## Introduction

Attempts to investigate the energy balance at the earth’s surface are often hindered by a significant uncertainty in the estimated magnitude of downwelling longwave radiation received at the surface (LW_{d}). Accurate estimates of LW_{d} are vitally important in determining the radiation budget, which, in turn, modulates the magnitude of the terms in the surface energy budget (e.g., evaporation). Many reasonably successful techniques have been developed in recent decades that estimate LW_{d} based on surface observations alone. These methods have had varying degrees of success, and new techniques continue to be developed. The main problem seems to be versatility; that is, the methods are developed empirically at one location and with one set of instruments. Often, those developing new methods later show that these methods do not fare well in other locations. In addition, previous methods were developed for daily or longer-term averages and therefore are usually much less accurate at shorter time intervals. Finally, practically all techniques are valid in clear skies only, greatly limiting their utility. We provide a formula for estimating atmospheric emissivity that can be used to calculate LW_{d} under any daytime sky condition.

Brunt (1932), based on a perceived similarity between heat conduction and radiative transfer, theorized that LW_{d} was related to the square root of near-surface vapor pressure, *e.* Using monthly averages of LW_{d} and *e,* he developed the first empirical relationship between the two quantities. Use of this technique resulted in a correlation coefficient of 0.97. Three decades later, Swinbank (1963) argued that LW_{d} was not related to *e* at all, but to the square of the temperature *T* alone and that the Brunt (1932) formula worked only due to the positive correlation between *e* and *T.* Swinbank’s new formula resulted in a correlation coefficient of 0.99 between observed and estimated LW_{d} and a root-mean-square error (rmse) of less than 5 W m^{−2}.

Idso and Jackson (1969) developed an equation, also dependent on *T*^{2}, which was tested against a much wider range of temperatures than the previous formulations. This formula was touted as being “valid at any latitude and for any air temperature reached on earth,” and had a correlation coefficient of 0.99 between measured and estimated LW_{d}. Staley and Jurica (1972) integrated emissivity over the entire atmosphere using vertical profiles of vapor pressure, carbon dioxide, and ozone and related it to *e.* This formula was much more theoretical than previous efforts and was successfully used by Deardorff (1978) in the development of a surface energy budget model. Finally, Satterlund (1979) found that previous formulations did not perform well in temperatures below 0°C and developed a new formulation that claimed to be more accurate at extreme temperatures and of similar accuracy at moderate temperatures.

Brutsaert (1975, hereafter B75) was the first to develop a more physically rigorous parameterization of atmospheric emissivity. This parameterization is based on Schwarzschild’s equation (Liou 1980, 22) and assumptions of standard atmospheric lapse rates of temperature and vapor pressure. Culf and Gash (1993) concluded that this method was superior to previous formulations since it is easily adjusted for locally measured lapse rates. Like so many of the other techniques, though, it was developed for clear skies alone. Deardorff (1978) developed a simple correction for cloudiness and applied it to the Staley and Jurica (1972) clear-sky parameterization for emissivity. When observations of fractional cloud cover are not available, however, the correction for cloudiness cannot be determined. Here, we have developed an improved method that incorporates the B75 clear-sky parameterization and the Deardorff (1978) cloudiness correction. A clear-sky model was used along with the observed magnitude of solar radiation to provide a proxy for fractional cloudiness. We have also added an annual sinusoidal modification to the clear-sky Brutsaert parameterization coefficient, which is representative of typical variations in vertical atmospheric profiles of water vapor pressure. We found that this new emissivity parameterization, when used to calculate LW_{d}, performs better than previous methods and performs well at any time of the year and in any sky condition, although its usefulness is restricted to daylight hours.

It also will be shown that atmospheric sounding data can be used, along with the derivations found in B75, to estimate the clear-sky coefficient directly. This method is even less empirical and shows great promise for use in estimating LW_{d} at any observing station collocated with an upper-air sounding system or any location at which vertical moisture profiles can be reasonably interpolated from nearby soundings.

## Previous formulations

_{atm}and effective temperature

*T*

_{atm}of the overlying atmosphere according to

_{d}

_{atm}

*σT*

^{4}

_{atm}

*σ*is the Stefan–Boltzmann constant. Since it is difficult to specify ε

_{atm}or

*T*

_{atm}for a vertical column of atmosphere, methods have been developed to parameterize LW

_{d}from the measured temperature and/or vapor pressure near the surface during clear skies such that

_{d}

_{c}

*T, e*

*σT*

^{4}

_{c}is the effective clear-sky atmospheric emissivity, and

*T*and

*e*are the near-surface temperature and vapor pressure, respectively. In the past, empirical formulations for ε

_{c}were developed based on least squares regression of observed LW

_{d}during periods of clear skies.

