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Fourier analysis of meteorological data for Seeb
Energy Conversion & Management 41 (2000) 1283±1291
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Fourier analysis of meteorological data for Seeb Atsu S.S. Dorvlo College of Science, Sultan Qaboos University, P.O. Box 36, P.C. 123, Al-Khod, Sultanate of Oman Received 3 May 1999; accepted 29 October 1999
Abstract Harmonic analysis methods are used to obtain typical annual time dependent functions for the solar radiation and temperature data for Seeb, Sultanate of Oman. The ®rst two harmonics adequately model the solar radiation and temperature data. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Amplitude; Annual time function; Harmonics; Meteorology; Phase angle; Solar radiation; Temperature
1. Introduction Meteorological data by its nature is time series data. An eective way of studying periodic data is by Fourier analysis [4]. Fourier analysis is a method that breaks a series into independent components called harmonics. The harmonics represent the important features of the particular series. Because they are independent, they are additive. Usually, the ®rst few harmonics are enough to explain the major features of any series [1±5,7]. In their study of 28 stations in Nigeria, Fagbenle and Karayiannis [5] showed that the annual series dominates the rest of the harmonics. Selcedo [8] used Fourier analysis methods to obtain typical annual time functions for Barcelona, Spain. In this article, we use the Fourier analysis method to model data on the solar radiation and temperature for Seeb, one of the major meteorological stations in Oman. Our ultimate goal is to develop typical annual time dependent functions for the solar radiation, monthly average daily temperature and monthly minimum and maximum temperatures.
E-mail address: [email protected] (A.S.S. Dorvlo). 0196-8904/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 9 9 ) 0 0 1 8 0 - 6
1284
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
2. Fourier analysis A Fourier series of a function, G(t ), is a linear combination of trigonometric functions. This can be written in the form: X N=2 N X 2pm 2pm 2pm t t G0 t fm am sin bm cos Rm cos G t G0 p p p m1 m1 m1 N=2 X
1
where t is the time of year, p is the period, N is the number of months (N = 12), the mth p amplitude Rm a2m b2m and the mth phase angle is fm tanÿ1 ÿam =bm : If G1 , . . . ,GN are particular monthly averaged daily global irradiation of a station, then the least squares estimates for the Fourier coecients am and bm are N 2X 2pm Gi sin i , a^ m N i1 p
m 1, 2, . . . , N2 ÿ 1
N X 2pm ^b 2 i , Gi cos m N i1 p
m 1, 2, . . . , N2 ÿ 1
a^ N=2 0 N 1X ÿ 1 i Gi : b^ N=2 N i1
q 2 ^ The mth amplitude is given by Rm a^ 2m b^ m and the mth phase angle is 8 > ÿ1 ^ > ^ ÿ am =bm , if b^ m > 0 > > tan > > > > > ÿ1 ^ ^ > ^ ^m > 0 tan ÿ a = b > m m ÿ p, if bm < 0, a > > > > < ÿ1 ÿ a^ m =b^ m p, if b^ m < 0, a^ m R0 f^ m tan > > > > > ÿp=2, if b^ m 0, a^ m > 0 > > > > > > if b^ m 0, a^ m < 0 > > p=2, > > : arbitrary, if b^ m 0, a^ m 0:
2
3
P 2 The variance of the model is s2 N=2 a2m b^ m : The square of the individual amplitudes i1 ^ determines the importance of the particular harmonics in the model.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
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3. Data for the study The data for the study is taken at the meteorological station at Seeb in the Sultanate of Oman. Seeb is on latitude 23835 'N and 58817 'E. It is only 14.6 m above sea level. The solar radiation data are recorded using a Pyranograph. It is the sum of the direct and diused radiation received by a unit horizontal surface for a day. The air temperature values, recorded every hour during the month, are used to compute the monthly mean. The monthly maximum and minimum temperatures are obtained from the daily values read from the maximum and minimum thermometers. The raw data are published monthly by the Department of Meteorology, Ministry of Communications [6]. The period of this analysis is January 1986 to December 1995. This period is chosen because it has complete data, and therefore, there was no need for data rehabilitation. 4. Analysis and discussion 4.1. Solar radiation data Applying Fourier analysis methods separately for each year, representative equations are derived. The estimates of the parameters of these are given in Table 1. Only the amplitude and phase angle of the ®rst and second harmonics are presented. The overall variance and the percent contribution of each of the harmonics are also provided. It can be seen that the ®rst harmonic, which represents the annual variations, dominates the equations. The ®rst harmonic's contribution to the variance is between 77 and 94% for all the years under consideration. The dierence between the amplitudes of the ®rst harmonic for the dierent years is small. The phase angles are almost the same for all the years. The semiannual series' contribution was small, the highest value being about 12.5% in 1994. Hence, the Table 1 Estimate of year harmonics: solar radiation First harmonic
Second harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
20.969 20.936 20.478 19.632 19.396 17.879 18.483 18.659 18.453 18.507
4.757 5.350 5.777 5.196 5.648 4.153 4.811 4.361 4.211 4.691
3.125 3.142 3.028 3.033 3.087 3.055 3.040 3.095 3.218 3.063
1.670 0.575 2.063 1.132 1.421 0.724 1.433 1.348 1.688 1.096
2.385 2.413 2.043 1.963 2.083 2.626 1.676 2.294 2.167 1.694
81.721 92.958 82.904 90.411 92.286 93.930 87.509 88.549 77.859 88.961
10.065 1.075 10.574 4.293 5.843 2.856 7.761 8.465 12.514 4.856
1.099 0.967 1.146 0.891 0.618 0.617 0.799 0.567 1.073 0.882
a
SD: standard deviation.
