Category Archives: EuroSun2008-13

Modeling and Analysis of Chinese Exposure to Solar Radiation Based on the Available Meteorological Data at CMA, China

Yu Qiang1* and Wang Zhifeng2

1,2 Institute of Electrical Engineering, CAS P. O.BOX 2703 Beijing 100190, China Corresponding Author, yuqiang1984@mail. iee. ac. cn


The distribution of solar radiation is very important for choosing sites for solar power tower plants. In this paper the observed data which include solar radiation and sunshine duration of eight typical cities of China during the period 1994-2003 are used to establish a correlation equation between monthly average daily values of clearness and relative sunshine. The model is used to estimate the global solar radiation of the whole country. And it is proved to be good results (greater than 94% in most cases). The predictive efficiency of this model is also compared with some other models which are believed to be applicable globally in terms of mean percentage error (MPE), mean bias error (MBE) and root mean square error (RMSE). And the results prove that it is also better than that of those models.

Key Words: global solar radiation, sunshine duration, MBE, MABE, RMSE.

1. Introduction

The design of a solar energy conversion system must always start with a study of solar radiation data at a site. One of the most important requirements in the design is the information on the intensity of solar radiation at a given location [1]. Unfortunately, there are very few meteorological stations that measure global solar radiation. Solar radiation data are still very scarce, especially in developing countries. So we must consider other methods to calculate relative solar data for places where they are not directly measured, many attempts have been made to develop models and empirical correlations that can predict the amount of solar radiation available at a given location from a few input parameters.

While it has been proved that a number of commonly measurable atmospheric and meteorological parameters such as turbidity, relative humidity, degree of cloudiness, temperature and sunshine duration taken severally or jointly, affect the magnitude of the global radiation incident on a given location. And the preponderance of data point to the fact that the greatest influence is exerted by sunshine hours.

There are several correlations [2-7] to have been developed that predict the correlation between the global radiation and the percentage of bright sunshine hours in a simple linear regression form (the Angstrom-Prescott type). And some authors have also developed quadratic correlation [2, 4, 6] model and multiple linear regression. The study of this paper is to establish a linear regression form which uses the data of eight typical cities of China for estimation global solar radiation for the cities where there are no meteorological stations but have similar meteorological conditions.

Materials, methods and models

In the present work the solar radiation is forecasted with the non-hydrostatic model Advanced Regional Prediction System (ARPS). This model is providing its forecast weather variables for a horizontal grid of (0.12 x 0.12)° resolution with a sampling interval of 10 min. The model is simulated at the LEPTEN laboratory (Laboratory of Energy Conversion Process Engineering and Energy Technology), former LABSOLAR, at the Federal University of Santa Catarina. The simulation assimilates the data of the global reanalysis delivered by the National Center for Environmental Prediction (NCEP) [12]. The analysis data characterize the initial condition at every 6 h, necessary to operate ARPS in actual time. The reanalysis data represent improved analysis data of the atmosphere. Both the analysis and reanalysis data are based on atmospheric measurements and their interpolations, as well as on the last forecasts of the GFS, which can accomplish forecasts until a ten days horizon. The operational forecast uncertainty includes both the analysis and the forecast uncertainty. In the present article only the uncertainty based on the reanalysis are verified. Therefore the reanalysis data, based on the GFS model, is assimilated with the regional ARPS model in a 6 h interval. For the uncertainty verification the 24 h mean value of the downward short wave radiation of the ARPS output is compared to the measured mean value of the global radiation. In a second step a statistical correction for the reanalysis uncertainty is accomplished. In the text that follows the daily energy E [Wh/m2] is equal to the daily mean radiation H [W/m2] multiplied by 24 hours.

Ordinary kriging

Kriging refers to a family of least-square linear regression algorithms that attempt to predict values of a variable at locations where data are not available based on the spatial pattern of the available data. The description of kriging theory and its application are given in detail by [5]. A semivariogram, y(h) represents the spatial variability in the data and is defined as Eq. (1):

1 N(h) 2

Y(h) = E [Z(xj + h) — Z(x )]2, (1)

2N(h) 7=1

where N(h) is the number of pairs of points separated by lag distance h, Z(x) and Z(x+h) are random values at locations x and x+h.

