Introduction
Longwave radiation is an important component of the radiation balance on
earth and it affects many phenomena, such as evapotranspiration, snowmelt
, glacier evolution
, vegetation dynamics
, plant respiration, and primary productivity
. Longwave radiation is usually measured with
pyrgeometers, but these are not normally available in basic meteorological
stations, even though an increasing number of projects has been developed to
fill the gap
. The
use of satellite products to estimate longwave solar radiation is increasing
(GEWEX, Global Energy and Water cycle Experiment; ISCCP, the International
Satellite Cloud Climatology Project), but they have too coarse a spatial
resolution for many hydrological uses. Therefore, models have been developed
to solve energy transfer equations and compute radiation at the surface
e.g.. These physically based and
fully distributed models provide accurate estimates of the radiation
components. However, they require input data and model parameters that are
not easily available. To overcome this issue, simplified models (SMs), which
are based on empirical or physical conceptualizations, have been developed to
relate longwave radiation to atmospheric proxy data such as air temperature,
water vapour deficit, and shortwave radiation. They are widely used and
provide clear-sky e.g. and all-sky
estimations of downwelling (L↓) and upwelling (L↑)
longwave radiation
e.g..
SM performances have been assessed in many studies by comparing measured and
modelled L↓ at hourly and daily time steps
e.g..
were among the first to present a comparison
of the most used SMs in an evaluation of their accuracy. They tested seven
clear-sky algorithms using atmospheric data from different stations in the
United States. In order to validate the SMs under different climatic
conditions, they performed linear regression analyses on the relationship
between simulated and measured L↓ for each algorithm. The
results of the study show that the best models were ,
and .
made a similar comparison using more
formulations (13) and a wider dataset from North America and China,
considering all possible sky conditions. Finally,
evaluated the performance of six SMs, with both
literature and site-specific formulations, under clear-sky conditions for the
sub-humid Pampean region of Argentina.
However, none of the above studies have developed a method to systematically
estimate site-specific model parameters for location where measurements are
not available using basic site characteristics.
This paper introduces the LongWave Radiation Balance package (LWRB) of the
JGrass-NewAGE modelling system . LWRB
implements 10 formulations for L↓ and 1 for L↑
longwave radiation. The package was systematically tested against measured
L↓ and L↑ longwave radiation data from 24 stations
across the contiguous USA, chosen from the 65 stations of the AmeriFlux
Network. Unlike all previous works, the LWRB component follows the
specifications of the Object Modeling System (OMS) framework
. Therefore, it can use all of the JGrass-NewAge
tools for the automatic calibration algorithms, data management and GIS
visualization, and it can be seamlessly integrated into various modelling
solutions for the estimation of water budget fluxes
. Moreover, differently from other studies,
all the tools used in this paper are open-source, well documented, and ready
for practical use by other researchers and practitioners.
Methodology
The SMs for L↑ (W m-2) and L↓ (W m-2) longwave radiation are
based on the Stefan–Boltzmann equation:
L↓=ϵall-sky⋅σ⋅Ta4,L↑=ϵs⋅σ⋅Ts4,
where σ = 5.670 × 10-8 (W m-2 K-4) is
the Stefan–Boltzmann constant, Ta (K) is the air temperature,
ϵall-sky (–) is the effective atmospheric emissivity,
ϵs (–) is the soil emissivity and Ts (K) is the
surface soil temperature. To account for the increase in L↓ in
cloudy conditions, ϵall-sky (–) is formulated according to
Eq. ():
ϵall-sky=ϵclear⋅1+a⋅cb,
where c (–) is the cloud cover fraction and a (–) and b (–) are two
calibration coefficients. Site-specific values of a and b are presented
in (a = 0.22 and b = 1),
(a ranges between 0.25 and 0.4 and b = 2)
and (a = 0.183 and b = 2.18).
In our modelling system a and b are calibrated to fit measurement data
under all sky conditions. The cloud cover fraction, c, can be estimated
from solar radiation measurements , from visual
observations , and from
satellite data , or it can be modelled as well.
