2001 Lopez (Agri for Meteorol_107)

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Agricultural and Forest Meteorology 107 (2001) 279291

Estimation of hourly global photosynthetically active radiationusing artificial neural network models

G. Lpez a, M.A. Rubio a, M. Martnez b, F.J. Batlles a,

a Facultad de Ciencias, Dpto. de F sica Aplicada, Universidad de Almer a, La Canada de San Urbano s/n, 04120 Almer a, Spainb Dpto. de Lenguajes y Computacin, Universidad de Almera, 04120 Almera, Spain

Received 5 July 2000; received in revised form 21 December 2000; accepted 3 January 2001

Abstract

Photosynthetically active radiation (PAR) reaching the earths surface is a major parameter controlling many biologicaland physical processes related with the evolution of plant canopies, agricultural and environmental fields. Unfortunately, PAR

is not often measured and therefore it must be estimated. The unavailability of measurements of global solar radiation at the

place of interest and different factors affecting the linear relation between PAR and global solar radiation can preclude the

estimation of PAR from global solar radiation. In this paper, a novel approach based on a simple multilayered feedforward

perceptron has been used to analyse the non-linear relationships between PAR and different meteorological and radiometric

variables in order to determinetheirrelative relevance. An artificialneural network based model forthe estimation of thehourly

PAR involving hourly global irradiance as only measured variable has been successfully developed. The model was tested

using data recorded at six radiometric stations covering a wide range of climates. The model performance has been compared

with other existing empirical complex models showing important improvements. Next, a second artificial neural network

based model involving only sunshine duration measurements has been developed and proved to be an acceptable alternative

to calculate hourly PAR when radiometric information is not available. 2001 Elsevier Science B.V. All rights reserved.

Keywords:Photosynthetically active radiation; Artificial neural network; Global solar radiation; Sunshine duration; Estimation

1. Introduction

One critical factor for crop energy conversion for

plant growth is the solar energy in the wavelength

region 400700 nm, referred to as photosynthetically

active radiation (PAR), received by the plant. Fur-

thermore, many of the exchange processes between

vegetation canopies and the atmosphere, as well

as dry matter yield, are regulated by photosynthe-

sis, which has often been related to the amount of

Corresponding author. Tel.: +34-950-015414;

fax: +34-950-215477.

E-mail address: [email protected] (F.J. Batlles).

absorbed PAR (Hanan and Bgu, 1995; Li et al.,

1997). Therefore, the amount of PAR absorbed by

green vegetation not only governs the processes of

the plant evolution but also influences the exchange

of energy and water between the land surface and the

atmosphere. Unfortunately, measurements of PAR are

not routinely carried out at radiometric stations. This

problem necessitates estimation from other commonly

measurable meteorological and radiometric variables

available at the location of interest. In the literature

there are two kinds of models for estimating the PAR:

(i) radiative transfer models (Gueymard, 1989a,b;Olseth and Skartveit, 1993) and (ii) empirical models

(Moon, 1940; Britton and Dodd, 1976; Howell et al.,

0168-1923/01/$ see front matter 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 1 9 2 3 ( 0 1 ) 0 0 2 1 7 - 9


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280 G. Lopez et al. / Agricultural and Forest Meteorology 107 (2001) 279291

Nomenclature

a zero intercept of the linear regression

fit between estimated and measured

values

b slope of the linear regression fit between

estimated and measured valuesD diffuse solar irradiance on horizontal

surface (W m2)

E sum of squares of differences between

network outputs and target outputs

G global solar irradiance on horizontal

surface (W m2)

h air relative humidity

I direct beam solar irradiance (W m2)

IIIid identity matrix

I0 extraterrestrial solar irradiance

(W m2)

I0h extraterrestrial solar irradiance

on horizontal surface (W m2)

JJJ Jacobian matrix

JJJT transpose of Jacobian matrix

k diffuse fraction

kt clearness index

mr relative optical air mass

MBE mean bias error expressed as a

percentage of the mean measured value

of the corresponding photosynthetically

active radiation

N number of nodes used by an artificial

neural network layer

Nh number of hidden nodes used by anartificial neural network

Ni number of input nodes used by an

artificial neural network

Np number of training patterns

oi artificial neural network output

forith pattern

Qp photosynthetically active radiation

on a horizontal surface (E m2 s1)

r residual vector

r T transpose vector ofr

R2 coefficient of determination of the

linear regression fit between estimated

and measured valuesS relative sunshine hours

ti target output forith pattern

T air temperature (C)

