and 5km. The calculations were made using the following interpolations,

N(z) =a(z)

VIS+ b(z)

We obtain for example :


134

τ = 0.152 for V = 50km,

τ = 0.520 for V = 8km.

Reference :

R.A. Mc CLATCHEY, R.W. FENN, J.E.A. SELBY, F.E. VOLZ, J.S. GARING. Optical properties

of the Atmosphere (revised), AFCRL 71-0279, Env. Research Paper 354, Bedford,

Massachusetts, U.S.A., 1971.

D. DEIRMENDJIAN, Electromagnetic Scattering on Spherical Polydispersions , 1969.

L. ELTERMAN , Rayleigh and Extinction Coefficient to 50km for the region 0.27µm to 0.55µm.

Applied Optics, 10, 1139-1145, 1964.


135

SUBROUTINE ODRAYL

Function: To compute the molecular optical depth as a function of wavelength for any

atmosphere defined by the pressure and temperature profiles.

Description: The optical depth is written

τλR = βλ (z) dz

0

+∞

∫where βλ (z) is the molecular extinction coefficient at altitude z and for wavelength λ. It can

be obtained from

βλ (z) = σλ Nr (z)105 ,

with Nr(z) is the molecules number/cm3 at altitude z, and σλ the extinction (or scattering)

cross section in cm2 .

These two quantities are defined by

σλ =8Π3(ns

2 −1)2

3λ4Ns2

6 + 3λ6 − 7λ

and

N r (z) = NsP(z)

1013.25

273.15

T(z)

where P(z) and T(z) are respectively the pressure and the temperature at the altitude z. Recallthat nS is the air refractive index, NS the molecular density at z=0 in STP conditions, and δ the

molecular depolarization factor.

We have taken:

* for refractive index

(ns −1).108 = 8342.13 +2406030

130 − ν2 +15997

38.9 −ν2

where ν is the frequency in cm-1


136

* NS = 2.54143.1019

* and the depolarization factor δ = 0.0279 following Young's (1980).

This depolarization factor is also used to compute the Rayleigh phase function (see routine

CHAND.f) according to:

P(Θ) =3

4

1 −γ1 + 2γ

(1+ cos2 Θ) +3γ

1 + 2γ

where Θ is the scattering angle, and γ = δ/(2-δ).

References :

B. EDLEN, The refractive Index of Air. Meteorologia, 2, 71-80, 1966

L. ETERMAN, Rayleigh and Extinction Coefficients to 50 km for the Region 0.27 µm to 0.55 µm

Appl. Opt., 10, p. 1139-1145, 1964

D.V. HOYT, A redetermination of the Rayleigh Optical Depth and its Application to selected Solar

Radiation Problems, J. Appl. Meteor, 16, p. 432-436, 1977.

A. T. YOUNG, Revised Depolarization Corrections for Atmospheric Extinction. Applied Optics,

19, 3427-3428, 1980


137

SUBROUTINE OS

Function: Compute the atmospheric intrinsic reflectance for the case of eihter satellite or

aircraft observation. Also compute the downward radiation field needed by the integral formula of

ρ and ρ ' (see Chapter I, §2.5.1, Eqs. 25 and 26) used in the computation in case of a non

lambertian target.

Description: The general purpose of the successive order of scattering is to solve

numericaly the equation of radiative transfer for upward (Eq. 01.) and downward radiation (Eq. 02.)

at any optical thickness τ. If τ1 is the total optical thickness and µ the cosine of the view angle, then

we can write:

I(τ;µ,φ) = I(τ1;µ, φ) −(τ1 −τ )/µe + J(τ'; µ,φ) −(τ' −τ )/µedτ'

µτ

τ1

∫ (1 ≥µ ≥ 0) (01)

I(τ; −µ,φ) = I(0; −µ,φ) −τ /µe + J(τ' ;−µ,φ) −(τ'−τ )/ µedτ'

µ0

τ

∫ (1≥ µ ≥ 0) (02)

where the source function, J(τ;µ,φ) accounts for the interaction of the present radiation field with

the particles of the layer located at τ, so that:

J(τ;µ,φ) =ω0

4ΠI(τ;µ' ,φ' ) P(µ,φ;µ', φ' ) dµ' dφ'

−1

1

∫0

2π

∫ +ω0

4ππ F0 P(µ,φ; −µ0 ,φ0) −τ /µ 0e (03)

The second term of equation (03) represents the sun source F0 transversing the path along

(µ0,φ0) directly to the level τ and then being scattered in direction (µ,φ) (primary scattering).

To solve this differential equation, one has to fix boundary conditions which are:

I(0;−µ,φ) =0 (04)

I(τ1;µ,φ) = 0 (05)

These express the fact that there is no diffuse downward nor upward radiation at the top and the

bottom of the finite atmosphere.


138

The convention is to describe the atmosphere with the top at τ=0 and the bottom at τ=τ1. The

upward radiation correspond to +µ and the downward to −µ with (1≥µ>0) as depicted by figure 1.

Figure 1: Schematic view of the radiative transfer problem for a plane parralel atmosphere.

Top

Bottom

τ = 0

τ = 0

τ = τ1

I(0;µ,φ)

I(τ1;µ,φ)

I(τ;µ,φ)

I(τ1;-µ,φ)

I(τ;-µ,φ)

The successive order of scattering consists of solving numerically the equation of transfer by

iteration. First, the equation is solved for each layer considering only the primary scattering

radiation (one interaction between the source (sun) and the atmosphere), giving for Eqs. (01) and

(02):

I (1)(τ;µ,φ) =

ω0

4ππ F0 P(µ, φ;−µ 0 ,φ0 ) −τ /µ0e (1≥ µ ≥ 0) (06)

I (1)(τ;−µ,φ) =

ω0

4ππ F0 P(−µ,φ; −µ0 ,φ0) −τ /µ 0e (1≥µ ≥ 0) (07)

Then for higher order of scattering we write:


139

I(n)

(τ j;µ,φ) =1

µJ(n)(τj ;µ,φ) − (τ j −τ )/µ[ ]e ∆τ

i= j

p

∑ (08)

I(n)

(τ j;−µ,φ) =1

µJ(n) (τ j;−µ, φ) − (τ−τ j)/ µ[ ]e ∆τ

i=1

j

∑ (09)

where p represents the number of layers used for the decomposition of the atmosphere, τj the optical

thickness at level j and ∆τ the increment in optical thickness between two successive layer. J(n) is

computed from I(n-1) by:

J(n)(τ;µ,φ) =ω0

4ΠI(n)(τ;µ', φ' )P( µ,φ;µ' ,φ' )dµ' dφ'

−1

1

∫0

2π

∫ (10)

In the code a numerical integration of Eq. 10 is performed using the decomposition in Fourier series

(for φ), the legendre polynomial and Gaussian quadrature (for µ).


140

SUBROUTINE SCATRA

Function: To compute the scattering transmission functions for the three atmospheric models,

rayleigh, aerosol and a mixture of both on the two paths (downward and upward). We also compute

the spherical albedo.

Description: As in ATMREF.f, we have to compute the transmission function and albedo in

three different cases and three sensor configurations. Again, the accuracy of the computation of 6S

was a concern, so the approximation adopted in 5S has been replaced by using the sucessive order

of scattering method (ISO.f) for the aerosol case and the mixed case, or when the sensor was inside

the atmosphere on board an aircraft. For the Rayleigh atmosphere we used an accurate analytical

formula which has sufficient accuracy and enables us to save computer time. The formula is

explicity coded into SCATRA.f for the transmission and call CSALBR.f for the albedo. For ground

measurements, the upward transmission is set to 1.0 and the spherical albedo to 0.0, because we

neglect the atmosphere between the sensor and the target.

We only give here the formula of the Rayleigh transmission which is based on the two stream

method adapted to the case of a single scattering albedo equal to 1.0 (Rayleigh case). The total

transmission on path of length µ, T(µ) can be approximated by:

T(µ) =[(2/ 3) +µ ] + [(2/ 3) −µ ]e

− τR

µ

(4 / 3) +τ R (01)

where τR is the Rayleigh optical thickness.

Figure 1 compares the accuracy of Eq. 01 to the "exact" computation (Successive Order of

Scattering).


141

References

J.H. JOSEPH, W.J. WISCOMBE, J.A. WEINMAN, The Delta-Eddington Approximation for

Radiative Flux Transfer,.J. Atmos. Sci. 33, p.2242-2459, 1976.

W.E. MEADOR, W.R. WEAVER, Two-Stream Approximations to Radiative Transfer in Planetary

Atmospheres: a Unified Description of Existing Methods and a New Improvement, J. Atmos.

Sci. 37, p. 630-643, 1980.

R.H.WELCH, W.G. ZDUNKOWSKI, Back Scattering Approximations and their Influence on

Eddington-Type Solar Flux Calculation, Beitr. Phys. Atmosph., 55, no 1, p. 28-42, 1982.

W.G. ZDUNKOWSKI, R.M. WELCH, G. KORB, An Investigation of the Structure of Typical

Two-Stream Methods for the Calculation of Solar Fluxes and Heating Rates in Clouds, Beitr.