In this study we use the formulation presented in
and , where c is related to the
clearness index s (–), i.e. the ratio between the measured incoming solar
radiation, Im (W m-2), and the theoretical solar radiation
computed at the top of the atmosphere, Itop (W m-2),
according to c = 1 - s . This type of
formulation needs a shortwave radiation balance model to estimate
Itop and meteorological stations to measure Im; also, it cannot
estimate c at night. In our application, the fact that the SMs are fully
integrated into the JGrass-NewAge system allows us to use the shortwave
radiation balance model to compute
Itop. Night-time values of c are computed with a linear
interpolation between its values at the last hour of daylight and the first
hour of daylight on consecutive days. The computation of the first and last
hours of the day is based on the model proposed in
that follows the approach proposed in
, Eqs. (4.23)–(4.25). The sunrise occurs at
t = 12 ⋅ (1 - ω/π) and the sunset will be at
t = 12 ⋅ (1 + ω/π), where ω is the hour
angle, i.e. the angle between the observer meridian and the solar meridian.
It is zero at noon and positive before noon. Those equations are based on the
assumption that sunrise and sunset occur at the time when the z coordinate
of the sun vector equals zero.
The LWRB component of JGrass-NewAge and the flowchart to model longwave
radiation.
The formulation presented in Eq. (3) was proposed by
applied in other studies
.
Evaluating the effectiveness of different formulations with respect to
Eq. (3) is still an open question which is not the object of the current
paper. It has been investigated in several studies i.e.and
references therein and
some of them recommended the one proposed by .
Ten SMs from the literature have been implemented for the computation of
ϵclear. Table specifies assigned component
number, component name, defining equation, and reference to the paper from
which it is derived. X, Y and Z are the parameters provided in the
literature for each model, listed in Table .
Clear-sky emissivity formulations: Ta is the air
temperature (K), w (kg m-2) is precipitable water = 4650
(e0/Ta) and e (kPa) is screen-level water-vapour pressure. The
models follow the formulations presented and used in
. The Angstrom and Brunt models were
presented as cited by . Konzelmann uses water
vapour pressure in Pa, not kPa.
ID
Name
Formulation
Reference
1
Angstrom
ϵclear = X - Y × 10Ze
2
Brunt's
ϵclear = X + Y ⋅ e0.5
3
Swinbank
ϵclear = (X × 10-13 ⋅ Ta6)/(σ ⋅ Ta4)
4
Idso and Jackson
ϵclear = 1 - X ⋅ exp(-Y × 10-4 ⋅ (273 - Ta)2)
5
Brutsaert
ϵclear = X ⋅ (e/Ta)1/Z
6
Idso
ϵclear = X + Y × 10-4 ⋅ e ⋅ exp (1500/Ta)
7
Monteith and Unsworth
ϵclear = X + Y ⋅ σ ⋅ Ta4
8
Konzelmann
ϵclear = X + Y ⋅ (e/Ta)1/8
9
Prata
ϵclear = [1 - (X + w) ⋅ exp(-(Y + Z ⋅ w)1/2)]
10
Dilley and O'Brien
ϵclear = (X + Y ⋅ (Ta/273.16)6 + Z ⋅ (w/25)1/2)/(σ ⋅ Ta4)
The models presented in Table were proposed with
coefficient values (X, Y, Z) strictly related to the location in which the
authors applied the model and where measurements of L↓
radiation were collected. Coefficients reflect climatic, atmospheric and
hydrological conditions of the sites, and are reported in Table .
Model parameter values as presented in their literature formulation.
ID
Name
X
Y
Z
1
Angstrom
0.83
0.18
-0.07
2
Brunt
0.52
0.21
-
3
Swinbank
5.31
-
-
4
Idso and Jackson
0.26
-7.77
-
5
Brutsaert
1.72
7
-
6
Idso
0.70
5.95
-
7
Monteith and Unsworth
-119.00
1.06
-
8
Konzelmann et al.
0.23
0.48
-
9
Prata
1.00
1.20
3.00
10
Dilley and O'brien
59.38
113.70
96.96
The formulation of the L↑ requires the soil emissivity, which
usually is a property of the nature of a surface, and the surface soil
temperature. Table shows the literature values
of the soil emissivity for different surface
types: ϵs varies from a minimum of 0.95 for bare soils to a maximum
of 0.99 for fresh snow.
It is well known that surface soil temperature measurements are only
available at a few measurement sites; therefore, under the hypothesis that
the difference between soil and air temperatures is not too big, it is
possible to simulate L↑ using the air temperature
. In our approach three different types of
temperature were used to simulate L↑, specifically, surface soil
temperature (where available), air temperature at 2 m height, and soil
temperature at 4 cm depth.
