Models applied to daily data
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 1T
ij
for its expression in terms of annual 1T in each location and the corre-
sponding Fourier series coefficients. Combining both the expressions, an equation for the residuals is ob-
tained. Residuals were found to be well represented by a normal distribution model, so the probability dis-
tribution of the errors was assumed known. The distri- bution 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 transfor- mation Jacobian Eq. 8. The residuals are given in
this case by Eq. 9 and the first derivative by Eq. 10. gR
Gij
= [J ]f
R
Gij
8 The residuals are given by the following equation ex-
pressed as a function of terms already defined:
E
ij
= −
ln 1 − R
Gij
0.7R
Aij
B
i 12.4
− 1T
i
− C
i
cos 2πj
12 −
D
i
sin 2πj
12 9
The first derivative is: |
J | = 1
B
i 12.4
1 6.8
− ln
1 − R
Gij
0.7R
Aij −
1.42.4
× 1
1 − R
Gij
0.7R
Aij
1 R
Aij
10 The cumulative distribution function CDF is ob-
tained 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 radia-
tion. 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 replac- ing 1T
ij
for its expression in terms of annual 1T in each location and the corresponding Fourier series co-
efficients 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:
E
ij
= R
Gij
R
Aij
K
rai
P P
0.5
− 1T
i
− C
i
cos 2πj
12 −
D
i
sin 2πj
12 11
The Jacobian is: |
J | = 2R
Gij
P K
2 ra i
R
Aij 2
P 12
The CDF is obtained integrating the probability den- sity function. It was evaluated numerically to define
confidence intervals for global solar radiation. Re- sults for Arica and Vallenar are shown graphically
in Fig. 2c,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.