Solar Radiative Transfer_2

8/3/2019 Solar Radiative Transfer_2

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22 SSOOLLAARRRRAADDIIAATTIIVVEE TTRRAANNSSFFEERR

2 SOLAR RADIATIVE TRANSFER ................................................................................ 1

2.1 RT EQUATION FOR SOLAR RADIATIVE TRANSFER ................................................... 2

2.2 THE SOURCE FUNCTION .......................................................................................... 5

2.3 MORE ON THE PHASE FUNCTION ............................................................................. 6

2.4 THE GENERAL SOLUTION OF THE SOLAR RT EQUATION ...................................... 10

2.5 THE REFLECTION AND TRANSMISSION FUNCTIONS FOR A SINGLE LAYER ............ 13

2.6 SINGLE SCATTERING APPROXIMATION................................................................... 15

2.7 EXACT SOLUTIONS TO THE RT EQUATION............................................................ 21

2.7.1 The Adding method ...............................................................................................212.7.2 Application of Adding Method to Inhomogeneous Atmospheres...................262.7.3 The Discrete Ordinates Method ..........................................................................29

2.7.4 Quadrature Rules ..................................................................................................302.7.5 Back to the discrete ordinates .........................................................................32

2.8 THE SUN ................................................................................................................. 35


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22..11 RRTT EEQQUUAATTIIOONN FFOORRSSOOLLAARRRRAADDIIAATTIIVVEE TTRRAANNSSFFEERR

We are interested in calculating instantaneous radiance in the solar spectral

range (wavelength less than about 5 microns) in any direction at any point in

the atmosphere. To render the problem manageable we assume that the planeparallel assumption can be followed.

The transfer of radiation can be divided into the direct and diffuse component.

The former represents the attenuation of the unscattered solar beam and

requires knowledge of the position and irradiance of the Sun, which is treated

as a point source. The latter arises from the photons that are multiply scattered

and, to a far lesser extent, from photons emitted by the atmosphere and surface.

In polar coordinated the directions of incoming and outgoing light beams aredenoted by ),( and ),( , where )cos( = , is the zenith angleand is the azimuthal angle. Positive indicate the upward direction of the

light beam. Therefore the position of the Sun is )0,( 0 as in figure Fig.2.1-1

Fig. 2.1-1 Geometry for solar radiative transfer.


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For a proper treatment of both upwelling (>0) and downwelling (


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+

+=

d

eJ

eIL

/)(

0

/

),,(

),,0(),,(

Eq. 2.1-4

In Eq. 1-3 and 1-4 the terms ),,( 1 I and ),,0( represent the radiances

entering the layer at the top and bottom levels

Fig. 2.1-2 Upwelling and downwelling radiance in a finite, plane parallelatmosphere


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22..22 TTHHEE SSOOUURRCCEE FFUUNNCCTTIIOONN

In general the source function can be expressed as:

+

++=

2

0

1

1

/00

'')',',,()',',(4

~

)()~1(),,,(4~),,( 0

ddPL

TBePSJ

Eq. 2.2-1

ev

sv

v k

k

=~

is the albedo for single scattering

)',',,( P is the Phase Function and describes the portion of

radiation entering the volume from direction)','( that is scattered into direction ),( .

SS = S is the solar constant and is the irradiance over a

unit horizontal area.S

The source function ),,( J (Eq. 2.2-1) is made of 3 terms:

1) the first term on the r.h.s. accounts for the direct solar irradiance on a

horizontal surface at TOA coming from direction (S ), 00 ,

transmitted through the atmosphere with transmittance , and

diffused in direction

0/e

),( within the volume of air;

2) the second term at the r.h.s. is the isotropic emission term;

3) the third term at the r.h.s is the radiance arriving from direction

)','( that interacts with the volume at optical depth and isscattered into direction ),( . The single scattering albedodetermines the amount of energy which is diffused and the phase

function determines its distribution within the 4 solid angle aroundthe volume of air.


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22..33 MMOORREE OONN TTHHEE PPHHAASSEE FFUUNNCCTTIIOONN

The Phase Function is a normalised quantity so that

1)',',,(4

1 2

0

1

1

=

ddP

Eq. 2.3-1

When scattering is due to spherical particles, or randomly oriented non

spherical particles, the phase function can be expressed using only one angle,

the scattering angle , which is related to azimuth and zenith angles by

)'cos()1()1(cos

)'cos('sinsin'coscoscos

2/122/12

+=

+=

Eq. 2.3-2

and the normalization is now written as:

1cos)(cos41

2

0

1

1

=

ddP

Eq. 2.3-3

Our objective is to solve Eq. 2.1-2 which in presence of multiple scattering

(Eq. 2.2-1) becomes an integro-differential equation. To this aim we could, for

example, expand the phase function using some orthogonal base function such

as the Legendre polynomials:

=

=N

lll PP

0

)(cos)(cos

Eq. 2.3-4

The Legendre polynomials Plcan be defined as:

[ ll

l

ll d

d

lP )1(

!2

1)( 2 =

]

