Fast scheme for determination of direct normal irradiance. Part I: New aerosol parameterization and performance assessment

Solar Energy 199 (2020) 268–277

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Fast scheme for determination of direct normal irradiance. Part I: New aerosol parameterization and performance assessment

T

Guoping Shia, Zhian Sunb, , Jiangnan Lic, Yongjian Hea ⁎

a

School of Geographic Sciences, Nanjing University of Information Science and Technology, Nanjing, China Branch of Science to Services, Australian Bureau of Meteorology, Melbourne, Victoria 3001, Australia c Centre for Climate Modelling and Analysis, Environment Canada University of Victoria, British Columbia, Canada b

ARTICLE INFO

ABSTRACT

Keywords: Aerosol optical depth Direct normal irradiance Bsrn Aeronet

A new aerosol direct transmittance parameterization is developed based on detailed radiative transfer model calculations for improving direct normal irradiance (DNI) for the solar energy forecast in the fast radiation scheme SUNFLUX. The modified SUNFLUX scheme is evaluated against observational datasets collected from the 14 Baseline Surface Radiation Network (BSRN) stations. The aerosol data from AERONET stations collocated with the BSRN network are used in the evaluations. The evaluation results have indicated that the accuracy of the calculated DNI using the new version of SUNFLUX has been improved significantly compared to the old version. The relative mean bias difference (MBD) and relative root mean square difference (RMSD) are −3.03% and 4.95%, respectively. The performance of the modified DNI model is also compared with those from the other five models. The results predicted by the current model are comparable with those from the best reference models. More importantly, the evaluation results show that the new aerosol scheme developed in this study can also be applied to other models to improve their performance.

1. Introduction A fast scheme (Sun and Liu, 2013) has been developed for estimating the direct normal irradiance (DNI) at the surface using Numerical Weather Prediction (NWP) model data or datasets from satellite retrieval or reanalysis. This scheme is a part of the “SUNFLUX” scheme for the determination of radiation at the surface (more details in Section 2). The motivation for the development of the SUNFLUX scheme has been described in our previous publications (Sun et al., 2013, 2012) and the main reason is to apply the SUNFLUX scheme in an NWP model so that the surface radiation can be updated at the model integration time step. Currently, the integration time step in most NWP global models is about 300–700 s, but the radiation time step in these models could be 1–3 h. The reason for the model's radiation to be updated at a longer time interval is to reduce the computational cost to meet the operational forecasting timeline. Use of a fast scheme such as the SUNFLUX will not be subject to this restriction because of its computational efficiency. This application also enables an NWP model to provide more frequent solar radiation outputs that are required by solar energy industries. A Fast All-sky Radiation Model for Solar applications (FARMS) was developed by (Xie et al., 2016). This model is similar to the SUNFLUX and can also be used in NWP models for the purpose described

⁎

above. However, another important issue related to NWP applications of such fast schemes is that the model energy field must be conserved. Thus, if these schemes are to be used to update the shortwave radiation at the surface that is further used to advance the model forecasts, the corresponding longwave radiation, as well as radiation at the top of the atmosphere, must be updated simultaneously. Unlike the FARMS scheme, which does not meet these criteria because it can only determine the solar radiation at the surface, the SUNFLUX package does meet them as it includes the extra components of the surface longwave radiation and the reflected solar radiation at the top of the atmosphere. The SUNFLUX scheme has been extensively evaluated using observational datasets, e.g. (He et al., 2017; Tang et al., 2017). Based on these evaluation studies, a number of deficiencies of the DNI scheme have been identified, which need further addressing to improve the performance of the scheme. The main problem is related to the treatment of aerosols. The original SUNFLUX uses the aerosol parameterization scheme developed by Kokhanovsky et al. (2005). There are two limitations to this scheme. The first is that the effects of aerosol absorption are not included because a single scattering albedo = 1 was used in its development. This may cause a large error for absorbing aerosols, such as black carbon. The second is that the direct transmittance is determined with Beer’s law which is only valid in the

Corresponding author. E-mail address: [email protected] (Z. Sun).

https://doi.org/10.1016/j.solener.2020.02.028 Received 12 April 2019; Received in revised form 4 February 2020; Accepted 7 February 2020 0038-092X/ © 2020 International Solar Energy Society. Published by Elsevier Ltd. All rights reserved.

