Derivative from Values of Solar Irradiance 181 The formula of the derivative is specially converted to form (5.12), (5.13). Written in this way, it looks as integral (2.20) directly calculated with the Monte-Carlo method according to (2.21).  Ψ a (u)B a (u)W a (u)du = M ξ (Ψ a (ξ)W a (ξ)) (5.14) That is to say, the calculation of the derivatives according to (5.14) is reduced to the multiplying of the value written to the counter by a certain “weight” function W a (ξ) (Marchuk et al. 1980). To construct the concrete algorithm of calculating W a (ξ)thederivative explicit form of the right part of series (5.10) is obtained. For that we are using the known expression of the derivative of the product through the sum of logarithm derivatives (xyz )  = (xyz )(x  |x + y  |y + z  |z + ).Thefollowing is obtained: ( Ψ a q a )  =  Ψ a (u)q a (u)  Ψ  a (u) Ψ a (u) + q  a (u) q a (u)  , ( Ψ a K n a q a )  =   dudu 1 du n Ψ a (u)q a (u 1 )K a (u 1 , u 2 ) K a (u n , u) ×  Ψ  a (u) Ψ a (u) + q  a (u) q a (u) + K  a (u 1 , u 2 ) K a (u 1 , u 2 ) + + K  a (u n , u) K a (u n , u)  . (5.15) After writing (5.15) to form (5.14) as it is more convenient for the Monte-Carlo method, finally derive: ( Ψ a K n a q a )  =   dudu 1 du n Ψ a (u)q a (u 1 )K a (u 1 , u 2 ) K a (u n , u)W a (u, u 1 , u 2 , ,u n ) , (5.16) W a (u, u 1 , u 2 , ,u n ) = Ψ  a (u) Ψ a (u) + q  a (u) q a (u) + K  a (u 1 , u 2 ) K a (u 1 , u 2 ) + + K  a (u n , u) K a (u n , u) . As it follows from (5.16), in the Monte-Carlo method the derivatives could be calculated using the same algorithms as desired values with multiplying value Ψ(ξ)byspecialweightW a (ξ)duringeachwritingtothecounter.Inaddition, if value Ψ(ξ) depends on the current magnitude of random value ξ only,i.e.of the current coordinates of t he p hoton, t hen W a (ξ) is the sum and depends on the whole history of its trajectory. Thus, to compute the derivatives of the irradiances, it is enough to dif- ferentiate the explicit expressions of functions Ψ a (u), q a (u)andK a (u, u  )with respect to the retrieved parameters. Then the following elementary changes are introduced to the algorithm of irradiance calculations described in Sect. 2.1: the counting of values W a (for entire set of parameters) at every modeling of theelementofthephotontrajectorywiththewritingtothespecialcounters of the derivatives simultaneously with writing to the counters of the irradi- ances. Although the irradiances are calculated as integrals with respect to 182 Determination of Parameters of the Atmosphere and the Surface in a Clear Atmosphere wavelength (5.4), the wavelength remains the fixed one while modeling every single trajectory. Hence, it is enough to consider the monochromatic case only during the differentiation and the derivative of integral (5.4) will be obtained automatically. It should be emphasized also that the optical thickness itself is the function of differentiated parameters. Thus, the atmospheric pressure is to be used as a vertical coordinate, while computing the derivatives. Nothing changesintherealmodelingbutforthederivationof(2.8)thephotonfreepath pr obability from altitude level P 1 (in the pressure scale) to level P is written as: 1−exp ⎛ ⎝ − 1 µ P  P 1 α(P  )dP  ⎞ ⎠ , where α(P) is the extinction coefficient, then probability density (2.8) trans- forms to the follo wing: ρ(P) = − α(P) |µ| exp ⎛ ⎝ − 1 µ P  P 1 α(P  )dP  ⎞ ⎠ . (5.17) It is just (5.17), which is to be used as a probability density of the photon free path, while differentiating. Now apply the algorithm of the irradiance calculation, described in Sect. 2.1 to the algorithm for the calculating of derivatives while taking into account the explicit form of the functions in (5.16). Counters W a are introduced for the whole set of parameters. Starting every trajectory of the counter W a := 0 is assumed. While modelingevery photon free path, the following value is assigned to the counter while taking into account (5.17): W a := W a + 1 α(P 2 ) ∂ ∂a ( α(P 2 )) − 1 |µ  | ∂ ∂a ( ∆τ  (P 1 , P 2 )) , (5.18) where ∆τ  (P 1 , P 2 ) is the photon free path from level P 1 to level P 2 (2.7). If the photon reaches the surface, then the item with value α(P 2 )willbeabsent.While modeling every act of the interaction between the photon and atmosphere, i. e. while multiplying the photon weight by ω 0 (τ  ),thefollowingvalueiswritten to the counter: W a := W a + 1 ω 0 (P  ) ∂ ∂a ( ω 0 (P  )) , (5.19) where P  is the current coordinate (in the atmospheric pressure scale) corre- sponding to optical thickness τ  . Analogously, the values for the interaction of thephotonwiththesurfaceiswrittentothecounterinaccordancewith(2.23): W a := W a + 1 A ∂ ∂a (A) . (5.20) Derivative from Values of Solar Irradiance 183 The value is written to the counter at ev ery step of m odeling the photon scattering in the atmosphere according to (2.9): W a := W a + 1 x(P  , χ) ∂ ∂a (x(P  , χ)) . (5.21) Finally,thefollowingvalueiswrittentothecounterofthederivativessimulta- neously with writing weight ψ to the counter of irradiances as per to (2.18): ψ  W a − 1 |µ  | ∂ ∂a ( ∆τ(P  , P))  , where P is the coordinate of the counter. The obtained algorithm is essentially simplified while taking into account that thefollowing sum is calculat ed simultaneously at thepoint of thescattering modeling P  = P 2 1 α(P  ) ∂ ∂a ( α(P  )) + 1 ω 0 (P  ) ∂ ∂a ( ω 0 (P  )) + 1 x(P  , χ) ∂ ∂a (x(P  , χ)) . (5.22) After substituting the expressions of the optical parameters through aerosol and molecular components (1.24) and (1.25) with the elementary algebraic manipulations, this sum is reduced to the following form: ∂ ∂a [σ m (P  )x m (χ)+σ a (P  )x a (P  , χ)] σ m (P  )x m (χ)+σ a (P  )x a (P  , χ) , (5.23) where σ m , x m , σ a , x a are the volume coefficients and phase functions of the molecular and aerosol scattering. In addition, remember that the phase func- tion of the molecular scattering determined by (1.25) does not depend on optical parameters. Finally, the only value is written to the counter in the algorithm of the photon free path modeling: W a := W a − 1 |µ  | ∂ ∂a ( ∆τ  (P 1 , P 2 )) , (5.24) and after modeling the scattering angle the only value is: W a := W a + x m (χ) ∂ ∂a σ m (P  )+x a (P  , χ) ∂ ∂a σ a (P  )+σ a (P  ) ∂ ∂a x a (P  , χ) σ m (P  )x m (χ)+σ a (P  )x a (P  , χ) . (5.25) Theexplicitexpressionsoftheabove-mentionedderivativesthroughthede- sired parameters of the inverse problem are presented further. The total set of the retrieved parameters has been defined in the previous section. There are the vertical profile of air temperature T(P i ), profiles of con tents o f four gases absorbing radiation Q H 2 O (P i ), Q O 3 (P i ), Q NO 2 (P i ), Q NO 3 (P i )(TheO 2 con- tent is con stant), volume coefficients of the aerosol absorption and scattering 184 Determination of Parameters of the Atmosphere and the Surface in a Clear Atmosphere σ a (P i , λ j ), κ a (P i , λ j ), and surface albedo A(λ j ). The concentrations of the atmo- spheric gases will be expressed through the volume-mixing ratio that gives the simple relation for their counting concentrations: n(P i ) = P i Q(P i ) kT(P i ) . (5.26) Let the sets of altitude levels P i and wavelengths λ j to be specified in a general form for the present, their concrete magnitudes will be obtained on the basis of the derivative analysis (S ect. 4.4). Note that in practice to simplify the derivatives computing (and to prevent the errors while programming) the deriva tives are to be written as a chain of the simplest formulas using the rule of the composite function differentiation. It is also useful even if the substituting of the derivatives to the general formulas causes the simplification of the expressions. The other approach effectively simplifying the calculations is application of the expression of the derivative of the product through the logarithmic derivatives. As intermed iate values in the grids P i and λ j are com puted with the linear interpolation according to the following: F(u) = F(u i ) u i+1 − u u i+1 − u i + F(u i+1 ) u − u i u i+1 − u i , the derivative of function ∂F(u)|∂F(u i ) is obtained as the following process: After determining number n from condition u n ≤ u ≤ u n+1 the following equalities are correct: ∂F(u) ∂F(u i ) = 0fori<n or i>n+1; ∂F(u) ∂F(u i ) = u i+1 − u u i+1 − u i for i = n , ∂F(u) ∂F(u i ) = u − u i u i+1 − u i for i = n +1. Asthederivativedependsontheargumentonly,specifyitas ∂F(u)|∂F(u i ) ≡ L i (u). Then the derivative with respect to the surface albedo is written as: ∂ ∂A(λ j ) (A) = L j (λ). Thephotonfreepath∆τ  (P 1 P 2 ), as per (2.1)–(2.4), is the quadratic function of volume extinction coefficient α(P i ). Hence the following algorithm is elab- orated for computing deriva tive ∂|∂α(P i )(∆τ  (P 1 , P 2 )), where inequity P 2 <P 1 is assumed for the definiteness: Derivative from Values of Solar Irradiance 185 1. Finding num bers n 1 and n 2 from conditions P n 1 ≥ P 1 ≥ P n 1 +1 , P n 2 ≥ P 2 ≥ P n 2 +1 2. Then three cases are considered depending on the magnitude of differ- ence n 2 − n 1 : n 2 >n 1 +1 ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = 0fori<n 1 or i>n 2 +1; ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = 1 2 (P 1 − P i+1 ) 2 (P i − P i+1 ) ,fori = n 1 ; ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = P 1 − P i − 1 2 (P 1 − P i ) 2 P i−1 − P i + 1 2 (P i − P i+1 ), for i = n 2 + 1 ; (5.27) ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = 1 2 (P i−1 − P i+1 ), for n 1 +2≤ i ≤ n 2 −1; ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = P i − P 2 − 1 2 (P i − P 2 ) 2 P i − P i+1 + 1 2 (P i−1 − P i ), for i = n 2 ; ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = 1 2 (P i−1 − P 2 ) 2 P i−1 − P i ,fori = n 2 +1. n 2 = n 1 +1.Thiscasediffersfromthelatterbythederivativebeingequalto: ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = P 1 − P 2 − 1 2 (P 1 − P i ) 2 P i−1 − P i − 1 2 (P i − P 2 ) 2 P i − P i+1 for i = n 1 +1= n 2 (5.28) n 2 = n 1 : ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = 0fori<n 1 or i>n 1 +1; ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = P i − P 2 + 1 2 (P 1 − P i+1 ) 2 −(P i − P 2 ) 2 P i − P i+1 , for i = n 1 = n 2 , ∂ ∂α(P i ) ( ∆τ  (P 1 , P 2 )) = P 1 − P i + 1 2 (P i−1 − P 2 ) 2 −(P 1 − P i ) 2 P i−1 − P i , for i = n 1 +1= n 2 +1. (5.29) Note that, the volume extinction coefficient in the described algorithm is applied after recalculating per the pressure unit α P (P i ), while it has been 186 Determination of Parameters of the Atmosphere and the Surface in a Clear Atmosphere calculated per the altitude unit initially. After the differentiation of (5.1) we obtained: ∂α P (P i ) ∂α z (P i ) = RT(P i ) g(P i )µ(P i )P i . (5.30) Owing to summarizing rules (1.24), analogous relations are also derived f or the volume coefficient of the aerosol scattering. Now the final formulas are presented for the derivatives of the radiative characteristics with respect to the desired parameters. The specifications, used in Chapter 1 and in the previous section are kept. Derivativ e s with respect to contents of the gases absorbing