Corresponding author: G. Sánchez, Department of Physics, University of Extremadura, Avda. de Elvas s/n, E-06006 Badajoz, Spain. (firstname.lastname@example.org)
 Reliable and accurate measurements of diffuse solar irradiance are needed in order to partition global irradiance into its direct and diffuse components. Diffuse irradiance is commonly measured using sun tracking systems or shadow rings. Data obtained using a shadow ring must be corrected for the portion of diffuse irradiance blocked by the ring. In this paper we have examined and evaluated six of the most widely used correction models. Approaches that account for radiation anisotropy perform notably better than those using only geometric corrections. Our results also argue for the need to adjust empirical models to local conditions. Empirical approaches developed by LeBaron et al. (1990) and by Batlles et al. (1995) perform best when compared with the more theoretical models.
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 Many solar radiation studies rely on a good knowledge of not only global radiation but on its direct and diffuse components as well. The accurate assessment of global and diffuse solar radiation is essential for estimating the radiation intercepted by horizontal and tilted surfaces such as hills, buildings, vegetation, and animals. In addition to meteorological studies, solar radiation data is used in many applications such as in architecture, engineering, agriculture and ecology.
 The amount of diffuse radiation reaching the earth's surface and its proportion to global radiation is mainly determined by the solar zenith angle, the composition of the atmosphere, and surface albedo. Most of the atmospheric constituents are relatively constant in time and space with the exception of aerosols and clouds. Their interaction with solar radiation is complex and any change of these constituents in a changing climate will affect the amount of direct and diffuse radiation reaching the earth's surface and consequently, the surface energy balance. Therefore, very accurate measurements of global, direct, and diffuse solar radiation at the earth's surface are required to suitably detect and quantify the effect of climate change on the earth's radiation balance [Hansen et al., 2005; Wild et al., 2005].
 Information on the relative amount of direct and diffuse solar irradiance that is available is also required in the rapidly developing field of renewable energy. There are presently considerable efforts being devoted to improving the efficiency of solar collectors, relying mainly on better collecting systems. An accurate knowledge of the direct and diffuse components of the radiation field is absolutely essential to achieve this goal.
 There are different instruments and methodologies aimed to measure solar diffuse radiation. The most precise method consists of shading the pyranometer with a small disc or ball synchronized with the sun's apparent motion. However, the technique is costly, requiring much maintenance and is unstable under strong winds. A more practical and widely used approach is to measure diffuse radiation using a shadow-ring/band. This is a robust mechanism consisting in a ring/band parallel to the sun path that blocks the direct irradiance and prevents it from reaching the sensor. It is an easy-to-operate stationary device that only requires manual adjustment of the sliding bar every few days to account for changing solar declination. The shadow band system is the most common method for measuring diffuse radiation at meteorological stations worldwide. Thus, long time series of diffuse radiation measurements using shading rings exist since the first half of the last century. Some studies have shown that measurements with shadow rings are comparable to those given by more sophisticated tracking devices under totally cloudy skies whereas in clear sky conditions some differences appear [Ineichen et al., 1984].
 The shadow ring screens not only the sun's disc but also a substantial portion of the sky and, therefore, its measurements must be corrected for the diffuse radiation intercepted by the ring. This error can, for example, result in a monthly average error of up to 24% [Drummond, 1956], significantly affecting any subsequent calculation of other radiometric variables. Thus, a 5% error in the measurement of the horizontal diffuse irradiance will propagate to more than 20% in the calculated direct normal irradiance at zenith angles greater than 75° [LeBaron et al., 1990]. The needed correction factor has been estimated to be between 8.9% and 37.7% [Kudish and Ianetz, 1993], depending on the latitude, the weather conditions and the type of the shadow ring [Steven, 1984]. This uncertainty limits the accuracy of diffuse radiation measurements and makes it difficult to compare measurements performed at different locations or different seasons [Drummond, 1956; Steven, 1984]. To emphasize the importance of this correction, it must be noted that it is higher than the variation in solar radiation observed between 1961 and 1990, estimated to be around the 4–6% [Liepert, 2002], and higher than the accuracy of ±3% required by the World Meteorological Organization for a particular measurement to be classified as high quality [World Meteorological Organization (WMO), 2008].
