Estimation of downwelling longwave irradiance under all-sky conditions


  • I. Alados,

    1. Departamento de Física Aplicada, Universidad de Málaga, Málaga, Spain
    2. Centro Andaluz de Medio Ambiente (CEAMA), Junta de Andalucía-Universidad de Granada, Av. Del Mediterráneo s/n, 18071, Granada, Spain
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  • I. Foyo-Moreno,

    1. Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, Fuentenueva s/n, 18071, Granada, Spain
    2. Centro Andaluz de Medio Ambiente (CEAMA), Junta de Andalucía-Universidad de Granada, Av. Del Mediterráneo s/n, 18071, Granada, Spain
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  • L. Alados-Arboledas

    Corresponding author
    1. Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, Fuentenueva s/n, 18071, Granada, Spain
    2. Centro Andaluz de Medio Ambiente (CEAMA), Junta de Andalucía-Universidad de Granada, Av. Del Mediterráneo s/n, 18071, Granada, Spain
    • Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain.
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This work is focused on the characterization and parameterisation of the downward atmospheric irradiance (LW) for clear and cloudy skies. LW is a component of the surface radiation budget that is present throughout the day. Unlike solar irradiance, LW is not measured routinely in extended networks, so it must be estimated indirectly. We evaluated five parameterisations for estimating LW under clear skies. After some consideration regarding the local fitting of the parameterisations, we analysed their different behaviour during day and night and propose a correction model for this effect. We use measurements registered at Tabernas (Spain) from 2001 to 2003. For the locally adjusted parameterisations the root mean square deviation (RMSD) and the mean bias deviation (MBD) are smaller than 5.7 and 0.6%, respectively. The combination of the more complex correction parameterisation of the day/night differences with the locally adjusted formula of Brutsaert and the original formula of Berdalh and Martin leads to estimations with RMSD below 3.1%. Using data registered at Palaiseau (France) the proposed parameterisations yields MBD close to 0% and RMSD below 3.2%. Cloudy conditions were analysed and two different approaches were used to estimate the cloud effect. Both approaches determine all sky LW using a clear-sky formulae and a cloud modification factor, computed with the solar global irradiance on a horizontal surface. The results show that LW can be estimated under all-sky conditions during the daytime with a RMSD of 5.8 and 6.2%, and a MBD of 1.6 and − 2.2% for the Crawford and Duchon scheme and the parameterisation in kt, respectively, at Tabernas. The application of the same parameterisations to Palaiseau yields RMSD of 6.7 and 7.7%, and MBD of − 2.5 and 0.7%. Copyright © 2011 Royal Meteorological Society

1. Introduction

The downward atmospheric radiation (LW) is of great importance in meteorological and climatic studies. Knowledge of LW is required for the forecast of nocturnal frosts, fogs, temperature variation, and cloudiness; energy balance studies; the design of radiant cooling systems as well as calculations on climate variability and global warming (Crawford and Duchon, 1999; Gröbner et al., 2009).

Owing to difficulties associated with its measurement, LW must be estimated from routinely collected meteorological data. There exist detailed methods that require measurements of atmospheric variables at several levels in the atmospheric column (Kneizys et al., 1988 and Snell et al., 1995), information that it is not always available. That is why estimation of the LW often relies on meteorological variables measured routinely at the surface level.

The simplicity of the empirical parameterisations implies some assumptions regarding the vertical structure of the atmosphere. In some cases these assumptions are explicitly presented while in other cases they are implicitly considered through the fitting of local coefficients. From the point of view of thermal atmospheric irradiance (4–100 µm) the atmosphere can be considered as a grey body with an effective emissivity defined as ε = LW/σTa4 where σ is the Stefan-Boltzman constant and Ta is the air temperature.

The clear-sky emissivity of the atmosphere, εo, can be modelled as a function of air temperature, Ta(K), and/or vapour pressure, e (hPa), measured at levels that are routinely measured in meteorological observatories and registered in automatic weather stations around the world. Most of these parameterisations were derived for night-time data using local empirical coefficients (Brunt, 1932; Idso and Jackson, 1969; Brutsaert, 1975; Idso, 1981). Several authors (Paltridge, 1970; Berdahl and Fromberg, 1982; Alados-Arboledas and Jiménez, 1988; Alados-Arboledas, 1993; Dupont et al., 2008) pointed out the differences between the day and night effective emissivity regimes. Results of this type suggest that the evaluation of various LW parameterisations covering the complete daily cycle would provide useful information for a variety of applications.

In recent years, a fair number of studies addressed the problem of estimating LW from simple and widely available atmospheric variables. These studies focused on different regions from high-altitude locations like the Andean Altiplano (Lhomme et al., 2007), Porta Grossa, Brazil (Duarte et al., 2006) and Nsukka, Nigeria (Ezekwe, 1986). Nevertheless, most of these studies used exclusively daytime measurements.

The presence of clouds substantially modifies the atmospheric radiation flux that is received at the surface, because the radiation emitted by water vapour and carbon dioxide in the lower atmosphere is supplemented by emission from clouds, and transmitted through the air in the 8–14 µm window. In this way, under cloudy conditions, the thermal atmospheric effective emittance increases with respect to the clear-sky value (Malek, 1997). A number of empirical corrections were proposed in an attempt to estimate the cloud contribution to LW. Most are based on a corrective term that must be added to LW estimated for clear skies. Alados-Arboledas et al. (1995) and Niemalä et al. (2001) presented several expressions developed to predict LW from cloudy skies. In these expressions the cloud effect was parameterised with total cloud cover. A different approach was followed by Crawford and Duchon (1999) and Bilbao and De Miguel (2007) who proposed expressions of the cloud effect in terms of the ratio of measured solar global irradiance to global irradiance under clear sky conditions.