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monochromatic transmission. In this paper, we introduce a newly developed aerosol parameterization. SUNFLUX is modified in the DNI modelling using this new aerosol scheme. The DNI determined using the modified scheme is then evaluated against observations and other DNI models. The paper is organized as follows. Section 2 briefly introduces the SUNFLUX scheme. Section 3 presents the development of the new aerosol parameterization. Section 4 introduces the data used for evaluation. Sections 5 present the evaluation results under clear-sky conditions using the BSRN dataset. Section 6 conducts a cross-comparison with other algorithms. Section 7 performs a sensitivity study and Section 8 summarizes results and draws conclusions.

sun-earth distance correction factor. SES2 was run to calculate the direct solar irradiance for AOD ranging between 0 and 4 with an increment of 0.01, and solar zenith angles between 0°and 89° with an increment of 1°. The reference calculations were performed for aerosol atmosphere separately, i.e. no gases or clouds are included. The varied AOD is integrated into the calculations by scaling the profiles of the model background AOD, to ensure that the scaled total optical depth is equal to the varied value. For the broadband irradiance, the direct transmittance can be expressed as an exponential function. Kocifaj (2016) developed an analytical approximation using the exponential rule to determine the broadband direct transmittance. Mueller et al. (2004) developed the socalled modified Lambert-Beer relation to fit results from a radiative transfer model in order to make their radiation scheme running fast yet accurate. Here we take an empirical approach to fit the band average transmittance determined by Eq. (1) using the following equation

2. Description of the SUNFLUX scheme SUNFLUX is a fast scheme for the determination of solar radiation at the earth's surface. The detailed development procedures have been described in the relevant references (Sun and Liu, 2013, Sun et al., 2013, 2012), and so only a brief description is presented here. SUNFLUX is a two-band spectral model covering the ultraviolet–visible (0.2–0.7 µm ) and near-infrared (0.7–5.0 µm). The scheme includes the effects of absorption and scattering of solar radiation by major absorbing gases, aerosols, and clouds. The effects of Rayleigh scattering and scattering by clouds and aerosols are represented by their albedos, while the effects of absorption by these constituents are determined by their transmittances. The transmittances and albedos for these atmospheric components are generated using the detailed radiative transfer scheme, Sun-Edwards-Slingo (SES2) (Sun, 2011) for various atmospheric conditions possibly occurring in the earth-atmosphere, and the results are parameterized using simplified expressions. This scheme has advantages in that it can be used to determine the surface solar radiation under both clear and cloudy conditions, and also used online in an NWP model or offline using data from satellite retrieval or reanalysis. The scheme is currently under a process of implementation in the operational forecasting systems used in the Australian Government Bureau of Meteorology and the UK Met Office. The input variables required for the determination of DNI under clear-sky conditions are listed in Table 3. The optical depths for ice and water clouds are further required for cloudy-sky conditions.

Ti = exp{ m (µ 0) }

where is the varied AOD from 0 to 4 and m (µ 0 ) is usually called the optical air mass that depends on the solar zenith angle. Several formulas can be used to determine m (µ 0 ), e.g. those used in Maxwell (1987), (Davies et al., 1988), Rigollier et al. (2000), and Gueymard (2001). Instead of using the conventional m (µ 0 ) definition, we determine it based on the numerical fitting to the aerosol transmittance determined by the reference radiation scheme SES2. This is to ensure the parameterized results consistent with the reference results. All SUNFLUX components are derived based on the reference calculations using SES2. The procedures for determining transmittances of the absorbing gases, clouds and aerosols are all treated in a similar manner, thus ensuring that SUNFLUX can reproduce the results closer to the reference results. The actual procedure for determining m (µ 0 ) follows the steps below:

• SES2 calculates aerosol transmittances for the AODs varying from 0 • •

We have developed a new aerosol parameterization scheme. We applied a similar process to that used in the development of SUNFLUX itself, first conducting reference calculations for aerosol using the SES2 radiative transfer model, to generate model reference data for use in the development of the parameterization. In these reference calculations, we used globally averaged dust aerosol mixing ratio profiles simulated by the Spectral Radiation-Transport Model for Aerosol Species (SPRINTARS) (Takemura et al., 2009) as an aerosol background. SPRINTARS is a numerical model developed by the Climate Change Science Section, Research Institute for Applied Mechanics, Kyushu University for simulating effects of atmospheric aerosols on the climate system and air pollution on the global scale. The model data used in this study were from SPRINTARS simulations for 2006–2008. The global mean mixing ratios of the SPRINTARS aerosol climatology are used in the SES2 model calculations. The effect of aerosols on the direct irradiance can be represented by the aerosol transmittance of the direct irradiance defined by

Fidir , µSi

to 4 with 0.01 increments and the solar zenith angles ranging from 0 to 90° with an increment of 1°. For each solar zenith angle, fit the reference transmittance to in Eq. (2) to obtain m (µ 0 ). Then fit the coefficient m (µ 0 ) to µ 0 in Eq. (3).

