Directory UMM :Data Elmu:jurnal:A:Agricultural & Forest Meterology:Vol102Issue1April2000:

Agricultural and Forest Meteorology 102 (2000) 39–50

Estimation of photosynthetically active radiation
under cloudy conditions
I. Alados a , F.J. Olmo b , I. Foyo-Moreno b , L. Alados-Arboledas b,∗
b

a Dpto de F´ısica Aplicada, Universidad de Malaga, Malaga, Spain
Dpto de F´ısica Aplicada, Universidad de Granada, 18071 Granada, Spain

Grupo de F´ısica de la Atmósfera
Received 2 August 1999; accepted 16 December 1999

Abstract
Clouds are the largest modulators of the solar radiative flux reaching the Earth’s surface. The amount and type of cloud
cover prevailing at a given time and location largely determines the amount and type of solar radiation received at the Earth’s
surface. This cloud radiative forcing is different for the different solar spectral bands. In this work, we analysed the influence
of cloud radiative forcing over the photosynthetically active radiation. Knowledge of the photosynthetically active radiation
is necessary in different applications, but due to the absence of widespread measurements of this radiometric flux, it must be
estimated from available variables. Cloudless sky parametric models compute the global photosynthetically active radiation
at surface level by addition of its direct beam and diffuse components. To compute this flux under all sky conditions one

must consider the influence of clouds. This could be done by defining a cloud transmittance function. We have developed
such a cloud transmittance function considering three different types of clouds. The efficacy of the cloud radiative forcing
scheme has been tested in combination with a cloudless sky parametric model using independent data sets. For this purpose,
data recorded at two radiometric stations are used. The combination of an appropriate cloudless sky parametric model with
the cloud transmittance scheme provides estimates of photosynthetically active radiation with mean bias deviation about 4%
that is close to experimental errors. Comparisons with similar formulations of the cloud radiative effect over the whole solar
spectrum shows the spectral dependency of the cloud radiative effect. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Photosynthetically active radiation; Solar irradiance; Cloud radiative effect; Modelling; Parametric models; Estimation model

1. Introduction
Incident photosynthetically active radiation (400–
700 nm) is required to model photosynthesis, of
single plant leaves to complex plant communities. Photosynthetically active radiation is the gen∗ Corresponding author. Tel.: +34-958-244024;
fax: +34-958-243214.
E-mail address: alados@ugr.es (L. Alados-Arboledas)

eral radiation term that covers both photon terms
and energy terms. Photosynthetic photon flux density, Qp , is defined as the photon flux density
(1 mmol photons m−2 s−1 =6.022×1017 photons
m−2 s−1 =1 mE m−2 s−1 ). This radiometric flux is

strongly affected by the presence of clouds. Clouds
reflect, absorb and transmit the incoming solar radiation, modifying in this way the amount and spectral
quality of the solar radiation reaching the Earth’s
surface. The cloud particles are responsible for scat-

0168-1923/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 1 9 2 3 ( 0 0 ) 0 0 0 9 1 - 5

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I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

tering processes that affect more markedly the shorter
wavelengths in the solar spectrum, which include
the photosynthetically active radiation spectral range.
These phenomena produce an effective reduction of
the global and direct components reaching the Earth’s
surface through the enhancement of the diffuse radiation scattered both to the surface and to space. Clouds
are also responsible for the absorption of solar radiation. This absorption affects the infrared wavelength
of the solar spectrum, leaving unaltered the photosynthetically active spectrum. The cloud transmittance

function analysed in this study tries to parameterise
this effect in a simple and operational way.
In a previous paper (Alados et al., 1999) we analysed parametric models that provide an estimate of
the different components of solar photosynthetically
active radiation under cloudless conditions. We have
considered broadband models, which are physically
based. These models use broadband transmittances of
the extinction process that takes place in the atmosphere, obtained by means of a parametric approach.
In order to improve the original models some modifications have been introduced concerning the diffuse component parameterisation. It appears that for
cloudless conditions the aerosol effect is the largest
one, being characterised by a higher variability in both
space and time. The findings in the parameterisation
of the cloudless sky photosynthetically active radiation (Alados et al., 1999) have been used in the present
work to derive a model that estimates this radiative
flux under all sky conditions. For this purpose, we
have defined a cloud transmittance function considering three different types of clouds: low, medium and
high level clouds. Although this classification is simplistic it provides a first distinction of cloud radiative
effects and allows the derivation of statistically significant cloud transmittance function using the available date base. Different authors (Kasten and Czeplak,
1980; Blumthaler et al., 1994; Davies, 1995) have followed similar procedures in their analysis of the cloud
radiative effect over the whole solar spectrum and over

the UV spectral range.
The cloud transmittance functions developed have
been tested in relation to their predictive capability of
global photosynthetically active radiation when they
are combined with a cloudless sky parametric model.
For this validation purpose, data recorded at two radiometric stations are used.

