# Assessments of Solar, Thermal and Net Irradiance from Simple Solar Geometry and Routine Meteorological Measurements in the Pannonian Basin

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## Abstract

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## 1. Introduction

## 2. The Methodology

#### 2.1. Terminology

^{−2}), including time-averaged values (for example, hourly or daily irradiation).

^{−2}).

#### 2.2. Description of Key Variables

${K}_{inp}$ | Global solar irradiance is radiant flux emitted from the Sun and received at the Earth’s surface separated in two basic components: direct and diffuse. Global solar irradiance is a measure of the rate of total incoming solar energy, both direct and diffuse, on a horizontal plane at the Earth’s surface. It depends on position of the Sun in the sky, season, time of the day and turbidity of the atmosphere. Turbidity mostly depends on the cloudiness, humidity, content of aerosol particles and, of course, from the pressure (amount of the air column). |

${K}_{out}$ | Reflected solar irradiance is part of global solar irradiance that is reflected from the Earth’s surface. It depends on the global solar irradiance and the surface albedo (function of the angle of solar elevation and characteristics of the ground surface). |

${L}_{inp}$ | Incoming longwave (LW) irradiance is a downward flux of thermal radiation emitted from the atmospheric molecules (such as H_{2}O, CO_{2} and O_{3}); aerosol particles and clouds per unit horizontal area in a given time period. It depends, first of all, on cloudiness, temperature precipitable water and turbidity of the atmosphere. |

${L}_{out}$ | Outgoing (upwelling) longwave irradiance represents a redistribution of the absorbed global solar irradiance. The power of this energy emitted from the Earth’s surfaces per unit area and in the given time is called thermal (terrestrial) irradiance. Besides the global solar irradiance, it depends on the temperature of the Earth’s surface (or atmospheric temperature). |

${Q}_{net}$ | Net irradiance is often convenient to split into four components: ${K}_{inp}$, ${K}_{out}$, ${L}_{inp}$ and ${L}_{out}.$ Therefore, net irradiance is a sum of these components.
$${Q}_{net}={K}_{inp}-{K}_{out}+{L}_{inp}-{L}_{out}$$
In our case, the signs of all the surface radiation balance components were positive or zero. |

ϕ | The angle of solar elevation (rad) |

φ | Latitude (rad) |

λ | Longitude (rad) |

h | Hour angle (rad) |

${h}_{H}$ | Hour angle used in [5] (rad) |

${h}_{F}$ | Hour angle used in [11] (rad) |

t | Time UTC |

DOY | Day of the year |

$\mathsf{\Delta}{t}_{d}$ | The duration of full rotation of the Earth (86,400 s) |

${t}_{h}$ | The time distance from $t$ to the culmination of the Sun in seconds (s) |

${t}_{CET}$ | Central European Time |

EQT | Equation of Time, the difference between the true and averaged local times |

N | Total cloud cover (octas) |

${N}_{CH}$ | Covering of clouds genera cirrus, cirrocumulus and cirrostratus (octa) |

${N}_{CM}$ | Covering of clouds genera altocumulus, altostratus and nimbostratus (octa) |

${N}_{CL}$ | Covering of clouds genera stratocumulus, stratus, cumulus and Cb. (octa) |

$A$ | Albedo |

${A}_{1}$ | Height solar elevation albedo for specific surface, according Geiger et al. [36] |

${Aee}^{\prime}$ | Calculated albedo depending on ${A}_{1}$ and $\varphi ,$ according to Nyren and Gryning [37] |

$Abb$ | Calculated albedo depending on $N$ and $\varphi ,$ according to Beljaars and Bosveld [38] |

${\epsilon}_{0}$ | Surface emissivity |

ρ | Air density (kg m^{−3}); ρ = f (p, T, e) |

$e$ | Water vapor pressure (hPa) |

$p$ | Pressure (hPa) |

$T$ | Temperature (K) |

${T}_{s}$ | Surface temperature (K) |

#### 2.3. Description of the Key Method

${}_{H}$ | Holstlag and Van Ulden methodology for ${K}_{0}\downarrow $ calculation [5] |

${}_{F}$ | Foken methodology for ${K}_{0}\downarrow $ calculation [11] |

${}_{kc}$ | Kasten and Czeplak [39] parameterization of cloudiness for incoming solar radiation |

${}_{bg}$ | Burridge and Gadd [40] parameterization of cloudiness for incoming solar radiation |

${}_{SBb}$ | Stefan–Boltzmann (SB) low with ${\epsilon}_{0}=1$ |

${}_{SBg}$ | Stefan–Boltzmann low with ${\epsilon}_{0}=0.98$ |

${}_{Jac}$ | Cloudiness parameterization for downwelling LW radiation based on Jacobs [49] |

${}_{M\&C}$ | Cloudiness parameterization for downwelling LW radiation based on Maykut and Church [50] |

${}_{M\&C}$ | Cloudiness parameterization for downwelling LW radiation based on Iziomon et al. [51] |

${}_{Izi}$ | Cloudiness parameterization for downwelling LW radiation based on Iziomon et al. [51] |

${}_{HVU}$ | Cloudiness parameterization for downwelling LW radiation based on Swinbank [47] and Dilley and O’Brien [48] |

${}_{Nie}$ | Cloudiness parameterization for downwelling LW radiation based on Niemelä et al. [43] |

## 3. The Description of the Datasets

- temperature, relative humidity and wind speed, at 1, 2, 4 and 10 m;
- infrared ground surface temperature;
- two levels of soil temperature and humidity (5 and 10 cm) and
- all radiation balance components: global solar radiation or ${K}_{inp}$, reflected solar radiation ${K}_{out}$, incoming longwave (LW) radiation ${L}_{inp}$ and outgoing LW radiation ${L}_{out}$.

_{2}and sensible and latent heat fluxes were also measured directly using the CSAT-3 and LI-7500 instruments with the Campbell data acquisition program.

^{−2}. Regarding the downward and upward longwave sensors, the differences, because the downward sensor was not shaded and ventilated, were between ±3.5% and 2%, respectively. The addition of the personal checking of measuring sites was done every 3 months when the leveling of the sensors was checked and cleaning was done. The detailed calculations of uncertainty of the calibration factors of the solar sensors is not part of the relevant working instructions.

## 4. Assessments of Solar Irradiance

#### 4.1. Assessments of Downwelling Shortwave Solar Radiation for Clear Sky

^{−2}in 11-h data, while Holtslag’s calculation had nine such situations. The errors over 50 W m

^{−2}took up about one-quarter of the hourly data, mostly during the winter period, and were fewer for Holtslag’s than Foken’s calculations.

#### 4.2. Assessments of Downwelling Shortwave Solar Irradiance for All Sky Condition

^{−2}, but about 500 of them had errors greater than 200 W m

^{−2}. There is no comment here whether these errors were the result of the crude method for estimation of the input solar irradiance or even sporadic errors in the measurements. Namely, the radiation balance components are measured with first-class radiation sensors separately using the Campbell CR1000 datalogger, so the measurements are quality-controlled and mostly reliable. Therefore, without analyzing the reason, all situations when the differences between the measured and estimated solar radiations were fewer than: 50, 100, 200,..., 700 W m

^{−2}were countered, and the results are shown in Table 5.

^{−2}. In that case, the MAE became smaller around 25% and RMSE around 50%, and the coefficient of correlation grew to 0.96, while the BIAS stayed similar. The few greatest absolute errors (>200 W m

^{−2}) seemed to be connect to the Kasten and Czeplak [39] method for describing cloudiness, because after excluding these situations, the RMSE and MAE became 10% smaller for this method than for the Burridge and Gadd [40]-type cloudiness parameterizations for both clear sky calculations.

^{−2}, although the few greatest AE relate to this parameterization. All methods have relative errors below 25% in about 45% of data hours. The usual errors are connected with the short periods when the sky fast becomes either overcast or clear, or when solar elevation is low, especially during the winter period.

- (1)
- (2)
- Holtslag’s clear sky model and Burridge and Gadd [40] cloudiness parameterization were the best for low solar elevation, when, especially during sunset, Foken’s calculation permanently overestimated the measurements.

#### 4.3. Assessments of Upwelling Shortwave Solar Irradiance

#### Assessments of Albedo

## 5. Assessments of Net Irradiance

#### 5.1. Daytime

^{−2})) from the routine meteorological measurements, an empirical equation was taken, such as Holtslag and Van Ulden [5].

^{−3}). Based on the ideal gas low,

^{−1}) that is calculated as $q=0.622\xb7\frac{e}{p-0.378\xb7e}$, where $e$ is the water vapor pressure in the same unit as the pressure (hPa), which is calculated based on the relative humidity ($Rh)$ and the saturated water vapor pressure (${e}_{s}$) in the temperature, $T,$ is calculated as follows:

^{−1}K

^{−1}).