Eq. 2.3-5

The computations ofPl can be also done using one of many useful recurrencerelations:


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)(12

)(12

1)( 11 + +

++

+= lll P

l

lP

l

lP

Eq. 2.3-6

The termsPlpossess the following orthogonal properties:

kl

kldPP

l

kl =

=

+

12

2

01

1

)()(

Eq. 2.3-7

The first 6 Legendre polynomials are given in next Table

n Pn(x) n Pn(x) n P n(x)

0 1 2 (3x2-1)/2 4 (35x4-30x2+3)/81 x 3 (5x

3-3x)/2 5 (63x

5-70x

3+15x)/8

The definition (Eq. 2.3-4 and Eq. 2.3-5) and orthogonal relations (Eq. 2.3-7)

allow to compute the expansion coefficients (multiply both sides ofEq. 2.3-4

byPk and integrate to obtain

12

2)()(

)()()(),(

1

10

0

1

1

1

1

+==

=

=

=

kdPP

dPPdPP

kkl

N

ll

k

N

lllk

Therefore

NldPPl ll K,1,0)(),(2

121

1

=+=

Eq. 2.3-8

From the normalization condition it follows that the zero-order coefficient is

1),(2

1 1

1

0 ==

dP

The first order coefficient is


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=

dP1

1

1 ),(2

3

The Associated Legendre and Legendre polynomials are used

extensively in the analysis of RT problems. The relation between the associated

Legendre and is:

m

l

Pl

P

mlP lP

m

lmm

ml

d

PdP

)()1()( 22=

Eq. 2.3-9

The possess the following orthogonal propertiesmlP

nm

mmdPP

kl

kldPP

ml

ml

m

nl

ml

ml

ml

l

mk

ml

=

=

=

=

+

+

+

)!(

)!(1

01

12

)!(

)!(

12

2

01

1

1)()(

)()(

Eq. 2.3-10

Fig. 2.3-1 Legendre polys (degree 1 to 5)


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Fig. 2.3-2 Higher degree Legendre polys

Fig. 2.3-3 Associated Legendre functions (m=0,,2) of ordern=2


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22..44 TTHHEE GGEENNEERRAALL SSOOLLUUTTIIOONN OOFF TTHHEE SSOOLLAARR RRTT

EEQQUUAATTIIOONN

The phase function can more usefully be expressed using the addition theorem

for spherical harmonics as in:

0

0

0

1

0;,,)!(

)!()2(

)(cos)()(),;,(

,0

,0

0

=

=

=+

=

= = =

m

m

MmNmlml

ml

mPPP

m

lmml

N

m

N

ml

ml

ml

ml

K

Eq. 2.4-1

The diffuse radiance appearing in Eq. 2.1-2 can be expanded also in a cosine

series of the azimuth angle (Fourier expansion in azimuth):

=

=N

m

m mLL0

0 )(cos),(),,(

Eq. 2.4-2We now insert Eq. 2.4-1 and Eq. 2.4-2 into Eq. 2.1-2 (omitting the emission

term) thus obtaining:

[ ]

Nm

ddmPPL

m

SePPm

Lmd

dLm

N

m

N

ml

ml

ml

ml

m

N

m

N

m

N

ml

m

l

m

l

m

l

mN

m

mN

m

K,1,0

'')(cos)()(),(4

~

)(cos

4

~

)()()(cos

),()(cos),(

)(cos

2

0

1

1 0

00

0

/

00

00

00

0

=

+

+=

= =

=

=

=

==


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The orthogonal properties of the expansion over azimuth transforms the

integro-differential RT equation into a set ofN+1 differential equation thatmust be satisfied:

==

=

+=

1

1

2

00

/0

')(),()(')(cos4

~

4~)()(),(),( 0

dPLPdm

SePPLd

dL

ml

mN

ml

ml

ml

N

m

N

ml

ml

ml

ml

mm

Eq. 2.4-3

Since the integral over azimuth is different from zero only form=0

0

0

0

2')(cos

2

0

=

=m

mdm

,

the multiple scattering term can be simplified to

=

=

+

+=

1

1

,0

/0

')(),()(4

~)1(

)()(4

~),(),( 0

dPLP

eSPPLd

dL

ml

mN

ml

ml

mlm

N

ml

ml

ml

ml

mm

Eq. 2.4-4

We have obtained the set of N+1 differential equations that is equivalent to the

starting integro-differential equation.

The equation for m=0 provides the solution for the azimuthally averaged

radianceL0:

=

=

+=

1

1

00

0

00

/

00

00000

')(),()(2~

)()(4

~),(

),(0

dPLP

SePPLd

dL

l

N

lll

N

llll

Eq. 2.4-5


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When the direct solar term is negligible (for example during nighttime or when

computing multiply scattered radiances in the infrared),L0

is the exact solution

since all source functions (at any height and at the surface) are not function ofazimuth. It is also the correct solution for zenith radiances (i.e. =1). As soon

as solar computations involve different from unity then it is necessary toobtain the solution for higherm orders.