2. Data and measurements
The data set used in this study came from two
radiometric stations. The first one is located at the
University of Almer´ıa (36.83◦ N, 2.41◦ W, 20 m
a.m.s.l.). This radiometric station is located on the
Mediterranean coast in south-eastern Spain and is
characterised by a greater frequency of cloudless
days, and high humidity. Measurements include
5 min values of various parameters. Solar global irradiance, Rs , was measured using a Kipp & Zonen
model CM-11 solarimeter (Delft, Netherlands), while
another Kipp & Zonen model CM-11 with a polar
axis shadowband was used to measure solar diffuse
irradiance, Rd . Photosynthetic active photon flux density, Qp , has been measured by means of a LICOR

model 190 SA quantum sensor (Lincoln, NE, USA).
Another quantum sensor has been equipped with a
polar axis shadowband in order to measure the diffuse
photosynthetic active photon flux density incident
on a horizontal surface, Qpd . Finally, air temperature
and relative humidity at 1.5 m, were recorded. Diffuse irradiance measurements have been corrected for
the effect of the shadow-band following the method
proposed by Batlles et al. (1995). This method has
also been applied to correct the diffuse photon flux
density. From this database, hourly values have been
generated covering 1990–1994. The corrected diffuse
horizontal values of both radiometric fluxes and the
global horizontal fluxes are used to obtain the normal
incidence direct beam components, both for the solar
broadband irradiance, Rb , and the photosynthetically
active photon flux density, Qpb .
Rb =

(Rs − Rd )
cos θ

Qpb =

(Qp − Qpd )
cos θ

(1)
(2)

where θ is the solar zenith angle. The broadband solar
direct radiation has been used in combination with
meteorological information for estimating the aerosol
contribution through the computation of the Angstrom
coefficient for aerosol, β (Alados et al., 1999),
following the procedure proposed by Gueymard
(1998).
A second station is located in the outskirts of
Granada (37.18◦ N, 3.58◦ W, 660 m a.m.s.l), an inland

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

location. Data collected at 1 min intervals during
1994–1995 has been used in the present study. The
radiometric sensors are similar to those used at
Almer´ıa. The diffuse irradiance, measured by a radiometer equipped with a shadowband, has been
corrected using the previously cited model developed
by Batlles et al. (1995). Mean hourly values have
been obtained for the radiometric and meteorological
variables. The Angstrom coefficient for aerosol extinction, β, have been obtained following a procedure
similar to that used at Almer´ıa. Granada is located
in the south-eastern of the Iberian Peninsula. Cool
winters and hot summers characterise its inland location. Their diurnal temperature range is rather wide
with the possibility of freezing on winter nights. Most
rainfalls occur in spring and winter. The summer is
very dry, with scarce rainfalls in July and August.
Considering the period used, a complete range of
seasonal conditions and solar angles is included among
the samples. Analytical checks, for measurement consistency, were carried out to eliminate problems associated with shadowband misalignments, and other
questionable data. Due to cosine response problems,
we have limited our studies to cases with solar zenith

angle less than 85◦ . Calibration constants of the radiometric devices used at Almer´ıa and Granada have been
checked periodically by our research team. Degradation of less than a few tenths per cent per year has
been observed in the CM-11 pyranometers. The drift
of the calibration constants of the Quantum sensors
have been evaluated both by means of a calibrated
standard lamp and by field comparison with measurements performed by a well-calibrated field spectroradiometer (LI-1800). Measurements of solar global and
diffuse irradiance have an estimated experimental error of about 2–3%, while the quantum sensor has a
relative error of less than 5%.
The cloud information, cloud type and cloud
amount, have been obtained from the Spanish Meteorological Service. At Almer´ıa, the Meteorological
Office, where the cloud observations have been made,
is located 1 km away from the radiometric station,
both places are located close to the coast line. The
frequency of the cloud observations used in this
study is hourly from 1990 to 1993 and every 2 h
since 1993. The registered information includes the
cloud amount in octas for three different levels of
clouds: low, medium and high. There is also informa-

