1. Introduction
Insolation at the earth’s surface [global incoming shortwave radiation (Rg)] is the primary energy source for the majority of biogeochemical or physical land surface processes as well as for the operation of photovoltaic (PV) power production systems. Therefore, it is one of the most important drivers for land surface models that calculate energy, water, and carbon balances, and site-specific information about Rg is essential to estimate the viability of PV systems. With knowledge of albedo and temperature, Rg is the starting point for estimates of net radiation, evapotranspiration, and the energy balance. Assumptions about the spectral composition of Rg lead to estimates of photosynthetically active radiation (PAR) and practical ecosystem-scale models of photosynthesis, biogeochemical cycling, carbon uptake, and plant growth (e.g., Wang and Jarvis 1990; Williams et al. 1996; Sellers et al. 1997; Baldocchi and Meyers 1998; Arora 2002; Bonan 2008). At the site level, such models are ideally driven with directly measured values of PAR, Rg, or net radiation at high temporal resolution. Despite this strong demand for radiation data, direct measurements are not standard at most climate stations, and even at flux network (FLUXNET) stations (e.g., Baldocchi et al. 2001) data series often suffer from missing values. If direct surface radiation measurements are unavailable, it is thus necessary to model or parameterize them from whatever data are available.
Many models for Rg exist (Besharat et al. 2013; Bindi and Miglietta 1991; Li et al. 2013; Woli and Paz 2012), and commonly they are based on the ratio of Rg (at the surface) over the value of downwelling solar radiation at the top of the atmosphere (extraterrestrial solar radiation RE), defining the overall atmospheric transmissivity 
Empirical models of Rg must address the fact that cloudiness has commonly both the strongest and the most variable effect on atmospheric transmissivity for shortwave radiation. Unfortunately, neither cloudiness nor sunshine duration are standard variables reported by climate stations, and thus, such models need to revert to using a suitable proxy for cloudiness. As cloudiness also affects the thermal regime at the surface, one obvious such proxy is the daily range of air temperatures (Baigorria et al. 2004; Bristow and Campbell 1984; Goodin et al. 1999; Lee 2010). Among the most frequently used Rg models of this kind are those by Hargreaves and Samani (1982) and Mahmood and Hubbard (2002), which require only the daily range (minimum and maximum) of air temperature as an input variable. Since only one daily value is used as input in these models, they are also limited to produce daily mean estimates of Rg only. Daily means can then be distributed over a daily course using prescribed (e.g., sinusoidal) functions (e.g., Berninger 1994). However, if surface radiation data are needed to parameterize processes at subdaily resolution (e.g., to gap fill eddy-covariance-based CO2 exchange time series; e.g., Reichstein et al. 2005), such estimates introduce considerable uncertainty, as they are unable to respond to short-term atmospheric variability.
At a mountaintop carbon exchange flux station on Lackenberg Mountain (1308-m elevation) in the Bavarian Forest National Park (southern Germany; see Lindauer et al. 2014), we found that cloud cover, and thus Rg and PAR, typically varied strongly over the course of the day. As the relationship between carbon assimilation by photosynthesis and shortwave radiation is strongly nonlinear (e.g., Reichstein et al. 2005), daily mean values of a cloudiness proxy were not sufficient to drive our carbon assimilation model at times when we were lacking all but standard climate station data. We also found that the atmospheric moisture regime that supports or suppresses the formation of clouds in elevated layers of the troposphere appears to be fairly well coupled to surface humidity—at least at the Lackenberg site. This very heuristic and speculative notion led us to use standard measurements of relative humidity as a proxy for cloud cover, haziness, and thus variability of atmospheric transmissivity. Surprisingly, we have found no evidence that a relation between relative humidity and atmospheric transmissivity has ever been explored before. In this work, we present a simple empirical model of Rg at subdaily (e.g., half hourly) resolution that requires only relative humidity as meteorological input. Relative humidity is a standard observation variable at most climate stations. We test the model at a wide range of observation sites and evaluate it against independent datasets. We also investigate its accuracy at hourly and daily resolutions.