In this study, the exponential model have been used to fit the sample semivariograms, this model parameterizes the semivariogram in the following way:

Y(h) = C0 + C1 [1 — exp(-h / a)], (2)

where C0, Cj, a are called nugget, sill, and range respectively.

The objective of ordinary kriging procedure is to estimate data values at unsampled locations x0 using information available elsewhere in the domain (x1, x2,……………………………………………………………………………………………… , xn) . This can be carried out

by expressing Z(x0)as a linear combination of the data Z(x1), Z(x2),…………………… Z(xn) :

Подпись:Z(x0) = EX7Z (x0).

The optimal weights “ A7 ” are calculated assuming that the estimation Z(x0) by Z(x0) is

unbiased, that is, the expected value of the estimates is the same as that of the known data. The


condition needed for unbiased estimator is 2 A = 1.

7=1 7

Monthly-averaged daily data

Подпись: 0

Figures 2 and 3 show the annual variation of global irradiation and sunshine hours for the stations considered in this work. Generally, they show a regular annual variation with a maximum between June and July and a minimum in December. The only exception is represented by the station of Pedro Murias, which has a maximum in May and a relative minimum in June. As expected, the highest value of global insolation is found at Ancares, due to its altitude and rural location. The evolution of the sunshine hours shows a similar trend, with maxima occurring in summertime and minima in winter.


Table 2 summarizes these results. Having a look at the mean values of insolation and sunshine hours, Corrubedo has the highest values. It should be noticed that this site, which is located at only 38 km west from Lourizan, has mean sunshine hours and insolation significantly higher than Lourizan, 1 hour and 2 MJ m-2, respectively. The two stations located in an urban environment (Lourizan and Ferrol) are characterized by similar levels of insolation and sunshine hours.

Fig.3: Monthly average daily sunshine hours for the stations analysed

On table 3 are represented the main features of the yearly and monthly mean values of temperature, precipitations and relative humidity. The trend of the temperature is similar in all the locations, with maxima in August and minima in February.

Table 2: Monthly and yearly averaged daily values of Insolation and Sunshine Hours in the stations.














(kJ m-2)

(kJ m-2)

(kJ m-2)






20307 Jun

4069 Dec


8.5 Jul

2.7 Dec



20383 Jun

4194 Dec


9.1 Jul

2.8 Dec

P. Murias


16628 May

3754 Dec


6.5 May

2.3 Jan




20962 Jun

4444 Dec


8.8 Jul

3.1 Dec




20697 Jun

5013 Dec


8.5 Jul

3.6 Jan



22807 Jun

5185 Dec


9.6 Jun

3.5 Dec



23748 Jul

5264 Dec


9.7 Jul

3.3 Dec




3809 Dec


8.1 Jul

2.3 Dec

As it was already mentioned, mean precipitations are higher than the rest of Spain [10]. The highest values are found in Monte Aloia, due to the westerly air masses with high water content that

channel into the outfall of Mino river and ascend to the slope of the local mountain range, Sierra do Galineiro. In all the stations, maxima occur in October and minima in summertime.



















26.3 Aug



22.7 Jul



23.3 Aug

20.1 Aug



7.2 Feb

5.8 Feb

6.4 Feb

2.3 Feb



8.6 Feb

0 Feb

4.3 Feb









85 Oct

83 Dec

92 Jun

91 Nov



88 Oct

87 Oct

90 Jan

79 Apr

73 Jun

84 Apr

77 Aug



72 Jun

72 Jun

78 Aug










331 Oct


220 Oct



150 Oct

230 Oct

261 Oct

45 Aug

46 Jul

39 Jun

41 Jul

62 Jul

30 Jun

44 Aug

24 Aug

Table 3: Monthly and yearly averaged daily values of Temperature, Relative Humidity and accumulated

Precipitations in the stations analysed.