_{err}< 25%) for the calculation for clear sky global solar radiation, cloudiness calculations and albedo, where albedo ${Aee}^{\prime}$ does not depend on the cloudiness, whereas $Abb$ does. The second numbers in the column represent statistical errors of the net radiation but only for periods with high solar elevation when the sensible heat flux ($H$) was positive. The moment when the sensible heat flux changed its direction was calculated, according to De Bruin and Holtslag [10], from equation:

#### 5.2. Nighttime

- (1)
- During the midday period, when sensitive heat flux is directed upwards, Foken’s calculation is clearly better than Holtslag’s calculation.
- (2)
- Holtslag’s method is slightly better for low solar elevation when a sensible heat flux is directed downwards.
- (3)
- The difference between the measured (${Q}_{net}$) and calculated (${Q}^{\ast}$) net irradiance is very small when we compared both methods for mathematically described cloudiness.
- (4)
- An albedo that includes cloudiness ($Abb$) is much better than one (${Aee}^{\prime}$) that does not.
- (5)
- Measured global solar radiation makes an estimation of the net irradiance significantly better in comparison to when this value is estimated.
- (6)
- Using the same value for global solar radiation, Foken’s estimation for net irradiance is slightly better than Holtslag’s.
- (7)
- The estimation of nighttime net thermal radiation is crude and has systematic errors.

## 6. Assessment of Longwave Irradiance

#### 6.1. Assessment of Upwelling LW Irradiance

- (a)
- by Stefan–Boltzmann: $L{\uparrow}_{SB};{\epsilon}_{0}=1$ and $L{\uparrow}_{SB};{\epsilon}_{0}=0.98$
- (b)
- by global solar irradiance, ($K\downarrow $) and (${K}_{inp}$): $L{\uparrow}_{KFkcAbb}$ and $L{\uparrow}_{{K}_{inp}Abb}$
- (c)
- by the assessment of ${Q}^{\ast}$: $L{\uparrow}_{{Q}_{FFkcAbb}^{\ast}}$ and $L{\uparrow}_{{Q}_{F{K}_{inp}Abb}^{\ast}}$
- (d)
- by ${Q}^{\ast}$ included in Equation (35): $L{\uparrow}_{{Q}_{FFkcAbb+{\alpha}_{PT}}^{\ast}}$ and $L{\xb7}_{{K}_{inp}Abb+{\alpha}_{PT}}$

#### 6.1.1. Assessment of Downwelling LW Irradiance for Clear Sky

#### 6.1.2. Assessment of Downwelling LW Irradiance for All Sky Conditions

^{−}

^{2}for ${\left({L}_{04}\downarrow \right)}_{Izi}$ and 17 W m

^{−}

^{2}for ${\left({L}_{03}\downarrow \right)}_{Izi}$, MAE between 18 W m

^{−}

^{2}for ${\left({L}_{06}\right)}_{Izi}$ and 26 W m

^{−}

^{2}for ${\left({L}_{05}\downarrow \right)}_{Izi}$, RMSE between 22 W m

^{−}

^{2}for ${\left({L}_{06}\downarrow \right)}_{Izi}$ and 32 W m

^{−2}for ${\left({L}_{05}\right)}_{Izi}$ and a correlation coefficient 0.91 for almost all parameterizations except ${\left({L}_{01}\downarrow \right)}_{Izi}\mathrm{and}{\left({L}_{05}\downarrow \right)}_{Izi},\mathrm{which}\mathrm{were}\mathrm{less}$. Slightly better statistical errors than all three above-mentioned assessments of downwelling LW irradiance were described by Niemela et al. [43] and shown in the grey column of Table 13.

^{−2}for Niemelä et al. [43] parameterization was very stable and varied between 24 for Dilley and O’Brien [48] to 29 for Swinbank [47] and Iziomon et al. [51] clear sky downwelling LW parameterization. According to the relative errors, it was noticed that, for almost all parameterizations, the relative errors were smaller than 25%. However, in order to distinguish slight differences between them, percentages for the relative errors smaller than 10% are offered in Table 14. The columns represent different clear sky parameterizations, while the rows represent additional cloudiness corrections.

## 7. The Statistical Errors and Validation of the Results

^{−2}. The second period was from 2008 to 2017 and included 61,880 data hours for LW radiation and 40,869 data hours with ${K}_{inp}$ > 1 W m

^{−2}(Table 15). We used the parameterization with the proven best results.

- Foken’s calculation for clear sky downwelling solar irradiance (${K}_{0F}\downarrow $)
- Kasten and Czeplak [39] cloudiness correction for downwelling solar radiation ($K{\downarrow}_{Fkc}$)
- Foken’s equation for assessment net irradiance [11] (${Q}_{FFkcAbb}^{\ast})$
- Dilley and O’Brien [48] for parameterization of clear sky downwelling LW irradiance (${L}_{06}\downarrow $)
- Holtslag and Van Ulden [5] cloudiness correction for downwelling LW radiation ${\left({L}_{06}\downarrow \right)}_{HVU}$

