20 Solar Radiation in the Atmosphere π|2 ≤ γ ≤ π) and it is very suitable for the theoretical consideration, as will be shown further. However, it describ es the real phase functions with a large uncertainty (Vasilyev O and Vasilyev V 1994). Therefore, the using of this function needs a careful evaluation of the errors. The detailed consideration of t his 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 –isproportionaltothespectrald λ and time dt i n tervals, to the arc of solid angle d Ω encircledaroundtheincidentdirectionandtothevalueofemitting volume dV = dSdl.Specifythevolumeemissioncoefficientε as a coefficient of this proportionality: ε = dE r dVdΩdλdt . (1.32) 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 afterpassingthevolumealongthesamedirection.Accordingtoenergydefi- nition in (1.1) incoming energy is equal to E 0 = IdSdΩdλdt then the change of energy after passing the volume is equal to dE = dIdSdΩdλdt.Accordingtothe law of the conservation of energy, this change is equal to the difference between enhancing dE r and extincting dE e energies. Then, taking into account the def- initions of the volume emitting coefficient (1.32) and the volume extinction coefficient, we can define the radiative transfer equation: dI dl = −αI + ε . (1.33) In spite of the simple form, (1.33)is thegeneral transfer equationwith accepting the coefficients α and ε as variable values. This derivation of the radiative transfer equation is phenomenological. The rigorous derivation must be done using the Maxwell equations. We will move to a consideration of particular cases of transfer (1.33) in conformity with shortwavesolarradiationintheEarthatmosphere. Within the shortwave spectral range we o m it the heat atmospheric radiation against the solaroneandseemtohavetherelation ε = 0. However, we are taking into account that the enhancing of emitted energy within the elementary volume could occur also owing to the scattering of external radiation coming to the Radiative Transfer in the Atmosphere 21 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 r 0 and scrutinize radiation scat- tering from direction r with scattering angle γ (Fig. 1.6). Encircling the similar volume aro und direction ~r (it is denoted as a dashed line), we are obtaining energy scattered to direction r 0 . Then employing precedent value of energy E 0 and definition (1.32), we are obtaining the yield to the emission coefficient corresponded to direction r: d ε(r) = σ 4π x(γ)I(r)dSdΩdλdtdΩdl dVdΩdλdt = σ 4π x(γ)I(r)dΩ . Then it is necessary to integrate value d ε(r) over all directions and it leads to the integro-differential tra nsfer equa tion with taking into accou nt the scattering: dI(r 0 ) dl = −αI(r 0 )+ σ 4π  4π x(γ)I(r)dΩ . (1.34) Considerthe geometry of solar radiation spreading throughoutthe atmosphere for concr etization (1.34) as Fig. 1.7 illustrates. Asdescribed above inSect. 1.1 we are presenting the atmosphere as amodel of the plane-parallel and horizontally homogeneous layer. The direction of the radiation spreading is characterized with the zenith angle ϑ and with the azim uth ϕ counted off an arbitrary direction at a horizontal plane. Set all coefficients in (1.34) depending on the altitude (it completely corresponds t o reality). Length element dl in the plane-parallel atmosphere is dl = −dz| cos ϑ.The groundsurfaceatthebottomoftheatmosphereisneglectedforthepresent(i.e. it is accounted that the radiation incoming to the bottom of the atmosphere is notreflectedbacktotheatmosphereanditisequivalenttothealmostabsorbing surface). Within this horizontally homogeneous medi um, the radiation field is also the horizontally homogeneous owing to the shift symmetry (theinvariance of all conditions of the problem r elatively to any horizontal displacement). 