Category Archives: EuroSun2008-13

External explanatory variables for the residual kriging

In this work, we have only considered explanatory variables that can be derived from DEM. This makes the results more general and useful, since these variables are readily available elsewhere. Particularly, in this study we have evaluated the effect of the elevation and shadows cast as external variables in the kriging procedure. To deal with the seasonality of the solar radiation variability, the study is carried out for each month independently. The topographic characteristics were derived based on a 1 km spatial resolution DEM and the IDRISI GIS software [6] was used to carry out the regression analysis and the whole residual kriging procedure.

Regarding the elevation, it is clear that this variable is related to the atmospheric attenuation of the incoming solar radiation at the earth surface. The higher the elevation, the lower the atmospheric layer thickness and, therefore, the lower the atmospheric attenuation. Nevertheless, the importance of this explanatory variable depends on the elevation gradient. In the present study, this gradient is relative small (minimum elevation is 4 m and maximum 1212 m).

The other external explanatory variable that has been explored in this study is related to the shadow cast caused by the topography. The slope and aspect effect on the measured values can not be evaluated in this study since the stations are located on horizontal surfaces. The sky-view factor influence, nevertheless, can be evaluated. Particularly, to assess this effect, a sky-view factor can be used. This sky-view factor can be defined as the ratio between radiation actually received by a planar surface to that received from the entire hemispheric radiating environment (without any obstruction). It represents the shadows cast by topographic features. Sky view factor can be derived from DEM using several available procedures. In the present study the sky-view factor was computed using the GRASS GIS [6] environment in the following way: given a location, the maximum elevation angle of sky obstruction were computed for 36 directions around this location. Then, a continuous curve of angle values was computed using linear interpolation. The area under this continuous curve, normalized to the entire hemisphere is considered the sky-view factor. Note that the value of the sky-view factor varies from 1, when the whole sky is obscured, to 0, when no obstructions are present. In actual cases, sky-view factor seldom are higher than 0.20.

In this study, a modified version of this sky-view factor was used. Particularly, the sky-view factor was computed just using the topographic information in the south direction. That is, the angles were only computed for the half of the sky view, from solar azimuth angles between 90° (eastern ) to -90° (western), instead of for the complete circumference (solar azimuth angles from -180° to 180°). We will call this new sky-view factor semi-sky-view factor hereinafter. As for the complete circumference, the semi-sky-view factor may take values between 0 and 1. Actual values for all the stations used in this study ranged between 0 and 0.195. This new explanatory variable reveled to be much more effective in explaining the solar radiation variability than a complete sky-view factor. The rationale behind this result is that, as it is known, in the northern hemisphere, the amount of radiation received from solar azimuth angles between below -90° and above 90° is scarce. Therefore, the topographic information associated with these angles is not relevant in explaining the solar radiation variability and just increases the noise of the explanatory variable, without adding new information. This new semi-sky-view factor was used as independent variable in the residual kriging procedure.

3. Results and Discussion

Heliosat Method

The Heliosat method is a technique of determining the global radiation at the ground by using data from a geostationary satellite.

In the Heliosat method it is assumed that the intensity of the visible solar irradiance which is scattered back to the satellite from the Earth and the atmosphere, behaves proportional to the atmospheric reflection. Also the isotropy of the atmospheric reflection is suppositional in order to disable the influence of the atmosphere’s heterogeneity on the reflection properties.

Due to the dominating dependence of the reflection on the cloudiness, it is feasible to derive an important quantity characterizing the degree of cloudiness existing within a solid angle from the radiation measured by the satellite. Via this quantity it is possible to deduce the transmission properties of the atmosphere and then determine the global irradiance.

Snow covers influence

Figure 1C shows a typical case of a clear day with the snow covering the Sierra Nevada Mountains. In these particular images, the method Heliosat-2 can lead to subestimations, because the pixels are considered to be covered by clouds instead of by snow. In this work, a preprocessing method is applied to perform a detection of possible snow covers. Consisting in a comparison between the mountain pixels versus the surrounding ones, where a very low albedo in the surrounding if compared with the mountain is considered to belong to a clear day with snow covers. The method offers a success of near 85% in the detection of snow covers and it will be used. Error reduction due to this detection is discussed in the Results section.