 As a first approach to the problem, Drummond  proposed a first correction factor as a function of the solid angle subtended by the shadow ring and its altitude above the pyranometer horizon. This model is exclusively based on solar geometry calculations and assumes an isotropic sky radiation distribution. It provides a fairly good estimation of diffuse irradiance occluded by the ring and may be applied anywhere on earth. However, the assumption of isotropy is not generally fulfilled since sky radiance directional distribution notably depends on sun elevation, atmospheric turbidity, cloudiness and usually changes with weather conditions during the day. Thus, Drummond  estimated differences of about 7% for cloudless skies and 3% for overcast conditions between its corrected diffuse irradiance values and the reference values due to the anisotropy. Other subsequent studies quantified the additional correction for anisotropic conditions to be between 14% and 30% above the isotropic correction [Stanhill, 1985]. Several models have been developed in order to correct for the anisotropy effect. A straightforward modification to the Drummond's model was proposed by Steven , who assumed a sky radiance distribution as the sum of a uniform background and a circumsolar component. On the other hand, LeBaron et al.  developed a model which classified the different sky conditions into 256 categories as a function of three parameters describing the anisotropic contribution and the fraction of the sky hemisphere occluded by the shadow ring as estimated by Drummond . Using the same parameters than Le Baron et al., Batlles et al.  proposed two correction factors as multivariate-lineal functions of these parameters. More recently, Muneer and Zhang  have developed a new model based on the anisotropic sky-diffuse distribution theory proposed by Moon and Spencer .
 Comparative studies are required in order to assess the performance and limitations of these models and their suitability at different locations in the world. However, to our knowledge, there are few comparative studies [López et al., 2004a, 2004b; Kudish and Evseev, 2008], and even fewer for regions with similar climatic characteristics as our study site in southwestern Europe. Additionally, most recent studies applied the models using the original values of local parameters as they were proposed by their respective authors, although Dehne  noted the need to use coefficients applicable to the local study areas. Failure to do so may create large errors since some of these models have been developed empirically and their correction factors depend on the geographical and climatic characteristics of each location.
 Therefore, the main objectives of this paper are to assess the performance of the models proposed by Drummond , Steven , LeBaron et al. , Batlles et al.  and Muneer and Zhang  in estimating diffuse irradiance measured with a pyranometer and a shadow-ring at Badajoz (southwestern Spain) and to examine how the coefficients of these empirical models need to be adapted to new sites. These models will be called DR (Drummond), ST (Steven), LB (LeBaron et al.), BA or BB (Models A and B by Batlles et al.), and MZ (Muneer and Zhang) hereafter. All the mentioned models have been extensively used in different climatic regions throughout the world [López et al., 2004a, 2004b; Kudish and Evseev, 2008] and are well considered by the scientific community. In this study we evaluate the suitability of the empirical models ST, LB, BA and BB to our study site and examine the sensitivity of the various coefficients to the estimate of diffuse radiation. Finally, these adjusted models are compared with the theoretical models DR and MZ.
 Data used in this study have been acquired at the radiometric station installed in Badajoz, southwestern of Spain (38.9° N; 7.01° W; 199 m a.s.l). The site has an open horizon, free of obstacles and is operated by the Physics Department of the University of Extremadura, guaranteeing careful maintenance. The local climate is characterized by high noon solar elevation, mainly in summer, and by a high number of sunshine hours per year, with our site having one of the highest instantaneous and annual irradiance values in Europe.
 The data set consists of one-minute measurements of horizontal global irradiance and two simultaneous measurements of diffuse irradiance using two different methods. The diffuse solar irradiance was measured using a Kipp & Zonen CMP11 pyranometer which was shaded using a Kipp & Zonen CM121 shadow-ring with 620 mm diameter and 55 mm width. Simultaneous measurements of diffuse solar irradiance were taken by another CMP11 pyranometer mounted on a solar tracker (Kipp & Zonen Solys 2). The solar tracker has a ball that prevents the direct solar irradiance to reach the sensor. The ball only shadows the pyranometer sensor and does not obstruct any other sky portion. The correction factor for these measurements is negligible [Ineichen et al., 1984] and therefore they have been used as a reference. A third Kipp & Zonen CMP11 pyranometer was used to measure global irradiance. The study period encompassed an entire year, from 7 August 2010 to 8 August 2011, therefore ensuring that a variety of seasonal processes and meteorological conditions were sampled.