*s,*

*s*is the ratio of the measured solar irradiance to the clear-sky irradiance.

*I*at the ground was calculated using a previously developed model based on the results of Paltridge and Platt (1976) and Meyers and Dale (1983). This quantity was approximated by

*I*

*I*

_{o}*Z*

*T*

_{R}

*T*

_{pg}

*T*

_{w}

*T*

_{a}

*I*

_{o}is the effective solar constant,

*Z*is the solar zenith angle, and

*T*

_{i}the transmission coefficients for Rayleigh scattering

*R,*absorption by permanent gases pg and water vapor

*w,*and absorption and scattering by aerosols

*a.*

^{−2}) is given by

*I*

_{o}

*r*

*r*

^{2}

*r*

*r*are the average and daily distances between the sun and the earth, respectively. The cosine of the solar zenith angle is represented by

*Z*

*γ*

*δ*

*γ*

*δ*

*H,*

*γ*is the latitude of the station,

*δ*is the solar declination, and

*H*is the hour angle. The hour angle is

*H*

*π*

*t*

_{noon}

*t*

*t*

_{noon}is local solar noon (∼12.5 in Oklahoma) and

*t*is the local solar time (e.g.,

*t*= 12.5 and

*H*= 0 at local solar noon). The empirical expression for the product of the first two transmission coefficients is (Atwater and Brown 1974)

*p*is the pressure in millibars and

*m*is the optical air mass at

*p*= 1013 mb given by

*m*= 35 cos

*Z*(1224 cos

^{2}

*Z*+ 1)

^{−1/2}. The third coefficient is (McDonald 1960)

*T*

_{w}

*um*

^{0.3}

*u*is the precipitable water given by

*u*= exp[0.1133 − ln(

*G*+ 1) + 0.0393

*T*

_{d}] (Smith 1966),

*T*

_{d}is the dewpoint (°F), and

*G*is an empirical constant dependent upon time of year and latitude. The fourth transmission coefficient is (Houghton 1954; Meyers and Dale 1983)

*T*

_{a}

^{m}

*I*was calculated from (4), direct observations of solar irradiance were used to calculate

*s*and clf in (3). Inclusion of the effects of clouds yields

_{d}

_{c}

*σT*

^{4}

*σT*

^{4}

_{c}) and the limiting (but unobserved) value (ε = 1). Calculated values of clf less than zero were adjusted back to zero so as to be physically realistic.

_{c}formulations, modified by the cloudiness correction and inserted into (11), were tested against LW

_{d}data collected at the Department of Energy Atmospheric Radiation Measurement (ARM) Southern Great Plains Cloud and Radiation Testbed Central Facility (CF) in Lamont, Oklahoma (Stokes and Schwartz 1994):

*T*is in degrees kelvin and

*e*is in millibars. In (15),

*q*is the specific humidity, which is a function of

*e*and the barometric pressure. Equation (12) is a modification of the formulation originally developed by Brunt (1932) and has been shown to provide acceptable clear-sky estimates of LW

_{d}[using the value of ε in (11)] in Oklahoma (Arnfield 1979).

## Results and discussion

The data needed for testing values of LW_{d} [using (12)–(17) in (11)] included near-surface observations of LW_{d}, downwelling shortwave radiation SW_{d}, barometric pressure, vapor pressure, and temperature.

At the ARM CF site (36.61°N, 97.49°W, altitude = 318 m), measurements of LW_{d} (SW_{d}) were made with an upward-looking hemispherical broadband Eppley pyrgeometer (pyranometer). The instruments were mounted at a height of 1.5 m and employed a 1-s sampling interval. Barometric pressure was measured at a height of 1 m at 1-min intervals and vapor pressure and temperature were measured at 2 m at 1-s intervals. All variables were averaged over 30 min.

Often, the most difficult part of obtaining an accurate measurement of downwelling longwave radiation is overcoming the problems associated with solar heating of the instrument dome, which can result in spuriously high LW_{d}. The upward-looking pyrgeometer at the CF site was shaded and ventilated to reduce these dome heating effects, and a correction was also made based on the measured dome and case temperatures (M. Splitt 1997, personal communication).

Calculations of LW_{d} using (11) and the six parameterizations of ε (12)–(17) were first compared to observed LW_{d} ARM data over four 1-month periods: November 1995, February 1996, May 1996, and August 1996. By doing this, seasonal biases in the performance of any of the formulas could be detected. Since the clear-sky model could be used only during daylight hours, the total amount of data available was restricted. For fall and winter (spring and summer), ARM data from 1400–2230 (1300–2330) UTC were used. Table 1 shows the results of these comparisons.