Phys. Atmosph.53, no 2, p. 147-166, 1980.

Figure 1: Accuracy of Eq. 01

0.00

0.20

0.40

0.60

0.80

1.00

0.000

0.002

0.004

0.006

0.008

0.010

0 10 20 30 40 50 60 70

exact resultsanalytical expression

absolute difference

Zenith observation angle (°)

Tra

nsm

issi

on f

un

ctio

n T

()

Ab

solute d

ifference


142

SUBROUTINE TRUNCA

Function: decompose the aerosol phase function in series of Legendre polynomial used in

OS.f and ISO.f and compute truncation coefficient f to modify aerosol optical thickness τ and single

scattering albedo ω0 according to:

τ' = (1−ω 0 f) τ

ω0 ' =ω0 (1− f)

(1−ω 0 f)

References

J. LENOBLE, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard computional

procedures, pp 83-84, A. Deepak Publishing, 1985


143

DESCRIPTION OF THE SUBROUTINES USED FOR BRDF GROUND


144


145

SUBROUTINE HAPKALBE

Function: To compute the spherical albedo following the the BRDF computed from Hapke

(1981) Model.

Description: The target spherical albedo s is equal to the flux reflected by the target divided

by the incoming flux for an isotropic source. It is defined by the relation:

s=

⌡⌠0

Π/2

a(θs)cos(θs)sin(θs)dθs

⌡⌠0

Π/2

cos(θs)sin(θs)dθs

where a(θs) is the directional albedo for a parallel solar beam, and is given by

a(θs ) =ρ θs ,θv ,∆φ = φs −φ v( )cos(θv )sin(θv)dθvd∆φ

0

π/2

∫0

2π

∫

cos(θv )sin(θv )dθvd∆φ0

π /2

∫0

2π

∫

with ρ(θs,θv,∆φ), the bidirectionnal reflectances generated by the users's inputs (see

HAPKBRDF).


146

SUBROUTINE IAPIALBE

Function: Same as HAPKALBE but for a BRDF from the subroutine IAPIBRDF.

SUBROUTINE MINNALBE

Function: Same as HAPKALBE but for a BRDF from the subroutine MINNBRDF.

SUBROUTINE OCEALBE (and GLITALBE)

Function: Same as HAPKALBE but for a BRDF from the subroutine OCEABRDF.

SUBROUTINE RAHMALBE

Function: Same as HAPKALBE but for a BRDF from the subroutine RAHMBRDF.

SUBROUTINE ROUJALBE

Function: Same as HAPKALBE but for a BRDF from the subroutine ROUJBRDF.

SUBROUTINE VERSALBE

Function: Same as HAPKALBE but for a BRDF from the subroutine VERSBRDF.

SUBROUTINE WALTALBE

Function: Same as HAPKALBE but for a BRDF from the subroutine WALTBRDF.


147

SUBROUTINE BRDFGRID

Function: To generate a BRDF following user's inputs.

Description: The user enters the value of ρ for a sun at a given zenith sun angle θs by steps of

10° for zenith view angles θv (from 0° to 80° and the value at 85°) and by steps of 30° for azimuth

view angles φv from 0° to 360°. The user does the same for a sun which would be at θv. In addition,

the spherical albedo of the surface has to be specified.

The parameters are:

1. for θs you have to enter ρ(θv,φv)

ρ(0°,00°), ρ(10°,00°)...ρ(80°,00°), ρ(85°,00°)

ρ(0°,30°), ρ(10°,30°)...ρ(80°,30°), ρ(85°,30°)

...

ρ(0°,360°), ρ(10°,360°)...ρ(80°,360°), ρ(85°,360°)

2. for θs=θv you have to enter ρ(θv,φv)

ρ(0°,00°), ρ(10°,00°)...ρ(80°,00°), ρ(85°,00°)

ρ(0°,30°), ρ(10°,30°)...ρ(80°,30°), ρ(85°,30°)

...

ρ(0°,360°), ρ(10°,360°)...ρ(80°,360°), ρ(85°,360°)

3. the spherical albedo of the surface


148

SUBROUTINE HAPKBRDF

Function: To generate a BRDF following the Hapke's (1981) model.

Description (from Pinty and Verstraete, 1991): From the fundamental principles of

radiative transfer theory, Hapke (1981) derived an analytical equation for the bidirectional

reflectance function of a medium composed of dimensionless particles. The singly scattered

radiance is derived exactly, whereas the multiply scattered radiance is evaluated from a two-stream

approximation, assuming that the scatterers making up the surface are isotropic. The bidirectional

reflectance ρ of a surface illuminated by the sun from a direction (θs,φs), observed from a direction

(θv,φv), and normalized with respect to the reflectance of a perfectly reflecting Lambertian surface

under the same condition is given by

ρ(θs,φs,θv,φv)=ω4

1

µs+µv [1+B(g)]P(g)+H(µs)H(µv)-1

where µs=cos(θs) and µv=cos(θv)

g, the phase angle between the incoming and the outgoing rays is defined as:

cos(g)=cos(θs)cos(θv)+sin(θs)sin(θv)cos(φs-φv)

B(g) is a backscattering function that accounts for the hot spot effect

B(g)=SH(0)

ωP(0)

1

[1+(1/h)tan(g/2)]

H(x) is a function to account for multiple scattering

H(x)=1+2x

1+2(1-w)1/2x

In these equations:


149

• ω is the average single scattering albedo of the particles,

• P(g) is the average of the phase function of the particles, it is computed here by the Heyney

and Greenstein's function:

P(g)=1-Θ2

1+Θ2+2Θcos(g)

with Θ is the asymmetry factor ranging from -1 (backward scattering) to +1.

• S(0) is the amplitude of the hot spot,

• h is the width of the hot spot.

The parameters are:1- ω, Θ, S(0), h,

where ω is the average single-scattering albedo of the scatterers, Θ the asymmetry factor for the

phase function, S(0) amplitude of the hot spot, and h width of the hot spot

0.00

0.05

0.10

0.15

0.20

0.25

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0

Cal

cula

ted

Ref

lact

ance

s

Viewing angle in the principal plane

backward forward

sun at 60°

sun at 00°

Clover Patch

ω=0.101

Θ=-0.263h=0.046S(0)=0.589

References

B.W. HAPKE, Bidirectional reflectance spectroscopy - 1. Theory, J. Geophys. Res. , 86, 3039-3054,

1981.

B.W. HAPKE, Bidirectional reflectance spectroscopy - 4. The extinction coefficient and the

opposition effect, Icarus, 67, 264-280, 1986.

B. PINTY, and M. VERSTRAETE, Extracting Information on Surface Properties From

Bidirectional Reflectance Measurements, J. Geophys. Res., 96, 2865-2874, 1991.


150

SUBROUTINE IAPIBRDF

Function: To generate a BRDF following the Iaquinta and Pinty's (1994) model.

Description (from Iaquinta and Pinty, 1994): The model presents an improvement of the

original model by Verstraete et al., 1990, in order to account for the effects due to an underlying

soil below the vegetation canopy. The singly scattered component is solved exactly using an

analytical hot-spot description. The multiply scattered component is approximated on the basis of

Discrete Ordinates Method reduced to a one-angle problem.

The reflectance field is split into three main components which are unscattered (0), singly

scattered (1) and multiply scattered (M) by the leaves:

ρtot(Ωo,Ω)=ρ0(Ωo,Ω)+ρ1(Ωo,Ω)+ρM(Ωo)

where the canopy is illuminated from the direction Ωo (µo=cos(θo),φo) by the direct solar radiation,

and observed from the direction Ω (µ=cos(θ),φ).

• The single scattering by canopy elements ρ1(Ωo,Ω) is given by

ρ1(Ωo,Ω)=Γ(Ωo→Ω)

| |µo µ ⌡⌠0

LT

To(L)T(L)dL

where • Γ(Ωo→Ω) is the area scattering phase function (bi-Lambertian)

• L is the leaf area index (0<L<LT)

• To(L) is the transmission of direct solar radiation through the canopy layers above the

level L, it can be written as follow

To(L)=exp(-G(Ωo)

| |µo )

• T(L) is the transmission of scattered radiation, it can be written as follow

T(L)=exp(-G(Ωo)

µV2(Ωo,Ω,L)

V(Ω,L) L)


151

whereV2(Ωo,Ω,L)

V(Ω,L) ≅ (1-

4

3Π )L

Li ifL<Li

=1-4

3ΠLiL ifL≥Li

with

Li=2rΛ

tan2(θo)+tan2(θ)-2tan(θo)tan(θ)cos(φo-φ)

Λ denotes the leaf area density [m2 m-3] and r [m] is the radius of the

sun-flecks on the illuminated leaf

• The second component of the reflectance is the uncollided radiation ρ0(Ωo,Ω) (first order

reflectance from the soil), which is written

ρ0(Ωo,Ω)=RsTo(LT)T(LT)=Rsexp-(G(Ωo)

| |µo +

G(Ω)

µV2(Ωo,Ω,LT)

V(Ω,LT) )LT

where Rs is the soil albedo

• Using a canopy transport equation reduced to a one-angle problem and assuming isotropic

scattering, the multiply scattered radiation exiting at the top of the canopy is given by

ρM(Ωo)=1

| |µo⌡⌠0

1

IM(0,µ')µ'dµ'

where IM is the multiply scattered intensity which means photons which have been scattered two or

more times in the finite canopy.