In this work, we present a study of LW in Tabernas (Almería, Spain) based on three years of radiative flux measurements under clear and cloudy conditions. First, we present the analysis and comparison of several parameterisations developed to predict LW under clear skies. Then we evaluate the differences between the effective emissivity for day and night, and propose a corrective formula that takes into account its daily cycle. Finally, we study two parameterisations for the estimation of LW under cloudy skies. These methods are based on the use of global solar irradiance as a surrogate for cloudiness. In this way, they can be used when cloud cover is not registered as in the case of extended automated networks operating in remote locations. For the development of the parameterisations we used data measured at Tabernas (Almería, Spain), while for the validation we used both an independent dataset from the same place and a dataset registered in the station of Palaiseau (France) during 2006, characterized by very different climatic conditions.

2. Site and instrumental setup

In this paper, we used data recorded at two different locations: Tabernas (37°8′N, 2°22′W, 630 m a.s.l., Almería, Spain) and Palaiseau (48.71°N, 2.21°E, 156 m a.s.l., France). The data registered at Tabernas were used to check different parameterisations of LW for clear and cloudy skies and to investigate the convenience of applying certain modifications. The applicability of the modified parameterisations to other locations was tested with the Palaiseau database.

The Tabernas dataset was registered at the Rambla Honda field site, Almeria, Spain from 2001 to 2003. A detailed description of the field site is given by Alados et al. (2003). Tabernas is partially surrounded by the Betic cordillera and lee of the Sierra de los Filabres, Sierra Nevada, and Sierra of Gádor. The climate is semi-arid with a mean annual temperature of 16 °C; mean rainfall is 279 mm (10-year record) which falls mainly in winter, followed by a dry period centred on the months of June–September.

The instruments were mounted on a mast at 3.8 m above ground level. The LW was measured with an Eppley pyrgeometer model PIR. The measurements were corrected for solar heating of the pyrgeometer following the procedure described in Alados-Arboledas and Jiménez (1988) and Pérez and Alados-Arboledas (1999). The experimental error in LW data is less than 3%. The measurements included horizontal solar global irradiance, measured using a Kipp & Zonen model CM-11 with an estimated experimental error of 2-3%. The automatic station of Rambla Honda measured air temperature and relative humidity with a Vaisala HMP35AC. The error of the temperature sensor is 0.4 °C over the range − 20 °C to + 48 °C. The error associated with the relative humidity measurement is of 2%. The calibration constants of the radiometers were periodically checked. All the data were recorded every 5 s and were stored as 5 min averages. In this sense, our analyses are based on 5 min values of the different variables.

The dataset was separated in clear and cloudy data. Different authors proposed alternative methods to estimate cloud cover when measurements are not available (Long et al., 2006, Dürr and Philipona, 2004). We used a previously developed approach (Alados et al., 2000) based on the use of hemispherical transmittance (kt = global horizontal irradiance/extraterrestrial horizontal irradiance) to separate clear and cloudy conditions during the daytime. Our approach has some similarities with the method proposed by Long et al. (2006), although these authors used as an additional input the diffuse component of the solar irradiance. During the night time our classification is based on the visual inspection of the smoothness of the temporal evolution of LW combined with the consideration of the cloud classification determined for the previous and subsequent daytime periods. The method proposed by Dürr and Philipona (2004) uses as input temperature and water vapour at screen level including LW irradiance.

The main features of the study area are summarized in Table I, where we include basic statistical parameters of air temperature, water vapour pressure, and relative humidity for the period analysed. Additional information concerning descriptive statistics of LW and global solar irradiance, G, are also included.

Table I. Annual average values, 25th and 75th percentile of air temperature (Ta), water vapour pressure (e), relative humidity (HR), downward atmospheric irradiance (LW) and global solar irradiance (G). Tabernas dataset for the period 2001–2003
ParametersYearAverage25th Percentil75th Percentil
Ta (K)2001290.6284.8295.8
e (hPa)200110.47.412.9
HR (%)2001543771
LW (Wm−2)2001320289348
G (Wm−2)2001484244715

The station of Palaiseau is located 25 km to the West of Paris (France). This station is part of the Baseline Surface Radiation Network (BSRN) (Ohmura et al., 1998). The site is a semi-urban environment divided equally in agricultural fields, wooded areas, and housing and industrial developments. The prevailing winds are westerlies, blowing air of maritime origin over the site. Northeasterly winds occur quite frequently, as well advecting polluted air from the Paris metropolitan area over the site (Haeffelin et al., 2005). The data used are LW, measured with a Pyrgeometer CG-4 by Kipp & Zonen, and horizontal solar global irradiance, measured with a Pyranometer CM-22 by Kipp & Zonen. Other parameters used are the air temperature at 2 m height and the relative humidity. The data correspond to 2006 and are 1 min averages. The station is included in the BSRN and yearly calibration of the radiometer is done at the World Radiometric Center (WRC).

3. Downward atmospheric irradiance parameterisations description

3.1. Clear skies

Surface radiative fluxes can be calculated fairly accurately using rather complex radiative transfer, RT, models such as LOWTRAN (Kneizys et al., 1988) or MODTRAN (Snell et al., 1995), which require detailed information on the atmospheric structure. In this sense, Viúdez-Mora et al. (2009) analysed the performance of SBDART radiative transfer code (Ricchiazzi et al., 1998) using different approaches for the required atmospheric profiles. They evidenced the degradation of the results when atmospheric soundings were not available. For these cases, they estimated an uncertainty of around 10 Wm−2.