2. Methods
Effective local transmissivity (Rgobs/R0) vs binned values of relative humidity (black diamonds). Only sparse data (gray bars show number of data in each bin) were available for relative humidity below 30%. Thus, the transmissivity for these values was set to 1 (circles). This figure is taken from Lindauer et al. (2014).
Citation: Journal of Applied Meteorology and Climatology 56, 7; 10.1175/JAMC-D-16-0085.1
For model development and parameterization, we looked for freely available weather information with high quality and resolution that cover a wide range of environmental conditions. Therefore, we used data derived from U.S. Surface Climate Observing Reference Networks (http://www.ncdc.noaa.gov/crn/qcdatasets.html—hourly data). For additional evaluation, we used hourly data from independent sites with consistent data that are not involved in the process of model development and parameterization. These sites are Alice Springs (data from 2011 to 2013) in Australia [part of the Australian Terrestrial Ecosystem Research Network (TERN; http://www.tern.org.au); Cleverly 2011]; Fendt (data from 2011 to 2013) in southern Germany [part of the German Terrestrial Environmental Observatories (TERENO; http://www.tereno.net); e.g., Eder et al. 2014]; Lackenberg (data from 2011 to 2013), also in southern Germany (Lindauer et al. 2014); Manaus in Brazil (Falge et al. 2005; data from 1996); Kruger National Park in South Africa (data from 2001 to 2002); and Bontioli (data from 2004 to 2005) in Burkina Faso (Grote et al. 2009).
High-resolution meteorological data that include humidity and radiation are available at all of these sites. Table 1 shows general information about the selected sites. The uncertainty for data from the U.S. sites is reported as ±3% (±5% at <10% and >90%), ±0.3°C for air temperature, and ±70 W m−2 for solar radiation (Diamond et al. 2013).
Parameters of sites selected for model development and evaluation. The first 15 sites are from the U.S. Surface Climate Observing Reference Networks (data from 2012) and data from these sites were used for model development and parameterization. The remaining sites are outside the United States, and parameters from these six are used for additional model evaluation. Station names in the first column are followed by the common codes for states (for the U.S. stations) and countries. The quantities Tavg, Tmax, and Tmin are the average, maximum, and minimum air temperatures for the data period used; Prec is the annual precipitation; rHavg and rHavg(daytime) are the overall average and daytime average relative humidities; and CI is the continentality index. Data source references are 1) Diamond et al. (2013); 2) Cleverley (2011); 3) Grote et al. (2009); 4) Eder et al. (2014); 5) Hanan et al. (2004); 6) Lindauer et al. (2014); and 7) Falge et al. (2005).
These sites cover a wide range of latitudes, longitudes, elevations, average air humidity, and continentality. A continentality index (CI) was calculated as the difference between the average monthly air temperature in January and July (e.g., Botta-Dukát et al. 2005; Holmlund and Schneider 1997).
Figure 2 illustrates the relation between Rgobs/R0 and rHf (Fig. 2a) and shows the resulting scatterplot of modeled Rg* versus observed Rg values (Fig. 2b) at Boulder, Colorado, in 2011. This analysis was done for every site listed in the first part of Table 1 (above the bold horizontal line). Parameters b [the shape parameter in Eq. (6)] and a (the slope of the regression) and the coefficient of determination R2, as well as the root-mean-square error (RMSE) and the normalized RMSE (NRMSE), are listed in Table 2. These parameters and model evaluation measures were estimated with 3 yr of data (2011–13) at each site to account for the year-to-year variability.
(a) The quantity Rgobs/R0 vs rHf at Boulder in 2011. The blue line shows the power-law regression with an estimated parameter of b = 0.38. (b) Modeled values of Rg* vs observed values of Rg. The blue line is the linear regression with slope a = 0.87.
Citation: Journal of Applied Meteorology and Climatology 56, 7; 10.1175/JAMC-D-16-0085.1
Parameters of the regression functions in Fig. 2 and measures of predictive power (see text for definitions).