In the above equations, the extinction coefficient of the radiation transport equation uses the G-

function. Physically G(Ωp) is the leaf area projected to the direction Ωp by a unit leaf area in the

canopy

G(Ωp)=1

2Π⌡⌠

2Π+gL(ΩL)| |ΩL.Ωp dΩL

where gL(ΩL) is the probability density of the distribution of leaf normals with respect to the

upward hemisphere (its computation is depending of the input parameter ild)


152

The area scattering phase function Γ(Ωo→Ω) is given by1

Π Γ(Ω'→Ω)=

1

2Π⌡⌠

2Π+gL(ΩL)| |Ω'.ΩL f(Ω'→Ω,ΩL)dΩL

where f(Ω'→Ω,ΩL) is the leaf scattering distribution function. Here, it is assuming that the

leaves follow a bi-Lambertian scattering model and then f(Ω'→Ω,ΩL) can be written

f(Ω'→Ω,ΩL)=rL| |Ω.ΩL

Πif(Ω.ΩL)(Ω'.ΩL)<0

tL| |Ω.ΩL

Πif(Ω.ΩL)(Ω'.ΩL)>0

with rL the leaf reflection coefficient and tL the leaf transmission coefficient

The parameters are :

1- ild, ihs

2- Lt, 2rΛ3- rL, tL, Rs

where ild is the leaf angle distribution (1=planophile, 2=erectophile, 3=plagiophile,

4=extremophile, 5=uniform), ihs a hot spot describter (0=no hot-spot, 1=hot spot), Lt the leaf area

index in [1. - 15.], 2rΛ a hot-spot parameter in [0. (no hot-spot) - 2.0], rL the leaf reflection

coefficient in [0., 0.99], tL the leaf transmission coefficient in [0., 0.99], Rs the soil albedo in [0.,

0.99]


153

0.015

0.020

0.025

0.030

0.035

0.040

0.045

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0

Cal

cula

ted

Ref

lect

ance

Viewing angle in principal planebackward forward

θs=30°

LT

=3.

Rs=0.2

rL

=0.0607 & tL

=0.0429

2rΛ=0.2

Erectophile

no hot-spot

hot-spot

References

J. IAQUINTA, and B. PINTY, Adaptation of a bidirectional reflectance model including the hot-

spot to an optically thin canopy, Proceedings of the VIe International Colloquium: Physical

measurements and signatures in remote sensing. Val d'Isère, France, January 17-21.

M. VERSTRAETE, B. PINTY, and R.E. DICKINSON, A Physical Model of the Bidirectional

Reflectance of Vegetation Canopies- 1. Theory, J. Geophys. Res., 95, 11755-11765, 1990.


154

SUBROUTINE MINNBRDF

Function: To generate a BRDF following the Minnaert's (1941) model.

Description: Using the optical reciprocity principle, Minnaert (1941) defined a simple model

to compute the BRDF. The reflectance is written

ρ(θs,θv,φ)=ρLk+12 (cos(θv)cos(θs)) k-1

where ρL is the albedo of the surface and k the surface parameter. For k=1, the BRDF

corresponds to an ideal Lambertian surface; k=0 corresponds to a minimum reflectance at nadir and

k=2 at a maximum reflectance at nadir.

The parameters are:1- k, ρL

0.00

0.05

0.10

0.15

0.20

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0

k=2k=1k=0

Cal

cula

ted

Ref

lect

ance

Viewing angles in principal plane

ρL

=0.10

θs=45°

References

M. MINNAERT, The reciprocity principle in lunar photometry, Astrophy. J., 93 , 403-410, 1941


155

SUBROUTINE OCEABRDF (and OCEATOOLS)

Function: To compute the directional reflectance of the ocean surface according to the

whitecaps, sun glint and pigment concentration influences.

Description: In the solar spectral range, the reflectance of the ocean surface ρos(λ) can be

assumed for a given set of geometrical condition θs, θv and φ as the sum of three components

dependent of the wavelength λ (Koepke, 1984):

ρos(θs ,θv ,φ,λ ) = ρwc (λ) +1− W.ρgl (θs,θv ,φ,λ ) +1−ρ wc (λ).ρsw (θs ,θv, φ,λ)

where •ρwc(λ) is the reflectance due to the whitecaps

•ρgl(λ) is the specular reflectance at the ocean surface

•ρsw(λ) is the scattered reflectance emerging from the sea water

•W is the relative area covered with whitecaps, for water temperature greater than

14°C it can be expressed from the wind speed ws (Monahan and O'Muircheartaigh,

1980) by W=2.9510-6.ws3.52

1-Reflectance of whitecaps wc( )

According to Koepke, 1984, "the optical influence of the whitecaps is given by the product

of the area of each individual whitecaps W with its corresponding reflectance ρf(λ). However, the

area of an individual whitecap increases with its age while its reflectance decreases. Since

whitecaps of different ages are taken into consideration in the W values, the combination of W with

ρf(λ) gives ρwc(λ) values that are too high". Thus Koepke defines, in place of ρf(λ), an effective

reflectance of ocean foam, patches ρef(λ) nearly independent of wind speed by the relation

ρwc(λ) = W.ρef (λ) = W.f ef .ρf (λ)

where fef is the efficiency factor slightly dependent of the wind speed but independent of the

wavelength (fef=0.4.± 0.2).

The figure 1 shows the reflectance of the whitecaps as a function of the wind speed in the

visible spectral range; in this range the effective reflectance ρef(λ) is a constant value (22 ± 11)%.

2-Reflectance of the sun glint gl( ) (SUBROUTINE SUNGLINT)


156

Cox and Munk (1954,1955) made measurements of the sun glitter from aerial photographs.

They defined a many-faceted model surface whose wave-slopes vary according to isotropic and

anisotropic Gaussian distribution with respect of the surface wind.

Let us consider the system of coordinates (P,X,Y,Z) where P is the observed point, Z the

altitude, PY is pointed to the sun direction and PX to the direction perpendicular to the sun plane. In

this system the surface slope is defined by its two components Zx and Zy by

xZ =∂Z

∂X= sin(α)tan(β) yZ =

∂Z

∂Y= cos(α)tan(β)

where α is the azimuth of the ascent (clockwise from the sun) and β the tilt. Using spherical

trigonometry Zx and Zy can be related to the incident and reflected directions (Π/2 ≥ θs and θv ≥ 0)

through

xZ =−sin( vθ )sin( sϕ − vϕ )

cos( sθ ) + cos( vθ )yZ =

sin( sθ ) + sin( vθ ) cos( sϕ − vϕ )

cos( sθ ) + cos( vθ )

In the case of an anisotropic distribution of slope components (dependent of the wind

direction), let us consider new principal axes (P;X',Y',Z'=Z) defined by a rotation of χ from the sun(P;X,Y,Z) system with PY' parallel to the wind direction (related clockwise from the North by wϕ ,

then χ = sϕ − wϕ ). The slope components are now expressed as

Zx'=cos(χ).Zx+sin(χ).Zy Zy'=-sin(χ).Zx+cos(χ).Zy

and the slope distribution is expressed by a Gram-Charlier series as

P( x'Z , y

'Z ) = 1

2Πσ x' σy

' exp(− ξ2 + η2

2) 1 − 1

2 C21(ξ2 − 1)− 16 C03(η3 − η) +

+ 124 C40 (ξ4 − 6ξ2 + 3) + 1

4 C22 (ξ2 −1)(η2 −1)+ 124 C04(η4 − 6η2 + 3)

where • ξ=Zx'/σx' and η=Zy'/σy'

• σx' and σy' are the rms values of Zx' and Zy', the skewness coefficients C21 and

C03, and the peakedness coefficients C40, C22 and C04 has been defined by Cox

and Munck for a clean (uncontaminated) surface as follows


157

x'2σ = 0.003 + 0.00192ws ± 0.002 y

'2σ = 0.00316ws ± 0.004

21C = 0.01 − 0.0086ws ± 0.03 03C = 0.04 − 0.033ws ± 0.12

40C = 0.40 ± 0.23 22C = 0.12 ± 0.06 04C = 0.23 ± 0.41

And the directional reflectance is written

glρ ( sθ , vθ , sϕ , vϕ ) =Π P(Z x

' ,Z y' ) R(n, sθ , vθ , sϕ , vϕ )

4 cos( sθ )cos( vθ ) cos4(β)

where R(n, sθ , vθ , sϕ , vϕ ) , defined below, is the Fresnel 's reflection coefficient (n is the complex

refractive index of the sea water, see below).

Figure 2 shows computations of ρgl for a wind speed of 05 and 15m/s, χ equal to 00, 90,

180, and 270°, a solar zenith angle of 30° and a wavelength of 0.550µm.