The numerical fit to the AOD for each of the 90 solar zenith angles yields accurate results with correlation coefficients are all close to one and mean relative fitting errors are all less than 0.54%. Fig. 1 shows the fitting coefficient m (µ 0 ) as a function of µ 0 in the visible band. Since the relationship between m (µ 0 ) and µ 0 is linear in the logarithm scale, it can be well represented by a power function.

3. Aerosol transmittance parameterization

Ti =

(2)

m (µ 0 ) = aµ 0b

(3)

The solid line in Fig. 1 shows the fitting result in the visible band from this equation. The fitting result is excellent and the mean relative fit error is only 0.05%. Similar fitting results are obtained in the near IR spectral band. The coefficients a and b are 1.00016, −0.998945 for the visible band and 1.00028, −0.999166 for the near IR band, respectively. The differences in the coefficients a and b between the two bands are very small, so one pair of coefficients can be used in the two bands without loss of accuracy in practice. Although Eqs. (2) and (3) are for the two-band model, they can be applied to a broadband scheme. To achieve this, the following spectral averaging procedure may be necessary. We now use subscript 1 and 2 to represent the visible and near IR spectral bands. The broadband transmittance can be defined by

(1)

T=

where Fidir represents the direct irradiance at the horizontal surface in the spectral band i (visible or near IR), Si is the spectral band solar constant, and µ is the cosine of the solar zenith angle. Fidir includes the

F dir + F2dir T µS + T2 µS2 F dir = 1 = 1 1 = f1 T1 + f2 T2 µS µS µS

(4)

where S is the solar constant, f1 = S1/ S and f2 = S2/S represent the fraction of the solar flux at the two-spectral bands. These two fractions 269

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spectral averaging procedure or just using one pair of coefficients as mentioned earlier. In Section 7, we will show the examples of applying Eqs. (2) and (4) to other models and examine their performance. 4. Data used for evaluations To evaluate the DNI model modified with the new aerosol scheme presented in the previous section, several observational datasets are used and briefly described in this section. The radiation data were collected from 14 BSRN stations (Driemel et al., 2018; König-Langlo et al., 2013). The AOD data and the total precipitable water (PW) data were obtained from AERONET stations collocated with the BSRN network. AERONET provides the spectral data of AOD and the total precipitable water (PW) retrieved from ground-based Sun photometer measurements (Dubovik et al., 2006; Holben et al., 1998; PérezRamírez et al., 2014). The measurements in the 940 nm channel are used to retrieve PW and those in other channels are used to determine AODs. The measurements are performed every 15 min with a treatment of the triplets to obtain 1-min averaged results (Giles et al., 2019). 14 stations were collocated with the BSRN network at the time when we conducted this study and this is the reason why we choose these 14 BSRN stations. The level 2.0 AOD from the AERONET version 3 dataset is used in this work. The level-2 AOD data have undergone cloudscreening and quality control. The total column daily ozone amount data are taken from the Total Ozone Monitoring Instrument (OMI, https://www.nasa.gov/mission_pages/aura/spacecraft/omi.html/; last accessed 22 November 2019). The radiometric observations are processed in two stages. The first stage is to perform quality control (QC) for the BSRN data. The QC is conducted using the BSRN Toolbox [a software package supplied by the world radiation monitoring centre (Schmithüsen et al., 2019)] for the physically possible limits, extremely rare limits and cross-comparisons. The checked results are flagged with quality codes between 0 and 63. A quality code of 0 means that all checks were passed successfully and only such observations of the highest quality are used. The number of data samples passing all QC checks and their fractions compared to the total number of raw data for 14 BSRN stations are listed in Table 1. These fractions (shown in the last column in Table 1) are very high and the passing rates are greater than 95% from 13 stations. The only exception is Tamanrasset, where the fraction is less than 90%. The second stage is to select those BSRN data points that correspond to clear-sky conditions, and synchronize them with the AERONET data points. Although a cloud-screen algorithm has been used to eliminate the effects of clouds on the AERONET AOD retrievals, this procedure is not perfect and these effects cannot be completely removed. Hence, some of the selected data points may still be affected by cloud

Fig. 1. Exponential coefficient m(µ0) in Eq. (2) as a function of the cosine of the solar zenith angle. The solid curve is the fitting result using Eq. (3) and the dots represent the reference m values.

