In addition, there is a strong dependence of the accuracy upon the degree of the
scattering anisotropy (the extension forward of the phase function defined by
the parameter g magnitude). The uncertainties of formulas for reflected and transmitted radiation are about 2% beginning from optical thickness t 0 ! 4/(1Àk). The
numerical analysis of formulas for the spherical albedo and radiation transmittance
for the wide set of parameters shows uncertainty not exceed 5% by values t ! 2.0
and o 0 ! 0.7.
The accuracy of formulas for irradiances was tested with closed numerical
experiments to provide relative errors less than 5% in the region plotted in
coordinates “t À o 0 ” in Fig. 11.2. Curves in Fig. 11.2a, b correspond to the level
of 5% error of the reflected irradiance (a) and radiative divergence (b) calculated for
parameters g ¼ 0.5; 0.75 and 0.9 and for two values of cosine m 0 ¼1 and 0.5.
The numerical analysis of the accuracy of the radiance calculation within the
optically thick layer gives the applicability region of the radiance (t > 15;
o 0 > 0.99) is restricted stronger than of the irradiance (t > 7; o 0 > 0.9), which
in turn is narrower than for spherical albedo and transmittance (t > 2; o 0 > 0.8).
Errors of asymptotic expansions (11.6) and (11.7) are defined by omitted items
proportional to s
3 or s
4
. The accuracy of the approximations was tested by comparison with the function values computed by the numerical methods. The relative
uncertainties of the escape function computed with approximations (11.8) are
presented in Table 11.5. It has been found that uncertainties are rather small as
far as o 0 ¼0.98 for magnitudes g ¼ 0.5 and m > 0.2. Table 11.5 are illustrating that
the errors of the escape function calculation are not exceeding 6% for value
s < 0.12.
0
5
10
15
20
25
30
35
0
0.005 0.025 0.075
0.1
1- ω 0
ω 0
Optical thickness
g=0,75
g=0,9
g=0,9
g=0,75
g=0,5
g=0,5
0
5
10
15
20
25
30
35
0.000
0.002
0.008
0.050
1Optical thickness
g=0,9
g=0,9
g=0,75
g=0,75
g=0,5
g=0,5
b
a
Fig. 11.2 The applicability ranges of the asymptotic formulas of the radiative transfer theory in
case of calculation reflected irradiance (a) and radiative divergence (b) for cloudy layer. Curves
corresponded to the relative uncertainty equal to 5%. Solid curve is a phase function parameter
g ¼ 0.5; dashed line – g ¼ 0.75 and dashed dotted line – g ¼ 0.9; curves with circles correspond
to m ¼ 1, with crosses – to m ¼ 0.5
11.6 Error Analysis
115
scattering anisotropy (the extension forward of the phase function defined by
the parameter g magnitude). The uncertainties of formulas for reflected and transmitted radiation are about 2% beginning from optical thickness t 0 ! 4/(1Àk). The
numerical analysis of formulas for the spherical albedo and radiation transmittance
for the wide set of parameters shows uncertainty not exceed 5% by values t ! 2.0
and o 0 ! 0.7.
The accuracy of formulas for irradiances was tested with closed numerical
experiments to provide relative errors less than 5% in the region plotted in
coordinates “t À o 0 ” in Fig. 11.2. Curves in Fig. 11.2a, b correspond to the level
of 5% error of the reflected irradiance (a) and radiative divergence (b) calculated for
parameters g ¼ 0.5; 0.75 and 0.9 and for two values of cosine m 0 ¼1 and 0.5.
The numerical analysis of the accuracy of the radiance calculation within the
optically thick layer gives the applicability region of the radiance (t > 15;
o 0 > 0.99) is restricted stronger than of the irradiance (t > 7; o 0 > 0.9), which
in turn is narrower than for spherical albedo and transmittance (t > 2; o 0 > 0.8).
Errors of asymptotic expansions (11.6) and (11.7) are defined by omitted items
proportional to s
3 or s
4
. The accuracy of the approximations was tested by comparison with the function values computed by the numerical methods. The relative
uncertainties of the escape function computed with approximations (11.8) are
presented in Table 11.5. It has been found that uncertainties are rather small as
far as o 0 ¼0.98 for magnitudes g ¼ 0.5 and m > 0.2. Table 11.5 are illustrating that
the errors of the escape function calculation are not exceeding 6% for value
s < 0.12.
0
5
10
15
20
25
30
35
0
0.005 0.025 0.075
0.1
1- ω 0
ω 0
Optical thickness
g=0,75
g=0,9
g=0,9
g=0,75
g=0,5
g=0,5
0
5
10
15
20
25
30
35
0.000
0.002
0.008
0.050
1Optical thickness
g=0,9
g=0,9
g=0,75
g=0,75
g=0,5
g=0,5
b
a
Fig. 11.2 The applicability ranges of the asymptotic formulas of the radiative transfer theory in
case of calculation reflected irradiance (a) and radiative divergence (b) for cloudy layer. Curves
corresponded to the relative uncertainty equal to 5%. Solid curve is a phase function parameter
g ¼ 0.5; dashed line – g ¼ 0.75 and dashed dotted line – g ¼ 0.9; curves with circles correspond
to m ¼ 1, with crosses – to m ¼ 0.5
11.6 Error Analysis
115