The LWRB package (see the flowchart in Fig. ) is part of the
JGrass-NewAge system and was first tested in . Model
inputs depend on the specific SM being implemented and the purpose of the run
being performed (calibration, verification, simulation). The inputs are
meteorological observations such as air temperature, relative humidity,
incoming solar radiation, and sky clearness index. The LWRB is also fed by
other JGrass-NewAGE components, such as the shortwave radiation
balance (SWRB) . To test model performances
(i.e. verification), the LWRB can be connected to the system's Verification
component; to execute the parameter calibration algorithm
, it can be connected to the LUCA (Let Us
CAlibrate) component. In turn, all these components can and/or need to be
connected to other ones, as the problem under examination may require. Model
outputs are L↓ and L↑. These can be provided in
single points of specified coordinates or over a whole geographic area,
represented as a raster map. For the latter case a digital elevation model
(DEM) of the study area is necessary in input.
Sections 2.1 and 2.2, respectively, present the calibration and the
verification procedure. Moreover, a model sensitivity analysis procedure is
presented in Sect. 2.3 and a multi-regression model to relate the optimal
parameter set and easily available meteorological data is proposed in
Sect. 2.4.
Calibration of L↓ longwave radiation models
Model calibration estimates the site-specific parameters of L↓
models by tweaking them with a specific algorithm in order to best fit
measured data. To this end, we use the LUCA calibration algorithm proposed in
, which is a part of the OMS core and is able to optimize
parameters of any OMS component. LUCA is a multiple-objective, stepwise, and
automated procedure. As with any automatic calibration algorithm, it is based
on two elements: a global search algorithm and the objective function(s) to
evaluate model performance. In this case, the global search algorithm is the
shuffled complex evolution, which has been widely used and described in the
literature (e.g. ). As the objective function we
use the Kling–Gupta efficiency (KGE, ),
which is described below, but LUCA could use other objective functions just
as well.
Soil emissivity for surface types .
Nature of surface
Emissivity
Bare soil (mineral)
0.95–0.97
Bare soil (organic)
0.97–0.98
Grassy vegetation
0.97–0.98
Tree vegetation
0.96–0.97
Snow (old)
0.97
Snow (fresh)
0.99
The calibration procedure for L↓ follows these steps.
The theoretical solar radiation at the top of the atmosphere (Itop)
is computed using the SWRB (see Fig. 1).
The clearness index, c, is calculated as the ratio between the measured
incoming solar radiation (Im) and Itop.
Clear-sky and cloud-cover hours are detected by a threshold on the clearness
index (equal to 0.6), providing two subsets of measured L↓,
which are L↓clear and L↓cloud. On
one side, a threshold of 0.6 to define the clear-sky conditions helps in the
sense that it allows us to define time series of measured clear-sky
L↓ with comparable length in all the stations, and this is
useful for a reliable calibration process. On the other side, it introduces a
small error in computing the emissivity in all-sky conditions using Eq. (3).
Although the effects of this small error would need further investigations,
they could be compensated by the optimization of the parameters a and b
that are non-linearly related to the emissivity in all-sky conditions.
The parameters X, Y, and Z for the models in Table are
optimized using the subset L↓clear and setting
a = 0 in Eq. ().
The parameters a and b for Eq. () are optimized using the
subset L↓cloud and using the X, Y, and Z values computed in
the previous step.
The calibration procedure provides the optimal set of parameters at a given
location for each of the 10 models.
As well as parameter calibration, we carry out a model parameter sensitivity
analysis and we provide a linear regression model relating a set of
site-specific optimal parameters to mean air temperature, relative humidity,
precipitation, and altitude.
Verification of L↓ and L↑ longwave radiation models
As presented in previous applications
e.g., we use the
SMs with the original coefficients from the literature (i.e. the parameters
of Table ) and compare the performances of the models against
available measurements of L↓ and L↑ for each site.
The goodness of fit is evaluated by using two goodness-of-fit estimators: the
Kling–Gupta efficiency (KGE) and the root mean square error (RMSE).
The KGE (Eq. ) is able to incorporate into one objective function
three different statistical measures of the relation between measured (M)
and simulated (S) data: (i) the correlation coefficient, r; (ii) the
variability error, a = σS/σM; and (iii) the
bias error, b = μS/μM. In these definitions
μS and μM are the mean values, while
σS and σM are the standard deviations of
measured and simulated time series.
