Agricultural and Forest Meteorology 100 (2000) 231–241
Estimation of mean monthly solar global radiation as
a function of temperature
Francisco Meza a,∗ , Eduardo Varasb,1
a
b
Departamento de Ciencias de Recursos Naturales, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
Departamento de Ingenierı́a Hidráulica y Ambiental, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
Received 14 December 1998; received in revised form 11 August 1999; accepted 13 August 1999
Abstract
Solar radiation is the primary energy source for all physical and biochemical processes that take place on earth. Energy
balances are a key feature of processes such as temperature changes, snow melt, carbon fixation through photosynthesis in
plants, evaporation, wind intensity and other biophysical processes. Solar radiation level is sometimes recorded, but generally
it needs to be estimated by empirical models based on frequently available meteorological records such as hours of sunshine
or temperature.
This paper evaluates the behavior of two empirical models based on the difference between maximum and minimum
temperatures and compares results with a model based on sunshine hours. This work concludes that empirical models based
on temperature have a larger coefficient of determination than the model based on cloud cover for the normal conditions of
Chile. These models are easy to use in any location if the parameters are correctly adjusted. In addition, probability distribution
functions and confidence intervals for solar radiation estimates using stochastic modeling of temperature differences were
calculated. ©2000 Published by Elsevier Science B.V. All rights reserved.
Keywords: Solar radiation; Temperature; Random variable; Fourier series
1. Introduction
In some cases a record of global solar radiation (RG )
using instruments such as pyranometers or actinometers is available, however, there are many meteorological stations which do not measure solar radiation, but
do register other variables such as precipitation, pressure and temperature. For this reason, this paper eval∗ Corresponding author. Fax: +56-2-553-92-31.
E-mail addresses: [email protected] (F. Meza), [email protected]
(E. Varas).
1 Fax +56-2-686-58-76.
uates proposed mathematical models to estimate solar radiation as a function of temperature differences
and compares their performance with models based
on sunshine hours.
Solar radiation is the principal energy source for
physical, biological and chemical processes, such as,
snow melt, plant photosynthesis, evaporation, crop
growth and is also a variable needed for biophysical
models to evaluate risk of forest fires, hydrological
simulation models and mathematical models of natural processes. Hence, in many occasions, a record of
observed solar radiation or an estimate of radiation is
required.
0168-1923/00/$ – see front matter ©2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 1 9 2 3 ( 9 9 ) 0 0 0 9 0 - 8
232
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
Table 1
Angström coefficients (a and b) recommended for Chilean localities. (Castillo and Santibáñez, 1981)
2. Model description
Extra-terrestrial solar radiation, also known as Angot radiation (RA , MJ m−2 day−1 ) can be calculated
as a function of the distance from the sun to earth (d,
km), the mean distance sun–earth (dm , km), latitude
(φ, rad), solar declination (δ, rad) and solar angle at
sunrise (sunset) (Hs , rad) using the following expression (Romo and Arteaga, 1983):
(86400)(1360) dm 2
RA =
π
d
× [(Hs )sin(φ)sin(δ) + cos(φ)cos(δ)sin(Hs )]
(1)
Using the preceding relationship, solar radiation can
be calculated for any point in the earth’s outer atmosphere for each day of the year as a function of latitude
and solar declination. However, gases and clouds introduce changes to both magnitude and spectral composition of solar radiation.
2.1. Angström model, 1924
Since the beginning of the century, efforts have
been made to estimate solar radiation as a function
of extra-terrestrial solar radiation and the state of the
atmosphere (Castillo and Santibáñez, 1981). The parameter most commonly used is hours of sunshine.
Usually, the ratio of global solar radiation to Angot radiation is correlated to the ratio of effective sunshine
hours to total possible sunshine hours.
Effective sunshine hours (n) are measured with a
heliograph (Martı́nez-Lozano et al., 1984). Although
this instrument has a threshold, under which sunshine
is not recorded, this error is not significant when estimating daily solar radiation.
