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Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
Pergamon
PII: S0038 – 092X( 01 )00009 – 3
Solar Energy Vol. 71, No. 1, pp. 11–21, 2001 2001 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X / 01 / $ - see front matter
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SPATIAL INTERPOLATION AND ESTIMATION OF SOLAR IRRADIATION BY CUMULATIVE SEMIVARIOGRAMS ~ ZEKAI~ S¸EN* , † and AHMET D. S¸AHIN** ~ *Istanbul Technical University, Faculty of Civil Engineering, Hydraulics Division, Maslak 80626, ~Istanbul, Turkey ~ **Istanbul Technical University, Meteorology Department, Maslak 80626, ~Istanbul, Turkey Received 29 July 1999; revised version accepted 23 January 2001 Communicated by RICHARD PEREZ
Abstract—The main purpose of this paper is to find a regional procedure for estimating the solar irradiation value of any point from sites where measurements of solar global irradiation already exist. The spatial weights are deduced through the regionalized variables theory and the cumulative semivariogram (CSV) approach. The CSV helps to find the change of spatial variability with distance from a set of given solar irradiation data. It is then employed in the estimation of solar irradiation value at any desired point through a weighted average procedure. The number of adjacent sites considered in this weighting scheme is based on the least squares technique which is applied spatially by incrementing nearest site numbers successively from one up to the total site number. The validity of the methodology is first checked with the cross validation technique prior to its application to sites with no solar irradiation records. Hence, after the cross-validation each site will have different number of nearest adjacent sites for spatial interpolation. The application is achieved for monthly solar irradiation records over Turkey by considering 29 measurements stations. It has been shown that the procedure presented in this paper is better than the classical techniques such as the inverse distance or inverse distance square approaches. 2001 Elsevier Science Ltd. All rights reserved. Keywords—Solar irradiation; Weighting factors; Regionalized variable; Regional interpolation; Cross validation.
estimate ore grades in gold mines the regionalized variables theory was developed by Matheron (1971). This theory is also termed as geostatistics which has been used to quantify the spatial variability of parameters. The basic idea in geostatistics is that for many natural phenomena, such as solar irradiation, samples taken close to each other have higher probability of being similar in magnitude than samples taken further apart. This implies spatial correlation structure in the phenomena. Especially in earth sciences, considerable effort has been directed towards the application of the statistical techniques leading to convenient regional interpolation and extrapolation methodologies (Matheron, 1965; Journel and Huijbregts, 1989; Cressie, 1993). The estimation of spatial solar irradiation estimation problem has been addressed first by Dooley and Hay (1983) and Hay (1986). They tried to evaluate the errors using solar irradiance data at a number of sites in Canada. The basis of their approach was the optimal interpolation techniques as suggested by Gandin (1963) in the meteorology literature. The main interest was to estimate the long-term average of all the sites considered and for each month irrespective of any
1. INTRODUCTION
Spatial variability of regionalized variables are very common in the physical sciences (Cressie, 1993). In practical applications, the spatial variation rates are of great significance in fields such as in solar engineering, agriculture, remote sensing and other earth and planetary sciences. A set of measurement stations during a fixed time interval (hour, day, month, etc.) provides records of the regionalized variable at irregular sites, and there are few methodologies to deal with this type of scattered data. There are various difficulties in making spatial estimations originating not only from the regionalized random behaviour of the solar irradiation but also from the irregular site configuration. Hence, the basic questions are: (1) how to transfer the influence of each of the neighboring measurement stations to the estimation point, and (2) how to combine these effects to make reliable regional estimates of solar irradiance. Based on empirical work by Krige (1951) to †
Corresponding author. Tel.: 190-212-285-3442; fax: 190122-285-3129; e-mail: [email protected] 11
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Z. S¸en and A. D. S¸ahin
particular year. Systematic interpolation evaluations have been carried out in solar irradiation networks by different authors (Hay, 1983, 1984; Zelenka, 1985; Zelenka and Lazic, 1987; Zelenka et al., 1992). It is possible to prepare solar irradiation maps of a region based on a set of measurements at different sites by using basic geostatistical techniques such as semivariograms and then the Kriging methodology, (Journel and Huijbregts, 1989). The success of Kriging maps is dependent on the suitability of the theoretical semivariogram to the data at hand. In fact, semivariograms are the fundamental ingredients in Kriging procedures because they represent the spatial correlation structure of the phenomenon concerned. There are, however, practical difficulties in the identification of semivariograms from available data (S¸en, 1989, 1991). The main purpose of this paper is to present the use of the cumulative semivariogram (CSV) technique for the solar irradiation spatial estimation. Emprical CSVs are adopted as representatives of spatial correlation structure of irradiation data. They are rendered into standard weighting functions which show the change of weighting factor with dimensionless distance values. As the dimensionless distance value increases the effect of weighting decreases. 2. EXISTING METHODOLOGIES
2.1. Linear interpolation The essence of the spatial interpolation is to transfer available information in the form of data from a number of adjacent irregular sites to the estimation site through a function that represents the spatial weights according to the distances between the sites. Generally, changes in the measurement site number or especially the location of the estimation site will cause changes in the weightings due to change in the distances. In the linear interpolation technique as presented by Gandin (1970), the value at an uninstrumented site is assumed to be the linear combination of the records at the adjacent sites which can be expressed as
OwS n
SE 5
i i
(1)
i 51
where SE is the solar irradiance estimation at a site with no measurement. In this paper, such a site will be referred to as the estimation site. Here, n is the number of the measurement site and w i is
the weighting factor which shows the contribution from the i-th site with the measured solar irradiation value of Si . Due to the unbiased estimation requirement, the summation of all weights must be equal to 1 as a restriction (Journel and Huijbregts, 1989). Of course, such an estimation will give rise to an error, E, defined as the difference between the solar irradiance estimation, SE , at site i and the measured values Si (i 5 1, 2, . . . , n). For least squares the best estimation is achieved when the estimation variance VE is minimized.
O
1 n VE 5 ] S 2 Sid 2 n i 51 s E
(2)
The same estimation variance may also be used for cross validation whereby the measured solar irradiation value at site I, is considered as if it is not measured. The analytical derivation of weightings for the data is found in Cressie (1993).The Kriging approach calculates the Best Linear Unbiased Estimate (BLUE) for the interpolation at hand. It is based on the linear estimator as in Eq. (1) and minimization of the estimation variance in Eq. (2). The Kriging technique (Matheron, 1965) has been applied for the first time in the earth sciences for ore body recovery in mining but it has had several applications in the atmospheric and hydrologic sciences (Delhomme, 1978; Gambolati and Volpi, 1979a,b; Volpi et al., 1979; Pucci and Murashie, 1987; Philip and Kitanidis, 1989; Subyani and S¸en, 1989). In this paper, the concept of the cumulative semivariogram (CSV) as suggested by S¸en (1989) will be applied for the representation of spatial dependence structure in solar irradiation data and then for finding a standard weighting function. This function is used for determining weighting factors depending on the dimensionless distances that lead to the spatial estimation of solar irradiance.
2.2. Classical weighting functions In any optimum analysis technique, the main idea is that the estimation at any point is considered as a weighted average of the measured values at irregular sites. Hence, if there are i 5 1, 2, . . . , n measurement sites with records Si then the estimated site solar irradiation, SE , can be calculated as
O W(r )S S 5 ]]]] O W(r ) n
E
i 51 n
i 51
i,E
i
(3)
i,E
where W(r i,E ) is the weighting function between the i-th site and the estimation site and r i,E is the
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
distance between the i-th solar irradiation measurement station and the estimation site. In fact, Eq. (3) is identical to Eq. (1) since W(r i,E ) / o in51W(r i,E ) 5 w i . However, Eq. (3) is most commonly used in different disciplines because of its explicit expression as the weighted average. For instance, in the application of inverse distance and inverse distance square methods W(r i,E ) is considered simply to be equal to 1 /r i,E and 1 /r 2i,E , respectively. Additionally, weighting functions proposed by Cressman (1959), Gandin (1970) and Barnes (1964) also appear as sole functions of the distances between the sites. Unfortunately, none of the aforementioned weighting functions are event dependent, but they are suggested on the basis of the logical and geometrical conceptualizations of site configuration. The weighting functions that are prepared on a rational and logical basis without consideration of regional data have the following major drawbacks: (a) They do not take into consideration the natural variability of the regional variability features. For instance, in meteorology, Cressman (1959) weightings are given as 2 R 2 2 r i,E ]]] 2 W(r i,E ) 5 R 2 1 r i,E 0
5
W(r i,E ) 5
5S 0
2
(6)
In reality, it is expected that weighting functions should reflect the spatial dependence behaviour of the phenomenon. To this end, regional covariance and semivariogram (SV) functions are among the early alternatives for the weighting functions that take into account the spatial correlation of the phenomenon considered. The former method requires a set of assumptions such as the Gaussian distribution of the regionalized variable. The latter technique, SV, does not always yield a clear pattern of regional correlation structure (S¸en, 1989). Hence, in this paper the cumulative semivariogram technique is used which does not suffer from these drawbacks. It is the main purpose of the following section to propose naturally flexible and event dependent weighting functions by using the cumulative semivariograms obtained directly from the solar irradiation data recorded at a set of irregular sites.