1.2. Horizon calculation

image130

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 spent in this process, without loosing much information about the actual horizon. The used DTM file contains around two million values of altitude, and the horizon for a specific point should be calculated by using all that information, but in this work most of the points are skipped and only around 70000 are utilized. The skipping step is defined as a function of the distance to the studied point, and a random contribution is also added. Finally, the horizon is only calculated for those selected points which altitude is higher than the studied point one, reducing the time consumption. Figure 3 shows the random and the selected points in the calculation of the horizon.

A comparison of the calculated horizon following the described algorithm and the real horizon (calculated by using the whole set of DTM points) appears in Figure 4, where it can be noted the little influence of the selection in the final calculation. Finally, for each day, the horizon will be overlaid with the sun path, obtaining the actual sunrise and sunset hours from the solar altitudes that will be introduced to the model.

image131

Fig. 4 . Differences between real and used horizons.

2. Results and discussion

Daily irradiation values were calculated with and without inclusion of local effects; afterwards, the error was calculated in terms of Mean Bias Error (MBE) and Root Mean Square Error (RMSE), as a percentage of the mean measured value. Table 2 shows the obtained errors for the different stations and the three considered methods: A. Heliosat-2 with no horizon and using a mean altitude for the pixel, B. Heliosat-2 considering the calculated horizon and the altitude of the point and C. same as B but including the detection of clear days with snow covers.

Table 2. Obtained errors for the 14 stations and three different methods

A

B

C

Station #

RMSE

(%)

MBE

(%)

RMSE

(%)

MBE

(%)

RMSE

(%)

MBE

(%)

1

15.7

1.8

13.1

-4.4

11.2

-1.5

2

15.6

2.7

12.0

-2.2

10.3

0.6

3

17.5

2.5

12.5

-4.4

10.5

-1.5

4

16.2

5.9

13.6

-2.2

13.2

-0.1

5

16.5

4.3

12.2

-3.1

10.9

-0.3

6

16.2

3.9

12.6

-2.6

11.7

-0.6

7

16.3

3.1

11.9

-3.0

10.5

-0.2

8

15.2

1.5

13.3

-5.6

11.5

-3.0

9

19.5

7.2

12.3

-0.5

11.5

2.4

10

14.9

0.5

13.6

-5.6

12.0

-3.0

11

15.3

3.8

11.6

-2.1

10.9

-0.4

12

14.1

1.6

12.0

-3.8

11.4

-2.2

13

15.9

8.7

11.7

2.1

12.2

3.0

14

15.5

4.1

10.9

-1.1

10.8

0.2

Mean

16.0

3.7

12.4

-2.8

11.3

-0.5

Error reduction is detected for all the stations, and the importance of horizon effects appears to be larger than the effect of the snow covers. In fact, the consideration of snow covers is almost negligible in the case of the stations 13 and 14, as they are located in the plane area, outside the mountains, where snow is not usually present, but it can decrease up to 2% in the upper stations.

In addition, the station 9 is the one with a major improvement in the estimates, this is ought to the fact that is placed in a gully, having the larger horizon obstructions compared with the other stations. The mean RMSE considering the whole set of stations is reduced from 16% to 11%, verifying the good behaviour of the proposed methodology.

Once the performance of the method has been analyzed and in view of the satisfactory results provided in the global radiation estimates for the 14 stations located on a complex topography area, we generated a map with daily radiation values for the zone. Figure 5 shows a map created using this methodology.

image132

Fig. 5 . Mapping of daily solar irradiation over a complex topography area.

3. Conclussions

Results obtained using the proposed method showed a good agreement with the measured ones and it is interesting to note that this procedure can be applied under all kind of sky conditions. A proposed method to calculate the horizon of a point was tested with satisfactory results.

The RMSE was diminished for the whole set of stations with a mean reduction of 4.7%. MBE was also improved, with almost no over or underestimation for most of the stations. It has been observed that the improvement in the estimates is small in plane areas with almost no horizon obstructions, but becomes significant in the locations inside the mountain. The effect of the snow was also studied and it was pointed out that for the stations in the upper area it could lead to a reduction of the RMSE up to 2%, being smaller or nil in the lower stations. Finally, the method was employed over a wide area allowing the generation of an irradiation map on a complex terrain. If no additional information would be considered, the HELIOSAT 2 model would give a fixed value for the whole pixel, taking into account a mean altitude of the pixel and no horizon. On the other hand, using this methodology, the topographic variability is included in the model and a map of irradiation can be made in an easy way for a complex topography site. In future works, this method is intended to be extended to the estimation of different components of the radiation (i. e. Direct Radiation, Photosynthetically Active Radiation PAR); and also to be used together with Artificial Neural Networks.