 To guarantee the quality and comparability of measurements, both pyranometers were previously inter-compared and calibrated in a field campaign which took place in September 2009 at the Atmospheric Sounding Station of the National Institute for Aerospace Techniques (ESAt/ INTA) located at “El Arenosillo,” Huelva, Spain (37.10° N, 7.06° W). The pyranometers were calibrated by intercomparison with a reference pyranometer (Kipp & Zonen CM11 pyranometer, #027771) which had been previously calibrated at the World Radiation Centre (WRC) in Davos, Switzerland. First, signals were corrected for thermal offsets estimated as the averages of values recorded for solar zenith angles higher than 100°. Then, ratios of our pyranometer signal to the signal from the reference pyranometer were estimated for a range of conditions. They proved to be largely constant so that calibration factors were calculated as average ratios obtained during the field campaign.
 Additionally, the data set corresponding to the study period was subjected to a quality control procedure in order to detect and eliminate possible erroneous measurements. The data was subsequently averaged on an hourly basis and used for the analysis. This time interval is widely used in solar radiation studies since it shows the daily variations without been affected by the very fast short-term fluctuations. The hourly data set was randomly divided into two subsets. One of them, containing 75% of all data, was used to construct the empirical models and their coefficients. The second subset, consisting of the remaining 25% of the data, was used for validation purposes as well as for the analysis and model comparison.
 The models DR and MZ result from theoretical studies which make certain assumptions regarding the sky radiance distribution. However, the only parameters involved in these models are the size of the shadow ring and the geographic coordinates, and therefore both models can be used at any location just as originally proposed by their authors. Conversely, the models ST, LB, BA and BB involve empirical local coefficients which were obtained for specific locations. In the present study, these local empirical models have been adjusted using the irradiance data acquired at the radiometric station of Badajoz. This allows us to estimate diffuse irradiance using the empirical models with and without local adjustments. Furthermore, adjusting the empirical models with local coefficients provides a better framework with which to judge their performance. For this analysis, the functional expressions for the correction factor proposed by Steven and by Batlles et al. (models ST, BA and BB) have been fitted to the 75% of the hourly data set and new regression coefficients have been estimated. In the case of the model LB, the correction factors have been calculated following the steps described in detail in the original study [LeBaron et al., 1990]. The differences between original and adjusted models have been tested on the remaining 25% of the data set and the models which performed best have been selected. Subsequently, these selected locally fitted empirical models have been included, together with the globally applicable models DR and MZ in a final comparison using the second subset consisting of the remaining 25% of the data.
 The evaluation of the performance of the models was undertaken by graphical and statistical means. For the statistical analysis the relative root mean square error (rRMSE) and the relative mean bias error (rMBE) were calculated in order to numerically quantify the performance of the models. These statistics are defined by the follow expressions:
where N is the total number of measurements, Ii,corr is the ith-corrected measurement and Ii,ref is the ith-reference value of diffuse irradiance as measured by the pyranometer mounted in the sun tracker device. The statistic rRMSE is a measure of the relative differences between corrected and reference values. On the other hand, the rMBE quantifies the mean bias between the corrected and reference data sets. Positive values indicate overestimation with respect to the reference measurements while negative values mean underestimation. Best models are those with these statistics close to zero.
 To complete the statistical analysis, the Taylor diagram [Taylor, 2001] was obtained. These diagrams provide a way of graphically summarizing how closely a model matches the reference observations.
 Differences between the reference diffuse irradiance measurements and the diffuse irradiance corrected using the original and adjusted models have been evaluated. In addition to the magnitude of the differences, their behavior versus time and other parameters such as the clearness index, the diffuse portion, and the solar zenith angle have been also analyzed.