## 8. Conclusions

- Measured global solar irradiance makes the estimation of the net irradiance significantly better in comparison to when this value is estimated.
- The estimation of nighttime net thermal irradiance is crude and has systematic errors.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- European Communities. Harmonisation of the Pre-Processing of Meteorological Data for Atmospheric Dispersion Models; COST Action 710: Final Report; EUR 18195; Fisher, B.E.A., Erbrink, J.J., Finardi, S., Joffre, S., Morselli, M.G., Pechinger, U., Seibert, P., Thomson, D.J., Eds.; European Communities: Brussels, Belgium, 2013; p. 431. ISBN 92-828-3302-X. [Google Scholar]
- Crawford, T.M.; Bluestein, H.B. An Operational, Diagnostic Surface Energy Budget Model. J. Appl. Meteorol.
**2000**, 39, 1196–1217. [Google Scholar] [CrossRef] - Wong, H.; Elleman, R.; Wolvovsky, E.; Richmond, K.; Paumier, J. AERCOARE: An overwater meteorological preprocessor for AERMOD. J. Air Waste Manag. Assoc.
**2016**, 66, 1121–1140. [Google Scholar] [CrossRef] - Rzeszutek, M.; Szulecka, A. Assessment of the AERMOD dispersion model in complex terrain with different types of digital elevation data. IOP Conf. Ser. Earth Environ. Sci.
**2021**, 642, 012014. [Google Scholar] [CrossRef] - Holtslag, A.A.M.; Van Ulden, A.P. A simple scheme for daytime estimates of the surface fluxes from routine weather data. J. Appl. Meteorol. Climatol.
**1983**, 22, 517–529. [Google Scholar] [CrossRef] - Beljaars, A.C.M.; Holtslag, A.A.M. Flux Parameterization over Land Surfaces for Atmospheric Models. J. Appl. Meteorol.
**1991**, 30, 327–339. Available online: http://www.jstor.org/stable/26186639 (accessed on 20 July 2021). [CrossRef] - Erbes, G.; Pechinger, U. Comparison of synoptic-based preprocessor estimates. Agric. For. Meteorol.
**1999**, 98–99, 509–519. [Google Scholar] [CrossRef] - Meyers, T.P.; Baldocchi, D.D. Current Micrometeorological Flux Methodologies with Applications in Agriculture. Micrometeorol. Agric. Syst. Agron. Monogr.
**2005**, 47, 381–396. Available online: http://digitalcommons.unl.edu/usdeptcommercepub/500 (accessed on 20 July 2021). - Foken, T. 50 Years of the Monin–Obukhov Similarity Theory. Bound. Layer Meteorol.
**2006**, 119, 431–447. [Google Scholar] [CrossRef] - De Bruin, H.A.R.; Holtslag, A.A.M. A simple parameterization of the surface fluxes of sensible and latent heat during daytime compared with the Penman-Monteith concept. J. Appl. Meteorol. Climatol.
**1982**, 21, 1610–1621. [Google Scholar] [CrossRef][Green Version] - Foken, T. Micrometeorology; Springer: Berlin/Heidelberg, Germany, 2008; p. 305. ISBN 978-3-540-74666-9. [Google Scholar]
- Van Ulden, A.P.; Holtslag, A.A.M. Estimation of Atmospheric Layer Parameters for Diffusion Applications. J. Clim. Appl. Meteorol.
**1985**, 24, 1196–1207. [Google Scholar] [CrossRef][Green Version] - Holtslag, A.A.M.; De Bruin, H.A.R. Applied modeling of the nighttime surface energy balance over land. J. Appl. Meteorol. Climatol.
**1988**, 27, 689–704. [Google Scholar] [CrossRef][Green Version] - Göckede, M.; Foken, T. Ein weitenerentwickelters Holtslag van Uldel schema zur Stabilitätsparametrisierung in der Bodenschicht. In DACH 2001, Wien. Österreichische Beiträge zu Meteorologie und Geophysik; 2001; ISSN Heft Nr. 27/Publ. Nr. 399 (CD-ROM); Available online: https://eref.uni-bayreuth.de/19623/ (accessed on 20 July 2021).
- Foken, T. Micrometeorology; Springer: Berlin/Heidelberg, Germany, 2017; p. 362. ISBN 978-3-642-25440-6. [Google Scholar]
- Das, Y.; Padmanabhamurty, B.; Murty, A.S.N. Some Parameterizations of Radiative Fluxes at Atmospheric Boundary Layer (ABL). J. Atmos. Pollut.
**2014**, 2, 1–5. [Google Scholar] [CrossRef] - Jiang, B.; Zhang, Y.; Liang, S.; Wohlfahrt, G.; Arain, A.; Cescatti, A.; Georgiadis, T.; Jia, K.; Kiely, G.; Lund, M.; et al. Empirical estimation of daytime net radiation from shortwave radiation and ancillary information. Agric. For. Meteorol.
**2015**, 211–212, 23–36. [Google Scholar] [CrossRef][Green Version] - Ishola, K.A.; Mills, G.; Fealy, R.M.; Choncubhair, O.N.; Fealy, R. Improving a land surface scheme for estimating sensible and latent heat fluxes above grasslands with contrasting soil moisture zones. Agric. For. Meteorol.
**2020**, 294, 108151. [Google Scholar] [CrossRef] - Zhu, M.; Yao, T.; Yang, W.; Xu, B.; Wang, X. Evaluation of Parameterizations of Incoming Longwave Radiation in the High-Mountain Region of the Tibetan Plateau. J. Appl. Meteorol. Climatol.
**2017**, 6, 5833–5848. [Google Scholar] [CrossRef] - Liu, M.; Zheng, X.; Zhang, J.; Xia, X. A revisiting of the parametrization of downward longwave radiation in summer over the Tibetan Plateau based on high-temporal-resolution measurements. Atmos. Chem. Phys.
**2020**, 20, 4415–4426. [Google Scholar] [CrossRef][Green Version] - Senese, A.; Maugeri, M.; Diolaiuti, G.A. Comparing Measured Incoming Shortwave and Longwave Radiation on a Glacier Surface with Estimated Records from Satellite and Off-Glacier Observations: A Case Study for the Forni Glacier, Italy. Remote Sens.
**2020**, 12, 3719. [Google Scholar] [CrossRef] - Stettz, S.; Zaitchik, B.F.; Ademe, D.; Musie, S.; Simane, B. Estimating variability in downwelling surface shortwave radiation in a tropical highland environment. PLoS ONE
**2019**, 14, e0211220. [Google Scholar] [CrossRef] - Lindauer, M.; Schmid, H.P.; Grote, R.; Steinbrecher, R.; Mauder, M.; Wolpert, B. A Simple New Model for Incoming Solar Radiation Dependent Only on Screen-Level Relative Humidity. J. Appl. Meteorol. Climatol.
**2017**, 56, 1817–1825. [Google Scholar] [CrossRef] - Acs, F.; Hantel, M. The land-surface flux model PROGSURF. Glob. Planet. Chang.
**1998**, 19, 19–34. [Google Scholar] [CrossRef][Green Version] - Mihailovic, D.T.; Lazic, J.; Leśny, J.; Olejnik, J.; Lalic, B.; Kapor, D. A new design of the LAPS land surface scheme for use over and through heterogeneous and non-heterogeneous surfaces: Numerical simulations and tests. Theor. Appl. Climatol.
**2010**, 100, 299–323. [Google Scholar] [CrossRef] - Weidinger, T.; Nagy, Z.; Baranka, G.; Mészáros, R.; Gyöngyösi, A.Z. Determination of meteorological preprocessor for air quality models in the New Hungarian Standards. Croathian Meteorol. J.
**2008**, 12, 460–464. [Google Scholar] - Badescu, V.; Gueymard, C.A.; Cheval, S.; Oprea, C.; Baciu, M.; Dumitrescu, A.; Iacobescu, F.; Milos, I.; Rada, C. Computing global and diffuse solar hourly irradiation on clear sky. Review and testing of 54 models. Renew. Sustain. Energy Rev.
**2012**, 16, 1636–1656. [Google Scholar] [CrossRef] - Kostić, R.; Mikulović, J. The empirical models for estimating solar insolation in Serbia by using meteorological data on cloudiness. Renew. Energy
**2017**, 114 Pt B, 1281–1293. [Google Scholar] [CrossRef] - Alboteanu, I.L.; Bulucea, C.A.; Degeratu, S. Estimating Solar Irradiation Absorbed by Photovoltaic Panels with Low Concentration Located in Craiova, Romania. Sustainability
**2015**, 7, 2644–2661. [Google Scholar] [CrossRef][Green Version] - Despotovic, M.; Nedic, V.; Despotovic, D.; Cvetanovic, S. Review and statistical analysis of different global solar radiation sunshine models. Renew. Sustain. Energy Rev.
**2015**, 52, 1869–1880. [Google Scholar] [CrossRef] - Maleki, S.A.M.; Hizam, H.; Gomes, C. Estimation of Hourly, Daily and Monthly Global Solar Radiation on Inclined Surfaces: Models Re-Visited. Energies
**2017**, 10, 134. [Google Scholar] [CrossRef][Green Version] - Choi, Y.; Suh, J.; Kim, S.-M. GIS-Based Solar Radiation Mapping, Site Evaluation, and Potential Assessment: A Review. Appl. Sci.
**2019**, 9, 1960. [Google Scholar] [CrossRef][Green Version] - Enríquez-Velásquez, E.A.; Benitez, V.H.; Obukhov, S.G.; Félix-Herrán, L.C.; de Lozoya-Santos, J.J. Estimation of Solar Resource Based on Meteorological and Geographical Data: Sonora State in Northwestern Territory of Mexico as Case Study. Energies
**2020**, 13, 6501. [Google Scholar] [CrossRef] - Liou, K.N. Radiation and Cloud Processes in the Atmosphere. Theory, Observation, and Modeling; United States, Oxford Monographs on Geology and Geophysics; 1992; p. 473. Available online: https://www.osti.gov/biblio/7081459 (accessed on 20 July 2021).
- Guderian, R. (Ed.) Band 1A: Atmosphäre: Anthropogene und Biogene Emissionen—Photochemie der Troposphäre—Chemie der Stratosphäre und Ozonabbau; Springer: Berlin/Heidelberg, Germany, 2011; p. 424. ISBN 3642570879. [Google Scholar]