41

tion concerning the cloud type following the World
Meteorological Organisation scheme.
At Granada the cloud observations have been made
at the same location as the radiometric station. In this
case, the observation frequency is lower, four observations per day (at 9:00, 12:00, 15:00 and 18:00 hours
GMT), and the registered information is the total
amount of clouds and that of the lowest cloud layer.
Thus, if there are three layers of clouds present the
cloud amount for the lowest one is completely determined but this is not the case for the higher ones.
Thus, it is necessary to define a criterion to distribute
the difference between the total cloud amount and
that of the lowest cloud layer between the medium
and high level clouds. We divided this cloud amount
equally between the two types of higher clouds.

3. Cloud transmittance functions
In order to define the cloud transmittance function
we have considered the following expression that relates the solar global radiation under all sky conditions, Rs , to the cloudless sky radiation, Rso , being F a
function that depends on the cloud features (amount,

location, type):
Rs = F Rso

(3)

Different authors have proposed empirical expressions
for this function parameterised in terms of the type and
amount of clouds (Atwater and Ball, 1981; Davies and
Mckay, 1989; Davies, 1995; Degaetano et al., 1995).
Usually this function is considered as a cloud transmittance τc :
Rs = τc Rso

(4)

Considering the different radiative effects associated
with different cloud types it seems convenient to define
different transmittance for the different cloud types.
The overall transmittance will be the combination of
the transmittances due to the different clouds present.
Obviously, the contribution of a given cloud layer must

depend also on its extension through the cloud amount
term. In this work, we have considered the following
expression for the total transmittance due to clouds:
τ c = τl τm τh

(5)

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I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

where τ l , τ m and τ h represents the cloud transmittance
associated with low, medium and high level clouds,
respectively. This transmittance is a function of the
cloud amount, ci , associated with each cloud layer.
The different radiative effects associated with different
cloud types are captured through the differences in
these functions.
To define the cloud transmittance function associated with each one of the three different types of clouds
considered we have used the Eq. (4) adapted to photosynthetically active radiation:
Qp = τc Qpo = τl (cl )τm (cm )τh (ch )Qpo

(6)

where Qp represents the photosynthetic active photon
flux density under all sky conditions and Qpo represents the same flux under cloudless sky conditions.
As a first step, we have classified the experimental cases in the Almer´ıa database, considering three
categories. The first category includes those cases
characterised by the presence of low level clouds
with exclusion of cases with more than one type of
cloud, this is the low cloud category. In order to have
larger number of cases in the medium and high level
categories we have included in these categories cases
in which the nominal type of cloud is combined with
only one octa of the remaining types of clouds.
It is obvious that the derivation of the cloud transmittance associated with a given experimental case
needs knowledge of the cloudless sky estimate of the
photosynthetically active radiation. For this purpose,
we have used the models proposed in previous work
(Alados et al., 1999), in this work it has been shown
that the aerosols are responsible for the greatest effect on the radiative flux under cloudless conditions.
The cloudless sky model selected has been that called
PARM that is based on a model proposed by Gueymard (1989). This model includes some modifications
in order to improve the parameterisation of the diffuse
component of the photosynthetically active radiation
and through this, of the global photosynthetically active radiation. Thus, in a previous work (Alados et al.,
1999) some modifications suggested by Bird and Riordan (1986) for the spectral code SPCTRAL2 have
been included in the cloudless sky parametric model.
As an additional modification step, the parameterisation of the diffuse component by aerosol scattering has
been modified in order to include the single scattering
albedo as a multiplicative factor. For the selectable