KGE=1-(r-1)2+(a-1)2+(b-1)2
The RMSE, on the other hand, is presented in Eq. ():
RMSE=1N∑i=1NMi-Si2,
where M and S represent the measured and simulated time series,
respectively, and N is their length.
Sensitivity analysis of L↓ models
For each L↓ model we carry out a model parameter sensitivity
analysis to investigate the effects and significance of parameters on
performance for different model structures (i.e. models with one, two, and
three parameters). The analyses are structured according to the following
steps:
we start with the optimal parameter set, computed by the optimization
process for the selected model;
all parameters are kept constant and equal to the optimal parameter set,
except for the parameter under analysis;
1000 random values of the analysed parameter are picked from a uniform
distribution centered on the optimal value with width equal to ±30 % of
the optimal value; in this way 1000 model parameter sets were defined and
1000 model runs were performed; and
1000 values of KGE are computed by comparing the model outputs with
measured time series.
The procedure was repeated for each parameter of each model and for each
station of the analysed dataset.
Regression model for parameters of L↓ models
The calibration procedure previously presented to estimate the site-specific
parameters for L↓ models requires measured downwelling longwave
data. Because these measurements are rarely available, we implement a
straightforward multivariate linear regression
to relate the site-specific
parameters X, Y and Z to a set of easily available site-specific
climatic variables, used as regressors ri. To perform the regression we
use the open-source R software (https://cran.r-project.org) and to
select the best regressors we use algorithms known as “best subsets
regression”, which are available in all common statistical software
packages. The regressors we have selected are mean annual air temperature,
relative humidity, precipitation, and altitude. The models that we use for
the three parameters are presented in Eqs. ()–():
X=iX+∑k=1Nαk⋅rk+ϵX,Y=iY+∑k=1Nβk⋅rk+ϵY,Z=iZ+∑k=1Nγk⋅rk+ϵZ,
where N = 4 is the number of regressors (annual mean air temperature,
relative humidity, precipitation, and altitude); rk with
k = 1, …, 4 are the regressors; iX, iY, and iZ are
the intercepts; αk, βk, and γk are the
coefficients; and ϵX, ϵY, and ϵZ are the
normally distributed errors. Once the regression parameters are determined,
the end-user can estimate site-specific X, Y and Z parameter values for
any location by simply substituting the values of the regressors in the model
formulations.
Test site locations in the USA.
Results
Verification of L↓ models with literature parameters
When implementing the 10 L↓ SMs using the literature
parameters, in many cases, they show a strong bias in reproducing measured
data. A selection of representative cases is presented in
Fig. , which shows scatterplots for four SMs in relation to
one measurement station. The black points represent the hourly estimates of
L↓ provided by literature formulations, while the solid red
line represents the line of optimal predictions. Model 1
shows a tendency to lie below the 1 : 1 line,
indicating a negative bias (percent bias of -9.8) and, therefore, an
underestimation of L↓. In contrast, model 9
shows an overestimation of L↓ with a
percent bias value of 26.3.
Figure presents the boxplot of KGE (first column) and RMSE
(second column) obtained for each model under clear-sky conditions, grouped
by classes of latitude and longitude. In general, all the models except
Model 8 provided values of KGE higher
than 0.5 and a RMSE lower than 100 W m-2 for all the latitude and
longitude classes. Model 8 is the less performing model for many of the
stations, likely because the model parameters were estimated for Greenland,
where snow and ice play a fundamental role in the energy balance. Its KGE
values range between 0.33 and 0.62 on average, while its RMSE values are
higher than 100 W m-2 except for latitude classes >40∘ N
and longitude classes > -70∘ W. Model 6 and
Model 2 provide the best results and the lower
variability, independently of the latitude and longitude ranges where they
are applied. Their average KGE values are between 0.75 and 0.92, while the
RMSE has a maximum value of 39 W m-2. Moreover, all the models
except 2 and 6 show a high variability of the goodness of fit through the
latitude and longitude classes.
Results of the clear-sky simulation for four literature models using
data from Howland Forest (Maine).
KGE and RMSE values for each clear-sky simulation using literature
formulations, grouped by classes of latitude and longitude. Only values of
KGE above 0.5 are shown. Only values of RMSE below 100 W m-2 are
shown.