Angström (1924), suggested a simple linear relationship to estimate global solar radiation (RG ,
MJ m−2 day−1 ) as a function of Angot radiation,
actual sunshine hours (n) and potential or theoretical
sunshine hours (N).
n
RG
=a+b
RA
N
(2)
Angström suggested values of 0.2 and 0.5 for
empirical coefficients a and b respectively. Other
authors, such as Bennett (1962), Davies (1965),
Locality
a
b
Latitude
(◦ S)
Longitude
(◦ W)
Altitude
(m)
Arica
Iquique
Antofagasta
Copiapó
Vallenar
La Serena
La Paloma
Quintero
Valparaiso
Santiago
Curicó
Constitución
Chillan
Concepción
Temuco
Osorno
Puerto Montt
Ancud
Puerto Aysén
Balmaceda
Punta Arenas
0.28
0.23
0.23
0.26
0.22
0.29
0.22
0.22
0.22
0.22
0.23
0.22
0.23
0.26
0.23
0.23
0.26
0.26
0.26
0.26
0.26
0.57
0.47
0.47
0.51
0.46
0.57
0.46
0.45
0.55
0.44
0.47
0.45
0.47
0.51
0.47
0.47
0.51
0.51
0.51
0.51
0.52
18.29
20.13
23.28
27.21
28.35
29.54
30.41
32.47
33.01
33.27
34.58
35.20
36.36
36.47
38.46
40.35
41.28
41.54
45.24
45.54
53.10
70.19
70.09
70.20
70.20
70.46
71.15
71.02
71.32
70.38
70.42
71.13
72.26
72.02
73.07
72.39
73.09
72.56
73.48
72.42
71.43
70.54
035
008
122
283
469
032
320
002
041
520
227
007
124
009
114
027
110
020
010
520
008
Monteith (1966), Penman (1948), and Turc (1961)
have calibrated this expression for different places.
Coefficients can vary significantly as Doorenbos
and Pruitt (1975) show. In Chile, Castillo and Santibáñez (1981), have recommended the values given in
Table 1.
2.2. Bristow–Campbell model, 1984
Incoming solar radiation is determined by the state
of the atmosphere. However, the dynamics of the
atmosphere is very difficult to predict. Considering
transformations experienced by solar radiation, one
can expect to find a relationship to express solar
radiation as a function of meteorological variables
commonly registered at climatological stations. When
solar radiations reaches the earth surface, part of it
is reflected and part is absorbed. The same occurs
with long-wave radiation that each body emits as a
function of its temperature. As Chang (1968), reports,
there is usually a good relation between net radiation
and global solar radiation, since the latter one is the
principal source of energy.
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
Furthermore, if the heat flow towards the soil is
neglected, one can find the ratio of sensible heat to
latent heat or Bowen ratio, on a daily basis (Campbell,
1977). Sensible heat is responsible for temperature
variations, so it is possible to obtain a relationship
between temperature differences and solar radiation,
being temperature a reflection of radiation balance.
Using this argument, Bristow and Campbell (1984),
suggested the following relationship for daily RG , as
a function of daily RA and the difference between
maximum and minimum temperatures (1T, ◦ C):
h
i
RG
= A 1 − exp(−B1T C )
(3)
RA
Athough coefficients A, B and C are empirical, they
have some physical meaning. Coefficient A represents
the maximum radiation that can be expected on a clear
day. Coefficients B and C control the rate at which A
is approached as the temperature difference increases.
Values most frequently reported for these coefficients
are 0.7 for A, the range 0.004 to 0.010 for B and 2.4
for C.
Since clear days present large temperature differences A tends to be the ratio between global solar radiation and Angot radiation, hence the sum of Angström
coefficients a and b tends to be similar to A.
2.3. Allen model, 1997
Allen (1997), suggested the use of a self-calibrating
model to estimate mean monthly global solar radiation
following the work of Hargreaves and Samani (1982).
He suggested that the mean daily RG can be estimated
as a function of RA , mean monthly maximum(TM , ◦ C)
and minimum temperatures (Tm , ◦ C).
RG
= Kr (TM − Tm )0.5
RA
(4)
Previously, Allen (1995), had expressed the empirical coefficient (Kr ) as a function of the ratio of atmospheric pressure at the site (P, kPa) and at sea level
(P0 , 101.3 kPa) as follows:
0.5
P
(5)
Kr = Kra
P0
In his work, Allen suggested values of 0.17 for interior
regions and 0.20 for coastal regions for the empirical
coefficient Kra .
233
3. Climatic data
In order to compare the behavior of the different
models, monthly climatological data of 21 stations
representing different climatic regions of Chile were
collected. Data ranged from Arica (latitude 18.3◦ S) to
Punta Arenas (53.1◦ S) and was registered between the
years 1971 and 1992.
Selected meteorological variables were TM , Tm , P,
mean monthly degree of cloud cover (x) and RG .