parameters related to the aerosol optical properties we
have used the values described in Alados et al. (1999).
In our previous work (Alados et al., 1999), we have
shown that the aerosol contribution has a marked influence in the cloudless sky case. For this purpose the use
of the parametric model requires the available information on the aerosol load, through the coefficient β.
Since the broadband procedure followed (Gueymard,
1998) for the derivation of this coefficient requires that
broadband data were acquired under cloudless conditions, we have computed monthly average values of
this coefficient from the available cloudless sky cases.
Shorter period averages could be used, but monthly
values seem appropriate if the interest is in the global
flux.
Following the procedure described in the preceding
paragraph, we have obtained for each hour the cloudless sky estimation of global photosynthetically active
radiation. This value has been combined with the measured photosynthetically active radiation to define the
cloud transmittance for each hour. In a second step,
these cases have been classified according to the cloud
fraction amount. The cloud amount has been characterised in terms of the fraction of the sky covered by
the particular type of clouds. For each cloud type, eight
different levels of fractional cloud coverage have been
considered (since the recording of these observations
is in octas). For each cloud coverage level, we have
computed the mean cloud transmittance and the associated standard deviation.
Fig. 1 shows cloud transmittance for low level
clouds, as a function of the cloud fractional coverage. The size of the bar, representing the standard
deviation, indicates the great spread of this transmittance factor for a given fractional amount. In a next
section we will consider the contribution of the solar
zenith angle to this scatter. Part of this scatter is obviously associated to the variety of clouds included
under each one of the three categories considered
in our rather simple classification. It is obvious that
the mean value of the cloud transmittance reveals
the higher efficiency of overcast skies to reduce effectively the photosynthetically active radiation. The
dependence of the cloud transmittance deviates from
the simpler linear function and thus we have tried to
fit this dependency through a power function.
a

τl = 1 − bl cl l

(7)

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

43

Fig. 1. Cloud transmittance for low level clouds. The square symbols represent the average values while the bars denote the standard
deviation for each one of the cloud coverage categories considered.

Fig. 1 displays the fitting function obtained by a
weighted fit using the standard deviation as the
weighting factor, the coefficients that provides the
best result for the χ 2 statistic are:
b1 = 0.334 ± 0.016 a1 = 1.66 ± 0.13
The value obtained for the exponent al reveals that
the effect of partially covered skies to reduce the

photosynthetically active radiation does not increases
linearly with the cloud fractional amount. Similar
non-linear relations between the cloud radiative effect
and the cloud coverage have been shown by other
authors for the whole solar spectrum (Kasten and
Czeplak, 1980; Davies, 1995) and for the thermal infrared emission of the atmosphere (Alados-Arboledas
et al., 1995).

Fig. 2. Cloud transmittance for medium level clouds. The square symbols represent the average values while the bars denote the standard
deviation for each one of the cloud coverage categories considered.

44

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

Our analysis of medium level clouds is shown in
Fig. 2, with similar features to that of Fig. 1. After
fitting to a power function
a

τm = 1 − bm cmm

(8)

1.65
)(1 − 0.104ch )
τ = (1 − 0.334cl1.66 )(1 − 0.290cm

(10)

we obtain the following coefficients
bm = 0.290 ± 0.026 am = 1.65 ± 0.21
The corresponding function is included in Fig. 2. It is
interesting to note that the differences between bl and
bm reveal a slightly greater radiative influence of low
level clouds, shown by other authors for the whole
solar spectrum (Atwater and Ball, 1981). On the other
hand, the decrease of the cloud transmittance with the
cloud fractional coverage follows similar patterns for
low and medium level clouds as the coincidence of
the exponents al and am reveals.
Finally, the high level clouds category has been considered in Fig. 3. In this case, after a first trial we have
selected a linear fit
τh = 1 − bh ch

present in the sky. Thus the general expression for the
cloud transmittance is:

(9)

with a coefficient bh =0.104±0.012 that reveals a
lower radiative effect of this last cloud category.
According to the previous results, for situations that
are more complex we must consider a cloud transmittance that combines the effect of the different clouds

In a next section, we will check this cloud transmittance expression using data acquired at the same
location where the parameterisation has been developed. The validation data set includes the cases used
in the development of the cloud transmittance function and additionally the cloudless sky cases and those
cloudy conditions cases that include more than one
cloud type. Anyway, in order to have a test against
an independent data set we use the data registered at
Granada. However, before the validation of the proposed cloud transmittance function, we have analysed
the convenience of including additional parameters
in it. One parameter considered has been the sun
position, described by its solar zenith angle. Through
this we have computed the optical air mass, m (Kasten, 1966). After several trials and considering the
available database, this analysis has been restricted to
perform the computation of the cloud transmittance
function considering cases with optical air mass above
and below 1.56. This optical air mass value corresponds to a solar zenith angle of 50◦ and separates the

Fig. 3. Cloud transmittance for high level clouds. The square symbols represents the average values while the bars denote the standard
deviation for each one of the cloud coverage categories considered.