Acknowledgements

This work has been financed by the Ministerio de Ciencia y Tecnologia of Spain (projects ENE/2004- 0786-C03-01 and ENE2007-67849-C02-02). We would like to thank CIEMAT for the use of the satellite images.

References

[1] Dubayah and Van Katwijk, 1992. The topographic distribution of annual incoming solar radiation in the grand River basin. Geophys Res. Leter, 19, 2231-2234.

[2] Fu and Rich, 2000. A geometric solar radiation model and its applications in the agriculture and forestry. Proceedings of Second International Conference on Geospatial Information in Agriculture and Forestry, Lake Buena Vista.

[3] Diabate, L., Demarcq, H., Michaud-Regas, N. and Wald, L., 1988. Estimating incident solar radiation at the surface from images of the Earth transmitted by geostationary satellites: the Heliosat Project. International Journal of Solar Energy 5, 261-278.

[4] Dribssa, E., Cogliani, E., Lavagno, E. and Petrarca, S., 1999. A modification of the Heliosat method to improve its performance. Solar Energy 65, 369-377.

[5] Zarzalejo, L. F., 2006. Irradiancia solar global horaria a partir de imagenes de satelite. Serie: Coleccion Documentos CIEMAT. Editorial CIEMAT, Madrid (Espana).

[6] ESRA, 2000. The European solar radiation atlas. Vol. 1: Fundamentals and maps. Editado por: Scharmer, K. and Reif, J. Les Presses de l’Ecole des Mines, Paris (France).

Benchmarking measures and rules

Benchmarking of solar radiation products can be done in different ways. If a kind of reference data is available which is assumed to be the “truth”, the modelled data sets can be compared and ranked how well they represent the reference data. But there is not always reference data available: e. g. for solar radiation spatial products (maps). Here benchmarking can assess the uncertainty of mapping products by their cross-comparison.

For site specific time series there are a number of different measures for benchmarking. A first set is based on first order statistics. These are the well known bias, root mean square deviations, standard deviations, their relative values to the average of the data set and the correlation coefficient. They compare how well data pairs at the same point of time compare with each other. They are important if one needs an exact representation of real data, e. g. for evaluations of real operating systems or forecasts of solar radiation parameters.

This exact match is not always important, e. g. for system design studies. Here the similarity of statistical properties as frequency distributions is more important than the exact match of data pairs. The MESoR project therefore suggests a number of parameters based on second order statistics [6].

Besides a common set of measures the selection of valid pairs of data is of importance to achieve comparable results. Valid data pairs should have passed the quality control procedure as described above, measured global horizontal ground data should be above zero (valid measurement, sun above the horizon) and the modelled data should be valid. Average values (e. g. for relative bias or standard deviations) are calculated based on the valid data pairs. If averages from multiple stations are to be calculated, all data pairs should be treated with the same weight. Averages should be calculated from the complete data set and not from the single stations results. This gets relevant if the stations have different numbers valid data pairs.

A benchmarking exercise applying the measures and rules to available data bases within Europe will be done in the second half of 2008.

Benchmarking of angular distributions is yet difficult, as there are a number of different instruments available with very different characteristics in terms of number of sensors, acquired parameters

(irradiance, radiance, luminance), geometry of the measurement directions in the sky hemisphere, spectral sensitivity of the sensors, aperture angle, size and shape and the sensors and the sensors linearity and dynamics. Further, there are only very few measurements available so far.

Solar maps can be benchmarked in two ways, either point based or map based. The point based benchmarking is similar to the time series benchmarking. Data is extracted from the maps and compared to the measurements (“ground truth”). First and second order statistics can be applied. Map based cross comparison of solar radiation 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. A sample evaluation has been done with five spatial data bases: ESRA, PVGIS, Meteonorm, Satel-light and NASE SSE.

image030

Fig. 1: Yearly sum of global horizontal irradiation, average of all five data bases [kWh/m2] (left), uncertainty at 95% confidence interval from the comparison of the five data bases. [%] (right).