1286
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global irradiation can be estimated adequately by a function consisting of only the ®rst harmonic. The annual functions are pooled to give the typical annual function using the ®rst two harmonics: pt pt 3:089 1:315 cos 2:135 : 4 G t 19:339 4:896 cos 6 3 Fig. 1 shows the representative annual equations for solar radiation. The graph illustrates that the typical annual function summarizes all the information in the representative equations. The typical annual time function is the dashed line. From the function obtained, the minimum radiation (13.4 MJ/m2/day) is expected during mid-December, while the maximum radiation (24.1 MJ/m2/day) is expected at the beginning of May. 4.2. Air temperature data Fourier analysis was applied to data on the average, minimum and maximum temperatures for the years 1986±1995. The estimated yearly harmonic parameters for the mean, minimum and maximum temperatures are given in Tables 2±4, respectively. The ®rst harmonic dominates for all three parameters. For the average air temperature, the ®rst harmonic retained between 86 and 97% of the variance. A typical annual function for estimating the mean temperature is
Fig. 1. Representative equations for solar radiation of each year 1986±1995 and typical annual function in broken line.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
1287
Table 2 Estimate of year harmonics: average temperature First harmonic
Second harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
28.768 28.820 28.923 28.073 28.853 28.149 28.165 28.640 28.416 27.948
6.940 7.197 6.445 6.112 6.740 5.604 7.708 6.815 6.204 6.409
2.641 2.708 2.739 2.606 2.741 2.717 2.649 2.697 2.728 2.579
2.352 0.852 2.050 2.010 1.758 1.269 1.957 1.093 1.795 0.842
1.554 1.385 1.398 1.167 0.917 1.066 0.649 0.690 1.169 1.247
86.987 96.979 90.094 88.546 89.822 91.742 90.784 96.641 91.204 95.793
9.993 1.358 9.117 9.577 6.115 4.703 5.849 2.486 7.633 1.654
0.965 0.736 0.436 0.657 1.014 0.780 1.053 0.459 0.499 0.757
a
SD: standard deviation.
pt pt Tmean t 28:476 6:617 cos 2:681 1:598 cos 1:124 : 6 3
5
This equation uses the ®rst two harmonics, but because of the dominance of the ®rst harmonic, the second harmonic can be ignored. The retention rate of the ®rst harmonic is between 91 and 99% for the minimum temperature for all the years (Table 3). Hence, the typical annual minimum temperature can be estimated using only the ®rst harmonic. A typical annual minimum temperature is:
Table 3 Estimate of year harmonics: minimum temperature First harmonic
Second harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
21.342 20.442 21.425 20.192 21.500 20.375 20.608 20.117 20.083 19.742
6.080 7.519 6.099 6.360 7.042 7.470 7.916 7.168 6.271 6.777
2.621 2.508 2.635 2.434 2.671 2.499 2.700 2.637 2.595 2.567
1.270 0.573 0.975 1.132 0.342 1.426 1.481 0.573 0.369 0.805
1.419 4.417 1.032 0.420 0.042 1.078 6.146 5.388 1.366 4.964
91.555 95.551 94.215 93.784 95.475 93.031 92.945 96.732 98.129 93.615
3.997 0.555 2.408 2.970 0.225 3.389 3.252 0.619 0.339 1.320
0.978 1.170 0.818 0.844 1.063 1.036 1.160 0.841 0.572 1.138
a
SD: standard deviation.
