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Author's personal copy Agricultural and Forest Meteorology 151 (2011) 1812–1822 Contents lists available at ScienceDirect Agricultural and Forest Meteorology journal homepage: www.elsevier.com/locate/agrformet A topographic geostatistical approach for mapping monthly mean values of daily global solar radiation: A case study in southern Spain J.A. Ruiz-Arias ∗ , D. Pozo-Vázquez, F.J. Santos-Alamillos, V. Lara-Fanego, J. Tovar-Pescador Department of Physics, Campus Lagunillas, Building A3, University of Jaén, 23071 Jaén, Spain a r t i c l e i n f o Article history: Received 22 November 2010 Received in revised form 25 July 2011 Accepted 26 July 2011 Keywords: Solar radiation Kriging Terrain effects Climate a b s t r a c t Local topography influences total incoming solar radiation at ground surface in mountainous areas, and so it becomes a key factor for the spatial distribution of plants. However, radiometric stations are often clustered only around farmland or populated areas, usually throughout valleys and flat regions. In this work, we use residual kriging methods to account for cloud- and terrain-related effects, especially when availability of measurements in mountains is scarce. Terrain-related effects have been considered through the terrain elevation and a topographic clear-sky solar radiation model that, additionally, also allow us to consider local clouds effects. Mesoscale-level phenomena were considered through the distance to the coast and the geographical longitude, that partially explain the atmospheric circulation in the studied region. The study has been conducted in the region of Andalusia, in southern Spain, using a target grid support of 1 km of grid-spacing and based on a 10-year length experimental dataset of 63 stations. Two different residual kriging approaches were evaluated and compared against ordinary kriging estimates. Overall, all kriging methods showed good skills in predicting the spatial regionalization of the monthly averages of daily solar radiation. The use of the distance to the coast and the geographical longitude enhanced the performance of residual kriging methods. Elevation proved to be important during summer months, while clear-sky solar radiation estimates were helpful especially during winter months. Overall, the RMSE value for ordinary kriging at the validation sites was about 3%. The residual kriging methods were able to outperform ordinary kriging around a 5% in winter and up to a 18% in summer, in relative terms. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Solar radiation is a major energy supporter of the physical and biological processes in our planet. Its spatial and temporal heterogeneity strongly affects the dynamic of the agricultural (Fu and Rich, 2002; Reuter et al., 2005), ecological (Kumar and Skidmore, 2000; Trivedi et al., 2008) and hydrological (O’Loughlin, 1990; McVicar et al., 2007) systems by influencing air temperature, soil moisture and evapotranspiration, snow cover and many photochemical processes. Hence, solar radiation drives plant productivity and vegetation distribution, being a key factor in agricultural and forestry sciences that must be known accurately. The amount of solar radiation available at the earth’ surface is firstly constrained at global scale, being primarily affected by the Sun–Earth geometry and the atmosphere. However, a detailed description of its space-time variability requires consideration of ∗ Corresponding author. Tel.: +34 953 212 474; fax: +34 953 212 838. E-mail addresses: [email protected] (J.A. Ruiz-Arias), [email protected] (D. Pozo-Vázquez), [email protected] (F.J. Santos-Alamillos), [email protected] (V. Lara-Fanego), [email protected] (J. Tovar-Pescador). 0168-1923/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2011.07.021 local processes which often become also relevant, as is the case in mountainous areas. Particularly, local terrain modifies the incoming solar radiation by shadow-casts, gradient of elevations, surface slope and orientation, or surface albedo (Dubayah et al., 1990; Tovar et al., 1995; Oliphant et al., 2003). Consequently, accurate spatial modelling of incoming solar radiation should consider the influence of the terrain surface. In the last years, several procedures to include the local terrain effects in the solar radiation field have been proposed, such as the use of Geographical Information Systems (GIS) (Wilson and Gallant, 2000; Fu and Rich, 2002; Šùri and Hofierka, 2004), artificial intelligence techniques (Bosch et al., 2008; Siqueira et al., 2010) or post-processing of satellite based methods (Ruiz-Arias et al., 2010; Bosch et al., 2010). Solar radiation can be also assessed using numerical weather prediction (NWP) models. However, the space and time scales resolved with them and the limited computational capabilities often prevent the consideration of terrain-related effects. Alternatively, interpolation techniques allow us to obtain spatially continuous databases from data recorded at isolated stations over wide regions. Although their reliability is strongly dependent on the gap distance between stations, they ultimately rely Author's personal copy J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 on observed data, which have a higher accuracy than other methods. Hence, when a sufficient recording spatial density is available, interpolation techniques are preferred. Traditionally, solar radiation has not been as densely sampled as other variables as temperature or rainfall, thus the availability of measurements is often scarce. However, in the last years, the number of experimental networks which register solar radiation has grown and interpolation has become a well-suited method for solar radiation assessment. Nonetheless, radiometric stations are often clustered around farmland or populated areas, usually throughout valleys and flat regions, whereas mountains still lack sufficient recording density. This fact is especially relevant provided the high spatial variability of solar radiation in these regions. As a consequence, particular interpolation techniques that allow incorporating external sources must be used to explain this additional spatial variability. Many different spatial interpolation methods can be found in the literature such as natural neighbour interpolation (Dinis et al., 2009), inverse functions of distance (Pons and Ninyerola, 2008), multiple linear regression (Daly et al., 1994), splines (Mitášová and Mitáš, 1993; McKenney et al., 2008) or kriging. Particularly, kriging methods (Cressie, 1993) make use of existing knowledge without any outer deterministic assumption by taking into account the way the field varies in space through, generally, the variogram model. The family of kriging methods has been succesfully used to estimate the rainfall (Buytaert et al., 2006), temperature (Benavides et al., 2007; Spadavecchia and Williams, 2009), snow cover (Erxleben et al., 2002) or solar radiation (Ertekin and Evrendilek, 2007; Alsamamra et al., 2009). Among the kriging methods, residual or regression kriging (Hengl, 2007) involves a convenient way to use external explanatory variables to explain the spatial heterogeneity of the target field. Particularly, Alsamamra et al. (2009) showed for southern Spain that the effect of terrain elevation and terrain shading on monthly averages of daily solar radiation can be partially considered using elevation and sky-view fraction as external covariates. However, data availability in mountainous areas is often very limited. Consequently, it is difficult to produce an accurate solar radiation climatology in mountainous regions to be used in ecology, forestry, hydrology or climate change studies (Díaz et al., 2003; Huber et al., 2005). In this work, we address the mapping of monthly and yearly averages of daily global solar radiation values from a 10-year length 1813 experimental dataset of 63 stations using two different residual kriging models. The external covariates incorporated in the procedures allow us to account for the spatial variability caused by the topography and the cloudiness. Particularly, terrain-related effects have been addressed by using the terrain elevation and the estimates of a clear-sky solar radiation model that already accounts for terrain-related effects. Additionally, the clear-sky model also enables us to consider local cloudiness by comparing its point estimates with the actual recorded values at the experimental sites through the clear-sky index (Section 3). Former works as Hutchinson et al. (1984) and, more recently, McKenney et al. (2008), have included the effect of clouds through a transformation of the monthly mean rainfall, traditionally, a much more densely registered variable than solar radiation. However, the clear-sky solar radiation model, besides the cloudiness-related information, provides more comprehensive information on the topographic influence. Additionally, cloud-related effects at mesoscale-level have been also gathered from the distance to the coast and the geographical longitude, which partially explain the atmospheric circulation of cloud fronts in the studied region. The proposed residual kriging methods have been evaluated against a simple interpolation using ordinary kriging, which has been intended as a skill measure of the expected residual kriging improvements. The work is organized as follows: Section 2 describes the study region and the experimental dataset. Section 3 presents the methodology and Section 4 shows the results. Finally, main conclusions are presented in Section 5. 2. Study region and dataset 2.1. Study region The study has been carried out in the region of Andalusia, in the southern part of the Iberian Peninsula (Fig. 1). The region comprises the southern-most part of the European continent, at the western end of the Mediterranean basin. It covers an area of about 87,000 km2 with 917 km of coastlines. It is located in the transition zone from temperate to warm climates. The average annual temperature is 17 ◦ C. Its climate is of Mediterranean variety, with temperate-warm weather, short mild winters and hot summers. Rain falls mainly from October to March with little rain at other Fig. 1. Study region and stations location. Author's personal copy 1814 J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 Fig. 2. Box-plot for the whole dataset along the region for each month and the annual average, using both the training and validation datasets together. The boxes extend from the lower to upper quartile values of the data, with a line at the median. The whiskers extend from the box to show the inner quartile range. times. It is mainly driven by the Azores High which, particularly, is responsible for an extremely dry summer season. From the topographic point of view, the region has two different parts: the flat western one and the rugged eastern. The former covers the Guadalquivir river basin, with an area of about 30,000 km2 and 100 m of mean elevation. The eastern part houses several mountain ridges, reaching 3482 m in the Mulhacen Peak, in the National Park of Sierra Nevada, the highest summit in the Iberian Peninsula. The combination of mesoscale circulation phenomena and the high topographic heterogeneity gives rise to the existence of several particular climates throughout the region, from pure mediterranean or oceanic climates near the coasts to continental inland or, even in some areas, high mountain climate. As an example of this climatic heterogeneity, the wettest point (Sierra de Grazalema, 2000 mm/year) and the dryest (Cabo de Gata, 250 mm/year) are separated by only 300 km. 2.2. Experimental dataset The Andalusian Regional Office of Agriculture and Fishing records the daily global solar radiation on horizontal surface by means of two different networks: the Alert and Phytosanitary Information Network, with 86 stations, and the Agroclimatic Network, with 102 stations. Data are measured with Kipp & Zonen CM-5 pyranometers. An instrumental error of 5% is expected in the best case. However, although the stations are regularly maintained and calibrated, typically, it is more realistic to assume a 7%. The networks were deployed to provide information of, mainly, agroclimatical interest. Therefore, the radiometric stations are clustered around the principal agricultural areas of the region. The range of elevations above mean sea level goes from 4 to 1212 m, with a mean elevation of 345 m. 2.2.1. Quality control Kriging performance is affected by quality of the experimental dataset. In principle, even a single bad point may affect the prediction over the whole area and lead to unexpected results (Hengl, 2007). Therefore, a