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    Evolution of the average daylight downward longwave irradiance values for the period from April 2001 to December 2004 at Valladolid.

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    Frequency distribution of the daylight downward longwave irradiance at Valladolid for the period from April 2001 to December 2004.

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    Hourly values of weather conditions and daylight downward longwave irradiance from 10 to 19 Jul 2002 at Valladolid.

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    As in Fig. 3, but from 20 Feb to 1 Mar 2002.

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    Comparison of measured and estimated hourly daylight downward longwave irradiance values for clear-sky conditions for four calibrated different schemes at Valladolid. The data period is from July to December 2004.

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    As in Fig. 5, but for all-sky conditions.

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Estimation of Daylight Downward Longwave Atmospheric Irradiance under Clear-Sky and All-Sky Conditions

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  • 1 Department of Applied Physics, Facultad de Ciencias, University of Valladolid, Valladolid, Spain
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Abstract

Daylight downward longwave irradiance data recorded over a flat place for the period between April 2001 and December 2004 in Valladolid, Spain, have been compared with estimates generated using four different schemes. The parameterization schemes of Brutsaert, Swinbank, Idso, and Brunt have been considered and calibrated for the comparison. Root-mean-square errors (rmse), mean bias errors, and linear regression correlations have been used to compare measured and estimated values. The results of this comparison show that, for clear-sky conditions, rmse values range between 19.57 and 8.85 W m−2 for calibrated schemes and between 39.78 and 11.13 W m−2 for original ones. The Idso and Brunt schemes give the best results with calibrated coefficients, and the Brunt scheme performs the best with original coefficients. A new scheme for estimating daylight downward longwave irradiance under “all-sky” conditions has been developed based on clear-sky schemes and solar global shortwave irradiance, and, after comparing measured and estimated values by calibrated schemes, it has been found that the Idso and Brunt schemes give the best results.

Corresponding author address: Julia Bilbao, Department of Applied Physics, Facultad de Ciencias, University of Valladolid, 47005 Valladolid, Spain. Email: juliab@fa1.uva.es

Abstract

Daylight downward longwave irradiance data recorded over a flat place for the period between April 2001 and December 2004 in Valladolid, Spain, have been compared with estimates generated using four different schemes. The parameterization schemes of Brutsaert, Swinbank, Idso, and Brunt have been considered and calibrated for the comparison. Root-mean-square errors (rmse), mean bias errors, and linear regression correlations have been used to compare measured and estimated values. The results of this comparison show that, for clear-sky conditions, rmse values range between 19.57 and 8.85 W m−2 for calibrated schemes and between 39.78 and 11.13 W m−2 for original ones. The Idso and Brunt schemes give the best results with calibrated coefficients, and the Brunt scheme performs the best with original coefficients. A new scheme for estimating daylight downward longwave irradiance under “all-sky” conditions has been developed based on clear-sky schemes and solar global shortwave irradiance, and, after comparing measured and estimated values by calibrated schemes, it has been found that the Idso and Brunt schemes give the best results.

Corresponding author address: Julia Bilbao, Department of Applied Physics, Facultad de Ciencias, University of Valladolid, 47005 Valladolid, Spain. Email: juliab@fa1.uva.es

1. Introduction

The radiation budget at the surface of the earth is fundamental to understanding the climate system, its variability, and the changes resulting from human influence. The upward and downward shortwave and longwave radiation are the fluxes involved in the exchange of energy between surface and atmosphere (Konzelmann et al. 1994; Gabathuler et al. 2001). Radiative fluxes are determined by the state and composition of the atmosphere. Measurements of energy radiative fluxes are essential in assessing theoretical treatments of radiative transfer in the atmosphere, verifying climate model computations, and studying trends in surface radiation at scales that are smaller than those normally associated with climatic regions.

Daylight downward longwave radiation from the atmosphere is a very important energy flux because it is a component of the radiation budget in land surface models (Sridhar and Elliot 2002). The understanding of the surface downward longwave irradiance energy components is necessary for improving weather predictions, global climate simulations, and energy budget evaluations.