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A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
Table 4 Estimate of year harmonics: maximum temperature First Harmonic
Second Harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
38.633 39.408 38.983 39.267 38.417 38.992 38.408 39.692 38.800 38.575
7.373 9.001 7.907 6.664 8.240 6.778 9.132 7.978 7.704 8.798
2.693 2.764 2.825 2.709 2.826 2.859 2.782 2.863 2.729 2.637
3.061 1.681 2.186 1.744 2.676 2.153 1.453 1.697 1.834 1.179
1.909 2.529 1.962 1.456 1.188 1.447 1.107 2.018 1.318 1.269
79.116 94.846 88.956 82.596 82.031 76.485 96.167 92.743 86.080 96.503
13.631 3.308 6.797 5.654 8.655 7.716 2.433 4.194 4.876 1.733
1.803 0.888 1.236 1.780 1.964 2.405 0.781 1.065 1.790 0.967
a
SD: standard deviation.
Fig. 2. Representative equations for average temperature of each year 1986±1995 and typical annual function in broken line.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
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Fig. 3. Representative equations for minimum temperature of each year 1986±1995 and typical annual function in broken line.
Fig. 4. Representative equations of maximum temperature of each year 1986±1995 and typical annual function in broken line
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Fig. 5. Typical annual time functions for solar radiation, average, minimum and maximum temperatures.
pt pt 2:587 0:895 cos 2:627 : Tmin t 20:583 6:870 cos 6 3
6
The contribution of the ®rst harmonic to estimating the maximum temperature (more than 76%) is not as high as in the case of the mean and minimum temperatures. However, for all the years, the ®rst two harmonics contributed more than 84%. A typical annual function for the maximum temperature is:
pt pt 2:769 1:966 cos 1:620 : Tmax t 38:918 7:958 cos 6 3
7
The lowest daily average temperature of 208C will occur in December as will the lowest minimum of 138C. The lowest maximum (298C), however, will occur in the period from the end of December to the beginning of January. The highest temperatures (highest average 368C and highest maximum 468C) are obtained in May. Figs. 2±4 show the year to year representative equations for the average, minimum and maximum temperatures, respectively, with their corresponding typical annual time functions also plotted in broken lines. The graphs show the adequacy of the typical annual functions. The uni-modal nature of the plots is because of the dominance of the annual harmonics. Fig. 5 gives a plot of the typical annual functions. The typical temperature ranges can be seen from this graph. The positive relationship between solar radiation and temperature can be observed.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
1291
5. Conclusion Typical annual functions are derived for global irradiation and mean, minimum and maximum temperatures for Seeb, Sultanate of Oman, using Fourier analysis. The equation given use the ®rst and second harmonics. However, since the contribution of the ®rst harmonic is over 76% for all the yearly equations, an equation using only the ®rst harmonic may be adequate. The minimum radiation levels will occur in mid-December, while the maximum levels will occur in May. Minimum temperatures are expected in December and maximum temperatures will occur in late May or early June. There appears to be no signi®cant lag time between the peaks of solar radiation and temperature. These new functions permit us to calculate the most probable values of the solar radiation and temperature for any date in the year. References [1] Baldasano JM, Clar J, Berna A. Fourier analysis of daily solar radiation data in Spain. Solar Energy 1988;41(4):327±33. [2] Balling RC. Harmonic analysis of monthly insolation levels in the US. Solar Energy 1983;31(3):293±8. [3] Balling RC, Cerceny RS. Spatial and temporal variations in long-term normal percent possible solar radiation levels in the United States. J Climate and Applied Meteorology 1983;22:1726±32. [4] Essenwager O. Applied statistics in atmospheric science. Amsterdam: Elsevier, 1976. [5] Fagbenle RO, Karayiannis TD. Harmonic analysis of monthly solar radiation in Nigeria. Renewable Energy 1994;4(5):551±9. [6] Oman Ministry of Communications. Annual climatic summaries. Sultanate of Oman: Directorate General of Civil Aviation and Meteorology, Department of Meteorology, 1986±1996. [7] Philips WF. Harmonic analysis of climatic data. Solar Energy 1984;32(3):319±28. [8] Salcedo AC, Baldasano JM. Fourier analysis of meterological data to obtain a typical annual time function. Solar Energy 1984;32(4):479±88.