*p*)

_{d,i}and LW(

*o*)

_{d,i}are the parameterized and observed values, respectively, and

*n*is the total number of half-hourly averaged observations for the month. A positive value means that the parameterization overestimates LW

_{d}. Only the Anderson (1954) and the B75 schemes had absolute mbe’s that were consistently less than 12 W m

^{−2}throughout the year.

*R.*” The B75 and Anderson (1954) schemes had the highest values throughout the year, and every scheme had its smallest value in summer. The

*y*intercept and the slope of the regression lines are also given in Table 1. A perfect scheme would produce a

*y*intercept of zero and a slope of one. The Idso–Jackson (1969) scheme performed best in these two categories but had the largest rmse’s of all the equations tested.

Based on their comparatively low mbe and rmse values, the Anderson (1954) and the B75 formulations were considered to be superior to the other four. Scatterplots of the two schemes for November 1995 are depicted in Figs. 1 and 2. One can see that the distributions of data points about the linear fits are similar. What distinguishes the formulations is the slope of each regression curve. Since the B75 equation has a physically based derivation while the Anderson (1954) equation is strictly empirical, the B75 scheme was chosen for further investigation.

## Seasonal adjustment to Brutsaert coefficient

Equation (16) was derived using Schwarzschild’s radiative transfer equation, in which standard atmosphere vertical profiles of temperature and vapor pressure were used to calculate the leading coefficient (1.24) [see (1)–(11) in B75]. Using measured profiles of water vapor pressure and temperature in Niger, Culf and Gash (1993) were able to rederive the original B75 equation and get a slightly different value of the leading coefficient. By doing this they were able to reduce the rmse by 50%. They also found that, during the dry season in Niger, the lapse rate of vapor pressure was significantly smaller than the standard atmospheric lapse rate assumed in B75. This means that for the given surface conditions there was more water vapor aloft, and thus higher emissivity, than expected using a standard atmosphere vapor pressure profile. Because of this, the leading coefficient was increased to 1.31 in order to properly represent LW_{d} in these conditions. On the other hand, for wet surface conditions, the vapor pressure lapse rate would be expected to be larger than in a standard atmosphere since vapor pressure varies much more near the surface than aloft. In this case, the B75 coefficient would have to be reduced to compensate. It is important to note that it is the magnitude of the lapse rate that determines the correct value of the leading coefficient, not the magnitude of the measured vapor pressure at the surface.

_{d}is least in winter and greatest in summer. However, the relatively low

*e*during the winter months more than offsets the larger leading coefficient, with the opposite relationship during the summer months.

Table 2 reveals that using (20) rather than (16) resulted in fairly uniform and small absolute mbe’s (less than 10 W m^{−2}) throughout the year. It is immediately apparent that the new scheme had a smaller mbe than the other five schemes (Table 1), with the exception of the Anderson (1954) scheme in August. The rmse’s are similar to the original B75 scheme, while correlation coefficients were slightly smaller. The *y* intercept and slope of the regression lines were also mostly similar to B75. The new scheme appears to be valid for all seasons and sky conditions with rmse’s less than 23 W m^{−2} for 30-min averages. This scheme will not work at night since there is then no way to ascertain cloud fraction.

Because of the empiricism involved in fitting the sinusoid to the CF data, the broader applicability of (20) may be in question. For this reason, data from the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE; Sellers et al. 1992) were acquired and used to test the new formulation. These data were available as 30-min averages. Barometric pressure and the wet- and dry-bulb temperatures were measured at 2 m at 10 different locations within the FIFE domain (39.05°N, 96.53°W, altitude = 410 m). These 10 values were averaged to obtain representative site values at each half-hourly observation time. The site-average mixing ratio was then calculated from the site-average barometric pressure and wet-bulb temperature data and converted to vapor pressure. The LW_{d} and SW_{d} data were measured by Eppley pyranometers and pyrgeometers, respectively, at two different sites located 14 km apart, then averaged to obtain representative site values for the area at 30-min intervals. These observations were compared to the parameterization represented by (20), and the error statistics are shown in Table 3. The FIFE comparisons show larger mbe’s but smaller rmse’s than the ARM data. Considering that the method was developed empirically for ARM data, the FIFE comparisons are very encouraging.

The results obtained in this study compare favorably to recent parameterizations of LW_{d} in all sky conditions. Sugita and Brutsaert (1993) were able to empirically calibrate the adjustable parameters in their formulation using 1987 FIFE data (summer and fall only) to reduce their rmse’s to around 15–17 W m^{−2}, which are larger than the summer and fall errors found by our improved technique. Moreover, they used only 325 data points and did not test the effectiveness of their method in winter and spring. Culf and Gash (1993), by tuning the Brutsaert coefficient to 1.31 during the dry season in Niger in 1990, were able to get an rmse of 12 W m^{−2}, but again, this is a limited dataset from summer and early fall and was only valid in clear skies.