Td air dew point temperature (C)

w precipitable water thickness (cm)

w vector of weights used by anartificial neural network

wij weight for the connection betweennodei and node j in an artificial

neural network

x real-valued variable

xmax maximum value ofx

xmin minimum value ofx

xscaled scaled value ofx

y activation function of the nodes

used by an artificial neural network

Greek symbols

weight correction parameter used in the

training process of an artificial

neural network brightness of the sky

clearness of the sky

z solar zenith angle

Marquardt parameter used in the

training process of an artificial

neural network

1983; Papaioannou et al., 1993; Alados et al., 1996;

Al-Shooshan, 1997). The first ones take into account

interactions on 400700 nm waveband radiation with

terrestrial atmosphere, such as Rayleigh scattering,

ozone absorption and aerosol extinction. The currentproblem in the use of these models is the large amount

of meteorological information required, which is not

often measured at meteorological stations. On the

other hand, the empirical models relate the PAR to

other solar radiation measurements, especially global

solar irradiance on horizontal surfaces, thus requiring

simpler input data than the parametric models and

providing an estimate of the PAR both for cloudless

and cloudy conditions. In this way, these models avoid

the difficulties associated with the parameterisation

of clouds.

The linear relationship between PAR and global

radiation (2902700 nm) is usually the basis ofthe empirical models. Some studies (Szeicz, 1974;

Papaioannou et al., 1993; Alados et al., 1996) tend


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G. Lopez et al. / Agricultural and Forest Meteorology 107 (2001) 279291 281

to show that the ratio between the PAR and global

radiation depends on solar elevation, as well as dew-

point temperature and sky conditions. These relations

appear to be non-linear and difficult to handle with

standard statistical techniques. Furthermore, sky con-

ditions are characterised by means of clearness of sky

and the brightness of skylight, which are functions

of direct beam and diffuse irradiance (Perez et al.,

1990). This dependence presents an additional prob-

lem since normal beam or diffuse components are not

measured as frequently as global irradiance.

This paper presents a novel technique for mod-

elling the hourly global PAR based on artificial neural

networks (ANNs). Neural networks are widely ac-

cepted as a technology offering an alternative way to

tackle complex and ill-defined problems. The power

of neural networks in modelling complex mappings

has been demonstrated (Kohonen, 1984; Lupo, 1989;

Hammerstrom, 1993) and successfully applied to a

wide range of agricultural and engineering applica-tions reporting higher accuracy compared to classical

methods (Bolte, 1989; Zhuang and Engel, 1990;

Muttiah and Engel, 1991; Thai and Shewfelt, 1991).

Furthermore, in the meteorological field, neural net-

work models have been developed to model radiation

variables, as global or diffuse solar irradiance (Lpez

et al., 2000a,b), showing important improvements

against traditional statistical models.

In this work, some radiometric and meteorolog-

ical variables derived from physical considerations

for estimating hourly PAR have been studied. An

analysis of the relevance of these inputs based on a

Bayesian method involving ANNs (MacKay, 1994;Neal, 1996) has been carried out. This first stage se-

lected the relevant inputs to develop the simplest and

optimal neural network model including radiometric

information. In addition to this, since throughout the

Table 1

Geographical locations of the stations, period of measurement and number of hours

Country Latitude (N) Longitude (W) Altitude (a.m.s.l.) Years N

Abisko Sweden 68.35 18.82 385 19951997 5573

Almera Spain 36.83 2.41 6 19901995 15200

Granada Spain 37.18 3.58 660 19941995 7442

Bondville USA 40.05 88.37 213 19971998 3123Desert Rock USA 36.63 116.02 1007 19981999 3274

Table Mountain USA 40.13 105.3 1689 19951997 7052

world more solar data are in the form of sunshine

measurements than in the form of radiation measure-

ments (Michalsky, 1992), a similar analysis has been

performed using sunshine duration rather than radia-

tion data inputs, so that a new and alternative model

may be applied in calculating PAR values without

data on global irradiance. Once the input variables

have been determined for both input sets, we have

developed the two respective ANN models. For that,

the ANNs have been trained using meteorological and

radiometric measurements obtained from only one

radiometric station, i.e. Almeras radiometric station.

Data collected at other radiometric stations operating

in climatologically diverse regions, have been used

to test the proposed models. In addition, for a better

analysis of the statistical results, the performance of

the proposed models has been compared with the

original versions of two empirical models by Alados

et al. (1996).