www.elsevier.com/locate/enconman
Fourier analysis of meteorological data for Seeb Atsu S.S. Dorvlo College of Science, Sultan Qaboos University, P.O. Box 36, P.C. 123, Al-Khod, Sultanate of Oman Received 3 May 1999; accepted 29 October 1999
Abstract Harmonic analysis methods are used to obtain typical annual time dependent functions for the solar radiation and temperature data for Seeb, Sultanate of Oman. The ®rst two harmonics adequately model the solar radiation and temperature data. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Amplitude; Annual time function; Harmonics; Meteorology; Phase angle; Solar radiation; Temperature
1. Introduction Meteorological data by its nature is time series data. An eective way of studying periodic data is by Fourier analysis [4]. Fourier analysis is a method that breaks a series into independent components called harmonics. The harmonics represent the important features of the particular series. Because they are independent, they are additive. Usually, the ®rst few harmonics are enough to explain the major features of any series [1±5,7]. In their study of 28 stations in Nigeria, Fagbenle and Karayiannis [5] showed that the annual series dominates the rest of the harmonics. Selcedo [8] used Fourier analysis methods to obtain typical annual time functions for Barcelona, Spain. In this article, we use the Fourier analysis method to model data on the solar radiation and temperature for Seeb, one of the major meteorological stations in Oman. Our ultimate goal is to develop typical annual time dependent functions for the solar radiation, monthly average daily temperature and monthly minimum and maximum temperatures.
E-mail address: [email protected] (A.S.S. Dorvlo). 0196-8904/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 9 9 ) 0 0 1 8 0 - 6
1284
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
2. Fourier analysis A Fourier series of a function, G(t ), is a linear combination of trigonometric functions. This can be written in the form: X N=2 N X 2pm 2pm 2pm t t G0 t fm am sin bm cos Rm cos G t G0 p p p m1 m1 m1 N=2 X
1
where t is the time of year, p is the period, N is the number of months (N = 12), the mth p amplitude Rm a2m b2m and the mth phase angle is fm tanÿ1 ÿam =bm : If G1 , . . . ,GN are particular monthly averaged daily global irradiation of a station, then the least squares estimates for the Fourier coecients am and bm are N 2X 2pm Gi sin i , a^ m N i1 p
m 1, 2, . . . , N2 ÿ 1
N X 2pm ^b 2 i , Gi cos m N i1 p
m 1, 2, . . . , N2 ÿ 1
a^ N=2 0 N 1X ÿ 1 i Gi : b^ N=2 N i1
q 2 ^ The mth amplitude is given by Rm a^ 2m b^ m and the mth phase angle is 8 > ÿ1 ^ > ^ ÿ am =bm , if b^ m > 0 > > tan > > > > > ÿ1 ^ ^ > ^ ^m > 0 tan ÿ a = b > m m ÿ p, if bm < 0, a > > > > < ÿ1 ÿ a^ m =b^ m p, if b^ m < 0, a^ m R0 f^ m tan > > > > > ÿp=2, if b^ m 0, a^ m > 0 > > > > > > if b^ m 0, a^ m < 0 > > p=2, > > : arbitrary, if b^ m 0, a^ m 0:
2
3
P 2 The variance of the model is s2 N=2 a2m b^ m : The square of the individual amplitudes i1 ^ determines the importance of the particular harmonics in the model.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
1285
3. Data for the study The data for the study is taken at the meteorological station at Seeb in the Sultanate of Oman. Seeb is on latitude 23835 'N and 58817 'E. It is only 14.6 m above sea level. The solar radiation data are recorded using a Pyranograph. It is the sum of the direct and diused radiation received by a unit horizontal surface for a day. The air temperature values, recorded every hour during the month, are used to compute the monthly mean. The monthly maximum and minimum temperatures are obtained from the daily values read from the maximum and minimum thermometers. The raw data are published monthly by the Department of Meteorology, Ministry of Communications [6]. The period of this analysis is January 1986 to December 1995. This period is chosen because it has complete data, and therefore, there was no need for data rehabilitation. 4. Analysis and discussion 4.1. Solar radiation data Applying Fourier analysis methods separately for each year, representative equations are derived. The estimates of the parameters of these are given in Table 1. Only the amplitude and phase angle of the ®rst and second harmonics are presented. The overall variance and the percent contribution of each of the harmonics are also provided. It can be seen that the ®rst harmonic, which represents the annual variations, dominates the equations. The ®rst harmonic's contribution to the variance is between 77 and 94% for all the years under consideration. The dierence between the amplitudes of the ®rst harmonic for the dierent years is small. The phase angles are almost the same for all the years. The semiannual series' contribution was small, the highest value being about 12.5% in 1994. Hence, the Table 1 Estimate of year harmonics: solar radiation First harmonic
Second harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
20.969 20.936 20.478 19.632 19.396 17.879 18.483 18.659 18.453 18.507
4.757 5.350 5.777 5.196 5.648 4.153 4.811 4.361 4.211 4.691
3.125 3.142 3.028 3.033 3.087 3.055 3.040 3.095 3.218 3.063
1.670 0.575 2.063 1.132 1.421 0.724 1.433 1.348 1.688 1.096
2.385 2.413 2.043 1.963 2.083 2.626 1.676 2.294 2.167 1.694
81.721 92.958 82.904 90.411 92.286 93.930 87.509 88.549 77.859 88.961
10.065 1.075 10.574 4.293 5.843 2.856 7.761 8.465 12.514 4.856
1.099 0.967 1.146 0.891 0.618 0.617 0.799 0.567 1.073 0.882
a
SD: standard deviation.