quality control check that filters out suspicious records is a required first step. Reliable and continuous solar radiation measurements have been made available throughout wide networks only recently because of the high operational maintenance cost of these sensors compared to the traditional temperature and rain gauges, for instance. In this work, only stations with less than 30% of data-gaps in the period from October 1999 to October 2009 were selected. Thus, the dataset was reduced from the initially available 188 stations to only 97. Following, the data underwent a visual inspection and 14 of the stations were ruled out since they presented suspicious inter-annual trends, likely caused by a deficient maintenance that caused incoherencies. Next, a quality assessment based upon physical limits was applied: an upper limit of 0.8 for the atmospheric clearness, calculated as the daily measured solar radiation to the daily potential extraterrestrial, and a lower limit of 0.01 MJ/m2 (Iqbal, 1983). Overall, excluded values did not reach 2%. Finally, remaining data gaps were replaced with the mean value of the same day along the data period if seven or more days were available. At this point, all the months for all the stations had more than 26 valid days and the monthly averages were obtained based on these records. Eighty-three stations passed the quality control procedure. However, they tended to be clustered around flat areas which, from the methodological point of view, may lead to wrong results by overweighting these areas. Therefore, 20 of these stations were carefully extracted trying to homogenize the spatial distribution of the stations, and reserved for independent validation of the results. Fig. 1 shows the distribution of the training and validation datasets along the study region. Fig. 2 shows a box-plot of the monthly averages for the whole region using the training and validation datasets together. The boxes extend from the lower to upper quartile values of the data, with a line at the median. The whiskers extend away from the box boundaries to show the inter-quartile range. Red markers are the values beyond the whiskers’ limits. They represent extreme values far away from the upper and lower quartiles. Note the annual pattern of the daily solar radiation along the months. It varies from around 7 MJ/m2 in December to almost 30 MJ/m2 in July. The increase of the spatial variability, presumably due to the cloudiness, is appreciated in the spring and autumn months by the presence of station outliers. Particularly, March, April and November present unusually low extreme values (data outliers) indicating the presence of unusually cloudy days. Contrarily, October and December have unusually high extreme values that show a predominance of overcast and cloudy days throughout the month. Additionally, it can be seen that May is the month with Author's personal copy J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 the highest relative variability since it has both low and high data outliers. 3. Methodology Ordinary kriging is the most common type of kriging. It assumes the process can be represented by a stationary random function with unknown mean. The estimates of a random variable Z at a point x0 are given as a weighted average of the sample data with weights determined by minimising the estimation variance. They are given in terms of the spatial semivariance of Z, which is estimated by the semivariogram model. It sketches up how the data vary spatially throughout the area of interest as a function of the separation distance between points. For a detailed revision of kriging methods see Cressie (1993). Particularly, following Alsamamra et al. (2009), the spatial variation was assumed isotropic and the sample variograms fitted using a theoretical exponential model. When the semivariogram modelling is intended for kriging it is generally interesting to give more weight to those data at shorter lag distances (Webster and Oliver, 2001). Particularly, we fitted the exponential curves by means of weigthed least-squares using the weighting scheme proposed by McBratney and Webster (1983). It enhances data at shorter distances and with a higher number of data pairs. While ordinary kriging (hereafter also referred as OK) assumes a constant mean for the random variable, natural processes are usually governed by deterministic trends. In these cases, nonstationary techniques as residual kriging should be used. Residual kriging separates the deterministic trend from the stochastic signal (residuals) by using multiple linear regression with the help of external covariates that must be available at every point of the grid support. Ideally, the residuals shall have an improved stationarity and normality with respect to the initial random field and be suitable for interpolation with ordinary kriging (see Hengl et al., 2003, for further details). This technique, although relatively simple, allows us to easily include multiple sources of external information that compensate for the lack of local data. The main shortcomings of the residual kriging are with the regression model. Ideally, it requires an even spatial distribution of the samples. Besides, the values of the external covariates at the sampling points should also span their whole range of variation throughout the grid support. Consequently, as it was commented in Section 2.2.1, the 20 withheld stations for independent validation were chosen to homogenize the spatial distribution of the remaining sample points. Additionally, both geographical longitude and distance to the coast cover the whole range of variation of the grid support, as can be seen in Fig. 1. Nevertheless, due to the lack of experimental stations in mountainous areas, both elevation and topographic shading at the experimental sites do not range the whole spectrum of values that take place throughout the target grid. However, the terrain-related phenomena that induce spatial gradients in the solar radiation at ground level and the way they interact with solar radiation are well-known (Dubayah et al., 1990; Corripio, 2003; Ruiz-Arias et al., 2010). In the last years, they have been profusely investigated in GIS frameworks by means of topographic solar radiation models, devised usually for climatic clear-sky conditions. See Ruiz-Arias et al. (2009) for a