- Geiger, R.; Aron, R.H.; Todhunter, P. The Climate Near the Ground; Friedrich Vieweg & Sohn Verlasges, GmbH: Brauschweig/Wiesbaden, Germany, 1995; p. 528. ISBN 978-3-322-86584-7. [Google Scholar]
- Nyren, K.; Gryning, S.E. Nomogram for the height of daytime mixed layer. Bound. Layer Meteorol.
**1999**, 91, 307–322. [Google Scholar] [CrossRef] - Beljaars, A.C.M.; Bosveld, F.C. Cabauw Data for the Validation of Land Surface Parameterization Schemes. J. Clim.
**1997**, 10, 1172–1193. [Google Scholar] [CrossRef] - Kasten, F.; Czeplak, G. Solar and terrestrial radiation dependent on the amount and type of cloud. Sol. Energy
**1980**, 24, 177–189. [Google Scholar] [CrossRef] - Burridge, D.M.; Gadd, A.J. The Meteorological Office Operational 10 Level Numerical Weather Prediction Model (December1974); Technical Notes; British Meteorological Office: Bracknell, UK, 1974; p. 57.
- Zheng, Z.; Wei, Z.; Wen, Z.; Dong, W.; Li, Z.; Wen, X.; Zhu, X.; Ji, D.; Chen, C.; Yan, D. Inclusion of solar elevation angle in land surface albedo parameterization over bare soil surface. J. Adv. Model. Earth Syst.
**2017**, 9, 3069–3081. [Google Scholar] [CrossRef] [PubMed][Green Version] - Angström, A. A study of the radiation of the atmosphere. Smith. Misc. Coll.
**1918**, 65, 159–161. [Google Scholar] - Niemelä, S.; Räisänen, P.; Savijärvi, H. Comparison of surface radiative flux parameterization Part I: Longwave radiation. Atmos. Res.
**2001**, 58, 1–18. [Google Scholar] [CrossRef] - Brutsaert, W. On a derivable formula for long-wave radiation from clear skies. Water Resour. Res.
**1975**, 11, 742–744. [Google Scholar] [CrossRef] - Idso, S.B. A set of equations for full spectrum and 8- to 14-mm and 10.5- to 12.5-mm thermal radiation from cloudlessskies. Water Resour. Res.
**1981**, 17, 295–304. [Google Scholar] [CrossRef] - Prata, A.J. A new long-wave formula for estimating downward clear-sky radiation at the surface. Q. J. R. Meteorol. Soc.
**1996**, 122, 1127–1151. [Google Scholar] [CrossRef] - Swinbank, W.C. Long-wave radiation from clear skies. Q. J. R. Meteorol. Soc.
**1963**, 89, 339–348. [Google Scholar] [CrossRef] - Dilley, A.C.; O’Brien, D.M. Estimating downward clear sky long-wave irradiance at the surface from screen temperature and precipitable water. Q. J. R. Meteorol. Soc.
**1998**, 124, 1391–1401. [Google Scholar] [CrossRef] - Jacobs, J.D. Radiation climate of Broughton Island. In Energy Budget Studies in Relation to Fast-ice Breakup Processes in Davis Strait Climatological Overview; Barry, R.G., Jacobs, J.D., Eds.; Institute of Arctic and Alpine Research, Occasional Paper No. 26; University of Colorado: Boulder, CO, USA, 1978; pp. 105–120. ISSN 0069-6145. [Google Scholar]
- Maykut, G.A.; Church, P.E. Radiation Climate of Barrow Alaska, 1962–1966. J. Appl. Meteorol. Climatol.
**1973**, 12, 620–628. [Google Scholar] [CrossRef][Green Version] - Iziomon, M.G.; Mayer, H.; Matzarakis, A. Downward atmospheric longwave irradiance under clear and cloudy skies: Measurement and parameterization. J. Atmos. Sol. Terr. Phys.
**2003**, 65, 1107–1116. [Google Scholar] [CrossRef] - Sellers, W.D. Physical Climatology; The University of Chicago Press: Chicago, IL, USA, 1965; p. 272. ISBN 10: 0226746992. [Google Scholar]
- Offerle, B.; Grimmond, S.B.; Oke, T.R. Parameterization of Net All-Wave Radiation for Urban Areas. J. Appl. Meteorol.
**2003**, 42, 1157–1173. [Google Scholar] [CrossRef] - Nagy, Z.; Szasz, G.; Weidinger, T.; Baranka, G.; Kovacs, N.; Decsi, A. Long term micrometeorological and energy budget measurements in Agrometeorological Observatory in Debrecen. Geophys. Res. Abstr.
**2012**, 14, EGU2012-8915. Available online: https://meetingorganizer.copernicus.org/EGU2012/EGU2012-8915.pdf (accessed on 20 July 2021). - Popov, Z.; Weidinger, T.; Baranka, G.; Nagy, Z. Assessment of net radiation from routine measurements in the Pannonian Region. In Gewex Workshop on the Climate System of the Pannonian Basin; Book of AbstractsOsijek: Osijek, Croatia, 2015; p. 53. Available online: http://www.opb.com.hr/literatura/GEWEX_Workshop_Osijek%202015_v02_20-02-2017-web.pdf (accessed on 20 July 2021).
- Neckel, T.; Montenbruck, O. Tables. In Ahnerts Kalender für Sternfreunde; Kleines Astronomisches Jahrbuch; 52. Jahrgang Sterne und Weltraum; Hüthig GmbH: Heidelberg, Germany, 1999; ISBN 3-87973-9331. [Google Scholar]
- Göckede, M. Das Windprofil in den Untersten 100 m der Atmosphäre unter Besonderer Berücksichtigung der Stabilität der Schictung. Master’s Thesis, Universitat Bayreuth, Bayreuth, Germany, 2000; p. 133. [Google Scholar]
- Younes, S.; Muneer, T. Comparison between solar radiation models based on cloud information. Int. J. Sustain. Energy
**2007**, 26, 121–147. [Google Scholar] [CrossRef] - Didari, S.; Zand-Parsa, S. Estimation of daily global solar irradiation under different skyconditions in central and southern Iran. Theor. Appl. Climatol.
**2017**, 127, 587–596. [Google Scholar] [CrossRef] - Pietras-Szewczyk, M. A GIS Open Source Software Application for Mapping Solar Energy Resources in Urban Areas. E3S Web Conf.
**2019**, 116, 00060. [Google Scholar] [CrossRef] - Stull, R.B. An Introduction to Boundary Layer Meteorology; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1988; p. 670. ISBN 978-94-009-3027-8. [Google Scholar]
- Liston, G.E.; Itkin, P.; Stroeve, J.; Tschudi, M.; Stewart, J.S.; Pedersen, S.H.; Reinking, A.K.; Elder, K. A Lagrangian snow-evolution system for sea-ice applications (SnowModel-LG): Part I—Model description. J. Geophys. Res. Oceans
**2020**, 125, e2019JC015913. [Google Scholar] [CrossRef] - Marsh, C.B.; Pomeroy, J.W.; Wheater, H.S. The Canadian Hydrological Model (CHM) v1.0: A multi-scale, multi-extent, variable-complexity hydrological model—Design and overview. Geosci. Model Dev.
**2020**, 13, 225–247. [Google Scholar] [CrossRef][Green Version] - Warren, S.G.; Eastman, R.M.; Hahn, C.J. A Survey of Changes in Cloud Cover and Cloud Types over Land from Surface Observations, 1971–1996. J. Clim.
**2007**, 20, 717–738. [Google Scholar] [CrossRef][Green Version] - World Meteorological Organization. Guide to Meteorological Instruments and Methods of Observation; CIMO Guide, 2014: World Meteorological Organization (2014) Obseration of Clouds. (WMO-No.8); World Meteorological Organization: Geneva, Switzerland, 2014; Chapter 15, p. 1128; Available online: http://hdl.handle.net/11329/365 (accessed on 20 July 2021).
- Paltridge, G.W.; Platt, C.M.R. Radiative Processes in Meteorology and Climatology; Elsevier: Amsterdam, The Netherlands, 1976; p. 318. ISBN 10:0444414444. [Google Scholar]
- De Rooy, W.C.; Holtslag, A.A.M. Estimation of Surface Radiation and Energy Flux Densities fromSingle-Level Weather Data. J. Appl. Meteorol.
**1999**, 38, 526–540. [Google Scholar] [CrossRef] - Iziomon, M.G.; Mayer, H.; Matzarakis, A. Empirical Models for Estimating Net Radiative Flux: A Case Study for Three Mid-Latitude Sites with Orographic Variability. Astrophys. Space Sci.
**2000**, 273, 313–330. [Google Scholar] [CrossRef] - Duynkerke, P.G. The roughness length for heat and othervegetation parameters for a surface of short grass. J. Appl. Meteorol.
**1992**, 31, 579–586. [Google Scholar] [CrossRef][Green Version] - Holtslag, A.A.M. Surface Fluxes and Boundary Layer Scaling Models and Applications; Scientific Report WR 87-2 of the Royal Netherlands Meteorological Institute; Royal Netherlands Meteorological Institute: De Bilt, The Netherlands, 1987; p. 185.
- Kleczek, M.A.; Steeneveld, G.-J.; Holtslag, A.A.M. Evaluation of the Weather Research and Forecasting Mesoscale Model for GABLS3: Impact of Boundary-Layer Schemes, Boundary Conditions and Spin-Up. Bound. Layer Meteorol.
**2014**, 152, 213–243. [Google Scholar] [CrossRef] - Lábó, E.; Geresdi, I. Numerical modeling of the transfer of longwave radiation in water clouds. Időjárás
**2019**, 123, 147–163. [Google Scholar] [CrossRef] - Agam, N.; Kustas, W.P.; Anderson, M.C.; Norman, J.M.; Colaizzi, P.D.; Howell, T.A.; Prueger, J.H.; Meyers, T.P.; Wilson, T.B. Application of the Priestley–Taylor Approach in a Two-Source Surface Energy Balance Model. J. Hydrometeorol.
**2010**, 11, 185–198. [Google Scholar] [CrossRef] - Iqubal, M. An Introduction to Solar Radiation; Academic Press: Don Mills, ON, Canada, 1983; p. 390. ISBN 0-12-373750-8. [Google Scholar]