22 Solar Radiation in the Atmosphere Fig. 1.7. Geometry of propagation of solar radiation in the plane parallel atmosphere Thus, the radiance is a function of only three coordinates: al titude z and two angles, defining direction ( ϑ, ϕ). Hence, (1.34) could be written as: dI(z, ϑ, ϕ) dz cos ϑ = α(z)I(z, ϑ, ϕ) − σ(z) 4π 2π  0 dϕ  π  0 x(z, γ)I(z, ϑ  , ϕ  ) sin ϑ  dϑ  (1.35) where scattering angle γ is an angle between directions (ϑ, ϕ)and(ϑ  ϕ  ). It is easy to express the scattering angle through ϑ, ϕ: to consider the scalar product of the orts in the Cartesian coordinate system and then pass to the spherical coordinates. This proced ure yields the following relation known as the Cosine law for the s pheroid triangles 7 : cos γ = cos ϑ cos ϑ  + sin ϑ sin ϑ  cos(ϕ − ϕ  ) . (1.36) To begin with, consider the simplest particular case of transfer (1.35). Negle ct the radiation scattering, i. e. the term with the integral. For atmospheric optics, 7 Usein(1.35)oftheplaneatmospheremodelinspiteoftherealsphericaloneisanapproximation. 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 co rrect to interpret the model of the horizontallyhomogeneousatmosphereasaresultoftheaveragingoftherealatmosphericparameters over the horizontal coordinate. Radiative Transfer in the Atmosphere 23 it conforms to the direction of the direct radiation spreading (ϑ 0 , ϕ 0 ). Actually in the cloudless atmosphere, the intensity of solar direct radiation is essentially greater than the intensity of scattered radiation. In this case, the direction of solar radiation is only one, the intensity depends only on the altitude, and the transfer equation (1.35) transforms to the following: dI(z) dz cos ϑ 0 = α(z)I(z) . (1.37) Markthatitisalwayscos ϑ 0 > 0 in (1.37). Differential equation (1.37) together with boundary condition I = I(z ∞ ), where z ∞ is the altitude of the top of the atmosphere (the level above which it is possible to neglect the interaction between solar radiation and atmosphere) is elementary solved that leads to: I(z) = I(z ∞ )exp ⎛ ⎝ 1 cos ϑ 0 z  z ∞ α(z  )dz  ⎞ ⎠ . It is convenient to rewrite this solu tion as: I(z) = I(z ∞ )exp ⎛ ⎝ − 1 cos ϑ 0 z ∞  z α(z  )dz  ⎞ ⎠ . (1.38) This relation illustrates the exponential decrease of the intensity in the extinct medium and it is called Beer’s Law. Introduce the dimensionless value: τ(z) = z ∞  z α(z  )dz  , (1.39) that is called the optical de pth oftheatmosphereataltitudez.Itsimportant particular case is the optical thickness of the whole atmosphere: τ 0 = z ∞  0 α(z  )dz  . (1.40) Then Beer’s Law is written as: I(z) = I(z ∞ ) exp(−τ(z)| cos ϑ 0 ) . (1.41) 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 the optical thickness: τ(z) = M  i=1 τ i (z), τ 0 = M  i=1 τ 0,i . 24 Solar Radiation in the Atmosphere Therefore, it is possible to specify the optical thickness of the molecular scat- tering, the optical thickness of the aerosol absorption etc. According to the condition accepted in Sect. 1.1 we are considering solar radiation incoming to the plane atmosphere top as an incident solar parallel flux F 0 from direction (ϑ 0 , ϕ 0 ). Then, deducing the intensity through delta- function (1.10) and substituting it to the formula of the link between the flux and intensity (1.5) it is possible to obtain Beer’s Law for the solar irradiance incoming to the horizontal surface at the level z: F d (z) = F 0 cos ϑ 0 exp(−τ(z)| cos ϑ 0 ) . (1.42) In particular, it is accomplished for the solar direct irradiance at the bottom of the atmosphere 8 : F d (0) = F 0 cos ϑ 0 exp(−τ 0 | cos ϑ 0 ) . (1.43) Returntothegeneralcaseofthetransferequationwithtakingintoaccount scattering (1.35). Accomplish the transformation to the dimensionless param- eters in the transfer equation for convenience of further analysis. In accordance with the optical thickness definition (1.39) the function τ(z) is monotonically decreasing with altitude that follows from condition α(z  ) > 0. In this case there is an inverse