Analytical Method of Inverse Problem Solution for Cloudy Atmospheres - Short-Wave Solar Radiation in the Earth's Atmosphere - page 227

6.3.1
Uncertainties of Derived Formulas

There are four main sources of uncertainties, while using the proposed formu-
las for the retrieval of the cloud optical parameters:

1. observational uncertainties;
2. a priori specification of parameter g ;
3. breakdown of the applicability region of the asymptotic formulas;
4. inhomogeneity of the cloud layer, while the derived expressions are as-
suming the cloud homogeneity (while consideration of the observations
within the cloud layer).

∆ | It is easy to deduce the corresponding formulas for relative uncertainties s s

∆τ 0 |τ 0 caused by observational uncertainty, as we have the analytical
expressions for the calculation of the optical parameters using the approach
described in Sect. 4.3, namely, if the vector of observations y

and

= f ( x 1 , x 2 ,..., x n ),

then:

∂ ∆

x 1 + ∂

∆

x 2 +...+ ∂

∆

f
∂ x 1

f
∂ x 2

f
∂ x n

∆

y ≤ x n ,

∆ where x i isthemeansquaredeviationcausedbytheobservationaluncertainty

or interpolation of the functions over look-up tables.
In particular, if irradiances F ↑ and F ↓ have beenmeasured with uncertainty

∆ F and the optical parameters have been calculated with (6.1), the expression

of the relative uncertainties are the following (Melnikova 1992; Melnikova and
Mikhailov 1994):
∆ ∆ ∆ µ 0 )+16 K 0 ( µ 0 ) ∆ ∆ s F

1− F ↑ − F ↓

+ 2

Fa 2 ( K 0 + F (0) a 2

≤ , (6.42)

µ 0 )−2 F (0) a 2 ( µ 0 )

s 16 K 0 (

∆τ 0 |τ 0 : and for relative uncertainty

30

F (0) 2 +

∆τ 0
τ 0 ≤

∆

1− g + ∆

∆

1
τ 0

F g s

∆ s + , (6.43)

s

where value 1 − F ↑ − F ↓ defines the radiative flux divergence in the cloud layer
in relative units

π

S . In the short-wave range it is about 0.05-0.2. Then the first
itemprovides the order of the magnitude of the uncertainty, namely ∆ s | s ≥ 4%

m 2 .
The uncertainties of functions

∆ |

for F ∼ 1-3W

∆ µ ∆ µ

0 )areinducedfortwo
reasons: the inaccurate measuring of the incident angle and the income of
partly scattered solar radiation to the cloud top. The first reason (measuring
of solar incident angle arccos

K 0 (

0 )and

a 2 (

µ 0 )couldnotgiveasignificanterrorasthevalue

µ 0 is defined by the moment and geographical site of the observation and of