revision of some of them. Hence, the variability caused by topography can be partially considered using the estimates of a topographic clear-sky solar radiation model. Particularly, we used these estimates as a covariate in the preliminary multiple linear regression procedure for residual kriging prediction (hereafter, RKv1). Moreover, clear sky solar radiation models have been extensively used in applications that require distinguishing among different sky conditions (Beyer et al., 1996; Marty and Philipona, 1815 2000; Rigollier et al., 2004; Müller et al., 2004; Šùri and Hofierka, 2004; Lorenz et al., 2009). In this case, they are used as an intermediate parameter to calculate the clear-sky index, k, which is defined as the ratio of the measured global irradiance, IG , to that received under clear-sky conditions, IGcs , usually estimated by means of a clear-sky solar radiation model. The clear-sky index lacks strong climatic and topographic trends, which have been mostly removed based on the physically-based clear-sky solar radiation model, and it is mostly made of the quasi-random signal due to cloudiness. This makes clear-sky index a good candidate for geostatistical methods. Particularly, we interpolated the clear-sky index calculated at the experimental station locations using residual kriging (hereafter, RKv2). The predicted solar irradiance values can be retrieved later as: ÎG = k̂IGcs , (1) where ÎG and k̂ are the predicted solar irradiance and clear-sky index, respectively. 3.1. Clear-sky index assessment Monthly averaged daily clear-sky solar radiation has been calculated using the module r.sun of the GRASS GIS platform (Šùri and Hofierka, 2004; GRASS, 2009). It computes the solar radiation using the model of the European Solar Radiation Atlas (ESRA; Rigollier et al., 2000), which is based on the turbidity coefficient of Linke. This parameter accounts for the climatic aerosols and atmospheric water vapor. In addition, r.sun includes routines to account for topographic shading and terrain surface orientation. The terrain has been represented with the Shuttle Radar Topographic Mission v3 digital elevation model (DEM) (Jarvis et al., 2006) upscaled to 30 arc seconds (approximately 1 km at the study region latitudes). The grid of this DEM has been used as support for the subsequent kriging interpolations. In order to assess the monthly averaged daily clear-sky solar radiation, the monthly turbidity coefficient of Linke was firstly retrieved for each month from the Solar Database (SODA) service (Gschwind et al., 2005) for 200 locations evenly spread over the study region. As the turbidity parameter depends on the site altitude, its values were initially retrieved at the site altitude estimated by the SODA service at every sampled site. Then, the turbidity values were move down to sea level following Remund et al. (2003) and interpolated to the target grid support using ordinary kriging. Finally, the interpolated values were moved back, now to the altitudes estimated by the target DEM. These turbidity maps were then used with r.sun to simulate the 15th day of every month that was used as an estimate of the monthly averaged daily clear sky solar radiation in the study region. Finally, the values at the experimental sites were extracted and used to calculate the clear-sky index for every month and experimental site. The annual period was calculated by previously averaging the monthly values at every site. 3.2. External explanatory variables and regression analyses Unlike ordinary kriging, where kriging applies directly to solar radiation, residual kriging requires a previous regression analysis to remove spatial trends and so derive the residuals to be kriged. Local terrain effects could be partially considered through the skyview fraction and elevation (Alsamamra et al., 2009). However, as the range of variation of the sky-view fraction in our experimental dataset is very reduced, it is not an appropriate covariate to account for the topography. Instead, terrain influence has been considered by means of the terrain elevation (for both RKv1 and RKv2) and the clear-sky solar radiation estimates obtained with Author's personal copy 1816 J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 r.sun (only for RKv1 because the clear-sky solar radiation is, by definition, embedded in the clear-sky index, the interpolated variable in RKv2). In addition, solar radiation field is also influenced by mesoscalelevel phenomena as cloud fronts or aerosol events. In order to consider their effects, the distance to the coastline (also named continentality) and the easting coordinate (increasing west to east) have been included as proxy explanatory variables in the regression analysis (for both RKv1 and RKv2). The rationale is that both cloudiness and aerosol events depend to some extent on the geographical location and the local topography and they are straightforwardly available everywhere from the DEM. Particularly, continentality aims to account for the differential effect of Saharan dust intrusion events, usually coming from the south, and the clouds associated to the frontal systems from the southwestern part of the region. The easting coordinate aims to account for the topographic forcing of terrain on cloud fronts. The rationale is that the study region is a flat and open area to the Atlantic Ocean in the west whereas the east contains important mountain ridges with altitudes over 3000 m above sea level. For both RKv1 and RKv2, and every month, different linear models were constructed by successively adding covariates. The Akaike’s Information Criterion (AIC) (Burnham and Anderson, 2004) was used to select the best model (with this criterion, the one with the lowest AIC value) among all the candidates for every month. It is based on information theory and represents a compromise between the goodness of the fit and the number of input parameters. 