2. Data and measurements

The present work is based on data sets collected

at six radiometric stations. Table 1 summarises their

geographical locations, the number of observations

and the recording period. Almeras station is lo-

cated on a seashore site on the Mediterranean coast

of south-eastern Spain. High frequency of cloudless

days, an average annual temperature of 17C, and

the persistence of high humidity regime characterise

the local climate. The remaining stations have been

chosen with the intention of best representing a widediversity of climates. Granadas station is located on

the outskirts of Granada (south-eastern Spain). Cool

winter and hot summers characterise its inland loca-

tion. Abiskos station belongs to and is administered


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by The Royal Swedish Academy of Sciences. It is sit-

uated about 200 km north of the arctic circle, on the

south shore of Lake Tornetrsk (Sweden). The aver-

age annual temperature is approximately 1.0C andthe annual precipitation varies from about 1000 to

400 mm. The stations at Bondville, Desert Rock and

Table Mountain are part of Surface Radiation Research

Branch (SRRB) of NOAAs Air Resources Labora-

tory, and they represent inland locations with different

degrees of continental climate.

The data sets contain measurements of global hori-

zontal photosynthetic active photon flux density,

which we refer to as PAR, global and diffuse solar

irradiances on a horizontal surface, temperature, and

relative humidity. In Abiskos database, PAR and

global irradiance are only recorded. PAR was mea-

sured by LI-COR silicon quantum sensors at all sta-

tions and expressed as E m2 s1. Kipp and Zonen

pyranometers were employed to measure the global

solar irradiance in Almeras, Granadas and Abiskosstations, whereas Eppley ventilated pyranometers

model PSP were employed at the stations in the USA.

At Almeras and Granadas station, diffuse irradi-

ance was measured by a Kipp and Zonen pyranometer

equipped with an Eppley shadow band model SBS.

Because of the shadow band screens the sensor from a

portion of the diffuse radiation coming in from the sky,

a correction has been made to the measurements fol-

lowing Batlles et al. (1995). The SRRB stations used

Eppley pyranometers mounted on Eppley automatic

solar trackers model SMT-3 equipped with shade

disks model SDK. The measurements of temperature

and relative humidity were registered by means ofstandard sensors exposed in a meteorological screen.

Data were recorded and averaged on different sam-

pling basis times (1, 3, 5 and 10 min). Next, hourly

averaged values were obtained for all variables. Due

to cosine response problems of radiometric sensors,

we have only used cases corresponding to solar zenith

angles less than 85.

3. Artificial neural network

3.1. Basic architecture

ANNs learn the relationship between the input and

output variables by studying previously recorded data.

Fig. 1. A simple three-layer ANN.

Our neural network correspond to a feedforward mul-

tilayered perceptron (MLP) (Rumelhart et al., 1986),

which is among the most widely used neural network

models that learn from examples. These neural net-

works perform a non-polynomial regression and were

found to be suitable for our task. In the model for the

MLP, there is an input layer, a hidden layer, and anoutput layer (Fig. 1). Every layer is formed from el-

emental units named neurons or nodes. The neurons

in the input layer only receive the input signals and

distribute them forward to the network. In the follow-

ing layers, each neuron receives a signal, which is a

weighted sum of the outputs of the nodes in the layer

below. To node i in a posterior layer, l, the total input

xis given by

x(l)i =

N(l1)

j=0

w(l)ij y

(l1)j (1)

whereNis the number of nodes, wijthe weight for theconnection between nodej and node i andyj the output

of node j. Each node i has a bias, represented by the

weightwi0from a node with a constant unit activation

(y0 1). The output of a node in a hidden or outputlayer is obtained through a nodal activation function,

which uses the node input as argument. The activation

function used for the hidden neurons is sigmoid from

0 to 1

y = 1

1 + ex (2)

whereas the linear function (y = x) has been used for

the output nodes.Whereas the number of input and output neurons is

determined by the respective number of independent


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and dependent variables involved in the current prob-

lem, the choice of the number of hidden neurons

is usually not as easy. In current applications, the

number of intermediate neurons is often decided in

a heuristic way. In this work, the optimal number of

intermediate neurons has been determined empiri-

cally as the minimum number of neurons for which

estimation performance on a test set is satisfying.

3.2. Network training

The network training is the procedure where the

unknowns, i.e. weights and bias terms, are adjusted

based on numerical training data. The training data

is a set of patterns consisting of input and corre-

sponding output values, so-called target values. The

training method used in this work is named as su-

pervised training. The patterns in the training set

are presented to the network one at a time, and fol-

lowing a random sequence for optimal learning. Foreach sample, we compare the outputs obtained by

the network with the desired outputs. After the entire

training pattern has been processed, the weights and

bias are updated. This updating is done in such a way

that a measure of the error in the networks results is

reduced.