Different authors have studied longwave irradiance: Cellier (1993) and Figuerola and Mazzeo (1997) predicted night temperature from downward and upward longwave irradiance as input variables. Konzelmann et al. (1994), Prata (1996), Guest (1998), and Gabathuler et al. (2001) proposed new parameterizations for estimating downward clear-sky radiation that have been extensively tested under different atmospheric conditions. Garrat et al. (1999) measured values of the longwave radiation component to understand the influence of carbon dioxide concentration in the atmosphere and for minimum air temperature prediction and climate risk evaluations. Crawford and Duchon (1999) improved parameterization for estimating the atmospheric emissivity.

Longwave irradiance data are not frequently available, however, because it is not a conventional measurement in meteorological stations in Spain; as a consequence, different parameterization schemes have to be developed that take available meteorological variable values as input. Equations have been developed to estimate the downward longwave irradiance from screen-level air temperature and humidity (e.g., Brunt 1932; Idso 1981; Idso and Jackson 1969). Many of the proposed equations are empirical and specific for the atmospheric conditions under which they were developed; therefore, these equations have to be redefined or calibrated for each place.

The cloud influence on longwave irradiance has been estimated by authors such as Alados-Arboledas (1993), Niemellä et al. (2001), Konzelmann et al. (1994), and Köning-Langlo and Augstein (1994), who compared, measured, and evaluated longwave irradiance values taking clear-sky and all-sky conditions into account and for different cloud fractions.

To estimate the cloud contribution to downward longwave irradiance, a variety of methods have been proposed, some of which are based on a corrective term that must be added to the downward longwave irradiance evaluated under clear-sky conditions, for which it is necessary to have knowledge of the total cloud cover. Crawford and Duchon (1999) generalized the effect of clouds by introducing a cloud fraction term “clf,” defined as clf = 1 − s, where s is the ratio of measured horizontal global solar irradiance to the horizontal global solar irradiance under clear-sky conditions (symbols defined in this paper are listed in the appendix). They also suggested that cloud effects on daylight downward longwave irradiance could be included by using the clf term and the clear-sky emissivity. Kasten and Czeplak (1979) also found that cloud cover is related to the s parameter. Marty and Philipona (2000) defined the clear-sky index to separate clear-sky from cloudy-sky conditions on the basis of longwave downward radiation, screen temperature, and water vapor pressure at the earth’s surface.

Authors such as Long and Ackermann (2000) and Dürr and Philipona (2004) show different methods to detect clouds that are based on shortwave flux modification or on longwave downward radiation, temperature, and relative humidity, respectively.

The aim of this paper is to select, implement, and evaluate downward longwave parameterization schemes using long-term irradiance data recorded at the Low-Atmosphere Research Center (Centro de Investigación de la Baja Atmósfera; CIBA) in Valladolid, Spain, to choose the most appropriate one for the region. The measured meteorological data series used are composed of hourly solar global and daylight downward longwave irradiance, air temperature, and relative humidity in electronic format for a period of 4 yr (2001–04).

In this study, the parameterization schemes of Brutsaert, Swinbank, Idso, and Brunt have been selected and then calibrated using independent data. They have been used to estimate daylight longwave irradiance, and the agreement between measured and estimated values has been evaluated. The results have been compared to decide which scheme could be recommended as the best for the continental Mediterranean region. After analysis of the existing clear-sky schemes, a new scheme for estimating daylight downward longwave irradiance under all-sky conditions based on clear-sky schemes and solar global shortwave irradiance is proposed.

The interest of the paper is that similar data series have not been recorded before and no similar work has previously been done in central Spain, and it has the advantage of using a large amount of recorded data. The results of this paper represent the first daylight downward longwave irradiance parameterization evaluation in the region, which is essential because of the fact that the existing schemes have been used in Mediterranean continental conditions, with dry and cold winters and high air temperatures and low water vapor pressure values in the central hours of the day during summers.

The theoretical base of the selected parameterization schemes, the fitted equations, the performance, and the recommended and proposed equations to be used in central Spain are given in the following sections, and special details can be seen at the corresponding references.