When scattering is neglected, the LW at the surface especially in cloud-free conditions can be expressed as:

equation image(1)

where p is the pressure, ps the surface pressure, T the temperature, λ the wavelength, Bλ the monochromatic Planck function and tλ is the monochromatic flux transmissivity from the pressure level, p, to the surface. Usually LW is expressed in terms of the atmospheric emissivity, ε, (e.g. Prata, 1996):

equation image(2)

where σ is the Stefan–Bolzmann constant, and To the screen-level temperature. It is evident from Equations (1) and (2) that the emissivity so defined depends both on the vertical temperature profile and on the vertical distribution of radiatively active constituents, which determines the transmissivity, tλ. Since the major part of the downwelling flux reaching the surface originates within a few hundred meters of the surface, the near-surface temperature profile is of primary importance in determining LW (e.g. Zhao et al., 1994).

Several parameterisations were developed that produce estimates for the surface radiative fluxes using synoptic observations only. Typically, these parameterisations use screen-level temperature and humidity information for the calculation of clear-sky fluxes. Given the relative complexity of the calculation and the fact that the data required are rarely available, simplified and empirical expressions of LW for clear skies are generally used. These take the general form LW = ε0(T, eTo4, where ε0 is the effective emissivity of a clear atmosphere, generally expressed as a function of air temperature and vapour pressure at screen level.

Several of the best known formulations to estimate the LW for clear skies were evaluated in this study. Table II includes the formulations of the effective emissivity of the atmosphere with the original coefficients proposed by their authors. These parameterisations are:

  • 1)The formula developed by Brunt (1932) expressesing LW in terms of an effective emissivity computed from vapour pressure measured at the screen level, 2 m; aBr and bBr are empirical constant to be determined from the local observational data. Thus, Alados-Arboledas et al. (1986) proposed the values aBr = 0.60 and bBr = 0.042, for an urban location in southeast Spain, characterized by a low humidity regime with a wider range of temperatures than our study area.
  • 2)Idso and Jackson (1969) (I&J) proposed an expression for LW in terms of screen level air temperature, To (K).The coefficients that appear in the expression (Table II) are proposed by these authors that claim general validity for them, because they were obtained using information from a variety of atmospheric conditions. The coefficients were calculated using night time values only.
  • 3)Brutsaert (1975) proposed an equation where emissivity is a function of vapour pressure, e, and temperature at screen level, To. Emissivity formula was obtained by an approximate integration of the Schwartzschild's transfer equation for a standard atmosphere.
  • 4)Berdahl and Martin (1984) (B&M) introduced some modifications to the formula developed by Berdahl and Fromberg (1982), based on the use of a more extensive dataset. Their final parameterisation was function of the dewpoint temperature, Td ( °C). This expression has general validity, according to the authors.
  • 5)Prata (1996) derived an equation based on radiative transfer theory with coefficients fitted empirically to observations. Prata's scheme is an emissivity formula with a continuum absorption correction. In this parameterisation the emissivity depends on the precipitable water content, ω, where ω is parameterised using screen level value of vapour pressure and temperature.
Table II. Formulation and original/local coefficients of the estimation's expressions for the effective emissivity under clear sky
ExpressionsFormulationOriginal coefficientsLocal coefficients
Bruntε0 = aBr + bBre1/2aBr range 0.34–0.71aBr = 0.612
  bBr range 0.023–0.110 (hPa−1/2)bBr = 0.044 hPa−1/2
I&Jε0 = 1 − aIJexp(−bIJ(273 − T0)2)aIJ = 0.261aIJ = 0.258
  bIJ = 7.77 × 10−4 K−2bIJ = 1.30 × 10−4 K−2
Brutsaertequation image  
  aB = 1.24(K/hPa)1/7aB = 1.225(K/hPa)1/7
B&Mequation imageaBM = 0.711aBM = 0.722
  bBM = 0.56 °C−1bBM = 0.37 °C−1
  cBM = 0.73 °C−2cBM = 0.81 °C−2
Prataε0 = 1 − (1 + ω)exp(−(aP + bPω)1/2)aP = 1.2aP = 1.19
  bP = 3 cm−1bP = 2.71 cm−1

3.2. Cloudy skies

The presence of clouds substantially modifies the atmospheric radiation flux that is received at the surface. Over the years a number of methods were proposed in an attempt to estimate the cloud contribution to thermal atmospheric radiation. Most are based on a corrective term that must be added to LW estimated under clear skies (Arnfield, 1979; Alados-Arboledas et al., 1995; Crawford and Duchon, 1999).

The use of parameterisations under cloudy conditions requires information routinely registered in most meteorological stations like cloud amount expressed as fractional cloud coverage and cloud type. When these measurements are not available we can use dimensionless ratios of radiative quantities to characterize sky conditions, such as kt (ratio of global horizontal irradiance to extraterrestrial horizontal irradiance), or the ratio of global horizontal irradiance (G) to global irradiance under clear skies (G0).

Alados-Arboledas et al. (1995) presented a comparison among several expressions developed to predict LW under cloudy skies using cloud observations. They used the expression developed by Boltz (Sellers, 1965; Monteith, 1973):

equation image(3)

where LW is for cloudy conditions, LW0 is under clear skies, N is the total cloud cover fraction (with values from 0 to 1), and q an empirical constant dependent on cloud type and cloud elevation. Alados-Arboledas et al. (1995) use this expression with the q coefficient proposed by Morgan et al. (1971).