**Figure 1.**Micrometeorological (radiation, surface energy budget and profile) measurements in the Agrometeorological Observatory Debrecen. (

**a**) Temperature, relative humidity and wind speed profile measurements on a 10-m tower; (

**b**) Eddy covariance measurements on a 4-m tower; (

**c**) radiation balance components and surface temperature measurements with WMO first class instruments (small photo with a Kipp&Zonen CGR4 pyrgeometer) and (

**d**) Debrecen, Novi Sad and Belgrade locations in the Pannonian Basin.

**Figure 2.**(

**a**) Shortwave downwelling solar irradiance for a clear sky obtained by Holtslag’s and Foken’s clear sky calculations for 2009 (

**upper**panel) and (

**b**) during 201 h in chronological order with clear sky conditions in Agrometeorological Observatory Debrecen. The red line represents the measurements (in the

**middle**). (

**c**) Comparison between the measured and calculated ${K}_{0}\downarrow $ for four clear sky days in different periods of the year (

**lower**panels).

**Figure 3.**Comparisons between the measurements and four models for the calculation of global solar irradiance can be seen with the (

**a**) mean annual daily variations of the global solar irradiance. (

**b**) The hours with the greatest errors.

**Figure 4.**Comparisons between the measurements (${K}_{out})$ and two models for the calculation of shortwave reflected solar irradiance (a) (green line) albedo calculations based on the Nyren and Gryning [37] model, $K{\uparrow}_{FkcAee\u2019}$ and (b) (violet line) using Beljaars and Bosveld [38] methodology, $K{\uparrow}_{FkcAbb}$. Foken’s calculations for clear skies and Kasten and Czeplak [39]-type cloud parameterization was used for the global solar radiation calculations.

**Figure 6.**Mean annual daily variation of $L\uparrow $ for four models and measurements. Models that used routine meteorological measurements included, separately: $K{\downarrow}_{Fkc}$ or ${K}_{inp}$. Abbreviations of the models are same as in Table 9.

**Figure 7.**Mean annual daily variation of all modeled sky downwelling LW irradiance Niemelä et al.

**[43]**combined with all the clear sky parameterizations $({L}_{0i,i=1,\dots ,8}\downarrow )$ (W m

^{−2}) described in Section 6.1.1 are represented with black and grey lines and marked from ${\left({L}_{01}\downarrow \right)}_{Nie}$ in Table 8. Two parameterizations: (${\left({L}_{05}\downarrow \right)}_{HVU}$ and ${\left({L}_{06}\downarrow \right)}_{HVU}$) based on Holtslag and Van Ulden [5] are represented in colored lines. See, also, Table 11, Table 12 and Table 14.

**Figure 8.**Mean annual daily variations of the measured ${Q}_{net}$ and calculated net irradiances: ${Q}_{1}^{\ast}$, ${Q}_{2}^{\ast}$, ${Q}_{3}^{\ast}$ and ${Q}_{4}^{\ast}$.

**Figure 9.**Mean annual daily variations of all four components of the radiation balance. The components are measured and calculated from routine meteorological measurements and observations: ${K}_{Fkc},{K}_{inp}$, $K{\uparrow}_{FkcAbb},{K}_{out}$, ${\left({L}_{06}\downarrow \right)}_{HVU},{L}_{inp}$, $L{\uparrow}_{{Q}_{FFkcAbb}^{\ast}}\mathrm{and}{L}_{out}$.

**Table 1.**List of applied parameterizations for radiation budget components. ($K\downarrow $, $K\uparrow $, $L\downarrow $, $L\uparrow $ and ${Q}^{\ast}$ are calculated, and ${K}_{inp}\mathrm{and}{Q}_{net}$ are the measured values.

Symbol | Radiation | Ref. | Abbr. | Eq. | Function of |
---|---|---|---|---|---|

${K}_{0}\downarrow $ | Global solar radiation for clear sky | [5] | ${K}_{0H}\downarrow $ | (7)–(10) and (12) | = f ($\varphi $,${h}_{H},\phi ,\lambda ,t)$ |

[11] | ${K}_{0F}\downarrow $ | (7)–(9), (11) and (13) | = f ($\varphi $,${h}_{F},\phi ,\lambda ,t)$ | ||

$K\downarrow $ | Global solar radiation | [5,39] | $K{\downarrow}_{Hkc}$ | (12) and (14) | = f ($\varphi $,${h}_{H},\phi ,\lambda ,t,N)$ |

[11,39] | $K{\downarrow}_{Fkc}$ | (13) and (14) | = f ($\varphi $,${h}_{F},\phi ,\lambda ,t,N)$ | ||

[5,40] | $K{\downarrow}_{Hbg}$ | (12) and (15) | = f ($\varphi $,${h}_{H},\phi ,\lambda ,t,$${N}_{CH},{N}_{CM},{N}_{CL})$ | ||

[11,40] | $K{\downarrow}_{Fbg}$ | (13) and (15) | = f ($\varphi $,${h}_{F},\phi ,\lambda ,t,$${N}_{CH},{N}_{CM},{N}_{CL})$ | ||

$K\uparrow $ | Reflected solar radiation | [36,37] | $K{\uparrow}_{FkcAee\u2019}$ | (16) | = f ($K\downarrow $,$A\left(\varphi ,{A}_{1}\right)$) |

[36,41] | $K{\uparrow}_{FkcAee\u2019}$ | (17) | = f ($K\downarrow $,$A\left(\varphi ,{A}_{1}\right)$) | ||

[36,38] | $K{\uparrow}_{FkcAbb}$ | (20) Without snow | = f ($K\downarrow $,$A\left(\varphi ,{A}_{1},N\right))$ | ||

[36,38] | $K{\uparrow}_{FkcAbb}$ | (21) With snow | = f ($K\downarrow $,$A\left(\varphi ,{A}_{1},N\right))$ | ||

${L}_{0}\downarrow $ | Incoming LW radiation for Clear sky | [42,43] | ${L}_{01}\downarrow $ | Table 11 row (1) | = f (T, e) |

[44] | ${L}_{02}\downarrow $ | Table 11 (2) | = f (T, e) | ||

[45] | ${L}_{03}\downarrow $ | Table 11 (3) | = f (T) | ||

[46] | ${L}_{04}\downarrow $ | Table 11 (4) | = f (T, e) | ||

[47] | ${L}_{05}\downarrow $ | Table 11 (5) | = f (T) | ||

[48] | ${L}_{06}\downarrow $ | Table 11 (6) | = f (T, e) | ||

[43] | ${L}_{07}\downarrow $ | Table 11 (7) | = f (T, e) | ||

[46] | ${L}_{08}\downarrow $ | Table 11 (8) | = f (T, e) | ||

$L\downarrow $ | Incoming | [43,49] | ${\left({L}_{0i}\downarrow \right)}_{Jac}$ | (36) | = f (N,${L}_{0i,i=1,\dots ,8}\downarrow $) |

LW | [43,50] | ${\left({L}_{0i}\downarrow \right)}_{M\&C}$ | (37) | = f (N,${L}_{0i,i=1,\dots ,8}\downarrow $) | |

radiation | [51] | ${\left({L}_{0i}\downarrow \right)}_{Izi}$ | (38) | f (N,${L}_{0i,i=1,\dots ,8}\downarrow $) | |

[5,47] | ${\left({L}_{05}\downarrow \right)}_{HVU}$ | (39) | = f (N,${L}_{05}\downarrow $) | ||

[5,48] | ${\left({L}_{06}\downarrow \right)}_{HVU}$ | (40) | = f (N,${L}_{06}\downarrow $) | ||

[43] | ${\left({L}_{0i}\downarrow \right)}_{Nie}$ | (41) | f (N,${L}_{0i,i=1,\dots ,8}\downarrow ,L\uparrow $) | ||

$L\uparrow $ | Outgoing | SB low | $L{\uparrow}_{SBb}$ | (30) | = f (${\epsilon}_{0},{T}_{s}$) |

LW | [5,52,53] | $L{\uparrow}_{KFkcAbb}$ | (31) and (33) | = f (T, $K{\downarrow}_{Fkc}$,$Abb)$ | |

radiation | [5,52,53] | $L{\uparrow}_{{K}_{inp}Abb}$ | (31) and (33) | = f ($T,{K}_{inp},Abb)$ | |

[5,52] | $L{\uparrow}_{{Q}_{F}^{\ast}FkcAbb}$ | (31) and (34) | = f (T,${Q}_{FFkcAbb}^{\ast})$ | ||

[5,52] | $L{\uparrow}_{Qnet}$ | (31) and (34) | = f (T,${Q}_{net}$) | ||

${Q}^{\ast}$ | Net radiation | [5] | ${Q}_{HHkcAe{e}^{\prime}}^{\ast}$ | (22) | = f ($K{\downarrow}_{Hkc},{Aee}^{\prime},T,N)$ |

