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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
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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
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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.
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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
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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,
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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
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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%
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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 .
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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
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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,
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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
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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
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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.
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