function z( τ) that is also d ecreasing monotonically. Using the formal substitution of function z( τ)rewritethetransferequationandpass from vertical coordinate τ to coordinate z, moreover , the boundary condition is at the top of the atmosphere τ = 0andatthebottomτ = τ 0 ,andthedirection of a xis τ is op posite to axis z. It follows from the definition (1.39): dτ = −α(z)dz. Specify µ = cos ϑ and pass from the zenith angle to its cosine (the formal substitution ϑ = arccos µ with taking into account sin ϑdϑ = −dµ). Finally, divide both parts of the equation to value α(τ), and instead (1.35) obtain the following equation: µ dI(τ, µ, ϕ) dτ = −I(τ, µ, ϕ)+ ω 0 (τ) 4π 2π  0 dϕ  1  −1 x(τ, χ)I(τ, µ  , ϕ  )dµ  , (1.44) where ω 0 (τ) = σ (τ) α(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 follo ws 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 of ten 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). Radiative Transfer in the Atmosphere 25 and the scattering angle cosine according to (1.36): χ = µµ  +  1−µ 2  1−µ 2 cos(ϕ − ϕ  ). (1.46) For the phase function it is also suitable to pass from scattering angle γ to its cosine χ with formal substitution γ = arccos χ. Dimensionless value ω 0 defined by(1.45)iscalledthesinglescatteringalbedo or otherwise the probability of the quan tu m surviving per the single scattering event. If there is no absorption ( κ = 0) then the case is called conservative scattering, ω 0 = 1. If the scattering is absent then the extinction is caused only by absorption, σ = 0, ω 0 = 0 and the solution of the transfer equation is reduced to Beer’s Law – (1.41)–(1.43). From consideration of these cases, the sense of value ω 0 is following: it defines the part of scattered radiation relatively to the total extinction, and corresponds to the probability of the quantum to survive and accepts the quantum absorption as its “death”. It is necessary to specify theboundary conditionsatthe topand bottomof the atmosphere. As it has been mentioned above, solar radiation is characterizing with values F 0 , ϑ 0 , ϕ 0 incomestothetop.Usuallyitisassumedϕ 0 = 0, i.e. all azimuths are counted off the solar azimuth. Additionally specify µ 0 = cos ϑ 0 and F 0 = πS. 9 As has been mentioned above, solar radiation in the Earth’s atmosphere consists of direct and scattered radiation. It is accepted not to include the direct radiation to the transfer equation and to write the equation only for thescatteredradiation.Thecalculationofthedirectradiationisaccomplished using Beer’s Law (1.41). Therefore, present the radiance as a sum of direct and scattered radianc e 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) = π Sδ(µ − µ 0 )δ(ϕ − 0), and it leads to I  (τ, µ, ϕ) = πSδ(mu − µ 0 )δ(ϕ)exp(−τ|µ 0 ) for Beer’s Law. Substitute the above sum to (1.44), with taking into a c co unt the validity of (1.37) for direct radiation and pr operties of the delta function (Kolmogorov and Fomin 1989). Then introducing the dependence upon value µ 0 and omitting primes I  (τ, µ, µ 0 , ϕ), we are obtaining the tr ansfer equation for scattered radiation. µ I(τ, µ, µ 0 , ϕ) dτ = −I(τ, µ, µ 0 , ϕ)+ ω 0 (τ) 4π 2π  0 dϕ  1  −1 x(τ, χ)I(τ, µ  , µ 0 , ϕ  )dµ  + ω 0 (τ) 4 Sx( τ, χ 0 ) exp(−τ|µ 0 ) (1.47) 9 Specifying πS has the following sense. Suppose that radiation equal to radiance S fromall 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 to πS.Thus,valueS is an isotropic radiance that corresponds to the same irradiance as a parallel solar beam normally incoming to the top of the atmosphere is. 26 Solar Radiation in the Atmosphere where value χ is defined by (1.46) and for χ 0 the following expression is correct according to the same equation: χ 0 = µµ 0 +  1−µ 2  1−µ 2 0 cos(ϕ) (1.48) 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 bo ttom 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 at the bottom of the a tmosphere