${Q}_{H}^{\ast}$ | [5] | ${Q}_{HHkcAbb}^{\ast}$ | (22) | = f ($K{\downarrow}_{Hkc},Abb,T,N)$ | |

[5] | ${Q}_{HHbgAe{e}^{\prime}}^{\ast}$ | (22) | = f ($K{\downarrow}_{Hbg},{Aee}^{\prime},T,N)$ | ||

[5] | ${Q}_{HHbgAbb}^{\ast}$ | (22) | = f ($K{\downarrow}_{Hbg}$,$Abb,T,N)$ | ||

[5] | ${Q}_{H{K}_{inp}Ae{e}^{\prime}}^{\ast}$ | (22) | = f (${K}_{inp}$, ${Aee}^{\prime},T,N)$ | ||

[5] | ${Q}_{H{K}_{inp}Abb}^{\ast}$ | (22) | = f (${K}_{inp}$, $Abb,T,N)$ | ||

Net radiation | [11] | ${Q}_{FFkcAe{e}^{\prime}}^{\ast}$ | (23) | = f ($K{\downarrow}_{Fkc},{Aee}^{\prime},T,N)$ | |

${Q}_{F}^{\ast}$ | [11] | ${Q}_{FFkcAbb}^{\ast}$ | (23) | = f ($K{\downarrow}_{Fkc}$, $Abb,T,N)$ | |

[11] | ${Q}_{FFbgAe{e}^{\prime}}^{\ast}$ | (23) | = f ($K{\downarrow}_{Fbg}$, ${Aee}^{\prime},T,N)$ | ||

[11] | ${Q}_{FFbgAbb}^{\ast}$ | (23) | = f ($K{\downarrow}_{Fbg}$, $Abb,T,N)$ | ||

[11] | ${Q}_{F{K}_{inp}Ae{e}^{\prime}}^{\ast}$ | (23) | = f (${K}_{inp}$, ${Aee}^{\prime},\rho )$ | ||

[11] | ${Q}_{F{K}_{inp}Abb}^{\ast}$ | (23) | = f (${K}_{inp}$, $Abb,\rho )$ |

**Table 2.**Instrumentation used on the micrometeorological measurements in Agrometeorological Observatory Debrecen (Figure 1) from 2008.

No | Instrument | Height | Variables | Comment |
---|---|---|---|---|

Eddy covariance measurements | ||||

1 | Campbell Scientific CSAT3 | 4 m | Wind speed components ($u,v,w$) (m/s), sound speed ($c$) (m/s), sonic temperature (${T}_{sonic}$) (K) | 10 Hz time resolution |

2. | LI-7500 | 4 m | Open path H_{2}O/CO_{2} sensor (H _{2}O concentration) (ppt)(CO _{2} concentration) (ppm)pressure ($p$) (hPa) | 10 Hz time resolution |

3 | Campbell Scientific CR1000 | 1.6 m | Collecting and calculating of Eddy covariance fluxes | 10 Hz sampling frequency and 30 min time period for flux calculations. |

Profile measurements | ||||

4 | Vaisala WAA151 | 10, 4, 2, 1 m | Wind speed ($U$) (mean, max, std.) (m/s) | Cup anemometer |

5. | Vaisala HMP155 | 10, 4, 2, 1 m | Temperature ($T$) (°C) and relative humidity (Rh) (%) | With Vaisala DTR13 shield |

Surface radiation balance components | ||||

6 | Kipp&Zonen CMP11 pyranometer | 2 m | Shortwave downward (downwelling) radiation (W m ^{−2}) | |

7 | Kipp&Zonen CMP6 pyranometer | 2 m | Shortwave upward (upwelling) radiation (W m ^{−2}) | |

8 | Kipp&Zonen CGR4 pyrgeometer | 2 m | Longwave downward (downwelling) radiation (W m ^{−2}) | |

9 | Kipp&Zonen CGR3 pyrgeometer | 2 m | Longwave upward (upwelling) radiation (W m ^{−2}) | |

10 | Apogee IRTS-P | 2 m | Soil surface temperature | |

Soil measurements | ||||

11 | Campbell Scientific TCAV | −0.04 m | Soil temperature (°C) | |

12 | Campbell Scientific CS616 | −0.04 m | Soil water content (%V) | |

13 | Hukseflux HFP01SC | −0.08 m | Soil heat flux plats (2) (W m ^{−2}) | |

Other | ||||

14 | PG200 weigting gauge | 1 m | precipitation (mm) | |

15 | Campbell Scientific CR1000 | 1.6 m | Collecting sensors output of profile, radiation budget components, surface, soil and precipitation measurements. | 0.5-Hz sampling frequency and 10 min averaging |

Statistical Error | BIAS | MAE | RMSE | Correlation $\mathbf{Coefficient}\left(\mathit{r}\right)$ |
---|---|---|---|---|

${K}_{0F}\downarrow $ | +14.6 W m^{−2} | 40.7 W m^{−2} | 52.8 W m^{−2} | 0.97 |

${K}_{0H}\downarrow $ | −17.2 W m^{−2} | 32.1 W m^{−2} | 44.0 W m^{−2} | 0.97 |

Method | BIAS [W m ^{−2}] | MAE [W m ^{−2}] | RMSE [W m ^{−2}] | $\mathbf{Correlation}\mathbf{Coefficient}\left(\mathit{r}\right)$ |
---|---|---|---|---|

$K{\downarrow}_{Fkc}$ | +5 | 71 | 109 | 0.90 |

$K{\downarrow}_{Fbg}$ | +8 | 74 | 108 | 0.91 |

$K{\downarrow}_{Hkc}$ | −12 | 69 | 109 | 0.91 |

$K{\downarrow}_{Hbg}$ | −10 | 77 | 111 | 0.91 (0.85, 0.80) |

**Table 5.**The total number of situations with absolute errors (AE) fewer than 50, 100,…, 700 W m

^{−2}.

AE (W m^{−2}) | <50 | <100 | <200 | <300 | <400 | <500 | <600 | <700 |
---|---|---|---|---|---|---|---|---|

$K{\downarrow}_{Fkc}$ | 2214 | 3174 | 3703 | 3914 | 3982 | 4009 | 4018 | 4022 |

$K{\downarrow}_{Fbg}$ | 2014 | 3022 | 3697 | 3920 | 4000 | 4019 | 4022 | 4022 |

$K{\downarrow}_{Hkc}$ | 2455 | 3207 | 3720 | 3916 | 3983 | 4011 | 4019 | 4022 |

$K{\downarrow}_{Hbg}$ | 2140 | 2988 | 3668 | 3931 | 4008 | 4022 | 4022 | 4022 |

**Table 6.**Statistical errors: BIAS, MAE and RMSE based on the Nyren and Gryning [37] model of albedo for all data and separately for nice weather without snow cover (${A}_{1}=0.25$), for weather with snow cover and without precipitation (${A}_{1}=0.85$) and, finally, for precipitation weather (${A}_{1}=0.11$).

Statistical Error for Albedo Aee’ Data Type | BIAS | MAE | RMSE | Correlation $\mathbf{Coefficient}\left(\mathit{r}\right)$ |
---|---|---|---|---|

All data | 0.055 | 0.076 | 0.118 | 0.08 |

Nice weather ${A}_{1}=0.25$ | 0.046 | 0.064 | 0.078 | −0.19 |

Snow caver ${A}_{1}=0.85$ | 0.150 | 0.179 | 0.269 | −0.16 |

Precipitation ${A}_{1}=0.11$ | −0.072 | 0.076 | 0.179 | <0.10 |

Statistical Error for Albedo Abb Data Type | BIAS | MAE | RMSE | Correlation $\mathbf{Coefficient}\left(\mathit{r}\right)$ |
---|---|---|---|---|

All data | −0.023 | 0.069 | 0.118 | 0.79 |

Nice weather ${A}_{1}=0.25$ | −0.024 | 0.053 | 0.085 | 0.22 |

Snow caver ${A}_{1}=0.85$ | 0.011 | 0.117 | 0.221 | 0.15 |

Precioitation ${A}_{1}=0.11$ | 0.108 | 0.168 | 0.287 | <0.10 |

Statistical Error Methodology | BIAS (W m ^{−2}) | MAE (W m ^{−2}) | RMSE (W m ^{−2}) | Correlation Coefficient (r) |
---|---|---|---|---|

$K{\uparrow}_{FkcAee\u2019}$ | 16 | 29 | 45 | 0.84 |

$K{\uparrow}_{FkcAbb}$ | 0 | 16 | 34 | 0.85 |

**Table 9.**Statistical errors and percentages of cases in which the relative errors were smaller than 25% (R

_{err}< 25%) in the calculations of the net irradiance, ${Q}^{\ast}$. We used the next symbols in the table: Foken [11] net radiation model ${Q}_{F}^{\ast}$ and Holtslag and Van Ulden [5] net radiation model ${Q}_{H}^{\ast}$. The low index used ${K}_{inp}$ for the measured global solar radiation and $Fkc$, $Fbg$, $Hkc$ and $Hbg$ for its assessment depending on Foken’s or Holtslag’s clear sky calculation and cloudiness correction. ${Aee}^{\prime}$ represent the Nyren and Gryning [37] and $Abb$ Beljaars and Bosveld [38]-type parameterizations of albedo. The second column inside the main columns represents statistical errors of the net radiation but only for period with high solar elevation when the estimated sensible heat flux based on De Bruin and Holtslag [10] was positive. Gray—methods with higher correlation coefficients.