For the locations mentioned in Table 1, monthly values of maximum and minimum temperatures, cloud
cover and atmospheric pressure for each year in the
period 1971 to 1992, were available. Unfortunately,
for global solar radiation only the average value for
each month in that period was available and monthly
radiation values for each year were impossible to obtain from Dirección Meteorológica de Chile.
In addition to the above, data from La Paloma station was collected to compare the behaviour of models
based on temperature differences when they are applied to estimate monthly global solar radiation. The
selected meteorological variables in this case were TM ,
Tm , P, and RG between the years 1971 and 1978.
Finally, data from Santiago station was used to
compare the behaviour of Bristow–Campbell and
Allen models when they are applied to estimate daily
global solar radiation. The meteorological variables
were daily TM , Tm , P, and RG .
4. Models applied to mean monthly data
The extension of the reviewed models to apply them
to monthly averages requires some explanation. The
Angström model was originally derived for daily solar radiation and hours of sunshine. Nonetheless, being a linear function it can be readily applied to mean
monthly data since the expected value of a sum is
equal to the summation of the expected values. Allen’s
model was derived for monthly data so it can readily
be used. However, the Bristow–Campbell model is defined for daily data and has no evident extrapolation
to mean monthly values. For this reason, one can expect to find a new set of coefficients when the same
expression is applied to monthly data.
With the values of temperature, atmospheric pressure and sunshine hours, mean monthly global solar
234
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
radiation was calculated at each site, using the expressions and empirical coefficients suggested by
Angström (1924), Bristow and Campbell (1984), and
Allen (1995). Results show that models using the
coefficients proposed in the literature do not estimate correctly the historical average in each location.
The slope of the relationship between calculated
and observed radiation is significantly different from
unity. This is especially notorious in the case of the
Bristow–Campbell model, although this result was expected since the coefficients suggested by the authors
are applicable to daily data.
Given the results, it was necessary to change the
Allen and Bristow–Campbell model coefficients to
obtain a better fit, following the idea suggested by
Castillo and Santibáñez (1981) for the Angström
model. Least squares coefficients, which minimize the
sum of square errors for each location were calculated
and included in Table 2.
Due to the fact that monthly solar radiation values,
were not available for each year, as mentioned in the
section about climatic data, the A and C coefficients
of Bristow–Campbell model were assumed fixed and
the B coefficient was adjusted to minimize the square
Table 2
Adjusted coefficients (Kra and B) of Allen and Bristow–Campbell
models
Locality
Kra
B
Arica
Iquique
Antofagasta
Copiapó
Vallenar
La Serena
La Paloma
Quintero
Valparaı́so
Santiago
Curicó
Constitución
Chillan
Concepción
Temuco
Osorno
Puerto Montt
Ancud
Puerto Aysén
Balmaceda
Punta Arenas
0.3354
0.2854
0.4717
0.2577
0.3457
0.2697
0.1589
0.2731
0.0114
0.2593
0.4348
0.2423
0.2316
0.3402
0.2583
0.3756
0.3252
0.2820
0.2870
0.3058
0.3471
0.01354
0.01619
0.01944
0.00203
0.00200
0.00677
0.00347
0.00589
0.01144
0.00202
0.00152
0.00555
0.00159
0.00242
0.00154
0.00150
0.00290
0.00493
0.00463
0.00348
0.00389
Table 3
Regression between calculated and observed mean monthly global
solar radiation using adjusted parameters of 20 Chilean localities
Model
Slope
Upper
limit. (95%)
Lower
limit. (95%)
R2
Angström
Allen
Bristow–Campbell
0.959
0.999
1.152
0.970
1.010
1.170
0.939
0.990
1.138
0.892
0.961
0.928
errors. The available data made it impossible to study
the contribution of coefficients A and C. However, A
represents the maximum radiation on a clear day and
its value represents the observed data reasonably well.
Moreover, a change in coefficient C does not affect
significantly the calculated global solar radiation.
Observed and calculated values for different
locations and models are shown in Fig. 1. In this
figure the improvement in the relationships when using locally calibrated coefficients can be appreciated.
The Angström model results using the coefficients
proposed by Castillo and Santibáñez (1981) are also
included for comparison. Slopes of the different models and the coefficients of determination are given in
Table 3.
Allen’s model presents the best relationship. It has
a higher coefficient of determination and the slope
is equal to unity with 90% confidence interval. The
Bristow–Campbell model tends to under-estimate
global solar radiation but explains a large proportion
of sample variance. The Angström model fit the data
poorer than the other two.