The goal is to find a network that describes the

inputoutput relation represented by the training

patterns. The LevembergMarquardt method (Mar-

quardt, 1963; Finschi, 1996) has been used to adjust

the unknown network parameters in order to minimise

the sum of squares of residuals, E, calculated as the

differences between network outputs, oi , and targetoutputs, ti , where i = 1Np with Np the number ofpatterns . The problem can be formulated as

minw

{E= r Tr} (3)

where r = (ti oi ; i = 1, . . . , N p) is the residualvector, rT the transpose vector ofr , and the networkunknowns are collected into a vector, w, according to

w = (w0i=1,Ni;j=1,Nh , wk=1,Nh , wbiasl=0,Nh

) (4)

wherew 0ij are the weights from input to hidden layer,

wk the weights from hidden to output layer, wbiasl theoutput and hidden bias, Ni the number of the input

nodes, and Nh the number of the hidden nodes. At

each iteration, m, the optimisation method adjusts the

unknowns according to

w(m+1) = w(m) + (m) (5)

where the correction, , is obtained from

(JJJ(m)

JJJ(m)T

+ (m)

IIIid)(m)

= JJJ(m)

rrr(m)

(6)

In the above equation,JJJis the Jacobian matrix with

first derivates of the residuals with respect to the

unknowns,JJJT the transpose matrix ofJJJid and III the

identity matrix. The Marquardt parameter is au-

tomatically adjusted during the training (Marquardt,

1963). The method approaches GaussNewton if

0 and steepest descent with a small step lengthif . Analytical expressions are derived for cal-culation of the Jacobian (Williams and Zipser, 1989).

3.3. Training and testing data

The performance of a trained network must be

evaluated by testing it on a different data set than

the one on which it was trained. This is an important

task, since a multilayer net can approximate any con-

tinuous multivariate function to any desired degree of

accuracy, provided that sufficiently many hidden neu-

rons are available. Thus, rather than learning the basic

structure of the data, enabling it to generalise well, it

learns irrelevant details of the individual cases.

The training file was created by randomly selecting

10% of the whole data set of Almeras station. The

ANN model is tested at each location. The tested file

used at Almera corresponds to the remaining 90%

of the whole data set whereas at the other sites the

corresponding whole data set is used. It should be

noted how few training patterns are required to develop

the ANN model against traditional statistical methods.

The input and output values were linearly scaled to lie

in the range 01 using

xscaled= x xmin

xmax xmin(7)

The constantsxmaxandxminare equal to the maximum

and minimum recorded value for each variable x. This

approach reduces the training time by eliminating thepossibility of reaching the saturation regions of the

sigmoid transfer function during training.


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4. Description of the input variables used

In the literature, there are models that use different

input sets for estimating PAR (Gueymard, 1989a,b;

Alados et al., 1996). We have tried to collect the most

common parameters used by empirical models and

analyse their input relevance as they are included in

the ANN.

Temperature, T, relative humidity, h, dew point

temperature,Td, and precipitable water thickness, w ,

influence the water vapour absorption in 400700 nm

band and solar broadband radiation on hourly PAR

estimation. Hourly precipitable water thickness has

been estimated from dew point temperature using

Reitans equation, which has been proved to be valid

for instantaneous values (Wright et al., 1989):

w = exp(0.0756 + 0.0693 Td) (8)

Concerning radiometric parameters, global and dif-

fuse solar irradiances have been considered. Thedimensionless indices kt, k, and have also been

introduced to account for the different sky conditions.

The hourly clearness index is defined as the ratio be-

tween the hourly horizontal global irradiance, G, to

the hourly horizontal extraterrestrial irradiance, I0h,

(kt = G/I0h). Hourly diffuse fraction is defined asthe ratio of hourly diffuse irradiance, D, to the hourly

horizontal global irradiance, (k = D/G). The clear-ness of the sky and the brightness of the sky are given

by Perez et al. (1990)

={(I+ D)/D} + 1.0413z

1 + 1.0413z(9)

= Dmr

I0(10)

where z is the solar zenith angle, mr the relative

optical airmass given by Kasten (1965), Ithe direct

beam solar irradiance and I0 the hourly extraterres-

trial irradiance. Values of direct solar irradiance were

obtained from measurements of global irradiance,

diffuse irradiance and solar zenith angle by means of

their relationship:

I = G D

cos z

(11)

It is clear that the kt k and representationsexpress equivalent information. The advantage of the

first representation is simpler computation whereas the

second provides a better characterisation of the clouds

and atmospheric aerosol amount. Finally, hourly sun-

shine duration represents the sum of minutes each

hour that the direct radiation,I, is above a threshold of

120Wm2 according to the WMO (Michalsky, 1992).