4. Shadow Ring Correction Models
 The error in the diffuse irradiance measurements caused by the use of a shadow ring has been extensively studied. This error causes an underestimation of the measurement because the ring blocks a certain amount of sky radiance that should be incident on the sensor. The corrected diffuse irradiance on an horizontal surface Id,c may then be written as:
where C is the correction factor and Id,u is the uncorrected horizontal diffuse irradiance measured by the pyranometer installed on a shadow ring.
Drummond's  model estimates the fraction F of diffuse irradiance that is blocked by a shadow ring with an axis parallel to the polar axis. This model was developed under the following assumptions: (1) the ring width is small compared to its radius, (2) the sensor is negligible in size compared to the other dimensions, (3) there is no back-reflection of radiation from the inner surface of the ring, and (4) the sky radiance distribution was isotropic. With these conditions the diffuse fraction F may be written as:
where r is the radius and b is the width of the ring, ωs the hour angle at sunset in radians, δ the solar declination, and ϕ the latitude of the location. The correction factor proposed in this model is:
4.2. Steven: Model ST
 Based on previous studies, Steven  proposed a new model for the correction factor to use under clear conditions. This model introduces an additional anisotropy factor independent of the geometrical correction. The main hypothesis is that the sky radiance distribution is the sum of a uniform background and a circumsolar component. The correction factor proposed is:
where F is the fraction of diffuse irradiance occluded by the ring given by the equation (4) and Q is the anisotropy correction factor calculated using the follow expressions:
where C′ expresses the relative strength of the circumsolar component and ξ is the angular width of the circumsolar region. These coefficients can be obtained by means of a regression analysis of Q values (calculated from equation (6) as a function of the reference diffuse irradiance, uncorrected diffuse irradiance and the portion of diffuse irradiance blocked by the ring) versus 1/f values.
4.3. LeBaron et al.: Model LB
 The model proposed by LeBaron et al.  describes anisotropic sky conditions by means of three parameters. One of these parameters, ε, describes the cloud conditions and is defined as:
where Id,u is the uncorrected diffuse irradiance and Inb is the uncorrected direct normal irradiance which can be calculated from the global and uncorrected diffuse irradiance.
 The second parameter Δ is a brightness index which is a function of the cloud thickness or aerosol loading [Perez et al., 1990]. It is defined as:
where m is the relative optical air mass and I0 is the irradiance at the top of the atmosphere.
 The third parameter is the solar zenith angle θ. These three parameters, together with the isotropy correction CD proposed by Drummond , are used to classify the different sky conditions into 256 categories. These categories are the results of the combination of the four possible values that each parameter can take. The range of each parameter was divided in four subintervals. Subsequently, the reference and uncorrected diffuse irradiance measurements were classified according to the categories defined by the combination of the parameters and, finally, the correction factor for each category was obtained as the average of the ratio between the reference diffuse irradiance measurements and the uncorrected diffuse irradiance measurements. More details can be found in LeBaron et al. .
4.4. Batlles et al.: Models BA and BB
 Based on the same parameters utilized in model LB, Batlles et al.  built two new models to correct the diffuse irradiance measured with shadow rings. Both models use a multiple linear regression of these parameters to obtain the correction factors. In this paper these models will be called BA and BB.
4.4.1. Model BA
 This first model proposes an unique expression of the correction factor for all sky conditions:
where CD, ε and Δ were defined in equations (5), (9) and (10) respectively and a, b, c, and d areempirical parameters.
4.4.2. Model BB
 After the geometric correction, the parameter ε is the second most significant parameter in the correction factor analysis. Based on this fact, Batlles et al.  proposed different multiple linear regressions with appropriate functions of CD, Δ, and θ for different ranges of ε. The resultant model is described by the following set of equations:
where a1, b1, d1, a2, b2, d2, a3, b3, a4, and b4 are obtained by regression analysis. More details can be found in Batlles et al. .