## Using sounding data to adjust Brutsaert coefficient

The Culf and Gash (1993) results also suggested that the leading coefficient could be adjusted, based on local sounding data, in order to provide accurate estimates of LW_{d}. To test this idea, soundings from 1730 UTC at the ARM CF were analyzed from November 1995–October 1996. The sounding time was chosen to represent the middle of daylight hours. Monthly averaged water vapor lapse rates were calculated, and the B75 derivation was reworked to provide a monthly averaged leading coefficient. A comparison of the leading coefficients derived from the soundings and those from the sinusoidal hypothesis are shown in Fig. 3. It is apparent that the most significant disagreement between the two methods occurred during the spring and summer months. Since the sinusoid represents a “best fit” to the data, the largest mbe’s will occur during these months if the sounding method is used, as seen in Fig. 4. Nevertheless, the absolute mbe’s were still no greater than 15 W m^{−2} using the sounding-derived coefficients, and Fig. 5 shows that the rmse’s were actually smaller. It is also interesting to note in Fig. 3 that the sounding-derived coefficients were close to the original 1.24 coefficient derived from the standard atmosphere assumptions employed by Brutsaert to derive (16).

## Summary and conclusions

An improved, practical technique for calculating effective atmospheric emissivity for use in estimating downwelling longwave radiation during the daytime has been investigated. The physically based model given in B75 has been modified using ideas from Deardorff (1978) and Culf and Gash (1993). Two improvements have been developed and tested. The first employs an empirical sinusoidal annual variation in the leading coefficient. This method was applied to two datasets with slightly different results. For the ARM data, the method resulted in absolute mbe’s less than 9 W m^{−2} and rmse’s less than 23 W m^{−2}. For the FIFE data, absolute mbe’s less than 15 W m^{−2} and rmse’s less than 12 W m^{−2} were found over a 4-month warm season period. The second improvement uses an objective method for determining the cloud fraction using observed solar radiation and modeled clear-sky radiation.

The use of upper-air sounding data to calculate the leading coefficient in B75 was also studied. The yearly variation of these coefficients hinted at a sinusoidal variation of similar phase but much smaller magnitude than the “best fit.” Although mbe’s were consequently larger using the sounding-derived coefficients, rmse’s were slightly smaller, and the empiricism of the best-fit sinusoid was removed. Because of this, the sounding method may be more portable than the sinusoid method, with the prerequisite that the site of interest have upper-air data available.

We recommend that if the climatology at a location where LW_{d} is needed but not measured is similar to that at another location where LW_{d} is measured, that the best-fit sinusoidal variation of the leading coefficient be employed. For example, the best-fit sinusoidal variation developed in this study based on observations of LW_{d} at the ARM sites in Oklahoma can be used to accurately estimate LW_{d} at many Oklahoma Mesonet (Brock et al. 1995) sites. In general, if the climatologies are sufficiently different, we suggest using the sounding-derived variation of the leading coefficients if there is a nearby upper-air station.

Future studies should include comparative analyses at the two other ARM sites in northern Alaska and the tropical western Pacific, where all components of the radiation budget in addition to standard meteorological variables are measured. Since these two climates are at opposite ends of the meteorological spectrum, these data would provide rigorous tests of the improved method. In addition, more tests using the sounding-derived variation of the leading coefficient should be performed using data from other field experiments (e.g., FIFE). These tests would act to verify or negate the promising results found in this study using ARM CF data.

## Acknowledgments

Data were obtained from the ARM program sponsored by the U.S. Department of Energy, Office of Energy Research, and Office of Health and Environmental Research, Environmental Sciences Division. We would like to thank many people for their contributions. Keith Brewster and Mark O’Malley at the University of Oklahoma provided the clear-sky model. Michael Splitt provided ARM–CART data support. Eric Smith and Harry Cooper at The Florida State University provided the FIFE data. Alistair Culf at the Institute of Hydrology in the United Kingdom helped us understand an important derivation. This study was funded by NSF research Grant ATM-9612674.

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Comparative LW_{d} statistics for the six *ϵ* formulations (12)–(17).

Statistics for the improved formulation based on a sinusoidal variation of the leading coefficient.

Comparisons of ARM and FIFE LW_{d} datasets using the sinusoidal variation in effective atmospheric emissivity in (20). ARM data are from 1996; FIFE data are from 1987. Units in W m^{−2}.