3.3. Validation procedure Validation has been accomplished by a leave-one-out crossvalidation with the training dataset and an independent validation with the 20 withheld stations (Section 2.2.1). Cross-validation is a commonly applied method in geostatistics (Cressie, 1993; Webster and Oliver, 2001) consisting of, repeatedly, removing each datum from the training dataset, then predicting at that site with the rest of the data and using the predicted values to estimate the error. The following statistical scores have been used: - mean bias error, 1 z(xi ) − ẑ(xi ) , N N MBE = (2) i=1 the process leads to errors, ignoring sign. RMSE, like MAE, ignores the sign but put higher emphasis on the error in outliers. Both are desired to be small. Kriging is partly insensitive to inaccuracies in the variogram. However, if the model for the variogram was accurate then NRMSE should be the unity. Then, it can be used as a measure of the variogram model performance. 4. Results Ordinary kriging and residual kriging methods have been applied to the monthly averaged daily solar radiation data recorded in the 63 stations that passed the quality control procedure. Thirteen maps with 1 km spatial resolution were assessed (twelve monthly maps and one for the annual period) with each methodology. In this section, we firstly show the preliminary steps that lead to the semivariograms and map derivation. Following, results of the validation are presented. Table 1 shows the explanatory variables used in the regression analyses and correlation results for RKv1. Overall, higher correlations occur during winter and summer months, being the highest in January (58% explained variance) and the lowest in September (14%). Among the predictors, continentality is the most significant, indicating a solar radiation decrease inland during winter months (note the negative sign) and an increase during summer months. This differential influence of the continentality can be explained based on the prevailing atmospheric circulation over the study region. Particularly, the atmospheric circulation over most of the Iberian Peninsula, but especially in the southern part, is driven by the semi-permanent subtropical high pressure center over the Azores islands, the southern center of action of the North Atlantic Oscillation (Pozo-Vázquez et al., 2001; Castro-Díez et al., 2002). The position and intensity of this semi-permanent center of pressure changes throughout the year. During winter the high is located in a lower latitude, allowing the Iberian Peninsula be affected by zonal circulations from the west. As a consequence, warm and humid maritime air masses enter the study region from the south-west. Clouds are generated by orographic forcing when these air masses encounter relevant topographic features, explaining why the solar radiation decreases inland in these months. On the other hand, in summer, the Azores High undergoes a northward displacement, blocking the western circulation over the study area. As a consequence, during this season, clear sky conditions predominates and only local circulations, thermally driven and of a low magnitude, are observed over the study region. Therefore, local conditions may - mean absolute error, 1 MAE = z(xi ) − ẑ(xi ) , N N (3) i=1 - root mean square error, RMSE = 2 1 z(xi ) − ẑ(xi ) N N 1/2 , (4) i=1 - and normalized mean square error NRMSE = 1 z(xi ) − ẑ(xi ) N (x ˆ i) N 2 1/2 . (5) i=1 where z(xi ) and ẑ(xi ) are the observed and estimated values at xi , respectively, and ˆ 2 (xi ) is the kriging variance (Webster and Oliver, 2001). MBE is a measure of the systematic error and should ideally be 0 because kriging is unbiased. MAE indicates the extent to which Table 1 Correlation results of the multiple linear regression analyses for the RKv1 method. The Akaike’s Information Criterion has been used to select the best set of predictors for each period. The columns show the fraction of linearly explained variance by each individual explanatory variable and all the selected variables together, in percentage. Coast refers to continentality and csrad refers to monthly averaged daily solar radiation calculated with the clear-sky solar radiation model. The statistically significant variables at 10% confidence level are in bold. The sign in parentheses indicates whether the correlation is positive (+) or negative (−). Month Coast Easting Csrad January February March April May June July August September October November (−)37% (−)33% (−)29% (−)17% (−)18% (+)8% (+)39% (+)20% Ann (+)8% (+)9% (−)11% (−)26% (−)18% (−)9% (−)20% (−)21% (−)9% Elev (+)1% (+)1% (+)1% (+)1% (+)1% (+)1% (+)13% (+)20% (+)8% (+)1% All 58% 37% 35% 17% 31% 15% 44% 43% 14% 17% 29% 13% Author's personal copy J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 Table 2 As in Table 1 but for the RKv2 method. Month Coast Easting January February March April May June July August September October November December (−)28% (−)22% (−)21% (−)11% (−)13% (+)1% (+)16% (+)22% (+)9% (+)5% (−)12% Elev (−)8% (−)21% (−)47% (−)12% (−)12% (+)1% (+)1% (+)4% (−)1% (−)5% (−)5% (−)7% (−)14% Ann All 38% 22% 23% 26% 50% 17% 34% 28% 20% 7% 12% 18% 14% explain the solar radiation spatial variability in the study region. Moreover, in this season, the relative humidity and aerosol load (principally, marine aerosols) are considerably higher in coastal areas. In addition, atmospheric total optical mass inland is usually lower. This may explain the increment of the solar radiation inland during summer months. As far as the easting variable is concerned, it presents an opposite trend compared with continentality which, is more important in summer months. The positive trend in January and December can be explained based on the same argument used for the case of the continentality. Particularly, note that the topographic complexity, and thus the orographic forcing effect on the air masses in the region, increases from west through east. On the other hand, the negative trend during the summer months is likely related with convective cloudiness and Saharan dust intrusions that are often trapped for many days in the mountains of the eastern part of the region (Lyamani et al., 2005). Clear-sky solar radiation is significant and positively correlated in winter months since the effect of the shading by relief is higher in this period. Finally, elevation is, in practice, uncorrelated. However, note that, regarding to AIC, its inclusion is helpful in some cases. Particularly, in June, easting explains 9% of variance and elevation explains only 1%. However, they together explain 15%, more than would be expected. Overall, according to AIC, elevation must be used as explanatory variable from June to September. This makes sense because in this period the solar height is higher so that the influence of the