can be determined by the sums of the spectral fractions of the solar flux from the SES2 bands 1 to 4 (0.2–0.7 µm ) and 5 to 9 (0.7–5.0 µm ), respectively. The results are f1 = 0.45389 and f2 = 0.54611. The explanation for this approach is given below. The spectral fraction of the solar flux in the SES2 model is defined as a ratio of the extraterrestrial spectral band flux to the total flux. The Kurucz (Chance and Kurucz, 2010) solar spectrum is used to determine the spectral fractions. The Kurucz spectrum covers the spectral interval between 0.2 and 200 µm . To include the contribution of the solar energy in the spectral region > 200 µm , the Rayleigh-Jeans law (https:// en.wikipedia.org/wiki/Rayleigh–Jeans_law) is used to estimate this part of the energy and the result is included in the total solar flux. It should be emphasized here that although the wavelength edge of the last band is 5.0 µm , the flux determined in this band actually includes those from the wavelengths between 5 and 200 µm . This is to ensure that the sum of the spectral fraction of the solar flux in each band is equal to one. For this reason, the solar tail flux beyond 200 µm estimated by the Rayleigh-Jeans law is also included in the last band flux calculation. Similarly, the total optical air mass can be obtained in the same

Table 1 Information of BSRN stations used in this study. The last column shows the number of data samples with quality code zero. The values in the parenthesis are the fraction in the percentage of “good” data points compared to the total number of data. Code

Station

Latitude

Longitude

Elevation

Surface type

Start date

Used data length

N QC0 (%)

BON CAB CAR FUA PAL PTR SBO SOV SMS SXF TAM TIK TOR XIA

Bondville Cabauw Carpentras Fukuoka Palaiseau, SIRTA Observatory Petrolina SedeBoqer Solar Village São Martinho da Serra Sioux Falls Tamanrasset Tiksi Toravere Xianghe

40.07 ° N 51.97° N 44.08° N 33.58° N 48.71° N 9.07 ° S 30.86 ° N 24.91 ° N 29.44 ° S 43.73 ° N 22.79 ° N 71.59 ° N 58.25 ° N 39.75 ° N

88.37 ° W 4.93 ° E 5.06 ° E 130.38 ° E 2.21 ° E 40.33 ° W 34.78 ° E 46.41 ° E 53.82 ° E 96.62 ° W 5.53 ° E 128.92 ° E 26.46 ° E 116.96 ° E

213 0 100 3 156 387 500 650 489 473 1385 48 70 32

Grass Grass Cultivated Asphalt Concrete Concrete Desert rock Desert, sand Concrete Grass Desert, rock Tundra Grass Desert, rock

Jan.1995 Feb.2005 Sep.1996 Apr.2010 Jun.2003 Dec.2006 Jan.2003 Oct.1998 Apr.2006 Jun.2003 Mar.2000 May.2010 Jan.1999 Jan.2005

Jun.1996-Dec.2008 Oct.2013-Feb.2014 Aug.2003-Mar.2014 Jun.2012-Jul.2014 Oct.2005-Sep.2013 Jun.2009-May.2013 Jan.2003-Dec.2012 Jan.1999-Dec.2001 Sep.2009-Jan.2013 Jun.2004-Jun.2009 Feb.2006-Dec.2013 May.2011-Oct.2012 Jun.2002-Apr.2014 Jan.2005-May.2012

3,361,077(96.8) 5,872,765 (99.8) 5,172,098 (99.5) 324,173 (96.0) 5,297,794(97.7) 2,913,844 (98.5) 78,823(95.2) 219,698 (99.1) 30,904,427 (97.9) 928,069 (95.5) 7,596,355 (88.8) 2,156,374 (97.8) 9,044,435 (98.3) 5,214,486 (97.7)

270

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interference removed and hence some evaluation data selected are still contaminated by cloud. For this reason, the method developed by Long and Ackerman (L&A) (Long and Ackerman, 2000) for identifying the condition of clear skies from broadband GHI and diffuse irradiance measurements is used to further filter the radiation data. The L&A algorithm is based on the four threshold-based tests and has been tested by Ruiz-arias et al. (2018) using the observations at the BSRN Tamanrasset station. They have found that the L&A method can effectively remove the DNI data contaminated by clouds. However, it also removes almost all cloudless observations below about 300 W m−2. They further identified the problem that the L&A method rejects most of the cloudless situations during which atmospheric turbidity is high, including dust storms. To solve this problem, they proposed a relaxed version of the L&A by removing the 4th test which, based on the diffuse-to-total solar irradiance ratio, is the largest source of problems in regions with frequent high turbidities. We thus adopted the relaxed version of the L& A method to filter the radiation data. The temporal resolution of the DNI data at the BSRN stations is generally higher than that of AERONET, hence in forming our evaluation datasets, we select the DNI data at the BSRN stations so as to match, in time, the AERONET data, i.e. the final dataset has the same temporal resolution as that of the AERONET. The BSRN irradiance data are regular 1-min averages whereas the times of AERONET data are highly irregular. The DNI data that are closest to the AERONET in time are selected to merge with the AERONET data.