Clearness Index: averaged values and frequency distribution


The monthly averaged daily data of KT are depicted in Fig. 4 for five stations. It can be noticed that the station of Ancares is the one with the highest values of the index all over the year; it can be explained because of its altitude, rural environment and relatively cloud-free atmosphere. These values have a maximum in summertime.

The lowest clearness indexes are found in Pedro Murias and Lourizan. The first station, located in a rural environment, is characterized by high values of relative humidity that produce, together with low values of temperature (Table 3) and persistent fogs, especially in summertime. The station of Lourizan, located in an urban environment, is characterized by the lowest values of relative

humidity (77% yearly average) and the highest value of temperature (table 3). These features suggest that low values of KT are produced by anthropogenic aerosols, generated by local factories and urban pollution.

The station of Ferrol, despite of its suburban location, close to a sea port, has values of KT higher than Lourizan, due to the presence of winds that clean up the atmosphere from aerosols and fogs.

3.1 Ground data vs. satellite observations

Yearly and monthly averaged daily values of global irradiation collected in the meteorological stations were compared with the averaged values derived from satellite images and collected in the Solar Atlas of Galicia [4].

Figure 5 shows, for the stations analyzed, the distribution of the yearly averaged daily values measured by pyranometers at five stations versus the estimated values by Vazquez et al. [4]. Global irradiances estimated from satellite images overestimates the data collected by the meteorological stations in every location, except in Ancares. In three cases (Ancares, Ferrol and Corrubedo) the agreement between ground and satellite is quite good, with relative differences less than 3% (Table 4). In the other sites, differences are higher, ranging from 10.7% (Alto do Rodicio) to 25.3% (Sergude). Relative differences in the monthly averaged daily values are greater, up till 76%. Mainly, overestimations by ground measurements occur during winter.


Fig. 5: Distribution of the yearly mean values of global irradiation for the stations analyzed

The reason for these disagreements may be found in the normalization of the satellite data adopted by Vazquez et al. [4]. To obtain monthly averaged daily values, the authors apply only one coefficient for the entire region, instead of dividing the territory in areas characterized by similar climatologic features and calculating different normalization coefficients for each area. In their work, Vazquez et al. [4], compare their results only with one station, obtaining a very good agreement. For that, they assume the goodness of the results achieved for the entire region that is, as previously stated, characterized by complex topography and high climatologic variability.

Table 4: Relative differences (satellite-pyranometer) between the yearly and monthly averaged daily values

of global irradiation in the location analysed.































































































































4. Conclusions

The evaluation of the solar resource in eight sites of Galicia has been carried out for the period 2001-2006. The analysis of global irradiation, sunshine hours, clearness index, together with other meteorological parameters — precipitations, relative humidity and temperature — allowed a characterization of the solar resource in this region.

Monthly averaged daily values of global radiation and sunshine hours point out the complex climatology of the Galician territory. In an area of less than 30.000 km2, distributed over 2° of latitude and longitude, differences in the yearly mean daily values of global irradiation of more than 3 MJm-2 per day were found between the locations considered. This range of variability in the values of global irradiation is comparable to that of Germany (Meteonorm; Bern, Switzerland).

In the same way, yearly averaged daily values of sunshine duration show differences of almost 2 hours per day between the sites.

The analysis of precipitations, temperature and relative humidity, combined with the study of solar radiation, evidence the presence of persistent fogs in particular zones of Galicia. The station of Pedro Murias represents the most remarkable example. In the same way, the analysis of the solar radiation combined with the mean values of precipitations points out that extremely rainy areas, as that represented by the station of Monte Aloia, are not necessarily associated with low insolation.

The complexity of the Galician climate comes also out analysing the mean values of the clearness index: while the highest values are found in high elevation sites, as a consequence of a clearer atmosphere, the lowest values occur in Lourizan and Pedro Murias due, respectively, to the urban pollution and to persistent fog episodes.