A set of benchmarks has been proposed to evaluate solar radiation forecasts. Depending on the application different methods to assess prediction quality are appropriate. For the users of the forecast the verification scheme should be kept as simple as possible and be reduced to a minimum set of accuracy measures. Statistical measures are similar to time series but need to be adapted to forecast specific issues. Forecast should not only be evaluated against the measured data but also against reference models as persistence or autoregressive models to show the advantage of a forecasting system. For forecasting not only the quality of the prediction of a single site is of interest but also the accuracy of prediction of an ensemble of distributed sites e. g. feeding into an electricity grid node.

Solar radiation forecasting with WRF model in the Iberian Peninsula

L. Martin1*, E. Lorenz2, A. Sood2, L. F. Zarzalejo1, K. Suselj2,J. Polo1, A. Navarro1, R.

Marchante3

1 CIEMAT, Department of Energy, Av. Complutense n°22, Madrid, 28040, Spain
2 Oldenburg University, Germany Energy and Semiconductor Research Laboratory Energy Meteorology Group
3 IrSOLaV, Calle Santiago Grisolia (PTM) 2, Tres Cantos, 28045, Madrid, Spain
* Corresponding Author, luis. martin@ciemat. es

Abstract

Although considerable effort has been done to make use of solar energy efficiently from industrial revolution, expecting fossil fuels would run out in the future, only minimal resources have been directed towards forecasting incoming energy at ground level. However, the necessity to have forecasting models which could optimize the integration of solar thermal power and photovoltaic into electric grid within different sources of electric power generation will grow up as they gain recognition as an energetic resource in the near future. In this work an evaluation of ERA-40 reanalysis data from ECMWF and Weather Research and Forecasting (WRF) mesoscale model is done over 40 stations from Spanish National Radiometrical Network which belongs to the Spanish National Weather Service (AEMet). The evaluaton is done in a hourly and daily resolution just for the next time step in the case of WRF evaluation and for daily resolution in the case of ERA-40 data. The results from WRF model are compared with predictions from European Center for Medium Range Forecasting (ECMWF).

Keywords: EuroSun 2008, proceedings, formatting guidelines, styles

1. Introduction

Although considerable effort has been done to make use of solar energy efficiently from industrial revolution, expecting fossil fuels would run out in the future, only minimal resources have been directed towards forecasting incoming energy at ground level [1]. However, the necessity to have forecasting models which could optimize the integration of solar thermal power and photovoltaic into electric grid within different sources of electric power generation will grow up as they gain recognition as an energetic resource in the near future.

Photovoltaic and solar thermoelectric power are main sources of solar energy for electricity generation. Currently the potential market is huge and its development is being supported by agreements in Kyoto protocol and by progressive series of regulations regarding green energy (feed-in tariff) established in several countries like Spain and German [2]. In the case of Spain, current legislation (Royal Decree 436/2004, 12th of March) allows to minimize investment risks to promoters and to contribute opening up great perspectives to solar energy development.

Energy stock market participation is regulated by two basic rules: on the one hand it is necessary to predict the amount of energy which will be produced, up to 72 hours before, and on the other hand deviations of energy produced compared to programmed one are strongly penalized.

In this work a skill evaluation of the Weather Research and Forecasting (WRF) mesoscale model and ERA-40 reanalysis data from ECMWF is done over 40 stations from Spanish National Radiometrical Network which belongs to the Spanish National Weather Service (AEMet). The prediction is done in a hourly and daily resolution just for the next time step. The results from WRF model are compared with predictions from European Center for Medium Range Forecasting (ECMWF).

Parameters measured

The SolData optics table was equipped to measure: global irradiance (3 instruments), ultraviolet B (UVB), illuminance (lux), barometric pressure (hPa), background radiation (counts/10 min), sky luminance (cd/m2) and PAR (einstein/m2). These measurements have been described in other work [5]. The focus of this paper is on further analysis of the global solar irradiance data. Other research groups on board the research vessel contributed to another database, and this database is also accessible (5-minute values) including relative humidity, a parameter which will also receive attention in this paper.