4.5. Muneer and Zhang: Model MZ
Muneer and Zhang  developed a methodology similar to the one applied in model DR but using a new irradiance distribution. This irradiance distribution was based on the model proposed by Moon and Spencer  and revised by Muneer . According to this model, the sky radiance distribution is two dimensional and function of any given sky patch geometry (sky patch altitude and azimuth) and the position of the sun. This new irradiance distribution can be used to estimate the total diffuse irradiance that is blocked by the shadow ring as follows:
where Lz is the zenith radiance, b is the width of the shadow ring, r is its radio, and δ is the solar declination. Parameters I1 and I2 are defined by the follow expressions:
Using this same irradiance distribution, the total diffuse irradiance Id is calculated by means of numerical integration which gives the following expression:
where b1 and b2 are radiance distribution indices for the sky quadrants containing the sun and the opposed quadrant, respectively, which are estimated as follows:
Finally, the correction factor given by this model is:
5. Results and Discussion
5.1. Fitting Local Models
 The empirical models ST, LB, BA and BB were fitted to local conditions and the resulting parameters compared with the original parameters listed by the authors in order to assess their general validity. The new correction factors of the models ST, BA and BB were obtained by means of regression analyses using 75% of the data measured at our radiometric station. Similarly, the model LB was adjusted to local conditions using the same 75% data set. Subsequently, the new adjusted models were compared with the original models using the remaining 25% of the data.
 Diffuse irradiance corrections using the original and modified methods are compared next. Differences in the estimation of diffuse irradiance between the original and the adjusted model ST can be up to 4.7% and with a mean of 3.1%. The mean difference is 3.2% for the model LB, although individual values can exceed 16.0% and differences are higher than 5% in 15.7% of the cases examined. Mean differences of 1.7% and 2.0% are obtained for models BA and BB with largest differences of up to 15% and 9.2% respectively being reached in high solar zenith angle conditions. The generally low values in the mean difference found for models BA and BB could be explained by the fact that these models were originally developed for two Spanish sites in a geographical region similar to ours. However, dispersion is high when considering individual differences in all of the above three models. One possibility could be that errors are introduced by the parameters ε and Δ, related to cloud cover and brightening conditions, and which might not be totally applicable and valid for our region.
 In the above paragraph we examined how the modified models differed from the originals. Here we assess the performance of these modified models against a reference diffuse measurement which is considered a true value. The rRMSE and rMBE values with respect to the reference measurements were obtained for the original and for the adjusted models (Table 1). The aim of this analysis is to check the improvement achieved by the adjusted models. It may be observed that the biases shown by the original models are largely eliminated in the adjusted models which approach zero. The rRMSE values improve by 0.67% for the model ST (from 4.02% to 3.35%) and by 2.06% for the model LB (from 4.67% to 2.61%). It may be concluded that use of empirical models with locally derived coefficients considerably improves the diffuse correction process in shadow-ring measurements. This supremacy of adjusted models relies on their ability to be adapted to locally specific atmospheric and surface albedo conditions.
Table 1. The rRMSE and rMBE Values for the Original and Adjusted Models
LeBaron et al.
Batlles et al.
Batlles et al.
Kudish and Ianetz  estimated the magnitude of the shadow-ring correction factor to be between 8.9% and 37.7%. In our study we obtained a mean difference of 10.8% between the uncorrected and the reference diffuse irradiance measurement. These differences are considerable and therefore any improvement in the correction methodology is valuable and justified. An overall assessment is performed in the next section where the locally fitted version of the empirical models are compared with the globally valid theoretical models DR and MZ.
5.2. Comparative Analysis of the Models
Figures 1a–1g show the uncorrected and corrected diffuse irradiances versus the reference diffuse irradiances for the six models. In general, the measurements exhibit a better performance when corrections are applied. However, although the correction applied by the model DR notably improves the estimation of diffuse irradiance, it still underestimates the reference values. Conversely, this underestimation is no longer evident in the correction models which consider anisotropy. However, all model/reference differences feature much scatter when diffuse irradiance values are high. This is likely due to the large variety of possible diffuse radiation values associated with cloudy conditions.
Table 2 shows the statistics rRMSE and rMBE for the uncorrected measurements and the various correction models versus the reference measurements. While the rRMSE for uncorrected measurements is about 7.48%, it decreases to values between 2.61% and 4.34% for the corrected measurements. According to this statistic, the model with the best performance is model LB (2.61%) followed by the model BB (3.20%). By contrast the models with the highest rRMSE values are the globally valid theoretical models DR and MZ. All models with the exception of DR estimate diffuse irradiance with a low rMBE. The model DR shows an underestimation in the rMBE of −1.4% as it does not consider the anisotropy in the spatial distribution of the radiation field.