The comparison of solar irradiation ground measurements vs. satellite observations gives the opportunity for making several remarks. The solar irradiation maps obtained by Vazquez et al. [4] evidence two different trends: a) the incoming solar radiation increases with the decreasing latitude and b) coastal zones receive more radiation than the inner nearby areas.

The ground measurements of global irradiation recorded in eight sites of Galicia depict a much more complex pattern not so easy to generalize. The latitude effect seems to be inexistent or, at least, is masked by more important local climatologic factors. Thus, Corrubedo and Lourizan, being at the same latitude, show yearly averaged daily differences of 2 MJ m-2. The same

differences are found in Ferrol and Pedro Murias, that have similar geographical features. In these areas, Vazquez et al. found differences of 0.36 MJ m-2 per day.

The stations of Ferrol and Alto do Rodicio, located respectively in the coast and in the interior, have similar mean values of irradiation even they are separated by 1° in latitude. The highest levels of radiation are found in Corrubedo and Ancares, located at the same latitude, but respectively on the western and eastern edges of Galicia. All these facts drive to the conclusion that the distribution of the solar resource in a region such as Galicia cannot be explained without the support of other climatologic and topographic features.

The present work was intended for characterizing the solar resource in eight sites of Galicia over a period of 5 years. In the future, the data record will increase also due to the installation of 17 more first class pyranometers and 30 pyranometers with Photovoltaic sensor. This will allow a more detailed characterization of the solar resource in time and space.

Due to the complex topography and climatology of Galicia, this will not be enough to obtain solar maps of the region since, in these cases, interpolation techniques do not provide sufficiently reliable results [11]. However, data from this solar radiation network will represent a very important tool to validate and calibrate the methods to estimate the solar resource.


Meteorological dataset provided by MeteoGalicia (Xunta de Galicia) from its web page is acknowledged This work was partially funded by Galician R&D Programme under project 07REM02CT.


[1] Institute Enerxetico de Galicia. Enema solar fotovoltaica na comunidade autonoma de Galicia. Conselleria de Inovacion, Industria e Comercio. Xunta de Galicia; 2003.

[2] Font Tullot I. Atlas de la radiacion solar en Espana. Instituto Nacional de Meteorologia, Ministerio de Transportes, Turismo y Comunicaciones. Madrid, Spain; 1984.

[3] Vera Mella N. Atlas climatico de irradiacion solar a partir de imagenes del satelite NOAA. Aplicacion a la peninsula iberica. PhD thesis. Univ. Politecnica de Catalunya, Barcelona; 2005.

[4] Vazquez Vazquez M., Santos Navarro J. M., Prado Cerqueira M. T., Vazquez Rios D., Rodrigues Fernandes F. M. Atlas de radiacion solar de Galicia. Universidad de Vigo. Vigo, Espana; 2005.

[5] Rigollier, C., Lefevre, M. and Wald, L. The method Heliosat-2 for deriving shortwave solar radiation from satellite images. Solar Energy 2004; 77: 159-169.

[6] Salson S., Souto J. A. Automatic weather stations network of the department of environment of Galicia: data acquisition, validation and quality control, Proceedings of the 3rd international conference on experiences with automatic weather stations, Torremolinos, Spain; 2003.

[7] Pettazzi A., Souto J. A., Salson S. EOAS, a shared joint atmospheric observation site of MeteoGalicia. Proceedings of 4th ICEAWS — International Conference on Experiences with Automatic Weather Stations, Lisbon, Portugal; 2006.

[8] Davies J. A. Validation of models for estimating solar radiation on horizontal surfaces. Report available from the IEA, Downsview, Ontario, Canada; 1988.

[9] Iqbal M. An introduction to solar radiation, Academic Press, San Diego, CA; 1983.

[10] Instituto Nacional de Meteorologia. Guia resumida del clima en Espana 1971-2000. Instituto Nacional de Meteorologia, D25.3, Ministerio de Medio Ambiente. Madrid, Espana; 2001.