3-Fresnel's reflection coefficient (SUBROUTINE FRESNEL and INDWAT)The coefficient R(n, s , v, s, v) is computed (Born and Wolf, 1975) involving the

absorption of the water (n nr in i ) as

R(n, sθ , vθ , sϕ , vϕ ) =1

2

(n r2 − n i

2)cos(θi ) − u[ ]2+ 2n rn i cos(θi ) + v[ ]2

(n r2 − n i

2)cos(θi ) + u[ ]2+ 2n rn i cos(θi ) − v[ ]2

with

u2 =1

2n r

2 − n i2 − sin2(θi ) + nr

2 − n i2 − sin2(θi )[ ]2 + 4n r

2n i2

v2 =1

2− nr

2 − n i2 − sin2(θi ) + n r

2 − n i2 − sin2(θi )[ ]2

+ 4n r2n i

2

cos(θi ) = 12[1+ cos(θs)cos(θv) + sin(θs)sin(θv)sin(ϕs −ϕv )]

sin(θi ) = 12[1− cos(θs)cos(θv) + sin(θs)sin(θv)sin(ϕs −ϕv )]

In 6S, the complex index of refraction of the sea water (we assume that the outside medium

is vacuum) is deduced from the complex index of refraction of the pure water given by Hale and

Querry, 1973. By default, we assume a typical sea water (Salinity=34.3ppt, Chlorinity=19ppt) as

reported by Sverdrup (1942, p. 173). McLellan (1965, p. 129) reported on measurements of the

index of refraction an increase as a function of the chlorinity. For a chlorinity of 19.0 ppt, the

increase was found to have a value of +0.006 and to be linear with the salt concentration Csal (see


158

also Friedman, 1969). For the extinction coefficient, Friedman, 1969, reported that no correction is

required between 1.5 and 9 µm.

Then in 6S we apply a correction δnr of +0.006 on the index of refraction of pure water and no

correction δni for the extinction coefficient. Therefore the user is able to enter his own salt

concentration. In that case, a linear interpolation is assumed to correct the index of refraction of the

pure water, so that δnr=0 for Csal=0ppt and δnr=+0.006 for Csal=34.3ppt.

4-Reflectance emerging from the sea water sw( ) (SUBROUTINE MORCASIWAT for Rw)

The reflectance emerging from sea water (also called remote sensing reflectance of the sea

water) ρsw(θs,θv,φ,λ) is the reflectance as observed just above the sea surface (level 0+). This

reflectance can be related to the reflectance Rw which is the ratio of the upwelling to downwelling

radiance just below the sea surface (level 0-). If we assume the ocean as a Lambertian reflector

ρsw(θs,θv,φ,λ) can be expressed by the relation:

ρsw (θs ,θv ,φ, λ) =1

n2Rw(λ).td (θs).t u (θv )

1 − a.R w(λ )

where:

•td is the transmittance for the downwelling radiance, and is expressed to the Fresnel reflectance

coefficient Ra-w(θs,θd,φ) for the air-water interface by the relation:

td (θs) = 1− Ra−w (θs ,θda ,φ).cos(θd

a ).sin(θda ).dθd

a .dφ0

Π /2

∫0

2Π

∫The angle θd

a represents (see Figure 3) the zenithal angle of the reflected solar beam according to

the wave-slopes distribution (Cox and Munk's model, see below).

•tu is the transmittance for the upwelling radiance, and is expressed to the Fresnel reflectance

coefficient Rw-a(θv,θu,φ) for the water-air interface by

t u(θv) =1 − Rw−a(θv ,θuw,φ).cos(θu

w).sin(θuw).dθu

w.dφ0

Π /2

∫0

2Π

∫The angle θu

w represents (see Figure 3) the zenith angle in the water of the upwelling beam

according to the Fresnel and Snell's law: nair sin(θair)=nseasin(θsea) and to the wave-slopes

distribution.

•a is defined by

a =1 − tu(θv).cos(θv).sin(θv ).d0

Π /2

∫ θv

In order to minimize computations we adopted a constant value of a=0.485. In theory, the value of a

depends on wind speed and water index of refraction. In practice, this value varies very little with


159

wind speed (see table 2 of Austin, 1974) and the index of refraction of water in that range of

wavelength (0.4µm-0.7µm) is almost constant and taken equal to 1.341.

As described above the irradiance reflectance Rw(λ) is the ratio of the upwelling spectral

irradiance Eu(λ) to the downwelling irradiance Ed(λ) just below the surface. This ratio is

particularly dependent on the inherent optical properties of the sea water: the total absorption

coefficients a(λ) [m -1] and the total backscattering coefficient bb(λ) [m -1]. For example, Morel and

Prieur, 1977, have shown within a good approximation (when a(λ) <<1) that it can be expressed as:

Rw(λ) = 0.33bb(λ)

a(λ )

According to Morel, 1988, "in many situations phytoplankton and their derivative, and

detrital products (mainly particulate, but also dissolved) play a predominant role in determining the

optical properties of oceanic waters. These waters are classified (by Morel) as "case I" waters and

are opposed to "case II" waters for which sediments, or dissolved yellow substance, make an

important or dominant contribution to the optical properties". Here we use the so called "case I

waters" (defined in a range from 0.4 to 0.7 µm) which roughly corresponds to the case I, case IA,

case IB, case II, and case III of the Jerlov's chart of optical water type (Jerlov, 1951, 1976). For the

so called "Case I waters", Morel splits the total backscattering coefficient into

bb(λ) =1

2bw (λ) + ˜ b b (λ).b

where:

•bw(λ) is the molecular scattering coefficient of water and is given Figure 4

• ˜ b b(λ) is the ratio backscattering/scattering coefficient of the pigments and is related to the

pigment concentration C (Chl a + Pheo a, in mg.m-3) and the wavelength (in µm) by

˜ b b(λ) = 0.002 + 0.02(0.5 − 0.25 log10 C)0.550

λ•b is the scattering coefficient of pigment expressed by

b=0.3C0.62

Also according to Morel's "Case I waters", the total absorption coefficient is written as

a(λ)=u(λ).Kd(λ)

where:

•u(λ) is computed as follow

u(λ) = 0.901− Rw (λ)

1+ 2.25 Rw (λ)

•Kd(λ) is the total diffuse attenuation coefficient for downwelling irradiance and is given by

Kd(λ)=Kw(λ)+χc(λ)Ce(λ)


160

with Kw(λ) (the diffuse attenuation coefficient for pure oceanic water), χc(λ) and e(λ) are tabular

values. We report Figure 5 the computations of Kd(λ) for several pigment concentrations.

Always according to the Morel's model (Case I waters) the computation of the reflectance

Rw(λ) is only dependent of the pigment concentration C. The Figure 6 shows the computed

reflectance Rw in the range from 0.4 to 0.7 µm for different concentrations C.

If the only information you have is the water type following the Jerlov's chart, you can use

the approximate C values given by Morel, 1988:

0-0.01 0.05 0.1 0.5 1.5-2 mg.m-3

I IA IB II III

The Parameters are:1- ws, wϕ , Csal, C

with: ws is the wind speed (in m/s)

wϕ is the direction of the wind (clockwise from the North)

Csal is the salt concentration (in ppt). If Csal < 0 then Csal=34.3ppt by default

C is the pigment concentration (Chl a + Pheo a in mg.m-3)


161

0.000

0.005

0.010

0.015

0.020

0 5 10 15 20

Whi

tcap

s R

efle

ctan

ce

Wind Speed

Figure 1: Whitecaps Reflectance as defined by Koepke, 1984,

in the visible spectral range.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-60 -40 -20 0 20 40 60

χ=00°

χ=90°-270°

χ=180°

ρ gl


ws = 05 m/s

ws = 15 m/s

Figure 2: Reflectance of the sunglint in the principal plane for a

wind speed of 5 and 15 m/s and for several χ=ϕs-ϕw. The solar zenith

angle has been up to 30° and the wavelength to 0.550µm.


162

θs θs θs

θd1θd3

θd2

SeaSurface

Level 0+

Level 0-

SeaWater

Rw

θv θv θv

a

aa

θu3wθu2

wθu1w

Ed

Eu

Figure 3

0.000

0.002

0.004

0.006

0.008

0.010

0.4 0.45 0.5 0.55 0.6 0.65 0.7spec

tral

mol

ecul

ar s

catte

ring

coe

ffic

ient

bw(λ

)

λ (µm)

Figure 4: spectral molecular scattering coefficients (m-1)


163

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

tota

l dif

fuse

atte

nuat

ion

coef

fici

ent

λ(µm)

10.0

3.0

1.0

0.0

Figure 5: Diffuse attenuation coefficients (m-1) as modeled in Morel's case I waters

for several pigment concentration (0.0-0.03-0.1-0.3-1-3-10 mg.m-3)

10- 4

10- 3

10- 2

10- 1

350 400 450 500 550 600 650 700 750

Rw(λ

)

λ (nm)

0.3

3.01.0

0.10.03

0.0

10.0

Figure 6: Spectral irradiance reflectances as defined by the Morel's model related to the "Case I

waters" for several pigment concentration (0.0-0.03-0.1-0.3-1-3-10 mg.m-3)


164

References

R.W. AUSTIN, and T.J. PETZOLD, Spectral dependence of the diffuse attenuation coefficient of

light in ocean water, Opt. Eng., 25, 471-479, 1986.