2.3

Подпись: Radiation model (red graph) and data (blue marks) for
image003

Global solar irradiance

Figure 2: Global solar irradiance from Accra, Ghana, October 2006.

Figure 2 shows typical global irradiance data — in this case when the expedition was near the Ghanan city of Accra off the western coast of Africa. We have focused on days which were “clear”, i. e. cloud cover was so insignificant that a clear day global irradiance model could be fitted to the global irradiance data for a given day. A significant number of days during the eight month voyage of Galathea III were so clear, that this procedure was possible.

Equation developed in this study

Chen zhi-jun has quoted four statistical methods [5] to explain the data characteristic which are based on the monthly data of single station or based on the whole year data of single station or based on the monthly data of all stations or based on the whole year data of all stations. And the results show that the correlation of clearness index and sunshine percentage is to some extent spatial-dependence and time-dependence. And he also proves that the statistical method based on the monthly data of all stations is the best.

image055 image056 image057

Followings are the figures of statistic result.

Fig. 1. The correlation of H / H о and S / S0 based on the monthly data of all stations in China

2.1. Comparative results

We can obtain the regression coefficient a and b (which are included in our model in the table 2) from the above Fig.1. To determine the predictive efficient of the model developed in this paper, it is pertinent to compare this model with those believed to be universally applicable (Eq10, 11, 12).And the result of these comparisons will determine the applicability of those models to China, as presented in Table 2.

Table.2. Comparison of error values for the estimated monthly average daily global solar radiation

from different models

Models

model expression

RMSE

MABE

MBE

our model

Jan:H/H0 =0.157+0.680S/S0

6.18 E-02

1.12 E-01

-3.50 E-03

Feb:H/H0=0.120+0.721S/S0

5.43 E-02

9.10 E-02

-3.30 E-03

Mar:H/H0=0.116+0.709S/S0

4.80 E-02

8.35 E-02

-1.55E-09

Apr:H/H0=0.111+0.678S/S0

4.33 E-02

7.75 E-02

-1.30 E-03

May:H/H0=0.132+0.624S/S0

4.70 E-02

8.16 E-02

-5.50 E-03

Jun:H/H0 =0.127+0.621S/S0

4.83 E-02

8.90 E-02

-3.82 E-02

Jul: H/H0 =0.193+0.478S/S0

5.45 E-02

9.35 E-02

-1.40 E-03

Aug:H/H0=0.192+0.487S/S0

5.36 E-02

9.38 E-02

-4.20 E-03

Sep:H/H0 =0.165+0.559S/S0

5.11 E-02

8.30 E-02

-3.80 E-03

Oct:H/H0 =0.102+0.685S/S0

4.52 E-02

7.66 E-02

-1.80 E-03

Nov:H/H0=0.078+0.754S/S0

5.33 E-02

8.77 E-02

-1.50 E-03

Dec:H/H0=0.160+0.656S/S0

7.22 E-02

1.23 E-01

-9.01E-05

Our model (Ave)

H/H0=0.139+0.637S/S0

5.27 E-02

9.10 E-02

-5.40 E-03

Rietveld

H/H0=0.180+0.620S/S0

7.00 E-02

1.33 E-01

2.55 E-00

Ogelman et al

H/H^=0.195+0.675S/S0-0.142(S/S0)2

7.14 E-02

1.46 E-01

2.71 E-00

Page

H/H0=0.230+0.480S/S0

6.55 E-02

1.22 E-01

0.77 E-00

2.2. Discussions

The errors of each model used in the estimation of global solar radiation are tested by calculating the mean bias errors (MBE), the root mean square errors (RMSE) and the mean absolute bias errors (MABE) from Equation (13), (14) and (15). The RMSE shows the discrete degree of the error, and the MABE shows the average status of the relative error, and MBE values obtained from the models are positive in some cases and negative in others, that means a positive MBE means over estimation and a negative MBE means under estimation. And it is observed that the lower the RMSE, the more the accuracy.

As can be seen in Table 2, our model has the best accuracy because the value of RMSE doesn’t exceed 6% which is the minimum in the four models, hence the estimation values of the global radiation can really reflect the real values. The performance of Page model (not exceed 7%)

is slighter better than that of Rietveld and Ogelman et al (they are 7% and 7.14%), the difference of accuracy is very little. With respect to MBE, the positive values of from the models (Rietveld, Ogelman et al, Page) indicate an over estimation, while the Ogelman et al’s model gives the worst over estimation, and our model only has very little under estimation. With respect to MABE, our model also has the minimum of these models and Ogelman et al’s model has the maximum.