[11] Batlles F. J., Martinez-Durban M., Miralles I., Ortega R., Barbero F. J., Tovar-Pescador J., Pozo — Vazquez D., Lopez G. Evaluacion de los recursos energeticos solares en zonas de topografia compleja. XII Congreso Iberico y VII Congreso Ibero Americano de Energia Solar. Vigo, Espana; 2004.

Mapping Solar Radiation over Complex Topography Areas Combining Digital Elevation Models and Satellite Images

J. L. Bosch1*, L. F. Zarzalejo2, F. J. Batlles1 and G. Lopez3

1 Universidad de Almeria, Departmento de Fisica Aplicada, Ctra. Sacramento s/n, 04120-Almeria, Espana
2 CIEMAT, Departamento de Energia, Madrid, Espana
3 EPS-Universidad de Huelva, Departamento de Ingenieria Electrica y Termica, Huelva, Espana

Corresponding Author, jlbosch@ual. es


The correlation of solar irradiation data in flat and homogeneous areas is relatively high and classic interpolation methods are very suitable for its estimation. However, in complex topography zones, a simple data interpolation is not adequate. On the other hand, spatial variability of solar irradiation is also affected by site latitude and cloud cover distribution. In this work, a methodology has been implemented consisting in daily solar irradiation estimation for all sky conditions, by means of Meteosat satellite images and additional information from a Digital Terrain Model (DTM) of the studied area. Solar irradiation is calculated following the HELIOSAT-2 methodology; and a method is presented to obtain the horizon of the studied points using the DTM. The effect of the snow covers is also studied. Model performance has been evaluated against data measured in 14 radiometric stations located in a mountainous area, offering good results, with a Root Mean Square Error (RMSE) around 11%. Finally, a daily solar irradiation map has been generated for the complex topography site.

Keywords: Daily Irradiation Mapping, DTM, Meteosat, Complex Terrain

1. Introduction

Incoming solar radiation, through its influence over the energy and water balances of the earth surface, affects processes like air and soil heating, photosynthesis, wind or snow thawing. Therefore, its knowledge is important in diverse fields and necessary for several applications. For most of these applications, global radiation measures are needed over wide regions, for long time periods and with a high spatial resolution.

At local scales the topography is the most important factor in the distribution of the solar radiation on the surface. In plane and homogeneous areas, classic interpolation methods can estimate the solar radiation accurately. However, in zones with a high topographical variability the spatial correlation is difficult to detect, and for distances between 300 and 1000 m is very small or disappear [1]. The use of interpolation in this kind of terrains can lead to large errors and more complex models that include topographical information are needed [2]. In recent years, digital models of terrain (DMT) have been utilized to develop radiation models incorporating topography and consequently, the spatial variability
of terrain mentioned before. In addition, spatial variability of solar radiation is affected by latitude and cloud cover distribution. This issue has been studied recently with the aid of satellite images. The geostationary satellites (METEOSAT) permanently occupy the same zone over the earth surface, acquiring several images per day, for this reason they are suitable for estimating the solar irradiation and evaluating the energy potential of a wide area. The Centre Energetique et Precedes (CEP), the School of Mines of Paris, in cooperation with other European Centers of Investigation, developed a statistical model to estimate the solar irradiation in the terrestrial surface from images of METEOSAT. The mentioned model is known as HELIOSAT [3]. The basic idea of the HELIOSAT is the interrelation between the cloud cover and the global incident irradiation on a point on the earth’s surface. This model was one of earliest used for the evaluation of the global irradiation from images of satellite. It was developed using measurements of French stations and its goal was the estimation of average monthly values of global irradiation. Afterwards different modifications were introduced [4] until a new version named HELIOSAT-2 was implemented.

In this work, a methodology is presented and tested, in which the estimation of solar irradiation is performed by using a modified HELIOSAT-2 [5], together with the information contained in a DTM. Instead of a single irradiation value for a pixel, the horizon of each inner point is used to estimate around 1000 irradiation values for every pixel. Computational cost of the horizon calculation can be a problem when dealing with large areas; that issue has been addressed by developing an algorithm that reduces drastically the time utilized in this process, without loosing much information about the actual horizon. Additionally, the happening of snow covers can lead to a subestimation of the model, because those pixels can be considered covered by very bright clouds instead of snow, this problem has been also addressed in this work with satisfactory results.