radiation (exclud- ing the water vapor). Volume coefficient of the molecular absorption κ m (P i ) depends on these contents and the volume extinction coefficient in its turn depends on the volume coefficient of the molecular absorption as per (1.24). Then specify the concrete gas with subscript k and obtain: ∂(∆τ  (P 1 , P 2 )) ∂Q k (P i ) = ∂ (∆τ  (P 1 , P 2 )) ∂α P (P i ) ∂α P (P i ) ∂α z (P i ) ∂α z (P i ) ∂κ m,k (P i ) ∂κ m,k (P i ) ∂n k (P i ) ∂n k (P i ) ∂Q k (P i ) , (5.31) where: ∂α z (P i ) ∂κ m,k (P i ) = 1 according to (1.23) and (1.24), ∂κ m,k (P i ) ∂n k (P i ) = C a,k according to (1.22) and ∂n k (P i ) ∂Q k (P i ) = P i kT(P i ) according to (5.18). The cross-sections of the molecular absorption depending on wavelength and (only for ozone) on temperature are computed by the linear interpolation with (1.28) and (5.7). Certainly, the derivatives with respect to gases content are not equal to zero within the spectral regions of these gases absorption only (Table 5.1). Deriva tive with respect to wa ter vapor content. In addition to the volume coefficient of the molecular absorption, the volume coefficient of the molecular scattering also depends on H 2 O content as per (1.27). It yields the following expression for the derivative of the free path: ∂(∆τ  (P 1 , P 2 )) ∂Q H 2 O (P i ) (5.32) = ∂ (∆τ  (P 1 , P 2 )) ∂α P (P i ) ∂α P (P i ) ∂α z (P i )  ∂α z (P i ) ∂κ m,k (P i ) ∂κ m,k (P i ) ∂n k (P i ) ∂n k (P i ) ∂Q k (P i ) + ∂σ z,m (P i ) ∂Q H 2 O (P i )  Derivative from Values of Solar Irradiance 187 and the following is obtained for the derivative with respect to the coefficient of the molecular scattering: ∂σ P,m (P  ) ∂Q H 2 O (P i ) = L i (P  ) ∂σ P,m (P i ) ∂σ z,m (P i ) ∂σ z,m (P i ) ∂Q H 2 O (P i ) . (5.33) The derivatives depending on volume coefficient of the molecular scattering have been calculated above, and the absorption cross-section for H 2 Oiscom- puted with (5.8). Theexpressionforthederivativeofthemolecularscatteringvolumecoeffi- cient is obtained as follows: ∂σ z,m (P i ) ∂Q H 2 O (P i ) = ∂σ z,m (P i ) ∂m ∂m ∂P w ∂P w ∂Q H 2 O (P i ) , (5.34) where: ∂σ z,m (P i ) ∂m = σ z,m (P i ) 4m m 2 −1 ∂m ∂P w = 10 −6 0.0624 − 0.00068λ −2 1 + 0.003661 T(P i ) ∂P w ∂Q H 2 O (P i ) = 0.7501P i . Derivative with respect to volume coefficient of the aerosol absorption. The volume extinctio n coefficient only depends on volume coefficient of the aerosol absorption tha t directly yields: ∂(∆τ  (P 1 , P 2 )) ∂κ z,a (P i , λ j ) = ∂ (∆τ  (P 1 , P 2 )) ∂α P (P i ) ∂α P (P i ) ∂α z (P i ) ∂α z (P i ) ∂κ z,a (P i ) L j (λ) , (5.35) where ∂α z (P i )|(∂κ z,a (P i )) = 1 with taking into account (1.23) and (1.24). Derivative with respect to volume coefficient of the aer osol scattering. The volume coefficients of the absorption and scattering and the phase function of the aerosol scattering as per (5.9) depend on the volume coefficient of the aerosol scattering. Therefore, we obtain: ∂(∆τ  (P 1 , P 2 )) ∂σ z,a (P i , λ j ) = ∂ (∆τ  (P 1 , P 2 )) ∂α P (P i ) ∂α P (P i ) ∂α z (P i ) ∂α z (P i ) ∂σ z,a (P i ) L j (λ) , (5.36) where ∂α z (P i )|∂σ z,a (P i ) = 1 with taking into account (1.23) and (1.24). Then we can write: ∂σ P,a (P  ) ∂σ z,a (P i , λ j ) = L i (P  ) ∂σ P,a (P i ) ∂σ z,a (P i , λ) L j (λ) . (5.37) 188 Determination of Parameters of the Atmosphere and the Surface in a Clear Atmosphere At last for the derivative of the phase function the following relations ar e correct: ∂x a (P  , χ) ∂σ z,a (P i , λ j ) = L i (P  ) ∂x a (P i , χ, λ) ∂σ z,a (P i , λ) L j (λ) (5.38) and with accounting for (5.9) after simple transformations we obtain: ∂x a (P i , χ, λ) ∂σ z,a (P i , λ) = x a (P i , χ, λ) σ z,a (P i , λ) ⎛ ⎝ D(P i , χ, λ)− 1 2 1  −1 D(P i , χ  , λ)x a (P i , χ  , λ)dχ  ⎞ ⎠ , (5.39) where D(P i , χ, λ) = b i (χ, λ)+2c i (χ, λ) ln(σ a,z (P i , λ)) . The derivative with respect to air temperature. A big quantity of values depends on temperature. Begin from the photon free path and obtain the following for it: ∂(∆τ  (P 1 , P 2 )) ∂T(P i ) = ∂ (∆τ  (P 1 , P 2 )) ∂α P (P i ) ∂α P (P i ) ∂T(P i ) (5.40) and for the volume coefficient of the molecular scattering: ∂σ P,m (P  ) ∂T(P i ) = L i (P  ) ∂σ P,m (P i ) ∂T(P i ) . (5.41) An important feature of calculating the derivatives with respect to temperature isthenecessityofaccountingforthetemperaturedependenceintheformulaof the recalculation of the volume extinction coefficients in terms of atmospheric pressure (5.1). It is obtained as follows: ∂α P (P i ) ∂T(P i ) = α P (P i )  1 α z (P i ) ∂α z (P i ) ∂T(P i ) + 1 T(P i )  . (5.42) The analogous relation is written for derivative ∂σ P,m (P i )|∂T(P i ), and for the aerosol scattering volume coefficient the following is obtained: ∂σ P,a (P i ) ∂T(P i ) = σ P,a (P i ) T(P i ) . Now the expression for the extinction coefficient is derived: ∂α z (P i ) ∂T(P i ) = ∂σ z,m (P i ) ∂T(P i ) + ∂κ z,m (P i ) ∂T(P i ) . (5.43) Derivative from Values of Solar Irradiance 189 Finally, the problem is reduced to the differentiation of the volume coefficients of the molecular scattering and absorption. The first coefficient is equal to the sum of the coefficients of absorbing gases (all, including O 2 ) by (1.22). The corresponding sum is inferred for the derivatives too. Specifying the concrete gas with subscript k, with accounting for (5.18) we get: ∂κ m,k (P i ) ∂T(P i ) = κ m,k (P i )  − 1 T(P i ) + 1 C a,k ∂C a,k ∂T(P i )  . (5.44) The absorption cross-sections of gases NO 2 ,NO 3 ,O 3 within the range 426– 848 nm don’t depend on temperature, hence, equality ∂C a,k |(∂T(P i )) = 0is correct. The following is obtained from (5.7) for O 3 within the range 330– 356 nm: ∂C a,k (λ, T(P i )) ∂T(P i ) = C 1 (λ)+2C 2 (λ)T(P i ) . (5.45) Equation (5.8) yields the following expression with taking into account the linear interpolation of cross-sections over wavelength: ∂C a,k (λ, P i , T(P i )) ∂T(P i ) = −C a,k (λ j , P, T(P i )) C 2 (λ j ) T(P i ) λ j+1 − λ λ j+1 − λ j − C a,k (λ j+1 , P, T(P i )) C 2 (λ j+1 ) T(P i ) λ − λ j λ j+1 − λ j . (5.46) The following is obtained for the derivative of the volume coefficient of the molecular scattering with (1.25) and (1.26): ∂σ z,m (P i ) ∂T(P i ) = σ z,m (P i )  4m m 2 −1 ∂m ∂T(P i ) + 1 T(P i )  , (5.47) and expression (1.27) yields the following: ∂m ∂T(P i ) = 10 −6 1 + 0.003661T(P i )  b( λ)  2.178 × 10 −11 P 2 i − 5.079 × 10 −6 P i (1 + 10 −6 P i (1.049 − 0.0157T(P i ))) 1 + 0.003661T(P i )  +10 −4 P w 2.284 − 0.0249λ −2 1 + 0.003661T(P i )  (5.48) After analyzingtheobtained derivatives with the methods descri bed inSect. 4.4 the concrete sets of altitudes and wavelengths are selected for the retrieval of the atmospheric parameters, namely: 190 Determination of Parameters of the Atmosphere and the Surface in a Clear Atmosphere – The grid over wavelengths: from 325 to 685 nm with step 20nm and from 725 to 985 nm with step 40 nm (28 points in a whole). – The grid over altitude: from 1000 to 800 mbar with step 10 mbar,from 800 to 500 mbar with step 20 mbar, 500 to 110 mbar with step 30mbar, 90 to 10 mbar with step 10 mbar and levels 5.2 and 0.5 mbar (61 points as a whole). The selection of the detailed grid in the lower atmospheric layers is caused by theirradiancesoundinglevelsandhavebeenmeasuredwithastepof100mbar. Note that the top of the atmosphere corresponding to 0.5 mbar (about 55 km) is in a good agreement with the altitude of the standard top atmospheric