By comparison, it is realized that the performance of the Ogelman et al’s model is the worst, and the performances of Rietveld’ (equationlO) and Page’ (equation 12) models are slight poor than that of the model developed in this paper.

The transmissivity of the atmosphere for the global radiation under perfectly clear sky conditions is given as the sum of the regression coefficients, a+b[10] in the figure.2. We can see that the value of the sum gradually becomes small until the July, after that it gradually rises until the November. We know that the probability of raining gradually increases in the first half of the year nearly in most areas of China, lots of clouds after raining often exist in the sky. So the transmissivity is lower than that in the second half of the year because the weather condition will become dry and there are few clouds in the sky. Hence, the result is quite consistent with the change trend of the climate. The average values of a and b are 0.1387 and 0.6367. It is observed that the sum (0.7754) of the values for the coefficient compares well with the value (0.80) reported in literature for the mesothermal forest climate (often dry season in winner) [9]. Because most areas of China are located in the mesothermal forest climate.

From the above comparisons, everything points to the fact that Angstrom-Prescott-one parameter-model is, to a large extent, locality dependent (It is expected that the performance of any general equation will always be poor than that of the model developed for that locality.)

image058

Fig.2. The sum of regression coefficient a and b with the month

3. Conclusion

The present work of this paper is that a linear correlation form of Angstrom-Prescott correlations has been developed for use in estimation global radiation of China. And the model is also compared with other models (Rietveld, Ogelman et al Page) in terms of different parameters. Comparative analysis shows that the predictive accuracy of our model is far better than the rest of

the models considered in this study. Hence the monthly average daily global radiation incident on the horizontal surfaces can be estimated by the correlation developed in this paper (the error doesn’t exceed 6%). Especially for the places where there are no stations for measurements but have similar meteorological conditions in China.

4. Acknowledgment

The author is grateful to the CMA (China Meteorological Administration) for providing the data.

References

[1] T. Muneer, Solar Radiation and Daylight Models for the Energy Efficient Design of Building(Architectural

Press, Oxford,1997)

[2] S. U.UDO, Contribution to the Relationship between Solar Radiation and Sunshine duration in the Tropics:

A Case Study of Experimental Data at Ilorin, Nigeria. Turk J Phys 26(2002)

[3] JIANG Ying-ni, Models for estimating monthly mean daily Diffuse Radiation. [J]

[4] Louis E. AKPABIO, Modeling Global Solar Radiation for a tropical location: onne, Nigeria

Turk J Phys 29(2005)

[5] Chen Zhi-jun, Exporing the monthly clearness index models in china, 2005.10, Journal of Nanjing Instutude of Meteorology

[6] Louis E. AKPABIO, Relationship Between Global Solar Radiation And Sunshine Duration for Onne, Nigeria, [J] Turk J Phys 27(2003)

[7] Zhou Jin, Sunshine-based model for estimation global solar radiation in China, [J], Journal of harbin

institute of technology

[8] Lin Wenxian, Ranking the overall performance of eight sunshine-based global solar radiation models

with a nonparametric statistical procedure,[J] New Energy, 1998. 20(2). 16-19

[9] William A. Beckman, Solar engineering of thermal processes, second edition 39-40, 70-71

[10] K. J.A. Revfeim, An interpretation of the coefficients of the Angstrom equation. Solar Energy. 31,

(1983), 415

Artificial neural network implementation

For each of the four {ss} time series a specific ANN is trained with an improved Back Propagation (BP) algorithm [9]. In the present work an ANN with a simple optional feedback connection was used (Figure 2, left line).The feedback line transforms the used Feed Forward (FF) ANN in a Recurrent Neural Network (RNN). The blank rectangles in figure 2 symbolize the activation functions, those one with z-1 a one day delay, and those one with the unity represent the unity inputs for the bias weight connections. Each layer of the ANN includes dendrite connections with its weights, designed by sloped lines. The dendrite summation point of the neurons is designed by circles and the output activation functions by blank rectangles. A bipolar sigmoidal activation function for the neurons in the hidden layers and a bipolar linear activation function for the output neuron were applied. The input and output signal were normalized to appear in the range (-1…1) utilizing the normalization equations in [9]. By a conventional BP algorithms, the weights [wu] at iteration step u are updated as a function of the matrix [n {5u} {yuT } ] of eqn. (4). Whereby {5u} is the propagated error at the output of an arbitrary layer of the ANN, n is the pre-adjustable learning rate and {yu} is the output of the previous layer.