Table 2 Statistical tests of modelled DNI against BSRN observations under clear-sky conditions. ST Code

Mean DNI

MBD %

MAD %

RMSD %

U95 %

CC

N

BON CAB CAR FUA PAL PTR SBO SMS SOV SXF TAM TIK TOR XIA Total

768.24 690.26 749.62 631.15 701.39 746.21 754.8 913.25 693.30 800.55 778.15 747.06 749.17 648.51 733.20

−3.60 −6.53 −5.08 −1.99 −4.90 −1.48 −2.44 −5.95 0.04 −2.044 −0.89 −3.20 −5.05 1.52 −3.03

3.89 6.16 5.15 4.95 5.06 2.96 3.77 6.36 1.92 3.30 2.07 3.86 5.20 3.71 4.04

4.79 6.96 5.81 6.14 6.20 3.75 4.71 7.53 2.41 4.14 2.49 4.66 5.80 4.63 4.95

11.01 14.01 12.26 16.05 13.82 9.66 11.77 16.68 6.49 10.3 6.47 10.50 12.29 12.10 11.98

0.987 0.987 0.989 0.973 0.985 0.991 0./983 0.962 0.997 0.976 0.997 0.982 0.990 0.989 0.986

4525 155 30,401 3413 8423 2500 545 8 12,235 3547 13,472 573 12,334 7580 99,711

Wm

2

• Root mean squared difference (RMSD)% • Uncertainty at 95% confidence (U95) • Correlation coefficient (CC). The definition of each of these statistics is provided by Gueymard (2014). Note the MBD, MAD, and RMSD are all expressed in percent of the mean observation. Results at each of the BSRN stations are presented in Table 2. The statistics show that the MBD of the modelled DNI from BSRN stations ranges between −6.53% and 1.52% with a negative bias at most stations, while the RMSD ranges between 2.41% and 7.53%. The MBD and RMSD of all samples are −3.03% and 4.95%, respectively As shown in Table 1, the 14 BSRN stations are spread over various climatic areas of the world, including the polar station of Tiksi at 71.59 ° N, the hot tropical station of Petrolina at 9.07° S, and the highly polluted station of Xianghe at 39.75 ° N. However, the statistical test results in Table 2 are very uniform among these stations. This implies that the current DNI model is not sensitive to different climate environments.

5. Evaluations under clear-sky conditions using BSRN data Fig. 2 shows the scatter density plot between the clear-sky DNI results derived using the modified scheme and observations from all 14 BSRN stations. The agreement between model and observation is much improved compared with the old model (results not shown here). Nevertheless, the model tends to underestimate the clear-sky DNI and this can be clearly seen in the histogram shown on the right-hand panel and Table 2 below. The overall performance of the current DNI model is evaluated with the following statistics and indices of performance:

• Mean bias difference (MBD)% • Mean absolute difference (MAD)%

Fig. 2. Density plot for comparison of modelled DNI with observations under clear-sky using the BSRN datasets. The number of data samples in the plots is 99711. The right-hand panel shows the histogram of the model DNI errors. 271

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6. Comparison with other schemes

Table 3 General input variables required by 6 models for DNI under clear-sky condition. p: surface pressure; pw: precipitable water; uo3 : total zone column amount; uco2 : carbon dioxide volum mixing ratio; a : AOD; a500 :AOD at 500 nm; Model

Inputs

Class in GR15

Solar constant

DISC ESRA MMAC REST2

GHI p, pw, p, pw, , uo3 , a p, pw, uo3 , un, , 1 , 2 p, pw, uo3,uco2 , 1, 2, a500 p, pw, uo3 , uco2 , a ,