L↓ models with site-specific parameters
The calibration procedure greatly improves the performances of all 10 SMs.
Optimized model parameters for each model are reported in the Supplement
(Table S1). Figure presents the boxplots of KGE and RMSE
values for clear-sky conditions grouped by classes of latitude and longitude.
The percentage of KGE improvement ranges from its maximum value of 70 % for
Model 8 (which is not, however, representative of the mean behaviour of the
SMs) to less than 10 % for Model 6, with an average improvement of around
35 %. Even though variations in model performances with longitude and
latitude classes still exist when using optimized model parameters, the
magnitude of these variations is reduced with respect to the use of
literature formulations. The calibration procedure reduces the RMSE values
for all the models to below 45 W m-2, even for Model 8, which also in
this case had the maximum improvement. Model 6 and
Model 2 provide the best results on average for all
the analysed latitude and longitude classes.
KGE (best is 1) and RMSE (best is 0) values for each optimized
formulation in clear-sky conditions, grouped by classes of latitude and
longitude. Only values of KGE above 0.5 are shown.
Figure presents the boxplots of KGE and RMSE values for
each model under all-sky conditions, grouped by latitude and longitude
classes. In general, for all-sky conditions we observe a deterioration of KGE
and RMSE values with respect to the clear-sky optimized case, with a decrease
in KGE values up to a maximum of 25 % on average for Model 10. This may be
due to uncertainty incorporated into the formulation of the cloudy-sky
correction model (Eq. ): it seems that sometimes the cloud
effects are not accounted for appropriately. This, however, is in line with
the findings of .
KGE and RMSE values for each model in all-sky conditions with the
optimized parameters; results are grouped by classes of latitude and
longitude. Only values of KGE above 0.5 are shown.
Results of the model parameters' sensitivity analysis. The variation
of the model performances due to a variation of one of the optimal parameters
and assuming constant the others is presented as a boxplot. The procedure is
repeated for each model and the blue line represents the smooth line passing
through the boxplot medians.
Sensitivity analysis of L↓ models
The results of the model sensitivity analysis are summarized in
Fig. a and b for Models 1 to 5 and Models 6 to 10,
respectively. Each figure presents three columns, one for each parameter.
Considering Model 1 and parameter X: the range of X is subdivided into 10
equal-sized classes and for each class the corresponding KGE values are
presented as a boxplot. A smooth blue line passing through the boxplot
medians is added to highlight any possible pattern to parameter sensitivity.
A flat line indicates that the model is not sensitive to parameter variation
around the optimal value. Results suggest that models with one and two
parameters are all sensitive to parameter variation, presenting a peak in KGE
in correspondence to their optimal values; this is more evident in models
with two parameters. Models with three parameters tend to have at least one
insensitive parameter, except for Model 1, which could reveal a possible
overparameterization of the modelling process.
Comparison between model performances obtained with regression and
classic parameters: the KGE values shown are those above 0.3 and results are
grouped by latitude classes.
Regression model for parameters of L↓ models
A multivariate linear regression model was estimated to relate the
site-specific parameters X, Y and Z to mean annual air temperature,
relative humidity, precipitation, and altitude. The script containing the
regression model is available, as specified in the Reproducible Research
section below.
The performances of the L↓ models using parameters assessed by
linear regression are evaluated through the leave-one-out cross validation
. We use 23 stations as training sets for
Eqs. ()–() and we perform the model verification on the
remaining station. The procedure is repeated for each of the 24 stations.
The cross validation results for all L↓ models and for all
stations are presented in Figs. and ,
grouped by classes of latitude and longitude, respectively. They report the
KGE comparison between the L↓ models with their original
parameters (in black) and with the regression model parameters (in black).
In general, the use of parameters estimated with the regression model gives a
good estimation of L↓, with KGE values of up to 0.92. With
respect to the classic formulation, model performance with regression
parameters improved for all the models independently of the latitude and
longitude classes. In particular for Model 8 the KGE improved from 0.26 for
the classic formulation to 0.92, on average. Finally, the use of the
parameters estimated by the regression model provides a reduction of the
model performances' variability for all the models except Models 5 and 8, for
longitude classes -125; -105∘ W and -105; -90∘ W,
respectively.
Comparison between model performances obtained with regression and
classic parameters: the KGE values shown are those above 0.3 and results are
grouped by longitude classes.