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

database into two categories with a similar number of
cases.
When the separation in optical air mass is applied
the transmittance functions obtained for the low level
clouds are:
τl = 1.0228cl1.27

for m < 1.56

τl = 1 − 0.376cl1.79

for m ≥ 1.56

(11a)
(11b)

The behaviour of the new cloud transmittance function
is presented in Fig. 4. It is evident that for higher
solar elevation, lower optical air mass, low level clouds
transmit the solar radiation in a more effective way for
fractional coverage higher than 40%. Below this limit,
there are no differences between both functions. This
result is associated with the increase of the effective
cross section of clouds for lower solar elevations.
For the medium level clouds we obtain the following
set of functions:
1.00
τm = 1 − 0.216 cm

for m < 1.56

(12a)

1.91
τm = 1 − 0.313 cm

for m ≥ 1.56

(12b)

Fig. 4 shows the behaviour of these transmittance functions. In this case, we find a slightly greater efficiency
of medium level clouds to provide radiative forcing
for higher solar elevation in cases with fractional coverage below 50%. For fractional coverage above this

45

value, there is a greater radiative effect of these clouds
for lower solar elevations. It is also interesting to note
that for the lower solar elevations the behaviour of the
low level and the medium level cloud transmittance
function shows a close agreement. It seems that the
geometric features of both cloud types provide similar
blocking effects of the solar radiative flux for lower
solar elevations.
Finally, for the high level clouds the differences in
the unique adjustable coefficient reveal a lower efficacy for radiative forcing for lower solar elevations:
τh = 1 − 0.143 ch

for m < 1.56

(13a)

τh = 1 − 0.063 ch

for m ≥ 1.56

(13b)

In this sense, Fig. 4 reveals a negligible contribution
of these clouds to the solar radiative forcing for lower
solar elevations. This behaviour is opposite to the one
encountered for the two other cloud types and shows
the higher transparency of these clouds to solar radiation and the less thick configuration of these cloud
layers.
The cloud transmittance functions obtained for the
photosynthetically active radiation can be compared
against similar functions developed for broadband solar radiation. Kasten and Czeplak (1980) and Davies
(1995) have developed similar functions using data
recorded at Hamburg (Germany) and Washington

Fig. 4. Comparison among the different cloud transmittance functions proposed in this study and the broadband cloud transmittance
functions proposed by Kasten and Czeplak (1980) and Davies (1995). The variable m represents the optical air mass.

46

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

(USA), respectively. In their studies, these authors
proposed a unique transmission function for all kind
of clouds following a functional dependence similar
to that used in our study. Kasten and Czeplak (1980)
obtained coefficients b=0.75 and a=3.4, while Davies
(1995) obtained coefficients b=0.674 and a=2.854.
Fig. 4 illustrates that these cloud transmittance functions despite the differences in the coefficients show
a similar behaviour. The comparison with our results
for the photosynthetically active radiation shows that
clouds transmit more effectively the shorter wavelengths of the solar spectrum. In a previous study
(Alados et al., 1996, 1999; Alados, 1997) we have
shown this fact by analysing the ratio of photosynthetically active radiation to broadband solar radiation. This ratio depends on the aerosol load, the solar
zenith angle and the cloud coverage. Its increase for
cloudy conditions is a result of the lower attenuation
suffered by the photosynthetically active radiation in
comparison with the whole solar spectrum.