2. Parameterization schemes

The parameterization schemes used in this study are based on empirical relationships, and the atmospheric longwave irradiance is estimated from screen-level values of air temperature and water vapor pressure. The set of schemes was selected as the most promising for further study. The criteria used for selecting schemes were 1) full availability of algorithms and numerical coefficients, 2) use of input data that are generally available or easily obtainable, and 3) the quality of the results reported by the original authors. The selected and proposed schemes can be classified as clear-sky and all-sky conditions, respectively, and they are given by the equations described below.

a. Clear-sky conditions

All reviewed schemes are empirical [see Eqs. (1)(4)] except Brutsaert’s model, which is derived from Schwarzchild’s equation (Brutsaert 1975). The scheme equations can be written by means of a general expression Lci = [ai + bif (e, T)]σT4, where e is the water vapor pressure (hPa), T is the screen air temperature (K), and f (e, T) is a function of e or T; the numerical coefficients ai and bi (i = 1, . . . , 4) are the coefficients of the parameterization schemes and have been fitted, taking into account that they characterize the atmospheric conditions of a measurement station (in this case, the Mediterranean continental climate of central Spain).

Brutsaert (1975) proposed an equation in which emissivity is a function of water vapor pressure and temperature at screen level. It is given by the following expression:
i1558-8432-46-6-878-e1
where Lc1 is the atmospheric longwave irradiance and σ is the Boltzmann constant (5.67 × 10−8 W m−2 K−4). Swinbank (1963) developed an equation for estimating downward longwave atmospheric irradiance that only depended on screen-level air temperature. It is given by the expression
i1558-8432-46-6-878-e2
Idso (1981) estimated longwave irradiance from air temperature and water vapor pressure with an equation of the form
i1558-8432-46-6-878-e3
Brunt’s (1932) equation estimated longwave atmospheric irradiance from screen-level temperature and water vapor pressure as
i1558-8432-46-6-878-e4
where e is the screen-level water vapor pressure.

b. All-sky conditions

Under all-sky conditions, downward longwave irradiance is increased by the radiation emission from liquid water and/or ice. An empirical correction that is widely used was developed by Bolz (1949) and can be seen in Alados-Arboledas et al. (1995) and Arnfield (1979); it is given by the expression
i1558-8432-46-6-878-e5
where L is the total downward longwave irradiance under all-sky conditions, Lc is the downward longwave irradiance under clear-sky conditions, N is the total cloud cover, and ξ is a function of height and type of clouds. The total cloud cover is not routinely measured at the CIBA site, and so this magnitude has to be estimated. To remove the influence of cloud cover in Bolz’s scheme, the following method is proposed in this paper: horizontal shortwave global solar irradiance is related to cloud cover; this relation has been investigated by many authors. Kasten and Czeplak (1979) considered 10 yr (1964–73) of continuous hourly data for Hamburg, Germany, in their research, and they showed that the ratio of horizontal global solar shortwave irradiance for any given cloud cover to horizontal global solar shortwave irradiance under clear-sky conditions is independent of solar elevation They found a relation given by the following expression:
i1558-8432-46-6-878-e6
where G is the horizontal shortwave global solar irradiance, Gc is the horizontal global solar shortwave irradiance under clear-sky conditions, N is the total cloud cover, and X and Z are empirical coefficients. Kasten and Czeplak (1979) found that X and Z coefficient values for Hamburg were 0.75 and 3.4, respectively, although these coefficients can vary from one site to another (Gul et al. 1998). Horizontal global irradiance under clear sky can be estimated from the following expression given by Haurwitz (1945):
i1558-8432-46-6-878-e7
where θz is the solar zenith angle. Equation (7) represents an average over all atmospheric turbidity conditions.
By taking into account Eqs. (5) and (6) and eliminating N, the following expression has been obtained and proposed in this paper to evaluate daylight downward longwave irradiance under all-sky conditions:
i1558-8432-46-6-878-e8
where L is daylight downward longwave irradiance under all-sky conditions, Lc is the daylight downward longwave irradiance under clear-sky conditions, and p and q are empirical coefficients. Considering Eq. (8) and the previous clear-sky model Eqs. (1)(4), four new models for estimating daylight downward longwave irradiance under all-sky conditions are proposed in this paper, with the general equation being given by the following expression:
i1558-8432-46-6-878-e9
where Li and Lci are the all-sky and clear-sky downward longwave irradiance values corresponding to each of the schemes and the coefficients pi and qi have to be estimated for each clear-sky model.

3. Site and measurements

Downward longwave irradiance, screen-level temperature, relative humidity, and horizontal global solar shortwave irradiance values were used in the study. Measurements were taken at a site 35 km from the city of Valladolid in the Castile and Leon region (41°49′N, 4°32′W, 845 m MSL). The measurement site occupies a flat area of 20 000 m2, without shady spots and providing a reasonably homogeneous field for recording meteorological and radiometric data.