The main goal is to perform an irradiation map of daily values from the satellite images, fitting the spatial resolution of pixels (~ 3.5 km) to the resolution of the DTM (100 m). Ground measurements registered at 14 stations located in a complex topography area have been used for validation purposes, observing an error reduction after the consideration of the horizon and snow effects. It is also interesting to note that this procedure can be applied under all kind of sky conditions.

User guidance

1.1 User survey

One of main objectives of the MESoR project is the involvement of key stakeholders throughout the project. Their first task was the participation in a user survey which was conducted as an interview via telephone calls or e-mail. The survey performed a comparative analysis of various solar radiation platforms concerning technical aspects but also addressing usability, integration and pre-commercial information. This analysis based on the evaluations expressed by actual “top-users” and “top- customers” of the platforms. The sample has been composed by current users of the services belonging to various academic, scientific, industrial and business categories, from public and private sectors. The organisations are active in the fields of architecture/building, PV and other solar applications. The initial sample of 53 was selected by each partner according to the criteria of importance, frequency of usage and attitude to scientific cooperation.

The collected answers indicate a very high degree of awareness about the analysed issues. This is witnessed by the fact that most of the respondents use multiple services, they have a deep knowledge of each service, are able to compare the various services and to highlight the related points of strength and weakness.

The survey detected a gap between expectations and the satisfaction as for quality and accuracy of some parameters, reliability of data measurement and calculation, comparability of data across the services and personalisation of services.

The users expect a truly new and integrated service that offers standardised data and protocols.

Specifications of the solar radiation databases and underlying methods

1.1. General specifications

Spatially-distributed (map) solar radiation databases are classified according to several factors:

• Input data from which they have been created: (a) observations from the meteorological stations (global, diffuse and direct irradiances, and other relevant climate data), (b) digital satellite images or (c) combination of both; here also ancillary atmospheric data used in the models are considered, such as water vapour, ozone and aerosols;

• Period of time (typically a number of years) which is represented by the input data;

• Spatial resolution, i. e geographical distribution of the meteorological sites, grid resolution of the satellite data and resulting outputs;

• Time resolution, which characterises periodicity of the measurement of the input data and of the resulting parameters. Thus a primary database may include time series with periodicity of a few minutes up to hourly and daily averages (sums), or it may contain only monthly and long-term averages.

• Methodical approach used for computation of the primary database: typically solar radiation models combined with interpolation methods (e. g. geostatistical methods or splines, in case that ground observations are used) or algorithms for satellite data processing (e. g. Heliosat). Primary database typically consists of global, direct normal or diffuse irradiances (irradiation in case of time-integrated products) and also some auxiliary parameters such as clear-sky index.

• Simulation models used for calculation of derived parameters, such as global irradiance for inclined and sun-tracking surface, spectral products (e. g., illuminances, UV and PV-related irradiances), estimation of terrain effects, derived statistical products (e. g. synthetic time series).

Quality of an individual data set is assessed for a set of locations by comparing them to ground measurements, where the first order statistics is calculated (bias, root mean square deviation, standard deviation, the correlation coefficient) and the frequency distribution is analysed. In this work we focus on the relative map-based cross comparison of several solar radiation products.

Such comparison provides means for improved understanding of regional distribution of the uncertainty by combining all existing resources (calculating the average of all) and quantifying their mutual agreement by the means of standard deviation.

Meteonorm program

Meteorological data can be generated with Meteonorm using a database with long term monthly average measurement data from different stations. In the recent software versions there are more than 7000 meteorological stations worldwide available. If no meteorological station is available in the database for a desired site, meteorological data will be interpolated based on the data of the nearest stations. The accuracy of the generated data depends on the accuracy of measurements of used stations in the database, the distance to the next stations and the interpolation method.