The phase function averaged over azimuth is defined as:

0

0

0

)()(

)(cos)()(2

1

),;,(2

1),(

0

2

00

2

0

=

=

=

==

=

= =

m

mPP

dmPP

dPP

N

llll

N

m

N

ml

ml

ml

ml

Eq. 2.4-6which is exactly the term appearing in Eq. 2.4-5.

Eq. 2.4-5 can now be expressed in terms of the azimuth-independent phase

function (using Eq. 2.4-6). Omitting for simplicity the superscript0

we obtain:

+=

1

1

/0

'),(),(2

~

),(4

~),(

),(0

dPL

eSPLd

dL

Eq. 2.4-7


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22..55 TTHHEE RREEFFLLEECCTTIIOONN AANNDD TTRRAANNSSMMIISSSSIIOONN FFUUNNCCTTIIOONNSS

FFOORRAA SSIINNGGLLEE LLAAYYEERR

Fig. 2.5-1 Definition of reflection and transmission functions and ray pathgeometry

For a light beam incident from above a layer of finite optical depth (see Fig.2.5-1) the reflected and transmitted radiance can be defined by the following

expressions

=

=

ddLTL

ddLRL

topinbottomout

topintopout

),(),;,(

1

),(

),(),;,(1

),(

,

1

0

2

0,

,

1

0

2

0

,

Eq. 2.5-1

When the light beam comes from below the layer:


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=

=

ddLTL

ddLRL

bottomintopout

bottominbottomout

),(),;,(1

),(

),(),;,(1

),(

,

1

0

2

0

,

,

1

0

2

0

,

Eq. 2.5-2

where R*

and T*

denote reflection and transmission of radiance coming from

below. There functions define the properties of the layer and are to be

considered total reflection and transmission properties, that include the effect

of all orders of scattering.


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22..66 SSIINNGGLLEE SSCCAATTTTEERRIINNGG AAPPPPRROOXXIIMMAATTIIOONN

Now we compute the reflection and transmission functions for the simple case

of:

1. monochromatic transfer

2. single layer with optical depth of zero and1 at the two boundary levels3. no diffuse contribution from above and below, that is only direct solar

radiation is incident onto the layer

4. single scattering

The solar direct radiation entering the volume can be described as:

SL topin )()(),( 0000, =

Eq. 2.6-1

and Eq. 2.5-1 and Eq. 2.5-2 become (setting : = SS0 )

=

=

STL

SRL

bottomout

topout

),;,(1

),(

),;,(1

),(

,

00,

Eq. 2.6-2from which the reflection and transmission functions are:

=

=

SLT

SLR

bottomout

topout

/),(),;,(

/),(),;,(

,00

,00

Eq. 2.6-3

In order to computeR and Twe need first to evaluateLout,top

andLout,bottom

. The

internal source of light (Eq. 2.2-1) under single scattering approximation is

0/00 ),,,(

4

~),,(

= ePSJ

Eq. 2.6-4

On the basis of Eq. 2.1-3 and Eq. 2.1-4 the upwelling reflected radiance and

the downwelling transmitted radiance are:


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( )[ ]

+

+=

+

d

ePS

eLL

0

1

1

),,,(4

~

),,(),,(

00

/)(1

( )[ ]

+

+=

+

d

ePS

eLL

0

0

00

/

),,,(4

),,0(),,(

Assumption (3) implies

),,( 1 L =0

),,0( L =0.Eq. 2.6-5

The solution forLout,top is:

( )[ ]

+

=

=

==

+

+

+

+

01

1

0

0

0

1

11

0

000

0

11

11

/0

00

0

00

,

1),,,(4

),,,(4

),,,(4

),,0(),(

ePS

ee

PS

dePS

LL topout

Eq. 2.6-6

The downwelling radianceLout,bottom is composed of diffuse radiance for0

and of the combination of direct and diffuse for=0. A general solution is not

possible and we must treat two separate cases.

When0 one obtains:


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[ ]

101

11

0

0

11

0

000

0

11

0

000

11

0

00

1,

),,,(4

),,,(4

),,,(4

),,(

=

=

=

==

eePS

ePS

dePS

LL bottomout

Eq. 2.6-7

When=0 one obtains rapidly:

01

10

1

/000

0

1

00

0000

001,

),,,(4

),,,(4

),,(

=

=

==

ePS

de

PS

LL bottomout

Eq. 2.6-8

Finally the reflection and transmission function can be computed putting Eq.

2.6-8 and Eq. 2.6-7 into Eq. 2.6-3 to obtain

+

==

+

01

11

0

00

,00

1),,,(

4

/),(),;,(

eP

SLR topout

Eq. 2.6-9


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[ ]

=

=

==

0000

0/

002

0

1

,00

011

01

),,,()(4

),,,(4

/),(),;,(

eeP

eP

SLT bottomout

Eq. 2.6-10

In is interesting to note that these equations simplify greatly in case the optical

depth of the layeris much smaller than unity (smaller than say 10-4

).