M. BORN, and E. WOLF, Principles of Optics - fifth edition, Pergamon Press, New-York, 1975

C. COX, and W. MUNK, Statistics of the sea surface derived from sun glitter, J. Marine Res., 13,

198-227, 1954.

C. COX, and W. MUNK, Measurement of the roughness of the sea surface from photographs of the

sun's glitter, J. Opt. Soc. Am., 44, 838-850, 1954.

C. COX, and W. MUNK, Some problems in optical oceanography, J. Marine Res., 14, 63-78, 1955.

D. FRIEDMAN, Infrared Characteristics of Ocean Water (1.5-15µ), Appl. Opt. , 8-10, 2073-2078,

1969.

G.M. HALE, and M.R. QUERRY, Optical Constants of Water in the 200 nm to 200 µm

Wavelength Region, Appl. Opt., 12-3, 555-563, 1973.

N.G. JERLOV, Optical Studies of Ocean Water, Rep. Swed. Deep. Sea. Exped., 1947 1948, 3, 1-19,

1951.

N.G. JERLOV, Marine Optics, Elsevier Oceanogr. Ser., vol. 14, 231 pp., Elsevier, Amsterdam,

1976.

P. KOEPKE, Effective Reflectance of Oceanic Whitecaps, Appl. Opt., 23-11, 1816-1824, 1984.

H.J. McLELLAN, Elements of physical Oceanography, Pergamon Press, Inc., New-York, 1965.

E.C. MONAHAN, and I. O'MUIRCHEARTAIGH, Optimal Power-Law Description of Oceanic

Whitecap Dependence on Wind Speed, J. Phys. Ocean, 10, 2094, 1980.

A. MOREL, Optical Modeling of the Upper Ocean in Relation to its Biogenous Matter Content

(Case I Waters), J. Geophys. Res., 93-C9, 10749-10768, 1988.

A. MOREL, and L. PRIEUR, Analysis of variations in ocean color, Limnol. Oceanogr., 22, 709-

722, 1977.

H. V. SVERDRUP, M.W. JOHNSON, and R.H. FLEMING, The Ocean, Prentice-Hall, Inc.,

Englewood Cliffs, N.J., 1942.


165

SUBROUTINE RAHMBRDF

Function: To generate a BRDF following the Rahman et al.'s model.

Description (from Raman et al., 1993) : The model follows a semi-empirical approach and is

designed to be applicable to arbitrary natural-surfaces both in visible and near-infrared using 3

parameters.

The bidirectional reflectance ρ(θs,θv,φs-φv) is assumed to be:

ρ(θs,θv ,φs ,φv) =ρ 0(cosθs )k−1(cosθv)k−1

(cosθs + cosθv)1−k F(g) 1 + R(G)[ ]

where: -ρ0 is, according to the authors, an arbitrary parameter characterizing the intensity of the

reflectance of the surface cover, but it should not be taken as a single scattering albedo

or as a normalized reflectance: the only constraint on it is ρ0 ≥0,

-k indicates the level of anisotropy of this surface,

-F(g) is a function to moderate the overall contributions in the forward and backward

scattering (modified Heyney and Greenstein's function):

F(g) =1−Θ2

[1+Θ 2 − 2Θcos(Π − g)]1.5

with g the phase angle defined by:cos(g) = cos(θs)cos(θv) + sin(θs)sin(θv)cos(φs −φ v)

and Θ an asymmetry factor controling the relative amount of forward

(0≤Θ≤+1 ) and backward scattering (-1≤Θ≤0)

-R(G) is a function to account for the hot spot

R(G) =1 −ρ 0

1+ G

with G a geometrical factor expressed by:

G = (tanθs )2 + (tan θv )2 − 2tan θs tanθv cos(φs −φ v)


166

Parameters are:1- ρ0, Θ, k

with ρ0=intnsity of the reflectance of the surface cover

Θ=asymmetry factor used in the modified Heyney and Greenstrein's function

k=structural parameter

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0

Cal

cula

ted

Ref

lect

ance


Sun at 60°

Sun at 00°

ρ0=0.012

Θ=-0.391k=0.811

Clover Patch

Backward Forward

References

H. RAHMAN, M. VERSTRAETE, and B. PINTY, A coupled Surface-Atmosphere Reflectance

(CSAR) Model. Part1: Model description and inversion on synthetic data, Submitted at J.

Geophys. Res., 1993

H. RAHMAN, B. PINTY, and M. VERSTRAETE, A coupled Surface-Atmosphere Reflectance

(CSAR) Model. Part2: A Semi-empirical Surface Model usable with NOAA/AVHRR Data,

Submitted at J. Geophys. Res., 1993


167

SUBROUTINE ROUJBRDF

Function: To generate a BRDF following the Roujean et al.'s (1992) model.

Description (from Roujean et al., 1992) : The model follows a semi-empirical approach and

is designed to be applicable to heterogeneous surfaces. It considers that the observed surface

bidirectional reflectance is the sum of two main processes operating at a local scale:

- A diffuse reflection component taking into account the geometrical structure of opaque

reflectors on the surface, and shadowing effects

- A volume scattering contribution by a collection of dispersed facets which simulates the

volume scattering properties of canopies and bare soils.

This combinaison is made by assuming that the bidirectional reflectance ρ(θs,θv,φ) can be

expressed as:

ρ(θs,θv,φ)=αρgeom+(1-α)ρvol

where α is an empirical coefficient which characterizes the relative weight of the geometric and

volume component in the final bidirectional signature.

Then the bidirectional reflectance model may be written

ρ(θs,θv,φ)=k0+k1f1(θs,θv,φr)+k2f2(θs,θv,φr)

where: • φ = | |φs-φv

• k0, k1 and k2 are related to basic macroscopic properties of the surface

k0=ρ0[α+(1-α)e-bF]+r3(1-e-bF)(1-α)

k1=hlS ρ0α

k2=r(1-e-bF)(1-α)

with ρ0, the background and protrusion reflectance,h, the average height of the surface protrusions,l, the average length of the surface protrusions,S, the horizontal surface associated with each protrusion,r, the facet reflectance,F, the facet area index.

• f1 and f2 are simple analytic functions of the solar and viewing angles


168

f1(θs,θv,φ)=1

2Π (Π-φ)cos(φ)+sin(φ)tg(θs)tg(θv)

-1

Π tg(θs)+tg(θv)+ tg2(θs)+tg2(θv)-2tg(θs)tg(θv)cos(φ)

f2(θs,θv,φ)=4

3Π1

cos(θs)+cos(θv) (

Π2 -ξ)cos(ξ)+sin(ξ)-1

3

with ξ, the phase angle defined by

cos(ξ) = cos(θs) cos(θv) + sin(θs) sin(θv) cos(φ)

The parameters are:1- k0, k1, k2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0

Cal

cula

ted

Ref

lect

ance

s


backward forward

sun at 45°

sun at 30°

Plowed fieldk

0 = 0.243

k1 = 0.073

k2 = 0.642

References

J.L. ROUJEAN, M. LEROY, and P.Y. DESCHAMPS, A bidirectional reflectance model of the

Earth surface for the correction of remote sensing data, J. Geophys. Res. , 97, 20455-20468,

1992.


169

SUBROUTINE VERSBRDF

Function: To generate a BRDF following the Verstraete et al.'s (1990) model.

Description (from Pinty and Verstraete, 1991): Using the basic framework previously

suggested by Hapke (1981) (see HAPKBRDF), Verstraete et al. (1990) developed a model for

predicting the bidirectional reflectance exiting from a simple vegetation canopy. They concentrate

on the case of a fully covered, homogeneous and semi-infinite canopy made of leaves only. With

the same notations as for Hapke's model, the parametric version of the derived model (see Pinty et

al., 1990) is as follow:

ρ(θs,φs,θv,φv)=ω4

κs

κvµs+κsµv [1+Pv(G)]P(g)+H(κs/µs)H(κv/µv)-1

where

• ω is average single-scattering albedo of the particle making up the surface

• µs=cos(θs) and µv=cos(θv)

• κs and κv describe the leaf orientation distribution for the illumination and viewing angles,respectively. There are 3 possible options (see below, line1-opt3)

-κs and κv are entered by the user

-κ is obtained from Goudriaan's (1977) parameterization

κx = Ψ1 + Ψ2 µx

with Ψ1 = 0.5 - 0.6333 χl - 0.33 χl2 and Ψ2 = 0.877 (1 - 2 Ψ1)

-κ is obtained from the Dickinson et al.'s (1990) correction to Goudriaan's (1977)parameterization

κx = Ψ1 + Ψ2 µx

with Ψ1 = 0.5 - 0.489 χl - 0.11 χl2 and Ψ2 = 1 - 2 Ψ1

where χl is a function of the leaf angle distribution in the canopy, and varies from -0.4for an erectophile canopy to +0.6 for a planophile canopy. The equal probability for allleaf orientations is given by χl=0.