gradient of elevations on the total optical path is larger. Table 2 shows the explanatory variables used in the regression analyses and correlation results for RKv2. Note that in this case the target variable is monthly averaged daily clear-sky index. Overall, 1817 the fraction of explained variance is slightly lower than for RKv1. The highest correlations are found in May (50%) and January (38%) and the lowest in October with only 7%. Continentality and easting show similar trends as in Table 1, although with lower correlations. Ideally, if the clear-sky solar radiation model was accurate, elevation should be uncorrelated. However, elevation has a negative trend with relatively high correlations, meaning that change of solar radiation with elevation in r.sun is too strong for this region. This is coherent with the results announced by Ruiz-Arias et al. (2010) for the parameterization of the vertical profile of solar radiation using the ESRA model. Nonetheless, this negative trend is accounted for by the regression analysis. 4.1. Semivariograms In order to derive the maps using OK, RKv1 and RKv2 methods, a total of 39 semivariograms (36 and 3 for the monthly and annual periods, respectively) were needed. For all the cases, the exponential model was well-suited. Fig. 3 shows the semivariograms for OK, RKv1 and RKv2 in February which, as will be seen in Section 4.3, and according to the NRMSE, is likely the month with the semivariogram with the lowest performance. Table 3 shows the semivariogram parameters for every analyzed period and kriging method. It can be seen that the nugget remains smaller than the sill, but not null. That is caused by the finite number of sampling stations and error measurements. The sill for OK is greater for summer months (up to 0.97 (MJ m−2 day−1 )2 for August) consistently with the box-plot in Fig. 2. After the multiple regression analyses for RKv1 and RKv2, both the sill and range decrease as a result of the variability explained by the external explanatory variables. Range, which represents the distance at which autocorrelation yields zero, is similar in the two residual kriging models with values over 20 km in October to about 65 km in February. 4.2. Solar radiation maps Fig. 4 shows the maps for January, March, May, July, September and November obtained with the three kriging methods. Overall, solar radiation presents a similar spatial pattern and regionalization throughout the study region. However, the RKv1 and RKv2 methods introduce additional spatial variability due to the terrain influence. The spatial regionalization changes throughout the year similarly with the three kriging methods. In January, OK, for instance, predicts a roughly homogeneous pattern with weak maximum and minimum values in the south and north-eastern areas, respectively. Next, in March, these extreme values are emphasized and an additional maximum appears in the western coastal, which increases Table 3 Semivariogram parameters for every analyzed period and kriging method. Nugget and sill units are (MJ m−2 day−1 )2 . Range is given in km. Month OK RKv1 RKv2 Nugget Sill Range Nugget Sill Range Nuggeta Silla Range January February March April May June July August September October November December 0.01 0.12 0.09 0.09 0.15 0.17 0.10 0.12 0.05 0.04 0.05 0.06 0.41 0.36 0.43 0.30 0.41 0.57 0.96 0.97 0.35 0.23 0.18 0.19 46.3 85.6 42.1 30.1 67.8 47.9 64.8 98.4 48.1 42.2 49.6 74.0 0.02 0.09 0.01 0.07 0.09 0.07 0.08 0.06 0.02 0.02 0.04 0.03 0.10 0.23 0.30 0.31 0.22 0.48 0.59 0.44 0.21 0.13 0.10 0.10 31.0 65.2 27.7 25.8 28.3 24.6 27.0 24.5 37.9 21.5 24.8 46.3 0.28 0.35 0.20 0.21 0.20 0.23 0.22 0.22 0.21 0.21 0.29 0.31 0.58 0.68 0.52 0.30 0.34 0.40 0.41 0.43 0.33 0.32 0.68 0.58 30.0 65.3 34.6 25.7 26.6 22.7 25.4 25.9 40.5 23.6 23.2 47.1 Ann 0.07 0.19 31.3 0.05 0.16 28.9 0.25 0.35 30.1 a ×10−3 . Author's personal copy 1818 J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 Fig. 3. February experimental semivariograms and fitting models for (a) ordinary kriging, (b) RKv1 and (c) RKv2. in May. Then, in July, this maximum moves inland and the minimum solar radiation is reached along the Mediterranean coastal area. Like in March and May, September and November represent a transition to the situation in January, with relative homogeneity and slight extremes in the south-eastern part of the region. The way terrain influences the spatial distribution of solar radiation varies throughout the year, as can be observed in the RKv1 and RKv2 maps. In January, terrain produces small shaded patches, especially visible in the RKv1 map, and, as summer approaches, the influence of the shading by the terrain vanishes and the effect of the site altitude increases. 4.3. Models evaluation The validation stage has been carried out by a leave-one-out cross-validation and an independent dataset of 20 stations. Estimates were evaluated in terms of MBE, MAE, RMSE and NRMSE. For the sake of conciseness, for the cross-validation procedure, only the MBE and the MAE are shown. 4.3.1. Ordinary kriging model evaluation Table 4 shows the monthly and annual validation results for ordinary kriging. Both cross-validation and independent validation yielded a negligible bias error regardless of the period. The MAE for the cross-validation was around 2.5% for all the months except in winter, when it increased to around 3%. For the independent validation, it was slightly higher with the maximum (3.1%) in January and the minimum (2.2%) in May and September. For the independent dataset, the NRMSE has been calculated as a measure of the goodness of the variogram model. If the variogram was accurate, NRMSE should be unity. In this case, for January, February and March, the initial spatial variance of the random field was underestimated by the variogram while for the rest of months it was overestimated. The RMSE varies from a 2.7% in May and September to a 3.8% in January. 