Hourly relative sunshine values are defined as the ratio

of actual minutes of sunshine to minutes of possible

sunshine for the hour.

5. Development of the ANN models

As we noted in Section 3, the number of input

and output units depends on the independent and

dependent variables involved in the current problem.

In our case, the output layer has had a single unit

representing the estimated hourly PAR. For the input

layer, we have considered two cases: the first one

takes into account all meteorological and radiometricinformation, whereas in the second one, the sunshine

duration and meteorological information are used in-

stead. For both cases, some input variables are not

independent of each other, and they could thus be

excluded from the ANN input layer. However, no

information about which input combination is most

relevant to the estimation of PAR, and thus, an analy-

sis of the input relevance for determining the optimal

input configuration is necessary.

We have used the Bayesian method of automatic

relevance determination (ARD) (MacKay, 1994;

Neal, 1996) for multilayer perceptron networks to ob-

tain the relevant inputs for each case. In this method,a hyperparameter is associated with each input. The

hyperparameters for an input corresponds to an in-

verse variance of the weights on connections from

that input, and represents the relevance of that input to

the task of predicting the measured output. Thus, the

smallest hyperparameter will indicate the input which

accounts for the largest part of the variability and

hence is the most relevant input, and the hyperparam-

eter with the next increase in value indicates the next

most relevant input and so on. Hence, 12 and 7 input

units form the input layer, respectively. With two neu-

rons in the hidden layer for both cases, the ANN archi-

tectures developed for the ARD are fully determined.The ARD was performed using all of Almeras

database.


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Table 2

Results of the automatic relevance determination (ARD) method

(for definition of the symbols see Nomenclature)

Input Hyperparameter

Using radiometric

information

Using sunshine

information

Random 4639980 155907T 457 6

h 1500 8

Td 111 16

w 58 24

G 3

D 29

kt 18

k 31

19

33

cos z 19 2

S 1

Table 2 summarises the results of the ARD method.In the left-hand column are the input parameters. Two

sets of results, one for radiometric and meteorological

inputs and one for sunshine duration and meteoro-

logical inputs, are given. In both cases, the artificial

random signal presents the largest hyperparameter as

it is independent of the PAR. On the other hand, the

smallest hyperparameter obtained in the first case is

associated to the global irradiance. The inputs with

relevance similar to each other succeeding to global

irradiance are cos z,kt, and. Next,D,kand have

shown to be the following set of relevant inputs, and

lastly, the meteorological variables involving precip-

itable water and dew point temperature. It is also noted

Table 3

Statistical results obtained from various input configurations used in developing the ANN model for estimating hourly PAR from radiometric

and meteorological information (for definition of the symbols see Nomenclature)a

Inputs a (E m2 s1) b R2 RMSE (%) MBE (%)

G, cos z, , , Td , w 2.8 1.00 0.999 2.1 0.0

G, cos z, kt , k, Td, w 2.3 1.00 0.999 2.0 0.0

G, cos z, kt , , h , Td 3.7 1.00 0.998 2.3 0.0

G, cos z, , Td 2.5 1.00 0.999 2.0 0.0

G, cos z, Td 2.2 1.00 0.999 2.0 0.0

G, cos z, 2.8 1.00 0.999 2.1 0.0

G, cos z 2.6 1.00 0.999 2.1 0.0

G, kt 2.8 1.00 0.999 2.3 0.1G, Td 2.0 1.00 0.998 2.6 0.0

a Mean value of measured hourly PAR: 938 E m2 s1.

the higher relevance of the dew point temperature and

precipitable water against temperature and relative

humidity when radiometric measurements are consid-

ered. Anyway, these results show that meteorological

information accounting for water vapour absorption

can be excluded from the ANN, and the variability

of the PAR values could be explained using only ra-

diometric information. These results agree with those

obtained by Gueymard (1989a,b) which shows that

global PAR in cloudless skies is essentially dependent

on the solar zenith in most conditions. In the second

case, relative sunshine duration appears to be the most

relevant input in conjunction with cos z. Tempera-

ture and relative humidity become more relevant than

dew point temperature and precipitable water with

hyperparameter values slightly higher than the lowest

ones. Thus, the inclusion of these parameters in the

ANN cannot be deleted and must be considered. Once

the relevance of the inputs has been analysed for the

two cases following the ARD method, the optimumANN configurations are to be determined. In the first

case, when radiometric measurements are available,

various ANNs with different combinations of input

parameters different to each other and 15 hidden units

were trained and tested according to Section 3.