Expanding the exponential to the first order, when necessary, shows that:

),,,(4

11

),,,()(4),;,(

),,,(4

),;,(

),,,(4

),;,(

00

0

000

000

002

0

000

00

0

00

=

=

+

P

PT

PT

PR

Eq. 2.6-11

It is seen that, for an optically thin homogeneous layer, the reflection and

transmission functions are the same for single scattering.

Moreover it can be shown that for a thin homogeneous layer also the reflection

and transmission functions for radiation coming from below the layer (i.e. R*

and T*) are the same asR and Tin Eq. 2.6-11.

In Fig. 2.6-1 to Fig. 2.6-4 the reflectivity and transmissivity of a layer are

shown for various sun zenith angles, as function of zenith angle and log101.


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Fig. 2.6-1 Reflectivity of a layer for a sun zenith angle of 84, as function of zenith

angle and log101.

Fig. 2.6-2 Transmissivity of a layer for a sun zenith angle of 84, as function of

zenith angle and log101.


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Fig. 2.6-3 Reflectivity of a layer for a sun zenith angle of 11.5, as function of


Fig. 2.6-4 Transmissivity of a layer for a sun zenith angle of 11.5 , as function of



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22..77 EEXXAACCTT SSOOLLUUTTIIOONNSS TTOO TTHHEE RRTT EEQQUUAATTIIOONN

2.7.1 THE ADDING METHOD

Consider two layers, one on top of the other. The problem is to compute thereflection and transmission of the two layers starting from the single scattering

properties of each layer.

Fig. 2.7-1 Configuration for the adding method

The reflection by the two layers combines the effects of multiple scattering

taking place within the layers. Similarly for the total transmission which

combines the direct and diffuse transmission

+= /~

eTT Eq. 2.7-1

where 0 = when transmission is associated with the incident solar beam

and = when it is associated to the scattered light beam in direction .


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The reflection and transmission of the upper layer are denoted by and1R

1

~T and and2R 2

~T for the lower layer.

*1

21

21

1

1

112

R

R

R

*1

1

R

R

R

1

1

1

~

1~

R

12

K

~~~~

R

RR

RT

RT

T

TTT

*1

*1

2

2

2

12

1

1

~

R

R

R

2*1

*1

2

T

1

1

1

1

1

U

~+

R

R

RD

We define U and~

the combined total reflection and transmission functions

between layer 1 and 2. The geometry is shown in Fig. 2.7-1 where a number of

paths are indicated of multiple reflections and transmissions that give rise tomultiply scattered light. In principle a light beam can undergo an infinite

number of scattering events. We now write the total reflection and transmission

functions for the two layers combined:

( )[ ]( ) 1

1

2*12

*1

2

2*12

*1

*12

*12

*121

*12

*121

*1

~1

~

~1

~

~~~~~~

TRRT

TRRRRT

TRRRRRTTRRRTTT

+=

++++=

++++=

K

Eq. 2.7-2

( )[ ]( ) 2

1

2*12

*2

2

2*12

12*12

*1122

*12

~1

~

~~~

TR

TRR

TRRRRTTR

=

+++=

+++=

K

K

Eq. 2.7-3

Using same method we can derive the equations for U and~

( )[ ]( ) 1

1

2

1

2

2*12

*12

*12

*1212

*1

~

~

~~

TRR

TRRRR

TRRRRRTRRR

=

+++=

+++=

K

K

Eq. 2.7-4

( )[ ]( ) 1

2

2*12

12*12

*11

*1

~

~

~~~

=

+++=

++=

RT

RRRT

TRRRRTRT

K

K

Eq. 2.7-5

It is seen that in all equations Eq. 2.7-2-Eq. 2.7-5 an infinite series

(corresponding to the infinite orders of scattering) is replaced by a single

inverse function.


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We have seen that it is possible to compute exactly the reflection and

transmission functions for two combined layers from the properties of each

layer using Eq. 2.7-2 to Eq. 2.7-5. We can consider then the combined

properties of a new two layer system, each layer composed of the original two

layers. This is the principle of the adding method.

Usually the numerical computation starts with two identical layers of optical

depth . The adding method applied to two identical layers is termed

adding-doubling method. The procedure is reiterated until the desired optical

depth is achieved for a finite layer.

810=

It is possible to rewrite Eq. 2.7-2-Eq. 2.7-5 in various forms to highlight the

structure of the solution for two layers in terms of diffuse and directcomponents.

Examination of these equations show that U DR~

2= .

We can also define Sand re-writeEq. 2.7-5:

( ) ( )

1

2

*

1

1

2

*

12

*

1 111

=+ RRSRRRRS

1

~)1(

~TSD +=

Using the new definitions Eq. 2.7-2 and Eq. 2.7-3 can be written as

DTT

UTRR~~~

~

212

*1112

=

+=

At this point it is possible to separate the diffuse and direct components of the

total transmission function, defined in Eq. 2.7-1, and manipulate the equations

to arrive at the solution for the diffuse component of combined transmittance.