3. CUMULATIVE SEMIVARIOGRAM
for r i,E # R
(4)
for r i,E $ R
where R is the radius of influence and is determined subjectively by personal experience. (b) Although weighting functions are considered universally applicable all over the world, they may show unreliable variabilities for small areas. For instance, within the same study area, neighbouring sites may have quite different weighting functions. (c) Geometric weighting functions cannot reflect the morphology, i.e., the regional variability of the phenomenon. They can only be considered as practical first approximation tools. A generalized form of the Cressman model with an extra exponent parameter a is suggested as 2 R 2 2 r i,E ]]] 2 R 2 1 r i,E
F S DG
r i,E W(r i,E ) 5 exp 2 4 ] R
13
D
a
for r i,E # R
(5)
for r i,E $ R
The inclusion of a has alleviated the aforesaid drawbacks to some extent, but its determination still presents difficulties in practical applications. Another form of geometrical weighting function was proposed by Sasaki (1960) and Barnes (1964) as
The cumulative semivariogram (CSV) method proposed by S¸en (1989) as an alternative to the classical semivariogram technique of Matheron (1965) has various advantages over any conventional procedure in depicting the regional variability and hence spatial dependence structure. The CSV is a graph that shows the variation of successive half-squared difference summations with distance. Hence, a non-decreasing CSV function is obtained which exhibits various significant clues about the regional behaviour of the meteorological factors. The CSV provides a measure of regional dependence whereby the closer the two sites are, the more correlated the regional event at these sites is and the smaller is the value of the CSV. The CSV can be obtained from a given set of solar irradiation data by executing the following steps: (i) Calculate the distance d i, j (i ± j 5 1,2, . . . , m) between every possible pair of sparse measurement sites. For instance, if the number of sample sites is n, then there are m 5 n(n 2 1) / 2 distance values. (ii) For each distance, d i, j calculate the corresponding half-squared differences, Di, j , of the solar irradiation data. For instance, if the solar irradiation variable has values of Si and Sj at two distinct sites at distance d i, j apart, then the halfsquared difference is
14
Z. S¸en and A. D. S¸ahin
1 Di, j 5 ]sSi 2 Sjd 2 2
(7)
(iii) Take the successive summation of the half-squared differences starting from the smallest distance to the largest in order. This procedure will yield a non-decreasing function as
OOD m
g (di, j ) 5
m
(8)
i, j
i 51 i 51
where g (d i, j ) represents CSV value at distance d i, j . (iv) Plot g (d i, j ) values versus the corresponding distance d i, j . The result will appear similar to the representative CSV functions as in Fig. 1. The sample CSV functions are free of subjectivity because no a priori selection of distance classes is involved in contrast to the analysis as suggested by Perrie and Toulany (1989) in which the distance axis is divided into subjective intervals, and subsequently, averages are taken within an individual interval which are regarded as the representative value for this interval.
3.1. Standard weighting function ( SWF) As already mentioned, calculation of a sample CSV leads to a non-decreasing function with distance. It is said in Section 2.2 that all the classical weighting functions appear as a nonincreasing function with distance. It is, therefore, logical to execute the following steps in order to
obtain a valid and standard weighting function from the sample CSV similar to the classical weighting functions. (a) Depict on the sample CSV the maximum distance, R M , and corresponding sample CSV value, VM . R M corresponds to the distance between the two farthest station locations in any study area, (see Fig. 1). (b) Divide all the distances (CSV values) by R M (by VM ) and the result appears as a scaled form of the sample CSV within limits of zero and one on both axis. This shows the change of dimensionless CSV with dimensionless distance. (c) Subtraction of the dimensionless CSV values from the maximum value of one appears as a non-increasing function of the dimensionless distance as shown in Fig. 2 which has similar pattern to all the classical weighting functions as explained in the previous section. This function is referred to as the standard weighting function (SWF). 4. DATA BASE AND APPLICATION
Although there are 50 solar irradiation measurement stations in Turkey, only 29 are found reliable for many studies (S¸ahin and S¸en, 1998; S¸en and S¸ahin, 2000; S¸en, 1998). These are included in the application of SWF and the subsequent local estimation in Eq. (1). All of the 29 station solar iradiation measurements have the same recording period. The solar irradiation sta-
Fig. 1. Representative cumulative semivariogram.
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
15
Fig. 2. A representative SWF.
tion locations are shown in Fig. 3 and recorded monthly average solar irradiation amounts in MJ / m 2 are presented in Table 1. In order to apply and indicate the reliability of the proposed approaches, stations are considered one by one for cross validation. Let us say that Ankara is chosen as the prediction site and January as the month of prediction. Fig. 4 shows the CSV and thereof obtained SWF for January at this station. Although according to Table 1, Ankara has an average January solar irradiation record of 5.88 MJ / m 2 , it will be assumed nonexistent for the cross-validation. The subsequent
step is to apply the estimation process as explained in the previous sections. For this purpose, it is necessary to consider the distances of Ankara to all other 28 stations. Table 2 includes these distances in the third column. For the sake of comparison, the fourth and fifth columns include ‘the inverse distance’ (ID) and ‘the inverse distance square’ (IDS). In the regional estimation of solar irradiation, ID and IDS values are taken as weights which are dependent on the stations configuration only. In the sixth column the dimensionless distances, which are necessary prerequisites for the SWF in Fig. 4b are given. Dimension-
Fig. 3. Solar irradiation measurement site locations.
16
Z. S¸en and A. D. S¸ahin
Table 1. Monthly solar irradiation values (MJ / m 2 ) Station
Location
Months
No. Name
Long. Lat.