[ w u+i ] = [ Wu ] + nx [ {5u} (VuT } ] + a [ w u — w u-i ] (4)

As the present ANN has only one neuron in the output layer, {5u} is a variable (5u) and the weights matrixes in eqn. 4 are all vectors. In order to improve the convergence speed online training, rather than batch training is used, where the ANN weights are updated for each daily mean [30].

+

image117

Figure 2 — Circuit of the utilized ANN during its training phase utilizing BP with fully dendrite connections in between the layers (Observation: In present article an additional hidden layers of neurons is used, but for the simplification of the scheme, the ANN is designed with only one hidden layer)

By the BP the error energy E (eqn. 3) is propagated back due to partial differentiations, hence sAsi = sAsu = 5u is obtained at the ANN output layer [30] (Figure 2). If during the training 5u = f(u) a local minimum of 5u is separated from the general minimum by high walls, with high A5u = f (Au) gradients, the algorithm may need too many steps to climb the walls moving out of the local

minimum and it runs the risk of being trapped [30]. Therefore were used as learning rates px, two distinct pre-adjustable values, one p(-A5u) = 0.008 for decreasing 5u residuals and another p(A5u) = 0.013 for increasing 5u. The former is used to minimize the uncertainties by learning and the latter enables the algorithm to climb the walls more quickly by increasing residuals in order to search the global minimum. An adjustable momentum factor a = 0.8, increases additionally the weight actualization (eqn. 4) and thus the learning speed, at locations where the learning process occurs with more success. These locations are identified by the weight modification gradient, of the last two learning steps [wt — wt-1]. For higher gradients the matrix a [w t — w t-1] accomplish higher weight modifications and vice versa. The decrease of the weight actualization avoids that the algorithm jumps over a narrow global minimum and therefore increases the stability of the learning process [30].

Regression analysis

A multiple regression analysis was carried out independently for each month using the elevation and the semi-sky-view factor as independent explanatory variables. A step-wise procedure was used for the regression parameter estimation. Particularly, the semi-sky-view factor was firstly regressed as independent variable and then the elevation was added. A t-test was carried for each step of the regression procedure and only parameters statistically significant at 5% level were further considered.

The most important explanatory variable is the semi-sky-view factor, which is statistically significant for all the months. This explanatory variable is able to explain from a minimum of 13% of the spatial variability (February) to a maximum of 45% in June. On the other hand, the elevation showed to be statically significant just from March to August and associated explained variance is considerable lower than for the semi-sky-view factor: values ranges from a minimum of 9.7% in August to a maximum of 15% in June. When considering both explanatory variables, explained variance ranges from a minimum of 13.2% in February to a maximum value of 46.7% in June, with most part related to the semi-sky-view factor, which is negatively correlated to the monthly solar radiation data.

Standardization

The satellite measures the sunlight in the visible spectral region reflected by the earth. Hence the measurement signal depends on the irradiation hitting the reflecting layers. After subtraction of the radiometer offset Co the remaining Count behaves proportionally to the irradiance I. Under utilization of these proportionalities the relative reflectivity p can be defined as standardized backscattering value:

p := C-Co /1. (1)

In the following only relative differences not absolute values of the reflecting properties are important therefore it is sufficient to divide by the cosine of the sun zenith.

1.1.1 Cloud Index

The standardized backscattering values of clouds usually exceed those of the Earth’s surface, excluding the case of snow. Thus it is possible to identify the occurrence of clouds. If the reflectivity for a completely cloudy pixel pc and the reflectivity of the unclouded ground (and ocean respectively) pg is known, the cloud index n as a degree of cloudiness can be defined as

n = (P — Pg)/(Pc — Pg). (2)

If the maximum and minimum of the standardized backscattering values of a pixel is selected as reference values, the Cloud index takes values in the range of 0<=n<=1.