should be taken into account. Ho wever , it is done otherwise in the radiative transfer theory. As will be shown in the following section, there are comparatively simple methods of calculating the reflection by the surface if the solution of the transfer equation for the atmosphere without the interaction between radiation and surface is obtained. Thus, neither direct nor reflected radia tion 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, µ 0 , ϕ) = 0 µ > 0, I( τ 0 , µ, µ 0 , ϕ) = 0 µ < 0. (1.49) Transfer equation (1.47) together w ith (1.46), (1.48) and boundary conditions (1.49) defines the problem of the solar diffused radiance in the plane parallel atmosphere. Nowadays different methods both analytical (Sobolev 1972; Hulst 1980; Minin 1988; Yanovitskij 1997) and numerical (Lenoble 1985; Marchuk 1988) are elaborated. Our interest to the transfer equation is concerning the processing and interpretation of the observational data of the semispherical solar irradiance inthe clear andovercast sky conditions.The specific numerical methods used for these cases will be exposed in Chap. 2. Now continue the analysis of the transfer equation to introduce some notions and rela tions, which will be used further. The diffused radiation within the elementary volume could be interpreted as a source o f radiation. It follows from the derivation of the v olume emission coefficient through the diffused radiance in (1.34) if the increasing of the radiance is linked with the existence of the radiation sources. Then introduce the source function: B( τ, µ, µ 0 , ϕ) = ω 0 (τ) 4π 2π  0 dϕ  1  −1 x(τ, χ)I(τ, µ  , µ 0 , ϕ  )dµ  + ω 0 (τ) 4 Sx( τ, χ 0 ) exp(−τ|µ 0 ), (1.50) Radiative Transfer in the Atmosphere 27 and the transfer equation is rewritten as follows: µ dI(τ, µ, µ 0 , ϕ) dτ = −I(τ, µ, µ 0 , ϕ)+B(τ, µ, µ 0 , ϕ) . (1.51) Equation (1.51) is the linear inhomogeneous differential equation of type dy(x) |dx = ay(x)+b(x). Its solution i s w ell known: y(x) = y(x 0 )exp(a(x − x 0 )) + x  x 0 b(x  )exp(a(x − x  ))dx  . Applying it to (1.51) under boundary co nditions (1.49), it is obtained: I(τ, µ, µ 0 , ϕ) = 1 µ τ  0 B(τ  , µ, µ 0 , ϕ)exp  − τ − τ  µ  dτ  µ > 0, I( τ, µ, µ 0 , ϕ) = − 1 µ τ 0  τ B(τ  , µ, µ 0 , ϕ)exp  − τ − τ  µ  d τ  µ < 0. (1.52) Certainly(1.52)arenottheproblem’ssolutionbecausesourcefunction B( τ, µ, µ 0 , 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, µ 0 , ϕ) (1.50). The expressions forthereflectedandtransmittedscatteredradianceofthefirstorderscattering in the homogeneous atmosphere (where the single scattering albedo does not depend on altitude) have been obtained (Minin 1988): I 1 (τ, µ, µ 0 , ϕ) = Sµ 0 ω 0 4 x( χ 0 ) 1−exp  − τ( 1 µ + 1 µ 0 )  µ + µ 0 µ < 0, I 1 (τ, µ, µ 0 , ϕ) = Sµ 0 ω 0 4 x( χ 0 ) exp( −τ µ ) − exp( −τ µ 0 ) µ − µ 0 µ > 0. Return to general expressions for the radiance (1.21), substitute them to source function definition (1.19), and deduce the following: B(τ, µ, µ 0 , ϕ) = ω 0 (τ) 4π 2π  0 dϕ   1  0 x(τ, χ) dµ  µ  τ  0 B(τ  , µ  , µ 0 , ϕ  ) × exp  − τ − τ  µ   dτ  − 0  −1 x(τ, χ) dµ  µ  τ 0  τ B(τ  , µ  , µ 0 , ϕ  )exp  − τ − τ  µ   dτ   + ω 0 (τ) 4 Sx( τ, χ 0 ) exp(−τ|ζ). (1.53) 28 Solar Radiation in the Atmosphere Equation (1.53) is the integral equation for the source function. Usually just this equation is analyzed in the radiative transfer theory but not (1.47). The desired radiance is linked with the solution of (1.53) with the simple expressions. It is possible otherwise to substitute definition (1.50) to expressions (1.52) and to obtain 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) B = KB + q , (1.54) where B = B(τ, µ, µ 0 , ϕ)isthesourcefunction,q istheabsoluteterm,K is the integral operator. The operator kernel and theabsolutetermare expressed according to (1.53) as: K = K(τ, µ, µ 0 , ϕ, τ  , µ  , ϕ  ) = ω 0 (τ) 4πη  x(τ, χ)exp  − τ − τ  µ   for 0 ≤ τ  ≤ τ0 ≤ µ  ≤ 