010101 //1

/1 )1())(1(

~ +++=++= eSeTSeTSD

Since by definition 0101/

1/

)1()(~ ++=+= SeTSDeDD

Eq. 2.7-6


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)(~

01 /22

+== eDRDRU Eq. 2.7-7

Finally forR12 :

UeTRR )(/*

11121 ++=

and T12 :

( )

02101020102

021

/)(/

2

/

2

//

212

/1212

))((~

~

+

+

+++=++=

+

eeTDeDTeDeTT

eTT

Finally

0102 /2

/212

++= eTDeDTT

and the direct transmission for the combined layer is ( ) 021 / +e .

The equation are fairly simple in this compact notation. We must howeverrecognize that the product of any of these functions implies an integration over

the solid angle to take into account all the possible multiple angular scattering

contributions.

For example the product for two layers of very small optical depth can

be computed as:2

*1RR

=

=

ddP

P

ddRRRR

),,,(4

),,,'(4

1

),;,(),;','(1

222

001

0

111

0

2

0

200*1

1

0

2

0

2*1

If we confine to an average over azimuth solution (i.e. analogous to Eq. 2.4-5)

and isotropic optical properties for the layer (optical depth independent of

zenith angle) a simplification of the preceding equation is possible


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= d

PPRR

1

0

201

0

21212

*1

),(),'(

)(16

2

Eq. 2.7-8

where the phase functions of Eq. 2.4-6 can be used.

Using similar reasoning it is possible to compute the combined transmission

and reflection functions when the light beam comes (along direction ) from

below the layer:

( )

++=

++=

+=

++=

==

=

/*1

/*112

/2

*2

*

/*1

*1

/*2

*2

11

2*12

*1

*

12

21

2

12

2

2

)1(1

eTUeUTT

DeDTRR

eRURD

SeSTTU

QQRRRRSRRQ

It can be shown that when polarization and azimuth dependence are neglected,

the transmission functions from above and below are the same, that is

),(),(* = TT


26/42

2.7.2 APPLICATION OF ADDING METHOD TO INHOMOGENEOUS

ATMOSPHERES

We have seen that the atmosphere is far from being homogeneous. One of the

major difficulties in any method to compute radiance is to account for such non

homogeneity.

The adding method can be applied to non-homogeneous atmospheres by using

a numerical procedure which is outlined in the following.

1. The atmosphere is divided into N homogeneous layers. Each layer is

characterised by a single value of

a. single scattering albedo, phase function, optical depth

b. temperaturec. concentration of all gases and materials that are optically active

2. Rland Tl(l=1,N) are the reflection and transmission function for each layer.Each layer must be sufficiently thin (from the optical depth point of view) so

that thenR*l= Rland T*

l=Tl.

3. The surface is treated as a layer of zero transmissivity and whose

reflection function is specified. For example a Lambertian

approximation yieldsRN+1=a, where a is the albedo.

With reference to Fig. 2.7-2 we wish to compute the upwelling and down-

welling flux at the interface between layerland l+1, where the optical depth

is . We must compute the transmission and reflection functions (both for up

and down radiation) for the composite layer from level 0 to level l, and the

same for the composite layer fromN+1 to l+1 (the numbering in Fig. 2.7-2refer to layers: the levels start with level 0 (the top) to level N for the

boundary).

4. Compute the functionsR1,2, T1,2, R*

1,2, T*

1,2, by combining the first two

layers, then adding the next layer downward until the properties of the

composite layer (R1,l, T1,l, R*

1,l, T*

1,l) are derived.

5. Compute the same for the composite layer extending from levelN+1 to

level l, starting with the properties of the lowest two layers and moving

upwards.6. We have obtained two composite layers (1,l) and (l+1,N+1) and can apply

the adding equations to compute the internal radiances denoted by U and D.


27/42

Fig. 2.7-2 The geometry for the computation of the upwelling and downwellingflux at the interface between level l and l+1 according to the adding principle.

From Eq. 2.7-6 we have

0,1 /

,1,1

lSeSTTD ll

++=

and from Eq. 2.7-7

01 /

1,11,1

++++ += eRDRU NlNl

where


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1

*,11,1

)1( ++

=

=

QQS

RRQ lNl

By definition functions UandD are proportional to the upwelling and down-welling diffuse radiances at level l(corresponding to optical depth ):

)(),(

)(),(

0

0

DS

L

US

L

=

=

Upwelling and downwelling fluxes can be computed by angular integration:

dDSE

dUSE

dif

dif

)(2)(

)(2)(

1

0

0

1

0

0

=

=

Total fluxes are therefore

=

+=

dif

dif

EE

EeSEl

)(

)( 0,1

0


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2.7.3 THE DISCRETE ORDINATES METHOD

We have seen in Section 2.4 that the orthogonal properties of the expansion

over azimuth transforms the integro-differential RT equation into a set ofN+1differential equation that must be satisfied:

Nm

dPLP

eSPPLd

dL

ml

mN

ml

ml

mlm

N

ml

ml

ml

ml

mm

,,0

')(),()(4

~)1(

)()(4

~),(

),(

1

1

,0

/0

0

K=

+

+=

=

=

The essence of the discrete ordinate method consist in discretising the RT

equation and solve it as a set of first-order differential equations.

