Short-Wave Solar Radiation in the Earth’s Atmosphere Part 2 ppsx

Geometry of propagation of solar radiation in the plane parallel atmosphereThus, the radiance is a function of only three coordinates: altitude z and two angles, defining direction ϑ,ϕ..

π|2 ≤γ ≤π) and it is very suitable for the theoretical consideration, as will

be shown further However, it describes the real phase functions with a largeuncertainty (Vasilyev O and Vasilyev V 1994) Therefore, the using of thisfunction needs a careful evaluation of the errors The detailed consideration

of this problem will be presented in Chap 5

1.3

Radiative Transfer in the Atmosphere

Within the elementary volume, the enhancing of energy along the length dl

could occur in addition to the extinction of the radiation considered above.Heat radiation of the atmosphere within the infrared range is an evident exam-ple of this process, though as will be shown the accounting of energy enhancing

is really important in the short-wave range Value dE r– the enhancing of energy

– is proportional to the spectral dλand time dt intervals, to the arc of solid angle dΩencircled around the incident direction and to the value of emitting

volume dV =dSdl Specify the volume emission coefficientεas a coefficient ofthis proportionality:

Consider now the elementary volume of medium within the radiation field

In general case both the extinction and the enhancing of energy of radiation

passing through this volume are taking place (Fig 1.6) Let I be the radiance incoming to the volume perpendicular to the side dS and I + dI be the radiance

after passing the volume along the same direction According to energy

defi-nition in (1.1) incoming energy is equal to E0=IdSdΩdλdt then the change of energy after passing the volume is equal to dE=dIdSdΩdλdt According to the

law of the conservation of energy, this change is equal to the difference between

enhancing dE r and extincting dE eenergies Then, taking into account the initions of the volume emitting coefficient (1.32) and the volume extinction

def-coefficient, we can define the radiative transfer equation:

dI

In spite of the simple form, (1.33) is the general transfer equation with acceptingthe coefficients αand ε as variable values This derivation of the radiativetransfer equation is phenomenological The rigorous derivation must be doneusing the Maxwell equations

We will move to a consideration of particular cases of transfer (1.33) in

conformity with shortwave solar radiation in the Earth atmosphere Within the

shortwave spectral range we omit the heat atmospheric radiation against thesolar one and seem to have the relationε = 0 However, we are taking intoaccount that the enhancing of emitted energy within the elementary volumecould occur also owing to the scattering of external radiation coming to the

Fig 1.6 To the derivation of the radiative transfer equation

volume along the direction of the transfer in (1.33) (i e along the direction

normal to the side dS) Specify this direction r0and scrutinize radiation

scat-tering from direction r with scatscat-tering angleγ(Fig 1.6) Encircling the similar

volume around direction ~ r (it is denoted as a dashed line), we are obtaining

energy scattered to direction r0 Then employing precedent value of energy

E0and definition (1.32), we are obtaining the yield to the emission coefficient

corresponded to direction r:

dε(r)= 4σ π x(γ)I(r)dSdΩdλdtdΩdl

dVdΩdλdt = σ4πx(γ)I(r)dΩ

Then it is necessary to integrate value dε(r) over all directions and it leads to the

integro-differential transfer equation with taking into account the scattering:

Length element dl in the plane-parallel atmosphere is dl =−dz|cosϑ Theground surface at the bottom of the atmosphere is neglected for the present (i e

it is accounted that the radiation incoming to the bottom of the atmosphere isnot reflected back to the atmosphere and it is equivalent to the almost absorbingsurface) Within this horizontally homogeneous medium, the radiation field isalso the horizontally homogeneous owing to the shift symmetry (the invariance

of all conditions of the problem relatively to any horizontal displacement)

Fig 1.7 Geometry of propagation of solar radiation in the plane parallel atmosphere

Thus, the radiance is a function of only three coordinates: altitude z and two

angles, defining direction (ϑ,ϕ) Hence, (1.34) could be written as:

cosγ =cosϑcosϑ
+ sinϑsinϑ
cos(ϕ−ϕ
) (1.36)

To begin with, consider the simplest particular case of transfer (1.35) Neglectthe radiation scattering, i e the term with the integral For atmospheric optics,