Подпись: Fig.1. Long-term monthly average ambient temperatures from Meteonorm 6.0 for Bishkek city (predefined in Meteonorm), weather station Frunze and Naryn available in the program database.

A density of weather stations in the program database is relatively low for Central Asia compared to Europe. In total, about 100 stations are available in the program database (5.1 and 6.0) for Central Asia with data for ambient temperature, humidity, velocity and wind direction, precipitation and only 3 stations with solar radiation data. The mentioned station Frunze in Bishkek is available in the program database, but without solar radiation data. The nearest database station to Bishkek with solar radiation data is about 600 km away in Tashkent, Uzbekistan. Unlike previous versions, Meteonorm 6.0 uses satellite-derived solar radiation data additionally to the weather stations for interpolation of solar radiation.

Bishkek city is already defined in Meteonorm 5.1 and 6.0 but with a wrong altitude of 2111 m instead of 760 m. This leads to significantly lower ambient temperatures compared to the measure

values at the weather station Frunze (Fig 1). Because of the wrong altitude, ambient temperature for Bishkek was obviously interpolated by Meteonorm using data from a station Naryn (about 300 km away) with a similar altitude of 2041 m. Thus, it is necessary for further investigations to define Bishkek in Meteonorm manually as a new site with the correct coordinates.


The extraterrestrial solar radiation on the horizontal surface is the function of Latitude of the location. As the solar radiation passes through the earth’s atmosphere, it is further modified by the processes of scattering and absorption due to the presence of the cloud and atmospheric particles. Hence, the solar radiation incident on the horizontal surface is to some extent locality-dependent and less than the extraterrestrial irradiation.

Several climatic parameters have been used to develop empirical relations to predict the solar radiation incident on the horizontal surface. Among these existing correlations, the Angstrom-Prescott type regression equation related monthly average daily radiation to clear-day radiation at the location in question and average fraction of possible sunshine hours is considered to widely accept by the scholars [9]:

h . S

= a+b (1)

H о ^

The Equation (1) has been proved to accurately predict the global solar radiation.

H : the monthly average daily global radiation on the horizontal surface(MJm-2 day-1)

H0: the monthly average daily extraterrestrial radiation on the horizontal surface (MJm-2day-1)

S : the monthly average daily number of hours bright sunshine

S0 : the monthly average daily maximum number of hours bright sunshine a, b : the regression constants to be determined

The extraterrestrial solar radiation [9] on a horizontal surface is determined by following equation.

24*3600 . 0 ■ п.

H0 = 10 f (cos л cos о sin a>s + a>s sin Asm0) (2)

n 180

Where I0 =1367 Wm 2 is the solar constant, Л is the latitude of the location, 0 is the declination angle.


and f = 1 + 0.033cos(360 ) (3)


Where n is the Julian number (for example Jan 1st is the number 1)




And a>s is the sunset hours angle given as

cos = cos 1 (- tan Л tan 0) The maximum possible sunshine duration

S 2

s0 =

0 15 s

H>, So in the equation are obtained from the equation(2) and (6).The regression coefficient a

Подпись:H S

and. The values of the monthly average daily H 0 S0

global radiation and the average number of sunshine duration are obtained from daily measurements for a period of ten years.

The regression coefficient (a, b) can be computed by the following formulas:







1 H bA S

a = z — z

n і=1 H 0 n і=1 S0

Where n is the number of measured data points.

Wavelet implementation of the MOS

The NWP uncertainties of the rainfall forecast is based on non-stationary, nonlinear and dynamic effects as stated by Todini [18]. As the solar radiation forecast is also a function of the cloud cover, it is probably subjected to the same underlying effects. For non-stationary signals the short-time Fourier transform, also named as Fast Fourier Transform (FFT) has the disadvantage that the information concerning the frequency content at a specific time interval can only be obtained with limited uncertainty. By the Heisenberg uncertainty theorem the method increases its uncertainty for the frequency, if the width of analyzing time window is small, and in the time location of a