4.3.2. RKv1 model evaluation Table 5 shows the monthly and annual validation results for RKv1. Both cross-validation and independent validation yielded, as for ordinary kriging, a negligible bias error regardless the period, albeit slightly smaller. In the cross-validation procedure, the MAE shows a seasonal pattern: smaller values in spring and summer (around a 2.3%) and higher values in autumn and winter, reaching the maximum (3.1%) in January. In the independent validation procedure, RKv1 outperforms ordinary kriging with relative improvements of around 7% for January or October, and up to 15% in July, for instance. This could be attributable to the higher explained variance for the winter and summer months in the previous regression analyses (Table 1). Overall, NRMSE in the independent validation procedure shows closer values to unity than ordinary kriging. This means that the variogram model in this case fits better the spatial variability of the random field. Relative RMSE was also slightly better than for ordinary kriging. The highest values were reached in winter months, with a maximum of 3.7% in January. May and June presented the lowest value (2.5%). Comparing these results with ordinary kriging, the highest relative improvement (18%) occurs in July, but from May to August it keeps around 15%. For January and December, it is about 5%. Again, this can be attributable to the higher explained variance for all these months (Table 1). 4.3.3. RKv2 model evaluation Table 6 shows the validation results for the RKv2 method. They are similar to those of RKv1 (Table 5). Again, both cross-validation Author's personal copy J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 1819 Fig. 4. Daily solar radiation maps (MJ m−2 day−1 ) obtained with ordinary kriging (first column), RKv1 (second column) and RKv2 (third column). The spatial resolution is 1 km (389 × 594 cells). Note that color scale varies according to the month. and independent validation yielded a negligible bias error regardless the period, though smaller than ordinary kriging. For the cross-validation, the MAE presents a seasonal pattern with higher values in winter months (the highest, 3.3%, in January) and lower in spring and summer. In the independent validation procedure, this method also outperforms ordinary kriging results with similar relative improvements over those of RKv1. Overall, NRMSE shows closer values to unity than ordinary kriging. Therefore, the variogram model in this case fits better the spatial variability of the random field. As far as RMSE is concerned, Author's personal copy 1820 J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 Table 4 Validation results for ordinary kriging. Scores are given in MJ m−2 day−1 except for NRMSE which is dimensionless. A negative MBE means that predictions underestimate the observed values. Values in parentheses are percentages relative to the mean observed value. Month Cross validation Independent dataset MBE MAE MBE MAE RMSE NRMSE January February March April May June July August September October November December 0.01 0.01 0.01 0.01 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.30(3.2) 0.36(2.9) 0.41(2.4) 0.51(2.4) 0.55(2.2) 0.66(2.3) 0.69(2.4) 0.62(2.4) 0.46(2.4) 0.36(2.6) 0.29(2.8) 0.24(2.9) −0.04 −0.02 −0.03 −0.02 0.03 0.01 −0.01 0.02 0.03 0.06 0.04 0.03 0.29(3.1) 0.36(2.9) 0.59(2.4) 0.52(2.4) 0.55(2.2) 0.68(2.4) 0.69(2.4) 0.61(2.4) 0.43(2.2) 0.37(2.7) 0.29(2.8) 0.25(3.0) 0.36(3.8) 0.45(3.6) 0.51(3.0) 0.65(3.0) 0.67(2.7) 0.82(2.9) 0.86(3.0) 0.76(3.0) 0.53(2.7) 0.45(3.3) 0.35(3.4) 0.30(3.7) 0.95 0.70 0.78 1.09 1.23 1.19 1.24 1.25 1.28 1.20 1.24 1.11 Ann 0.01 0.40(2.2) 0.01 0.40(2.2) 0.49(2.7) 1.20 Table 5 As for Table 4 but for the RKv1 method. Month Cross validation Independent dataset MBE MAE MBE MAE RMSE NRMSE January February Mar April May June July August September October November December −0.01 −0.01 −0.00 −0.00 0.00 0.02 0.00 0.01 0.01 0.01 0.00 0.00 0.29(3.1) 0.33(2.7) 0.41(2.4) 0.52(2.4) 0.50(2.0) 0.65(2.3) 0.66(2.3) 0.61(2.4) 0.47(2.4) 0.36(2.6) 0.26(2.5) 0.24(2.9) −0.05 −0.03 −0.05 −0.03 0.03 −0.03 −0.07 −0.03 −0.01 0.06 0.04 0.02 0.29(3.0) 0.32(2.6) 0.39(2.3) 0.50(2.3) 0.50(2.0) 0.59(2.1) 0.60(2.1) 0.56(2.2) 0.43(2.2) 0.33(2.4) 0.27(2.6) 0.25(3.0) 0.35(3.7) 0.43(3.5) 0.51(3.0) 0.63(2.9) 0.62(2.5) 0.71(2.5) 0.74(2.6) 0.69(2.7) 0.51(2.6) 0.41(3.0) 0.34(3.3) 0.30(3.6) 1.15 0.93 1.11 1.16 1.10 1.12 1.16 1.16 1.14 1.02 1.10 1.11 Ann 0.00 0.40(2.2) −0.01 0.38(2.1) 0.46(2.5) 1.18 higher values were reached in winter, around 3.5%. Again, the best results were obtained in spring and summer months, around 2.5%. When these results are compared with ordinary kriging, the highest relative improvement (15%) is achieved in June and July. For the rest of months, except March, a relative improvement in the range 3–11% is achieved. 4.3.4. Statistical distribution of the predicted values In order to discern whether the predicted values using the three models come from the same distribution than the observed values (null hypothesis), a two-sample Kolmogorov–Smirnov (KS) test has been carried out between the observed dataset (training and vali- dation stations together) and the predicted values for every studied period and interpolating method. Table 7 shows the p-values calculated with the null hypothesis that the observed and predicted values are drawn from the same distribution. Results show that, overall, the RKv2 reproduces better the distribution of the observed data along the year, especially in winter months, where p-values are greater than 40% and there is no way to reject the null hypothesis. RKv1 has higher p-values in spring and autumn. 4.3.5. Models comparison The external explanatory variables in RKv1 and RKv2 constrain the distribution of the predicted values throughout the region. Fig. 5 Table 6 As for Table 4 but for the RKv2 method. Month Cross validation Independent dataset MBE MAE MBE MAE RMSE NRMSE January February March April May June July August September October November December −0.01 −0.01 −0.00 −0.00 −0.00 0.02 −0.00 0.01 0.01 −0.01 −0.00 −0.01 0.31(3.3) 0.36(2.9) 0.40(2.4) 0.52(2.4) 0.52(2.1) 0.65(2.3) 0.66(2.3) 0.61(2.4) 0.46(2.4) 0.37(2.7) 0.29(2.8) 0.25(3.0) −0.05 −0.04 −0.05 −0.03 0.03 −0.03 −0.07 −0.03 −0.01 0.05 0.04 0.02 0.28(2.9) 0.35(2.8) 0.41(2.4) 0.50(2.3) 0.52(2.1) 0.59(2.1) 0.57(2.0) 0.56(2.2) 0.45(2.3) 0.33(2.4) 0.28(2.7) 0.24(2.9) 0.34(3.6) 0.45(3.6) 0.51(3.0) 0.65(3.0) 0.60(2.4) 0.71(2.5) 0.72(2.5) 0.66(2.6) 0.51(2.6) 0.42(3.1) 0.35(3.4) 0.29(3.5) 1.12 0.93 1.03 1.11 1.09 1.21 1.24 1.18 1.10 1.08 1.10 1.16 Ann −0.00 0.40(2.2) −0.02 0.37(2.0) 0.46(2.5) 1.31 Author's personal copy J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 1821 Fig. 5. Minimum value (a) and standard deviation (b) of the observed data (training + validation stations) and the