Table 3 shows the statistical results obtained with

some of these ANNs for estimating hourly PAR. The

accuracy of the ANN models were evaluated based on

the regression analysis of estimated versus measured

values, in terms of the intercept, a, and slope, b, of

the linear fit and the determination coefficient,R2. We

have also included the root mean square error (RMSE)

and the mean bias error (MBE). The RMSE is a


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measure of the variation of predicted values around the

measured values, while the MBE is an indication of

the average deviation of the predicted values from the

measured values. Both indicators have been expressed

as a percentage of the mean value of the measured

hourly PAR.

It is noted that all ANN model performances are

similar to each other with RMSE values that range

from 2.0 to 2.6% and R2 values around 0.999. None

of the cases exhibit marked deviations, all the slope

values are equal to 1.00 and the intercept values

are very similar to each other with values around

2.5 E m2 s1. These results show that a ANN based

model using only global irradiance and solar zenith

angle is suitable to estimate PAR and additional input

variables as those given in Table 3 do not provide an

improvement in the ANN model performance. Thus,

we have chosen the simplest ANN model, which in-

volves only global irradiance and cos z, as MODEL

1. Once the input layer is determined, several ex-periments were conducted to find the combination

of number of hidden nodes which gave the greatest

accuracy in predicting the validation data set. As the

number of hidden nodes increased, the R2 value of

the estimated versus measured hourly PAR increased

until the number of hidden units reached the value of

10. For values higher than 10, the number of hidden

units did not seem to improve the estimates.

In a similar way, the above procedure was applied

to develop the ANN model when only relative sun-

shine duration and meteorological measurements are

available. Table 4 displays the statistical results for

various models that use different ANN input config-urations. The best model performance corresponds to

Table 4

Statistical results obtained from various input configurations used in developing the ANN model for estimating hourly PAR from sunshine

duration and meteorological information (for definition of the symbols see Nomenclature)a

Inputs a (E m2 s1) b R2 RMSE (%) MBE (%)

S, cos z, T, h , Td , w 24.5 0.98 0.981 8.0 0.2

S, cos z, T, h 25.1 0.98 0.980 8.2 0.2

S, cos z, T 27.2 0.97 0.978 8.3 0.2

S, cos z, Td 32.6 0.97 0.979 8.3 0.5

S, cos z, w 30.4 0.97 0.973 8.8 0.0

S, Td 556.8 0.40 0.397 44.3 0.8

cos z, Td 138.3 0.86 0.855 21.7 0.8S, cos z 25.4 0.97 0.977 8.6 0.1

a Mean value of measured hourly PAR: 938 E m2 s1.

that input configuration that uses all variables and the

worse ones correspond to those models that exclude

either sunshine duration or cos z. These results agree

with those obtained by the ARD method. In those

cases involving sunshine duration and cos z, all sta-

tistical tests were only slightly different to each other.

These results encourage us to use the relative sun-

shine duration and cos zas the only input parameters

in the ANN model, since the inclusion of additional

meteorological information leads to a reduction in

RMSE of only around 0.6%, and the remaining statis-

tical tests present values which are very close to each

other. Lastly, 10 hidden units have also been found to

be the optimal number of hidden units. We will refer

to this second ANN model as MODEL 2.

6. Performance of the models and discussion

To further test the performances of the proposed

models, they are compared with the original versionsof two empirical models by Alados et al. (1996). These

two models have been chosen because they were also

developed using Almeras database and use various

combinations of input parameters. The first Alados

model, which we will refer to as Alados1, calculates

the hourly PAR, Qp, from global irradiance, , , Tdand z using the following equation:

Qp

G= 1.786 0.192 ln 0.202 ln + 0.005Td

+0.032 cos2z (12)

This parameterisation tries to explain the observeddependencies of the ratio between PAR to global


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Table 5

Statistical comparison between observed and estimated PAR by MODEL 1, MODEL 2, Alados1 and Alados2, for each individual data set

(for definition of the symbols see Nomenclature)

a (E m2 s1) b R2 RMSE (%) MBE (%)