Crawford and Duchon (1999) proposed a different approach for the estimation of the cloud effect on LW, using a cloud modification factor based on measurements of solar global irradiance that reads as follows:

equation image(4)

where ε0 is the emissivity in clear skies and clf is a fairly simple cloud modification factor (Deardorff, 1978) defined as:

equation image(5)

In their work, Crawford and Duchon (1999) used for ε0 a modified version of Brutsaert‘s expression that includes seasonal dependence. Bilbao and De Miguel (2007) use also the factor clf to estimate LW under cloudy skies.

In this work, we propose the use of global solar irradiance data as a proxy of the cloudy conditions. This approach is based on the idea that the solar hemispheric transmittance, kt, includes information on cloud cover. In this sense, LW under all sky conditions can be estimated using the following linear function of kt:

equation image(6)

where a and m are intercept and slope values obtained empirically when we calculate the linear fit of the ratio of experimental effective emissivity (LW/σTo4) to the effective emissivity estimated under clear skies, εo, versuskt.

Obviously, the use of radiative quantities to characterize the presence of clouds does not provide information on the cloud base height, being this last variable relevant in the quantification of the cloud contribution to the LW incoming at surface level. Nevertheless, the quantification of the cloud radiative effect on LW through the use of dimensionless ratios of solar radiation quantities, like kt, is based on the fact that we can relate the effect of clouds on different radiative fluxes. In this sense, clouds affect the solar global irradiance through a reduction in its value that leads to a reduction in kt. So, kt presents low values when the clouds are optically thick and large values when the clouds are optically thin. According to the morphology of clouds, high level clouds are usually optically thin, so the linear dependence of the cloud modification factor proposed in Equation (6) suggests a reduced effect due to these clouds, with rather cold cloud base and a reduced effective emissivity. On the other hand, low and middle clouds usually are optically thicker than high level clouds and lead to a substantial reduction of kt, that implies, according to Equation (6), a strong effect of low and middle level clouds on LW.

4. Parameterisations evaluation

The performance of the different expressions for clear and cloudy skies was evaluated using the root mean square deviation (RMSD) and the mean bias deviation (MBD). These statistics allow for the detection of the differences between experimental data and formula estimates, as well as systematic over- or underestimation tendencies. A linear regression between estimated and measured values was also computed. The linear fit was forced through zero, thus the slope, b, provides information about the relative underestimation or overestimation associated with the expression. The coefficient of determination, R2, provides an evaluation of the experimental data variance explained by the expression.

4.1. Clear skies

4.1.1. Original parameterisations

Expressions were tested using the 2002 and 2003 data sets at Tabernas (Spain). Table III shows results from the evaluation with the 2002 dataset including the statistics previously described. Similar results, not shown, were obtained with the 2003 dataset.

Table III. Statistical results for the clear-sky formulae of LW with original coefficients, local coefficients (subscript p) and our diurnal corrections (indicate with Dc). Total number of data, N = 25 733. Mean experimental values for LW, LW0ave = 312 Wm−2. Tabernas dataset for 2002
ExpressionsMBD Wm−2RMSD Wm−2MBD (%)RMSD (%)bR2
Brunt− 7.815.3−
Brunt p0.313.4−
I&J p1.917.
Brutsaert p0.0613.
B&M p− 0.313.4−
Prata p− 0.0312.8−
Brutsaert Dc3.
B&M Dc− 0.59.1−
Prata Dc8.612.
Brutsaertp Dc− 0.69.7−
B&Mp Dc− 0.98.6−
Pratap Dc− 0.78.5−

All parameterisations overestimate measured values but the original version of Brunt. In fact, the B&M estimate has negligible bias and the lowest RMSD. The Brutsaert (1975) parameterisation also provides good results, with an overestimation of only 1%. I&J presents a marked overestimation of nearly 8%. All parameterisations except that of I&J present MBD less than 3% and RMSD lower than 5.2%. In particular, the parameterisations of Brutsaert and B&M estimate our data set with rather low deviations; in fact we obtain RMSD below 5% and negligible MBD, with the slope of estimated versus measured values close to unity.

Additionally, we compared our results with those obtained by other authors (Table IV). It is interesting to note that we used 5-min values while most of the previous studies used hourly values. Niemalä et al. (2001), split their data into two categories, summer and winter, to study seasonal variations at Sodankylä Observatory, Finland. The climate in Sodankylä is characterized by temperate summers (July mean temperature, 14.1 °C), subarctic cold winters (January mean temperature, − 15.1 °C) and moderate precipitation amounts (annual mean, 501 mm). They obtained better results for the summer category than for winter. Their results for the summer season are similar to those that we obtained for the whole year.