Net Irradiance | BIAS (W m ^{−2}) | MAE (W m ^{−2}) | RMSE (W m ^{−2}) | Correlation Coefficient | Rerr < 25% (%) | |||||
---|---|---|---|---|---|---|---|---|---|---|

${Q}_{{FFkcAee}^{\prime}}^{\ast}$ | −25 | −35 | 42 | 73 | 61 | 102 | 0.93 | 0.80 | 41 | 84 |

${Q}_{{FFbgAee}^{\prime}}^{\ast}$ | −25 | −37 | 41 | 75 | 61 | 97 | 0.93 | 0.81 | 31 | 82 |

${Q}_{{HHkcAee}^{\prime}}^{\ast}$ | −24 | −22 | 43 | 72 | 65 | 100 | 0.92 | 0.84 | 52 | 74 |

${Q}_{{HHbgAee}^{\prime}}^{\ast}$ | −24 | −24 | 42 | 74 | 63 | 100 | 0.92 | 0.83 | 51 | 73 |

${Q}_{F{K}_{inp}{Aee}^{\prime}}^{\ast}$ | −23 | −28 | 29 | 34 | 38 | 46 | 0.98 | 0.98 | 59 | 97 |

${Q}_{FFkcAbb}^{\ast}$ | −18 | 0 | 40 | 67 | 61 | 94 | 0.94 | 0.83 | 50 | 71 |

${Q}_{FFbgAbb}^{\ast}$ | −17 | −4 | 40 | 67 | 57 | 90 | 0.93 | 0.84 | 50 | 72 |

${Q}_{HHkcAbb}^{\ast}$ | −12 | −10 | 43 | 72 | 64 | 98 | 0.92 | 0.81 | 50 | 69 |

${Q}_{HHbgAbb}^{\ast}$ | −22 | −21 | 40 | 67 | 59 | 93 | 0.93 | 0.84 | 52 | 75 |

${Q}_{F{K}_{inp}Abb}^{\ast}$ | −15 | 1 | 25 | 21 | 32 | 26 | 0.99 | 0.99 | 58 | 93 |

${Q}_{H{K}_{inp}Abb}^{\ast}$ | −13 | −5 | 30 | 34 | 37 | 41 | 0.98 | 0.97 | 55 | 84 |

${Q}_{H{K}_{inp}{Aee}^{\prime}}^{\ast}$ | −22 | −14 | 32 | 39 | 40 | 49 | 0.97 | 0.96 | 55 | 86 |

**Table 10.**Statistical errors and coefficients of correlation for different types of models of $L\uparrow $. (The best model is shaded).

$\mathbf{Types}\mathbf{of}\mathbf{Models}\mathbf{of}\mathit{L}{\uparrow}_{}$ | AS (W m^{−2}) | MAE (W m^{−2}) | RMSE (W m ^{−2}) | Correlation Coefficient (r) |
---|---|---|---|---|

$L{\uparrow}_{SB}{\epsilon}_{0}=1$ | 17 | 17 | 26 | 0.99 |

$L{\uparrow}_{SB}{\epsilon}_{0}=0.98$ | 10 | 12 | 20 | 0.99 |

$L{\uparrow}_{{K}_{FkcAbb}}$ | 7 | 12 | 16 | 0.98 |

$L{\uparrow}_{{K}_{inp}Abb}$ | 7 | 11 | 15 | 0.98 |

$L{\uparrow}_{{Q}_{FFkcAbb}^{\ast}}$ | 3 | 8 | 13 | 0.98 |

$L{\uparrow}_{{Q}_{F{K}_{inp}Abb}^{\ast}}$ | 3 | 8 | 12 | 0.99 |

$L{\uparrow}_{{Q}_{FFkcAbb+{\alpha}_{PT}}^{\ast}}$ | −1 | 14 | 24 | 0.94 |

$L{\uparrow}_{{Q}_{F{K}_{inp}Abb+{\alpha}_{PT}}^{\ast}}$ | −13 | 14 | 22 | 0.96 |

**Table 11.**Parameterizations of clear sky downwelling LW radiation, ${L}_{01}\downarrow $ (W m

^{−2}), as a function of the screen level temperature, $T,$ expressed in (°K), and water vapor pressure, $e,$ expressed in (hPa).

Model | References |
---|---|

${L}_{01}\downarrow =\left(0.83-0.118\xb7{10}^{-0.6\xb7e}\right)\xb7\sigma \xb7{T}^{4}$ | Angström [42], recalculated 1997 for summertime in Finland (Niemelä et al. [43]) |

${L}_{02}\downarrow =1.24\xb7{\left(e/T\right)}^{1/7}\xb7\sigma \xb7{T}^{4}$ | Brutsaert [44] |

${L}_{03}\downarrow =\left[0.7+5.95\xb7{10}^{-5}\xb7\mathrm{exp}\left(1500/T\right)\right]\xb7\sigma \xb7{T}^{4}$ | Idso [45] |

${L}_{04}\downarrow =\left[1-\left(1+w\right)\xb7\mathrm{exp}\left(-1.2+3.0\xb7{w}^{1/2}\right)\right]\xb7\sigma \xb7{T}^{4}$ $w=46.5\xb7\left(e/T\right)$ is the precipitable water content [cm] | Prata [46] |

${L}_{05}\downarrow =5.31\xb7{10}^{-13}\xb7{T}^{6}$ | Swinbank [47] |

${L}_{06}\downarrow =59.38+113.7\xb7{\left(T/{T}_{0}\right)}^{6}+96.96\xb7\sqrt{w/2.5}$ ${T}_{0}=273.16\mathrm{K}$, $w=46.5\xb7\left(e/T\right)$ is the precipitable water content [cm] | Dilley and O’Brein [48] |

${L}_{07}\downarrow =\left\{\begin{array}{c}\left[0.72+0.009\xb7\left(e-2\right)\right]\xb7\sigma \xb7{T}^{4}ife\ge 2hPa\\ \left[0.72+0.076\xb7\left(e-2\right)\right]\xb7\sigma \xb7{T}^{4}ife2hPa\end{array}\right.$ | Niemelä et al. [43] |

${L}_{08}\downarrow =\left[1-0.35\xb7\mathrm{exp}\left(-10\xb7e/T\right)\right]\xb7\sigma \xb7{T}^{4}$ | Iziomon et al. [51] |

**Table 12.**Statistical errors and coefficients of correlation for clear sky conditions. (all—all cases, n.—nighttime and d.—daytime). Notations are similar to Table 10. (The best model is shaded).

LW Irradiance | BIAS (W m^{−2}) | MAE (W m^{−2}) | RMSE (W m^{−2}) | Correlation Coefficient (r) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

all | n. | d. | ||||||||

${L}_{01}\downarrow $ | –9 | 2 | −13 | 15 | 12 | 17 | 21 | 17 | 22 | 0.93 |

${L}_{02}\downarrow $ | −16 | −2 | −20 | 18 | 10 | 20 | 27 | 18 | 29 | 0.94 |

${L}_{03}\downarrow $ | 4 | 15 | 0 | 15 | 19 | 13 | 19 | 22 | 18 | 0.94 |

${L}_{04}\downarrow $ | −11 | 1 | −15 | 15 | 10 | 17 | 22 | 16 | 23 | 0.94 |

${L}_{05}\downarrow $ | −18 | 2 | −25 | 24 | 19 | 25 | 31 | 25 | 32 | 0.92 |

${L}_{06}\downarrow $ | −18 | −10 | −20 | 19 | 12 | 21 | 25 | 17 | 27 | 0.94 |

${L}_{07}\downarrow $ | –3 | –8 | 11 | 14 | 3 | 17 | 19 | 19 | 20 | 0.95 |

${L}_{08}\downarrow $ | −19 | −23 | –7 | 20 | 23 | 10 | 26 | 29 | 19 | 0.94 |

**Table 13.**Statistical errors for all sky conditions. Each error is represented in three separated columns for Jacobs [49], Maykut and Church [50] cloudiness and for Niemelä et al. [43] parameterization. Shown errors are calculated in 2$\xb78+8=24$ cases. In the last two rows, we show Holtslag and Van Ulden [5] cloudiness corrections with Swinbank [47] ${\left({L}_{05}\downarrow \right)}_{HVU}$ and with Dilley and O’Brien [48] ${\left({L}_{06}\right)}_{HVU}$ parameterizations (gray belt, best solution).