1286
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
global irradiation can be estimated adequately by a function consisting of only the ®rst harmonic. The annual functions are pooled to give the typical annual function using the ®rst two harmonics: pt pt 3:089 1:315 cos 2:135 : 4 G t 19:339 4:896 cos 6 3 Fig. 1 shows the representative annual equations for solar radiation. The graph illustrates that the typical annual function summarizes all the information in the representative equations. The typical annual time function is the dashed line. From the function obtained, the minimum radiation (13.4 MJ/m2/day) is expected during mid-December, while the maximum radiation (24.1 MJ/m2/day) is expected at the beginning of May. 4.2. Air temperature data Fourier analysis was applied to data on the average, minimum and maximum temperatures for the years 1986±1995. The estimated yearly harmonic parameters for the mean, minimum and maximum temperatures are given in Tables 2±4, respectively. The ®rst harmonic dominates for all three parameters. For the average air temperature, the ®rst harmonic retained between 86 and 97% of the variance. A typical annual function for estimating the mean temperature is
Fig. 1. Representative equations for solar radiation of each year 1986±1995 and typical annual function in broken line.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
1287
Table 2 Estimate of year harmonics: average temperature First harmonic
Second harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
28.768 28.820 28.923 28.073 28.853 28.149 28.165 28.640 28.416 27.948
6.940 7.197 6.445 6.112 6.740 5.604 7.708 6.815 6.204 6.409
2.641 2.708 2.739 2.606 2.741 2.717 2.649 2.697 2.728 2.579
2.352 0.852 2.050 2.010 1.758 1.269 1.957 1.093 1.795 0.842
1.554 1.385 1.398 1.167 0.917 1.066 0.649 0.690 1.169 1.247
86.987 96.979 90.094 88.546 89.822 91.742 90.784 96.641 91.204 95.793
9.993 1.358 9.117 9.577 6.115 4.703 5.849 2.486 7.633 1.654
0.965 0.736 0.436 0.657 1.014 0.780 1.053 0.459 0.499 0.757
a
SD: standard deviation.
pt pt Tmean t 28:476 6:617 cos 2:681 1:598 cos 1:124 : 6 3
5
This equation uses the ®rst two harmonics, but because of the dominance of the ®rst harmonic, the second harmonic can be ignored. The retention rate of the ®rst harmonic is between 91 and 99% for the minimum temperature for all the years (Table 3). Hence, the typical annual minimum temperature can be estimated using only the ®rst harmonic. A typical annual minimum temperature is:
Table 3 Estimate of year harmonics: minimum temperature First harmonic
Second harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
21.342 20.442 21.425 20.192 21.500 20.375 20.608 20.117 20.083 19.742
6.080 7.519 6.099 6.360 7.042 7.470 7.916 7.168 6.271 6.777
2.621 2.508 2.635 2.434 2.671 2.499 2.700 2.637 2.595 2.567
1.270 0.573 0.975 1.132 0.342 1.426 1.481 0.573 0.369 0.805
1.419 4.417 1.032 0.420 0.042 1.078 6.146 5.388 1.366 4.964
91.555 95.551 94.215 93.784 95.475 93.031 92.945 96.732 98.129 93.615
3.997 0.555 2.408 2.970 0.225 3.389 3.252 0.619 0.339 1.320
0.978 1.170 0.818 0.844 1.063 1.036 1.160 0.841 0.572 1.138
a
SD: standard deviation.