The performance of the models was evaluated using different statistics suggested by Willmott et al.
(1985). These include the root mean square deviation
(RMSD ) and the mean bias deviation (MBD). We have
analysed also the linear regression between estimated
and measured values, providing information about
correlation coefficient, R, slope, a, and intercept, b.
The first one gives an evaluation of the experimental data variance explained by the model while the
last two provide information about over or underestimation tendency, in a particular range. The root
mean square deviation, RMSD , under certain assumptions can be separated into systematic (RMSDs ) and
unsystematic deviations (RMSDu ):
#1/2
"
n
1X
(14)
(Pi − Oi )2
RMSD =
N
i=1

RMSDs =

4. Performance of models
RMSDu =
As indicated the developed cloud transmittance
function has been checked against experimental data
in order to test their success in predicting the global
photosynthetically active radiation in combination
with a cloudless sky model. Two different data sets
have been considered for this task. The first one is that
recorded at Almer´ıa, that includes the data used to
develop the cloud transmittance functions. Obviously,
this data set also includes cases not used in the development of the cloud transmittance functions like the
cloudless sky cases and those cases including a mixture of different cloud types. The second data set has
been that recorded at Granada, where we have some
limitation concerning the available information. As
previously indicated the cloud information included
in this data set is limited to the total cloud amount
and that of the lower cloud type. Thus when there are
three different types of clouds present, according to
our classification, it is not possible to know exactly
the specific amount of high and medium level clouds.
We have solved this problem considering that in such
cases the difference between the total amount and
that of the lower level clouds is partitioned equally
between the high and medium level clouds.

n

#1/2

(15)

#1/2

(16)

"

1X
(Pave − Oi )2
N

"

1X
(Pi − Pave )2
N

i=1
n

i=1

where Pi and Oi refer respectively to the predicted and
observed values, N is the number of cases and Pave
represents the average of the predicted values.
The MBD and the different terms related to RMSD
have been presented as a percentage of the average
value of the measured variable in order to facilitate the
comparisons. In our analysis we have also included
the index of agreement, d, that is a dimensionless index bounded between 0 and 1. This index is a better
measure of the model performance than the correlation statistics such as R and is defined as:
#
"
PN
2
i=1 (Pi − Oi )
(17)
d = 1 − PN
2
i=1 (|Pi − Oave | + |Oi − Oave |)
where the term Oave represents the average of the observed values.
Separate results are presented for the whole set of
data and for those cases considered as cloudy skies.
Table 1 shows the results for Almer´ıa where we have
included also the results for cloudless conditions obtained by using the monthly average value for the
Angstrom coefficient for aerosol extinction, β. The

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

47

Table 1
Statistical results for the combination of the cloudless sky parametric model PARM (Alados et al., 1999) and the cloud transmittance
functions without explicit dependence on sun position — Almer´ıa data seta

Cloudless skies
N=2936
Cloudy skies
N=4396
All conditions
N=7334

Qpave (mE m−2 s−1 )

a (mE m−2 s−1 )

b

R

MBD (%)

1085

15

1.026

0.985

4.0

9.1

5.0

8.2

0.990

1057

74

0.946

0.958

1.5

13.4

3.0

13.1

0.978

1068

50

0.979

0.968

2.5

11.8

2.8

11.5

0.983

RMSD (%)

RMSDs (%)

RMSDu (%)

d

a

N represents the total number of cases in each analysed category. Qpave correspond to the average value of photosynthetically active
radiation for the indicated sky conditions. Other variables included in Table 1 are the linear regression statistics, intercept, a, slope, b, and
correlation coefficient, R. The Mean Bias Deviation (MBD) and the Root Mean Square Deviation (RMSD ) are expressed as percentages
of the average value Qpave . The systematic, RMSDs , and unsystematic, RMSDu , parts of RMSD defined by Willmott (1984) are expressed as
percentages. Finally, d means the index of agreement as defined by Willmott (1981).

use of an average value in spite of that estimated for
each one of the experimental cases is responsible of
a slightly degraded model behaviour. In any case the
MBD is lower than the experimental error and the
RMSD is lower than 10% with a lower contribution of
systematic deviation. The negligible intercept, a, and
the slope, b, close to unity reveal that the cloudless
sky model have a similar behaviour for the complete
range of the photosynthetically active radiation values.
The goodness of this parameterisation when monthly
average values of the Angstrom coefficient for aerosol
extinction are used is also shown by the value of the
index of agreement, d, that is only slightly lower than
unity.
The combination of cloudless sky model with the
developed cloud transmittance functions is its simpler
form, that is without consideration of solar position
dependence, has been tested. Considering the coincidence of the low and medium level cloud exponents,
al and am , and the non significant difference between
the coefficients bl and bm , due to the size of the associated errors, we have tested the behaviour of a simplified cloud transmittance function:
1.66
)×(1 − 0.104 Ch )
τ =(1 − 0.317 cl1.66 )(1−0.317 cm

