Geoscience Reference
In-Depth Information
Appendix B: Formulas Derivation
Derivation of formulas for the determination of the cloud optical parameters
from the data of observations of solar irradiances at its boundaries. The initial
formulas for the irradiances reflected and transmitted by the cloud layer:
µ 0 ) ml exp(−2 k
τ 0 )
µ 0 )− nK (
µ 0 ) F 0 =
F (0,
a (
,
(B.1)
1− ll exp(−2 k
τ 0 )
µ 0 ) m exp(− k
τ 0 )
nK (
F (
τ
µ 0 )
=
,
.
(B.2)
1− ll exp(−2 k
τ 0 )
From (B.1) the following is easy to derive:
µ 0 )− F 0 ]
µ 0 )+ l [ a (
l mnK (
τ 0 )
=
exp(2 k
.
(B.3)
µ 0 )− F 0
a (
According to the definition, values n , assuming the ground albedo influence,
are expressed with the following:
Amn 2
1− Aa
n
1− Aa
l
=
=
l
and
n
.
Substituting (B.3) to (B.2) the intermediate relation is obtained:
τ
µ 0 )− F 0 ) exp(− k
)
1
1− Aa
F =
( a (
.
(B.4)
l
τ
Raise to the squarebothparts of (B.4), substituting the expression for exp(−2 k
),
and deleting the common multilayer, obtain the following:
F 2 [1 − Aa ][ l (1 − Aa )− Amn 2 ]
=
(B.5)
µ 0 )− F 0 ]+ l [ a (
µ 0 )− F 0 ] 2
µ 0 )[ a (
mnK (
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