J
F
M
A
M
J
J
A
S
O
N
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
36.98 37.75 38.75 40.65 36.10 39.95 36.88 37.85 39.65 40.18 40.13 40.60 37.92 38.67 39.73 39.77 40.97 37.57 37.70 38.40 40.60 41.37 38.72 36.13 37.87 38.35 36.82 41.28 41.10
7.68 6.42 7.40 5.31 7.99 5.88 9.31 8.41 4.78 5.96 6.32 6.35 6.02 6.18 7.48 6.38 4.87 7.04 7.25 8.46 5.30 6.62 7.80 7.28 6.60 8.62 5.61 5.21 9.29
9.95 8.73 10.46 8.26 10.80 8.81 11.93 11.06 7.14 7.64 8.95 9.80 9.74 9.41 10.89 9.83 7.52 9.60 10.24 11.86 8.21 9.29 11.08 10.84 9.48 11.38 8.01 7.72 12.93
13.72 12.19 15.00 12.67 14.23 12.83 16.61 15.13 10.07 10.43 12.66 14.18 14.29 13.28 14.44 14.47 11.12 12.87 14.16 16.41 11.59 12.90 15.20 14.69 13.00 16.19 10.92 10.78 16.40
17.57 15.62 18.17 16.18 18.82 15.34 20.44 19.42 13.67 13.37 16.70 16.31 18.68 16.58 16.64 17.04 15.18 15.18 17.98 17.80 13.27 15.61 17.95 17.75 18.14 20.02 14.27 13.42 19.91
20.05 17.75 20.55 19.01 21.58 19.15 23.25 22.59 17.11 16.55 20.44 19.80 22.99 20.10 18.46 20.31 19.18 17.27 21.01 18.94 15.94 18.50 20.69 20.83 21.21 22.19 17.99 15.82 22.39
21.41 19.64 23.89 21.99 23.22 21.54 25.87 25.46 19.63 19.11 22.20 21.94 26.64 23.77 21.94 22.70 21.80 19.33 23.98 22.09 17.85 21.35 22.94 23.58 24.74 24.11 20.83 18.18 25.22
21.88 19.12 23.43 21.82 22.62 21.82 25.26 25.44 20.01 18.87 22.28 22.02 25.67 23.70 22.05 23.02 21.04 19.10 23.49 22.83 18.20 22.29 23.05 23.73 25.17 23.78 20.52 15.50 24.28
19.83 17.16 22.22 20.10 20.52 20.52 22.40 23.38 18.17 17.74 20.37 20.53 23.19 21.14 19.69 21.15 18.81 17.56 21.34 20.50 16.97 19.60 21.05 21.83 22.42 22.25 18.49 14.58 22.39
16.42 14.58 18.40 15.91 17.73 17.08 20.24 19.51 14.99 14.67 16.77 17.25 19.63 17.57 16.70 17.91 14.83 15.14 18.52 17.62 13.68 16.73 18.01 17.84 18.51 19.33 14.63 11.78 19.16
11.99 10.63 12.27 10.09 12.54 11.28 14.61 14.01 9.76 9.96 11.21 11.77 13.90 11.67 11.43 12.12 9.68 10.87 12.91 11.91 8.65 11.02 12.39 12.74 12.18 14.26 9.54 8.58 13.21
8.47 7.18 8.01 6.09 8.85 7.37 9.94 9.01 5.64 5.89 7.22 6.90 8.47 6.82 7.00 7.94 6.16 7.42 8.46 7.87 5.79 7.23 8.29 8.37 7.65 9.60 6.20 5.85 9.53
6.83 5.48 6.31 4.79 7.04 4.76 8.00 7.35 3.91 4.62 5.12 4.87 6.00 5.10 6.23 5.55 4.51 5.81 6.57 6.87 4.11 5.79 6.74 6.16 5.28 7.56 4.96 4.76 7.48
ADANA ADIYAMAN AFYON AMASYA ANAMUR ANKARA ANTALYA AYDIN BALIKESIR BURSA CANAKKALE CANKIRI DIYARBAKIR ELAZIG ERZINCAN ESKISEHIR ISTANBUL ISPARTA IZMIR KARS KASTAMONU KAYSER KIRSEHIR KONYA MALATYA MERSIN SAMSUN TRABZON VAN
35.30 38.28 30.53 35.85 32.83 32.88 30.70 27.83 27.87 29.07 26.40 33.62 40.20 39.22 39.50 30.52 29.08 43.77 30.55 27.17 43.08 33.77 35.48 34.17 32.50 38.30 34.60 36.33 39.72
less distances are calculated by dividing each distance value in the third column by the maximum distance value, which appears between Ankara and Van sites as 916.4 km. Consequently, in the sixth column, the location of maximum distance has dimensionless distance value equal to one. In the seventh column, the SWF values are included as found for January from Fig. 4b corresponding to the dimensionless distances. In the application of SWF for the cross-validation estimation, the available measurement sites are considered in the weighting procedure according to Eq. (1) with n528. The plots of the cross-validation estimation and the actual measurement values are presented in Fig. 5. Great differences between the two values are observed. From this graph, it is obvious that consideration of all stations in the estimation procedure appears to be successful on the average. This is tantamount to saying that consideration of all the measurement sites without any distinction causes smoothing in the solar irradiation spatial estimation. Relative error percentages of more than 10% appeared excessively at almost all the sites. In order to improve the situation, it is suggested in this paper to restrict the number of adjacent sites for the weighting procedure during the crossvalidation so that the spatial estimation error
becomes minimum. For this purpose, during the estimation at a particular site, the number of adjacent stations is increased from one (the nearest site) to the total number of sites in the order of increasing distance. Consequently, it is observed that each site has its special number of adjacent sites for the best interpolation depending on the regional variability of the solar irradiation. The consequent estimates resulting from the ID, IDS and SWF are shown in Table 3 where the number of adjacent sites are also presented. In the same table comparison of relative errors from these three approaches indicates that the SWF has the least values – almost all of which fall within the acceptable limit of 10%. Last but not least, Fig. 6 shows the variation in the measured and estimated values of solar irradiation. Even the visual comparison of Figs. 5 and 6 shows clearly that the restrictive number of adjacent site consideration in the spatial solar irradiation estimation leads to great improvements. Table 3 provides the necessary adjacent site numbers for each of the 29 sites considered in this study for Turkey. A practical question is how many solar irradiation stations should be considered in the spatial estimation at any given unmeasured site? In order to provide objective answers, it is necessary to provide equal adjacent site
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
17
Fig. 4. Ankara station, (a) CSV, (b) SWF.
contours for the whole region of the study area. Fig. 7 indicates such a map and the following features can be depicted. (a) In general the eastern Mediterranean and the south eastern parts of Turkey require the least adjacent stations for the solar irradiation spatial estimation. In fact, in the south eastern parts two measurement stations are sufficient for estimation. The regional climatology of Turkey indicates that
these regions have less rainfall, long sunshine duration hours and relatively high temperatures. (b) In the southwestern part, up to five adjacent site numbers are necessary. This is well correlated with the topographic heights in this region which include several lakes that effect the evaporation and rainfall regimes. (c) The adjacent site number increases towards the east near the Iranian border. These regions are
18
Z. S¸en and A. D. S¸ahin
Table 2. Distances between Ankara and other stations and weightings for January Station No. Name
Distance (km)
ID (310 3 ) (1 / km)
IDS (310 6 ) (1 / km 2 )
Dimensionless distance a
SWF value
Estimation (MJ / m 2 ) CSV ID IDS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
391.7 527.6 242.1 263.6 428.1 0.0 390.6 495.4 429.2 325.2 551.9 95.7 671.9 563.6 565.6 202.4 340.9 315.3 521.4 867.9 174.8 262.1 231.0 233.6 500.0 378.9 326.6 590.1 916.4
2.553 1.895 4.130 3.794 2.336 0.000 2.560 2.019 2.330 3.075 1.812 10.446 1.488 1.774 1.768 4.940 2.934 3.171 1.918 1.152 5.720 3.815 4.329 4.281 2.000 2.639 3.062 1.695 1.091
6.518 3.592 17.057 14.391 5.456 0.000 6.554 4.075 5.427 9.455 3.283 109.115 2.215 3.148 3.126 24.407 8.606 10.056 3.678 1.328 32.719 14.556 18.738 18.324 4.000 6.964 9.377 2.872 1.191
0.427 0.576 0.264 0.288 0.467 0.000 0.426 0.541 0.468 0.355 0.602 0.104 0.733 0.615 0.617 0.221 0.372 0.344 0.569 0.947 0.191 0.286 0.252 0.255 0.546 0.414 0.356 0.644 1.000
0.730 0.466 0.922 0.909 0.646 0.000 0.733 0.534 0.640 0.836 0.452 0.996 0.260 0.443 0.427 0.951 0.818 0.854 0.467 0.053 0.972 0.910 0.929 0.927 0.533 0.754 0.831 0.376 0.000
5.60 2.99 6.82 4.82 5.16 0.00 6.82 4.49 3.06 4.98 2.86 6.32 1.57 2.74 3.19 6.06 3.98 6.01 3.38 0.45 5.15 6.03 7.24 6.74 3.52 6.50 4.66 1.95 0.00
ADANA ADIYAMAN AFYON AMASYA ANAMUR ANKARA ANTALYA AYDIN BALIKESIR BURSA CANAKKALE CANKIRI DIYARBAKIR ELAZIG ERZINCAN ESKISEHIR ISTANBUL ISPARTA IZMIR KARS KASTAMONU KAYSER KIRSEHIR KONYA MALATYA MERSIN SAMSUN TRABZON VAN a
19.60 12.17 30.57 20.14 18.66 0.00 23.84 16.97 11.14 18.33 11.44 66.33 8.96 10.97 13.23 31.50 14.28 22.31 13.90 9.75 30.31 25.27 33.77 31.14 13.19 22.74 17.17 8.82 10.14
50.04 23.07 126.24 76.40 43.59 0.00 61.03 34.26 25.96 56.38 20.74 692.89 13.33 19.47 23.39 155.60 41.90 70.74 26.66 11.24 173.35 96.41 146.20 133.31 26.39 60.02 52.59 14.95 11.07
Dimensionless distance5distance / 916.4.
very rugged and have the highest mountain chains in Turkey. Winter seasons are long about 5 to 7 months each year. (d) Another high adjacent site requirement
appears in the north along the middle Black sea coast where the elevations reach up to almost 3000 meters. Severe winter conditions occur in this same region.