• Pv(G) is the function that accounts for the joint transmission of the incoming and outgoing

radiation and thereby also for the hot spot phenomenon.

Pv(θs,φs,θv,φv)=1

1+Vp(θs,φs,θv,φv)


170

with Vp(θs,φs,θv,φv) is a function defined by

4(1-4

3Π ) 1

2rΛµv

κvtan2(θs)+tan2(θv)-2tan(θs)tan(θv)cos(φs-φv)

r is the radius of the Sun flecks on the inclined scatterers (in m)

Λ is the scatterer area density of the canopy (in m2.m-3)

• P(g) is the average of the phase function of the particles. There are 3 options to compute

P(g) (see below, line1-opt4):

-the case of an isotropic phase function

P(g) = 1

-the empirical function introduced by Henyey and Greenstein (1941)

P(g)=1-Θ2

1+Θ2+2Θcos(g)

-the phase function is approximated by a Legendre polynomial function

P(g)=1+Θcos(g)+L23cos2(g)-1

2

where Θ is the asymmetry factor ranging from -1 (backward scattering) to +1, g the

phase angle between the incoming and the outgoing rays defined as

cos(g) = cos(θs) cos(θv) + sin(θs) sin(θv) cos(φs-φv),

and L2 the second coefficients of the Legendre polynomial.

• H(x) is a function to account for multiple scattering (see below, line1-opt5)

- for single scattering

H(x)=0

- for multiple scattering

H(x)=1+2x

1+2(1-ω)1/2x


171

There are three lines of input parameters:

line 1- choice of options : opt3, opt4, opt5

line 2- structural parameters : str1, str2, str3, str4

line 3- optical parameters : optics1, optics2, optics3

line 1 :

opt 3- 0 for given values of κ1 for Goudriaan's parameterization of κ2 for Dickinson et al.'s correction to Goudriaan's parameterization of κ

opt 4- 0 for isotropic phase function

1 for Heyney and Greenstein's phase function

2 for Legendre polynomial phase function

opt 5- 0 for single scattering only

1 for Dickinson et al.'s parameterization of multiple scattering

line 2 :

str1- leaf area density (in m2 m-3)

str2- radius of the sun flecks on the scatterer (in m)

str3- leaf orientation parameter

if opt3=0 then str3=κs

if opt3=1 or 2 then str3=χl

str4- leaf orientation parameter (continued)

if opt3=0 then str4=κv

if opt3=1 or 2 then str4 is not used

line 3 :

optics 1- single scattering albedo, value between 0.0 and 1.0

optics 2- phase function parameter

if opt4=0 then this input is not used

if opt4=1 then asymmetry factor, value between -1.0 and 1.0

if opt4=2 then first coefficient of Legendre polynomial

optics 3- second coefficient of Legendre polynomial (if opt4=2)


172

0.00

0.05

0.10

0.15

0.20

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0

Cal

cula

ted

Ref

lect

ance

Viewing Angle in the principal plane

Backward forward

Sun at 00°

Sun at 60°Clover Patch

ω=0.099

Θ=-0.391

χl=0.115

2rΛ=0.277

References

R.E. DICKINSON, B. PINTY, and M. VERSTRAETE, Relating surface albedos in gcm to

remotely sensed data, agricultural and forest meteorology, 52, 109-131, 1990.

J. GOUDRIAAN, Crop micrometeorology: a simulation study (Wageningen: Wageningen Centre

for Agricultural Publishing and Documentation), 1977.

L. G. HENYEY AND J. L. GREENSTEIN,, Diffuse Radiation in the Galaxy. Astrophysic Journal,

93, 70, 1941.

B. PINTY, M. VERSTRAETE, and R.E. DICKINSON, A Physical Model of the Bidirectional

Reflectance of Vegetation Canopies- 1. Inversion and Validation, J. Geophys. Res., 95,

11767-11775, 1990.

B. PINTY, and M. VERSTRAETE, Extracting Information on Surface Properties From

Bidirectional Reflectance Measurements, J. Geophys. Res., 96, 2865-2874, 1991.

M. VERSTRAETE, B. PINTY, and R.E. DICKINSON, A Physical Model of the Bidirectional

Reflectance of Vegetation Canopies- 1. Theory, J. Geophys. Res., 95, 11755-11765, 1990.


173

SUBROUTINE WALTBRDF

Function: To generate a BRDF following the Walthall et al.'s (1985) model.

Description (from Walthall et al., 1985): Using a deterministic model, Walthall et al. (1985)

have simulated different canopy reflectance distributions. The 2-D contours of this distribution

appearsto be similar to the shape of the limacon of Pascal. Using the simple limacon equation other

equation forms were used in an attempt to fit the 3-D reflectance surface directly. They found

satisfactory results with the following equation:

ρ(θs,θv,φs,φv) = a θv2 + b θv cos(φv-φs) + c

where ρ is the reflectance at a given view zenith θv and view azimuth φv look angles; a, b, and c arecoefficients derived using a linear least-squares fitting procedure.

The model has to be slightly modified to match the reciprocity principle. The reflectance is written:

ρ(θs,θv,φs,φv) = a θs2 θv2 + a' (θs2+θv2) + b θs θv cos(φv-φs) + c

The parameters are:1- a, a', b, c

0.02

0.03

0.04

0.05

0.06

-80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0


backward forward

a = 0.00606a'= 0.00523b = 0.01127c = 0.02425

sun at 17°

sun at 40°

LAI=4

References

C.L. WALTHALL, J.M. NORMAN, J.M. WELLES, G. CAMPBELL, and B.L. BLAD, Simple

equation to approximate the bidirectional reflectance from vegetative canopies and bare soil

surfaces, Applied Optics, Vol 24, n°3, 383-387, 1985.


174


175

DESCRIPTION OF THE SUBROUTINES USED TO UPDATE ATMOSPHERIC PROFILE (PLANE OR ELEVATED TARGET SIMULATION)


176


177

SUBROUTINE PRESPLANE

Function: Update the atmospheric profile (P(z),T(z),H2O(z),O3(z)) in case the observer is on

board an aircraft.

Description: Given the altitude or pressure at aircraft level as input, the first task is to

compute the altitude (in case the pressure has been entered) or the pressure (in case the altitude has

been entered) at plane level. Then, a new atmospheric profile is created (Pp,Tp,H2Op,O3p) for which

the last level is located at the plane altitude. This profile is used in the gaseous absorption

computation (ABSTRA.f) for the path from target to sensor (upward transmission). The ozone and

water vapor integrated content of the "plane" atmospheric profile are also an output of this

subroutine. The last output is the proportion of molecules below plane level which is useful in

scattering computations (OS.f,ISO.f).


178

SUBROUTINE PRESSURE

Function: Update the atmospheric profile (P(z),T(z),H2O(z),O3(z)) in case the target is not at

sea level.

Description: Given the altitude of the target in kilometers as input, we transform the

original atmospheric profile (Pressure, Temperature, Water Vapor, Ozone) so that first level of the

new profile is the one at the target altitude. We also compute the new integrated content in water

vapor and ozone, that are used as outputs or in computations when the user chooses to enter a

specific amount of Ozone and Water Vapor.


179

DESCRIPTION OF THE SUBROUTINES

USED TO READ THE DATA


180


181

SUBROUTINE SOLIRR

Function: To read the solar irradiance (in Wm-2) from 250 nm to 4000 nm by steps of 5 nm,

The total solar irradiance is put equal to 1372 Wm-2. Between 250 and 4000 nm we have 1358 Wm -

2.

0

500

1000

1500

2000

0 0.5 1 1.5 2 2.5 3 3.5 4

Sol

ar I

rrad

ian

ce [

W/m

2 ]

Wavelength [ m]

Reference:

H. NECKEL, D. LABS, The solar radiation between 3300 and 12500 Solar Physics 90, p. 205-258,

1984.


182

SUBROUTINE VARSOL

Function: To take into account the variation of the solar constant as a function of the Julian

day.

Description: We apply a simple multiplicative factor DS to the solar constant CS .DS is

written as,

DS=1

(1-ecosM)2

with

M = 0.9856 (J-4) π

180

where e = 0.01673 and J is the julian day.

Reference:

G.W. PALTRIDGE, C.M.R. PLATT, Radiative Processes in Meteorology and Climatology,

Development in Atmospheric Science, 5, Elsevier Scientific Publishing Company, New-York,

N.Y. 10017, 1977.


183

SUBROUTINE AVHRR

Function: To read the two spectral bands (red and near infrared ) of Advanced Very High

Resolution Radiometer (AVHRR) on NOAA 6,7,8,9,10 and 11 (extreme wavelengths and spectral

response of the filter function).