(18)
The use of this simplified transmittance function
provides estimations under cloudy conditions with
negligible MBD although there is an increase in RMSD
with reference to the cloudless analysis. It is interesting to note that the results obtained with this equation
presents non significant differences with that obtained

using Eq. (8), that is using different coefficients for
the low and medium level cloud transmittances. The
relative contribution of RMSDs as quantified by the
ratio RMSDs 2 /RMSD 2 is close to 4%. The index of
agreement shows the good behaviour of the model.
It is interesting to note that under cloudy conditions
the cloud radiative forcing damps the relative importance of the aerosol radiative forcing. Thus, the good
results obtained for cloudy conditions can be considered as evidence of the goodness of the developed
cloud transmittance functions.
Fig. 5 shows the scatter plot of estimated versus
measured values including all kinds of conditions. The
symbols try to distinguish between cloudy and cloudless conditions. As a general comment, we obtain a
good agreement between estimated and experimental
values. As the statistics in Table 1 reveal the global behaviour is similar to that described both for the cloudless and cloudy conditions. That is both components
of the model perform adequately and thus the cloud
transmittance function can be applied in combination
with any other good parameterisation of the cloudless
sky conditions to provide good estimation under all
sky conditions.
As indicated in a previous section, the Sun’s position can be responsible for differences in the cloud
transmittance for a given type and amount of clouds.
Thus, the cloud transmittance functions that include
the sun position dependence have been tested in combination with the cloudless sky model. Table 2 presents
the results both for the cloudy conditions and for the
whole set of data, including all kinds of sky conditions. The comparison with Table 1 reveals that the

48

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

Fig. 5. Scatter plot of estimated, Qpe , vs measured values, Qpm , of global photosynthetically active photon density flux using the Almer´ıa
data set.

differences between the results obtained with the cloud
transmittance function including the sun position dependence and that excluding this dependence are negligible. There is a slight increase in MBD and RMSD ,
being shown by an increase in the systematic component, RMSDs . The index of agreement also shows this
situation showing a slight reduction. Thus, from an
operational point of view the simpler approach for the
cloud transmittance function parameterised in terms of
cloud amount and type seems good enough and convenient.
Table 3 shows the results obtained for the Granada
data set. This database has not been used in the development of the cloud transmittance function and thus
this test can provide a independent verification of the
model validity. In this case, we use the cloudless sky
model, with the aerosol model adjusted to Granada

conditions (Alados et al., 1999), in conjunction with
the cloud transmittance functions that do not consider
explicit dependence on sun position. As in Almer´ıa,
Table 3 includes the results for the cloudless sky model
when executed using monthly average values of the
Angstrom coefficient for aerosols, β. The consideration of cloudy conditions and all sky conditions reveals
a degradation of the predictive capability of the model,
see for example the index of agreement, d. Nevertheless, the MBD values are close to or less than the experimental error and the index of agreement is close
to unity, similar to that encountered at Almer´ıa. Under
cloudy conditions, the model provides a RMSD value
with a high contribution of the systematic component
that is consistent with the MBD values obtained. In
any case, the global behaviour shows a good agreement between the experimental and the predicted data

Table 2
Statistical results for the combination of the cloudless sky parametric model PARM (Alados et al., 1999) and the cloud transmittance
functions with explicit dependence on sun position — Almer´ıa data seta

Cloudy Skies
N=4396
All conditions
N=7334
a

Qpave (mE m−2 s−1 )

a (mE m−2 s−1 )

b

R

MBD (%)

RMSD (%)

RMSDs (%)

RMSDu (%)

d

1057

89

0.928

0.958

1.2

13.3

6.0

11.8

0.932

1068

59

0.968

0.968

2.3

11.8

5.2

10.9

0.946

The symbols used have been defined in Table 1.

I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

49

Table 3
Statistical results for the combination of the cloudless sky parametric model PARM (Alados et al., 1999) and the cloud transmittance
functions without explicit dependence on sun position — Granada data seta
Qpave (mEm−2 s−1 )
Cloudless skies
N=399
Cloudy Skies
N=1116
All conditions
N=1515
a

a (mE m−2 s−1 )

b

R

MBD (%)

RMSD (%)

RMSDs (%)

RMSDu (%)

d

1261

−60

1.061

0.995

1.2

5.3

2.9

4.5

0.996

988

112

0.945

0.936

5.8

19.5

6.5

18.3

0.964

1060

78

0.970

0.954

4.3

15.9

4.6

15.2

0.975

The symbols used have been defined in Table 1.