Horizontal global solar shortwave irradiance was measured using a Kipp and Zonen, Inc., CM6B pyranometer that is sensitive over the range from 0.3 to 2.5 μm, downward longwave irradiance was measured with an Eppley Laboratory, Inc., precision infrared radiometer (pyrgeometer), and relative humidity and air temperature were measured by means of an HMP35AC solid-state probe (Campbell Scientific, Inc.) with accuracy of ±1% and ±0.2%, respectively. The associated recording instruments were a Campbell Datalogger CR10X model connected to the CM6B pyranometer and HMP35AC probe meters and a CR23X model connected to the pyrgeometer sensor. Dataloggers CR23X and CR10X were programmed at 30- and 10-s sampling rates, respectively, and both provided measurements at every 10- and 60-min integration time.

Measurements used in this paper were recorded for the period from 1 April 2001 to 30 December 2004. Data taken between April 2001 and June 2004 were used to fit schemes, and data recorded during the period from July to December 2004 were used to estimate the performance and validation of fitted schemes.

To ensure the high quality of observations and to eliminate spurious errors that could arise by incidental shading of an instrument, all observations for which horizontal solar global irradiance was less than 20 W m−2 were rejected. However, only downward longwave irradiance measurements within 200–400 W m−2 were accepted with a 99% confidence level. Furthermore, because of the cosine response of the pyranometer, no measurements made at solar zenith angles larger than 80° were included in the computations of solar and longwave irradiance measurements (Miskolczi et al. 1997). Some necessary quality-control tests were performed before the data were used (Bilbao et al. 2002, 2004). They are given by the following expressions:
i1558-8432-46-6-878-e10
i1558-8432-46-6-878-e11
i1558-8432-46-6-878-e12
where G is the horizontal hourly solar global shortwave irradiance, T is the hourly air temperature, H is the relative humidity, and Go is the horizontal hourly extraterrestrial solar irradiance. For the number of data in this study, 32 599 hourly values, 207 564 ten-minute values, and 1443 daily values of longwave irradiance were used, and no substitution of missing data was carried out.
Saturated water vapor pressure es was calculated by means of the Magnus expression (Guyot 1997), and the water vapor pressure e was estimated from relative humidity and es by the following expression:
i1558-8432-46-6-878-e13
where T is the air temperature (K) and H is the relative humidity (%).
Two thermistor circuits were built into the pyrgeometer sensor for the sake of monitoring the dome and body temperatures (the thermistor model is YSI, Inc., precision thermistor YSI 44031). Thermopile voltages were measured with a precision of 0.33 μV in a 10-mV full scale. Thermistor resistances were measured using a tension division half bridge with 2500 mV of excitation and a 0.10-s delay. Temperatures were computed from the thermistor resistances by using the Steinhart–Hart equation (Miskolczi et al. 1997):
i1558-8432-46-6-878-e14
where TS is the thermistor temperature (K), RS is the thermistor resistance (Ω), and the c0, c1, and c2 coefficients have the values 1.0295 × 10−3, 2.391 × 10−4, and 1.568 × 10−7, respectively. The data were corrected by using the following expression given by Albrecht and Cox (1977):
i1558-8432-46-6-878-e15
where L is the measured downward longwave irradiance (W m−2), x is the calibration constant, Q is the thermopile voltage (μV), σ is the Stefan–Boltzmann constant, Th and Tb are the dome and body temperatures (K), respectively, and K′ is the dome heating constant.

4. Results and discussion

The evolution of measured values is shown in Fig. 1, and it can be seen that the values increased during spring, reaching a maximum in summertime and a minimum in winter. Daylight downward longwave monthly minimum varied between 210 and 330 W m−2, and the maximum values were between 280 and 400 W m−2. An initial analysis of the data characteristics was made on the basis of the frequency histograms and the longwave irradiance levels as a function of time of year. From the frequency histogram of the data it is observed that the 10-min data series shows smooth evolution (Fig. 2), and the most frequent values are between 300 and 360 W m−2.