Discretising implies replacing the variable by a sequence ofi,

,...2,1;,...,

,...,1,0);,(),(),(

==

==

nnni

NmJL

d

dLi

mi

mim

i

Moreover integrals appearing in the source function need to be computed

using a quadrature rule, that is are replaced by a summation

where a=

=n

ni

ii afdf )()(1

1

i are appropriate weights.

We confine our attention to the transfer of solar fluxes (no emission term) and

consider the azimuth-independent component in the diffuse intensity, that is the

equation for m=0 Eq. 2.4-7, which is repeated here

+=

1

1

/0

'),(),(2

~

),(4

~),(

),(0

dPL

eSPLd

dL


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2.7.4 QUADRATURE RULES

In numerical analysis, a quadrature rule is an approximation of the definite

integral of a function, usually stated as a weighted sum of function values at

specified points within the domain of integration.

An n-point Gaussian quadrature is stated as

=

1

1

)()( jj

n

nj

fwdf

for any function f(). For the Gaussian quadrature the iare the roots of theLegendre polynomials. Therefore there is a symmetry about the roots andweights:

nnjaa jjjj ,...,; === . Moreover for odd n the central point

is at the origin with highest weight.

Some roots and weights are given in Table 0-1. This quadrature rule is

constructed to yield an exact result for polynomials of degree 2n 1 by a

suitable choice of the n pointsi and weights wi, and is therefore very accurate.

The integration problem can be expressed in a slightly more general way by

introducing a weight function W into the integrand, and allowing an intervalother than [-1, 1]. That is, the problem is to calculate

b

a

dfW )()(

for some choices ofa, b, and W.

Fora = -1, b = 1, and W() = 1, the problem is the same as that consideredabove. Other choices lead to other integration rules.

Some of these are tabulated in Table 0-2.


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Table 0-1

n a

1 1= 0 a1= 2

2 1= 0.577350269189626D0 a1= 13 2= 0.774596669241483D0

1= 0

a2= 0.555555555555556D0

a1= 0.888888888888889D0

4 2= 0.861136311594053D0

1= 0.339981043584856D0

a2= 0.347854845137454D0

a1= 0.652145154862546D0

5 3= 0.906179845938664D0

2= 0.538469310105683D0

1= 0

a3= 0.236926885056189D0

a2= 0.478628670499366D0

a1= 0.568888888888889D0

6 3= 0.932469514203152D0 2= 0.661209386466265D0

1= 0.238619186083197D0

a3= 0.171324492379170D0a2= 0.360761573048139D0

a1= 0.467913934572691D0

Table 0-2

Interval W(x) Orthogonal polynomials]1,1[ 1 Legendre]1,1[

21

1

x

Chebyshev

],0[ xe Laguerre

],[ 2xe Hermite


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2.7.5 BACK TO THE DISCRETE ORDINATES

After (A) discretization (replacing with i (i=-n,,n; n=1,2,)), and(B) replacement of the angular integration by a summation

),(),(2~'),(),(

2~ 1

1

jijj

n

nji PLadPL

=

=

the equation becomes

=

+=

n

njjijj

iii

i

PLa

eSPLd

dL

),(),(2

~

),(4

~),(

),(0/

0

Eq. 0-1

It is possible to manipulate this equation in a number of ways to highlight

particular symmetries. For example

(1) introducing the following definitions

nnjPac jijji ,...,);,(2

~

, ==

The properties of the phase function imply that

jijijiji cccc == ,,,, ; and Eq. 0-1 becomes

),(4

~),(),(

),(0

/,

0

=

= ijjin

nji

ii PSeLcL

d

dL

(2) we may also define

ji

ji

c

cb

iji

iji

ji

=

=

/)1(

/

,

,

,

from which definitions follows jijijiji bbbb ,,,, ; == Eq. 0-2


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and Eq. 0-1 is now written as

),(4

~),(

),(0

/,

0

=

= ijjin

nj

i PSeLbd

dL

or

jijji

n

nj

i PSeLbd

dL,

/,

0

4

~

=

=

Eq. 0-3

This is a system of 2n first-order total differential equations. The completesolution is the sum of the general solution for the associated homogeneous

system and of the particular solution

The homogeneous system can be written in compact notation after separation

of the upward and downward radiances

niLbLbd

dL

niLbLbd

dL

jji

n

jjji

n

j

i

jji

n

jjji

n

j

i

,1;

,1;

,1

,1

,1

,1

=+=

=+=

=

=

==

Eq. 0-4

In terms of matrix notation we may write

=

+

+

+

+

L

L

bb

bb

L

L

d

d

Eq. 0-5

where vectors and matrices appearing in Eq. 0-4 are defined, taking into

account Eq. 0-2, as


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=

=

+

nn L

L

L

L

L

L

L

L

.

.;

.

.

2

1

2

1

, and the matrix b derives directly from Eq. 0-5.