7 Use in (1.35) of the plane atmosphere model in spite of the real spherical one is an approximation.

It has been shown, that it is possible to neglect the sphericity of the atmosphere with a rather good accuracy if the angle of solar elevation is more than 10 ◦ Then the refraction (the distortion) of the

solar beams, which has been neglected during the deriving of the transfer equation is not essential Mark that the horizontal homogeneity is not evident This property is usually substantiated with the great extension of the horizontal heterogeneities compared with the vertical ones However, this condition could be invalid for the atmospheric aerosols It is more correct to interpret the model of the horizontally homogeneous atmosphere as a result of the averaging of the real atmospheric parameters over the horizontal coordinate.

it conforms to the direction of the direct radiation spreading (ϑ0,ϕ0) Actually

in the cloudless atmosphere, the intensity of solar direct radiation is essentiallygreater than the intensity of scattered radiation In this case, the direction ofsolar radiation is only one, the intensity depends only on the altitude, and thetransfer equation (1.35) transforms to the following:

dI(z)

Mark that it is always cosϑ0> 0 in (1.37) Differential equation (1.37) together with boundary condition I = I(z∞), where z∞ is the altitude of the top ofthe atmosphere (the level above which it is possible to neglect the interactionbetween solar radiation and atmosphere) is elementary solved that leads to:

I(z)=I(z∞) exp

⎛

⎝ 1cosϑ0

It is convenient to rewrite this solution as:

I(z)=I(z∞) exp

Then Beer’s Law is written as:

As it follows from definitions (1.39) and (1.40) and from “summarizing rules”(1.23), the analogous rules are correct for the optical deepness and for theoptical thickness:

Therefore, it is possible to specify the optical thickness of the molecular tering, the optical thickness of the aerosol absorption etc.

scat-According to the condition accepted in Sect 1.1 we are considering solarradiation incoming to the plane atmosphere top as an incident solar parallel

flux F0 from direction (ϑ0,ϕ0) Then, deducing the intensity through function (1.10) and substituting it to the formula of the link between the fluxand intensity (1.5) it is possible to obtain Beer’s Law for the solar irradiance

delta-incoming to the horizontal surface at the level z:

F d (z)=F0cosϑ0exp(−τ(z)|cosϑ0) (1.42)

In particular, it is accomplished for the solar direct irradiance at the bottom ofthe atmosphere8:

F d(0)=F0cosϑ0exp(−τ0|cosϑ0) (1.43)Return to the general case of the transfer equation with taking into accountscattering (1.35) Accomplish the transformation to the dimensionless param-eters in the transfer equation for convenience of further analysis In accordancewith the optical thickness definition (1.39) the functionτ(z) is monotonically

decreasing with altitude that follows from conditionα(z
) > 0 In this case

there is an inverse function z(τ) that is also decreasing monotonically Using

the formal substitution of function z(τ) rewrite the transfer equation and passfrom vertical coordinateτto coordinate z, moreover, the boundary condition

is at the top of the atmosphereτ =0 and at the bottomτ = τ0, and the direction

of axisτis opposite to axis z It follows from the definition (1.39): dτ =−α(z)dz.

Specifyµ = cosϑand pass from the zenith angle to its cosine (the formalsubstitutionϑ = arccosµ with taking into account sinϑdϑ = −dµ) Finally,divide both parts of the equation to valueα(τ), and instead (1.35) obtain thefollowing equation:

ω0(τ)= σ α((τ τ)) = σ(τ σ) +(τ)κ(τ) , (1.45)

8 Point out that according to Beer’s Law the radiance in vacuum (α = 0) does not change (the same conclusion follows immediately from the radiance definition) It contradicts to the everyday identification of radiance as a brightness of the luminous object Actually, it is well known that the viewing brightness of stars decreases with the increasing of distance It is evident that as the star is further, then the solid angle, in which the radiation incomes to a receiver (an eye, a telescope objective),

is smaller, hence energy perceived by the instrument is smaller too Just this energy is often identified with the brightness (and it is called radiance sometimes), although in accordance to definition (1.1) it is necessary to normalize it to the solid angle Thus, the essence of the contradiction is incorrect using of the term “radiance” In astronomy, the notion equivalent to radiance (1.1) is the absolute star quantity (magnitude).