E D C B

1367.0 1367.0 1353.0 1361.1

SMARTS SUNFLUX

In this section, we compare our method with other schemes. Many DNI schemes have been developed over the past decades. Gueymard and co-authors have conducted a number of validations to test existing and newly developed models. Their most recent validation project was conducted in 2018 (Ruiz-Arias and Gueymard, 2018). Here we compare our method with the models evaluated in their previous project (Gueymard and Ruiz-Arias, 2015) (hereafter referred to as GR15). 24 models have been validated in the GR15 paper and they were grouped in five classes, based on the type of aerosol-related input data they require, from simple to more detailed. It may be natural to compare our model results with these well-established models. It may not be necessary to compare the current scheme with all these models in this

1361.1 1360.8

Fig. 3. Comparison of clear-sky DNI determined using six models with observations from 14 BSRN stations. The upper-colored panels are scattered density plots. The lower black-white panels show the histogram distribution of the relative frequency of the difference between the model and observation. 272

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study, which is beyond the scope of this paper. Alternatively, we pick up the best models identified in GR15 from each class to compare with the current scheme. Class A has only one reference model, RRTMG which is a radiative transfer scheme used in the NWP and climate models and cannot be used in this comparison due to the restriction of the input data. Therefore, four models are selected from classes B to E and these are the DISC model (Maxwell, 1987) from class E; ESRA model (Rigollier et al., 2000) from class D; MMAC model (Davies et al., 1975; 1988) from class C, and REST2 (Gueymard, 2008) from class B. The REST2 is a two-band model for the DNI prediction and the spectral band limits are the same as those of the SUNFLUX. These schemes have been briefly described by Gueymard and Ruiz-Arias (2015) and Gueymard (2012a). Here, we want to emphasize two points. The first one, as mentioned by Gueymard (2012a), is that the MMAC model was originally named as MAC (Davies et al., 1988, 1975) and has evolved slightly from its original version. The modified version described by Gueymard et. al., (2003) is used in this work. The second one is that the REST2 model was developed based on a reference calculation using a more accurate scheme SMARTS (Simple Model of the Radiative Transfer of Sunshine) (Gueymard, 2019, 2001, 1995). The SMARTS code has been implemented in our system, it is thus included in the comparisons presented below. SMARTS is a highly parameterized scheme and can produce most solar radiative variables using only relatively limited data including variables of PW, total column ozone, aerosol optical depth, surface albedo, etc. Hence, the dataset collected from the BSRN and AERONET stations basically meets the input requirement by SMARTS code. The option of user-defined aerosol model ‘USER’ for the input card 8 of the SMARTS code is selected so that more realistic aerosol optical parameters can be used. The option 'USER' requires the input of Ångström turbidity coefficients, 1 , 2 and , where, for spectral regions separated at 1 , 2 are the respective values of 500 nm. In this work, the AERONET AOD data at the number of spectral and using a linearization of channels are used to determine Ångström law:

ln

= ln

ln

modified models are clearly better than the original ones. To present a clear comparison, the statistical testing parameters in the predicted DNI by 6 models for all BSRN stations are plotted in Fig. 5. The stacked bars show the differences in these statistics from the two original (upper red bar) and modified models (colour bar below red). Consider the overall results excluding the modifications first; the results from REST2 are the best, followed by SMARTS and SUNFLUX. The performances from the other three models are slightly worse. With the applications of Eqs. (3 to 4), the statistics from ESRA and MMAC models are substantially improved. The statistical parameters for these two models are comparable to those of SMARTS, REST2, and SUNFLUX. This result indicates that the optical air mass Eq. (3) and the aerosol transmittance Eq. (4) not only improve the current DNI model but are also useful to improve the representation of aerosol in other similar models. 7. Sensitivity test The evaluations presented in the previous sections are subject to a range of uncertainties. Uncertainties associated with observational radiation data can be found on the manufacturer's websites and related studies in the literature. However, quantification of uncertainty in the model input data is difficult to assess, given the absence of reporting via standard references. In order to examine the possible error responses of modelled DNI to uncertainties of the input data, we conducted two sensitivity tests. The first one is to test the sensitivity of MBD against the individual input variable of aerosol, water vapour, and ozone amount using the dataset of BSRN. To show the model sensitivity to a specified variable, the effects from the changes in other variables must be removed. To achieve this goal, we choose the modelled MBD results against a specified input variable for a condition that the other variables are limited to a very narrow range. The input data from all BSRN stations range from 0.05 to 0.73 for AOD at 500 nm, 0.04 to 5.42 cm for PW, and 252 to 387 Dobson Unit (DU) for ozone. The limited variable ranges are then defined 1.4 ± 0.01 cm for PW, 340 ± 1 DU for ozone, and 0.2 ± 0.01 for AOD. We specify these limited ranges rather than fixed constants to obtain a certain number of data samples. For example, to identify a relationship between MBD and AOD, we choose the modelled results that meet the limited ranges of PW in 1.4 ± 0.01 cm and ozone in 340 ± 1 DU. The similar processes apply for searching the relationships with the PW and ozone. The second one is to compare the daily aerosol variability index (AVI) defined by (Gueymard, 2012b):