Table 2. Values of rRMSE and rMBE for Uncorrected Measurement and for Corrected Measurements Using the Correction Models Shown Here
 Results also indicate that the locally fitted empirical models (ST, LB, BA and BB) perform better than the globally valid theoretical models (DR and MZ). We also note the fairly good performance shown by the model ST despite being originally developed for cloudless conditions. This good general behavior may be related to the fact that cloudless conditions are very frequent in our region. The best performance is achieved by the model LB followed by the models Ba and BB. These three models feature the parameters ε and Δ that seem to describe the anisotropy effect better than other model parameters such kt in the model MZ, or the circumsolar region characteristics, C and ξ, in the model ST. The main difference between these two best-performing models is that, while the model LB estimates the correction factor as a direct average of a certain subset of the experimental data, models BA and BB propose specific functional expressions which are fitted by regression. On the negative side we note that the model LB needs a considerable amount of computation to obtain a calibration factor for each of the 256 categories.
 In order to have a complete comparison of the performance of the different models, the Taylor diagram was plotted (Figure 2). This diagram provides a graphical summary of different aspects of the performance of a model, such as the centered root-mean square error, the correlation and the magnitude of its differences with respect to the reference values [Taylor, 2001]. Therefore, it is a suitable tool for model comparison. Figure 2 shows the very good performance of the models LB, BB and BB, followed by the models ST, DR and MZ, and ending with the uncorrected measurements.
 An additional analysis was performed in order to better understand the reasons for the differences between the various models and the reference measurement. These differences were analyzed as a function of the solar zenith angle, the clearness index, and the ratio of diffuse-to-global solar irradiance kd. No significant dependence was found with the solar altitude or with the clearness index, but certain differences between models appeared in the case of kd (Figure 3). While the models LB, BB and BA suitably account for the dependence with kd, the models DR, ST and MZ correct for this dependence only partially.
 In this study, the six most widely used models for correcting horizontal diffuse irradiance measured with a shadow ring have been analyzed and compared. First, the parameters of the empirical models ST, LB, BA and BB were adjusted for local conditions as determined by our radiometric measurements in Badajoz, southwestern Europe. The locally fitted versions of the original empirical models significantly improved the estimation as determined by the central tendency (quantified by rMBE) and the dispersion (quantified by rRMSE). We conclude that adjusting the empirical coefficients for local atmospheric and surface albedo conditions is warranted and recommend this procedure in other environments that may differ from ours or from the environment associated with the original model development.
 These locally fitted versions of the empirical models were subsequently compared with the globally valid theoretical models DR and MZ. Results show the remarkable improvement achieved by the models which account for radiation anisotropy as contrasted with the model DR which considers only a geometrical correction factor. The locally fitted empirical models perform notably better than the globally valid theoretical models. In particular, the models LB, BB and BA performed best, exhibiting the lowest values of both the rRMSE and rMBE statistics. All these results are also confirmed by the global comparison performed by means of the Taylor diagram.
 The analysis of residuals shows that they are independent of both solar zenith angle and clearness index in all models. However, the models ST, DR and MZ proposed by Steven, Muneer and Zhang, and by Drummond show certain dependence of their residuals on kd, while the models LB, BB and BA show no dependence.
 All these results argue for the use of locally adjusted empirical models in order to correct for errors caused by the shadow ring when measuring solar diffuse irradiance. The model LB developed by LeBaron et al., and the models BB and BA by Battles et al. performed best for our specific location.
 This study has made a positive contribution to the methodology of measuring accurately the diffuse solar irradiance at the earth's surface. Therefore, it provides useful information for future studies which focus on estimating precisely the earth's radiation balance or studies which attempt to quantify the solar radiation resource for renewable energy applications.
 This study was partially supported by the research projects CGL2008-05939-C03-02/CLI and CGL2011-29921-C02-01 granted by the “Ministerio de Ciencia e Innovación” from Spain. The authors thank Manuel Nunez for proofreading the text.