Fig. 5. Spatial estimation with all sites.
Table 3. Actual and estimated solar irradiation values Actual
No.
Name
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
ADANA ADIYAMAN AFYON AMASYA ANAMUR ANKARA ANTALYA AYDIN BALIKESIR BURSA CANAKKALE CANKIRI DIYARBAKIR ELAZIG ERZINCAN ESKISEHIR ISTANBUL ISPARTA IZMIR KARS KASTAMONU KAYSER KIRSEHIR KONYA MALATYA MERSIN SAMSUN TRABZON VAN
7.68 6.42 7.40 5.31 7.99 5.88 9.31 8.41 4.78 5.96 6.32 6.35 6.02 6.18 7.48 6.38 4.87 7.04 7.25 8.46 5.30 6.62 7.80 7.28 6.60 8.62 5.61 5.21 8.29
No. of adjacent sites 3 2 5 3 2 3 5 5 4 5 6 7 2 2 7 5 2 8 6 3 4 8 4 2 5 3 2 3 2
Estimation Adjacent sites, CSV
All sites, CSV
ID
IDS
7.69 6.39 7.18 5.75 7.95 5.83 7.61 7.14 6.00 5.93 6.25 6.10 6.30 6.31 6.52 6.23 5.38 7.52 6.65 7.36 5.79 6.85 7.55 7.42 6.54 7.82 5.30 5.55 7.27
6.89 6.88 6.75 6.63 6.98 6.70 6.82 6.72 6.83 6.71 6.65 6.59 6.95 6.86 6.71 6.83 6.71 6.86 6.76 6.40 6.70 6.74 6.77 6.84 6.82 6.86 6.55 6.80 6.71
7.16 6.77 6.78 6.56 7.08 6.69 6.90 6.77 6.75 6.44 6.56 6.45 6.85 6.77 6.58 6.73 6.60 7.12 6.86 6.58 6.69 6.84 6.90 7.01 6.69 7.03 6.49 6.86 6.76
7.87 6.63 6.79 6.13 7.48 6.53 7.07 6.89 6.58 5.87 6.14 5.99 6.73 6.67 6.30 6.63 6.24 7.62 7.21 6.39 6.48 6.95 7.08 7.27 6.49 7.42 5.91 6.92 6.88
Relative error (Adjacent CSV)
Relative error (All sites CSV)
Relative error (ID)
Relative error (IDS)
0.21 0.48 2.95 7.69 0.50 0.90 18.28 15.10 20.27 0.50 1.07 3.93 4.51 1.97 12.87 2.28 9.51 6.43 8.28 13.00 8.55 3.31 3.28 1.95 0.82 9.24 5.44 6.22 12.36
10.21 6.66 8.80 19.92 12.60 12.24 26.75 20.12 29.95 11.11 5.00 3.64 13.39 9.79 10.38 6.66 27.48 2.43 6.78 24.35 20.97 1.67 13.18 5.93 3.31 20.38 14.33 23.41 19.05
6.75 5.11 8.34 19.11 11.33 12.04 25.94 19.52 29.14 7.41 3.73 1.62 12.14 8.63 12.01 5.27 26.18 1.21 5.37 22.29 20.78 3.12 11.53 3.63 1.41 18.37 13.52 24.15 18.43
2.42 3.10 8.20 13.46 6.34 9.91 24.03 18.00 27.29 1.58 2.75 5.69 10.52 7.29 15.78 3.77 21.99 7.71 0.59 24.45 18.18 4.64 9.26 0.04 1.64 13.85 5.11 24.82 17.07
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
Station
19
20
Z. S¸en and A. D. S¸ahin
Fig. 6. Spatial estimation with restricted number of adjacent sites.
5. CONCLUSION
A regional estimation procedure through an interpolation technique in the form of weighted averages is proposed for determining the solar irradiation regional variations. The basis of the
methodology is first to obtain a cumulative semi variogram (CSV) function which is then converted into a standard weighting function (SWF) that has similar features with the classical weighting functions available in the literature. However, in addition to the site configuration, the SWF
Fig. 7. Equal adjacent site contours.
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
procedure takes into account the recorded data values at the sites. The proposed methodology has a general spatial and temporal basis because SWFs can be presented monthly. In order to show the validity of the methodology presented, a cross-validation procedure is applied with the actual solar irradiation measurements at a given set of sites. At this stage two different spatial estimation procedures are presented. The first one takes into consideration all of the measurement sites available, but the results are not within the acceptable practical error limits. The second alternative is to consider a restricted number of adjacent stations such that the spatial estimation error becomes a minimum. Hence, each site has a different number of nearest adjacent sites for consideration in the spatial solar irradiation estimation procedure. Equal adjacent site number contours are given for the solar irradiation data over Turkey which provide basic information in estimation of solar irradiation values at sites with no measurement. The application of the SWF methodology is completed numerically for global solar irradiation records at 29 sides scattered all over Turkey. The interpolation procedure based on the SWFs is very general and it can be easily applicable for any other regional variable. REFERENCES Barnes S. L. (1964) A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor. 3, 369. Cressie N. A. L. (1993) Statistics for spatial data. In Statistics For Spatial Data, p. 898, John Wiley and Sons, Inc., New York. Cressman G. D. (1959) An operational objective analysis system. Mon. Wea. Rev. 87, 367. Delhomme J. P. (1978) Kriging in the hydrosciences. Advances in Water Resources 1, 251. Dooley J. E. and Hay J. E. (1983) Structure of the global solar radiation field in canada. In Report To Atmospheric Environment Service, Downsview, Vol. Contact No. DSS– 39SS–KM601.0.1101, 2 volumes. Gambolati G. and Volpi G. (1979a) Groundwater contour mapping in Venice by stochastic interpolation. 1. Water Resour. Res. 15, 281–297. Gambolati G. and Volpi G. (1979b) A conceptual deterministic analysis of the Kriging technique in hydrology. Water Resour. Res. 15, 625–629. Gandin L. S. (1963) Objective Analysis of Meteorological Fields, Translated from Russian by the Israel Programme for Scientific Translations, Jerusalem.
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Gandin L. S. (1970) The planning of meteorological station networks. In Technical Note, Vol. No. 111, World Meteorological Organization, Geneva. Hay J. E. (1983) Solar energy system design. The impact of mesoscale variations in solar radiation. Atmos. Ocean 24, 138. Hay J. E. (1984) An assessment of the mesoscale variability of solar radiation at the Earth’s surface. Solar Energy 32(3), 425–434. Hay J. E. (1986) Errors associated with the spatial interpolation of mean solar irradiance. Solar Energy 37, 135. Journel A. G. and Huijbregts Ch. J. (1989) In Mining Geostatistics, p. 600, Academic Press, New York. Krige D. G. (1951) A statistical approach to some basic mine evaluation problems on the Witwateround. J. Chimic. Min. Soc. South-Africa 52, 119–139. Matheron G. (1965) In Les Variables Regionalisees Et Leur Estimation, p. 306, Masson, Paris. Matheron G. (1971) In The Theory of Regionalized Variables and Its Applications, Ecole de Mines, Fontainbleau, France. Philip R. D. and Kitanidis P. K. (1989) Geostatistical estimation of hydraulic head gradients. Ground Water 27, 855– 865. Perrie W. and Toulany B. (1989) Correlation of sea level pressure field for objective analysis. Mon. Wea. Rev. 17, 1965. Pucci A. A. and Murashie J. A. E. (1987) Applications of universal Kriging to an aquifer study in New Jersey. Ground Water 25, 672–678. Sasaki Y. (1960) An objective analysis for determining initial conditions for the primative equations. In Tech. Report, Vol. Ref. 60–16T, Texas A / M University, College Station. Subyani A. M. and S¸en Z. (1989) Geostatistical modeling of the Wasia aquifer in Central Saudi Arabia. Journal of Hydrology 110, 295. S¸ahin A. D. and S¸en Z. (1998) Statistical analysis of the ¨ formula coefficients and application for Turkey. Angstrom Solar Energy 62, 29–38. S¸en Z. (1989) Cumulative semivariogram model of regionalized variables. Int. J. Math. Geol. 21, 891. S¸en Z. (1991) Standard cumulative semivariograms of stationary stochastic processes and regional correlation. Int. J. Math. Geol. 24, 417–435. S¸en Z. (1998) Fuzzy algorithm for estimation of solar irradiation from sunshine duration. Solar Energy 63, 39–49. S¸en Z. and S¸ahin A. D. (2000) Solar irradiation polygon concept and application in Turkey. Solar Energy 68(1), 57–68. Volpi G., Campolati G., Carbonin L., Gatto P. and Mazzi G. (1979) Groundwater contour mapping in Venice by Stochastic interpolation. 2. Water Resour. Res. 15, 291–297. Zelenka A. (1985) Satellite versus ground observation based model for global irradiation. In INTERSOL 58, Proceeding of the 9th Biennial Congress ISES, Vol. 4, p. 2513, Pergamon. Zelenka A. and Lazic D. (1987) Supplementing network global irradiance data. Advances in Solar Energy Technology, Proceedings 1987 Biennial Congress ISES 4, 3861– 3865. Zelenka A., Czeplak G., D’Agostino V., Weine J., Maxwell E. and Perez R. (1992) Techniques for supplimenting solar radiation network data. In Report IEA Task 9, Vol. 2, Report No. IEA-SHCP-9D-1.