4.1. Models applied to monthly data.
Since the available data of global solar radiation
for most stations is only the average value for each
month, it was necessary to examine if the relationships
with the adjusted coefficients represent accurately the
monthly values for each year. One station available
with monthly global solar radiation data, is La Paloma.
In this case the models with the adjusted coefficients
derived with the average monthly values were used to
estimate monthly global solar radiation for each year.
A comparison between estimated monthly values for
La Paloma, compared to observed monthly values is
shown in Table 4.
Results show that monthly global radiation for each
year can be adequately estimated with the derived
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
235
Fig. 1. (a) Comparison between observed and measured mean monthly global solar radiation using Angström parameters from the literature
(see text); (b) Comparison between observed and measured mean monthly global solar radiation using Angström adjusted parameters;
(c) Comparison between observed and measured mean monthly global solar radiation using Allen parameters from the literature; (d)
Comparison between observed and measured mean monthly global solar radiation using Allen adjusted parameters; (e) Comparison between
observed and measured mean monthly global solar radiation using Bristow-Campbell parameters from the literature; and (f) Comparison
between observed and measured mean monthly global solar radiation using Bristow–Campbell adjusted parameters.
236
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
Fig. 1 (Continued).
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
Table 4
Regression between calculated and observed monthly global solar
radiation at La Paloma station
Model
Slope
Upper
limit. (95%)
Lower
limit. (95%)
R2
Allen
Bristow–Campbell
1.000
0.994
1.010
1.006
0.990
0.982
0.97
0.96
models. Allen’s model presents the best relationship
between observed and calculated monthly solar global
radiation because it explains a large proportion of the
sample variance. In both models the slope is equal to
unity with 90 % confidence interval. This verifies that
the models can be used to estimate monthly values for
different years.
4.2. Global solar radiation distribution functions
A probability distribution function for global solar
radiation was obtained as a derived distribution, when
radiation is expressed as a function of temperature differences and temperature differences are expressed as
a Fourier series with a random component. This random error was found to be a random variable with normal distribution. This hypothesis was tested in both for
the Bristow–Campbell and the Allen models using the
Anderson–Darling test for normal distribution. Once
a distribution model for solar radiation is calculated,
confidence intervals for estimates can be computed.
237
Table 5
Average temperatures (1Ti ) and Fourier series coefficients Ci and
Di of 20 Chilean localities
Locality
1Ti
Ci
Arica
Iquique
Antofagasta
Copiapó
Vallenar
La Serena
La Paloma
Quintero
Valparaı́so
Santiago
Curicó
Constitución
Chillan
Concepción
Temuco
Osorno
Puerto Montt
Ancud
Puerto Aysén
Balmaceda
Punta Arenas
06.344
05.629
06.489
14.545
13.193
07.856
14.143
08.366
05.549
13.917
14.612
08.397
13.802
10.073
11.494
11.031
08.592
07.255
06.823
09.078
07.019
0.722
0.629
0.269
0.640
1.264
0.220
0.330
0.670
0.804
2.539
4.235
0.723
3.991
2.134
3.052
3.140
1.862
1.636
1.427
2.130
1.829
2πj
1Tij = 1Ti + Ci cos
12
2πj
+ Eij
+Di sin
12
Di
1.412
1.162
0.945
−0.292
0.177
−0.044
0.345
0.495
0.330
1.830
2.851
0.188
2.910
1.365
2.111
1.592
0.773
1.051
0.345
1.151
0.665
(6)
The coefficients Ci and Di are given in Table 5 for the
sites used in this work.
4.3. Temperature difference modelling.
Temperature has a marked seasonal variation due to
periodicity in the earth’s orbit about the sun. For this
reason temperature variations can be represented using mathematical cyclic functions. In this paper, differences between maximum and minimum temperatures were modelled using a Fourier series once the
stationary component was removed, as suggested by
Van Wijk and De Vries (1966) and Campbell and Norman (1997). These authors applied Fourier series with
one term to represent air temperatures.
The 1T in location i and month j (1Tij , ◦ C) can be
expressed as a function of mean annual 1T in location
i (1Ti , ◦ C), Fourier series coefficients at location i (Ci ,
Di ) and an error or residual in location i and month j
(Eij , ◦ C) as follows:
4.4. Probability distribution functions
If X is a continuous random variable with a probability density function f(x) and Y is a monotonic function of X, then the probability function of Y can be
obtained multiplying the inverse function by the absolute value of the Jacobian of the transformation (J) or
determinant of the first derivative of w(y) with respect
to X (Walpole and Myers, 1992):
g(y) = f [w(y)]|J |
(7)
Using this procedure probability density and probability distribution functions for RG estimated by Allen
and Bristow–Campbell models were derived.