and the scattering angle cosine according to (1.36):

Dimensionless valueω0defined by (1.45) is called the single scattering albedo

or otherwise the probability of the quantum surviving per the single scattering

event If there is no absorption (κ = 0) then the case is called conservative scattering, ω0 = 1 If the scattering is absent then the extinction is causedonly by absorption,σ =0,ω0=0 and the solution of the transfer equation isreduced to Beer’s Law – (1.41)–(1.43) From consideration of these cases, thesense of valueω0is following: it defines the part of scattered radiation relatively

to the total extinction, and corresponds to the probability of the quantum tosurvive and accepts the quantum absorption as its “death”

It is necessary to specify the boundary conditions at the top and bottom of theatmosphere As it has been mentioned above, solar radiation is characterizing

with values F0,ϑ0,ϕ0incomes to the top Usually it is assumedϕ0=0, i e allazimuths are counted off the solar azimuth Additionally specifyµ0 = cosϑ0

and F0= πS.9

As has been mentioned above, solar radiation in the Earth’s atmosphereconsists of direct and scattered radiation It is accepted not to include thedirect radiation to the transfer equation and to write the equation only forthe scattered radiation The calculation of the direct radiation is accomplishedusing Beer’s Law (1.41) Therefore, present the radiance as a sum of direct and

scattered radiance I(τ,µ,ϕ)=I
(τ,µ,ϕ) + I

(τ,µ,ϕ) From expression for the

direct radiance of the parallel beam (1.10) the following is correct I
(0,µ,ϕ)=

πSδ(µ−µ0)δ(ϕ− 0), and it leads to I
(τ,µ,ϕ)= πSδ(mu −µ0)δ(ϕ) exp(−τ|µ0)for Beer’s Law Substitute the above sum to (1.44), with taking into accountthe validity of (1.37) for direct radiation and properties of the delta function(Kolmogorov and Fomin 1989) Then introducing the dependence upon value

µ0and omitting primes I

(τ,µ,µ0,ϕ), we are obtaining the transfer equation for scattered radiation.

9 Specifyingπ S has the following sense Suppose that radiation equal to radiance S from all directions

incomes to the top of the atmosphere, and this radiation is called isotropic Then, according to (1.6) linking the irradiance and radiance, the incoming to the top irradiance is equal toπ S Thus, value S is an

isotropic radiance that corresponds to the same irradiance as a parallel solar beam normally incoming

to the top of the atmosphere is.

where valueχis defined by (1.46) and forχ0the following expression is correctaccording to the same equation:

Point out that (1.47) is written only for the diffuse radiation The boundary

conditions are taking into account by the third term in the right part of (1.47)

The sense of this term is the yield of the first order of the scattering to the radiance and the integral term describes the yield of the multiple scattering The ground surface at the bottom of the atmosphere is usually called the underlying surface or the surface Solar radiation interacts with the surface

reflecting from it Hence, the laws of the reflection as a boundary condition atthe bottom of the atmosphere should be taken into account However, it is doneotherwise in the radiative transfer theory As will be shown in the followingsection, there are comparatively simple methods of calculating the reflection bythe surface if the solution of the transfer equation for the atmosphere withoutthe interaction between radiation and surface is obtained Thus, neither directnor reflected radiation is included in (1.47) As there is no diffused radiation

at the atmospheric top and bottom, the boundary conditions are following

I(0,µ,µ0,ϕ)=0 µ> 0 ,

Transfer equation (1.47) together with (1.46), (1.48) and boundary conditions(1.49) defines the problem of the solar diffused radiance in the plane parallelatmosphere Nowadays different methods both analytical (Sobolev 1972; Hulst1980; Minin 1988; Yanovitskij 1997) and numerical (Lenoble 1985; Marchuk1988) are elaborated Our interest to the transfer equation is concerning theprocessing and interpretation of the observational data of the semisphericalsolar irradiance in the clear and overcast sky conditions The specific numericalmethods used for these cases will be exposed in Chap 2 Now continue theanalysis of the transfer equation to introduce some notions and relations,which will be used further