PII: S0038 – 092X( 01 )00009 – 3
Solar Energy Vol. 71, No. 1, pp. 11–21, 2001 2001 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X / 01 / $ - see front matter
www.elsevier.com / locate / solener
SPATIAL INTERPOLATION AND ESTIMATION OF SOLAR IRRADIATION BY CUMULATIVE SEMIVARIOGRAMS ~ ZEKAI~ S¸EN* , † and AHMET D. S¸AHIN** ~ *Istanbul Technical University, Faculty of Civil Engineering, Hydraulics Division, Maslak 80626, ~Istanbul, Turkey ~ **Istanbul Technical University, Meteorology Department, Maslak 80626, ~Istanbul, Turkey Received 29 July 1999; revised version accepted 23 January 2001 Communicated by RICHARD PEREZ
Abstract—The main purpose of this paper is to find a regional procedure for estimating the solar irradiation value of any point from sites where measurements of solar global irradiation already exist. The spatial weights are deduced through the regionalized variables theory and the cumulative semivariogram (CSV) approach. The CSV helps to find the change of spatial variability with distance from a set of given solar irradiation data. It is then employed in the estimation of solar irradiation value at any desired point through a weighted average procedure. The number of adjacent sites considered in this weighting scheme is based on the least squares technique which is applied spatially by incrementing nearest site numbers successively from one up to the total site number. The validity of the methodology is first checked with the cross validation technique prior to its application to sites with no solar irradiation records. Hence, after the cross-validation each site will have different number of nearest adjacent sites for spatial interpolation. The application is achieved for monthly solar irradiation records over Turkey by considering 29 measurements stations. It has been shown that the procedure presented in this paper is better than the classical techniques such as the inverse distance or inverse distance square approaches. 2001 Elsevier Science Ltd. All rights reserved. Keywords—Solar irradiation; Weighting factors; Regionalized variable; Regional interpolation; Cross validation.
estimate ore grades in gold mines the regionalized variables theory was developed by Matheron (1971). This theory is also termed as geostatistics which has been used to quantify the spatial variability of parameters. The basic idea in geostatistics is that for many natural phenomena, such as solar irradiation, samples taken close to each other have higher probability of being similar in magnitude than samples taken further apart. This implies spatial correlation structure in the phenomena. Especially in earth sciences, considerable effort has been directed towards the application of the statistical techniques leading to convenient regional interpolation and extrapolation methodologies (Matheron, 1965; Journel and Huijbregts, 1989; Cressie, 1993). The estimation of spatial solar irradiation estimation problem has been addressed first by Dooley and Hay (1983) and Hay (1986). They tried to evaluate the errors using solar irradiance data at a number of sites in Canada. The basis of their approach was the optimal interpolation techniques as suggested by Gandin (1963) in the meteorology literature. The main interest was to estimate the long-term average of all the sites considered and for each month irrespective of any
1. INTRODUCTION
Spatial variability of regionalized variables are very common in the physical sciences (Cressie, 1993). In practical applications, the spatial variation rates are of great significance in fields such as in solar engineering, agriculture, remote sensing and other earth and planetary sciences. A set of measurement stations during a fixed time interval (hour, day, month, etc.) provides records of the regionalized variable at irregular sites, and there are few methodologies to deal with this type of scattered data. There are various difficulties in making spatial estimations originating not only from the regionalized random behaviour of the solar irradiation but also from the irregular site configuration. Hence, the basic questions are: (1) how to transfer the influence of each of the neighboring measurement stations to the estimation point, and (2) how to combine these effects to make reliable regional estimates of solar irradiance. Based on empirical work by Krige (1951) to †
Corresponding author. Tel.: 190-212-285-3442; fax: 190122-285-3129; e-mail: [email protected] 11
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Z. S¸en and A. D. S¸ahin
particular year. Systematic interpolation evaluations have been carried out in solar irradiation networks by different authors (Hay, 1983, 1984; Zelenka, 1985; Zelenka and Lazic, 1987; Zelenka et al., 1992). It is possible to prepare solar irradiation maps of a region based on a set of measurements at different sites by using basic geostatistical techniques such as semivariograms and then the Kriging methodology, (Journel and Huijbregts, 1989). The success of Kriging maps is dependent on the suitability of the theoretical semivariogram to the data at hand. In fact, semivariograms are the fundamental ingredients in Kriging procedures because they represent the spatial correlation structure of the phenomenon concerned. There are, however, practical difficulties in the identification of semivariograms from available data (S¸en, 1989, 1991). The main purpose of this paper is to present the use of the cumulative semivariogram (CSV) technique for the solar irradiation spatial estimation. Emprical CSVs are adopted as representatives of spatial correlation structure of irradiation data. They are rendered into standard weighting functions which show the change of weighting factor with dimensionless distance values. As the dimensionless distance value increases the effect of weighting decreases. 2. EXISTING METHODOLOGIES
2.1. Linear interpolation The essence of the spatial interpolation is to transfer available information in the form of data from a number of adjacent irregular sites to the estimation site through a function that represents the spatial weights according to the distances between the sites. Generally, changes in the measurement site number or especially the location of the estimation site will cause changes in the weightings due to change in the distances. In the linear interpolation technique as presented by Gandin (1970), the value at an uninstrumented site is assumed to be the linear combination of the records at the adjacent sites which can be expressed as
OwS n
SE 5
i i
(1)
i 51
where SE is the solar irradiance estimation at a site with no measurement. In this paper, such a site will be referred to as the estimation site. Here, n is the number of the measurement site and w i is
the weighting factor which shows the contribution from the i-th site with the measured solar irradiation value of Si . Due to the unbiased estimation requirement, the summation of all weights must be equal to 1 as a restriction (Journel and Huijbregts, 1989). Of course, such an estimation will give rise to an error, E, defined as the difference between the solar irradiance estimation, SE , at site i and the measured values Si (i 5 1, 2, . . . , n). For least squares the best estimation is achieved when the estimation variance VE is minimized.
O
1 n VE 5 ] S 2 Sid 2 n i 51 s E
(2)
The same estimation variance may also be used for cross validation whereby the measured solar irradiation value at site I, is considered as if it is not measured. The analytical derivation of weightings for the data is found in Cressie (1993).The Kriging approach calculates the Best Linear Unbiased Estimate (BLUE) for the interpolation at hand. It is based on the linear estimator as in Eq. (1) and minimization of the estimation variance in Eq. (2). The Kriging technique (Matheron, 1965) has been applied for the first time in the earth sciences for ore body recovery in mining but it has had several applications in the atmospheric and hydrologic sciences (Delhomme, 1978; Gambolati and Volpi, 1979a,b; Volpi et al., 1979; Pucci and Murashie, 1987; Philip and Kitanidis, 1989; Subyani and S¸en, 1989). In this paper, the concept of the cumulative semivariogram (CSV) as suggested by S¸en (1989) will be applied for the representation of spatial dependence structure in solar irradiation data and then for finding a standard weighting function. This function is used for determining weighting factors depending on the dimensionless distances that lead to the spatial estimation of solar irradiance.