Verification of the L↑ model
Figure presents the results of the L↑ simulations
obtained using the three different temperatures available at experimental
sites: soil surface temperature (skin temperature), air temperature, and soil
temperature (measured at 4 cm below the surface). The figure shows the
performances of the L↑ model for the three different temperatures
used in terms of KGE, grouping all the stations for the whole simulation
period according to season. This highlights the different behaviours of the
model for periods where the differences in the three temperatures are larger
(winter) or negligible (summer). The values of soil emissivity are assigned
according the soil surface type, according to Table
. Although many studies investigated the
influence of snow covered area on longwave energy balance
e.g., the SMs do not
explicitly take it into account. As presented in
, the effect of snow could be implicitly
taken into account by tuning the emissivity parameter.
Boxplots of the KGE values obtained by comparing modelled upwelling
longwave radiation, computed with different temperatures (soil surface
temperature – SKIN, air temperature – AIR, and soil temperature – SOIL),
against measured data. Results are grouped by seasons.
The best fit between measured and simulated L↑ is obtained with
the surface soil temperature, with an all-season average KGE of 0.80.
Unfortunately, the soil surface temperature is not an easily available
measurement. In fact, it is available only for 8 sites of the 24 in the study
area. Very good results are also obtained using the air temperature, where
the all-season average KGE is around 0.76. The results using air temperature
present much more variance compared to those obtained with the soil surface
temperature. However, air temperature (at 2 m height) is a readily available
measure; in fact, it is available for all 24 sites.
The use soil temperature at 4 cm depth provides the least accurate results
for our simulations, with an all-season average KGE of 0.46. In particular,
the use of soil temperature at 4 cm depth during the winter is not able to
capture the dynamics of L↑. It does, however, show a better fit
during the other seasons. This could be because during the winter there is a
substantial difference between the soil and skin temperatures, as also
suggested in .
Conclusions
This paper presents the LWRB package, a new modelling component integrated
into the JGrass-NewAge system to model upwelling and downwelling longwave
radiation. It includes 10 parameterizations for the computation of
L↓ longwave radiation and 1 for L↑. The package
uses all the features offered by the JGrass-NewAge system, such as algorithms
to estimate model parameters and tools for managing and visualizing data in
GIS.
The LWRB is tested against measured L↓ and L↑ data
from 24 AmeriFlux test sites located all over the contiguous USA. The
application for L↓ longwave radiation involves model parameter
calibration, model performance assessment, and parameter sensitivity
analysis. Furthermore, we provide a regression model that estimates optimal
parameter sets on the basis of local climatic variables, such as mean annual
air temperature, relative humidity, and precipitation. The application for
L↑ longwave radiation includes the evaluation of model
performance using three different temperatures.
The main achievements of this work include (i) a broad assessment of the
classic L↓ longwave radiation parameterizations, which clearly
shows that the and models are the
more robust and reliable for all the test sites, confirming previous results
; (ii) a site-specific assessment of the
L↓ longwave radiation model parameters for 24 AmeriFlux sites
that improved the performances of all the models; (iii) the set-up of a
regression model that provides an estimate of optimal parameter sets on the
basis of climatic data; and (iv) an assessment of L↑ model
performances for different temperatures (skin temperature, air temperature,
and soil temperature at 4 cm below the surface), which shows that the skin
and the air temperature are better proxies for the L↑ longwave
radiation. Regarding longwave downwelling radiation, the
model is able to provide on average the best
performances with the regression model parameters independently of the
latitude and longitude classes. For the model the
formulation with a regression parameter provided lower performances with
respect to the literature formulation for latitudes between 25 and
30∘ N.
The integration of the package into JGrass-NewAge will allow users to build
complex modelling solutions for various hydrological scopes. In fact, future
work will include the link of the LWRB package to the existing components of
JGrass-NewAge to investigate L↓ and L↑ effects on
evapotranspiration, snow melting, and glacier evolution. Finally, the
methodology proposed in this paper provides the basis for further
developments such as the possibility of (i) investigating the effect of
different all-sky emissivity formulations and quantifying the influence of
the clearness index threshold, (ii) verifying the usefulness of the
regression models for climates outside the contiguous USA, and
(iii) analysing in a systematic way the uncertainty due to the quality of
meteorological input data on the longwave radiation balance in scarce
instrumented areas.