Table IV. Statistical results for the clear-sky LW formulae obtained for different authors
AuthorsSitesPeriodMBD (Wm−2)MBD %RMSD (Wm−2)RMSD (%)
Niemalä et al(2001)Sodankylä, (Finland)Summer97//Winter973.2//7.510//22
Iziomon et al(2003)SW Germany     
 Lowland//MountainJan91-Sep96//Jul91-Sep96− 6//711//12
Jiménez et al. (1987)Barcelona (Spain)197514.4
Alados-Arboledas et al(1986)Granada (Spain)May-Nov 838.4
Viudez-Mora et al. (2009)Girona (Spain)12 days (Jul05-Ap06)9.3
Iziomon et al(2003)SW, Germany     
 Lowland//MountainJan91-Sep 96//Jul91-Sep96− 4//111//19
Jiménez et al. (1987)Barcelona (Spain)197511.3
Niemalä et al(2001)Sodankylä, (Finland)Summer 97//Winter 973.5//115.5//25
Sriddhar et Elliott (2002)Oklahoma:     
 10 OASIS sitesJun99-Dec9915to2327 to 41
Iziomon et al(2003)SW, Germany     
 Lowland//MountainJan91- Sep96//Jul91-Sep96− 6//710//12
Jiménez et al. (1987)Barcelona (Spain)197516.5
Alados-Arboledas et al(1986)Granada (Spain)May-Nov 8322.7
Alados-Arboledas (1993)Granada (Spain)Jan83-Dec8521.230.8
Dupont et al(2008)SIRCA (France)May04-Oct05− 13.522.3
Viudez-Mora et al. (2009)Girona (Spain)12 days (Jul05-Ap06)12.2
Alados-Arboledas et al(1986)Granada (Spain)May-Nov 8317.5
Alados-Arboledas (1993)Granada (Spain)Jan83-Dec8515.626.0
Niemalä et al(2001)Sodankylä, (Finland)Summer 97//Winter 973.2//7.63.7//12
Dupont et al(2008)SIRTA (France)May04-Oct05− 11.517.6

Sridhar and Elliott (2002) used up to nine stations located in Oklahoma in their study. Oklahoma’s geographical position in the southern Great Pains has a diverse climatic regime, with average annual precipitation ranging from less than 406 mm in the western panhandles to more than 1321 mm in the mountains of southeast Oklahoma. The freeze-free period varies by 9 weeks across the state, from 24 weeks to 33 weeks. Analyses by Sridhar and Elliott using Brutsaert's parameterisation showed MBD and RMSD greater than those obtained in the present study. Iziomon et al. (2003) analysed a pair of stations with different altitudes, at Bremgarten in the Upper Rhine plain, at 212 m a.s.l. and Feldberg mountain site at 1489 m a.s.l. They also checked several parameterisations with best results corresponding to Brutsaert and Brunt. Results were better for the lowland site than for the high mountain site.

Jiménez et al. (1987) used information recorded at Barcelona, in the western Mediterranean, with mean annual temperature of 15.5 °C and 640 mm mean annual rainfall throughout the year. In their study of the parameterisations of Brunt, I&J, and Brutsaert, they obtained RMSD values similar to those we obtained in this study. Alados-Arboledas (1993) analysed data registered at Granada; characterized by a mean annual temperature of 15.2 °C and 361 mm mean annual rainfall (dry season from June to September). In this study the author obtained worse results for the expressions of Brutsaert as well as Berdahl and Fromberg than those obtained in here.

Dupont et al. (2008) using 1-min data registered at Pailaseau (France) showed that the Brutsaert and Prata formulae underestimate experimental values, with RMSD values similar to those obtained in our study. Viúdez-Mora et al. (2009) used 10-min averaged data measured at Girona (Spain). According to their study Brunt and Brutsaert formulae provide worse estimates than those shown in our study.

4.1.2. Parameterisations with locally fitted coefficients

Considering the empirical or semi-empirical nature of the analysed expressions, we opted for a local fitting of their coefficients. To do that we used the data recorded at Tabernas (Spain) during 2001. The local fitting coefficients were obtained using a nonlinear least squares fit that uses the Levenberg-Marquardt (L-M) algorithm to adjust the parameter values in the iterative procedure. For each parameterisation, we used the function and the variables proposed by the corresponding authors.

The locally fit formulae were tested using the dataset measured at Tabernas during 2002 and 2003. Table II shows both the new coefficients and the original coefficients for the different formulae. As a general rule, excluding the Brunt formula, the new coefficients are slightly lower than the original ones. Table III includes results from the validation of parameterisations with locally adjusted coefficients, those with the subscript p. Table III reveals that the original and locally adjusted version of B&M expression provided similar results.

It must be pointed out that in the local fit we used both day time and night time data, as opposed to the original fit of some parameterisations that were only adjusted for night time measurements.

The most significant variation of the coefficients was for I&J, where the coefficient b presented a decrease of 80%. According to Table III the original version of I&J presents a marked overestimation when using data for the whole daily period. We must point out that this overestimation was larger for day time values, likely as a result of the night time fitting of the original coefficients.

Table II evidences that the difference between the locally adjusted and the original Brutsaert's coefficient is of order 1%. This small difference is in the range of that encountered by Lhomme et al. (2007) in the Andean plateau, 5%. Therefore, these results confirm the extended applicability of this expression in rather different environments (Jiménez et al., 1987; Alados-Arboledas, 1993; Niemalä et al., 2001; Sridhar and Elliott, 2002; Iziomon et al., 2003). The evaluation of the original and locally adjusted versions of Brutsaert against experimental values, Table III, leads to similar statistics with a negligible MBD. The locally adjusted expressions of Prata and Brutsaert showed low values and the same range for MBD and RMSD.

To test the applicability of these local fittings to other locations we used data measured at Palaiseau Cedex (France) during year 2006. For this purpose, 5-min averages have been computed from the original 1-min data base. Table V shows values of MBD and RMSD using the clear-sky parameterisations that provided better results at Tabernas. The expressions with locally fitted coefficients at Tabernas provided estimation of clear-sky LW with RMSD values close to 10 Wm−2. It is interesting to note that these results are similar to those obtained at Tabernas, where the local fitting of the coefficients was performed.