238
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
4.5. Bristow–Campbell model
In this case the distribution function is calculated
using Eq. (3) and replacing 1Tij for its expression in
terms of annual 1T in each location and the corresponding Fourier series coefficients. Combining both
the expressions, an equation for the residuals is obtained. Residuals were found to be well represented
by a normal distribution model, so the probability distribution of the errors was assumed known. The distribution hypothesis was tested using Anderson–Darling
test.
The probability density function for solar radiation
following Eq. (7), is equal to the product of the normal
density function evaluated at the residuals for location
i and month j and the absolute value of the transformation Jacobian (Eq. (8)). The residuals are given in
this case by Eq. (9) and the first derivative by Eq. (10).
(8)
g(RGij ) = [J ]f (RGij )
The residuals are given by the following equation expressed as a function of terms already defined:
!1/2.4
−ln 1 − RGij /0.7RAij
− 1Ti
Eij =
Bi
2πj
2πj
− Di sin
(9)
−Ci cos
12
12
The first derivative is:
−1.4/2.4
1 1/2.4 1 RGij
−ln 1 −
|J | = Bi
6.8
0.7RAij
1
1 ×
(10)
1 − RGij /0.7RAij RAij The cumulative distribution function (CDF) is obtained by integrating the probability density function.
The CDF was evaluated numerically, using very small
intervals and the trapezoidal integration method, to
define confidence intervals for global solar radiation. Results for two locations Arica and Vallenar are
shown graphically in Fig. 2 (a,b).
4.6. Allen’s model
Similarly, for Allen’s model, 1997, the probability
density function is obtained using Eq. (4) and replacing 1Tij for its expression in terms of annual 1T in
each location and the corresponding Fourier series coefficients (Eq. (6)). Residuals in this case were also
found to be well represented by a normal distribution
model, so the probability distribution of errors was
assumed known.
The probability density function for solar radiation
is shown in Eq. (8).The residuals are given in this case
by Eq. (11) and the first derivative by Eq. (12):
RGij
Eij =
− 1Ti
(RAij )Krai (P /P0 )0.5
2πj
2πj
− Di sin
(11)
−Ci cos
12
12
The Jacobian is:
2R (P ) Gij 0
|J | = 2
Kra i (RAij )2 P (12)
The CDF is obtained integrating the probability density function. It was evaluated numerically to define
confidence intervals for global solar radiation. Results for Arica and Vallenar are shown graphically
in Fig. 2(c,d). The expected value for global solar
radiation given by the CDF using Allen’s model
are higher than the Angot radiation because the
limits of integration derived in this case were zero
and infinite. On the other hand, the CDF using
Bristow-Campbell model have clear and defined limits
which are zero and A times the Angot radiation. For
this reason the CDF obtained with Bristow–Campbell
model is more accurate and has smaller confidence
intervals.
5. Models applied to daily data
5.1. Allen’s model
Allen’s model, 1997 includes a correction term for
barometric pressure which in fact represents the altitude of the station above sea level, since the pressure
as a function of elevation can be expressed in terms of
the pressure at sea level, the temperature gradient, the
temperature at station elevation and the Avogadro air
constant. This correction term is small compared to the
influence of the temperature difference on radiation.
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
Fig. 2. (a) Expected values and confidence limits (5 and 95%) of daily mean global radiation using Bristow-Campbell model and
radiation for Arica; (b) Expected value and confidence limits (5 and 95 %) of daily mean global radiation using Bristow-Campbell
and Angot radiation for Vallenar; (c) Expected value and confidence limits (5 and 95 %) of daily mean global radiation using Allen
and Angot radiation for Arica; (d) Expected value and confidence limits (5 and 95%) of daily mean global radiation using Allen
and Angot radiation for Vallenar.
This model tends to over estimate global solar radiation in a daily basis, and frequently estimates radiation in excess of the extra-terrestrial radiation, since
the condition expressed by Eq. (13) is fulfilled. This
model does not have a limit for the estimated solar
radiation.
1T >
P0
(Kra )2 P
(13)
This condition is frequently true when the model is
applied to points located in interior regions which
usually experience large daily temperature variations.
239
Angot
model
model
model
Even though Allen’s model has a larger coefficient of determination, the slope is clearly less than
unity, indicating that the model over-estimates solar
radiation.