We seek a solution to equation Eq. 0-4 (o Eq. 0-5) of the type

keL = , and substituting into Eq. 0-5 we obtain

=

+

+

+

+

k

bb

bb

which can be solved as a standard eigenvalue problem with the simplification

that since matrix b is symmetrical the eigenvalues are all real and occur insymmetric pairs of different sign.

The complete solution of Eq. 0-3 is

0/

=

+= eZeCL ik

ijjn

nji

j

where the unknown coefficients Cj are determined using the boundaryconditions. We will not dwell into the definition of the function Zi or further

mathematics of the solution. Approximate solutions will however be discussed

further.


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22..88 TTHHEE SSUUNN

The Sun is classified as a main sequence star, which means it is in a state of

hydrostatic balance, neither contracting or expanding, and is generating its

energy through nuclear fusion of hydrogen nuclei into helium. The Sun has aspectral class of G2V, with the G2 meaning that its color is yellow and its

spectrum contains spectral lines of ionized and neutral metals as well as very

weak hydrogen lines, and the V signifying that it, like most stars, is a "dwarf"

star on the main sequence.

The Sun has a predicted main sequence lifetime of about 10 billion years. Its

current age is thought to be about 4.5 billion years. The Sun orbits the center of

the Milky Way galaxy at a distance of about 25,000 to 28,000 light-years fromthe galactic centre, completing one revolution in about 226 million years. The

orbital speed is 217 km/s, equivalent to one light year every 1400 years, and

one AU every 8 days.

The Sun is a near-perfect sphere, with an oblateness estimated at about 9

millionths, which means the polar diameter differs from the equatorial by about

10 km. This is because the centrifugal effect of the Sun's slow rotation is 18

million times weaker than its surface gravity (at the equator). Tidal effectsfrom the planets do not significantly affect the shape of the Sun, although the

Sun itself orbits the center of mass of the solar system, which is offset from the

Sun's center mostly because of the large mass of Jupiter. The mass of the Sun

is so comparatively great that the center of mass of the solar system is

generally within the bounds of the Sun itself.

The solar interior is not directly observable and the Sun itself is opaque to

electromagnetic radiation. However the study of pressure (or acoustic) wavesthat travel through the Sun's interior has contributed greatly to our

understanding of the Sun's structure. Computer modeling of the Sun is also

used as a theoretical tool to investigate its deep layers.
http://en.wikipedia.org/wiki/Main_sequencehttp://en.wikipedia.org/wiki/Hydrostatic_balancehttp://en.wikipedia.org/wiki/Stellar_classificationhttp://en.wikipedia.org/wiki/Orbital_speedhttp://en.wikipedia.org/wiki/Computer_modelinghttp://en.wikipedia.org/wiki/Computer_modelinghttp://en.wikipedia.org/wiki/Orbital_speedhttp://en.wikipedia.org/wiki/Stellar_classificationhttp://en.wikipedia.org/wiki/Hydrostatic_balancehttp://en.wikipedia.org/wiki/Main_sequence


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Core

At the center of the Sun, where

its density reaches up to

150,000 kg/m3

(150 times the

density of water on Earth),thermonuclear reactions

(nuclear fusion) convert

hydrogen into helium,

producing the energy that

keeps the Sun in a state of

equilibrium. About 8.91037

protons (hydrogen nuclei) are converted to helium nuclei every second,

releasing energy at the matter-energy conversion rate of 4.26 million tonnes persecond or 91016

tons of TNT per second.

Models predict that the high-energyphotons released in fusion reactions take

about a million years to reach the Sun's surface, where they escape as visible

light.Neutrinos are also released in the fusion reactions in the core, but unlike

photons they very rarely interact with matter, and so almost all are able to

escape the Sun immediately.

The core extends from the center of the Sun to about 0.2 solar radii, and is the

only part of the Sun where an appreciable amount of heat is produced by

fusion: the rest of the star is heated by energy that is transferred outward. All of

the energy of the interior fusion must travel through the successive layers to the

solar photosphere, before it escapes to space.

Radiative zone

From about 0.2 to about 0.7 solar radii, the material is hot and dense enough

that thermal radiation is sufficient to transfer the intense heat of the core

outward. In this zone, there is no thermal convection: while the material grows

cooler with altitude, this temperature gradient is not strong enough to drive

convection. Heat is transferred by ions of hydrogen and helium emitting

photons, which travel a brief distance before being re-absorbed by otherions.
http://en.wikipedia.org/wiki/Hydrogenhttp://en.wikipedia.org/wiki/Heliumhttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Photonhttp://en.wikipedia.org/wiki/Visible_lighthttp://en.wikipedia.org/wiki/Visible_lighthttp://en.wikipedia.org/wiki/Neutrinohttp://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Ionshttp://en.wikipedia.org/wiki/Photonshttp://en.wikipedia.org/wiki/Ionshttp://en.wikipedia.org/wiki/Ionshttp://en.wikipedia.org/wiki/Photonshttp://en.wikipedia.org/wiki/Ionshttp://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Neutrinohttp://en.wikipedia.org/wiki/Visible_lighthttp://en.wikipedia.org/wiki/Visible_lighthttp://en.wikipedia.org/wiki/Photonhttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Heliumhttp://en.wikipedia.org/wiki/Hydrogen


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Convection zone

From about 0.7 solar radii to 1.0 solar radii,

the material in the Sun is not dense enough

or hot enough to transfer the heat energy ofthe interior outward via radiation. As a

result, thermal convection occurs as thermal

columns carry hot material to the surface

(photosphere) of the Sun. Once the material

cools off at the surface, it plunges back

downward to the base of the convection zone, to receive more heat from the

top of the radiative zone. Convective overshoot is thought to occur at the base

of the convection zone, carrying turbulent downflows into the outer layers ofthe radiative zone.