1, K = K(τµ, µ 0 , ϕ, τ  , µ  , ϕ  ) = − ω 0 (τ) 4πη  x(τ, χ)exp  − τ − τ  µ   for τ ≤ τ  ≤ τ 0 −1≤ µ ≤ 0, K = 0outofthepointedranges, q = q(τ, µ, µ 0 , ϕ) = ω 0 (τ) 4 Sx( τ, χ 0 ) exp(−τ|µ 0 ). (1.55) Remember that according to Kolmogorov and Fomin (1989) the operator recording is: Ky ≡ b  a K(x, x  )y(x  )dx  . Equation (1.54) is the Fredholm equation of the second kind. The mathematical theory of these equations is perfectly developed, e. g. Kolmogorov and Fomin (1989). The formal solution of the Fredholm equation of the second kind is presented with the Neumann series: B = q + Kq + K 2 q + K 3 q + (1.56) Expression (1.56)concerning thetransfer theory is an expansionof thesolution (the source function) over powers of the scattering order. Actually, the item q isayieldofthefirstorderscatteringtothesourcefunction,theitemKq is the second order, K 2 q = K(Kq) is the third order etc. As kernel K is proportional tothesinglescatteringalbedo,thevelocityoftheseriesconvergenceislinked with this parameter: the higher ω 0 (the scattering is greater) the higher order Radiative Transfer in the Atmosphere 29 of the scattering is necessary to account in the series. Mark that, according to (1.56), source function B linearly depends on q.Hence,sourcefunction B (and the desired radiance) is directly proportional to value S,i.e.tothe extraterrestrial solar flux. So it is often assumed S = 1 a nd finally the obtained radiance multiplied by the real val ue S = F 0 |π. As per (1.55) q = µ 0 BI 0 ,whereI 0 = I(0, µ, µ 0 , ϕ) = πδ(µ − µ 0 )δ(ϕ)isthe extraterrestrial radiance. Consequently the desired radiance I = I(τ, µ, µ 0 , ϕ) also linearly depends on I 0 and it is possible to formally write the following: I = TI 0 , (1.57) where T is the linear operator and the problem of calculating the radiance is reduced to the finding of the operator. As function I 0 is the delta-function of direction ( µ 0 , ϕ 0 ) (where the azimuth of extraterrestrial radiation is assumed arbitrary) the radiance could be calculated for no matter how complicated an incident radiation field I 0 (µ 0 , ϕ 0 ) after obtaining the operator T as a function of all possible directions T( µ 0 , ϕ 0 ) due to the linearity of (1.57). The following relation is used for that: I = 2π  0 dϕ 0 1  0 T(µ 0 , ϕ 0 )I 0 (µ 0 , ϕ 0 )dµ 0 . (1.58) The linearity of (1.57) is widely used in the modern radiative transfer the ory including the applied calculation s. It is especially convenient for describing the reflection from the surface that will be considered in the following section. The presentation of the solution of the differential and integral equation as a series expansion over the orthogonal functions is the standard mathematical method. Certain simplification is succeeded afterexpanding thephase function over the series of Legendre Polynomials in the case of the radiative transfer equation. Legendre Polynomials are defined, e.g. (Kolmogorov and Fomin 1999) as, P n (z) = 1 2 n n! d(z 2 −1) n dz . However, during the practical calculation the following recurrent formula is more appropriated: P n (z) = 2n −1 n zP n−1 (z)− n −1 n P n−2 (z) (1.59) where P 0 (z) = 1, P 1 (z) = z. With (1.59) the relations P 1 (z) = z, P 2 (z) = 1|2(3z 2 −1)etc.areobtained. Legendre P olynomials constitute the orthogonal function system within the in terval [−1, 1]: 1  −1 P n (z)P m (z)dz = 0, for n = m and 1  −1 P 2 n (z)dz = 2 2n +1 [...]... differentiating with respect to τ the following is obtained: ρ(∆τ ) = 1 exp(−∆τ |µ ) |µ | (2. 