3. Results
Equation (6) was used in a first estimate of Rg*, and the results were compared by linear regression with observed data at all sites of Table 1. We used a constrained linear regression that is forced through the origin here, even though data from some sites clearly exhibit nonzero intercepts (see Fig. 2b). The reason for this choice is to keep the model as simple as possible and to avoid introducing an additional free parameter. Table 2 summarizes the results of the regressions and shows the averages of b, a, R2, RMSE, and NRMSE of the 3 yr for every site. The first part shows again the sites in the United States, which were used for model development and parameterization. The shape parameter b ranges between 0.17 and 0.53, with an average of 0.34, and the slope parameter, a, ranges from 0.74 to 0.99, with an average of 0.92. The average R2 value of all sites is 0.92, indicating a very good overall performance of the model. The NRMSE ranges between 24% and 74%, with an average of 40%.
We did not detect any trend in the variation of shape parameter b. In contrast, the slope parameter a as well as the random uncertainty (NRMSE) shows a linear relation to the annual average daytime relative humidity 
(a) Slope of linear regressions and (b) NRMSE vs annual average daytime humidity 
Citation: Journal of Applied Meteorology and Climatology 56, 7; 10.1175/JAMC-D-16-0085.1
To reduce the systematic bias of the estimates due to nonunity slopes (parameter a) in the modeled versus observed radiation values, we exploited the linear dependence of parameter a on 
To examine the applicability, we used this generalized model [Eq. (9)] to the sites listed in the lower part of Table 1 to test the model performance at independent sites, which have not been used in the model development.
Table 3 shows parameters of the linear regression functions after using Eq. (9) at the independent sites. The slope parameter a ranges from 0.96 to 1.13, with an average of 1.02. The average R2 value of all sites is 0.91, and the NRMSE ranges between 26% and 74%, with an average of 65%. As shown in Fig. 3b (for Rg*), the uncertainty exhibits some heteroscedastic behavior with increasing NRMSD toward higher values of 
Figure 4 compares observed and modeled hourly Rg values at Champaign, Illinois, from 13 to 18 September 2001. The gray and orange dotted lines are derived following Hargreaves and Samani (1982) and by the method of Mahmood and Hubbard (2002), respectively. These daily values were transformed to hourly values according to the method described in Berninger (1994). The time series in Fig. 4 contains days with high as well as low relative humidity (sunny as well as cloudy days) and demonstrates the satisfactory performance of the present model [Eq. (9)] in comparison with the observed values and also in comparison with other models.
Time series plot of hourly Rg values at Champaign from 13 to 18 Sep 2011. The solid black line shows the observed values, and the solid blue line shows Rg values derived using Eq. (9), as proposed in the present work. Dotted gray and orange lines are derived from the models of Hargreaves and Samani (1982) and Mahmood and Hubbard (2002), respectively.
Citation: Journal of Applied Meteorology and Climatology 56, 7; 10.1175/JAMC-D-16-0085.1
Apart from model performance at an hourly temporal resolution, the question could arise if the relationship between global radiation and air humidity also holds for more temporally aggregated data (e.g., daily values). Therefore, we tested the relationship with daily input values of air humidity and compared the results with those obtained with a conventional method. For this exercise, we used the same model as described before for all sites in the investigation but restrict ourselves to the year 2012. The quantities R0 and rH are simply replaced by daily values instead of hourly values. It should be noted that relative humidity is calculated from daytime values only to be consistent with the period where the radiation data are originating from. We compared the daily sum of measured Rg values with the daily sum of modeled hourly Rg values (Fig. 4a), with the results of this daily model (Fig. 5b), with Rg values derived after the method of Hargreaves and Samani (1982; Fig. 5c) and with Rg values derived after the method of Mahmood and Hubbard (2002; Fig. 5d).