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m ]

NOAA6Channel 1

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA6Channel 2

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA7Channel 1

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA7Channel 2


184

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA8Channel 1

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA8Channel 2

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Sp

ectr

al

Res

pon

se

[%]

Wavelength [ m]

NOAA9Channel 1

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA9Channel 2

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA10Channel 1

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA10Channel 2


185

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA11Channel 1

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

NOAA11Channel 2

References:

NOAA POLAR ORBITER DATA USERS GUIDE, (1985) U.S. Dept. of Commerce, NOAA,

National Environment Satellite, National Climatic Data Center, Satellite Data Service Division,

World Weather Building, Room 100, Washington D.C., 202333, U.S.A.

DUE C.T., 1982, Optical-Mechanical Active/passive imaging Systems - Volume II, Report number

153200-2-TIII- ERIM Infrared information and Analysis Center, P.O. BOX 8518, Ann. Arbor.,

MI.98107.

SCHNEIDER S.R. and Mc GINNIS D.F., 1982, The NOAA/AVHRR: A new satellite sensor for

monitoring crop growth, Proc. 8th Inter. Symp.on Machine Processing of Remotely Sensed

data, Purdue University, Indiana, p. 250-281.


186

SUBROUTINE GOES

Function: To read the visible spectral bands of the Visible Infrared Spin-Scan Radiometer

(VISSR) on GOES 5 (East) and GOES 4 (West), (extreme wavelengths and spectral response of the

filter function).

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Sp

ectr

al R

esp

onse

[%

]

Wavelength [ m]

GOESEAST

0

20

40

60

80

100

120

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Sp

ectr

al R

esp

onse

[%

]

Wavelength [ m]

GOESWEST

References:

STAFF MEMBERS, 1980, Visible infrared Spin Scan Radiometer Atmospheric sounder system

description, Santa Barbara Research center, Goleta, Cali., U.S.A.

CORBELL R.P., C.J. CULLAHAN and W.J. KOTSCH, 1976, The GOES/SMS user's Guide: U.S.

Dept. of Commerce, NOAA, NESS, Washington D.C., U.S.A.


187

SUBROUTINE HRV

Function: To read the four spectral bands of High Resolution Visible (HRV1 and 2) on Spot

1 (extreme wavelengths and spectral response of the filter function).

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV1Channel 1

0

20

40

60

80

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV1Channel 2

0

20

40

60

80

100

0.7 0.75 0.8 0.85 0.9 0.95 1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV1Channel 3

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV1Panchromatique


188

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV2Channel 1

0

20

40

60

80

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV2Channel 2

0

20

40

60

80

100

0.7 0.75 0.8 0.85 0.9 0.95 1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV2Channel 3

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

HRV2Panchromatique

Reference:

CHEVREL M., M. COURTOIS, and G. WEILL, 1981, The Spot Satellite Remote Sensing Mission

Photogrammetric Engineering and Remote Sensing, Vol. 47, no 8, p. 1163-1171.


189

SUBROUTINE METEO

Function: To read the visible spectral band of the radiometer on Meteosat 2 (extreme

wavelengths and spectral response of the filter function).

0

20

40

60

80

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Sp

ectr

al r

esp

onse

[%

]

Wavelenght [ m]

METEOSAT 2

Reference:

MORGAN, 1981, Introduction to the Meteosat System, ESOC, Darmstadt, R.F.A.


190

SUBROUTINE MSS

Function: To read the four spectral bands of the Multispectral Scanner System (MSS) on

Landsat 5 (extreme wavelengths and spectral response of the filter function).

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

MSS4

0

20

40

60

80

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

MSS5

0

20

40

60

80

100

0.6 0.65 0.7 0.75 0.8 0.85 0.9

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

MSS6

0

20

40

60

80

100

0.7 0.8 0.9 1 1.1 1.2

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

MSS7

References:

LANDSAT DATA USERS HANDBOOK (revised) (1979), U.S. Geol. Survey, EROS Data Center,

Sioux Falls, SD 57198.

LANDSAT DATA USERS NOTES (1982), International Land Satellite programs, ibid.


191

SUBROUTINE TM

Function: To read the six visible spectral bands of Thematic Mapper (TM) on Landsat 5

(extreme wavelengths and spectral response of the filter function).

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

TM 1

0

20

40

60

80

100

0.4 0.45 0.5 0.55 0.6 0.65 0.7Sp

ectr

al

Res

pons

e [%

]Wavelength [ m]

TM 2

0

20

40

60

80

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

TM 3

0

20

40

60

80

100

0.7 0.75 0.8 0.85 0.9 0.95 1

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

TM 4

0

20

40

60

80

100

1.4 1.5 1.6 1.7 1.8 1.9 2

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

TM 5

0

20

40

60

80

100

1.9 2 2.1 2.2 2.3 2.4 2.5

Spec

tral

R

espo

nse

[%]

Wavelength [ m]

TM 7

Reference:

MARKHAM B.L. and J.L. BARKER, 1985, Spectral Characterization of the LANDSAT thematic

Mapper sensors, Int. J. Remote Sensing,


192

SUBROUTINE CLEARW

Function: To read a typical spectral reflectance of clear water (sea) from 250 to 4000 nm by

steps of 5 nm.

0

1

2

3

4

5

6

7

0.4 0.5 0.6 0.7 0.8 0.9 1

Clear water (sea)

Wavelengths (microns)

Reference:

M. VIOLLIER, Télédétection des concentrations de seston et pigments chlorophylliens contenus

dans l'Océan, Thèse de Doctorat d'Etat, no 503, 1980.


193

SUBROUTINE LAKEW

Function: To read a typical spectral reflectance of water (lake) from 250 to 4000 nm by steps

of 5 nm.

0

2

4

6

8

10

0.4 0.5 0.6 0.7 0.8 0.9 1

Clear water (lake)

Wavelength [ m]

Reference:

KONDRATYEV K. Ya (1969), Radiation in the atmosphere, Academic Press, N.Y. 10003, U.S.A.


194

SUBROUTINE SAND

Function: To read a typical spectral reflectance of sand from 250 to 4000 nm by steps of 5

nm.

0

5

10

15

20

25

30

35

40

0.4 0.8 1.2 1.6 2

Dry sand

Wavelength [ m]

Reference:

R. STAETTER, M. SCHROEDER, Spectral characteristics of natural surfaces, Proceeding of ten

Int. Conf. on Earth Obs. from Space, 6-11 March 1978, (ESA-SP, 134).


195

SUBROUTINE VEGETA

Function: To read a typical spectral reflectance of a mean green vegetation surface from 250

to 4000 nm by steps of 5 nm.

0

10

20

30

40

50

60

0.5 1 1.5 2 2.5

Clear water (sea)Clear water (lake)VegetationDry sand

Ref

lect

ance

[%

]

Wavelength [ m]

Reference:

Manual of Remote Sensing, Falls Church, Virginia, American Society of Photogrammetry, 1983.


196

SUBROUTINE DICA1

Function: To read the eight coefficients necessary to compute the CO2 transmission

according to the Malkmus model (see the text). The frequency interval is 2500-5050 cm-1, step of

10 cm-1.

SUBROUTINE DICA2

Function: Same as DICA1 but for frequency interval 5060-7610 cm-1.

SUBROUTINE DICA3

Function: Same as DICA1 but for frequency interval 7620-10170 cm-1.


197

SUBROUTINE METH1

Function: To read the eight coefficients necessary to compute the methane transmission


10 cm-1.

SUBROUTINE METH2

Function: Same as METH1 but for frequency interval 5060-7610 cm-1.

SUBROUTINE METH3


SUBROUTINE METH4


SUBROUTINE METH5


SUBROUTINE METH6



198

SUBROUTINE MOCA1

Function: To read the eight coefficients necessary to compute the CO transmission according

to the Malkmus model (see the text). The frequency interval is 2500-5050 cm-1, step of 10 cm-1.

SUBROUTINE MOCA2

Function: Same as MOCA1 but for frequency interval 5060-7610 cm-1.

SUBROUTINE MOCA3


SUBROUTINE MOCA4


SUBROUTINE MOCA5


SUBROUTINE MOCA6



199

SUBROUTINE NIOX1

Function: To read the eight coefficients necessary to compute the nitrous oxyde transmission


10 cm-1.

SUBROUTINE NIOX2

Function: Same as NIOX1 but for frequency interval 5060-7610 cm-1.

SUBROUTINE NIOX3


SUBROUTINE NIOX4


SUBROUTINE NIOX5


SUBROUTINE NIOX6



200

SUBROUTINE OXYG3

Function: To read the eight coefficients necessary to compute the O2 transmission according

to the Malkmus model (see the text). The frequency interval is 2500-5050 cm-1, step of 10 cm-1.

SUBROUTINE OXYG4

Function: Same as OXYG3 but for frequency interval 10180-12730 cm-1.

SUBROUTINE OXYG5


SUBROUTINE OXYG6



201

SUBROUTINE OZON 1

Function: To read the eight coefficients necessary to compute the O3 transmission according to

the Malkmus Model (see the text). The frequency interval is 2500-5050 cm-1, steps of 10 cm-1.


202

SUBROUTINE WAVA1

Function: To read the eight coefficients necessary to compute the H2O transmission

according to the Goody Model (see the text ). The frequency interval is 2500-5050 cm-1, steps of 10

cm-1.