particular shape if the windows width is large [19]. A high resolution in time and frequency is obtained by the wavelet convolution, also referred as mathematical microscope [20], where the analyzing time window with is variable in a single transformation. With digital computers the Discrete Wavelet Transform (DWT), has the advantage to reconstruct the decomposed signal with lower uncertainties than the Continuous Wavelet Transformation (CWT) [19]. Also the amount of convolutions is reduced with the DWT which increase the transform speed. This transform is based on the members of a family of functions [20]. One has to begin with the selection of the family of wavelets, e. g. the bi-orthogonal wavelet family, and one of the mother wavelets within the selected family. While the orthogonal DWT uses the inverse filters for the reconstruction of the signal, the bi-orthogonal transform introduced by Cohen [21] permits the utilization of distinct filters for the decomposition of the signal and its reconstruction (Souza [22] citing [21]) in order to obtain symmetric wavelet functions. The mother wavelet function determines the order and specifies the time window or support length of the convolution at the first time scale (m = 1). Also each mother wavelet has its own function shape and degrees of freedom [19]. A TDW transform is accomplished at different time scales (m = 1 … mx), using different functions, named by the members of a family, which are all specifically related to the mother wavelet function. If at a specific time location the signal shape is similar to the wavelet shape, one obtains high wavelet convolution coefficients. At each of the m time scales the signal is convoluted by the DWT with distinct wavelet functions. The daughter wavelet functions ym, n(t) (eqn.1) are equal to the expanded and translated mother wavelet functions у [19].

ym, n(t) = 2-m/2 y( 2-m t — n) ; m, n є Z; t є ^ (1)

Where m defines the scaling or expansion of the mother wavelet and n defines the translation of у, relatively to the time t of the time series values from the signal to be analyzed. Due to the expansion, the convolution support lengths are increased by the factor two from scale m to m+1. For the DWT, the wavelet convolutions are obtained by a filter bank of Finite Impulse Response (FIR) digital filters [19] (Figure 1a). The filter bank separate by low and high pass filters the signal to be analyzed in signals with distinct frequency bands. The mother wavelet (Figure 1a — first bk filter) represents the FIR high pass which separates the highest frequencies appearing within the bandwidth of (SL -1 … ГО). SL is the support length of the mother wavelet. The low pass filters ck, also named as scaling function, represent on its output the signal with the complementary low frequency band until to zero frequency. At m = 1, e. g. the complementary frequency bandwidth is (0 … SL-1) and for m = 2 its frequency content decrease to (0 .. .(2SL)’1). The frequency band of the high pass filter at this scale is ((2 SL)-1… SL-1) and from scale m to (m+1) its band width is reduced by the factor two. Where in the Fourier transform the frequency bins are hold constant, in DWT the energy is hold constant to obtain nearly complete reconstruction of the original time series signal. The signal details and approximations at distinct time scales or filter bands are obtained by the bk and ck filters (Figure 1a). The last scaling function is also known as father scaling function [20].

The downsampling function (2f) after each filter reduces the vector length by two, avoiding a redundant representation of the decomposed signal and due to the upsampling (2t) the signal is reconstructed to its original vector length. The decomposed signal can be represented by the wavelet and scaling coefficient vectors T(m, n) and S(m, n) (Figure 1), or by equal length partially reconstructed sub-signals. If during the reconstruction of the original signal, utilizing the inverse filter bank (Figure 1 b), only one of these vectors is supplied to its input, the signal which corresponds to the supplied vector, is reconstructed to the length of the original time series. This wavelet transform is also referred as Non Decimated Wavelet Transform (NDWT), or a trous WT and its partially reconstructed signal vectors are here named as sub-signals. Beside the



reconstruction based on the wavelet and scaling coefficients (Figure 1 b), with the NDWT one can reconstruct the original signal by the sum of the complete sub-signal set.

Подпись: S(3,n)Cf(1,k)


Figure 1 — Wavelet digital filter bank for the decomposition of a signal (a) and its reconstruction (b), where ( 2i ) stands for the downsampling process and ( 2t ) stands for the upsampling process