(6)

where is a wavelength in µm . Three pairs of and are determined using Eq. (6). The first pairs are broad-band values of and that are determined using the AOD data available from all spectral channels. The second pairs are 1 and 1 that are determined using the spectral 0.5µm . The third pairs are 2 and 2 and determined AOD data for using the spectral AOD data for > 0.5µm . Note that the REST2 model also uses 1, 2 , and to determine AOD in the two spectral bands, but values are defined for spectral regions separated at 700 nm. To distinguish the values from those used in SMARTS model, we use 1 and 2 to represent the values used in the REST2 model and they are determined for spectral regions separated at 700 nm. Table 3 lists the input variables required by 6 models evaluated in this section. Fig. 3 shows the scatter density plots of the modelled DNI from the six models against observations for all 14 BSRN stations. It is seen that the results from DISC scheme scatter considerably and these are apparently due to its simple parametric nature. The scatter plots for the rest five models are comparable, but those from the MMAC and ESRA models show negative biases and this can be clearly seen in the histogram plot on the lower set of the plots. The histogram also indicates that the REST2 and SMARTS models perform better because the maximum frequencies fall in zero interval. Since the performance of SUNFLUX is improved significantly by the new aerosol direct transmittance Eq. (2), it is worth examining whether this equation can be applied to the other models to improve the performance in these models. For this reason, the optical air mass in the ESRA model is replaced by Eq. (3). The full-band aerosol transmittance from Eq. (4) is used in the MMAC model. Fig. 4 shows the results. It is seen that the negative biases in DNI from the original models (shown in the left-hand panels) are reduced (right-hand panels). While the scatter pattern from the modified ESRA model remains, the results from the

AVI =

1 M

m = 12 N

| m=1 i=1

i

¯ |/ N , m

(7)

where i is the Ångström turbidity coefficient during day i of month (daily average), ¯m is the long-term daily average (climatology) for that month, N is the number of days, and M = 12 is the number of months. To further evaluate the sensitivity of the instantaneous DNI to concomitant changes in , Gueymard (2012b) defined an aerosol sensitivity index (ASI) that is obtained as the ratio between the relative anomaly in DNI and the anomaly in . Both the AVI and ASI concepts are used here to evaluate the aerosol variability and aerosol-induced variability in DNI. However, our evaluations are performed on a daily basis and hence we use the terminology Daily Sensitivity Index (DSI) to replace ASI in the following discussion. Note that our DSI calculation is slightly different from the ASI defined by Gueymard (2012b). The daily relative anomaly in DNI is used in the DSI calculations to obtain the DSI values with a similar magnitude to those of aerosols so that both the AVI and DSI results can be shown in the same figure (Fig. 7 below). The first tests were conducted for 5 models. The DISC model is not included because it only depends on GHI. The ESRA model is also not included in the test for ozone because it does not use ozone as the input 273

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Fig. 4. Comparison of modelled DNI using the original ESRA and MMAC models (a, c) with those using the modified models with Eqs. (3) and (4) (b, d).

sorted in ascending order and the DSIs are plotted following the aerosol AVI orders. It is seen that the aerosol AVIs vary from 0.026 at São Martinho da Serra, Brazil to 0.143 at Xianghe, China. Clearly, the aerosol AVIs at the observational sites with the desert or heavy pollutant environments are generally larger compared with those at the other sites. The large aerosol AVIs also cause large DSIs in DNI, indicating the sensitivity of DNI to the variability of aerosols. These results are generally consistent with those presented by Gueymard (2012b). The DSIs from the Sunflux, SMARTS, REST2, and MMAC are generally comparable with that of the observations. It is surprising to see that most DSI values for the DISC model are small, especially for those of larger aerosol AVIs. This result is probably due to the fact that the DISC model uses the observed GHI as an input and therefore the effects of aerosols have been largely included in the GHI values. 8. Conclusions In this paper, we have discussed the necessity for modification of the SUNFLUX scheme for the DNI calculations. The original scheme has errors in the reference calculations and aerosol transmission calculations. These errors led to significant overestimations in the modelled DNI. To correct these errors, the DNI parameterization in the SUNFLUX scheme has been modified using the new reference results determined by the updated radiative transfer model SES2. The new aerosol transmittance scheme for the DNI is developed and used to replace the original scheme. The modifications described in this paper have been evaluated using the observational datasets from the BSRN network. The evaluation results under clear-sky conditions using the BSRN dataset are encouraging. All 14 stations have MBD ranging between −6.53% and 1.52% and the RMSD ranges between 2.41% and 7.53%. The MBD of DNI for total samples is −3.03% and the RMSD is 4.95%. The improved model was also compared with five other parametrized models using the input dataset of BSRN. The results show that the performance of the current model is comparable with that of the SMARTS and REST2 models that are ones of the recognized best radiative models available for estimating the surface solar radiation. The more useful result from these cross-comparisons is that the performances of the two selected models: ESRA, and MMAC can be improved using the optical air mass scheme developed in this study. The improvements in the ESRA and MMAC models are significant and the performance of statistics in these models is very close to that of the SMARTS model.