To know the influence of meteorological conditions and longwave irradiance levels, Fig. 3 shows the hourly values of overall conditions for 10 days, from 10 to 19 July 2002. The synoptic weather was remarkably sunny for 6 days in that period. Global radiation G shows daily cycles with maximum values between 950 and 500 W m−2, sometimes disturbed by clouds.

Downward longwave irradiance showed an evolution similar to temperature, with a relative minimum value in the early morning, values that increased until late afternoon, and then values that decreased in the final hours of the day. On clear days, it can be seen that minima of temperature and longwave irradiance take place at the same time. On cloudy days, the relation between temperature and longwave irradiance is similar but more maximum and minimum relative values appear. The highest value of longwave irradiance was on 14 July with 380 W m−2, and it was a day with high humidity and vapor pressure but low global solar irradiation and air temperature.

Relative humidity ranged from 23% to 100%. During the first 4 days, minimum air temperatures T were over 10°C at night and maximum values were 30°C. The following days, daily temperature oscillated from 8° to 25°C; as can be seen, this period was ideal in terms of sunshine. That changed on 14 July, with the maximum temperature up to 22°C and minimum over 12°C. The following days, the minimum temperatures decreased and the maximum increased.

Figure 4 shows the overall conditions for 10 days in winter, and a comparison of Figs. 3 and 4 shows the correlation and influence of meteorological variables on solar and longwave irradiance levels.

a. Clear-sky condition results

Daylight downward longwave irradiance hourly values have been classified according to the clearness index value k, which is defined as the relation between the horizontal solar global irradiation and the extraterrestrial horizontal solar irradiation during the same period of time. Taking k into account (Iqbal 1983), the sky can be considered as clear (k ≥ 0.7), partly cloudy (0.3 ≤ k < 0.7), and very cloudy (k < 0.3). From this index, skies were classified in clear- and all-sky conditions, and, from measured longwave irradiance data, the coefficients of the selected schemes were evaluated using Eqs. (1)(4). The obtained coefficients are shown in Table 1. It can be seen that there are differences between original and calibrated coefficients. Brutsaert (1975) derived his formula by an approximate integration of the Schwartzschild transfer equation, assuming a nearly standard atmosphere. The difference between original and calibrated values indicates that the climatological characteristics of CIBA do not agree with the standard atmosphere pattern used in the derivation of Brutsaert’s formula. For Swinbank’s formula coefficient, the calibrated value is lower than the original one; the reason for this could be that Swinbank (1963) used data from Australia and the Indian Ocean, where the conditions were warm. As pointed out by Alados-Arboledas et al. (1986), differences between calibrated and original Idso’s coefficients could be attributed to the atmospheric turbidity because data used for the development of the Idso equation were obtained at a very dusty place (the variations in atmospheric dust content can significantly alter the downward longwave irradiance). In our evaluations, the reduction of calibrated coefficient values with respect to original ones could also be due to the fact that the area where downward longwave irradiance measurements were made is a little dusty. Last, Brunt-model calibrated coefficients a4 and b4 are included in the ranges 0.34–0.71 and 0.020–0.110, respectively; similar results were found by Alados-Arboledas et al. (1986).

Daylight longwave irradiance data corresponding to the period from July to December 2004 have been used for evaluating the performance of original and calibrated schemes. The accuracy was assessed by means of two widely used statistical estimators [root-mean-square error (rmse) and mean bias error (mbe)] and by scatterplots of the estimated values as a function of the measured values (Bilbao et al. 2002). The following expressions for rmse and mbe were used:
i1558-8432-46-6-878-e16
i1558-8432-46-6-878-e17
where n is the number of data pairs, Ei is the ith estimated downward longwave irradiance value with a given scheme, and Mi is the ith measured longwave irradiance value.

Table 2 shows the comparison of calibrated models with respect to original ones, and it can be seen that calibrated models show rmse and mbe values that are lower than those of the original models, indicating that the former schemes present a better performance than the latter. Rmse values for calibrated schemes range between 8.85 and 19.57 W m−2, and all of them (calibrated and original) overestimate daylight downward longwave irradiance except the Brunt scheme with original coefficients. Idso’s and Brunt’s schemes give the lowest rmse and mbe with calibrated coefficients. The diminution of mbe values has been very important, and Swinbank’s scheme shows the worst performance with both original and calibrated coefficients. This result means that sky emissivity is a function of water vapor pressure.