LW Irradiance | BIAS (W m^{−2}) | MAE (W m^{−2}) | RMSE (W m^{−2}) | Correlation Coefficient (r) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Jac | M&C | Nie | Jac | M&C | Nie | Jac | M&C | Nie | Jac | M&C | Nie | |

(${L}_{01}\downarrow $)… | 6 | −1 | 2 | 18 | 17 | 16 | 24 | 22 | 24 | 0.88 | 0.90 | 0.88 |

(${L}_{02}\downarrow $)… | −3 | −8 | −1 | 21 | 21 | 17 | 27 | 27 | 25 | 0.89 | 0.91 | 0.90 |

(${L}_{03}\downarrow $)… | 31 | −13 | 10 | 34 | 21 | 18 | 42 | 27 | 25 | 0.90 | 0.92 | 0.90 |

(${L}_{04}\downarrow $)… | 13 | −3 | 1 | 24 | 18 | 16 | 31 | 23 | 23 | 0.90 | 0.91 | 0.90 |

(${L}_{05}\downarrow $)… | 5 | −11 | −2 | 28 | 26 | 20 | 35 | 32 | 28 | 0.87 | 0.88 | 0.88 |

(${L}_{06}\downarrow $)… | 6 | −10 | −4 | 18 | 18 | 18 | 24 | 22 | 23 | 0.90 | 0.92 | 0.88 |

(${L}_{07}\downarrow $)… | 3 | −12 | −4 | 22 | 20 | 17 | 28 | 26 | 24 | 0.90 | 0.92 | 0.89 |

(${L}_{08}\downarrow $)… | 22 | 5 | 6 | 29 | 20 | 18 | 38 | 26 | 25 | 0.90 | 0.91 | 0.90 |

${\left({L}_{05}\downarrow \right)}_{HVU}$ | −3 | 24 | 28 | 0.87 | ||||||||

${\left({L}_{06}\downarrow \right)}_{HVU}$ | −2 | 15 | 19 | 0.91 |

**Table 14.**Number of situations in percentages (%), with relative errors (R

_{err}) smaller than 10%. Columns represent: $\left({L}_{0i,i=1,\dots ,8}\downarrow \right)$, respectively, as in Table 11. Rows represent cloudiness corrections: Jacobs [49], Maykut and Church [50], Niemelä et al. [43] and Holtslag and Van Ulden [5]-type corrections $\left(60\xb7\frac{N}{8}{\mathrm{W}\mathrm{m}}^{-2}\right),$ which work only with ${L}_{CL5}\downarrow $ and ${L}_{CL6}\downarrow .$ For cloudiness corrections, these abbreviations are used for the (${L}_{0i}$) index: Jacobs (Jac), Maykut and Church (M&C), Niemelä (Nie) and Holtslag and Van Ulden (HVU), respectively.

$\mathbf{Rerr}({\mathit{L}}_{0\mathit{i}}\downarrow )10\%(\%)$ Method | ${\mathit{L}}_{01}\downarrow $ | ${\mathit{L}}_{02}\downarrow $ | ${\mathit{L}}_{03}\downarrow $ | ${\mathit{L}}_{04}\downarrow $ | ${\mathit{L}}_{05}\downarrow $ | ${\mathit{L}}_{06}\downarrow $ | ${\mathit{L}}_{07}\downarrow $ | ${\mathit{L}}_{08}\downarrow $ |
---|---|---|---|---|---|---|---|---|

Jacobs | 73% | 70% | 57% | 73% | 65% | 81% | 76% | 66% |

Maykut & Church | 85% | 77% | 80% | 84% | 67% | 87% | 77% | 65% |

Niemelä | 87% | 85% | 85% | 86% | 80% | 87% | 85% | 86% |

Holtslag and Van Ulden | 85% | 90% |

**Table 15.**Comparison of statistical errors between the year 2009 and the periods 2008–2010 and 2008–2017 for all components of radiation using the best parameterizations: $K\downarrow $, $K\uparrow $, $L\downarrow $ and $L\uparrow $.

BIAS (W m ^{−2}) | MAE (W m ^{−2}) | RMSE (W m ^{−2}) | Correlation Coefficient (r) | Time Periods | |
---|---|---|---|---|---|

$K{\downarrow}_{Fkc}$ | 4 | 76 | 105 | 0.90 | 2008–2010 |

5 | 71 | 109 | 0.91 | 2009 | |

4 | 60 | 96 | 0.93 | 2008–2017 | |

$K{\uparrow}_{FkcAbb}$ | −2 | 14 | 33 | 0.88 | 2008–2010 |

0 | 16 | 34 | 0.84 | 2009 | |

−7 | 19 | 34 | 0.82 | 2008–2017 | |

${\left({L}_{06}\downarrow \right)}_{HVU}$ | 9 | 16 | 21 | 0.91 | 2008–2010 |

−3 | 14 | 19 | 0.95 | 2009 | |

−1 | 14 | 19 | 0.92 | 2008–2017 | |

$L{\uparrow}_{{Q}_{FFkcAbb}^{\ast}}$ | 7 | 10 | 15 | 0.98 | 2008–2010 |

3 | 8 | 13 | 0.98 | 2009 | |

7 | 10 | 16 | 0.98 | 2008–2017 |

**Table 16.**Statistical errors and coefficients of correlation for ${Q}_{1}^{\ast}$, ${Q}_{2}^{\ast}$, ${Q}_{3}^{\ast}$ and ${Q}_{4}^{\ast}$ in Agrometeorological Observatory Debrecen for the 3-year time period from 2008 to 2010. The applied parameterizations are presented in Figure 8.

Net Irradiance | BIAS (W m^{−2}) | MAE (W m^{−2}) | RMSE (W m^{−2}) | Correlation Coefficient (r) |
---|---|---|---|---|

${Q}_{1}^{\ast}$ | −20 | 46 | 66 | 0.93 |

${Q}_{2}^{\ast}$ | 9 | 29 | 46 | 0.96 |

${Q}_{3}^{\ast}$ | 0 | 37 | 62 | 0.93 |

${Q}_{4}^{\ast}$. | −6 | 21 | 31 | 0.96 |

**Table 17.**Statistical errors and coefficients of correlation for ${Q}_{1}^{\ast}$, ${Q}_{2}^{\ast}$, ${Q}_{3}^{\ast}$ and ${Q}_{4}^{\ast}$ in Agrometeorological Observatory Debrecen for the 10-year time period from 2008 to 2017.

Net Irradiance | BIAS (W m^{−2}) | MAE (W m^{−2}) | RMSE (W m^{−2}) | Correlation Coefficient (r) |
---|---|---|---|---|

${Q}_{1}^{\ast}$ | −15 | 36 | 58 | 0.94 |

${Q}_{2}^{\ast}$ | −17 | 27 | 36 | 0.98 |

${Q}_{3}^{\ast}$ | −11 | 37 | 56 | 0.95 |

${Q}_{4}^{\ast}$. | −13 | 26 | 34 | 0.98 |

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**MDPI and ACS Style**

Popov, Z.; Nagy, Z.; Baranka, G.; Weidinger, T. Assessments of Solar, Thermal and Net Irradiance from Simple Solar Geometry and Routine Meteorological Measurements in the Pannonian Basin. *Atmosphere* **2021**, *12*, 935.
https://doi.org/10.3390/atmos12080935

**AMA Style**

Popov Z, Nagy Z, Baranka G, Weidinger T. Assessments of Solar, Thermal and Net Irradiance from Simple Solar Geometry and Routine Meteorological Measurements in the Pannonian Basin. *Atmosphere*. 2021; 12(8):935.
https://doi.org/10.3390/atmos12080935

**Chicago/Turabian Style**

Popov, Zlatica, Zoltán Nagy, Györgyi Baranka, and Tamás Weidinger. 2021. "Assessments of Solar, Thermal and Net Irradiance from Simple Solar Geometry and Routine Meteorological Measurements in the Pannonian Basin" *Atmosphere* 12, no. 8: 935.
https://doi.org/10.3390/atmos12080935