Table 4
Statistical results for the combination of the cloudless sky parametric model PARM (Alados et al., 1999) and the cloud transmittance
functions with explicit dependence on sun position — Ganada data seta
Qpave (mE m−2 s−1 )
Cloudy skies
N=1116
All conditions
N=1515
a

a (mE m−2 s−1 )

b

R

MBD (%)

RMSD (%)

RMSDs (%)

RMSDu (%)

d

988

130

0.916

0.936

4.9

18.9

12.2

14.6

0.889

1060

89

0.953

0.955

3.7

15.5

8.7

12.8

0.916

The symbols used have been defined in Table 1.

sets for the whole range of photosynthetically active
radiation values under consideration. Also the results
obtained at Granada are only slightly worse than those
obtained at Almer´ıa. In this sense a separate analysis
of the lower level clouds has shown that the criterion
adopted in the separation of the cloud amount for the
higher cloud layers can be improved. Nevertheless,
this task will be accomplished when additional data
are available.
Finally, as in Almer´ıa, the application of the cloud
transmittance function depending on Sun’s position
does not imply important changes as Table 4 shows.
At this point, it is interesting to note that on some
occasions simpler models are more appropriate than
the complex ones, due to possible amplification of
input parameter errors.

5. Concluding remarks
Our study reveals that for low and medium level
clouds the dependence of the cloud transmittance on
the cloud amount is far from linear. These results are
similar to those obtained in previous studies for the
whole solar spectrum and the thermal infrared spectrum. For the high level clouds the relationship can

be well represented by a simple linear equation. Fittings of the cloud transmittance functions indicate that
low and medium level clouds affect more markedly
the photosynthetically active radiation than high level
clouds. High level clouds provides the minor radiative effect. Comparisons with the results obtained by
other authors for the whole solar spectrum suggest that
clouds transmit more effectively the short-wave part
of the spectrum.
The sun position influences the radiative effect of
all types of clouds but this effect is different for high
level clouds when compared with that associated
with medium and low level clouds. Both altitude and
thickness of the different cloud layer controls this behaviour. For the lower solar elevation range we found
that the cloud transmittance associated with low and
medium level clouds are similar. This indicates the
similar influence of the geometric features of these
types of clouds when the sun is closer to the horizon.
The parameterisations of the cloud transmittance
have been tested in combination with a cloudless sky
parametric model about their predictive capability of
the global photosynthetically active photon density
flux. Separate analysis have been done for both sets
of cloud transmittance functions, the one depending
on cloud amount and type and the one that includes

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I. Alados et al. / Agricultural and Forest Meteorology 102 (2000) 39–50

additional dependence on sun position. For this purpose, we have used the complete database of Almer´ıa
that partially have been used in the development of
the cloud transmittance functions and an independent
database collected at Granada.
The results of these tests indicate that the cloudless sky parametric model combined with the cloud
transmittance scheme can estimate the global photosynthetically active radiation with a high confidence
level. Thus, the MBD shows values about 4% and
the index of agreement reveals the goodness of the
modelling, showing values higher than 0.90 in all
cases. The use of a simplified transmittance function with same coefficients for the low and medium
level cloud transmittances provides estimations close
to that obtained using different formulations for this
two cloud types. This good performance is obtained
with both sets of data, indicating the applicability of
the developed cloud transmittance scheme to different
locations.

Acknowledgements
This work was supported by La Dirección General de Ciencia y Tecnolog´ıa from the Education
and Research Spanish Ministry through the project
N◦ CLI-99-0835-C02-01. We are very grateful to
the Armilla Air Base Meteorological Office Staff
and specially to Guillermo Ballester Valor, Meteorologist Chief of the Meteorological Office for the
maintenance of the radiometric devices. The Instituto
Nacional de Meteorolog´ıa kindly provided the cloud
observation information for the two-radiometric stations. The authors are indebted to the Regional editor
Dr. J.B. Stewart, to Dr. Juhan Ross and the anonymous referee who read the manuscript and made
valuable suggestions.
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