2.2. Classical weighting functions In any optimum analysis technique, the main idea is that the estimation at any point is considered as a weighted average of the measured values at irregular sites. Hence, if there are i 5 1, 2, . . . , n measurement sites with records Si then the estimated site solar irradiation, SE , can be calculated as
O W(r )S S 5 ]]]] O W(r ) n
E
i 51 n
i 51
i,E
i
(3)
i,E
where W(r i,E ) is the weighting function between the i-th site and the estimation site and r i,E is the
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
distance between the i-th solar irradiation measurement station and the estimation site. In fact, Eq. (3) is identical to Eq. (1) since W(r i,E ) / o in51W(r i,E ) 5 w i . However, Eq. (3) is most commonly used in different disciplines because of its explicit expression as the weighted average. For instance, in the application of inverse distance and inverse distance square methods W(r i,E ) is considered simply to be equal to 1 /r i,E and 1 /r 2i,E , respectively. Additionally, weighting functions proposed by Cressman (1959), Gandin (1970) and Barnes (1964) also appear as sole functions of the distances between the sites. Unfortunately, none of the aforementioned weighting functions are event dependent, but they are suggested on the basis of the logical and geometrical conceptualizations of site configuration. The weighting functions that are prepared on a rational and logical basis without consideration of regional data have the following major drawbacks: (a) They do not take into consideration the natural variability of the regional variability features. For instance, in meteorology, Cressman (1959) weightings are given as 2 R 2 2 r i,E ]]] 2 W(r i,E ) 5 R 2 1 r i,E 0
5
W(r i,E ) 5
5S 0
2
(6)
In reality, it is expected that weighting functions should reflect the spatial dependence behaviour of the phenomenon. To this end, regional covariance and semivariogram (SV) functions are among the early alternatives for the weighting functions that take into account the spatial correlation of the phenomenon considered. The former method requires a set of assumptions such as the Gaussian distribution of the regionalized variable. The latter technique, SV, does not always yield a clear pattern of regional correlation structure (S¸en, 1989). Hence, in this paper the cumulative semivariogram technique is used which does not suffer from these drawbacks. It is the main purpose of the following section to propose naturally flexible and event dependent weighting functions by using the cumulative semivariograms obtained directly from the solar irradiation data recorded at a set of irregular sites.
3. CUMULATIVE SEMIVARIOGRAM
for r i,E # R
(4)
for r i,E $ R
where R is the radius of influence and is determined subjectively by personal experience. (b) Although weighting functions are considered universally applicable all over the world, they may show unreliable variabilities for small areas. For instance, within the same study area, neighbouring sites may have quite different weighting functions. (c) Geometric weighting functions cannot reflect the morphology, i.e., the regional variability of the phenomenon. They can only be considered as practical first approximation tools. A generalized form of the Cressman model with an extra exponent parameter a is suggested as 2 R 2 2 r i,E ]]] 2 R 2 1 r i,E
F S DG
r i,E W(r i,E ) 5 exp 2 4 ] R
13
D
a
for r i,E # R
(5)
for r i,E $ R
The inclusion of a has alleviated the aforesaid drawbacks to some extent, but its determination still presents difficulties in practical applications. Another form of geometrical weighting function was proposed by Sasaki (1960) and Barnes (1964) as
The cumulative semivariogram (CSV) method proposed by S¸en (1989) as an alternative to the classical semivariogram technique of Matheron (1965) has various advantages over any conventional procedure in depicting the regional variability and hence spatial dependence structure. The CSV is a graph that shows the variation of successive half-squared difference summations with distance. Hence, a non-decreasing CSV function is obtained which exhibits various significant clues about the regional behaviour of the meteorological factors. The CSV provides a measure of regional dependence whereby the closer the two sites are, the more correlated the regional event at these sites is and the smaller is the value of the CSV. The CSV can be obtained from a given set of solar irradiation data by executing the following steps: (i) Calculate the distance d i, j (i ± j 5 1,2, . . . , m) between every possible pair of sparse measurement sites. For instance, if the number of sample sites is n, then there are m 5 n(n 2 1) / 2 distance values. (ii) For each distance, d i, j calculate the corresponding half-squared differences, Di, j , of the solar irradiation data. For instance, if the solar irradiation variable has values of Si and Sj at two distinct sites at distance d i, j apart, then the halfsquared difference is
14
Z. S¸en and A. D. S¸ahin
1 Di, j 5 ]sSi 2 Sjd 2 2
(7)
(iii) Take the successive summation of the half-squared differences starting from the smallest distance to the largest in order. This procedure will yield a non-decreasing function as
OOD m
g (di, j ) 5
m
(8)
i, j
i 51 i 51
where g (d i, j ) represents CSV value at distance d i, j . (iv) Plot g (d i, j ) values versus the corresponding distance d i, j . The result will appear similar to the representative CSV functions as in Fig. 1. The sample CSV functions are free of subjectivity because no a priori selection of distance classes is involved in contrast to the analysis as suggested by Perrie and Toulany (1989) in which the distance axis is divided into subjective intervals, and subsequently, averages are taken within an individual interval which are regarded as the representative value for this interval.
3.1. Standard weighting function ( SWF) As already mentioned, calculation of a sample CSV leads to a non-decreasing function with distance. It is said in Section 2.2 that all the classical weighting functions appear as a nonincreasing function with distance. It is, therefore, logical to execute the following steps in order to
obtain a valid and standard weighting function from the sample CSV similar to the classical weighting functions. (a) Depict on the sample CSV the maximum distance, R M , and corresponding sample CSV value, VM . R M corresponds to the distance between the two farthest station locations in any study area, (see Fig. 1). (b) Divide all the distances (CSV values) by R M (by VM ) and the result appears as a scaled form of the sample CSV within limits of zero and one on both axis. This shows the change of dimensionless CSV with dimensionless distance. (c) Subtraction of the dimensionless CSV values from the maximum value of one appears as a non-increasing function of the dimensionless distance as shown in Fig. 2 which has similar pattern to all the classical weighting functions as explained in the previous section. This function is referred to as the standard weighting function (SWF). 4. DATA BASE AND APPLICATION
Although there are 50 solar irradiation measurement stations in Turkey, only 29 are found reliable for many studies (S¸ahin and S¸en, 1998; S¸en and S¸ahin, 2000; S¸en, 1998). These are included in the application of SWF and the subsequent local estimation in Eq. (1). All of the 29 station solar iradiation measurements have the same recording period. The solar irradiation sta-
Fig. 1. Representative cumulative semivariogram.
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
15
Fig. 2. A representative SWF.
tion locations are shown in Fig. 3 and recorded monthly average solar irradiation amounts in MJ / m 2 are presented in Table 1. In order to apply and indicate the reliability of the proposed approaches, stations are considered one by one for cross validation. Let us say that Ankara is chosen as the prediction site and January as the month of prediction. Fig. 4 shows the CSV and thereof obtained SWF for January at this station. Although according to Table 1, Ankara has an average January solar irradiation record of 5.88 MJ / m 2 , it will be assumed nonexistent for the cross-validation. The subsequent
step is to apply the estimation process as explained in the previous sections. For this purpose, it is necessary to consider the distances of Ankara to all other 28 stations. Table 2 includes these distances in the third column. For the sake of comparison, the fourth and fifth columns include ‘the inverse distance’ (ID) and ‘the inverse distance square’ (IDS). In the regional estimation of solar irradiation, ID and IDS values are taken as weights which are dependent on the stations configuration only. In the sixth column the dimensionless distances, which are necessary prerequisites for the SWF in Fig. 4b are given. Dimension-
Fig. 3. Solar irradiation measurement site locations.
16
Z. S¸en and A. D. S¸ahin
Table 1. Monthly solar irradiation values (MJ / m 2 ) Station
Location
Months
No. Name
Long. Lat.