Estimated values with calibrated schemes and measured data have also been compared for assessment of calibrated schemes, in the interest of clarity, and the following expression has been proposed for the regression:
i1558-8432-46-6-878-e18
where Lc,meas and Lc,est are the measured and estimated daylight longwave irradiance, respectively, under clear-sky conditions. Table 3 shows the linear regression results, and it can be seen that the correlation coefficients r2 for each scheme are similar to 1, the slope values Bc are near 1, and Idso’s scheme obtains the highest correlation coefficient.

The graphical comparison of estimated schemes with calibrated schemes and measured daylight downward longwave irradiance for clear-sky conditions using each of the mentioned calibrated schemes is shown in Fig. 5; the figure consists of four scattered graphs, and the diagonal line represents the ideal match between the estimated and measured values. In comparing these scattered graphs and taking into account Table 3 results, it can be seen that Brunt’s and Idso’s schemes show similar assessment but Swinbank’s scheme shows more scattering for low and high irradiance values than the others. A general characteristic of Brutsaert’s, Brunt’s, and Idso’s equations is that all of them perform well for high irradiance levels and that the difference between measured and estimated values increases for small irradiance levels.

b. All-sky condition results

In a previous section, a parameterization for all-sky conditions was proposed by the authors of this paper; the coefficients p and q of these equations have been evaluated for each of the clear-sky schemes and from the hourly longwave measured values L during the period from April 2001 to June 2004.

Table 4 shows the calibrated coefficient values obtained by Eq. (9). It can be seen that depending on the clear-sky model used to estimate the daylight downward longwave irradiance different values of the pairs (pi, qi) have been obtained. Idso’s and Brunt’s schemes show similar results; Swinbank’s scheme shows different values. This means, for clear-sky conditions, that all schemes except Swinbank’s have the highest value of the qi coefficient and that Brutsaert’s scheme obtained the lowest value for the pi coefficient.

Linear regressions of estimated and measured values were performed for each scheme, using Eq. (18) and the measured data corresponding to the period from July to December 2004. Table 5 shows the number of data, the linear regression results, the correlation coefficients, and the rmse and mbe error values. Rmse and mbe errors have increased in comparison with clear-sky results; Idso’s scheme obtained the best correlation coefficient and rmse values. All schemes except Swinbank’s underestimate longwave solar irradiance.

The comparison of estimated and measured values of daylight downward longwave irradiance with calibrated coefficients for all-sky conditions is shown graphically in Fig. 6; the ideal match between estimated and measured values is represented by the diagonal line. Taking into account Table 5 and Fig. 6 results, it can be seen that all scatterplots are similar except Swinbank’s scheme, which shows the highest dispersion, and that Idso’s scheme shows the best performance in this region.

5. Conclusions

A series of daylight longwave solar irradiance recorded in central Spain has been analyzed, the first time that work of this nature has been done for that place. For this study, 45 months of hourly daylight downward longwave irradiance data were measured and modeled for clear- and cloudy-sky conditions. The objective of this study was to evaluate the performance of calibrated schemes for estimating daylight downward longwave irradiance, especially in cloudy conditions. Schemes for calculating the downward longwave irradiance have been reviewed and calibrated, taking into account the atmospheric characteristics of the measurement site. From the results, it may be concluded that Idso’s and Brunt’s schemes perform equally well and, therefore, may be used to obtain accurate downward longwave irradiance for clear-sky conditions. Under all-sky conditions, downward longwave irradiance was estimated from calibrated clear-sky equations that include a corrective term that depends on global horizontal irradiance and global solar irradiance × under clear skies. Considering this corrective term, it is concluded that Idso’s calibrated equation is the scheme that shows the best performance for all-sky conditions at the CIBA site in Valladolid, with an rmse value of 17.08 W m−2.

With the proposed parameterization, longwave irradiance can be evaluated in places where measurements of global solar irradiance, temperature, and relative humidity are available. The results of this study are interesting for climatology, radiative interaction of the atmosphere with ecosystems, and energy efficiency evaluation performance systems, among other things.

Acknowledgments

The authors gratefully acknowledge the financial support extended by the Spanish Science and Technology Ministry and by the Autonomous Government of the Castile and Leon region under the Projects REN2000-1103/CLI and CO05/103, respectively. The authors also thank the anonymous reviewers and the editor for their useful comments and suggestions in improving the paper.