5.2. Bristow–Campbell model
This model is defined solely in terms of temperature
differences and is thus simpler to apply. The value for
A coefficient is 0.7, which is a reasonable value for
clear days. This type of day usually is associated to
large temperature differences.
240
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
Fig. 2 (Continued).
Table 6
Regression between calculated and observed daily global solar
radiation at Santiago station
Model
Slope
Upper
limit. (95%)
Lower
limit. (95%)
R2
Allen
Bristow–Campbell
0.561
1.090
0.549
0.979
0.571
1.202
0.85
0.79
The behavior of the Bristow–Campbell model is
more consistent and reliable, since it has an upper limit
given by parameter A. The regression analysis shows
that the Bristow–Campbell model performs better
(Table 6). On the other hand, Bristow–Campbell
model gives consistently a better estimate when
applied to daily data.
6. Conclusions
Empirical models to estimate global solar radiation are a convenient tool if the parameters can be
calibrated for different locations. These models have
the advantage of using meteorological data which are
commonly available.
For Chile, the models proposed by Allen and
Bristow–Campbell are adequate and allow estimates of
mean average global solar radiation as a function of air
temperature variation. Allen’s model has a larger coefficient of determination but requires both atmospheric
pressure and temperature variation measurements.
Models were calibrated for 20 locations in Chile which
represent a wide variation in climatic characteristics
and hence the procedure described is considered to be
F. Meza, E. Varas / Agricultural and Forest Meteorology 100 (2000) 231–241
of general application. Temperature variation can be
modelled by Fourier series and confidence intervals for
global solar radiation estimates can be obtained using
derived distribution procedures. Both the models have
limitations when applied to daily data. Solar radiation
at locations with large temperature differences are
not correctly modelled using Allen procedure and the
Bristow–Campbell model had a better performance.
References
Allen, R., 1995. Evaluation of procedures of estimating mean
monthly solar radiation from air temperature. FAO, Rome.
Allen, R., 1997. Self-calibrating method for estimating solar
radiation from air temperature. J. Hydrol. Eng. 2, 56–67.
Angström, A., 1924. Solar and terrestrial radiation. Q.J.R.
Meteorol. Soc. 50, 121–125.
Bennett, I., 1962. A method of preparing maps of mean daily
global radiation. Arch. Meteorol. Geophys. Bioklimatol Ser. B
13, 216–248.
Bristow, K., Campbell, G., 1984. On the relationship between
incoming solar radiation and daily maximum and minimum
temperature. Agric. For. Meteorol. 31, 159–166.
Castillo, H., Santibáñez, F., 1981. Evaluación de la radiación solar
global y luminosidad en Chile I. Calibración de fórmulas para
241
estimar radiación solar global diaria. Agricultura Técnica 41,
145–152.
Campbell, G., 1977. An Introduction to Environmental Biophysics.
Springer, New York.
Campbell, G., Norman, J., 1997. An Introduction to Environmental
Biophysics. Soils/AOS 532. Pullman WA.
Chang, J.-H., 1968. Climate and Agriculture. Aldine Publishing.
Chicago.
Davies, J.A., 1965. Estimation of insolation for West Africa. Q.J.R.
Meteorol. Soc. 91, 359–363.
Doorenbos, J., Pruitt, W., 1975. Crop water requierements,
Irrigation and drainage paper, 24. FAO, Rome.
Hargreaves, G., Samani, Z., 1982. Estimating potential
evapotranspiration. J. Irrig. Drain. Eng. ASCE. 108, 225–230.
Martı́nez-Lozano, J., Tena, F., Onrubia, J., De la Rubia, J.,
1984. The historical evolution of Angström formula and its
modifications: review and bibliography. Agric. For. Meteorol.
33, 109–128.
Monteith, J.L., 1966. Local differences in the attenuation of solar
radiation over Britain. Q.J.R. Meteorol. Soc. 92, 254–262.
Penman, H.L., 1948. Natural evaporation from open water, bare
soil and grass. Proc. R. Soc. London Ser. A. 193, 120–145.
Romo, J., Arteaga, R., 1983. Meteorologı́a Aplicada. Universidad
Autónoma de Chapingo. 442 pp.
Turc, L., 1961. Evaluation des besoins en eau d’irrigation.
Evapotranspiration potentielle. Ann. Agron. 12, 13–49.
Van Wijk, W., De Vries, D., (1966). Physics of Plant Environment,
2nd ed. North-Holland, The Netherlands, 382 pp.