Theoretical Base of Solar Irradiance and Radiance Calculation in the Earth Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 71

Fig. 2.2a,b. The applicability ranges of the asymptotic formulas of radiative transfer theory
in the case of calculation reflected irradiance ( a )andradiativefluxdivergence( b )forthe
cloudy layer. Curves correspond to the relative uncertainty equal to 5%. The solid curve is
for phase function parameter g

= =

0.5; dashed line - g 0.75 and the dashed dotted line -

= µ = µ =

g 0.9; curves with circles correspond to 1, with crosses -to 0. 5

τ 0 )wereanalyzedinstud-
ies by Konovalov (1974,1975). The accuracy of the formulas for

ρ µ µ 0 , τ 0 )and σ µ µ 0 , and transmission functions ( , ( ,

ρ µ µ 0 , τ 0 )and ( ,

σ τ 0 ) has turned out roughly equal to each other. The uncertainties of the

formulas for reflected and transmitted radiation are about 2% beginning from
optical thickness

µ µ 0 , ( ,

τ 0 ≥ 4 | (1 − k ). The numerical analysis of the formulas of the

spherical albedo and transmittance (the values of reflected and transmitted
irradiances integrated with respect to the cosine of the solar zenith angle) for
the wide set of parameters has been accomplished in the study by Harshvard-
han and King (1986). It has been shown there that their uncertainty does not
exceed 5% by values

ω 0 ≥ 0.7.
The accuracyof the formulas for irradianceswas tested toprovide the relative
errors less than 5% in the region plotted in coordinates “

τ ≥ 2.0 and

ω 0 ”inFig.2.2
(Demyanikov andMelnikova 1986). The curves in Fig. 2.2a,b correspond to the
levelof5%errorofthereflectedirradiance(a)andradiativefluxdivergence(b)
calculated for asymmetry parameters g

τ −

= 0.5, 0.75 and 0.9 and for two values

µ 0 = of cosine 1 and 0.5. The result for the transmitted irradiance is similar to

the result shown in Fig. 2.2 for the reflected irradiance.
The numerical analysis of the accuracy of the radiance calculation in the
optically thick layer has been accomplished in the topic by Yanovitskij (1997).
According to this topic the applicability region of the radiance ( τ ω 0 > > 15;

ω 0 > 0. 9),
which in their turn is narrower than for the integral over the zenith angle
characteristics (

τ 0.99) is more strongly restricted than of the irradiance ( > 7;

ω 0 > 0. 8).
The accuracy of asymptotic expansions (2.29) and (2.30) is defined by the
omitted items proportional to s 3 or s 4 .Theaccuracyoftheapproximations
was tested by comparison with the function values computed by the numeri-

τ > 2;