Table V. Statistical results for the clear-sky LW formulae with data of Palaiseau Cedex (France) using the parameterisations with original coefficients, locally fitted coefficients at Tabernas (subscript p) and our diurnal corrections (indicate with Dc)
ExpressionsMBD (Wm2)RMSD (Wm−2)MBD (%)RMSD (%)bR2
Brutsaert− 1.010.4−
B&M− 3.29.4−
Brutsaert p− 4.310.8−
B&M p− 1.49.0−
Prata p− 0.78.5−
Brutsaert Dc0.
B&M Dc− 1.36.7−
Prata Dc8.610.
Brutsaertp Dc− 2.58.7−
B&Mp Dc0.
Pratap Dc1.

In the next step, the Day-Night differences has also been analysed. Figure 1 describes the daily pattern of the measured effective emissivity under clear sky derived from the whole set of clear days registered in 2001 at Tabernas (Spain). It was evident that daytime effective emissivity is lower than that obtained at night. Alados-Arboledas and Jiménez (1988) evidenced similar behaviour using a reduced dataset registered under clear sky and this was explained in terms of day-night differences in the effective emissivity regime due to differences in the vertical structure of the atmosphere. The effective emissivity depends on the vertical structure of the atmosphere and those formulae that only rely on surface information must be applied with caution.

Figure 1.

Daily pattern of the mean effective emissivity under clear sky and standard deviation (bars). Data 2001 (Tabernas, Spain)

Several authors evidenced different behaviour of the empirical parameterisations when applied to day time or night time data (Paltridge, 1970; Arnfield, 1979; Alados-Arboledas and Jiménez, 1988; Tang et al., 2004; Dupont et al., 2008), with an overestimation effect during the day time. Alados-Arboledas (1993) showed that the use of a single expression based on screen level variables for the whole day requires the inclusion of a day-night correction term, taking into account the day-night differences in the vertical structure of the atmosphere and thus the effective emissivity regime.

Figure 2 shows the daily evolution of the differences between experimental and estimated effective emissivity using the locally fitted formula of Brutsaert. It is evident that this formula clearly overestimates the experimental values during the day but yields underestimation at night.

Figure 2.

Daily pattern of the difference between experimental and estimated effective emissivity using Brutsaert's expression with local coefficients. The lines correspond to diurnal correction expressions proposed by: Berdhal & Martin (dash line), Alados & Jiménez (dot line) and our parameterisation (Alados) (solid line). Data summer 2001 (Tabernas, Spain). This figure is available in colour online at

For all the original parameterisations considered we found a clear daytime overestimation and night time underestimation. The former was especially large for those expressions developed for night time data; while the opposite is true for the latter. For the locally fitted parameterisations of this study, the local fitting of the coefficients, based on day and night time data, led to similar absolute values of the night time underestimation and day time overestimation.

This problem has been addressed by different authors. Berdahl and Martin (1984) proposed the addition of a corrective term evaluated as a function of the hour of the day:

equation image(7)

where ε is effective emissivity determined experimentally, εi the estimated effective emissivity, and t the solar time in hours.

Alados-Arboledas and Jiménez (1988) showed that the daily cycles of emissivity have some similarities with the daily variations of the vertical profiles of temperature. In this way, they evidenced that the day time decrease of the effective emissivity presents some delay in winter that may be related to the presence of a stronger night time temperature inversion that remains for several hours after sunrise. During summer thermal inversions presented shorter duration and were weaker and thus the reduced day time values of emissivity were reached earlier. This indicates the importance of the vertical structure of the atmosphere on the effective emissivity and implies a limitation in the applicability of those formulae relying only on surface information. Thus Alados-Arboledas and Jiménez (1988), selecting a few clear days, proposed the following equation for Δε:

equation image(8)

where t′ is the approximate hour of dawn in solar time, Δt the maximum day time duration, and f is a term that takes into account the delay of the daily cycle in the effective emittance with respect to the solar cycle. Alados-Arboledas (1993) successfully applied this correction to two years of data measured in South-eastern Iberian Peninsula.

The availability, in this study, of five-minute values of the relevant variables allowed us to model the daily cycle of effective emissivity. Basing on the previous developments we tested the following correction function:

equation image(9)

This function approaches in a simple way the diurnal cycle of the differences between experimental and modelled emissivity values. The coefficient P1 is related to the amplitude of the differences in the emissivity daily cycle. The coefficient P2 is related to a delay of the diurnal cycle following Alados-Arboledas and Jiménez (1988). The fitting coefficients of the proposed correction function were computed for the original and locally fitted versions of the parameterisations with best results in the previous section. Table VI includes the values of coefficients P1 and P2, calculated with the data measured at Tabernas in 2001.

Table VI. Values of coefficients P1 and P2, and Chi-square χ2, calculated for diurnal corrections. Tabernas dataset for 2001
Brutsaert0.0325 ± 0.00031.171 ± 0.0080.0006
B&M0.0342 ± 0.00021.195 ± 0.0070.0004
Prata0.0325 ± 0.00031.24 ± 0.0030.0009
Brutsaert p0.0325 ± 0.00031.174 ± 0.0080.0005
B&M p0.0339 ± 0.00021.250 ± 0.0060.0004
Prata p0.0325 ± 0.00021.253 ± 0.0060.0004

Figure 2 visualizes the behaviour of the new correction Equation (9) (Alados) and those of the previous proposals: Equation (7), Berdhal & Martin and Equation (8), Alados & Jiménez. The new Equation accounts for the variability of Δε better than Berdhal & Martin proposal (Berdhal and Martin, 1984). Furthermore the new correction is simpler and more efficient than that proposed by Alados & Jimenez (Alados-Arboledas and Jimenez, 1988).