SUBROUTINE WAVA2

Function: Same as WAVA1 but for frequency interval 5060-7610 cm-1.

SUBROUTINE WAVA3


SUBROUTINE WAVA4


SUBROUTINE WAVA5


SUBROUTINE WAVA6



203

SUBROUTINE DUST

Function: To read the scattering phase function for the Dust-Like component. Computations

have been performed for 83 phase angles (80 Gauss Angles and 0°, 90°, 180°) and 10 wavelengths,

0.400, 0.515, 0.550, 0.633, 0.694, 0.860, 1.536, 2.250, 3.750 µm.

SUBROUTINE OCEA

Function: Same as DUST but for the oceanic component.

SUBROUTINE SOOT

Function: Same as DUST but for the soot component.

SUBROUTINE WATE

Function: Same as DUST but for the water-soluble component.

SUBROUTINE BBM

Function: Same as DUST but for biomass burning model.

SUBROUTINE BDM

Function: Same as DUST but for desertic background model.

SUBROUTINE STM

Function: Same as DUST but for stratospheric model.

Reference:

R.A. Mc CLATCHEY, M.J. BOLLE, K.Ya. KONDRATYEV, "A preliminary cloudless standard

atmosphere for radiation computation", Boulder, Colorado, U.S.A. (1982).

E. P. SHETTLE, Optical and radiative properties of a desert aerosol model. Symposium of

Radiation in the atmosphere, 1 Deepak publishing), pp. 74-77, 1984.

M. KING, HARSHVARDHAN, and ARKING, A, A model of the Radiative Properties of the El

Chichon Stratospheric Aerosol Layer, J. Appl. Meteor., 23, (7), pp. 1121-1137, 1984.


204

SUBROUTINE MIDSUM

Function: To read the midlatitude summer model atmosphere, i.e.pressure (mb), temperature

(K), water vapor and ozone concentrations (g/m3) as a function of the altitude (34 levels)

• Z = 1 km for 0 < Zkm < 25

• Z = 5 km for 25 < Zkm < 50

• Z = 70, 100 km and ∞ (p = 0).

200 220 240 260 280 300

-4 -3 -2 -1 0 1 2 3 4

0

20

40

60

80

100

Log(P [mb] )

Temperature [K]

Alt

itud

e [K

m]

P

T

Mid. Sum. atm.

-11 -10 -9 -8 -7 -6 -5 -4 -3

-10 -8 -6 -4 -2 0 2

0

20

40

60

80

100

Log(H 2O dens. [g/m3])

Log(O3 dens [g/m 3])

Alt

itud

e [K

m]

H2O

O3

Mid. Sum. atm.

Reference:

Mc CLATCHEY R.A., FENN R.W., SELBY J.E.A., VOLZ F.E.and GARING J.S., Optical

properties of the Atmosphere, AFCRL-TR- 71-0279, Enviro. Research papers, No 354, L.G.

HANCOM FIEL Bedford, Mass. U.S.A., 1971.


205

SUBROUTINE MIDWIN

Function: To read the midlatitude winter model atmosphere, i.e., pressure (mb), temperature


• Z = 1 km for 0 < Zkm < 25

• Z = 5 km for 25 < Zkm < 50

• Z = 70, 100 km and ∞ (p = 0).

180 200 220 240 260 280 300

-4 -3 -2 -1 0 1 2 3 4

0

20

40

60

80

100

Log(P [mb] )

Temperature [K]

Alt

itud

e [K

m]

P

T

Mid. Win. atm.

-11 -10 -9 -8 -7 -6 -5 -4 -3

-10 -8 -6 -4 -2 0 2

0

20

40

60

80

100

Log(H 2O dens. [g/m3])

Log(O3 dens [g/m 3])

Alt

itud

e [K

m]

H2O

O3

Mid.Win. atm.


properties of the Atmosphere, AFCRL-TR- 71-0279, Enviro. Research papers, No 354, L. G.



206

SUBROUTINE SUBSUM

Function: To read the subarctic summer model atmosphere, i.e., pressure (mb), temperature


• Z = 1 km for 0 < Zkm < 25

• Z = 5 km for 25 < Zkm < 50

• Z = 70, 100 km and ∞ (p = 0).

180 200 220 240 260 280 300

-4 -3 -2 -1 0 1 2 3 4

0

20

40

60

80

100

Log(P [mb] )

Temperature [K]

Alt

itud

e [K

m]

P

T

Sub. Sum. atm.

-11 -10 -9 -8 -7 -6 -5 -4 -3

-10 -8 -6 -4 -2 0 2

0

20

40

60

80

100A

ltit

ude

[Km

]

H2O

O3

Sub. Sum. atm.

Log(O3 dens. [g/m3])

Log(H2O dens. [g/m3])

Reference:

Mc CLATCHEY R.A., FENN R.W., SELBY J.E.A., VOLZ F.E. AFCRL-TR- 71-0279, Enviro.

Research papers, No 354, L.G. HANCOM FIEL Bedford, Mass. U.S.A., 1971.


207

SUBROUTINE SUBWIN

Function: To read the subarctic model atmosphere, i.e., pressure (mb), temperature (K), water

vapor and ozone concentrations (g/m3) as a function of the altitude (34 levels)

• Z = 1 km for 0 < Zkm < 25

• Z = 5 km for 25 < Zkm < 50

• Z = 70, 100 km and ∞ (p = 0).

180 200 220 240 260 280 300

-4 -3 -2 -1 0 1 2 3 4

0

20

40

60

80

100

Log(P [mb] )

Temperature [K]

Alt

itud

e [K

m]

P

T

Sub. Win. atm.

-11 -10 -9 -8 -7 -6 -5 -4 -3

-10 -8 -6 -4 -2 0 2

0

20

40

60

80

100



Alt

itud

e [K

m]

H2O

O3

Sub. Win. atm.

Reference:

Mc CLATCHEY R.A., FENN R.W., SELBY J.E.A., VOLZ F.E. and GARING J.S., Optical




208

SUBROUTINE TROPIC

Function: To read the tropical model atmosphere, i.e., pressure (mb), temperature (K), water


• Z = 1 km for 0 < Zkm < 25

• Z = 5 km for 25 < Zkm < 50

• Z = 70, 100 km and ∞ (p = 0).

180 200 220 240 260 280 300

-4 -3 -2 -1 0 1 2 3 4

0

20

40

60

80

100

Log(P [mb] )

Temperature [K]

Alt

itud

e [K

m]

P

T

Trop. atm.

-11 -10 -9 -8 -7 -6 -5 -4 -3

-10 -8 -6 -4 -2 0 2

0

20

40

60

80

100A

ltit

ude

[Km

]

H2O

O3

Trop. atm.



Reference:





209

SUBROUTINE US 62

Function: To read the U.S. Standard Atmosphere, i.e., pressure (mb), temperature (K), water


• Z = 1 km for 0 < Zkm < 25

• Z = 5 km for 25 < Zkm < 50

• Z = 70, 100 km and ∞ (p = 0).

180 200 220 240 260 280 300

-4 -3 -2 -1 0 1 2 3 4

0

20

40

60

80

100

Log(P [mb] )

Temperature [K]

Alt

itud

e [K

m]

P

T

US62 atm.

-11 -10 -9 -8 -7 -6 -5 -4 -3

-10 -8 -6 -4 -2 0 2

0

20

40

60

80

100A

ltit

ude

[Km

]

H2O

O3

U62 atm.



Reference:

Mc CLATCHEY R.A., FENN R.W., SELBY J.E.A., VOLZ F.E. and GARING J.S., Optical




210


211

MISCELLANEOUS


212


213

SUBROUTINE EQUIVWL

Function: To compute the equivalent wavelength needed for the calculation of the

downward radiation field used in the computation of the non lambertian target contribution

(main.f).

Description: The input is the spectral response of the selected sensor as well as the solar

irradiance spectrum. The output is the equivalent wavelength which is computed by averaging the

spectral response of the sensor over the solar irradiance using increment of 2.5nm,


214

SUBROUTINE PRINT_ERROR

Function: provide centralized error handling for the code and output specific error

messages.


215

SUBROUTINE SPECINTERP

Function: To compute the atmospheric properties at the equivalent wavelength (see

EQUIVWL.f) needed for the calculation of the downward radiation field used in the computation of

the non lambertian target contribution (main.f).

Description: The input is the equivalent wavelength for which coupling between BRDF and

atmosphere is to be computed. Using the actual aerosol model, the ouput are the atmospheric

properties at this wavelength that is the aerosol optical thickness, the aerosol single scattering

albedo and the aerosol phase function. In addition, this routine modifies the thickness and single

scattering albedo of the aerosol layer below an aircraft.


216

SUBROUTINE SPLIE2,SPLIN2,SPLINE,SPLINT

Function: Perform interpolation of a furnished BDRF discrete dataset (BRDFGRID.f,

option 1 of BRDF model) to compute the albedo (ρ= ) as well as the BRDF value at the gaussian

quadrature points.

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