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Chapter 20
The umbra IPDF,
is determined in the interval
and
reaches its maximum value at the point
the penumbra IPDF,
is
determined in the interval
and the sun flecks IPDF,
in the
and
can be estimated from measurements with a PAR
pyranometer, but the estimation of
and
needs special measurements or
calculations.
Proportional areas of umbra
penumbra and sun flecks are given
by the following formulae
and
Within the umbra and the sun flecks, the shape of IPDF is relatively
constant and can be fitted with a normal distribution. The situation is more
complicated in penumbra, in which the shape of
varies rapidly. Among
the various theoretical distributions available, only the beta-distribution is
able, more or less satisfactorily, to describe the variability of the penumbra
IPDF. The beta-distribution is given by formula
where
the parameters
and
and where
is the beta
function given by
To fit the IPDF in umbra, we can only use the values of Q in the interval
use the interval between
and
the interval between
and
being
an overlapping area of penumbra and sun flecks.
between
and
while the interval between
and
is an overlapping
area of umbra and penumbra. Similarly, to fit the IPDF in sun flecks, we can
interval
with its maximum value at the point
The values