Fig. 5. Comparison of statistical parameters in the predicted DNI by 6 models for 14 BSRN. Stacked bars represent the differences of statistics between results from the two original models and modified models. MBDs are plotted as absolute values, but indicated with ‘–’sign for the negative values and ‘+’ for the positive values.

variable. Fig. 6 shows the results of this test. The left column panels present the scatter plots from SUNFLUX model. The slopes of the linear fit of MBD to each variable for each model are printed in the right column panels. These slopes represent the variation trends of MBD with each variable. It is seen that the MBDs from SUNFLUX tend to increase with the increasing AOD, but decrease with the increasing amounts of PW and ozone. Looking at the plotted slope values in the left column panels for all models, we can see that the MBD slopes relative to the AOD are positive for all 5 models (top right panel) and those relative to ozone are negative (bottom right panel). These results indicate that the model's behaviors relative to these two variables are consistent. This is not true for the sensitivity to PW. As shown in the middle right panel, the slopes for SUNFLUX and SMARTS are negative while whose for the other three models are positive. The MMAC model has the least sensitivity to AOD while ESRA has the largest sensitivity. For the sensitivity to PW, the SMARTS model has the least slope while ESRA again has the largest. The second test results are displayed in Fig. 7. The aerosol AVIs are 274

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Fig. 6. Variations of MBD of SUNFLUX scheme with AOD (top left panel for limited ranges of PW and ozone), PW (middle left panel for limited ranges of AOD and ozone), and ozone (bottom left panel for limited ranges of PW and AOD). The slopes of a linear fit of MBD to each variable for each model are displayed in the right-hand panels.

The evaluation results have shown that the modelled DNIs have negative biases at most stations. These negative biases can be partially attributed to ignoring the scattering contribution of the circumsolar radiation in the DNI model. To further improve the DNI model, we have developed a new method to estimate the contribution of the circumsolar radiation, which has been presented in part II of this study

(Sun et al., 2020). Sensitivity tests were performed to examine the sensitivities of modelled DNI to the variability of input data. The variation of MBDs with the selected variable while the other variables are held roughly constant is tested using the slope parameter of the least square fit of the two variables. For the model sensitivity to AOD, all 5 models tested Fig. 7. Daily aerosol variability index (AVI) and daily sensitivity index (DSI) determined using AERONET AOD data and observational DNI data and model predictions from 14 BSRN stations. The legend 'AOD' represents the AVI of aerosol; 'OBS' denotes the DSI of observed DNI, and the others are the DSI of DNI from 6 models. The observed and modelled DSIs are determined using the daily relative anomaly in DNI.

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have positive slopes with the maximum slope occurring with the ESRA model and the minimum with the MMAC model. For the sensitivity to ozone, all four models tested have very small negative slopes. These two tests indicate that all models have consistent sensitivities relative to AOD and ozone. This argument, however, does not hold for the model sensitivities to PW. The MBD slopes relative to PW for SUNFLUX and SMARTS are negative while those for the rest three models are positive. The sensitivity of modelled and observed DNI to the variability of AOD is further examined using the Aerosol Variability Index (AVI) and daily sensitivity index (DSI), which are calculated from Ångström turbidity coefficient and observed and modelled DNI. The results show that DSI from both modelled and observed DNI is sensitive to that of AOD, i.e. the higher the aerosol AVI the greater the DSI in DNI. This result is consistent with that previously identified by Gueymard (2012b). Finally, the Fortran source codes of SUNFLUX are available from the second author via email request to [email protected].

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