1288
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
Table 4 Estimate of year harmonics: maximum temperature First Harmonic
Second Harmonic
Percent of variance
Year
Average
Amplitude
Phase
Amplitude
Phase
First
Second
SDa
1986 1987 1988 1989 1990 1991 1992 1993 1994 1995
38.633 39.408 38.983 39.267 38.417 38.992 38.408 39.692 38.800 38.575
7.373 9.001 7.907 6.664 8.240 6.778 9.132 7.978 7.704 8.798
2.693 2.764 2.825 2.709 2.826 2.859 2.782 2.863 2.729 2.637
3.061 1.681 2.186 1.744 2.676 2.153 1.453 1.697 1.834 1.179
1.909 2.529 1.962 1.456 1.188 1.447 1.107 2.018 1.318 1.269
79.116 94.846 88.956 82.596 82.031 76.485 96.167 92.743 86.080 96.503
13.631 3.308 6.797 5.654 8.655 7.716 2.433 4.194 4.876 1.733
1.803 0.888 1.236 1.780 1.964 2.405 0.781 1.065 1.790 0.967
a
SD: standard deviation.
Fig. 2. Representative equations for average temperature of each year 1986±1995 and typical annual function in broken line.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
1289
Fig. 3. Representative equations for minimum temperature of each year 1986±1995 and typical annual function in broken line.
Fig. 4. Representative equations of maximum temperature of each year 1986±1995 and typical annual function in broken line
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Fig. 5. Typical annual time functions for solar radiation, average, minimum and maximum temperatures.
pt pt 2:587 0:895 cos 2:627 : Tmin t 20:583 6:870 cos 6 3
6
The contribution of the ®rst harmonic to estimating the maximum temperature (more than 76%) is not as high as in the case of the mean and minimum temperatures. However, for all the years, the ®rst two harmonics contributed more than 84%. A typical annual function for the maximum temperature is:
pt pt 2:769 1:966 cos 1:620 : Tmax t 38:918 7:958 cos 6 3
7
The lowest daily average temperature of 208C will occur in December as will the lowest minimum of 138C. The lowest maximum (298C), however, will occur in the period from the end of December to the beginning of January. The highest temperatures (highest average 368C and highest maximum 468C) are obtained in May. Figs. 2±4 show the year to year representative equations for the average, minimum and maximum temperatures, respectively, with their corresponding typical annual time functions also plotted in broken lines. The graphs show the adequacy of the typical annual functions. The uni-modal nature of the plots is because of the dominance of the annual harmonics. Fig. 5 gives a plot of the typical annual functions. The typical temperature ranges can be seen from this graph. The positive relationship between solar radiation and temperature can be observed.
A.S.S. Dorvlo / Energy Conversion & Management 41 (2000) 1283±1291
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5. Conclusion Typical annual functions are derived for global irradiation and mean, minimum and maximum temperatures for Seeb, Sultanate of Oman, using Fourier analysis. The equation given use the ®rst and second harmonics. However, since the contribution of the ®rst harmonic is over 76% for all the yearly equations, an equation using only the ®rst harmonic may be adequate. The minimum radiation levels will occur in mid-December, while the maximum levels will occur in May. Minimum temperatures are expected in December and maximum temperatures will occur in late May or early June. There appears to be no signi®cant lag time between the peaks of solar radiation and temperature. These new functions permit us to calculate the most probable values of the solar radiation and temperature for any date in the year. References [1] Baldasano JM, Clar J, Berna A. Fourier analysis of daily solar radiation data in Spain. Solar Energy 1988;41(4):327±33. [2] Balling RC. Harmonic analysis of monthly insolation levels in the US. Solar Energy 1983;31(3):293±8. [3] Balling RC, Cerceny RS. Spatial and temporal variations in long-term normal percent possible solar radiation levels in the United States. J Climate and Applied Meteorology 1983;22:1726±32. [4] Essenwager O. Applied statistics in atmospheric science. Amsterdam: Elsevier, 1976. [5] Fagbenle RO, Karayiannis TD. Harmonic analysis of monthly solar radiation in Nigeria. Renewable Energy 1994;4(5):551±9. [6] Oman Ministry of Communications. Annual climatic summaries. Sultanate of Oman: Directorate General of Civil Aviation and Meteorology, Department of Meteorology, 1986±1996. [7] Philips WF. Harmonic analysis of climatic data. Solar Energy 1984;32(3):319±28. [8] Salcedo AC, Baldasano JM. Fourier analysis of meterological data to obtain a typical annual time function. Solar Energy 1984;32(4):479±88.