8) The probability of the photon scattering in the atmosphere is ω0 (τ ) (directly to the terminology – the probability of the quantum surviving) Thus if β ≤ ω0 (τ ), then the photon scattering is occurring in the opposite case the absorption is happening, i e at the end of the trajectory The cosine of the scattering... The clouds in the high latitudes don’t increase the light reflection because the snow albedo is also high and, in this case, the clouds play a prevailing role in atmospheric heating However, it has been elucidated in the last few decades that the situation is more complicated: the clouds themselves absorb a certain part of incoming radiation providing the atmospheric heating in all latitudes Thus, the. .. angle χ and the azimuth of the scattering φ are to be obtained in the scattering case As the phase function does not depend on the azimuth it is uniformly distributed within the interval [0, 2 ] that gives φ = 2 β The density of the probability of the scattering to the angle with cosine χ is phase function x(χ) [according to the definition (1 .2) in Sect 1.1] As this value is specified in the look-up... (2. 12) does not depend on value µ In result, the following algorithm of the irradiance calculation is obtained: 1 In the beginning of every trajectory it can be written τ = 0, µ = µ0 , ϕ = 0 2 Further, the photon free path is simulated according to (2. 7) with the writing to the counters (2. 12) 3 If the photon is going out of the atmosphere (τ ≤ 0), its trajectory will finish and the trajectory of the. .. simulated 7 If the photon stays in the atmosphere (0 < τ < τ0 ), its interaction (scattering or absorption) with the atmosphere is simulated 8 The absorption leads to the end of the trajectory 9 In the case of the scattering, the new direction of the photon is determined using (2. 9) and (2. 10) and the photon following free path is simulated The desired values of the irradiances are found with (2. 13) after... Gidrometeoizdat, Leningrad (in Russian) Matveev VS (19 72) The approximate presentation of the absorption coefficient and equivalent linewidth with the Voigt function (Bilingual) J Appl Spectrosc 16 :22 8 23 3 Matveev YuL (1984) Physics-statistical analysis of the global cloud field (Bilingual) Izv RAS Atmosphere and Ocean Physics 20 : 120 5– 121 8 Minin IN (1988) The theory of radiation transfer in the planets atmospheres Nauka,... transfer equation (1.35), but the multiple scattering plays the main 40 Solar Radiation in the Atmosphere role unlike the clear atmosphere Only the horizontally homogeneous atmosphere will be considered below Applying to the cloudy atmosphere, it means the considering of the model of the in nitely extended and horizontally homogeneous cloud layer In reality, it is the stratus cloudiness, which corresponds... mentioned above, the process of radiative transfer in the MonteCarlo method is simulated as a photon motion Coming to the atmosphere the photon is moving along a certain trajectory, which finishes either with a photon outgoing from the atmosphere or with its absorption in the atmosphere or at the surface The position of the photon in the atmosphere is determined completely with three coordinates: τ , µ... problem of the interaction between the clouds and radiation comes to the foreground in the stratus clouds study The climate simulation requires inputting the adequate optical models of the clouds and so it is necessary to obtain the real cloud optical parameters (volume scattering and absorption coefficients) The including of atmospheric aerosols in the processes of the interaction between short-wave radiation. .. also that the reflection processes depend on the incident radiation 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 . 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. normally incoming to the top of the atmosphere is. 26 Solar Radiation in the Atmosphere where value χ is defined by (1.46) and for χ 0 the following expression is correct according to the same. marks. Then the solution of the radiative transfer problem, written in the operator form (1.57), will be the following: I = TI 0 where I 0 is the radiance incoming to 36 Solar Radiation in the Atmosphere thetopoftheatmosphere.IntroduceoperatorT ↓ ,sothatI ↓ =