Almera (938 E m2 s1)a

Alados1 1.9 1.01 0.998 2.7 0.9

Alados2 3.6 1.02 0.998 3.0 1.4

MODEL 1 2.6 1.00 0.999 2.0 0.0MODEL 2 25.6 0.97 0.977 8.5 0.1

Granadab (936 E m2 s1)a

Alados1 9.5 0.96 0.998 4.8 3.4

Alados2 7.0 0.99 0.998 2.7 0.7

MODEL 1 11.7 0.97 0.998 3.7 2.2

Desert Rock (970 E m2 s1)a

Alados1 33.8 0.99 0.998 5.1 4.4

Alados2 31.5 0.98 0.997 6.0 5.3

MODEL 1 19.9 0.98 0.998 4.5 3.7

MODEL 2 3.4 0.89 0.976 14.3 10.4

Bondville (912 E m2 s1)a

Alados1 13.6 0.99 0.997 5.5 3.1

Alados2 13.0 0.99 0.996 5.9 3.5

MODEL 1 7.3 0.98 0.997 5.3 3.2

MODEL 2 83.7 0.88 0.945 18.8 1.3

Table Mountain (806 E m2 s1)a

Alados1 15.2 1.03 0.995 4.8 0.6

Alados2 13.7 1.02 0.995 4.8 0.1

MODEL 1 2.6 1.01 0.996 4.5 1.0

MODEL 2 66.3 0.90 0.948 15.3 2.1

Abiskob (414 E m2 s1)a

Alados2 14.1 1.03 0.993 9.5 6.0

MODEL 1 22.1 0.99 0.991 9.2 4.0

a Mean value of the measured hourly PAR.b Sunshine duration is not available.

irradiance with these variables and to exclude the ne-

cessity of local and seasonal calibration of this ratio.

The second Alados model, which we will refer to

as Alados2, estimates the hourly PAR from global irra-

diance,G, the clearness index, kt, and the solar zenith

angle,z. This model reads as follows:

Qp

G= 1.832 0.191 ln kt + 0.099cos z (13)

A statistical summary of the overall performance of

the four models is indicated in Table 5. The statistical

tests have been the same ones that we used in the

previous section. MODEL 1 appears to present thebest global results. In fact, MODEL 1 shows the low-

est RMSE percentages for all stations, excepting for

Granadas database. Moreover, MODEL 1 provides

estimates with errors below the experimental ones, for

Almeras station. Nevertheless, the statistical results

for the models that include radiometric information

present no significant differences to each other. This

outcome shows that measurements of global solar ir-

radiance are sufficient to obtain accurate estimates of

the hourly PAR in all sky conditions. Moreover, the

ANN model, MODEL 1, capture in a more real and

robust way the spread of PAR versus global irradiance

values. In this sense, Fig. 2 displays the scatter plot of

measured PAR values versus global irradiance values.

Additionally, we have plotted MODEL 1 evaluated forfour different fixed values of cos z. It is noted how

the spread of the PAR versus global irradiance values


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Fig. 2. Measured hourly PAR versus hourly global irradiance

values. Solid lines correspond to MODEL 1 evaluated for four

different fixed values of cos z (for a better legibility of the figure,

MODEL 1 has been evaluated from initial values of measured

global irradiance equal to 470, 450 and 310W m2, for the values

of cos z equal to 0.950, 0.650 and 0.500, respectively).

is explained by MODEL 1. Furthermore, MODEL 1

describes the slight increase of the PAR proportion of

broadband radiation as sky conditions change from

clear to cloudy, as Papaioannou et al. (1993) reported.

It is observed how the line corresponding to PAR val-

ues ofcos zequal to 0.950 tends to the upper bound of

the measured ones as global irradiance decrease, e.g.,

the sky becomes cloudy. Addition of meteorological

variables as temperature, relative humidity, dew point

temperature or precipitable water, or other radiometric

parameters as diffuse solar irradiance, clearness index,

diffuse transmittance, clearness or brightness of the

sky, do not lead to an improvement in PAR estimation.Among the multiple regression models, the statis-

tical tests show that the model Alados1 presents the

best global performance in agreement to Alados et al.

(1996). It should be noted that the model Alados1 uses

and as input variables and thus direct or diffuse

irradiance measurements are needed. However, these

variables are not often recorded. Table 5 shows that

the model MODEL 1 reduces both RMSE and MBE

against the model Alados1 at all locations other than

Table Mountain. In addition, the zero intercept values,

a, of the model MODEL 1 exhibit a slight improve-

ment against those obtained by the model Alados1 at

the USA stations. It is also noted from Table 5 that thestatistical results concerning each model as they are

evaluated with data collected at locations different to

that where the models were developed show an equiv-

alent deterioration. One advantage using the ANN

based model, MODEL 1, is the lower amount of in-

put information, since it needs only global irradiance

measurements which are available at many sites.

On the other hand, the statistical results corre-

sponding to the second ANN based model, MODEL

2, present small deviations in Almeras and SRRB

databases, with the exception of Desert Rocks one

where the MBD reaches a value of about 10% . The

variance of the PAR values explained is around 96%

and the RMSD values are around 8.5 and 15.5% for

Almeras database and the SRRB databases, respec-

tively. Although these values are relatively higher than

the RMSD values corresponding to MODEL 1, this

model provides an acceptable hourly PAR estimation

if only hourly sunshine duration measurements are

available.