Table III evidences that after the application of the diurnal correction expression there is a clear improvement in all formulae, original and locally fitted. MBD below 1 Wm−2 and RMSD below 10 Wm−2 were obtained using local fitting and diurnal correction. This improvement can be evidenced by the histograms of the differences between experimental values and those estimated by the different versions of the analysed formulae. According to Figure 3(a) that shows these histograms for different versions of Brutsaert proposal, the combination of local fitting and diurnal correction implies a symmetric distribution of the differences around zero. Furthermore for this parameterisation up to 88% of the data have differences below 3%.

Figure 3.

Histograms of the differences between experimental and estimated LW values for the different versions of the proposed expression by Brutsaert: with original coefficients, local fitting and diurnal correction: (a) Tabernas (Spain), and (b) Palaiseau (France)

It must be noted that the application of the diurnal correction to the original B&M expression yields results similar to those obtained by application of the diurnal correction to the expressions of Brutsaert and Prata with locally fitted coefficients. This evidences the importance of the correction of the diurnal cycle of the effective emissivity.

To test the applicability of the proposed diurnal correction procedure to other locations we used data acquired at Palaiseau. Table V evidences an improvement of the formulae when the diurnal correction is applied. Thus, we obtain MBD values close to 0 Wm−2 and RMSD around 8 Wm−2. Furthermore according to Figure 3(b), up to 94% of the data showed differences below 3%.

4.2. Cloudy skies

4.2.1. Scheme of Crawford and Duchon

In this work, we used the scheme proposed by Crawford and Duchon (1999), described by Equation (4). The cloud modification factor, clf (Equation (5)), was computed by means of global solar irradiance measurements and the estimated clear-sky value of this variable, Go. Five different alternatives were considered for the estimation of Go. These parameterisations compute the global solar irradiance using screen level input data. Four parameterisations were included in the work of Niemalä et al. (2001)—Bennett, Paltridge and Platt, Moritz and Zillman - to which we added a fifth (Haurwitz, 1945).

These parameterisations were checked against the experimental values measured at Tabernas in 2002 and 2003 under clear-sky conditions for zenith angles lower than 80°. The parameterisation that provides the best results was that proposed by Zillman (1972) with locally fitted coefficients:

equation image(10)

where Isc is the solar constant (Isc = 1367 Wm−2), θz is solar zenith angle and e is the water vapour partial pressure (hPa).

To estimate the clear-sky emissivity we selected two parameterisations: B&M with original coefficients, ε0BMT, and Brutsaert with locally fitted coefficient, ε0BpT, both corrected for daily effects with the expression proposed in Equation (9). These parameterisations presented low values of MBD and RMSD and represent a formula with original coefficients and the local fit for a rather simple expression of ample use in the field, respectively.

Thus the parameterisations for the LW for all-sky conditions read as follows:

equation image(11)
equation image(12)

As clf increases from 0 to 1, computed values of clf less than zero were adjusted back to zero so as to be physically realistic.

Table VII shows the behaviour of these parameterisations when checked against the experimental values measured at Tabernas in 2002. For all-sky conditions, MBD were below 2% and RMSD were below 6%, while for cloudy skies MBD were below 3% and RMSD were below 7%. Although this represents an increase in the deviation between measured and estimated values in comparison with the results obtained for clear skies, it can be observed that these deviations are in the range of the experimental errors. The slope of the linear fit of estimated versus measured values was close to unity.

Table VII. Statistical results for the cloudy sky LW formulae under all-sky conditions and cloudy sky conditions. Total number of data N = 41.593 (all-sky), and N = 19.033 (cloudy sky). Mean experimental values for LW, 326 Wm−2 (all-sky) and 338 Wm−2 (cloudy sky). Tabernas dataset for 2002
ExpressionsMBD (Wm−2)RMSD (Wm−2)MBD (%)RMSD (%)bR2
All sky conditions
C& D
LWktBMT− 7.320.1−
LWktBpT− 6.920.2−
Cloudy sky
C& D
LWktBMT− 8.323.1−
LWktBpT− 7.623.1−

This analysis was also done at Palaiseau (France). Table VIII shows the statistical analysis, obtaining MBD values lower than − 3% and for RMSD below 7% for all-sky conditions. The slope of the linear fit for estimated versus measured values was close to unity. According to Table VIII the differences associated to the use of different clear-sky parameterisations of the effective emissivity were negligible. In comparison with Tabernas, the results at Palaiseau suggested a slight underestimation. In this sense the different results at each location can be attributed to different climate and instrumentation.

Table VIII. Statistical results for the cloudy sky LW formulae under all-sky conditions and cloudy sky with data of Palaiseau Cedex (France)
ExpressionsMBD (Wm−2)RMSD (Wm−2)MBD (%)RMSD (%)bR2
All sky conditions
C& D
LWCDBMT− 8.623.0−
LWCDBpT− 7.822.4−
LWktBMT− 21.031.9−
LWktBpT− 20.131.1−
Cloudy sky
C& D
LWCDBMT− 8.823.1−
LWCDBpT− 7.922.5−
LWktBMT− 21.532.0−
LWktBpT− 20.231.2−

4.2.2. Parameterisation in kt

We tested Equation (6), using the same parameterisations to estimate effective emissivity under clear skies that were used testing the Crawford and Duchon scheme. Thus, the expressions are:

equation image(13)
equation image(14)

The coefficients a and m were obtained by fitting the data set measured in Tabernas during 2001. Thus for ktBpT we obtained a = 1.202 ± 0.001 and m = 0.303 ± 0.002, while for ktBMT the fitting coefficients were a = 1.208 ± 0.001 and m = 0.315 ± 0.002. It is evident that both pairs of coefficients are rather similar, suggesting negligible differences between the two selected clear-sky parameterisations for the effective emissivity.

