APPENDIX A: DETAILS OF EQUATIONS 3 AND 4
Suppose we have a series Ti of N surface-layer air temperature measurements.
The average temperature is
1 N
T =
∑ Ti ,
(A1)
N i =1
and the standard deviation is
1/2
(
)
1 N
2
σT
=
∑ Ti - T
.
(A2)
N - 1 i =1
Since the Ti are approximately normally distributed,
T -T
~
Ti = i
(A3)
σT
is approximately normal with mean 0 and variance 1. That is, the probability den-
~
sity function for T is
()
(
)
1
~
~
φ1 T =
exp -T 2 / 2 ,
(A4)
2π
and the cumulative distribution function is
()
Φ1 T = ∫-∞ φ1(T ') dT ' .
~
~
T
(A5)
We have already discussed the distribution function for total cloud amount n;
call this φ2(n), where
( ) n
^^
Γ α +β
(1 - n)
^
β -1
φ2 (n) =
α -1
^
Γ(α) Γ(β)
for 0 ≤ n ≤ 1,
(A6a)
^
^
= 0
otherwise.
(A6b)
^
Again, Table 4 lists appropriate α and β values to use in winter. The cumulative
^
distribution function for total clouds is thus
Φ 2 (n) = ∫0 φ2 (n') dn' .
n
(A7)
We seek a function that predicts total cloud amount from surface-layer air tem-
~
perature alone. In other words, in terms of T , we want ψ such that
()
~
n = ψT .
(A8)
~
Based on Tables 13 and physical intuition, we can reasonably assume that ψ ( T )
~
is a monotonically increasing function of T . From mathematical statistics, we know
we can then approximate (Ventcel 1964, p. 263 ff.)
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