Daily values of Rg derived from (a) hourly air humidity (present study), (b) daily air humidity (present study), (c) minimum and maximum air temperature with the method of Hargreaves and Samani (1982), and (d) minimum and maximum temperature with the method of Mahmood and Hubbard (2002) shown vs daily aggregated observations. The data are from the U.S. stations that are listed in Table 1.
Citation: Journal of Applied Meteorology and Climatology 56, 7; 10.1175/JAMC-D-16-0085.1
These results show that using subdaily values performs better than using daily values and that using daily values is still superior to using the method of Hargreaves and Samani (1982) or the model of Mahmood and Hubbard (2002).
4. Discussion and conclusions
To our knowledge, this work is the first study to present a relation between relative humidity and atmospheric transmissivity as a basis for modeling downwelling solar radiation (global radiation) at the land surface.
The model has been tested for a variety of sites that represent a range of global terrestrial microclimates. Thus, we are confident to propose it for general application. However, a rigorous cause-and-effect chain between screen-level relative humidity, turbidity, cloud cover, and transmissivity over the entire atmosphere above a given site is not straightforward to establish, and we do not attempt to try. At our Lackenberg site, the necessity of filling large gaps in radiation data was “the mother of invention” to guide our intuition toward exploring relative humidity as a proxy. As a variable, relative humidity combines moisture and temperature characteristics of air and thus expresses the state of the air relative to condensation conditions. Turbidity is not likely well related to direct measures of humidity content (such as mixing ratio or due point temperature) but to the relative proximity to saturation. This quality is unique for rH, and thus, we did not pursue other avenues in our study. The fact that relative humidity near the surface correlates well with haze and cloud conditions aloft [e.g., Walcek (1994) and references therein] indicates that the general shape of atmospheric profiles of moisture and temperature is fairly robust, and relative shifts due to airmass changes appear to be anchored well to their surface values for given local climatic conditions. We were surprised that our simple method worked as well as it did and even more so when we found that it performs well for a wide range of elevations and latitudes globally.
Nevertheless, it should be noted that the relation between transmissivity and relative humidity near the surface is likely uncoupled under certain conditions. Such conditions include the presence of surface advection (e.g., near coastlines or in katabatic flows), nonconvective lifting (e.g., orographic or frontal lifting), or strong dust–aerosol loading in arid–semiarid environments. Some of our test sites may be affected by such factors at times. Because the sensitivity of modeled radiation to changes in relative humidity is particularly strong under moist conditions, the uncertainty of our radiation model is less at dry (continental) sites than in humid regions (see Tables 1–3 and Fig. 3b).
We have shown that the accuracy of our model increases when subdaily resolution values are used, which underlines the importance of the nonlinearity in the relation between transmissivity and humidity. Although the model performance statistics reported by Vuichard and Papale (2015) are different from the ones used here and are not given for individual sites, it appears that the present model performs nearly as well as their ERA-Interim-based method (they report a mean relative error after bias correction for global radiation of 34%). However, as mentioned, retrieval and downscaling of site-specific reanalysis time series is quite demanding. In addition, Vuichard and Papale’s method includes an intermediate step of debiasing that requires preexisting representative site-level data of global radiation. We thus conclude that the comparative simplicity and general applicability of the present model, without recourse to preexisting data other than relative humidity, comes at a relatively small cost of uncertainty.
All data used in this work are traceable through the citations listed in Table 2 or available by e-mail request to the corresponding author (m-lindauer@web.de). This research was supported, in part, by the Bavarian Ministry of the Environment and Public Health (UGV06080204000), the German Helmholtz Association with its research program Atmosphere and Climate (ATMO), and the KIT Graduate School for Climate and Environment (GRACE). The support by the administration of the Bavarian Forest National Park is very much appreciated. We thank Matthias Zeeman, Carsten Jahn, and Elisabeth Eckart for scientific and technical support. We also thank Trevor Keenan (Lawrence Berkeley National Laboratory, Berkeley, California) for referring us to the Australian dataset, and we gratefully acknowledge the constructive remarks by Ankur Desai (University of Wisconsin–Madison) and two anonymous reviewers.
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