Geoscience Reference
In-Depth Information
data are available from national weather services and agricultural agencies. In the event
that suitable data are not available, it may be necessary to make an estimate by means of
one of several theoretical models or simpler empirical formulae that relate short-wave
radiation with other physical factors, such as extraterrestrial radiation, optical air mass,
turbidity, water vapor content of the air, amount and type of cloud cover. However, these
should be used with caution.
A simple equation which can be used for daily averages, was proposed by Prescott
(1940) in terms of daily total extraterrestrial radiation
Q
se
as follows
Q
s
=
Q
se
[
a
+
b
(
n
/
N
)]
(2.74)
where
a
and
b
are constants which depend on the location, the season and the state of the
atmosphere; their values have been determined for many locations and on average they
appear to be around
a
=
0.25 and
b
=
0.50. In Equation (2.74)
n
is the actual number of
hours of bright sunshine and
N
the number of daylight hours; as a first approximation for
steady weather conditions
n
/
N
can be related to the mean fractional cloud cover
m
c
by
a
(
n
/
N
)
+
bm
c
=
1
(2.75)
in which
a
and
b
are different constants, for which values averaging around 1.1 and 0.85,
respectively, have been observed in the Netherlands and in Japan (e.g. De Vries, 1955;
Kondo, 1967).
Many other regression equations like (2.74), also for instantaneous values
R
s
, have been
proposed in the literature, but such simple equations can be only poor substitutes for direct
measurements. Nevertheless, it is possible to obtain fairly accurate radiation estimates by
better empirical and partly theoretical methods, which are, however, more difficult to apply.
Examples of such methods, which can give useful results, are those presented by among
others, Kondo (1967; 1976), Paltridge and Platt (1976, p. 137) and Meyers and Dale (1983).
Because such approaches often rely on the extraterrestrial radiation, it is appropriate to take
a quick look at it.
Extraterrestrial radiation
The extraterrestrial radiation
R
se
can readily be calculated for a given latitude, time of day
and day of the year from the solar constant. For a horizontal surface, instantaneous values
can be calculated from
R
se
=
R
so
(
d
so
/
d
s
)
2
cos
β
(2.76)
in which
β
is the zenith angle, that is the angular distance between the sun and the vertical,
and in which
d
s
and
d
so
are the instantaneous distance and the annual mean distance of the
Earth from the Sun, respectively; however,
d
s
and
d
so
differ by at most 3.5%, so that this
effect is often neglected in hydrologic applications. It can be readily shown that the zenith
angle can be calculated as follows
β
=
φ
δ
+
φ
δ
cos
cos
cos
h
cos
sin
sin
(2.77)
where
φ
is the latitude and
h
is the hour angle, such that its origin
h
=
0 is local noon
or 1200, and 24 h
=
2
π
. The angle
δ
is the solar declination, that is the angular distance
of the Sun north (or south when negative) of the Equator. Daily values of Equation (2.76)