J
F
M
A
M
J
J
A
S
O
N
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
36.98 37.75 38.75 40.65 36.10 39.95 36.88 37.85 39.65 40.18 40.13 40.60 37.92 38.67 39.73 39.77 40.97 37.57 37.70 38.40 40.60 41.37 38.72 36.13 37.87 38.35 36.82 41.28 41.10
7.68 6.42 7.40 5.31 7.99 5.88 9.31 8.41 4.78 5.96 6.32 6.35 6.02 6.18 7.48 6.38 4.87 7.04 7.25 8.46 5.30 6.62 7.80 7.28 6.60 8.62 5.61 5.21 9.29
9.95 8.73 10.46 8.26 10.80 8.81 11.93 11.06 7.14 7.64 8.95 9.80 9.74 9.41 10.89 9.83 7.52 9.60 10.24 11.86 8.21 9.29 11.08 10.84 9.48 11.38 8.01 7.72 12.93
13.72 12.19 15.00 12.67 14.23 12.83 16.61 15.13 10.07 10.43 12.66 14.18 14.29 13.28 14.44 14.47 11.12 12.87 14.16 16.41 11.59 12.90 15.20 14.69 13.00 16.19 10.92 10.78 16.40
17.57 15.62 18.17 16.18 18.82 15.34 20.44 19.42 13.67 13.37 16.70 16.31 18.68 16.58 16.64 17.04 15.18 15.18 17.98 17.80 13.27 15.61 17.95 17.75 18.14 20.02 14.27 13.42 19.91
20.05 17.75 20.55 19.01 21.58 19.15 23.25 22.59 17.11 16.55 20.44 19.80 22.99 20.10 18.46 20.31 19.18 17.27 21.01 18.94 15.94 18.50 20.69 20.83 21.21 22.19 17.99 15.82 22.39
21.41 19.64 23.89 21.99 23.22 21.54 25.87 25.46 19.63 19.11 22.20 21.94 26.64 23.77 21.94 22.70 21.80 19.33 23.98 22.09 17.85 21.35 22.94 23.58 24.74 24.11 20.83 18.18 25.22
21.88 19.12 23.43 21.82 22.62 21.82 25.26 25.44 20.01 18.87 22.28 22.02 25.67 23.70 22.05 23.02 21.04 19.10 23.49 22.83 18.20 22.29 23.05 23.73 25.17 23.78 20.52 15.50 24.28
19.83 17.16 22.22 20.10 20.52 20.52 22.40 23.38 18.17 17.74 20.37 20.53 23.19 21.14 19.69 21.15 18.81 17.56 21.34 20.50 16.97 19.60 21.05 21.83 22.42 22.25 18.49 14.58 22.39
16.42 14.58 18.40 15.91 17.73 17.08 20.24 19.51 14.99 14.67 16.77 17.25 19.63 17.57 16.70 17.91 14.83 15.14 18.52 17.62 13.68 16.73 18.01 17.84 18.51 19.33 14.63 11.78 19.16
11.99 10.63 12.27 10.09 12.54 11.28 14.61 14.01 9.76 9.96 11.21 11.77 13.90 11.67 11.43 12.12 9.68 10.87 12.91 11.91 8.65 11.02 12.39 12.74 12.18 14.26 9.54 8.58 13.21
8.47 7.18 8.01 6.09 8.85 7.37 9.94 9.01 5.64 5.89 7.22 6.90 8.47 6.82 7.00 7.94 6.16 7.42 8.46 7.87 5.79 7.23 8.29 8.37 7.65 9.60 6.20 5.85 9.53
6.83 5.48 6.31 4.79 7.04 4.76 8.00 7.35 3.91 4.62 5.12 4.87 6.00 5.10 6.23 5.55 4.51 5.81 6.57 6.87 4.11 5.79 6.74 6.16 5.28 7.56 4.96 4.76 7.48
ADANA ADIYAMAN AFYON AMASYA ANAMUR ANKARA ANTALYA AYDIN BALIKESIR BURSA CANAKKALE CANKIRI DIYARBAKIR ELAZIG ERZINCAN ESKISEHIR ISTANBUL ISPARTA IZMIR KARS KASTAMONU KAYSER KIRSEHIR KONYA MALATYA MERSIN SAMSUN TRABZON VAN
35.30 38.28 30.53 35.85 32.83 32.88 30.70 27.83 27.87 29.07 26.40 33.62 40.20 39.22 39.50 30.52 29.08 43.77 30.55 27.17 43.08 33.77 35.48 34.17 32.50 38.30 34.60 36.33 39.72
less distances are calculated by dividing each distance value in the third column by the maximum distance value, which appears between Ankara and Van sites as 916.4 km. Consequently, in the sixth column, the location of maximum distance has dimensionless distance value equal to one. In the seventh column, the SWF values are included as found for January from Fig. 4b corresponding to the dimensionless distances. In the application of SWF for the cross-validation estimation, the available measurement sites are considered in the weighting procedure according to Eq. (1) with n528. The plots of the cross-validation estimation and the actual measurement values are presented in Fig. 5. Great differences between the two values are observed. From this graph, it is obvious that consideration of all stations in the estimation procedure appears to be successful on the average. This is tantamount to saying that consideration of all the measurement sites without any distinction causes smoothing in the solar irradiation spatial estimation. Relative error percentages of more than 10% appeared excessively at almost all the sites. In order to improve the situation, it is suggested in this paper to restrict the number of adjacent sites for the weighting procedure during the crossvalidation so that the spatial estimation error
becomes minimum. For this purpose, during the estimation at a particular site, the number of adjacent stations is increased from one (the nearest site) to the total number of sites in the order of increasing distance. Consequently, it is observed that each site has its special number of adjacent sites for the best interpolation depending on the regional variability of the solar irradiation. The consequent estimates resulting from the ID, IDS and SWF are shown in Table 3 where the number of adjacent sites are also presented. In the same table comparison of relative errors from these three approaches indicates that the SWF has the least values – almost all of which fall within the acceptable limit of 10%. Last but not least, Fig. 6 shows the variation in the measured and estimated values of solar irradiation. Even the visual comparison of Figs. 5 and 6 shows clearly that the restrictive number of adjacent site consideration in the spatial solar irradiation estimation leads to great improvements. Table 3 provides the necessary adjacent site numbers for each of the 29 sites considered in this study for Turkey. A practical question is how many solar irradiation stations should be considered in the spatial estimation at any given unmeasured site? In order to provide objective answers, it is necessary to provide equal adjacent site
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
17
Fig. 4. Ankara station, (a) CSV, (b) SWF.