An analysis of the behaviour of the residual PAR

values, which are obtained as the difference betweenestimated and measured values, has also been carried

out. Fig. 3 shows the averaged residual PAR values

for MODEL 1 and Alados1 evaluated in Almera

versus (a) global irradiance, (b) cos z, and (c) dew

point temperature. These models have been selected

because they present better performances than the

other ones. It may be seen that PAR residuals corre-

sponding to MODEL 1 do not exhibit any deviation

against global irradiance and cos z and a very slight

(0.5%) and quasi constant deviation against thedew point temperature. Dependences of PAR residual

differences against the other meteorological and ra-

diometric variables are also practically nil. In contrastthe results using the empirical model Alados1 exhibit

significant dependence of the PAR residuals on these

variables. So these results demonstrate the success of

using ANNs methodology.

Lastly, an analysis of the residual differences, sim-

ilar to the above one, using MODEL 1 and Alados1

evaluated for the data from Table Mountain has been

performed. From Fig. 4 it is firstly noted that PAR

values estimated by the ANN based model, MODEL

1, present a lower dependence on global irradiance,

cos z, and dew point temperature than those esti-

mated by Alados1, and second, the averaged residual

differences for both models show trends similar toeach one for each corresponding variable. Analysis

of the residual differences using other meteorological


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Fig. 3. Averaged residuals from estimates of the hourly PAR by

MODEL 1 and Alados1 versus: (a) G, (b) cos z and (c) Td . Avera-

ged residuals have been set as a percentage of the mean measured

PAR value. The error bars denote one standard deviation from the

mean values of the averaged residuals. Almeras data are used.

and radiometric variables and using other databases

have shown similar behaviour. In order to developa non-local model, PAR dependencies on global

radiation and solar zenith angle for locations with

Fig. 4. Averaged residuals from estimates of the hourly PAR by

MODEL 1 and Alados1 versus: (a) G, (b) cos z and (c) Td . Avera-

ged residuals have been set as a percentage of the mean measured

PAR value. The error bars denote one standard deviation from the

mean values of the averaged residuals. Table Mountains data are

used.

meteorological conditions different to each other

should be analysed. However, a thorough descriptionof this subject is outside the scope of this paper; but

will be presented in a future paper.


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7. Conclusions

In this work, a new method for estimating hourly

global PAR has been developed and tested. This

method is based on ANNs, particularly, on a multilayer

feedforward perceptron trained with the Levenberg

Marquardt algorithm. From this procedure, two mod-

els have been derived successfully. The first one in-

volves only global irradiance and solar zenith angle as

input variables, whereas the second one uses sunshine

duration and solar zenith angle. The models have been

extensively validated using data representative from

various climatic environments. These range from mar-

itime to high altitude deserts. For the model that uses

global irradiance and solar zenith angle as inputs, a

noticeable performance improvement is found over

multiple regression models that use a larger number

of input variables, as direct or diffuse irradiance, dew

point temperature, which are not often measured.

This result also demonstrates the viability of neu-ral network methods to solve such problems when

compared with existing methods that accomplish the

same task. The model involving relative sunshine du-

ration and solar zenith angle as input parameters also

produces acceptable results considering the limited

input information. This model provides an alternative

way to estimate hourly global PAR at many locations

where radiometric measurements are not available and

where PAR cannot be accurately calculated. It should

also be noted the relative small data sample needed

to train the ANN and to obtain a successful model.

Our study has revealed that the inclusion of other

radiometric input variables to the ANN models, suchas diffuse irradiance, clearness index, or Perezs pa-

rameters do not lead to more accurate estimations of

hourly global PAR. Similarly, meteorological infor-

mation such as temperature, relative humidity, dew

point temperature or precipitable water have very

little effect on the accuracy of PAR estimation.

Acknowledgements

This work was supported through the collaboration

convene between the Plataforma Solar de Almera

which belongs to the Centro de InvestigacionesEnergticas y Medioambientales (CIEMAT) and the

University of Almera. The authors are grateful to

SRRBs staff and The Royal Swedish Academy of

Sciences (particularly to Sir Martin Tjus) for the

facilities offered for providing the SRRB and Abisko

Research Station databases and the technical speci-

fication of the sensors, respectively. The authors are

indebted to Dr. Arahal for his helpful introductory

comments on ANNs. Lastly, the authors thank the

regional Editor Dr. J.B. Stewart, Prof. Juhan Ross and

an anonymous referee.

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