level, usually used in calculations of the radiative transfer in the shortwave region (Rozanov et al. 1995; Kneizis et al. 1996). Consider briefly the specific features of the calculated derivatives of the irradiances and their magnitudes. This analysis allows estimating the mech- anisms of the parameter influences on the measured characteristics of solar radiation and concluding the possibility of the retrieval of certain atmospheric parameters. Dependence of the upwelling irradiance upon the surface albedo is well studied (Kondratyev et al. 1971,1977). Theinhomogeneouslinear function (y = ax + b) has been proposed for its description, where the multiplicative item is thepartofirradiancedirectlyreflectedfromthesurfaceproportionaltoalbedo, and the additive item is connected with diffused radiation in the atmosphere. Correspondingly, the greater albedo is the stronger is the upwelling irradiance dependent on it. The dependence of the surface albedo is also elucidated in the downwelling irradiance (Sect. 3.4). The corresponding derivative is greater, when the albedo is greater and the scattering in the atmosphere is stronger. It could reach decimals of percent of the irradiance variation to one percent of the albedo variation as it follows from the calculations with the bright surfaces like snow. Thus, the influence of surface albedo on the downwelling irradiance could exceed the uncertainty of the irradiance observation. Out of O 2 and H 2 Oabsorptionbands,the dependence o f the irradiance upon tempera ture is extremely weak: it conserves c lose to the value of the observational uncertainty even if the a priori variations of the temperature are maximal. The same is valid in the case of the ozone absorption bands. Thus, the temperature dependence of the irradiances could be igno red out of the absorption bands and the corresponding derivativ es could be assumed equal to zero. At the same time, the temperature dependence is essential within the O 2 and H 2 O absorption bands including the weak bands also. In addition, within some spectral regions, for example in wavelength 932 nm in the center of the H 2 O band, it is strong and reaches the percent of the irradiance variation to one-degree variation of the whole temperature profile. Deriva tive wi th respect to water vapor content are also essential only within its absorption bands, hence the relationship between the volume coefficient of the molecular scattering and H 2 Ocontentcouldbeneglected.Thesederivatives are maximal within the absorption band 910–980 nm,wheretheirradiance variation reaches 40% to the a priori variations of the vertical profile of H 2 O conten t as a whole. [...]... low because the information about the remote points is “forgotten” due to the multiple scattering within the cloud Therefore, the photon, registered by the instrument, brings information only about the last collision Thus, while measuring the diffused irradiance, the information collected by the instrument mainly concerns the points, remote from the instrument for the photon free path In the case of stratus... constitute the set of formulas for solving the inverse problem in the case of radiation observations within the cloud layer They allow more correct data interpretation than has been done in the study by King (19 87) , because these expressions are derived while taking into account the absorption of solar radiation and provide the data processing independently at every wavelength 6.1.3 Problem Solution in the. .. Dotted line indicates the a priori profile temperature profile are inevitable because of the existence of the observational uncertainties within the oxygen band In this connection, the question