contours for the whole region of the study area. Fig. 7 indicates such a map and the following features can be depicted. (a) In general the eastern Mediterranean and the south eastern parts of Turkey require the least adjacent stations for the solar irradiation spatial estimation. In fact, in the south eastern parts two measurement stations are sufficient for estimation. The regional climatology of Turkey indicates that
these regions have less rainfall, long sunshine duration hours and relatively high temperatures. (b) In the southwestern part, up to five adjacent site numbers are necessary. This is well correlated with the topographic heights in this region which include several lakes that effect the evaporation and rainfall regimes. (c) The adjacent site number increases towards the east near the Iranian border. These regions are
18
Z. S¸en and A. D. S¸ahin
Table 2. Distances between Ankara and other stations and weightings for January Station No. Name
Distance (km)
ID (310 3 ) (1 / km)
IDS (310 6 ) (1 / km 2 )
Dimensionless distance a
SWF value
Estimation (MJ / m 2 ) CSV ID IDS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
391.7 527.6 242.1 263.6 428.1 0.0 390.6 495.4 429.2 325.2 551.9 95.7 671.9 563.6 565.6 202.4 340.9 315.3 521.4 867.9 174.8 262.1 231.0 233.6 500.0 378.9 326.6 590.1 916.4
2.553 1.895 4.130 3.794 2.336 0.000 2.560 2.019 2.330 3.075 1.812 10.446 1.488 1.774 1.768 4.940 2.934 3.171 1.918 1.152 5.720 3.815 4.329 4.281 2.000 2.639 3.062 1.695 1.091
6.518 3.592 17.057 14.391 5.456 0.000 6.554 4.075 5.427 9.455 3.283 109.115 2.215 3.148 3.126 24.407 8.606 10.056 3.678 1.328 32.719 14.556 18.738 18.324 4.000 6.964 9.377 2.872 1.191
0.427 0.576 0.264 0.288 0.467 0.000 0.426 0.541 0.468 0.355 0.602 0.104 0.733 0.615 0.617 0.221 0.372 0.344 0.569 0.947 0.191 0.286 0.252 0.255 0.546 0.414 0.356 0.644 1.000
0.730 0.466 0.922 0.909 0.646 0.000 0.733 0.534 0.640 0.836 0.452 0.996 0.260 0.443 0.427 0.951 0.818 0.854 0.467 0.053 0.972 0.910 0.929 0.927 0.533 0.754 0.831 0.376 0.000
5.60 2.99 6.82 4.82 5.16 0.00 6.82 4.49 3.06 4.98 2.86 6.32 1.57 2.74 3.19 6.06 3.98 6.01 3.38 0.45 5.15 6.03 7.24 6.74 3.52 6.50 4.66 1.95 0.00
ADANA ADIYAMAN AFYON AMASYA ANAMUR ANKARA ANTALYA AYDIN BALIKESIR BURSA CANAKKALE CANKIRI DIYARBAKIR ELAZIG ERZINCAN ESKISEHIR ISTANBUL ISPARTA IZMIR KARS KASTAMONU KAYSER KIRSEHIR KONYA MALATYA MERSIN SAMSUN TRABZON VAN a
19.60 12.17 30.57 20.14 18.66 0.00 23.84 16.97 11.14 18.33 11.44 66.33 8.96 10.97 13.23 31.50 14.28 22.31 13.90 9.75 30.31 25.27 33.77 31.14 13.19 22.74 17.17 8.82 10.14
50.04 23.07 126.24 76.40 43.59 0.00 61.03 34.26 25.96 56.38 20.74 692.89 13.33 19.47 23.39 155.60 41.90 70.74 26.66 11.24 173.35 96.41 146.20 133.31 26.39 60.02 52.59 14.95 11.07
Dimensionless distance5distance / 916.4.
very rugged and have the highest mountain chains in Turkey. Winter seasons are long about 5 to 7 months each year. (d) Another high adjacent site requirement
appears in the north along the middle Black sea coast where the elevations reach up to almost 3000 meters. Severe winter conditions occur in this same region.
Fig. 5. Spatial estimation with all sites.
Table 3. Actual and estimated solar irradiation values Actual
No.
Name
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
ADANA ADIYAMAN AFYON AMASYA ANAMUR ANKARA ANTALYA AYDIN BALIKESIR BURSA CANAKKALE CANKIRI DIYARBAKIR ELAZIG ERZINCAN ESKISEHIR ISTANBUL ISPARTA IZMIR KARS KASTAMONU KAYSER KIRSEHIR KONYA MALATYA MERSIN SAMSUN TRABZON VAN
7.68 6.42 7.40 5.31 7.99 5.88 9.31 8.41 4.78 5.96 6.32 6.35 6.02 6.18 7.48 6.38 4.87 7.04 7.25 8.46 5.30 6.62 7.80 7.28 6.60 8.62 5.61 5.21 8.29
No. of adjacent sites 3 2 5 3 2 3 5 5 4 5 6 7 2 2 7 5 2 8 6 3 4 8 4 2 5 3 2 3 2
Estimation Adjacent sites, CSV
All sites, CSV
ID
IDS
7.69 6.39 7.18 5.75 7.95 5.83 7.61 7.14 6.00 5.93 6.25 6.10 6.30 6.31 6.52 6.23 5.38 7.52 6.65 7.36 5.79 6.85 7.55 7.42 6.54 7.82 5.30 5.55 7.27
6.89 6.88 6.75 6.63 6.98 6.70 6.82 6.72 6.83 6.71 6.65 6.59 6.95 6.86 6.71 6.83 6.71 6.86 6.76 6.40 6.70 6.74 6.77 6.84 6.82 6.86 6.55 6.80 6.71
7.16 6.77 6.78 6.56 7.08 6.69 6.90 6.77 6.75 6.44 6.56 6.45 6.85 6.77 6.58 6.73 6.60 7.12 6.86 6.58 6.69 6.84 6.90 7.01 6.69 7.03 6.49 6.86 6.76
7.87 6.63 6.79 6.13 7.48 6.53 7.07 6.89 6.58 5.87 6.14 5.99 6.73 6.67 6.30 6.63 6.24 7.62 7.21 6.39 6.48 6.95 7.08 7.27 6.49 7.42 5.91 6.92 6.88
Relative error (Adjacent CSV)
Relative error (All sites CSV)
Relative error (ID)
Relative error (IDS)
0.21 0.48 2.95 7.69 0.50 0.90 18.28 15.10 20.27 0.50 1.07 3.93 4.51 1.97 12.87 2.28 9.51 6.43 8.28 13.00 8.55 3.31 3.28 1.95 0.82 9.24 5.44 6.22 12.36
10.21 6.66 8.80 19.92 12.60 12.24 26.75 20.12 29.95 11.11 5.00 3.64 13.39 9.79 10.38 6.66 27.48 2.43 6.78 24.35 20.97 1.67 13.18 5.93 3.31 20.38 14.33 23.41 19.05
6.75 5.11 8.34 19.11 11.33 12.04 25.94 19.52 29.14 7.41 3.73 1.62 12.14 8.63 12.01 5.27 26.18 1.21 5.37 22.29 20.78 3.12 11.53 3.63 1.41 18.37 13.52 24.15 18.43
2.42 3.10 8.20 13.46 6.34 9.91 24.03 18.00 27.29 1.58 2.75 5.69 10.52 7.29 15.78 3.77 21.99 7.71 0.59 24.45 18.18 4.64 9.26 0.04 1.64 13.85 5.11 24.82 17.07
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
Station
19
20
Z. S¸en and A. D. S¸ahin
Fig. 6. Spatial estimation with restricted number of adjacent sites.
5. CONCLUSION
A regional estimation procedure through an interpolation technique in the form of weighted averages is proposed for determining the solar irradiation regional variations. The basis of the
methodology is first to obtain a cumulative semi variogram (CSV) function which is then converted into a standard weighting function (SWF) that has similar features with the classical weighting functions available in the literature. However, in addition to the site configuration, the SWF
Fig. 7. Equal adjacent site contours.
Spatial interpolation and estimation of solar irradiation by cumulative semivariograms
procedure takes into account the recorded data values at the sites. The proposed methodology has a general spatial and temporal basis because SWFs can be presented monthly. In order to show the validity of the methodology presented, a cross-validation procedure is applied with the actual solar irradiation measurements at a given set of sites. At this stage two different spatial estimation procedures are presented. The first one takes into consideration all of the measurement sites available, but the results are not within the acceptable practical error limits. The second alternative is to consider a restricted number of adjacent stations such that the spatial estimation error becomes a minimum. Hence, each site has a different number of nearest adjacent sites for consideration in the spatial solar irradiation estimation procedure. Equal adjacent site number contours are given for the solar irradiation data over Turkey which provide basic information in estimation of solar irradiation values at sites with no measurement. The application of the SWF methodology is completed numerically for global solar irradiation records at 29 sides scattered all over Turkey. The interpolation procedure based on the SWFs is very general and it can be easily applicable for any other regional variable. REFERENCES Barnes S. L. (1964) A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor. 3, 369. Cressie N. A. L. (1993) Statistics for spatial data. In Statistics For Spatial Data, p. 898, John Wiley and Sons, Inc., New York. Cressman G. D. (1959) An operational objective analysis system. Mon. Wea. Rev. 87, 367. Delhomme J. P. (1978) Kriging in the hydrosciences. Advances in Water Resources 1, 251. Dooley J. E. and Hay J. E. (1983) Structure of the global solar radiation field in canada. In Report To Atmospheric Environment Service, Downsview, Vol. Contact No. DSS– 39SS–KM601.0.1101, 2 volumes. Gambolati G. and Volpi G. (1979a) Groundwater contour mapping in Venice by stochastic interpolation. 1. Water Resour. Res. 15, 281–297. Gambolati G. and Volpi G. (1979b) A conceptual deterministic analysis of the Kriging technique in hydrology. Water Resour. Res. 15, 625–629. Gandin L. S. (1963) Objective Analysis of Meteorological Fields, Translated from Russian by the Israel Programme for Scientific Translations, Jerusalem.
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