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APPENDIX

Nomenclature

  • a  Coefficient of the parameterization schemes
  • b  Coefficient of the parameterization schemes
  • c0  Coefficient of the thermistor temperature
  • c1  Coefficient of the thermistor temperature
  • c2  Coefficient of the thermistor temperature
  • Ei  Longwave solar irradiance estimated value
  • e  Screen-level water vapor pressure (hPa)
  • es  Saturated screen-level water vapor pressure (hPa)
  • G  Horizontal global solar irradiance (W m−2)
  • Gc  G for clear skies
  • Go  Horizontal extraterrestrial irradiance (W m−2)
  • H  Relative humidity (%)
  • K′  Dome heating constant
  • k  Hourly clearness index
  • L  Longwave solar irradiance (W m−2)
  • n  Number of data pairs
  • N  Total cloud cover (octa)
  • Mi  Longwave solar irradiance measured value
  • p  Fitting coefficient
  • q  Fitting coefficient
  • Q  Thermopile voltage (μV)
  • R  Resistance (Ω)
  • s  The ratio of measured horizontal global solar irradiance to the horizontal global solar irradiance under clear-sky conditions
  • T  Screen-level temperature (K)
  • X  Empirical coefficient
  • x  Calibration constant
  • Z  Empirical coefficient
  • ξ  Bolz coefficient (octa−1)
  • σ  Stefan–Boltzmann constant (W m−2 K−4)
  • θz  Zenith solar angle (rad)
  • b  Pyrgeometer-case subscript
  • c  Clear-sky subscript
  • est  Estimated-value subscript
  • h  Pyrgeometer-dome subscript
  • i  Index of proposed model
  • meas  Measured-value subscript
  • S  Thermistor subscript

Fig. 1.
Fig. 1.

Evolution of the average daylight downward longwave irradiance values for the period from April 2001 to December 2004 at Valladolid.

Citation: Journal of Applied Meteorology and Climatology 46, 6; 10.1175/JAM2503.1

Fig. 2.
Fig. 2.

Frequency distribution of the daylight downward longwave irradiance at Valladolid for the period from April 2001 to December 2004.

Citation: Journal of Applied Meteorology and Climatology 46, 6; 10.1175/JAM2503.1

Fig. 3.
Fig. 3.

Hourly values of weather conditions and daylight downward longwave irradiance from 10 to 19 Jul 2002 at Valladolid.

Citation: Journal of Applied Meteorology and Climatology 46, 6; 10.1175/JAM2503.1

Fig. 4.
Fig. 4.

As in Fig. 3, but from 20 Feb to 1 Mar 2002.

Citation: Journal of Applied Meteorology and Climatology 46, 6; 10.1175/JAM2503.1

Fig. 5.
Fig. 5.

Comparison of measured and estimated hourly daylight downward longwave irradiance values for clear-sky conditions for four calibrated different schemes at Valladolid. The data period is from July to December 2004.

Citation: Journal of Applied Meteorology and Climatology 46, 6; 10.1175/JAM2503.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for all-sky conditions.

Citation: Journal of Applied Meteorology and Climatology 46, 6; 10.1175/JAM2503.1

Table 1.

Calibrated and original coefficient values of daylight downward longwave irradiance schemes under clear-sky conditions at Valladolid using hourly data for the period from April 2001 to June 2004.

Table 1.
Table 2.

Comparison of statistical estimators of four original and calibrated daylight downward longwave irradiance schemes under clear-sky conditions at Valladolid using hourly values for the period from April 2001 to June 2004.

Table 2.
Table 3.

Comparison of measured and estimated daylight downward longwave irradiance values by means of linear regression for clear-sky condition schemes at Valladolid. The data period is from July to December 2004. Scatterplot coefficients are Bc (slope coefficient) and r (correlation coefficient).

Table 3.
Table 4.

Calibrated and original coefficients of daylight downward longwave irradiance schemes for clear- and all-sky conditions at Valladolid for the period from April 2001 to June 2004. The ai and bi scheme coefficients are for clear-sky schemes; pi and qi scheme coefficients are for all-sky schemes.

Table 4.
Table 5.

Comparison of statistical estimators of four calibrated daylight downward longwave irradiance schemes under all-sky conditions at Valladolid for the period from July to December 2004.

Table 5.
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