of the possibility of using the radiosounding data for the irradiance data processing was discussed even in the 70 th, while accomplishing the described experiments (e g Kondratyev et al 1 977 ) However, the geographical... within the cloud layer was solved in the study by King et al (1990) again with the assumption of the conservative scattering The important exact expression for scaled optical thickness τ = 3(1 − g)τ0 through the reflected radiance was derived in the study by King (19 87) as a result of transforming the first of (2.24) Regretfully the author of the study (King 19 87) continued the further application of the. .. cause the extinction of the downwelling irradiance to 1–2% Thus, even with the low systematic uncertainty in the observed irradiances the algorithm of the inverse problem solving could cause the false conclusion about the existence of the aerosol layers in the upper troposphere and in the stratosphere Hence, the results of the retrieval of the aerosol scattering and absorption volume coefficients obtained... bands In particular, it concerns oxygen narrow band 76 0 nm However, as has been mentioned in Chap 3, while describing the observations with the K-3 spectrometer the large systematic uncertainty could appear within the oxygen band connected with the shift of the wavelength scale owing to the mechanical scanning of the K-3 instrument Besides, the instrumental function obtained from the measurements in the. .. results The authors of the study by Rozenberg et al (1 974 ) took the first step by using the observation of the reflected solar radiation from satellites for obtaining the small parameter connected with the single scattering albedo ω0 of the cloud while assuming its in nite optical thickness and the expansion analogous to (2.29) Only the first power of the expansion was taken into account, and the optical... likely caused by the surface inhomogeneity (the snow was melting on 29th April) The surface inhomogeneity has been smoothed during the second stage of the data processing, but the spectral distortions of the upwelling irradiances have remained and they cause the systematic uncertainty of the retrieved albedo, which does not exceed the interval of three SD and statistically can be assumed the insignificant... follows even from the results of the study by Rozenberg et al (1 974 ) Note, that introducing the atmospheric aerosols to the cloud model provides certain radiation absorption in the calculations of the radiative characteristics (direct problem solving), but this is rather weak Hence, many investigators have been still assuming the conservative scattering of solar radiation in clouds within the VD spectral... ) − F ↑ (τ0 ) are the net solar fluxes at the top (τ = 0) and at the bottom (τ = τ0 ) of the cloud layer; a2 (µ0 ) is the second coefficient in the expansion for the plane albedo of the semiinfinite atmosphere, a∞ is the spherical albedo of the in nite atmosphere; ρ = ρ(0, µ, µ0 ) and σ = σ(τ0 , µ, µ0 ) are the radiances (in relative units of the incident solar flux πS) measured during the radiative experiments: . P is the coordinate of the counter. The obtained algorithm is essentially simplified while taking into account that thefollowing sum is calculat ed simultaneously at thepoint of thescattering modeling. is determined with the in uence of the stratospheric ozone on the solar irradiance value because the in uence of all other components (including the aerosols) is negligibly weak at the high altitudes to the revealing of themethodological algorithm short- comings. Therefore, we are presenting the analysis of all retrieved parameters of the atmosphere including even the parameters whose obtaining