The thermal columns in the convection zone form an imprint on the surface of

the Sun, in the form of the

solar granulation and

supergranulation. The

turbulent convection of this

outer part of the solar interiorgives rise to a 'small-scale'

dynamo that produces

magnetic north and south

poles all over the surface of

the Sun.

Photosphere

The visible surface of the

Sun, the photosphere, is the

layer below which the Sun

becomes opaque to visible

light. Above the photosphere, sunlight is free to propagate into space and its

energy escapes the Sun entirely. Sunlight has approximately a black-body

spectrum that indicates its temperature is about 6,000 K, interspersed with

atomic absorption lines from the tenuous layers above the photosphere (see

Fig. 2.8-1). The photosphere has a particle density of about 1023

/m3

(this is

about 1% of the particle density of Earth's atmosphere at sea level).
http://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Thermalhttp://en.wikipedia.org/wiki/Thermalhttp://en.wikipedia.org/wiki/Convective_overshoothttp://en.wikipedia.org/wiki/Granule_%28solar_physics%29http://en.wikipedia.org/w/index.php?title=Supergranulation&action=edithttp://en.wikipedia.org/wiki/Black-bodyhttp://en.wikipedia.org/wiki/Black-bodyhttp://en.wikipedia.org/w/index.php?title=Supergranulation&action=edithttp://en.wikipedia.org/wiki/Granule_%28solar_physics%29http://en.wikipedia.org/wiki/Convective_overshoothttp://en.wikipedia.org/wiki/Thermalhttp://en.wikipedia.org/wiki/Thermalhttp://en.wikipedia.org/wiki/Convection


38/42

Fig. 2.8-1 Extra-terrestrial irradiance spectrum of the Sun (resolution of 5 cm

-1),

observed with the Fourier Transform Spectrometer at the McMath-Pierce Solar

Facility at Kitt Peak National Observatory, near Tucson, Arizona.


39/42

Fig. 2.8-2 Extra-terrestrial irradiance spectrum of the Sun (resolution of 5 cm

-1),

observed with the Fourier Transform Spectrometer at the McMath-Pierce Solar

Facility at Kitt Peak National Observatory, near Tucson, Arizona.


40/42

Chromosphere

It is a thin (about 10,000 km deep) layer

above the visible surface of the Sun,

that is dominated by a spectrum ofemission and absorption lines. The

chromosphere is visible as a colored

flash at the beginning and end of total

eclipses of the Sun.

Corona

The corona is the extended outer

atmosphere of the Sun, which is much

larger in volume than the Sun itself.

The corona merges smoothly with the

solar wind that fills the solar system

and heliosphere. The low corona,

which is very near the surface of the

Sun, has a particle density of 1011

/m3

(Earth's atmosphere near sea level has

a particle density of about 2x1025

/m3).

The temperature of the corona is about

one to three million degrees K. The high temperature of the corona suggests

that it is heated by something other than the photosphere.

It is thought that the energy necessary to heat the corona is provided by

turbulent motion in the convection zone below the photosphere. Two main

mechanisms have been proposed to explain coronal heating: Wave heating, in

which sound, gravitational and magnetohydrodynamic waves are produced by

turbulence in the convection zone. These waves travel upward and dissipate in

the corona, depositing their energy in the ambient gas in the form of heat. The

other proposed mechanism is flare heating at small scales, but this is still an

open topic of investigation.
http://en.wikipedia.org/wiki/Solar_eclipsehttp://en.wikipedia.org/wiki/Solar_eclipsehttp://en.wikipedia.org/wiki/Solar_windhttp://en.wikipedia.org/wiki/Solar_systemhttp://en.wikipedia.org/wiki/Solar_systemhttp://en.wikipedia.org/wiki/Solar_windhttp://en.wikipedia.org/wiki/Solar_eclipsehttp://en.wikipedia.org/wiki/Solar_eclipse


41/42

Fig. 2.8-3 Large solarflare recorded by the

Solar and HeliosphericObservatory (SOHO)EIT304 instrument in

the ultraviolet.

(Some text and pictures taken from Wikipedia, the free encyclopedia)

http://en.wikipedia.org/wiki/Sun
http://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatoryhttp://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatoryhttp://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatoryhttp://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatoryhttp://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatoryhttp://en.wikipedia.org/wiki/Solar_and_Heliospheric_Observatory


42/42

Documents

Solar Radiative Transfer_2