The fitted parameterisations were validated using the data registered during 2002 at Tabernas. Table VII summarizes the results, under all sky conditions MBD and RMSD are of − 2.2 and 6.2%, respectively, with a slight tendency to underestimate. Under cloudy skies RMSD increased up to 6.8%, while MBD was − 2.5%. With data from Palaiseau Cedex (France) for all sky conditions (Table VIII), the MBD is around − 6% and RMSD was 9.7%. This parameterisation underestimates the experimental values for both stations.

The results at Palaiseau suggested that local fitting of the cloudy formula could lead to some improvements. Thus after local fitting at Palaiseau the new coefficients for ktBpT were a = 1.267 ± 0.001 and m = 0.274 ± 0.002, while for ktBMT were a = 1.278 ± 0.001 and m = 0.280 ± 0.002. The use of these locally fitted coefficients led to estimates with MBD values lower than 1% and for RMSD lower than 8%. According to these results, it is clear that the cloudy scheme based on kt presents local dependence, while that proposed by Crawford and Duchon (1999) showed a greater generality. The reason for this could be that in the latter case the formulation, due to the use of a cloud factor based on the normalization of the solar global irradiance to its clear-sky value, takes into account the influence of climatic differences that are lacking in the kt formulation. In particular, the humidity regime at the two analysed stations is very different and this has a direct effect on the clear sky, global irradiance computed using Equation (10).

We compared our results with those obtained in previous studies of LW under all-sky conditions. Thus, Alados-Arboledas et al. (1995) compared several expressions to estimate LW under all-sky conditions using the total cloud cover fraction as an input variable. The best results were obtained for Boltz's expression, Equation (3), with a RMSD of 5%. Niemelä et al. (2001) estimated LW under all-sky conditions using simple cloud correction factors depending on total cloud cover, with RMSD and MBD of 4%. Iziomon et al. (2003) fitted the coefficient q for lowland and mountain sites using hourly data. In their study, these authors obtained estimates for LW with MBD and RMSD of − 1 and 7%, respectively, for the lowland site, and − 0.5 and 9% for the mountain site. Bilbao and De Miguel (2007) presented all-sky conditions estimates with RMSD of 17 Wm−2 and MBD of − 4 Wm−2.

5. Conclusions

We analysed five parameterisations proposed in the bibliography to estimate downward atmospheric irradiance (LW) under clear skies (expressions of Brunt, I&J, Brutsaert, B&M, and Prata) using data measured at Tabernas (Almería, Spain). The best results were obtained with the expression proposed by B&M (MBD = 0.05% and RMSD = 4.4%), an empirical parameterisation with dependence on dewpoint temperature.

In a second step, we applied a local fitting of coefficients for the evaluated parameterisations. After being locally adjusted, all the parameterisations presented negligible MBD and RMSD lower than 4.5%, excluding I&J with RMSD around 6%.

Furthermore, we studied the different behaviour of the clear-sky parameterisations during day and night time. A simple expression based on the time of the day was proposed to correct the daily cycle of deviations of the empirical parameterisations. The use of this correction implies an improvement of the parameterisations estimation. Thus, using the daily correction scheme, the expressions of Brutsaert, B&M and Prata with local coefficients present RMSD close to 3% and MBD lower than 0.3%, similar to the results that we obtained using the expression B&M with original coefficients.

To test the applicability of our proposal, we used data from another station with different climatology (Palaiseau, France). For the locally fitted parameterisations of the expressions proposed by Brutsaert, B&M and Prata the use of the daily correction yields MBD below 1% and RMSD below 3%.

Moreover, we tested two parameterisations for the estimation of LW under all-sky conditions. The cloud effect on LW was modelled through the use of solar global irradiance data, avoiding the requirement of cloud cover observations. In the first parameterisation we followed the scheme of Crawford and Duchon, 1999 with a cloud factor defined as a function of global solar irradiance, measured, G, and estimated under clear skies, G0. For the last one, we used the expression proposed by Zillman (1972) with local coefficients. A second approach was tested using the hemispheric solar transmittance, kt, to estimate the cloud effect.

Both parameterisations require the estimation of LW under clear skies. For this purpose we used the expressions of B&M with original coefficients and diurnal correction (BMT) and an expression of Brutsaert locally fitted with diurnal correction (BpT). Using an independent data set measured at Tabernas for testing the proposed parameterisations, we obtained for all sky conditions (clear and cloudy) MBD of 1.5% and RMSD of 5.8% for the Crawford and Duchon scheme, while the results for the kt parameterisations present MBD of − 2.2% and RMSD of 6.2%.

Testing the cloudy parameterisations at Palaiseau, we obtained for Crawford and Duchon scheme results close to those obtained at Tabernas. Nevertheless, the results obtained at Palaiseau for the kt parameterisation reveal clear differences from those obtained at Tabernas. These results suggest the wider generality of Crawford and Duchon scheme and put in evidence greater local influence on the simplest kt parameterisation.

These parameterisations represent a useful alternative for the estimation of LW under all-sky conditions at those locations where cloud cover observations are not available. Nevertheless, it is worthy to remember that, due to their dependence on solar global irradiance, these parameterisations are restricted to daytime.


This work was supported by the Spanish Ministry of Science and Technology through Projects Nos CGL2010-18782 and CSD2007-00067, and by the Andalusian Regional Government through Projects Nos P10-RNM-6299 and P08-RNM-3568. The authors would like to thank the BSRN network and the anonymous reviewers that have helped us to improve the manuscript.