predicted values using OK, RKv1 and RKv2 for all the studied periods. Table 7 Two-samples Kolmogorov–Smirnov test between the observed data (training + validation stations) and the predicted values using OK, RKv1 and RKv2. Table shows the p-values for all the studied periods. Values in bold indicate that the null hypothesis can be accepted at a 95% confidence level. Month OK RKv1 RKv2 January February March April May June July August September October November December 0.243 0.093 0.149 0.039 0.085 0.016 0.110 0.190 0.029 0.072 0.120 0.068 0.109 0.102 0.276 0.047 0.088 0.014 0.001 0.009 0.042 0.062 0.088 0.054 0.471 0.549 0.148 0.012 0.013 0.067 0.061 0.051 0.121 0.025 0.024 0.431 Ann 0.032 0.014 0.057 shows the minimum and standard deviation values for every month and the annual period for the observed dataset (training and validation stations) and the three kriging methods. Maximum and mean values are accurately reproduced by all the methods. Fig. 5a shows that, in winter, the minimum predicted values using OK match the minimum observed values. However, using RKv1 and RKv2, the minimum predicted values are smaller. This is coherent with the fact that OK does not account for shadows caused by local topography. In addition, note that RKv2 predicts slightly smaller minimum values in summer than RKv1, which accurately follows the minima of the observed dataset. This seems to indicate that RKv2 is able to account for the small fraction of shadowing in mountainous regions in summer. Fig. 5b shows the standard deviation. It can be seen that only RKv2 in winter yields approximately a similar dispersion to that of the observed dataset. Particularly, the standard deviation in January using RKv1 is too high while that of OK is too low. However, for the rest of the year, none of the methods is able to reproduce the dispersion of the observational dataset, although RKv2 is the best one. In the annual period, only RKv1 appears able to reproduce the dispersion of the measured values. 5. Conclusions Solar radiation is a scarcely sampled variable with respect to other environmental variables such as temperature or precipitation, in part due to the high maintenance cost of the required radiometric sensors. It is highly sensitive to environmental factors from regional to local scales. Particularly, terrain surface challenges the traditional interpolation techniques when predictions with high spatial resolution are sought, especially because of the lack of measurement stations in mountainous regions. Geostatistics pose a stochastic approach to solve the spatial prediction problem that prevents reliance on previously hypothetized deterministic models and allows us to include the effect of external information sources based on experimental datasets. In this work, we have tackled the problem of using geostatistics to account for terrainrelated and cloudiness effects on monthly averages of daily solar radiation when availability of measurements in mountains is scarce and does not allow tracking of the topographic signal. Tipically, this is the case when long datasets with records spanning several years are employed. In this case, topographic characteristics derived from a digital elevation model at the experimental sites are not enough. Therefore, we have evaluated the usefulness of the estimates of a clear-sky solar radiation model that account for terrain-related effects as an external explanatory source. Additionally, mesoscale phenomena have been considered through the distance to the coast, the geographical longitude and, indirectly, the Linke’s turbidity coefficient in the clear-sky solar radiation model. Three different kriging procedures have been evaluated: ordinary kriging, residual kriging with clear-sky solar radiation as a predictor and residual kriging using the clear-sky index as a proxy to solar radiation. Ordinary kriging has been used as a skill reference for the residual kriging methods. All three methodologies have shown a high performance in predicting the spatial regionalization of the monthly averaged daily solar radiation in the study region, the southern part of the Iberian Peninsula. The use of external explanatory variables in the residual kriging methods has allowed improve of the reliability of the spatial modelling with the variogram model, as is evident by the NRMSE values in Tables 4–6. Particularly, continentality explains around a 30% of the variance for solar radiation and clear sky index (Tables 1 and 2). This variable and the easting coordinate have allowed us to improve the results by acting as a proxy to the regional distribution of cloudiness along the year. The independent validation procedure yielded a relative RMSE ranging from about 3% to 4% using ordinary kriging and from about 2.5% to 3.5% using RKv1 and RKv2. In all the cases, the best performance was reached during spring and summer seasons. In relative terms, residual kriging methods achieved improvements around a 5% in winter and around a 18% in summer with respect to ordinary kriging. Overall, these approaches have proven to be very useful for mapping solar radiation from long datasets that, usually, do not Author's personal copy 1822 J.A. Ruiz-Arias et al. / Agricultural and Forest Meteorology 151 (2011) 1812–1822 have available observations in mountainous areas. Both methods, RKv1 and RKv2, performed similarly in the validation procedures (Tables 5 and 6). However, RKv2 reproduced slightly better the statistical distribution of the observed dataset (Table 7) and also provided better results for the minimum and standard deviation values throughout the study region (Fig. 5). Acknowledgements This work was supported by the Spanish Ministry of Science and Technology (Project ENE2007-67849-C02-01) and the Andalusian Ministry of Science and Technology (Project P07-RNM-02872). The data were kindly provided by the regional office of Agriculture and Fishing of Andalusia. References Alsamamra, H., Ruiz-Arias, J.A., Pozo-Vázquez, D., Tovar-Pescador, J., 2009. A comparative study of ordinary and residual kriging techniques for mapping global solar radiation over southern Spain. Agric. Forest Meteorol. 149 (8), 1343–1357. Beyer, H.G., Costanzo, C., Heinemann, D., 1996. Modifications of the Heliosat procedure for irradiance estimates from satellite images. 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