The diffused radiation within the elementary volume could be interpreted

as a source of radiation It follows from the derivation of the volume emissioncoefficient through the diffused radiance in (1.34) if the increasing of theradiance is linked with the existence of the radiation sources Then introduce

the source function:

and the transfer equation is rewritten as follows:

µdI(τ,µ,µ0,ϕ)

dτ =−I(τ,µ,µ0,ϕ) + B(τ,µ,µ0,ϕ) (1.51)Equation (1.51) is the linear inhomogeneous differential equation of type

dy(x)|dx=ay(x) + b(x) Its solution is well known:

y(x)=y(x0) exp(a(x − x0)) +

Certainly (1.52) are not the problem’s solution because source function

B(τ,µ,µ0,ϕ) itself is expressed through the desired radiance However, (1.52)allows the calculation of the radiance if the source function is known, for exam-ple in the case of the first order scattering approximation when only the second

term exists in the definition of function B(τ,µ,µ0,ϕ) (1.50) The expressionsfor the reflected and transmitted scattered radiance of the first order scattering

in the homogeneous atmosphere (where the single scattering albedo does notdepend on altitude) have been obtained (Minin 1988):

Equation (1.53) is the integral equation for the source function Usually just thisequation is analyzed in the radiative transfer theory but not (1.47) The desiredradiance is linked with the solution of (1.53) with the simple expressions It ispossible otherwise to substitute definition (1.50) to expressions (1.52) and toobtain the integral equations for the radiance used in the numerical methods

of the radiative transfer theory

It is possible to write the integral equation for the source function (1.53)through the operator form (Hulst 1980; Lenoble 1985; Marchuk et al 1980)

where B = B(τ,µ,µ0,ϕ) is the source function, q is the absolute term, K is

the integral operator The operator kernel and the absolute term are expressed

Expression (1.56) concerning the transfer theory is an expansion of the solution

(the source function) over powers of the scattering order Actually, the item q

is a yield of the first order scattering to the source function, the item Kq is the second order, K2q=K(Kq) is the third order etc As kernel K is proportional

to the single scattering albedo, the velocity of the series convergence is linkedwith this parameter: the higherω0(the scattering is greater) the higher order

of the scattering is necessary to account in the series Mark that, according

to (1.56), source function B linearly depends on q Hence, source function

B (and the desired radiance) is directly proportional to value S, i e to the

extraterrestrial solar flux So it is often assumed S=1 and finally the obtained

radiance multiplied by the real value S=F0|π

As per (1.55) q = µ0BI0, where I0 =I(0,µ,µ0,ϕ)= πδ(µ−µ0)δ(ϕ) is the

extraterrestrial radiance Consequently the desired radiance I= I(τ,µ,µ0,ϕ)

also linearly depends on I0and it is possible to formally write the following:

where T is the linear operator and the problem of calculating the radiance is reduced to the finding of the operator As function I0is the delta-function ofdirection (µ0,ϕ0) (where the azimuth of extraterrestrial radiation is assumedarbitrary) the radiance could be calculated for no matter how complicated an

incident radiation field I0(µ0,ϕ0) after obtaining the operator T as a function

of all possible directions T(µ0,ϕ0) due to the linearity of (1.57) The followingrelation is used for that:

a series expansion over the orthogonal functions is the standard mathematicalmethod Certain simplification is succeeded after expanding the phase functionover the series of Legendre Polynomials in the case of the radiative transferequation Legendre Polynomials are defined, e g (Kolmogorov and Fomin1999) as,

because any function within the interval could be expanded to the series overLegendre Polynomials The following is deduced for the phase function:

From the normalizing condition of the phase function (1.18) and from equality

P0=1 it always follows x0=1 The first coefficient of the expansion x1is of animportant physical sense:

From the phase function interpretation as a probability density of the

scat-tering to the certain angle it follows that value g =x1|3 is the mean cosine of scattering angle It determines the elongation of the phase function, namely,

as g is closer to unity then the phase function is more extended to the forward

direction and weaker extended to the backscatter direction In the context of

parameter g the Henyey-Greenstein approximation (1.31) is appropriate It is

easy testing that its mean cosine is just equal to the parameter of the

approx-imation and it is specified with the same sign g (but it is not otherwise, the using of sign g for the mean cosine does not imply the Henyey-Greenstein ap-

proximation is obligatory) Other expansion items of the Henyey-Greensteinfunction over Legendre Polynomials are also simply expressed through its pa-

rameter: x i =(2i + 1)g i This very reason determines the wide application ofthe Henyey-Greenstein function but not an accuracy of the real phase functionapproximation

Practically the series is to break at the certain item with number N The value N was shown in the study by Dave (1970) to reach hundreds and even

thousands to approximate the phase function with the necessary accuracy

It is not appropriate for expansion (1.60) using for the radiance calculationeven with modern computers It is the essential problem of the application ofthe described methodology We would like to point out that for the molecu-lar scattering determined by (1.25) the phase function is much more simple

(N=2):

x m(χ)=P0(χ) +1 −δ

2 +δP2(χ) The phase function cosineχin transfer equation (1.47) (and in all consequencesfrom it) is a function of directions of incident and scattered radiation (1.46)

For such a function the theorem of Legendre Polynomials addition (Smirnov1974; Korn and Korn 2000) is known According to it the following is correct:

analogous relations) There are known recurrence relations for the practical

calculation of function P i m (z) (Korn G and Korn T 2000) Applying relation

(1.31) to expansion of the phase function (1.60) it is inferred:

After changing the summation order in the second item of (1.63) and

account-ing that for m=1 it is valid i = 1, , N, and m = 2− is i = 2, , N etc., we

finally obtain the following:

where I m(τ,µ,µ0) and B m(τ,µ,µ0) for m=0, , N are certain unknown

func-tions Substitute expansions (1.64) and (1.65) to expression for the source

function (1.50), compute the integral over the azimuth and level items with

the equal numbers m in the left-hand and right-hand parts of the equation Only the items with equal numbers m will be nonzero in the product of the

series (the phase function by the radiance) due to the orthogonality of thetrigonometric functions:

The substitution of (1.69) to (1.66) yields the integral equation again for thesource function:

are called expansions over the azimuthal harmonics and the method is called

a method of the azimuthal harmonics.

1.4

Reflection of the Radiation from the Underlying Surface

The ratio of the irradiances reflected from the surface to the irradiances

in-coming to the surface is called an albedo of the surface and it is one of the most

important characteristics of the underlying surface:

A= F↑(τ0)

This characteristic has a clear physical meaning – it corresponds to the part

of the incoming radiation energy reflected back to the atmosphere Actually,

if value A = 0 then the surface absorbs all radiation (the absolutely black

surface), if value A = 1 then, otherwise, the surface absorbs nothing andreflects all radiation (the absolutely white surface) Generalizing the notion

of the albedo, we are introducing the albedo of the system of atmosphere plus surface, specifying it at arbitrary levelτ:

Remember that here and below we are considering values defining the singlewavelength, i e the spectral characteristics of the radiation field and surface.The integral albedo that is called just “albedo” for briefness (do not confuse it

with the spectral albedo) is of great importance in atmospheric energetics.10

10 It is necessary to point out that the albedo (like other reflection characteristics) is formally defined only for the surface without the atmosphere In transfer theory, they are often called “true” Taking

The albedo of the surface characterizes the reflection process of radiationonly as a description of the energy transformation, but doesn’t tell us aboutthe dependence of the radiance upon the reflection angle and azimuth Ifthe surface were an ideal plane, such dependence would be defined with thewell-known laws of reflection and refraction (Sivukhin 1980) However, allnatural surfaces are rough, i e they have different scales of the roughness andeven the water surface is practically always not smooth Therefore, consideringthe incoming parallel beam is more complicated in reality, notwithstandingthe reflection from every micro-roughness is ordered to the classical laws

of geometric optics In particular, reflected radiation extends to all possibledirections and not only to the direction according to the law: “the reflectionangle is equal to the incident angle” This light reflection from natural surfaces

is usually called the diffused.

It is possible to select three main types of diffused reflection The orthotropic (or isotropic) reflection, when the diffused reflected radiance does not depend

on the direction The mirror reflection, when the maximum of the reflected

radiance coincides with the direction of the mirror reflection (the reflection

angle is equal to the incident angle) and the backward reflection when the

maximum is situated along the direction opposite to the incident radiationdirection The mirror reflection evidently characterizes the surfaces close tothe ideally smooth surface and otherwise the backward one characterizes thesurfaces close to the strongly rough surface because it is formed by a largeamount of the micro-grounds oriented perpendicular to the incident direction

of radiation The observations some of them we will consider in detail in Chap 3indicate that the cloud and snow are the closest to the orthotropic surface, thewater is the most mirror surface and others are mainly backward reflectedsurfaces However, the reference to the observation is excessive because of themirror reflection of the banks from the water that everybody has seen and thebackward reflection maximum is clearly observed from the airplane board.The orthotropic surfaces are especially convenient for the theoretical anal-ysis and practical calculations because they are characterized with only oneparameter – the albedo and because of the simplicity of the mathematicaldescription We would like to point out that the assumption concerning theorthotropic reflection is an approximation and its accuracy is necessary toevaluate in a concrete problem It is said that the anisotropic reflection fromother surfaces needs some additional values for its description The rathervariable characteristics of the anisotropic reflection are considered in differ-ent studies, however here we are describing the general problem without itsconcretization Note also that the reflection processes depend on the incidentradiation polarization accompanying its change (Sivukhin 1980) Therefore,the consideration of the reflection without an account of polarization is an

into account the atmosphere, the other characteristics of the system “atmosphere plus surface” are analogously defined For example, the incoming irradiance to the surface from the diffusing atmosphere depends itself on the surface albedo (true) because it contains the part of the reflected radiation that scattered back to the surface Mark that on the one hand, only true values are used in formulas of the transfer theory for the reflection characteristics, and on the other hand, only characteristics of the system “atmosphere plus surface” are available for the observation.

approximation Considering different orientations of the micro-roughness ofthe natural surfaces it is possible to assert that as reflection is closer to theorthotropic, then reflected radiation is less polarized The homogeneous dis-tribution of reflected radiation over directions is corresponded to the fullychaotic orientation of the micro-reflectors that causes the chaotic distribution

of the polarization ellipses, i e the unpolarized light Thus, the orthotropicreflection means also the absence of the dependence upon the polarization.Otherwise, when the anisotropy is stronger the dependence is clearer Thewater surface is the most anisotropic surface, therefore, in this case the ques-tion about the exactness of the approximation of unpolarized radiation needsspecial study

The function R(µ,ϕ,µ
,ϕ
), defined from the relation between the radiances

incoming on the surface I(τ0,µ,ζ,ϕ
), (µ
> 0) and reflected from the surface

I(τ0,µ,ζ,ϕ), (µ< 0), characterizes the radiation reflection from the surface: I(τ0,µ,µ0,ϕ)= π1

It is easy to test that for the orthotropic surface (1.71) and (1.73) yield the

equality R(µ,ϕ,µ
,ϕ
)= A and just it defines the existence of the factorµ
|π

for normalizing in (1.73) Equation (1.73) in the operator form is written as:

where: I↑ = I(τ0,µ,µ0,ϕ) is the reflected radiance, I↓ = I(τ0,µ,µ0,ϕ) is the

incoming radiance, and r = µ π
R(µ,ϕ,µ
,ϕ
) is the operator of the reflectionfrom the surface

The necessity of accounting the reflection from the surface in the radiativetransfer theory is based on the evident assumption that the reflection is equal

to the illumination of the atmosphere from the bottom (i e from the bottom

boundary of the atmosphereτ = τ0) Thus, it is enough to solve the radiativetransfer problems for diffused radiation in the atmosphere first with the illu-mination from the top and then with the illumination from the bottom, andafter all, it is necessary to add both results

Introduce the following notation system:

1 the values related to the system “atmosphere plus surface” are specifiedwith the upper line;

2 the values related to the atmosphere illuminated from the bottom withoutsurface are specified with the symbol∼;

3 the values related to the atmosphere illuminated from the top withoutsurface are specified without special marks

Then the solution of the radiative transfer problem, written in the operator

form (1.57), will be the following: I=TI0where I0is the radiance incoming to